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--- abstract: 'We review the present status of our research and understanding regarding the dynamics and the statistical properties of earthquakes, mainly from a statistical physical viewpoint. Emphasis is put both on the physics of friction and fracture, which provides a “microscopic” basis of our understanding of an earthquake instability, and on the statistical physical modelling of earthquakes, which provides “macroscopic” aspects of such phenomena. Recent numerical results on several representative models are reviewed, with attention to both their “critical” and “characteristic” properties. We highlight some of relevant notions and related issues, including the origin of power-laws often observed in statistical properties of earthquakes, apparently contrasting features of characteristic earthquakes or asperities, the nature of precursory phenomena and nucleation processes, the origin of slow earthquakes, [etc]{}.' author: - Hikaru Kawamura - Takahiro Hatano - Naoyuki Kato - Soumyajyoti Biswas - 'Bikas K. Chakrabarti' bibliography: - 'article.bib' nocite: '[@]' title: 'Statistical Physics of Fracture, Friction and Earthquake' --- Introduction ============ Earthquakes are large scale mechanical failure phenomena, which have still defied our complete understanding. In this century, we already experienced two gigantic earthquakes: 2004 Sumatra-Andaman earthquake (M9.1) and 2011 East Japan earthquake (M9.0). Given the disastrous nature of the phenomena, the understanding and forecasting of earthquakes have remained to be the most important issue in physics and geoscience (Carlson, Langer and Shaw, 1996; Rundle, Turcotte and Klein, 2000; Scholz, 2002; Rundle et al, 2003; Bhattacharyya and Chakrabarti, 2006; Ben-Zion, 2008; Burridge, 2006; De Rubies et al, 2006; Kanamori, 2009; Daub and Carlson, 2010). Although there is some recent progress in our understanding of the basic physics of fracture and friction, it is still at a primitive stage (Marone, 1998; Scholz, 1998, 2002; Dieterich, 2009; Tullis, 2009; Daub and Carlson, 2010). Furthermore, our lack of a proper understanding of the dynamics of earthquakes poses an outstanding challenge to both physicists and seismologists. While earthquakes are obviously complex phenomena, certain empirical laws have been known concerning their statistical properties, [e.g.]{}, the Gutenberg-Richter (GR) law for the magnitude distribution of earthquakes, and the Omori law for the time evolution of the frequency of aftershocks (Scholz, 2002; Rundle, 2003; Turcotte, 2009). The GR law states that the frequency of earthquakes of its energy (seismic moment) $E$ decays with $E$ obeying a power-law, [i.e.]{}, $\propto E^{-(1+B)}=E^{-(1+\frac{2}{3}b)}$ where $B$ and $b=\frac{3}{2}B$ are appropriate exponents, whereas the Omori law states that the frequency of aftershocks decays with the time elapsed after the mainshock obeying a power-law. These laws, both of which are power-laws possessing a scale-invariance, are basically of statistical nature, becoming evident only after examining large number of events. Although it is extremely difficult to give a definitive prediction for each individual earthquake event, clear regularity often shows up when one measures its statistical aspect for an ensemble of many earthquake events. This observation motivates statistical physical study of earthquakes due to the following two reasons: First, a law appearing after averaging over many events is exactly the subject of statistical physics. Second, a power law or a scale invariance has been a central subject of statistical physics over years in the context of critical phenomena. Indeed, Bak and collaborators proposed the concept of “self-organized criticality (SOC)” (Bak, Tang and Wiesenfeld, 1987). According to this view, the Earth’s crust is always in the critical state which is self-generated dynamically (Turcotte 1997; Hergarten, 2002; Turcotte, 2009; Pradhan, Hansen and Chakrabarti, 2010). One expects that such an SOC idea might possibly give an explanation of the scale-invariant power-law behaviors frequently observed in earthquakes, including the GR law and the Omori law. However, one should also bear in mind that real earthquakes often exhibit apparently opposite features, [i.e.]{}, the features represented by “characteristic earthquakes” where an earthquake is regarded to possess its characteristic energy or time scale (Scholz, 2002; Turcotte, 2009). Earthquakes also possess strong relevance to material science. It is now established that earthquakes could be regarded as a stick-slip frictional instability of a pre-existing fault, and statistical properties of earthquakes are governed by the physical law of rock friction (Marone, 1998; Scholz, 1998; 2002, Dieterich, 2009; Tullis, 2009). The physical law describing rock friction or fracture is often called “constitutive law”. As most of the major earthquakes are caused by rubbing of faults, such friction laws give the “microscopic” basis in analyzing the dynamics of earthquakes. One might naturally ask: How statistical properties of earthquakes depend on the material properties characterizing earthquake faults, [e.g.]{}, the elastic properties of the crust or the frictional properties of the fault, [etc]{}. Answering such questions would give us valuable information in understanding the nature of earthquakes. In spite of some recent progress, we still do not have precise knowledge of the constitutive law characterizing the stick-slip dynamics of earthquake faults. In fact, law of rock friction is often quite complicated, depending not just on the velocity or the displacement, but on the previous history and the “state” of contact surface, [etc]{}. The rate-and-state friction (RSF) law currently occupies the standard position among friction laws in the field of tectonophysics. Although the RSF law is formulated empirically three decades ago to account for certain aspects of rock friction experiments (Dieterich, 1979; Ruina 1983), the underlying physics was not known until very recently. While the RSF law shows qualitatively good agreement with numerous experiments, it is only good at aseismic slip velocities (slower than mm/sec). Among some progress made recently in the study of friction process, the most fascinating findings might be the rich variety of mechano-chemical phenomena, which comes into play at seismic slip velocities. Another important progress might be the understanding of the friction law of granular matter. This is also a very important point in understanding the friction law of faults as they consist of fine rock powder that are ground up by the fault motion of the past. The investigations on friction phenomenon at seismic slip velocities is now a frontier in tectonophysics. The RSF law no longer applies to this regime, where many mechano-chemical phenomena have been observed in experiments. The most illustrating examples are melting due to frictional heat, thermal decomposition of calcite, silica-gel lubrication and so on. There have not been any friction laws that can describe such varied class of phenomena, which significantly affect the nature of sliding friction. In this review article, we wish to review the recent development concerning the basic physics of friction and fracture. Statistical physical study of earthquakes is usually based on models of various levels of simplification. There are several advantages in employing simplified models in the study of earthquakes. First, it is straightforward in the model study to control various material parameters as input parameters. A systematic field study of the material-parameter dependence of real earthquakes meets serious difficulties, because it is difficult to get precise knowledge of, or even to control, various material parameters characterizing real earthquake faults. Second, since an earthquake is a large-scale natural phenomenon, it is intrinsically not “reproducible”. Furthermore, large earthquakes are rare, say, once in hundreds of years for a given fault. If some observations are to be made for a given large event, it is often extremely difficult to see how universal it is and to put reliable error bars to the obtained data. In the model, on the other hand, it is often quite possible to put reliable error bars to the data under well controlled conditions, say, by performing extensive computer simulations. An obvious disadvantage of the model study is that the model is not the reality in itself, and one has to be careful in elucidating what aspect of reality is taken into account or discarded in the model under study. While numerous earthquake models of various levels of simplifications have been studied in the past, one may classify them roughly into two categories: The first one is of the type possessing an equation of motion describing its dynamics where the constitutive relation can be incorporated as a form of “force”. The so-called spring-block or the Burridge-Knopoff (BK) model, which is a discretized model consisting of an assembly of blocks coupled via elastic springs, belong to this category (Burridge and Knopoff, 1967). Continuum models also belong to this category (Tse and Rice, 1986; Rice, 1993). The second category encompasses further simplified statistical physical models, coupled-lattice models, most of which were originally introduced as a model of SOC. This category includes the so-called Olami-Feder-Christensen (OFC) model (Olami, Feder and Christensen, 1992), the fiber bundle model (Pradhan, Hansen and Chakrabarti, 2010), and the two fractal overlap model [@bkc1; @bkc2; @bkc26]. These models possesses extremely simplified evolution rule, instead of realistic dynamics and constitutive relation. Yet, one expects that its simplicity enables one to perform exact or precise analysis which might be useful in extracting essential qualitative features of the phenomena. It often happens in practice that, even when the adopted model looks simple in its appearance, it is still highly nontrivial to reveal its statistical properties. Then, the strategy in examining the model properties is often to perform numerical computer simulations on the model, together with the analytical treatment. In this review article, we wish to review the recent developments concerning the properties of these models mainly studied in statistical physics. Earthquake forecast is an ultimate goal of any earthquake study. A crucially important ingredient playing a central role there might be various kinds of precursory phenomena. We wish to touch upon the following two types of precursory phenomena in this review article: The first type is a possible change in statistical properties of earthquakes which might occur prior to mainshocks. The form of certain spatiotemporal correlations of earthquakes might change due to the proximity effect of the upcoming mainshock. For example, it has been pointed out that the power-law exponent describing the GR law might change before the mainshock, or a doughnut-like quiescence phenomenon might occur around the hypocenter of the upcoming mainshock, etc. The second type of precursory phenomena is a possible nucleation process which might occur preceding mainshocks (Dieterich, 2009). Namely, prior to seismic rupture of a mainshock, the fault might exhibit a slow rupture process localized to a compact “seed” area, with its rupture velocity orders of magnitude slower than the seismic wave velocity. The fault spends a very long time in this nucleation process, and then at some stage, exhibits a rapid acceleration process accompanied by a rapid expansion of the rupture zone, finally getting into a final seismic rupture of a mainshock. These possible precursory phenomena preceding mainshocks are of paramount importance in their own right as well as in possible connection to earthquake forecast. We note that similar nucleation process is ubiquitously observed in various types of failure processes in material science and in engineering. The purpose of the present review article is to help researchers link different branches of earthquake studies. First, we wish to link basic physics of friction and fracture underlying earthquake phenomenon to macroscopic properties of earthquakes as a large-scale dynamical instability. These two features should be inter-related as “input versus output” or as “microscopic versus macroscopic” relation, but its true connection is highly nontrivial and still remains largely unexplored. To understand an appropriate constitutive law describing an earthquake instability and to make a link between such constitutive relations and the macroscopic properties of earthquakes is crucially important in our understanding of earthquakes. Second, we wish to promote an interaction between statistical physicists and seismologists. We believe that the cooperation of scientists in these two areas would be very effective, and in some sense, indispensable in our proper understanding of earthquakes. Recently, there has been some progress made by statistical physicists in characterizing the statistical aspects of the earthquake phenomena. These efforts are of course based on established literature in seismology, physics of fracture and friction. Also, there has been considerable fusion and migration of the scientists and the established knowledge bases between physics and seismology. In this article, we intend to review the present state of our understanding regarding the dynamics of earthquakes and the statistical physical modelling of such phenomena, starting with the same for fracture and friction. The article is organized as follows. In section II, we deal with the basic physics of fracture and friction. After reviewing the classic Griffith theory of fracture in II.A, we review a theory of fracture as a dynamical phase transition in II.B. Rate and state dependent friction (RSF) law is reviewed in II.C, while the recent development beyond the RSF law is discussed in II.D. Section II.E is devoted to some microscopic statistical mechanical theories of friction. In section III, we deal with statistical properties of the model of our first type which includes the spring-block Burridge-Knopoff model (III.A) and the continuum model (III.B). In III.A, we examine statistical properties of earthquakes including precursory phenomena with emphasis on both their critical and characteristic properties, while, in III.B, we mainly examine characteristic properties of earthquakes and various slip behaviors including slow earthquakes. Implications of RSF laws to earthquake physics are also discussed in this section (III.B). In section IV, we deal with statistical properties of our second type of models which include the OFC model (IV.A), the fiber bundle model (IV.B) and the two fractal overlap model (IV.C). We also provide a a Glossary of some interdisciplinary terms as an Appendix. Fracture and friction ===================== Griffith energy balance and brittle fracture strength of solids --------------------------------------------------------------- In a solid, stress ($\sigma $) and strain ($S$) bear a linear relation in the Hookean region (small stress). Non-linearity appears for further increase of stress, which finally ends in fracture or failure of the solid. In brittle solids, failure occurs immediately after the linear region. Hence linear elastic theory can be applied to study this essentially non-linear and irreversible phenomena. The failure process has strong dependence on, among other things, the disorder properties of the material [@bkc66]. Often, stress gets concentrated around the disorder [@bkc5; @bkc6; @bkc65] where microcracks are formed. The stress values at the notches and corners of the microcracks can be several times higher than the applied stress. Therefore the scaling properties of disorder plays an important role in breakdown properties of solids. Although the disorder properties tell us about the location of instabilities, it does not tell us about when a microcrack propagates. For that detailed energy balance study is needed. Griffith in 1920, equating the released elastic energy (in an elastic continuum) with the energy of the surface newly created (as the crack grows), arrived at a quantitative criterion for the equilibrium extension of the microcrack already present within the stressed material [@bkc27]. The following analysis is valid effectively for two-dimensional stressed solids with a single pre-existing crack, as for example the case of a large plate with a small thickness. Extension to three-dimensional solids is straightforward. ![A portion of a plate (of thickness $w$) under tensile stress $\sigma$ (Model I loading) containing a linear crack of length $2l$. For a further growth of the crack length by $2{\rm d}l$, the elastic energy released from the annular region must be sufficient to provide the surface energy $4 \Gamma w {\rm d} l$ (extra elastic energy must be released for finite velocity of crack propagation).[]{data-label="bkc-slitsigma"}](sigma-bkc.eps){width="3.0cm"} Let us assume a thin linear crack of length $2l$ in an infinite elastic continuum subjected to uniform tensile stress $\sigma$ perpendicular to the length of the crack (see Fig. \[bkc-slitsigma\]). Stress parallel to the crack does not affect the stability of the crack and has not, therefore, been considered. Because of the crack (which can not support any stress field, at least on its surfaces), the strain energy density of the stress field ($\sigma^2 /2Y$; where $Y$ represents the elasticity modulus) is perturbed in a region around the crack, having dimension of the length of the crack. We assume here this perturbed or stress-released region to have a circular cross-section with the crack length as the diameter. The exact geometry of this perturbed region is not important here, and it determines only an (unimportant) numerical factor in the Griffith formula (see e.g. @bkc5). Assuming for the purpose of illustration that half of the stress energy of the annular or cylindrical volume, having the internal radius $l$ and outer radius $l + {\rm d}l$ and length $w$ (perpendicular to the plane of the stress; here the width $w$ of the plate is very small compared to the other dimensions), to be released as the crack propagates by a length ${\rm d}l$, one requires this released strain energy to be sufficient for providing the surface energy of the four new surfaces produced. This suggests $$\frac{1}{2} (\sigma^2 /2Y) (2 \pi w l {\rm d}l) \ge \Gamma (4 w {\rm d}l).$$ Here $\Gamma$ represents the surface energy density of the solid, measured by the extra energy required to create unit surface area within the bulk of the solid. We have assumed here, on average, half of the strain energy of the cylindrical region having a circular cross-section with diameter $2l$ to be released. If this fraction is different or the cross-section is different, it will change only some of the numerical factors, in which we are not very much interested here. Also, we assume here linear elasticity up to the breaking point, as in the case of brittle materials. The equality holds when energy dissipation, as in the case of plastic deformation or for the propagation dynamics of the crack, does not occur. One then gets $$\label{bkc-sigma_f} \sigma_f = \frac{\Lambda}{\sqrt{2l}}; \ \ \Lambda = \left( \frac{4}{\sqrt \pi} \right) \sqrt{Y \Gamma}$$ for the critical stress at and above which the crack of length $2l$ starts propagating and a macroscopic fracture occurs. Here $\Lambda$ is called the critical stress-intensity factor or the fracture toughness. In a three-dimensional solid containing a single elliptic disk-shaped planar crack parallel to the applied tensile stress direction, a straightforward extension of the above analysis suggests that the maximum stress concentration would occur at the two tips (at the two ends of the major axis) of the ellipse. The Griffith stress for the brittle fracture of the solid would therefore be determined by the same formula (\[bkc-sigma\_f\]), with the crack length $2l$ replaced by the length of the major axis of the elliptic planar crack. Generally, for any dimension therefore, if a crack of length $l$ already exists in an infinite elastic continuum, subject to uniform tensile stress $\sigma$ perpendicular to the length of the crack, then for the onset of brittle fracture , Griffith equates (the differentials of) the elastic energy $E_l$ with the surface energy $E_s$: $$\label{bkc-E_l} E_l \simeq \left( \frac{\sigma^2}{2Y}\right) l^d = E_s \simeq \Gamma l^{d-1},$$ where $Y$ represents the elastic modulus appropriate for the strain, $\Gamma$ the surface energy density and $d$ the dimension. Equality holds when no energy dissipation (due to plasticity or crack propagation) occurs and one gets $$\label{bkc-sigma_f1} \sigma_f \sim \frac{\Lambda}{\sqrt l}; \ \Lambda \sim \sqrt{Y\Gamma}$$ for the breakdown stress at (and above) which the existing crack of length $l$ starts propagating and a macroscopic fracture occurs. It may also be noted that the above formula is valid in all dimensions ($d \ge 2$). This quasistatic picture can be extended [@bkc31] to fatigue behavior of crack propagation for $\sigma<\sigma_f$. At any stress $\sigma$ less than $\sigma_f$, the cracks (of length $l_0$) can still nucleate for a further extension at any finite temperature $k_BT$ with a probability $\sim \exp[-E/k_BT]$ and consequently the sample fails within a failure time $\tau$ given by $$\tau^{-1}\sim \exp[-E(l_0)/k_BT],$$ where $$E(l_0)=E_s+E_l\sim \Gamma l_0^2-\frac{\sigma^2}{Y}l_0^3$$ is the crack (of length $l_0$) nucleation energy. One can therefore express $\tau$ as $$\tau \sim \exp[A(1-\frac{\sigma^2}{\sigma_f^2})],$$ where (the dimensionless parameter) $A\sim l_0^3\sigma_f^2/(Yk_BT)$ and $\sigma_f$ is given by Eq. (\[bkc-sigma\_f1\]). This immediately suggests that the failure time $\tau$ grows exponentially for $\sigma<\sigma_f$ and approaches infinity if the stress $\sigma$ is much smaller than $\sigma_f$ when the temperature $k_BT$ is small, whereas $\tau$ becomes vanishingly small as the stress $\sigma$ exceeds $\sigma_f$; see, e.g., @bkc32 and @bkc33. For disordered solids, let us model the solid by a percolating system. For the occupied bond/site concentration $p > p_c$, the percolation threshold, the typical pre-existing cracks in the solid will have the dimension ($l$) of correlation length $\xi \sim \Delta p^{-\nu}$ and the elastic strength $Y \sim \Delta p^{T_e}$ [@bkc7]. Assuming that the surface energy density $\Gamma$ scales as $\xi^{d_B}$, with the backbone (fractal) dimension $d_B$ [@bkc7], equating $E_l$ and $E_s$ as in (\[bkc-E\_l\]), one gets $\left( \frac{\sigma_f^2}{2Y}\right) \xi^d \sim \xi^{d_B}$. This gives $$\sigma_f \sim (\Delta p)^{T_f}$$ with $$\label{bkc-T_f} T_f = \frac{1}{2} [T_e + (d-d_B)\nu]$$ for the ‘average’ fracture strength of a disordered solid (of fixed value) as one approaches the percolation threshold. Careful extensions of such scaling relations (\[bkc-T\_f\]) and rigorous bounds for $T_f$ has been obtained and compared extensively in @bkc6 [@bkc28; @bkc29]. ### Extreme statistics of the fracture stress {#extrm .unnumbered} The fracture strength $\sigma_f$ of a disordered solid does not have self-averaging statistics; most probable and the average $\sigma_f$ may not match because of the extreme nature of the statistics. This is because, the ‘weakest point’ of a solid determines the strength of the entire solid, not the average weak points! As we have modelled here, the statistics of clusters of defects are governed by the random percolation processes. We have also discussed, how the linear responses, like the elastic moduli of such random networks, can be obtained from the averages over the statistics of such clusters. This was possible because of the self-averaging property of such linear responses. This is because the elasticity of a random network is determined by all the ‘parallel’ connected material portions or paths, contributing their share in the net elasticity of the sample. The fracture or breakdown property of a disordered solid, however, is determined by only the weakest (often the longest) defect cluster or crack in the entire solid. Except for some indirect effects, most of the weaker or smaller defects or cracks in the solid do not determine the breakdown strength of the sample. The fracture or breakdown statistics of a solid sample is therefore determined essentially by the extreme statistics of the most dangerous or weakest (largest) defect cluster or crack within the sample volume. We discuss now more formally the origin of this extreme statistics. Let us consider a solid of linear size $L$, containing $n$ cracks within its volume. We assume that each of these cracks have a failure probability $f_i(\sigma), i=1,2,\ldots,n$ to fail or break (independently) under an applied stress $\sigma$ on the solid, and that the perturbed or stress-released regions of each of these cracks are separate and do not overlap. If we denote the cumulative failure probability of the entire sample, under stress $\sigma$, by $F(\sigma)$ then [@bkc60; @bkc6] $$\begin{aligned} \label{bkc-1-Fsigma} 1-F(\sigma)&&=\prod_{i=1}^{n} (1-f_i(\sigma)) \simeq \exp \left[ -\sum_i f_i(\sigma) \right] \\ \nonumber &&= \exp \left[ -L^d \tilde{g}(\sigma) \right]\end{aligned}$$ where $\tilde{g}(\sigma)$ denotes the density of cracks within the sample volume $L^d$ (coming from the sum $\sum_i$ over the entire volume), which starts propagating at and above the stress level $\sigma$. The above equation comes from the fact that the sample survives if each of the cracks within the volume survives. This is the essential origin of the above extreme statistical nature of the failure probability $F(\sigma)$ of the sample. Noting that the pair correlation $g(l)$ of two occupied sites at distance $l$ on a percolation cluster decays as $\exp\left(-l/\xi(p)\right)$, and connecting the stress $\sigma$ with the length $l$ by using Griffith’s law (Eq. (\[bkc-sigma\_f\])) that $\sigma \sim \frac{\Lambda}{l^a}$, one gets $\tilde{g}(\sigma) \sim \exp \left( -\frac{\Lambda^{1/a}}{\xi \sigma^{1/a}} \right)$ for $p \to p_c$. On substituting this, Eq. (\[bkc-1-Fsigma\]) gives the Gumbel distribution [@bkc6]. If, on the other hand, one assumes a power law decay of $g(l)$: $g(l) \sim l^{-b}$, then using the Griffith’s law (\[bkc-sigma\_f\]), one gets $\tilde{g}(\sigma) \sim \left( \frac{\sigma}{\Lambda} \right)^m$, giving the Weibull distribution, from eqn. (\[bkc-1-Fsigma\]), where $m=b/a$ gives the Weibull modulus [@bkc6]. The variation of $F(\sigma)$ with $\sigma$ in both the cases have the generic form shown in Fig. \[bkc-weibullcurve\]. $F(\sigma)$ is non-zero for any stress $\sigma > 0$ and its value (at any $\sigma$) is higher for larger volume ($L^d$). This is because, the possibility of a larger defect (due to fluctuation) is higher in a larger volume and consequently, its failure probability is higher. Assuming $F(\sigma_f)$ is finite for failure, the most probable failure stress $\sigma_f$ becomes a decreasing function of volume if extreme statistics is at work. The precise ranges of the validity of the Weibull or Gumbel distributions for the breakdown strength of disordered solids are not well established yet. However, analysis of the results of detailed experimental and numerical studies of breakdown in disordered solids seem to suggest that the fluctuations of the extreme statistics dominate for small disorder [@bkc28; @bkc29]. Very near to the percolation point, the percolation statistics takes over and the statistics become self-averaging. One can argue [@bkc27], that arbitrarily close to the percolation threshold, the fluctuations of the extreme statistics will probably get suppressed and the percolation statistics should take over and the most probable breaking stress becomes independent of the sample volume (its variation with disorder being determined, as in Eqn.(\[bkc-T\_f\]), by an appropriate breakdown exponent). This is because the appropriate competing length scales for the two kinds of statistics are the Lifshitz scale $\ln L$ (coming from the finiteness of the volume integral of the defect probability: $L^d(1 - p)^l$ finite, giving the typical defect size $l \sim \ln L$) and the percolation correlation length $\xi$. When $\xi < \ln L$, the above scenario of extreme statistics should be observed. For $\xi > \ln L$, the percolation statistics is expected to dominate. Fracture as dynamical phase transition -------------------------------------- When a material is stressed, according to the linear elastic theory discussed above, it develops a proportional amount of strain. Beyond a threshold, cracks appear and on further application of stress, the material is fractured as it breaks into pieces. In a disordered solid, however, the advancing cracks may be stopped or [pinned]{} by the defect centers present within the material. So a competition develops between the pinning force due to disorder and the external force. Upto a critical value of the external force, the average velocity of the crack-front will disappear in the long time limit, i.e., the crack will be pinned. However, if the external force crosses this critical value, the crack front moves with a finite velocity. This depinning transition can be viewed as a [dynamical critical phenomena]{} in the sense that near the criticallity universal scaling is observed which are independent of the microscopic details of the materials concerned [@bkc25]. The order parameter for this transition is the average velocity $\overline{v}$ of the crack front. When the external force $f^{ext}$ approaches the critical value $f^{ext}_c$ from a higher value, the order parameter vanishes as $$\overline{v} \sim (f^{ext}-f^{ext}_c)^{\theta},$$ where $\theta$ denotes the velocity exponent. ![The average velocity of the crack front is plotted against external force ($f=G-\Gamma$, where $G$ is the mechanical energy release rate and $\Gamma$ is the fracture energy). For $T=0$ the depinning transition is seen. For finite temperature sub-critical creep is shown [@bkc64]. From [@bkc64].[]{data-label="depinning"}](depinning.eps){width="9.0cm"} It is to be mentioned here that the pinning of a crack-front by disorder potential can occur at zero temperature. At finite temperature, there can be healing of cracks due to diffusion or there can be sub-critical crack propagation (in the so called creep regime) [@bkc25]. In the later case, the velocity is expected to scale as $$\overline{v}\sim\exp\left(-C\left(\frac{f^{ext}_c}{f}\right)^{\phi}\right).$$ This sub-critical scaling agrees well with experiments [@bkc15; @bkc16]. In Fig. \[expt\], the experimental result of the crack propagation in the Botucatu sandstone [@bkc15] is shown. The average velocity of the crack is plotted against the mechanical energy release rate $G$ ($f=G-\Gamma$, where $\Gamma$ is the fracture energy). The subcritical creep regime and the supercritical power-law variations are clearly seen (insets), which gives the velocity exponent close to $\theta \approx 0.81$ . ![Variation of average crack-front velocity against the mechanical energy release rate is shown for Botucatu sandstone [@bkc15]. The sub-critical creep region and supercritical power-law variations are shown in top-left and bottom right insets respectively. For the sub-critical regime, the data is fitted with a function $v\sim e^{-C/(G-\langle \Gamma\rangle)^{\mu}}$ for $\mu=0.60$ and $\langle \Gamma\rangle=65 Jm^{-2}$. For the power-law variation ($v\sim (G-G_c)^{\theta}$) in the super-critical region, $G_c=140 Jm^{-2}$ and $\theta=0.80$. From [@bkc15]. []{data-label="expt"}](expt.eps){width="8.0cm"} Theoretical predictions of this exponent using Functional Renormalisations Group methods have placed its value around $\theta=0.59$ [@bkc17], where the experimental findings differ significantly ($\theta\approx0.80\pm0.15$). Here we mention the numerical study of a model of the elastic crack-front propagation in a disordered solid. The basic idea is to consider the propagation of the crack front as an elastic string driven through a random medium. The crack front is characterised by an array of integral height (measured in the direction of the crack propagation) $\{h_1,h_2,\dots,h_L\}$ with periodic boundary conditions, where the unique values for the height profile suggest that any overhangs in the height profile is neglected. The forces acting on a site can be written as $$f_i(t)=f^{el}_i+f^{ext}+g\eta_i(h_i)$$ where $f^{el}$ is the elastic force due to stretching, $f^{ext}$ is the applied external force and $\eta$ is due to disorder. The dynamics of the driven elastic chain is then given by the simple rule $$\begin{aligned} h_i(t+1)-h_i(t)=v_i(t)&=& 1 \qquad \mbox{if}\quad f_i(t)>0 \nonumber \\ &=& 0 \qquad \mbox{otherwise}.\end{aligned}$$ The elastic force may have different forms in various contexts. When this force is short ranged (nearest neighbours) the well studied models are Edwards -Wilkinson (EW) [@bkc39] (see also @bkc53 [@bkc54]) and KPZ models [@bkc40; @bkc55; @bkc56] (see [@bkc41] for extensive analysis). The long range versions includes the ones where the force decays as inverse square (see e.g., [@bkc18]). The velocity exponent $\theta$, as is defined before, turns out to be $0.625\pm0.005$ [@bkc18]. Also, mean field models (infinite range) are studied in this context [@bkc42; @bkc43; @bkc44] (with $\theta=1/2$; exactly). An infinite range model, where the elastic force only depends upon the total stretching of the string, has also been studied recently [@bkc45], where the observed velocity exponent value ($\theta=0.83\pm0.01$) is rather close to that found in some experiments [@bkc15]. Rate- and state-dependent friction law -------------------------------------- ### General remarks In a simplified view, an earthquake may be regarded as the rubbing of a fault. From this standpoint, friction laws of faults play a vital role in understanding and predicting the earthquake dynamics. In addition, it should be noted that the celebrated Coulomb-Mohr criterion for brittle fracture involves the (internal) friction coefficient and thus the role of a friction law in earthquake physics is considerable. In this section, the phenomenology of friction and its underlying physical processes are briefly reviewed focusing on the recent developments. Some recent remarkable progress in experiments shall be also introduced, whereas, unfortunately, theoretical understanding of such experiments is rather poor. Thus, we try to propose the problems to be solved by physicists. Before explaining the knowledge obtained in the 20th and 21st centuries, it is instructive to see the ancient (16-17th centuries) phenomenology, which has been referred to as the Coulomb-Amonton’s law: (i) Frictional force is independent of the apparent area of contact. (ii) Frictional force is proportional to the normal load. (iii) Kinetic friction does not depend on the sliding velocity and is smaller than static friction. Among these three, the first two laws do not need any modification to this date, whereas the third law is to be modified and to be replaced by the celebrated rate- and state-dependent friction law, which shall be introduced in the following sections. The Coulomb-Amonton law in its original form is just a phenomenology involving only the macroscopic quantities such as apparent contact area and normal load. It is generally instructive to consider the sub-level (or microscopic) ingredients that underlie such a macroscopic phenomenology. The essential microscopic ingredient in friction is [asperity]{}, which is a junction of protrusions of the surfaces [@Bowden2001; @Rabinowicz1965]. In other words, the two macroscopic surfaces in contact are indeed detached almost everywhere except for asperities. The total area of asperities defines the [true contact area]{}, which is generally much smaller than the apparent contact area. Thus, the macroscopic frictional behavior is mainly determined by the rheological properties of asperity. We write the area of asperity $i$ as $A_i$. Then the total area of true contact reads $$A_{\rm true}=\sum_{i\in{\cal S}} A_i,$$ where ${\cal S}$ denotes the set of the asperities. This set depends on the state of the surfaces such as topography, and is essentially time dependent because the state of the surface is dynamic due to sliding and frictional healing. Due to the stress concentration at asperities, molecules or atoms are directly pushed into contact so that an asperity may be viewed as a grain boundary possibly with some inclusions and impurities [@Bowden2001; @Rabinowicz1965]. Suppose that each asperity has its own shear strength $\sigma_i$, above which the asperity undergoes sliding. It may depend on the degree of grain-boundary misorientation and on the amount of impurities at asperity. For simplicity, however, here we assume $\sigma_i =\sigma_Y$; i.e., the yield stress or shear strength of each asperity is the same. Then the frictional force needed to slide the surface reads $$\label{truecontact} F=\sum_{i\in{\cal S}} A_i \sigma_i \simeq \sigma_Y\sum_{i\in{\cal S}} A_i = \sigma_Y A_{\rm true}.$$ The frictional force is thus proportional to the area of true contact. Dividing Eq. (\[truecontact\]) by the normal force $N$, one obtains the friction coefficient $\mu\equiv F/N$. Using $N=A_{\rm a} P$, where $A_{\rm a}$ is the apparent area of contact and $P$ is the normal pressure, one gets $$\label{frictioncoefficient} \mu=\sum_{i\in{\cal S}} \frac{A_i}{A_{\rm a}}\frac{\sigma_i}{P} \simeq \frac{A_{\rm true}\sigma_Y}{A_{\rm a} P}.$$ Alternatively, one can have $A_{\rm true} / A_{\rm a} = \mu P / \sigma_Y$. This means that the fraction of true contact area is proportional to the pressure normalized by the yield stress, where the friction coefficient is the proportionality coefficient. Assuming that the yield stress of asperity is the same as that of the bulk, we may set $\sigma_Y\sim 0.01G$, where $G$ is the shear modulus. Inserting this and $\mu\simeq0.6$ into Eq. (\[frictioncoefficient\]), one has $A_{\rm true} / A_{\rm a} \sim 60 P/G$. This rough estimate can be confirmed in experiment and numerical simulation [@Dieterich1996; @Hyun2004], where the proportionality coefficient is on the order of $10$. For example, at the normal pressure on the order of kPa, the fraction of true contact is as small as $10^{-5}$. In view of Eq. (\[truecontact\]), the first two laws of Coulomb-Amonton can be recast in the form that [frictional force is proportional to the true contact area, which is independent of the apparent contact area but proportional to the normal load.]{} This constitutes the starting point of a theory on friction, which shall be discussed in the following subsections. The third law of Coulomb-Amonton is just a crude approximation of what we know of today. This should be replaced by the modern law, which is now referred to as the rate- and state- dependent friction (RSF) law. In the next subsection, we discuss the RSF law based on the first two laws of Coulomb-Amonton. ### Formulation Extensive experiments on rock friction have been conducted in 1970s and 80s in the context of earthquake physics. An excellent review on these experimental works is done by Marone (1998). Importantly, these experiments reveal that kinetic friction is indeed not independent of sliding velocity. Thus, the third law of Coulomb-Amonton must be modified. Dieterich devises an empirical law that describes the behavior of friction coefficient (for both steady state and transient state) based on his experiments on rock friction [@Dieterich1979]. Later, the formulation is to some extent modified by Ruina (1983) by introducing additional variable(s) other than the sliding velocity. A new set of variables describes the state of the frictional surfaces so that they are referred to as the [state variables]{}. Although in general state variable(s) may be a set of scalars, in most cases a single variable is enough for the purpose. Hereafter the state variable is denoted by $\theta'$. Using the state variable, the friction law reads $$\label{rsf} \mu = c' + a' \log \frac{V}{V_} + b' \log\frac{V_\theta'}{{\cal L}},$$ where $a'$ and $b'$ are positive nondimensional constants, $c'$ is a reference friction coefficient at a reference sliding velocity $V_$, and ${\cal L}$ is a characteristic length scale interpreted to be comparable to a typical asperity length. In typical experiments, $a'$ and $b'$ are on the order of $0.01$, and ${\cal L}$ is of the order of micrometers. Note that the state variable $\theta'$ has a dimension of time. The state variable $\theta'$ is in general time-dependent so that one must have a time evolution law for $\theta'$ together with Eq. (\[rsf\]). Many empirical laws have been proposed so far in order to describe time-dependent properties of friction coefficient. One of the commonly-used equations is the following [@Ruina1983]. $$\label{dieterich} \dot{\theta'} = 1 - \frac{V}{{\cal L}}\theta',$$ which is now referred to as the Dieterich’s law or the aging law. This describes the time-dependent increase of the state variable even at $V=0$. Meanwhile, other forms of evolution law may also be possible due to the empirical nature of Eq. (\[rsf\]). For example, the following one is also known to be consistent with experiments [@Ruina1983]. $$\label{ruina} \dot{\theta'} = -\frac{V\theta'}{{\cal L}} \log\frac{V\theta'}{{\cal L}},$$ which is referred to as the Ruina’s law or the slip law. In a similar manner, a number of other evolution laws have been proposed so far, such as the composite of the slowness law and the slip law (Kato and Tullis, 2001). Although there have been many attempts to clarify which evolution law is the most suitable, no decisive conclusions have been made. As most of them give the identical result if linearized around steady-sliding state, the difference between each evolution law becomes apparent only at far from steady-sliding state. One may immediately notice that in Eq. (\[ruina\]) the state variable is time-independent at $V=0$ so that it is not very quantitative in describing friction processes to which the healing is relevant. On the other hand, Eq. (\[ruina\]) can describe a relaxation process after the instantaneous velocity switch ($V=V_1$ to $V_2$) better than Eq. (\[dieterich\]), while Eq. (\[dieterich\]) predicts different responses for $V=V_1$ to $V_2$ and for $V=V_2$ to $V_1$, respectively. (Experimental data suggests that they are symmetric.) Also, it is known that Eq. (\[ruina\]) can describe a nucleation process better than Eq. (\[dieterich\]) [@Rubin]. However, we would not go further into the details of the experimental validation of evolution laws and leave it to the review by Marone (1998). Irrespective of the choice of evolution law, a steady state is characterized by $\theta'_{\rm ss}={\cal L}/V$ so that the steady-state friction coefficient at sliding velocity $V$ reads $$\label{steadystate} \mu_{\rm ss} = c' + (a'-b') \log\frac{V}{V_}.$$ Note that, as the nondimensional constants $a'$ and $b'$ are typically on the order of $0.01$, the velocity dependence of steady-state friction is very small; change in sliding velocity by one order of magnitude results in $\sim0.01$ (or even less) change in friction coefficient. It is thus natural that people in the 17th century overlooked this rather minor velocity dependence. However, this velocity dependence is indeed not minor at all but very important to the sliding instability problem: e.g., earthquakes. We also remark that Eq. (\[rsf\]) together with an evolution law such as Eqs. (\[dieterich\]) or (\[ruina\]), well describe the behavior of friction coefficient not only for rock surfaces but also metal surfaces [@Popov2010], two sheets of paper [@Heslot1994], etc. In this sense, the framework of Eq. (\[rsf\]) is rather universal. This universality is partially because the deformation of asperities involves atomistic processes (i.e., creep). One can assume for creep of asperities $\sigma_Y = k_BT/\Omega \log(V/V_0)$, where $\Omega$ is an activation volume and $V_0$ is a characteristic velocity involving the activation energy. Then Eq. (\[frictioncoefficient\]) leads to $$\label{creep_RSF} \mu \simeq \frac{k_BT}{\Omega P_{\rm true}}\log(V/V_0)$$ where $P_{\rm true} = P A_{\rm a}/ A_{\rm true}$ is the actual pressure acting on the asperities. Comparing Eqs. (\[creep\_RSF\]) and (\[rsf\]) with $b'=0$ (no healing), one can infer that $a'=\frac{k_BT}{\Omega P_{\rm true}}$, as previously derived by some authors [@Heslot1994; @Nakatani2001; @Rice2001]. However, we are unaware of a microscopic expression for $b'$ to this date. We are also unaware of any microscopic derivations of evolution laws, such as Eqs. (\[dieterich\]), and (\[ruina\]), whereas an interesting effort to understand the physical meaning of evolution laws can be found in [@Yoshioka1997]. ### Stability of a steady state within the framework of the RSF As we discuss the earthquake dynamics based on the RSF, it is essential to discuss the frictional instability within the framework of the RSF. For simplicity, we consider a body on the frictional surface. The body is pulled by a spring at a constant velocity $V$. $$\label{onebody} M\ddot{X} = -k (X - Vt) - \mu N,$$ where $X$ is the position, $M$ is the mass, $k$ is the spring constant, and $N$ is the normal load. This may be regarded as the simplest model of frictional instability driven by tectonic loading. Suppose that the friction coefficient $\mu$ is given by the RSF law Eq. (\[rsf\]) together with an evolution law. The choice of the evolution law, i.e., Dieterich’s or Ruina’s, does not affect the following discussions as they are identical if linearized around steady state. The motion of the block is uniform in time if the surface is steady-state velocity strengthening ($a' > b'$) or if the spring constant is sufficiently large. For a steady-state velocity weakening surface the steady-sliding state undergoes Hopf bifurcation below a critical spring constant. A linear stability analysis [@Ruina1983; @Heslot1994] shows that the steady-sliding state is unstable if $$\label{kc} k < k_{\rm crt} \equiv \frac{N}{{\cal L}} (b'-a').$$ This relation plays a central role in various earthquake models, in which a constitutive law is given by the RSF law. This shall be discussed in section III. An important consequence of Eq. (\[kc\]) is that the tectonic motion is essentially stable if $a'-b'>0$. Namely, steady sliding is realized in the region where $a'-b' > 0$, whereas the motion may be unstable if $a'-b'<0$. In addition, smaller ${\cal L}$ widens the parameter range of unstable motion. Although the above analyses involve a one-body system, the stability condition Eq. (\[kc\]) appears to be essentially the same in many-body systems and continuum systems. Thus, provided that Eq. (\[onebody\]) applies, it is widely recognized in seismology that seismogenic zone has negative a′ − b′ and smaller L, whereas aseismic zone has the opposite tendency. Beyond the RSF law ------------------ It should be remarked that the RSF law has certain limit of its application. Many experiments reveal that the RSF no longer holds at high sliding velocities. This may be due to various mechano-chemical reactions that are induced by the frictional heat, which typically lubricate surfaces to a considerable degree; the friction coefficient becomes as low as $0.2$ or even less than $0.1$ [@Tsutsumi1997; @Goldsby2002; @DiToro2004; @Hirose2005; @Mizoguchi2006]. If such lubrication occurs in a fault, the fault motion is accelerated to a considerable degree and thus such effects have been paid much attention to during the last decade. Feedback of frictional heat may be indeed very important to faults, because the normal pressure in a seismogenic zone is of the order of $100$ MPa. (Note that, however, the presence of high-pressure pore fluid may reduce the effective pressure.) As this area of study is relatively new, the current status of our understanding on such mechano-chemical effects is rather incomplete. Taking the rapid development of this area into account, here we wish to mention some of the important experiments briefly. ### Flash heating Friction under such high pressure may lead to melting of rock. There have been some reports of molten rock observed in fault zones, which implies that the temperature is elevated up to $2000$ K during earthquakes. A series of pioneering works on frictional melting in the context of earthquake has been conducted by Shimamoto and his coworkers. They devised a facility for rock friction at high speed under high pressure to find a behavior very different from that of the RSF law. Steady-state friction coefficient typically shows remarkable negative dependence on sliding velocity and the relaxation to steady state is twofold [@Tsutsumi1997; @Hirose2005]. At higher sliding velocity (e.g., $1$ m/sec), friction coefficient decreases as low as $0.2$ (or even less), whereas the typical value at quasistatic regime is around $0.7$. We wish to stress that such a drastic decrease of friction coefficient cannot be explained in terms of the RSF law, where the change of steady-state friction coefficient is of the order of $0.01$ even if the sliding velocity changes by a few orders of magnitude. (recall Eq. (\[steadystate\]), where $a'$ and $b'$ are both on the order of $0.01$.) Thus, the mechanism of weakening must be qualitatively different from that of the RSF law. Indeed, in such experiments, molten rock is produced on surfaces due to the frictional heat. It is considered that so produced melt lubricates the surfaces to result in unusually low friction coefficient. In view of Eq. (\[truecontact\]), frictional melting must take place at asperities, where the frictional heat is produced. Thus, before the entire surface melts, asperities experience very high temperature, which may change the constitutive law. Such asperity heating has been also known in tribology and is referred to as [flash heating]{}. Rice applied this idea to fault friction in order to estimate the feasibility of flash heating in earthquake dynamics. His argument is as follows [@Rice2006]: The power input to asperity $i$ is $\sigma_Y A_i V$, which is to be stored in the proximity of asperity. As discussed later, it is essential to assume here that heat conduction is one-dimensional; i.e., temperature gradient is normal to the surface, whereas uniform along the transverse directions. The produced heat invades toward the bulk over the distance $\sqrt{D_{\rm th} t}$, where $D_{\rm th}$ is thermal diffusivity. Thus, frictional heat is stored in the small volume of $A_i\sqrt{\alpha t}$. Writing the average temperature of this hot volume as $T(t)$, the deposited thermal energy reads $c_P \rho A_i\sqrt{D_{\rm th} t}(T(t)- T_0)$, where $c_P$ is the isobaric specific heat, $\rho$ is the mass density, and $T_0$ is the ambient temperature. Then the energy balance leads to $$T(t) - T_0 \simeq \frac{\sigma_Y V}{\rho c_P}\sqrt{\frac{t}{D_{\rm th}}}.$$ This indicates that the surface temperature increases with time as $\sqrt{t}$. Writing $T_w$ as the critical temperature above which an asperity looses its shear strength, then the duration $t_w$ for the temperature to be elevated up to the critical temperature reads $$\label{tw} t_w = D_{\rm th} \left[\frac{\rho c_P(T_w - T_0)}{\sigma_Y V}\right]^2.$$ This heating process is limited to the duration or lifetime of an asperity contact. If we write the longitudinal dimension of each asperity as ${\cal L}_i$, the lifetime of an asperity is estimated as ${\cal L}_i/V$. Thus, weakening of an asperity occurs if and only if $t_w \le {\cal L}_i/V$. Taking Eq. (\[tw\]) into account, this condition may be written as $$\label{fh_condition} V \ge \frac{D_{\rm th}}{{\cal L}_i} \left[\frac{\rho c_P(T_w - T_0)}{\sigma_Y}\right]^2.$$ Neglecting the statistics of ${\cal L}_i$, one gets the characteristic sliding velocity $V_w$ above which weakening occurs. $$\label{Vw} V_w = \frac{D_{\rm th}}{{\cal L}} \left[\frac{\rho c_P(T_w - T_0)}{\sigma_Y}\right]^2.$$ Alternatively, from Eq. (\[fh\_condition\]), the maximum size of asperity that does not melt at the sliding velocity $V$ is given by $$\label{Lmax} {\cal L}_{\rm max} = \frac{D_{\rm th}}{V} \left[\frac{\rho c_P(T_w - T_0)}{\sigma_Y}\right]^2.$$ The proportion of non-melting asperity may be approximated by ${\cal L}_{\rm max}/ {\cal L}$. Assuming that the friction coefficients of a molten asperity and a non-melting one are given as $$\label{weakening} \mu=\left\{ \begin{array}{@{\,}ll} f_1, \ \ (T < T_w) \\ f_2. \ \ (T > T_w), \end{array} \right.$$ the average friction coefficient reads $$\begin{aligned} \mu &=& f_1 \frac{{\cal L}_{\rm max}}{{\cal L}} + f_2 \left(1-\frac{{\cal L}_{\rm max}}{{\cal L}} \right) \\ \label{fh_mu} &=& f_2 + (f_1-f_2) \frac{V_w}{V}.\end{aligned}$$ The friction coefficient decreases as $V^{-1}$ at high slip velocity $V\ge V_w$. Taking $\alpha=1 {\rm mm^2/s}$, $\rho c_P =4$ ${\rm MJ /m}^3K$, $D=5 \mu{\rm m}$, $T_w - T_0=700$K, and $\sigma_Y = 0.02$ to $0.1$ G (shear modulus) = $0.6$ to $3$ GPa, the characteristic velocity $V_w$ is $0.5$ to $14$ m/s. This does not contradict rock experiments on melting-induced weakening. Also, comparison of Eq. (\[fh\_mu\]) with experiments is not inconsistent, although $f_1$ and $f_2$ are fitting parameters. Note that the above discussion does not depend on the apparent normal pressure, as the pressure on asperity is approximately the yield stress (of uniaxial compression) irrespective of the apparent normal pressure. Thus, flash melting could occur in principle even when the apparent pressure is very low as long as the sliding velocity is larger than $V_w$ given by Eq. (\[Vw\]). However, in an experiment conducted at relatively low pressures, the threshold velocity is an order of magnitude smaller than the prediction of Eq. (\[Vw\]) [@Kuwano2011]. This may be because other relevant mechanisms are responsible for dynamic weakening observed in experiments, but the answer is yet to be given. It is also important to notice that in the above discussion the assumption of one dimensional heat conduction is essential; i.e., the frictional heat is not transferred in the horizontal directions but only to the normal direction. This assumption implies that the thermal diffusion length $\sqrt{\alpha t_w}$ must be smaller than the height of a protrusion that constitutes an asperity. Assuming that the height of a protrusion is proportional to a horizontal dimension $L_i$, this condition leads to $$\label{fh_condition2} {\cal L}_i \ge \frac{D_{\rm th} \rho c_P (T_w-T_0)}{\sigma_Y V}.$$ Because it is estimated in general that $\rho c_P (T_w-T_0) > \sigma_Y$, Eq. (\[fh\_condition2\]) immediately follows from Eq. (\[fh\_condition\]). Thus, the assumption of one-dimensional heat conduction may be sound. If the asperities are sufficiently small so that the thermal diffusion length exceeds the height of protrusions, the assumption of one-dimensional heat conduction is violated. A good example is friction of nanopowders, in which a typical size of the true contact area is on the order of nanometers [@Han2011]. Interestingly, one can still observe dynamic weakening similar to those caused by flash melting, but they did not attribute this behavior to flash melting, because the duration of contact between nano-grains was too short to cause the significant temperature increase. The physical mechanism of such weakening is still not clear. (Silica-gel lubrication may be ruled out as the material they used is silica-free.) ### Frictional melting and thermal pressurization There is yet another class of weakening phenomena called frictional melting; the melt is squeezed out of asperities to fill the aperture between the two surfaces. Such a situation can occur if the surfaces are rubbed for sufficiently long time. If this process occurs, the melt layer supports the apparent normal pressure to reduce the effective pressure at asperities and ultimately hinders solid-solid contact. This leads to the disappearance of the asperities; i.e., no solid-solid contacts between the surfaces but a thin layer of melt under shear. There are some analyses of such systems assuming the Arrhenius-type viscosity [@Fialko2005; @Nielsen2008]. In doing so, one can predict the shear traction is proportional to $P^{1/4}$, where $P$ is the normal pressure. The quantitative validation of such theories is yet to be done. It may be noteworthy here that the viscosity of such a liquid film involves rather different problem; nanofluidics. The melt may be regarded as nanofluid, the viscosity of which may be very different from the ordinary ones. The shear flow of very thin layers of melt (under very high pressure) may be unstable due to the partial crystallization [@Thompson1992] Till this date, the effect of nanofluidics on frictional melting is not taken into account and left open to physicists. Meanwhile, the evidence of frictional melting of a fault is not very often found in core samples or in outcrops. As faults generally contain fluid, frictional heat increases the fluid temperature as well. As a result, the fluid pressure increases and the effective pressure on solid-solid contact decreases. Therefore, the frictional heat production generally decreases in the presence of fluid. In the simplest cases where the fault zone is impermeable, the effective friction (and the produced frictional heat) may vanish as the fluid pressure can be as large as the rock pressure [@Sibson1973]. This is referred to as thermal pressurization, and a large number of work has been devoted to such dynamic interaction between frictional heat and the fluid pressure. More detailed formulations incorporate the effect of fluid diffusion with nonzero permeability of host rocks[@Lachenbruch1980; @Mase1987]. In these analyses, the extent of weakening is enhanced if a fault zone has smaller compressibility and permeability. Although this behavior is rather trivial in a qualitative viewpoint, some nontrivial behaviors are found in a model where the permeability is assumed to be a dynamic quantity coupled with the total displacement [@Suzuki2010]. However, it is generally difficult to judge the validity of a model from observations and thus we do not discuss this problem further. ### Other mechanochemical effects In some systems, anomalous weakening of friction ($\mu\sim0.2$) can be observed at sliding velocities much lower than the critical velocity for flash heating (Eq. (\[Vw\])). Typically, one can observe weakening at sliding velocity of the order of mm/s. Thus, there might be mechanisms for drastic weakening other than frictional melting. Such experiments are typically conducted with complex materials like fault gouge taken from a natural fault so that there may be many different mechanisms of weakening depending on the specific compositions of rock species. Among them, the mechanism that might bear some robustness is the lubrication by silica-gel production [@Goldsby2002; @DiToro2004]. In several experiments on silica-rich rock such as granite, SEM observation of the surfaces reveals a silica-gel layer that experienced shear flow. The generation of silica-gel may be attributed to chemical reactions between silica and water in the environment. This silica-gel is intervened between the surfaces to result in the lubrication of fault. Although the details of the chemical reactions is not very clear, the mechanics of weakening may be essentially the same as that of flash heating and melting, because in the both cases the cause of weakening is some soft materials (or liquids) that are produced by shear and intervened at asperities. However, in the case of silica-gel formation, the thixotropic nature of silica-gel may result in peculiar behaviors of friction, as observed in experiment by Di Toro et al (2004). In addition, we wish to add several other mechanisms that lead to anomalous weakening. Han et al. (2007) found friction coefficient as low as $0.06$ in marble under relatively high pressure ($1.1$ to $13.4$ MPa) and high sliding velocity ($1.3$ m/s). Despite the utilization of several techniques for microstructural observation, they could not observe any evidence for melting such as glass or amorphous texture but only a layer of nanoparticles produced by thermal decomposition of calcite due to frictional heating. Mizoguchi et al. (2006) also found friction coefficient as low as $0.2$ in fault gouge taken from a natural fault, where they also could not find any evidences for melting. To this date, the mechanism of such frictional weakening at higher sliding velocity is not clear. It might be important to notice that these samples inevitably include a large amount of sub-micron grains that are worn out by high-speed friction [@Han2007; @Hayashi2010]. They may play an important role in weakening at high sliding velocities. The grain size distribution of fault gouge is typically well fitted by a power law with exponent $-2.6$ to $-3.0$ [@Chester1998] so that smaller grains cannot be neglected in terms of volume fraction. The exponent appears to be common to laboratory [@Marone1989] or numerical experiments of wear [@Abe2009]. Rheology of such fractal grains has not been investigated in a systematic manner, notwithstanding a pioneering computational work [@Morgan1999]. The influence of grains to friction shall be discussed in detail in the next subsection. ### Effect of the third body: granular friction Previously, we considered the situation where two surfaces were in contact only at asperities. This is generally not the case if the asperities are worn out to be free particles that are intervened between the two surfaces. In this case, a system can be regarded as granular matter that is sheared by the two surfaces. The core of a natural fault always consists of powdered rock [@Chester1998], which is produced by the fault motion of the past. Thus, friction on fault is closely related to the rheology of granular rock. As is briefly mentioned in the previous subsection, earthquake physics involves a wide range of sliding velocities (or shear rate) ranging from tectonic time scale (e.g., nm/s) to coseismic scale (m/s). It is thus plausible that the rheological properties of granular matter is qualitatively different depending on the range of sliding velocities. Here we define two regimes for granular friction: quasititatic and dynamic regimes. In the quasistatic regime, the frictional properties of granular matter is described by the RSF. However, some important properties will be remarked that are not observed for bare surfaces. In the dynamic regime, one may expect dynamic strengthening as observed in numerical simulations [@GDR2004; @DaCruz2005]. However, at the same time one may also expect weakening due to various mechanochemical reactions [@Mizoguchi2006; @Hayashi2010]. The rheological properties that are experimentally observed are determined by the competition of these two ingredients. Here we review the essential rheological properties of granular matter in these two regimes. In experiments on quasistatic deformation, friction of granular matter seems to obey the RSF law. However, some important properties that are different from those of bare surfaces should be remarked. 1. Velocity dependence of steady-state friction appears to be affected by the layer thickness. In particular, the value of $a'-b'$ in Eq. (\[steadystate\]) is an increasing function of the layer thickness. 2. The value of $a'-b'$ appears to have negative dependence on the total displacement applied to a system. This is true both for granular matter and bare surfaces. 3. Transient behaviors can be described by either Dieterich’s or Ruina’s laws, as in the case of bare surfaces. The characteristic length in an evolution law is proportional to the layer thickness. These experimental observations are well summarized and discussed in detail by Marone (1998). We thus shall not repeat them here and just remark the essential points described above. As to the first point, there is no plausible explanation to this date. It appears that the second point could be merged into the first point if the effective layer thickness (i.e., the width of shear band) decreases as the displacement increases. However, we wish to remark that it is also true in the case of bare surfaces, where the effective layer thickness is not a simple decreasing function of the displacement. Thus, the second point cannot be explained in terms of the thickness. The third point indicates that the shear strain is a more appropriate variable than the displacement of the boundary for the description of the time evolution of friction coefficient. This may be reasonable as the duration of contact between grains is inversely proportional to the shear rate. However, the derivation of evolution laws (either Dieterich’s or Ruina’s) from the grain dynamics is not known to this date. To construct a theory that can explain these three properties based on the nature of granular matter is still a challenge to statistical physicists. Then we discuss the dynamic regime. Rheology of granular matter in the dynamic regime is extensively investigated in statistical physics [@GDR2004; @DaCruz2005]. As to the steady-state friction coefficient, the shear-rate dependence is one of the main interests also in statistical physics. There are many ingredients that potentially affect the friction coefficient of granular matter: the grain shape, degree of inelasticity (coefficient of restitution), friction coefficient between grains, the stiffness, pore-fluid, etc. Shape dependence is very important to granular friction, but theoretical understanding of this effect is still very poor. Thus, for simplicity, we neglect the shape effect and involve only spherical grains. Furthermore, we limit ourselves to the effects of shear rate, stiffness, mass and diameter of grains, coefficient of restitution, and intergrain friction. This means that we neglect time- or slip-dependent deformation of the grain contacts, such as wear [@Marone1989] or frictional healing [@Bocquet1998]. The effects of pore-fluid is also neglected; i.e., we discuss only the dry granular matter here. With such idealization, one can make a general statement on a constitutive law by dimensional analysis. The friction coefficient of granular matter is formally written as $$\label{dimension1} \mu = \mu(P, m, d, \dot{\gamma}, Y, \mu_e, e),$$ where $P$ is the normal pressure, $m$ is the mass, $d$ is the diameter, $\dot{\gamma}$ is shear rate, $Y$ is the Young’s modulus of grains, $\mu_e$ is the intergrain friction coefficient, and $e$ is the coefficient of restitution. (It should be noted that one assumes a single characteristic diameter $d$ in Eq. (\[dimension1\]).) From the viewpoint of dimensional analysis, the arguments on the left hand side of Eq. (\[dimension1\]) must be nondimensional numbers. $$\mu = \mu(I, \kappa, \mu_e, e),$$ where $I= \dot{\gamma}\sqrt{m/Pd}$ and $\kappa = Y/P$. Thus, the friction coefficient of granular matter depends in principle on these four nondimensional parameters. Many numerical simulations reveal that $\mu$ is rather insensitive to $\kappa$ and $\epsilon$, and the shear-rate dependence is mainly described by $I$. This nondimensional number $I$ is referred to as the inertial number. Importantly, the dependence on $I$ is positive in numerical simulations [@GDR2004; @DaCruz2005; @Hatano2007]; namely, the shear-rate dependence is positive. It is important to remark that the negative shear-rate dependence, which is ubiquitously observed in experiments, cannot be reproduced in numerical simulation. This is rather reasonable because the origin of the negative velocity dependence is the time dependent increase of true contact, whereas in simulation the parameters are time-independent. Experiments in the context of earthquake physics are conducted at relatively high pressures at which the frictional heat affects the physical state of granular matter. In some experiments [@Mizoguchi2006; @Hayashi2010], remarkable weakening ($\mu\sim0.1$) is observed. Because such anomalous behaviors may involve shear-banding as well as various chemical reactions such as thermal decomposition or silica-gel formation, the frictional properties should depend on the detailed composition of rock species contained in granular matter. These weakening behaviors must be further investigated by extensive experiments. So far we discussed the steady-state friction coefficient, but the description of transient states are also important in understanding the frictional instability (and earthquake dynamics). An evolution law for quasistatic regime is indeed essentially the same as that for bare surfaces; namely, aging law or slip law [@Marone1998]. An evolution law in the dynamic regime is well described by the linear relaxation equation even for relatively large velocity change [@Hatano2009]. $$\label{granular_evolution} \dot{\mu} = - \tau ^{-1} \left[\mu(t) -\mu_{\rm ss}\right],$$ where $\mu_{\rm ss}$ is the steady-state friction coefficient that depends on the sliding velocity and $\tau$ is the relaxation time. It is found that $\tau$ is the relaxation time of the velocity profile inside granular matter and is scaled with $\sqrt{m/Pd}$. Thus, importantly, the inertial number, which describes steady-state friction may be written using $\tau$ as $I\simeq\tau\dot{\gamma}$; i.e., the shear rate multiplied by the velocity relaxation time. It may be noteworthy that the inertial number is an example of Deborah number, which is in general the internal relaxation time normalized by the experimental time scale. Note the difference from the conventional evolution law in the framework of the RSF law; Eqs. (\[dieterich\]) and (\[ruina\]). It is essential that Eq. (\[granular\_evolution\]) does not contain any length scale but only the time scale. This means that the relaxation process of high-speed granular friction takes time rather than the slip distance. However, we wish to stress that the validity of Eq. (\[granular\_evolution\]) is found only in simulation on dry granular matter and is not verified in physical experiment. Microscopic theories of friction -------------------------------- Many attempts were made in explaining friction from an atomistic point of view. Of course, such effort are meaningful only when the surfaces are smooth and the atomistic properties determine friction. This approach has gained importance in recent years, because of advancement of technology in this field. Due to Atomic Force Microscopy (AFM) etc., sliding surfaces can now be probed upto atomic scales. Also, present day computers allow large scale molecular dynamics simulation that helps in understanding the atomic origin of friction. In this approach, the atomic origin of friction forces are investigated (see also [@bkc50; @bkc48; @bkc51]). In this purpose, two atomically smooth surfaces are taken and by writing down the equations of motion, friction forces are calculated. Effect of inhomogeneity, impurity, lubrication and disorder in terms of vacancies of atoms are also considered. One of the foremost attempts to model friction from atomic origin was of Tomlinson’s [@bkc19]. In this model, only one atomic layer of the surfaces in contact are considered. In particular, the lower surface in considered to be rigid and provide a periodic (sinusoidal) potential for the upper body. The contact layer of the upper body is modelled by mutually disconnected beads (atoms) which are attached elastically to the bulk above. This model is, of course, oversimplified. The main drawback is that no interaction between the atoms of the upper body is considered. ### Frenkel Kontorova model Frenkel-Kontorova [@bkc20] model overcomes some of these difficulties. In this model the surface of the sliding object is modelled by a chain of beads (atoms) connected harmonically by springs. The base is again represented by a sinusoidal potential. The Hamiltonian of the system can, therefore, be written as $$H=\sum\limits_{i=1}^N[\frac{1}{2}K(x_{i+1}-x_i-a)^2+V(x_i)],$$ where, $x_i$ is the position of the $i$-th atom, $a$ is the equilibrium spacing of the chain and $V(x)=-V_0cos(\frac{2\pi x}{b})$. Clearly, there are two competing lengths in this model, viz. the equilibrium spacing of the upper chain ($a$) and the period of the substrate potential ($b$). While the first term tries to keep the atoms in their original positions, the second term tries to bring them in the local minima of the substrate potential. Simultaneous satisfaction of these two forces is possible when the ratio $a/b$ is commensurate. The chain is then always pinned to the substrate in the sense that a finite force is always required to initiate sliding. Below that force, average velocity vanishes at large time. However, interesting phenomena takes place when the ratio $a/b$ is incommensurate. In that case, upto a finite value of the amplitude of the substrate potential, the chain remains “free”. In that condition, for arbitrarily small external force, sliding is initiated. The hull function [@bkc21] remains analytic. Beyond the critical value of the amplitude, the hull function is no longer analytic and a finite external force is now required to initiate sliding. This transition is called the breaking of analyticity transition or the Aubry transition [@bkc21] (for extensive details see @bkc49). ### Two-chain model The Frenkel-Kontorova model has been generalised in many ways viz., extension in higher dimensions, effect of impurity, the Frenkel-Kontorova-Tomlinson model and so on (see @bkc48 and references therein). But one major shortcoming of the Frenkel-Kontorova model is that the substrate or the surface atoms of the lower substance are considered to be rigidly fixed in their equilibrium position. But for the same reason why the upper surface atoms should relax, the lower surface atoms should relax too. In the two chain model of friction [@bkc22] this question is addressed. In this model, a harmonically connected chain of atoms is being pulled over another. The atoms have only one degree of freedom in the direction parallel to the external force. The equations of motion of the two chains are $$\begin{aligned} m_a\gamma_a(\dot{x_i}-\langle\dot{x_i}\rangle)&=& K_a(x_{i+1}+x_{i-1}-2x_i)\nonumber \\ &&+\sum\limits_{j\in b}^{N_b}F_I(x_i-y_j)+F_{ex},\end{aligned}$$ $$\begin{aligned} m_b\gamma_b(\dot{y_i}-\langle \dot{y_i}\rangle)&=& K_b(y_{i+1}+y_{i-1}-2y_i)\nonumber \\&+&\sum\limits_{j\in a}^{N_a}F_I(y_i-x_j) -K_s(y_i-ic_b)\end{aligned}$$ where, $x_i$ and $y_i$ denotes the equilibrium positions of the upper and lower chain respectively, $m$’s represent the atomic masses, $\gamma$’s represent the dissipation constant, $K$’s the strength of inter-atomic force and $N$’s the number of atoms in each chain, $c$’s the lattice spacing, while suffix $a$ denotes upper chain and suffix $b$ denotes the lower chain. $F_{ex}$ is the external force and $F_I$ is the inter-chain force between the atoms, which is derived from the following potential $$U_I=-\frac{K_I}{2}\exp(-4(\frac{x}{c_b})),$$ where $K_I$ is the interaction strength. It is argued that the frictional force is of the form $$-\sum\limits_i\sum\limits_j\langle F_I(x_i-y_j)\rangle_t=N_a\langle F_{ex}\rangle_t.$$ It is then shown by numerical analysis that the velocity dependence of the kinetic frictional force becomes weaker as the static friction increases (tuned by different $K$’s). The velocity dependence essentially vanishes when static frictional force is increased, giving one of the Amonton-Coulomb laws. ![The variations of the maximum static friction with the amplitude ($K_I$) of the inter-chain potential for different values of the lower-chain stiffness ($K_s$). The limit $K_s\to\infty$ corresponds to Frenkel-Kontorova model. But it is clearly seen that even for finite $K_s$ (i.e., when the lower chain can relax) there is a finite value of inter-chain potential amplitude upto which the static friction is practically zero and it increases afterwards, signifying Aubry transition [@bkc22]. From [@bkc22].[]{data-label="twochain-bkc"}](twochain-bkc.eps){width="6.0cm"} In this case, the lower chain atoms, which forms the substrate potential, is no longer rigidly placed. Still, the breaking of analyticity transition is observed. Fig. \[twochain-bkc\] shows the variation of the maximum static frictional force with interaction potential strength. For different values of the rigidity with which the lower chain is bound ($K_s$), different curves are obtained. This indicates a pinned state even for finite rigidity of the lower chain. ### Effect of fractal disorder Effects of disorder and impurity have been studied in the microscopic models of friction. Also there have been efforts to incorporate the effect of self-affine roughness in friction. In Ref. [@bkc23], the effect of disorder on static friction is considered. A two-chain version of the Tomlinson model is considered. The self-affine roughness is introduced by removing atoms and keeping the remaining ones arranged in the form of a Cantor set. The Cantor set, as is discussed before, is a simple prototype of fractals. Instead of considering the regular Cantor set, here a random version of it is used. A line segment \[0,1\] is taken. In each generation, it is divided into $s$ equal segments and $s-r$ of those are randomly removed. In this way, a self similar disorder is introduced, which is present only in the statistical sense, rather than strict geometric arrangement. ![Schematic representation of the two chain version of the Tomlinson model with (a) no disorder, (b) Cantor set disorder, (c)the effective substrate potential [@bkc23].[]{data-label="model"}](model.ps){width="6.0cm"} ![The overlap distribution for $s=9, r=8$ is shown. The dotted curve shows the average distribution with random off-set and the continuous curve is that without random off set. The distribution is qualitatively different from the Gaussian distribution expected for random disorder [@bkc23].[]{data-label="overlap"}](overlap.ps){width="6.0cm"} This kind of roughness is introduced in both the chains. Then the inter-chain interaction is taken to be very short range type. Only when there is one atom exactly over the other (see Fig. \[model\]) there is an attractive interaction. In this way, the maximum static friction force can be calculated by estimating the overlap of these two chains. It turns out that the static friction force has a distribution, which is qualitatively different from what is expected if a random disorder or no disorder is present. The scaled (independent of generation) distribution of overlap or static friction looks like [@bkc23] $$f^{s,r}(x/R)/R=\sum\limits_{j=1}^r\tilde{c}^{s,r}(j)(f^{s,r}\dots \mbox{$j-1$ terms}\dots f^{s,r})(x),$$ where $R=r^2/s$ and $\tilde{c}^{s,r}(x)=\frac{{}^rC_x{}^{s-r}C_{r-x}}{{}^sC_r}$. For a particular ($s,r$) combination (9,8), the distribution function is shown in Fig. \[overlap\]. It clearly shows that the distribution function is qualitatively different from the Gaussian distribution expected if the disorder were random. Earthquake models and statistics I: Burridge-Knopoff and Continuum models ========================================================================= In the previous section, we reviewed the basic physics of friction and fracture, which constitutes a “microscopic” basis for our study of “macroscopic” properties of an earthquake as a large-scale frictional instability. Some emphasis was put on the RSF law now regarded as the standard constitutive law in seismology. In this and following sections, we wish to review the present status of our research on various types of statistical physical models of earthquakes introduced to represent their “macroscopic” properties. Statistical properties of the Burridge-Knopoff model ---------------------------------------------------- ### The model One of the standard models widely employed in statistical physical study of earthquakes might be the Burridge-Knopoff (BK) model (Rundle, 2003; Ben-Zion, 2008). The model was first introduced in Burridge and Knopoff (1967). Then, Carlson, Langer and collaborators performed a pioneering study of the statistical properties of the model (Carlson and Langer, 1989a; Carlson and Langer, 1989b; Carlson et al., 1991; Carlson, 1991a; Carlson, 1991b; Carlson, Langer and Shaw, 1994), paying particular attention to the magnitude distribution of earthquake events and its dependence on the friction parameter. In the BK model, an earthquake fault is simulated by an assembly of blocks, each of which is connected via the elastic springs to the neighboring blocks and to the moving plate. Of course, the space discretization in the form of blocks is an approximation to the continuum crust, which could in principle give rise to an artificial effect not realized in the continuum. Indeed, such a criticism against the BK model employing a certain type of friction law, [e.g.]{}, the purely velocity-weakening friction law to be defined below in 2\[A\], was made in the past (Rice, 1993), which we shall return to later. ![ The Burridge-Knopoff (BK) model in one dimension. []{data-label="BKmodel"}](BKmodel.eps) In the BK model, all blocks are assumed to be subjected to friction force, the source of nonlinearity in the model, which eventually realizes an earthquake-like frictional instability. As mentioned in section II, the standard friction law in modern seismology might be the RSF law. In order to facilitate its computational efficiency, even simpler friction law has also been used in simulation studies made in the past. We first introduce the BK model in one dimension (1D). Extension to two dimensions (2D) is straightforward. The 1D BK model consists of a 1D array of $N$ identical blocks, which are mutually connected with the two neighboring blocks via the elastic springs of the elastic constant $k_c$, and are also connected to the moving plate via the springs of the elastic constant $k_p$, and are driven with a constant rate: See Fig. \[BKmodel\]. All blocks are subjected to the friction force $\Phi$, which is the only source of nonlinearity in the model. The equation of motion for the $i$-th block can be written as $$m \ddot U_i=k_p (\nu ' t'-U_i) + k_c (U_{i+1}-2U_i+U_{i-1})-\Phi_i, \label{eq-1DBK}$$ where $t'$ is the time, $U_i$ is the displacement of the $i$-th block, $\nu '$ is the loading rate representing the speed of the moving plate, and $\Phi_i$ is the friction force at the $i$-th block. In order to make the equation dimensionless, the time $t'$ is measured in units of the characteristic frequency $\omega =\sqrt{k_p/m}$ and the displacement $U_i$ in units of the length $L^=\Phi_0/k_p$, $\Phi_0$ being a reference value of the friction force. Then, the equation of motion can be written in the dimensionless form as $$\ddot u_i=\nu t-u_i+l^2(u_{i+1}-2u_i+u_{i-1})-\phi_i, \label{eq-1DBK-normalized}$$ where $t=t'\omega $ is the dimensionless time, $u_i\equiv U_i/L^$ is the dimensionless displacement of the $i$-th block, $l \equiv \sqrt{k_c/k_p}$ is the dimensionless stiffness parameter, $\nu =\nu '/(L^\omega)$ is the dimensionless loading rate, and $\phi_i\equiv \Phi_i/\Phi_0$ is the dimensionless friction force at the $i$-th block. The corresponding equation of motion of the 2D BK model is given in the dimensionless form by $$\begin{array}{ll} \ddot u_{i,j}=\nu t-u_{i,j}+l^2(u_{i+1,j}+u_{i,j+1} \ \ \ \ \ \\ \ \ \ \ \ +u_{i-1,j}+u_{i,j-1}-4u_{i,j})-\phi_{i,j} , \end{array}$$ where $u_{i,j}\equiv U_{i,j}/L^$ is the dimensionless displacement of the block ($i,j$). It is assumed here that the displacement of each block occurs only along the direction of the plate drive. The motion perpendicular to the plate motion is neglected. Often (but not always), the motion in the direction opposite to the plate drive is also inhibited by imposing an infinitely large friction for $\dot u_i<0$ (or $\dot u_{i,j}<0$) in either case of 1D or 2D. It is also often assumed both in 1D and 2D that the loading rate $\nu$ is infinitesimally small, and put $\nu=0$ during an earthquake event, a very good approximation for real faults (Carlson et al., 1991). Taking this limit ensures that the interval time during successive earthquake events can be measured in units of $\nu^{-1}$ irrespective of particular values of $\nu$. Taking the $\nu \rightarrow 0$ limit also ensures that, during an ongoing event, no other event takes place at a distant place independently of this ongoing event. ### The friction law The friction force $\Phi$ causing a frictional instability is a crucially important element of the model. Here, we refer to the following two forms for $\Phi$; \[A\] a velocity weakening friction force (Carlson and Langer, 1989a), and \[B\] a rate-and-state dependent friction (RSF) law (Dieterich, 1979; Ruina, 1983; Marone, 1998; Scholz, 1998; Scholz, 2002). \[A\] In this velocity-weakening friction force, one simply assumes that the friction force $\phi=\phi(\dot u_i)$ is a unique function of the block velocity $\dot u_i$. In order for the model to exhibit a frictional instability corresponding to earthquakes, one needs to assume a velocity-weakening force, [i.e.]{}, $\phi (\dot u_i)$ needs to be a decreasing function of $\dot u_i$. The detailed form of $\phi(\dot u_i)$ would be irrelevant. The form originally introduced by Carlson and Langer has widely been used in many subsequent works, that is (Carlson et al, 1991), $$%\[ \phi(\dot u_i) = \left\{ \begin{array}{ll} (-\infty, 1], & \ \ \ \ {\rm for}\ \ \dot u_i\leq 0, \\ \frac{1-\delta}{1+2\alpha \dot u_i/(1-\delta )}, & \ \ \ \ {\rm for}\ \ \dot u_i>0, \end{array} \right. %\]$$ where the maximum value corresponding to the static friction has been normalized to unity. This normalization condition $\phi(\dot u_i=0)=1$ has been utilized to set the length unit $L^$. The friction force is characterized by the two parameters, $\delta$ and $\alpha$. The former, $\delta$, introduced in (Carlson et al.,1991) as a technical device facilitating the numerics of simulations, represents an instantaneous drop of the friction force at the onset of the slip, while the latter, $\alpha$, represents the rate of the friction force getting weaker on increasing the sliding velocity. As emphasized by Rice (Rice, 1993), this purely velocity-weakening friction law applied to the discrete BK model did not yield a sensible continuum limit. To achieve the sensible continuum limit, one then needs to introduce an appropriate short-length cutoff by introducing, [e.g.]{}, the viscosity term as was done in (Mayers and Langer, 1993): See also the discussion below in subsecton 6. We note that, in several simulations on the BK model, the slip-weakening friction force (Ida, 1972; Shaw, 1995; Myers et al., 1996), where the friction force is assumed to be a unique function of the slip distance $\phi(u_i)$, was utilized instead of the velocity-weakening friction force. Statistical properties of the corresponding BK model, however, seem not so different from those of the velocity-weakening friction force. Real constitutive relations is of course more complex, neither purely velocity-weakening nor slip-weakening. As discussed in section II, the RSF friction law was introduced to account for such experimental features, which we now refer to. \[B\] From Eq. (\[rsf\]), friction force in the BK model is given by $$\begin{aligned} \phi_i=\{c' +a'\log(\frac{v'_i}{v'_}) +b'\log\frac{v'_ \theta'_i}{{\cal L}}\}{\cal N},\end{aligned}$$ where ${\cal N}$ is an effective normal load. See section II.C for the other quantities and parameters. Among the several evolution laws, we use the aging (slowness) law (Eq.(\[dieterich\])). $$\begin{aligned} \frac{d\theta'_i}{dt'}=1-\frac{v'_i\theta'_i}{{\cal L}}. \label{slowness}\end{aligned}$$ Under the evolution law above, the state variable $\theta'_i$ grows linearly with time at a complete halt $v'_i=0$ reaching a very large value just before the seismic rupture, while it decays very rapidly during the seismic rupture. The equation of motion can be made dimensionless by taking the length unit to be the characteristic slip distance ${\cal L}$ and the time unit to be the rise time of an earthquake $\omega^{-1}=(m/k_p)^{1/2}$. Then, one has, $$\begin{aligned} \frac{d^2u_i}{dt^2} &=& (\nu t-u_i)+ l^2(u_{i+1}-2u_i+u_{i-1}) \nonumber \\ &-& (c+a\log(v_i/v^)+b\log (v^\theta_i)) \label{eq-eqmotion1} \\ \frac{d\theta_i}{dt} &=& 1-v_i\theta_i \label{eq-eqmotion2}\end{aligned}$$ where the dimensionless variables are defined by $t=\omega t'$, $u_i=u'_i/{\cal L}$, $v_i=v'_i/({\cal L}\omega )$, $v^=v'_/({\cal L}\omega )$, $\theta_i=\omega \theta'_i$, $\nu=\nu'/({\cal L}\omega )$, $a=a'{\cal N}/(k_p{\cal L})$, $b=b'{\cal N}/(k_p{\cal L})$, $c=c'{\cal N}/(k_p{\cal L})$, while $l \equiv (k_c/k_p)^{1/2}$ is the dimensionless stiffness parameter defined above. In some numerical simulations, a slightly different form is used for the $a$-term, where the factor inside the $a$-term, $v/v^$, is replaced by $1+(v/v^)$, [i.e.]{}, $$\begin{aligned} \frac{d^2u_i}{dt^2} &=& (\nu t-u_i)+ l^2(u_{i+1}-2u_i+u_{i-1}) \nonumber \\ &-& (c+a\log(1+\frac{v_i}{v^})+b\log \theta_i), \label{eq-eqmotion1'}\end{aligned}$$ where the constant factor $c$ in Eq.(\[eq-eqmotion1’\]) is shifted by $b\log v^$ from $c$ in Eq.(\[eq-eqmotion2\]). This replacement enables one to describe the system at a complete halt, whereas, without this replacement, the system cannot stop because of the logarithmic anomaly occurring at $v=0$. Similar replacement is sometimes made also for the $b$-term, [i.e.]{}, $\theta$ to $1+\theta$. The values of various parameters of the model describing natural faults were estimated (Ohmura and Kawamura, 2007). Typically, $\omega ^{-1}$ corresponds to a rise time of an earthquake event and is estimated to be a few seconds from observations. Though the characteristic slip distance ${\cal L}$ remains to be largely ambiguous, an estimate of order a few mm or cm was given by Tse and Rice (Tse and Rice, 1986) and by Scholz (Scholz, 2002). The loading rate associated with the plate motion is typically a few cm/year, and the dimensionless loading rate $\nu=\nu '/({\cal L}\omega)$ is of order $\nu \simeq 10^{-8}$. The dimensionless quantity $k_p{\cal L}/{\cal N}$ was roughly estimated to be of order $10^{-4}$. The dimensionless parameter $c$ should be of order $10^3 \sim 10^4$, and the $a$ and $b$ parameters are one or two orders of magnitude smaller than $c$. ### The 1D BK model with short-range interaction The simplest version of the BK model might be the 1D model with only nearest-neighbor inter-block interaction. Since this model was reviewed in an earlier RMP review article by Carlson, Langer and Shaw, 1994, we keep the discussion here to be minimum, focusing mainly on recent results obtained after the above review article. Earlier studies on the 1D BK model have revealed that, while smaller events persistently obeyed the GR law, [i.e.]{}, staying critical or near-critical, larger events exhibited a significant deviation from the GR law, being off-critical or “characteristic” (Carlson and Langer, 1989a; Carlson and Langer, 1989b; Carlson et al., 1991; Carlson, 1991a; Carlson, 1991b; Schmittbuhl, Vilotte and Roux, 1996). In Fig. \[magnitude-1DBK\], we show the recent data of the magnitude distribution (Mori and Kawamura, 2005; 2006). The magnitude of an event, $M$, is defined by $$M = \ln \left( \sum_i \Delta u_i \right).$$ where the sum is taken over all blocks involved in the event. ![ The magnitude distribution of earthquake events of the 1D BK model with nearest-neighbor interaction for various values of the friction parameter $\alpha$; (a) for larger $\alpha =1,2,3,5$ and 10, and (b) for smaller $\alpha =0.25, 0.5, 0.75$ and 1. The parameters $l$ and $\delta$ are fixed to be $l=3$ and $\delta=0.01$. The system size is $N=800$. Taken from (Mori and Kawamura, 2006). []{data-label="magnitude-1DBK"}](magnitude-a-1DBK-rev.eps "fig:") ![ The magnitude distribution of earthquake events of the 1D BK model with nearest-neighbor interaction for various values of the friction parameter $\alpha$; (a) for larger $\alpha =1,2,3,5$ and 10, and (b) for smaller $\alpha =0.25, 0.5, 0.75$ and 1. The parameters $l$ and $\delta$ are fixed to be $l=3$ and $\delta=0.01$. The system size is $N=800$. Taken from (Mori and Kawamura, 2006). []{data-label="magnitude-1DBK"}](magnitude-b-1DBK-rev.eps "fig:") As can be seen from Fig. \[magnitude-1DBK\], the form of the calculated magnitude distribution $R(M)$ depends on the value of the velocity-weakening parameter $\alpha$. The data for $\alpha =1$ lie on a straight line fairly well, apparently satisfying the GR law, which may be called “near-critical” behavior. The values of the exponent $B$ describing the power-law behavior is estimated to be $B\simeq 0.50$ corresponding to the $b$-value, $b=\frac{3}{2}B\simeq 0.75$. By contrast, the data for larger $\alpha$ deviate from the GR law at larger magnitudes, exhibiting a pronounced peak structure, while the power-law feature still remains for smaller magnitudes: See Fig. \[magnitude-1DBK\](a). These features of the magnitude distribution were observed in many simulations in common (Carlson and Langer,1989a; Carlson and Langer, 1989b; Carlson et al, 1991). It means that, while smaller events exhibit self-similar critical properties, larger events tend to exhibit off-critical or characteristic properties, which may be called “supercritical”. Te data for smaller $\alpha <1$ exhibit still considerably different behaviors from those for $\alpha >1$. Large events are rapidly suppressed, which may be called “subcritical” behavior. For $\alpha =0.25$, in particular, all events consist almost exclusively of small events only: See Fig. \[magnitude-1DBK\](b). Here the words “critical”, “supercritical” and “subcritical” have been defined on the basis of the shape of the magnitude-frequency relationship. As an example of properties other than the magnitude distribution, we show in Fig. \[recurrence-1DBK\] the recurrence-time distribution (Mori and Kawamura, 2005; 2006). The recurrence time $T$ is defined here locally for large earthquakes with $M \geq M _c=3$ or $M_c=4$, [i.e.]{}, the subsequent large event is counted when a large event occurs with its epicenter in the region within 30 blocks from the epicenter of the previous large event. As can be seen from the figure, the tail of the distribution is exponential at longer $T$ irrespective of the value of $\alpha$. Such an exponential tail of the distribution has also been reported for real seismicity (Corral, 2004). By contrast, the distribution at shorter $T$ is non-exponential and largely differs between for $\alpha =1$ and for $\alpha >1$. For $\alpha>1$, the distribution has an eminent peak corresponding to a characteristic recurrence time, which suggests the near-periodic recurrence of large events. Such a near-periodic recurrence of large events was reported for several real faults (Nishenko and Buland, 1987; Scholz, 2002). For $\alpha =1$, by contrast, the peak located close to the mean $\bar T$ is hardly discernible. Instead, the distribution has a pronounced peak at a shorter time, just after the previous large event. In other words, large events for $\alpha=1$ tend to occur as “twins”. A large event for the case of $\alpha=1$ often occurs as a “unilateral earthquake” where the rupture propagates only in one direction, hardly propagating in the opposite direction. ![ The local recurrence-time distribution of the 1D BK model with nearest-neighbor interaction for various values of the frictional parameter $\alpha$. Large events of $M > M_c=3$ or 4 are considered. The parameters are $l$ and $\delta$ are $l=3$ and $\delta=0.01$. The recurrence time $T$ is normalized by its mean $\bar T$. The total number of blocks is $N=800$. The insets represent the semi-logarithmic plots including the tail part of the distribution. Taken from (Mori and Kawamura, 2005). \[recurrence-1DBK\] ](recurrence-1DBK-rev2.eps) Possible precursory phenomenon exhibited by the model is of much interest, since it might open a way to an earthquake forecast. In fact, certain precursory features were observed in the 1D BK model. Shaw, Carlson and Langer examined the spatio-temporal patterns of seismic events preceding large events, observing that the seismic activity accelerates as the large event approaches (Shaw, Carlson and Langer, 1992). Mori and Kawamura observed that the frequency of smaller events was gradually enhanced preceding the mainshock, whereas, just before the mainshock, it is suppressed in a close vicinity of the epicenter of the upcoming event (Mori and Kawamura, 2005; 2006), a phenomenon closely resembling the “Mogi doughnut” (Mogi, 1969; 1979; Scholz, 2002). Fig. \[mogidoughnut-1DBK\] represents the space-time correlation function between the large events and the preceding events of arbitrary size (dominated in number by smaller events): It represents the conditional probability that, provided that a large event of $M >M_c =3$ occurs at a time $t_0$ and at a spatial point $r_0$, an event of arbitrary size occurs at a time $t_0-t$ and at a spatial point $r_0\pm r$. As can be seen from the inset of Fig. \[mogidoughnut-1DBK\], seismic activity is gradually accerelated toward the mainshock either spatially or temporally. As can be seen from the main panel, however, the seismic activity is supressed just before the mainshock in a close vicinity of the epicenter of the mainshock: See the dip developing around $r=0$ for $t\leq 0.01$. It turned out that the size of the quiescence region was always of only a few blocks, independent of the size of the upcoming mainshock (Mori and Kawamura, 2006). This may suggest that the quiescence is closely related to the discrete nature of the BK model: See subsection III.A.6 below. As such, the size of the quiescence region cannot be used in predicting the size of the upcoming mainshock. Instead, certain correlation was observed between the size of the upcoming mainshock and the size of the seismically active “ring” region surrounding the quiescence region (Pepke, Carlson and Shaw, 1994; Mori and Kawamura, 2006). Such a correlation was also reported in real seismic catalog (Kossobokov and Carlson, 1995). ![The event frequency preceding the large event of $M >M_c=3$ versus the distance from the epicenter of the upcoming mainshock of the 1D BK model with nearest-neighbor interaction. The parameters $\alpha$, $l$ and $\delta$ are $\alpha=1$, $l=3$ and $\delta=0.01$. The data are shown for several time periods before the mainshock. The insets represent similar plots with longer time intervals. The system size is $N=800$. Taken from (Mori and Kawamura, 2006). []{data-label="mogidoughnut-1DBK"}](mogidoughnut-1DBK.eps) An aftershock sequence obeying the Omori law, although a common observation in real seismicity, is not observed in the BK model, at least in its simplest version (Carlson and Langer, 1989a, 1989b; Mori and Kawamura, 2006). Interestingly, Pelletier reported that the inclusion of the viscosity effect in the form of “dashpot” in the 2D BK model, together with the introduction of inhomogeneity of friction parameters, could realize an aftershock sequence obeying the Omori law (Pelletier, 2000). The frictional force employed by Pelletier was a very simple one, [i.e.]{}, a constant dynamical vs. static friction coefficient. Further analysis will be desirable to establish the occurrence of the aftershock sequence obeying the Omori law in the BK model. We note in passing that the 1D BK model has also been extended in several ways, [e.g.]{}, taking account of the effect of the viscosity (Myers and Langer, 1993; Shaw, 1994; De and Ananthakrisna, 2004; Mori and Kawamura, 2008b), modifying the form of the friction force (Myers and Langer, 1993; Shaw, 1995; Cartwright, 1997; De and Ananthakrisna, 2004), and driving the system only at one end of the system (Vieira, 1992; 1996a). The effect of the long-range interactions introduced between blocks was also analyzed, which we will review in subsection III.A.4. ### The 2D BK model with short-range interaction Real earthquake faults are 2D rather than 1D. Hence, it is clearly desirable to study the 2D version of the BK model in order to further clarify the statistical properties of earthquakes. The 2D BK model taken up here is to be understood as representing a 2D fault plane itself, where the direction orthogonal to the fault plane is not considered explicitly in the model (Carlson, 1991b). The other possible version is the one where the second direction of the model is taken to be orthogonal to the fault plane (Myers et al, 1996). Extensive numerical studies have revealed that statistical properties of the 2D BK model are more or less similar to those of the 1D BK model reviewed in the previous subsection, at least qualitatively. The magnitude distribution $R(M)$ of the 2D BK model was studied by several groups (Carlson and Langer, 1989a; Carlson and Langer, 1989b; Carlson et al, 1991; Kumagai, et al, 1999; Mori and Kawamura, 2007). In Fig. \[magnitude-2DBK\], we show typical behaviors of the magnitude distribution of the 2D BK model with varying the frictional parameter $\alpha$ (Mori and Kawamura, 2007). For smaller $\alpha \lsim 0.5$, $R(M)$ bends down rapidly at larger magnitudes, exhibiting a “subcritical” behavior. Only small events of $M \lsim 2$ occur in this case. At $\alpha \gsim 0.5$, large earthquakes of their magnitudes $M \simeq 8$ suddenly appear, while earthquakes of intermediate magnitudes, say, $2\lsim M \lsim 6$, remain rather scarce. Such a sudden appearance of large earthquakes at $\alpha =\alpha_{c1}\simeq 0.5$ coexisting with smaller ones has a feature of a discontinuous or “first-order” transition. In this context, it might be interesting to point out that Vasconcelos observed that a single block system exhibited a “first-order transition” at $\alpha =0.5$ from a stick-slip to a creep (Vasconcelos, 1996), whereas this discontinuous transition becomes apparently continuous in many-block system (Vieira et al, 1993; Clancy and Corcoran, 2005). A “first-order” transition observed at $\alpha=\alpha_{c1}\simeq 0.5$ in the 2D model may have some relevance to the first-order transition of a single-block system observed by Vasconcelos, although events observed at $\alpha<\alpha_{c1}$ in the present 2D model are not really creeps, but rather are stick-slip events of small sizes. With increasing $\alpha$ further, earthquakes of intermediate magnitudes gradually increase their frequency. Fig. \[magnitude-2DBK\](b) exhibits $R(M)$ for larger $\alpha$. In the range of $1\lsim \alpha \lsim 10$, $R(M)$ exhibits a pronounced peak structure at a larger magnitude, deviating from the GR law, while it exhibits a near straight-line behavior corresponding to the GR law at smaller magnitudes (“supercritical” behavior). As $\alpha$ increases further, the peak at a larger magnitude becomes less pronounced. At $\alpha =\alpha_{c2}\simeq 13$, $R(M)$ exhibits a near straight-line behavior for a rather wide magnitude range, though $R(M)$ falls off rapidly at still larger magnitudes $M \gsim 7$, indicating that the “near-critical” behavior observed for $\alpha=\alpha_{c2}\simeq 13$ cannot be regarded as a truly asymptotic one, since this rapid fall-off of $R(M)$ at very large magnitudes is a bulk property, not a finite-size effect. ![ The magnitude distribution $R(M)$ of the 2D BK model with nearest-neighbor interaction for various values of the friction parameter $\alpha$. The other parameters are $l=3$ and $\delta=0.01$. Fig.(a) represents $R(M)$ for smaller values of the friction parameter $0\leq \alpha \leq 3$, while Fig.(b) represents $R(M)$ for larger values of the friction parameter $3\leq \alpha \leq \infty$. The system size is $60\times 60$. Taken from (Mori and Kawamura, 2008a). []{data-label="magnitude-2DBK"}](magnitude-a-2DBK-rev2.eps "fig:") ![ The magnitude distribution $R(M)$ of the 2D BK model with nearest-neighbor interaction for various values of the friction parameter $\alpha$. The other parameters are $l=3$ and $\delta=0.01$. Fig.(a) represents $R(M)$ for smaller values of the friction parameter $0\leq \alpha \leq 3$, while Fig.(b) represents $R(M)$ for larger values of the friction parameter $3\leq \alpha \leq \infty$. The system size is $60\times 60$. Taken from (Mori and Kawamura, 2008a). []{data-label="magnitude-2DBK"}](magnitude-b-2DBK-rev2.eps "fig:") A “phase diagram” of the model in the elasticity parameter $l$ versus the friction parameter $\alpha$, as reported by Mori and Kawamura, 2007 is shown in Fig. \[phasediagram-2DBK\]. The region or the “phase”, called “supercritical”, “near-critical” and “subcritical” are observed. The straight-line behavior of $R(M)$, [i.e.]{}, the GR law is realized only in the restricted region in the phase diagram along the phase boundary between the supercritical and subcritical regimes. Even along the phase boundary, the GR relation is characterized by a finite cutoff magnitude above which larger earthquakes cease to occur. Hence, the GR relation, as observed in a ubiquitous manner in real faults, is not realized in this model. Since each phase boundary has a finite slope in the $\alpha-l$ plane, one can also induce the “subcritical”-“supercritical” transition with varying the $l$-value for a fixed $\alpha$ (Espanol, 1994; Vieira, 1996b). ![ Phase diagram of the 2D BK model with nearest-neighbor interaction in the friction parameter $\alpha$ versus the elastic-parameter $l$ plane. The parameter $\delta$ is $\delta=0.01$. Taken from (Mori and Kawamura, 2008a). []{data-label="phasediagram-2DBK"}](phasediagram-2DBK.eps) As for other quantities, the recurrence-time distribution of the 2D model exhibits a behavior similar to that of the 1D model (Mori and Kawamura, 2007). As in case of 1D, an aftershock sequence obeying the Omori law is not observed even in the 2D model, at least in its simplest version. The 2D model also exhibits precursory phenomena similar to the ones observed in the 1D model (Mori and Kawamura, 2007). Acceleration of seismic activity prior to mainshock is observed in the supercritical regime, while it is not realized in the subcritical regimes. As in case of 1D, mainshocks are accompanied by the “Mogi doughnut”-like quiescence in both supercritical and subcritical regimes. As an other signature of the precursory phenomena, we show in Fig. \[magnitude-2DBK-before\] the “time-resolved” local magnitude distribution calculated for time periods before the large event in the supercritical regime of $\alpha=1$ and $l=3$ (Mori and Kawamura, 2007). Only events with their epicenters lying within 5 blocks from the upcoming mainshock of its magnitude $M\geq M_c=5$. As can be seen from the figure, an apparent $B$-value describing the smaller magnitude region gets [smaller]{} as the mainshock is approached, [i.e.]{}, it changes from $B\simeq 0.89$ of the long-time value to $B\simeq 0.65$ in the time range $t\nu \leq 0.1$ before the mainshock. In real seismicity, an appreciable decrease of the $B$-value has been reported preceding large earthquakes (Suyehiro, Asada and Ohtake, 1964; Jaume and Sykes, 1999; Kawamura, 2006). Obviously, a possible change in the magnitude distribution preceding the mainshock possesses a potential importance in earthquake fo recast. ![ The local magnitude distribution preceding the mainshock of $M >M _c=5$ of the 2D BK model with nearest-neighbor interaction. The parameters are $\alpha=1$, $l=3$ and $\delta=0.01$. The data are shown for several time periods before the mainshock. The system size is $60\times 60$. Taken from (Mori and Kawamura, 2008a).[]{data-label="magnitude-2DBK-before"}](magnitude-before-2DBK-rev2.eps) ### The BK model with long-range interaction So far, we assumed that the interaction between blocks worked only between nearest-neighboring blocks. This may correspond to the situation where a thin isolated plate is subject to friction force and is driven by shear force (Clancy and Corcoran, 2006). However, a real fault is not necessarily a thin isolated plate, and the elastic body extends in a direction away from the fault plane. Indeed, the BK model extended in the direction orthogonal to the fault plane was also studied (Myers, Shaw and Langer, 1996). Considering the effect of such an extended elastic body adjacent to the fault plane under certain conditions amounts to considering the effective inter-block interaction to be [long-ranged]{}. Thus, taking account of the effect of long-range interaction might make the model more realistic. Rundle [et al]{} studied the properties of the 2D cellular automaton version of the BK model with the long-range interaction decaying as $1/r^3$ (Rundle, et al, 1995). Xia [et al]{} studied the 1D BK model with a variable range interaction where a block is connected to its $R$ neighbors with a rescaled spring constant proportional to $1/R$ (Xia et al, 2005; Xia et al, 2007). The type of the long-range model considered by Xia [et al]{} may be regarded as a mean-field type, since the model reduces to the mean-field infinite-range model in the $R\rightarrow \infty$ limit. One can also derive the relevant long-range interaction based on an elastic theory (Mori and Kawamura, 2008a). Suppose that the 3D elastic body in which the 2D BK model lies is isotropic, homogeneous and infinite, and a fault surface is a plane lying in this elastic body and slips along one direction only. Then, a static approximation for an elastic equation of motion for the elastic body would give rise to a spring constant between blocks decaying with their distance $r$ as $1/r^3$. This static assumption is justified when the velocity of the seismic-wave propagation is high enough compared with the velocity of the seismic-rupture propagation. Properties of the 2D BK model with the long-range power-law interaction derived from an elastic theory, [i.e.]{}, the one decaying as $1/r^3$, was investigated (Mori and Kawamura, 2008a). The interaction between the two blocks at sites ($i,j$) and ($i^{\prime},j^{\prime}$) is given in the dimensionless form by $$\left(l^2_x\frac{\|i^{\prime}-i\|^2}{r^5}+l^2_z\frac{\|j^{\prime}-j\|^2} {r^5}\right)(u_{i^{\prime},j^{\prime}}-u_{i,j}),$$ which falls off with distance $r$ as $1/r^3$. Then, the dimensionless equation of motion of the 2D long-range can be written as $$\begin{array}{ll} \ddot{u}_{i,j}=\nu t-u_{i,j} \ \ \ \ \\ \ \ \ \ + \sum_{(i^{\prime},j^{\prime}) \ne (i,j)} \left(l^2_x\frac{\|i^{\prime}-i\|^2}{r^5}+l^2_z\frac{\|j^{\prime}-j\|^2} {r^5}\right)(u_{i^{\prime},j^{\prime}}-u_{i,j}) \\ \ \ \ \ - \phi _{i,j}. \end{array} \label{eq-1DBK-normalized2}$$ If one restricts the range of interaction to nearest neighbors and takes the spatially anisotropic spring constant to be isotropic, $l_x=l_z=l$, one recovers the isotropic nearest-neighbor model described by Eq. \[eq-1DBK-normalized\]. The “isotropy” assumption $l_x=l_z$ is equivalent to putting the Lame’s constant to vanish. In fact, in the short-range model, such a spatial anisotropy of the 2D BK model turned out to hardly affect the statistical properties of the model in the sense that the properties of the anisotropic model was quite close to the corresponding isotropic model characterized by the [mean]{} spring constant $l=(l_x+l_z)/2$ (Mori and Kawamura, 2008a). One might also consider the 1D BK model with the long-range interaction (Mori and Kawamura, 2008a). One possible way to construct the 1D model might be to impose the condition on the corresponding 2D model that the systems is completely rigid along the $z$-direction corresponding to the depth direction, [i.e.]{}, $u(x,z,t)=u(x,t)$. This yields an effective inter-block interaction decaying with distance $r$ as $1/r^2$, $$l^2\frac{1}{\|i-i^{\prime}\|^2}(u_{i^{\prime}}-u_{i}),$$ with the dimensionless equation of motion $$\begin{array}{ll} \ddot{u}_i=\nu t-u_i+ l^2 \sum_{i^{\prime} \ne i} \frac{u_{i^{\prime}}-u_{i}}{\|i-i^{\prime}\|^2} -\phi_i. \end{array}$$ In Figs. \[magnitude-BKLR\](a) and (b), we show the magnitude distribution $R(M)$ of the long-range 2D BK model for smaller and larger values of $\alpha$, [i.e.]{}, (a) $0\leq \alpha \leq 10$ and (b) $10\leq \alpha \leq \infty$ (Mori and Kawamura, 2008a). Similarly to the short-range case, three distinct regimes are observed depending on the $\alpha$-value. The intermediate-$\alpha$ region corresponds to the supercritical regime where $R(M)$ exhibits a pronounced peak at a larger magnitude, showing a characteristic behavior. Major difference from the short-range case is that the subcritical behavior realized in the short-range model in the smaller- and larger-$\alpha$ region is now replaced by the near-critical behavior in the long-range model. Namely, for smaller $\alpha< \alpha_{c1}\sim 2$ and for larger $\alpha > \alpha_{c2}\sim 25$, $R(M)$ exhibits a near straight-line behavior over a rather wide magnitude range, and drops off sharply at larger magnitudes. The associated $B$-value is estimated to be $B\simeq 0.59$ ($\alpha<\alpha_{c1}$) and $B\simeq 0.55$ ($\alpha>\alpha_{c2}$), which is rather insensitive to the $\alpha$-value. This straight-line behavior of $R(M)$ cannot be regarded as a truly critical one, since $R(M)$ drops off sharply at very large magnitudes. As in the short-range case, the change from the supercritical to the near-critical behaviors at $\alpha=\alpha_{c2}\simeq 25$ is continuous, while it is discontinuous at $\alpha=\alpha_{c1}\simeq 2$. ![ The magnitude distribution $R(M)$ of the 2D BK model with long-range interaction for various values of the friction parameter $\alpha$. The other parameters are $l=3$ and $\delta=0.01$. Fig.(a) represents $R(M)$ for smaller values of the frictional parameter $0\leq \alpha \leq 10$, while Fig.(b) represents $R(M)$ for larger values of the frictional parameter $10\leq \alpha \leq \infty$. The system size is $60\times 60$. Taken from (Mori and Kawamura, 2008a). []{data-label="magnitude-BKLR"}](magnitude-a-BKLR-rev2.eps "fig:") ![ The magnitude distribution $R(M)$ of the 2D BK model with long-range interaction for various values of the friction parameter $\alpha$. The other parameters are $l=3$ and $\delta=0.01$. Fig.(a) represents $R(M)$ for smaller values of the frictional parameter $0\leq \alpha \leq 10$, while Fig.(b) represents $R(M)$ for larger values of the frictional parameter $10\leq \alpha \leq \infty$. The system size is $60\times 60$. Taken from (Mori and Kawamura, 2008a). []{data-label="magnitude-BKLR"}](magnitude-b-BKLR-rev2.eps "fig:") Such a near-critical behavior realized over a wide parameter range is in sharp contrast to the behavior of the corresponding short-range model where $R(M)$ at smaller and larger $\alpha$ exhibits only a down-bending subcritical behavior, while a straight-line near-critical behavior is realized only by fine-tuning the $\alpha$-value to a special value $\alpha\simeq \alpha _{c2}$. The robustness of the near-critical behavior of $R(M)$ observed in the 2D long-range model might have an important relevance to real seismicity, since the GR law is ubiquitously observed for different types of faults. Note also that the associated $B$-value observed here turns out to be close to the one observed in real seismicity (Mori and Kawamura, 2008a). ![ The phase diagram of the 2D BK models with long-range interaction in the friction parameter $\alpha$ versus elastic-parameter $l$ plane, which is compared with the one of the 2D BK model with short-range interaction. The parameter $\delta$ is set $\delta=0.01$. Taken from (Mori and Kawamura, 2008a). []{data-label="phasediagram-BKLR"}](phasediagram-BKLR.eps) In Fig. \[phasediagram-BKLR\], the behavior of $R(M)$ is summarized in the form of a “phase diagram” in the friction parameter $\alpha$ versus the elastic-parameter $l$ plane (Mori and Kawamura, 2008a). As can be seen from the figure, the phase diagram of the long-range model consists of three distinct regimes, two of which are near-critical regimes and one is a supercritical regime. The “phase boundary” between the smaller-$\alpha$ near-critical regime and the supercritical regime represents a “discontinuous transition”, while the one between the larger-$\alpha$ near-critical regime and the supercritical regime represents a “continuous transition”. For comparison, the corresponding phase boundary of the short-range model is also shown. The near-critical phases in the long-range model are replaced by the subcritical phases in the short-range model. It might be interesting to notice that the system at different “phases” of Fig. \[phasediagram-BKLR\] really show different properties. For example, we show in Fig. \[displacement-BKLR\] the magnitude dependence of the mean displacement $\Delta \bar u$ at a seismic event (Mori and Kawamura, 2008a). As can be seen from the figure, the data in the two near-critical regimes (the data in blue and in green) are grouped into two distinct branches, while the data in the supercritical regime (the data in red) exhibit a significantly different behavior. Interestingly, the mean displacement in the near-critical regimes hardly depends on the event magnitude. It was observed that the mean stress drop at a seismic event also hardly depends on the event magnitude in the near-critical regimes of the 2D long-range BK model (Mori and Kawamura, 2008a). A similar independence was also reported in the mean-field-type 1D long-range BK model (Xia [et al]{}, 2005; 2008) and in the 1D long-range BK model (Mori and Kawamura, 2008a). ![ The magnitude dependence of the mean displacement $\Delta \bar u$ at each seismic event of the 2D BK model with long-range interaction. In the main panel, the friction parameter $\alpha$ is varied with fixing the system size $60\times 60$, while in the inset the system-size $N$ is varied for the case of $\alpha=30$. The parameters $l$ and $\delta$ are fixed to $l=3$ and $\delta=0.01$. Taken from (Mori and Kawamura, 2008a). []{data-label="displacement-BKLR"}](displacement-BKLR-rev2.eps) ### Continuum limit of the BK model Although the BK model has widely been used as a useful tool to investigate statistical properties of earthquakes, the block discretization inherent to the model construction is a crude approximation of the originally continuum earthquake fault. It [introduces the short-length cutoff scale into the problem]{}. Therefore, in order to check the validity of the model, it is crucially important to examine the continuum limit of the BK model carefully. Indeed, Rice criticized that the discrete BK model with the velocity-weakening friction law was “intrinsically discrete”, lacking in a well-defined continuum limit (Rice, 1993). Rice argued that the spatiotemporal complexity observed in the discrete BK model was due to the inherent discreteness of the model, which should disappear in continuum. Indeed, he applied the RSF law, which possessed an intrinsic length scale corresponding to the characteristic slip distance, and showed that the system tended to exhibit a quasi-periodic behavior, if the grid spacing $d'$ was taken smaller than the characteristic length scale, while if the grid spacing $d'$ was taken longer than it, the system exhibited an apparently complex or critical behavior. This problem of the continuum limit of the BK model was also addressed by Myers and Langer (Myers and Langer, 1993) within the velocity-weakening friction law, who introduced the Kelvin viscosity term to produce a small length scale which allowed a well-defined continuum limit. Myers and Langer, and subsequently Shaw (Shaw, 1994), observed that the added viscosity term smoothed the rupture dynamics, apparently giving rise to the continuum limit accompanied by the spatiotemporal complexity. More recently, the continuum limit of the 1D BK model with and without the viscosity was examined by Mori and Kawamura within the velocity-weakening friction law (Mori and Kawamura, 2008b). Thus, two different ways of taking the continuum limit of the BK model were tried so far, each introducing the short length scale via (A) the viscosity term, or (B) the RSF law. In this subsection, we examine the former (A), while the latter (B) will be discussed in the next subsection. As mentioned, the naive continuum limit of the discrete BK model with a velocity-weakening friction force without viscosity has a problem in that the pulse of slip tends to become increasingly narrow in width in the limit, [i.e.]{}, the dynamics becomes sensitive to the grid spacing $d^{\prime}\rightarrow 0$. One way to circumvent this problem is to introduce the viscosity term $\eta^{\prime} \partial^3 U_i/(\partial {x^{\prime}}^2 \partial t^{\prime})$ into Eq.\[eq-1DBK\] to produce a small length scale, where $\eta^{\prime}$ is the viscosity coefficient. Myers and Langer showed that, owing to the added viscosity term, the system became independent of the grid spacing $d^{\prime}$ as long as a new small length scale $\epsilon^{\prime}$, defined by $$\epsilon^{\prime} =\pi \sqrt{\frac{\eta^{\prime}}{\alpha \omega}},$$ is sufficiently larger than the grid spacing $d^{\prime}$ (Myers and Langer, 1993). With $\xi^{\prime}$ being the wave velocity in the continuum limit, this small length scale $\epsilon^{\prime}$ can also be given in the dimensionless form as $$\epsilon \equiv \epsilon^{\prime}/(\xi^{\prime}/\omega) =\pi \sqrt{\frac{\eta}{\alpha}}, \label{eq-smalllength}$$ where $\eta \equiv \eta^{\prime}/(\xi^{\prime 2}/\omega)$ is the dimensionless viscosity coefficient. The dimensionless distance $r$ between the block $i$ and $i^{\prime}$ is measured by $$r=d\|i-i^{\prime}\|,$$ where $d\equiv d^{\prime}/(\xi^{\prime}/\omega)$ is the dimensionless grid spacing. The continuum limit corresponds to taking the limit $d\rightarrow 0$ with fixing $L=Nd$ and $r$, which means $N\rightarrow \infty$ and $l\rightarrow \infty$. Thus, taking the continuum limit in the BK model corresponds to making the model to be infinitely rigid $l\rightarrow \infty$. Numerically, various observables were calculated with successively smaller $d$ to examine its asymptotic $d\rightarrow 0$ limit. Shaw showed, by adding the viscosity term to the 1D BK model, that the magnitude distribution became independent of the grid spacing $d^{\prime}$ for sufficiently small $d^{\prime}$ (Shaw, 1994). Mori and Kawamura studied the 1D BK model with successively smaller grid spacings $d^{\prime}$ to examine how various statistical properties of the model changed and approached the continuum limit for both cases of nonzero ($\eta>0$) and zero ($\eta=0$) viscosity (Mori and Kawamura, 2008b). It was then observed that, in the former viscous case, the results converged to the continuum limit when the condition $d < \epsilon $ was met, whereas, in the latter non-viscous case, such a convergence was obscure. As an example, we show in Fig. \[magnitude-continuumBK\] the way of convergence of the magnitude distribution function $R(M)$ for $\alpha=1$ (a) and for $\alpha=3$ (b), in the viscous case ($\eta=0.02$). For both cases of $\alpha=1$ and 3, the continuum limit seems to be well reached, [i.e.]{}, $R(M)$ seems to converge to an asymptotic form for smaller $d$, except that the minimum magnitude continuously gets lower as the grid spacing $d$ gets smaller. A similar result was reported by Shaw, 1994. From Fig. \[magnitude-continuumBK\](a), one also sees that a nonzero viscosity tends to weaken the GR character of the magnitude distribution somewhat. Such a deviation from the GR law at smaller magnitudes is probably originated from the fact that the viscosity tends to make the relative displacement of neighboring blocks being smoother, enhancing the correlated motion of neighboring blocks, which makes the frequency of smaller events of one or a few blocks considerably reduced (Mori and Kawamura, 2008b). The small-length cutoff scale $\epsilon$ as given by Eq. \[eq-smalllength\] is estimated here to be $\epsilon \simeq 0.44$ and 0.26 for $\alpha=1$ and 3, respectively. As can be seen from Figs. \[magnitude-continuumBK\](a) and (b), $R(M)$ converges to an asymptotic form for the $\alpha$-values smaller than $d \simeq 1/4$ and 1/8 for $\alpha=1$ and 3, respectively, which is consistent with the expected condition of the continuum limit $d < \epsilon$. ![ The magnitude distribution $R(M)$ of earthquake events of the 1D viscous BK model ($\eta=0.02$) with $\delta=0.01$. The dimensionless grid spacing $d$ is varied in the range $1 \geq d \geq 1/32$. Figs.(a) and (b) represent the cases of $\alpha=1$ and 3, respectively. The system size is $L=dN=200$. Taken from (Mori and Kawamura, 2008b). []{data-label="magnitude-continuumBK"}](magnitude-a-continuumBK-rev2.eps "fig:") ![ The magnitude distribution $R(M)$ of earthquake events of the 1D viscous BK model ($\eta=0.02$) with $\delta=0.01$. The dimensionless grid spacing $d$ is varied in the range $1 \geq d \geq 1/32$. Figs.(a) and (b) represent the cases of $\alpha=1$ and 3, respectively. The system size is $L=dN=200$. Taken from (Mori and Kawamura, 2008b). []{data-label="magnitude-continuumBK"}](magnitude-b-continuumBK-rev2.eps "fig:") As mentioned in subsection III.A.3, the BK model generally gives rise to a seismic quiescence phenomenon prior to mainshock, [i.e.]{}, the Mogi-doughnut. Then, a natural question is whether the doughnut-like quiescence observed in the discrete BK model survives the continuum limit, or it is a phenomenon intrinsically originated from the short cutoff length scale of the model. This question was addressed in (Mori and Kawamura, 2008b). Fig. \[mogidoughnut-continuumBK\] exhibits the time-dependent spatial correlation functions before the mainshock in the case of the viscous model of $\alpha=1$. As the grid spacing $d$ gets smaller, the spatial range of the quiescence gets narrower, tending to vanish for small enough $d$: See the inset of Fig. \[mogidoughnut-continuumBK\]. This observation strongly suggests that the doughnut-like quiescence might vanish altogether in the continuum limit $d \to 0$. Thus, the doughnut-like quiescence observed in the discrete BK model is likely to be a phenomenon closely related to the short-length cutoff scale of the model. This seems fully consistent with the observation that the one-block events are responsible for the observed doughnut-like quiescence (Mori and Kawamura, 2006; 2008a). ![ The event frequency in the time period $t\nu=0\sim0.01$ immediately before the mainshock of $M >M_c=2$ of the 1D viscous BK model ($\eta=0.02$) with $\alpha=1$ plotted versus $r$, the distance $r$ from the epicenter of the upcoming mainshock. The dimensionless grid spacing $d$ is varied in the range $1/4 \geq d \geq 1/32$. The parameter $\delta$ is fixed to $\delta=0.01$. The system size is $L=dN=200$. The insets represent the peak position of the event frequency, corresponding to the range of the doughnut-like quiescence, as a function of the dimensionless grid spacing $d$. The doughnut-like quiescence vanishes in the continuum limit $d\rightarrow 0$. Taken from (Mori and Kawamura, 2008b). []{data-label="mogidoughnut-continuumBK"}](mogidoughnut-continuumBK-rev2.eps) The observation might have some implications to real seismicity. While the real crust is obviously a continuum, it is often not so uniform, possibly with a short-length cutoff. In any case, in real earthquakes the Mogi-doughnut is occasionally reported to occur (Mogi, 1969; 1979; Scholz, 2002), although establishing its statistical significance is sometimes not easy. Then, our present result may suggest that, if the real crust possesses a cutoff length scale due to the inhomogeneity of the crust, the “Mogi-doughnut” quiescence might occur at such a length scale. In other words, spatial inhomogeneity might be an essential ingredient for the Mogi-doughnut to occur in real seismicity (Mori and Kawamura, 2008b). ### The BK model with RSF law So far, we have mostly assumed a simple velocity-weakening friction law where the friction force is a single-valued function of the velocity. As detailed in section II and in subsection III.A.2, the RSF law is now regarded in seismology as the standard consititutive law. Tse and Rice employed this RSF constitutive relation in their numerical simulations of earthquakes (Tse and Rice, 1986). These authors studied the stick-slip motion of the two-dimensional strike-slip fault within an elastic continuum theory, assuming that the fault motion is rigid along strike. It was then observed that large events repeated periodically. Since then, similar RSF constitutive laws have widely been used in numerical simulations (Stuart, 1988; Horowitz and Ruina, 1989; Rice, 1993; Ben-Zion and Rice, 1997; Kato and Hirasawa, 1999; Kato, 2004; Bizzarri and Cocco, 2006). Somewhat different type of slip- and state-dependent constitutive law was also used (Cochard and Madariaga, 1996). Cao and Aki performed a numerical simulation of earthquakes by combining the 1D BK model with the RSF law in which various constitutive parameters were set nonuniform over blocks (Cao and Aki, 1986). Ohmura and Kawamura extended an earlier calculation by Cao and Aki to study the statistical properties of the 1D BK model combined with the RSF constitutive law with uniform constitutive parameters (Ohmura and Kawamura, 2007). Clancy and Corcoran also performed a numerical simulation of the 1D BK model based on a modified version of the RSF law (Clancy and Corcoran, 2009). Rice and collaborators argued that the slip complexity of the BK model might be caused by its intrinsic discreteness (Rice, 1993; Ben-Zion and Rice, 1997). In this context, it is important to clarify the statistical properties of the model where the discrete BK structure is combined with the RSF law, to compare its statistical properties with those of the standard BK model with the velocity-weakening or slip-weakening friction law reviewed in the previous subsections. Recent study by Morimoto and Kawamura has revealed that the model exhibits largely different behaviors depending on whether the frictional instability is either “strong” or “weak” (Morimoto and Kawamura, 2011). The condition of strong or weak frictional instability is given by $b>2l^2+1$ or $b<2l^2+1$, respectively, for the 1D BK model. In the case of a weaker frictional instability, the model exhibits a [precursory process]{} where a slow nucleation process occurs prior to mainshock. In the next subsection, we discuss such a precursory process realized in the BK model in more detail. Interestingly, presence or absence of such a nucleation process also affects statistical properties of the model. From a simulation point of view, the case of a weaker friction instability is much harder to deal with, since slow and long-standing nucleation process prior to mainshock generally requires a lot of CPU time. Statistical properties of the 1D BK model with the RSF law Eq.\[eq-eqmotion1\] (or Eq.\[eq-eqmotion1’\]) and Eq.\[eq-eqmotion2\] was investigated by Ohmura and Kawamura for the case of a strong frictional instability (Ohmura and Kawamura, 2007), and by Yamamoto and Kawamura for the case of a weak frictional instability (Yamamoto and Kawamura, 2011). Typical behaviors of the magnitude distribution are respectively shown in Figs. \[magnitude-1DBKRSF\](a) and (b). As can be seen from the figure, when the frictional instability is strong, almost flat distribution spanning from small to large magnitudes is realized, while, as the critical value is approached, a peak at a larger magnitude becomes more pronounced giving rise to an enhanced characteristic behavior. In the weak frictional instability regime, the distribution has no weight at smaller magnitudes, with a pronounced peak only at a large magnitude. It means that only large earthquakes of more or less similar magnitude occ ur in the regime of a weak frictional instability. ![ (Color online) The magnitude distribution of the 1D BK model with the RSF law, for the case of (a) a strong frictional instability $b>b_c$, and of (b) a weak frictional instability $b<b_c$, with $b_c=2l^2+1$. The parameter values are $a=0$, $c=1000$, $\nu=10^{-8}$, $v^=1$ and $l =3$ in (a), and $a=1$, $b=5$, $c=1000$ $v^=1$ and $l =5$ in (b). The borderline $b$-value is $b_c=19$ in (a), and $b_c=51$ in (b). The system size is $N=800$ in (a), and $N=1200$ in (b). (a) Taken from (Ohmura and Kawamura, 2007). (b) Taken from (Morimoto and Kawamura, 2011). []{data-label="magnitude-1DBKRSF"}](magnitude-a-1DBKRSF-rev.eps "fig:") ![ (Color online) The magnitude distribution of the 1D BK model with the RSF law, for the case of (a) a strong frictional instability $b>b_c$, and of (b) a weak frictional instability $b<b_c$, with $b_c=2l^2+1$. The parameter values are $a=0$, $c=1000$, $\nu=10^{-8}$, $v^=1$ and $l =3$ in (a), and $a=1$, $b=5$, $c=1000$ $v^=1$ and $l =5$ in (b). The borderline $b$-value is $b_c=19$ in (a), and $b_c=51$ in (b). The system size is $N=800$ in (a), and $N=1200$ in (b). (a) Taken from (Ohmura and Kawamura, 2007). (b) Taken from (Morimoto and Kawamura, 2011). []{data-label="magnitude-1DBKRSF"}](magnitude-b-1DBKRSF-rev.eps "fig:") Statistical properties of the corresponding 2D model were investigated by Kakui and Kawamura for both cases of weak and strong frictional instabilities (Kakui and Kawamura, 2011). In the 2D BK model, the condition of strong or weak frictional instability is given by $b>4l^2+1$ or $b<4l^2+1$, respectively. Typical behaviors of the magnitude distribution are shown in Figs. \[magnitude-2DBKRSF\](a) and (b) for the cases of strong and weak instabilities, respectively. As can be seen from the figure, when the frictional instability is strong, a behavior more or less close to the GR law, characterized by the exponent close to $B\sim 2/3$, is realized, although there is a weak shoulder-like structure superimposed at larger magnitudes. The observation of a near-critical behavior close to the GR law would be of much interest in conjunction with real seismicity. As the critical value is approached, on the other hand, a peak at a larger magnitude is further developed, giving rise to an enhanced characteristic behavior. In the weak frictional instability regime, the distribution has double peaks exhibiting more characteristic behavior: See Fig. \[magnitude-2DBKRSF\](b). ![ (Color online) The magnitude distribution of the 2D BK model with the RSF law, for the case of (a) a strong frictional instability $b>b_c$, and of (b) a weak frictional instability $b<b_c$, with $b_c=4l^2+1$. The parameter values are $a=1$, $c=1000$, $\nu=10^{-8}$, $v^=1$ and $l =2$ in (a), and $a=1$, $c=1000$, $\nu=10^{-8}$, $v^=1$ and $l =2$ in (b). The borderline value is $b_c=17$ in both (a) and (b). The system size is $N=60\times 60$ in (a), and $N=30\times 30$ in (b). Taken from (Kakui and Kawamura, 2011). []{data-label="magnitude-2DBKRSF"}](magnitude-a-2DBKRSF-rev.eps "fig:") ![ (Color online) The magnitude distribution of the 2D BK model with the RSF law, for the case of (a) a strong frictional instability $b>b_c$, and of (b) a weak frictional instability $b<b_c$, with $b_c=4l^2+1$. The parameter values are $a=1$, $c=1000$, $\nu=10^{-8}$, $v^=1$ and $l =2$ in (a), and $a=1$, $c=1000$, $\nu=10^{-8}$, $v^=1$ and $l =2$ in (b). The borderline value is $b_c=17$ in both (a) and (b). The system size is $N=60\times 60$ in (a), and $N=30\times 30$ in (b). Taken from (Kakui and Kawamura, 2011). []{data-label="magnitude-2DBKRSF"}](magnitude-b-2DBKRSF-rev.eps "fig:") ### Nucleation process of the BK model In this subsection, we touch upon the nucleation process as a precursory phenomenon prior to mainshock as realized in the BK model obeying the RSF law. It was observed that the nucleation process is realized even in the BK model with the RSF law for both cases of 1D and 2D, if the model lies in the regime of a weak frictional instability (Morimoto and Kawamura, 2011; Kakui and Kawamura, 2011). Namely, prior to seismic rupture, the system exhibits a slow rupture process localized to a compact “seed” area with its rupture velocity orders of magnitude slower than the seismic wave velocity. The system spends a very long time in this nucleation process, and then at some stage, exhibits a rapid acceleration process accompanied by a rapid growth of the rupture velocity and a rapid expansion of the rupture zone, finally getting into a final seismic rupture or a mainshock (Dieterich, 2009). Such a nucleation process has also been observed and extensively studied in the continuum model: See, [e.g.]{}, (Ampuero and Rubin, 2008). We illustrate in Fig. \[nucleation\] typical example of seismic events realized in the 1D BK model with the RSF law for each case of a weak frictional instability (b), and of a strong frictional instability (a). As can be seen from the figure, a slow nucleation process with a long duration time is observed only in (b), while such a nucleation process is absent in (a). ![ (Color online) The typical rupture process realized in the 1D BK model with the RSF law for (a) a strong and (b) a weak frictional instability, each corresponding to (a) $b>b_c$ and (b) $b<b_c$ with $b_c=2l^2+1$. The color represents the rupture velocity. The parameter values are $a=1$, $c=1000$, $\nu=10^{-2}$, $v^=1$ and $l =5$ for both (a) and (b) corresponding to $b_c=51$, whereas $b=60$ in (a) and $b=3$ in (b). Taken from (Morimoto and Kawamura, 2011). []{data-label="nucleation"}](nucleation-a-rev.eps "fig:") ![ (Color online) The typical rupture process realized in the 1D BK model with the RSF law for (a) a strong and (b) a weak frictional instability, each corresponding to (a) $b>b_c$ and (b) $b<b_c$ with $b_c=2l^2+1$. The color represents the rupture velocity. The parameter values are $a=1$, $c=1000$, $\nu=10^{-2}$, $v^=1$ and $l =5$ for both (a) and (b) corresponding to $b_c=51$, whereas $b=60$ in (a) and $b=3$ in (b). Taken from (Morimoto and Kawamura, 2011). []{data-label="nucleation"}](nucleation-b-rev.eps "fig:") As mentioned, the condition for the appearance of such a nucleation process is given by $b<b_c=2l^2 +1$ in 1D, and by $b<b_c=4l^2 +1$ in 2D (for a square array of blocks). Indeed, Morimoto and Kawamura found that the critical nucleation size at which the slow nucleation process ends getting into the acceleration stage is given by $X_c=\pi/[\arccos(1-\frac{b-1}{2l^2})]-1$ in units of block size (Morimoto and Kawamura, 2011). Indeed, this length $X_c$ corresponds in its physical meaning to the length $h^$ of Rice (Rice, 1993), although its detailed functional form, [e.g.]{}, the dependence on $b$, is somewhat different from the standard one. The condition of this critical nucleation size being greater than the block size $X_c > 1$ yields the condition of the weak frictional instability $b<b_c$. In other words, when $b>b_c$, the nucleation process cannot be realized in the BK model due to its intrinsic discreteness. Indeed, this is exactly the situation as discussed by Rice (Rice, 1993). The above observation means that, if one takes the continuum limit of the BK model with the RSF law, the system should necessarily lie in the limit of a weak frictional instability, since the continuum limit means $l\rightarrow \infty$. Hence, at least [as long as one considers a uniform fault obeying the RSF law without any discretization short-length scale, earthquakes should exhibit characteristic properties rather than critical properties]{}. This fully corroborates an earlier criticism by Rice against the SOC view of earthquakes based on the BK model (Rice, 1993). Indeed, in seismology the concept of earthquake cycle has been used in long-term probabilistic earthquake forecasts (Scholz, 2002; Nishenko, 1987; Working Group on California Earthquake Probabilities, 1995). Of course, a big issue to understand is what is then the true origin of the GR law widely observed in real seismicity. Continuum models ---------------- As discussed in III.A.6, @Rice1993 criticized inherently discrete models, where simulated earthquake sequences depend on computation grid size. He confirmed in numerical simulations that complex earthquake sequences disappear when the grid size is sufficiently smaller than the critical size of slip nucleation zone for almost spatially uniform frictional properties. Moreover, he argued that geometrical and/or material disorder is the origin of complexity of earthquakes. The models with sufficiently small grid sizes may be called continuum models, which generate simulation results independent of the grid size, in contrast to inherently discrete models. Note that if a model does not have a finite critical size for nucleating unstable slip, such as a model with constant static and dynamic friction, it is always inherently discrete. In this subsection, we discuss continuum models of earthquakes, especially models using the rate- and state-dependent friction (RSF) law. In the RSF law, the critical size of slip nucleation can be defined as a function of frictional constitutive parameters, and the computation grid sizes are sufficiently smaller than the critical size in the studies mentioned below. We use elastic continuum models below, in contrast to spring-block models in the previous section. “Continuum model” is thus used to express two senses. The RSF law has commonly been used in models for understanding earthquake phenomena [@Scholz2002; @Dieterich2007].These models were sometimes constructed for reproducing and understanding particular earthquakes, earthquake cycles, or sliding processes observed by seismometers, strainmeters, Global Positioning System (GPS), etc. We will see deterministic aspects of earthquake phenomena, in addition to statistical characteristics of earthquakes. Note that comprehensive reviews were presented by @Rundle2003 [@Turcotte_etal2007; @Ben-Zion2008] for models of statistical properties of earthquakes using friction laws other than the RSF law. ### Earthquake cycles, asperities, and aseismic sliding Before introducing earthquake models, we briefly review observational facts about earthquakes and fault slip behavior. Earthquakes repeatedly occur at the same fault segment. At the Parkfield segment along the San Andreas fault, California, magnitude of about 6 interplate earthquakes have occurred at recurrence intervals of 23 $\pm$ 9 years since 1857 [@SykesMenke2006]. Great earthquakes of magnitude 8 class repeatedly occurred along the Nankai trough, where the Philippine Sea plate subducts beneath southwestern Japan, every one hundred years [@SykesMenke2006]. Quasi-periodic earthquake recurrence has been used for long-term forecasts of earthquakes (Working Group on California Earthquake Probabilities, 1995; Matthews et al., 2002). One of the most remarkable examples of regularity of earthquakes was found off Kamaishi, where the Pacific plate subducts beneath northern Honshu, Japan. Magnitude of 4.8 $\pm$ 0.1 earthquakes have repeatedly occurred at recurrence intervals of 5.5 $\pm$ 0.7 years at the same region since 1957. @Okada_etal2003 estimated coseismic slip distributions of recent Kamaishi earthquakes from seismic waveform data and found that they overlap with each other (Fig. \[kamaishi\]). Although many smaller earthquakes occur around the source area of the Kamaishi earthquakes, no comparable or larger earthquakes occur there. This observation suggests that aseismic sliding surrounds the source area of the Kamaishi earthquakes, where stick-slip motion occurs, and steady loading by the surrounding aseismic sliding to the source area leads to the quasi-periodic recurrence of almost the same magnitude earthquakes. The variance in the recurrence interval was suggested to come from temporal variation of aseismic sliding rate surrounding the earthquake source [@Uchida_etal2005]. Significant afterslip of the 2011 great Tohoku-oki earthquake (M=9.0) rapidly loaded the source area of the Kamaishi earthquake, generating earthquakes at much shorter recurrence intervals. Recurrences of small earthquakes at the same source areas in mainly creeping (aseissmic sliding) regions were found in many places and these earthquakes are called small repeating earthquakes [@NadeauJohnson1998; @Igarashi_etal2003]. Although small earthquakes occur, most strain is released by aseismic sliding on these fault planes. The seismic coupling coefficient is defined by the long-term average of the ratio of seismic slip amount to total (seismic and aseismic) slip expected from relative plate motion. The seismic coupling coefficient is variable, dependent on localities. It is close to unity at some segments along Chile and Aleutians, indicating little aseismic sliding and nearly complete locking during interseismic periods, and is nearly equal to zero at Marianas, indicating no or few large interplate earthquakes [@Pacheco_etal1993]. These facts show that aseismic sliding is common phenomenon and it plays an important part in strain release at plate boundaries and that frictional properties differ from place to place. ![\[kamaishi\] (a) Recurrence of Kamaishi earthquakes of nearly the same magnitudes and recurrence intervals. (b) Cumulative seismic moment of Kamaishi earthquakes. (c) Coseismic slip distribution of the 1995 and 2001 Kamaishi earthquakes estimated from seismic waveforms. Red broken contours and blue contours denote seismic slip of the 1995 and 2001 earthquakes, respectively [@Okada_etal2003].](kamaishi.eps){width="6.0cm"} A patch where stick-slip motion occurs, that is, a fault region where earthquakes repeatedly occur, is often called an asperity, which comes from the rock mechanics term for a contact spot between sliding surfaces as used in II.C. Note that an asperity of an earthquake occupies a considerable part of the earthquake fault area and its size is orders of magnitude larger than seismic slip amount. In contrast, an asperity of a sliding surface is much smaller and its size may be comparable to slip amount. The asperity model has been developed for explaining spatial heterogeneity in seismic slip on faults and complex source processes of earthquakes [@KanamoriMcNally1982; @Lay_etal1982; @Thatcher1990]. When the asperity model was developed around 1980, sliding behavior surrounding asperities was not clarified from observations because aseismic sliding cannot be detected by seismometers. To detect aseismic sliding, geodetic observations such as GPS are required. Since dense GPS networks were established in 1990s [@SegallDavis1997], many aseismic sliding phenomena have been reported such as afterslip (postseismic sliding) and slow (silent) earthquakes. The source areas of afterslip are usually located near coseismic slip areas (asperities), and the afterslip area and the asperity do not overlap as shown in Fig. \[tokachi\] [@Yagi_etal2003; @Miyazaki_etal2004; @Johnson_etal2006], which also support spatial heterogeneity of frictional properties. The locations of asperities of large earthquakes were confirmed to be locked during interseismic periods from geodetic observations [@Chlieh_etal2008; @Hashimoto_etal2009; @Perfettini_etal2010]. For instance, Figure \[sumatra\] clearly shows that seismic slip areas of large interplate earthquakes off the Sumatra island coincide with the locked areas during interseismic periods. For the 2011 great Tohoku-oki earthquake (M = 9.0), a significant peak of seismic slip larger than 30 m was estimated from inversions of seismic waveform and tsunami data [@Koketsu_etal2011]. This also suggests nonuniform frictional property on the plate interface. ![\[tokachi\] Spatial distribution of cumulative slip for 30 days of afterslip of the 2003 Tokachi-oki earthquake (M = 8.0), off Hokkaido, northern Japan, estimated from GPS data (color contours) by @Miyazaki_etal2004. Black contours with 0.5m interval show seismic slip in the 1973 Nemuro-oki (right), 1968 Tokachi-oki (left), and 2003 Tokachi-oki (center) earthquakes [@YamanakaKikuchi2004]. The black star and small circles denote the epicenter and aftershocks of the 2003 earthquake.](tokachi.eps){width="6.0cm"} ![\[sumatra\] Spatial distribution of interplate coupling estimated from geodetic data (colored circles) along the Sunda trench, where the Australian plate subducts beneath the Sumatra island. Red and orange circles indicate that the plate interface is nearly locked and strain is accumulated during an interseismic period, and white and yellow circles indicate that continuous aseismic sliding occurs and strain is not accumulated. Red and green contours with 5m interval show seismic slip in the 2004 Sumatra-Andaman (M = 9.1) and the 2005 Nias-Simeulue (M = 8.7) earthquakes. Blue and black lines show the approximate source areas of the 1797 and 1833 great earthquakes [@Chlieh_etal2008]. ](sumatra.eps){width="7.0cm"} Spatial distribution of asperities on plate boundaries has been estimated from source areas of past large interplate earthquakes, and earthquakes repeatedly occurred on the same asperities [@YamanakaKikuchi2004]. This suggest that the locations of asperities are unchanged at least a few earthquake cycles. Apparently complex earthquake cycle, where earthquake rupture areas are variable, may be understood by a change in combination of simultaneously ruptured asperities. For example, two adjacent asperities are simultaneously ruptured, resulting in a large earthquake in some cases, and one of them is ruptured to generate a smaller event in other cases. Note that some researchers object against persistent asperities on the basis of seismic waveform analyses [@ParkMori2007]. ### Models for nonuniform fault slip using the RSF law The asperity model indicates that spatial heterogeneity of material property is important, and it is compatible with the RSF law discussed in II.C. Regions of velocity-weakening frictional property ($a-b < 0$) correspond to asperities, where stick-slip occurs, and aseismic sliding occurs at regions of velocity-strengthening frictional property ($a-b > 0$). Afterslip occurs in velocity-strengthening areas, and it slowly relaxes stress increases generated by nearby earthquakes. Using a single-degree-of-freedom spring-block model, @Marone_etal1991 obtained theoretical slip time function $u(t)$ of afterslip, which occurs on a fault with velocity-strengthening friction ($a-b > 0$), as follows: $$\label{uniformslip} u(t) = \frac {(a-b)\sigma_n}{k} \ln \biggl[\frac{kV_{cs}}{(a-b)\sigma_n}t + 1\biggr] +V_0 t,$$ where $\sigma_n$ is normal stress on the fault plane, $k$ is spring stiffness, $V_{cs}$ is coseismic slip velocity, $V_0$ is preseismic slip rate, time $t$ is measured from the earthquake occurrence time. Quantitative comparison between afterslip observations and models indicate that the RSF law well explains afterslip [@PerfettiniAvouac2004; @Freed2007]. In case the stiffness is larger than the critical stiffness defined by Eq. (\[kc\]) for a velocity-weakening fault, it is called conditionally stable (Scholz, 1988). Although aseismic sliding usually occurs under quasi-static loading for conditionally stable case, rapid stress increase may generate seismic slip [@Gu_etal_1984]. This fact indicates that sliding behavior at a fault is not determined only by the fault properties but by a loading condition, suggestive of variable sliding behavior of a fault. Note that the effective stiffness of a fault may be related to fault size as will be shown in the next subsection. Since the RSF law takes into consideration time-dependent healing process, it can be used in simulations of earthquake cycles. @TseRice1986 first published an earthquake cycle model for a strike-slip fault in an elastic continuum using the RSF law to successfully explain stick-slip behavior at a shallower part of a fault, continuous stable sliding at a deeper part, and afterslip at intermediate depths. In the simulation, quasi-dynamic equilibrium between frictional stress and elastic stress generated by fault slip and relative plate motion is numerically solved. Their assumption on depth dependence of $a-b$ is consistent with laboratory data, which indicate $a-b$ changes from negative to positive at about 300$^\circ$C [@Blanpied_etal1995]. Similar models have been presented for earthquake cycles at particular regions to compare the simulations with observed earthquake recurrence and/or crustal deformation. Figure \[nankai\] shows an example simulation result of spatiotemporal evolution of slip velocity on a model plate interface, where great interplate earthquakes repeatedly occur at a shallower part and stable sliding on a deeper part [@Hori_etal2004]. ![image](nankai.eps){width="16.0cm"} If a single asperity exists on a fault plane without any interactions with other asperities, regular stick-slip at a constant recurrence interval is expected to occur. Note that when the asperity size is close to the critical nucleation zone size, irregular stick-slip cycle is observed even for a single asperity model [@LiuRice2007]. When some asperities exist within short distances, they interact with each other, resulting in complex earthquake sequences including single asperity ruptures and multiple asperity ruptures. Numerical simulations of complex earthquake sequences due to interactions between some asperities have been carried out by @KatoHirasawa1999, @Kato2004, @LapustaLiu2009, and @Kaneko_etal2010. In these studies, friction obeying the RSF law was assumed and different values of friction parameters ($a', b', {\cal L}$) are assigned for model asperities with velocity-weakening friction, to reproduce compound earthquakes, where some asperities are ruptured simultaneously or with some time delays, which resembles some observations. @Kato2008, for instance, reproduced a complex earthquake cycle similar to that observed at the Sanriku-oki region, northeastern Japan, where simulated earthquakes included the 1968 Tokachi-oki earthquake (M=8.2), the 1994 Sanriku-oki earthquake (M=7.7) and its largest aftershock (M=6.9) and afterslip. These studies suggest that spatial distribution of asperities or friction parameters controls regularity and complexity of earthquake recurrence. This further suggests that numerical forecasts of earthquakes may be possible if we can obtain detailed map of friction parameters on a fault. Friction parameters have actually been estimated through comparison of observed data and simulations at California [@Johnson_etal2006] and Japan [@Miyazaki_etal2004; @Fukuda_etal2009] from afterslip data. Preseismic sliding, which is aseismic sliding during a slip nucleation process, is expected before earthquake occurrence from the RSF law. It is almost ubiquitously observed in laboratory experiments, where the amount of preseismic sliding is of the order of micrometers [@OhnakaShen1999]. Using a spring-block system implemented with the RSF law, one can show that the preseismic sliding amount is approximately given by ${\cal L}$ [@Popov2010]. Some model studies with the RSF law discussed crustal deformation expected from preseismic sliding for particular earthquakes [@StuartTullis1995; @Kuroki_etal2002]. However, it is difficult to predict precise amplitudes of crustal deformation, because friction parameters that influence preseismic sliding are not well constrained from presently available data. There are some reports of observations of preseismic sliding, though insignificant or questionable observations are included [@Wyss1997]. For example, the close and dense geodetic observation of the Parkfield segment of the San Andreas fault could not detect any precursory slip prior to the 2004 earthquake, although it should be remarked that an observation of the tremor may suggest the accelerated creep on the fault $\sim 16$ km beneath the eventual earthquake hypocenter [@Shelly2009]. @KanamoriCipar1974 detected precursory signals in long-period strain seismogram before the occurrence of the 1960 great Chilian earthquake (M=9.5). Since no earthquake that could explain the observed strain signals was detected, they inferred that the signals were caused by preseismic sliding on a deeper extension of the mainshock fault plane. @LindeSacks2002 examined crustal deformation data before the occurrence of the 1944 Tonankai (M=8.0) and 1946 Nankai (M=8.1) earthquakes, southwestern Japan to construct a model for preseismic sliding of these earthquakes. Their model indicates that preseismic sliding took place at a deeper extension of the main shock fault plane. However, in models with the common RSF law, accelerating preseismic sliding just before earthquake occurrence takes place within the source area of seismic slip because spontaneous accelerating slip can be nucleated only in velocity-weakening region, being inconsistent with these models of preseismic sliding. @Kato2003b proposed a model for earthquake cycles at a subduction zone to explain large preseismic sliding at the deeper extension of the seismogenic plate interface. He assumed velocity-weakening friction ($d \mu_{\rm ss}/d \ln V < 0$) at low velocities and velocity-strengthening friction ($d \mu_{\rm ss}/d \ln V > 0$) at high velocities, where $\mu_{\rm ss}$ is a steady-state friction coefficient given by Eq. (\[steadystate\]). Preseismic sliding relaxes regional stresses, which may decrease seismic activity, while it increases stresses around the edges of the slipped region, which tends to increase seismic activity [@Kato_etal1997]. This may explain precursory seismic quiescence observed for some large earthquakes [@Kanamori1981; @Wyss_etal1981]. Preseismic sliding perturbs regional stress field, resulting in an increase or decrease of seismicity. Taking into consideration this effect,@Ogata2005 systematical searched seismicity changes in Japan to find possible crustal stress changes due to preseismic sliding. ### Slow earthquakes Slow earthquakes are episodic fault slip events that generate little or no seismic waves because their source durations are longer than the periods of observable seismic waves. Slip events without seismic wave radiation are often called silent earthquakes or slow slip events. Slow earthquakes have been studied by using records of very-long-period seismographs [@KanamoriStewart1979], creepmeters that directly detect fault creep at the ground surface [@King_etal1973], and strainmeters [@Linde_etal1996]. Afterslip and preseismic sliding mentioned earlier may be included in slow earthquakes. Recent development of dense geodetic observation networks including GPS and borehole tiltmeters accelerates studies of slow earthquakes [@SchwartzRokosky2007]. @Hirose_etal1999 found that episodic aseismic slip with duration of about 300 days took place in 1997 on the plate boundary in the Hyuganada region, southwestern Japan, from GPS data. The estimated slip and source area indicated that it released seismic moment corresponding to magnitude of 6.6. Later, almost the same size aseismic slip events occurred at the same area in 2003 and 2010. In the Tokai region, central Japan, another large slow earthquake from 2000 to 2005 released seismic moment nearly equal to that of an M=7.0 earthquake [@Ozawa_etal2002; @Miyazaki_etal2006]. The source area of this slow earthquake was estimated at the deeper extension of the locked plate boundary, where a magnitude 8 class interplate earthquake is expected to occur. At almost the same area, smaller slow earthquakes, corresponding to moment magnitude of about 6.0, with shorter durations of a few days were found to occur repeatedly [@HiroseObara2006]. These slow earthquakes are often called short-term slow slip events (SSEs) to be discriminated from long-term SSEs of durations of several months or longer. Furthermore, @HiroseObara2006 found low-frequency tremors, which radiate seismic waves with long durations, from high-sensitivity borehole seismometer array data. These events are clearly distinguished from long durations of wave trains and lack of high-frequency components of seismic waves. Short-term SSEs and low-frequency tremors occur simultaneously almost at the same locations. Synchronized occurrence of short-term SSEs and low-frequency tremors were observed in other regions such as the Cascadia subduction zone, North America [@RogersDragert2003] and Shikoku, southwestern Japan [@Obara_etal2004]. These findings of slow earthquakes and low-frequency tremors force us to reconsider simple view of earthquakes as brittle fracture. Many mechanical models for slow earthquakes has been proposed. Since both seismic and aseismic slip can be easily modeled with the RSF law, it is natural to consider that slow earthquakes can be modeled with the RSF law. In fact, sustaining aseismic oscillation, similar to recurrence of slow earthquakes, occurs in a single-degree-of-freedom spring-block model if the spring stiffness $k$ is equal to the critical stiffness $k_{\rm crt}$ given by Eq. (\[kc\]) [@Ruina1983]. Using a more realistic elastic continuum model, @Kato2004 showed that slow earthquakes occur when the size of velocity-weakening region is close to the critical size of slip nucleation zone. An effective stiffness $k_{\rm eff}$ of a fault may be defined by $$k_{\rm eff} = \Delta \tau/\Delta u,$$ where $\Delta \tau$ is shear-stress change on the fault due to slip $\Delta u$ [@Dieterich1986]. For a circular fault of radius $r$ with a constant stress drop in an infinite uniform elastic medium with Poission ratio = 0.25, $k_{\rm eff}$ is given by $$k_{\rm eff} = \frac{7 \pi}{24} \frac{G}{r},$$ where $G$ is rigidity. Recalling that unstable slip occurs for $k < k_{\rm crt}$ for a spring-block model, unstable slip is expected to occur for $k_{\rm eff} < k_{\rm crt}$ on a fault in an elastic medium. This leads to the condition for occurrence of unstable slip is that the fault radius $r$ is larger than the crtical fault size $r_c$ given by $$r_c = \frac{7 \pi}{24} \frac{G}{(b'-a')\sigma_n} {\cal L},$$ where $\sigma_n$ is the normal stress. Note that the critical nucleation zone size $r_c$ obtained by considering the stability around steady-state sliding may not be realistic in natural conditions during earthquake cycles. Other forms of critical nucleation zone sizes were obtained by considering more realistic conditions [@Dieterich1992; @RubinAmpuero2005]. It is confirmed in numerical simulations that usual earthquakes with short slip duration occurs for a circular fault with $r > r_c$, continuous stable sliding for $r \ll r_c$, and slow earthquakes for $r \sim r_c$, where slip duration increases with a decrease in $r/r_c$ as shown in Fig. \[sloweq\] (Kato, 2003b, 2004). The same idea was adopted by @LiuRice2007 in their model for slow earthquakes at a subduction zone, where they showed that high pore fluid pressure in the fault zone is required to explain observed recurrence intervals and slip amounts of slow earthquakes. Although these models are simple and plausible, slow earthquakes may occur under limited conditions of $r \sim r_c$. This seems to be inconsistent with the observations that slow earthquakes are common phenomena at some regions. Using a two-degree-of-freedom spring-block model, @YoshidaKato2003 showed that slow earthquakes may occur for wider conditions by considering interaction between unstable block where usual earthquakes repeatedly occur and a conditionally stable block where slow earthquakes occur. @ShibazakiIio2003 and @ShibazakiShimamoto2007 introduced a cut-off velocity to the state evolution effect in Eq.(\[rsf\]) to obtain frictional property of velocity weakening ($d \mu_{ss}/d \ln V < 0$) at low velocities and of velocity strengthening ($d \mu_{ss}/d \ln V > 0$) at high velocities, which is similar to the model by @Kato2003b for deep preseismic sliding. Similar complex frictional behavior that $d \mu_{ss}/d \ln V$ depends on velocity was actually observed in the laboratory for halite [@Shimamoto1986] and for serpentine [@Moore_etal1997]. In this case, slip is accelerated at low velocities with $d \mu_{ss}/d \ln V < 0$ and is decelerated at high velocities with $d \mu_{ss}/d \ln V > 0$, leading to slow earthquakes. Repeating slow earthquakes at transition depths from shallow locked zone to deeper stable sliding zone were simulated in @ShibazakiIio2003 and @ShibazakiShimamoto2007. This kind of model was further extended to simulate short- and long-term SSEs and their interaction with shallower large interplate earthquakes [@Matsuzawa_etal2010]. Weakness of these models is that experimental data for velocity dependence of $d \mu_{ss}/d \ln V$ are insufficient and frictional properties at depths where slow earthquakes occur are unknown. @Rubin2008 reviewed models for slow earthquakes based on the RSF law and pointed out that the existing models seem to be difficult to explain common occurrence of slow earthquakes at subduction zone s. He suggested that the variation of pore fluid pressure due to inelastic dilation of fault zone and fluid diffusion is required for generating slow earthquakes. ![\[sloweq\]The duration of slip events versus $r/r_c$ in numerical simulations using the RSF law, where a circular asperity of radius $r$ with velocity-weakening friction is embedded in a fault of velocity-strengthening frictional property. $r_c$ denotes the critical fault radius for unstable slip defined in the text [@Kato2003a].](sloweq.eps){width="6.0cm"} ### Origin of complexities of earthquakes and aftershock decay law @Rice1993 claimed that complex earthquake sequences simulated in inherently discrete models may be artifact and geometrical and/or material heterogeneity is required to explain observed complexity of earthquakes. Continuum models with relatively homogeneous frictional properties produce simple patterns of earthquakes such as periodic recurrence of large earthquakes that break the entire seismogenic zone. Using a continuum model with the RSF law, @Ben-ZionRice1995 introduced heterogeneity in effective normal stress on the fault and successfully produces moderately complex earthquake sequences. They pointed out that abrupt change in the effective normal stress is necessary to produce complex earthquakes. @Hillers_etal2007 introduced spatial heterogeneity in characteristic slip distance ${\cal L}$ in the model of a vertical strike-slip fault to produce complex earthquake sequences that include a wide range of earthquake magnitude. The obtained relation between earthquake magnitude and frequency mimics the Gutenberg-Richter (GR) law and the statistical properties of simulated earthquakes depend on the degree of heterogeneity in ${\cal L}$. They also argued temporal clustering of simulated earthquakes and tendency of nucleation sites of smaller ${\cal L}$. @HillersMiller2007 introduced spatial variation of pore pressure to generate complex earthquake sequences. An important fact about the relation between magnitude and frequency of earthquakes obtained in observations is that the GR law may not always be valid for each individual fault. For some faults and plate boundaries, the number of small earthquakes is too few than that expected from the GR law and the frequency of large earthquakes that rupture the entire fault, indicating violation of the GR law [@Stirling_etal1996; @IshibeShimazaki2009]. This behavior of fewer small earthquakes than that expected from the frequency of large earthquakes is referred to the characteristic earthquake model. Highly coupled plate interface in the Tokai region, central Japan, is nearly quiescent, while many small earthquakes occur in the overriding plate and subducting oceanic plate [@Matsumura1997]. This suggests that except for great earthquakes few small earthquakes occur at the plate interface in the Tokai region. Considering large earthquakes along the San Andreas fault, California, and smaller earthquakes at secondary faults around the San Andreas, @Turcotte1997 argued that the observed GR law comes from a fractal distribution of faults and characteristic earthquakes at each fault. Another important empirical law that demonstrates complexities of earthquakes is the modified Omori (Omori-Utsu) law for decay in aftershock occurrence rate [@Utsu_etal1995]. Aftershock rate $n$ at time $t$ from the occurrence of the mainshock is well approximated by $$n(t) = \frac{K}{(t+T_{\rm MOL})^p},$$ where, $K$, $T_{\rm MOL}$, and $p$ are constants. Constant $p$ takes $\sim 1$ for many cases. In the case of $T_{\rm MOL}=0$, this relation is simply referred to as the Omori law. Aftershocks have been thought to be manifestation of relaxation of stress generated by the mainshock. To explain delay times of aftershocks, subcritical cracking due to stress corrosion [@YamashitaKnopoff1987] and the variation of effective normal stress due to diffusion of pore fluid, whose pressure is perturbed by the mainshock [@BoslNur2002], were invoked. @Dieterich1994b considered responses of many fault patches, where friction is assumed to obey the RSF law, to instantaneous stress change due to the mainshock. He furtehr assumed that a constant seismicty rate is achieved under a constant loading rate without any stress perturbation. This model successfully explains the power law decay of seismicity rate with $p = 1$, being consistent with observations, and has been applied to analyses of aftershocks of some large earthquakes [@Toda_etal1998]. Another important model for aftershocks using the RSF law is related to afterslip. Afterslip perturbs stresses around its source area, causing aftershocks. Differentiating the slip function Eq.(\[uniformslip\]) of afterslip with respect to time, we have a slip rate approximately proportional to $(t+c)^{-1}$, which may related to a stress rate and therefore seismicity rate [@PerfettiniAvouac2004]. This expected seismicity rate coincides with the Omori-Utsu formula with $p=1$. Moreover, afterslip propagates outward from a mainshock slip area, leading to expansion of aftershock area [@Kato2007]. Aftershock expansion pattern obtained from a numerical model with the RSF law is consistent with observed expansions of aftershock ares [@TajimaKanamori1985; @PengZhao2009]. ### Earthquake dynamics: critical slip distance Here we consider the dynamics of unstable motion. The unstable slip of a spring-block system given by Eq. (\[onebody\]) is accompanied by the drop of frictional force. If one plots the frictional force as a function of the slip distance (Fig. \[slipweakening\]), one can define the distance $D_c$ over which the frictional force drops. This behavior of decreasing frictional force with increasing slip is referred to the slip-weakening model [@Ida1972; @Andrews1976], and the slip distance $D_c$ is called the critical slip distance in seismology. $D_c$ is on the same order of (or at most several tens of) the characteristic length ${\cal L}$ in an evolution law [@BizzarriCocco2003]. This is so irrespective of the number of degrees of freedom: discrete or continuum. ![\[slipweakening\] Schematic diagram of the relation between frictional force and slip distance during slip-weakening process, where $F_i$, $F_p$, $F_d$, and $D_c$ denote the initial force, the peak frictional force, the dynamic frictional force, and the critical slip distance, respectively. The shaded area indicates the fracture energy $G_c$.](slipweakening.eps){width="6.0cm"} Importantly, one can estimate $D_c$ of earthquakes by analyzing seismic wave. Such analyses show that $D_c$ is on the order of several tens of centimeters or a meter [@Ide1997]. Note that fracture energy $G_c$, which is equal to twice the surface energy density $\Gamma$, can rather stably be estimated from seismic waveform data, though accurate estimate of $D_c$ is difficult because of poor resolution of rupture process modeling from seismic waves [@GuatteriSpudish2000]. The characteristic slip distance ${\cal L}$ estimated for afterslip of a large interplate earthquake by GPS data is on the order of mm (Fukuda et al. 2009). This makes a quite contrast to laboratory experiments, where ${\cal L}$ is typically estimated as several micrometers. Because ${\cal L}$ is a typical longitudinal dimension of true contact patch, application of the RSF law to a natural fault implies that a natural fault also consists of true contact patches, a linear dimension of which is several tens of centimeters. Although the aperture of a fault is not empty but filled with fluid and gouge, a fault generally has the non-planer structure (e.g., jogs) that can interlock to resist the displacement. Such jogs may effectively act as the true contact area. However, it is not obvious at all if the RSF law still holds for such true contact area of macroscopic scale. At least, we believe that the RSF law should not be used except for very low speed friction. Namely, the RSF law no longer holds at seismic slip rate due to physical processes caused by frictional heat: flash heating, melting, mechanochemical effects, etc. In such cases, the critical slip distance $D_c$ is proportional to $\epsilon_c/P$, where $\epsilon_c$ is the critical energy per unit area for a weakening process (e.g. melting) to occur and $P$ is the normal stress. (As the frictional force is proportional to $P$, the produced heat is proportional to $P$ and to the slip distance $D$. Thus, the weakening process may occur if $PD$ is on the order of $\epsilon_c$.) Namely, the critical slip distance is inversely proportional to the normal pressure. This implies that the critical slip distance must be smaller for deeper faults. However, unfortunately, such depth dependence has not been observed so that the mechanism that determines the critical slip distance is different. Another important process that affect the critical slip distance is the off-fault fracture accompanied by the crack propagation on fault. Andrews (2005) analyzes a model for the slip propagation on a fault supplemented with the Coulomb yield condition for off-fault material. He finds that the effective critical slip distance depends on the distance from the crack initiation point. This is because the plastic zone is wider for larger crack. Thus, the critical slip distance is essentially scale dependent, which is consistent with the observation facts. Earthquake models and statistics II: SOC and other models ========================================================= Statistical properties of the OFC model --------------------------------------- ### The model In the previous section, we reviewed the properties of statistical physical models of earthquakes such as the spring-block BK model and the continuum model. In the present section, we deal with further simplified statistical physical models of earthquakes (Turcotte 1997; Hergarten, 2002; Turcotte, 2009). Many of them were coupled map lattice models originally introduced as the SOC models (Bak, Tang and Wiesenfeld, 1987; Bak and Tang, 1989, Ito and Matsuzaki, 1990; Nakanishi, 1990; Brown, Scholz and Rundle, 1991; Olami, Feder and Christensen, 1992; Hainzl, Zöller and Kurths, 1999; 2000; Hergarten, and Neugebauer, 2000; Helmstetter, Hergarten and Sornette, 2004). The one introduced by Olami, Feder and Christensen (OFC) as a further simplification of the BK model, now called the OFC model, is particularly popular (Olami, Feder and Christensen, 1992). It is a two-dimensional coupled map lattice model where the rupture propagates from lattice site to its nearest-neighboring sites in a non-conservative manner, often causing multi-site “avalanches”. Extensive numerical studies have also been devoted to this model, mainly in the field of statistical physics, which we wish to review in the present section (Christensen and Olami, 1992; Grassberger, 1994; Middleton and Tang, 1995; Bottani and Delamotte, 1997; de Carvalho and Prado, 2000; Pinho and Prado, 2000; Lise and Paczuski, 2001; Miller and Boulter, 2002; Hergarten and Neugebauer, 2002; Boulter and Miller, 2003; Helmstetter, Hergarten and Sonette, 2004; Peixoto and Prado, 2006; Wissel and Drossel, 2006; Ramos, 2006; Kotani, Yoshino and Kawamura, 2008; Kawamura et al, 2010, Jagla, 2010). In the OFC model, “stress” variable $f_i$ ($f_i\geq 0$) is assigned to each site on a square lattice with $L\times L$ sites. Initially, a random value in the interval \[0,1\] is assigned to each $f_i$, while $f_i$ is increased with a constant rate uniformly over the lattice until, at a certain site $i$, the $f_i$ value reaches a threshold, $f_c=1$. Then, the site $i$ “topples”, and a fraction of stress $\alpha f_i$ ($0<\alpha<0.25$) is transmitted to its four nearest neighbors, while $f_i$ itself is reset to zero. If the stress of some of the neighboring sites $j$ exceeds the threshold, [i.e.]{}, $f_j\geq f_c=1$, the site $j$ also topples, distributing a fraction of stress $\alpha f_j$ to its four nearest neighbors. Such a sequence of topplings continues until the stress of all sites on the lattice becomes smaller than the threshold $f_c$. A sequence of toppling events, which is assumed to occur instantaneously, corresponds to one seismic event or an avalanche. After an avalanche, the system goes into an interseismic period where uniform loading of $f$ is resumed, until some of the sites reaches the threshold and the next avalanche starts. The transmission parameter $\alpha$ measures the extent of non-conservation of the model. (This $\alpha$ should not be confused with $\alpha$ describing the velocity-weakening friction force employed in the study of the BK model of subsection III.A. We are using $\alpha$ as a conservation parameter of the OFC model throughout this subsection IV.A). The system is conservative for $\alpha =0.25$, and is non-conservative for $\alpha <0.25$. A unit of of time is taken to be the time required to load $f$ from zero to unity. In the OFC model, boundary conditions play a crucial role. For example, SOC state is realized under open or free boundary conditions, but is not realized under periodic boundary conditions. Thus, most of the studies made in the past employed open (or free) boundary conditions. ### Properties of the homogeneous model Earlier studies concentrated mostly on the event size distribution of the model (Olami, Feder and Christensen, 1992; Christensen and Olami, 1994; Grassberger, 1994; de Carvalho and Prado, 2000; Lise and Paczuski, 2001; Miller and Boulter, 2002; Boulter and Miller, 2003; Drossel, 2006). The avalanche size $s$ is defined by the total number of “topples” in a given avalanche, which could be larger than the number of toppled sites because multi-toppling is possible in a given avalanche. (In fact, it is observed that multi-toppling rarely occurs in the model except in the conservation limit or in the regime very close to it.) It turned out that the size distribution of the model exhibited a power-law-like behavior close to the GR law. Yet, there still remains controversy concerning whether the model is strictly critical (Lise and Paczuski, 2001) or only approximately so (de Carvalho and Prado, 2000; Miller and Boulter, 2002; Boulter and Miller, 2003; Drossel, 2006). In Fig.\[magnitude-OFC\], we show the size distribution of the model under open boundary conditions for several values of the transmission parameter $\alpha$ (Kawamura et al, 2010). As can be seen from the figure, a near straight-line behavior corresponding a power-law is observed. The slope representing the $B$-value is not universal varying from $\simeq 0.90$ to $\simeq 0.22$ as $\alpha$ is varied from 0.17 to 0.245. The power-law feature is weakened as one approaches the conservation limit. ![ (Color online) The size distribution of the OFC model under open boundary conditions for various values of the transmission parameter $\alpha $. The slope of the data gives the value of $1+B$, which is shown in the figure. Taken from (Kawamura et al, 2010). []{data-label="magnitude-OFC"}](magnitude-OFC.eps) Hergarten [et al]{} observed that the OFC model also exhibited another well-known power-law feature of seismicity, [i.e.]{}, the Omori law (or the inverse Omori law) describing the time evolution of the frequency of aftershocks (foreshocks) (Hergarten and Neugebauer, 2002; Helmstetter, Hergarten and Sonette, 2004). We show in Fig.\[Omori-OFC\](a) on a log-log plot the frequency of aftershocks as a function of the time elapsed after the mainshock $t$ (Kawamura et al, 2010). The slope representing the Omori exponent $p$ is again not universal depending on the parameter $\alpha$ as $p=0.84$, 0.69 and 0.03 for $\alpha=0.17$,0.20 and 0.23, respectively. Since the $p$-value is known to come around unity in real seismicity, the $p$ value of the OFC model is not necessarily close to real observation. Similar results are obtained also for foreshocks: See Fig.\[Omori-OFC\](b). Aftershocks and foreshocks are defined here as events of arbitrary sizes which occur in the vicinity of mainshock with its epicenter lying with the distance $r\leq r_c$ (the range parameter $r_c$ it taken to be $r_c=10$ in Fig.\[Omori-OFC\]). As one approaches the conservation limit $\alpha=0.25$, both aftershocks and foreshocks tend to go away. ![ (Color online) The time dependence of the frequency of aftershocks (a), and of foreshocks (b), of the OFC model under open boundary conditions on a log-log plot for several values of the transmission parameter $\alpha$. Mainshocks are the events of their size greater than $s\geq s_c=100$. The time $t$ is measured with the occurrence of a mainshock as an origin. The range parameter is $r_c=10$. Taken from (Kawamura et al, 2010). []{data-label="Omori-OFC"}](Omori-a-OFC.eps "fig:") ![ (Color online) The time dependence of the frequency of aftershocks (a), and of foreshocks (b), of the OFC model under open boundary conditions on a log-log plot for several values of the transmission parameter $\alpha$. Mainshocks are the events of their size greater than $s\geq s_c=100$. The time $t$ is measured with the occurrence of a mainshock as an origin. The range parameter is $r_c=10$. Taken from (Kawamura et al, 2010). []{data-label="Omori-OFC"}](Omori-b-OFC.eps "fig:") As mentioned, the properties of the model depends on applied boundary conditions. Middleton and Tang observed that the model under open boundary conditions went into a special transient state where events of size 1 (single-site events) repeated periodically with period $1-4\alpha$ (Middleton and Tang, 1995). These single-site events occur in turn in a spatially random manner, but after time $1-4\alpha$, the same site topples repeatedly. Although such a periodic state consisting of single-site events is a steady state under periodic boundary conditions, it is not a steady state under open boundary conditions because of the boundary. Indeed, clusters are formed near the boundary, within which the stress values are more or less uniform, and gradually invades the interior destroying the periodic state. Eventually, such clusters span the entire lattice, giving rise to an SOC-like steady state. Middleton and Tang pointed out that such clusters might be formed via synchronization between the interior site and the boundary site, the latter having a slower effective loading rate due to the boundary. Large-scale synchronization occurring in the steady state of the OFC model was further investigated by Bottani and Delamotte (Bottani and Delamotte, 1997). In contrast to the aforementioned critical properties of the model, recent studies also unraveled the [characteristic]{} features of the OFC model (Ramos, 2006; Kotani, Yoshino and Kawamura, 2008; Kawamura et al, 2010). By investigating the time series of events, Ramos found the nearly periodic recurrence of large events (Ramos, 2006). Kotani [et al]{} studied the spatiotemporal correlations of the model and identified in the OFC model a phenomenon resembling the “asperity” (Kotani, Yoshino and Kawamura, 2008; Kawamura et al, 2010). These authors computed the local recurrence-time distribution, $P(T)$, of the model. The computed $P(T)$, shown in Fig. \[recurrence-OFC\], exhibited a sharp $\delta$-function-like peak at $T=T^=1-4\alpha$, indicating that many (though not all) events of the OFC model were repeated with a fixed time-interval $T=T^$. While the peak at $T=T^$ is sharp, it is not infinitely sharp with a finite intrinsic width: See the inset. The peak position turned out to be independent of the range parameter $r_c$, the size threshold $s_c$, and the lattice size (as long as it was not too small). As $\alpha$ is increased toward $\alpha=0.25$, the $\delta$-function peak is gradually suppressed with keeping its position strictly at $T=1-4\alpha$. The $\delta$-function peak of $P(T)$ goes away toward the conservation limit $\alpha=0.25$: See Fig. \[recurrence-OFC\]. ![ (Color online) Log-log plots of the local recurrence-time distributions of large avalanches of their size $s\geq s_c=100$ for a fixed range parameter $r_c=10$, with varying the transmission parameter $\alpha$. The arrow in the figure represents the expected peak position for $\alpha=0.245$ corresponding to the period $T^=1-4\alpha =0.02$. The inset is a magnified view of the main peak for the case of $\alpha=0.17$. Taken from (Kawamura et al, 2010). []{data-label="recurrence-OFC"}](recurrence-OFC.eps) In the longer time regime $T>T^$, $P(T)$ exhibits behaviors close to power laws (Kotani, Yoshino and Kawamura, 2008; Kawamura et al, 2010). Furthermore, the periodic events contributing to a sharp peak of $P(T)$ (“peak events”) possess a power-law-like size distribution very much similar to those observed for other aperiodic events (Kotani, Yoshino and Kawamura, 2008; Kawamura et al, 2010). Hence, in earthquake recurrence of the model, the characteristic or periodic feature, [i.e.]{}, a sharp peak in $P(T)$ at $T=T^$, and the critical feature, [i.e.]{}, power-law-like behaviors of $P(T)$ at $T>T^$ and power-law-like size distribution, coexist. ### Asperity-like phenomena In fact, it has turned out that the $\delta$-function peak of $P(T)$ is borne by “asperity-like” events, [i.e.]{}, the events which rupture repeatedly with almost the same period $1-4\alpha$ and with a common rupture zone and a common epicenter. In seismology, the concept of asperity is now quite popular. A typical example might be the one observed along the subduction zone in northeastern Japan, particularly repeating earthquakes off Kamaishi (Matsuzawa, Igarashi and Hasegawa, 2002; Okada, Matsuzawa and Hasegawa, 2003). In Fig. \[asperity-OFC\], we show typical examples of such asperity-like events as observed in the OFC model (Kawamura et al, 2010). In the upper panel, we show for the case of $\alpha=0.2$ typical snapshots of the stress-variable distribution immediately before and after a large event which occurs at time $t=t_0$. Discontinuous drop of the stress associated with a rupture of a synchronized cluster is discernible. Then, at time $t=t_0+T^$, the same cluster (except for a minor difference) ruptures again. In the lower panel, we show snapshots of the stress-variable distribution immediately before and after this subsequent avalanche occurring at $t=t_0+T^$. In this particular example, a rhythmic rupture of essentially the same cluster has repeated more than ten times. The asperity-like events go away in the conservation limit $\alpha \rightarrow 1/4$ (Kawamura et al, 2010). It is also observed that an epicenter site tends to lie at the tip or at the corner of the rupture zone rather than in its interior (Kawamura et al, 2010). The asperity-like events observed in the OFC model closely resemble those familiar in seismology (Scholz, 2002), in the sense that almost the same spatial region ruptures repeatedly with a common epicenter site and with a common period. ![ (Color online) Snapshots of the stress-variable distribution of the OFC model under open boundary conditions for the case of $\alpha=0.2$; (a) immediately before a large event at time $t=t_0$, (b) immediately after this event, (c) immediately before the following event which occurs at time $t=t_0+T^* (T^=0.2)$, and (d) immediately after this second event. Two events are of size $s=15891$ and $s=15910$ on a $L=256$ lattice. The region surrounded by red bold lines represents the rupture zone, while the star symbol represents an epicenter site which is located at the tip of the rupture zone. Taken from (Kawamura et al, 2010). []{data-label="asperity-OFC"}](asperity-OFC.eps) In fact, not all large events of the OFC model occur in the form of asperity. Many clusters forming large events are left out of the rhythmic recurrence, and rupture more critically with widely-distributed recurrence time, thereby bearing the observed power-law-like part of $P(T)$. A key ingredient in the asperity formation is a self-organization of the highly concentrated stress state (Kawamura et al, 2010). The stress-variable distribution in the asperity region tends to be “discretized” to certain values. In Fig. \[stress-OFC\], we show for the case of $\alpha=0.17$ the stress-variable distribution $D(f)$ of the asperity sites immediately (a) before and (b) after an avalanche, averaged over asperity events. As can be seen from the figure, $D(f)$ now consists of several “spikes” located at appropriate multiples of the transmission parameter $\alpha$, [i.e.]{}, at $1-n\alpha$ before the rupture, and at $f=n\alpha$ after the rupture, with $n$ being an integer. Furthermore, as the asperity events repeat, the tendency of the stress-variable concentration is more and more enhanced. In Fig. \[stress-evolution-OFC\], we show the time sequence of the stress-variable distribution at the time of toppling for the asperity events. As the asperity events repeat, the stress-variable distribution tends to be narrower, being more concentrated on the threshold value $f_c=1$ (Kawamura et al, 2010; Hergarten and Krenn, 2011). In fact, one can prove that the stress-variable distribution at the time of toppling tends to be more concentrated on the threshold value $f_c=1$ as the asperity events repeat (Kawamura et al, 2010). Namely, once each site starts to topple more or less at similar stress values close to the threshold value $f_c=1$, this tendency is more and more evolved as the asperity events repeat. [The stress-variable concentration tends to be self-organized]{}. Such a stress-variable concentration immediately explains why the interval time of the asperity events is equal to $1-4\alpha$, and why the same site becomes an epicenter in the asperity sequence (Kawamura et al, 2010). For example, the reason why the interval time is $1-4\alpha$ when all sites topple at the stress value close to the threshold $f_c= 1$ in the asperity events can easily be seen just by remembering the conservation law of the stress, [i.e.]{}, the stress-variable dissipated at the time of toppling, which is $1-4\alpha$ per site if the toppling occurs exactly at $f=1$, should match the stress loaded during the interval time $T$. See (Kawamura et al, 2010), for further details. More recently, Hergarten and Krenn made further analysis of this stress-concentration phenomenon, demonstrating that the mean stress excess representing the extent of the stress concentration approaches zero exponentially with a certain decay time which is dependent on the number of “internal” sites (the sites contained in the rupture zone) connected to an epicenter site (Hergarten and Krenn, 2011). Then, the epicenter site with the smallest number of internal nearest-neighbor sites, [i.e.]{}, the one lying at the tip of the rupture zone, has the longest decay time and turns out to be most stable. This observation gives an explanation of the finding of (Kawamura et al, 2010) that the majority of epicenter sites of the asprity-like events are located at the tip of the rupture zone. ![ (Color online) The stress-variable distribution $D(f)$ of each site contained in the rupture zone of the asperity event of the OFC model under open boundary conditions, just before (a) and after (b) the asperity event. An asperity event is defined here as an event of its size greater than $s\geq s_c=100$ belonging to the main peak of the local recurrence-time distribution function. The transmission parameter is $\alpha=0.17$. The inset is a magnified view of the main peak. Taken from (Kawamura et al, 2010). []{data-label="stress-OFC"}](stress-a-OFC.eps "fig:") ![ (Color online) The stress-variable distribution $D(f)$ of each site contained in the rupture zone of the asperity event of the OFC model under open boundary conditions, just before (a) and after (b) the asperity event. An asperity event is defined here as an event of its size greater than $s\geq s_c=100$ belonging to the main peak of the local recurrence-time distribution function. The transmission parameter is $\alpha=0.17$. The inset is a magnified view of the main peak. Taken from (Kawamura et al, 2010). []{data-label="stress-OFC"}](stress-b-OFC.eps "fig:") ![ (Color online) The time sequence of the stress-variable distribution $D(f)$ at the time of toppling of each site contained in the rupture zone of the asperity events. An asperity event is defined here as an event of its size greater than $s\geq s_c=100$ belonging to the main peak of the local recurrence-time distribution function. The transmission parameter is $\alpha=0.17$. As the events repeat, the stress-variable distribution at the time of toppling gets more and more concentrated on the borderline value $f_c=1$. Taken from (Kawamura et al, 2010). []{data-label="stress-evolution-OFC"}](stress-evolution-OFC.eps) Although the origin of the asperity is usually ascribed in seismology to possible inhomogeneity of the material property of the crust or of the external conditions of that particular region, we stress that, in the present OFC model, there is no built-in inhomogeneity in the model parameters nor in the external conditions. [The “asperity” in the OFC model is self-generated from the spatially uniform evolution-rule and model parameters.]{} As mentioned, the asperity in the OFC model is not a permanent one: In long terms, its position and shape change. After all, the model is uniform. Nevertheless, recovery of spatial uniformity often takes a long time, and the asperity exists stably over many earthquake recurrences. Although one has to be careful in immediately applying the present result for the OFC model to real earthquakes, it might be instructive to recognize that the observation of asperity-like earthquake recurrence does not immediately mean that the asperity region possesses different material properties nor different external conditions from other regions. Thus, critical and characteristic features coexist in the OFC model in an intriguing manner. Although the critical features were emphasized in earlier works, the model certainly involves the eminent characteristic features in it as well. Thus, the OFC model, though an extremely simplified model, may capture some of the essential ingredients necessary to understand apparent coexistence of critical and characteristic properties in real earthquakes. ### Effects of inhomogeneity It should be noticed that the original OFC model is a spatially homogeneous model, where homogeneity of an earthquake fault is implicitly assumed. Needles to say, real earthquake fault is more or less spatially inhomogeneous, which might play an important role in real seismicity. Then, a natural next step is to extend the original homogeneous OFC model to the inhomogeneous one where the evolution rule and/or the model parameters are taken to be random from site to site. Spatial inhomogeneity could be either static or dynamical. As a cause of possible temporal variation of spatial inhomogeneity, one may consider the two distinct processes, [i.e.]{}, the fast dynamical process during an earthquake rupture changing the fault state via, [e.g.]{}, wear, frictional heating, melting, [etc]{} and many slower processes taking place during a long interseismic period until the next earthquake, [e.g.]{}, water migration, plastic deformation, chemical reactions, [etc]{} (Scholz, 2002). Thus, in introducing the spatial inhomogeneity into the OFC model, there might be two extreme ways: In one, one may assume that the randomness is quenched in time, namely, spatial inhomogeneity is fixed over many earthquake recurrences. In the other extreme, spatial inhomogeneity is assumed to vary with time in an uncorrelated way over earthquake recurrences. Several studies have been made on the inhomogeneous OFC model for both types of inhomogeneities. For the first type of inhomogeneity, [i.e.]{}, the quenched or static randomness, Janosi and Kertesz introduced spatial inhomogeneity into the stress threshold and found that the inhomogeneity destroyed the SOC feature of the model (Janosi and Kertesz, 1993). Torvund and Froyland studied the effect of spatial inhomogeneity in the stress threshold, and observed that the inhomogeneity induced a periodic repetition of system-size avalanches (Torvund and Froyland, 1995). Ceva introduced defects associated with the transmission parameter $\alpha$, and observed that the SOC feature was robust against small number of defects (Ceva, 1995). Mousseau and Bach et al introduced inhomogeneity into the transmission parameter at each site. These authors observed that the bulk sites fully synchronized in the form of a system-wide avalanche over a wide parameter range of the model (Mousseau , 1996; Bach, Wissel and Dressel, 2008). For the second type of inhomogeneity, [i.e.]{}, the dynamical randomness, Ramos considered the randomness associated with the stress threshold, and observed that the nearly periodic recurrence of large events persisted (Ramos, 2006). More recently, Jagla studied the same stress-threshold inhomogeneity, to find that the GR law was weakened by randomness (Jagla, 2010). Very interesting observation by Jagla is that, once the slow structural relaxation process is added to the inhomogeneous OFC model, both the GR law and the Omori law are realized with the exponents which are stable against the choice of the model parameter values and are close to the observed values. Yamamoto [et al]{} studied the dynamically inhomogeneous model with a variety of implementations of the form of inhomogeneities to find the general tendency that critical features found in the original homogeneous OFC model, [e.g.]{}, the Gutenberg-Richter law and the Omori law, were weakened or suppressed in the presence of inhomogeneity, whereas the characteristic features of the original homogeneous OFC model, [e.g.]{}, the near-periodic recurrence of large events and the asperity-like phenomena, tended to persist (Yamamoto, Yoshino and Kawamura, 2010). Thus, the properties of the dynamically inhomogeneous models are quite different from those of the static or quenched inhomogeneous models. In the latter case, introduced inhomogeneity often gives rise to a full synchronization and a periodic repetition of system-size events. Such a system-wide synchronization is never realized in the dynamically homogeneous models. Presumably, temporal variation of the spatial inhomogeneity may eventually average out the inhomogeneity over many earthquake recurrences, giving rise to the behavior similar to that of the homogeneous model. Fiber bundle models ------------------- The fiber bundle model, initiated by @bkc46 in the context of testing strength of cotton yarns, represents various aspects of fracture processes of disordered systems, through its self-organised dynamics (for detailed review see @bkc34). The fiber bundle (see Fig. \[bkc-fbm-schematic\]) consists of $N$ fibers or Hook springs, each having identical spring constant $\kappa$. The bundle supports a load $W=N\sigma$ and the breaking threshold $\left( \sigma _{th}\right) _{i}$ of the fibers are assumed to be different for different fiber ($i$). For the equal load sharing model we consider here, the lower platform is absolutely rigid, and therefore no local deformation and hence no stress concentration occurs anywhere around the failed fibers. This ensures equal load sharing, i.e., the intact fibers share the applied load $W$ equally and the load per fiber increases as more and more fibers fail. The strength of each of the fiber $\left( \sigma_{th}\right)_{i}$ in the bundle is given by the stress value it can bear, and beyond which it fails. The strength of the fibers are taken from a randomly distributed normalised density $\rho (\sigma _{th})$ within the interval $0$ and $1$ such that $$\int _{0}^{1}\rho (\sigma _{th})d\sigma _{th}=1.$$ The equal load sharing assumption neglects ‘local’ fluctuations in stress (and its redistribution) and renders the model as a mean-field one. The breaking dynamics starts when an initial stress $ \sigma $ (load per fiber) is applied on the bundle. The fibers having strength less than $ \sigma $ fail instantly. Due to this rupture, total number of intact fibers decreases and rest of the (intact) fibers have to bear the applied load on the bundle. Hence effective stress on the fibers increases and this compels some more fibers to break. These two sequential operations, namely the stress redistribution and further breaking of fibers continue till an equilibrium is reached, where either the surviving fibers are strong enough to bear the applied load on the bundle or all fibers fail. This breaking dynamics can be represented by recursion relations in discrete time steps. For this, let us consider a very simple model of fiber bundles where the fibers (having the same spring constant $\kappa$) have a white or uniform strength distribution $\rho(\sigma_{th})$ upto a cutoff strength normalized to unity, as shown in Fig. \[bkc-uniform\]: $\rho (\sigma_{th}) = 1$ for $0 \le \sigma_{th} \le 1$ and $\rho(\sigma_{th})=0$ for $\sigma_{th} > 1$. Let us also define $U_t(\sigma)$ to be the fraction of fibers in the bundle that survive after (discrete) time step $t$, counted from the time $t=0$ when the load is put (time step indicates the number of stress re-distributions). As such, $U_t(\sigma=0)=1$ for all $t$ and $U_t(\sigma)=1$ for $t=0$ for any $\sigma$; $U_t(\sigma)=U^(\sigma) \ne 0$ for $t \to \infty$ and $\sigma < \sigma_f$, the critical or failure strength of the bundle, and $U_t(\sigma)=0$ for $t \to \infty$ if $\sigma > \sigma_f$. Therefore $U_{t}(\sigma)$ follows a simple recursion relation (see Fig. \[bkc-uniform\]) $$U_{t+1}= 1-\sigma_t;\ \ \sigma_t = \frac{W}{U_t N}$$ $$\label{bkc-recU_t} {\rm or,} \ \ U_{t+1}=1-\frac{\sigma }{U_{t}}.$$ At the equilibrium state ($ U_{t+1}=U_{t}=U^{} $), the above relation takes a quadratic form of $ U^{} $ : $$U^{^{2}}-U^{}+\sigma =0.$$ The solution is $$U^{}(\sigma )=\frac{1}{2}\pm (\sigma_{f}-\sigma )^{1/2};\sigma_{f}=\frac{1}{4}.$$ Here $ \sigma_{f} $ is the critical value of initial applied stress beyond which the bundle fails completely. The solution with ($ + $) sign is the stable one, whereas the one with ($ -) $ sign gives unstable solution [@bkc8; @bkc9; @bkc10]. The quantity $ U^{}(\sigma ) $ must be real valued as it has a physical meaning: it is the fraction of the original bundle that remains intact under a fixed applied stress $ \sigma $ when the applied stress lies in the range $ 0\leq \sigma \leq \sigma_{f} $. Clearly, $ U^{}(0)=1 $. Therefore the stable solution can be written as $$\label{bkc-Ustarsigma_c} U^{}(\sigma )=U^{}(\sigma_{f})+(\sigma_{f}-\sigma )^{1/2}; \ U^(\sigma_f) = \frac{1}{2} \ {\rm and}\ \sigma_{f}=\frac{1}{4}.$$ For $ \sigma >\sigma_{f} $ we can not get a real-valued fixed point as the dynamics never stops until $ U_{t}=0 $ when the bundle breaks completely. .2in (a) At $\sigma <\sigma_{f} $ .2in It may be noted that the quantity $ U^{}(\sigma )-U^{}(\sigma_{f}) $ behaves like an order parameter that determines a transition from a state of partial failure ($ \sigma \leq \sigma_{f} $) to a state of total failure ($ \sigma >\sigma_{f} $): $$\label{bkc-Ustar} O\equiv U^{}(\sigma )-U^{}(\sigma_{f})=(\sigma_{f}-\sigma )^{\beta };\beta =\frac{1}{2}.$$ To study the dynamics away from criticality ($\sigma \rightarrow \sigma_{f}$ from below), we replace the recursion relation (\[bkc-recU\_t\]) by a differential equation $$-\frac{dU}{dt}=\frac{U^{2}-U+\sigma }{U}.$$ Close to the fixed point we write $ U_{t}(\sigma )=U^{}(\sigma ) $ + $ \epsilon $ (where $ \epsilon \rightarrow 0 $). This, following Eq. (\[bkc-Ustar\]), gives $$\label{bkc-epsilon} \epsilon =U_{t}(\sigma )-U^{}(\sigma )\approx \exp (-t/\tau ),$$ where $ \tau =\frac{1}{2}\left[ \frac{1}{2}(\sigma_{f}-\sigma )^{-1/2}+1\right] $. Near the critical point we can write $$\label{bkc-dec19} \tau \propto (\sigma_{f}-\sigma )^{-\alpha };\alpha =\frac{1}{2}.$$ Therefore the relaxation time diverges following a power-law as $ \sigma \rightarrow \sigma_{f} $ from below. One can also consider the breakdown susceptibility $ \chi $, defined as the change of $ U^{}(\sigma ) $ due to an infinitesimal increment of the applied stress $ \sigma $ $$\label{bkc-sawq} \chi =\left\| \frac{dU^{}(\sigma )}{d\sigma }\right\| =\frac{1}{2}(\sigma_{f}-\sigma )^{-\gamma };\gamma =\frac{1}{2}$$ from Eq. \[bkc-Ustar\]. Hence the susceptibility diverges as the applied stress $ \sigma $ approaches the critical value $ \sigma_{f}=\frac{1}{4} $. Such a divergence in $ \chi $ had already been observed in the numerical studies. .2in (b) At $\sigma =\sigma_{f} $ .2in At the critical point ($ \sigma =\sigma_{f} $), we observe a different dynamic critical behavior in the relaxation of the failure process. From the recursion relation (\[bkc-recU\_t\]), it can be shown that decay of the fraction $ U_{t}(\sigma_{f}) $ of unbroken fibers that remain intact at time $ t $ follows a simple power-law decay: $$\label{bkc-qqq} U_{t}=\frac{1}{2}(1+\frac{1}{t+1}),$$ starting from $ U_{0}=1 $. For large $ t $ ($ t\rightarrow \infty $), this reduces to $ U_{t}-1/2\propto t^{-\delta } $; $ \delta =1 $; a strict power law which is a robust characterization of the critical state (see, however, @bkc62). ### Universality class of the model The universality class of the model has been checked taking two other types of fiber strength distributions: (I) linearly increasing density distribution and (II) linearly decreasing density distribution within the ($\sigma_{th}$) limit $0$ and $1$. One can show that while $\sigma_{f}$ changes with different strength distributions ($\sigma_f= \sqrt{4/27}$ for case (I) and $\sigma_f=4/27$ for case II), the critical behavior remains unchanged: $\alpha =1/2=\beta =\gamma$, $\delta =1$ for all these equal load sharing models [@bkc34]. ### Precursors of global failure in the model In any such failure case, it is important to know [when]{} the failure will take place. In this model, there exist several precursors. The growth of susceptibility $\chi$ with $\sigma$, following Eq. \[bkc-sawq\] indeed suggests one such possibility: $\chi^{-1/2}$ decreases linearly with $\sigma$ to $0$ at $\sigma=\sigma_f$ from below. @bkc47 studied the rate $R(t)$ ($\equiv -\frac{dU_t}{dt}$) of failure of fibers following the dynamics like in Eq. \[bkc-recU\_t\] for $\sigma>\sigma_f$ and found that the rate becomes minimum at a time $t_0$, which is half of the failure time $t_f$ of the bundle. This relation is shown to be independent of the breaking strength distribution of the fibers. A similar relation was also found [@bkc52] for the rate of energy released in a bundle. This is, of course, easier to measure using accoustic emmisions. ### Strength of the local load sharing fiber bundles So far, we studied models with fibers sharing the external load equally. This type of model shows (both analytically and numerically) existence of a critical strength (non zero $\sigma_{f}$) of the macroscopic bundle beyond which it collapses. The other extreme model, i.e., the local load sharing model has been proved to be difficult to tackle analytically. It is clear, however, that the extreme statistics comes into play for such local load sharing models, for which the strength $\sigma_f \to 0$ as the bundle size ($N$) approaches infinity. Essentially, for any finite load ($\sigma$), depending on the fiber strength distribution, the size of the defect cluster can be estimated using Lifshitz argument (see section \[extrm\]) as $\ln N$, giving the failure strength $\sigma_f \sim 1/(\ln N)^a$, where the exponent $a$ assumes a value appropriate for the model (see e.g., @bkc14). If a fraction $f$ of the load of the failed fiber goes for global redistribution and the rest (fraction $1-f$) goes to the fibers neighboring to the failed one, then we see that there is a crossover from extreme to self-averaging statistics at a finite value of $f$ (see e.g., @bkc34). ### Burst distribution: crossover behavior In fiber bundle models, when the load is slowly increased until a new failure occurs, a burst can be defined as the number ($\Delta$) of fiber failures following that failure. The distribution of such bursts ($D(\Delta)$) shows power-law behavior. It was shown for a generic case (independent of threshold distribution) that the form of this distribution (for continuous loading) is $$D(\Delta)/N=C\Delta^{-\zeta}$$ in the limit $N\to \infty$. ![The burst size distribution for different values of $x_0$ in the equal load sharing model with uniform threshold distribution. The number of fibers is $N=50000$ [@bkc36].[]{data-label="cross1"}](rmp-cross-dist.eps){width="6.0cm"} The burst exponent $\zeta$ has a value $\frac{5}{2}$ for average over all $\sigma(=0 \quad\mbox{to} \quad \sigma_f)$ and it is universal [@bkc37]. However, the burst exponent value depends, for e.g., on the details of loading process and also from which point of the loading the burst statistics are recorded. If the burst distribution is recorded only near the critical point ($\sigma \lesssim \sigma_f$), the burst exponent ($\zeta$) value becomes $3/2$ [@bkc35]. For equal load sharing model model with uniform strength distribution, the burst distribution is shown (Fig. \[cross1\]) for recording that starts from different points of effective loading which is denoted by $x_0$, where $x_t=\sigma/U_t$ is the elongation or the effective loading (for linear elastic behavior) at any point $t$). The crossover behavior is clearly seen. In these studies, the load increase rate is extremely slow and the increase is assumed to stop once a fiber fails. The consequent avalanches are studied at that load. Once the avalanche stops, the load is increased again. This process is realistic in the case of earthquakes where stress accumulation takes place over years. However, if the increase in load is fixed ($d\sigma$), then the above exponent value of $\zeta$ becomes $3$: $\Delta \sim \frac{d(1-U^)}{d\sigma}$, giving $\Delta^{-2}=\sigma-\sigma_f$ (from Eq. \[bkc-Ustarsigma\_c\]) and since $D(\Delta)d\Delta\sim d\sigma$, $D(\Delta)\sim\frac{d\sigma}{d\Delta}\sim \Delta^{-\zeta}$, $\zeta=3$ [@bkc9]. ![Crossover signature in the local magnitude distribution of earthquakes in Japan. During the 100 days before mainshock the exponent is 0.60; much smaller than the average value 0.88. [@bkc38][]{data-label="cross2"}](ah.eps){width="7.0cm"} In fact, the earthquake frequency statistics may indeed show the crossover behavior mentioned above: If event frequency is denoted by $D(M)$, then it is known that $D(M)\sim M^{-\zeta}$, where $M$ denotes the magnitude (may be assumed to be related to avalanche size $\Delta$ in the models) and $\zeta$ value is found [@bkc38] to be more ($\zeta\approx 0.9$) for statistics over a smaller time period (before the mainshock), compared to the long time average value ($\zeta\approx 0.6$); see Fig. \[cross2\]. Two fractal overlap model ------------------------- The common geometrical property observed in seismic faults is its fractal nature. It is now well known that, like other fractured surfaces, fault surfaces also posses self-affine roughness (see e.g., @bkc61 and references therein). Therefore, it is worth investigating if earthquake phenomena can be modelled as an outcome of relative movement of two self-affine surfaces over each other. @bkc1, in a simplistic model, studied the overlap statistics of two Cantor sets in order to understand the underlying physics of such phenomena. Cantor set is a prototype example of fractal. In order to construct a triadic Cantor set, in the first step the middle third of a base interval \[0,1\] is removed. In the successive steps, the middle thirds of the remaining intervals (\[0,1/3\] and \[2/3,1\] and so on) are removed. After $n$ such steps, the remaining set is called a Cantor set of generation $n$. When this process is continued ad infinitum i.e., in the limit $n\to\infty$, it becomes a true fractal. ![(a) Schematic representation of the rough earth’s surface and the tectonic plate. (b) The one-dimensional projection of the surfaces form overlapping Cantor sets.[]{data-label="tfo0-bkc"}](tfo0-bkc.eps){width="9.0cm"} In this model, the solid-solid contact surfaces of both the earth’s crust and the tectonic plate are considered as average self-affine surfaces (see Fig. \[tfo0-bkc\]). The strain energy grown between the two surfaces due to a stick period is taken to be proportional to the overlap between them. During a slip event, this energy is released. Considering that such slips occur at intervals proportional to the length corresponding to that area, a power-law for the frequency distribution of the energy release is obtained. This compares well with the GR law (see e.g. @bkc30). ### Renormalisation group study: continuum limit Let the sequence of generators $G_n$ define the Cantor set at the $n$-th generation within the interval \[0,1\]: $ G_0=[0,1]$, $G_1 \equiv RG_0=[0,a] \cup [b,1]$, ... ,$G_{n+1}=RG_n,... $ . The mass density of the set $G_n$ is represented by $D_n(r)$ i.e., $D_n(r)=1$ if $r$ is in any of the occupied intervals of $G_n$ and 0 elsewhere. The overlap magnitude between the sets at any generation $n$ is then given by the convolution form $s_n(r)=\int dr^{\prime}D_n(r^{\prime})D_n(r-r^{\prime})$ (for symmetric fractals). ![(a) Two Cantor sets along the axes $r$ and $r-r^{\prime}$. (b) The overlap $s_1(r)$ along the diagonal. (c) The corresponding density $\rho_1(s)$.[]{data-label="tfo1-bkc"}](tfo1-bkc.eps){width="9.0cm"} ![The overlap densities (probabilities) $\rho(s)$ at various generations; (a) zeroth, (b) first, (c) second, and (d) infinite generation.[]{data-label="tfo2-bkc"}](tfo2-bkc.eps){width="9.0cm"} One can express the overlap integral $s_1$ in the first generation by the projection of the shaded region along the vertical diagonals in Fig. \[tfo1-bkc\](a). That gives the form shown in Fig. \[tfo1-bkc\](b). For $a=b \le 1/3$, the non-vanishing $s_1(r)$ regions do not overlap and are symmetric on both sides with slope of the middle curve being exactly double those on the sides. One can then easily check that the distribution $\rho_1(s)$ of overlap $s$ at this generation is given by Fig. \[tfo1-bkc\], with both $c$ and $d$ greater than unity, maintaining the normalisation condition with $cd=5/3$. The successive generations of the density $\rho_n(s)$ may therefore be represented by Fig. \[tfo2-bkc\], where $$\rho_{n+1}(s)=\tilde{R}\rho_n(s)\equiv \frac{d}{5}\rho_n(\frac{s}{c})+\frac{4d}{5}\rho_n(\frac{2s}{c}).$$ In the infinite generation limit of the renormalisation group (RG) equation, if $\rho^(s)$ denotes the fixed point distribution such that $\rho^(s)=\tilde{R}\rho^(s)$, then assuming $\rho^(s)\sim s^{-\gamma}\tilde{\rho}(s)$, one gets $(d/5)c^{\gamma}+(4d/5)(c/2)^{\gamma}=1$. Here $\tilde{\rho}(s)$ represents an arbitrary modular function, which also includes a logarithmic correction for large $s$. This agrees with the above mentioned normalisation condition $cd=5/3$ for the choice $\gamma=1$ giving $$\rho^(s)\equiv \rho(s)\sim s^{-\gamma}; \qquad \gamma=1$$ The above analysis is for the continuous relative motion of the overlapping fractals. For discrete steps, the contact area distribution can be found exactly for two Cantor sets having same dimension ($log 2/log 3$) [@bkc2]. The step size is taken as the minimum element in the generation at which the distribution is found. ### Discrete limit Let $s_n(t)$ represent the amount of overlap between the two Cantor sets of generation $n$ at time $t$. Initially ($t=0$) the two identical Cantor sets are placed on top of each other, generating the maximum overlap ($2^n$ for the $n$-th generation sets). Then in every time step (discrete) the length of the shift is chosen to be $1/3^n$ for the $n$-th generation, such that a line segment in one set either completely overlaps with one such segment on the other set or does not overlap at all, i.e., partial overlap of two segments of the two sets are not allowed. Periodic boundary conditions are assigned in both the sets. The magnitude of overlap ($s_n(t)$), therefore, in this discrete version, is given by the number of overlapping pairs if line segment of the two sets. Because of the structure of the Cantor sets, the overlap magnitudes can only have certain discrete values which are in geometric progression: $s_n=2^{n-k}$, $k=0,\dots,n$. Let $Nr(s_n)$ denote the the number of times an overlap $s_n$ has occurred in one period of the time series for the $n$-th generation (i.e. $3^n$ time steps). It can be shown that [@bkc30] $$Nr(2^{n-k})={}^nC_k2^k, \qquad k=0,\dots,n$$ Now, if $Prob(s_n)$ denotes the probability that after time $t$ there are $s_n$ overlapping segments, then for the general case of $s_n=2^{n-k}$, $k=0,\dots,n$ it is given by $$\begin{aligned} Prob(2^{n-k})&=&\frac{Nr(2^{n-k})}{\sum\limits_{k=0}^{n}Nr(2^{n-k})} \nonumber \\ &=& \frac{2^k}{3^n}{}^nC_k \nonumber \\ &=& {}^nC_{n-k}\left(\frac{1}{3}\right)^{n-k}\left(\frac{2}{3}\right)^k\end{aligned}$$ ### Gutenberg-Richter law Since the allowed values of the overlap are $s_n=2^{n-k}$, $k=0,\dots,n$, one can write $\log_2s_n=n-k$. Then the above equation becomes $$\begin{aligned} Prob(s_n)&=&{}^nC_{\log_2 s_n}\left(\frac{1}{3}\right)^{\log_2 s_n}\left(\frac{2}{3}\right)^{n-\log_2s_n}\nonumber \\ &&\equiv F(\log_2 s_n).\end{aligned}$$ Near the maxima it may be written as $$F(M)=\frac{3}{2\sqrt{n\pi}}\exp[-\frac{9}{4}\frac{(M-n/3)^2}{n}],$$ where $M=\log_2 s_n$. To obtain the GR law analog from this distribution we have to integrate $F(M)$ from $M$ to $\infty$. $$\begin{aligned} F_{cum}(M) &=& \int\limits_{M}^{\infty}F(M^{\prime})dM^{\prime} \nonumber \\ &=& \int\limits_{M}^{\infty}\frac{3}{2\sqrt{n\pi}}\exp(-\frac{9}{4}\frac{(M^{\prime}-n/3)^2}{n})dM^{\prime}.\end{aligned}$$ Substituting $p=\frac{3}{2\sqrt{n}}(M^{\prime}-n/3)$ we get $$F_{cum}(M)=\frac{1}{\sqrt{\pi}}\int\limits_{\frac{3}{2\sqrt{n}}(M-n/3)}^{\infty}\exp(-p^2)dp.$$ On simplification, it gives $$\label{bkc-f-cum} F_{cum}(M)=\frac{1}{3}\sqrt{\frac{n}{\pi}}\exp\left[-\frac{9}{4}\frac{(M-n/3)^2}{n}\right](M-n/3)^{-1}.$$ $F_{cum}(M)$ in the above equation suggests that the ‘average’ quakes are of magnitude $n/3$, while $$F_{cum}(M)\sim \exp\left[-(9/4)(M-n/3)^2/n\right]$$ can be simplified for large $M$. Using $e^{-a^2}=\left(1/\sqrt{2\pi}\right) \int\limits_{-\infty}^{+\infty}e^{-x^2/2+\sqrt{2}ax}dx$ and $\int\limits_{-\infty}^{+\infty}e^{-f(x)}dx\sim e^{-f(x_0)}$ where $x_0$ refers to the extremal point with $\partial f/\partial x \|_{x=x_0}=0$, one finds $F_{cum}(M)\sim e^{-(9/4)[M(m_0/n)-2M/3]}\sim e^{-3M/4}$ where $x_0=\left(\frac{3}{\sqrt{2n}}\right)m_0$; $m_0=n$. It gives [@bkc26] $$\log F_{cum}(M)=A-\frac{3}{4}M,$$ where $A$ is a constant depending on $n$. This is the Gutenberg-Richter law in the model and clearly holds for the high magnitude end of the distribution. Also, one can equate easily the magnitude $M$ with the released energy $E$ by noting that $M=\log_2 s$ here. The overlap $s$ is related to energy $E$ and hence the relation $M\sim \log E$, giving $F_{cum}\sim E^{-3/4}$. Similar to outcome of the simple fractal models considered here, a power law behavior for the overlap distribution also occurs for two overlapping random Cantor sets, Sierpinsky gasket and Sierpinsky carpet overlapping on their respective replica [@bkc3], and a fractional Brownian profile overlapping on another [@bkc4]. In view of the generality of the power law distribution and the fractal geometry of the fault surfaces, it is suggested that the GR law owes its origin significantly to the fractal geometry of the fault surfaces. It may be noted that identifying the aftershocks as these adjusted overlaps, with average size given by Eq. ([\[bkc-f-cum\]]{}), one can define an average magnitude ($n/3$) dependent on the fractal geometry generator fraction ($=1/3$ here) and the genration number ($n$). This agrees with the observed data quite satisfactorily (see @bkc63). ### Omori law Let $N^{(M_0)}(t)$ denote the cumulative number of aftershocks (of magnitude $M \ge M_0$, where $M_0$ is some threshold) after the mainshock. Then the Omori law states that $$\frac{dN^{(M_0)}(t)}{dt}=\frac{1}{t^p}.$$ The value of the exponent $p$ is close to unity for tectonically active region, although a range of $p$ values are also observed (for review see @bkc26). In practice, a particular value of $p$ is observed when the threshold $M_0$ is given. For this model, when the threshold is fixed at the minimum (i.e., $M_0=1$), then $p=0$ due to the fact that aftershock occurs at every step in this model. However, interesting facts are seen when the threshold is set at the second highest possible value $n-1$ (recall that the second highest overlap was $2^{n-1}$). Then for $t=2.3^{r_1}$ (where, $r_1=0,\dots,n-1$) there is an aftershock of magnitude $n-1$. Therefore, neglecting the prefactor 2, an aftershock of magnitude $n-1$ occurs in geometric progression with common ratio 3. Therefore we get the general rule $N(3t)=N(t)+1$, leading to $$N(t)=\log_3(t).$$ On integration, Omori law gives $N(t)=t^{1-p}$, and therefore from this model we get $p=1$, which is the Omori law suggested value for $p$. The model therefore gives a range of $p$ values between 0 and 1 which systematically increases within the range of threshold values. Discussions and conclusions =========================== Earthquakes, due to their devastating consequences, have been a subjected of extensive studies in various diciplines, ranging from seismology to physics. Although the effeorts were not always commensurate (see also @bkc59 for a critical view of the inherent difficulties of the present approach of theretical physics), in the last decade considerable progress have been made in studying different aspects of this vast topic. In this review, the progresses in such studies are discussed from the point of view of statistical physics. Principally being a large scale dynamic failure process, it is necessary to formulate the background of friction and fracture in order to understand the physics of earthquake. In Sec. II such issues are discussed: After the Griffith’s theory for crack nucleation and the fracture stress statistics of disordered solids, we discuss the RSF law and microscopic models for solid-solid friction. Also, the effects that could lead to violations of RSF laws are also discussed. Several statistical approaches to model earthquake dynamics are discussed. The BK model is discussed in one and two spatial dimensions as well as its long range version. In Sec. IIIA6, the continuum limit is also discussed, which gives ‘characteristic’ earthquakes. BK model has also been discussed in terms of RSF law. Apart from relatively complex modelling like that of BK models and continuum models, we also discuss simplistic models such as OFC models, fiber bundle models and purely geometrical models like the Two Fractal Overlap model. While many details are lost in any such model, they still captures the complex nature of the dynamics and the different statistical aspects, helping us to gain new insights. As one can easily see that inspite of considerable progress in the study of such an important and complex dynamical phenomenon as earthquake, our knowledge is far short of any satisfactory level. We believe, major collaborative efforts, involving physicist and seismologiests in particular, are urgently necessary to unfold the dynamics and employ our knowledge of the precursor events to save us from catastrophic disasters in future. We acknowledge collaborations, at various stages, with P. Bhattacharya, P. Bhattacharyya, J. A. Eriksen, A. Hansen, S. Kakui, T. Kotani, T. Mori, S. Morimoto, S. Pradhan, P. Ray, R. B. Stinchcombe, T. Yamamoto and H. Yoshino. Glossary ======== [aftershocks:]{} Small earthquakes that follow a large earthquake (main shock). [afterslip:]{} Aseismic sliding that follows an earthquake. [asperity:]{} (a) A region where stick-slip motion occurs on a fault or a plate boundary. Strain energy is accumulated at an asperity during a stick stage between earthquakes and it is released by seismic slip at the occurrence of an earthquake. (b) Junction of protrusions of the two contacting surfaces. [Cantor set:]{} One starts with the set of real numbers in the interval \[$0:1$\] and divide the set in a few subsets and remove one of the subsets in the first step. As the removal scheme is repeated ad infinitum, one is left with a dust of real numbers called the Cantor set. It is a fractal object. [characteristic earthquakes:]{} Earthquakes that repeatedly rupture approximately the same segment of a fault. The magnitudes and slip distributions of characteristic earthquakes are similar to one another. [dynamical critical phenomena:]{} Critical behaviors, which are associated with the dynamical properties of the system, rather than the equilibrium properties (e.g., thermal transition is Ising model) are called dynamical critical phenomena (e.g., depinning transition of a fracture front, time dependent field induced transitions in Isng model etc.). [fiber bundle model: ]{} Originating from texttile engineering, fiber bundle model is often used as a prototype model for fracture dynamics. In its simplest form, it consists of a large number of fibers or Hooke-springs. The bundle hangs from a rigid ceiling and supports, via a platform at the bottom, a load. Each fiber has got identical spring constants but the breaking stress for each differs. Depending on the breaking stress of the fibers, the fibers fail and successive failure occur due to load redistribution, showing complex failure dynamics. [fractals:]{} A fractal is a geometrical object having self-similarity in its internal structure. [fractional Brownian profile :]{} Fractional Brownian motion (fBm) is a continuous time random walk with zero mean. However, the directions of the subsequent steps of an fBm are correlated (positively or negatively). A fractional Brownian profile is the trajectory of such a walk. It is self-similar. [Gutenberg-Richter (GR) law:]{} The power law describing the magnitude-frequency relation of earthquakes. The frequency of earthquakes of its energy (seismic moment) $E$ decays with $E$ according to $\propto E^{-(1+B)}=E^{-(1+\frac{2}{3}b)}$ where $B$ and $b=\frac{3}{2}B$ are appropriate exponents. [Hamiltonian:]{} It is essentially the total energy of a system. For a closed system, it would be the sum of kinetic and potential energies. [Omori law:]{} The power law describing the decay of the number (frequency) of aftershocks with the time elapsed after the mainshock. [power-law distribution:]{} (Also called ‘scale free distribution’) A distribution of the generic form $P(x)\sim x^{\alpha}$. Note that there is no length-scale associated with this type of distribution, since a transformation like $x\to x/b$ would keep the functional form unchanged. Observables (e.g., magnetisation, susceptibility etc.) show power-law behavior near criticality. Therefore it is often considered as a signature of critical behavior. [slow earthquakes:]{} Fault slip events that radiate little or no seismic wave radiations. Rupture propagation velocities and slip velocities of slow earthquakes are much smaller than those of ordinary earthquakes. Slow earthquakes without seismic wave radiations are often called silent earthquakes. [quenched randomness:]{} The randomness in the system which is not in thermal equilibrium with the same reservoir as the system and does not fluctuate are called quenched randomness. [rate-and-state friction (RSF) law:]{} An empirical constitutive law describing the dynamic friction coefficient either at steady states or transient states. [self-organized criticality (SOC):]{} When the dynamics of a system leads it to a state of criticality (where scale invariance in time and space are observed) without any need of external tuning parameter, the system is said to have self-organized to a critical state. This phenomenon, where a critical point is an attractor of the dynamics, is called self-organized criticality. [self-similarity and self-affinity:]{} Self similarity refers to the property of an object that it is similar (exactly or approximately) to one or more of its own part(s). Self-affinity refers to the properties of those objects which, in order to be self-similar, are to be scaled by different factor in x and y direction (for 2-d object). [Sierpinski gasket and Sierpinski carpet:]{} Sierpinski carpet is a fractal object, embedded in a 2-d surface. Its construction is as follows: First a square is taken and it is divided into 9 equal squares. Then the square in the middle is removed. then similar operation is performed upon the 8 remaining squares. This process is continued ad infinitum to obtain what is called a Sierpinski carpet. Sierpinski gasket (also called Sierpinski triangle) is again a fractal object. Its construction is as follows: First a equilateral triangle is taken. Then it is divided into four equilateral triangle of same sizes and the middle one is removed. Then same operation is performed upon the three remaining triangles. When this process is continued ad infinitum, one is left with what is called the Sierpinski gasket. [universality class:]{} Phase transitions are characterised by a set of critical exponent values. The values of these exponents are independent of the microscopic details of the system and only depend on the symmetry and dimensionality of the order parameter. Therefore, a large class of systems often have same critical exponent values. A Universality class is a group of systems having same critical exponent values.
--- abstract: 'In mixed multi-view data, multiple sets of diverse features are measured on the same set of samples. By integrating all available data sources, we seek to discover common group structure among the samples that may be hidden in individualistic cluster analyses of a single data-view. While several techniques for such integrative clustering have been explored, we propose and develop a convex formalization that will inherit the strong statistical, mathematical and empirical properties of increasingly popular convex clustering methods. Specifically, our Integrative Generalized Convex Clustering Optimization (iGecco) method employs different convex distances, losses, or divergences for each of the different data views with a joint convex fusion penalty that leads to common groups. Additionally, integrating mixed multi-view data is often challenging when each data source is high-dimensional. To perform feature selection in such scenarios, we develop an adaptive shifted group-lasso penalty that selects features by shrinking them towards their loss-specific centers. Our so-called iGecco+ approach selects features from each data-view that are best for determining the groups, often leading to improved integrative clustering. To fit our model, we develop a new type of generalized multi-block ADMM algorithm using sub-problem approximations that more efficiently fits our model for big data sets. Through a series of numerical experiments and real data examples on text mining and genomics, we show that iGecco+ achieves superior empirical performance for high-dimensional mixed multi-view data.' author: - 'Minjie Wang[^1] and Genevera I. Allen[^2] ^,^[^3]' bibliography: - 'main.bib' title: 'Integrative Generalized Convex Clustering Optimization and Feature Selection for Mixed Multi-View Data' --- [Keywords: Integrative clustering, convex clustering, feature selection, convex optimization, sparse clustering, GLM deviance, Bregman divergences]{} Introduction ============ As the volume and complexity of data grows, statistical data integration has gained increasing attention as it can lead to discoveries which are not evident in analyses of a single data set. We study a specific data-integration problem where we seek to leverage common samples measured across multiple diverse sets of features that are of different types (e.g., continuous, count-valued, categorical, skewed continuous and etc.). This type of data is often called mixed, multi-view data [@hall1997introduction; @acar2011all; @lock2013joint; @tang2018integrated; @baker2019feature]. While many techniques have been developed to analyze each individual data type separately, there are currently few methods that can directly analyze mixed multi-view data jointly. Yet, such data is common in many areas such as electronic health records, integrative genomics, multi-modal imaging, remote sensing, national security, online advertising, and environmental studies. For example in genomics, scientists often study gene regulation by exploring only gene expression data, but other data types, such as short RNA expression and DNA methylation, are all part of the same gene regulatory system. Joint analysis of such data can give scientists a more holistic view of the problem they study. But, this presents a major challenge as each individual data type is high-dimensional (i.e., a larger number of features than samples) with many uninformative features. Further, each data view contains different data types: expression of genes or short RNAs measured via sequencing is typically count-valued or zero-inflated plus skewed continuous data whereas DNA methylation data is typically proportion-valued. In this paper, we seek to leverage multiple sources of mixed data to better cluster the common samples as well as select relevant features that distinguish the inherent group structure. We propose a convex formulation which integrates mixed types of data with different data-specific losses, clusters common samples with a joint fusion penalty and selects informative features that separate groups. Due to the convex formulation, our methods enjoy strong statistical, mathematical and empirical properties. We make several methodological contributions. First, we consider employing different types of losses for better handling non-Gaussian data with Generalized Convex Clustering Optimization (Gecco), which replaces Euclidean distances in convex clustering with more general convex losses. We show that for different losses, Gecco’s fusion penalty forms different types of centroids which we call loss-specific centers. To integrate mixed multi-view data and perform clustering, we incorporate different convex distances, losses, or divergences for each of the different data views with a joint convex fusion penalty that leads to common groups; this gives rise to Integrative Generalized Convex Clustering (iGecco). Further, when dealing with high-dimensional data, practitioners seek interpretability by identifying important features which can separate the groups. To facilitate feature selection in Gecco and iGecco, we develop an adaptive shifted group-lasso penalty that selects features by shrinking them towards their loss-specific centers, leading to Gecco+ and iGecco+ which performs clustering and variable selection simultaneously. To solve our methods in a computationally efficient manner, we develop a new general multi-block ADMM algorithm using sub-problem approximations, and make an optimization contribution by proving that this new class of algorithms converge to the global solution. Related Literature ------------------ Our goal is to develop a unified, convex formulation of integrative clustering with feature selection based on increasingly popular convex clustering methods. [@pelckmans2005convex; @lindsten2011just; @hocking2011clusterpath] proposed convex clustering which uses a fusion penalty to achieve agglomerative clustering like hierarchical clustering. This convex formulation guarantees a global optimal solution, enjoys strong statistical and mathematical theoretical properties, and often demonstrates superior empirical performance to competing approaches. Specifically, in literature, @pelckmans2005convex [@chi2017convex] showed it yields stable solutions to small perturbations on the data or tuning parameters; [@radchenko2017convex] studied statistical consistency; [@tan2015statistical] established its link to hierarchical clustering as well as prediction consistency; and many others have studied other appealing theoretical properties [@zhu2014convex; @sui2018convex; @chi2019recovering]. Despite these advantages, convex clustering has not yet gained widespread popularity due to its intensive computation. Recently, some proposed fast and efficient algorithms to solve convex clustering and estimate its regularization paths [@chi2015splitting; @weylandt2019dynamic]. Meanwhile, convex clustering has been extended to biclustering [@chi2017convex] and many other applications [@chi2018provable; @choi2019regularized]. One potential drawback to convex clustering however, is that thus far, it has only been studied employing Euclidean distances between data points and their corresponding cluster centers. As is well known, the Euclidean metric suffers from poor performance with data that is highly non-Gaussian such as binary, count-valued, skewed data, or with data that has outliers. While [@wang2016robust] studied robust convex clustering and [@sui2018convex] investigated convex clustering with metric learning, there has not been a general investigation of convex clustering for non-Gaussian data and data integration on mixed data has not been studied. But, many others have proposed clustering methods for non-Gaussian data in other contexts. One approach is to perform standard clustering procedures on transformed data [@anders2010differential; @bullard2010evaluation; @marioni2008rna; @robinson2010scaling]. But, choosing an appropriate transformation that retains the original cluster signal is a challenging problem. Another popular approach is to use hierarchical clustering with specified distance metrics for non-Gaussian data [@choi2010survey; @fowlkes1983method]. Closely related to this, [@banerjee2005clustering] studied different clustering algorithms utilizing a large class of loss functions via Bregman divergences. Yet, the proposed methods are all extensions of existing clustering approaches and hence inherit both good and bad properties of those approaches. There has also been work on model-based clustering, which assumes that data are generated by a finite mixture model; for example @banfield1993model [@si2013model] propose such a model for the Poisson and negative binomial distributions. Still these methods have a non-convex formulation and local solutions like all model-based clustering methods. We propose to adopt the method similar to [@banerjee2005clustering] and study convex clustering using different loss functions; hence our method inherits the desirable properties of convex clustering and handles non-Gaussian data as well. More importantly, there is currently no literature on data integration using convex clustering and we achieve this by integrating different types of general convex losses with a joint fusion penalty. Integrative clustering, however, has been well-studied in the literature. The most popular approach is to use latent variables to capture the inherent structure of multiple types of data. This achieves a joint dimension reduction and then clustering is performed on the joint latent variables [@shen2009integrative; @shen2012integrative; @shen2013sparse; @mo2013pattern; @mo2017fully; @meng2015mocluster]. Similar in nature to the latent variables approach, matrix factorization methods assume that the data has an intrinsic low-dimensional representation, with the dimension often corresponding to the number of clusters [@lock2013joint; @hellton2016integrative; @zhang2012discovery; @chalise2017integrative; @zhang2011novel; @yang2015non]. There are a few major drawbacks of latent variable or dimension reduction approaches, however. First it is often hard to directly interpret latent factors or low-dimensional projections. Second, many of these approaches are based on non-convex formulations yielding local solutions. And third, choosing the rank of factors or projections is known to be very challenging in practice and will often impact resulting clustering solutions. Another approach to integrative clustering is clustering of clusters (COC) which performs cluster analysis on every single data set and then integrates the primary clustering results into final group assignments using consensus clustering [@hoadley2014multiplatform; @lock2013bayesian; @kirk2012bayesian; @savage2013identifying; @wang2014similarity]. This, however, has several potential limitations as each individual data set might not have enough signal to discern joint clusters or the individual cluster assignments are too disparate to reach a meaningful consensus. Finally, others have proposed to use distance-based clustering for mixed types of data by first defining an appropriate distance metric for mixed data (for example, the Gower distance by [@gower1971general]) and then applying an existing distance-based clustering algorithm such as hierarchical clustering [@ahmad2007k; @ji2012fuzzy]. Consequently, this method inherits both good and bad properties of distance-based clustering approaches. Notice that all of these approaches are either two-step approaches or are algorithmic or non-convex problems that yield local solutions. In practice, such approaches often lead to unreliable and unstable results. Clustering is known to perform poorly for high-dimensional data as most techniques are highly sensitive to uninformative features. One common approach is to reduce the dimensionality of the data via PCA, NMF, or t-SNE before clustering [@ghosh2002mixture; @bernardo2003bayesian; @tamayo2007metagene]. A major limitation of such approaches is that the resulting clusters are not directly interpretable in terms of feature importance. To address this, several have proposed sparse clustering for high-dimensional data. This performs clustering and feature selection simultaneously by iteratively applying clustering techniques to subsets of features selected via regularization [@witten2010framework; @sun2012regularized; @chang2014sparse]. The approach, however, is non-convex and is highly susceptible to poor local solutions. Others have proposed penalized model-based clustering that selects features [@raftery2006variable; @wang2008variable; @pan2007penalized]. Still, these methods inherit several disadvantages of model-based clustering approaches. Moreover, sparse integrative clustering is relatively under-studied. @shen2013sparse [@mo2013pattern] extended iCluster using a penalized latent variable approach to jointly model multiple omics data types. They induce sparsity on the latent variable coefficients via regularization. As feature selection is performed on the latent variables, however, this is less interpretable in terms of selecting features directly responsible for distinguishing clusters. Recently, and most closely related to our own work, @wang2018sparse proposed sparse convex clustering which adds a group-lasso penalty term on the cluster centers to shrink them towards zero, thus selecting relevant features. This penalty, however, is only appropriate for Euclidean distances when the data is centered; otherwise, the penalty term shrinks towards the incorrect cluster centers. For feature selection using different distances and losses, we propose an adaptive shifted group-lasso penalty that will select features by shrinking them towards their appropriate centroid. Integrative Generalized Convex Clustering with Feature Selection ================================================================ In this section, we introduce our new methods, beginning with the Gecco and iGecco and then show how to achieve feature selection via regularization. We also discuss some practical considerations for applying our methods and develop an adaptive version of our approaches. Generalized Convex Clustering Optimization (Gecco) -------------------------------------------------- In many applications, we seek to cluster data that is non-Gaussian. In the literature, most do this using different distance metrics other than Euclidean distances [@choi2010survey; @fowlkes1983method; @de2004clustering]. Some use losses based on exponential family or deviances closely related to Bregman divergences [@banerjee2005clustering]. To account for different types of losses for non-Gaussian data, we propose to replace the Euclidean distances in convex clustering with more general convex losses; this gives rise to Generalized Convex Clustering Optimization (Gecco). $$\operatorname{minimize}_{\U} \hspace{2mm}\sum_{i=1}^n \curl(\X_{i.},\U_{i.}) + \gamma \sum_{i < i'} w_{ii'} \\| \U_{i.} - \U_{i'.} \\|_q$$ Here, our data $\X$ is an $n \times p$ matrix consisting of $n$ observations and $p$ features; $\U$ is an $n \times p$ centroid matrix with the $i^{th}$ row, $\U_{i.}$, the cluster centroid attached to point $\X_{i.}$. The general loss $\curl(\X_{i.},\U_{i.}) $ refers to a general loss metric that measures dissimilarity between the data point $\X_{i.}$ and assigned centroids $\U_{i.}$. $\\|\cdot\\|_q$ is the $\ell_q$-norm of a vector and usually $q \in \{1,2,\infty\}$ is considered [@hocking2011clusterpath]. Here we prefer using the $\ell_2$-norm in the fusion penalty $(q=2)$ as it encourages the entire rows of similar observations to be fused together simultaneously and is also rotation-invariant; but one could use $\ell_1$ or $\ell_{\infty}$ norms as well. $\gamma$ is a positive tuning constant and $w_{ij}$ is a nonnegative weight. When $\gamma$ equals zero, each data point occupies a unique cluster. As $\gamma$ increases, the fusion penalty encourages some of the rows of the cluster center $\U$ to be exactly fused, forming clusters. When $\gamma$ becomes sufficiently large, all centroids eventually coalesce to a single cluster centroid, which we define as the loss-specific center associated with $\curl(\cdot)$. Hence $\gamma$ regulates both the cluster assignment and number of clusters, providing a family of clustering solutions. The weight $w_{ij}$ should be specified by the user in advance and is not a tuning parameter; we discuss choices of weights for various convex losses in Section \[practical\]. Going beyond Euclidean distances, we propose to employ convex distance metrics as well as deviances associated with exponential family distributions and Bregman divergences, which are always convex. Interestingly, we show that each of these possible loss functions shrink the cluster centers, $\U$, to different loss-specific centers, instead of the mean-based centroid as in convex clustering with Euclidean distances. For example, one may want to use least absolute deviations ($\ell_{1}$-norm or Manhattan distances) for skewed data or for data with outliers; with this loss, we show that all observations fuse to the median when $\gamma$ is sufficiently large. We emphasize loss-specific centers here as they will be important in feature selection in the next section. For completeness, we list common distances and deviance-based losses, as well as their loss-specific centers $\tilde x_j$ respectively in Table \[loss-table\]. (See Appendix \[centroidcal\] for all calculations associated with loss-specific centers, and we provide a formal proof when studying the properties of our approaches in Section \[property\].) 0.15in -0.1in Integrative Generalized Convex Clustering (iGecco) {#igecco} -------------------------------------------------- In data integration problems, we observe data from multiple sources and would like to get a holistic understanding of the problem by analyzing all the data simultaneously. In our framework, we integrate mixed multi-view data and perform clustering by employing different convex losses for each of the different data views with a joint convex fusion penalty that leads to common groups. Hence we propose Integrative Generalized Convex Clustering (iGecco) which can be formulated as follows: $$\begin{aligned} \operatorname{minimize}_{\U^{(k)}} \hspace{2mm} \sum_{k=1}^K \pi_k \curl_k(\X^{(k)},\U^{(k)}) + \gamma \sum_{i < j} w_{ii'} \sqrt{ \sum_{k=1}^K \\| \U_{i.}^{(k)} - \U_{i'.}^{(k)} \\|^2 }\end{aligned}$$ Here, we have $K$ data sources. The $k^{th}$ data-view $\X^{(k)}$ is an $n \times p_k$ matrix consisting of $n$ observations and $p_k$ features; $\U^{(k)}$ is also an $n \times p_k$ matrix and the $i^{th}$ row, $\U_{i.}^{(k)}$, is the cluster center associated with the point $\X_{i.}^{(k)}$. And, $\curl_k(\X_{i.}^{(k)},\U_{i.}^{(k)}) $ is the loss function associated with the $k^{th}$ data-view. Each loss function is weighted by $\pi_k$, which is fixed by the user in advance. We have found that setting $\pi_k$ to be inversely proportional to the null deviance evaluated at the loss-specific center, i.e., $\pi_k = \frac{1}{\curl_k(\X^{(k)},\tilde \X^{(k)})}$, performs well in practice. Note that $\tilde \X = \begin{pmatrix} \tilde \X^{(1)} \cdots \tilde \X^{(K)} \end{pmatrix}$ where each $j^{th}$ column of $\tilde \X^{(k)}$ denotes the loss-specific center $\tilde x_j^{(k)}$. We employ this loss function weighting scheme to ensure equal scaling across data sets of different types. Finally, notice that we employ a joint group-lasso penalty on all of the $\U^{(k)}$’s; this incorporates information from each of the data sources and enforces the same group structure amongst the shared observations. We study this further and prove these properties in Section \[property\]. Feature Selection: Gecco+ and iGecco+ ------------------------------------- In high dimensions, it is important to perform feature selection both for clustering purity and interpretability. Recently, @wang2018sparse proposed sparse convex clustering by imposing a group-lasso-type penalty on the cluster centers which achieves feature selection by shrinking noise features towards zero. This penalty, however, is only appropriate for Euclidean distances when the data is centered; otherwise, the penalty term shrinks towards the incorrect cluster centers. For example, the median is the cluster center with the $\ell_1$ or Manhattan distances. Thus, to select features in this scenario, we need to shrink them towards the median, and we should enforce “sparsity" with respect to the median and not the origin. To address this, we propose adding a shifted group-lasso-type penalty which forces cluster center $\U_{\cdot j}$ to shrink toward the appropriate loss-specific center $\tilde x_j$ for each feature. Both the cluster fusion penalty and this new shifted-group-lasso-type feature selection penalty will shrink towards the same loss-specific center. To facilitate feature selection with the adaptive shifted group-lasso penalty for one data type, our Generalized Convex Clustering Optimization with Feature Selection (Gecco+) is formulated as follows: $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{2mm} &\sum_{i=1}^n \curl(\X_{i.},\U_{i.}) + \gamma \sum_{ i < i'}^n w_{ii'} \\| \U_{i.} - \U_{i'.} \\|_2 \\ &+ \alpha \sum_{j=1}^p \zeta_j \\| \U_{.j} - \tilde x_j \cdot \textbf 1_n \\|_2 \end{aligned}$$ Again, $\U$ is an $n \times p$ matrix and $\tilde x_j$ is the loss-specific center for the $j^{th}$ feature introduced in Table \[loss-table\]. The tuning parameter $\alpha$ controls the number of informative features and the feature weight $\zeta_j$ is a user input which plays an important role to adaptively penalize the features. (We discuss choices of $\zeta_j$ in Section \[adaptive\] when we introduce the adaptive version of our method.) When $\alpha$ is small, all features are selected and contribute to defining the cluster centers. When $\alpha$ grows sufficiently large, all features coalesce at the same value, the loss-specific center $\tilde{x}_j$, and hence no features are selected and contribute towards determining the clusters. Another way of interpreting this is that the fusion penalty exactly fuses some of the rows of the cluster center $\U$, hence determining groups of rows. On the other hand, the shifted group-lasso penalty shrinks whole columns of $\U$ towards their loss-specific centers, thereby essentially removing the effect of uninformative features. Selected features are then columns of $\U$ that were not shrunken to their loss-specific centers, $\U_{.j} \neq \tilde x_j \cdot \textbf 1_n$. These selected features, then, exhibit differences across the clusters determined by the fusion penalty. Clearly, sparse convex clustering of @wang2018sparse is a special case of Gecco+ using Euclidean distances with centered data. Our approach using both a row and column penalty is also reminiscent of convex biclustering [@chi2017convex] which uses fusion penalties on both the rows and columns to achieve checker-board-like biclusters. Building upon integrative generalized convex clustering in Section \[igecco\] and our proposed feature selection penalty above, our Integrative Generalized Convex Clustering Optimization with Feature Selection (iGecco+) is formulated as follows: $$\begin{aligned} \operatorname{minimize}_{\U^{(k)}} \hspace{2mm} &\sum_{k=1}^K \pi_k \curl_k(\X^{(k)},\U^{(k)}) + \gamma \sum_{i < i'} w_{ii'} \sqrt{ \sum_{k=1}^K \\| \U_{i.}^{(k)} - \U_{i'.} ^{(k)} \\|^2 } \nonumber \\ & + \alpha \sum_{k=1}^K \sum_{j=1}^{p_k} \zeta_j^{(k)} \\| \U_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \\|_2 \label{eq:1}\end{aligned}$$ Again, $\U^{(k)}$ is an $n \times p_k$ matrix and $\tilde x_j^{(k)}$ is the loss-specific center for the $j^{th}$ feature for $k^{th}$ data type. By construction, iGecco+ directly clusters mixed multi-view data and selects features from each data view simultaneously. Similarly, adaptive choice of $\zeta_j^{(k)}$ gives rise to adaptive iGecco+ which will be discussed in Section \[adaptive\]. Detailed discussions on practical choices of tuning parameters and weights can be also found in Section \[practical\]. Properties {#property} ---------- In this section, we develop some properties of our methods, highlighting several advantages of our convex formulation. The corresponding proofs can be found in Section \[propprove\] of the Appendix. Define the objective function in as $F_{\gamma,\alpha} (\U)$ where $\U = \begin{pmatrix} \U^{(1)} \cdots \U^{(K)} \end{pmatrix}$. Then due to convexity, we have the following: [proposition]{}[prop1]{} \[theorem:diff1\](Global solution) If $\curl_k$ is convex for all $k$, then any minimizer of $F_{\gamma,\alpha}(\U)$, $\U^$, is a global minimizer. If $\curl_k$ is strictly convex for all $k$, then $\U^$ is unique. [proposition]{}[prop2]{} \[theorem:diff2\](Continuity with respect to data and input parameters) The global minimizer $\U^{}_{{\textbf w}, \mathbf \pi, {\boldsymbol \zeta},\X}(\gamma, \alpha)$ of iGecco+ exists and depends continuously on the data, $\X$, tuning parameters $\gamma$ and $\alpha$, the weight matrix ${\textbf w}$, the loss weight $\pi_k$, and the feature weight $\zeta_j^{(k)}$. [proposition]{}[prop3]{} \[theorem:diff3\](Loss-specific center) Define $\tilde \X = \begin{pmatrix} \tilde \X^{(1)} \cdots \tilde \X^{(K)} \end{pmatrix}$ where each $j^{th}$ column of $\tilde \X^{(k)}$ equals the loss-specific center $\tilde x_j^{(k)}$. Suppose each observation corresponds to a node in a graph with an edge between nodes $i$ and $j$ whenever $w_{ij}>0$. If this graph is fully connected, then $F_{\gamma,\alpha}(\U)$ is minimized by the loss-specific center $\tilde \X$ when $\gamma$ is sufficiently large or $\alpha$ is sufficiently large. Remark:* As Gecco, Gecco+ and iGecco are special cases of iGecco+, it is easy to show that all of our properties hold for these methods as well. These properties illustrate some important advantages of our convex clustering approaches. Specifically, many other widely used clustering methods are known to suffer from poor local solutions, but any minimizer of our problem will achieve a global solution. Additionally, we show that iGecco+ is continuous with respect to the data, tuning parameters, and other input parameters. Together, these two properties are very important in practice and illustrate that the global solution of our method remains stable to small perturbations in the data and input parameters. Stability is a desirable property in practice as one would question the validity of a clustering result that can change dramatically with small changes to the data or parameters. Importantly, most popular clustering methods such as k-means, hierarchical clustering, model-based clustering, or low-rank based clustering, do not enjoy these same stability properties. Finally in Proposition \[theorem:diff3\], we verify that when the tuning parameters are sufficiently large, full fusion of all observations to the loss-specific centers is achieved. Hence, our methods indeed behave as intended, achieving joint clustering of observations. We illustrate this property in Figure \[viz-path-all\] where we apply Gecco+ to the authors data set (described fully in Section \[Simulation\]). Here, we illustrate how our solution, $\hat \U({\gamma,\alpha})$, changes as a function of $\gamma$ and $\alpha$. This so-called “cluster solution path" begins with each observation as its own cluster center when $\gamma$ is small and stops when all observations are fused to the loss-specific center when $\gamma$ is sufficiently large. In between, we see that observations are fusing together as $\gamma$ increases. Similarly, when $\alpha$ is small, all features are selected and as $\alpha$ increases, some of the features get fused to their loss-specific center. Practical Considerations and Adaptive iGecco+ {#practical} --------------------------------------------- In this section, we discuss some practical considerations for applying our method to real data. We discuss choosing user-specific inputs such as weights as well as how to select tuning parameters. In doing so, we introduce an adaptive version of our method as well. ### Choice of Weights and Tuning Parameters In practice, a good choice of fusion weights ($w_{ij}$) has been shown to enhance both computational efficiency and clustering quality of convex clustering [@chi2015splitting]. It has been empirically demonstrated that using weights inversely proportional to the distances yields superior clustering performance; this approach is widely adopted in practice. Further, setting many of the weights to zero helps reduce computation cost. Considering these two, the most common weights choice for convex clustering is to use $K$-nearest-neighbors method with a Gaussian kernel. Specifically, the weight between the sample pair ($i,j$) is set as $w_{ij} = I_{ij}^k \exp(- \phi d(\X_{i.} , \X_{j.}))$, where $I_{ij}^k$ equals 1 if observation $j$ is among observation $i$’s $K$ nearest neighbors or vice versa, and 0 otherwise. However, this choice of weights based on Euclidean distances may not work well for non-Gaussian data in Gecco(+) or for mixed data in iGecco(+). To account for different data types and better measure the similarity between observations, we still adopt $K$-nearest-neighbors method with an exponential kernel, but further extend this by employing appropriate distance metrics for specific data types in the exponential kernel. In particular, for weights in Gecco and Gecco+, we suggest using the same distance functions or deviances in the loss function of Gecco and Gecco+. For weights in iGecco and iGecco+, the challenge is that we need to employ a distance metric which measures mixed types of data. In this case, the Gower distance, which is a distance metric used to measure the dissimilarity of two observations measured in different data types [@gower1971general], can address our problem. To be specific, the Gower distance between observation $i$ and $i'$ overall can be defined as $d(\X_{i.},\X_{i'.}) = \sum_{k=1}^K \sum_{j=1}^{p_k} d_{ii'j}^{(k)} \big/ \sum_{k=1}^K p_k$ where $d_{ii'j}^{(k)} = \frac{ \| \X_{ij}^{(k)} - \X_{i'j}^{(k)} \|}{R_j^{(k)}}$ refers to the Gower distance between observation $i$ and $i'$ for feature $j$ in data view $k$ and $ R_j^{(k)} = \max_{i , i'} \| \X_{ij}^{(k)} - \X_{i'j}^{(k)} \|$ is the range of feature $j$ in data view $k$. In the literature, Gower distance has been commonly used as distance metrics for clustering mixed types of data [@wangchamhan2017efficient; @hummel2017clustering; @akay2018clustering] and shown to yield superior performance than other distance metrics [@ali2013k; @dos2015categorical]. Alternatively, we also propose and explore using stochastic neighbor embedding weights based on symmetrized conditional probabilities [@maaten2008visualizing]. These have been shown to yield superior performance in high-dimensions and if there are potential outliers. Specifically, the symmetrized conditional probabilities are defined as $p_{ij} = \frac{p_{j\|i} + p_{i\|j}}{2n}$, where $p_{j\|i} = \frac{\exp(-\phi d(\X_{i.} , \X_{j.}))}{\sum_{k \neq i} \exp(-\phi d(\X_{i.} , \X_{k.}))}$. We propose to use the weights $w_{ij} = I_{ij}^k \cdot p_{ij}$ where $I_{ij}^k$ still equals 1 if observation $j$ is among observation $i$’s $K$ nearest neighbors or vice versa, and 0 otherwise. Again, we suggest using distance metrics appropriate for specific data types or the Gower distance for mixed data. In empirical studies, we experimented with both weight choices and found that stochastic neighbor embedding weights tended to work better in high-dimensional settings and if there are outliers. Hence, we recommend these and employed them in our empirical investigations in Section \[Simulation\] and \[realdata\]. Estimating the number of clusters in a data set is always challenging. Current literature for tuning parameter selection mainly focuses on stability selection or consensus clustering [@wang2010consistent; @fang2012selection] and hold-out validation [@chi2017convex]. In this paper, we adopt hold-out validation approach for tuning parameter selection and we follow closely the approach described in [@chi2017convex]; we have found that this performs well in empirical studies. For the choice of the feature selection tuning parameter, $\alpha$, we find that clustering result is fairly robust to choices of $\alpha$. Hence, we suggest using only a few possibilities for $\alpha$ and to choose the combination of $\alpha$ and $\gamma$ which jointly minimizes hold-out error or within-cluster deviance. In many cases, we know the number of clusters a priori (or have an idea of an appropriate range for the number of clusters) and we can directly choose $\alpha$ which minimizes the hold-out error or within cluster deviance for that number of clusters. ### Adaptive Gecco+ and iGecco+ to Weight Features {#adaptive} Finally, we consider how to specify the feature weights, $\zeta_j$ used in the shifted group-lasso penalty. While employing these weights are not strictly necessary, we have found, as did @wang2018sparse, that like the fusion weights, well-specified $\zeta_j$’s can both improve performance and speed computation. But unlike the fusion weights where we can utilize the pairwise distances, we don’t have prior information on which features may be relevant in clustering. Thus, we propose to use an adaptive scheme that first fits the iGecco+ with no feature weights and uses this initial estimate to define feature importance for use in weights. This is similar to many adaptive approaches in the literature [@zou2006adaptive; @wang2018sparse]. Our adaptive iGecco+ approach is given in Algorithm \[alg:adaptive-feature\]; this applies to adaptive Gecco+ as a special case as well. We assume that the number of clusters (or a range of the number of clusters) is known a priori. We begin by fitting iGecco+ with $\alpha = 1$ and uniform feature weights $\zeta_j^{(k)}=1$. We then find the $\gamma$ which gives the desired number of clusters, yielding the initial estimate, $\hat \U^{(k)}$. (Alternatively, we can use hold-out validation to select $\gamma$.) Next, we use this initial estimate to adaptively weight features by proposing the following weights: $\zeta_j^{(k)} = 1/\\| \hat \U_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \\|_2$. These weights place a large penalty on noise features as $\\| \hat \U_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \\|_2$ is close to zero in this case. We also notice that noise features impact the distances used in the fusion weights as well. Hence, we suggest updating the distances adaptively by using the selected features to better measure the similarities between observations. To this end, we propose a new scheme to compute weighted Gower distances. First, we scale the features within each data view so that informative features in different data views contribute equally and on the same scale. Then, we employ the inverse of $\pi_k$, i.e., the null deviance, to weight the distances from different data types, resulting in an aggregated and weighted Gower distance, $\hat d(\X_{i.},\X_{i'.})$ as further detailed in Algorithm \[alg:adaptive-feature\]. Note that if the clustering signal from one particular data type is weak and there are few informative features for this data type, then our weighting scheme will down-weight this entire data type in the weighted Gower distance. In practice, our adaptive iGecco+ scheme works well as evidenced in our empirical investigations in the next sections. \ iGecco+ Algorithm ================= In this section, we introduce our algorithm to solve iGecco+, which can be easily extended to Gecco, Gecco+ and iGecco. We first propose a simple, but rather slow ADMM algorithm as a baseline approach. To save computation cost, we further develop a new multi-block ADMM-type procedure using inexact one-step approximation of the sub-problems. Our algorithm is novel from optimization perspective as we extend the multi-block ADMM to higher number of blocks and combine it with the inexact sub-problem solve ADMM literature, which often results in major computational savings. Full ADMM to Solve iGecco+ (Naive Algorithm) -------------------------------------------- Given the shifted group-lasso and fusion penalties along with general losses, developing an optimization routine for iGecco+ method is less straight-forward than convex clustering or sparse convex clustering. In this section, we propose a simple ADMM algorithm to solve iGecco+ as a baseline algorithm and point out its drawbacks. The most common approach to solve problems with more than two non-smooth functions is via multi-block ADMM [@lin2015global; @deng2017parallel], which decomposes the original problem into several smaller sub-problems and solves them in parallel at each iteration. [@chen2016direct] established a sufficient condition for the convergence of three-block ADMM. We develop a multi-block ADMM approach to fit our problem for certain types of losses and prove its convergence. We first recast iGecco+ problem as the equivalent constrained optimization problem: $$\begin{aligned} &\operatorname{minimize}_{\U^{(k)},\V} \hspace{5mm} \sum_{k=1}^K \pi_k \curl_k(\X^{(k)},\U^{(k)}) + \gamma \underbrace{ \bigg(\sum_{l \in \mathcal E} w_l \\|\V_{l.}\\|_2\bigg)}_{P_1(\V;{\textbf w})}+ \alpha \sum_{k=1}^K \sum_{j=1}^{p_k} \zeta_j^{(k)} \\| \U_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \\|_2 \\ & \operatorname{subject \hspace{2mm} to}\hspace{5mm} \D \begin{bmatrix} \U^{(1)} & \cdots & \U^{(K)} \end{bmatrix} - \V = \mathbf 0 \end{aligned}$$ Recently, [@weylandt2019dynamic] derived the ADMM for convex clustering in matrix form and we adopt similar approach. We index a centroid pair by $l = (l_1, l_2)$ with $l_1 < l_2$, define the set of edges over the non-zero weights $\mathcal E = \{l = (l_1,l_2) : w_l > 0\}$, and introduce a new variable $\V = \begin{bmatrix} \V^{(1)} & \cdots & \V^{(K)} \end{bmatrix} \in \mathbb R^{\|\mathcal E \| \times \sum {p_k}}$ where $\V^{(k)}_{l.} = \U^{(k)}_{l_1.} - \U^{(k)}_{l_2.} $ to account for the difference between the two centroids. Hence $\V^{(k)}$ is a matrix containing the pairwise differences between connected rows of $\U^{(k)}$ and the constraint is equivalent to $\D \U^{(k)} - \V^{(k)} = \mathbf 0$ for all $k$; $\D \in \mathbb R^{ \|\mathcal E \| \times n} $ is the directed difference matrix corresponding to the non-zero fusion weights. It is clear the $\V$ sub-problem has closed-form solution for each iteration. We give general-form multi-block ADMM (Algorithm \[alg:full-admm\]) to solve iGecco+. Here $\text{prox}_{h(\cdot)} (\operatorname{\textbf x}) = \operatorname{argmin}_{\operatorname{\textbf z}} \frac{1}{2} \\|\operatorname{\textbf x}- \operatorname{\textbf z}\\|_2^2 + h(\operatorname{\textbf z})$ is the proximal mapping of function $h$. $\U^{ (k) } = \operatorname{argmin}\limits_{\U} \hspace{1mm} \pi_k \curl_k(\X^{(k)},\U) + \frac{\rho}{2} \\| \D \U - \V^{(k)} + \operatorname{\boldsymbol \Lambda}^{(k)} \\|_F^2 + \alpha \sum_{j=1}^{p_k} \zeta_j^{(k)} \\| \U_{.j} - \tilde x_j^{(k)} \cdot \textbf 1_n \\|_2 $ $\V = \text{prox}_{\gamma /\rho P_1(\cdot; {\textbf w})} (\begin{bmatrix} \D \U^{(1)} + \operatorname{\boldsymbol \Lambda}^{(1)} & \cdots & \D \U^{(K)} + \operatorname{\boldsymbol \Lambda}^{(K)} \end{bmatrix} )$ $\operatorname{\boldsymbol \Lambda}^{(k)} = \operatorname{\boldsymbol \Lambda}^{(k)} + ( \D \U^{(k)} - \V^{(k)} ) $ for all $k$ Notice that there is no closed-form solution for the $\U^{(k)}$ sub-problem for general losses. Typically we need to apply an inner optimization routine to solve the $\U^{(k)}$ sub-problem until full convergence. In the next section, we seek to speed up this algorithm by using $\U^{(k)}$ sub-problem approximations. But, first we propose two different approaches to fully solve the $\U^{(k)}$ sub-problem based on specific loss types and then use these to develop a one-step update to solve the sub-problem approximately with guaranteed convergence. iGecco+ Algorithm ----------------- We have introduced Algorithm \[alg:full-admm\], a simple baseline ADMM approach to solve iGecco+. In this section, we consider different ways to solve the $\U^{(k)}$ sub-problem in Algorithm \[alg:full-admm\]. First, based on specific loss types (differentiable and non-differentiable), we propose two different algorithms to solve the $\U^{(k)}$ sub-problem to full convergence. These approaches, however, are rather slow for general losses as there is no closed-form solution which results in nested iterative updates. To address this and in connection with current literature on variants of ADMM with sub-problem approximations, we propose iGecco+ algorithm, a multi-block ADMM which solves the sub-problems approximately by taking a single one-step update. We prove convergence of this general class of algorithms, a novel result in the optimization literature. ### Differentiable Case When the loss $\curl_k$ is differentiable, we consider solving the $\U^{(k)}$ sub-problem with proximal gradient descent, which is often used when the objective function can be decomposed into a differentiable and a non-differentiable function. While there are many other possible optimization routines to solve the $\U^{(k)}$ sub-problem, we choose proximal gradient descent as there is existing literature proving convergence of ADMM algorithms with approximately solved sub-problems using proximal gradient descent [@liu2013linearized; @lu2016fast]. We will discuss in detail how to approximately solve the sub-problem by taking a one-step approximation in Section \[inexact\]. Based upon this, we propose Algorithm \[alg:full-diff\], which solves the $\U^{(k)}$ sub-problem by running full iterative proximal gradient descent to convergence. Here $P_2(\tilde \U^{(k)};{\boldsymbol \zeta}^{(k)}) = \sum_{j=1}^{p_k} \zeta_j^{(k)} \\|\tilde \U^{(k)}_{.j}\\|_2$. $\U^{(k)} = \operatorname{prox}_{s_{k} \cdot \alpha P_2(\cdot;{\boldsymbol \zeta}^{(k)})} \begin{footnotesize} \big( \U^{(k)} - \tilde \X^{(k)} - s_{k} \cdot [ \pi_k \nabla \curl_{k}(\X^{(k)},\U^{(k)} ) + \rho \D^T (\D \U^{(k)} - \V^{(k)} + \operatorname{\boldsymbol \Lambda}^{(k)} )] \big) + \tilde \X^{(k)} \end{footnotesize} $ In Algorithm \[alg:full-diff\] and typically in general (proximal gradient) descent algorithms, we need to choose an appropriate step size $s_k$ to ensure convergence. Usually we employ a fixed step size by computing the Lipschitz constant as in the squared error loss case; but in our method, it is hard to compute the Lipschitz constant for most of our general losses. Instead, we suggest using backtracking line search procedure proposed by [@beck2009gradient; @parikh2014proximal], which is a common way to determine step size with guaranteed convergence in optimization. Further, we find decomposing the $\U^{(k)}$ sub-problem to $p_k$ separate $\U_{.j}^{(k)}$ sub-problems brings several advantages such as (i) better convergence property (than updating $\U^{(k)}$’s all together) due to adaptive step size for each $\U_{.j}^{(k)}$ sub-problem and (ii) less computation cost by solving each in parallel. Hence, in this case, we propose to use proximal gradient for each separate $\U_{.j}^{(k)}$ sub-problem. To achieve this, we assume that the loss is elementwise, which is satisfied by every deviance-based loss. Last, as mentioned, there are many other possible ways to solve the $\U^{(k)}$ sub-problem than proximal gradient, such as ADMM. We find that when the loss is squared Euclidean distances or the hessian of the loss can be upper bounded by a fixed matrix, this method saves more computation. We provide all implementation details discussed above in Section \[diffdetail\] of the Appendix. ### Non-differentiable Case When the loss $\curl_k$ is non-differentiable, we can no longer adopt the proximal gradient method to solve the $\U^{(k)}$ sub-problem as the objective is now a sum of more than one separable non-smooth function. To address this, as mentioned, we can use multi-block ADMM; in this case, we introduce new blocks for the non-smooth functions and hence develop a full three-block ADMM approach to fit our problem. To augment the non-differentiable term, we assume that our loss function can be written as $\curl_k(\X^{(k)},\U^{(k)}) = f_k(g_k(\X^{(k)},\U^{(k)}))$ where $f_k$ is convex but non-differentiable and $g_k$ is affine. This condition is satisfied by all distance-based losses with $g_k(\X^{(k)},\U^{(k)}) = \X^{(k)} - \U^{(k)}$; for example, for Manhattan distances, we have $f_k(\Z) = \sum_{j=1}^p \\|\operatorname{\textbf z}_j\\|_1 = \\|\text{vec}(\Z)\\|_1$, and $g_k(\X,\U) = \X -\U$. The benefit of doing this is that now the $\U^{(k)}$ sub-problem has closed-form solution. Particularly, we can rewrite the $\U^{(k)}$ sub-problem as:$$\begin{aligned} &\operatorname{minimize}_{\U^{(k)},\V} \hspace{5mm} \sum_{k=1}^K \pi_k f_k(\Z^{(k)} ) + \frac{\rho}{2} \\| \D \U - \V^{(k)} + \operatorname{\boldsymbol \Lambda}^{(k)} \\|_F^2 + \alpha \sum_{k=1}^K \bigg( \underbrace{ \sum_{j=1}^{p_k} \zeta_j^{(k)} \\| {\textbf r}_j^{(k)} \\|_2}_{P_2(\R^{(k)};{\boldsymbol \zeta}^{(k)})} \bigg) \\ & \operatorname{subject \hspace{2mm} to}\hspace{5mm} \X^{(k)} - \U^{(k)} = \Z^{(k)} , \hspace{5mm} \U^{(k)} - \tilde \X^{(k)} = \R^{(k)}\end{aligned}$$ where $ \tilde \X^{(k)}$ is an $n \times p_k$ matrix with $j^{th}$ columns equal to scalar $\tilde x_j^{(k)}$. It is clear that we can use multi-block ADMM to solve the problem above and each primal variable has simple update with closed-form solution. We propose Algorithm \[alg:full-non-diff\], a full, iterative multi-block ADMM, to solve the $\U^{(k)}$ sub-problem when the loss is a non-differentiable distance-based function. Algorithm \[alg:full-non-diff\] applies to iGecco+ with various distances such as Manhattan, Minkowski and Chebychev distances and details are given in Section \[nondiffdetail\] of the Appendix. Difference matrix $\D$, $\M = (\D^T \D + 2 \mathbf I)^{-1}$. $\U^{(k)} = \M(\D^T (\V^{(k)} - \operatorname{\boldsymbol \Lambda}^{(k)} ) + \tilde \X^{(k)} + \R^{(k)} - \N^{(k)} + \X^{(k)} - \Z^{(k)} + \bKSI^{(k)} )$ $\Z^{(k)} = \text{prox}_{\pi_k f_k/\rho} (\X^{(k)} - \U^{(k)} + \bKSI^{(k)} )$ $\R^{(k)} = \text{prox}_{\alpha /\rho P_2(\cdot; {\boldsymbol \zeta}^{(k)} )} (\U^{(k)} - \tilde \X^{(k)} + \N^{(k)} ) $ $\bKSI^{(k)} = \bKSI^{(k)} + (\X^{(k)} - \U^{(k)} - \Z^{(k)}) $ $\N^{(k)} = \N^{(k)} + ( \U^{(k)} - \tilde \X^{(k)} - \R^{(k)} ) $ ### iGecco+ Algorithm: Fast ADMM with Inexact One-step Approximation to the Sub-problem {#inexact} Notice that for both Algorithm \[alg:full-diff\] and \[alg:full-non-diff\], we need to run them iteratively to full convergence in order to solve the $\U^{(k)}$ sub-problem for each iteration, which is dramatically slow in practice. To address this in literature, many have proposed variants of ADMM with guaranteed convergence that find an inexact, one-step, approximate solution to the sub-problem (without fully solving it); these include the generalized ADMM [@deng2016global], proximal ADMM [@shefi2014rate; @banert2016fixing] and proximal linearized ADMM [@liu2013linearized; @lu2016fast]. Thus, we propose to solve the $\U^{(k)}$ sub-problem approximately by taking a single one-step update of the algorithm for both types of losses and prove convergence. For the differentiable loss case, we propose to apply the proximal linearized ADMM approach while for the non-differentiable case, we show that taking a one-step update of Algorithm \[alg:full-non-diff\] is equivalent to applying a four-block ADMM to the original problem and we provide a sufficient condition for the convergence of four-block ADMM. Our algorithm, to the best of our knowledge, is the first to incorporate higher-order multi-block ADMM and inexact ADMM with a one-step update to solve sub-problems for general loss functions. When the loss is differentiable, as mentioned in Algorithm \[alg:full-diff\], one can use full iterative proximal gradient to solve the $\U_{.j}^{(k)}$ sub-problem, which however, is computationally burden-some. To avoid this, many proposed variants of ADMM which find approximate solutions to the sub-problems. Specifically, closely related to our problem here, @liu2013linearized [@lu2016fast] proposed proximal linearized ADMM which solves the sub-problems efficiently by linearizing the differentiable part and then applying proximal gradient due to the non-differentiable part. We find their approach fits into our problem and hence develop a proximal linearized 2-block ADMM to solve iGecco+ when the loss $\curl_k$ is differentiable and gradient is Lipschitz continuous. It can be shown that applying proximal linearized 2-block ADMM to Algorithm \[alg:full-admm\] is equivalent to taking a one-step update of Algorithm \[alg:full-diff\] along with $\V$ and $\operatorname{\boldsymbol \Lambda}$ update in Algorithm \[alg:full-admm\]. In this way, we avoid running full iterative proximal gradient updates to convergence for the $\U^{(k)}$ sub-problem as in Algorithm \[alg:full-diff\] and hence save computation cost. When the loss is non-differentiable, we still seek to take an one-step update to solve the $\U^{(k)}$ sub-problem. We achieve this by noticing that taking a one-step update of Algorithm \[alg:full-non-diff\] along with $\V$ and $\operatorname{\boldsymbol \Lambda}$ update in Algorithm \[alg:full-admm\] is equivalent to applying multi-block ADMM to the original iGecco+ problem recast as follows (for non-differentiable distance-based loss): $$\begin{aligned} &\operatorname{minimize}_{\U^{(k)},\V} \hspace{5mm} \sum_{k=1}^K \pi_k f_k(\Z^{(k)} ) + \gamma \underbrace{ \bigg(\sum_{l \in \mathcal E} w_l \\|\V_{l.}\\|_2\bigg)}_{P_1(\V;{\textbf w})}+ \alpha \sum_{k=1}^K \bigg( \underbrace{ \sum_{j=1}^{p_k} \zeta_j^{(k)} \\| {\textbf r}_j^{(k)} \\|_2}_{P_2(\R^{(k)};{\boldsymbol \zeta}^{(k)})} \bigg) \\ & \operatorname{subject \hspace{2mm} to}\hspace{5mm} \X^{(k)} - \U^{(k)} = \Z^{(k)} , \hspace{5mm} \D \begin{bmatrix} \U^{(1)} & \cdots & \U^{(K)} \end{bmatrix} - \V = \mathbf 0, \hspace{5mm} \U^{(k)} - \tilde \X^{(k)} = \R^{(k)}\end{aligned}$$ Typically, general higher-order multi-block ADMM algorithms do not always converge, even for convex functions [@chen2016direct]. We prove convergence of our algorithm and establish a novel convergence result by casting the iGecco+ with non-differentiable losses as a four-block ADMM, proposing a sufficent condition for convergence of higher-order multi-block ADMMs, and finally showing that our problem satisfies this condition. (Details are given in the proof of Theorem \[theorem:inexact-full\] in Appendix \[algcovproof\].) Therefore, taking a one-step update of Algorithm \[alg:full-non-diff\] converges for iGecco+ with non-differentiable losses. So far, we have proposed inexact-solve one-step update approach for both differentiable loss and non-differentiable loss case. For mixed type of losses, we combine those two algorithms and this gives Algorithm \[alg:inexact-admm\], a multi-block ADMM algorithm with inexact one-step approximation to the $\U^{(k)}$ sub-problem to solve iGecco+. We also establish the following convergence result. Update $\U^{(k)}$: Take a one-step update of Algorithm \[alg:full-diff\] Take a one-step update of Algorithm \[alg:full-non-diff\] $\V = \text{prox}_{\gamma /\rho P_1(\cdot; {\textbf w})} (\begin{bmatrix} \D \U^{(1)} + \operatorname{\boldsymbol \Lambda}^{(1)} & \cdots & \D \U^{(K)} + \operatorname{\boldsymbol \Lambda}^{(K)} \end{bmatrix} )$ $\operatorname{\boldsymbol \Lambda}^{(k)} = \operatorname{\boldsymbol \Lambda}^{(k)} + ( \D \U^{(k)} - \V^{(k)} ) $ for all $k$ [theorem]{}[inexact-full]{} \[theorem:inexact-full\](iGecco+ convergence) If $\curl_k$ is convex for all $k$, Algorithm \[alg:inexact-admm\] converges to a global solution. In addition, if each $\curl_k$ is strictly convex, it converges to the unique global solution. Remark:* Our corresponding Theorem \[theorem:inexact-full\] establishes a novel convergence result as it is the first to show the convergence of four-block or higher ADMM using approximate sub-problems for both differentiable and non-differentiable losses. It is easy to see that Algorithm \[alg:inexact-admm\] can be applied to solve other Gecco-related methods as special cases. When $K=1$, Algorithm \[alg:inexact-admm\] gives the algorithm to solve Gecco+. When $\alpha= 0$, Algorithm \[alg:inexact-admm\] gives the algorithm to solve iGecco+. When $K=1$ and $\alpha = 0$, Algorithm \[alg:inexact-admm\] gives the algorithm to solve Gecco. To conclude this section, we compare the convergence results of both full ADMM and inexact ADMM with one-step update in the sub-problem to solve Gecco+ ($n=120$ and $p=210$) in Figure \[conv\_plot\]. The left plots show the number of iterations needed to yield optimization convergence while the right plots show computation time. We see that Algorithm \[alg:inexact-admm\] (one-step update to solve the sub-problem) saves much more computational time than Algorithm \[alg:full-admm\] (full updates of the sub-problem). It should be pointed out that though Algorithm \[alg:inexact-admm\] takes more iterations to converge due to inexact approximation for each iteration, we still reduce computation time dramatically as the computation time per iteration is much less than the full-solve approach. 0.2in -0.2in Simulation Studies {#Simulation} ================== In this section, we first evaluate performance of Gecco+ against existing methods on non-Gaussian data. Next we compare iGecco+ with other methods on mixed multi-view data. Non-Gaussian Data {#Gecco+_sim} ----------------- In this subsection, we evaluate the performance of Gecco and (adaptive) Gecco+ by comparing it with k-means, hierarchical clustering and sparse convex clustering. For simplicity, we have the following naming convention for all methods: loss type name + Gecco(+). For example, Poisson Deviance Gecco+ refers to Generalized Convex Clustering with Feature Selection using Poisson deviance. Sparse CC refers to sparse convex clustering using Euclidean distances. We measure the accuracy of clustering results using adjusted Rand index [@hubert1985comparing]. The adjusted Rand index is the corrected-for-chance version of the Rand index, which is used to measure the agreement between the estimated clustering assignment and the true group label. A larger adjusted Rand index implies a better clustering result. For all methods we consider, we assume oracle number of clusters for fair comparisons. For our Gecco+, we choose the largest $\alpha$ which minimizes within cluster variance or hold-out error. Each simulated data set is comprised of $n=120$ observations with 3 clusters. Each cluster has an equal number of observations. Only the first 10 features are informative while the rest are noise. We consider the following simulation scenarios. - S1: Spherical data with outliers The first 10 informative features in each group are generated from a Gaussian distribution with different $\mu_k$’s for each class. Specifically, the first 10 features are generated from $N(\mu_k, \mathbf I_{10})$ where $\mu_1= (-2.5\cdot \mathbf 1_{5}^T, \mathbf 0_{5}^T )^T$, $\mu_2= (\mathbf 0_{5}^T , 2.5\cdot \mathbf 1_{5}^T)^T$, $\mu_3= (2.5\cdot \mathbf 1_{5}^T,\mathbf 0_{5}^T)^T$. The outliers in each class are generated from a Gaussian distribution with the same mean centroid $\mu_k$ but with higher variance, i.e, $N(\mu_k, 5 \cdot \mathbf I_{10})$. The remaining noise features are generated from $N(0,1)$. In the first setting (S1A), the number of noise features ranges in $25,50,75,\cdots$ up to 225 with the proportion of the number of outliers fixed ( = 5%). We also consider the setting when the variance of noise features increases with number of features fixed $p=200$ and number of outliers fixed (S1B) and high-dimensional setting where $p$ ranges from $250,500,750$ to 1000 (S1C). - S2: Non-spherical data with three half moons Here we consider the standard simulated data of three interlocking half moons as suggested by [@chi2015splitting] and [@wang2018sparse]. The first 10 features are informative in which each pair makes up two-dimensional three interlocking half moons. We randomly select 5% of the observations in each group and make them outliers. The remaining noise features are generated from $N(0,1)$. The number of noise features ranges from $25,50,75,\cdots$ up to 225. In both S1 and S2, we compare Manhattan Gecco+ with other existing methods. - S3: Count-valued data The first 10 informative features in each group are generated from a Poisson distribution with different $\mu_k$’s $(i=1,2,3)$ for each class. Specifically, $\mu_1 = 1 \cdot \mathbf 1_{10}$, $\mu_2 = 4 \cdot \mathbf 1_{10}$, $\mu_3 = 7 \cdot \mathbf 1_{10}$. The remaining noise features are generated from a Poisson distribution with the same $\mu$’s which are randomly generated integers from 1 to 10. The number of noise features ranges from $25,50,75,\cdots$ up to 225. 0.2in -0.2in We summarize simulation results in Figure \[sim\_plot\]. We find that for spherical data with outliers, adaptive Manhattan Gecco+ performs the best in high dimensions. Manhattan Gecco performs well in low dimensions but poorly as number of noisy features increases. Manhattan Gecco+ performs well as the dimension increases, but adaptive Manhattan Gecco+ outperforms the former as it adaptively penalizes the features, meaning that noisy features quickly get zeroed out in the clustering path and that only the informative features perform important roles in clustering. We see that, without adaptive methods, we do not achieve the full benefit of performing feature selection. As we perform adaptive Gecco+, we show vast improvement in clustering purity as the number of noise features grows where regular Gecco performs poorly. Sparse convex clustering performs the worst as it tends to pick outliers as singleton classes. Our simulation results also show that adaptive Manhattan Gecco+ works well for non-spherical data by selecting the correct features. For count data, all three adaptive Gecco+ methods perform better than k-means, hierarchical clustering and sparse convex clustering. We should point out that there are several linkage options for hierarchical clustering. For visualization purposes, we only show the linkage with the best and worst performance instead of all the linkages. Also we use the appropriate data-specific distance metrics in hierarchical clustering. Table \[vs-accuracy-F\] shows the variable selection accuracy of sparse convex clustering and adaptive Gecco+ in terms of F$_1$ score. In all scenarios, we fix $p=225$. We see that adaptive Gecco+ selects the correct features, whereas sparse convex clustering performs poorly. 0.15in -------------------------- --------------- --------------- --------------- Sparse Convex Clustering 0.37 (3.1e-2) 0.25 (2.4e-2) 0.14 (7.2e-3) Adaptive Gecco+ 0.97 (1.9e-2) 0.99 (1.0e-2) 0.81 (8.0e-2) -------------------------- --------------- --------------- --------------- : Comparisons of F$_1$ score for adaptive Gecco+ and sparse convex clustering[]{data-label="vs-accuracy-F"} -0.1in Multi-View Data --------------- In this subsection, we evaluate the performance of iGecco and (adaptive) iGecco+ on mixed multi-view data by comparing it with hierarchical clustering, iCluster+ [@mo2013pattern] and Bayesian Consensus Clustering [@lock2013bayesian]. Again, we measure the accuracy of clustering results using the adjusted Rand index [@hubert1985comparing]. As before, each simulated data set is comprised of $n=120$ observations with 3 clusters. Each cluster has an equal number of observations. Only the first 10 features are informative while the rest are noise. We have three data views consisting of continuous data, count-valued data and binary/proportion-valued data. We investigate different scenarios with different dimensions for each data view and consider the following simulation scenarios: - S1: Spherical data with $p_1 = p_2 = p_3 = 10$ - S2: Three half-moon data with $p_1 = p_2 = p_3 = 10$ - S3: Spherical data with $p_1 = 200$, $p_2= 100$, $p_3 = 50$ - S4: Three half-moon data with $p_1 = 200$, $p_2= 100$, $p_3 = 50$ - S5: Spherical data with $p_1 = 50$, $p_2= 200$, $p_3 = 100$ - S6: Three half-moon data with $p_1 = 50$, $p_2= 200$, $p_3 = 100$ We employ a similar simulation setup as in Section \[Gecco+\_sim\] to generate each data view. The difference is that here for informative features, we increase the within-cluster variance for Gaussian data and decrease difference of cluster mean centroids $\mu_k$’s for binary and count data so that there is overlap between different clusters. Specifically, for spherical cases, Gaussian data is generated from $N(\mu_k, 3 \cdot \mathbf I_{10})$; count data is generated from Poisson with different $\mu_k$’s ($\mu_1 = 2$, $\mu_2 = 4$, $\mu_3 = 6$, etc); binary data is generated from Bernoulli with different $\mu_k$’s ($\mu_1 = 0.5$, $\mu_2 = 0.2$, $\mu_3 = 0.8$, etc). For half-moon cases, continuous data is simulated with larger noise and the count and proportion-valued data is generated via a copula transform. In this manner, we have created a challenging simulation scenario where accurate clustering results cannot be achieved by considering only a single data-view. Again, for fair comparisons across methods, we assume the oracle number of clusters. When applying iGecco methods, we employ Euclidean distances for continuous data, Manhattan distances for count-valued data and Bernoulli log-likelihood for binary or proportion-valued data. We use the latter two losses as they perform well compared with counterpart losses in Gecco+ and demonstrate faster computation speed. Again, we choose the largest $\alpha$ that minimizes within-cluster deviance. Simulation results in Table \[iGecco\_rand\] and Table \[iGecco+\_rand\] show that our methods perform better than existing methods. In low dimensions, iGecco performs comparably with iCluster and Bayesian Consensus Clustering for spherical data. For non-spherical data, iGecco performs much better. For high dimensions, iGecco+ performs better than iGecco while adaptive iGecco+ performs the best as it achieves the full benefit of feature selection. 0.15in Method Scenario 1 Scenario 2 -------- ------------------- ------------------- 0.35 (2.9e-2) 0.54 (1.3e-2) 0.53 (4.6e-2) 0.61 (4.0e-2) 0.52 (2.2e-2) 0.70 (3.0e-2) 0.68 (4.7e-2) 0.66 (4.4e-2) 0.86 (1.5e-2) 0.84 (4.0e-2) 0.90 (1.6e-2) 0.70 (8.0e-3) 0.95 (1.2e-2) 0.63 (1.0e-2) 0.93 (4.7e-3) 1.00 (0.0e-0) : Comparisons of adjusted Rand index for mixed multi-view data[]{data-label="iGecco_rand"} -0.1in 0.15in -0.1in Also we show the variable selection results in Table \[iGecco+\_vsa\] and compare our method to that of iClusterPlus. Our adaptive iGecco+ outperforms iClusterPlus for all scenarios. 0.15in -0.1in Real Data Examples {#realdata} ================== In this section, we apply our methods to various real data sets and evaluate our methods against existing ones. We first evaluate the performance of Gecco+ for several real data sets and investigate the features selected by various Gecco+ methods. Authors Data {#author} ------------ The authors data set consists of word counts from $n = 841$ chapters written by four famous English-language authors (Austen, London, Shakespeare, and Milton). Each class contains an unbalanced number of observations with 69 features. The features are common “stop words" like “a", “be" and “the" which are typically removed before text mining analysis. We use Gecco+ not only to cluster book chapters and compare the clustering assignment with true labels of authors, but also to identify which key words help distinguish the authors. We choose tuning parameters using hold-out validation. 0.05in Method Adjusted Rand Index -------------------------- --------------------- K-means 0.73 Hierarchical Clustering 0.73 Sparse Convex Clustering 0.50 Manhattan Gecco+ 0.96 Poisson LL Gecco+ 0.96 Poisson Deviance Gecco+ 0.96 : Adjusted Rand index of different methods for authors data set[]{data-label="author-rand"} -0.1in In Table \[author-rand\], we compare Gecco+ with existing methods in terms of clustering quality. For hierarchical clustering, we only show the linkage with the best performance (in this whole section). Our method outperforms k-means and the best hierarchical clustering method. Secondly, we look at the word texts selected by Gecco+. As shown in Table \[author-feature\], Jane Austen tended to use the word “her" more frequently than the other authors; this is expected as the subjects of her novels are typically females. The word “was" seems to separate Shakespeare and Jack London well. Shakespeare preferred to use present tense more while Jack London preferred to use past tense more. To summarize, our Gecco+ not only has superior clustering performance but also selects interpretable features. 0.05in [lcccr]{} Method & Features\ Manhattan Gecco+ & ---------------------- “be" ,“had" ,“her",\ “the" ,“to", “was" ---------------------- : Features selected by different Gecco+ methods for authors data set[]{data-label="author-feature"} \ Poisson LL Gecco+ & “an" , “her" , “our", “your"\ Poisson Deviance Gecco+ & ----------------------------- “an", “be" , “had", “her",\ “is", “my" , “the", “was" ----------------------------- : Features selected by different Gecco+ methods for authors data set[]{data-label="author-feature"} \ -0.1in TCGA Breast Cancer Data ----------------------- The TCGA data set consists of log-transformed Level III RPKM gene expression levels for 445 breast-cancer patients with 353 features from The Cancer Genome Atlas Network [@cancer2012comprehensive]. Five PAM50 breast cancer subtypes are included, i.e, Basal-like, Luminal A, Luminal B, HER2-enriched, and Normal-like. Only 353 genes out of 50,000 with somatic mutations from COSMIC [@forbes2010cosmic] are retained. The data is Level III TCGA BRCA RNA-Sequencing gene expression data that have already been pre-processed according to the following steps: i) reads normalized by RPKM, and ii) corrected for overdispersion by a log-transformation. We remove 7 patients, who belong to the normal-like group and the number of subjects $n$ becomes 438. We also combine Luminal A with Luminal B as they are often considered one aggregate group [@choi2014identification]. Method Adjusted Rand Index -------------------------- --------------------- K-means 0.44 Hierarchical Clustering 0.26 Sparse Convex Clustering 0.01 Manhattan Gecco+ 0.76 Poisson LL Gecco+ 0.72 Poisson Deviance Gecco+ 0.72 : Adjusted Rand index of different methods for TCGA data set[]{data-label="TCGA-rand"} From Table \[TCGA-rand\], our method outperforms k-means and the best hierarchical clustering method. Next, we look at the genes selected by Gecco+ in Table \[TCGA-feature\]. FOXA1 is known to be a key gene that characterizes luminal subtypes in DNA microarray analyses [@badve2007foxa1]. GATA binding protein 3 (GATA3) is a transcriptional activator highly expressed by the luminal epithelial cells in the breast [@mehra2005identification]. ERBB2 is known to be associated with HER2 subtype and has been well studied in breast cancer [@harari2000molecular]. Hence our Gecco+ not only outperforms existing methods but also selects genes which are relevant to biology and have been implicated in previous scientific studies. 0.05in [lcccr]{} Method & Features\ Manhattan Gecco+ & ---------------------------- “BCL2" , “ERBB2" ,“GATA3"\ “HMGA1", “IL6ST" ---------------------------- : Features selected by different Gecco+ methods for TCGA data set[]{data-label="TCGA-feature"} \ Poisson LL Gecco+ & “ERBB2" “FOXA1" “GATA3"\ Poisson Deviance Gecco+ & ----------------------------- “ERBB2" , “FOXA1", “GATA3"\ “RET", “SLC34A2" ----------------------------- : Features selected by different Gecco+ methods for TCGA data set[]{data-label="TCGA-feature"} \ -0.1in Next we evaluate the performance of iGecco+ for mixed multi-view data sets and investigate the features selected by iGecco+. Multi-omics Data {#omics} ---------------- One promising application for integrative clustering for multi-view data lies in integrative cancer genomics. Biologists seek to integrate data from multiple platforms of high-throughput genomic data to gain a more thorough understanding of disease mechanisms and detect cancer subtypes. In this case study, we seek to integrate four different types of genomic data to study how epigenetics and short RNAs influence the gene regulatory system in breast cancer. We use the data set from [@cancer2012comprehensive]. [@lock2013bayesian] analyzed this data set using integrative methods and we followed the same data pre-processing procedure: i) filter out genes in expression data whose standard deviation is less than 1.5, ii) take square root of methylation data, and iii) take log of miRNA data. We end up with a data set of 348 tumor samples including: - RNAseq gene expression (GE) data for 645 genes, - DNA methylation (ME) data for 574 probes, - miRNA expression (miRNA) data for 423 miRNAs, - Reverse phase protein array (RPPA) data for 171 proteins. The data set contains samples used on each platform with associated subtype calls from each technology platform as well as integrated cluster labels from biologists. We use the integrated labels from biologists as true label. Also we merged the subtypes 3 and 4 in the integrated labels as those two subtypes correspond to Luminal A and Luminal B respectively from the predicted label given by gene expression data (PAM50 mRNA). Figure \[fig:hist\] in Appendix \[genohist\] gives the distribution of data from different platforms. For our iGecco+ methods, we use Euclidean distances for gene expression data and protein data as the distributions of those two data sets appear gaussian; binomial deviances for Methylation data as the value is between $[0,1]$; Manhattan distances for miRNA data as the data is highly-skewed. We compare our adaptive iGecco+ with other existing methods. From Table \[real-data-multi\], we see that our method outperforms all the existing methods. Method Adjusted Rand Index --------------------------------------------- --------------------- Hclust: $\X_1$ GE 0.51 Hclust: $\X_2$ Meth 0.39 Hclust: $\X_3$ miRNA 0.21 Hclust: $\X_4$ Protein 0.24 Hclust: $[\X_1 \X_2 \X_3 \X_4]$ - Euclidean 0.51 Hclust: $[\X_1 \X_2 \X_3 \X_4]$ - Gower 0.40 iCluster+ 0.36 Bayesian Consensus Clustering 0.35 Adaptive iGecco+ 0.71 : Adjusted Rand index of different methods for multi-omics TCGA data set[]{data-label="real-data-multi"} We also investigate the features selected by adaptive iGecco+, shown in Table \[TCGA-feature-igecco\], and find that our method is validated as most are known in the breast cancer literature. For example, FOXA1 is known to segregate the luminal subtypes from the others [@badve2007foxa1], and AGR3 is a known biomarker for breast cancer prognosis and early breast cancer detection from blood [@garczyk2015agr3]. Several well-known miRNAs were selected including MIR-135b, which is upregulated in breast cancer and promotes cell growth [@hua2016mir] as well as MIR-190 which suppresses breast cancer metastasis [@yu2018mir]. Several known proteins were also selected including ERalpha, which is overexpressed in early stages of breast cancer [@hayashi2003expression] and GATA3 which plays an integral role in breast luminal cell differentiation and breast cancer progression [@cimino2013gata3]. 0.05in [lcccr]{} Data view & Features\ Gene Expression & --------------------------------------------- “AGR3", “FOXA1", “AGR2", “ROPN1",\ “ROPN1B", “ESR1", “C1orf64", “ART3",“FSIP1" --------------------------------------------- : Features selected by adaptive iGecco+ methods for multi-omics TCGA data set[]{data-label="TCGA-feature-igecco"} \ miRNA & -------------------------------------------------------------- “hsa-mir-135b", “hsa-mir-190b", “hsa-mir-577", “hsa-mir-934" -------------------------------------------------------------- : Features selected by adaptive iGecco+ methods for multi-omics TCGA data set[]{data-label="TCGA-feature-igecco"} \ Methylation & -------------------------------------------------------- “cg08047457", “cg08097882", “cg00117172", “cg12265829" -------------------------------------------------------- : Features selected by adaptive iGecco+ methods for multi-omics TCGA data set[]{data-label="TCGA-feature-igecco"} \ Protein & “ER.alpha", “GATA3", “AR", “Cyclin\_E1"\ -0.1in We also visualize resulting clusters from adaptive iGecco+. In Figure \[viz-omics\], we see that there is a clear separation between groups and adaptive iGecco+ identifies meaningful subtypes. Discussion ========== In this paper, we develop a convex formulation of integrative clustering for high-dimensional mixed multi-view data. We propose a unified, elegant methodological solution to two critical issues for clustering and data integration: (i) dealing with mixed types of data and (ii) selecting interpretable features in high-dimensional settings. Specifically, we show that clustering for mixed, multi-vew data can be achieved using different data specific convex losses with a joint fusion penalty. We also introduce a shifted group-lasso penalty that shrinks noise features to their loss-specific centers, hence selecting features that play important roles in separating groups. In addition, we make an optimization contribution by proposing and proving the convergence of a new general multi-block ADMM algorithm with sub-problem approximations that efficiently solves our problem. Empirical studies show that iGecco+ outperforms existing clustering methods and selects interpretable features in separating clusters. This paper focuses on the methodological development for integrative clustering and feature selection, but there are many possible avenues for future research related to this work. For example, we expect in future work to be able to show that our methods inherit the strong theoretical properties of other convex clustering approaches such as clustering consistency and prediction consistency. An important problem in practice is choosing which loss function is appropriate for a given data set. While this is beyond the scope of this paper, an interesting direction for future research would be to learn the appropriate convex loss function in a data-driven manner. Additionally, many have shown block missing structure is common in mixed data [@yu2019optimal; @xiang2013multi]. A potentially interesting direction for future work would be to develop an extension of iGecco+ that can appropriately handle block-missing multi-view data. Additionally, @weylandt2019dynamic developed a fast algorithm to compute the entire convex clustering solution path and used this to visualize the results via a dendrogram and pathwise plot. In future work, we expect that algorithmic regularization path approaches can also be applied to our method to be able to represent our solution as a dendrogram and employ other dynamic visualizations. Finally, while we develop an efficient multi-block ADMM algorithm, there may be further room to speed up computation of iGecco+, potentially by using distributed optimization approaches. In this paper, we demonstrate that our method can be applied to integrative genomics, yet it can be applied to other fields such as multi-modal imaging, national security, online advertising, and environmental studies where practitioners aim to find meaningful clusters and features at the same time. In conclusion, we introduce a principled, unified approach to a challenging problem that demonstrates strong empirical performance and opens many directions for future research. Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to thank Michael Weylandt and Tianyi Yao for helpful discussions. GA and MW also acknowledge support from NSF DMS-1554821, NSF NeuroNex-1707400, and NSF DMS-1264058. [Integrative Generalized Convex Clustering Optimization and Feature Selection for Mixed Multi-View Data: Supplementary Materials]{} [Minjie Wang, Genevera I. Allen]{} The supplementary materials are organized as follows. In Appendix \[propprove\], we prove properties of our methods discussed in Section \[property\]. In Appendix \[algcovproof\], we provide detailed proof for Theorem \[theorem:inexact-full\]. We provide implementation details for Gecco+ with differentiable losses in Appendix \[diffdetail\]. In Appendix \[nondiffdetail\], we discuss implementation details for Gecco+ with non-differentiable losses. We introduce multinomial Gecco(+) in Appendix \[multin\]. In Appendix \[centroidcal\], we show how to calculate the loss-specific center in Table \[loss-table\]. In Appendix \[authorvizone\], we visualize the results of authors data discussed in Section \[author\]. We show the distribution of data from different platforms in Section \[omics\] in Appendix \[genohist\]. Proof of Propositions {#propprove} ===================== Proposition \[theorem:diff1\] and \[theorem:diff2\] are direct extension from Proposition 2.1 in @chi2015splitting. Notice they proved the solution path depends continuously on the tuning parameter $\gamma$ and the weight matrix ${\textbf w}$. It follows that the argument can be also applied to tuning parameter $\alpha$, the loss weight $\pi_k$, and feature weight $\zeta_j^{(k)}$. Also it is obvious that the loss $\curl(\cdot)$ is continuous with respect to the data, $\X$. We show in detail how to prove Proposition \[theorem:diff3\] in the following. First we rewrite $ F_{\gamma,\alpha} (\U) $ as: $$\begin{aligned} F_{\gamma,\alpha} (\U) &= \sum_{k=1}^K \pi_k \curl_k(\X^{(k)},\U^{(k)}) + \gamma \sum_{i < i'} w_{ii'} \sqrt{ \sum_{k=1}^K \\| \U_{i.}^{(k)} - \U_{i'.} ^{(k)} \\|^2 } + \alpha \sum_{k=1}^K \sum_{j=1}^{p_k} \zeta_j^{(k)} \\| \U_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \\|_2 \\ & = \sum_{k=1}^K \sum_{i=1}^n \pi_k \curl_k(\X_{i.}^{(k)},\U_{i.}^{(k)}) + \gamma \sum_{i < i'} w_{ii'} \sqrt{ \sum_{k=1}^K \\| \U_{i.}^{(k)} - \U_{i'.} ^{(k)} \\|^2 } + \alpha \sum_{k=1}^K \sum_{j=1}^{p_k} \zeta_j^{(k)} \\| \U_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \\|_2 \end{aligned}$$ [By definition, loss-specific cluster center is $\tilde \operatorname{\textbf x}^{(k)} = \operatorname{argmin}\limits_{\operatorname{\textbf u}} \sum_{i=1}^n \curl_k(\X_{i.}^{(k)},\operatorname{\textbf u})$. Since $\curl_k$ is convex, it is equivalent to $\operatorname{\textbf u}$ such that $\partial \sum_i \curl_k(\X_{i.}^{(k)},\operatorname{\textbf u}) = 0$. Hence, $\partial \sum_i \curl_k(\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)} ) = 0$]{} We use the similar proof approach in [@chi2015splitting]. A point $\X$ furnishes a global minimum of the convex function $F_Y(\X)$ if and only if all forward directional derivatives $d_{\theta} F_Y(\X)$ at $\X$ are nonnegative. We calculate the directional derivatives: $$\begin{aligned} d_{\theta} F_{\gamma,\alpha} (\tilde \X ) &= \sum_{k=1}^K \sum_{i=1}^n \pi_k \langle \partial \curl_k (\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)}), \Theta_{i.}^{(k)} \rangle + \gamma \sum_{i < i'} w_{ii'} \sqrt{ \sum_{k=1}^K \\| \Theta_{i.}^{(k)} - \Theta_{i'.} ^{(k)} \\|^2 } \\ & + \alpha \sum_{k=1}^K \sum_{j=1}^{p_k} \zeta_j^{(k)} \\| \Theta_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \\|_2 \end{aligned}$$ Note: $\sum_{i=1}^n \langle \partial \curl_k (\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)}), \Theta_{i'.}^{(k)} \rangle = 0 $ The generalized Cauchy-Schwartz inequality therefore implies $$\begin{aligned} \sum_{i=1}^n \langle \partial \curl_k (\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)}), \Theta_{i.}^{(k)} \rangle &= \frac{1}{n} \sum_{i=1}^n \sum_{i'=1}^n \langle \partial \curl_k (\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)}), \Theta_{i.}^{(k)} \rangle \\ &= \frac{1}{n} \sum_{i=1}^n \sum_{i'=1}^n \langle \partial \curl_k (\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)}), \Theta_{i.}^{(k)} - \Theta_{i'.}^{(k)} \rangle \\ &\geq -\frac{2}{n} \sum_{i < i'} \\| \partial \curl_k (\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)}) \\|_2 \cdot \\| \Theta_{i.}^{(k)} - \Theta_{i'.}^{(k)} \\|_2 \\ &\geq -\frac{2}{n} \sum_{i < i'} \\| \partial \curl_k (\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)}) \\|_2 \cdot \sqrt{ \sum_{k=1}^K \\| \Theta_{i.}^{(k)} - \Theta_{i'.} ^{(k)} \\|^2 } \end{aligned}$$ Hence $$\begin{aligned} \sum_{k=1}^K \sum_{i=1}^n \pi_k \langle \partial \curl_k (\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)}), \Theta_{i.}^{(k)} \rangle \geq -\frac{2}{n} \sum_{k=1}^K \sum_{i < i'} \pi_k \\| \partial \curl_k (\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)}) \\|_2 \cdot \sqrt{ \sum_{k=1}^K \\| \Theta_{i.}^{(k)} - \Theta_{i'.} ^{(k)} \\|^2 }\end{aligned}$$ Take $\gamma$ sufficiently large such that: $$\begin{aligned} \gamma \sum_{i < i'} w_{ii'} \sqrt{ \sum_{k=1}^K \\| \Theta_{i.}^{(k)} - \Theta_{i'.} ^{(k)} \\|^2 } \geq -\frac{2}{n} \sum_{k=1}^K \sum_{i < i'} \pi_k \\| \partial \curl_k (\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)}) \\|_2 \cdot \sqrt{ \sum_{k=1}^K \\| \Theta_{i.}^{(k)} - \Theta_{i'.} ^{(k)} \\|^2 }\end{aligned}$$ When all $w_{ii'} > 0$, one can take any $\gamma$ that exceeds $$\begin{aligned} K \cdot \frac{2}{n} \max_{i,i',k} \frac{ \pi_k \\| \partial \curl_k (\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)}) \\|_2 }{w_{ii'}}\end{aligned}$$ In general set $$\begin{aligned} \beta = K \cdot \frac{2}{n \min_{w_{ii'} >0} w_{ii'}} \max_{i,i',k} \pi_k \\| \partial \curl_k (\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)}) \\|_2 \end{aligned}$$ For any pair $i$ and $i'$ there exists a path $i \to k \to \cdots \to l \to i'$ along which the weights are positive. It follows that $$\begin{aligned} \frac{2}{n} \sum_{k=1}^K \pi_k \\| \partial \curl_k (\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)}) \\|_2 \cdot \sqrt{ \sum_{k=1}^K \\| \Theta_{i.}^{(k)} - \Theta_{i'.} ^{(k)} \\|^2 } \leq \beta \sum_{i < i'} w_{ii'} \sqrt{ \sum_{k=1}^K \\| \Theta_{i.}^{(k)} - \Theta_{i'.} ^{(k)} \\|^2 } \end{aligned}$$ We have $$\begin{aligned} \frac{2}{n} \sum_{k=1}^K \sum_{i < i'} \pi_k \\| \partial \curl_k (\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)}) \\|_2 \cdot \sqrt{ \sum_{k=1}^K \\| \Theta_{i.}^{(k)} - \Theta_{i'.} ^{(k)} \\|^2 } \leq \binom{n}{2} \beta \sum_{i < i'} w_{ii'} \sqrt{ \sum_{k=1}^K \\| \Theta_{i.}^{(k)} - \Theta_{i'.} ^{(k)} \\|^2 } \end{aligned}$$ Hence the forward directional derivative test is satisfied for any $\gamma \geq \binom{n}{2} \beta$. On the other hand, for fixed $\gamma$, the generalized Cauchy-Schwartz inequality implies $$\begin{aligned} \sum_{i=1}^n \langle \partial \curl_k (\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)}), \Theta_{i.}^{(k)} \rangle &= \sum_{i=1}^n \langle \partial \curl_k (\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)}), \Theta_{i.}^{(k)} - \tilde \operatorname{\textbf x}^{(k)} \rangle \\ & = \sum_{j=1}^{p_k} \langle \partial \curl_k (\X_{.j}^{(k)},\tilde x_j^{(k)} \cdot \textbf 1_n ), \Theta_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \rangle \\ & \geq - \sum_{j=1}^{p_k} \\| \partial \curl_k (\X_{.j}^{(k)},\tilde x_j^{(k)} \cdot \textbf 1_n ) \\|_2 \cdot \\| \Theta_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \\|_2 \end{aligned}$$ Hence $$\begin{aligned} \sum_{k=1}^K \sum_{i=1}^n \pi_k \langle \partial \curl_k (\X_{i.}^{(k)},\tilde \operatorname{\textbf x}^{(k)}), \Theta_{i.}^{(k)} \rangle \geq - \sum_{k=1}^K \sum_{j=1}^{p_k} \pi_k \\| \partial \curl_k (\X_{.j}^{(k)},\tilde x_j^{(k)} \cdot \textbf 1_n ) \\|_2 \cdot \\| \Theta_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \\|_2 \end{aligned}$$ Take $\alpha$ sufficiently large so that $$\begin{aligned} \alpha \sum_{k=1}^K \sum_{j=1}^{p_k} \zeta_j^{(k)} \\| \Theta_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \\|_2 \geq - \sum_{k=1}^K \sum_{j=1}^{p_k} \pi_k \\| \partial \curl_k (\X_{.j}^{(k)},\tilde x_j^{(k)} \cdot \textbf 1_n ) \\|_2 \cdot \\| \Theta_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \\|_2 \end{aligned}$$ When all $\zeta_j^{(k)} > 0$, one can take any $\alpha$ that exceeds $$\begin{aligned} \max_{j,k} \frac{ \pi_k \\| \partial \curl_k (\X_{.j}^{(k)},\tilde x_j^{(k)} \cdot \textbf 1_n ) \\|_2 }{\zeta_j^{(k)}}\end{aligned}$$ In general, set $\alpha \geq \frac{1}{\min_{\zeta_j^{(k)} > 0} \zeta_j^{(k)}} \cdot \max_{j,k} \pi_k \\| \partial \curl_k (\X_{.j}^{(k)},\tilde x_j^{(k)} \cdot \textbf 1_n ) \\|_2 $. It is easy to check the forward directional derivative test is satisfied. $\square$ Proof of Theorem \[theorem:inexact-full\] {#algcovproof} ========================================= Recall the iGecco+ problem is: $$\begin{aligned} \min_{\U^{(k)}} \hspace{2mm} &\sum_{k=1}^K \pi_k \curl_k(\X^{(k)},\U^{(k)}) + \gamma \sum_{i < i'} w_{ii'} \sqrt{ \sum_{k=1}^K \\| \U_{i.}^{(k)} - \U_{i'.} ^{(k)} \\|^2 } \\ & + \alpha \sum_{k=1}^K \sum_{j=1}^{p_k} \zeta_j^{(k)} \\| \U_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \\|_2 \end{aligned}$$ We can recast the orginal iGecco+ problem as a multi-block ADMM form: $$\begin{aligned} &\operatorname{minimize}_{\U^{(k)},\V} \hspace{5mm} \sum_{k=1}^K \pi_k \curl_k(\X^{(k)},\U^{(k)}) + \gamma \underbrace{ \bigg(\sum_{l \in \mathcal E} w_l \\|\V_{l.}\\|_2\bigg)}_{P_1(\V;{\textbf w})} \nonumber \\ & \hspace{20mm}+ \alpha \sum_{k=1}^K \sum_{j=1}^{p_k} \zeta_j^{(k)} \\| \U_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \\|_2 + \sum_{k'=1}^K \pi_{k'} f_{k'}(\Z^{(k')} ) + \alpha \sum_{k'=1}^K \bigg( \underbrace{ \sum_{j=1}^{p_k'} \zeta_j^{(k')} \\| {\textbf r}_j^{(k')} \\|_2}_{P_2(\R^{(k')};{\boldsymbol \zeta}^{(k')})} \bigg) \nonumber \\ & \operatorname{subject \hspace{2mm} to}\hspace{3mm} \D \U^{(k)} - \V^{(k)} = 0, \hspace{2mm} \D \U^{(k')} - \V^{(k')} = 0, \hspace{2mm} \X^{(k')} - \U^{(k')} = \Z^{(k')}, \hspace{2mm} \U^{(k')} - \tilde \X^{(k')} = \R^{(k')} \label{eq:2}\end{aligned}$$ where $\curl_k$ refers to the differentiable losses and $\curl_{k'}$ refers to the non-differentiable losses. Hence we have the multi-block ADMM algorithm (Algorithm \[alg:full-igecco+\]) to solve the problem above: $\U^{(k)} = \operatorname{argmin}\limits_{\U} \hspace{1mm} \pi_k \curl_k(\X^{(k)},\U) + \frac{\rho}{2} \\| \D \U - \V^{(k)} + \operatorname{\boldsymbol \Lambda}^{(k)} \\|_F^2 + \alpha \sum_{j=1}^{p_k} \zeta_j^{(k)} \\| \U_{.j} - \tilde x_j^{(k)} \cdot \textbf 1_n \\|_2 $ $\U^{(k')} = \operatorname{argmin}\limits_{\U} \hspace{1mm} \frac{\rho}{2} \\| \X^{(k')} - \U - \Z^{(k')} + \bKSI^{(k')} \\|_F^2 + \frac{\rho}{2} \\| \U - \tilde \X^{(k')} - \R^{(k')} + \N^{(k')} \\|_F^2 + \frac{\rho}{2} \\| \D \U - \V^{(k')} + \operatorname{\boldsymbol \Lambda}^{(k')} \\|_F^2 $ $\Z^{(k')} = \operatorname{argmin}\limits_{\Z} \hspace{1mm} \pi_{k'} f_{k'}(\Z ) + \frac{\rho}{2} \\| \X^{(k')} - \U^{(k')} - \Z + \bKSI^{(k')} \\|_F^2 $ $\R^{(k')} = \operatorname{argmin}\limits_{\R} \hspace{1mm} \alpha \sum_{j=1}^{p_{k'}} \zeta_j^{(k')} \\| \U_{.j}^{(k')} - \tilde x_j^{(k')} \cdot \textbf 1_n \\|_2 + \frac{\rho}{2} \\| \U^{(k')} - \tilde \X^{(k')} - \R+ \N^{(k')} \\|_F^2 $ $\bKSI^{(k')} = \bKSI^{(k')} + (\X^{(k')} - \U^{(k')} - \Z^{(k')}) $ $\N^{(k')} = \N^{(k')} + ( \U^{(k')} - \tilde \X^{(k')} - \R^{(k')} ) $ $\V = \operatorname{argmin}\limits_{\V} \hspace{1mm} \frac{\rho}{2} \\| \D \U^{(k)} - \V^{(k)} + \operatorname{\boldsymbol \Lambda}^{(k)} \\|_F^2 + \frac{\rho}{2} \\| \D \U^{(k')} - \V^{(k')} + \operatorname{\boldsymbol \Lambda}^{(k')} \\|_F^2 + \gamma \bigg(\sum_{l \in \mathcal E} w_l \\|\V_{l.}\\|_2\bigg) $ $\operatorname{\boldsymbol \Lambda}^{(k)} = \operatorname{\boldsymbol \Lambda}^{(k)} + ( \D \U^{(k)} - \V^{(k)} ) $ for all $k$ and $k'$ To prove convergence of Algorithm \[alg:inexact-admm\], we first show that multi-block ADMM Algorithm \[alg:full-igecco+\] converges to a global minimum. Then we show that we can proximal-linearize the sub-problems in the primal updates of Algorithm \[alg:full-igecco+\] with proved convergence and this is equivalent to Algorithm \[alg:inexact-admm\]. Without loss of generality, we assume we have one differentiable loss $\curl_1(\cdot)$ and one non-differentiable distance-based loss $\curl_2(\cdot)$. To prove convergence of Algorithm \[alg:full-igecco+\], we first propose a sufficient condition for the convergence of four-block ADMM and prove it holds true. This is an extension of the convergence results in Section 2 of the work by [@chen2016direct]. Suppose the convex optimization problem with linear constraints we want to minimize is $$\begin{aligned} \min \hspace{2mm} &\theta_1(\operatorname{\textbf x}_1) + \theta_2(\operatorname{\textbf x}_2) + \theta_3(\operatorname{\textbf x}_3) + \theta_4(\operatorname{\textbf x}_4) \nonumber \\ \text{s.t} \hspace{2mm} & \A_1 \operatorname{\textbf x}_1 + \A_2 \operatorname{\textbf x}_2 + \A_3 \operatorname{\textbf x}_3 + \A_4 \operatorname{\textbf x}_4 = \operatorname{\textbf b}\label{eq:3}\end{aligned}$$ The multi-block ADMM has the following form. Note here, the superscript $\operatorname{\textbf x}_i^{(k+1)}$ refers to the $(k+1)^{th}$ iteration in the ADMM updates. $$\begin{aligned} \begin{cases}\label{eq:4} \operatorname{\textbf x}_1^{(k+1)} = \operatorname{argmin}\hspace{1mm} \{ L_\A(\operatorname{\textbf x}_1,\operatorname{\textbf x}_2^{(k)},\operatorname{\textbf x}_3^{(k)},\operatorname{\textbf x}_4^{(k)},\operatorname{\boldsymbol \lambda}^{(k)})\} \\ \operatorname{\textbf x}_2^{(k+1)} = \operatorname{argmin}\hspace{1mm} \{ L_\A(\operatorname{\textbf x}_1^{(k+1)},\operatorname{\textbf x}_2,\operatorname{\textbf x}_3^{(k)},\operatorname{\textbf x}_4^{(k)},\operatorname{\boldsymbol \lambda}^{(k)})\} \\ \operatorname{\textbf x}_3^{(k+1)} = \operatorname{argmin}\hspace{1mm} \{ L_\A(\operatorname{\textbf x}_1^{(k+1)},\operatorname{\textbf x}_2^{(k+1)},\operatorname{\textbf x}_3,\operatorname{\textbf x}_4^{(k)},\operatorname{\boldsymbol \lambda}^{(k)})\} \\ \operatorname{\textbf x}_4^{(k+1)} = \operatorname{argmin}\hspace{1mm} \{ L_\A(\operatorname{\textbf x}_1^{(k+1)},\operatorname{\textbf x}_2^{(k+1)},\operatorname{\textbf x}_3^{(k+1)},\operatorname{\textbf x}_4,\operatorname{\boldsymbol \lambda}^{(k)})\} \\ \operatorname{\boldsymbol \lambda}^{(k+1)} = \operatorname{\boldsymbol \lambda}^{(k)} - (\A_1 \operatorname{\textbf x}_1^{(k+1)} + \A_2 \operatorname{\textbf x}_2^{(k+1)} + \A_3 \operatorname{\textbf x}_3^{(k+1)} + \A_4 \operatorname{\textbf x}_4^{(k+1)} - \operatorname{\textbf b}) \end{cases} \end{aligned}$$ where $$\begin{aligned} L_\A = \sum_{i=1}^4 \theta_i (\operatorname{\textbf x}_i) - \operatorname{\boldsymbol \lambda}^T (\A_1 \operatorname{\textbf x}_1 + \A_2 \operatorname{\textbf x}_2 + \A_3 \operatorname{\textbf x}_3 + \A_4 \operatorname{\textbf x}_4 - \operatorname{\textbf b}) + \frac{1}{2} \\| \A_1 \operatorname{\textbf x}_1 + \A_2 \operatorname{\textbf x}_2 + \A_3 \operatorname{\textbf x}_3 + \A_4 \operatorname{\textbf x}_4 - \operatorname{\textbf b}\\|_2^2\end{aligned}$$ We establish Lemma \[theorem:four\_block\], a sufficient condition for convergence of four-block ADMM: [lemma]{}[lemma-four\_block]{} \[theorem:four\_block\](Sufficient Condition for Convergence of Four-block ADMM) A sufficient condition ensuring the convergence of to a global solution of is: $\A_2^T \A_3 = \mathbf 0$, $\A_2^T \A_4 = \mathbf 0$, $\A_3^T \A_4 = \mathbf 0$. Proof of Lemma \[theorem:four\_block\]: According to the first-order optimality conditions of the minimization problems in , we have: $$\begin{aligned} \begin{cases} \theta_1(\operatorname{\textbf x}_1) - \theta_1(\operatorname{\textbf x}_1^{(k+1)}) + (\operatorname{\textbf x}_1 - \operatorname{\textbf x}_1^{(k+1)})^T \big\{ -\A_1^T [ \operatorname{\boldsymbol \lambda}^{(k)} - (\A_1 \operatorname{\textbf x}_1^{(k+1)} + \A_2 \operatorname{\textbf x}_2^{(k)} + \A_3 \operatorname{\textbf x}_3^{(k)} + \A_4 \operatorname{\textbf x}_4^{(k)} - \operatorname{\textbf b})] \big\} \geq 0 \\ \theta_2(\operatorname{\textbf x}_2) - \theta_2(\operatorname{\textbf x}_2^{(k+1)}) + (\operatorname{\textbf x}_2 - \operatorname{\textbf x}_2^{(k+1)})^T \big\{ -\A_2^T [ \operatorname{\boldsymbol \lambda}^{(k)} - (\A_1 \operatorname{\textbf x}_1^{(k+1)} + \A_2 \operatorname{\textbf x}_2^{(k+1)} + \A_3 \operatorname{\textbf x}_3^{(k)} + \A_4 \operatorname{\textbf x}_4^{(k)} - \operatorname{\textbf b})] \big\} \geq 0 \\ \theta_3(\operatorname{\textbf x}_3) - \theta_3(\operatorname{\textbf x}_3^{(k+1)}) + (\operatorname{\textbf x}_3 - \operatorname{\textbf x}_3^{(k+1)})^T \big\{ -\A_3^T [ \operatorname{\boldsymbol \lambda}^{(k)} - (\A_1 \operatorname{\textbf x}_1^{(k+1)} + \A_2 \operatorname{\textbf x}_2^{(k+1)} + \A_3 \operatorname{\textbf x}_3^{(k+1)} + \A_4 \operatorname{\textbf x}_4^{(k)} - \operatorname{\textbf b})] \big\} \geq 0 \\ \theta_4(\operatorname{\textbf x}_4) - \theta_4(\operatorname{\textbf x}_4^{(k+1)}) + (\operatorname{\textbf x}_4 - \operatorname{\textbf x}_4^{(k+1)})^T \big\{ -\A_4^T [ \operatorname{\boldsymbol \lambda}^{(k)} - (\A_1 \operatorname{\textbf x}_1^{(k+1)} + \A_2 \operatorname{\textbf x}_2^{(k+1)} + \A_3 \operatorname{\textbf x}_3^{(k+1)} + \A_4 \operatorname{\textbf x}_4^{(k+1)} - \operatorname{\textbf b})] \big\} \geq 0 \\ \end{cases}\end{aligned}$$ Since $\A_2^T \A_3 = \mathbf 0$, $\A_2^T \A_4 = \mathbf 0$, $\A_3^T \A_4 = \mathbf 0$, we have: $$\begin{aligned} \begin{cases} \theta_1(\operatorname{\textbf x}_1) - \theta_1(\operatorname{\textbf x}_1^{(k+1)}) + (\operatorname{\textbf x}_1 - \operatorname{\textbf x}_1^{(k+1)})^T \big\{ -\A_1^T [ \operatorname{\boldsymbol \lambda}^{(k)} - (\A_1 \operatorname{\textbf x}_1^{(k+1)} + \A_2 \operatorname{\textbf x}_2^{(k)} + \A_3 \operatorname{\textbf x}_3^{(k)} + \A_4 \operatorname{\textbf x}_4^{(k)} - \operatorname{\textbf b})] \big\} \geq 0 \\ \theta_2(\operatorname{\textbf x}_2) - \theta_2(\operatorname{\textbf x}_2^{(k+1)}) + (\operatorname{\textbf x}_2 - \operatorname{\textbf x}_2^{(k+1)})^T \big\{ -\A_2^T [ \operatorname{\boldsymbol \lambda}^{(k)} - (\A_1 \operatorname{\textbf x}_1^{(k+1)} + \A_2 \operatorname{\textbf x}_2^{(k+1)} - \operatorname{\textbf b})] \big\} \geq 0 \\ \theta_3(\operatorname{\textbf x}_3) - \theta_3(\operatorname{\textbf x}_3^{(k+1)}) + (\operatorname{\textbf x}_3 - \operatorname{\textbf x}_3^{(k+1)})^T \big\{ -\A_3^T [ \operatorname{\boldsymbol \lambda}^{(k)} - (\A_1 \operatorname{\textbf x}_1^{(k+1)} + \A_3 \operatorname{\textbf x}_3^{(k+1)} - \operatorname{\textbf b})] \big\} \geq 0 \\ \theta_4(\operatorname{\textbf x}_4) - \theta_4(\operatorname{\textbf x}_4^{(k+1)}) + (\operatorname{\textbf x}_4 - \operatorname{\textbf x}_4^{(k+1)})^T \big\{ -\A_4^T [ \operatorname{\boldsymbol \lambda}^{(k)} - (\A_1 \operatorname{\textbf x}_1^{(k+1)} + \A_4 \operatorname{\textbf x}_4^{(k+1)} - \operatorname{\textbf b})] \big\} \geq 0 \end{cases}\end{aligned}$$ which is also the first-order optimality condition of the scheme: $$\begin{aligned} \begin{cases} \label{eq:5} \operatorname{\textbf x}_1^{(k+1)} = \operatorname{argmin}\hspace{1mm} \{ \theta_1(\operatorname{\textbf x}_1) - (\operatorname{\boldsymbol \lambda}^{(k)})^T (\A_1 \operatorname{\textbf x}_1) + \frac{1}{2} \\| \A_1 \operatorname{\textbf x}_1 + \A_2 \operatorname{\textbf x}_2^{(k)} + \A_3 \operatorname{\textbf x}_3^{(k)} + \A_4 \operatorname{\textbf x}_4^{(k)} - \operatorname{\textbf b}\\|_2^2 \} \\ (\operatorname{\textbf x}_2^{(k+1)}, \operatorname{\textbf x}_3^{(k+1)},\operatorname{\textbf x}_4^{(k+1)}) = \operatorname{argmin}\hspace{1mm} \{ \theta_2(\operatorname{\textbf x}_2) + \theta_3(\operatorname{\textbf x}_3) + \theta_4(\operatorname{\textbf x}_4) - (\operatorname{\boldsymbol \lambda}^{(k)})^T (\A_2 \operatorname{\textbf x}_2 + \A_3 \operatorname{\textbf x}_3 + \A_4 \operatorname{\textbf x}_4) \\ \hspace{35mm} + \frac{1}{2} \\| \A_1 \operatorname{\textbf x}_1^{(k+1)} + \A_2 \operatorname{\textbf x}_2 + \A_3 \operatorname{\textbf x}_3 + \A_4 \operatorname{\textbf x}_4 - \operatorname{\textbf b}\\|_2^2 \} \\ \operatorname{\boldsymbol \lambda}^{(k+1)} = \operatorname{\boldsymbol \lambda}^{(k)} - (\A_1 \operatorname{\textbf x}_1^{(k+1)} + \A_2 \operatorname{\textbf x}_2^{(k+1)} + \A_3 \operatorname{\textbf x}_3^{(k+1)} + \A_4 \operatorname{\textbf x}_4^{(k+1)} - \operatorname{\textbf b}) \end{cases} \end{aligned}$$ Clearly, is a specific application of the original two-block ADMM to by regarding $(\operatorname{\textbf x}_2, \operatorname{\textbf x}_3, \operatorname{\textbf x}_4)$ as one variable, $[\A_2,\A_3,\A_4]$ as one matrix and $\theta_2 (\operatorname{\textbf x}_2) + \theta_3(\operatorname{\textbf x}_3) + \theta_4 (\operatorname{\textbf x}_4)$ as one function. $\square$. Note that Lemma \[theorem:four\_block\] is stated in vector form and therefore we need to transform the constraints in the original iGecco+ problem from matrix form to vector form in order to apply Lemma \[theorem:four\_block\]. Note that $\D \U^{(k)} = \V^{(k)} \Leftrightarrow {\U^{(k)}}^T \D^T = {\V^{(k)}}^T \Leftrightarrow (\D \otimes \textbf I_{p_k}) \text{vec} ({\U^{(k)}}^T ) = \text{vec} ({\V^{(k)}}^T )$. Hence we can write the constraints in as: $$\begin{aligned} \begin{pmatrix} \A_1 & \textbf 0 \\ \textbf 0 & \textbf I \\ \textbf 0 & \A_2 \\ \textbf 0 & \textbf I \end{pmatrix} \operatorname{\textbf u}+ \begin{pmatrix} \textbf 0 \\ \textbf I \\ \textbf 0 \\ \textbf 0 \end{pmatrix} \operatorname{\textbf z}+ \begin{pmatrix} \textbf 0 \\ \textbf 0 \\ \textbf 0 \\ - \textbf I \end{pmatrix} {\textbf r}+ \begin{pmatrix} - \textbf I & \textbf 0 \\ \textbf 0 & \textbf 0 \\ \textbf 0 & - \textbf I \\ \textbf 0 & \textbf 0 \end{pmatrix} \operatorname{\textbf v}= \textbf b \end{aligned}$$ where $\operatorname{\textbf u}= \begin{pmatrix} \operatorname{\textbf u}_1 \\ \operatorname{\textbf u}_2 \end{pmatrix} = \begin{pmatrix} \text{vec}( {\U^{(1)}} ^T) \\ \text{vec}({\U^{(2)}}^T) \end{pmatrix} $. $\A_1 = \D \otimes \textbf I_{p_1}$, $\A_2 = \D \otimes \textbf I_{p_2}$. $\operatorname{\textbf z}= \text{vec}(\Z^T)$, ${\textbf r}= \text{vec}(\R^T)$. $\operatorname{\textbf v}= \text{vec} (\V^T) = \begin{pmatrix} \text{vec}( {\V^{(1)}}^T) \\ \text{vec}( {\V^{(2)}} ^T) \end{pmatrix} $, $\textbf b = \begin{pmatrix} \textbf 0_{p_1 \times \|\mathcal E\|} \\ \text{vec}( {\X^{(2)}} ^T) \\ \textbf 0_{p_2 \times \|\mathcal E\|} \\ \tilde \operatorname{\textbf x}^{(2)} \\ \vdots \\ \tilde \operatorname{\textbf x}^{(2)} \end{pmatrix} $, $\tilde \operatorname{\textbf x}\in \mathbb{R}^{p_2}$ is a column vector consisting of all $\tilde x^{(2)}_j$ and is repeated $n$ times in $\textbf b$. Next we show that the constraint sets in for our problem satisfies the condition in Lemma \[theorem:four\_block\] and hence the multi-block ADMM Algorithm \[alg:full-igecco+\] converges. By construction, $\E_2 = \begin{pmatrix} \textbf 0 \\ \textbf I \\ \textbf 0 \\ \textbf 0 \end{pmatrix}$, $\E_3 = \begin{pmatrix} \textbf 0 \\ \textbf 0 \\ \textbf 0 \\ - \textbf I \end{pmatrix}$ and $\E_4 = \begin{pmatrix} - \textbf I & \textbf 0 \\ \textbf 0 & \textbf 0 \\ \textbf 0 & - \textbf I \\ \textbf 0 & \textbf 0 \end{pmatrix} $. It is easy to verify that: $\E_2^T \E_3 = \textbf 0$, $\E_2^T \E_4 = \textbf 0$, $\E_3^T \E_4 = \textbf 0$. Hence our setup satisfies the sufficient condition in Lemma \[theorem:four\_block\] and hence the multi-block ADMM Algorithm \[alg:full-igecco+\] converges. Next, we see that each primal update in Algorithm \[alg:inexact-admm\] is equivalent to the primal update by applying proximal linearized ADMM to the sub-problems in Algorithm \[alg:full-igecco+\]. (We will show this in detail in Theorem \[theorem:diff\].) It is easy to show that those updates with closed-form solutions are special cases of proximal-linearizing the sub-problems. [@lu2016fast; @liu2013linearized] showed the convergence of proximal linearized multi-block ADMM. Hence Algorithm \[alg:inexact-admm\] converges to a global minimum if $\curl_k$ is convex for all $k$ and has Lipschitz gradient when it is differentiable. Further, if each $\curl_k$ is strictly convex, it converges to the unique global solution. $\square$. Gecco+ for Differentiable Losses {#diffdetail} ================================ In this section, we propose algorithms to solve Gecco+ when the loss $\curl$ is differentiable and gradient is Lipschitz continuous. In this case, we develop a fast two-block ADMM algorithm without fully solving the $\U$ sub-problem. Our result is closely related to the proximal linearized ADMM literature [@liu2013linearized; @lu2016fast]. Also solving the sub-problem approximately is closely connected with the generalized ADMM literature [@deng2016global]. In the following sections, we discuss algorithms to solve Gecco+ instead of iGecco+ for notation purposes as we would like to include iteration counter indices in the algorithm for illustrating backtracking; but we can easily extend the algorithm to solve iGecco+. To begin with, we clarify different notations in Gecco+ and iGecco+: $\U^{(k)}$ in iGecco+ refers to the $k^{th}$ data source while $\U^{(k)}$ in Gecco+ refers to the $k^{th}$ iteration counter in the ADMM updates. We omit iteration counter indices in all iGecco+ algorithm for notation purposes and use the most current iterates. Two-block ADMM in Matrix Form ----------------------------- Suppose the loss $\curl(\X,\U)$ is differentiable. Similar to the formulation in convex clustering, we can recast the Gecco+ problem as the equivalent constrained problem: $$\begin{aligned} &\operatorname{minimize}_{\U,\V} \hspace{5mm} \curl(\X,\U) + \gamma \underbrace{ \bigg(\sum_{l \in \mathcal E} w_l \\|\V_{l.} \\|_2 \bigg)}_{P_1(\V;{\textbf w})} + \alpha \sum_{j=1}^p \zeta_j \\| \U_{.j} - \tilde x_j \cdot \textbf 1_n \\|_2 \\ & \operatorname{subject \hspace{2mm} to}\hspace{5mm} \D \U - \V = 0\end{aligned}$$ Like in convex clustering [@chi2015splitting; @weylandt2019dynamic], we index a centroid pair by $l = (l_1, l_2)$ with $l_1 < l_2$, define the set of edges over the non-zero weights $\mathcal E = \{l = (l_1,l_2) : w_l > 0\}$, and introduce a new variable $\V_{l.} = \U_{l_1.} - \U_{l_2.} $ to account for the difference between the two centroids. Hence $\V$ is a matrix containing the pairwise differences between connected rows of $\U$. $\D$ is the difference matrix defined in the work of [@weylandt2019dynamic]. We can show that the augmented Lagrangian in scaled form is equal to: $$\begin{aligned} L(\U,\V,\operatorname{\boldsymbol \Lambda}) &= \curl(\X,\U) + \frac{\rho}{2} \\| \D \U - \V + \operatorname{\boldsymbol \Lambda}\\|_F^2 + \alpha \sum_{j=1}^p \zeta_j \\| \U_{.j} - \tilde x_j \cdot \textbf 1_n \\|_2 + \gamma \sum_{l \in \mathcal E} w_l \\|\V_{l.} \\|_2 \end{aligned}$$ where $\D$ is the difference matrix and the dual variable is denoted by $\operatorname{\boldsymbol \Lambda}$. To update $\U$, we need to solve the following sub-problem: $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \curl(\X,\U) + \frac{\rho}{2} \\| \D \U - \V + \operatorname{\boldsymbol \Lambda}\\|_2^2 + \alpha \sum_{j=1}^p \zeta_j \\| \U_{.j} - \tilde x_j \cdot \textbf 1_n \\|_2 \end{aligned}$$ Let $\tilde \U = \U - \tilde \X$. The sub-problem becomes: $$\begin{aligned} \operatorname{minimize}_{\tilde \U} \hspace{5mm} \curl(\X,\tilde \U +\tilde \X) + \frac{\rho}{2} \\| \D ( \tilde \U +\tilde \X) - \V + \operatorname{\boldsymbol \Lambda}\\|_2^2 + \alpha \sum_{j=1}^p \zeta_j \\|\tilde \operatorname{\textbf u}_j \\|_2\end{aligned}$$ where $\tilde \operatorname{\textbf u}_j$ is the $j^{th}$ column of $\tilde \U$. For each ADMM iterate, we have: $$\begin{aligned} \tilde \U^{(k)} = \operatorname{argmin}_{\tilde \U} \hspace{5mm} \curl(\X,\tilde \U +\tilde \X) + \frac{\rho}{2} \\| \D ( \tilde \U +\tilde \X) - \V^{(k-1)} + \operatorname{\boldsymbol \Lambda}^{(k-1)} \\|_2^2 + \alpha \underbrace{\sum_{j=1}^p \zeta_j \\|\tilde \operatorname{\textbf u}_j \\|_2}_{P_2(\tilde \U;{\boldsymbol \zeta})}\end{aligned}$$ This can be solved by running iterative proximal gradient to full convergence: $$\begin{aligned} \tilde \U^{(k,m)} = \operatorname{prox}_{s_k \cdot \alpha P_2(\cdot;{\boldsymbol \zeta})} \begin{footnotesize} \big( \tilde \U^{(k,m-1)} - s_k \cdot [ \nabla \curl(\X,\tilde \U^{(k,m-1)} + \tilde \X ) + \rho \D^T (\D (\tilde \U^{(k,m-1)} + \tilde \X ) - \V^{(k-1)} + \operatorname{\boldsymbol \Lambda}^{(k-1)} )] \big) \end{footnotesize} \end{aligned}$$ which is equivalent to: $$\begin{aligned} \U^{(k,m)} = \operatorname{prox}_{s_k \cdot \alpha P_2(\cdot;{\boldsymbol \zeta})} \begin{footnotesize} \big( \U^{(k,m-1)} - \tilde \X - s_k \cdot [ \nabla \curl(\X,\U^{(k,m-1)} ) + \rho \D^T (\D \U^{(k,m-1)} - \V^{(k-1)} + \operatorname{\boldsymbol \Lambda}^{(k-1)} )] \big) + \tilde \X \end{footnotesize} \end{aligned}$$ Here $\U^{(k,m)}$ refers to the $m^{th}$ inner iteration counter in the $\U$ sub-problem out of the $k^{th}$ outer iteration counter of the ADMM update. It is straightforward that this is computationally expensive. To address this, we propose to solve the $\U$ sub-problem approximately using just a one-step proximal gradient update and prove convergence in the next section. This approach is based on proximal linearized ADMM [@liu2013linearized; @lu2016fast], which solves the sub-problems efficiently by linearizing the differentiable part and then applying proximal gradient due to the non-differentiable part. To ensure convergence, the algorithm requires that gradient shoule be Lipschitz continuous. The $\V$ and $\operatorname{\boldsymbol \Lambda}$ updates are just the same as in regular convex clustering. We adopt such an approach and develop the proximal linearized 2-block ADMM (Algorithm \[alg:gecco+\_diff\_matrix\]) to solve Gecco+ when the loss is differentiable and gradient is Lipschitz continuous. $\U^{(k)} = \operatorname{prox}_{s_k \cdot \alpha P_2(\cdot;{\boldsymbol \zeta})} \begin{footnotesize} \big( \U^{(k-1)} - \tilde \X - s_k \cdot [ \nabla \curl(\X,\U^{(k-1)} ) + \rho \D^T (\D \U^{(k-1)} - \V^{(k-1)} + \operatorname{\boldsymbol \Lambda}^{(k-1)} )] \big) + \tilde \X \end{footnotesize} $ $\V^{(k)} = \text{prox}_{\gamma /\rho P_1(\cdot; {\textbf w})} (\D \U^{(k)} + \operatorname{\boldsymbol \Lambda}^{(k-1)})$ $\operatorname{\boldsymbol \Lambda}^{(k)} = \operatorname{\boldsymbol \Lambda}^{(k-1)} + ( \D \U^{(k)} - \V^{(k)} ) $ Further, if the $\U$ sub-problem can be decomposed to $p$ separate $\U_{.j}$ sub-problems where the augmented Lagrangian for each now is a sum of a differentiable loss, a quadratic term and a sparse group-lasso penalty, we propose to use proximal gradient descent for each separate $\U_{.j}$ sub-problem. In this way, we yield adaptive step-size for each $\U_{.j}$ sub-problem and hence our algorithm enjoys better convergence property than updating $\U$’s all together. (In the latter case, the step size becomes fairly small as we are moving all $\U$ to some magnitude in the direction of negative gradient.) To achieve this, we assume that the loss is elementwise, which means we can write the loss function as a sum of $p$ terms. (The loss can be written as $\sum_i \curl(\X_{i.},\U_{i.}) = \sum_j \curl(\X_{.j},\U_{.j}) = \sum_i \sum_j q(x_{ij},u_{ij})$ where $q$ is the element-wise version of the loss while $\curl$ is the vector-wise version of the loss.) We see that every deviance-based loss satisfies this assumption. Moreover, by decomposing to $p$ sub-problems, we can solve each in parallel which saves computation cost. We describe in detail how to solve each $\U_{.j}$ sub-problem in the next subsection. Two-block ADMM in Vector Form in Parallel ----------------------------------------- Suppose the $\U$ sub-problem can be decomposed to $p$ separate $\U_{.j}$ sub-problems mentioned above. The augmented Lagrangian now becomes: $$\begin{aligned} L(\U,\V,\operatorname{\boldsymbol \Lambda}) &= \sum_{j=1}^p \curl(\X_{.j},\U_{.j}) + \frac{\rho}{2} \sum_{j=1}^p \\| \D \U_{.j} - \V_{.j} + \operatorname{\boldsymbol \Lambda}_{.j} \\|_2^2 \\ &+ \alpha \sum_{j=1}^p \zeta_j \\| \U_{.j} - \tilde x_j \cdot \textbf 1_n \\|_2 + \gamma \sum_{l \in \mathcal E} w_l \\|\V_{l.} \\|_2 \end{aligned}$$ In this way we can perform block-wise minimization. Now minimizing the augmented Lagrangian over $\U$ is equivalent to minimizing over each $\U_{.j}$, $j = 1,\cdots,p$: $$\begin{aligned} \operatorname{minimize}_{\U_{.j}} \hspace{5mm} \curl(\X_{.j},\U_{.j}) + \frac{\rho}{2} \\| \D \U_{.j} - \V_{.j} + \operatorname{\boldsymbol \Lambda}_{.j} \\|_2^2 + \alpha \zeta_j \\| \U_{.j} - \tilde x_j \cdot \textbf 1_n \\|_2 \end{aligned}$$ Let $\tilde \operatorname{\textbf u}_j = \U_{.j} - \tilde x_j \cdot \textbf 1_n$. The problem above becomes: $$\begin{aligned} \operatorname{minimize}_{\tilde \operatorname{\textbf u}_j} \hspace{5mm} \curl(\X_{.j},\tilde \operatorname{\textbf u}_j +\tilde x_j \cdot \textbf 1_n) + \frac{\rho}{2} \\| \D ( \tilde \operatorname{\textbf u}_j +\tilde x_j \cdot \textbf 1_n) - \V_{.j} + \operatorname{\boldsymbol \Lambda}_{.j} \\|_2^2 + \alpha \zeta_j \\|\tilde \operatorname{\textbf u}_j \\|_2\end{aligned}$$ Similarly, this can be solved by running iterative proximal gradient to full convergence. However, as mentioned above, we propose to solve the $\U$ sub-problem approximately using just a one-step proximal gradient update and prove convergence. Still this approach is based on proximal linearized ADMM [@liu2013linearized; @lu2016fast], which solves the sub-problems efficiently by linearizing the differentiable part and then applying proximal gradient due to the non-differentiable part. To ensure convergence, the algorithm requires that gradient shoule be Lipschitz continuous. We propose Algorithm \[alg:diff\_vec\] to solve Gecco+ when $\curl$ is differentiable and gradient is Lipschitz continuous in vector form. Note the $\U$ update in Algorithm \[alg:diff\_vec\] is a just a vectorized version of that in Algorithm \[alg:gecco+\_diff\_matrix\] if we use fixed step size $s_k$ for each feature $j$. We use the vector form update here since it enjoys better convergence property mentioned above and we use this form for proof of convergence. Next we prove the convergence of Algorithm \[alg:diff\_vec\]. $\mathbf{X}$, $\gamma$, $\mathbf w$, $\alpha$, ${\boldsymbol \zeta}$ $\U^{(0)},\V^{(0)},\mathbf \Lambda^{(0)}$\ Difference matrix $\D$, $\tilde x_j$ $\U_{.j}^{(k)} = \operatorname{prox}_{s_k \cdot \alpha \zeta_j \\|\cdot \\|_2} \big (\U_{.j}^{(k-1)} - \tilde x_j \cdot \textbf 1_n - s_k \cdot [ \nabla \curl(\X_{.j},\U_{.j}^{(k-1)} ) + \rho \D^T (\D \U_{.j}^{(k-1)} - \V_{.j}^{(k-1)} + \operatorname{\boldsymbol \Lambda}_{.j}^{(k-1)} )] \big) + \tilde x_j \cdot \textbf 1_n$ $\V^{(k)} = \text{prox}_{\gamma /\rho P_1(\cdot; {\textbf w})} (\D \U^{(k)} + \operatorname{\boldsymbol \Lambda}^{(k-1)})$ $\operatorname{\boldsymbol \Lambda}^{(k)} = \operatorname{\boldsymbol \Lambda}^{(k-1)} + ( \D \U^{(k)} - \V^{(k)} ) $ $\U^{(k)}$. Proof of Convergence -------------------- [theorem]{}[theoremdiff]{} \[theorem:diff\] If $\curl$ is convex and differrentiable and $\nabla \curl$ is Lipschitz continuous, then Algorithm \[alg:diff\_vec\] converges to a global solution. Further, if $\curl$ is strictly convex, it converges to the unique global solution. Proof:* We will show that the $\U$ sub-problem update in Algorithm \[alg:diff\_vec\] is equivalent to linearizing the $\U$ sub-problem and then applying a proxmial operator, which is proximal linearized ADMM. Note that each $\U_{.j}$ sub-problem is: $$\begin{aligned} \U_{.j}^{(k+1)} = \operatorname{argmin}_{\U_{.j}} \hspace{5mm} \curl(\X_{.j},\U_{.j}) + \frac{\rho}{2} \\| \D \U_{.j} - \V_{.j}^{(k)} + \operatorname{\boldsymbol \Lambda}_{.j}^{(k)} \\|_2^2 + \alpha \zeta_j \\| \U_{.j} - \tilde x_j \cdot \textbf 1_n \\|_2 \end{aligned}$$ For simplicity of notation, we replace $\U_{.j}$ with $\operatorname{\textbf u}_j$ in the following. $$\begin{aligned} \operatorname{\textbf u}_j^{(k+1)} = \operatorname{argmin}_{\operatorname{\textbf u}_j} \hspace{5mm} \curl(\X_{.j}, \operatorname{\textbf u}_j ) + \frac{\rho}{2} \\| \D \operatorname{\textbf u}_j - \V_{.j}^{(k)} + \operatorname{\boldsymbol \Lambda}_{.j}^{(k)} \\|_2^2 + \alpha \zeta_j \\| \operatorname{\textbf u}_j - \tilde x_j \cdot \textbf 1_n \\|_2\end{aligned}$$ Rearranging terms, we have: $$\begin{aligned} \operatorname{\textbf u}_j^{(k+1)} = \operatorname{argmin}_{\operatorname{\textbf u}_j} \hspace{5mm} \curl(\X_{.j}, \operatorname{\textbf u}_j) + \frac{\rho}{2} \\| \D \operatorname{\textbf u}_j - \V_{.j}^{(k)} \\|_2^2 + \rho {\operatorname{\boldsymbol \Lambda}^{(k)}_{.j}}^T \big(\D \operatorname{\textbf u}_j - \V_{.j}^{(k)} \big) + \alpha \zeta_j \\| \operatorname{\textbf u}_j - \tilde x_j \cdot \textbf 1_n \\|_2\end{aligned}$$ According to the proximal linearized ADMM with parallel splitting algorithm (see Algorithm 3 in @liu2013linearized and Equation (14) in @lu2016fast), we can linearize the first two terms and add a quadratic term in the objective: $$\begin{aligned} \operatorname{\textbf u}_j^{(k+1)} = \operatorname{argmin}_{\operatorname{\textbf u}_j} \hspace{5mm} &\curl(\X_{.j}, \operatorname{\textbf u}_j^{(k)} ) + \langle \nabla \curl(\X_{.j}, \operatorname{\textbf u}_j^{(k)} ) , \operatorname{\textbf u}_j - \operatorname{\textbf u}_j^{(k)} \rangle \\ &+ \langle \rho \D^T \big( \D \operatorname{\textbf u}_j^{(k)} - \V_{.j}^{(k)} \big) , \operatorname{\textbf u}_j - \operatorname{\textbf u}_j^{(k)} \rangle \\ & + \rho {\operatorname{\boldsymbol \Lambda}^{(k)}_{.j}}^T \big(\D \operatorname{\textbf u}_j - \V_{.j}^{(k)} \big) + \alpha \zeta_j \\| \operatorname{\textbf u}_j - \tilde x_j \cdot \textbf 1_n \\|_2 + \frac{1}{2 s_k} \\| \operatorname{\textbf u}_j - \operatorname{\textbf u}_j^{(k)} \\|_2^2\end{aligned}$$ Rearranging terms and removing irrelevant terms, we have: $$\begin{aligned} \operatorname{\textbf u}_j^{(k+1)} & = \operatorname{argmin}_{\operatorname{\textbf u}_j} \hspace{5mm} (\nabla g(\operatorname{\textbf u}_j ^{(k)}))^T (\operatorname{\textbf u}_j - \operatorname{\textbf u}_j^{(k)}) + \frac{1}{2 s_k} \\|\operatorname{\textbf u}_j - \operatorname{\textbf u}_j^{(k)}\\|_2^2 + \alpha \zeta_j \\| \operatorname{\textbf u}_j - \tilde x_j \cdot \textbf 1_n \\|_2 \end{aligned}$$ where $\nabla g(\operatorname{\textbf u}_j ^{(k)}) = \nabla \curl(\X_{.j},\operatorname{\textbf u}_j^{(k)}) + \rho \D^T (\D \operatorname{\textbf u}_j^{(k)} - \V_{.j}^{(k)} + \operatorname{\boldsymbol \Lambda}_{.j}^{(k)} ) )$. Let $\hat \operatorname{\textbf u}_j = \operatorname{\textbf u}_j - \tilde x_j \cdot \textbf 1_n $. We have: $$\begin{aligned} \hat \operatorname{\textbf u}_j^{(k+1)} & = \operatorname{argmin}_{\hat \operatorname{\textbf u}_j} \hspace{5mm} (\nabla g(\operatorname{\textbf u}_j ^{(k)}))^T (\hat \operatorname{\textbf u}_j + \tilde x_j \cdot \textbf 1_n - \operatorname{\textbf u}_j^{(k)}) + \frac{1}{2 s_k} \\|\hat \operatorname{\textbf u}_j + \tilde x_j \cdot \textbf 1_n - \operatorname{\textbf u}_j^{(k)}\\|_2^2 + \alpha \zeta_j \\| \hat \operatorname{\textbf u}_j \\|_2 \end{aligned}$$ Recall the definition of proximal operator: $$\begin{aligned} \operatorname{\textbf x}^{(k+1)} & = \operatorname{prox}_{th} (\operatorname{\textbf x}^{(k)} - t \nabla g(\operatorname{\textbf x}^{(k)})) \\ &= \operatorname{argmin}_{\operatorname{\textbf u}} \big( h(\operatorname{\textbf u}) + g(\operatorname{\textbf x}^{(k)}) + \nabla g(\operatorname{\textbf x}^{(k)})^T (\operatorname{\textbf u}-\operatorname{\textbf x}^{(k)}) + \frac{1}{2t} \\| \operatorname{\textbf u}- \operatorname{\textbf x}^{(k)} \\|_2^2 \big)\end{aligned}$$ Therefore, the $\hat \operatorname{\textbf u}_j$ update is just a proximal gradient descent update: $$\begin{aligned} \hat \operatorname{\textbf u}_j^{(k+1)} = \operatorname{prox}_{s_k \cdot \alpha \zeta_j \\|\cdot \\|_2} \big (\operatorname{\textbf u}_{j}^{(k)} - \tilde x_j \cdot \textbf 1_n - s_k \cdot [ \nabla \curl(\X_{.j},\operatorname{\textbf u}_{j}^{(k)} ) + \rho \D^T (\D \operatorname{\textbf u}_{j}^{(k)} - \V_{.j}^{(k)} +\operatorname{\boldsymbol \Lambda}_{.j}^{(k)} )] \big)\end{aligned}$$ Now we plug back and get the $\operatorname{\textbf u}_j$, i.e, $(\U_{.j})$ update: $$\begin{aligned} \U_{.j}^{(k)} = \operatorname{prox}_{s_k \cdot \alpha \zeta_j \\|\cdot \\|_2}& \big (\U_{.j}^{(k-1)} - \tilde x_j \cdot \textbf 1_n - s_k \cdot [ \nabla \curl(\X_{.j},\U_{.j}^{(k-1)} ) + \rho \D^T (\D \U_{.j}^{(k-1)} - \V_{.j}^{(k-1)} + \operatorname{\boldsymbol \Lambda}_{.j}^{(k-1)} )] \big) \\ & + \tilde x_j \cdot \textbf 1_n\end{aligned}$$ which is equivalent to the $\U_{.j}$ update in Algorithm \[alg:diff\_vec\]. The $\V$ and $\operatorname{\boldsymbol \Lambda}$ update is just the same as the one in convex clustering. Hence Algorithm \[alg:diff\_vec\] satisfies the condition of proximal linearized ADMM by [@liu2013linearized; @lu2016fast] and hence converges to a global solution as long as $\nabla \curl$ is Lipschitz continuous. Note that Algorithm \[alg:gecco+\_diff\_matrix\] is equivalent to Algorithm \[alg:diff\_vec\] with a fixed step size $s_k$. (We can choose $s_k$ to be the minimunm step size $s_k$ over for feature $j$.) Therefore, Algorithm \[alg:gecco+\_diff\_matrix\] also converges to a global solution as long as $\nabla \curl$ is Lipschitz continuous. Further, if $\curl$ is strictly convex, the optimization problem has unique minimum and hence Algorithm \[alg:gecco+\_diff\_matrix\] and \[alg:diff\_vec\] converges to the global solution. $\square$ Note: - In @liu2013linearized [@lu2016fast], the algorithm requires that $\nabla \curl$ is Lipschitz continuous to guarantee convergence. We know that $\nabla \curl$ is Lipschitz continuous is equivalent to $\curl$ is strongly smooth. It is easy to show that the hessian of log-likelihood of exponential family and GLM deviance is upper bounded since in (generalized) convex clustering, the value of $\U$ is bounded as it moves along the regularization path from $\X$ to the loss-specific center; also to avoid numerical issues, we add trivial constraint that $u_{ij}>0$ as zero is not defined in the log-likelihood/deviance. Hence the condition for convergence of proximal linearized ADMM is satisfied. - To obtain a reasonable step size $s_k$, we need to compute the Lipschitz constant. However, it is non-trivial to calculate the Lipschitz constant for most of our general losses. Instead, we suggest using backtracking line search procedure proposed by [@beck2009gradient; @parikh2014proximal], which is a common way to determine step size with guaranteed convergence in optimization. Empirical studies show that choosing step size with backtracking in our framework also ensures convergence. The details for backtracking procedure are discussed below. - For proximal linearized ADMM, [@liu2013linearized; @lu2016fast] established convergence rate of $O(1/K)$. An interesting future direction might be establishing the linear convergence rate of proximal linearized ADMM when the objective is strongly convex. Backtracking Criterion ---------------------- In this section we discuss how to choose the step size $s_k$ in Algorithm \[alg:diff\_vec\]. As mentioned, usually we employ a fixed step size by computing the Lipschitz constant as in the squared error loss case; but in our method, it is hard to compute the Lipschitz constant for most of our general losses. Instead, we propose using backtracking line search procedure proposed by [@beck2009gradient; @parikh2014proximal], which is a common way to determine step size with guaranteed convergence in optimization. Recall the objective function we want to minimize in the $\U$ sub-problem is: $$\begin{aligned} f(\tilde \operatorname{\textbf u}_j) = \curl(\X_{.j},\tilde \operatorname{\textbf u}_j +\tilde x_j \cdot \textbf 1_n) + \frac{\rho}{2} \\| \D ( \tilde \operatorname{\textbf u}_j +\tilde x_j \cdot \textbf 1_n) - \V_{.j} + \operatorname{\boldsymbol \Lambda}_{.j} \\|_2^2 + \alpha \zeta_j \\|\tilde \operatorname{\textbf u}_j \\|_2\end{aligned}$$ where $\tilde \operatorname{\textbf u}_j = \U_{.j} - \tilde x_j \cdot \textbf 1_n$. Define: $$\begin{aligned} g(\tilde \operatorname{\textbf u}_j) & = \curl(\X_{.j},\tilde \operatorname{\textbf u}_j +\tilde x_j \cdot \textbf 1_n) + \frac{\rho}{2} \\| \D ( \tilde \operatorname{\textbf u}_j +\tilde x_j \cdot \textbf 1_n) - \V_{.j} + \operatorname{\boldsymbol \Lambda}_{.j} \\|_2^2 \\ h(\tilde \operatorname{\textbf u}_j) & = \alpha \zeta_j \\|\tilde \operatorname{\textbf u}_j \\|_2 \\ G_t(\tilde \operatorname{\textbf u}_j) & = \frac{ \tilde \operatorname{\textbf u}_j - \operatorname{prox}_{t \cdot \alpha \zeta_j \\| \cdot \\|_2 } ( \tilde \operatorname{\textbf u}_j - t \nabla g (\tilde \operatorname{\textbf u}_j) ) } {t} \end{aligned}$$ We adopt the backtracking line search procedure proposed by [@beck2009gradient; @parikh2014proximal]. At each iteration, while $$\begin{aligned} g( \tilde \operatorname{\textbf u}_j - t G_t(\tilde \operatorname{\textbf u}_j)) &> g(\tilde \operatorname{\textbf u}_j) - t \nabla g(\tilde \operatorname{\textbf u}_j)^T G_t(\tilde \operatorname{\textbf u}_j) + \frac{t}{2} \\| G_t(\tilde \operatorname{\textbf u}_j) \\|_2^2 \hspace{10mm} \text{i.e.,}\\ g( \operatorname{prox}_t ( \tilde \operatorname{\textbf u}_j - t \nabla g (\tilde \operatorname{\textbf u}_j) ) ) &> g(\tilde \operatorname{\textbf u}_j) - \nabla g(\tilde \operatorname{\textbf u}_j)^T ( \tilde \operatorname{\textbf u}_j - \operatorname{prox}_t ( \tilde \operatorname{\textbf u}_j - t \nabla g (\tilde \operatorname{\textbf u}_j) ) ) \\&+ \frac{1}{2t} \\| \tilde \operatorname{\textbf u}_j - \operatorname{prox}_t ( \tilde \operatorname{\textbf u}_j - t \nabla g (\tilde \operatorname{\textbf u}_j) ) \\|_2^2\end{aligned}$$ shrink $t = \beta t$. We still adopt the one-step approximation and hence suggest taking a one-step proximal update to solve the $\U$ sub-problem with backtracking. To summarize, we propose Algorithm \[alg:diff-backtracking\], which uses proximal linearized 2-block ADMM with backtracking when the loss is differentiable and gradient is Lipschitz continuous. $\mathbf{X}$, $\gamma$, $\mathbf w$, $\alpha$, ${\boldsymbol \zeta}$ $\U^{(0)},\V^{(0)},\mathbf \Lambda^{(0)},t$\ Difference matrix $\D$, $\tilde x_j$ $t = 1$ $\tilde \operatorname{\textbf u}_j^{(k-1)} = \U_{.j}^{(k-1)} - \tilde x_j \cdot \textbf 1_n$ $\nabla g(\tilde \operatorname{\textbf u}_j^{(k-1)}) = \nabla \curl(\X_{.j},\tilde \operatorname{\textbf u}_j^{(k-1)} +\tilde x_j \cdot \textbf 1_n) + \rho \D^T \big( \D ( \tilde \operatorname{\textbf u}_j^{(k-1)} +\tilde x_j \cdot \textbf 1_n) - \V_{.j}^{(k-1)} + \operatorname{\boldsymbol \Lambda}_{.j}^{(k-1)} \big) $ $\operatorname{\textbf z}= \operatorname{prox}_{t \alpha \zeta_j \\|\cdot \\|_2} \big(\tilde \operatorname{\textbf u}_j^{(k-1)} - t \nabla g(\tilde \operatorname{\textbf u}_j^{(k-1)}) \big)$ $t = \beta t$ $\operatorname{\textbf z}= \operatorname{prox}_{t \alpha \zeta_j \\|\cdot \\|_2} \big(\tilde \operatorname{\textbf u}_j^{(k-1)} - t \nabla g(\tilde \operatorname{\textbf u}_j^{(k-1)}) \big)$ $\U_{.j}^{(k)} = \operatorname{\textbf z}+ \tilde x_j \cdot \textbf 1_n $ $\V^{(k)} = \text{prox}_{\gamma /\rho P_1(\cdot; {\textbf w})} (\D \U^{(k)} + \operatorname{\boldsymbol \Lambda}^{(k-1)})$ $\operatorname{\boldsymbol \Lambda}^{(k)} = \operatorname{\boldsymbol \Lambda}^{(k-1)} + ( \D \U^{(k)} - \V^{(k)} ) $ $\U^{(k)}$. Alternative Algorithm for Differentiable Losses ----------------------------------------------- It should be pointed out that there are many other methods to solve the $\U$ sub-problem when the loss $\curl$ is differentiable. We choose to use proximal gradient descent algorithm as there is existing literature on approximately solving the sub-problem using proximal gradient under ADMM with proved convergence [@liu2013linearized; @lu2016fast]. But there are many other optimization techniques to solve the $\U$ sub-problem such as ADMM. In this subsection, we show how to apply ADMM to solve the $\U$ sub-problem and specify under which conditions this method is more favorable. Recall to update $\U$, we need to solve the following sub-problem: $$\begin{aligned} \operatorname{minimize}\limits_{\U} \hspace{5mm} \curl(\X,\U) + \frac{\rho}{2} \\| \D \U - \V + \operatorname{\boldsymbol \Lambda}\\|_F^2 + \alpha \sum_{j=1}^{p} \zeta_j \\| \U_{.j} - \tilde x_j \cdot \textbf 1_n \\|_2\end{aligned}$$ We use ADMM to solve this minimization problem and can now recast the problem above as the equivalent constrained problem: $$\begin{aligned} &\operatorname{minimize}_{\U,\V,\operatorname{\boldsymbol \Lambda},\R} \hspace{5mm} \sum_{j=1}^p \curl(\X_{.j},\U_{.j}) + \frac{\rho}{2} \\| \D \U - \V + \operatorname{\boldsymbol \Lambda}\\|_F^2 + \alpha \underbrace{ \bigg(\sum_{j=1}^p \zeta_j \\| {\textbf r}_j \\|_2 \bigg)}_{ P_2(\R;{\boldsymbol \zeta})}\\ & \operatorname{subject \hspace{2mm} to}\hspace{5mm} \U - \tilde \X = \R \end{aligned}$$ The augmented Lagrangian in scaled form is: $$\begin{aligned} L(\U,\V,\R,\operatorname{\boldsymbol \Lambda},\N) = & \sum_{j=1}^p \curl(\X_{.j},\U_{.j}) + \frac{\rho}{2} \\| \D \U - \V + \operatorname{\boldsymbol \Lambda}\\|_F^2 + \frac{\rho}{2} \\| ( \U - \tilde \X) - \R + \N \\|_F^2 \\ & + \alpha \sum_{j=1}^p \zeta_j \\| {\textbf r}_j \\|_2 \end{aligned}$$ where the dual variable for $\V$ is denoted by $\operatorname{\boldsymbol \Lambda}$; the dual variable for $\R$ is denoted by $\N$. The $\U_{.j}$ sub-problem in the inner nested ADMM is: $$\begin{aligned} \U_{.j}^{(k)} = \operatorname{argmin}_{\U_{.j} } \hspace{2mm} \curl(\X_{.j},\U_{.j}) & + \frac{\rho}{2} \\| \D \U_{.j} - \V_{.j}^{(k-1)} + \operatorname{\boldsymbol \Lambda}_{.j}^{(k-1)} \\|_F^2 \\ &+ \frac{\rho}{2} \\| ( \U_{.j} - \tilde x_j \cdot \textbf 1_n ) - {\textbf r}_j^{(k-1)} + \N_{.j}^{(k-1)} \\|_F^2 \end{aligned}$$ Now, we are minimizing a sum of a differentiable loss $\ell$ and two quadratic terms which are all smooth. Still the $\U_{.j}$ sub-problem does not have closed-form solution for general convex losses and we need to run an iterative descent algorithm (such as gradient descent, Newton method) to full convergence to solve the problem. Similarly, to reduce computation cost, we take a one-step update by applying linearized ADMM [@lin2011linearized] to the $\U_{.j}$ sub-problem. The $\U_{.j}$ update in the inner ADMM now becomes: $$\begin{aligned} \U_{.j}^{(k)} = \U_{.j}^{(k-1)} - s_k \bigg[ \nabla \curl(\X_{.j},\U_{.j}^{(k-1)}) &+ \rho \D^T ( \D \U_{.j}^{(k-1)} - \V_{.j}^{(k-1)} + \operatorname{\boldsymbol \Lambda}_{.j}^{(k-1)} ) \\ &+ \rho ( \U_{.j}^{(k-1)} - \tilde x_j \cdot \textbf 1_n - {\textbf r}_j^{(k-1)} + \N_{.j}^{(k-1)} ) \bigg] \end{aligned}$$ In this case, empirical studies show that taking a one-step Newton update is favored than a one-step gradient descent update as the former enjoys better convergence properties and generally avoids backtracking. However, inverting a hessian is computationally burdensome at each iteration when $n$ is large. Exceptions are for Euclidean distances case where there is a closed-form solution for the $\U_{.j}$ update and for Bernoulli log-likelihood case where the hessian of the loss can be upper bounded by a fixed matrix. In the latter case, we propose to pre-compute the inverse of that fixed matrix instead of inverting a hessian matrix at each iteration. To illustrate this, we write out the $\U_{.j}$ sub-problem of Gecco+ with Bernoulli log-likelihood: $$\begin{aligned} \U_{.j} = \operatorname{argmin}_{\U_{.j} } \hspace{2mm} \bigg( \sum_{i=1}^n - x_{ij} u_{ij} + \log(1+e^{u_{ij}}) \bigg) &+ \frac{\rho}{2} \\| \D \U_{.j} - \V_{.j} + \operatorname{\boldsymbol \Lambda}_{.j} \\|_F^2 \\ &+ \frac{\rho}{2} \\| ( \U_{.j} - \tilde x_j \cdot \textbf 1_n ) - {\textbf r}_j + \N_{.j} \\|_F^2 \end{aligned}$$ The hessian is $\text{diag} \bigg \{ \frac{ e^{u_{ij}}}{(1 + e^{u_{ij}})^2} \bigg\} + \rho \D^T \D + \rho \mathbf I $ which can be upper bounded by $\frac{1}{4} \textbf I + \rho \D^T \D + \rho \mathbf I$. We propose to replace hessian with this fixed matrix in Newton method and use its inverse as step size. This is closely related to the approximate hessian literature [@krishnapuram2005sparse; @simon2013blockwise]. In this way, we just pre-compute this inverse matrix instead of inverting the hessian matrix at each iteration, which dramatically saves computation. We give Algorithm \[alg:log-likelihood-full\] to solve Gecco+ for Bernoulli log-likelihood with a one-step update to solve the $\U$ sub-problem. Empirical studies show that this is faster than taking the inner nested proximal gradient approach as we generally don’t need to perform the backtracking step. Difference matrix $\D$, $\M_1 = (\frac{1}{4} \textbf I + \rho \D^T \D + \rho \mathbf I)^{-1}$. $\U^{(k)} = \U^{(k-1)} - \M_1 \bigg[ \nabla \curl(\X,\U^{(k-1)}) + \rho \D^T (\D \U^{(k-1)} - \V^{(k-1)} + \operatorname{\boldsymbol \Lambda}^{(k-1)} ) + \rho( \U^{(k-1)} - \tilde \X - \R^{(k-1)} + \N^{(k-1)} ) \bigg]$ $\R^{(k)} = \text{prox}_{\alpha /\rho P_2(\cdot; {\boldsymbol \zeta})} (\U^{(k)} - \tilde \X + \N^{(k-1)} ) $ $\N^{(k)} = \N^{(k-1)} + ( \U^{(k)} - \tilde \X - \R^{(k)} ) $ $\V^{(k)} = \text{prox}_{\gamma /\rho P_1(\cdot; {\textbf w})} (\D \U^{(k)} + \operatorname{\boldsymbol \Lambda}^{(k-1)})$ $\operatorname{\boldsymbol \Lambda}^{(k)} = \operatorname{\boldsymbol \Lambda}^{(k-1)} + ( \D \U^{(k)} - \V^{(k)} ) $ Yet, Algorithm \[alg:log-likelihood-full\] is slow as we need to run iterative inner nested ADMM updates to full convergence. To address this, as mentioned, we can take a one-step update of the inner nested iterative ADMM algorithm. To see this, we can recast the original Gecco+ problem as: $$\begin{aligned} &\operatorname{minimize}_{\U^{(k)},\V} \hspace{5mm} \curl(\X,\U) + \gamma \underbrace{ \bigg(\sum_{l \in \mathcal E} w_l \\|\V_{l.}\\|_2\bigg)}_{P_1(\V;{\textbf w})}+ \alpha \bigg( \underbrace{ \sum_{j=1}^{p} \zeta_j \\| {\textbf r}_j \\|_2}_{P_2(\R;{\boldsymbol \zeta})} \bigg) \\ & \operatorname{subject \hspace{2mm} to}\hspace{5mm} \D \U - \V = 0 , \hspace{5mm} \U - \tilde \X = \R\end{aligned}$$ We apply multi-block ADMM to solve this optimization problem and hence get Algorithm \[alg:log-likelihood\]. As discussed above, we take a one-step update to solve the $\U$ sub-problem with linearized ADMM and use the inverse of fixed approximate hessian as step size. Difference matrix $\D$, $\M_1 = (\frac{1}{4} \textbf I + \rho \D^T \D + \rho \mathbf I)^{-1}$. $\U^{(k)} = \U^{(k-1)} - \M_1 \bigg[ \nabla \curl(\X,\U^{(k-1)}) + \rho \D^T (\D \U^{(k-1)} - \V^{(k-1)} + \operatorname{\boldsymbol \Lambda}^{(k-1)} ) + \rho( \U^{(k-1)} - \tilde \X - \R^{(k-1)} + \N^{(k-1)} ) \bigg]$ $\R^{(k)} = \text{prox}_{\alpha /\rho P_2(\cdot; {\boldsymbol \zeta})} (\U^{(k)} - \tilde \X + \N^{(k-1)} ) $ $\N^{(k)} = \N^{(k-1)} + ( \U^{(k)} - \tilde \X - \R^{(k)} ) $ $\V^{(k)} = \text{prox}_{\gamma /\rho P_1(\cdot; {\textbf w})} (\D \U^{(k)} + \operatorname{\boldsymbol \Lambda}^{(k-1)})$ $\operatorname{\boldsymbol \Lambda}^{(k)} = \operatorname{\boldsymbol \Lambda}^{(k-1)} + ( \D \U^{(k)} - \V^{(k)} ) $ Similarly, we adopt this approach to solve Gecco+ with Euclidean distances (sparse convex clustering). We first recast the original problem as: $$\begin{aligned} &\operatorname{minimize}_{\U^{(k)},\V} \hspace{5mm} \frac{1}{2} \\| \X - \U \\|_2^2 + \gamma \underbrace{ \bigg(\sum_{l \in \mathcal E} w_l \\|\V_{l.}\\|_2\bigg)}_{P_1(\V;{\textbf w})}+ \alpha \bigg( \underbrace{ \sum_{j=1}^{p} \zeta_j \\| {\textbf r}_j \\|_2}_{P_2(\R;{\boldsymbol \zeta})} \bigg) \\ & \operatorname{subject \hspace{2mm} to}\hspace{5mm} \D \U - \V = 0 , \hspace{5mm} \U - \tilde \X = \R\end{aligned}$$ Still, we use multi-block ADMM to solve this optimization problem and hence get Algorithm \[alg:scc\]. Note that $\U$ sub-problem now has closed-form solution. Difference matrix $\D$, $\M_2 = ( \textbf I + \rho \D^T \D + \rho \mathbf I)^{-1}$. $\U^{(k)} = \M_2 \big[ \X + \rho \D^T ( \V^{(k-1)} - \operatorname{\boldsymbol \Lambda}^{(k-1)} ) + \rho( \tilde \X + \R^{(k-1)} - \N^{(k-1)} ) \big]$ $\R^{(k)} = \text{prox}_{\alpha /\rho P_2(\cdot; {\boldsymbol \zeta})} (\U^{(k)} - \tilde \X + \N^{(k-1)} ) $ $\N^{(k)} = \N^{(k-1)} + ( \U^{(k)} - \tilde \X - \R^{(k)} ) $ $\V^{(k)} = \text{prox}_{\gamma /\rho P_1(\cdot; {\textbf w})} (\D \U^{(k)} + \operatorname{\boldsymbol \Lambda}^{(k-1)})$ $\operatorname{\boldsymbol \Lambda}^{(k)} = \operatorname{\boldsymbol \Lambda}^{(k-1)} + ( \D \U^{(k)} - \V^{(k)} ) $ Typically, we do not have approximate hessian or closed-form solution to the sub-problem for each loss and we have to use one-step gradient descent with backtracking to solve the $\U$ sub-problem. Empirical study shows that this approach converges slower than Algorithm \[alg:diff-backtracking\] which uses one-step proximal gradient descent with backtracking. Gecco+ for Non-Differentiable Losses {#nondiffdetail} ==================================== In this section, we propose algorithm to solve Gecco+ when the loss $\curl$ is non-differentiable. In this case, we develop a multi-block ADMM algorithm to solve Gecco+ and prove its algorithmic convergence. Gecco+ Algorithm for Non-Differentiable Losses ---------------------------------------------- Suppose the non-differentiable loss $\curl$ can be expressed as $\curl(\X,\U) = f(g(\X,\U))$ where $f$ is convex but non-differentiable and $g$ is affine. This expression is reasonable as it satisfies the affine composition of a convex function. For example, for the least absolute loss, $f(\Z) = \sum_{j=1}^p \\|\operatorname{\textbf z}_j \\|_1 = \\| \text{vec} (\Z) \\|_1$ and $g(\X,\U) = \X -\U$. We specify the affine function $g$ as we want to augment the non-differentiable term in the loss function $\curl$. We can rewrite the problem as: $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} f(g(\X,\U)) + \gamma \sum_{1\leq i < i' \leq n} w_{ii'} \\| \U_{i.} - \U_{i'.} \\|_2 + \alpha \sum_{j=1}^p \zeta_j \\| \U_{.j} - \tilde x_j \cdot \textbf 1_n \\|_2\end{aligned}$$ We can now recast the problem above as the equivalent constrained problem: $$\begin{aligned} &\operatorname{minimize}_{\U,\V,\Z,\R} \hspace{5mm} f(\Z) + \gamma \underbrace{ \bigg(\sum_{l \in \mathcal E} w_l \\|\V_{l.}\\|_2\bigg)}_{P_1(\V;{\textbf w})} + \alpha \underbrace{ \bigg(\sum_{j=1}^p \zeta_j \\| {\textbf r}_j \\|_2 \bigg)}_{ P_2(\R;{\boldsymbol \zeta})}\\ & \operatorname{subject \hspace{2mm} to}\hspace{5mm} g(\X,\U) = \Z \\ & \hspace{25mm} \D \U - \V = 0 \\ & \hspace{25mm} \U - \tilde \X = \R \end{aligned}$$ where $ \tilde \X$ is an $n \times p$ matrix with $j^{th}$ columns equal to scalar $\tilde x_j$. The augmented Lagrangian in scaled form is: $$\begin{aligned} L(\U,\V,\Z,\R,\operatorname{\boldsymbol \Lambda},\N,\bKSI) = & \frac{\rho}{2} \\| \D \U - \V + \operatorname{\boldsymbol \Lambda}\\|_F^2 + \frac{\rho}{2} \\| ( \U - \tilde \X) - \R + \N \\|_F^2 \\ & + \frac{\rho}{2} \\| g(\X,\U) - \Z + \bKSI \\|_F^2 + f(\Z) + \gamma \sum_{l \in \mathcal E} w_l \\|\V_{l.} \\|_2 + \alpha \sum_{j=1}^p \zeta_j \\| {\textbf r}_j \\|_2 \end{aligned}$$ where the dual variable for $\V$ is denoted by $\operatorname{\boldsymbol \Lambda}$; the dual variable for $\Z$ is denoted by $\bKSI$; the dual variable for $\R$ is denoted by $\N$. Since we assume $g$ to be affine, i.e, $g(\X,\U) = \A \X + \B\U + \C$, the augmented Lagrangian in scaled form can be written as: $$\begin{aligned} L(\U,\V,\Z,\R,\operatorname{\boldsymbol \Lambda},\N,\bKSI) &= \frac{\rho}{2} \\| \D \U - \V + \operatorname{\boldsymbol \Lambda}\\|_F^2 + \frac{\rho}{2} \\| ( \U - \tilde \X) - \R + \N \\|_F^2 \\ & + \frac{\rho}{2} \\| \A \X + \B\U + \C - \Z + \bKSI \\|_F^2 + f(\Z) + \gamma \sum_{l \in \mathcal E} w_l \\|\V_{l.} \\|_2 + \alpha \sum_{j=1}^p \zeta_j \\| {\textbf r}_j \\|_2 \end{aligned}$$ It can be shown that the $\U$ sub-problem has closed-form solution. Note that, hinge loss is also non-differentiable and we can write $g(\X,\U) = \textbf 1- \U \circ \X$ where $\textbf 1$ is a matrix of all one and “$\circ$" is the Hadamard product. Now the $\U$ sub-problem does not have closed-form solution and we will discuss how to solve this problem in the next section. For distance-based losses, the loss function can always be written as: $\curl(\X,\U) = f( \X - \U)$, which means $g(\X,\U) = \X - \U$. Then the augmented Lagrangian in scaled form can be simplified as: $$\begin{aligned} L(\U,\V,\Z,\R,\operatorname{\boldsymbol \Lambda},\N,\bKSI) = & \frac{\rho}{2} \\| \D \U - \V + \operatorname{\boldsymbol \Lambda}\\|_F^2 + \frac{\rho}{2} \\| ( \U - \tilde \X) - \R + \N \\|_F^2 \\ & + \frac{\rho}{2} \\| \X - \U - \Z + \bKSI \\|_F^2 + f(\Z) + \gamma \sum_{l \in \mathcal E} w_l \\|\V_{l.} \\|_2 + \alpha \sum_{j=1}^p \zeta_j \\| {\textbf r}_j \\|_2 \end{aligned}$$ Now the $\U$ sub-problem has closed-form solution: $$\begin{aligned} \U^{(k)} = (\D^T \D + 2 \mathbf I)^{-1} (\D^T (\V^{(k-1)} - \operatorname{\boldsymbol \Lambda}^{(k-1)} ) + \tilde \X + \R^{(k-1)} - \N^{(k-1)} + \X - \Z^{(k-1)} + \bKSI^{(k-1)} )\end{aligned}$$ This gives us Algorithm \[alg:non-diff\] to solve Gecco+ for non-differentiable distance-based loss. Difference matrix $\D$, $\M = (\D^T \D + 2 \mathbf I)^{-1}$. $\U^{(k)} = \M(\D^T (\V^{(k-1)} - \operatorname{\boldsymbol \Lambda}^{(k-1)} ) + \tilde \X + \R^{(k-1)} - \N^{(k-1)} + \X - \Z^{(k-1)} + \bKSI^{(k-1)} )$ $\Z^{(k)} = \text{prox}_{f/\rho} (\X - \U^{(k)} + \bKSI^{(k-1)} )$ $\R^{(k)} = \text{prox}_{\alpha /\rho P_2(\cdot; {\boldsymbol \zeta})} (\U^{(k)} - \tilde \X + \N^{(k-1)} ) $ $\bKSI^{(k)} = \bKSI^{(k-1)} + (\X - \U^{(k)} - \Z^{(k)}) $ $\N^{(k)} = \N^{(k-1)} + ( \U^{(k)} - \tilde \X - \R^{(k)} ) $ $\V^{(k)} = \text{prox}_{\gamma /\rho P_1(\cdot; {\textbf w})} (\D \U^{(k)} + \operatorname{\boldsymbol \Lambda}^{(k-1)})$ $\operatorname{\boldsymbol \Lambda}^{(k)} = \operatorname{\boldsymbol \Lambda}^{(k-1)} + ( \D \U^{(k)} - \V^{(k)} ) $ Algorithm \[alg:non-diff\] can be used to solve Gecco+ problem with various distances such as Manhattan, Minkowski and Chebychev distances by applying the corresponding proximal operator in the $\Z$ update. For example, for Gecco+ with Manhattan distances, the $\Z$ update is just applying element-wise soft-thresholding operator. For Gecco+ with Chebychev distances, the proximal operator in the $\Z$-update can be computed separately across the rows of its argument and reduces to applying row-wise proximal operator of the infinity-norm. For Gecco+ with Minkowski distances, we similarly apply row-wise proximal operator of the $\ell_q$-norm. Next we prove the convergence of Algorithm \[alg:non-diff\]. Proof of Convergence for Algorithm \[alg:non-diff\] --------------------------------------------------- [theorem]{}[theoremnondiff]{} \[theorem:non-diff\] If $\curl$ is convex, Algorithm \[alg:non-diff\] converges to a global minimum. Proof: Note we have provided a sufficient condition for the convergence of four-block ADMM in Lemma \[theorem:four\_block\]. Next we show that the constraint set in our problem satisfies the condition in Lemma \[theorem:four\_block\] and hence the multi-block ADMM Algorithm \[alg:non-diff\] converges. Recall our problem is: $$\begin{aligned} &\operatorname{minimize}_{\U,\V,\Z,\R} \hspace{5mm} f(\Z) + \gamma \sum_{l \in \mathcal E} w_l \\|\V_{l.}\\|_2 + \alpha \sum_{j=1}^p \zeta_j \\| {\textbf r}_j \\|_2 \\ & \operatorname{subject \hspace{2mm} to}\hspace{5mm} \X - \U = \Z \\ & \hspace{25mm} \D \U - \V = 0 \\ & \hspace{25mm} \U - \tilde \X = \R\end{aligned}$$ Note that Lemma \[theorem:four\_block\] is stated in vector form. Hence we transform the constraints above from matrix form to vector form. Note that $\D \U = \V \Leftrightarrow \U^T \D^T = \V^T \Leftrightarrow (\D \otimes \textbf I_p) \text{vec} (\U^T ) = \text{vec} (\V^T )$. Hence we can write the constraints as: $$\begin{aligned} \begin{pmatrix} \textbf I \\ \A \\ \textbf I \end{pmatrix} \operatorname{\textbf u}+ \begin{pmatrix} \textbf I \\ \textbf 0 \\ \textbf 0 \end{pmatrix} \operatorname{\textbf z}+ \begin{pmatrix} \textbf 0 \\ \textbf 0 \\ - \textbf I \end{pmatrix} {\textbf r}+ \begin{pmatrix} \textbf 0 \\ - \textbf I \\ \textbf 0 \end{pmatrix} \operatorname{\textbf v}= \textbf b \end{aligned}$$ where $\operatorname{\textbf u}= \text{vec}(\U^T)$, $\A = \D \otimes \textbf I_p$, $\operatorname{\textbf z}= \text{vec}(\Z^T)$, ${\textbf r}= \text{vec}(\R^T)$, $\operatorname{\textbf v}= \text{vec} (\V^T$), $\textbf b = \begin{pmatrix} \text{vec}(\X^T) \\ \textbf 0_{p \times \|\mathcal E\|} \\ \tilde \operatorname{\textbf x}\\ \vdots \\ \tilde \operatorname{\textbf x}\end{pmatrix} $, $\tilde \operatorname{\textbf x}\in \mathbb{R}^p$ is a column vector consisting of all $\tilde x_j$ and is repeated $n$ times in $\textbf b$. By construction, $\A_2 = \begin{pmatrix} \textbf I \\ \textbf 0 \\ \textbf 0 \end{pmatrix}$, $\A_3 = \begin{pmatrix} \textbf 0 \\ \textbf 0 \\ - \textbf I \end{pmatrix} $ and $\A_4 = \begin{pmatrix} \textbf 0 \\ - \textbf I \\ \textbf 0 \end{pmatrix} $. It is easy to verify that: $\A_2^T \A_3 = \mathbf 0$, $\A_2^T \A_4 = \mathbf 0$, $\A_3^T \A_4 = \mathbf 0$. Hence our setup satisfies the sufficient condition in Lemma \[theorem:four\_block\] and hence the multi-block ADMM Algorithm \[alg:non-diff\] converges. $\square$. D.3. Examples: Gecco+ with Minkowski/Chebychev distances {#d.3.-examples-gecco-with-minkowskichebychev-distances .unnumbered} -------------------------------------------------------- Gecco+ with Chebychev distances is: $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_{i=1}^n \max_j \{\|x_{ij} - u_{ij}\|\} + \gamma \sum_{1\leq i < i' \leq n} w_{ii'} \\| \U_{i.} - \U_{i'.} \\|_2 + \alpha \sum_{j=1}^p \zeta_j \\| \U_{.j} - \tilde x_j \cdot \textbf 1_n \\|_2\end{aligned}$$ which is equivalent to $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_{i=1}^n \\|\X_{i.} - \U_{i.}\\|_{\infty} + \gamma \sum_{1\leq i < i' \leq n} w_{ii'} \\| \U_{i.} - \U_{i'.} \\|_2 + \alpha \sum_{j=1}^p \zeta_j \\| \U_{.j} - \tilde x_j \cdot \textbf 1_n \\|_2\end{aligned}$$ Still, we recast the problem above as the equivalent constrained problem: $$\begin{aligned} &\operatorname{minimize}_{\U,\V,\Z,\R} \hspace{5mm} \underbrace{\sum_{i=1}^n \\| \Z_{i.} \\|_{\infty}}_{P_3(\Z)} + \gamma \underbrace{ \bigg(\sum_{l \in \mathcal E} w_l \\|\V_{l.}\\|_2\bigg)}_{P_1(\V;{\textbf w})} + \alpha \underbrace{ \bigg(\sum_{j=1}^p \zeta_j \\| {\textbf r}_j \\|_2 \bigg)}_{ P_2(\R;{\boldsymbol \zeta})}\\ & \operatorname{subject \hspace{2mm} to}\hspace{5mm} \X - \U = \Z \\ & \hspace{25mm} \D \U - \V = 0 \\ & \hspace{25mm} \U - \tilde \X = \R\end{aligned}$$ The augmented Lagrangian in scaled form is: $$\begin{aligned} L(\U,\V,\Z,\R,\operatorname{\boldsymbol \Lambda},\N,\bKSI) = & \frac{\rho}{2} \\| \D \U - \V + \operatorname{\boldsymbol \Lambda}\\|_F^2 + \frac{\rho}{2} \\| ( \U - \tilde \X) - \R + \N \\|_F^2 \\ & + \frac{\rho}{2} \\| \X - \U - \Z + \bKSI\\|_F^2 + \sum_{i=1}^n \\| \Z_{i.} \\|_{\infty} + \gamma \sum_{l \in \mathcal E} w_l \\|\V_{l.} \\|_2 + \alpha \sum_{j=1}^p \zeta_j \\| {\textbf r}_j \\|_2 \end{aligned}$$ Still the $\U$ sub-problem has closed-form solution: $$\begin{aligned} \U^{(k)} = (\D^T \D + 2 \mathbf I)^{-1} (\D^T (\V^{(k-1)} - \operatorname{\boldsymbol \Lambda}^{(k-1)} ) + \tilde \X + \R^{(k-1)} - \N^{(k-1)} + \X - \Z^{(k-1)} + \bKSI^{(k-1)} )\end{aligned}$$ By the row-wise structure of $P_3$, the proximal operator in the $\Z$-update can be computed separately across the rows of its argument and reduces to applying row-wise proximal operator of the infinity-norm. $$\begin{aligned} \Z^{(k)} = \text{prox}_{1/\rho P_3(\cdot)} (\X - \U^{(k)} + \bKSI^{(k-1)} ) \end{aligned}$$ For Gecco+ with Minkowski distances: $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_{i=1}^n \\|\X_{i.} - \U_{i.}\\|_{q} + \gamma \sum_{1\leq i < i' \leq n} w_{ii'} \\| \U_{i.} - \U_{i'.} \\|_2 + \alpha \sum_{j=1}^p \zeta_j \\| \U_{.j} - \tilde x_j \cdot \textbf 1_n \\|_2\end{aligned}$$ we follow the same approach by changing the $\ell_\infty$-norm above with $\ell_q$-norm. Special Case: Gecco+ with Hinge Losses -------------------------------------- As mentioned, we cannot directly apply Algorithm \[alg:non-diff\] to solve Gecco+ with hinge losses as the function $g(\X,\U)$ in this case is not the same as the one in distance-based losses. Recall Gecco+ with hinge losses is: $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_{i=1}^n \sum_{j=1}^p \max(0,1-u_{ij} x_{ij}) + \gamma \sum_{1\leq i < i' \leq n} w_{ii'} \\| \U_{i.} - \U_{i'.} \\|_2 + \alpha \sum_{j=1}^p \zeta_j \\| \U_{.j} - \tilde x_j \cdot \textbf 1_n \\|_2\end{aligned}$$ Like before, we can rewrite the problem as: $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} f(g(\X,\U)) + \gamma \sum_{1\leq i < i' \leq n} w_{ii'} \\| \U_{i.} - \U_{i'.} \\|_2 + \alpha \sum_{j=1}^p \zeta_j \\| \U_{.j} - \tilde x_j \cdot \textbf 1_n \\|_2\end{aligned}$$ We can now recast the problem above as the equivalent constrained problem: $$\begin{aligned} &\operatorname{minimize}_{\U,\V,\Z,\R} \hspace{5mm} f(\Z) + \gamma \underbrace{ \bigg(\sum_{l \in \mathcal E} w_l \\|\V_{l.}\\|_2\bigg)}_{P_1(\V;{\textbf w})} + \alpha \underbrace{ \bigg(\sum_{j=1}^p \zeta_j \\| {\textbf r}_j \\|_2 \bigg)}_{ P_2(\R;{\boldsymbol \zeta})}\\ & \operatorname{subject \hspace{2mm} to}\hspace{5mm} \textbf 1- \U \circ \X = \Z \\ & \hspace{25mm} \D \U - \V = 0 \\ & \hspace{25mm} \U - \tilde \X = \R \end{aligned}$$ Here, $f(\Z) = \max(0,\Z)$. With a slight abuse of notation, we refer $f$ to applying element-wise maximum to all entries in the matrix. We set $g(\X,\U) = \textbf 1- \U \circ \X$ where $\textbf 1$ is a matrix of all one and “$\circ$" is the Hadamard product. $ \tilde \X$ is an $n \times p$ matrix with $j^{th}$ columns equal to scalar $\tilde x_j$. The augmented Lagrangian in scaled form is: $$\begin{aligned} L(\U,\V,\Z,\R,\operatorname{\boldsymbol \Lambda},\N,\bKSI) = & \frac{\rho}{2} \\| \D \U - \V + \operatorname{\boldsymbol \Lambda}\\|_F^2 + \frac{\rho}{2} \\| ( \U - \tilde \X) - \R + \N \\|_F^2 \\ & + \frac{\rho}{2} \\| \textbf 1- \U \circ \X - \Z + \bKSI \\|_F^2 + f(\Z) + \gamma \sum_{l \in \mathcal E} w_l \\|\V_{l.} \\|_2 + \alpha \sum_{j=1}^p \zeta_j \\| {\textbf r}_j \\|_2 \end{aligned}$$ The $\U$ sub-problem now becomes: $$\begin{aligned} \U^{(k+1)} = \operatorname{argmin}_{\U} \hspace{2mm} & \\| \D \U - \V^{(k)} + \operatorname{\boldsymbol \Lambda}^{(k)} \\|_F^2 + \\| \U - \tilde \X - \R^{(k)} + \N^{(k)} \\|_F^2 \\ & + \\| \textbf 1- \U \circ \X - \Z^{(k)} + \bKSI^{(k)} \\|_F^2 \end{aligned}$$ The first-order optimality condition is: $$\begin{aligned} \D^T ( \D \U - \V^{(k)} + \operatorname{\boldsymbol \Lambda}^{(k)} ) + \U - \tilde \X - \R^{(k)} + \N^{(k)} + \X \circ ( \U \circ \X + \Z^{(k)} - \textbf 1 - \bKSI^{(k)} ) = \textbf 0 \end{aligned}$$ which can be written as: $$\begin{aligned} & (\X \circ \X) \circ \U + \X \circ (\Z^{(k)} - \textbf 1 - \bKSI^{(k)} ) + \D^T \D \U - \D^T (\V^{(k)} - \operatorname{\boldsymbol \Lambda}^{(k)} ) + \U - \tilde \X - \R^{(k)} + \N^{(k)} = \textbf 0 \\ & (\X \circ \X) \circ \U + (\D^T \D + \textbf I ) \U = \D^T (\V^{(k)} - \operatorname{\boldsymbol \Lambda}^{(k)} ) + \tilde \X + \R^{(k)} - \N^{(k)} + \X \circ ( \textbf 1 + \bKSI^{(k)} - \Z^{(k)} ) \end{aligned}$$ To solve $\U$ from the above equation, one way is to first find the SVD of the leading coefficient: $$\begin{aligned} \X \circ \X = \sum_k \sigma_k \tilde \operatorname{\textbf u}_k \tilde \operatorname{\textbf v}_k = \sum_k \tilde {\textbf w}_k \tilde \operatorname{\textbf v}_k\end{aligned}$$ From this decomposition we create two sets of diagonal matrices: $$\begin{aligned} \tilde \W_k &= \text{Diag}(\tilde {\textbf w}_k) \\ \tilde \V_k &= \text{Diag}(\tilde \operatorname{\textbf v}_k)\end{aligned}$$ The Hadamard product can now be replaced by a sum $$\begin{aligned} \sum_k \tilde \W_k \U \tilde \V_k + (\D^T \D + \textbf I) \U = \C\end{aligned}$$ where $\C = \D^T (\V^{(k)} - \operatorname{\boldsymbol \Lambda}^{(k)} ) + \tilde \X + \R^{(k)} - \N^{(k)} + \X \circ ( \textbf 1 + \bKSI^{(k)} - \Z^{(k)} ) $. Now we can solve this equation via vectorization: $$\begin{aligned} \text{vec}(\C) &= \big( \textbf I \otimes (\D^T \D + \textbf I) + \sum_k \tilde \V_k \otimes \tilde \W_k \big) \text{vec}(\U) \\ \text{vec}(\U) &= \big( \textbf I \otimes (\D^T \D + \textbf I) + \sum_k \tilde \V_k \otimes \tilde \W_k \big)^{+} \text{vec}(\C) \\ \U &= \text{Mat} \bigg( \big( \textbf I \otimes (\D^T \D + \textbf I) + \sum_k \tilde \V_k \otimes \tilde \W_k \big)^{+} \text{vec}(\C) \bigg)\end{aligned}$$ where $\B^+$ denotes the pseudo-inverse of $\B$, and () is the inverse of the () operation. Here we have to compute the pseudo-inverse of a matrix which is computationally expensive in practice. To avoid this, we adopt Generalized ADMM approach proposed by [@deng2016global] where the $\U$ sub-problem is augmented by a positive semi-definite quadratic operator. In our case, our modified $\U$ sub-problem becomes: $$\begin{aligned} \operatorname{argmin}_{\U} \hspace{2mm} \\| \D \U - \V^{(k)} + \operatorname{\boldsymbol \Lambda}^{(k)} \\|_F^2 & + \\| ( \U - \tilde \X) - \R^{(k)} + \N^{(k)} \\|_F^2 + \\| \textbf 1- \U \circ \X - \Z^{(k)} + \bKSI^{(k)} \\|_F^2 \\ & + \\| \big (\textbf 1 - \X \circ \X \big) \circ (\U - \U^{(k)}) \\|_F^2 \end{aligned}$$ The first-order optimality condition now becomes: $$\begin{aligned} & \D^T ( \D \U - \V^{(k)} + \operatorname{\boldsymbol \Lambda}^{(k)} ) + \U - \tilde \X - \R^{(k)} + \N^{(k)} + \X \circ ( \U \circ \X + \Z^{(k)} - \textbf 1 - \bKSI^{(k)} ) \\ &+ \big (\textbf 1 - \X \circ \X \big) \circ (\U - \U^{(k)}) = \textbf 0 \end{aligned}$$ We have: $$\begin{aligned} \mathbf H \U = \D^T ( \V^{(k)} - \operatorname{\boldsymbol \Lambda}^{(k)} ) + \tilde \X + \R^{(k)} - \N^{(k)} - \X \circ ( \Z^{(k)} - \textbf 1 - \bKSI^{(k)} ) + (\textbf 1 - \X \circ \X ) \circ \U^{(k)}\end{aligned}$$ where $\mathbf H = ( \D^T \D + \textbf I + \textbf 1 )$. Hence we have analytical update: $$\begin{aligned} \U^{(k+1)} = \mathbf H^{-1} \big( \D^T ( \V^{(k)} - \operatorname{\boldsymbol \Lambda}^{(k)} ) + \tilde \X + \R^{(k)} - \N^{(k)} - \X \circ ( \Z^{(k)} - \textbf 1 - \bKSI^{(k)} ) + (\textbf 1 - \X \circ \X ) \circ \U^{(k)} \big)\end{aligned}$$ It is easy to see that the $\V$, $\Z$ and $\R$ updates all have closed-form solutions. Multinomial Gecco+ {#multin} ================== In this section, we briefly demonstrate how Gecco+ with multinomial losses is formulated, which is slightly different from the original Gecco+ problems. Suppose we observe categorical data as follows ($K=3$): $$\begin{aligned} \X_{n \times p} = \begin{pmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 3 \end{pmatrix}\end{aligned}$$ We can get the indicator matrix $\X^{(k)}$ for each class $k$ as: $$\begin{aligned} \X^{(1)} = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \hspace{5mm} \X^{(2)} = \begin{pmatrix} 0 & 0 & 0 \\ 1 & 1 & 1 \\ 0 & 0 & 0 \end{pmatrix} \hspace{5mm} \X^{(3)} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 1 & 1 \end{pmatrix}\end{aligned}$$ Then we concatenate $\X^{(1)},\X^{(2)},\X^{(3)}$ and get $\hat \X_{n \times (pK)} = \begin{pmatrix} \X^{(1)} & \X^{(2)} & \X^{(3)} \end{pmatrix} $. This is equivalent to the dummy coding of the categorical matrix $\tilde \X$ after some row/column shuffle. $$\begin{aligned} \tilde \X = \begin{pmatrix} 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \end{pmatrix}_{n \times (pK)}\end{aligned}$$ It is obvious that measuring the difference of two observations by comparing rows of $\hat \X$ is better than simply comparing the Euclidean distances of rows of original data matrix $\X$. Also parameterizing in $\hat \X$ is beneficial for computing the multinomial log-likelihood or deviance. Hence we concatenate all columns of $\X^{(k)}$ as input data. Similarly, we concatenate all columns of the corresponding $\U^{(k)}$ and then fuse $\hat \U$ in row-wise way. Gecco with Multinomial Log-likelihood ------------------------------------- Gecco with multinomial log-likelihood can be formulated as: $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_i \sum_j \bigg\{ \sum_{k=1}^{K} -x_{ijk} u_{ijk} + \log(\sum_{k=1}^K e^{u_{ijk}}) \bigg \} + \gamma \sum_{i < i'} w_{ii'} \bigg \\| \begin{bmatrix} \U_{i.}^{(1)} \\ \vdots \\ \U_{i.}^{(K)} \end{bmatrix} - \begin{bmatrix} \U_{i'.}^{(1)} \\ \vdots \\ \U_{i'.}^{(K)} \end{bmatrix} \bigg \\|_2\end{aligned}$$ where $x_{ijk}$ refers to the elements of indicator matrix $\X_{ij}^{(k)}$ discussed previously and $\U_{i.}^{(k)} = \begin{bmatrix} u_{i1k} \\ u_{i2k} \\ \vdots \\ u_{ipk} \end{bmatrix}$. Gecco+ with Multinomial Log-likelihood -------------------------------------- Gecco+ with multinomial log-likelihood can be formulated as: $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_i \sum_j &\bigg\{ \sum_{k=1}^{K} -x_{ijk} u_{ijk} + \log(\sum_{k=1}^K e^{u_{ijk}}) \bigg \} + \gamma \sum_{i < i'} w_{ii'} \bigg \\| \begin{bmatrix} \U_{i.}^{(1)} \\ \vdots \\ \U_{i.}^{(K)} \end{bmatrix} - \begin{bmatrix} \U_{i'.}^{(1)} \\ \vdots \\ \U_{i'.}^{(K)} \end{bmatrix} \bigg \\|_2 \\ &+ \alpha \sum_{j=1}^p \sum_{k=1}^K \\| \U_{.j}^{(k)} - \tilde x_j^{(k)} \cdot \textbf 1_n \\|_2\end{aligned}$$ where $\U_{i.}^{(k)} = \begin{bmatrix} u_{i1k} \\ u_{i2k} \\ \vdots \\ u_{ipk} \end{bmatrix}$, $\U_{.j}^{(k)} = \begin{bmatrix} u_{1jk} \\ u_{2jk} \\ \vdots \\ u_{njk} \end{bmatrix}$ and $\tilde x_j^{(k)}$ is the loss-specific center for $j$ variable in $k^{th}$ class. Loss-specific Center Calculation {#centroidcal} ================================ In this section, we show how to calculate the loss-specific center in Table \[loss-table\]. Continuous Data --------------- For continuous data, we consider Gecco with Euclidean distances. ### Euclidean Distance $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \frac{1}{2} \sum_{i=1}^n \\|\X_{i.} - \U_{i.}\\|_2^2 + \gamma \sum_{i < i'} w_{ii'} \\|\U_{i.} - \U_{i'.} \\|_2\end{aligned}$$ When total fusion, $\U_{i.}= \U_{i'.}$, $\forall i \neq i'$. Let $\U_{i.} = \U_{i'.} = \operatorname{\textbf u}$. The problem above becomes: $$\begin{aligned} \operatorname{minimize}_{\operatorname{\textbf u}} \hspace{5mm} \frac{1}{2} \sum_{i=1}^n \\|\X_{i.} - \operatorname{\textbf u}\\|_2^2 \end{aligned}$$ Taking derivative, we get: $$\begin{aligned} \sum_{i=1}^n (\X_{i.} - \operatorname{\textbf u}) = 0 \Rightarrow \operatorname{\textbf u}= \bar \operatorname{\textbf x}\end{aligned}$$ Count Data ---------- For count-valued data, we consider Gecco with Poisson log-likelihood/deviance, negative binomial log-likelihood/deviance and Manhattan distances. ### Poisson Log-likelihood $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p -x_{ij} u_{ij} + \exp(u_{ij}) + \gamma \sum_{i < i'} w_{ii'} \\|\U_{i.} - \U_{i'.} \\|_2\end{aligned}$$ When total fusion, $\operatorname{\textbf u}_{ij} = \operatorname{\textbf u}_{i'j}$, $\forall i \neq i' $. Let $\operatorname{\textbf u}$ be the fusion vector and $\operatorname{\textbf u}= (u_1,\cdots,u_p)$. The problem above becomes: $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p -x_{ij} u_{j} + \exp(u_{j})\end{aligned}$$ Taking derivative, we get: $$\begin{aligned} \sum_{i=1}^n -x_{ij} + n \exp(u_{j}) = 0 \Rightarrow \exp(u_{j}) = \bar x_j \Rightarrow u_{j} = \log(\bar x_j) \Rightarrow \operatorname{\textbf u}= \log(\bar \operatorname{\textbf x})\end{aligned}$$ ### Poisson Deviance $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p -x_{ij} \log u_{ij} + u_{ij} + \gamma \sum_{i < i'} w_{ii'} \\|\U_{i.} - \U_{i'.}\\|_2\end{aligned}$$ Let $\operatorname{\textbf u}$ be the fusion vector and $\operatorname{\textbf u}= (u_1,\cdots,u_p)$. The problem above becomes: $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p -x_{ij} \log u_{j} + u_{j}\end{aligned}$$ Taking derivative, we get: $$\begin{aligned} \sum_{i=1}^n \frac{-x_{ij}}{u_j} + n = 0 \Rightarrow u_j = \bar x_j \Rightarrow \operatorname{\textbf u}= \bar \operatorname{\textbf x}\end{aligned}$$ ### Negative Binomial Log-likelihood $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p - x_{ij} u_{ij} + (x_{ij} + \frac{1}{\alpha} ) \log (\frac{1}{\alpha} + e^{u_{ij}} ) + \gamma \sum_{i < i'} w_{ii'} \\|\U_{i.} - \U_{i'.}\\|_2\end{aligned}$$ When total fusion, $\operatorname{\textbf u}_{ij} = \operatorname{\textbf u}_{i'j}$, $\forall i \neq i $. Let $\operatorname{\textbf u}$ be the fusion vector and $\operatorname{\textbf u}= (u_1,\cdots,u_p)$. The problem above becomes: $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p - x_{ij} u_{j} + (x_{ij} + \frac{1}{\alpha} ) \log (\frac{1}{\alpha} + e^{u_{j}} ) \end{aligned}$$ Taking derivative, we get: $$\begin{aligned} \sum_{i=1}^n -x_{ij} + (x_{ij} + \frac{1}{\alpha} ) \frac{e^{u_{j}}}{\frac{1}{\alpha} + e^{u_{j}} } &= 0\\ \sum_{i=1}^n x_{ij} &= \sum_{i=1}^n (x_{ij} + \frac{1}{\alpha} ) \frac{e^{u_{j}}}{\frac{1}{\alpha} + e^{u_{j}} } \\ \sum_{i=1}^n x_{ij} \cdot \frac{1}{\alpha} + \sum_{i=1}^n x_{ij} \cdot e^{u_{j}} & = \sum_{i=1}^n x_{ij} \cdot e^{u_{j}} + \frac{n}{\alpha} e^{u_{j}} \\ e^{u_{j}} &= \sum_{i=1}^n x_{ij} /n \\ \exp(u_{j}) &= \bar x_j \Rightarrow u_{j} = \log(\bar x_j) \Rightarrow \operatorname{\textbf u}= \log(\bar \operatorname{\textbf x})\end{aligned}$$ ### Negative Binomial Deviance $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p x_{ij} \log(\frac{x_{ij}}{u_{ij}}) - (x_{ij} + \frac{1}{\alpha}) \log (\frac{1+\alpha x_{ij}}{1+\alpha u_{ij} } ) + \gamma \sum_{i < i'} w_{ii'} \\|\U_{i.} - \U_{i'.} \\|_2\end{aligned}$$ The formulation above is equivalent to $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p - x_{ij} \log u_{ij} + (x_{ij} + \frac{1}{\alpha}) \log (1+\alpha u_{ij} ) + \gamma \sum_{i < i'} w_{ii'} \\|\U_{i.} - \U_{i'.}\\|_2\end{aligned}$$ Let $\operatorname{\textbf u}$ be the fusion vector and $\operatorname{\textbf u}= (u_1,\cdots,u_p)$. The problem above becomes: $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p - x_{ij} \log u_{j} + (x_{ij} + \frac{1}{\alpha}) \log (1+\alpha u_{j} )\end{aligned}$$ Taking derivative, we get: $$\begin{aligned} &\sum_{i=1}^n \frac{-x_{ij}}{u_j} + \frac{ x_{ij} + \frac{1}{\alpha}}{1+\alpha u_{j} } \cdot \alpha = 0 \\ & \sum_{i=1}^n x_{ij} + \alpha u_{j} \sum_{i=1}^n x_{ij} = n u_{j} + \alpha u_{j} \sum_{i=1}^n x_{ij} \\ & u_j = \bar x_j \Rightarrow \operatorname{\textbf u}= \bar \operatorname{\textbf x}\end{aligned}$$ ### Manhattan Distance $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_{i=1}^n \\|\operatorname{\textbf x}_i - \operatorname{\textbf u}_i\\|_1 + \gamma \sum_{i < i'} w_{ii'} \\|\U_{i.} - \U_{i'.}\\|_2\end{aligned}$$ When total fusion, $\U_{i.} = \U_{i'.}$, $\forall i \neq i'$. Let $\U_{i.} = \U_{i'.} = \operatorname{\textbf u}$. We have: $$\begin{aligned} \operatorname{minimize}_{\operatorname{\textbf u}} \hspace{5mm} \sum_{i=1}^n \\|\operatorname{\textbf x}_i - \operatorname{\textbf u}\\|_1\end{aligned}$$ For each $j$, we have: $$\begin{aligned} \operatorname{minimize}_{u_j} \hspace{5mm} \sum_{i=1}^n \\|x_{ij} - u_j\\|_1\end{aligned}$$ We know that $u_j$ is just the median of $x_{ij}$ for each $j$. Binary Data ----------- For binary data, we consider Gecco with Bernoulli log-likelihood, binomial deviance and hinge loss. ### Bernoulli Log-likelihood $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p - x_{ij} u_{ij} + \log(1+e^{u_{ij}}) + \gamma \sum_{i < i'} w_{ii'} \\|\U_{i.} - \U_{i'.}\\|_2\end{aligned}$$ When total fusion, $\operatorname{\textbf u}_{ij} = \operatorname{\textbf u}_{i'j}$, $\forall i \neq i' $. Let $\operatorname{\textbf u}$ be the fusion vector and $\operatorname{\textbf u}= (u_1,\cdots,u_p)$. The problem above becomes: $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p -x_{ij} u_{j} + \log(1+e^{u_{j}})\end{aligned}$$ Taking derivative, we get: $$\begin{aligned} \sum_{i=1}^n -x_{ij} + n \frac{\exp(u_{j})}{1+\exp(u_{j})} = 0 &\Rightarrow \frac{\exp(u_{j})}{1+\exp(u_{j})} = \bar x_j \\ &\Rightarrow \text{logit}^{-1} (u_{j}) = \bar x_j \Rightarrow u_{j} = \text{logit} (\bar x_j) \Rightarrow \operatorname{\textbf u}= \text{logit} (\operatorname{\textbf x})\end{aligned}$$ ### Binomial Deviance $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p - x_{ij} \log{u_{ij}} - (1-x_{ij})\log({1-u_{ij}}) + \gamma \sum_{i < i'} w_{ii'} \\|\U_{i.} - \U_{i'.}\\|_2\end{aligned}$$ Let $\operatorname{\textbf u}$ be the fusion vector and $\operatorname{\textbf u}= (u_1,\cdots,u_p)$. The problem above becomes: $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p - x_{ij} \log{u_{j}} - (1-x_{ij})\log({1-u_{j}})\end{aligned}$$ Taking derivative, we get: $$\begin{aligned} \sum_{i=1}^n \frac{-x_{ij}}{u_j} + \frac{1-x_{ij}}{1-u_{j}} = 0 &\Rightarrow \sum_i^n - x_{ij} (1-u_j) + (1-x_{ij}) u_j = 0 \\ &\Rightarrow \sum_i^n - x_{ij} + u_j = 0 \Rightarrow u_j = \bar x_j \Rightarrow \operatorname{\textbf u}= \bar\operatorname{\textbf x}\end{aligned}$$ ### Hinge Loss $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_{i=1}^n \sum_{j=1}^p \max(0,1-u_{ij} x_{ij}) + \gamma \sum_{i < i'} w_{ii'} \\|\U_{i.} - \U_{i'.} \\|_2\end{aligned}$$ Let $\operatorname{\textbf u}$ be the fusion vector and $\operatorname{\textbf u}= (u_1,\cdots,u_p)$. The problem above becomes: $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_{i=1}^n \sum_{j=1}^p \max(0,1-u_{j} x_{ij}) \end{aligned}$$ For each feature $j$, the problem becomes: $$\begin{aligned} \operatorname{minimize}_{u_j} \hspace{5mm} \sum_{i=1}^n \max(0,1-u_{j} x_{ij}) \end{aligned}$$ Note in hinge loss, $x_{ij} \in \{-1,1\}$. Suppose we have $n_1$ observations for class “1" and $n_2$ observations for class “-1". The problem now becomes $$\begin{aligned} \operatorname{minimize}_{u_j} \hspace{5mm} n_1 \max(0,1-u_{j} ) + n_2 \max(0,1+u_{j} ) \end{aligned}$$ Define $h(t) = n_1 \max(0,1-t ) + n_2 \max(0,1+t )$. We have: $$\begin{aligned} h(t) = \begin{cases} n_1 (1-t) & \text{if} \hspace{2mm} t \leq -1 \\ n_1 (1-t) + n_2 (1+t) & \text{if} \hspace{2mm} -1 < t < 1 \\ n_2 (1+t) & \text{if} \hspace{2mm} t \geq 1 \end{cases}\end{aligned}$$ Clearly, if $n_2 > n_1$ (more “-1"), $h(t)$ is minimized by $t=-1$; if $n_1 > n_2$ (more “1"), $h(t)$ is minimized by $t=1$; if $n_1 = n_2$, $h(t)$ is minimized by any $t$ between $[-1,1]$. Therefore, $u_j$ should be the mode of all observations for feature $j$: $$\begin{aligned} u_j = \text{mode}_i (x_{ij})\end{aligned}$$ Categorical Data ---------------- For categorical data, we consider Gecco with multinomial log-likelihood and deviance. ### Multinomial Log-likelihood $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p \big\{ \sum_{k=1}^{K} -x_{ijk} u_{ijk} + \log(\sum_{k=1}^K e^{u_{ijk}}) \big \} + \gamma \sum_{i < i'} w_{ii'} \bigg \\| \begin{bmatrix} \U_{i.}^{(1)} \\ \vdots \\ \U_{i.}^{(K)} \end{bmatrix} - \begin{bmatrix} \U_{i'.}^{(1)} \\ \vdots \\ \U_{i'.}^{(K)} \end{bmatrix} \bigg \\|_2\end{aligned}$$ When total fusion, $u_{ijk} = u_{i'jk}$, $\forall i \neq i'$. Let $\operatorname{\textbf u}$ be the fusion vector and $\operatorname{\textbf u}_k = (u_{1k},\cdots,u_{pk})$. The problem above becomes: $$\begin{aligned} \operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p \big\{ \sum_{k}^K -x_{ijk} u_{jk} + \log( \sum_{k}^K e^{u_{jk}}) \big\}\end{aligned}$$ Taking derivative with respect to $u_{jk}$, we get: $$\begin{aligned} \sum_{i=1}^n -x_{ijk} + n \frac{\exp(u_{jk})}{ \sum_k^K \exp(u_{jk})} &= 0 \\ u_{jk} = \log \frac{ \bar x_{.jk}} {\sum_k \bar x_{.jk} } & = \text{mlogit}( \bar x_{.jk} )\end{aligned}$$ ### Multinomial Deviance $$\begin{aligned} &\operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p \big\{ \sum_{k=1}^{K} -x_{ijk} \log(u_{ijk}) \big \} + \gamma \sum_{i < i'} w_{ii'} \bigg \\| \begin{bmatrix} \U_{i.}^{(1)} \\ \vdots \\ \U_{i.}^{(K)} \end{bmatrix} - \begin{bmatrix} \U_{i'.}^{(1)} \\ \vdots \\ \U_{i'.}^{(K)} \end{bmatrix} \bigg \\|_2 \\ & \text{subject to} \hspace{5mm} \sum \limits_{k=1}^K u_{ijk} = 1\end{aligned}$$ When total fusion, $\operatorname{\textbf u}_{ijk} = \operatorname{\textbf u}_{i'jk}$, $\forall i \neq i $. Let $\operatorname{\textbf u}$ be the fusion vector and $\operatorname{\textbf u}_k = (u_{1k},\cdots,u_{pk})$. The problem above becomes: $$\begin{aligned} & \operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p \sum_{k}^K -x_{ijk} \log u_{jk} \\ & \text{subject to} \hspace{5mm} \sum \limits_{k=1}^K u_{jk} = 1\end{aligned}$$ We can write the constraint in Lagrangian form: $$\begin{aligned} & \operatorname{minimize}_{\U} \hspace{5mm} \sum_i^n \sum_j^p \sum_{k}^K -x_{ijk} \log u_{jk} + \lambda( \sum \limits_{k=1}^K u_{jk} - 1)\end{aligned}$$ Taking derivative with respect to $u_{jk}$, we get: $$\begin{aligned} \sum_{i=1}^n \frac{-x_{ijk}}{u_{jk}} + \lambda &= 0 \\ \sum_{i=1}^n x_{ijk} = \lambda u_{jk} \end{aligned}$$ We have: $$\begin{aligned} n = \sum_{i=1}^n \sum_{k=1}^K x_{ijk} = \sum_{k=1}^K \sum_{i=1}^n x_{ijk} = \sum_{k=1}^K \lambda u_{jk} = \lambda\end{aligned}$$ Therefore, $$\begin{aligned} u_{jk} = \sum_{i=1}^n x_{ijk} / n \end{aligned}$$ Visualization of Gecco+ for Authors Data {#authorvizone} ======================================== Figure \[viz-selected-feature\] illustrates selected features and cluster assignment for authors data set with one combination of $\alpha$ and $\gamma$. We select meaningful features and achieve satisfactory clustering results. We have already discussed the results and interpretation in detail in Section \[author\]. Multi-omics Data {#genohist} ================ In this section, we show the distribution of data from different platforms in Section \[omics\]. We see that both gene expression data and protein data appear gaussian; Methylation data is between $[0,1]$; miRNA data is highly-skewed. ![Histograms of data from different platforms for multi-omics TCGA data set. Both gene expression data and protein data appear gaussian; Methylation data is proportion-valued; miRNA data is highly-skewed.[]{data-label="fig:hist"}](int_data_hist_all.pdf){width="12cm"} [^1]: Department of Statistics, Rice University, Houston, TX [^2]: Departments of Electrical and Computer Engineering, Statistics, and Computer Science, Rice University, Houston, TX [^3]: Jan and Dan Duncan Neurological Research Institute, Baylor College of Medicine, Houston, TX
--- abstract: 'In this paper we prove that periodic boundary-value problems (BVPs) for delay differential equations are locally equivalent to finite-dimensional algebraic systems of equations. We rely only on regularity assumptions that follow those of the review by Hartung et al. (2006). Thus, the equivalence result can be applied to differential equations with state-dependent delays (SD-DDEs), transferring many results of bifurcation theory for periodic orbits to this class of systems. We demonstrate this by using the equivalence to give an elementary proof of the Hopf bifurcation theorem for differential equations with state-dependent delays. This is an alternative and extension to the original Hopf bifurcation theorem for SD-DDEs by Eichmann (2006).' author: - Jan Sieber bibliography: - 'delay.bib' title: 'Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations' --- Introduction {#sec:intro} ============ If a dynamical system is described by a differential equation where the derivative at the current time may depend on states in the past one speaks of delay differential or, more generally, functional differential equations (FDEs). A reasonably general formulation of an autonomous dynamical system of this type looks like this: $$\label{eq:ivp} \dot x(t)=f(x_t,\mu)$$ where $\tau>0$ is an upper bound for the delay. On the right-hand side $f$ is a functional, mapping $C^0([-\tau,0];{\mathbb{R}}^n)$ (the space of continuous functions on the interval $[-\tau,0]$ with values in ${\mathbb{R}}^n$) into ${\mathbb{R}}^n$. The dependent variable $x$ is a function on $[-\tau,T_{\max})$ for some $T_{\max}>0$, and $x_t$ is the current function segment: $x_t(s)=x(t+s)$ for $s\in[-\tau,0]$ such that $x_t\in C^0([-\tau,0];{\mathbb{R}}^n)$. The second argument $\mu\in{\mathbb{R}}^\nu$ is a system parameter. For a system of the form one would have to prescribe a continuous function $x$ on the interval $[-\tau,0]$ as the initial value and then extend $x$ toward time $T_{\max}$ (see textbooks on functional differential equations such as [@DGLW95; @HL93; @S89]). A long-standing problem with certain types of FDEs is that they do not fit well into the general framework of smooth infinite-dimensional dynamical system theory. The problem occurs whenever the functional $f$ invokes the evaluation operation in a non-trivial way, that is, for example, if one has a state-dependent delay. A prototypical caricature example would be the functional $$\begin{aligned} \label{eq:example} f:&\ U\times{\mathbb{R}}\mapsto {\mathbb{R}}\mbox{,}& f(x,\mu)&=\mu-x(-x(0))\mbox{,} \intertext{where $U=\{x\in C^0([-\tau,0];{\mathbb{R}}): 0<x(0)<\tau\}$ is an open set in $C^0([-\tau,0];{\mathbb{R}})$. The corresponding FDE is} \dot x(t)&=\mu-x(t-x(t))\mbox{.}\label{eq:examplede}\end{aligned}$$ Here, $f$ evaluates its first argument $x$ at a point that itself depends on $x$. We restrict ourselves to solutions $x$ of with $x(t)\in(0,\tau)$ for $t\geq0$ to avoid problems with causality and to limit the maximal delay to $\tau$ (always keeping $x_t$ in $U$). The difficulty with stems from the fact that $f$ as a map is only as smooth as its argument $x$. Specifically, the derivative of $f$ with respect to its first argument in this example exists only for $x\in C^1([-\tau,0];{\mathbb{R}})$ (the space of all continuously differentiable functions on $[-\tau,0]$): $$\label{eq:exdf} \begin{split} \partial^1 f&:C^1([-\tau,0];{\mathbb{R}})\times{\mathbb{R}}\times C^1([-\tau,0];{\mathbb{R}})\mapsto{\mathbb{R}}\mbox{,}\\ \partial^1f&(x,\mu,y)=x'(-x(0))\,y(0)-y(-x(0))\mbox{.} \end{split}$$ So, if we choose $U$ as the phase space for initial-value problems (IVPs) in example then the functional $f$ is not differentiable for all elements of $U$. In fact, it is not even locally Lipschitz continuous in $U$. Indeed, Winston [@W70] gave an example of an initial condition in $U$ for (with $\mu=0$ and $\tau>1$), for which the IVP did not have a unique solution. This counterexample is not surprising since the right-hand side $f$ does not fit into the framework that the textbooks [@DGLW95; @HL93; @S89] assume to be present. A result of Walther [@W04] rescues IVPs with state-dependent delays (such as ) by restricting the phase space in general to the closed submanifold $C_c$ of $C^1([-\tau,0];{\mathbb{R}}^n)$: $$C_c=\{x\in C^1([-\tau,0];{\mathbb{R}}^n): x'(0)=f(x)\}\mbox{.}$$ Walther [@W04] could prove the existence of a semiflow inside this manifold that is continuously differentiable with respect to its initial conditions. However, this result is restricted to a single degree of differentiability. Results about higher degrees of smoothness are lacking for the semiflow [@HKWW06]. A typical task one wants to perform for problems of type , or example , is bifurcation analysis of equilibria and periodic orbits. Equilibria are solutions $x$ of that are constant in time, and periodic orbits are solutions $x\in C^1({\mathbb{R}};{\mathbb{R}}^n)$ of that satisfy $x(t+T)=x(t)$ for some $T>0$ and all $t\in{\mathbb{R}}$. Equilibria of the general FDE can be determined by finding the solutions $(p,\mu)\in{\mathbb{R}}^n\times{\mathbb{R}}^\nu$ of the algebraic system of equations $$\label{eq:eqsys} 0=f(E_0p,\mu)$$ where $E_0$ is the trivial embedding $$E_0:{\mathbb{R}}^n\mapsto C^0([-\tau,0];{\mathbb{R}}^n)\mbox{,\quad} [E_0p](s)=p\mbox{\quad for all $s\in[-\tau,0]$.}$$ We observe that, even though the FDE is an infinite-dimensional system, its equilibria can be found as roots of the finite-dimensional system of algebraic equations. Moreover, the regularity problems of the semiflow do not affect : in the example , the algebraic equation reads $0=\mu-p$, which is smooth to arbitrary degree, and can be solved even for negative $\mu$ (near equilibria with $\mu=p<0$ the semiflow does not exist). In this paper we establish a system similar to , but for periodic orbits: we find a finite-dimensional algebraic system of equations that does not suffer from the regularity problems affecting the semiflow, and an equivalence between solutions of this algebraic system and periodic orbits of . In comparison, for ordinary differential equations (ODEs) of the form $\dot x(t)=f(x(t),\mu)$ with a smooth $f:{\mathbb{R}}^n\times{\mathbb{R}}^\nu\mapsto{\mathbb{R}}^n$, the fact that the problem of finding periodic orbits can be reduced to algebraic root-finding is well known [@GH83]. For example, in ODEs one can use the algebraic system $0=X(T;p,\mu)-p$ where $t\mapsto X(t;p,\mu)$ is the trajectory defined by the IVP starting from $p\in{\mathbb{R}}^n$ and using parameter $\mu\in{\mathbb{R}}^\nu$. A central notion in the construction of the equivalent algebraic system for periodic orbits of FDEs are periodic boundary-value problems (BVPs) for FDEs on the interval $[-\pi,\pi]$ with periodic boundary conditions (which we identify with the unit circle ${\mathbb{T}}$). Periodic orbits of can then be found as solutions of periodic BVPs. If one wants to make the equivalence result useful in practical applications, one has to find a regularity (smoothness) condition on the right-hand side $f$ that includes the class of state-dependent delay equations reviewed in [@HKWW06], while still ensuring that it is possible to prove the existence of an equivalent algebraic system. We use exactly the same condition as used by Walther in [@W04] to prove the existence of a continuously differentiable semiflow, the so-called extendable continuous differentiability (originally introduced as “almost Frech[é]{}t differentiability” in [@MNP94]), which implies a restricted form of local Lipschitz continuity. We generalize restricted continuous differentiability to higher degrees of restricted smoothness (which we call $EC^k$ smoothness) in a similar fashion as Krisztin [@K03] did for the proof of the existence and smoothness of local unstable manifolds of equilibria. Our definition of $EC^k$ smoothness is comparatively simple to state and check, and lends itself easily to inductive proofs. After introducing the notation for periodic BVPs and $EC^k$ smoothness we state the main result, an equivalence theorem between periodic BVPs and algebraic systems of equations in Section \[sec:theorem\]. The equivalence theorem reduces statements about existence and smooth dependence of periodic orbits of FDEs to root-finding problems of smooth algebraic equations. The result is weaker than the corresponding results for equilibria of FDEs and for periodic orbits of ODEs because the equivalence is only valid locally. For any given periodic function $x_0$ with Lipschitz continuous time derivative we construct an algebraic system that is equivalent to the periodic BVP in a sufficiently small open neighborhood of $x_0$. However, the result is still useful, as we then demonstrate in Section \[sec:po\]. We apply the equivalence theorem in the vicinity of equilibria for which the linearization of has eigenvalues on the imaginary axis (for example, near $x_0=\mu=\pi/2$ in example ) to prove the Hopf Bifurcation Theorem. The equivalence theorem reduces the proof of the Hopf Bifurcation Theorem to an application of the Algebraic Branching Lemma [@AG79]. This provides a complete proof for the Hopf Bifurcation Theorem for FDEs with state-dependent delays, including the regularity of the emerging periodic orbits. We discuss differences to the first version of the proof by Eichmann [@E06] and the approach of Hu and Wu [@HW10] in Section \[sec:po\]. The equivalence is applicable in other scenarios where one would expect branching of periodic solutions. Examples are period doublings, the branching from periodic orbits with resonant Floquet multipliers on the unit circle in Arnol’d tongues, and branching scenarios in FDEs with symmetries. We give a tentative list of straightforward applications and generalizations of the equivalence theorem in the conclusion (Section \[sec:conc\]). We note that the theorem stated in Section \[sec:theorem\] differs from statements about numerical approximations. As part of the theorem we also provide a map $X$ that maps the root of the algebraic system back into a function space to give the exact solution of the periodic BVP, and a projection $P$ that maps functions to finite-dimensional vectors (and, hence, periodic orbits to roots of the algebraic system). In numerical methods one typically has to increase the dimension of the algebraic system in order to get more and more accurate approximations of the true solution whereas the dimension of the algebraic system constructed in Section \[sec:theorem\] is finite. The Equivalence Theorem {#sec:theorem} ======================= This section states the assumptions and conclusions of the main result of the paper, the Equivalence Theorem stated in Theorem \[thm:main\]. Before doing so, we introduce some basic notation (function spaces on intervals with periodic boundary conditions and projections onto the leading Fourier modes). ### Periodic BVPs {#periodic-bvps .unnumbered} We first state precisely what we mean by periodic BVP and introduce the usual hierarchy of continuous, continuously differentiable and Lipschitz continuous functions on the compact interval $[-\pi,\pi]$ with periodic boundary conditions. For $j\geq0$ we will use the notation $C^j({\mathbb{T}};{\mathbb{R}}^n)$ for the spaces of all functions $x$ on the interval $[-\pi,\pi]$ with continuous derivatives up to order $j$ (including order $0$ and $j$) satisfying the periodic boundary conditions $x^{(l)}(-\pi)=x^{(l)}(\pi)$ for $l=0\ldots j$. Elements of $C^0({\mathbb{T}};{\mathbb{R}}^n)$ are continuous and satisfy $x(-\pi)=x(\pi)$. For derivatives of order $j>0$, $x^{(j)}(-\pi)$ is the right-sided $j$th derivative of $x$ in $-\pi$, and $x^{(j)}(\pi)$ is the left-sided $j$th derivative of $x$ in $\pi$. The norm in $C^j({\mathbb{T}};{\mathbb{R}}^n)$ is $$\\|x\\|_j=\max_{t\in[-\pi,\pi]}\left\{\|x(t)\|,\|x'(t)\|,\ldots,\|x^{(j)}(t)\|\right\}\mbox{.}$$ We can extend any function $x$ in $C^j({\mathbb{T}};{\mathbb{R}}^n)$ to arguments in ${\mathbb{R}}$ by defining $x(t)=x(t-2k\pi)$ where $k$ is an integer chosen such that $-\pi\leq t-2k\pi<\pi$ (we will write $t_{{\operatorname{mod}}[-\pi,\pi)}$ later). Thus, every element of $C^j({\mathbb{T}};{\mathbb{R}}^n)$ is also an element of $BC^j({\mathbb{R}};{\mathbb{R}}^n)$, the space of functions with bounded continuous derivatives up to order $j$ on the real line. We use the notation $t\in{\mathbb{T}}$ for arguments $t$ of $x$, and also call ${\mathbb{T}}$ the unit circle. This make sense because the parametrization of the unit circle by angle provides a cover, identifying ${\mathbb{T}}$ with ${\mathbb{R}}$ where we use $[-\pi,\pi)$ as the fundamental interval. Additional useful function spaces are the space of Lipschitz continuous functions and, correspondingly, spaces with Lipschitz continuous derivatives, denoted by $C^{j,1}({\mathbb{T}};{\mathbb{R}}^n)$, which are equipped with the norm $$\label{eq:cj1def} \\|x\\|_{j,1}=\max\left\{\\|x\\|_j, \sup_{ \begin{subarray}{c} t,s\in{\mathbb{R}}\\[0.2ex] t\neq s \end{subarray} }\frac{\|x^{(j)}(s)-x^{(j)}(t)\|}{\|s-t\|} \right\}$$ ($x^{(0)}(t)$ refers to $x(t)$). Note that we used the notation $t,s\in{\mathbb{R}}$ in the index of the supremum, as we can apply arbitrary arguments in ${\mathbb{R}}$ to a function $x\in C^0({\mathbb{T}};{\mathbb{R}}^n)$ by considering it as an element of $BC^0({\mathbb{R}};{\mathbb{R}}^n)$, as explained above. We use the same notation ($C^j(J;{\mathbb{R}}^n)$ and $C^{j,1}(J;{\mathbb{R}}^n)$) also for functions on an arbitrary compact interval $J\subset{\mathbb{R}}$ without periodic boundary conditions (and one-sided derivatives at the boundaries). As any function $x\in C^j({\mathbb{T}};{\mathbb{R}}^n)$ is also an element of $BC^j({\mathbb{R}};{\mathbb{R}}^n)$, it is also an element of $C^j(J;{\mathbb{R}}^n)$ for any compact interval $J$ (and the norm of the embedding operator equals unity). On the function spaces $C^j({\mathbb{T}};{\mathbb{R}}^n)$ we define the time shift operator $$\Delta_t:C^j({\mathbb{T}};{\mathbb{R}}^n)\mapsto C^j({\mathbb{T}};{\mathbb{R}}^n)\mbox{,\qquad} [\Delta_tx](s)=x(t+s)\mbox{.}$$ The operator $\Delta_t$ is linear and has norm $1$ in all spaces $C^j({\mathbb{T}};{\mathbb{R}}^n)$. Similarly, $\Delta_t$ maps also $C^{j,1}({\mathbb{T}};{\mathbb{R}}^n)\mapsto C^{j,1}({\mathbb{T}};{\mathbb{R}}^n)$, and has unit norm there as well. Let $f$ be a continuous functional on the space of continuous periodic functions, that is, $$f:C^0({\mathbb{T}};{\mathbb{R}}^n)\mapsto{\mathbb{R}}^n\mbox{.}$$ The right-hand side $f$, together with the shift $\Delta_t$, creates an operator in $C^0({\mathbb{T}};{\mathbb{R}}^n)$, defined as $$\begin{aligned} \label{eq:fdef} F&:C^0({\mathbb{T}};{\mathbb{R}}^n)\mapsto C^0({\mathbb{T}};{\mathbb{R}}^n) & [F(x)](t)&=f(\Delta_tx)\mbox{.}\end{aligned}$$ The operator $F$ is invariant with respect to time shift by construction: $F(\Delta_tx)=\Delta_tF(x)$. We consider autonomous periodic boundary-value problems for differential equations where $f$ is the right-hand side: $$\label{eq:perbvp} \begin{split} \dot x(t)&=f(\Delta_tx)=F(x)(t)\mbox{.} \end{split}$$ A function $x\in {\mathbb{C}}^1({\mathbb{T}};{\mathbb{R}}^n)$ is a solution of if $x$ satisfies equation for all $t\in{\mathbb{T}}$ (for each $t\in{\mathbb{T}}$ equation is an equation in ${\mathbb{R}}^n$). In contrast to the introduction we do not expressly include a parameter $\mu$ as an argument of $f$. This does not reduce generality as we will explain in Section \[sec:po\]. The main result, the Equivalence Theorem \[thm:main\], will be concerned with equivalence of the periodic BVP to an algebraic system of equations. The notion of the shift $\Delta_t$ on the unit circle and the operator $F$, combining $f$ with the shift, is specific to periodic BVPs such that the BVP looks different from the IVP in the introduction. Several results stating how regularity of $f$ transfers to regularity of $F$ are collected in Appendix \[sec:basicprop\]. ### Definition of $EC^k$ smoothness and local (restricted) $EC$ Lipschitz continuity {#definition-of-eck-smoothness-and-local-restricted-ec-lipschitz-continuity .unnumbered} Continuity of the functional $f$ is not strong enough as a condition to prove the Equivalence Theorem. Rather, we need a notion of smoothness for $f$. However, as explained in the introduction, we cannot assume that $f$ is continuously differentiable with degree $k\geq1$, if we want to include examples such as $f(x)=-x(-x(0))$ (see FDE for $\mu=0$) into the class under consideration. The review by Hartung et al. [@HKWW06] observed the following typical property of functionals $f$ appearing in equations of type : the derivative ${\partial}^1f(x)$ of $f$ in $x$ as a linear map from $C^1({\mathbb{T}};{\mathbb{R}}^n)$ into ${\mathbb{R}}^n$ can be extended to a bounded linear map from $C^0({\mathbb{T}};{\mathbb{R}}^n)$ into ${\mathbb{R}}^n$, and the mapping $${\partial}^1f: C^1({\mathbb{T}};{\mathbb{R}}^n)\times C^0({\mathbb{T}};{\mathbb{R}}^n) \mapsto {\mathbb{R}}^n \mbox{\quad defined by\quad} (x,y)\mapsto{\partial}^1 f(x,y)$$ is continuous as a function of both arguments. In other words, the derivative of $f$ may depend on $x'$ but not on $y'$. For the example $f(x)=-x(-x(0))$ this is true (see ). Most of the fundamental results establishing basic dynamical systems properties for FDEs with state-dependent delay in [@HKWW06] rest on this extendability of $\partial^1f$. We also rely strongly on this notion of extendable continuous differentiability. The precise definition is given below in Definition \[def:extdiff\]. In this definition we permit the argument range $J$ to be any compact interval or ${\mathbb{T}}$. We use the notation of a subspace of higher-order continuous differentiability not only for $C^j(J;{\mathbb{R}}^n)$ but also for products of such spaces in a natural way. Say, if $$\label{eq:Dspacedef} D=C^{k_1}(J;{\mathbb{R}}^{m_1})\times\ldots\times C^{k_\ell}(J;{\mathbb{R}}^{m_\ell}) \mbox{,}$$ where $\ell\geq 1$, and $k_j\geq0$ and $m_j\geq1$ are integers, and denoting the natural maximum norm on the product $D$ by $$\\|x\\|_D=\\|(x_1,\ldots,x_\ell)\\|_D=\max_{j\in\{1,\ldots,\ell\}}\\|x_j\\|_{k_j}\mbox{,}$$ then for integers $r\geq0$ the space $D^r$ is defined in the natural way as $$\begin{aligned} D^r&=C^{k_1+r}(J;{\mathbb{R}}^{m_1})\times\ldots\times C^{k_\ell+r}(J;{\mathbb{R}}^{m_\ell}) \mbox{,\quad with \ }\\ \\|x\\|_{D,r}&=\max_{ \begin{subarray}{c} 0\leq j\leq r\\[0.2ex] 1\leq i\leq \ell \end{subarray} } \\|x_i^{(j)}\\|_{k_i}\mbox{.}\end{aligned}$$ For the simplest example, $D=C^0(J;{\mathbb{R}}^n)$, $D^k$ is $C^k(J;{\mathbb{R}}^n)$. If $J={\mathbb{T}}$ then the time shift $\Delta_t$ extends naturally to products of spaces: $$\Delta_tx=(\Delta_tx_1,\ldots,\Delta_tx_\ell) \mbox{\quad for $x=(x_1,\ldots,x_\ell)\in D$.}$$ \[def:extdiff\] Let $D$ be a product space of the type , and let $f:D\mapsto{\mathbb{R}}^n$ be continuous. We say that $f$ has an extendable continuous derivative if there exists a map $\partial^1f$ $$\partial^1f:D^1\times D\mapsto {\mathbb{R}}^n$$ that is continuous in both arguments $(u,v)\in D^1\times D$ and linear in its second argument $v\in D$, such that for all $u\in D^1$ $$\label{eq:ass:contdiff:j} \lim_{ \begin{subarray}{c} v\in D^1\\[0.2ex] \\|v\\|_{D,1}\to 0 \end{subarray} } \frac{\|f(u+v)-f(u)-\partial_1f(u,v)\|}{\\|v\\|_{D,1}}=0\mbox{.}$$ We say that $f$ is $k$ times continuously differentiable in this extendable sense if the map $\partial^kf$, recursively defined as $\partial^kf=\partial^1[\partial^{k-1}f]$, exists and satisfies the limit condition for $\partial^{k-1}f$. We abbreviate this notion by saying that $f$ is $EC^k$ smooth in $D$. The limit in is a limit in ${\mathbb{R}}$. For $k=1$ the definition is identical to property (S) in the review [@HKWW06], one of the central assumptions for fundamental results on the semiflow. Extendable continuous differentiability requires the derivative to exist only in points in $D^1$ and with respect to deviations in $D^1$, but it demands that the derivative must extend in its second argument to $D$ ($\partial^1f$ is linear in its second argument). This is the motivation for calling this property extendable continuous differentiability. The definition of $EC^k$ smoothness for $k>1$ uses the notation that a functional (say, $\partial^1f$) of two arguments (say, $u\in D^1$ and $v\in D$) for which one would write $\partial^1f(u,v)$, is also a functional of a single argument $w=(u,v)\in D^1\times D$, such that one can also write $\partial^1f(w)$. When using this notation we observe that the space $D^1\times D$ is again a product of type such that $\partial^1f$ is again a functional of the same structure as $f$. For example, let us consider the functional $f:x\mapsto -x(-x(0))$ from example (setting $\mu=0$). The functional is well defined and continuous also on $D=C^0({\mathbb{T}};{\mathbb{R}})$. Moreover, $f$ is $EC^k$ smooth in $D$ to arbitrary degree $k$. Its first two derivatives are: $$\begin{aligned} &\partial^1f: C^1({\mathbb{T}};{\mathbb{R}})\times C^0({\mathbb{T}};{\mathbb{R}})\mbox{,}\nonumber\\ &\partial^1f(u,v)=u'(-u(0))\,v(0)-v(-u(0))\mbox{, and}\label{eq:exampledf1}\\ &\partial^2f: \left[C^2({\mathbb{T}};{\mathbb{R}})\times C^1({\mathbb{T}};{\mathbb{R}})\right]\times \left[C^1({\mathbb{T}};{\mathbb{R}})\times C^0({\mathbb{T}};{\mathbb{R}})\right]\mbox{,}\nonumber\\ & \begin{aligned} \partial_2f(u,v,w,x)=&-u''(-u(0))\,w(0)\,v(0)+u'(-u(0))\,x(0)\\ &+w'(-u(0))\,v(0)- v'(-u(0))\,w(0)-x(-u(0))\mbox{.} \end{aligned}\label{eq:exampledf2}\end{aligned}$$ As one can see, the first derivative $\partial^1f$ has the same structure as $f$ itself if we replace $D=C^0({\mathbb{T}};{\mathbb{R}}^n)$ by $D^1\times D$. So, it is natural to apply the definition again to $\partial^1f$ on the space $D^1\times D$. Assuming that $f$ is $EC^1$ smooth on $C^0(J;{\mathbb{R}}^n)$ implies classical continuous differentiability of $f$ as a map from $C^1(J;{\mathbb{R}}^n)$ into ${\mathbb{R}}^n$ and is, thus, strictly stronger than assuming that $f$ is continuously differentiable on $C^1({\mathbb{T}};{\mathbb{R}}^n)$. Since every element of $C^j({\mathbb{T}};{\mathbb{R}}^n)$ is also an element of $C^j(J;{\mathbb{R}}^n)$ for any compact interval $J$ (and the embedding operator has unit norm), any $EC^k$ smooth functional $f:C^0(J;{\mathbb{R}}^m)\mapsto{\mathbb{R}}^n$ is also a $EC^k$ smooth functional from $C^0({\mathbb{T}};{\mathbb{R}}^m)$ into ${\mathbb{R}}^n$. It is worth comparing Definition \[def:extdiff\] with the definition for higher degree of regularity used by Krisztin in [@K03]. With Definition \[def:extdiff\] the $k$th derivative has $2^k$ arguments. In contrast to this, the $k$th derivative as defined in [@K03] has only $k+1$ arguments (the first argument is the base point, and the derivative is a $k$-linear form in the other $k$ arguments). The origin of this difference can be understood by looking at the example $f(x)=-x(-x(0))$ and its derivatives in –. Krisztin’s definition applied to the second derivative does not include the derivative of $\partial^1f$ with respect to the linear second argument $v$ (as is often convention, because it is the identity). One would obtain the second derivative according to Krisztin’s definition by setting $x=0$ in . Indeed, the terms containing the argument $x$ in are simply $\partial^1f(u,x)$, as one expects when differentiating $\partial^1f(u,v)$ with respect to $v$, calling the deviation $x$. While in practical examples it is often more economical to use the compact notation with $k$-forms, inductive proofs of higher-order differentiability using the full derivative only require the notion of at most bi-linear forms, making them less complex. If $f$ is $EC^1$ smooth then it automatically satisfies a restricted form of local Lipschitz continuity [@HKWW06], which we call local $EC$ Lipschitz continuity: \[def:loclip\] We say that $f:C^0({\mathbb{T}};{\mathbb{R}}^n)\mapsto {\mathbb{R}}^n$ is locally $EC$ Lipschitz continuous if for every $x_0\in C^1({\mathbb{T}};{\mathbb{R}}^n)$ there exists a neighborhood $U(x_0)\subset C^1({\mathbb{T}};{\mathbb{R}}^n)$ and a constant $K$ such that $$\label{eq:loclip} \|f(y)-f(z)\|\leq K\\|y-z\\|_0$$ holds for all $y$ and $z$ in $U(x_0)$. That $EC^1$ smoothness implies local $EC$ Lipschitz continuity has been shown, for example, in [@W04] (but see also Lemma \[thm:flip\] in Appendix \[sec:basicprop\]). Note that the estimate uses the $\\|\cdot\\|_0$-norm for the upper bound. This is a sharper estimate than one would obtain using the expected $\\|\cdot\\|_1$-norm. The constant $K$ may depend on the derivatives of the elements in $U(x_0)$ though. For example, for $f(x)=-x(-x(0))$ as in with $\mu=0$, one would have the estimate $$\left\|f(x+y)-f(x)\right\|\leq \left[1+\\|x'\\|_0\right]\\|y\\|_0 \mbox{\quad such that\quad} K\leq1+\max_{x\in U(x_0)}\\|x\\|_1\mbox{.}$$ This means that in this example, the neighborhood $U(x_0)$ can be chosen arbitrarily large as long as it is bounded in $C^1({\mathbb{T}};{\mathbb{R}}^n)$. The following lemma states that we can extend the neighborhood $U(x)$ in Definition \[def:loclip\] into the space of Lipschitz continuous functions ($C^{0,1}$ instead of $C^1$) and include time shifts (which possibly increases the bound $K$). \[thm:loclip\] Let $f$ be locally $EC$ Lipschitz continuous, and let $x_0$ be in $C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)$. Then there exists a bounded neighborhood $U(x_0)\subset C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)$ and a constant $K$ such that $$\|f(\Delta_ty)-f(\Delta_tz)\|\leq K\\|\Delta_ty-\Delta_tz\\|_0=K\\|y-z\\|_0$$ holds for all $y$ and $z$ in $U(x_0)$, and for all $t\in{\mathbb{T}}$. Thus, $ \\|F(y)-F(z)\\|_0\leq K\\|y-z\\|_0 $ for all $y$ and $z$ in $U(x_0)$. Recall that $F(x)(t)=f(\Delta_tx)$. See Lemma \[thm:flip\] and Lemma \[thm:Flipbound\] in Appendix \[sec:basicprop\] for the proof of Lemma \[thm:loclip\]. A consequence of Lemma \[thm:loclip\] is that the time derivative of a solution $x_0$ of the periodic BVP is also Lipschitz continuous (in time): if $\dot x_0(t)=f(\Delta_tx_0)$ then there exists a constant $K$ such that $$\label{eq:lipx0t} \\|x_0'(t)-x_0'(s)\\|_0\leq K\|t-s\|$$ Thus, $x_0\in C^{1,1}({\mathbb{T}};{\mathbb{R}}^n)$. This follows from Lemma \[thm:loclip\] by inserting $\Delta_tx_0$ and $\Delta_sx_0$ for $y$ and $z$ and using that $x_0'(t)=f(\Delta_tx_0)$ (it is enough to show for $\|t-s\|$ small). ### Projections onto subspaces spanned by Fourier modes {#projections-onto-subspaces-spanned-by-fourier-modes .unnumbered} The variables of the algebraic system in the Equivalence Theorem will be the coefficients of the first $N$ Fourier modes (where $N$ will be determined as sufficiently large later) of elements of $C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)$ (the space of Lipschitz continuous functions on ${\mathbb{T}}$). Consider the functions on ${\mathbb{T}}$ $$b_0=t\mapsto\frac{1}{2}\mbox{,\quad}b_k=t\mapsto\cos(kt) \mbox{,\quad}b_{-k}=t\mapsto\sin(kt)$$ for $k=1,\ldots,\infty$ (which is the classical Fourier basis of ${\mathbb{L}}^2({\mathbb{T}};{\mathbb{R}})$). For any $m\geq1$ we define the projectors and maps $$\label{eq:proj} \begin{aligned} P_N&:C^j({\mathbb{T}};{\mathbb{R}}^m)\mapsto C^j({\mathbb{T}};{\mathbb{R}}^m)\mbox{,}& [P_Nx](t)_i&=\sum_{k=-N}^N \left[\frac{1}{\pi}\int_{-\pi}^\pi b_k(s)x_i(s){\mathop{}\!\mathrm{d}}s\right]\,b_k(t)\mbox{,} \allowdisplaybreaks\\ Q_N&={I}-P_N\mbox{,}\\ E_N&:{\mathbb{R}}^{m\times (2N+1)}\mapsto C^j({\mathbb{T}};{\mathbb{R}}^m)\mbox{,}& [E_Np](t)_i&=\sum_{k=-N}^Np_{i,k}b_k(t)\mbox{,}\allowdisplaybreaks\\ R_N&:C^j({\mathbb{T}};{\mathbb{R}}^m)\mapsto {\mathbb{R}}^{m\times(2N+1)}\mbox{,} & [R_Nx]_{i,k}\ \ &=\frac{1}{\pi}\int_{-\pi}^\pi b_k(s)x_i(s){\mathop{}\!\mathrm{d}}s\mbox{,} \allowdisplaybreaks\\ L_{\phantom{N}}&:C^j({\mathbb{T}};{\mathbb{R}}^m)\mapsto C^j({\mathbb{T}};{\mathbb{R}}^m)\mbox{,}& [Lx](t)\ \ \,&=\int_0^t x(s)- R_0x\,{\mathop{}\!\mathrm{d}}s=\int_0^t Q_0[x](s){\mathop{}\!\mathrm{d}}s\mbox{.} \end{aligned}$$ The projector $P_N$ projects a periodic function onto the subspace spanned by the first $2N+1$ Fourier modes, and $Q_N$ is its complement. The map $E_N$ maps a vector $p$ of $2N+1$ Fourier coefficients (which are each vectors of length $n$ themselves) to the periodic function that has these Fourier coefficients. The map $R_N$ extracts the first $2N+1$ Fourier coefficients from a function. The simple relation $P_N=E_NR_N$ holds. The vector $R_0x$ is the average of a function $x$, and $Q_0$ subtracts the average from a periodic function. The operator $L$ takes the anti-derivative of a periodic function after subtracting its average (to ensure that $L$ maps back into the space of periodic functions). In all of the definitions the degree $j$ of smoothness of the vector space $C^j$ can be any non-negative integer. The operator $L$ not only maps $C^j$ back into itself, but it maps $C^j({\mathbb{T}};{\mathbb{R}}^m)$ into $C^{j+1}({\mathbb{T}};{\mathbb{R}}^m)$. We do not attach an index $m$ to the various maps to indicate how many dimensions the argument and, hence, the value has because there is no room for confusion: for example, if $x\in C^0({\mathbb{T}};{\mathbb{R}}^2)$ then $P_Nx\in C^0({\mathbb{T}};{\mathbb{R}}^2)$ such that we use the same notation $P_Nx$ for $x:{\mathbb{T}}\mapsto{\mathbb{R}}^m$ with arbitrary $m$. Similarly, we apply all maps also on product spaces $D$ of the type $C^{k_1}({\mathbb{T}};{\mathbb{R}}^{m_1})\times\ldots\times C^{k_\ell}({\mathbb{T}};{\mathbb{R}}^{m_\ell})$ introduced in Equation by applying the maps element-wise. For example, $$\begin{aligned} P_Nx&=(P_Nx_1,\ldots,P_Nx_\ell) &&\mbox{for $x=(x_1,\ldots,x_\ell)\in D$,}\\ E_Np&=(E_Np_1,\ldots,E_Np_\ell) &&\mbox{for $p=(p_1,\ldots,p_\ell)\in {\mathbb{R}}^{m_1\times(2N+1)}\times\ldots\times{\mathbb{R}}^{m_\ell\times(2N+1)}$.}\end{aligned}$$ ### Equivalent integral equation {#sec:inteq .unnumbered} We note the fact that a function $x\in C^1({\mathbb{T}};{\mathbb{R}}^n)$ solves the periodic BVP $\dot x(t)=f(\Delta_tx)=F(x)(t)$ if and only if it satisfies the equivalent integral equation $$\label{eq:inteq} x(t)=x(0)+\int_0^tF(x)(s){\mathop{}\!\mathrm{d}}s\mbox{\quad for all $t\in{\mathbb{T}}$.}$$ For each $t\in{\mathbb{T}}$, Equation is an equation in ${\mathbb{R}}^n$. In particular, the term $x(0)$ is in ${\mathbb{R}}^n$. Thus, the integral equation is very similar to the corresponding integral equation used in the proof of the Picard-Lindel[ö]{}f Theorem for ODEs [@CL55]. This is in contrast to the abstract integral equations used by Diekmann et al. [@DGLW95] to construct unique solutions to IVPs, in which equality at every point in time is an equality in function spaces. It is the similarity of to its ODE equivalent that makes the reduction of periodic BVPs to finite dimensional algebraic equations possible. One minor problem is that the Picard iteration for cannot be expected to converge. In fact, the integral term $\int_0^tF(x)(s){\mathop{}\!\mathrm{d}}s$ does not even map back into the space $C^0({\mathbb{T}};{\mathbb{R}}^n)$ of periodic functions, even if $x$ is in $C^0({\mathbb{T}};{\mathbb{R}}^n)$. However, a simple algebraic manipulation using the newly introduced maps $L$, $P_N$, $Q_N$, $E_N$ and $R_N$ removes this problem (remember that $F(x)(t)=f(\Delta_tx)$): \[thm:split\] Let $N\geq0$ be an arbitrary integer. A function $x\in C^0({\mathbb{T}};{\mathbb{R}}^n)$ and a vector $p\in{\mathbb{R}}^{n\times(2\,N+1)}$ satisfy $$\begin{aligned} \dot x(t)&=f(\Delta_tx)\mbox{\quad and\quad} p=R_Nx\mbox{,}\label{eq:parbvp}\\ \intertext{if and only if they satisfy the system} x&=E_Np+Q_NLF(x)\mbox{,}\label{eq:fixp:intro}\\ 0&=R_N\left[P_0F(x)+Q_0\left(E_Np-P_NLF(x)\right)\right] \label{eq:lowmodes:intro}\mbox{.} \end{aligned}$$ Note that the map $R_N$ extracts the lowest $2N+1$ Fourier coefficients from a periodic function. Equation can be viewed as a fixed-point equation in $C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)$, parametrized by $p$. We will apply the Picard iteration to this fixed-point equation instead of . Equation is an equation in ${\mathbb{R}}^{n\times(2\,N+1)}$. If the Picard iteration converges then the fixed-point equation can be used to construct (for sufficiently large $N$) a map $X:U\subset{\mathbb{R}}^{n\times(2\,N+1)}\mapsto C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)$, which maps the parameter $p$ to its corresponding fixed point $x$. Inserting this fixed point $x=X(p)$ into turns into a system of $n\times(2\,N+1)$ algebraic equations for the $n\times(2\,N+1)$-dimensional variable $p$, making the periodic BVP for $x$ equivalent to an algebraic system for its first $2N+1$ Fourier coefficients, $p$. The proof of Lemma \[thm:split\] is simple algebra, see Section \[sec:splitproof\]. ### Statement of the Equivalence Theorem {#statement-of-the-equivalence-theorem .unnumbered} Using the Splitting Lemma \[thm:split\] we can now state the central result of the paper. The intention to treat as a fixed-point equation motivates the introduction of the map $$\begin{aligned} M_N&:C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)\times {\mathbb{R}}^{n\times(2\,N+1)}\mapsto C^{0,1}({\mathbb{T}};{\mathbb{R}}^n) \mbox{\quad given by}\\ M_N&(x,p)=E_Np+Q_NLF(x)\mbox{.}\end{aligned}$$ This means that we will look for fixed points of the map $M_N(\cdot,p)$ for given $p$ and sufficiently large $N$. We will do this in small closed balls in $C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)$ (the space of Lipschitz continuous functions) such that it is useful to introduce the notation $$B_\delta^{0,1}(x_0)=\left\{x\in C^{0,1}({\mathbb{T}};{\mathbb{R}}^n): \left\\|x-x_0\right\\|_{0,1}\leq\delta\right\}\mbox{,}$$ for $\delta>0$ and $x_0\in C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)$. That is, $B_\delta^{0,1}(x_0)$ is the closed ball of radius $\delta$ around $x_0\in C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)$ in the $\\|\cdot\\|_{0,1}$-norm (the Lipschitz norm on ${\mathbb{T}}$). \[thm:main\] Let $f$ be $EC^{j_{\max}}$ smooth, and let $x_0$ have a Lipschitz continuous derivative, that is, $x_0\in C^{1,1}({\mathbb{T}};{\mathbb{R}}^n)$. Then there exist a $\delta>0$ and a positive integer $N$ such that the map $M_N(\cdot,p)$ has a unique fixed point in $B_{6\delta}^{0,1}(x_0)$ for all $p$ in the neighborhood $U$ of $R_Nx_0$ given by $$U=\left\{p\in{\mathbb{R}}^{n\times(2\,N+1)}: \left\\|E_N\left[p-R_Nx_0\right]\right\\|_{0,1}\leq2\delta\right\}\mbox{.}$$ The maps $$\begin{aligned} X&:U\mapsto C^0({\mathbb{T}};{\mathbb{R}}^n)\mbox{,}& X(p)&=\mbox{\ fixed point of $M_N(\cdot,p)$ in $B_{6\delta}^{0,1}(x_0)$,}\\ g&:U\mapsto{\mathbb{R}}^{n\times(2\,N+1)}\mbox{,} & g(p)&=R_N\left[P_0F(X(p))+Q_0\left(E_Np-P_NLF(X(p))\right)\right]\mbox{,} \end{aligned}$$ are $j_{\max}$ times continuously differentiable with respect to their argument $p$, and $X(p)$ is an element of $C^{j_{\max}+1}({\mathbb{T}};{\mathbb{R}}^n)$. Moreover, for all $x\in B_{\delta}^{0,1}(x_0)$ the following equivalence holds: $$\begin{aligned} \dot x(t)&=f(\Delta_tx)\\ \intertext{if and only if $p=R_Nx$ is in $U$ and satisfies} g(p)&=0\mbox{\quad and\quad} x=X(p)\mbox{.} \end{aligned}$$ Theorem \[thm:main\] is the central result of the paper. It implies that, for any $x_0\in C^{1,1}({\mathbb{T}};{\mathbb{R}}^n)$ all solutions of the periodic BVP in a sufficiently small neighborhood of $x_0$ lie in the graph $X(U)$ of a finite-dimensional manifold. Moreover, these solutions can be determined by finding the roots of $g$ in $U\subset {\mathbb{R}}^{n\times(2\,N+1)}$. We note that Theorem \[thm:main\] is different from statements about numerical approximations. Even though the integer $N$ is finite, solving the algebraic system $g(p)=0$ and then mapping the solutions with the map $X$ into the function space $C^0({\mathbb{T}};{\mathbb{R}}^n)$ gives an exact solution $x=X(p)$ of the periodic BVP $\dot x(t)=f(\Delta_tx)$. The size of the radius $\delta$ of the ball in which the equivalence holds depends on how large one can choose $\delta$ such that a local $EC$ Lipschitz constant $K$ for $F$ exists for $B_{6\delta}^{0,1}(x_0)$ (such neighborhoods exist according to Lemma \[thm:loclip\]). In many applications (in particular, in the example ) this can be any closed ball in which the right-hand side $f$ is well defined (at the expense of increasing $K$ for larger balls). Once the local $EC$ Lipschitz constant $K$ is determined, one can find a uniform upper bound $R$ for the norm $\\|F(x)\\|_{0,1}$ for all $x\in B_{6\delta}^{0,1}(x_0)$ (see Lemma \[thm:Flipbound\]). The integer $N$, which determines the dimension of the algebraic system, is then chosen depending on $R$, $K$ and $\\|x_0'\\|_{0,1}$. Section \[sec:remproofs\] contains the complete proof of Theorem \[thm:main\]. The first step of the proof of Equivalence Theorem \[thm:main\] is the existence of the fixed point of $M_N$ in $B_{6\delta}^{0,1}(x_0)$ for $p\in U$. This is achieved by applying Banach’s contraction mapping principle to the map $M_N(\cdot,p)$ in the closed ball $B_{6\delta}^{0,1}(x_0)$. The only peculiarity in this step is that we apply the principle to $B_{6\delta}^{0,1}(x_0)$, which is a closed bounded set of Lipschitz continuous functions, using the (weaker) maximum norm ($\\|\cdot\\|_0$). This is possible because closed balls in $C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)$ are complete also with respect to the norm $\\|\cdot\\|_0$. With respect to the maximum norm the map $M_N(\cdot,p):x\mapsto E_Np+Q_NLF(x)$ becomes a contraction for sufficiently large $N$ (because the norm of the operator $Q_NL$ is bounded by $C\log(N)/N$, and $F$ has a Lipschitz constant $K$ with respect to $\\|\cdot\\|_0$ in $B_{6\delta}^{0,1}(x_0)$). After the existence of the fixed point of $M_N(\cdot,p)$ is established in Section \[sec:fixproof\] the equivalence between the algebraic system $g(p)=0$ and the periodic BVP $\dot x(t)=f(\Delta_tx)$ in the smaller ball $B_\delta^{0,1}(x_0)$ follows from the Splitting Lemma \[thm:split\]. The smoothness (in the classical sense) of the maps $X$ and $g$ follows, colloquially speaking, from implicit differentiation of the fixed-point problem $x=E_Np+Q_NLF(x)$ with respect to $p$. Section \[sec:algdiff1\] checks the uniform convergence of the difference quotient in detail, Section \[sec:smooth\] uses the higher degrees of $EC^{j_{\max}}$ smoothness of $f$ to prove higher degrees of smoothness for $X$ and $g$. For proving higher-order smoothness one has to check only if the spectral radius of a linear operator is less than unity, but the inductive argument requires more elaborate notation than the first-order continuous differentiability. Application to periodic orbits of autonomous FDEs — Hopf Bifurcation Theorem {#sec:po} ============================================================================ Let us come back to the original problem, the parameter-dependent FDE $\dot x = f(x_t,\mu)$, where $\mu\in{\mathbb{R}}^\nu$ is a system parameter and the functional $f:C^0(J;{\mathbb{R}}^n)\times{\mathbb{R}}^\nu\mapsto{\mathbb{R}}^n$ is defined for first arguments that exist on an arbitrary compact interval $J$. Periodic orbits are solutions $x$ of $\dot x = f(x_t,\mu)$ that are defined on ${\mathbb{R}}$ and satisfy $x(t)=x(t+T)$ for some $T>0$ and all $t\in{\mathbb{R}}$. Let $x$ be a periodic function of period $T=2\pi/\omega$. Then the function $y(s)=x(s/\omega)$ is a function of period $2\pi$ ($s\in{\mathbb{T}}$). This makes it useful to define the map $$\begin{aligned} S:&BC^0({\mathbb{R}};{\mathbb{R}}^n)\times{\mathbb{R}}\mapsto BC^0({\mathbb{R}};{\mathbb{R}}^n) & [S(y,\omega)](s)=y(\omega s)\mbox{,}\end{aligned}$$ such that $S(y,\omega)(t)=x(t)$ for all $t\in{\mathbb{R}}$ (remember that $BC^0({\mathbb{R}};{\mathbb{R}}^n)$ is the space of bounded continuous functions on the real line). Then $x\in C^1({\mathbb{R}};{\mathbb{R}}^n)$ satisfies the differential equation $$\label{eq:ft} \dot x(t)=f(x_t,\mu)$$ on the real line and has period $2\pi/\omega$ if and only if $y=S(x,1/\omega)\in C^1({\mathbb{T}};{\mathbb{R}}^n)$ satisfies the differential equation $$\dot y(s)=\frac{1}{\omega}f(S(\Delta_sy,\omega),\mu)\mbox{.}$$ Let us define an extended differential equation $$\begin{aligned} \label{eq:ftext} \dot x_\mathrm{ext}(s)&=f_\mathrm{ext}(\Delta_sx_\mathrm{ext})\mbox{,}\end{aligned}$$ where $f_\mathrm{ext}$ maps $C^0({\mathbb{T}};{\mathbb{R}}^{n+1+\nu})$ into ${\mathbb{R}}^{n+1+\nu}$ and is defined by $$\begin{aligned} f_\mathrm{ext} \begin{pmatrix} y \\ \omega\\ \mu \end{pmatrix}&= \begin{bmatrix} f\left(S(y,R_0\omega),R_0\mu\right)/{\operatorname{cut}}(R_0\omega)\\ 0\\ 0 \end{bmatrix}\mbox{,\quad where}\\ {\operatorname{cut}}(\omega)&= \begin{cases} \omega &\mbox{if $\omega>\omega_\mathrm{cutoff}>0$}\\ \mbox{smooth, uniformly non-negative extension} & \mbox{for $\omega<\omega_\mathrm{cutoff}$} \end{cases}\end{aligned}$$ for $y\in C^0({\mathbb{T}};{\mathbb{R}}^n)$, $\omega\in C^0({\mathbb{T}};{\mathbb{R}})$ and $\mu\in C^0({\mathbb{T}};{\mathbb{R}}^\nu)$ (recall that $R_0$ takes the average of a function on ${\mathbb{T}}$). We have used in our definition that any functional $f$ defined for $x\in C^0(J;{\mathbb{R}}^n)$ is also a functional on $C^0({\mathbb{T}};{\mathbb{R}}^n)$ (periodic functions have a natural extension $x(t)=x(t_{{\operatorname{mod}}[-\pi,\pi)})$ if $t\in {\mathbb{R}}$ is arbitrary). The extended system has introduced the unknown $\omega$ and the system parameter $\mu$ as functions of time, and the additional differential equations $\dot \omega=0$, $\dot\mu=0$, which force the new functions to be constant for solutions of . We have also introduced a cut-off for $\omega$ close to zero to keep $f_\mathrm{ext}$ globally defined. The extended BVP is in the form of periodic BVPs covered by the Equivalence Theorem \[thm:main\]. Thus, if $f_\mathrm{ext}$ is $EC^{j_{\max}}$ smooth then BVP satisfies the assumptions of Theorem \[thm:main\] in the vicinity of every periodic function $x_{0,\mathrm{ext}}\in C^{1,1}({\mathbb{T}};{\mathbb{R}}^{n+\nu+1})$. Any solution $x_\mathrm{ext}=(y,\omega,\mu)$ that we find for corresponds to a periodic solution $t\mapsto y(\omega t)$ of period $2\pi/R_0\omega$ at parameter $R_0\mu$ for and vice versa, as long as $R_0\omega>\omega_\mathrm{cutoff}$. The condition of $EC^{j_{\max}}$ smoothness has to be checked only for the first $n$ components of the function $f_\mathrm{ext}$ since its final $\nu+1$ components are zero. Application of the Equivalence Theorem \[thm:main\] results in a system of algebraic equations that has $(n+\nu+1)(2N+1)$ variables and equations, where $N$ is the positive integer proven to exist in Theorem \[thm:main\]. Let us denote as $F=(F_y,F_\omega,F_\mu)$ the components of the right-hand side $F_\mathrm{ext}$ (given by $F_\mathrm{ext}(x_\mathrm{ext})(t)=f_\mathrm{ext}(\Delta_tx_\mathrm{ext})$), of which $F_\mu$ and $F_\omega$ are identically zero. Let $p=(p_y,p_\omega,p_\mu)$ be the $2N+1$ leading Fourier coefficients of $y$, $\omega$ and $\mu$, respectively (these are the variables of the algebraic system constructed via Theorem \[thm:main\]), and $X(p)=(X_y(p),X_\omega(p),X_\mu(p))$ be the map from $R^{(n+\nu+1)(2N+1)}$ into $C^{j_{\max}}({\mathbb{T}};{\mathbb{R}}^{n+\nu+1})$. Then several of the components of $p$ can be eliminated as variables, and the equations for $p$ resulting from Theorem \[thm:main\] correspondingly simplified. Since $F$ is identically zero in its last $\nu+1$ components we have $$\begin{aligned} X_\omega(p)&=E_Np_\omega\mbox{,} & X_\mu(p)&=E_Np_\mu\mbox{.}\end{aligned}$$ Hence, the right-hand side $$g(p)=R_N\left[P_0F(X(p))+Q_0\left(E_Np-P_NLF(X(p))\right)\right]\mbox{,}$$ defined in Theorem \[thm:main\], has $\nu+1$ components that are identical to zero (since $P_0F(X(p))=0$ for the equations $\dot\omega=0$ and $\dot\mu=0$). Furthermore, $g(p)=0$ contains the equations $R_NQ_0E_Np_\omega=0$ and $R_NQ_0E_Np_\mu=0$, which require that all Fourier coefficients (except the averages $R_0\omega$ and $R_0\mu$) of $\mu$ and $\omega$ are equal to zero. This means (unsurprisingly) that the algebraic system forces $\omega$ and $\mu$ to be constant. Thus, we can eliminate $R_NQ_0E_Np_\omega$ and $R_NQ_oE_Np_\mu$ (which are $2N(\nu+1)$ variables), replacing them by zero, and drop the corresponding equations. Since $\omega$ and $\mu$ must be constant, we can replace the arguments $p_\omega$ and $p_\mu$ of $X$ by the scalar $R_0E_Np_\omega$ (which we re-name back to $\omega$) and the vector $R_0E_Np_\mu\in{\mathbb{R}}^\nu$ (which we re-name back to $\mu$). This leaves the first $n(2N+1)$ algebraic equations $$\begin{aligned} \label{eq:lowmodes:par} 0=&R_N\left(P_0F_y(X_y(p_y,\omega,\mu),\omega,\mu)+ Q_0\left[E_Np_y- P_NLF_y(X_y(p_y,\omega,\mu),\omega,\mu)\right]\right)\mbox{,}\end{aligned}$$ which depend smoothly (with degree $j_{\max}$) on the $n(2N+1)$ variables $p_y$ and the parameters $\omega\in{\mathbb{R}}$ and $\mu\in{\mathbb{R}}^\nu$. Overall, is a system of $n\times(2\,N+1)$ equations. ### Rotational Invariance {#rotational-invariance .unnumbered} The original nonlinearity $F$, defined by $[F(x)](t)=f(\Delta_tx)$ is equivariant with respect to time shift: $\Delta_tF(x)=F(\Delta_tx)$ for all $t\in{\mathbb{T}}$ and $x\in C^0({\mathbb{T}};{\mathbb{R}}^n)$. Furthermore, $\Delta_t$ commutes with the following operations: $$\begin{aligned} \Delta_tQ_NL&=Q_NL\Delta_t \mbox{\quad (if $N\geq0$) and } &\Delta_tP_N&=P_N\Delta_t\mbox{.}\end{aligned}$$ This property gets passed on to the algebraic equation in the following sense: let us define the operation $\Delta_t$ for a vector $p$ in ${\mathbb{R}}^{n(2N+1)}$, which we consider as a vector of Fourier coefficients of the function $E_Np\in C^0({\mathbb{T}};{\mathbb{R}}^n)$, by $$\Delta_tp=R_N\Delta_tE_Np\mbox{.}$$ With this definition $\Delta_t$ commutes with $R_N$ and $E_N$. It is a group of rotation matrices: $\Delta_t$ is regular for all $t$, and $\Delta_{2k\pi}$ is the identity for all integers $k$. The definition of $X(p)$ as a fixed point of $x\mapsto E_Np+Q_NLF(x)$ implies that $\Delta_tX(p)=X(\Delta_tp)$. From this it follows that the algebraic system of equations is also equivariant with respect to $\Delta_t$. If we denote the right-hand-side of the overall system by $G(p_y,\omega,\mu)$ then $G$ satisfies $$\begin{aligned} \Delta_t G(p_y,\omega,\mu)=G(\Delta_tp_y,\omega,\mu)\mbox{\quad for all $t\in{\mathbb{T}}$, $p_y\in{\mathbb{R}}^{n(2N+1)}$, $\omega>0$ and $\mu\in{\mathbb{R}}^\nu$.}\end{aligned}$$ ### Application to Hopf bifurcation {#application-to-hopf-bifurcation .unnumbered} One useful aspect of the Equivalence Theorem is that it provides an alternative approach to proving the Hopf Bifurcation Theorem for equations with state-dependent delays. The first proof that the Hopf bifurcation occurs as expected is due to Eichmann [@E06]. The reduction of periodic boundary-value problems to smooth algebraic equations reduces the Hopf bifurcation problem to an equivariant algebraic pitchfork bifurcation. Let us consider the equation $$\label{eq:dynsys} 0=f(E_0x_0,\mu)$$ where $f:C^0(J;{\mathbb{R}}^n)\times{\mathbb{R}}\mapsto{\mathbb{R}}^n$, $\mu\in{\mathbb{R}}$, $J$ is a compact interval, $x_0\in{\mathbb{R}}^n$, and the operator $E_0$ (as defined in in Section \[sec:theorem\]) extends a constant to a function on ${\mathbb{T}}$ (and thus, on $J$). This means that is a system of $n$ algebraic equations for the $n+1$ variables $(x_0,\mu)$. The definition of $EC^k$ smoothness does not cover functionals that depend on parameters. We avoid the introduction of a separate definition of $EC^k$ smoothness for parameter-dependent functionals that distinguishes between parameters and functional arguments. We rather extend Definition \[def:extdiff\]: \[def:extdiff:par\] Let $J=[a,b]$ be a compact interval (or $J={\mathbb{T}}$), and $D$ be a product space of the form $D=C^{k_1}(J;{\mathbb{R}}^{m_1})\times\ldots\times C^{k_\ell}(J;{\mathbb{R}}^{m_\ell})$ where $\ell\geq 1$, and $k_j\geq0$ and $m_j\geq1$ are integers. We say that $f:D\times{\mathbb{R}}^\nu\mapsto {\mathbb{R}}^n$ is $EC^k$ smooth if the functional $$\label{eq:ass:pareck} (x,y)\in D\times C^0(J;{\mathbb{R}}^\nu)\mapsto f(x,y(a))\in{\mathbb{R}}^n$$ is $EC^k$ smooth (if $J={\mathbb{T}}$ we use $a=-\pi$). Requiring $EC^k$-smoothness of the parameter-dependent functional $f$ in this sense, implies that the algebraic system $0=f(E_0x_0,\mu)$ is $k$ times continuously differentiable. Let us assume that the algebraic system $0=f(E_0x_0,\mu)$ has a regular solution $x_0(\mu)\in{\mathbb{R}}^n$ for $\mu$ close to $0$. Without loss of generality we can assume that $x_0(\mu)=0$, otherwise, we introduce the new variable $x_\mathrm{new}=x_\mathrm{old}-E_0x_0(\mu)\in C^0(J;{\mathbb{R}}^n)$. Hence, $f(0,\mu)=0$ for all $\mu$ close to $0$. The $EC^1$ derivative of $f$ in $(0,\mu)$ is a linear functional, mapping $C^0(J;{\mathbb{R}}^{n+\nu})$ into ${\mathbb{R}}^n$. Let us denote its first $n$ components (the derivative with respect to the first argument $x$ of $f$) by $a(\mu)$. The linear operator $a(\mu)$ can easily be complexified by defining $a(\mu)[x+iy]=a(\mu)[x]+ia(\mu)[y]$ for $x+iy\in C^0([-\tau,0];{\mathbb{C}}^n)$. If $f$ is $EC^k$ smooth with $k\geq2$ then the $n\times n$-matrix $K(\lambda,\mu)$ (called the characteristic matrix), defined by $$\label{eq:charmdef} K(\lambda,\mu)\,v=\lambda v-a(\mu)[v\exp(\lambda t)]$$ is analytic in its complex argument $\lambda$ and $k-1$ times differentiable in its real argument $\mu$ (since the functions $t\mapsto v\exp(\lambda t)$ to which $a(\mu)$ is applied are all elements of $C^k(J;{\mathbb{R}}^n)$). The Hopf Bifurcation Theorem states the following: \[thm:hopf\] Assume that $f$ is $EC^k$ smooth ($k\geq 2$) in the sense of Definition \[def:extdiff:par\] and that the characteristic matrix $K(\lambda,\mu)$ satisfies the following conditions: 1. \[ass:hopf:imag\](Imaginary eigenvalue) there exists an $\omega_0>0$ such that $\det K(i\omega_0,0)=0$ and $i\omega_0$ is an isolated root of $\lambda\mapsto \det K(\lambda,0)$. We denote the corresponding null vector by $v_1=v_r+iv_i\in{\mathbb{C}}^n$ (scaling it such that $\|v_r\|^2+\|v_i\|^2=1$). 2. \[ass:hopf:nonresonant\] (Non-resonance) $\det K(ik\omega,0)\neq 0$ for all integers $k\neq \pm1$. 3. \[ass:hopf:cross\] (Transversal crossing) The local root curve $\mu\mapsto \lambda(\mu)$ of $\det K(\lambda,\mu)$ that corresponds to the isolated root $i\omega_0$ at $\mu=0$ (that is, $\lambda(0)=i\omega_0$) has a non-vanishing derivative of its real part: $$0\neq\left.\frac{\partial}{\partial\mu} {\operatorname{Re}}\lambda(\mu)\right\vert_{\textstyle\mu=0}\mbox{.}$$ Then there exists a $k-1$ times differentiable curve $$\beta\in(-{\varepsilon},{\varepsilon})\mapsto (x,\omega,\mu)\in C^1({\mathbb{T}};{\mathbb{R}}^n)\times{\mathbb{R}}\times{\mathbb{R}}$$ such that for sufficiently small ${\varepsilon}>0$ the following holds: 1. \[thm:hopf:isperiodic\] $x(\omega\cdot)$ (or $S(x,\omega)$) is a periodic orbit of $\dot x(t)=f(x_t,\mu)$ of period $2\pi/\omega$, that is, $x\in C^1({\mathbb{T}};{\mathbb{R}}^n)$ and $$\dot x(t)= \frac{1}{\omega}f(S(\Delta_tx,\omega),\mu)\mbox{,}\label{eq:hopf:po}$$ 2. \[thm:hopf:phaseamp\] the first Fourier coefficients of $x$ are equal to $(0,\beta)$, that is, $$\begin{split} 0&=\frac{1}{\pi}\int_{-\pi}^\pi {\operatorname{Re}}\left[v_1\exp(it)\right]^T x(t){\mathop{}\!\mathrm{d}}t\mbox{,\quad and}\\ \beta&=-\frac{1}{\pi} \int_{-\pi}^\pi{\operatorname{Im}}\left[v_1\exp(it)\right]^T x(t){\mathop{}\!\mathrm{d}}t\mbox{,} \end{split} \label{eq:hopfphase}$$ 3. \[thm:hopf:pars\] $x\vert_{\beta=0}=0$, $\mu\vert_{\beta=0}=0$ and $\omega\vert_{\beta=0}=\omega_0$, that is, the solution $x$, the system parameter $\mu$ and the frequency $\omega$ of $x$, which are differentiable functions of the amplitude $\beta$, are equal to $x=0$, $\mu=0$, $\omega=\omega_0$ for $\beta=0$. The statement is identical to the classical Hopf Bifurcation Theorem for ODEs in its assumptions and conclusions apart from the regularity assumption on $f$ specific to FDEs. Note that the existence of the one-parameter family (parametrized in $\beta$) automatically implies the existence of a two-parameter family for $\beta\neq0$ due to the rotational invariance: if $x$ is a solution of then $\Delta_sx$ is also a solution of for every fixed $s\in{\mathbb{T}}$. Condition fixes the time shift $s$ of $x$ such that $x$ is orthogonal to ${\operatorname{Re}}[v_1\exp(it)]$ using the ${\mathbb{L}}^2$ scalar product on ${\mathbb{T}}$. The proof of the Hopf Bifurcation Theorem is a simple fact-checking exercise. We have to translate the assumptions on the derivative of $f:C^0(J;{\mathbb{R}}^n)\mapsto{\mathbb{R}}^n$ into properties of the right-hand side of the nonlinear algebraic system near $(x,\omega,\mu)=(0,\omega_0,0)$, and then apply algebraic bifurcation theory to the algebraic system. The only element of the proof that is specific to functional differential equations comes in at the linear level: the fact that the eigenvalue $i\omega_0$ is simple implies that the right nullvector $v_1\in{\mathbb{C}}^n$ and any non-trivial left nullvector $w_1$ satisfy $$\begin{aligned} w_1^H\left[\left.\frac{\partial}{\partial \lambda}K(\lambda,0)\right\vert_{ \textstyle\lambda=i\omega_0}\right] v_1\neq 0\mbox{.}\end{aligned}$$ This is the generalization of the orthogonality condition $w_1^Hv_1\neq 0$, known from ordinary matrix eigenvalue problems, to exponential matrix eigenvalue problems of the type $K(\lambda,\mu)\,v=0$. The proof of Theorem \[thm:hopf\] is entirely based on the standard calculus arguments for branching of solutions to algebraic systems as can be found in textbooks [@AG79]. The details of the proof can be found in Section \[sec:hopf:proof\]. The statements in Theorem \[thm:hopf\] should be compared to two previous works considering the same situation: the branching of periodic orbits from an equilibrium losing its stability in FDEs with state-dependent delays. The PhD thesis of Eichmann (2006, [@E06]) proves the existence of the curve $\beta\mapsto(x(\beta),\mu(\beta),\omega(\beta))$ and that it is once continuously differentiable (assuming only $EC^2$ smoothness of $f$). Since $\mu'(0)=0$ due to rotational symmetry (see proof in Section \[sec:hopf:proof\]) this is not enough to determine if the non-trivial periodic solutions exist for $\mu>0$ or for $\mu<0$ for small $\beta$ (the so-called criticality of the Hopf bifurcation, which is of interest in applications [@IBS08; @IST05]). Moreover, the non-resonance condition in [@E06] is slightly too strong, requiring that $i\omega_0$ is the only purely imaginary root of $\det K(\lambda,0)$ (this assumption is different in the summary given in the review by Hartung et al. [@HKWW06]; note that the publicly available version of [@E06] has a typo in the corresponding assumption L1), and only the pure-delay case (where the time interval $J$ equals $[-\tau,0]$) was considered. However, the techniques employed in [@E06], based on the Fredholm alternative, are likely to yield exactly the same result as stated in Theorem \[thm:hopf\] if one assumes general $EC^k$ smoothness with $k\geq2$ (the formulation of $EC^2$ smoothness is already rather convoluted in [@E06]). Hu and Wu [@HW10] use $S^1$-degree theory [@EWG92; @KW97] to prove the existence of a branch of non-trivial periodic solutions near $(x,\mu,\omega)=(0,0,\omega_0)$. This type of topological methods gives generally weaker results concerning the local uniqueness of branches of periodic orbits or their regularity, but they require only weaker assumptions ([@HW10] still needs to assume $EC^2$ smoothness, though). Degree methods also give global existence results by placing restrictions on the number of branches that can occur. Conclusion, applications and generalizations {#sec:conc} ============================================ Periodic boundary-value problems for functional differential equations (FDEs) are equivalent to systems of smooth algebraic equations if the functional $f$ defining the right-hand side of the boundary-value problem satisfies natural smoothness assumptions. These assumptions are identical to those imposed in the review by Hartung et al. [@HKWW06] and do not exclude FDEs with state-dependent delay. There are several immediate extensions of the results presented in this paper. The list below indicates some of them. ### Further potential applications of the Equivalence Theorem \[thm:main\] {#further-potential-applications-of-the-equivalence-theoremthmmain .unnumbered} Theorem \[thm:hopf\] on the Hopf bifurcation is not the central result of the paper, even though it is a moderate extension of the theorem proved in [@E06]. Rather, it is a demonstration of the use of the Equivalence Theorem \[thm:main\]. The main strength of the equivalence result stated in Theorem \[thm:main\] is that it permits the straightforward application of continuation and Lyapunov-Schmidt reduction techniques to FDE problems involving periodic orbits of finite period, regardless if the delay is state dependent, or if the equation is of so-called mixed type (that is, positive and negative delays are present). A source of complexity, for example, in [@E06; @EWG92; @G11; @HW10; @KW97; @W98], is that techniques such as Lyapunov-Schmidt reduction or $S^1$-degree theory had to be applied in Banach spaces. Theorem \[thm:main\] removes the need for this, reducing the analysis of periodic orbits to root-finding in ${\mathbb{R}}^{n\times(2\,N+1)}$. For example, Humphries et al. [@HDMU11] study periodic orbits in FDEs with two state-dependent delays numerically using DDE-Biftool [@ELR02], alluding to theoretical results about bifurcations of periodic orbits that have been proven only for constant delay. Fist of all, Humphries et al. [@HDMU11] continue branches of periodic orbits. Theorem \[thm:main\] makes clear when these branches as curves of points $(x,\omega,\mu)$ in the extended space $C^0({\mathbb{T}};{\mathbb{R}}^n)\times{\mathbb{R}}\times{\mathbb{R}}$ are smooth to arbitrary degree: the Jacobian of with respect to $(p_y,\omega,\mu)$ along the curve has to have full rank. Along these branches Humphries et al. [@HDMU11] encounter degeneracies of the linearization and conjecture the existence of the corresponding bifurcations (backed up with numerical evidence) such as: fold bifurcations, period-doubling bifurcations, or the branching off of resonance surfaces (Arnol’d tongues) when resonant Floquet multipliers of the linearized equation cross the unit circle [@K04]. The Equivalence Theorem \[thm:main\] provides a straightforward route to proofs that these scenarios occur as expected. Similarly, Theorem \[thm:main\] will likely not only simplify the proofs about bifurcations of symmetric periodic orbits such as those of Wu [@W98], but also extend them to the case of FDEs with state-dependent delays. As long as one considers branching of periodic orbits with finite periods, the problem can be reduced locally (and often in every ball of finite size) to a finite-dimensional root-finding problem. This transfers also a list of results of symmetric bifurcation theory found in textbooks [@GSS88] to FDEs with state-dependent delays. ### Globally valid algebraic system {#globally-valid-algebraic-system .unnumbered} The main result was formulated locally in the neighborhood of a given $x_0\in C^{1,1}({\mathbb{T}};{\mathbb{R}}^n)$ and required only local Lipschitz continuity. The proof makes obvious that the domain of definition for the map $X$, which maps between the function space and the finite-dimensional space is limited by the size of the neighborhood of $x_0$ for which one can find a uniform ($EC$) Lipschitz constant of the right-hand side $F$. In problems with delay the right-hand side is typically a combination of Nemytskii operators generated by smooth functions and the evaluation operator ${\operatorname{ev}}:C^0({\mathbb{T}};{\mathbb{R}}^n)\times{\mathbb{T}}\mapsto{\mathbb{R}}^n$, given by ${\operatorname{ev}}(x,t)=x(t)$. These typically satisfy a (semi-)global Lipschitz condition (see also condition (Lb) in [@HKWW06]): for all $R$ there exists a constant $K$ such that $$\|f(x)-f(y)\|\leq K\\|x-y\\|_0$$ for all $x$ and $y$ satisfying $\\|x\\|_1\leq R$ and $\\|y\\|_1\leq R$. Under this condition one can choose for any bounded ball an algebraic system that is equivalent to the periodic boundary-value problem in this bounded ball. For periodic orbits of autonomous systems this means that one can find an algebraic system such that all periodic orbits of amplitude less than $R$ and of period and frequency at most $R$ are given by the roots of the algebraic system. ### Implicitly given delays {#implicitly-given-delays .unnumbered} In practical applications the delay is sometimes given implicitly, for example, the position control problem considered in [@W02] and the cutting problem in [@IBS08; @IST05] contain a separate algebraic equation, which defines the delay implicitly. In simple cases these problems can be reduced to explicit differential equations using the standard Baumgarte reduction [@B72] for index-1 differential algebraic equations. For example, in the cutting problem the delay $\tau$ depends on the current position $x$ via the implicit linear equation $$\label{eq:cutdelay} \tau(t)=a-bx(t)-bx(t-\tau(t))\mbox{,}$$ which can be transformed into a differential equation by differentiation with respect to time: $$\label{eq:cutdelaydiff} \dot\tau(t)= \frac{-bv(t)-\left[\tau(t)-a+bx(t)+bx(t-\tau(t))\right]}{1+bv(t-\tau(t))}$$ ($v(t)=\dot x(t)$ is explicitly present as a variable in the cutting model, which is a second-order differential equation). The original model accompanied with the differential equation instead of the algebraic equation fits into the conditions of the Equivalence Theorem \[thm:main\]. The regularity statement of the Equivalence Theorem guarantees that the resulting periodic solutions have Lipschitz continuous derivatives with respect to time. This implies that the defect $d=\tau-(a-bx-bx(t-\tau))$ occurring in the algebraic condition satisfies $\dot d(t)=-d(t)$ along solutions. Since the solutions are periodic the defect $d$ is periodic, too, and, hence, $d$ is identically zero. The denominator appearing in Equation becomes zero exactly in those points in which the implicit condition cannot be solved for the delay $\tau$ with a regular derivative. The same argument can be applied to the position control problem as long as the object, at position $x$, does not hit the base at position $-w$ (the model contains the term $\|x+w\|$ in the right-hand side). ### Neutral equations {#neutral-equations .unnumbered} The index reduction works only if the delay $\tau$, which is itself a function of time, is not evaluated at different time points. For example, changing $bx(t-\tau(t))$ to $bx(t-\tau(t-1))$ on the right-hand-side of would make the index reduction impossible. However, certain simple neutral equations permit a similar reduction directly on the function space level. Consider $$\label{eq:neutral} \frac{{\mathop{}\!\mathrm{d}}}{{\mathop{}\!\mathrm{d}}t} \left[\Delta_t(x+g(x))\right]=f(\Delta_t x)$$ where the functional $f$ satisfies the local $EC$ Lipschitz condition, defined in Definition \[def:loclip\], in a neighborhood $U$ of a point $x_0\in{\mathbb{C}}^{1,1}({\mathbb{T}};{\mathbb{R}}^n)$, and $g:C^0({\mathbb{T}};{\mathbb{R}}^n)\mapsto{\mathbb{R}}^n$ has a global (classical) Lipschitz constant less than unity (this excludes state-dependent delays in the essential part of the neutral equation). Then one can define the map $X_g(y)$ as the unique solution $x$ of the fixed point problem $$x(t)=y(t)-g(\Delta_tx)\mbox{\quad for $y$ near $y_0=x_0+g(x_0)$,}$$ which reduces to the equation $$\label{eq:neutralred} \dot y(t)=f(\Delta_tX_g(y))=f(X_g(\Delta_ty))\mbox{.}$$ Equation satisfies the conditions of the Equivalence Theorem \[thm:main\]. One implication of this reduction is that periodic solutions of are $k$ times continuously differentiable if the functional $f$ is $EC^k$ smooth in the sense of Definition \[def:extdiff\] and $g$ is $k$ times continuously differentiable as a map from $C^0({\mathbb{T}};{\mathbb{R}}^n)$ into ${\mathbb{R}}^n$. Proof of the Equivalence Theorem \[thm:main\] {#sec:remproofs} ============================================= Theorem \[thm:main\] is proved in three steps. First, we establish the existence of a locally unique fixed point of the map $M_N(\cdot,p)$ using Banach’s contraction mapping principle. This step requires only local $EC$ Lipschitz continuity in the sense of Definition \[def:loclip\]. In the second step we prove continuous differentiability of the map $X$ and the right-hand side $g$ of the algebraic system assuming that $f$ is $EC^1$ smooth. In the final step we prove higher-order differentiability, assuming that $f$ is $EC^k$ smooth for degrees $k$ up to $j_{\max}$. Decay of Fourier coefficients for integrals and smooth functions {#sec:qnl} ---------------------------------------------------------------- The following preparatory lemma states the well-known fact that, colloquially speaking, integrating a function makes its high-frequency Fourier coefficients smaller. In the fixed-point equation of Theorem \[thm:main\] the term $Q_NL$ occurs, and we need this term to be small for large $N$. Recall that $Q_N$ removes the first $N$ Fourier modes from a periodic function and $Lx$ is the anti-derivative of $x$ (after subtracting the average of $x$), see Equation for the precise definitions. \[thm:qnl\] The norm of the linear operator $Q_NL$, mapping the space $C^j({\mathbb{T}};{\mathbb{R}}^n)$ back into itself, is bounded by $$\\|Q_NL\\|_j\leq C\frac{\log N}{N}$$ where $C$ is a constant. The same holds in the Lipschitz norm (with the same constant $C$): $$\\|Q_NL\\|_{0,1}\leq C\frac{\log N}{N}\mbox{.}$$ #### Proof {#proof .unnumbered} We find the norm $\\|Q_NL\\|_0$ first, and start out with the well-known estimate for interpolating trigonometric polynomials for continuous functions on ${\mathbb{T}}$. Let $x$ be a continuous function on ${\mathbb{T}}$ with modulus of continuity $\omega$. Then (see [@J30]) $$\\|Q_Nx\\|_0\leq C_0\omega\left(\frac{2\pi}{N}\right)\log N$$ where $C_0$ is a constant that does not depend on $x$ or $N$. A function $\omega:[0,\infty)\mapsto[0,\infty)$ is called a modulus of continuity of a continuous function $x$ if $$\|x(t)-x(s)\|\leq \omega(\|t-s\|)\mbox{.}$$ holds for all $s$ and $t\in{\mathbb{T}}$. For a function $x\in C^0({\mathbb{T}};{\mathbb{R}}^n)$ the anti-derivative $$[Lx](t)=\int_0^tx(s)-R_0x\,{\mathop{}\!\mathrm{d}}s$$ has the Lipschitz constant $\\|x\\|_0=\max\{\|x(t)\|:t\in{\mathbb{T}}\}$ such that a modulus of continuity for $Lx$ is $\omega(h)=\\|x\\|_0 h$. Consequently, $$\label{eq:qnlf0} \\|Q_NLx\\|_0\leq C_0\frac{2\pi\\|x\\|_0}{N}\log N\mbox{,}$$ where $C_0$ does not depend on $x$ or $N$. This proves the claim of the lemma for $j=0$. For $x\in C^j({\mathbb{T}};{\mathbb{R}}^n)$ we notice that all derivatives of $x$ up to order $j$ are continuous. Applying estimate to each of the derivatives of $x$ we get $$\\|Q_NLx^{(l)}\\|_0\leq \frac{2\pi C_0}{N}\log N\,\\|x^{(l)}\\|_0 \mbox{\quad for $l=0,\ldots j$.}$$ Consequently, the maximum of the left-hand sides over all $l\in\{0,\ldots,j\}$ must be less than the maximum of the right-hand sides: $$\\|Q_NLx\\|_j=\max_{l=0,\ldots,j}\\|Q_NLx^{(l)}\\|_0\leq \frac{2\pi C_0}{N}\log N\,\max_{l=0,\ldots,j}\\|x^{(l)}\\|_0= 2\pi C_0\frac{\log N}{N}\\|x\\|_j\mbox{,}$$ which implies the desired estimate for $\\|Q_NL\\|_j$ using the constant $C=2\pi C_0$. The estimate of $Q_NL$ in the Lipschitz norm is a continuity argument. The operator $Q_NL$ is bounded (and, thus, continuous) on $C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)$. For every element $y$ of $C^1({\mathbb{T}};{\mathbb{R}}^n)$ (which is a dense subspace of $C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)$) the Lipschitz constant is identical to $\\|y'\\|_0=\max_{t\in{\mathbb{T}}}\|y'(t)\|$, and, thus, $\\|y\\|_1=\\|y\\|_{0,1}$. Let $x_n\in C^1({\mathbb{T}};{\mathbb{R}}^n)$ be a sequence of continuously differentiable functions that converges to $x\in C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)$ in the $\\|\cdot\\|_{0,1}$-norm: $\\|x_n-x\\|_{0,1}\to0$ for $n\to\infty$. Then $$\\|Q_NLx_n\\|_{0,1}=\\|Q_NLx_n\\|_1\leq C\frac{\log N}{N}\\|x_n\\|_1= C\frac{\log N}{N}\\|x_n\\|_{0,1}\mbox{.}$$ On both sides of the inequality the limit for $n\to\infty$ exists, resulting in the desired estimate for $\\|Q_NL\\|_{0,1}$. [$\square$]{} A direct consequence of Lemma \[thm:qnl\] is that the Lipschitz norm of $Q_Nx$, $\\|Q_Nx\\|_{0,1}$, goes to zero for $N\to\infty$ for elements of $C^{1,1}({\mathbb{T}};{\mathbb{R}}^n)$, so, for example, for a solution $x$ of a periodic BVP: $$\label{eq:qn} \\|Q_Nx\\|_{0,1}=\\|Q_NLx'\\|_{0,1}\leq C\frac{\log N}{N}\\|x'\\|_{0,1}\leq C\frac{\log N}{N}\\|x\\|_{1,1}\mbox{.}$$ Proof of Splitting Lemma \[thm:split\] {#sec:splitproof} -------------------------------------- For any given integer $N\geq0$ we have to show that the pair $(x,p)\in C^0({\mathbb{T}};{\mathbb{R}}^n)\times{\mathbb{R}}^{n\times(2\,N+1)}$ satisfies $$\begin{aligned} \label{eq:inteq:app} x(t)&=x(0)+\int_0^tF(x)(s){\mathop{}\!\mathrm{d}}s\mbox{\quad for all $t\in{\mathbb{T}}$, and}\\ p&=R_Nx\mbox{,\quad (or, equivalently, $E_Np=P_Nx$)}\label{eq:peqpnx:app}\\ \intertext{ if and only if it satisfies the system} x&=E_Np+Q_NLF(x)\mbox{,}\label{eq:fixp:app}\\ 0&=R_N\left[P_0F(x)+Q_0\left(E_Np-P_NLF(x)\right)\right] \label{eq:lowmodes:app}\mbox{,} \end{aligned}$$ “$\Rightarrow$”: Assume that $x\in C^0({\mathbb{T}};{\mathbb{R}}^n)$ satisfies , and let $p=R_Nx$. Subtracting equation for $t=-\pi$ from for $t=\pi$ implies that the average of $F(x)$ is zero. Thus, $R_0F(x)=0$ and $P_0F(x)=0$. Since $Ly=\int_0^ty(s)-R_0y{\mathop{}\!\mathrm{d}}s$, the identity implies (in combination with $R_0F(x)=0$) $$\label{eq:intident:app} x(t)=x(0)+[LF(x)](t)\mbox{.}$$ Applying projection $Q_N$ to this identity we obtain $Q_Nx=Q_NLF(x)$. Adding to this we obtain equation . Applying projection $P_NQ_0$ (which is the same as $Q_0P_N$) to we obtain $Q_0P_Nx=Q_0P_NLF(x)$. Inserting $E_Np$ for $P_Nx$ into this identity leads to $Q_0[E_Np-P_NLF(x)]=0$. Since $P_0F(x)=0$, this in turn implies . “$\Leftarrow$”: Applying $P_N$ to implies $P_Nx=E_Np$ (and $p=R_Nx$) immediately. The expression inside the parentheses of $R_N$ in equation is a sum of two parts that each have to be zero (since they are both in the image of $P_N$ on which $R_N$ in injective). The projection $Q_0$ subtracts the average from its argument. Hence, $(x,p)\in C^0({\mathbb{T}};{\mathbb{R}}^n)\times{\mathbb{R}}^{n\times(2\,N+1)}$ satisfies – if and only if there exists a constant $c\in{\mathbb{R}}^n$ such that the triple $(c,x,p)$ satisfies the system of equations consisting of and $$\begin{aligned} 0&=R_0F(x)\label{eq:avg:app}\\ E_0c&=E_Np-P_NLF(x)\label{eq:othermodes:app}\mbox{.}\end{aligned}$$ Note that $E_0$ maps the constant $c\in{\mathbb{R}}^n$ to a function that equals this constant for all $t\in{\mathbb{T}}$. In this system, ensures that the average of $F(x)$ is zero. Equation is an equation in the finite-dimensional space ${\operatorname{rg}}P_N$. Subtracting from gives $$x=E_0c+LF(x)\mbox{.}$$ This equals , keeping in mind that $[Ly](t)=\int_0^ty(s)-R_0y{\mathop{}\!\mathrm{d}}s$ ($[Ly](0)=0$ for all $y\in C^0({\mathbb{T}};{\mathbb{R}}^n)$, hence, $x(0)=c$), and using $R_0F(x)=0$ (see equation ).[$\square$]{} Unique solvability of the fixed point problem {#sec:fixproof} ---------------------------------------------- Let $x_0$ be an element of $C^{1,1}({\mathbb{T}};{\mathbb{R}}^n)$, for example, a solution of the periodic boundary value problem $\dot x(t)=f(\Delta_tx)=F(x)(t)$. Consider a closed ball $B_\delta^{0,1}(x_0)$ of radius $\delta$ around $x_0$ in the Lipschitz norm: $$B_\delta^{0,1}(x_0)=\{x\in C^{0,1}({\mathbb{T}};{\mathbb{R}}^n): \\|x-x_0\\|_{0,1}\leq \delta\}\mbox{.}$$ The superscript “$0,1$” indicates which norm is used to measure the distance from $x_0$ and that only elements of $C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)$ are included. Lemma \[thm:loclip\] implies that $F$ is Lipschitz continuous with respect to the $\\|\cdot\\|_0$-norm in $B_\delta^{0,1}(x_0)$ if we choose $\delta$ sufficiently small (thus, $F$ is also called $EC$ Lipschitz continuous in $B_\delta^{0,1}(x_0)$): $$\label{eq:flip0} \\|F(x)-F(y)\\|_0\leq K\\|x-y\\|_0$$ for all $x$ and $y$ in $B_\delta^{0,1}(x_0)$ and a fixed $K>0$. In any ball $B_\delta^{0,1}$, in which $F$ is $EC$ Lipschitz continuous, $F$ is also bounded in the Lipschitz norm: $$\label{eq:Fboundlip} \\|F(x)\\|_{0,1}\leq R\mbox{\quad for all $x\in B_\delta^{0,1}(x_0)$.}$$ See Lemma \[thm:Flipbound\] in Appendix \[sec:basicprop\] for the proof. We can now formulate a lemma about the unique solvability of the fixed point problem $$x=E_Np+Q_NLF(x)\mbox{.}$$ This unique solvability and the Splitting Lemma \[thm:split\] allow us to reduce the periodic BVP $\dot x(t)=f(\Delta_tx)$ to a system of algebraic equations. Remember that $E_Np$ takes a vector $p$ of $2N+1$ Fourier coefficients and maps it to the periodic function having these Fourier coefficients, $R_Nx$ extracts the first $2N+1$ Fourier coefficients from a periodic function $x$, $P_Nx$ projects the periodic function $x$ onto the space spanned by the basis $b_{-N},\ldots,b_N$ and $Q_N={I}-P_N$ sets the first Fourier modes of a function to zero. ($P_N$ and $Q_N$ are projections in the function space, and $R_N$ and $E_N$ map between the finite-dimensional subspace ${\operatorname{rg}}P_N$ and ${\mathbb{R}}^{n\times(2N+1)}$.) \[thm:fixp\] Let $x_0$ be in $C^{1,1}({\mathbb{T}};{\mathbb{R}}^n)$, and let $\delta>0$ be such that $$\label{eq:fbounds} \\|F(x)\\|_{0,1}\leq R\mbox{\quad and\quad} \\|F(x)-F(y)\\|_0\leq K\\|x-y\\|_0$$ for all $x$ and $y\in B_{6\delta}^{0,1}(x_0)$ and for some constants $K>0$ and $R>0$ depending on $\delta$. Then for any sufficiently large $N$ the fixed point problem $$\label{eq:fixp} x=E_Np+Q_NLF(x)$$ has a unique solution $x\in B_{6\delta}^{0,1}(x_0)$ for all vectors $p\in{\mathbb{R}}^{n\times(2N+1)}$ in the neighborhood $U$ of $R_Nx_0$ given by $$\label{eq:pclose} U=\left\{p\in{\mathbb{R}}^{n\times(2\,N+1)}: \left\\|E_N\left[p-R_Nx_0\right]\right\\|_{0,1}<2\delta\right\}\mbox{.}$$ Moreover, if $x\in B_\delta^{0,1}(x_0)$ is continuously differentiable and satisfies $x'=F(x)$ then its projection $p=R_Nx$ is in the neighborhood $U$, and $x$ and $p$ satisfy . Note that $U$ is an open set of ${\mathbb{R}}^{n\times(2N+1)}$ since $E_N$ is an isomorphism between ${\operatorname{rg}}P_N$, equipped with the $\\|\cdot\\|_{0,1}$-norm, and ${\mathbb{R}}^{n\times(2N+1)}$. We have to prove the unique solvability of the fixed-point problem in a slightly larger ball (radius $6\delta$) and for a slightly larger range of parameters $p$ (note the $2\delta$ in ) in order to establish one-to-one correspondence in the ball of radius $\delta$. #### Proof {#proof-1 .unnumbered} The idea is, of course, that the function $$\begin{aligned} M_N(\cdot,p): x\mapsto E_Np+Q_NLF(x)\end{aligned}$$ maps the closed ball $B_{6\delta}^{0,1}(x_0)$ back into itself and is uniformly contracting for suitably large $N$ and vectors $p\in U$. First, any closed ball $B_r^{0,1}(x_0)$ is closed (and, thus, forms a complete metric space) with respect to the $\\|\cdot\\|_0$-norm. This completeness is a simple continuity argument: let $y_n=x_0+z_n$ be a Cauchy sequence in $B_r^{0,1}(x_0)$ with respect to the $\\|\cdot\\|_0$-norm. Then $z_n$ converges to a continuous function $z$, and, since $\\|z_n\\|_0\leq\\|z_n\\|_{0,1}\leq r$, for all $n$, the maximum norm of $z$ is also bounded by $r$: $\\|z\\|_0\leq r$. We only have to show that the Lipschitz constant of $z$ is bounded by $r$, too. Let ${\varepsilon}>0$ be arbitrary and let $t\neq s$ be arbitrary in ${\mathbb{T}}$. We select some $n$ such that $\\|z-z_n\\|_0< {\varepsilon}\|t-s\|/2$. Then $$\begin{aligned} \|z(t)-z(s)\|&\leq\|z(t)-z_n(t)\|+\|z_n(t)-z_n(s)\|+\|z_n(s)-z(s)\|\\ &< {\varepsilon}\|t-s\|+r\|t-s\|\leq (r+{\varepsilon})\|t-s\|\mbox{.}\end{aligned}$$ Thus, the Lipschitz constant of $z$ is less than $r+{\varepsilon}$ for arbitrary ${\varepsilon}>0$. Hence, $\\|z\\|_{0,1}\leq r$, completing the argument for completeness of $B_r^{0,1}(x_0)$ with respect to the $\\|\cdot\\|_0$-norm. This completeness implies that we can apply Banach’s contraction mapping principle in a ball $B_r^{0,1}(x_0)$, a ball of Lipschitz continuous functions, using the weaker maximum norm in the following. We choose the radius $r$ of the ball equal to $6\delta$ ($\delta$ was chosen in the lemma such that the estimates are true for the constants $K$ and $R$), Thus, $B_{6\delta}^{0,1}(x_0)$ is the set to which we want to apply Banach’s contraction mapping principle. To ensure that the map $M_N(\cdot,p)$ maps into $B_{6\delta}^{0,1}(x_0)$ for $p\in U$, and that $M_N(\cdot,p)$ is a contraction we pick $N$ large enough. Specifically, we pick $N$ such that $$\label{eq:nchoice} \begin{aligned} \\|Q_Nx_0\\|_{0,1}&\leq 2\delta\mbox{,}& \\|Q_NL\\|_{0,1}&\leq \frac{2\delta}{R}\mbox{,}\\ \\|Q_NL\\|_0&\leq \frac{1}{2K}\mbox{,}& C\frac{\log N}{N}&< 1/\max\left\{1,\left(R+\\|x_0\\|_{1,1}\right)/\delta\right\}\mbox{,} \end{aligned}$$ where $R$ and $K$ are the bounds on $F$ given in the conditions of the lemma, in Equation . We know that these bounds exist due to Lemma \[thm:loclip\] (see Equation ) and Lemma \[thm:Flipbound\] (see Equation ). We know that choosing $N$ according to is possible from Lemma \[thm:qnl\] and estimate following Lemma \[thm:qnl\]. Let us check first that $x\mapsto E_Np+Q_NLF(x)$ maps the closed ball $B_{6\delta}^{0,1}(x_0)$ back into itself: $$\begin{aligned} \lefteqn{\left\\|E_Np+Q_NLF(x)-x_0\right\\|_{0,1}\leq}&\\ &\leq\left\\|E_N\left[p-R_Nx_0\right]-Q_Nx_0+Q_NLF(x)\right\\|_{0,1}\\ &\leq \left\\|E_N\left[p-R_Nx_0\right]\right\\|_{0,1}+ \left\\|Q_Nx_0\right\\|_{0,1}+ \left\\|Q_NL\right\\|_{0,1}\left\\|F(x)\right\\|_{0,1}\\ &< 2\delta+2\delta+\frac{2\delta}{R}R=6\delta\mbox{.}\end{aligned}$$ Here we used the bounds implied by our choice of $N$ and the definition of the set $U$ of permitted $p$, and the bound on $\\|F(x)\\|_{0,1}$, which is determined in by our choice of $\delta$. Second, let us check that $x\mapsto E_Np+Q_NLF(x)$ is a uniform contraction in $B_{6\delta}^{0,1}$ with respect to the $\\|\cdot\\|_0$-norm: $$\begin{aligned} \left\\|Q_NL\left[F(x)-F(y)\right]\right\\|_0\leq \left\\|Q_NL\right\\|_0\left\\|F(x)-F(y)\right\\|_0\leq \frac{1}{2K}K\\|x-y\\|_0 \leq \frac{1}{2}\\|x-y\\|_0\mbox{.}\end{aligned}$$ Again, we exploited the bounds , implied by our choice of $N$, and the Lipschitz constant $K$ of $F$ determined in by our choice of $\delta$. Since $B_{6\delta}^{0,1}(x_0)$ is complete with respect to the $\\|\cdot\\|$-norm Banach’s contraction mapping principle implies that the fixed point problem has a unique solution $x\in B_\delta^{0,1}(x_0)$ for $p\in U$. Finally, let us check that for $x\in B_\delta^{0,1}(x_0)\cap C^1({\mathbb{T}};{\mathbb{R}}^n)$ satisfying the periodic BVP $x'=F(x)$ the projection $p=R_Nx$ is in $U$. For this we have to prove that if $\\|x-x_0\\|_{0,1}\leq \delta$ and $x'=F(x)$ then $\\|P_N(x-x_0)\\|_{0,1}< 2\delta$. We can estimate $\\|P_N(x-x_0)\\|_{0,1}$ via $$\begin{aligned} \\|P_N(x-x_0)\\|_{0,1} &\leq \\|(I-Q_N)(x-x_0)\\|_{0,1} \leq \\|x-x_0\\|_{0,1}+\\|Q_N(x-x_0)\\|_{0,1} \label{eq:pnxx0:tri}\\ &\leq \delta +C\frac{\log N}{N}\\|x-x_0\\|_{1,1} \label{eq:pnxx0:estqnl}\\ &\leq\delta +C\frac{\log N}{N}\max\{\|x-x_0\\|_{0,1},\\|x'-x_0'\\|_{0,1}\} \label{eq:pnxx0:defn11}\\ &\leq \delta +C\frac{\log N}{N}\max\{\delta,\\|x'\\|_{0,1}+\\|x_0'\\|_{0,1}\} \label{eq:pnxx0:balltri}\\ &= \delta +C\frac{\log N}{N}\max\{\delta,\\|F(x)\\|_{0,1}+\\|x_0\\|_{1,1}\} \label{eq:pnxx0:xpFx}\\ &\leq \delta+C\frac{\log N}{N}\max\{\delta,R+\\|x_0\\|_{1,1}\}<2\delta\mbox{.} \label{eq:pnxx0:fbound}\end{aligned}$$ The inequality follows from the definition of $P_N$ and $Q_N$ and the triangular inequality for the $\\|\cdot\\|_{0,1}$-norm. The step from to uses the estimate for the norm $\\|Q_Ny\\|_{0,1}$ for elements $y$ of $C^{1,1}({\mathbb{T}};{\mathbb{R}}^n)$. It also bounds $\\|x-x_0\\|_{0,1}$ by the radius $\delta$ of the ball. Step splits up the $\\|\cdot\\|_{1,1}$ norm into its two parts which are estimated separately in the following steps. One part, $\\|x-x_0\\|_{0,1}$ is bounded by $\delta$ (the radius of the ball), the difference of the derivatives is bounded by a triangular inequality for its parts, $\\|x'\\|_{0,1}$ and $\\|x_0'\\|_{0,1}$ in . To get to we use that $x$ satisfies the BVP $x'=F(x)$. We also bound the norm of $x_0'$ by $\\|x_0\\|_{1,1}$. Finally, in we estimate the Lipschitz norm of $F(x)$, $\\|F(x)\\|_{0,1}$ by the bound $R$ determined in by our choice of $\delta$. The right-hand side of is (strictly) less than $2\delta$ by our choice of $N$, see . [$\square$]{} Lipschitz continuity of the algebraic system {#sec:algred} -------------------------------------------- The Splitting Lemma \[thm:split\] guarantees in combination with the unique existence of the fixed point of $M_N(\cdot,p)$, proven in Lemma \[thm:fixp\], the equivalence between the periodic BVP $\dot x(t)=f(\Delta_tx)$ and the algebraic equation $g(p)=0$ for $x$ inside the ball $B_\delta^{0,1}(x_0)$, where $g$ is given in by $$\begin{aligned} \label{eq:lowmodes} g&:p\in U\mapsto R_N\left[P_0F(X(p))+Q_0\left(E_Np-P_NLF(X(p))\right)\right] \in{\mathbb{R}}^{n\times(2\,N+1)}\mbox{, where}\\ \label{eq:xpdef} X&:p\in U\mapsto C^0({\mathbb{T}};{\mathbb{R}}^n) \mbox{,\quad and $X(p)$ is the fixed point of $M_N(\cdot,p)$ in $B_{6\delta}^{0,1}(x_0)$}\mbox{.}\end{aligned}$$ The relation between $p\in U$ and $x\in B_\delta^{0,1}(x_0)$ is given via $p=R_Nx$ and $x=X(p)$: if $x$ satisfies the periodic BVP then $p=R_Nx$ satisfies $g(p)=0$, and, vice versa, if $p\in U$ satisfies $g(p)=0$ then $x=X(p)$ satisfies the periodic BVP. The domain of definition, $D(X)=U$ is an open set, however the map $X$ (and, thus, $g$) can be extended continuously to the boundary of $U$: $M_N(\cdot,p)$ maps into the closed ball $B_{6\delta}^{0,1}$ back into itself also for $p$ on the boundary of $U$ and it still has contraction rate $1/2$ with respect to the $\\|\cdot\\|_0$-norm. The remainder of the section addresses the remaining open claim of the Equivalence Theorem \[thm:main\], namely the regularity of the maps $X$ and $g$. Using only local $EC$ Lipschitz continuity (Definition \[def:loclip\]) we can prove the Lipschitz continuity of $g$ and $X$: \[thm:lipalg\] 1. \[thm:Xsmooth1\] For all $p$ in the neighborhood $U=D(X)$, defined in , the image $X(p)$ is in $C^{1,1}({\mathbb{T}};{\mathbb{R}}^n)$ (that is, $X(p)\in C^1({\mathbb{T}};{\mathbb{R}}^n)$ and its time derivative is Lipschitz continuous), 2. \[thm:Xlip1\] $X$ is Lipschitz continuous with respect to the $\\|\cdot\\|_1$-norm for its images[:]{.nodecor} there exists a constant $C_N$ such that $$\\|X(p)-X(q)\\|_1\leq C_N\|p-q\|\mbox{ for all $p$ and $q$ in $U$,}$$ 3. \[thm:algnonlip\] the map $p\in U\mapsto \left[R_0F(X(p)),P_NLF(X(p))\right]\in{\mathbb{R}}^n\times{\mathbb{R}}^{n\times(2\,N+1)}$ is Lipschitz continuous in $U$. #### Proof {#proof-2 .unnumbered} For a function $y\in{\operatorname{rg}}P_N$, differentiation is a bounded operator: $y'=D_Ny$. The vector $R_Ny$ of the first $2N+1$ Fourier coefficients of a function $y$ and the vector $R_N[y']$ satisfy $R_N[y']=\tilde D_NR_Ny$ where $\tilde D_N$ is a matrix (independent of $y$). Hence, $y'=E_N\tilde D_NR_Ny$ for all $y\in{\operatorname{rg}}P_N$ such that we can define $D_N=E_N\tilde D_NR_N$. Denote $X(p)$ as $x$. By definition of the map $X$, $x=E_Np+Q_NLF(x)$. The right-hand side of this fixed-point equation is differentiable with respect to time, giving $$\label{eq:xpform} x'=D_NE_Np+Q_0F(x)-D_NP_NLF(x)\mbox{.}$$ This guarantees that $x\in C^1({\mathbb{T}};{\mathbb{R}}^n)$. Equation ensures that $\\|F(x)\\|_{0,1}\leq R$, which implies that the right-hand side of is Lipschitz continuous in time. This in turn implies that $x'$ is Lipschitz continuous in time (thus, $x\in C^{1,1}({\mathbb{T}};{\mathbb{R}}^n)$), and $$\\|x'\\|_{0,1} \leq\\|D_NE_N\\|_{0,1}\|p\|+\\|Q_0\\|_{0,1}R+\\|D_NP_NL\\|_{0,1}R\mbox{.}$$ Representation also implies point \[thm:Xlip1\]: let $x=X(p)$ and $y=X(q)$ be two functions in the image of $X$: $$\label{eq:xlip1norm} \\|x'-y'\\|_0\leq\\|D_NE_N\\|_0\|p-q\|+(\\|Q_0\\|_0+\\|D_NP_NL\\|_0)K\\|x-y\\|_0\mbox{,}$$ where $K$ was the $EC$ Lipschitz constant of $F$ in $B_{6\delta}^{0,1}(x_0)$. The difference $x-y$ in the $\\|\cdot\\|_0$-norm is bounded due to the contractivity of the right-hand side in fixed point problem defining $X$ (the $\\|\cdot\\|_0$-norm was the metric used to apply the contraction mapping principle): $$\begin{aligned} \\|x-y\\|_0&\leq \\|E_N\\|_0\|p-q\|+\\|Q_NL[F(x)-F(y)\\|_0\leq \\|E_N\\|_0\|p-q\|+\frac{1}{2}\\|x-y\\|_0\mbox{.} \intertext{Thus,} \\|x-y\\|_0&\leq 2\\|E_N\\|_0\|p-q\|\mbox{,}\end{aligned}$$ which, combined with , gives Lipschitz continuity of $X$ as a map from $U$ into $C^1({\mathbb{T}};{\mathbb{R}}^n)$: $$\label{eq:xlip10norm} \\|x'-y'\\|_0\leq\left[\\|D_NE_N\\|_0+(\\|Q_0\\|_0+\\|D_NP_NL\\|_0)2K\\|E_N\\|_0\right] \|p-q\|=:C_N\|p-q\|\mbox{.}$$ Point \[thm:algnonlip\] is a direct consequence of the Lipschitz continuity of $F$ with respect to $\\|\cdot\\|_0$-norm in $B_{6\delta}^{0,1}(x_0)$, the Lipschitz continuity of $X$ on $U$ in the $\\|\cdot\\|_0$-norm, and the fact that $X$ maps into $B_{6\delta}^{0,1}(x_0)$. [$\square$]{} First-order differentiability of the algebraic system {#sec:algdiff1} ----------------------------------------------------- Until now we have only used the $EC$ Lipschitz continuity (in the sense of Definition \[def:loclip\]) of the right-hand side $F$ in the ball $B_{6\delta}^{0,1}(x_0)$ with respect to the $\\|\cdot\\|_0$-norm. We can expect that the right-hand side $g$ of the algebraic system, defined in , is smooth only if we require more smoothness of the right-hand side $f$ (which enters $F$ in the algebraic system). We first discuss first-order differentiability of the map $X$ and the right-hand side $g$, defined in and . For this we assume $EC^1$ smoothness of $f$ as defined in Definition \[def:extdiff\]. For $x\in C^1({\mathbb{T}};{\mathbb{R}}^n)\cap B_{6\delta}^{0,1}(x_0)$ the norm of $\partial^1f(x,\cdot)$ as an element of $L(C^0({\mathbb{T}};{\mathbb{R}}^n);{\mathbb{R}}^n)$ (the space of linear functionals mapping $C^0({\mathbb{T}};{\mathbb{R}}^n)$ into ${\mathbb{R}}^n$) is less than or equal to $K$, the $EC$ Lipschitz constant of $F$ (and, hence, $f$) in $B_{6\delta}^{0,1}(x_0)$ assumed to exist in the conditions of Lemma \[thm:fixp\]. Let us define the map $$\begin{aligned} \partial^1F:&C^1({\mathbb{T}};{\mathbb{R}}^n)\times C^0({\mathbb{T}};{\mathbb{R}}^n)\mapsto C^0({\mathbb{T}};{\mathbb{R}}^n)\mbox{,} & \left[\partial^1F(v,w)\right](t)&= \partial^1f(\Delta_t v,\Delta_tw)\mbox{.}\end{aligned}$$ If $v\in C^1({\mathbb{T}};{\mathbb{R}}^n)$ and $w\in C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)$ then the map $\partial^1F$ defined above is indeed the derivative of $F$ in $v$ with respect to the deviation $w$ (see Lemma \[thm:Fdiff\] in Appendix \[sec:basicprop\]): $$\label{eq:Fdiff1} \lim_{ \begin{subarray}{c} w\in C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)\\[0.2ex] \\|w\\|_{0,1}\to0 \end{subarray} } \frac{\\|F(v+w)-F(v)-\partial^1F(v,w)\\|_0}{\\|w\\|_{0,1}}=0\mbox{.}$$ Part of the definition of $EC^1$ smoothness for $f$ is that the map $\partial^1f$ is continuous in both arguments, $v\in C^1({\mathbb{T}};{\mathbb{R}}^n)$ and $w\in C^0({\mathbb{T}};{\mathbb{R}}^n)$. One can then apply Lemma \[thm:Fcont\] to $\partial^1f$ to conclude that the map $\partial^1F$ (a composition of $\Delta_t$ and $\partial^1f$) is continuous with respect to the $\\|\cdot\\|_0$-norm in its image space as a map of both arguments (in their respective norm), too. For $v\in B_{6\delta}^{0,1}(x_0)$ the norm of the linear map $\partial^1F(v,\cdot)$ as an element of $L(C^0({\mathbb{T}};{\mathbb{R}}^n);C^0({\mathbb{T}};{\mathbb{R}}^n))$, the space of continuous linear functionals from $C^0({\mathbb{T}};{\mathbb{R}}^n)$ back to itself, is bounded by the $EC$ Lipschitz constant $K$ of $F$ in $B_{6\delta}^{0,1}(x_0)$. The additional regularity assumption on $f$ and its implications for $F$ permit us to improve our statements about regularity of $X$ and the algebraic system: \[thm:Xdiff\] Assume that the right-hand side $f$ is $EC^1$ smooth in the sense of Definition \[def:extdiff\]. Then the regularity statements about the map $X$, defined in , and the right-hand side of the algebraic system, defined in , can be extended[:]{.nodecor} 1. $X(p)$ is in $C^2({\mathbb{T}};{\mathbb{R}}^n)$ for all $p\in U=D(X)$, the domain of definition of $X$, and $p\mapsto X(p)$ is continuous with respect to the $\\|\cdot\\|_2$-norm for its images. 2. The map $X$, which maps $U$ into $C^1({\mathbb{T}};{\mathbb{R}}^n)$ according to Lemma \[thm:lipalg\], is continuously differentiable with respect to its argument $p$ using the $\\|\cdot\\|_1$-norm for its images. 3. The map $p\in U\mapsto \left[R_0F(X(p)),P_NLF(X(p))\right]\in{\mathbb{R}}^n\times{\mathbb{R}}^{n\times(2\,N+1)}$ is continuously differentiable with respect to $p$. #### Proof {#proof-3 .unnumbered} Let $p\in U=D(X)\subset {\mathbb{R}}^{n\times(2\,N+1)}$, where $U$ is defined in , and let us denote $X(p)$ by $x$. Lemma \[thm:lipalg\] ensures already that $x$ is in $C^{1,1}({\mathbb{T}};{\mathbb{R}}^n)$. Lemma \[thm:Fimage\] in Appendix \[sec:basicprop\] proves that $F(x)\in C^1({\mathbb{T}};{\mathbb{R}}^n)$ for $x\in C^1({\mathbb{T}};{\mathbb{R}}^n)$ (choosing $D=C^0({\mathbb{T}};{\mathbb{R}}^n)$ and $k=0$ in the assumptions of Lemma \[thm:Fimage\]). This implies the first statement, that $X(p)\in C^2({\mathbb{T}};{\mathbb{R}}^n)$: since $$\label{eq:Xdiff:proof:xpc2} X(p)=E_Np+Q_NLF(X(p))$$ and $X(p)\in C^{1,1}({\mathbb{T}};{\mathbb{R}}^n)$ (see Lemma \[thm:lipalg\]), $F(X(p))$ is in $C^1({\mathbb{T}};{\mathbb{R}}^n)$, and, thus, $LF(X(p))$ is in $C^2({\mathbb{T}};{\mathbb{R}}^n)$. Hence, $X(p)$ is an element of $C^2({\mathbb{T}};{\mathbb{R}}^n)$, too. Furthermore, Lemma \[thm:Fimage\] states that $F$ is continuous as a map from $C^1({\mathbb{T}};{\mathbb{R}}^n)$ into $C^1({\mathbb{T}};{\mathbb{R}}^n)$. Since $X$ is continuous as a map from $U$ into $C^1({\mathbb{T}};{\mathbb{R}}^n)$ (in fact, it is Lipschitz continuous, see Lemma \[thm:lipalg\]), the right-hand side of in $p$ is continuous with respect to the $\\|\cdot\\|_1$-norm. This proves the first point. Concerning the second statement: again, let $p_0$ be in $U=D(X)$, and choose a small open neighborhood $U(p_0)$ which has a positive distance to the boundary of $U$. We will prove point two for all $p\in U(p_0)$. Let us choose an initial ${\varepsilon}_0$ sufficiently small such that $p+hq$ is still in $U$ for $h\in(-{\varepsilon}_0,{\varepsilon}_0)$, all $p\in U(p_0)$, and all $q$ with $\|q\|\leq 1$. Let us introduce the difference quotient for $h\in(-{\varepsilon}_0,{\varepsilon}_0)\setminus\{0\}$: $$\label{eq:Xdiffproof:zdef} z(h,q,p)=\frac{1}{h}\left[X(p+hq)-X(p)\right] \mbox{.}$$ The maps $z$ maps $\left[(-{\varepsilon}_0,{\varepsilon}_0)\setminus\{0\}\right]\times B_1(0)\times U(p_0)\subset {\mathbb{R}}\times {\mathbb{R}}^{n\times(2\,N+1)}\times {\mathbb{R}}^{n\times(2\,N+1)}$ into $C^1({\mathbb{T}};{\mathbb{R}}^n)$. We first prove that $z$ has a limit for $h\to0$ in $C^1({\mathbb{T}};{\mathbb{R}}^n)$, and that this limit is achieved uniformly for all $p\in U(p_0)$ and $\|q\|\leq 1$. By definition of $X$, $z$ satisfies the fixed point equation (dropping all arguments from $z$) $$\label{eq:gfixpd0} z=E_Nq+Q_NL\frac{1}{h}\left[F(X(p)+hz)-F(X(p))\right]$$ for $h\in(-{\varepsilon}_0,{\varepsilon}_0)\setminus\{0\}$. Let us introduce $$\label{eq:Xdiffproof:tA1def} \tilde A_1(p,z,h)= \begin{cases} \frac{1}{h}\left[F(X(p)+hz)-F(X(p))\right] & \mbox{if $h\neq0$}\\ \partial^1F(X(p),z) & \mbox{if $h=0$,} \end{cases}$$ which maps $U(p_0)\times C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)\times{\mathbb{R}}$ into $C^0({\mathbb{T}};{\mathbb{R}}^n)$. The limit implies that $\tilde A_1$ is continuous in all arguments (insert $v=x$, $w=hz$ into ). Using $\tilde A_1$ we extend the fixed point problem to $h=0$: $$\label{eq:gfixpd1} z=E_Nq+Q_NL\tilde A_1(X(p),z,h)\mbox{.}$$ The following intermediate lemma proves that the fixed point problem has a unique solution: \[thm:fixplin\] There exists an ${\varepsilon}>0$ and constants $C_0>0$ and $C_1>0$ such that the map $$\gamma:z\in C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)\mapsto E_Nq+Q_NL\tilde A_1(X(p),z,h)\in C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)\mbox{,}$$ which depends on the additional parameters $p$, $q$ and $h$, has a unique fixed point $z_$ in $$B=\{z\in C^{0,1}({\mathbb{T}};{\mathbb{R}}^n):\\|z\\|_0\leq C_0 \mbox{ and } \\|z\\|_{0,1}\leq C_1\}$$ for all $h\in(-{\varepsilon},{\varepsilon})$, all $p\in U(p_0)\subset U=D(X)\subset{\mathbb{R}}^{n\times(2\,N+1)}$ and all $q\in{\mathbb{R}}^{n(2N+1)}$ with $\|q\|<1$. The fixed point $z_$ is an element of $C^1({\mathbb{T}};{\mathbb{R}}^n)$ and depends continuously on $h$, $p$ and $q$ with respect to the $\\|\cdot\\|_1$-norm. Note that the ${\varepsilon}$ we have to choose in Lemma \[thm:fixplin\] is smaller than the initial ${\varepsilon}_0$ for which the difference quotient $z$ is defined. #### Proof of Lemma \[thm:fixplin\] {#proof-of-lemmathmfixplin .unnumbered} First of all, since $\tilde A_1$ is continuous in all arguments, the map $\gamma$ is continuous. Moreover, since $x'=(X(p))'$ and $x=X(p)$ depend continuously on $p$ (see Lemma \[thm:lipalg\] and expression ), the map $\gamma$ also depends continuously on the parameters $p$, $q$ and $h$ (that is, the expression $E_nq+Q_NL\tilde A_1(X(p),z,h)$, defining $\gamma$, depends continuously on $z$, $p$, $q$ and $h$ with respect to the $\\|\cdot\\|_{0,1}$-norm). We choose the constants $C_0>0$ and $C_1>0$ such that $$\begin{aligned} \label{eq:c0choice} C_0&\geq2\\|E_N\\|_0\\ \label{eq:c1choice} C_1&\geq \\|D_NE_N\\|_0+ \left(\\|Q_0\\|_0+\\|D_NP_NL\\|_0\right)KC_0\mbox{,}\end{aligned}$$ where $K$ is the Lipschitz constant of $F$ with respect to the $\\|\cdot\\|_0$-norm in $B_{6\delta}^{0,1}$. We choose ${\varepsilon}\leq{\varepsilon}_0$ such that for all $z$ satisfying $\\|z\\|_{0,1}\leq C_1$ and all $p\in U(p_0)$ the function $X(p)+hz$ is in $B_{6\delta}^{0,1}(x_0)$ for all $h\in(-{\varepsilon},{\varepsilon})$. This implies that for any $z_1$ and $z_2$ satisfying $\\|z_1\\|_{0,1}\leq C_1$ and $\\|z_2\\|_{0,1}\leq C_1$ we have $$\begin{split} \frac{1}{h}\\|F(X(p)+hz_1)-F(X(p)+hz_2)\\|_0&\leq K\\|z_1-z_2\\|_0\mbox{,}\\ \\|\partial^1F(X(p),z_1-z_2)\\|_0&\leq K\\|z_1-z_2\\|_0\mbox{,} \end{split}\label{eq:g0lip}$$ where $K$ was the Lipschitz constant for $F$ in $B_{6\delta}^{0,1}(x_0)$, and, thus, $$\begin{aligned} \\|\gamma(z_1)-\gamma(z_2)\\|_0&\leq \frac{1}{2}\\|z_1-z_2\\|_0\label{eq:g0contract}\end{aligned}$$ for all $h\in(-{\varepsilon},{\varepsilon})$ by choice of $N$ ($N$ was such that $\\|Q_NL\\|_0\leq (2K)^{-1}$). This estimate for $\gamma$ implies $$\label{eq:g0bound} \\|\gamma(z)\\|_0\leq \\|E_N\\|_0+\frac{1}{2}\\|z\\|_0 \mbox{\qquad if $\\|z\\|_{0,1}\leq C_1$,}$$ since $\gamma(0)=E_Nq$ and $\|q\|\leq1$. Moreover, the two inequalities imply that for $h\in(-{\varepsilon},{\varepsilon})$, $\\|z\\|_{0,1}\leq C_1$ and $p\in U(p_0)$ the maximum norm of $\tilde A_1(p,z,h)$ is bounded by $K\\|z\\|_0$: $$\label{eq:tA1bound} \\|\tilde A_1(p,z,h)\\|_0\leq K\\|z\\|_0$$ The time derivative of $\gamma(z)$ exists and its $\\|\cdot\\|_0$-norm can be estimated by differentiating the expression $E_nq+Q_NL\tilde A_1(X(p),z,h)$, defining $\gamma$, with respect to time in the same manner as we obtained (we insert to bound $\\|\tilde A_1(p,z,h)\\|_0$): $$\label{eq:g1bound} \left\\|\frac{{\mathop{}\!\mathrm{d}}}{{\mathop{}\!\mathrm{d}}t}\gamma(z)\right\\|_0\leq \\|D_NE_N\\|_0+(\\|Q_0\\|_0+\\|D_NP_NL\\|_0)K\\|z\\|_0\mbox{.}$$ The combination of the bounds and and the definition of the constants $C_0$ and $C_1$ guarantee that $\gamma(z)$ maps the set $$B=\{z\in C^{0,1}({\mathbb{T}};{\mathbb{R}}^n):\\|z\\|_0\leq C_0 \mbox{ and } \\|z\\|_{0,1}\leq C_1\}$$ back into itself. The contraction estimate for the $\\|\cdot\\|_0$-norm and the completeness of $B$ with respect to the $\\|\cdot\\|_0$-norm make the contraction mapping principle applicable with a uniform contraction rate for all $p\in U(p_0)$, all $\|q\|\leq1$ and $h\in(-{\varepsilon},{\varepsilon})$. This ensures that the fixed point $z_$ depends continuously on $p$, $q\in{\mathbb{R}}^{n(2N+1)}$ and $h\in(-{\varepsilon},{\varepsilon})$ with respect to the $\\|\cdot\\|_0$-norm (since $\gamma$ is continuous with respect to $z$, $h$, $q$ and $p$). The time derivative $z_'$ of $z_$ also exists and is continuous in $p$, $q$ and $h$: we differentiate the fixed point equation for $z_$ with respect to time (in the same way as done in ) to get $$\begin{aligned} \label{eq:zpform} z_'=&D_NE_Nq+Q_0\tilde A_1(X(p),z_,h)- D_NP_NL\tilde A_1(X(p),z_,h)\mbox{,}\end{aligned}$$ which is a continuous function in $p$, $q$ and $h$ with respect to the $\\|\cdot\\|_0$-norm (note that $z_$ depends on $p\in U(p_0)$, $q$ and $h$). Thus, the fixed point $z_$ is in $C^1({\mathbb{T}};{\mathbb{R}}^n)$ and depends continuously on $p$, $q$ and $h$ with respect to the $\\|\cdot\\|_1$-norm.[$\square$]{} #### Proof of Lemma \[thm:Xdiff\] continued {#proof-of-lemmathmxdiff-continued .unnumbered} As a consequence of Lemma \[thm:fixplin\] we may write the fixed point $z_$ of $\gamma$ as a function of $h$, $q$ and $p$: $z_(h,q,p)$ maps $h\in(-{\varepsilon},{\varepsilon})$, $q$ in the unit ball of ${\mathbb{R}}^{n\times(2\,N+1)}$ and $p\in U(p_0)$ continuously into $C^1({\mathbb{T}};{\mathbb{R}}^n)$. It is also identical to $z(h,q,p)$, defined in as the directional difference quotient of $X$. Thus, the directional difference quotient $z(h,q,p)$ has a limit for $h\to0$ in the $\\|\cdot\\|_1$-norm, and this limit equals $z_(0,q,p)$. Moreover, this limit $z_(0,q,p)$ depends continuously on $p$ and $q$ in the $\\|\cdot\\|_1$-norm (as proved in Lemma \[thm:fixplin\]), and it is linear in $q$ (since $\tilde A_1(p,z,0)$ is linear in $z$). Thus, $z_(0,q,p)$ is the Frech[é]{}t derivative: $$\label{eq:Xdiff1proof:Frechet0} \lim_{q\to0}\frac{\\|X(p+q)-X(p)-z_(0,q,p)\\|_1}{\|q\|}=0\mbox{.}$$ Consequently, the map $(p,q)\mapsto z_(0,q,p)=\partial^1X(p)\,q$ is continuous in the $\\|\cdot\\|_1$-norm as claimed in the lemma. The third statement of Lemma \[thm:Xdiff\] is a consequence of the second statement and the fact that the difference quotient of $F$ has a limit in the $\\|\cdot\\|_0$-norm if it is taken between arguments in $C^1({\mathbb{T}};{\mathbb{R}}^n)$ (see ). We split the difference quotients into two parts: $$\begin{aligned} \label{eq:Fd1expr1} \frac{F(X(p+hq))-F(X(p))}{h}=&\ \frac{F(X(p)+h\partial^1 X(p)q)-F(X(p))}{h} +\\ &\ +\frac{F(X(p+hq))-F(X(p)+h\partial^1 X(p)q)}{h}\label{eq:Fd1expr2}\end{aligned}$$ The right-hand side in converges in the $\\|\cdot\\|_0$-norm to $\partial^1F(X(p),\partial^1X(p)q)$ for $h\to0$, since $X(p)$ and $\partial^1X(p)q$ are in $C^1({\mathbb{T}};{\mathbb{R}}^n)$ because $F$ is $EC^1$ continuous (see the second point of the lemma for the regularity of $\partial^1X(p)q$ and Lemma \[thm:lipalg\] for the regularity of $X(p)$). For the term in we can apply the local $EC$ Lipschitz continuity (all arguments are in $B_{6\delta}^{0,1}(x_0)$ for $p\in U(p_0)$, $\|q\|\leq 1$ and $h\in({\varepsilon},{\varepsilon})$) such that we get $$\left\\|\frac{F(X(p+hq))-F(X(p)+h\partial X(p)q)}{h}\right\\|_0\leq K\left\\|\frac{X(p+hq)-X(p)}{h}- \partial X(p)q\right\\|_0\mbox{,}$$ which converges to $0$ for $h\to0$ due to the second statement of the lemma ($K$ is the $EC$ Lipschitz constant of $F$ in $B_{6\delta}^{0,1}(x_0)$). Consequently, we obtain from the limit of for $h\to0$ that the directional derivative of $F(X(p))$ in direction $q$ is equal to $\partial^1F(X(p),\partial^1 X(p)q)$, which is continuous with respect to $p$ and $q$ and linear in $q$. Thus, $$\label{eq:Fd1expr} \left[\frac{\partial}{\partial p}F(X(p))\right]q =\partial^1F(X(p),\partial X(p)q)\mbox{,}$$ and $p\mapsto F(X(p))$ is continuously differentiable with respect to $p$ in the $\\|\cdot\\|_0$-norm. Note that we use the notation not enclosing $q$ in the bracket in to highlight that this is a classical derivative with respect to a finite-dimensional variable. The linear operators $R_0$ and $P_NL$ preserve the continuity (and the linearity in $q$) of . [$\square$]{} \[thm:specradius\] For $x=X(p)$ (where $p\in U=D(X)$) consider the linear map $$M:z\mapsto Q_NL\partial^1F(x,z)\mbox{.}$$ The spectral radius of $M$ as a map from $C^0({\mathbb{T}};{\mathbb{R}}^n)$ back into itself, or as a map from $C^1({\mathbb{T}};{\mathbb{R}}^n)$ back into itself, is less or equal $1/2$. #### Proof {#proof-4 .unnumbered} Since $M$ is compact as an element of $L(C^k({\mathbb{T}};{\mathbb{R}}^n);C^k({\mathbb{T}};{\mathbb{R}}^n))$ (the space of linear functionals from $C^k({\mathbb{T}};{\mathbb{R}}^n)$ back to itself) for $k=0$ and $k=1$, the spectral radius is identical to the modulus of the maximal (in modulus) eigenvalue, which is of finite algebraic multiplicity if it is different from zero. An eigenvector $z$ corresponding to this maximal eigenvalue is an element of $C^1({\mathbb{T}};{\mathbb{R}}^n)$ such that the spectral radius of $M$ is the same for $k=0$ and $k=1$. Since $x$ and $z$ are both in $C^1({\mathbb{T}};{\mathbb{R}}^n)$ we have that $$\label{eq:sradproof:d1f} \partial^1F(x)z=\lim_{h\to0}\frac{1}{h}\left[F(x+hz)-F(x)\right]$$ For $x=X(p)$ where $p\in U$, and $h$ sufficiently small the arguments of $F$, $x+hz$ and $x$, both lie inside $B_{6\delta}^{0,1}$ such that the $EC$ Lipschitz constant $K$ applies to the difference: $$\label{eq:sradproof:flip} \frac{1}{h}\\|F(x+hz)-F(x)\\|_0\leq K\\|z\\|_0\mbox{.}$$ Since $\\|Q_NL\\|_0\leq 1/(2K)$, and combine to $$\\|Mz\\|_0\leq \frac{1}{2}\\|z\\|_0\mbox{.}$$ As $z$ is an eigenvector corresponding to the largest eigenvalue, the spectral radius of $M$ is less or equal $1/2$.[$\square$]{} Thus, the derivative $z={\partial}X(p)\,q$ of $X$ in $p$ is the unique solution of the contractive linear fixed point problem in $C^1({\mathbb{T}};{\mathbb{R}}^n)$ $$\label{eq:dxfixp} z=E_Nq+Q_NL\partial^1F(X(p),z)\mbox{.}$$ Higher degrees of smoothness {#sec:smooth} ---------------------------- We observe that $(x,y)=(X(p),\partial^1X(p)\,q)$ satisfies the system of equations $$\label{eq:fixp:ext} \begin{split} x&=E_Np+Q_NLF(x)\\ y&=E_Nq+Q_NL\partial^1F(x,y)\mbox{.} \end{split}$$ This has a similar structure to the original fixed point problem but in dimension $n_1=2n$ with the variables $(x,y)$ and parameters $(p,q)$. Thus, we aim to apply a linear version of the arguments of Section \[sec:algdiff1\] recursively, assuming that $f$ is $EC^k$ smooth as recursively defined in Definition \[def:extdiff\]. Throughout this section we assume that $f$ is $EC^k$ smooth for all degrees up to order $j_{\max}$. For higher-order derivatives, we introduce the spaces $D_j$ and the operators $\partial^jF$ for $j\geq0$ recursively: $$\begin{aligned} D_0&=C^0({\mathbb{T}};{\mathbb{R}}^n) & D_j&=D_{j-1}^1\times D_{j-1}\\ \partial^jF:&D_j\mapsto C^0({\mathbb{T}};{\mathbb{R}}^n)\mbox{,} & [\partial^jF(x)](t)&=\partial^jf(\Delta_tx)\mbox{.}\end{aligned}$$ The spaces $D_j$ are products of the type , and the argument $x$ of $\partial^jF$ and $\partial^jf$ is in $D_j$, a product of $2^j$ spaces. We also recall that the notion of subspaces $D_j^k$ of higher-oder ($k\geq0$) differentiability for product spaces such as $D_j$ was introduced in Section \[sec:theorem\]. For example, $$\begin{aligned} D_0^k&=C^k({\mathbb{T}};{\mathbb{R}}^n)\mbox{,}\\ D_1^k&=D_0^{k+1}\times D_0^k=C^{k+1}({\mathbb{T}};{\mathbb{R}}^n)\times C^k({\mathbb{T}};{\mathbb{R}}^n)\mbox{,}\\ D_2^k&=D_1^{k+1}\times D_1^k=C^{k+2}({\mathbb{T}};{\mathbb{R}}^n)\times C^{k+1}({\mathbb{T}};{\mathbb{R}}^n)\times C^{k+1}({\mathbb{T}};{\mathbb{R}}^n)\times C^k({\mathbb{T}};{\mathbb{R}}^n)\mbox{, etc.,}\end{aligned}$$ all with their natural maximum norms. The maps $\partial^jF$ are all continuous and map indeed into $C^0({\mathbb{T}};{\mathbb{R}}^n)$ due to the continuity of $\partial^jf$ and $\Delta_t$ (applying Lemma \[thm:Fcont\] to $D_j$, $\partial^jF$ and $\partial^jf$). It is also clear from the definition that $\partial^{j+k}F=\partial^j[\partial^kF]$ if $j+k\leq j_{\max}$. We will also use the notation $L(D_j^k;D_i^\ell)$ for the space of linear bounded functionals mapping from $D_k^k$ into $D_i^\ell$. The following lemma is a consequence of the $EC^k$ smoothness of $f$. \[thm:rhs:smooth\] For $j+k\leq j_{\max}$ the operator $\partial^jF$ is a continuous map from $D_j^k$ into $C^k({\mathbb{T}};{\mathbb{R}}^n)$. #### Proof of Lemma {#proof-of-lemma .unnumbered} We have to apply Lemma \[thm:Fimage\] from Appendix \[sec:basicprop\] inductively over the order of differentiability ($k$). To start the induction for $k=0$ we can apply Lemma \[thm:Fcont\] to $D_j$, $\partial^jF$ and $\partial^jf$. For the inductive step let us assume that for $k$ we know that $\partial^jF:D_j^k\mapsto C^k({\mathbb{T}};{\mathbb{R}}^n)$ is continuous for all $j\leq j_{\max}-k$. Let us fix a $j\leq j_{\max}-k-1$. We have to show that $\partial^jF$ maps $D_j^{k+1}$ continuously into $C^{k+1}({\mathbb{T}};{\mathbb{R}}^n)$. We know (by inductive assumption) that $\partial^jF$ maps $D_j^k$ continuously into $C^k({\mathbb{T}};{\mathbb{R}}^n)$ and that $\partial^{j+1}F$ maps $D_{j+1}^k=D_j^{k+1}\times D_j^k$ continuously into $C^k({\mathbb{T}};{\mathbb{R}}^n)$. Thus, we can apply Lemma \[thm:Fimage\] to $\partial^jF$ (this takes the place of the operator $F$ in Lemma \[thm:Fimage\]) and $D=D_j^k$, obtaining that $\partial^jF:D_j^{k+1}\mapsto C^{k+1}({\mathbb{T}};{\mathbb{R}}^n)$ is continuous. [$\square$]{} An immediate consequence of Lemma \[thm:rhs:smooth\] is that $X(p)$ and $\partial X(p)\,q$, as constructed in Section \[sec:algdiff1\], are as smooth as the right-hand-side: \[thm:xtsmooth\] Let $f$ be $EC^{j_{\max}}$ smooth. For every $p\in U=D(X)$ and every $q\in R^{n(2N+1)}$ the functions $X(p)$ and $\partial X(p)\,q$ satisfy $X(p)\in C^{j_{\max}+1}({\mathbb{T}};{\mathbb{R}}^n)$ and $\partial X(p)\,q\in C^{j_{\max}}({\mathbb{T}};{\mathbb{R}}^n)$. Moreover, the maps $$\begin{aligned} p&\mapsto X(p)\in C^{j_{\max}+1}({\mathbb{T}};{\mathbb{R}}^n)\mbox{\quad and\quad} [p,q]\mapsto \partial X(p)\,q\in C^{j_{\max}}({\mathbb{T}};{\mathbb{R}}^n) \end{aligned}$$ are continuous. #### Proof {#proof-5 .unnumbered} The function $x=X(p)$ satisfies $x=E_Np+Q_NLF(x)$. Since $F$ maps $D_0^k=C^k({\mathbb{T}};{\mathbb{R}}^n)$ back into itself continuously for all $k\leq j_{\max}$, $Q_NL$ maps $D_0^k$ into $D_0^{k+1}$ continuously for all $k$, and $E_Np\in C^\infty({\mathbb{T}};{\mathbb{R}}^n)$, the fixed point equation implies the following: if $x\in D_0^k$ then $F(x)\in D_0^k$, thus, $x=E_Np+Q_NLF(x)\in D_0^{k+1}$(for all $k\leq j_{\max}$). Similarly, $z=E_Nq+Q_NL\partial^1F(x)\,z$, and $\partial^1F$ maps $D_1^k$ into $D_0^k$ for all $k\leq j_{\max}-1$. Thus, the fixed point equation implies: if $z\in D_0^k$ and $x\in D_0^{k+1}$ then $(x,z)\in D_1^k$, thus, $\partial^1F(x,z)\in D_0^k$, thus, $z=E_Nq+Q_NL\partial^1F(x,z)\in D_0^{k+1}$ for all $k\leq j_{\max}-1$. All of the above dependencies are continuous such that the continuous dependence on $p$ and $q$ in the norms of $D_0^{j_{\max}+1}$ and $D_0^{j_{\max}}$, respectively, follows.[$\square$]{} We plan to find the derivatives of the map $X$ inductively through fixed point equations of the form . In order to set up the recursion we define inductively the operators $F_j$ by $$\begin{aligned} F_0(x)&=F(x) &&\mbox{for $x\in D_0$}\label{eq:F0def}\\ F_j \begin{pmatrix} x\\ y \end{pmatrix} &= \begin{bmatrix} F_{j-1}(x)\\ \partial^1F_{j-1}(x,y) \end{bmatrix}\mbox{,} &&\mbox{for\ } \begin{bmatrix} x\\ y \end{bmatrix}\in D_j=D_{j-1}^1\times D_j\mbox{.}\label{eq:Fjdef}\end{aligned}$$ Note that $F_j$ is always linear in its second argument, $y$, since $\partial^1F_{j-1}$ is linear in its second argument. The operators $F_j$ are combinations of derivatives of $F$. The plan is to study fixed-point problems of the type $x=E_Np+Q_NLF_j(x)$ (with $j=1$ we obtain ). Before doing so, we establish which spaces the operators $F_j$ map into: \[thm:rhs:image\] For $j+l+k\leq j_{\max}$ the operator $\partial^lF_j$ maps $D_{j+l}^k$ continuously into $D_j^k$. In particular, $F_j$ maps $D_j$ continuously back into itself. #### Proof {#proof-6 .unnumbered} The statement of the lemma follows inductively from the definition of $F_j$ and $D_j^k$. We apply Lemma \[thm:rhs:smooth\] to start our induction over $j$ (for $j=0$ the statement is identical to Lemma \[thm:rhs:smooth\]). For the inductive step let us assume that we know that $\partial^lF_{j-1}$ maps $D_{j+l-1}^k$ continuously into $D_{j-1}^k$ for all $k$ and $l$ satisfying $l+k\leq j_{\max}-j+1$. By definition of $F_j$ the derivative $\partial^lF_j$ for $l\leq j_{\max}-j$ is $$\begin{aligned} \partial^lF_j \begin{pmatrix} x\\ y \end{pmatrix}= \begin{bmatrix} \partial^lF_{j-1}(x)\\ \partial^{l+1}F_{j-1}(x,y) \end{bmatrix} &&\mbox{for\ } \begin{bmatrix} x\\ y \end{bmatrix}\in D_{l+j}=D_{l+j-1}^1\times D_{l+j-1}\mbox{.}\end{aligned}$$ The first component, $\partial^lF_{j-1}$ maps $D_{l+j-1}^{k+1}$ continuously into $D_{j-1}^{k+1}$ for all $k$ from $0$ to $j_{\max}-l-j$ (this is the assumption of the inductive step when one shifts the index $k$ down by $1$). Similarly, $\partial^{l+1}F_{j-1}$ maps $D_{j+l-1}^{1+k}\times D_{j+l-1}^k=D_{j+l}^k$ continuously into $D_{j-1}^k$ for all $k$ from $0$ to $j_{\max}-l-j$, again due to the assumption of the inductive step. Consequently, $\partial^lF_j$ maps $D_{j+l-1}^{k+1}\times D_{j+l-1}^k=D_{j+l}^k$ continuously into $D_{j}^k$ for all $k$ from $0$ to $j_{\max}-l-j$, which is the statement we had to prove for the inductive step.[$\square$]{} Even though the map $x\in D_j^1\mapsto \partial^1F_j(x,\cdot)\in L(D_j;D_j)$ is in general not continuous, the following map is: \[thm:speccont\] For $j<j_{\max}$ the map $x\in D_j^1\mapsto Q_NL\partial^1F_j(x,\cdot)\in L(D_j^1;D_j^1)$ is continuous with respect to $x\in D_j^1$. #### Proof of Lemma \[thm:speccont\] {#proof-of-lemmathmspeccont .unnumbered} The $EC^k$ smoothness of $f$ (for $k\leq j_{\max}$) implies that $F_j$ is continuously differentiable (in the classical sense) as a map from $D_j^1$ into $D_j$. Thus, the map $x\mapsto\partial^1F_j(x,\cdot)$ as a map from $D_j^1$ into $L(D_j^1;D_j)$ is continuous. Recall that the operator $L$ involves taking the anti-derivative of its argument such that $L:D_j\mapsto D_j^1$. Since $Q_NL$ maps $D_j$ continuously into $D_j^1$, the map $x\mapsto Q_NL\partial^1F_j(x,\cdot)$ is continuous as a map from $D_j^1$ into $L(D_j^1;D_j^1)$. [$\square$]{} The following theorem provides continuous differentiability of order $j_{\max}$ for $X$ and the map $p\mapsto F(X(p))$ if the right-hand side is $EC^k$ smooth in the sense of Definition \[def:extdiff\] for $k\leq j_{\max}$: \[thm:smooth\] Define $n_0=n(2N+1)$ and $n_j=2^jn_0$, and the maps $$\begin{aligned} X_0&: p\in U=D(X)\subseteq {\mathbb{R}}^{n_0}\mapsto X(p)\in D_0\mbox{\ and}\\ Y_0&:p\in U=D(X)\subseteq {\mathbb{R}}^{n_0}\mapsto F(X(p))\in D_0\mbox{,} \end{aligned}$$ and assume that $f:D_0=C^0({\mathbb{T}};{\mathbb{R}}^n)\mapsto{\mathbb{R}}^n$ is $EC^{j_{\max}}$ smooth. Then the following maps exist and are continuous for all $j$ up to $j_{\max}$: $$\begin{aligned} X_j&:[p,q]\in D(X_j):=D(X_{j-1})\times{\mathbb{R}}^{n_{j-1}}\subseteq{\mathbb{R}}^{n_j} \mapsto [X_{j-1}(p),\partial X_{j-1}(p)\,q]\in D_j\mbox{,}\\ Y_j&:[p,q]\in D(X_j) \phantom{\ :=D(X_{j-1})\times{\mathbb{R}}^{n_{j-1}}\subseteq{\mathbb{R}}^{n_j}} \mapsto [Y_{j-1}(p),\partial Y_{j-1}(p)\,q]\in D_j\mbox{.} \end{aligned}$$ The proof of Theorem \[thm:smooth\] does not require the application of the contraction mapping principle for nonlinear maps. It uses only Lemma \[thm:specradius\], Lemma \[thm:rhs:image\] and Lemma \[thm:speccont\] inductively. #### Proof of Theorem \[thm:smooth\] {#proof-of-theoremthmsmooth .unnumbered} The main work is the proof of the existence and continuity of $X_j$, which we will do first.The assumption of the inductive step is comprised of the following two statements. We assume for $j$: 1. \[p:ass:exfix0\] The map $(p_1,p_2)\in D(X_{j-1})\times {\mathbb{R}}^{n_{j-1}}\mapsto X_j(p_1,p_2)\in D_j$ exists and is continuous. Moreover, the pair $(x_1,x_2)=X_j(p_1,p_2)$ satisfies $$\begin{aligned} x_1&=E_Np_1+Q_NLF_{j-1}(x_1)\label{eq:ivf}\\ x_2&=E_Np_2+Q_NL\partial^1F_{j-1}(x_1,x_2)\mbox{.}\label{eq:ivderiv} \end{aligned}$$ 2. \[p:ass:specradius\] The linear map $z\mapsto Q_NL\partial^1F_{j-1}(x_1,z)$ maps $D_{j-1}^1$ back into itself and has spectral radius less or equal $1/2$. Both statements of the assumption of the inductive step have been proven for $j=1$ in Lemma \[thm:Xdiff\] and Lemma \[thm:specradius\] . Let $j$ be smaller than $j_{\max}$. ### Regularity of $X_j(p)$ {#regularity-of-x_jp .unnumbered} Let us first establish that the map $$p= \begin{bmatrix} p_1\\ p_2 \end{bmatrix}\mapsto x= \begin{bmatrix} x_1\\ x_2 \end{bmatrix} =X_j \begin{pmatrix} p_1\\ p_2 \end{pmatrix} = \begin{bmatrix} X_{j-1}(p_1)\\ \partial^1X_{j-1}(p_1)\,p_2 \end{bmatrix}$$ does not only map continuously into $D_j$ but into $D_j^k$ for all $k\leq j_{\max}-j+1$. The argument is the same as in the proof of Lemma \[thm:xtsmooth\]: the map $F_j$ maps $D_j^k$ continuously back into $D_j^k$ for all $k\leq j_{\max}-j$. If $x\in D_j^k$ then $F_j(x)\in D_j^k$, thus, $x=E_Np+Q_NLF(x)\in D_j^{k+1}$ for all $k\leq j_{\max}-j$ (and the dependence on $p$ is continuous because all dependencies are continuous). ### Proof of existence and continuity of $\partial^1X_j(p)\,q$ {#proof-of-existence-and-continuity-of-partial1x_jpq .unnumbered} Let us use the notation $p=(p_1,p_2)$ and $x=(x_1,x_2)=X_j(p)$. Let $p_0\in D(X_j)$ be arbitrary. We first show that $\partial^1X_j(p)\,q$ exists for all $p$ in a neighborhood $U(p_0)$ with positive distance to the boundary of $D(X_j)$. We can choose ${\varepsilon}>0$ sufficiently small such that $p+hq\in D(X_j)$ for all $h\in(-{\varepsilon},{\varepsilon})$, all $q=(q_1,q_2)$ with $\|q\|<1$ and all $p\in U(p_0)$. Consider the difference quotient $$\frac{X_j(p+hq)-X_j(p)}{h}= \frac{1}{h} \begin{bmatrix} X_{j-1}(p_1+hq_1)-X_{j-1}(p_1)\\ \partial^1X_{j-1}(p_1+hq_1)\,[p_2+hq_2]-\partial^1X_{j-1}(p_1)\,p_2 \end{bmatrix}=: \begin{bmatrix} z_1\\ z_2 \end{bmatrix} \mbox{.}$$ By assumption of the inductive step, $X_{j-1}$ is continuously differentiable such that the first row of this difference quotient has the form $$\label{eq:xjm1p1q1} z_1(h,p_1,q_1)=\frac{1}{h}\left[X_{j-1}(p_1+hq_1)-X_{j-1}(p_1)\right]= \int_0^1\partial^1X_{j-1}(p_1+hsq_1)\,q_1{\mathop{}\!\mathrm{d}}s$$ for $h\neq0$. As established above $(p_1,q_1)\mapsto \partial^1X_{j-1}(p_1)\,q_1 \in D_{j-1}^k$ is continuous for all $k\leq j_{\max}-j+1$ such that $$z_1(h,p_1,q_1)\in D_{j-1}^{j_{\max}-j+1} \subseteq D^2_{j-1}$$ ($j_{\max}-j+1\geq2$ since $j<j_{\max}$), and $z(h,p_1,q_1)$ depends continuously on its arguments, also when $h=0$. Let us use the abbreviations $$\begin{aligned} x_1(p_1)&=X_{j-1}(p_1)\mbox{,} \\ x_2(p_1,p_2)&=\partial^1X_{j-1}(p_1)p_2\mbox{,}\\ z_1(h,p_1,q_1)&=\frac{1}{h}\left[X_{j-1}(p_1+hq_1)-X_{j-1}(p_1)\right]= \int_0^1\partial^1X_{j-1}(p_1+hsq_1)\,q_1{\mathop{}\!\mathrm{d}}s\\ z_2(h,p_1,p_2,q_1,q_2)&=\frac{1}{h}\left[\partial^1X_{j-1}(p_1+hq_1)\,[p_2+hq_2] -\partial^1X_{j-1}(p_1)\,p_2\right]\mbox{\quad for $h\neq0$.}\end{aligned}$$ With these notations we have $X_{j-1}(p_1+hq_1)=x_1+hz_1$ and, for non-zero $h$, $\partial^1X_{j-1}(p_1+hq_1)[p_2+hq_2]=x_2+hz_2$. The fixed-point equations and imply a fixed-point equation for the difference quotient $z_2$ for non-zero $h$: $$\begin{aligned} z_2&=E_Nq_2+\frac{1}{h}Q_NL\left[\partial^1F_{j-1}(x_1+hz_1,x_2+hz_2)- \partial^1F_{j-1}(x_1,x_2)\right]\nonumber\\ &=E_Nq_2+\tilde z(p_1,p_2,q_1,h)+Q_NL\partial^1F_{j-1}(x_1+hz_1,z_2) \mbox{\quad where}\label{eq:z2fixp}\\ \tilde z(p_1,p_2,q_1,h)&=Q_NL\frac{\partial^1F_{j-1}(x_1+hz_1,x_2)- \partial^1F_{j-1}(x_1,x_2)}{h}\nonumber\end{aligned}$$ The regularity of $x_1$, $x_2$ and $z_1$ is: $$\label{eq:x1x2z1reg} \begin{split} x_1&\in D_{j-1}^{j_{\max}-j+2}\subseteq D_{j-1}^3\mbox{,} \\ x_2&\in D_{j-1}^{j_{\max}-j+1}\subseteq D_{j-1}^2\mbox{\quad and} \\ z_1&\in D_{j-1}^{j_{\max}-j+1}\subseteq D_{j-1}^2\mbox{.} \end{split}$$ We can apply the mean value theorem to the difference quotient appearing in $\tilde z$ since $x_1$ and $z_1$ are at least in $D_{j-1}^2$ and $x_2$ is at least in $D_{j-1}^1$ (see Lemma \[thm:rhs:smooth\], and Lemma \[thm:fmeanval\] and Lemma \[thm:Fdiff\] in Appendix \[sec:basicprop\]): $$\begin{aligned} \tilde z(p_1,p_2,q_1,h)&= Q_NL\int_0^1\partial^2F_{j-1}(x_1+shz_1,x_2,z_1,0){\mathop{}\!\mathrm{d}}s\mbox{.}\end{aligned}$$ The map $(x_1,x_2,z_1,h)\mapsto \partial^2F_{j-1}(x_1+shz_1,x_2,z_1,0)$ maps $x_1$, $x_2$, $z_1$ and $h$ continuously into the space $D_{j-1}^{j_{\max}-j-1}$ (we see this by applying Lemma \[thm:rhs:smooth\] to $\partial^2F_{j-1}$, setting $k$ in Lemma \[thm:rhs:smooth\] to $j_{\max}-j-1$). Thus, the quantity $\tilde z(p_1,p_2,q_1,h)$ is in $D_{j-1}^{j_{\max}-j}\subseteq D_{j-1}^1$ (since $j\leq j_{\max}-1$). It depends continuously on $p_1$, $p_2$, $q_1$ and $h$ in this space, and can be extended to $h=0$ continuously (such that $\tilde z(p_1,p_2,q_1,0)\in D_{j-1}^{j_{\max}-j}$, too). Hence, is a linear fixed-point problem for $z_2$ where the inhomogeneity is in $D_{j-1}^{j_{\max}-j}$ and depends continuously on $(p,q,h)$. The linear map $M(h):z_2\mapsto Q_NL\partial^1F_{j-1}(x_1+hz_1)\,z_2$ in front of $z_2$ on the right-hand side of depends continuously on $h$ as an element of $L(D_{j-1}^1;D_{j-1}^1)$ (see Lemma \[thm:speccont\] and note that $x_1$ and $z_1$ are in $D_{j-1}^1$). Since the spectral radius of the map $M(0)$ (for $h=0$) is less or equal than $1/2$ by assumption of our inductive step, the spectral radius of $M(h)$ is less than unity if we choose $h$ sufficiently small. Thus, for all $p\in D(X_j)$ and $q\in {\mathbb{R}}^{n_j}$ and sufficiently small $h$, $z_2$ satisfies a contractive linear fixed point equation with an inhomogeneity in $D_{j-1}^1$ and a contractive linear map that maps into $D_{j-1}^1$ where all coefficients depend continuously on $(h,p,q)$. Consequently, $z_2$ has a limit in $D_{j-1}^1$ for $h\to0$ that depends continuously on $(p,q)$. For $h=0$ the fixed point equation for $(z_1,z_2)$ simplifies to $$\label{eq:isz} \begin{split} z_1&=E_Nq_1+Q_NL\partial^1F_{j-1}(x_1,z_1)\\ z_2&=E_Nq_2+Q_NL\left[\partial^2F_{j-1}(x_1,x_2,z_1,0)+ \partial^1F_{j-1}(x_1,z_2)\right]\mbox{.} \end{split}$$ Both equations are linear in $q$ and $z=(z_1,z_2)$. Consequently, $z(0,p,q)$, which is by definition the directional derivative of $X_j$ in $p$ in direction $q$, depends linearly on $q$ and continuously on $p$ and $q$. Consequently, $$z(0,p,q)=\left[\frac{\partial}{\partial p}X_{j}(p)\right]q$$ is the Frech[é]{}t derivative of $X_j$. ### Collection to finish proof of statement 1 of inductive step {#collection-to-finish-proof-of-statement-1-of-inductive-step .unnumbered} The functions $x=X_j(p)$ and $z=\partial^1X_j(p)q$ satisfy $$\label{eq:ivfjp1} \begin{aligned} x=&E_Np+Q_NLF_j(x) &&\mbox{by inductive assumption \eqref{eq:ivf}--\eqref{eq:ivderiv},}\\ z=&E_Nq+Q_NL\partial^1F_j(x,z) &&\mbox{by \eqref{eq:isz} and definition of $F_j$.} \end{aligned}$$ The variable $x=X_j(p)$ depends continuously on $p$ with respect to the norm of $D_j^1$ by the assumption of the inductive step and the step “Regularity of $X_j(p)$”. The variable $z=\partial^1X(p)\,q$ depends continuously on $p$ and $q$ as shown in the previous step, “Existence and continuity of $\partial^1X(p)\,q$”. Thus $(x,z)=(X_j(p),\partial^1X(p)\,q)=X_{j+1}(p,q)\in D_j^1\times D_j=D_{j+1}$ depends continuously on $(p,q)$, and satisfies – for $j+1$ (which is identical to system ). This completes the proof of statement \[p:ass:exfix0\] of the inductive assumption for $j+1$. ### Spectral radius of map $z\mapsto Q_NL\partial^1F_j(x,z)$ {#spectral-radius-of-map-zmapsto-q_nlpartial1f_jxz .unnumbered} The map $\partial^1F_j$ maps $D_{j+1}$ continuously into $D_j$ (by Lemma \[thm:rhs:smooth\]). Thus, for fixed $x$ the linear map $z\mapsto \partial^1F_j(x,z)$ maps $D_j$ continuously into $D_j$, and, hence, the map $M_j(x): z\mapsto Q_NL\partial^1F_j(x,z)$ maps $D_j$ continuously into $D_j^1$, making $M_j(x)$ a compact linear operator. Thus, the spectral radius of $M_j(x)$ is determined by its largest eigenvalue (which has finite modulus and algebraic multiplicity if it is non-zero). Splitting $M_j(x)$ into its two components we get $$M_j(x):\begin{bmatrix} z_1\\ z_2 \end{bmatrix}\mapsto \begin{bmatrix} \begin{aligned} &Q_NL\phantom{[\ }\partial^1F_{j-1}(x_1,z_1)\\ &Q_NL\left[\partial^2F_{j-1}(x_1,x_2,z_1,0)+ \partial^1F_{j-1}(x_1,z_2)\right] \end{aligned} \end{bmatrix}$$ If $(\lambda,(z_1,z_2))$ is an eigenpair of $M_j(x)$ then the first row of the definition of $M_j(x)$ implies that, either $(\lambda,z_1)$ is an eigenpair of $z_1\mapsto Q_NL\partial^1F_{j-1}(x_1,z_1)$, or $z_1=0$. If $(\lambda,z_1)$ is an eigenpair of $z_1\mapsto Q_NL\partial^1F_{j-1}(x_1,z_1)$ then, by inductive assumption, $\|\lambda\|\leq 1/2$. If $z_1=0$ then the term $\partial^2F_{j-1}(x_1,x_2,z_1,0)$ vanishes in the second row, such that $(\lambda,z_2)$ is an eigenpair of $z_2\mapsto Q_NL\partial^1F_{j-1}(x_1,z_2)$. Thus, by inductive assumption, $\|\lambda\|\leq 1/2$ in this case, too. Consequently, the spectral radius of $M_j(x)$ is also less or equal to $1/2$, which proves statement \[p:ass:specradius\] of the inductive assumption for $j+1$. ##### Existence of $Y_j$ {#existence-of-y_j .unnumbered} We show inductively that $Y_j(p)=F_j(X_j(p))$. For $j=1$ this statement was proven in Lemma \[thm:Xdiff\]. Let $j<j_{\max}$ and assume that $Y_j=F_j(X_j(p))$ for $p\in D(X_j)$. Since $$X_j(p)=E_Np+Q_NLF_j(X_j(p))$$ and $F_j$ maps $D_j^1$ into $D_j^1$, $X_j$ is an element of $D_j^1$. Let $q\in {\mathbb{R}}^{n_j}$ be arbitrary, and let us denote $(x_1,x_2)=(X_j(p),\partial^1X_j(p)\,q)=X_{j+1}(p,q)$. The component $x_2$ satisfies $$x_2=E_Nq+Q_NL\partial^1F_j(x_1,x_2)$$ such that $x_2$ is in $D_j^1$, too. Consequently, $$\begin{aligned} \frac{Y_j(p+hq)-Y_j(p)}{h}&= \frac{F_j(X_j(p+hq))-F_j(X_j(p))}{h}\nonumber\\ &=\frac{F_j(x_1+hx_2)-F_j(x_1)}{h}+ \frac{F_j(X_j(p+hq))-F_j(x_1+hx_2)}{h}\mbox{.}\label{eq:Yjsplit}\end{aligned}$$ Since $F_j$ is continuously differentiable for $x_1\in D_j^1$ and deviations $hx_2\in D_j^1$ the first quotient in the expression converges to $\partial^1F_j(x_1,x_2)$. Since $F_j$ as a map from $D_j^1$ into $D_j$ is locally Lipschitz continuous the second term in can be bounded by $$\begin{aligned} \lefteqn{\left\\| \frac{F_j(X_j(p+hq))- F_j(X_j(p)+h\partial^1X_j(p)q)}{h}\right\\|_{D_j}\leq}\\ &\leq& K_1\left\\|\frac{X_j(p+hq)-X_j(p)}{h}-\partial^1X_j(p)q\right\\|_{D_j}\mbox{,}\end{aligned}$$ with some constant $K_1$, for sufficiently small $h$, which converges to zero for $h\to0$ because $X_j$ is differentiable. Consequently, the directional derivative of $Y_j$ in $p$ in direction $q$ is $\partial^1F_j(X_j(p))[\partial X_j(p)\,q]$, which is continuous in $p$ and $q$ and linear in $q$. Therefore, the Frech[é]{}t derivative of $Y_j$ exists and $$\left[\frac{\partial}{\partial p}Y_j(p)\right]q= \partial^1F_j(X_j(p),\partial X_j(p)\,q)\mbox{,}$$ which implies by definition of $F_j$ and $X_j$ that $Y_{j+1}=F_{j+1}(X_{j+1}(p,q))$. [$\square$]{} We can refine the statement of Theorem \[thm:smooth\] slightly by noting that $X_j:D(X_j)\mapsto D_j^1$ is continuous for all $j\leq j_{\max}$ (instead of $X_j:D(X_j)\mapsto D_j$). This follows from the continuity of $Y_j=F_j(X_j(p))$ as a map into $D_j$ and the relation $$X_j(p)=E_Np+Q_NLY_j(p)\mbox{.}$$ Theorem \[thm:smooth\] completes the proof of the Equivalence Theorem \[thm:main\]. The refinement (that $X_j$ maps into $D_j^1$) ensures that the image $X(p)$ is in $C^{j_{\max}+1}({\mathbb{T}};{\mathbb{R}}^n)$, as claimed in Theorem \[thm:main\] Proof of Hopf Bifurcation Theorem {#sec:hopf:proof} ================================= First, we note that $x\mapsto S(x,\omega)^{-1}=x(\omega^{-1}\cdot)$ maps $C^k({\mathbb{T}};{\mathbb{R}}^n)$ into a closed subspace of $C^k([-\tau,0];{\mathbb{R}}^n)$, if we extend functions $x$ on ${\mathbb{T}}$ to the whole real line by setting $x(t)=x(t_{{\operatorname{mod}}[-\pi,\pi)})$. This implies that, if the functional $f:C^0([-\tau,0];{\mathbb{R}}^n)\times{\mathbb{R}}\mapsto{\mathbb{R}}^n$ is $EC^k$ smooth then the functional $$(x,\mu,\omega)\in C^0({\mathbb{T}};{\mathbb{R}}^n)\times{\mathbb{R}}^2\mapsto\frac{1}{\omega} f(S(x,\omega),\mu)\in{\mathbb{R}}^n$$ is $EC^k$ smooth, too, such that we can reduce the problem of finding periodic orbits of frequency $\omega$ to the algebraic system . The right-hand side $F_y$ in is defined by $$\left[F_y(x,\omega,\mu)\right](t)= \frac{1}{\omega}f(S(\Delta_tx,\omega),\mu)\mbox{.}$$ Let us choose the periodic orbit $x_0=(x,\omega,\mu)$ with $x=0$, $\omega=\omega_0$, $\mu=0$ as the solution in the neighborhood of which we construct the equivalent algebraic system. We choose the number $N$ of Fourier modes and the size $\delta$ of the neighborhood $B_\delta^{0,1}(x_0)$ in $C^{0,1}({\mathbb{T}};{\mathbb{R}}^{n+2})$ such that the conditions of Theorem \[thm:main\] are satisfied in $B_\delta^{0,1}(x_0)$. The full algebraic system then reads (after multiplication with $\omega$ and mapping it onto the space ${\operatorname{rg}}P_N$ from ${\mathbb{R}}^{n(2N+1)}$ by applying $R_N^{-1}$) $$\begin{aligned} 0=&P_0F_y(X_y(p,\omega,\mu),\omega,\mu)+ \omega Q_0P_NE_Np- Q_0P_NL F_y(X_y(p,\omega,\mu),\omega,\mu) \end{aligned} \label{eq:hopf:modes}$$ The variables are $p\in{\mathbb{R}}^{n(2N+1)}$ (which was called $p_y$ in ), $\mu$ and $\omega$. We know from Theorem \[thm:main\] that $$\begin{aligned} Y_y:&(p,\omega,\mu)\in{\mathbb{R}}^{n(2N+1)}\times{\mathbb{R}}\times{\mathbb{R}}\mapsto F(X_y(p,\omega,\mu),\omega),\mu)\in C^0({\mathbb{T}};{\mathbb{R}}^n)\mbox{,}\\ X_y:&(p,\mu,\omega)\in{\mathbb{R}}^{n(2N+1)}\times{\mathbb{R}}\times{\mathbb{R}}\mapsto X_y(p,\omega,\mu) \in C^0({\mathbb{T}};{\mathbb{R}}^n)\end{aligned}$$ are $k$ times differentiable, and note that $$\label{eq:hopf:fzero} F_y(X_y(0,\omega,\mu),\omega),\mu)=0$$ for all $\omega\approx\omega_0$ and $\mu\approx0$ (because $x_0=(0,\omega,\mu)$ is a solution). The derivative of the right-hand side $F_y$ in $x=0$, $\omega\approx\omega_0$ and $\mu\approx0$ with respect to $x$ is $A(\omega,\mu)x$, defined by $$\begin{aligned} \left[A(\omega,\mu)x\right](t)=a(\mu)\left[x(t+\omega\cdot)\right]\mbox{,}\end{aligned}$$ where $a(\mu)$ is the same linear functional as used in the definition of the characteristic matrix $K(\lambda,\mu)$ in (the derivatives of $F$ with respect to $\omega$ and $\mu$ are zero due to ). We observe that $A(\omega,\mu)$ commutes with $P_j$ and $Q_j$ for all $j\geq0$. Let us now determine the linearization of $X_y(p,\omega,\mu)$ in $(p,\omega,\mu)=(0,\omega,\mu)$. Due to $X_y(0,\omega,\mu)$ is equal to zero for all $\omega\approx\omega_0$ and $\mu\approx0$: since $0$ is a solution to the periodic BVP and $P_N0=0$, the zero solution must also be equal to $X_y(0,\omega,\mu)$. Thus, we have $$\begin{aligned} 0&=\left.\frac{\partial}{\partial\omega} X_y(p,\omega,\mu) \right\vert_{\textstyle p=0}\mbox{\quad and} & 0&=\left.\frac{\partial}{\partial\mu} X_y(p,\omega,\mu)\right\vert_{\textstyle p=0}\mbox{.}\end{aligned}$$ Moreover, the fixed point equation defining $z=[\partial X_y/\partial p] (p,\omega,\mu)\,q$, evaluated in $(p,\omega,\mu)=(0,\omega,\mu)$ reads $$\label{eq:hopf:qnz} z=E_Nq+Q_NLA(\mu,\omega)\,z= E_Nq+Q_NLA(\mu,\omega)\,Q_Nz\mbox{,}$$ exploiting that $Q_NL=Q_NLQ_N$ and $Q_NA(\omega,\mu)=A(\omega,\mu)\,Q_N$. In the neighborhood $B_\delta^{0,1}(x_0)$ the spectral radius of $Q_NLA(\mu,\omega)$ is less than unity (see Lemma \[thm:specradius\]). Application of $Q_N$ to gives $Q_Nz=Q_NLA(\mu,\omega)Q_Nz$. Since the spectral radius of $Q_NLA(\mu,\omega)$ is less than unity this implies that $Q_Nz=0$, and, thus $$\left[\left.\frac{\partial}{\partial p} X_y(p,\omega,\mu)\right\vert_{\textstyle p=0}\right]q= E_Nq\mbox{.}$$ Consequently, the linearization of the algebraic system in $(0,\omega,\mu)$ with respect to the first variable is $$\label{eq:hopf:lin} 0=P_0A(\omega,\mu)E_Np+\omega Q_0P_NE_Np-Q_0P_NLA(\omega,\mu)E_Np$$ for all $\omega\approx \omega_0$ and $\mu\approx 0$ (also using $p$ for the argument of the linearization in ). We observe that the linear system decouples into equations for $$\begin{aligned} y_0&=P_oE_Np=E_0p=p_0 &&\mbox{(the average of $E_Np$),}\\ y_1&=Q_0E_1p=p_{-1}\sin t+p_1\cos t &&\mbox{(the first Fourier component of $E_Np$),}\\ y_j&=Q_{j-1}E_jp=p_{-j}\sin(jt)+p_j\cos(jt) &&\mbox{(the $j$-th Fourier component of $E_Np$,}\\ &&&\mbox{$2\leq j\leq N$),}\end{aligned}$$ where we denote the components of $p$ by $p_j\in {\mathbb{R}}^n$ ($j=-N\ldots N$). This decoupling is achieved by pre-multiplication of with $P_0$ and $Q_{j-1}P_j$ for $j=1\ldots N$: $$\begin{aligned} P_0\cdot\mbox{\eqref{eq:hopf:lin}}:&& 0&=A(\omega,\mu)y_0=a(\mu)\,p_0 \label{eq:hopf:lin:dec0}\\ Q_0P_1\cdot\mbox{\eqref{eq:hopf:lin}}:&& 0&=\omega y_1-Q_0LA(\omega,\mu)\,y_1 \label{eq:hopf:lin:dec1}\\ Q_{j-1}P_j\cdot\mbox{\eqref{eq:hopf:lin}}:&& 0&=\omega y_j-Q_0LA(\omega,\mu)\,y_j && \mbox{\ for $j=2\ldots N$.} \label{eq:hopf:lin:decj}\end{aligned}$$ Inserting the definition of $y_j$ into the equations and gives for $j\geq 1$ $$\begin{aligned} 0=&\omega\left[p_{-j}\sin(jt)+p_j\cos(jt)\right]- Q_0\int_0^ta(\mu)[p_j\sin(js+j\omega\cdot)+p_j\cos(js+j\omega\cdot)]{\mathop{}\!\mathrm{d}}s\\ =&\omega\left[p_{-j}\sin(jt)+p_j\cos(jt)\right]- \frac{1}{j}\sin(jt)a(\mu)[p_{-j}\sin(j\omega\cdot)+p_j\cos(j\omega\cdot)]\\ &\phantom{x}-\frac{1}{j}\cos(jt)a(\mu) [-p_{-j}\cos(j\omega\cdot)+p_j\sin(j\omega\cdot)]\mbox{.}\end{aligned}$$ These equations are satisfied if and only if the coefficients in front of $\sin(jt)$ and $\cos(jt)$ are zero. The resulting system of equations reads in complex notation (splitting up again into the cases $j=1$ and $j>1$) $$\begin{aligned} \label{eq:hopf:red1} i\omega u_1-a(\mu)\left[u_1\exp(i\omega s)\right]&=K(i\omega,\mu)\,u_1=0\mbox{,}\\ \label{eq:hopf:redj} ij\omega u_j-a(\mu)\left[u_j\exp(ij\omega s)\right]&=K(ij\omega,\mu)\,u_j=0 \mbox{\quad ($2\leq j\leq N$),}\end{aligned}$$ that is, $u_j=p_{-j}+ip_j\in{\mathbb{C}}^n$ is a solution of (or , respectively) if and only if $y_j=p_{-j}\sin(jt)+p_j\cos(jt)$ is a solution of (or , respectively). The non-resonance assumption of the theorem guarantees that equation is a regular linear system for $p_0$, and that is a regular linear algebraic system for $p_{-j}$ and $p_j$ ($j\geq 2$) at $\mu=0$ and $\omega=\omega_0$ (and, hence, for all $\omega$ and $\mu$ near-by). The condition on the simplicity of the eigenvalue $i\omega_0$ of $K$ ensures that equation (and, thus, ) has a one-dimensional (in complex notation) subspace of solutions for $\omega=\omega_0$ and $\mu=0$, spanned by the nullvector $v_1$ of $K(i\omega,0)$. Let us denote the adjoint nullvector of $K(i\omega_0,0)$ by $w_1\in{\mathbb{C}}^n$ (again, using complex notation, $w_1^HK(i\omega_0,0)=0$). Since $i\omega_0$ is simple, the relationship $$w_1^H\frac{\partial K}{\partial\lambda}(i\omega,0)\,v_1\neq0$$ holds which implies that we can choose $w_1\in{\mathbb{C}}^n$ without loss of generality such that $$w_1^H\frac{\partial K}{\partial\lambda}(i\omega,0)\,v_1=1\mbox{.}$$ With this convention we observe that $$\begin{aligned} w_1^H\frac{\partial K}{\partial\mu}(i\omega,0)\,v_1&= -\left.\frac{\partial\lambda}{\partial\mu}\right\vert_{\textstyle\mu=0} =:c_\mu\in{\mathbb{C}}\mbox{, and}& w_1^H\frac{\partial}{\partial\omega}K(i\omega,0)\,v_1&= i\in{\mathbb{C}}\label{eq:hopf:derivs}\end{aligned}$$ where ${\operatorname{Re}}c_\mu\neq0$ by the transversal crossing assumption of the theorem. In complex notation any scalar multiple of the nullvector $v_1=v_r+iv_i$ is also a nullvector. Thus, the complex scalar factor $\alpha+i\beta$ in front of $v_1$ makes up two components of the variable $p$ (in real notation): in short, $p$ solves the linearized algebraic system if and only if all $p_j$ with $\|j\|\neq1$ are zero and $p_{-1}\sin t+p_1\cos t={\operatorname{Re}}\left[(\alpha+i\beta)v_1\exp(it)\right]$ for some $\alpha,\beta\in{\mathbb{R}}$, that is, $$\label{eq:hopf:abintro} \begin{bmatrix} p_{-1}\\ p_{1\phantom{-}} \end{bmatrix}= \alpha \begin{bmatrix} -v_i\\ \phantom{-}v_r \end{bmatrix}+\beta \begin{bmatrix} -v_r\\ -v_i \end{bmatrix}=:\alpha b_r+\beta b_i\mbox{.}$$ Let us collect the statements so far and introduce coordinates. We collect all components $p_j$ with $\|j\|\neq1$ and the orthogonal complement in ${\mathbb{R}}^{2n}$ of the space spanned by $\{b_1,b_2\}$ into a single variable $q$ (of real dimension $n_q=n(2N-1)+2(n-1)$). Then a set of coordinates for $p$ are the variables $$\begin{aligned} (\alpha,\beta)&=:r\in{\mathbb{R}}^2\mbox{,\quad and\quad} q\in{\mathbb{R}}^{n_q}\mbox{.}\end{aligned}$$ We split up the full algebraic system of equations in the same way as we did for the linearized problem, by pre-multiplication with $P_0$ and $Q_{j-1}P_j$ for $j=1\ldots N$: $$\begin{aligned} P_0\cdot\mbox{\eqref{eq:hopf:modes}}:&& 0&=P_0F(X_y(p,\omega,\mu),\omega,\mu) \label{eq:hopf:nlin0}\\ Q_0P_1\cdot\mbox{\eqref{eq:hopf:modes}}:&& 0&=\omega Q_0E_1p-Q_0P_1LF(X_y(p,\omega,\mu),\omega,\mu) \label{eq:hopf:nlin1}\\ Q_{j-1}P_j\cdot\mbox{\eqref{eq:hopf:modes}}:&& 0&=\omega Q_{j-1}E_jp-Q_{j-1}P_jLF(X_y(p,\omega,\mu),\omega,\mu)\mbox{.} \label{eq:hopf:nlinj}\end{aligned}$$ We split equation further using $w_1^H$ and its orthogonal complement, the projection $w_1^\perp={I}-w_1w_1^H/(w_1^Hw_1)$. This gives rise to a splitting into two real equations ($w_1^H\cdot$) and $2(n-1)$ real equations ($w_1^\perp\cdot$). Collecting $w_1^\perp\cdot$ and the equations and into a subsystem of $n(2N-1)+2(n-1)=n_q$ equations the full algebraic system in the coordinates $(r,q)$ has the form $$\label{eq:hopf:nlin:matrixform} 0=\begin{bmatrix} M_{rr}(r,q,\omega,\mu) & M_{rq}(r,q,\omega,\mu)\\ M_{qr}(r,q,\omega,\mu) & M_{qq}(r,q,\omega,\mu) \end{bmatrix} \begin{bmatrix} r\\ q \end{bmatrix}\mbox{.}$$ By our choice of coordinates the matrices $M_{rr}\in{\mathbb{R}}^{2\times2}$, $M_{rq}\in{\mathbb{R}}^{2\times n_q}$ and $M_{qr}\in{\mathbb{R}}^{n_q\times2}$ are identically zero in $r=0$, $q=0$, $\mu=0$, $\omega=i\omega_0$ such that the system matrix has the form $$\begin{bmatrix} \begin{bmatrix} 0&0\\ 0&0 \end{bmatrix} & \begin{bmatrix} 0 & \ldots & 0\\ 0 & \ldots & 0 \end{bmatrix}\\ \begin{bmatrix} 0&0\\ \vdots&\vdots\\ 0&0 \end{bmatrix} & \begin{matrix} M_{qq}(0,0,i\omega,0)\\ \mbox{(regular)} \end{matrix} \end{bmatrix}$$ $(r,q,\mu,\omega)=(0,0,0,\omega_0)$. Thus, we can perform a Lyapunov-Schmidt reduction: we eliminate $q$ by solving the $n_q$ lower equations for $q$, obtaining a graph $q(r,\omega,\mu)\,r$ locally in a neighborhood of $(r,q,\mu,\omega)=(0,0,0,\omega_0)$. This graph respects rotational invariance: $q(\Delta_sr,\omega,\mu)\,\Delta_sr=\Delta_s[q(r,\omega,\mu)\,r]$. Note that the application of $\Delta_s$ to $r=(\alpha,\beta)$ corresponds to the rotation of $r$ by angle $s$ (the same as the multiplication $\exp(is)(\alpha+i\beta)$). The Lyapunov-Schmidt reduction of , replacing $q$ by the graph $q(r,\omega,\mu)\,r$, then reads $$\label{eq:hopf:nlin:reduced} 0=M_{rr}(r,q(r,\omega,\mu)\,r,\omega,\mu)\,r=:M_r(r,\omega,\mu)\,r\mbox{,}$$ where $M_r$ is still rotationally symmetric in $r$: $M_r(\Delta_sr,\omega,\mu)\,\Delta_sr=\Delta_sM_r(r,\omega,\mu)\,r$. Equation in real notation implies that $$\begin{aligned} \frac{\partial M_r}{\partial\omega}(0,\omega_0,0)&= \frac{\partial M_{rr}}{\partial\omega}(0,0,\omega_0,0)= \begin{bmatrix} 0&{-1}\\ 1 &\phantom{-}0 \end{bmatrix}\mbox{,} \\ \frac{\partial M_r}{\partial\mu}(0,\omega_0,0)&= \frac{\partial M_{rr}}{\partial\mu}(0,0,\omega_0,0)= \begin{bmatrix} {\operatorname{Re}}c_\mu&-{\operatorname{Im}}c_\mu\\ {\operatorname{Im}}c_\mu &\phantom{-}{\operatorname{Re}}c_\mu \end{bmatrix}\mbox{.}\end{aligned}$$ Equation is a system of two equations with four unknowns ($r=(\alpha,\beta)$, $\omega$ and $\mu$). We now fix one of the unknowns setting $$\alpha=0$$ such that we can expect one-parametric families of solutions $(\beta,\omega,\mu)$. Introducing $M_\beta$ as the second column of $M_r$ and dropping the dependence on $\alpha$ (which is zero), the first derivative of $M_\beta(\beta,\omega,\mu)$ in $(0,\omega_0,0)$ with respect to the pair $\omega$ and $\mu$ is: $$\begin{aligned} \begin{bmatrix} {\displaystyle\frac{\partial M_\beta}{\partial \omega}}& {\displaystyle\frac{\partial M_\beta}{\partial \mu}} \end{bmatrix} (0,\omega_0,0)= \begin{bmatrix} -1 &-{\operatorname{Im}}c_\mu\\ \phantom{-}0& \phantom{-}{\operatorname{Re}}c_\mu \end{bmatrix}\mbox{,}\end{aligned}$$ which is regular (as established in , since ${\operatorname{Re}}c_\mu\neq0$ due to the assumption that the eigenvalue crosses the imaginary axis transversally). Note that $M_\beta$ itself is a projection of the first derivative of the original right-hand side of the full algebraic system . Thus, $M_\beta$ is $k-1$ times continuously differentiable, and we end up with a system of two equations for three scalar variables $(\beta,\omega,\mu)$: $$\begin{aligned} 0=M_\beta(\beta,\omega,\mu)\,\beta\mbox{.}\end{aligned}$$ Hence, either $\beta=0$, which corresponds to the trivial solution or (after division by $\beta$) $$\begin{aligned} 0=&M_\beta(\beta,\omega,\mu)\mbox{,}\label{eq:hopf:nlin:factor}\end{aligned}$$ where $M_\beta(0,\omega_0,0)=(0,0)$ and the derivative with respect to the pair $(\omega,\mu)$ is regular in $(0,\omega_0,0)$. Thus, we can apply the Implicit Function Theorem to to obtain a unique graph $(\omega(\beta),\mu(\beta))$ solving . The graph satisfies $(\omega(0),\mu(0))=(\omega_0,0)$, and, thus, branches off from the trivial solution (which has $\beta=0$ and $\omega$ and $\mu$ arbitrary). The rotational symmetry of $M_r$ implies reflection symmetry of $M_\beta$ in $\beta$ such that $M_\beta(-\beta,\omega,\mu)=M_\beta(\beta,\omega,\mu)$ for all $\beta$, $\omega$ and $\mu$. Hence, the solution graph is reflection symmetric, too: $\omega(-\beta)=\omega(\beta)$ and $\mu(-\beta)=\mu(\beta)$. Thus, for small $\beta$ there is a unique non-trivial solution of the full algebraic system of the form $r=(0,\beta)$, $q=q(r,\omega(\beta),\mu(\beta))\,r$. As Equation shows, we can extract the coordinates $\alpha$ (which is zero) and $\beta$ from the full solution $x\in C^k({\mathbb{T}};{\mathbb{R}}^n)$ by applying the projections $$\begin{aligned} \frac{1}{\pi}\int_{-\pi}^\pi\cos(t)v_r^Tx(t)-\sin(t)v_i^Tx(t){\mathop{}\!\mathrm{d}}t&= \frac{1}{\pi}\int_{-\pi}^\pi{\operatorname{Re}}\left[v_1\exp(it)\right]^Tx(t){\mathop{}\!\mathrm{d}}t=\alpha\mbox{,}\\ \frac{1}{\pi}\int_{-\pi}^\pi\sin(t)v_r^Tx(t)+\cos(t)v_i^Tx(t){\mathop{}\!\mathrm{d}}t&= \frac{1}{\pi}\int_{-\pi}^\pi{\operatorname{Im}}\left[v_1\exp(it)\right]^Tx(t){\mathop{}\!\mathrm{d}}t=-\beta\mbox{,}\end{aligned}$$ which determines the First Fourier coefficients of $x$ as claimed in in Theorem \[thm:hopf\]. (Recall that the vector $v_1=v_r+v_i$ was scaled to have unit length and that the decomposition was orthogonal.) [$\square$]{} Basic differentiability properties of the right-hand side {#sec:basicprop} ========================================================= Let $J$ be a compact interval or ${\mathbb{T}}$. Let $(D,\\|\cdot\\|_D)$ be a Banach space of the form $$D=C^{k_1}(J;{\mathbb{R}}^{m_1})\times\ldots\times C^{k_\ell}(J;{\mathbb{R}}^{m_\ell})$$ where $\ell\geq 1$, the integers $k_j$ are non-negative and the integers $m_j$ are positive. We use the natural maximum norm on the product $D$: $$\\|x\\|_D=\\|(x_1,\ldots,x_\ell)\\|_D=\max_{j\in\{1,\ldots,\ell\}}\\|x_j\\|_{k_j}\mbox{,}$$ and use the notation $$\begin{aligned} D^k&=C^{k_1+k}(J;{\mathbb{R}}^{m_1})\times\ldots\times C^{k_\ell+k}(J;{\mathbb{R}}^{m_\ell})\mbox{,} & \\|x\\|_{D,k}&=\max_{0\leq j\leq k} \\|x^{(j)}\\|_D\mbox{,}\\ D^{0,1}&=\left\{x\in D: L(x)<\infty\right\}\mbox{, with the norm}& \\|x\\|_{D,L}&=\max\left\{\\|x\\|_D,L(x)\right\}\mbox{,} \intertext{where $x^{(j)}$ is the component-wise $j$th derivative and the Lipschitz constant $L(x)$ is defined as} L(x)&=\sup_{ \begin{subarray}{c} t\neq s \end{subarray} }\ \max_{j=1\ldots,\ell}\ \frac{\|x_j^{(k_j)}(t)-x_j^{(k_j)}(s)\|}{\|t-s\|} \mbox{,}\end{aligned}$$ where $t$ and $s$ in the index of $\sup$ are taken from $J$, if $J$ is a compact interval, and from ${\mathbb{R}}$ if $J={\mathbb{T}}$. Balls that are closed and bounded in $D^{0,1}$ are complete with respect to the norm of $D$. Basic properties of $f$ {#sec:fprop} ----------------------- This section proves three properties that $EC^1$ smooth functionals $f$ have: first that the derivative limit exists also for Lipschitz continuous deviations, second a weaker form of the mean value theorem, and third, local $EC$ Lipschitz continuity. \[thm:weakcontdiff\] Let $f:D\mapsto{\mathbb{R}}^n$ be $EC^1$ smooth in the sense of Definition \[def:extdiff\]. Then the limit required to exist in Definition\[def:extdiff\] exists also in the $\\|\cdot\\|_{D,L}$-norm: for all $x\in D^1$ $$\begin{aligned} \allowdisplaybreaks \label{eq:ass:contdifflip} \lim_{ \begin{subarray}{c} y\in D^{0,1}\\[0.2ex] \\|y\\|_{D,L}\to0 \end{subarray} }& \frac{\|f(x+y)-f(x)-\partial^1f(x,y)\|}{\\|y\\|_{D,L}}=0\mbox{.} \end{aligned}$$ Note that in the norm in which $y$ goes to zero is $\\|\cdot\\|_{D,L}$ instead of $\\|\cdot\\|_{D,1}$. #### Proof {#proof-7 .unnumbered} This is a simple continuity argument. Let ${\varepsilon}>0$ be arbitrary. We pick $\delta>0$ such that $$\label{eq:ydsmall} \|f(x+\tilde y)-f(x)-\partial^1f(x,\tilde y)\|<{\varepsilon}\\|\tilde y\\|_{D,1}$$ for all $\tilde y\in D^1$ satisfying $\\|\tilde y\\|_{D,1}<\delta$. Let $y\in D^{0,1}$ be such that $\\|y\\|_{D,L}<\delta$. We can choose a $\tilde y\in D^1$ that satisfies $$\begin{aligned} \\|\tilde y\\|_{D,1}&<\min\{\delta,2\\|y\\|_{D,L}\}\label{eq:yydsmall}\\ \|f(x+y)-f(x+\tilde y)\|&<{\varepsilon}\\|y\\|_{D,L}\label{eq:fyydsmall}\\ \|\partial^1f(x,y-\tilde y)\|&<{\varepsilon}\\|y\\|_{D,L}\label{eq:ayydsmall}\mbox{.}\end{aligned}$$ Condition can be achieved because $D^1$ is a dense subspace in $D^{0,1}$, and for every element $\tilde y$ of $D^1$ the $\\|\cdot\\|_{D,1}$-norm is not larger than the $\\|\cdot\\|_{D,L}$-norm: $\\|\tilde y\\|_{D,1}\leq\\|\tilde y\\|_{D,L}$. follows from the continuity of $f$ and the density of $D^{0,1}$ in $D^1$, and follows from the continuity of $\partial^1f$ as a map on $D^1\times D$, and the density of $D^{0,1}$ in $D^1$. Combining estimate with – we obtain $$\|f(x+y)-f(x)-\partial^1f(x,y)\|<4{\varepsilon}\\|y\\|_{D,L}\mbox{.}$$ [$\square$]{} \[thm:fmeanval\] There exists a continuous function $$\tilde a:D^1\times D^1\times D\mapsto {\mathbb{R}}^n$$ which is linear in its third argument and satisfies for all $x,y\in D^1$ $$\label{eq:meanval} f(x+y)-f(x)=\tilde a(x,y,y)\mbox{.}$$ Moreover, $\tilde a(x,0,y)=\partial^1f(x,y)$ for all $x\in D^1$ and $y\in D$. #### Proof {#proof-8 .unnumbered} The argument for the existence of a mean value follows exactly the proof of the general mean value theorem [@HKWW06]: the candidate for $\tilde a(u,v,w)$ is $$\label{eq:meandiff} \tilde a(u,v,w)=\int_0^1\partial^1f(u+sv,w){\mathop{}\!\mathrm{d}}s\mbox{.}$$ Since $\partial^1f$ is assumed to be continuous in its arguments the integral is well defined and continuous in its arguments $u\in D^1$, $v\in D^1$, $w\in D$. All one has to show is that the $\tilde a$ defined in satisfies : let $x,y \in D^1$ and ${\varepsilon}>0$ be arbitrary, and choose $m$ such that uniformly for all $s\in[0,1]$ $$\begin{aligned} \left\|\int_0^1\partial^1f(x+sy,y){\mathop{}\!\mathrm{d}}s- \frac{1}{m}\sum_{k=0}^{m-1} \partial^1f\left(x+\frac{k}{m}y,y\right)\right\|&<{\varepsilon}\mbox{,}\\ \left\|f\left(x+sy+\frac{y}{m}\right) -f(x+sy)-\partial^1f\left(x+sy,\frac{y}{m}\right)\right\|&<\frac{{\varepsilon}}{m}\mbox{.}\end{aligned}$$ Then it follows that $$\left\|f(x+y)-f(x)-\int_0^1\partial^1f(x+sy,y){\mathop{}\!\mathrm{d}}s\right\|<2{\varepsilon}\mbox{.}$$ Since ${\varepsilon}>0$ was arbitrary the left-hand side must be zero. [$\square$]{} \[thm:flip\] For all $x\in D^{0,1}$ there exists a neighborhood $U(x)\subseteq D^{0,1}$ and a constant $K_x>0$ such that for all $y_1$ and $y_2\in U(x)$ the following Lipschitz estimate holds: $$\|f(y_1)-f(y_2)\|\leq K_x\\|y_1-y_2\\|_D\mbox{.}$$ Note that the upper bound depends only on the $\\|\cdot\\|_D$-norm, not on the $\\|\cdot\\|_{D,L}$-norm, which would be a weaker statement. #### Proof {#proof-9 .unnumbered} We prove the Lipschitz continuity first for $y_1$ and $y_2$ from a sufficiently small neighborhood $U(x)\cap D^1\subseteq D^1$ of $x\in D^1$. Let $x$ be an element of $D^1$. Since the mean value $\tilde a$ is continuous in $(x,0,0)$, and $\tilde a(x,0,0)=0$, we have a $\delta>0$ such that for all $u,v\in D^1$ and $w\in D$ satisfying $\\|u\\|_{D,1}<\delta$, $\\|v\\|_{D,1}<\delta$ and $\\|w\\|_D<\delta$ $$\|\tilde a(x+u,v,w)\|<{\varepsilon}\mbox{.}$$ This implies that $\|\tilde a(x+u,v,w)\|<[{\varepsilon}/\delta]\\|w\\|_D$ for $u$ and $v$ with $\max\{\\|u\\|_{D,1},\\|v\\|_{D,1}\}<\delta$ and $w\in D$ (since $\tilde a$ is linear in its third argument). Thus, $\\|\tilde a(x+u,v,\cdot)\\|_D\leq{\varepsilon}/\delta$ for $\tilde a(x+u,v,\cdot)$ as an element of $L(D;D)$ in the operator norm corresponding to $D$. Consequently, if $\\|y_1-x\\|_{D,1}<\delta/2$ and $\\|y_2-x\\|_{D,1}<\delta/2$ $$\|f(y_1)-f(y_2)\|=\left\|\int_0^1\tilde a(y_2,y_1-y_2,y_1-y_2){\mathop{}\!\mathrm{d}}s\right\|\leq \frac{{\varepsilon}}{\delta}\\|y_1-y_2\\|_D\mbox{,}$$ such that we can choose $K_x={\varepsilon}/\delta$. The extension of the statement to $D^{0,1}$ follows from the continuity of $f$ in $D$: $U(x_0)\cap D^1$ is dense in $U(x_0)\subset D^{0,1}$ using the $\\|\cdot\\|_{D,L}$-norm. Pick two sequences $y_n$ and $z_n$ in $U(x_0)\cap D^1$ that converge to $y$ and $z$ in $U(x_0)$ in the Lipschitz norm. Then $f(y_n)\to f(y)$ and $f(z_n)\to f(z)$ since $f$ is continuous in $D$. Moreover, $\\|y_n-z_n\\|_D\to\\|y-z\\|_D$ for $n\to\infty$. Since $$\|f(y_n)-f(z_n)\|\leq K_x\\|y_n-z_n\\|_D$$ for all $n$ the inequality also holds for the limit for $n\to\infty$. [$\square$]{} Basic properties of $F$ {#sec:Fprop} ----------------------- In this section we restrict ourselves to the periodic case: $J={\mathbb{T}}$. Let $F:D\mapsto C^0({\mathbb{T}};{\mathbb{R}}^n)$ be defined as $F(x)(t)=f(\Delta_tx)$. \[thm:Fcont\] Let $f:D\mapsto{\mathbb{R}}^n$ be continuous. Then $F:D\mapsto C^0({\mathbb{T}};{\mathbb{R}}^n)$ is also continuous. #### Proof {#proof-10 .unnumbered} This is a simple consequence of the continuity of $f$, the continuity of $(t,x)\mapsto \Delta_tx$ with respect to both arguments ($t$ and $x$) in the $\\|\cdot\\|_0$-norm, and the compactness of ${\mathbb{T}}$. Let ${\varepsilon}>0$ and $x\in D$ be arbitrary. We want to prove continuity of $F$ in $x$. So, we have to find a $\delta>0$ such that $$\label{eq:Fcont:epsdelta} \left\|f(\Delta_sx+h)-f(\Delta_sx)\right\|<{\varepsilon}\mbox{\quad for all $s\in{\mathbb{T}}$ and $h\in D$, satisfying $\\|h\\|_D<\delta$.}$$ (Since $\\|\Delta_sh\\|_D=\\|h\\|_D$ we can replace $\Delta_sh$ by $h$.) The continuity of $f$ implies that for every $r>0$ and every $t\in{\mathbb{T}}$ we find a $\delta_x(t,r)$ such that $$\|f(\Delta_tx+h)-f(\Delta_tx)\|<r \mbox{\quad whenever $\\|h\\|_D<\delta_x(t,r)$.}\label{eq:Fcont:fdelta}$$ For every $t\in{\mathbb{T}}$ there exists an open neighborhood $U(t)\subset {\mathbb{T}}$ such that $$\\|\Delta_sx-\Delta_tx\\|_D<\delta_x(t,{\varepsilon}/2)/2 \mbox{\quad for all $s\in U(t)$,}$$ because the function $t\in{\mathbb{T}}\mapsto \Delta_tx$ is continuous in $t$. These neighborhoods $U(t)$ are an open cover of the compact set ${\mathbb{T}}$, so there exist finitely many $t_1,\ldots,t_m\in{\mathbb{T}}$ such that the union of the neighborhoods $U(t_j)$ contains all points $s\in{\mathbb{T}}$. We choose $$\delta=\min_{j=1,\ldots,m}\delta_x(t_j,{\varepsilon}/2)/2\mbox{,}$$ which is a positive quantity. Let $s\in{\mathbb{T}}$ be arbitrary and let $h\in D$ satisfy $\\|h\\|_D<\delta$. We have to check the inequality . The point $s$ is in one of the neighborhoods $U(t_j)$, say without loss of generality, $s\in U(t_1)$. Thus, $\\|\Delta_sx-\Delta_{t_1}x\\|_D<\delta_x(t_1,{\varepsilon}/2)/2$, and, consequently, $\\|\Delta_sx-\Delta_{t_1}x+h\\|_D<\delta_x(t_1,{\varepsilon}/2)$ (because also $\\|h\\|_D<\delta\leq \delta_x(t_1,/{\varepsilon}/2)/2$). Therefore, we can split up the difference $\|f(\Delta_sx+h)-f(\Delta_sx)\|$: $$\begin{aligned} \|f(\Delta_sx+h)-f(\Delta_sx)\|\leq&\ \left\|\left[f\left(\Delta_{t_1}x+(\Delta_sx-\Delta_{t_1}x+h)\right) -f(\Delta_{t_1}x)\right]\right\|\\ &\ +\left\|\left[f\left(\Delta_{t_1}x+(\Delta_sx-\Delta_{t_1}x)\right) -f(\Delta_{t_1}x)\right]\right\|\\ <&\ {\varepsilon}/2+{\varepsilon}/2={\varepsilon}\end{aligned}$$ Note that the deviations from $\Delta_{t_1}x$ in the arguments of $f$ in both terms of the sum are less than or equal to $\delta_x(t_1,{\varepsilon}/2)$ such that we can apply for $t=t_1$, $r={\varepsilon}/2$. [$\square$]{} The following lemma lists properties that $F$ has if $f$ satisfies local $EC$ Lipschitz continuity in the sense of Definition \[def:loclip\]. That is, we do not assume that $f$ is $EC^1$ smooth in the sense of Definition \[def:extdiff\] for Lemma \[thm:Flipbound\]. Since Lemma \[thm:flip\] was proved using only the assumption of $EC^1$ smoothness of $f$, local $EC$ Lipschitz continuity is a weaker condition. \[thm:Flipbound\] Assume that $f:D\mapsto{\mathbb{R}}^n$ is locally $EC$ Lipschitz continuous in the sense of Definition \[def:loclip\]. Then $F$ has the following properties: 1. \[thm:Flip\] for all $x\in D^{0,1}$ there exists a neighborhood $U(x)\subseteq D^{0,1}$ and a constant $K_x>0$ such that for all $y_1$ and $y_2\in U(x)$ $$\\|F(y_1)-F(y_2)\\|_0\leq K_x\\|y_1-y_2\\|_D\mbox{.}$$ 2. \[thm:Fd01bound\] $F$ maps elements of $D^{0,1}$ into $C^{0,1}({\mathbb{T}};{\mathbb{R}}^n)$. Moreover, for every $x\in D^{0,1}$, any bounded neighborhood $U(x)\subseteq D^{0,1}$ for which the Lipschitz constant $K_x$ exists has a bounded image under $F$: there exists a bound $R>0$ such that $\\|F(y)\\|_{0,1}\leq R$ for all $y\in U(x)$ ($R$ depends on $U(x)$). #### Proof {#proof-11 .unnumbered} Statement \[thm:Flip\] is a consequence of the local $EC$ Lipschitz continuity of $f$ and the compactness of ${\mathbb{T}}$ (which allows one to choose a uniform Lipschitz bound $K_x$ for all $t\in{\mathbb{T}}$). Concerning statement \[thm:Fd01bound\]: let $x\in D^{0,1}$ be arbitrary, and let the neighborhood $U(x)$ be bounded (say, $\\|y-x\\|_{D,L}\leq \delta$) such that $F$ has a Lipschitz constant $K_x$ in $U(x)$. Then we have for all $y,z\in U(x)$ and $t,s\in{\mathbb{T}}$ the estimate $$\|f(\Delta_ty)-f(\Delta_sz)\|\leq K_x\\|\Delta_ty-\Delta_sz\\|_D= K_x\\|\Delta_{t-s}y-z\\|_D\mbox{.}$$ Initially setting $z=x$ and $s=t$ we get a bound on $\\|F(y)\\|_0$: $\\|F(y)\\|_0\leq \\|F(x)\\|_0+K_x\delta=:R_0$ for all $y\in U(x)$. It remains to be shown that the Lipschitz constant of $F(y)$ is bounded for $y\in U(x)$: $$\begin{aligned} \|F(y)(t)-F(y)(s)\|=\|f(\Delta_ty)-f(\Delta_sy)\| \leq K_x\\|\Delta_ty-\Delta_sy\\|_D \leq K_x\\|y\\|_{D,L}\|t-s\|\mbox{.}\end{aligned}$$ Since $\\|y-x\\|_{D,L}\leq \delta$ for $y\in U(x)$, choosing $$R=\max\left\{R_0,K_x\left(\\|x\\|_{D,L}+\delta\right)\right\}$$ ensures that $\\|F(y)\\|_{0,1}\leq R$. [$\square$]{} Define the maps $$\begin{aligned} \partial^1F(u,v)(t)&=\partial^1f(\Delta_tu,\Delta_tv) &&\mbox{for $u\in D^1$, $v\in D$,}\\ \tilde A(u,v,w)(t)&=\tilde a(\Delta_tu,\Delta_tv,\Delta_tw) &&\mbox{for $u\in D^1$, $v\in D^1$, $w\in D$.}\end{aligned}$$ The following Lemma \[thm:Fdiff\], and Lemma \[thm:Fimage\] assume that $f$ is $EC^1$ smooth in $D$ in the sense of Definition \[def:extdiff\]. \[thm:Fdiff\] Let $f:D\mapsto{\mathbb{R}}^n$ be $EC^1$ smooth. Then $F$, $\partial^1F$ and $\tilde A$ have the following properties. 1. \[thm:Fcontdiff\] The map $(u,v)\mapsto\partial^1F(u,v)$ is continuous in both arguments (and linear in its second argument) as a map from $D^1\times D$ into $C^0({\mathbb{T}};{\mathbb{R}}^n)$. It satisfies for all $x\in D^1$ $$\label{eq:Fcontdiff} \lim_{ \begin{subarray}{c} y\in D^{0,1}\\[0.2ex] \\|y\\|_{D,L}\to0 \end{subarray} }\frac{\\|F(x+y)-F(x)- \partial^1F(x,y)\\|_0}{\\|y\\|_{D,L}}=0\mbox{.}$$ 2. \[thm:Fmeanval\] The map $\tilde A(u,v,w)$ is continuous in all three arguments (and linear in its third argument) as a map from $D^1\times D^1\times D$ into $C^0({\mathbb{T}};{\mathbb{R}}^n)$. It satisfies for all $x,y\in D^1$ $$F(x+y)-F(x)=\tilde A_1(x,y,y)\mbox{.}$$ Moreover, $\tilde A(x,0,y)=\partial^1F(x,y)$ for all $x\in D^1$ and $y\in D$. Note that in the limit we allow for deviations $y\in D^{0,1}$. #### Proof {#proof-12 .unnumbered} The continuity of $\partial^1F$ follows from the continuity of $\partial^1f$ by applying Lemma \[thm:Fcont\] to $\partial^1f:D^1\times D\mapsto{\mathbb{R}}^n$ instead of $f$. The linearity of $\partial^1F$ in its second argument follows from the linearity of $\partial^1f$ in its second argument. The limit also follows from the corresponding limit : let $x\in D^1$ and ${\varepsilon}>0$ be arbitrary. For every fixed $t$ there exists a $\delta(t)>0$ such that $$\label{eq:Fcontdiffproof:ineq} \frac{\|f(\Delta_tx+\Delta_ty)-f(\Delta_tx)- \partial^1f(\Delta_tx,\Delta_ty)\|}{\\|y\\|_{D,L}}<{\varepsilon}$$ for all $y$ with $\\|y\\|_{D,L}<\delta(t)$. As $f$ and $\partial^1f$ are continuous in their arguments $x\in D^1$ and $y\in D^{0,1}$, the inequality also holds for all $s$ in a sufficiently small open neighborhood of $t$, $U(t)$. The set of neighborhoods $U(t)$ for all $t\in{\mathbb{T}}$ are a cover of the compact set ${\mathbb{T}}$ by open sets. Choosing a finite subcover from this cover, and labeling the times $t_1,\ldots,t_m$, we can choose $$\delta=\min_{k=1,\ldots,m}\delta(t_k)$$ such that holds for all uniformly $t\in{\mathbb{T}}$. This proves statement \[thm:Fcontdiff\] of the lemma. Concerning statement \[thm:Fmeanval\]: for the continuity of $\tilde A$ we invoke again Lemma \[thm:Fcont\], this time for $\tilde a$ on $D^1\times D^1\times D$. The linearity of $\tilde A$ in its third argument follows from the linearity of $\tilde a$ in its third argument. The relations $F(x+y)(t)-F(x)(t)=\tilde A(x,y,y)(t)$ and $\tilde A(x,0,y)(t)=\partial^1F(x,y)(t)$ in every $t\in{\mathbb{T}}$ follow from the corresponding relations for $f$ and $\tilde a$, as stated in Lemma \[thm:fmeanval\]. [$\square$]{} \[thm:Fimage\] Let $f:D\mapsto{\mathbb{R}}^n$ be $EC^1$ smooth and let $k\geq0$ be some integer. We assume that $F:D\mapsto C^k({\mathbb{T}};{\mathbb{R}}^n)$ and $\partial^1F:D^1\times D\mapsto C^k({\mathbb{T}};{\mathbb{R}}^n)$ are continuous maps. Then $F$ maps elements of $D^1$ into $C^{k+1}({\mathbb{T}};{\mathbb{R}}^n)$, and $F$ is continuous as a map from $D^1$ to $C^{k+1}({\mathbb{T}};{\mathbb{R}}^n)$. #### Proof {#proof-13 .unnumbered} Let $x$ be in $D^1$, that is, $x'\in D$. If $\partial^1F:D^1\times D\mapsto C^k({\mathbb{T}};{\mathbb{R}}^n)$ is continuous then $\tilde A:D^1\times D^1\times D\mapsto C^k({\mathbb{T}};{\mathbb{R}}^n)$, which is given by $\tilde A(u,v,w)=\int_0^1\partial^1F(u+sv,w){\mathop{}\!\mathrm{d}}s$, is continuous, too. Using statement \[thm:Fmeanval\] of Lemma \[thm:Fdiff\] we have $$\begin{aligned} \frac{F(\Delta_hx)-F(x)}{h}&=\tilde A\left(x,\Delta_hx-x,\frac{\Delta_hx-x}{h}\right)\mbox{.} \label{eq:Fimage:meanval}\end{aligned}$$ On the right side $\\|\Delta_hx-x\\|_{D,1}$ converges to $0$ for $h\to0$. Also, $$\left\\|\frac{\Delta_hx-x}{h}-x'\right\\|_D\to0\mbox{\quad for $h\to0$,}$$ because $x\in D^1$. Since $\tilde A$ is continuous in its arguments the right side converges to $\tilde A(x,0,x')=\partial^1F(x,x')\in C^k({\mathbb{T}};{\mathbb{R}}^n)$ for $h\to0$. Thus, the limit of the left-hand side in for $h\to0$ exists, too, such that $F(x)\in C^{k+1}({\mathbb{T}};{\mathbb{R}}^n)$ and the time derivative $(F(x))'$ equals $\partial^1F(x,x')$. Since $(v,w)\in D^1\times D\mapsto \partial^1F(v,w)\in C^k({\mathbb{T}};{\mathbb{R}}^n)$ is continuous in $(u,v)$, the time derivative of $F(x)$, $(F(x))'=\partial^1F(x,x')$ is also continuous in $x$ if we use the norm $\\|\cdot\\|_{D,1}$ for the argument and $\\|\cdot\\|_k$ for the image. [$\square$]{}
--- author: - \| [ Xiaoxiao Zhao, and Shiqing Zhang]{}\ [Yangtze Center of Mathematics and College of Mathematics, Sichuan University,]{}\ [Chengdu 610064, People’s Republic of China]{} title: 'Periodic Solutions for Circular Restricted 4-body Problems with Newtonian Potentials [^1]' --- > [Abstract:]{} We study the existence of non-collision periodic solutions with Newtonian potentials for the following planar restricted 4-body problems: Assume that the given positive masses $m_{1},m_{2},m_{3}$ in a Lagrange configuration move in circular obits around their center of masses, the sufficiently small mass moves around some body. Using variational minimizing methods, we prove the existence of minimizers for the Lagrangian action on anti-T/2 symmetric loop spaces. Moreover, we prove the minimizers are non-collision periodic solutions with some fixed wingding numbers. > > [Keywords:]{} Restricted 4-body problem; non-collision periodic solution; variational minimizer; wingding number. > > 2000 AMS Subject Classification 34C15, 34C25, 58F. Introduction and Main Results ============================= In this paper, we study the planar circular restricted 4-body problems with Newtonian potentials. Suppose points of positive masses $m_{1},m_{2},m_{3}$ move in a plane of their circular orbits $q_{1}(t),q_{2}(t), q_{3}(t)$ and the center of masses is at the origin; suppose the sufficiently small mass point does not influence the motion of $m_{1},m_{2},m_{3}$, and moves in the plane for the given masses $m_{1},m_{2},m_{3}$. It is well-known that $q_{1}(t),q_{2}(t), q_{3}(t)$ satisfy the Newtonian equations: $$\label{e1} m_{i}\ddot{q_{i}}=\frac{\partial U}{\partial q_{i}},\ \ \ \ i=1,2,3,$$ where $$\label{e2} U=\sum\limits_{1\leq i< j\leq 3}\frac{m_{i}m_{j} }{\| q_{i}-q_{j}\|}.$$ Without loss of generality, we assume that there exists $\theta_{1},\theta_{2},\theta_{3}\in[0,2\pi)$ such that the planar circular orbits are $$\label{e3} q_{1}(t)=r_{1}e^{\sqrt{-1}\frac{2\pi}{T} t}e^{\sqrt{-1}\theta_{1}},\ \ q_{2}(t)=r_{2}e^{\sqrt{-1}\frac{2\pi}{T} t}e^{\sqrt{-1}\theta_{2}}, \ \ q_{3}(t)=r_{3}e^{\sqrt{-1}\frac{2\pi}{T} t}e^{\sqrt{-1}\theta_{3}},$$ where the radius $r_{1}, r_{2},r_{3}$ are positive constants depending on $m_{i}(i=1,2,3)$ and $T$ (see Lemma 2.6). We also assume that $$\label{e4} m_{1}q_{1}(t)+m_{2} q_{2}(t)+m_{3} q_{3}(t)=0$$ and $$\label{e5} \|q_{i}-q_{j}\|=l, \ \ 1\leq i\neq j\leq3,$$ where the constant $l>0$ depends on $m_{i}(i=1,2,3)$ and $T$ (see Lemma 2.5). The orbit $q(t)\in R^{2}$ for sufficiently small mass is governed by the gravitational forces of $m_{1},m_{2},m_{3}$ and therefore it satisfies the following equation $$\label{e6} \ddot{q}=\sum\limits_{i=1}^{3}\frac{m_{i}(q_{i}-q) }{\| q_{i}-q\|^{3}}.$$ For $N$-body problems, there are many papers concerned with the periodic solutions by using variational methods, see \[1-9,13-16,18\] and the references therein. In [@r3], Chenciner-Montgomery proved the existence of the remarkable figure-“8” type periodic solution for planar Newtonian 3-body problems with equal masses. Marchal[@r6] studied the fixed end problem for Newtonian n-body problems and proved the minimizer for the Lagrangian action has no interior collision. Especially, in [@r8], Simó used computer to discover many new periodic solutions for Newtonian n-body problems. Zhang-Zhou\[13-15\] decomposed the Lagrangian action for n-body problems into some sum for two body problems and \[14,15\] avoid collisions by comparing the lower bound for the Lagrangian action on the symmetry collision orbits and the upper bound for the Lagrangian action on test orbits in some cases. Motivated by the above works, we use variational methods to study the circular restricted 3+1-body problem with some fixed wingding numbers and some masses. For the readers’ conveniences, we recall the definition of the winding number, which can be found in many books on the classical differential geometry. Definition 1.1 Let $\Gamma: x(t), t \in [a,b]$ be an given oriented continuous closed curve, and $p$ be a point of the plane not on the curve. Then, the mapping $\varphi: \Gamma\rightarrow S^{1}$, given by $$\varphi(x(t)) = \frac{x(t)-p}{\|x(t)-p\|},\ \ \ t \in [a,b]$$ is defined to be the position mapping of the curve $\Gamma$ relative to $p$, when the point on $\Gamma$ goes around the curve once, its image point $\varphi(x(t))$ will go around $S^{1}$ a number of times, this number is called the winding number of the curve $\Gamma$ relative to $p$, and we denote it by $deg(\Gamma,p)$. If $p$ is the origin, we write $deg\Gamma$. Define $$W^{1,2}(R/TZ,R^{2})=\bigg\{x(t)\Big\| x(t),\dot{x}(t)\in L^{2}(R,R^{2}), \ x(t+T)=x(t) \bigg\}.$$ The norm of $W^{1,2}(R/TZ,R^{2})$ is $$\label{e7} \\|x\\|=\Big[\int_{0}^{T}\|x\|^{2}dt\Big]^{\frac{1}{2}}+\Big[\int_{0}^{T} \|\dot{x}\|^{2}dt\Big]^{\frac{1}{2}}.$$ The functional corresponding to the equation (\[e6\]) is $$\label{e8} f(q)=\int_{0}^{T}\Big[\frac{1}{2}\|\dot{q}\|^{2}+\sum\limits_{i=1}^{3}\frac{m_{i} }{\| q-q_{i}\|}\Big]dt,\ \ \ \ q\in \Lambda_{\pm},$$ where $$\Lambda_{-}=\left\{ q\in W^{1,2}(R/TZ,R^{2})\bigg\| \begin{array}{c} q(t+\frac{T}{2})=-q(t),\ deg(q-q_{1})=-1,\\ q(t)\neq q_{i}(t),\ \forall t\in[0,T],i=1,2,3\ \ \ \ \ \ \ \end{array} \right\}.$$ and $$\Lambda_{+}=\left\{ q\in W^{1,2}(R/TZ,R^{2})\bigg\| \begin{array}{c} q(t+\frac{T}{2})=-q(t),\ deg(q-q_{1})=1,\\ q(t)\neq q_{i}(t),\ \forall t\in[0,T],i=1,2,3\ \ \ \ \ \ \ \end{array} \right\}.$$ Our main results are the following: Theorem 1.1 Let $T=1$, for the values of $m_{1},m_{2},m_{3}$ given in Table 1 with $M=1$, the minimizer of $f(q)$ on the closure $\overline{\Lambda}_{-}$ of $\Lambda_{-}$ is a non-collision 1-periodic solution of (\[e6\]); for the values of $m_{1},m_{2},m_{3}$ given in Table 2 with $m_{1}=m_{2}=m_{3}=1$, the minimizer of $f(q)$ on $\overline{\Lambda}_{-}$ is a non-collision 1-periodic solution of (\[e6\]). In proving Theorem 1, we need to use test functions. We find that if the test functions are circular orbits, we can not get the desired results on $\overline{\Lambda}_{-}$. Therefore, we select elliptic orbits as test functions. Theorem 1.2 Let $T=1$, for the values of $m_{1},m_{2},m_{3}$ given in Table 3 with $M=1$, the minimizer of $f(q)$ on the closure $\overline{\Lambda}_{+}$ of $\Lambda_{+}$ is a non-collision 1-periodic solution of (\[e6\]); for the values of $m_{1},m_{2},m_{3}$ given in Table 4 with $m_{1}=m_{2}=m_{3}=1$, the minimizer of $f(q)$ on $\overline{\Lambda}_{+}$ is a non-collision 1-periodic solution of (\[e6\]). When we take elliptic orbits as test functions, we find that the biggest symmetric space is the anti-T/2 symmetric loop space if the wingding number $n$ is odd($n=\pm1,\pm3,\cdots$); we can not find suitable symmetric space if the wingding number is even. When the wingding number $n\neq\pm1$ and we take circular orbits as test functions, we find that the biggest symmetric space is $$\Lambda=\left\{ q\in W^{1,2}(R/TZ,R^{2})\bigg\| \begin{array}{c} q(t+\frac{T}{\|n-1\|})=R(\|n-1\|)q(t),\ deg(q-q_{1})=n,\\ q(t)\neq q_{i}(t),\ \forall t\in[0,T],i=1,2,3\ \ \ \ \ \ \ \end{array} \right\},$$ where $$R(\|n-1\|)=\left(\begin{array}{ccc} cos\frac{2\pi}{\|n-1\|} & -sin\frac{2\pi}{\|n-1\|}\\ sin\frac{2\pi}{\|n-1\|} & cos\frac{2\pi}{\|n-1\|} \end{array} \right)\in SO(2)$$ is a counter-clockwise rotation of angle $\frac{2\pi}{\|n-1\|}$ in $R^{2}$. But the Lagrangian actions on the circular test orbits are bigger than the lower bound for the Lagrangian actions on collision symmetric orbits. Hence we consider the anti-T/2 symmetric loop spaces $\Lambda_{\pm}$. Preliminaries ============= In this section, we will list some basic Lemmas and inequality for proving our Theorems 1.1 and 1.2. (Tonelli[@r1],[@r11]) Let $X$ be a reflexive Banach space, $S$ be a weakly closed subset of $X$, $f:S\rightarrow R\cup \{+\infty\}$. If $f\not\equiv +\infty$ is weakly lower semi-continuous and coercive($f(x)\rightarrow +\infty$ as $\\|x\\|\rightarrow +\infty$), then $f$ attains its infimum on $S$. Lemma 2.2(Poincare-Wirtinger Inequality[@r10]) Let $q\in W^{1,2}(R/TZ,R^{K})$ and $\int_{0}^{T}q(t)dt=0$, then $$\int_{0}^{T}\|q(t)\|^{2}dt\leq\frac{T^{2}}{4\pi^{2}}\int_{0}^{T}\|\dot{q}(t)\|^{2}dt.$$ (Palais’s Symmetry Principle([@r12])) Let $\sigma$ be an orthogonal representation of a finite or compact group $G$, $H$ be a real Hilbert space, $f:H\rightarrow R$ satisfies $f(\sigma\cdot x)=f(x),\forall\sigma\in G,\forall x\in H$. Set $F=\{x\in H\|\sigma\cdot x=x,\ \forall \sigma\in G\}$. Then the critical point of $f$ in $F$ is also a critical point of $f$ in $H$. By Palais’s Symmetry Principle and the perturbation invariance for wingding numbers, we know that the critical point of $f(q)$ in $\Lambda_{\pm}$ is a periodic solution of Newtonian equation (\[e6\]). (1)(Gordon’s Theorem[@r17]) Let $x\in W^{1,2}([t_{1}, t_{2}],R^{K})$ and $x(t_{1}) = x(t_{2}) = 0$. Then for any $a > 0$, we have $$\int_{t_{1}}^{t_{2}}(\frac{1}{2}\|\dot{x}\|^{2}+\frac{a}{\|x\|})dt \geq\frac{3}{2}(2\pi)^{2/3}a^{2/3}(t_{2}-t_{1})^{1/3}.$$ (2)(Long-Zhang[@r18]) Let $x\in W^{1,2}(R/TZ,R^{K}),\int_{0}^{T} xdt = 0$, then for any $a > 0$, we have $$\int_{0}^{T}(\frac{1}{2}\|\dot{x}\|^{2}+\frac{a}{\|x\|})dt \geq\frac{3}{2}(2\pi)^{2/3}a^{2/3}T^{1/3}.$$ Let $M=m_{1}+m_{2}+m_{3}$, we have $l=\sqrt[3]{\frac{MT^{2}}{4\pi^{2}}}$. It follows from (\[e1\]) and (\[e2\]) that $$\ddot{q_{1}}=m_{2}\frac{q_{2}-q_{1}}{\| q_{2}-q_{1}\|^{3}}+m_{3}\frac{q_{3}-q_{1}}{\| q_{3}-q_{1}\|^{3}}.$$ Then by (\[e3\])-(\[e5\]), we obtain $$\begin{aligned} -\frac{4\pi^{2}}{T^{2}}q_{1}&=\frac{1}{l^{3}}(m_{2}q_{2}+m_{3}q_{3}-m_{2}q_{1}-m_{3}q_{1})\\ &=\frac{1}{l^{3}}(-m_{1}q_{1}-m_{2}q_{1}-m_{3}q_{1}), \end{aligned}$$ which implies $$l^{3}=\frac{MT^{2}}{4\pi^{2}},\ \ \ \ \ $$ that is, $$\label{e17} l=\sqrt[3]{\frac{MT^{2}}{4\pi^{2}}}.\ \ \ \Box$$ The radius $r_{1},r_{2},r_{3}$ of the planar circular orbits for the masses $m_{1},m_{2},m_{3}$ are $$r_{1}=\frac{\sqrt{m^{2}_{2}+m_{2}m_{3}+m^{2}_{3}}}{M}l,\ \ \ \ \ $$ $$r_{2}=\frac{\sqrt{m^{2}_{1}+m_{1}m_{3}+m^{2}_{3}}}{M}l,\ \ \ \ \ $$ $$r_{3}=\frac{\sqrt{m^{2}_{1}+m_{1}m_{2}+m^{2}_{2}}}{M}l.\ \ \ \ \ $$ Choose the geometrical center of the initial configuration ($q_{1}(0),q_{2}(0),q_{3}(0)$) as the origin of the coordinate (x,y). Without loss of generality, by (\[e5\]), we suppose the location coordinates of $q_{1}(0),q_{2}(0),q_{3}(0)$ are $A_{1}(\frac{\sqrt{3}l}{3},0),A_{2}(-\frac{\sqrt{3}l}{6},\frac{l}{2}), A_{3}(-\frac{\sqrt{3}l}{6},-\frac{l}{2})$. Then we can get the coordinate of the center of masses $m_{1},m_{2},m_{3}$ is $C(\frac{\frac{\sqrt{3}}{3}m_{1}l-\frac{\sqrt{3}}{6}m_{2}l -\frac{\sqrt{3}}{6}m_{3}l}{M},\frac{\frac{m_{2}}{2}l-\frac{m_{3}}{2}l}{M})$. To make sure the Assumption (\[e4\]) holds, we introduce the new coordinate $$\left\{\begin{array}{ll} X=x- \frac{\frac{\sqrt{3}}{3}m_{1}l-\frac{\sqrt{3}}{6}m_{2}l -\frac{\sqrt{3}}{6}m_{3}l}{M},\\ Y=y-\frac{\frac{m_{2}}{2}l-\frac{m_{3}}{2}l}{M} . \end{array}\right.$$ Hence in the new coordinate (X,Y), the location coordinates of $q_{1}(0),q_{2}(0),q_{3}(0)$ are $A_{1}(\frac{\frac{\sqrt{3}}{2}m_{2}l+\frac{\sqrt{3}}{2}m_{3}l}{M},$ $\frac{-\frac{m_{2}}{2}l+\frac{m_{3}}{2}l}{M}),$ $A_{2}(-\frac{\frac{\sqrt{3}}{2}m_{1}l}{M},\frac{\frac{m_{1}}{2}l+m_{3}l}{M}), A_{3}(-\frac{\frac{\sqrt{3}}{2}m_{1}l}{M},-\frac{\frac{m_{1}}{2}l+m_{2}l}{M})$ and the center of masses $m_{1},m_{2},m_{3}$ is at the origin $O(0,0)$. Then compared with (\[e3\]), we have $$r_{1}=\|A_{1}O\|=\frac{\sqrt{m^{2}_{2}+m_{2}m_{3}+m^{2}_{3}}}{M}l,\ \ \ \ \ $$ $$r_{2}=\|A_{2}O\|=\frac{\sqrt{m^{2}_{1}+m_{1}m_{3}+m^{2}_{3}}}{M}l,\ \ \ \ \ $$ $$r_{3}=\|A_{3}O\|=\frac{\sqrt{m^{2}_{1}+m_{1}m_{2}+m^{2}_{2}}}{M}l,\ \ \ \ \ $$ and $$\sin\theta_{1}=\frac{-m_{2}+m_{3}}{2\sqrt{m^{2}_{2}+m_{2}m_{3}+m^{2}_{3}}},\ \ \ \ \ \ \ \cos\theta_{1}=\frac{\sqrt{3}(m_{2}+m_{3})}{2\sqrt{m^{2}_{2}+m_{2}m_{3}+m^{2}_{3}}},\ \ \ \ \ \ $$ $$\sin\theta_{2}=\frac{m_{1}+2m_{3}}{2\sqrt{m^{2}_{1}+m_{1}m_{3}+m^{2}_{3}}},\ \ \ \ \ \ \ \cos\theta_{2}=-\frac{\sqrt{3}m_{1}}{2\sqrt{m^{2}_{1}+m_{1}m_{3}+m^{2}_{3}}},\ \ \ \ $$ $$\sin\theta_{3}=-\frac{m_{1}+2m_{2}}{2\sqrt{m^{2}_{1}+m_{1}m_{2}+m^{2}_{2}}},\ \ \ \ \ \cos\theta_{3}=-\frac{\sqrt{3}m_{1}}{2\sqrt{m^{2}_{1}+m_{1}m_{2}+m^{2}_{2}}}.\ \ \ \Box$$ Proof of Theorems ================= In order to get Theorems, we need two steps to complete the proof. Step 1: We will establish the existence of variational minimizers of $f(q)$ in (\[e8\]) on $\bar{\Lambda}_{\pm}$. $f(q)$ in (\[e8\]) attains its infimum on $\bar{\Lambda}_{\pm}$. By using Lemma 2.2, for $\forall q\in \Lambda_{\pm}$, we can get that the equivalent norm of (\[e7\]) in $\bar{\Lambda}_{\pm}$ is $$\\|q\\|\cong\Big[\int_{0}^{T}\|\dot{q}\|^{2}dt\Big]^{\frac{1}{2}}.$$ Hence by the definition of $f(q)$, $f$ is coercive on $\bar{\Lambda}_{\pm}$. Next, we claim that $f$ is weakly lower semi-continuous on $\bar{\Lambda}_{\pm}$. In fact, for $\forall q^{k}\in \Lambda_{\pm}$, if $q^{k}\rightharpoonup q$ weakly, by compact embedding theorem, we have the uniformly convergence: $$\max\limits_{0\leq t\leq T}\|q^{k}(t)-q(t)\|\rightarrow 0,\ \ \ \ k\rightarrow\infty,$$ which implies $$\label{e9} \int_{0}^{T}\sum\limits_{i=1}^{3}\frac{m_{i}}{\| q^{k}-q_{i}\|}dt\rightarrow\int_{0}^{T}\sum\limits_{i=1}^{3}\frac{m_{i} }{\| q-q_{i}\|}dt.$$ It is well-known that the norm and its square are weakly lower semi-continuous. Therefore, combined with (\[e9\]), we obtain $$\liminf\limits_{k\rightarrow\infty}f(q^{k})\geq f(q),$$ that is, $f$ is weakly lower semi-continuous on $\bar{\Lambda}_{\pm}$. By Lemma 2.1, we can get that $f(q)$ in (\[e8\]) attains its infimum on $\bar{\Lambda}_{\pm}$. $\Box$ Step 2: We will prove the variational minimizers in Lemma 3.1 is the noncollision T-period solution of (\[e6\]). For any collision generalized solution $q$, we can estimate the lower bound for the value of Lagrangian action functional. For $\partial\Lambda_{\pm}=\{q\in W^{1,2}(R/TZ,R^{2})\|q(t+\frac{T}{2})=-q(t),\ \exists 1\leq i^{\pm}_{0}\leq 3, t_{i^{\pm}_{0}}\in[0,T] \ s.t. \ q_{i^{\pm}_{0}}(t_{i^{\pm}_{0}})=q(t_{i^{\pm}_{0}})\}$, we have $$\inf\limits_{q\in\partial\Lambda_{\pm}}f(q)\geq\frac{3}{2}(2\pi)^{2/3}C M^{-1/3}T^{1/3} \triangleq d_{1},$$ where $$C=\min\left\{ \begin{array}{c} 2^{\frac{2}{3}}m_{1}+m_{2}+m_{3}-\frac{1}{3M}(m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3}),\\ 2^{\frac{2}{3}}m_{2}+m_{1}+m_{3}-\frac{1}{3M}(m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3}),\\ 2^{\frac{2}{3}}m_{3}+m_{1}+m_{2}-\frac{1}{3M}(m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3}) \end{array} \right\}.$$ It follows from (\[e4\]) that $$\sum\limits_{i=1}^{3}m_{i}\dot{q}_{i}=0,$$ which implies $$\begin{aligned} \sum\limits_{i=1}^{3}m_{i}\|\dot{q}-\dot{q}_{i}\|^{2}&=&\sum\limits_{i=1}^{3}m_{i} \Big(\|\dot{q}\|^{2}+\|\dot{q}_{i}\|^{2}-2\langle\dot{q},\dot{q}_{i}\rangle\Big)\nonumber\\ &=&M\|\dot{q}\|^{2}+\sum\limits_{i=1}^{3}m_{i}\|\dot{q}_{i}\|^{2}-2\Big\langle\dot{q}, \sum\limits_{i=1}^{3}m_{i}\dot{q}_{i}\Big\rangle\nonumber\\ &=&M\|\dot{q}\|^{2}+\sum\limits_{i=1}^{3}m_{i}\|\dot{q}_{i}\|^{2}.\end{aligned}$$ Therefore $$\|\dot{q}\|^{2}=\frac{1}{M}\sum\limits_{i=1}^{3}m_{i}\Big(\|\dot{q}-\dot{q}_{i}\|^{2}-\|\dot{q}_{i}\|^{2}\Big).$$ Hence $$\begin{aligned} f(q)&=&\int_{0}^{T}\Big[\frac{1}{2}\|\dot{q}\|^{2}+\sum\limits_{i=1}^{3}\frac{m_{i} }{\| q-q_{i}\|}\Big]dt\nonumber\\ &=&\frac{1}{M}\int_{0}^{T}\sum\limits_{i=1}^{3}m_{i}\Big[\frac{1}{2}\|\dot{q}-\dot{q}_{i}\|^{2} +\frac{M}{\|q-q_{i}\|}\Big]dt-\frac{1}{2M}\int_{0}^{T}\sum\limits_{i=1}^{3}m_{i}\|\dot{q}_{i}\|^{2}dt.\end{aligned}$$ If $q\in\bar{\Lambda}_{-}$ is a collision generalized solution, then there exists $t_{i^{-}_{0}}\in [0,T]$ and $1\leq i^{-}_{0}\leq 3$ such that $q(t_{i^{-}_{0}}) = q_{i^{-}_{0}}(t_{i^{-}_{0}})$. Since $q_{i}(t+\frac{T}{2})=-q_{i}(t)$, we obtain $q(t_{i^{-}_{0}}+\frac{kT}{2})=q_{i^{-}_{0}}(t_{i^{-}_{0}}+\frac{kT}{2}), \ \ \forall 0\leq k\leq 2$. So, by (1) of Lemma 2.4, we get $$\begin{aligned} \label{e10} \frac{1}{M}\int_{0}^{T}m_{i^{-}_{0}}\Big[\frac{1}{2}\|\dot{q}-\dot{q}_{i^{-}_{0}}\|^{2} +\frac{M}{\|q-q_{i^{-}_{0}}\|}\Big]dt&=&\frac{2}{M}m_{i^{-}_{0}}\int_{0}^{\frac{T}{2}}\Big[\frac{1}{2}\|\dot{q}-\dot{q}_{i^{-}_{0}}\|^{2} +\frac{M}{\|q-q_{i^{-}_{0}}\|}\Big]dt\nonumber\\ &\geq&\frac{3}{2}(2\pi)^{2/3}2^{2/3}m_{i^{-}_{0}}M^{-1/3}T^{1/3}.\end{aligned}$$ For noncollision pair $q,q_{i}(i\neq i^{-}_{0})$, we have $\int_{0}^{T}q(t)dt=0$, $\int_{0}^{T}q_{i}(t)dt=0$. Therefore $\int_{0}^{T}\big(q(t)-q_{i}(t)\big)dt=0$. Hence by (2) of Lemma 2.4, we can get $$\label{e11} \frac{1}{M}\int_{0}^{T}\sum\limits_{i\neq i^{-}_{0}}m_{i}\Big[\frac{1}{2}\|\dot{q}-\dot{q}_{i}\|^{2} +\frac{M}{\|q-q_{i}\|}\Big]dt\geq\frac{3}{2}(2\pi)^{2/3}(M-m_{i^{-}_{0}})M^{-1/3}T^{1/3}.$$ For the other term of $f$, using the expression for the orbits $q_{1},q_{2},q_{3}$ as in (\[e3\]), Lemma 2.5 and Lemma 2.6, we obtain $$\label{e12} -\frac{1}{2M}\int_{0}^{T}\sum\limits_{i=1}^{3}m_{i}\|\dot{q}_{i}\|^{2}dt =-\frac{1}{2}(2\pi)^{2/3}(m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3})M^{-4/3}T^{1/3}.$$ Therefore, it follows from (\[e10\]) - (\[e12\]) that $$\inf\limits_{q\in\partial\Lambda_{-}}f(q)\geq\frac{3}{2}(2\pi)^{2/3}C M^{-1/3}T^{1/3} \triangleq d_{1},$$ where $$C=\min\left\{ \begin{array}{c} 2^{\frac{2}{3}}m_{1}+m_{2}+m_{3}-\frac{1}{3M}(m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3}),\\ 2^{\frac{2}{3}}m_{2}+m_{1}+m_{3}-\frac{1}{3M}(m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3}),\\ 2^{\frac{2}{3}}m_{3}+m_{1}+m_{2}-\frac{1}{3M}(m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3}) \end{array} \right\}.$$ Similarly, if $q\in\bar{\Lambda}_{+}$ is a collision generalized solution, we have $$\inf\limits_{q\in\partial\Lambda_{+}}f(q)\geq\frac{3}{2}(2\pi)^{2/3}C M^{-1/3}T^{1/3} \triangleq d_{1},\ \ \Box$$ Proof of Theorem 1.1 In order to get Theorem 1.1, we are going to find a test loop $\tilde{q}\in\Lambda_{-}$ such that $f(\tilde{q})\leq d_{2}$. Then the minimizer of $f$ on $\bar{\Lambda}_{-}$ must be a noncollision solution if $d_{2}<d_{1}$. Let $a > 0, b>0$, $\theta\in[0,2\pi)$ and $$\tilde{q}-q_{1}=\bigg(a\cos\Big(-\frac{2\pi}{T} t+\theta\Big),b\sin\Big(-\frac{2\pi}{T} t+\theta\Big)\bigg)^{T}.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ Hence $$\begin{aligned} \tilde{q}-q_{2}&=&\tilde{q}-q_{1}+q_{1}-q_{2}\nonumber\\ &=&(q_{1}-q_{2})+(\tilde{q}-q_{1})\nonumber\\ &=&\bigg(r_{1}\cos\Big(\frac{2\pi}{T} t+\theta_{1}\Big)-r_{2}\cos\Big(\frac{2\pi}{T} t+\theta_{2}\Big)+a\cos\Big(-\frac{2\pi}{T} t+\theta\Big),r_{1}\sin\Big(\frac{2\pi}{T} t+\theta_{1}\Big)\ \ \ \ \ \ \ \nonumber\\ & &-r_{2}\sin\Big(\frac{2\pi}{T} t+\theta_{2}\Big)+b\sin\Big(-\frac{2\pi}{T} t+\theta\Big)\bigg)^{T},\end{aligned}$$ $$\begin{aligned} \tilde{q}-q_{3}&=&\bigg(r_{1}\cos\Big(\frac{2\pi}{T} t+\theta_{1}\Big)-r_{3}\cos\Big(\frac{2\pi}{T} t+\theta_{3}\Big)+a\cos\Big(-\frac{2\pi}{T} t+\theta\Big),r_{1}\sin\Big(\frac{2\pi}{T} t+\theta_{1}\Big)\ \ \ \ \ \ \ \nonumber\\ & &-r_{3}\sin\Big(\frac{2\pi}{T} t+\theta_{3}\Big)+b\sin\Big(-\frac{2\pi}{T} t+\theta\Big)\bigg)^{T}.\end{aligned}$$ It is easy to see that $\tilde{q}\in\Lambda_{-}$ and $$\begin{aligned} \label{e13} \|\dot{\tilde{q}}-\dot{q_{1}}\|^{2}&=&\Big(\frac{2\pi}{T}\Big)^{2}\bigg[\frac{a^{2}+b^{2}}{2}-\frac{a^{2}-b^{2}}{2}cos\Big(\frac{4\pi}{T} t-2\theta\Big)\bigg],\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\end{aligned}$$ $$\begin{aligned} \label{14} \|\tilde{q}-q_{1}\|=\sqrt{\frac{a^{2}+b^{2}}{2}+\frac{a^{2}-b^{2}}{2}cos\Big(\frac{4\pi}{T} t-2\theta\Big)},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\end{aligned}$$ $$\begin{aligned} \label{e15} \|\dot{\tilde{q}}-\dot{q_{2}}\|^{2}&=&\Big(\frac{2\pi}{T}\Big)^{2}\bigg\{\frac{a^{2}+b^{2}}{2}-\frac{a^{2}-b^{2}}{2}cos\Big(\frac{4\pi}{T} t-2\theta\Big)+r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}cos(\theta_{2}-\theta_{1})\nonumber\\ & &-(a+b)\Big[r_{1}cos\Big(\frac{4\pi}{T} t+\theta_{1}-\theta\Big) -r_{2}cos\Big(\frac{4\pi}{T} t+\theta_{2}-\theta\Big)\Big]\nonumber\\ & &+(a-b)\big[r_{1}cos(\theta_{1}+\theta)-r_{2}cos(\theta_{2}+\theta)\big]\bigg\},\end{aligned}$$ $$\begin{aligned} \label{e16} \|\tilde{q}-q_{2}\|&=&\bigg\{\frac{a^{2}+b^{2}}{2}+\frac{a^{2}-b^{2}}{2}cos\Big(\frac{4\pi}{T} t-2\theta\Big)+r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}cos(\theta_{2}-\theta_{1})\ \ \ \ \ \ \ \ \nonumber\\ & &+(a+b)\Big[r_{1}cos\Big(\frac{4\pi}{T} t+\theta_{1}-\theta\Big) -r_{2}cos\Big(\frac{4\pi}{T} t+\theta_{2}-\theta\Big)\Big]\nonumber\\ & &+(a-b)\big[r_{1}cos(\theta_{1}+\theta)-r_{2}cos(\theta_{2}+\theta)\big]\bigg\}^{\frac{1}{2}},\end{aligned}$$ $$\begin{aligned} \label{e17} \|\dot{\tilde{q}}-\dot{q_{3}}\|^{2}&=&\Big(\frac{2\pi}{T}\Big)^{2}\bigg\{\frac{a^{2}+b^{2}}{2}-\frac{a^{2}-b^{2}}{2}cos\Big(\frac{4\pi}{T} t-2\theta\Big)+r_{1}^{2}+r_{3}^{2}-2r_{1}r_{3}cos(\theta_{3}-\theta_{1})\nonumber\\ & &-(a+b)\Big[r_{1}cos\Big(\frac{4\pi}{T} t+\theta_{1}-\theta\Big) -r_{3}cos\Big(\frac{4\pi}{T} t+\theta_{3}-\theta\Big)\Big]\nonumber\\ & &+(a-b)\big[r_{1}cos(\theta_{1}+\theta)-r_{3}cos(\theta_{3}+\theta)\big]\bigg\},\end{aligned}$$ $$\begin{aligned} \label{e18} \|\tilde{q}-q_{3}\|&=&\bigg\{\frac{a^{2}+b^{2}}{2}+\frac{a^{2}-b^{2}}{2}cos\Big(\frac{4\pi}{T} t-2\theta\Big)+r_{1}^{2}+r_{3}^{2}-2r_{1}r_{3}cos(\theta_{2}-\theta_{1})\ \ \ \ \ \ \ \ \nonumber\\ & &+(a+b)\Big[r_{1}cos\Big(\frac{4\pi}{T} t+\theta_{1}-\theta\Big) -r_{3}cos\Big(\frac{4\pi}{T} t+\theta_{3}-\theta\Big)\Big]\nonumber\\ & &+(a-b)\big[r_{1}cos(\theta_{1}+\theta)-r_{3}cos(\theta_{3}+\theta)\big]\bigg\}^{\frac{1}{2}},\end{aligned}$$ $$\label{e19} \|\dot{q_{1}}\|^{2}=\Big(\frac{2\pi}{T}\Big)^{2}r_{1}^{2},\ \ \ \ \|\dot{q_{2}}\|^{2}=\Big(\frac{2\pi}{T}\Big)^{2}r_{2}^{2},\ \ \ \ \|\dot{q_{3}}\|^{2}=\Big(\frac{2\pi}{T}\Big)^{2}r_{3}^{2}.$$ Therefore by (\[e13\])-(\[e19\]), we get $$\begin{aligned} \label{e20} f(\tilde{q})&=&\frac{1}{M}\int_{0}^{T}\sum\limits_{i=1}^{3}m_{i}\Big[\frac{1}{2}\|\dot{\tilde{q}}-\dot{q}_{i}\|^{2} +\frac{M}{\|\tilde{q}-q_{i}\|}\Big]dt-\frac{1}{2M}\int_{0}^{T}\sum\limits_{i=1}^{3}m_{i}\|\dot{q}_{i}\|^{2}dt\nonumber\\ &=&\frac{2\pi^{2}}{T}\Big\{\frac{a^{2}+b^{2}}{2}+\frac{m_{2}+m_{3}-m_{1}}{M}r_{1}^{2}-\frac{2m_{2}r_{2} cos(\theta_{2}-\theta_{1})+2m_{3}r_{3}cos(\theta_{3}-\theta_{1})}{M}r_{1}\nonumber\\ & &+\frac{m_{2}(a-b)}{M} [r_{1}cos(\theta_{1}+\theta)-r_{2}cos(\theta_{2}+\theta)]+\frac{m_{3}(a-b)}{M} [r_{1}cos(\theta_{1}+\theta)\nonumber\\ & &-r_{3}cos(\theta_{3}+\theta)]\Big\}+m_{1}\int_{0}^{T}\bigg[\frac{a^{2}+b^{2}}{2}+\frac{a^{2}-b^{2}}{2}cos\Big(\frac{4\pi}{T} t-2\theta\Big)\bigg]^{-\frac{1}{2}}dt\nonumber\\ & &+\sum\limits_{i=2}^{3}\int_{0}^{T}m_{i} \bigg\{\frac{a^{2}+b^{2}}{2}+\frac{a^{2}-b^{2}}{2}cos\Big(\frac{4\pi}{T} t-2\theta\Big)+r_{1}^{2}+r_{i}^{2}-2r_{1}r_{i}cos(\theta_{i}-\theta_{1})\nonumber\\ & &+(a+b)\Big[r_{1}cos\Big(\frac{4\pi}{T} t+\theta_{1}-\theta\Big) -r_{i}cos\Big(\frac{4\pi}{T} t+\theta_{i}-\theta\Big)\Big]\nonumber\\ & &+(a-b)\big[r_{1}cos(\theta_{1}+\theta)-r_{i}cos(\theta_{i}+\theta)\big]\bigg\}^{-\frac{1}{2}}dt\nonumber\\ &=&d_{2}(a,b,\theta).\end{aligned}$$ In order to estimate $d_{2}$, we have computed the numerical values of $d_{2} = f(q)$ over some selected test loops. The computation of the integral that appears in (\[e20\]) has been done using the function $\{quad\}$ of Mathematica 7.1 with an error less than $10^{-6}$. Let $T=1$, the results of the numerical explorations are given in Table 1 with $M=1$ and Table 2 with $m_{1}=m_{2}=m_{3}=1$. $$ \begin{tabular}{\|c\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline a & b & $\theta$ &m1 &m2 &m3 & d1 &d2 \\ \hline 0.13 &0.49 &$\frac{\pi}{20}$ &0.29 &0.42 &0.29 &5.419669 &5.417862 \\ \hline 0.15 &0.49 &$\frac{\pi}{20}$ &0.29 &0.41 &0.30 &5.417626 &5.416591 \\ \hline 0.15 &0.49 &$\frac{\pi}{20}$ &0.29 &0.42 &0.29 &5.419669 &5.413794 \\ \hline 0.15 &0.49 &$\frac{\pi}{20}$ &0.30 &0.35 &0.35 &5.441499 &5.436767 \\ \hline 0.15 &0.51 &$\frac{\pi}{20}$ &0.30 &0.36 &0.34 &5.441669 &5.437985 \\ \hline 0.15 &0.51 &$\frac{\pi}{20}$ &0.30 &0.37 &0.33 &5.442180 &5.433615 \\ \hline 0.15 &0.51 &$\frac{\pi}{20}$ &0.30 &0.38 &0.32 &5.443031 &5.429587 \\ \hline 0.15 &0.51 &$\frac{\pi}{20}$ &0.30 &0.39 &0.31 &5.444223 &5.425898 \\ \hline 0.15 &0.51 &$\frac{\pi}{20}$ &0.30 &0.40 &0.30 &5.445755 &5.422550 \\ \hline 0.15 &0.53 &$\frac{\pi}{20}$ &0.31 &0.35 &0.34 &5.470820 &5.467576 \\ \hline 0.15 &0.53 &$\frac{\pi}{20}$ &0.31 &0.36 &0.33 &5.471160 &5.462971 \\ \hline 0.15 &0.53 &$\frac{\pi}{20}$ &0.31 &0.37 &0.32 &5.471841 &5.458707 \\ \hline 0.15 &0.53 &$\frac{\pi}{20}$ &0.31 &0.38 &0.31 &5.472863 &5.454784 \\ \hline 0.17 &0.45 &$\frac{\pi}{20}$ &0.32 &0.32 &0.36 &5.500992 &5.488608 \\ \hline 0.17 &0.47 &$\frac{\pi}{20}$ &0.32 &0.33 &0.35 &5.500481 &5.454518 \\ \hline 0.17 &0.47 &$\frac{\pi}{20}$ &0.32 &0.34 &0.34 &5.500311 &5.449987 \\ \hline 0.17 &0.47 &$\frac{\pi}{20}$ &0.33 &0.34 &0.33 &5.530142 &5.444254 \\ \hline 0.45 &0.15 &$\pi$ &0.33 &0.31 &0.36 &5.471160 &5.456006 \\ \hline 0.45 &0.15 &$\pi$ &0.33 &0.32 &0.35 &5.500481 &5.455325 \\ \hline 0.45 &0.15 &$\pi$ &0.33 &0.33 &0.34 &5.530142 &5.454984 \\ \hline 0.47 &0.13 &$\pi$ &0.34 &0.30 &0.36 &5.441669 &5.439671 \\ \hline 0.47 &0.13 &$\pi$ &0.34 &0.31 &0.35 &5.470820 &5.438820 \\ \hline 0.47 &0.13 &$\pi$ &0.34 &0.32 &0.34 &5.500311 &5.438309 \\ \hline 0.47 &0.15 &$\pi$ &0.35 &0.30 &0.35 &5.441499 &5.417900 \\ \hline 0.49 &0.15 &$\pi$ &0.36 &0.29 &0.35 &5.412519 &5.411552 \\ \hline 0.49 &0.15 &$\pi$ &0.36 &0.32 &0.32 &5.500992 &5.410020 \\ \hline 0.49 &0.15 &$\pi$ &0.37 &0.29 &0.34 &5.412859 &5.411962 \\ \hline 0.49 &0.15 &$\pi$ &0.37 &0.30 &0.33 &5.442180 &5.411281 \\ \hline 0.49 &0.15 &$\pi$ &0.37 &0.31 &0.32 &5.471841 &5.410940 \\ \hline 0.49 &0.15 &$\pi$ &0.38 &0.29 &0.33 &5.413540 &5.412712 \\ \hline 0.49 &0.15 &$\pi$ &0.38 &0.30 &0.32 &5.443031 &5.412201 \\ \hline 0.49 &0.15 &$\pi$ &0.38 &0.31 &0.31 &5.472863 &5.412031 \\ \hline 0.49 &0.15 &$\pi$ &0.39 &0.29 &0.32 &5.414562 &5.413803 \\ \hline 0.49 &0.15 &$\pi$ &0.39 &0.30 &0.31 &5.444223 &5.413462 \\ \hline 0.49 &0.17 &$\pi$ &0.40 &0.29 &0.31 &5.415924 &5.415807 \\ \hline 0.49 &0.17 &$\pi$ &0.40 &0.30 &0.30 &5.445755 &5.415637 \\ \hline 0.49 &0.17 &$\pi$ &0.41 &0.30 &0.29 &5.417626 &5.416078 \\ \hline 0.49 &0.17 &$\pi$ &0.42 &0.29 &0.29 &5.419669 &5.416689 \\ \hline \end {tabular}$$ \[Table 1\] $$ \begin{tabular}{\|c\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline a & b & $\theta$ &m1 &m2 &m3 & d1 &d2 \\ \hline 0.15 &0.67 &$\frac{\pi}{30}$ &1.00 &1.00 &1.00 &11.523843 &11.505860 \\ \hline 0.15 &0.67 &$\frac{\pi}{30}$ &1.00 &1.00 &1.00 &11.523843 &11.505860 \\ \hline 0.15 &0.69 &$\frac{\pi}{30}$ &1.00 &1.00 &1.00 &11.523843 &11.444212 \\ \hline 0.17 &0.65 &$\frac{\pi}{30}$ &1.00 &1.00 &1.00 &11.523843 &11.493238 \\ \hline 0.17 &0.67 &$\frac{\pi}{20}$ &1.00 &1.00 &1.00 &11.523843 &11.452135 \\ \hline 0.17 &0.69 &$\frac{\pi}{20}$ &1.00 &1.00 &1.00 &11.523843 &11.400124 \\ \hline 0.19 &0.63 &$\frac{\pi}{30}$ &1.00 &1.00 &1.00 &11.523843 &11.519350 \\ \hline 0.19 &0.65 &$\frac{\pi}{20}$ &1.00 &1.00 &1.00 &11.523843 &11.455969 \\ \hline 0.19 &0.67 &$\frac{\pi}{20}$ &1.00 &1.00 &1.00 &11.523843 &11.386608 \\ \hline 0.19 &0.69 &$\frac{\pi}{20}$ &1.00 &1.00 &1.00 &11.523843 &11.344747 \\ \hline 0.61 &0.23 &$\pi$ &1.00 &1.00 &1.00 &11.523843 &11.516685 \\ \hline 0.63 &0.19 &$\pi$ &1.00 &1.00 &1.00 &11.523843 &11.489791 \\ \hline 0.63 &0.21 &$\pi$ &1.00 &1.00 &1.00 &11.523843 &11.436105 \\ \hline 0.65 &0.17 &$\pi$ &1.00 &1.00 &1.00 &11.523843 &11.461786 \\ \hline 0.65 &0.19 &$\pi$ &1.00 &1.00 &1.00 &11.523843 &11.392115 \\ \hline 0.65 &0.21 &$\pi$ &1.00 &1.00 &1.00 &11.523843 &11.349366 \\ \hline 0.67 &0.15 &$\pi$ &1.00 &1.00 &1.00 &11.523843 &11.472422 \\ \hline 0.67 &0.17 &$\pi$ &1.00 &1.00 &1.00 &11.523843 &11.383978 \\ \hline 0.67 &0.19 &$\pi$ &1.00 &1.00 &1.00 &11.523843 &11.324970 \\ \hline 0.67 &0.21 &$\pi$ &1.00 &1.00 &1.00 &11.523843 &11.291915 \\ \hline 0.69 &0.13 &$\pi$ &1.00 &1.00 &1.00 &11.523843 &11.522980 \\ \hline 0.69 &0.15 &$\pi$ &1.00 &1.00 &1.00 &11.523843 &11.412094 \\ \hline 0.69 &0.17 &$\pi$ &1.00 &1.00 &1.00 &11.523843 &11.334189 \\ \hline 0.69 &0.19 &$\pi$ &1.00 &1.00 &1.00 &11.523843 &11.284714 \\ \hline \end {tabular}$$ \[Table 2\] For the parameters $a,b,\theta$ given in Table 1 and Table 2, we all have $d_{2}< d_{1}$. This completes the Proof of Theorem 1.1. $\Box$ Proof of Theorem 1.2 To get Theorem 1.2, we are going to find a test loop $\bar{q}\in\Lambda_{+}$ such that $f(\bar{q})\leq d_{3}$. Then the minimizer of $f$ on $\bar{\Lambda}_{+}$ must be a noncollision solution if $d_{3}<d_{1}$. Let $a > 0$, $\theta\in[0,2\pi)$ and $$\bar{q}-q_{1}=ae^{\sqrt{-1}(\frac{2\pi}{T} t+\theta)}.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ Hence $$\begin{aligned} \bar{q}-q_{2}&=&q_{1}+ae^{\sqrt{-1}(\frac{2\pi}{T} t+\theta)}-q_{2}\nonumber\\ &=&r_{1}e^{\sqrt{-1}(\frac{2\pi}{T} t+\theta_{1})}-r_{2}e^{\sqrt{-1}(\frac{2\pi}{T} t+\theta_{2})}+ae^{\sqrt{-1}(\frac{2\pi}{T} t+\theta)},\end{aligned}$$ $$\begin{aligned} \bar{q}-q_{3}&=&q_{1}+ae^{\sqrt{-1}(\frac{2\pi}{T} t+\theta)}-q_{3}\nonumber\\ &=&r_{1}e^{\sqrt{-1}(\frac{2\pi}{T} t+\theta_{1})}-r_{3}e^{\sqrt{-1}(\frac{2\pi}{T} t+\theta_{3})}+ae^{\sqrt{-1}(\frac{2\pi}{T} t+\theta)}.\end{aligned}$$ It is easy to see that $\bar{q}\in\Lambda_{+}$ and $$\begin{aligned} \label{e21} \|\dot{\bar{q}}-\dot{q_{1}}\|^{2}&=&\Big(\frac{2\pi}{T}\Big)^{2}a^{2},\ \ \ \ \|\bar{q}-q_{1}\|=a,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\end{aligned}$$ $$\begin{aligned} \label{e22} \|\dot{\bar{q}}-\dot{q_{2}}\|^{2}=\Big(\frac{2\pi}{T}\Big)^{2}\big[a^{2}+r_{1}^{2}+ r_{2}^{2}-2r_{1}r_{2}cos(\theta_{2}-\theta_{1})+2ar_{1}cos(\theta_{1}-\theta) -2ar_{2}cos(\theta_{2}-\theta)\big],\end{aligned}$$ $$\begin{aligned} \label{e23} \|\bar{q}-q_{2}\|=\big[a^{2}+r_{1}^{2}+ r_{2}^{2}-2r_{1}r_{2}cos(\theta_{2}-\theta_{1})+2ar_{1}cos(\theta_{1}-\theta) -2ar_{2}cos(\theta_{2}-\theta)\big]^{\frac{1}{2}},\ \ \ \ \ \\end{aligned}$$ $$\begin{aligned} \label{e24} \|\dot{\bar{q}}-\dot{q_{3}}\|^{2}=\Big(\frac{2\pi}{T}\Big)^{2}\big[a^{2}+r_{1}^{2}+ r_{3}^{2}-2r_{1}r_{3}cos(\theta_{3}-\theta_{1})+2ar_{1}cos(\theta_{1}-\theta) -2ar_{3}cos(\theta_{3}-\theta)\big],\end{aligned}$$ $$\begin{aligned} \label{e25} \|\bar{q}-q_{3}\|=\big[a^{2}+r_{1}^{2}+ r_{3}^{2}-2r_{1}r_{3}cos(\theta_{3}-\theta_{1})+2ar_{1}cos(\theta_{1}-\theta) -2ar_{3}cos(\theta_{3}-\theta)\big]^{\frac{1}{2}},\ \ \ \ \ \\end{aligned}$$ $$\label{e26} \|\dot{q_{1}}\|^{2}=\Big(\frac{2\pi}{T}\Big)^{2}r_{1}^{2},\ \ \ \ \|\dot{q_{2}}\|^{2}=\Big(\frac{2\pi}{T}\Big)^{2}r_{2}^{2},\ \ \ \ \|\dot{q_{3}}\|^{2}=\Big(\frac{2\pi}{T}\Big)^{2}r_{3}^{2}.$$ Therefore by (\[e21\])-(\[e26\]), we get $$\begin{aligned} \label{e27} f(\bar{q})&=&\frac{1}{M}\int_{0}^{T}\sum\limits_{i=1}^{3}m_{i}\Big[\frac{1}{2}\|\dot{\bar{q}}-\dot{q}_{i}\|^{2} +\frac{M}{\|\bar{q}-q_{i}\|}\Big]dt-\frac{1}{2M}\int_{0}^{T}\sum\limits_{i=1}^{3}m_{i}\|\dot{q}_{i}\|^{2}dt\nonumber\\ &=&\frac{2\pi^{2}}{T}\Big[a^{2}+\frac{m_{2}+m_{3}-m_{1}}{M}r_{1}^{2}-\frac{2m_{2}r_{2} cos(\theta_{2}-\theta_{1})+2m_{3}r_{3}cos(\theta_{3}-\theta_{1})}{M}r_{1}\nonumber\\ & &+\frac{2(m_{2}+m_{3})}{M}ar_{1}cos(\theta_{1}-\theta)-\frac{2m_{2}r_{2} cos(\theta_{2}-\theta)+2m_{3}r_{3}cos(\theta_{3}-\theta)}{M}a\Big]\nonumber\\ & &+\frac{m_{1}T}{a}+m_{2}\int_{0}^{T} \big[a^{2}+r_{1}^{2}+ r_{2}^{2}-2r_{1}r_{2}cos(\theta_{2}-\theta_{1})+2ar_{1}cos(\theta_{1}-\theta)\nonumber\\ & &-2ar_{2}cos(\theta_{2}-\theta)\big]^{-\frac{1}{2}}dt+m_{3}\int_{0}^{T} \big[a^{2}+r_{1}^{2}+ r_{3}^{2}-2r_{1}r_{3}cos(\theta_{3}-\theta_{1})\nonumber\\ & &+2ar_{1}cos(\theta_{1}-\theta) -2ar_{3}cos(\theta_{3}-\theta)\big]^{-\frac{1}{2}}dt\nonumber\\ &=&d_{3}(a,\theta).\end{aligned}$$ In order to estimate $d_{3}$, we have computed the numerical values of $d_{3} = f(q)$ over some selected test loops. The computation of the integral that appears in (\[e27\]) has been done using the function $\{quad\}$ of Mathematica 7.1 with an error less than $10^{-6}$. Let $T=1$, the results of the numerical explorations are given in Table 3 with $M=1$ and Table 4 with $m_{1}=m_{2}=m_{3}=1$. $$ \begin{tabular}{\|c\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline a & $\theta$ &m1 &m2 &m3 & d1 &d3 \\ \hline 0.17 &$\frac{\pi}{2}$ &0.10 &0.75 &0.15 &5.062791 &5.060773 \\ \hline 0.17 &$\frac{\pi}{2}$ &0.10 &0.77 &0.13 &5.083903 &5.071551 \\ \hline 0.17 &$\frac{\pi}{2}$ &0.10 &0.78 &0.12 &5.094969 &5.077450 \\ \hline 0.17 &$\frac{\pi}{2}$ &0.10 &0.80 &0.10 &5.118123 &5.090270 \\ \hline 0.17 &$\frac{\pi}{2}$ &0.15 &0.53 &0.32 &5.051742 &5.050040 \\ \hline 0.17 &$\frac{\pi}{2}$ &0.15 &0.57 &0.28 &5.068768 &5.046398 \\ \hline 0.17 &$\frac{\pi}{2}$ &0.15 &0.60 &0.25 &5.085112 &5.047242 \\ \hline 0.17 &$\frac{\pi}{2}$ &0.15 &0.65 &0.20 &5.119162 &5.055458 \\ \hline 0.17 &$\frac{\pi}{2}$ &0.15 &0.70 &0.15 &5.161725 &5.072186 \\ \hline 0.17 &$\frac{\pi}{2}$ &0.15 &0.72 &0.13 &5.121130 &5.081261 \\ \hline 0.17 &$\frac{\pi}{2}$ &0.20 &0.31 &0.49 &5.176554 &5.175168 \\ \hline 0.17 &$\frac{\pi}{2}$ &0.20 &0.35 &0.45 &5.167020 &5.144967 \\ \hline 0.17 &$\frac{\pi}{2}$ &0.20 &0.40 &0.40 &5.162763 &5.114876 \\ \hline 0.17 &$\frac{\pi}{2}$ &0.20 &0.50 &0.30 &5.179789 &5.080232 \\ \hline 0.17 &$\frac{\pi}{2}$ &0.20 &0.55 &0.25 &5.201070 &5.075680 \\ \hline 0.17 &$\frac{\pi}{2}$ &0.20 &0.60 &0.20 &5.230864 &5.079639 \\ \hline 0.19 &$\frac{\pi}{2}$ &0.25 &0.22 &0.53 &5.249837 &5.237465 \\ \hline 0.19 &$\frac{\pi}{2}$ &0.25 &0.25 &0.50 &5.325541 &5.202291 \\ \hline 0.19 &$\frac{\pi}{2}$ &0.25 &0.30 &0.45 &5.308516 &5.150479 \\ \hline 0.19 &$\frac{\pi}{2}$ &0.25 &0.35 &0.40 &5.300003 &5.107178 \\ \hline 0.19 &$\frac{\pi}{2}$ &0.25 &0.62 &0.13 &5.041112 &5.020454 \\ \hline 0.19 &$\frac{\pi}{2}$ &0.30 &0.22 &0.48 &5.230258 &5.222385 \\ \hline 0.19 &$\frac{\pi}{2}$ &0.30 &0.25 &0.45 &5.308516 &5.189765 \\ \hline 0.19 &$\frac{\pi}{2}$ &0.30 &0.30 &0.40 &5.445755 &5.142208 \\ \hline 0.19 &$\frac{\pi}{2}$ &0.30 &0.35 &0.35 &5.441499 &5.103164 \\ \hline 0.19 &$\frac{\pi}{2}$ &0.30 &0.56 &0.14 &5.036553 &5.032137 \\ \hline 0.21 &$\frac{\pi}{2}$ &0.35 &0.21 &0.44 &5.192935 &5.184596 \\ \hline 0.21 &$\frac{\pi}{2}$ &0.35 &0.29 &0.36 &5.412519 &5.092092 \\ \hline 0.21 &$\frac{\pi}{2}$ &0.35 &0.39 &0.26 &5.327621 &5.007107 \\ \hline 0.21 &$\frac{\pi}{2}$ &0.35 &0.48 &0.17 &5.091316 &4.959734 \\ \hline 0.21 &$\frac{\pi}{2}$ &0.35 &0.53 &0.12 &4.971952 &4.945333 \\ \hline 0.21 &$\frac{\pi}{3}$ &0.40 &0.28 &0.32 &5.386433 &5.342981 \\ \hline 0.21 &$\frac{\pi}{3}$ &0.40 &0.32 &0.28 &5.386433 &5.287294 \\ \hline 0.21 &$\frac{\pi}{3}$ &0.40 &0.36 &0.24 &5.271874 &5.237055 \\ \hline 0.21 &$\frac{\pi}{3}$ &0.40 &0.38 &0.22 &5.216638 &5.213978 \\ \hline 0.23 &$\frac{\pi}{2}$ &0.45 &0.19 &0.36 &5.139742 &5.127834 \\ \hline 0.23 &$\frac{\pi}{2}$ &0.45 &0.29 &0.26 &5.337836 &5.003006 \\ \hline 0.23 &$\frac{\pi}{2}$ &0.45 &0.37 &0.18 &5.112805 &4.927660 \\ \hline 0.23 &$\frac{\pi}{2}$ &0.45 &0.46 &0.09 &4.885693 &4.868944 \\ \hline 0.23 &$\frac{\pi}{2}$ &0.50 &0.18 &0.32 &5.123871 &5.108878 \\ \hline 0.23 &$\frac{\pi}{2}$ &0.50 &0.23 &0.27 &5.266218 &5.044762 \\ \hline 0.23 &$\frac{\pi}{2}$ &0.50 &0.29 &0.21 &5.208258 &4.979058 \\ \hline 0.23 &$\frac{\pi}{2}$ &0.50 &0.37 &0.13 &4.990036 &4.910522 \\ \hline 0.23 &$\frac{\pi}{2}$ &0.50 &0.41 &0.09 &4.889098 &4.884426 \\ \hline \end {tabular}$$ \[Table 3\] $$ \begin{tabular}{\|c\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline a & $\theta$ &m1 &m2 &m3 & d1 &d3 \\ \hline 0.21 &$\frac{\pi}{2}$ &1.00 &1.00 &1.00 &11.523843 &11.327950 \\ \hline 0.23 &$\frac{\pi}{2}$ &1.00 &1.00 &1.00 &11.523843 &11.036769 \\ \hline 0.23 &$\frac{\pi}{3}$ &1.00 &1.00 &1.00 &11.523843 &11.336568 \\ \hline 0.25 &$\frac{\pi}{2}$ &1.00 &1.00 &1.00 &11.523843 &10.821272 \\ \hline 0.25 &$\frac{\pi}{3}$ &1.00 &1.00 &1.00 &11.523843 &11.187475 \\ \hline 0.25 &$\frac{\pi}{4}$ &1.00 &1.00 &1.00 &11.523843 &11.453195 \\ \hline 0.27 &$\frac{\pi}{2}$ &1.00 &1.00 &1.00 &11.523843 &10.667031 \\ \hline 0.27 &$\frac{\pi}{3}$ &1.00 &1.00 &1.00 &11.523843 &11.107374 \\ \hline 0.27 &$\frac{\pi}{4}$ &1.00 &1.00 &1.00 &11.523843 &11.411685 \\ \hline 0.29 &$\frac{\pi}{2}$ &1.00 &1.00 &1.00 &11.523843 &10.563849 \\ \hline 0.29 &$\frac{\pi}{3}$ &1.00 &1.00 &1.00 &11.523843 &11.085761 \\ \hline 0.29 &$\frac{\pi}{4}$ &1.00 &1.00 &1.00 &11.523843 &11.430090 \\ \hline 0.31 &$\frac{\pi}{2}$ &1.00 &1.00 &1.00 &11.523843 &10.504424 \\ \hline 0.31 &$\frac{\pi}{3}$ &1.00 &1.00 &1.00 &11.523843 &11.114860 \\ \hline 0.31 &$\frac{\pi}{4}$ &1.00 &1.00 &1.00 &11.523843 &11.500414 \\ \hline 0.33 &$\frac{\pi}{2}$ &1.00 &1.00 &1.00 &11.523843 &10.483477 \\ \hline 0.33 &$\frac{\pi}{3}$ &1.00 &1.00 &1.00 &11.523843 &11.188786 \\ \hline 0.35 &$\frac{\pi}{2}$ &1.00 &1.00 &1.00 &11.523843 &10.497161 \\ \hline 0.35 &$\frac{\pi}{3}$ &1.00 &1.00 &1.00 &11.523843 &11.302997 \\ \hline 0.37 &$\frac{\pi}{2}$ &1.00 &1.00 &1.00 &11.523843 &10.542652 \\ \hline 0.37 &$\frac{\pi}{3}$ &1.00 &1.00 &1.00 &11.523843 &11.453926 \\ \hline 0.39 &$\frac{\pi}{2}$ &1.00 &1.00 &1.00 &11.523843 &10.617860 \\ \hline \end {tabular}$$ \[Table 4\] For the parameters $a,\theta$ given in Table 3 and Table 4, we all have $d_{3}< d_{1}$. 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Coti Zelati, Symmetries and noncollision closed orbits for planar N-body-type problems, Nonlinear Anal. TMA, 16(1991), 587-598. W. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99(5)(1977), 961-971. Y. M. Long, S. Q. Zhang, Goemetric characterizations for variational minimization solutions of the 3-body problems, Acta Math. Sinica, 16(4)(2000), 579-592. [^1]: Supported by National Natural Science Foundation of China.
--- abstract: \| Stellar collisions have long been envisioned to be of great importance in the center of galaxies where densities of $10^6\,\mbox{stars}/\mbox{pc}^{3}$ or larger are attained. Not only can they play a unique dynamical role by modifying stellar masses and orbits, but high velocity disruptive encounters occurring in the vicinity of a massive black hole can also be an occasional source of fuel for the starved central engine. In the past few years, we have been building a comprehensive table of SPH (Smoothed Particle Hydrodynamics) collision simulations for main sequence stars. This database is now integrated as a module into our H[é]{}non-like Monte Carlo code. The combination of SPH collision simulations with a Monte Carlo cluster evolution code seems ideally suited to study the frequency, characteristics and effects of stellar collisions during the long term evolution of galactic nuclei. author: - Marc Freitag - Willy Benz title: 'A Monte Carlo Code to Investigate Stellar Collisions in Dense Galactic Nuclei.' --- \#1\#2[=\#1]{} Introduction ============ Compact massive dark objects, with masses $10^6-10^9\,M_\odot$, have been found in the center of nearly every bright galaxy where they have been searched through measurements and modeling of the gas or stellar kinematics (see reviews by [Kormendy]{} & [Richstone]{} 1995; [Richstone]{} [et al.]{} 1998; [Ho]{} 1999; [Moran]{}, [Greenhill]{}, & [Herrnstein]{} 1999; [Kormendy]{} 2000). In the two cases with the highest resolution, i.e. the Milky Way and NGC 4258, the size of the central object is observationally constrained to be so small that models resorting to compact cluster of small dark objects (Neutron stars, stellar black holes, brown dwarfs,…) seem very unlikely as such concentrations would not survive evaporation or run-away merging for many $10^9$ years ([Maoz]{} 1998). It it thus widely believed that these objects are “super-massive” black holes (SBHs). Our work is devoted to an exploration of some intriguing consequences a SBH should have on a surrounding stellar cluster and of the long-term evolution of such a system. Of particular interest to us are two kinds of disruptive events which could release stellar gas in the vicinity of the SBH and thus lead to bright accretion phases, even in otherwise non-active nuclei. These processes are tidal disruptions and stellar collisions. Other mechanisms through which a stellar cluster can contribute to the feeding of a SBH include stellar winds ([Shull]{} 1983; [David]{}, [Durisen]{}, & [Cohn]{} 1987b; [Norman]{} & [Scoville]{} 1988; [Coker]{} & [Melia]{} 1997), envelope-stripping when stars cross a pre-existing accretion disk (mainly relevant to red giants, see [Armitage]{}, [Zurek]{}, & [Davies]{} 1996) and inspiraling induced by strong emission of gravitational waves (mainly relevant to compact remnants, see [Hils]{} & [Bender]{} 1995; [Sigurdsson]{} & [Rees]{} 1997; [Miralda-Escud[é]{}]{} & [Gould]{} 2000; [Freitag]{} 2000). However, in this conference paper, we naturally focus on collisions between main sequence (MS) stars. To treat collisions with as much realism as possible, we decided to determine their outcome through a comprehensive set of SPH simulations. This important part of our work, described in Sec. \[sec:coll\] ([Freitag]{} & [Benz]{} 2000a), resulted in a database incorporated, as a module, in a new Monte Carlo (MC) cluster evolution code, presented in Sec. \[sec:MCcode\] ([Freitag]{} & [Benz]{} 2000b). This provides us with a numerical tool which seems ideally suited to investigate collisions in dense stellar clusters. Although our simulations can potentially produce detailed lists of collisionally formed objects (such as blue-stragglers) and not only overall rates, so far we have mainly addressed the question of the global influence of collisions on the SBH $+$ cluster system. Previous works relied on highly simplified prescriptions to account for collisional effects in the stellar dynamics of galactic nuclei[^1]. This situation stemmed not only from the limited knowledge of the collision itself, to be acquired from 3D hydrodynamical simulations, but also from intrinsic limitations of the stellar dynamics codes. Direct Fokker-Planck integrations, while very fast, treat the stellar system as a set of continuous distribution functions, one for each stellar mass. Thus, the mass spectrum is discretized into a few mass classes and collision products have to be re-distributed into these bins in a rather unphysical way. This shortcoming is required for mergers ([Lee]{} 1987; [Quinlan]{} & [Shapiro]{} 1990) or for collisions leading to partial mass loss ([David]{}, [Durisen]{}, & [Cohn]{} 1987a; [David]{} [et al.]{} 1987b; [Murphy]{}, [Cohn]{}, & [Durisen]{} 1991); only if complete disruption is assumed ([McMillan]{}, [Lightman]{}, & [Cohn]{} 1981; [Duncan]{} & [Shapiro]{} 1983), can it be avoided. But this latter assumption is, by itself, a gross over-simplification. On the other hand, $N$-body simulations can in principle incorporate realistic collisions, but as their results can not safely be scaled to larger $N$,[^2] they are still presently restricted to systems containing a few $10^4$ stars at most (see the work on open clusters by [Portegies Zwart]{} [et al.]{} 1999). Even though the computer hardware and software dedicated to $N$-body integration progress at high pace, this kind of simulation will still be limited to about $10^6$ stars in a near future ([Makino]{} 2000). In these proceedings, D. De Young presents an historical review of the researches on stellar collisions in galactic nuclei so we need only mention here a few issues appearing in the literature onto which we can cast new light with our simulations. As further reading about the role of collisions in stellar systems, we refer to [Davies]{} (1996), for instance. - Can stellar collisions amount to a significant gas source to fuel the central SBH? - Can repeated stellar mergers lead to run-away build-up of a very massive star, a possible precursor for a $10^2-10^3\,M_\odot$ seed BH? Or would this process be caught up by stellar evolution or come to saturation as small, relatively compact stars run across the low density massive star without being stopped ([Colgate]{} 1967)? - Is there any distinctive imprint of the collisions on the cluster’s central density profile? Previous works predict $\rho \propto R^{-\alpha}$, with $\alpha \simeq 0.5$, a noticeably lower value than the $\alpha \simeq 1.75$ cusp expected in a non-collisional relaxed cluster around a SBH ([Bahcall]{} & [Wolf]{} 1976, 1977). - Do stellar collisions produce particular stellar population in the center-most parts of the cluster? Can blue stragglers form through mergers in spite of the high relative velocities? Can collisions be efficient in stripping the envelopes of red giants (see Davies, these proceedings and [Davies]{} [et al.]{} 1998; [Bailey]{} & [Davies]{} 1999)? Our simulations still lack important features (mainly stellar evolution and binaries) to address some of these questions but we hope to demonstrate in Sec. \[sec:simul\] that they already produce interesting results when applied to simple models and, most importantly, that the potential of the MC code in this field is high. A Monte Carlo code for cluster dynamics {#sec:MCcode} ======================================= In the past few years, we wrote a new code in order to study the long-term ($10^9-10^{10}$years) evolution of galactic nuclei consisting of $10^6-10^9$ stars. We developed a Monte Carlo scheme based on the pioneering work of H[é]{}non (1973). This method, although adopted with deep modifications by Stodo[ł]{}kiewicz (1982, 1986) and now revived by Giersz (1998, 2000) and by Joshi and collaborators ([Joshi]{}, [Rasio]{}, & [Portegies Zwart]{} 2000; [Watters]{}, [Joshi]{}, & [Rasio]{} 2000; [Joshi]{}, [Nave]{}, & [Rasio]{} 1999), is not widely used. In particular, as far as we know, ours is the first MC code designed to treat galactic nuclei rather than globular clusters.[^3] The MC numerical scheme is nonetheless very attractive as a good compromise between computational efficiency and physical realism (not to mention ease of adaptation to new physical processes). By “efficiency”, we mean that integrating the evolution of a typical dense central cluster with $0.5-2\times 10^6$ particles over a Hubble time requires a few hours to a few days on a standard 400MHz CPU. This allows to carry out many simulations while varying initial conditions and simulated physics to investigate the interplay of various processes in such complex systems as galactic nuclei. The CPU needed time increases with the number $N$ of particles like $T_{\mathrm{CPU}} \propto N\log(N)$, a relation to be contrasted with the $N^{2-3}$ scaling of “exact” $N$-body calculations. According to simple extrapolations, a galactic nucleus simulation with $10^7$ particles would take only $\sim$10 CPU-days but we are presently limited to lower $N$ by the available computer memory (160 bytes/particle). By “realism”, we mean that we can incorporate many important physical processes into the simulation. Beyond 2-body relaxation which is the core of the MC code, the “micro-physics” include stellar collisions and tidal disruptions. Recently we added accretion of whole stars induced by emission of gravitational radiation and a preliminary treatment of stellar evolution (not covered here, see [Freitag]{} 2000). Furthermore, the MC code copes with the cluster’s self-gravitation, the growth of a central BH, an arbitrary stellar mass spectrum and velocity distribution. As demonstrated by [Stodo[ł]{}kiewicz]{} (1986), [Giersz]{} (1998) and [Rasio]{} (2000), the dynamical effects of binaries can also be included in MC codes. However, in the center of a SBH-hosting cluster, the velocity dispersion is so large that most binaries, being “soft” are likely to be disrupted in gravitational encounters with other stars, instead of acting as a heat source like they would do in globular clusters (see, e.g., [Binney]{} & [Tremaine]{} 1987; [Spitzer]{} 1987). Unfortunately, the MC scheme also suffers from a few shortcomings. The main limitations stem from the very simplifying assumptions that make MC codes so efficient: namely those of spherical symmetry and constant dynamical equilibrium. Consequently, it seems very difficult if not impossible to include such effects like BH wandering, cluster rotation, triaxiality, interaction between stars and an accretion disk, resonant or violent relaxation… Our code is described in detail in [Freitag]{} & [Benz]{} (2000b). Here we just outline its basics. The stellar cluster is represented as a set of “particles”; each of them can be seen as a spherical shell of stars that share the same properties, namely the stellar mass $M_{\ast}$, orbital energy $E$, modulus of angular momentum $J$ and instantaneous distance to the center $R$ (the radius of the shell). Together, these particles define a smooth spherical potential $\Phi(R)$ which is stored in a binary tree structure, for the sake of efficiency. Note that the number of stars per particle can be set to any value but has to be the same in each particle to ensure perfect energy conservation in 2-body processes. In $\Phi$, no relaxation occurs (except for a very small spurious numerical relaxation for $N < 1000$) and the cluster is in dynamical equilibrium. To simulate the slow relaxation-induced evolution of the system, “super-encounters” (SE) are computed between particles of adjacent ranks. A SE is a 2-body gravitational encounter between stars from the two particles. Its deflection angle is imposed to be the RMS value resulting, during time-step $\delta t$, from all the small angle scatterings between stars with the properties (masses $M_1$, $M_2$ and relative velocity $V_{\mathrm{rel}}$) of the interacting particles, i.e., $$\label{eq:thetase} \theta_{\mathrm{SE}} = \frac{\pi}{2} \sqrt{ \frac{\delta t} {T_{\mathrm{rel}}^{(1,2)}} } \mbox{\ \ with\ \ } T_{\mathrm{rel}}^{(1,2)} \propto \frac{V_{\mathrm{rel}}^3} {\ln\Lambda G^2 n_{\ast} \left(M_1+M_2\right)^2}$$ where $\ln\Lambda$ is the Coulomb logarithm. A Lagrangian radial mesh is used to evaluate the local stellar density $n_{\ast}$. To detect stellar collisions between two neighboring particles, we compare a random number of uniform $[0,1[$-variate with the probability for such an event, $$\label{eq:pcoll} P_{\mathrm{coll}}^{(1,2)} = \frac{\delta t}{T_{\mathrm{coll}}^{(1,2)}} \mbox{\ \ with\ \ } T_{\mathrm{coll}}^{(1,2)} = \frac{1}{ n_{\ast}V_{\mathrm{rel}}S_{\mathrm{coll}}^{(1,2)} }.$$ The collisional cross section reads $$\label{eq:scoll} S_{\mathrm{coll}}^{(1,2)} = \pi (R_1+R_2)^2 \left[ 1 + \left(\frac{V^{(1,2)}_{\ast}}{V_{\mathrm{rel}}}\right)^2 \right] \mbox{\ \ with\ \ } V^{(1,2)}_{\ast} =\left( \frac{2G(M_1+M_2)}{R_1+R_2} \right)^{1/2}$$ where $R_{1,2}$ are the stellar radii. To increase code speed, we use $R$-variable time-steps, that are a small fraction $\eta$ of the local relaxation and/or collision time, $\delta t(R) \leq \eta\left( T_{\mathrm{rel}}^{-1} + T_{\mathrm{coll}}^{-1} \right)^{-1}$. To check whether a particle is tidally disrupted by the central SBH or plunges directly through the horizon, we simulate the random walk of the tip of particle’s velocity vector due to small angle scatterings during $\delta t$. This is necessary because, as the “loss cone” aperture $\theta_{\mathrm{LC}}$ is tiny ([Lightman]{} & [Shapiro]{} 1977), the time scale for entering or leaving it, of order $\theta_{\mathrm{LC}}^2 T_{\mathrm{rel}}$, is generally much smaller than $\delta t$. Without a procedure to “over-sample” the time step, we would miss a lot of loss cone events as the velocity vector would just jump over $\theta_{\mathrm{LC}}$. Finally, if the particle avoided disruption, we randomly select a position $R$ on its new orbit with a probability density that matches the fraction of time spent at each radius, ${\mathrm{d}} P/ {\mathrm{d}} R \propto V_{\mathrm{rad}}(R)^{-1} $. This concludes a simulation step. The next one starts with the random selection of another pair of particles according to probability $\propto \delta t(R)^{-1}$. A comprehensive set of collision simulations between MS-stars {#sec:coll} ============================================================= Approach {#subsec:collappr} -------- In the MC scheme, the orbital and stellar properties of any given particle are independent of those of any other particle. This means that these quantities can be modified in any physically reasonable way. In particular, any prescription can be used for the outcome of stellar collisions so that we decided to describe them as realistically as possible through results of an important set of hydrodynamical computations of collisions between MS stars. The numerical algorithm we use is the so-called “Smoothed Particle Hydrodynamics” method (SPH, for a description see [Benz]{} 1990). As a genuinely 3D Lagrangian scheme that allows large density contrasts and imposes no spatial symmetries or limits, it is the method of choice to tackle this problem. This explains why the vast majority of previous investigations in this domain were done with SPH ([Benz]{} & [Hills]{} 1987, 1992; [Lai]{}, [Rasio]{}, & [Shapiro]{} 1993; [Lombardi]{}, [Rasio]{}, & [Shapiro]{} 1996, amongst others), with the noticeable exception of the early work of [Seidl]{} & [Cameron]{} (1972) who used a 2D finite difference algorithm. Our goal was to sample a region in the space of collisions’ initial conditions large enough so that most collisions happening in the course of the simulation of a galactic nuclei would be comprised in that domain. A reasonably good description of a collisions between MS stars imposes a four dimensional parameter space. The first two quantities to be specified are the stellar masses $M_1$ and $M_2$, with $M_1\le M_2$. If MS stars with different masses had homologous internal structures (which would require a power-law mass-radius relation, in particular), we could scale out the absolute mass and use $q=M_1/M_2$ as the only mass parameter. But we use realistic stellar structure models from [Schaller]{} [et al.]{} (1992) and [Charbonnel]{} [et al.]{} (1999) for $M_{\ast}=0.4-85\,M_{\odot}$ and $n=1.5$ polytropes for 0.1–0.3$M_{\odot}$ so that we have to specify both absolute masses independently. In globular clusters, the velocity dispersion is of order a few 10kms$^{-1}$ which is much lower than the escape velocity from the surface of MS stars ($\sim$ 600–1200kms$^{-1}$) so that the relative velocity at infinity plays virtually no role in collisions. This is not true in galactic nuclei, where $V_{\mathrm{rel}}$ can be nearly arbitrarily high near a SBH. For instance, velocities of 1000–1500kms$^{-1}$ have been measured ([Genzel]{} [et al.]{} 1997; [Ghez]{}, [Morris]{}, & [Becklin]{} 1999) at the Galactic center. So, $V_{\mathrm{rel}}$ is the next initial parameter of importance. Finally, we have to specify the impact parameter $b$, i.e. the distance between the trajectories of the two stars if they were straight lines. It is often more convenient to use $d_{\mathrm{min}}$, the periastron separation for the corresponding 2 point-mass hyperbolic encounter. If we can neglect tidal deformation until contact, $d_{\mathrm{min}}/(R_1+R_2) \in [0;1]$ is necessary for physical collision. We restrict ourselves to this domain as our resolution is probably too low to treat tidal interactions properly.[^4] The 4D initial parameter space is thus $(M_1,M_2,V_{\mathrm{rel}},d_{\mathrm{min}})$. Other quantities that could affect the collisions’ outcomes, like stellar rotation, metallicity, age on MS and so on are neglected as they are probably of second order importance. A related question of interest in highly collisional systems where run-away merging could occur is how collisions themselves affect the structure of stars and how these modifications could affect further collisions. We leave any study of this somewhat far-fetched issue, considering that it is more useful to first assess the physical conditions required for such collisional run-away to set in. In our cluster simulations, we assume that, after a collision, a star immediately returns to a “standard” MS structure. In fact, it takes a Kelvin-Helmholtz time scale ($T_{\mathrm{KH}}\simeq 1.6\times 10^7$yrs for the sun) for thermal equilibrium to be recovered, but, in most environments $T_{\mathrm{coll}} \gg T_{\mathrm{KH}}$, so the short post-collision swollen phase can be neglected. Although we chose to consider only collisions between MS stars, they may not dominate the total collision rate in many astrophysical environments. Indeed, as can be seen from Eq. \[eq:scoll\], when $V_{\mathrm{rel}} > V_{\ast}$, as it would occur close to a SBH, the cross section scales like $R_{\ast}^2$ so that red giants (RG) could participate in most collisions in spite of their low relative number ([Davies]{} 1996). An extension of the present work, taking into account MS-RG collisions, is thus desirable to complement the simulations by [Bailey]{} & [Davies]{} (1999). Of course, compact remnants can also collide with MS stars or even with other compact stars. In dense nuclei, such collisions should occur at low but non-vanishing rates, as Fig. \[fig:cumncoll\] testifies. They are particularly interesting as channels to form “exotic” objects. The initial parameter space to be explored being so huge, we had to limit the number of SPH particles per star to a relatively low value (1000–15000) to save computer time. However, we used initial structures with low mass particles in the stellar envelope and more and more massive ones toward the center in order to get a satisfactory resolution of the outer parts of the stars where the action takes place in most collisions. Thus fractional mass loss rates as low as $10^{-4}$ can reliably be predicted. More than $14000$ collision simulations have been computed on a local network of workstations. Such a high number could only be attained thanks to an automatic software package we developed to run simulation jobs on idle computers and analyze their outcome with nearly no human intervention needed. The result of a collision is described through a small set of quantities: the fractional mass loss $(M_1+M_2-{M_1}^{\prime}-{M_2}^{\prime})/(M_1+M_2)$, the new mass ratio, the fractional loss of orbital energy and the angle of deviation of $\vec{V}_{\mathrm{rel}}$. Note that these values completely describe the kinematical outcome of a collision only if the center-of-mass reference frame for the resulting star(s) (not including ejected gas) is the same as before the collision. Asymmetrical mass ejection violates this simplifying assumption by giving the stars a global kick but we neglect this, in order to reduce the complexity of the situation. We have kept the final SPH particle configuration for (nearly) all our simulations. This would allow us to re-analyze these files and extract other quantities of interest, like the induced rotation, a possible tell-tale sign of past collisions ([Alexander]{} & [Kumar]{} 2000). Another interesting issue is the resulting internal stellar structure. This is key to a prediction of the subsequent evolution and observational detectability of collision products (see Sills, these proceedings and [Sills]{} [et al.]{} 1997, 2000). Unfortunately, according to [Lombardi]{} [et al.]{} (1999), low resolution and use of particles of unequal masses can lead to important spurious particle diffusion in SPH simulations so that our models are probably not well suited for a study of the amount of collisional mixing, for instance. Results ------- Typical collisions will be described in [Freitag]{} & [Benz]{} (2000a). Here we want to give an overview of the results that may be extracted from our database. First, looking at Fig. \[fig:comppoly\], we note that using realistic stellar structure instead of the traditional polytropic stars has quite an important effect on the outcome of collisions. This is mainly due to the fact that massive stars are more concentrated than $n=3$ polytropes. Next, we ask whether computing such a high number of simulations was worth the trouble by confronting our results to those of the literature. In particular, we want to know how they compare to fitting formulae devised by [Lai]{} [et al.]{} (1993) and Davies (used by [Rauch]{} 1999) to describe the results of similar but limited sets of SPH simulations. Considering diagrams like those of Fig. \[fig:complit\], we can draw the following conclusions: - Surprisingly, the simple semi-analytical prescription from [Spitzer]{} & [Saslaw]{} (1966) usually gives quite accurate results for the fractional mass loss in the regime with $V_{\mathrm{rel}}/V_{\ast}\geq 1$ and $d_{\mathrm{min}}/(R_1+R_2)>0.4$. - As could be foreseen, empirical fitting formula must [never]{} bee used to extrapolate to initial conditions outside the (restricted) range they originate from. - In particular, the stellar structure has a central role in determining $\delta M$. This appears clearly in (dis-)agreement between our results and those of [Benz]{} & [Hills]{} (1987, 1992) in Fig. \[fig:complit\]. Another way to state the second point is that only a mathematical description grounded on well understood physical arguments has a chance to have any sound predictive power when applied to a wider set of collisions than those it derives from. For instance, it could be that a parameterization of the “closeness” of the interaction that accounts for the mass distribution inside the stars (contrary to $d_{\mathrm{min}}/(R_1+R_2)$) could result in a good agreement between simulations done with various stellar structures. Unfortunately, due to the complexity of the physical processes at play during collisions, such a “unifying” description seems very difficult to find and we have failed to figure it out so far. Consequently, we tried to cover as completely as possible the relevant domain of initial conditions and we use an interpolation algorithm to determine the outcome of any given collision that happens in a cluster simulation run. A zero-order description of the outcome of a stellar collision consists in the number of surviving stars. This is what we show in Fig. \[fig:netoiles\]. Collisions with similar values of $q=M_1/M_2$ have been grouped on the same plot, regardless of the absolute values of the masses. Interestingly, in a initial conditions plane parameterized by “half-mass” quantities (see caption of figure), well defined borders appear that separate various outcome regimes. It also appears that, unless $V_{\mathrm{rel}}$ is very low, collisions that lead to coalescence (at low relative velocities) or complete disruptions (at high $V_{\mathrm{rel}}$) must be nearly head-on. By far the most likely outcome at velocities in excess of 100kms$^{-1}$ is the preservation of both stars with only a small amount of mass loss. For small $M_1/M_2$ ratio, even head-on collisions do not necessarily result in mergers; the small star can fly through the large one without being stopped or destroyed. Such an effect, due to the high-mass star being of lower density than its small impacter was already proposed by [Colgate]{} (1967) to predict an upper mass limit to the process of run-away mergings. Whether this limiting mechanism really operates in dense stellar clusters has to be tested in dynamical simulations. It can be suppressed by mass segregation effects that drive most massive stars toward the center so that most important collisions take place between two high-mass stars. A more quantitative view on the results of 3 sets of collision simulations is given in panel (a) of Fig. \[fig:deltam\] where we show the (interpolated) fractional mass loss in the $(d_{\mathrm{min}},V_{\mathrm{rel}})$ plane. Note how the “landscape” changes from one choice of $(M_1,M_2)$ to another one. This is another indication of the difficulty of finding a universal set of fitting formulae. The upper left white region of each diagram indicates $\delta M/M > 85$%. The small surface of this zone (in particular for unequal masses) means that such highly destructive events are unlikely. This is to be compared to the extent of the black regions for which the fractional mass loss is less than $10^{-4}$. Integration of collisions into the MC code ------------------------------------------ Being unable to distillate the results of our SPH simulation into any compact mathematical formulation without losing most of the information, we resorted to the following interpolation strategy. In the 4D initial parameter space, the simulations form a irregular grid of points. We compute a Delaunay triangulation of this set using the program `Qhull`[^5] ([Barber]{}, [Dobkin]{}, & [Huhdanpaa]{} 1996) which allows us to interpolate the results onto a regular 4D grid. Three slices in this grid are presented in Fig. \[fig:deltam\](b). This table is used in MC simulations to determine – through a second interpolation – the outcome of collisions. Of course, extrapolation prescriptions have to be specified for events whose initial conditions fall outside the convex hull of the SPH simulation points. Most commonly, this happens when a collisionally produced star with mass outside the $0.1$–$74\,M_{\odot}$ range experience a further collision. In such cases, we try to re-scale both masses while preserving $M_1/M_2$ to get a “surrogate collision” lying in the domain covered by the SPH simulations. If $V_{\mathrm{rel}}$ is too low or too high, we increase or decrease it to enter the simulation domain[^6]. In many cases (for instance merging at low $V_{\mathrm{rel}}$ or complete destruction at high $V_{\mathrm{rel}}$), this method gives very sensible results. Encountyers with too high $d_{\mathrm{min}}$ are treated as purely Keplerian hyperbolic deflections with no mass loss. Simulation of galactic nuclei evolution with stellar collisions {#sec:simul} =============================================================== To illustrate the capacities of our “MC+SPH” approach and the role of collisions in the dynamics of dense galactic nuclei, we review some results from stellar dynamical simulations of simple nuclei models. The nominal model is a Plummer cluster with a scale radius of $R_{\mathrm{Plum}}=0.3$pc that contains $5.09\times 10^7$ MS stars with a Salpeter mass spectrum: $dN/dM_{\ast} \propto M_{\ast}^{-2.35}$, $M_{\ast} \in [0.2,20]\,M_{\odot}$. In its center, we put a seed black hole ($M_{\mathrm{BH}}=10^{-6}M_{\mathrm{clst}}$) which is allowed to grow through accretion of stellar gas released in stellar collisions and tidal disruptions. Accretion is assumed to be complete and instantaneous. Stellar evolution is not simulated. The simulations were realized with 512000 particles. Fig. \[fig:compdens\] shows snapshots of the density profile during the evolution of this model. When collisions are treated realistically, using our SPH grid, a steep central cusp with slope $\sim -1.75$ develops. This result is very similar to what is obtained when collisions are switched off and tidal disruptions are the only channel to consume stars ([Bahcall]{} & [Wolf]{} 1976, 1977). A milder slope of about $-1$ is obtained when collisions are assumed to result in complete stellar disruption. Even though we start with a model with very high central density, after a Hubble time, the mass density at 0.1–1pc from the BH has reached a value similar to what is measured in the center of the Milky Way ([Genzel]{} [et al.]{} 1997). However, at that time, the BH’s mass in our model is about $10^7\,M_{\odot}$, nearly 4 times larger than Milky Way’s value. In Fig. \[fig:compdmdt\], we compare the rates of mass accretion onto the central BH. When treated realistically, collisions dominate over tidal disruptions only during a short initial phase before massive stars segregate toward the center. We can not only explore the structure and evolution of the stellar cluster as a whole but also investigate some processes in more detail. For instance, it is possible to study the properties of individual collisions. In Fig. \[fig:collhist\], we follow a selection of stars that experienced a large number of collisions. We report the distance to the center and the stellar mass before each collision. In a cluster without a central BH (panel (a)), the typical evolution of one of these frequently colliding stars is to sink toward the center while growing through a few mergers. In this simulation, the merging process is not allowed if the colliding star already has a mass beyond $\sim 60\,M_{\odot}$ but there is no doubt that it would otherwise lead to very massive stars. Of course such results may be significantly altered when stellar evolution is introduced. For instance, the star represented by the dotted line would not be able to wait during $\sim 7\times 10^7$years between two successive mergers as the lifetime of a $20\,M_{\odot}$ star on the MS is only of order $10^7$years. When a seed BH is present initially (panel (b)), it rapidly grows and leads to such an important increase in the stellar velocities near the center that mergers are totally quenched. Most collisions are then disruptive and the average stellar mass in the central regions actually decreases. Conclusions =========== When assumptions of spherical symmetry and dynamical equilibrium are reasonable, the Monte Carlo code for cluster dynamics appears as the method of choice to get detailed statistical predictions about the role and characteristics of collisions (and other physical processes) during the evolution of a stellar system. The use of SPH-based prescriptions to include collisions enables us to take the best advantage of the flexibility of the MC scheme in terms of realism. Our models still lack other important and/or interesting physical aspects (stellar evolution, role of red giants and binaries,…). Other ingredients could be treated with more rigor. For instance, in the same spirit of our approach of stellar collisions, we could easily use the results of SPH simulations of tidal interactions between a star and the SBH (e.g. [Fulbright]{} 1996) to determine the outcome of these events. For the time being, we assume them to result in complete disruption of the star. More “realistic” prescriptions would not necessarily yield more reliable results, though, as the fraction of stripped stellar mass that is eventually accreted on the SBH is still a matter of debate ([Ayal]{}, [Livio]{}, & [Piran]{} 2000, and references therein). This illustrates the fact that many “improvements” could actually amount to adding more and more sources of uncertainty in the simulations. In such a context, it is all the more useful to dispose of a numerical tool flexible enough to allow changes in the treatment of various physical effects and fast enough to allow large sets of simulations to be conducted to test for the influence of these changes and the interplay between the many physical aspects of the problem. Concerning the role of stellar collisions in the evolution of galactic nuclei, our present results may be considered disappointing. Indeed, even in cluster with quite extreme initial conditions (high stellar density), collisions do not leave any strong imprint on the overall structure of the stellar cluster. Neither do they feed the central BH more efficiently than tidal disruption or, presumably, stellar evolution. However, it must be stressed that collisions could have played a role of greater importance in the past if the present day nuclei have evolved from denser configurations. Further, more systematic sets of simulations will allow us to delineate the conditions leading to a collisional phase in the evolution of a cluster. Furthermore, even if not efficient enough to rule the dynamics, collisions are interesting per se. More work is required to determinate the observational consequences of these events (creation of “exotic” stars, accretion of gas onto the central BH) but our code will stand as the central backbone for these future, more complete, studies. One of us (MF) wants to thank the organizers of this conference for partial financial support to attend it. This work has been supported in part by the Swiss National Science Foundation. , T. & [Kumar]{}, P. 2000, “Tidal Spin-up of Stars in Dense Stellar Cusps Around Black Holes”, preprint, astro-ph/0004240 , P. J., [Zurek]{}, W. 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[^2]: This is due to the fact that various processes, e.g., relaxation, evaporation, collisions, …have time scales with different dependencies on $N$. [^3]: The MC code used by Shapiro and collaborators (see, e.g., [Duncan]{} & [Shapiro]{} 1983) in a context similar to ours, was of quite different nature, somewhere in between H[é]{}non’s method and direct Fokker-Planck integrations. [^4]: According to [Kim]{} & [Lee]{} (1999), the cross-section for formation of a tidal binary without physical contact vanishes at $V_{\mathrm{rel}}/V_{\ast} \simeq 0.1$ for $n=3$ polytropic structures and at $V_{\mathrm{rel}}/V_{\ast} \simeq 1$ for $n=1.5$. Consequently, such processes could occur at a significant rate in galactic nuclei only between stars that both have $M_{\ast} < 0.7\,M_{\odot}$. Furthermore, the main uncertainties in our understanding of a tidal binary don’t reside in the conditions for its formation at first periastron passage (this can be delineated using linear oscillation theory) but in its longer term evolution and fate. The main issues are about the impact of the deposition of tidal energy on the stellar structure and the interplay between the stellar oscillations and the orbit of the binary. The fraction of tidal binaries that will quickly merge is still unknown. [^5]: Available at ` http://www.geom.umn.edu/software/qhull/` [^6]: All this fiddling does not violate mass or energy conservation as collision results are coded in a dimensionless fashion in the interpolation grid and are scaled back to the real physical masses and velocities before they are applied to the particles.
--- abstract: 'A variation to the usual formulation of Grassmann representation path integrals is presented. Time-indexed anticommuting partners are introduced for each Grassmann coherent state variable and a general method for handling the effect of these introduced Grassmann partners is also developed. These Grassmann partners carry the nilpotency and anticommutivity of the Grassmann coherent state variables into the propagator and allow the propagator to be written as a path integral. Two examples are introduced in which this variation is shown to yield exact results. In particular, exact results are demonstratated for a spin$-\frac{1}{2}$ in an time-dependent magnetic field, and for a spin-boson system. The stationary path approximation is then shown to be exact for each example.' address: 'Department of Physics, University of Maryland, College Park, MD 20742, USA' author: - S Shresta title: 'Discrete path integral for spin-$\frac{1}{2}$ in the Grassmannian representation' --- Introduction ============ In the use of coherent state path integrals to describe spin systems three approaches have been prevalent. One approach has been to describe the spin degrees of freedom via coherent states of SU(2) or on the sphere[@alscher1; @alscher2]. Another approach is to use a boson mapping or a stereograhic projection of the sphere onto the complex plane[@boudjeda]. A third approach, which is the one taken here, is to use Grassmannian coherent states to represent the spin. One drawback of this approach is that it is restricted to representations of spin-$\frac{1}{2}$ or two-level systems. Within the Grassmannian representation there are also two variations. One approach is to generate Grassmannian coherent states by an exponentation of the single fermion creation operators. The other approach is to generate them by exponentiation of spin increasing operators. The advantage of the first approach is that the Hamiltonian so defined always has definite even parity since the fermion operators always appear in pairs, thereby avoiding nonlinear terms in the action due to the mixed Grassmann parity of the Hamiltonian. As has been pointed out, this fact has caused the first approach to be predominant[@smirnov]. However an advantage to the second approach is that the interaction terms in many Hamiltonians of interest remain bilinear. Thereby allowing more straightforward evaluation of the propagator. When the advantages of these two approaches are combined consistently an altered formulation of the Grassmann coherent state path integral emerges which is more useful. The further contribution that this work hopes to make to the field is simply to suggest an addition to the formulation of Grassmann coherent state path integrals that help make that framework more useful and consistent. The addition is the introduction of a time-indexed anticommuting partner to each Grassmann variable, which carries the nilpotency and anticommutivity of the Grassmann variables into the propagator. It makes the Grassmann representation more useful and consistent in that it extends its range of validity beyond previous formulations[@anastopoulos], offers alternative expansions, and provides a simpler route to non-Markovian reduced dynamics in that representation. In the first section the framework is sketched roughly. In the subsequent sections two simple examples are developed and shown to yield exact results. The two examples are the one of a spin$-\frac{1}{2}$ in a time dependent B-field, and of a spin$-\frac{1}{2}$ interacting with a single quantized mode of the EM field (the Jaynes-Cummings Hamiltonian). Both cases are taken for an infinitely heavy particle (i.e. motional degrees of freedom are ignored). For the two simple examples shown, each is analysed exactly and then demonstrated to also be exact under the stationary phase approximation. Grassmannian coherent states were first formulated for use in a path integral by Ohnuki and Kashiwa [@ohnuki]. An excellent review of their properties is available from Cahill and Glauber [@cahill]. Those of the bosonic coherent states are detailed by Perelomov [@perelomov]. Some of the most important properties are the following. The bosonic and fermionic coherent states are constructed by exponentiation of their respective creation operators acting on their respective reference vacuum states. Due to the algebra which their creation and annihilation operators satisfy, the coherent states are then eigenstates of their respective annihilation operators. $$\begin{array}{lr} \| {z}\rangle = {\rm e}^{a^\dagger z} \| 0 \rangle & a \| {z}\rangle = {z}\| {z}\rangle \\ \| {\eta}\rangle = {\rm e}^{S_+ {\eta}} \| \downarrow \rangle & S_- \| {\eta}\rangle ={\eta}\| {\eta}\rangle \end{array}$$ The coherent states are an overcomplete set of states with a resolution of unity and measure. $$\begin{array}{lr} 1 = \int {\rm d}\mu(z) \| {z}\rangle \langle {\bar{z}}\| & {\rm d}\mu({z}) = d{z}d{\bar{z}}{\rm e}^{-{\bar{z}}{z}} \\ 1 = \int {\rm d}\mu({\eta}) \| {\eta}\rangle \langle {\bar{\eta}}\| & {\rm d}\mu({\eta}) = d{\eta}d{\bar{\eta}}{\rm e}^{-{\bar{\eta}}{\eta}} \end{array}$$ Note that these Grassmann coherent states are not single fermion coherent states that are generated by the fermionic creation operator. As a result, Hamiltonians which contain single spin up/down operator terms will contain odd terms in this representation. Current approach ================ The approach taken in this paper follows the standard path integral approach with a few differences. In particular, the guiding principle has been to evaluate all quantities discretely as far as possible. The continous limit is used only for the final expressions. It begins by writing the propagator in a coherent state representation and breaking it into a product of discrete time infinitesmal propagators, $$\begin{aligned} \fl K(t,0) = \langle {\bar{\alpha}}_f \| U(t,0) \| {\alpha}_i \rangle \nonumber\\ \lo{=} \int \prod_{j=1}^{N-1} {\rm d}\mu({\alpha}_j) \langle {\bar{\alpha}}_f \| U_{N,N-1} \| {\alpha}_{N-1} \rangle \langle {\bar{\alpha}}_{N-1} \| U_{N-1,N-2} \| {\alpha}_{N-2} \rangle ... \langle {\bar{\alpha}}_1 \| U_{1,0} \| {\alpha}_0 \rangle ,\end{aligned}$$ where ${\alpha}$ is a generic symbol for a set of even and odd coherent state variables. The boundary condition applied is ${\alpha}_0 = {\alpha}_i$ and ${\bar{\alpha}}_N = {\bar{\alpha}}_f$. This choice of boundary condition corresponds to the perspective of unitarily evolving the initial(final) ket(bra) vector and then taking the inner product of the evolved vector with the final(initial) bra(ket) vector. Therefore no additional terms in the action are needed to impose boundary conditions. It is interesting to note that for an orthogonal representation the inner product between the final state and the evolved initial state is zero unless the initial state evolves exactly to the final state. The distinction between final states and evolved initial states is thus unneccessary. At this point the product of infinitesmal propagators can not be naturally combined into a single exponential, as is desirable in a path integral formulation. The reason is that there can be odd terms which anticommute in the infinitesmal propagators. With bosonic path integrals that is not a problem since c-numbers commute. In order to avoid this problem a time-indexed anticommuting partner is introduced to all fermionic coherent state variables. Then the propagator becomes a discrete time path integral of a single exponential, $$\label{dtcspi1} K(t,0) =\int \prod_{j=1}^N {\rm d}\mu({\alpha}_j) \exp( {\bar{\alpha}}_f {\alpha}_N - \frac{{\rm i}{\epsilon}}{\hbar} \sum_{i=1}^N H_{i,i-1} ).$$ Although the introduction of the Grassmann partners clearly allow the propagator to be written as above in (since they make each infinitesmal propagator even), that does not justify their introduction or elucidate their use. The justification for introducing the anticommuting partners is that they are a counting tool that helps to preserve the trucations and signs of formal expressions. This is most clearly illustrated in a recursive evaluation of the propagator. For example, the propagator for an infinitesmal interval is $$K({\epsilon},0) = \exp( {\bar{\alpha}}_1 {\alpha}_0 - \frac{{\rm i}{\epsilon}}{\hbar} H_{1,0} ) = \exp({\bar{\alpha}}_1 \psi_1 +\phi_1)$$ with $\psi_1$ and $\phi_1$ being Hamiltonian dependent and containing a mixture of even and odd terms. For two infinitesmal intervals the propagator is $$\begin{aligned} \fl K(2{\epsilon},0) = \int {\rm d}\mu({\alpha}_1) \exp({\bar{\alpha}}_2 {\alpha}_1 - \frac{{\rm i}{\epsilon}}{\hbar} H_{2,1}) K({\epsilon},0) \nonumber\\ \lo{=} \int {\rm d}\mu({\alpha}_1) \exp({\bar{\psi}}_2 {\alpha}_1 +{\bar{\phi}}_2) \exp({\bar{\alpha}}_1 \psi_1 +\phi_1) \nonumber\\ \lo{\neq} \int {\rm d}\mu({\alpha}_1) \exp({\bar{\psi}}_2 {\alpha}_1 +{\bar{\alpha}}_1 \psi_1 +{\bar{\phi}}_2 +\phi_1) .\end{aligned}$$ The last inequality is due to the odd parts of the exponents. However, if one introduces anticommuting partners to the Grassmann coherent state variables then the inequality becomes an equality when done with standard Grassmann integration techniques. The anticommuting partners change the exponents above from Grassmann odd to Grassmann even. A key point is that if anticommuting partners were not introduced to the Grassmann coherent state variables then nonlinear terms would appear in the exponent[@smirnov]. The distinction may seem small, but one must at later points respect the anticommutation properties of the Grassmann partners. More explicit examples are given later. A recursive evaluation which continues the above single step to succesive infinitesmal unitary evolutions can now be performed. After each evolution anticommuting Grassmann partners are introduced at that time index and the propagator is rewritten in a standard form to facilitate the next evolution, $$\begin{aligned} \fl K({\epsilon},0) = \exp({\bar{\alpha}}_1 {\alpha}_0 - \frac{{\rm i}{\epsilon}}{\hbar} H_{1,0}) = \exp({\bar{\alpha}}_1 \psi_1 +\phi_1) \\ \fl K(2{\epsilon},0) = \int {\rm d}\mu({\alpha}_1) \exp({\bar{\alpha}}_2 {\alpha}_1 - \frac{{\rm i}{\epsilon}}{\hbar} H_{2,1}) K({\epsilon},0) = \exp({\bar{\alpha}}_2 \psi_2 +\phi_2) \\ . \nonumber\\ . \nonumber\end{aligned}$$ By continuing this process one finds the full propagator to be, $$\begin{aligned} \fl K(N{\epsilon},0) = \int {\rm d}\mu({\alpha}_{N-1}) \exp({\bar{\alpha}}_N {\alpha}_{N-1} - \frac{{\rm i}{\epsilon}}{\hbar} H_{N,N-1}) K((N-1){\epsilon},0) \nonumber\\ \lo{=} \exp({\bar{\alpha}}_N \psi_N +\phi_N) .\end{aligned}$$ At each step new exponents are defined from the previous ones, $$\begin{aligned} \psi_j = f( \psi_{j-1}, \phi_{j-1} )\\ \phi_j = g( \psi_{j-1}, \phi_{j-1} ) ,\end{aligned}$$ with $f$ and $g$ some functions specific to the Hamiltonian. This method explicitly gives only exact expressions. The propagator in this form is only formally valid because the introduction of the time-indexed anticommuting parts to the couplings causes various truncations in the polynomial expansion that must be respected. The next step is thus to expand the propagator in a polynomial series $$\label{genprop1} K(t,0) = \exp({\bar{\alpha}}_f \psi_N +\phi_N) = \sum_{m=0}^\infty \frac{1}{m!} ({\bar{\alpha}}_f \psi_N +\phi_N)^m ,$$ and use the finite difference equations for the exponents and the anticommutation properties to find finite difference equations for the terms in the polynomial series. $$\begin{aligned} \label{geneom1} [\psi^m]_j = [f(\psi_{j-1},\phi_{j-1})]^m \\ \label{geneom2} [\phi^m]_j = [g(\psi_{j-1},\phi_{j-1})]^m \\ \label{geneom3} [\psi^m]_j [\phi^n]_j = [f(\psi_{j-1},\phi_{j-1})]^m [g(\psi_{j-1},\phi_{j-1})]^n .\end{aligned}$$ These new equations can then be taken to the continous limit and solved. Finally, putting the solutions back into the expansion of the propagator and resumming gives the propagator. Only in the form of with - does the propagator cease to be a formal expression and allow explicit calculation. Examples ======== Spin-$\frac{1}{2}$ in a general time-dependent classical magnetic field ----------------------------------------------------------------------- To illustrate the use of time-indexed anticommuting couplings the following is a calculation of the evolution of a spin-$\frac{1}{2}$ in a classical magnetic field. The simplest non-trivial case is that of a spin in a $B_z$ field with the addition of a possibly time-dependent $B_x$ and $B_y$ field. The Hamiltonian for this system is $$\fl H =\gamma \vec{S} \cdot \vec{B} = \frac{1}{2} \hbar {\omega}S_z + \hbar B_x S_x + \hbar B_y S_y = \hbar {\omega}S_+ S_- -\frac{1}{2} \hbar {\omega}+ \hbar (S_+ B + B^* S_-).$$ Here it is written in a “hermitian” form in anticipation of the addition of a Grassmann part to the classical field. The propagator between initial and final Grassmann coherent states is $$K(t,0) = \langle {\bar{\eta}}_t \| \exp[-\frac{{\rm i}}{\hbar} \int_0^t H(s) ds] \| {\eta}_0 \rangle .$$ In the usual way ($t=N {\epsilon}$) the propagator can be time sliced into a discrete time formulation. The propagator for one infinitesmal time step is (up to $O({\epsilon})$ and dropping the constant term) $$\label{1stepprop1} \fl K(j,j-1)=\langle {\bar{\eta}}_j \| \exp(-\frac{{\rm i}}{\hbar} H {\epsilon}) \| {\eta}_{j-1} \rangle = \exp[(1-{\rm i} {\omega}{\epsilon}) {\bar{\eta}}_j {\eta}_{j-1} -{\bar{\eta}}_j ({\rm i} B_j {\epsilon}) - ({\rm i} B_j^* {\epsilon}) {\eta}_{j-1}].$$ With eqn(\[1stepprop1\]) the propagator for a single infinitesmal step can be written down, $$\fl K({\epsilon},0) = \exp[(1-{\rm i} {\omega}{\epsilon}) {\bar{\eta}}_1 {\eta}_0 -{\bar{\eta}}_1 ({\rm i} B_1 {\epsilon}) - ({\rm i} B_1^* {\epsilon}) {\eta}_0] = \exp({\bar{\eta}}_1 {\eta}_1 -\phi_1)$$ and the propagator for two infinitesmal time steps is $$\begin{aligned} \fl K(2 {\epsilon},0) =\int {\rm d}\mu ({\eta}_1) \langle {\bar{\eta}}_2 \| \exp(-\frac{{\rm i}}{\hbar} H_2 {\epsilon}) \| {\eta}_1 \rangle \langle {\bar{\eta}}_1 \| \exp(-\frac{{\rm i}}{\hbar} H_1 {\epsilon}) \| {\eta}_0 \rangle \nonumber\\ \lo{=} \exp[(1-{\rm i} {\omega}{\epsilon})^2 {\bar{\eta}}_2 {\eta}_0 ] [ 1 -{\eta}_2 ({\rm i} B_2 {\epsilon}) - ({\rm i} B_1^* {\epsilon}) {\eta}_0 - B_2 B_1^* {\epsilon}^2 {\bar{\eta}}_2 {\eta}_0 - B_2^* B_1 {\epsilon}^2 \nonumber\\ -{\bar{\eta}}_2 ({\rm i} B_1 {\epsilon}) (1-{\rm i} {\omega}{\epsilon}) -(1-{\rm i} {\omega}{\epsilon}) ({\rm i} B_2^* {\epsilon}) {\eta}_0 + B_2 B_1^* {\epsilon}^2 (1-{\rm i} {\omega}{\epsilon})^2 {\bar{\eta}}_2 {\eta}_0 ].\end{aligned}$$ These two propagators have very different forms. However if at this point a time-indexed anticommuting part is given to the classical field such that $\{ B_n, B_m \} = 0$ and $\{ B_n, {\eta}\} =\{ B_n, {\bar{\eta}}\} =0$ then the $2 {\epsilon}$ propagator can be rewritten as a single exponential, $$\begin{aligned} \fl K(2{\epsilon},0)= \exp\{{\bar{\eta}}_2 [-{\rm i} B_2 {\epsilon}+ (1-{\rm i} {\omega}{\epsilon})(-{\rm i} B_1 {\epsilon}+ (1-{\rm i} {\omega}{\epsilon}) {\eta}_0)] \nonumber\\ + [ -{\rm i} B_1^* {\epsilon}{\eta}_0 - ({\rm i} B_2^* {\epsilon})(-{\rm i} B_1 {\epsilon}+(1-{\rm i} {\omega}{\epsilon}) {\eta}_0)]\} \nonumber\\ \lo{=} \exp({\bar{\eta}}_2 {\eta}_2 + \phi_2)\end{aligned}$$ with the definitions $$\begin{aligned} {\eta}_2 = (1-{\rm i} {\omega}{\epsilon}) {\eta}_1 -{\rm i} B_2 {\epsilon}\\ \phi_2 = \phi_1 - {\rm i} B_2^* {\epsilon}{\eta}_1.\end{aligned}$$ Now the 2${\epsilon}$ propagator is in the same form as the ${\epsilon}$ propagator. This facilitates a recursive evaluation, so that process can be continued to find the propagator for any number of steps, $$K(j{\epsilon},0) = \exp({\bar{\eta}}_j {\eta}_j + \phi_j)$$ with the recursive definitions $$\label{ex1eom1} \begin{array}{ll} {\eta}_j = (1-{\rm i} {\omega}{\epsilon}) {\eta}_{j-1} -{\rm i} B_j {\epsilon}& {\eta}_0 = {\eta}_i \\ \phi_j = \phi_{j-1} - {\rm i} B_j^* {\epsilon}{\eta}_{j-1} & \phi_0 = 0 . \end{array}$$ Inserting the boundary condition ${\bar{\eta}}_N={\bar{\eta}}_f$, one gets for the full propagator $$K(t=N{\epsilon},0)=\exp({\bar{\eta}}_f {\eta}_N +\phi_N),$$ with the variables ${\eta}_N$ and $\phi_N$ defined by eq(\[ex1eom1\]). The propagator in the above form can not yet be shown to satify the Schroedinger equation because it hides a major pitfall. The pitfall is that it is a formal expression and has meaning only as a polynomial expansion. Due to the introduction of the time-indexed anticommuting part in the magnetic field, many terms in the polynomial expansion truncate due to the nilpotency of the Grassmann variables. However this is not a weakness, but a strength, since the truncation of polynomial expansions is the reason Grassmann variables were introduced. If the continous limit were taken at this point the correct expansion of the exponential propagator would be lost. Expanding the propagator gives $$\label{bprop} K(t,0) = \exp({\bar{\eta}}_f {\eta}_N +\phi_N) = \sum_{m=0}^\infty \frac{[\phi^m]_N}{m!} (1+ {\bar{\eta}}_f {\eta}_N).$$ In the expansion above the Grassmann variable ${\bar{\eta}}_f$ causes a truncation. Analogously, in the $m^{th}$ order terms such as $[\phi^m]_N$, the time-indexed Grassmann parts of the magnetic field cause a truncation. That is, $\phi_N$ is a sum of terms containing many products of Grassmann variables. Products of these coefficients have many terms that are truncated due to nilpotency of the Grassmann variables. Keeping track of the truncations in the final coefficients would be a formidable task, however doing so in the infinitesmal equations of motion is sufficient. For example, instead of calculating $[\phi^m]_N$ by calculating $\phi_N$ first, one can find a differential equation for $[\phi^m]_N$ and calculate it directly. The functions that need to be calculated are thus $[\phi^m]_N$ and $[\phi^m{\eta}]_N$. Adhering to the anticommutation rules one finds (up to $O({\epsilon})$), $$\begin{aligned} \label{bexeom1} [\phi^m]_j = [\phi^m]_{j-1} - {\rm i} m B_j^* {\epsilon}[\phi^{m-1} {\eta}]_{j-1} \\ \label{bexeom2} [\phi^m {\eta}]_j = (1-{\rm i} {\omega}{\epsilon})[\phi^{m} {\eta}]_{j-1} - {\rm i} B_j {\epsilon}[\phi^m]_{j-1}.\end{aligned}$$ The above equations can now safely be taken to the continous limit, $$\begin{aligned} \label{bconteom1} \frac{{\rm d}}{{\rm d}t}[\phi^m]_t &= - {\rm i} m B^(t) [\phi^{m-1} {\eta}]_t \\ \label{bconteom2} \frac{{\rm d}}{{\rm d}t}[\phi^m {\eta}]_t &= -{\rm i} {\omega}[\phi^m {\eta}]_t - {\rm i} B(t) [\phi^m]_t,\end{aligned}$$ and used to show that the propagator satisfies the Schroedinger equation (see Appendix A). The propagator and - give a novel expansion of the propagator and equations for the terms in its expansion. The Schroedinger equation can be reformed from it, but in the expanded form it may be possible to apply new approximations. This issue is addressed in future work. Having introduced and justified the introduction of the Grassmann partners, they can now be used to rewrite the propagator as a true path integral. The propagator for finite time is $$\begin{aligned} \fl K(t,0) =\int \prod_{j=1}^{N-1} {\rm d}\mu ({\eta}_j) \langle {\bar{\eta}}_N \| \exp(-\frac{{\rm i}}{\hbar} H {\epsilon}) \| {\eta}_{N-1} \rangle \langle {\bar{\eta}}_{N-1} \| \exp(-\frac{{\rm i}}{\hbar} H {\epsilon}) \| {\eta}_{N-2} \rangle ... \nonumber\\ \times \langle {\bar{\eta}}_{2} \| \exp(-\frac{{\rm i}}{\hbar} H {\epsilon}) \| {\eta}_{1} \rangle \langle {\bar{\eta}}_{1} \| \exp(-\frac{{\rm i}}{\hbar} H {\epsilon}) \| {\eta}_{0} \rangle.\end{aligned}$$ Due to the anti-commuting properties of the Grassmann variables, the infinitesmal propagators in the above expression could not be combined into a single exponential if a time-indexed anticommuting part were not introduced. After their introduction the propagator becomes, $$\begin{aligned} \fl K(t,0)=\int \prod_{j=1}^{N-1} {\rm d}^2 {\eta}_j \mbox{ } \exp \{{\bar{\eta}}_N {\eta}_N \nonumber\\ +\sum_{j=1}^N \left[ -{\bar{\eta}}_j {\eta}_j + (1-{\rm i} {\omega}{\epsilon}) {\bar{\eta}}_j {\eta}_{j-1} -{\rm i} {\bar{\eta}}_j B_j {\epsilon}-{\rm i} B_j^ {\epsilon}{\eta}_{j-1} \right] \}.\end{aligned}$$ One may now evaluate this discrete path integral at the saddle point. Varying discretely, the discrete equation for the stationary path is found to be $$\label{beom1} {\eta}_j = (1-{\rm i} {\omega}{\epsilon}) {\eta}_{j-1} -{\rm i} B_j {\epsilon}$$ and the propagator is $$K(t,0)=\exp \{{\bar{\eta}}_N {\eta}_N +\sum_{j=1}^N \left[-({\rm i} B_j^* {\epsilon}) {\eta}_{j-1} \right] \}.$$ Or, defining again the variable $$\label{beom2} \phi_j = -{\rm i} \sum_{i=1}^j B_i^* {\epsilon}{\eta}_{i-1} = \phi_{j-1} -{\rm i} B_j^* {\epsilon}{\eta}_{j-1},$$ and inserting the correct boundary conditions ${\bar{\eta}}_N={\bar{\eta}}_f$ and ${\eta}_0={\eta}_i$, one gets for the propagator $$K(t,0)=\exp({\bar{\eta}}_f {\eta}_t +\phi_t),$$ with the variables ${\eta}_N$ and $\phi_N$ defined by and . This is the same as the exact result previously derived. This example was handled, in the stationary path approximation, using a boson mapping in [@boudjeda] and using the SU(2) representation in [@alscher1]. The result found here of exactness of the stationary path approximation agrees with the same result found in those references. Spin-Boson ---------- The Jaynes-Cummings Hamiltonian for a spin$-\frac{1}{2}$ interacting with a single em field mode is $$H = \hbar {\omega}_o S_+ S_- + \hbar {\omega}a^{\dagger} a + \hbar (S_+ {\lambda}a + a^{\dagger}{\lambda}S_-).$$ Here again it is written in a “hermitian” form in anticipation of the addition of a Grassmann part to the spin-boson coupling constant. The propagator between initial and final coherent states is $$K(t,0) = \langle {\bar{\eta}}_t {\bar{z}}_t \| \exp[-\frac{{\rm i}}{\hbar} \int_0^t H(s) ds] \| {\eta}_0 z_0 \rangle .$$ In the usual way ($t=N {\epsilon}$) the propagator can be time sliced into a discrete time formulation. The propagator for one infinitesmal time step is (up to $O({\epsilon})$) $$\begin{aligned} \fl K(j,j-1)=\langle {\bar{\eta}}_j {\bar{z}}_j \| \exp(-\frac{{\rm i}}{\hbar} H {\epsilon}) \| {\eta}_{j-1} z_{j-1} \rangle \nonumber\\ \lo{=} \exp[(1-{\rm i} {\omega}{\epsilon}) {\bar{z}}_j {z}_{j-1} +(1-{\rm i} {\omega}_o {\epsilon}) {\bar{\eta}}_j {\eta}_{j-1} -{\bar{\eta}}_j ({\rm i} {\lambda}_j {\epsilon}) z_{j-1} - {\bar{z}}_j ({\rm i} {\lambda}_j {\epsilon}) {\eta}_{j-1}].\end{aligned}$$ Using this equation the single infinitesmal step propagator is, $$K(\epsilon,0) = \langle {\bar{\eta}}_1 {\bar{z}}_1 \| U_{\epsilon} \| {\eta}_0 {z}_0 \rangle = \exp[{\bar{\eta}}_1 (\psi_1 +{g}_1) +{\bar{z}}_1 ({f}_1 +\phi_1)]$$ with the definitions, $$\begin{array}{ll} {g}_1 = ({\rm i} {\lambda}_2 {\epsilon}) {z}_0 & \psi_1 = (1-{\rm i} {\omega}_o {\epsilon}) {\eta}_0 \\ {f}_1 = (1-{\rm i} {\omega}{\epsilon}) {z}_0 & \phi_1 = ({\rm i} {\lambda}_2 {\epsilon}) {\eta}_0 \end{array}$$ The 2${\epsilon}$ propagator is then computed from the above to be, $$\label{onemode} \fl K(2 \epsilon,0) = \langle {\bar{\eta}}_2 {\bar{z}}_2 \| U_{2 \epsilon} \| {\eta}_0 {z}_0 \rangle = \int {\rm d}\mu (z_1) {\rm d}\mu ({\eta}_1) \langle {\bar{\eta}}_2 {\bar{z}}_2 \| U_{\epsilon} \| {\eta}_1 {z}_1 \rangle \langle {\bar{\eta}}_1 {\bar{z}}_1 \| U_{\epsilon} \| {\eta}_0 {z}_0 \rangle$$ which gives the following unwieldy expression, $$\begin{aligned} \fl K(2 \epsilon,0)= \exp[(1-{\rm i} \omega_o {\epsilon}) {\bar{\eta}}_2 \psi_1 + (1- {\rm i} {\omega}{\epsilon}) {\bar{z}}_2 f_1] \nonumber\\ \lo{\times} \bigg [ 1+ {\rm i} {\lambda}{\epsilon}{\bar{z}}_2 \psi_1 +(1-{\rm i} \omega_o {\epsilon}) {\bar{\eta}}_2 g_1 + {\rm i} {\lambda}{\epsilon}{\bar{\eta}}_2 f_1 + (1- {\rm i} {\omega}{\epsilon}) {\bar{z}}_2 \phi_1 + {\rm i} {\lambda}{\epsilon}{\bar{z}}_2 {g}_1\nonumber\\ + {\rm i} {\lambda}{\epsilon}{\bar{\eta}}_2 \phi_1 + {\rm i} {\lambda}{\epsilon}{\bar{\eta}}_2 \phi_1 (1- {\rm i} {\omega}{\epsilon}) {\bar{z}}_2 f_1 - {\rm i} {\lambda}{\epsilon}{\bar{z}}_2 g_1 (1-{\rm i} {\omega}_o {\epsilon}) {\bar{\eta}}_2 \psi_1 \bigg ]\end{aligned}$$ At this point a time-indexed anticommuting part is given to the coupling constants such that $\{ {\lambda}_n, {\lambda}_m \} = 0$ and $\{ {\lambda}_n, {\eta}\} =\{ {\lambda}_n, {\bar{\eta}}\} =0$. The $2 {\epsilon}$ propagator can be rewritten as a single exponential in the same form as the ${\epsilon}$ propagator, $$K(2 \epsilon,0) = \exp[{\bar{\eta}}_2 (\psi_2 + g_2) + {\bar{z}}_2 (\phi_2 +f_2)]$$ with the definitions, $$\label{ex2eom1} \begin{array}{ll} g_2 = (1-{\rm i} {\omega}_o {\epsilon}) g_1 + ({\rm i} {\lambda}_2 {\epsilon}) f_1 & \psi_2 = (1-{\rm i} {\omega}_o {\epsilon}) \psi_1 + ({\rm i} {\lambda}_2 {\epsilon}) \phi_1 \\ f_2 = ({\rm i} {\lambda}_2 {\epsilon}) g_1 + (1-{\rm i} {\omega}{\epsilon}) f_1 & \phi_2 = ({\rm i} {\lambda}_2 {\epsilon}) \psi_1 + (1-{\rm i} {\omega}{\epsilon}) \phi_1 \end{array}.$$ Or for greater ease of use, $$K(2 \epsilon,0) = \exp({\bar{\eta}}_2 {\eta}_2 + {\bar{z}}_2 {z}_2)$$ with the definitions, $$\begin{aligned} {\eta}_2 = (1-{\rm i} {\omega}_o {\epsilon}) {\eta}_1 + ({\rm i} {\lambda}_2 {\epsilon}) {z}_1 \\ {z}_2 = ({\rm i} {\lambda}_2 {\epsilon}) {\eta}_1 + (1-{\rm i} {\omega}{\epsilon}) {z}_1 .\end{aligned}$$ This process can be continued to find the propagator for any number of infinitesmal steps, with the result, $$K(j \epsilon,0) = \exp({\bar{\eta}}_j {\eta}_j + {\bar{z}}_j {z}_j)$$ and the definitions, $$\begin{aligned} \label{ex2eom2} {\eta}_j &= (1-{\rm i} {\omega}_o {\epsilon}) {\eta}_{j-1} + ({\rm i} {\lambda}_j {\epsilon}) {z}_{j-1} \\ \label{ex2eom3} {z}_j &= ({\rm i} {\lambda}_j {\epsilon}) {\eta}_{j-1} + (1-{\rm i} {\omega}{\epsilon}) {z}_{j-1} .\end{aligned}$$ After inserting the correct boundary conditions ${\bar{\eta}}_N={\bar{\eta}}_f$, ${\eta}_0={\eta}_i$, ${\bar{z}}_N={\bar{z}}_f$, and ${z}_0={z}_i$ the propagator for time $t=N{\epsilon}$ is $$K(t=N \epsilon,0) = \exp({\bar{\eta}}_f {\eta}_N + {\bar{z}}_f {z}_N)$$ with the variables ${\eta}_N$, and ${z}_N$ defined by and . As in the previous example the propagator in the above form is only a formal expression and has meaning only as a polynomial expansion. Many terms in the polynomial expansion truncate due to the nilpotency of the Grassmann variables. Expanding the propagator gives $$\label{sbprop} K(t,0) = \exp({\bar{\eta}}_f {\eta}_N + {\bar{z}}_f {z}_N) = \sum_{m=0}^\infty \frac{({\bar{z}}_f)^m [{z}^m]}{m!} (1 +{\bar{\eta}}_f {\eta}_N).$$ As before differential equations are found for the functions in the expansion of the propagator. The functions that need to be calculated are $[{z}^m]_N$ and $[{z}^m{\eta}]_N$. Adhering to the anticommutation rules one finds (up to $O({\epsilon})$), $$\begin{aligned} [{z}^m]_j = (1 -{\rm i} m{\omega}{\epsilon})[{z}^m]_{j-1} - {\rm i} m {\lambda}_j {\epsilon}[{z}^{m-1}{\eta}]_{j-1} \\ \big [{z}^m{\eta}]_j = (1-{\rm i} m{\omega}{\epsilon}-{\rm i} {\omega}_o {\epsilon})[{z}^{m}{\eta}]_{j-1} - {\rm i} {\lambda}_j {\epsilon}[{z}^{m+1}]_{j-1}.\end{aligned}$$ Or in the continous limit, $$\begin{aligned} \label{sbconteom1} \frac{{\rm d}}{{\rm d}t}[{z}^m]_t = -{\rm i} m{\omega}[{z}^m]_t - {\rm i} m {\lambda}[{z}^{m-1} {\eta}]_t \\ \label{sbconteom2} \frac{{\rm d}}{{\rm d}t}[{z}^{m}{\eta}]_t = (-{\rm i} m{\omega}-{\rm i} {\omega}_o)[{z}^{m}{\eta}]_t - {\rm i} {\lambda}[{z}^{m+1}]_t.\end{aligned}$$ The propagator of can now be shown to satisfy the Schroedinger equation (see Appendix B). As in the previous example the propagator and - give a novel expansion of the propagator and equations for the terms in its expansion. However, in this case the unexpanded expression may offer an advantage when seeking the reduced dynamics. In that case the final state of the e.g. boson can be traced out using the formal exponential version of , leaving a formal expression for the reduced propagator. Equations - can then be used to find solutions for terms in the expansion of the reduced propagator. It remains to show that the stationary path approximation yields the same exact result in this example. The propagator for finite time in this case is $$\begin{aligned} \fl K(t,0) =\int \prod_{j=1}^{N-1} {\rm d}\mu ({\eta}_j) {\rm d}\mu (z_j) \langle {\bar{\eta}}_N {\bar{z}}_N \| \exp(-\frac{{\rm i}}{\hbar} H {\epsilon}) \| {\eta}_{N-1} z_{N-1} \rangle \nonumber\\ \times \langle {\bar{\eta}}_{N-1} {\bar{z}}_{N-1} \| \exp(-\frac{{\rm i}}{\hbar} H {\epsilon}) \| {\eta}_{N-2} z_{N-2} \rangle ... \langle {\bar{\eta}}_{1} {\bar{z}}_1 \| \exp(-\frac{{\rm i}}{\hbar} H {\epsilon}) \| {\eta}_{0} z_0 \rangle.\end{aligned}$$ As for the previous example the infinitesmal propagators in the above expression can be combined into a single exponential only after the introduction of a Grassmann partners to the coupling constants. The propagator is then written, $$\begin{aligned} \fl K(t,0)=\int \prod_{j=1}^{N-1} {\rm d}^2 {\eta}_j {\rm d}^2 z_j \mbox{ } \exp({\bar{z}}_N z_N +{\bar{\eta}}_N {\eta}_N) \exp \Bigg \{ \sum_{j=1}^N \bigg[-{\bar{z}}_j z_j -{\bar{\eta}}_j {\eta}_j \nonumber\\ + (1-{\rm i} {\omega}{\epsilon}) {\bar{z}}_j {z}_{j-1} + (1-{\rm i} {\omega}_o {\epsilon}) {\bar{\eta}}_j {\eta}_{j-1} -{\bar{\eta}}_j ({\rm i} {\lambda}_j {\epsilon}) z_{j-1} -{\bar{z}}_j ({\rm i} {\lambda}_j {\epsilon}) {\eta}_{j-1} \bigg] \Bigg \}.\end{aligned}$$ Varying discretely about the saddle point, equations for the stationary path are found to be $$\begin{aligned} \label{sb2eom1} z_j = (1-{\rm i} {\omega}{\epsilon}) {z}_{j-1} -{\rm i} {\lambda}_j {\epsilon}{\eta}_{j-1} \\ \label{sb2eom2} {\eta}_j = (1-{\rm i} {\omega}_o {\epsilon}) {\eta}_{j-1} -{\rm i} {\lambda}_j {\epsilon}z_{j-1}\end{aligned}$$ and the propagator after inserting the correct boundary conditions ${\bar{\eta}}_N={\bar{\eta}}_f$, ${\eta}_0={\eta}_i$, ${\bar{z}}_N={\bar{z}}_f$, and ${z}_0={z}_i$ is $$K(t,0)=\exp({\bar{\eta}}_f {\eta}_N + {\bar{z}}_f {z}_N).$$ with the variables ${\eta}_N$ and $\phi_N$ defined by -. This again is the same as the exact result, thereby demonstrating that the stationary path approximation is exact for the Jaynes-Cummings Hamiltonian. This specific example was computed with the stationary path approximation in [@anastopoulos] using a Grassmannian path integral and in [@alscher2] using the SU(2) representation. The results here agree with those found in [@alscher2], where it was also found that the stationary path approximation yielded exact results. The range of validity for the Grassmannian path integral method in [@anastopoulos] was resticted to an initial bosonic vaccum state, but for that restricted range they also found the stationary path approximation to be exact. Discussion ========== In this paper the path integral formulation of a spin$-\frac{1}{2}$ propagator in the Grassmann representation was modified by the introduction of time-indexed anticommuting partners to the Grassmann variables. These anticommuting partners make the formulation more consistent by carrying the effect of the nilpotency and anticommutation of the Grassmann variables in the infinitesmal propagator into the full propagator. That is, when the Grassmann variables at intermediate times are integrated over all values, the effect of their anticommutivity will be lost unless anticommuting partners were introduced which carry that effect. The resulting propagator then looks similar to a usual boson coherent state propagator. The difference is that the anticommutivity of the Grassmann partners must be respected during explicit calculation of the propagator. The introduction of the Grassmann partners in the propagator gives three main advantages. First it allows one to write the propagator as a path integral and thereby reconnect intuitive notions of integration over all paths with actual computation in the Grassmann representation. The concept of the stationary path can then be applied with greater meaning, becoming a specific sequence of mixed c-number/Grassmann values. The range of applicability of the stationary path approximation in the Grassmann representation then widens to include non-vacuum initial states, as shown by the second example. The second advantage is that it creates a novel expansion of the propagator as, for example, in . The expansion in turns out to be an expansion in the final coherent state. So for small $\| {z}_f \|$ (small energy) the first few terms are a good approximation. Alternatively, in the same example, $K(t,0)$ can also be written as an expansion in $\| {z}_i \|$ such that the expansion is one near initial vacuum. In that regime the Grassmann representation can be used to find the full quantum dynamics of a spin$-\frac{1}{2}$ (or two-level) system. An example of such a situation would for an atom in a cavity which is near vacuum. As a model, the Hamiltonian of the Jaynes-Cummings example could be extended to include the quantum vacuum, with one mode populated by a few photons. This is the focus of ongoing work. The third advantage is that it facilitates finding the reduced dynamics since the propagator is in an exponential form. Inserting an initial state and integrating out certain degrees of freedom then requires only a few gaussian integrations. The proviso is that the final result must be expanded in a polynomial series with the anticommutivity of the Grassmann partners respected. This would be especially suited to finding the evolution, including decoherence, of a spin$-\frac{1}{2}$ or two-level system interacting with a bath in some arbitrary initial state. Such an evolution would be non-Markovian, although reduced, since no Markovian approximation is in principle necessary. Spin-$\frac{1}{2}$ in a magnetic field details ============================================== First demonstrate that that the propagator with the equations of motion - satisfies the Schroedinger equation. The Schroedinger equation is $$\label{sch1} {\rm i} \hbar \frac{{\rm d}}{{\rm d}t} K(t,0) = H K(t,0).$$ In the Grassmannian representation the left hand side(LHS) is from $${\rm i} \hbar \frac{{\rm d}}{{\rm d}t} \langle {\bar{\eta}}_f \| K(t,0) \| {\eta}_0 \rangle = {\rm i} \hbar \frac{{\rm d}}{{\rm d}t}\sum_{m=0}^\infty \frac{1}{m!} ([\phi^m]_t+ {\bar{\eta}}_f [\phi^m {\eta}]_t).$$ Substituting - this becomes $$LHS = \hbar \sum_{m=0}^\infty \frac{1}{m!} (B^* [\phi^m {\eta}]_t +{\bar{\eta}}_f B [\phi^m]_t + {\omega}{\bar{\eta}}_f [\phi^m {\eta}]_t).$$ The right hand side(RHS) of is, in the Grassmannian representation $$\langle {\bar{\eta}}_f \| H K(t,0) \| {\eta}_0 \rangle = \int {\rm d}\mu({\eta}) \langle {\bar{\eta}}_f \| H \| {\eta}\rangle \langle {\bar{\eta}}\| K(t,0) \| {\eta}_0 \rangle.$$ After substituting the Hamiltonian and the propagator the right hand side is $$RHS = \hbar \sum_{m=0}^\infty \frac{1}{m!} (B^* [\phi^m {\eta}]_t +{\bar{\eta}}_f B [\phi^m]_t + {\omega}{\bar{\eta}}_f [\phi^m {\eta}]_t).$$ Thus in LHS=RHS so the Schroedinger equation is satisfied. Spin-boson details ================== The Schroedinger equation is . For the spin-boson propagator in the Grassmannian representation the left hand side is $${\rm i} \hbar \frac{{\rm d}}{{\rm d}t} \langle {\bar{\eta}}_f {\bar{z}}_f \| K(t,0) \| {\eta}_0 {z}_0 \rangle = {\rm i} \hbar \frac{{\rm d}}{{\rm d}t}\sum_{m=0}^\infty \frac{({\bar{z}}_f)^m}{m!} ( [{z}^m]_t +{\bar{\eta}}_f [{z}^{m}{\eta}]_t).$$ Substituting the equations of motion in the spin-boson example - gives $$\fl LHS = \hbar \sum_{m=0}^\infty \frac{({\bar{z}}_f)^m}{m!} \{ (m{\omega}[{z}^m]_t +m {\lambda}[{z}^{m-1}{\eta}]_t)+ {\bar{\eta}}_f ( (m{\omega}+{\omega}_o)[{z}^{m}{\eta}]_t + {\lambda}[{z}^{m+1}]_t ) \}.$$ The right hand side of the propagator for a system with both a Grassmannian and a bosonic coherent representation is $$\langle {\bar{\eta}}_f {\bar{z}}_f \| H K(t,0) \| {\eta}_0 {z}_o \rangle = \int {\rm d}\mu({\eta}) {\rm d}\mu({z}) \langle {\bar{\eta}}_f {\bar{z}}_f \| H \| {\eta}{z}\rangle \langle {\bar{\eta}}{\bar{z}}\| K(t,0) \| {\eta}_0 {z}_0 \rangle.$$ Inserting the Jaynes-Cummings Hamiltonian and evaluating the integral gives $$\fl RHS = \hbar \sum_{m=0}^\infty \frac{({\bar{z}}_f)^m}{m!} \{ (m{\omega}[{z}^m]_t +m {\lambda}[{z}^{m-1}{\eta}]_t)+ {\bar{\eta}}_f ( (m{\omega}+{\omega}_o)[{z}^{m}{\eta}]_t + {\lambda}[{z}^{m+1}]_t ) \}$$ By inpection LHS=RHS so the Schroedinger equation is satisfied in the spin-boson case. References {#references .unnumbered} ========== [9]{} Alscher A and Grabert H 1999 Semi-classical dynamics of a spin-$\frac{1}{2}$ in an arbitrary magnetic field [J. Phys. A:Math. Gen.]{} [32]{} 4907\ (Alscher A and Grabert H 1999 [Preprint]{} quant-ph/9904102) Alscher A and Grabert H 2001 Semi-classical dynamics of the Jaynes-Cummings model [Preprint]{} quant-ph/0006072 Boudjeda T, Hammann T F and Nouicer Kh 1995 Path integral for a spinning particle in a magnetic field [J. Math. Phys.]{} [36 4]{} 1602 Smirnov V V 1999 Path integral for system with spin [J. Phys. A:Math. Gen.]{} [32]{} 1285 Anastopoulos C and Hu B L 2000 Two-level atom-field interaction: Exact master equation for non-Markovian dynamics, decoherence, and relaxation [Phys. Rev. A]{} [62]{} 033821\ (Anastopoulos C and Hu B L 1999 [Preprint]{} quant-ph/9901078) Ohnuki Y and Kashiwa T 1978 Coherent States of fermi operators and the path integral [Coherent states: applications in physics and mathematical physics]{} eds J Klauder and B Skagerstam (Singapore: World Scientific) pp 449-465 Cahill K E and Glauber R J 1999 Density operators for fermions [Phys. Rev. A]{} [59 2]{} 1538\ (Cahill K E and Glauber R J 1998 [Preprint]{} physics/9808029) Perelomov A 1986 [Generalized coherent states and their applications]{} (Berlin: Springer)
--- abstract: 'A gamma-ray excess over background has been claimed in the inner regions of the Galaxy, triggering some excitement about the possibility that the gamma rays originate from the annihilation of dark matter particles. We point out that the existence of such an excess depends on how the diffuse gamma-ray background is defined, and on the procedure employed to fit such background to observations. We demonstrate that a gamma-ray emission with spectral and morphological features closely matching the observed excess arises from a population of cosmic ray protons in the inner Galaxy, and provide proof of principle and arguments for the existence of such a population, most likely originating from local supernova remnants. Specifically, the “Galactic center excess” is readily explained by a recent cosmic-ray injection burst, with an age in the 1-10 kilo-year range, while the extended inner Galaxy excess points to mega-year old injection episodes, continuous or impulsive. We conclude that it is premature to argue that there are no standard astrophysical mechanisms that can explain the excess.' author: - Eric Carlson - Stefano Profumo bibliography: - 'galcenter.bib' title: 'Cosmic Ray Protons in the Inner Galaxy and the Galactic Center Gamma-Ray Excess' --- Introduction ============ The Galactic center is a promising location to search for non-gravitational signals from particle dark matter such as gamma rays from dark matter pair annihilation. Any model for the density distribution of dark matter in the Galaxy predicts a high concentration of dark matter in the Galactic center, with a resulting large number density of dark matter particle pairs. Barring the possibility of a large, nearby dark matter “clump”, the Galactic center direction is the direction in the sky where the line-of-sight integral of the dark matter density squared is maximal. As a result, the Galactic center is the location where one of the brightest photon signals from dark matter annihilation is expected. On the downside, the center of the Galaxy hosts a complex combination of “standard” astrophysical $\gamma$-ray sources. The region contains numerous resolved and many unresolved $\gamma$-ray point sources; in addition, the diffuse Galactic emission is brightest in the center of the Galaxy, where the largest macroscopic concentrations of gas, cosmic rays and interstellar radiation energy density are found. This dense environment copiously sources $\gamma$ rays from hadronic inelastic interactions as well as from inverse Compton scattering and bremsstrahlung. Such complex background structure can be hardly reconstructed from first principles, and non-trivial extrapolations and inference often, if not always, define how the predicted background emission is calculated. The combination of such an appealing target with such a treacherous background has contributed to much debate about the existence and nature of excess $\gamma$-ray emission from the Galactic center region. Ever since the years of the EGRET telescope, claims of an excess diffuse $\gamma$-ray emission (extending even beyond the Galactic center) have been made [@Hunger:1997we; @deBoer:2005tm], and proved premature, with several groups proposing a Galactic cosmic-ray spectra differing from local values [@Strong:2004de; @Kamae:2004xx]. The EGRET excess was subsequently shown to be systematic in origin, relating instead to a miscalculation of EGRET’s sensitivity above a few GeV [@Stecker:2008], and was later shown to be conclusively unfounded [@Abdo:2010nz] using data from the Fermi Large Area Telescope (LAT) [@Atwood:2009ez]. Shortly after LAT data were made public, claims of a Galactic center Excess (GCE) have been put forward, pointing to differing particle dark matter properties (including the preferred mass, pair-annihilation rate, and annihilation pathway) depending on the background model employed in the analysis, see e.g. [@Goodenough:2009gk; @Hooper:2010mq; @Hooper:2011ti]. Several immediate issues have been raised following the identification of excess of $\gamma$ rays over background and with associations to new physics. These include the question of $\gamma$-ray point source modeling associated with the radio source Sgr A\, see e.g. [@Boyarsky:2010dr; @Chernyakova:2011zz; @Linden:2012bp; @Linden:2012iv], and the role of unresolved populations of $\gamma$-ray emitters such as millisecond pulsars [@2011JCAP...03..010A] (see however [@Hooper:2013nhl]). One of the key elements in assessing the presence of a genuine $\gamma$-ray excess in the Galactic center region is, naturally, that of modeling $\gamma$-ray sources in the region. Critical to this is the role of unidentified point sources, including population models for unidentified source classes, and of sources whose spectrum and even source extension is unclear (for example the $\gamma$-ray counterpart to Sgr A\). A second key element is the diffuse $\gamma$-ray emission induced by Galactic cosmic rays. It has long been known [@Stecker1977a; @Abdo:2010nz] that the key components of such emission, in the 0.1-100 GeV range are (i) hadronic emission from neutral pion decay produced by inelastic proton collision with the interstellar gas, (ii) inverse Compton up-scattering of background interstellar radiation by cosmic-ray electrons and positrons, and (iii) bremsstrahlung. We review below how the two key ingredients to the background model employed to extract the Galactic center excess have been handled in the three most recent and comprehensive analyses. What we believe is a crucial point to make is that the general procedure, in those studies, has been to employ background templates that make crucial assumptions about the Galactic diffuse emission. One of us had pointed out in Ref. [@Linden:2010ea], with Linden, that important systematic effects in extracting a diffuse $\gamma$-ray excess originate from neglecting the cosmic-ray density distribution and in utilizing templates where the diffuse hadronic emission, item (i) in the list above, follows the morphology of the target gas density. In the present study, we point out that (a) very little is known about cosmic rays in the Galactic center region; that (b) more or less young populations of cosmic rays are likely to inhabit that region and to importantly contribute to the hadronic emission in a way that would be completely missed by a current template analysis; and, finally, that (c) such a population(s) is likely to source the claimed $\gamma$-ray excess. Let us first briefly review three recent studies devoted to the Galactic center excess, Ref. [@gordon_macias:2013], [@Abazajian:2014fta] and [@Daylan:2014rsa]. The study presented in Ref. [@gordon_macias:2013] focuses on the $7^\circ\times7^\circ$ region centered around the Galactic center (GC, $b=0,\ l=0$), and employs the recommended LAT Collaboration diffuse background model [gal\_2yearp7v6\_v0]{} (we will comment below on the implicit assumptions included in this model), plus isotropic backgrounds, and known $\gamma$-ray sources in the second year Fermi catalogue (2FGL). The study confirms evidence for a spherically symmetric extended source, as obtained in previous studies [@Abazajian:2012pn], with a spectrum consistent both with emission from millisecond pulsars and with dark matter annihilation. Ref. [@gordon_macias:2013] also attempts to assess systematic uncertainties in the background modeling, concluding that such uncertainty is in the vicinity of the 20% level. The analysis of Ref. [@Abazajian:2014fta] also considers a region of interest of $7^\circ\times7^\circ$ centered around the GC, and employs two choices for the energy range, photon source class, pixel size, and energy binning. Ref. [@Abazajian:2014fta] then fits a variety of templates to the observed $\gamma$-ray data. These templates include, in addition to point sources, the recommended Galactic diffuse emission model [gal\_2yearp7v6\_v0]{} and isotropic background model [iso\_p7v6source]{}, a template (MG) that intends to map the bremsstrahlung emission associated with high-energy electrons interacting with molecular gas clouds as traced by the 20 cm radio map of the GC [@YusefZadeh:2012nh], a Galactic Center Excess (GCE) source, and a “new diffuse” component associated with a central stellar cluster, with varying spatial profiles. The two key findings of Ref. [@Abazajian:2014fta] are that (i) an extended emission in the GC region associated with the GCE template is present with any combination of templates and with both choices for the pixel and energy binning etc.; and that (ii) the fluxes and spectra associated with both the $\gamma$-ray emission from the central point source Sgr A\* and with the GC extended emission are significantly affected by the choice of the background model, especially in the low-energy range. The GC excess is found to have a spatial distribution consistent with a profile $\propto r^{-2.2}$. The study of Ref. [@Daylan:2014rsa], which appeared less than 10 days after Ref. [@Abazajian:2014fta], focused on an “Inner Galaxy” region, which masks out the Galactic plane ($\|b\|<1^\circ$) and includes a large region of several tens of degrees, and on a “Galactic center” region, defined by $\|b\|<5^\circ$ and $\|l\|<5^\circ$. Both studies use a novel cut on photon events based on the CTBCORE variable, producing higher resolution maps. In the “Inner Galaxy” analysis, Ref. [@Daylan:2014rsa] makes use of three templates (the Fermi collaboration [p6v11]{} Galactic diffuse model, an isotropic background and a uniform-brightness template matching the Fermi bubbles) plus a “dark matter” template of variable inner slope. In the “Galactic center” analysis, the templates used include a Galactic diffuse emission provided by the Fermi collaboration ([gal\_2yearp7v6\_v0]{}, the same choice as Ref. [@Abazajian:2014fta]), a template tracing the 20 cm emission, along the lines of Ref. [@Abazajian:2014fta], an isotropic component, and all 2FGL point sources [@Fermi-LAT:2011iqa]. As in Ref. [@Abazajian:2014fta] it is found that the isotropic component needed to provide an optimal fit is considerably brighter than the extragalactic $\gamma$-ray background. Ref. [@Daylan:2014rsa] indicates a strong preference for the existence of a Galactic center excess, and finds a similar preferred spatial distribution profile to Ref. [@Abazajian:2014fta] and, generically, a similar preferred spectral shape. Ref. [@Daylan:2014rsa] points out that the excess is approximately spherically symmetric. From both spectral and morphological considerations, Ref. [@Daylan:2014rsa] argues that a population of unresolved millisecond pulsars (MSP) in the relevant Galactic region is strongly disfavored. Also, as pointed out in Ref. [@Linden:2010ea], based on the population of [resolved]{} MSPs, the contribution from an unresolved population should account for less than $\sim5-10$ % of the $\gamma$-ray excess (see also [@Hooper:2013nhl]). It is apparent that a central issue to the determination of the existence of any diffuse $\gamma$-ray excess is whether or not the background model for the Galactic diffuse emission accurately reproduces the expected $\gamma$-ray emission. All recent studies reviewed above employ a diffuse Galactic model recommended by the Fermi collaboration for use with Pass 7 LAT data [@diffusemodel]. Interestingly, the Collaboration explicitly (and in bold face) discourages the use of one the most recent such model for Pass 7 reprocessed data “for analyses of spatially extended sources in the region defined in Fig. 1”, a region which includes the Galactic center region (as noted in Ref. [@Daylan:2014rsa]). While the key concern is the inclusion, in the reprocessed data background model, of sources with extension more than 2$^\circ$, it is also apparent that such background models are not designed with the purpose of establishing the existence of a diffuse emission. One of the key issues with using the diffuse model recommended by the Fermi Collaboration for the purposes of establishing a diffuse excess is the set of templates employed to reproduce the morphology of the hadronic and inverse-Compton Galactic diffuse emission. Employing gas column-density map templates to reproduce the diffuse $\gamma$-ray intensity entirely neglects the possibility of a significantly enhanced cosmic-ray abundance in the inner Galaxy, which almost certainly exists. Similarly, the inverse-Compton template is based, and sensitively depends, on specific choices for the input parameters in the [Galprop]{} code, most significantly source distribution, diffusive halo geometry and source spectrum (see e.g. [@CASANDJIAN]). Other quite relevant issues with the Fermi Collaboration recommended diffuse model have been discussed and tackled in the recent studies of Ref. [@Abazajian:2014fta] and [@Daylan:2014rsa]. These include a component of bremsstrahlung emission corresponding, and traced by, molecular gas [@Abazajian:2014fta; @Daylan:2014rsa]; a diffuse component with a density profile tracing the Milky Way Central Stellar Cluster [@Abazajian:2014fta]; and the so-called Fermi bubbles [@Daylan:2014rsa], whose intensity however quite likely deviates from the uniform-brightness assumption of Ref. [@Daylan:2014rsa]. With all the mentioned caveat in mind, in the present study we show that simple Galactic cosmic-ray models exist that naturally explain the observed excess. The origin of such cosmic rays is likely associated either with supernova remnants in the inner Galactic region, or with past activity of Sgr A\, or both. We demonstrate that there is no spectral or morphological preference for dark matter over such cosmic-ray models, whose existence in the inner Galaxy is more than plausible. Based on Occam’s razor principle, we argue that the Galactic center excess finds a much more compelling interpretation in the context of cosmic-ray models for the inner Galaxy rather than in that of dark matter annihilation. Cosmic-ray protons in the inner Galaxy ====================================== Morphological properties {#sec:morphology} ------------------------ There exist two key potential sources of cosmic rays in the inner Galaxy within the energy range relevant here: (i) supernova remnants and (ii) past activity of the central supermassive black hole associated with the radio source Sgr A\. For simplicity, we assume that both sources injected cosmic rays at the center of the Galaxy ($l=0,\ b=0$) at one or more points in time in the past. We will assume both an impulsive and a continuous injection for the sources, the former arguably more plausible for Sgr A\* or for isolated star-formation bursts, and the latter closer to what expected for a population of supernova remnants. We first feature a qualitative analytic discussion (sec. \[sec:ana\]), and we then present detailed results obtained with a full cosmic-ray propagation simulation with the [Galprop]{} package (sec. \[sec:nummorph\]). ### Analytic Estimates {#sec:ana} In the case of an impulsive source, the spatial distribution of the protons after a time $t_i$ can be approximated as follows [@aharonian]: $$\label{eq:imp} f(r)\propto\frac{\exp[-r^2/R_{\rm dif}^2(t_i)]}{R_{\rm dif}^3(t_i)},$$ where the diffusion radius $$\label{eq:rdif} R_{\rm dif}(E,t)=2\sqrt{D(E)t\frac{\exp[t\delta/\tau_{\rm pp}]-1}{t\delta/\tau_{\rm pp}}},$$ with $D(E)=D_0(E/4~\rm{GeV})^\delta$, and where $\tau_{\rm pp}$ is the approximately energy-independent proton cooling time. For timescales $t\ll\tau_{\rm pp}$, $R_{\rm dif}\approx2\sqrt{D(E)t}$, indicative of a purely Brownian process. Note that the particle spectrum clearly depends on position unless $\delta=0$, since the quantity $R_{\rm dif}$, driving the spatial dependence, depends on energy. In the narrowly peaked energy range of interest for the Galactic Center excess, such effect is, however, limited. For example, to first order and for $\delta\sim0.33$, $R_{\rm dif}\propto E^{0.16}$ which is less than a factor 1.5 difference from 10 GeV to 100 GeV. The counterpart to Eq. (\[eq:imp\]) for a continuous source is given by $$f(r)\propto\frac{{\rm erfc}[r/R_{\rm dif}]}{r},$$ where erfc is the error-function, and with the same $R_{\rm dif}$ as in Eq. (\[eq:rdif\]) but with $t$, this time, referring to the time at which the continuous cosmic-ray source started injecting particles. In the limit $r\ll R_{\rm dif}$, the cosmic-ray flux saturates to a density $\propto1/r$. For a diffusion coefficient $D(E)=D_0(E_{\rm p}/4\ {\rm GeV})^\delta$, with $D_0=6.1 \times 10^{28}\ {\rm cm}^2{\rm s}^{-1}$, we then consider a variety of impulsive and continuous sources, with ages listed in Table \[tab:sources\] along with their physical and angular diffusion radii at 2 GeV, where the GCE peaks. \[1\][>[\ ]{}m[\#1]{}]{} [C[1.5cm]{} C[1.5cm]{} C[1.5cm]{} C[1.5cm]{} C[1.5cm]{}]{} & & &\ Im1 & Impulse & .5 Kyr & 18 pc & $0.12^\circ$\ Im2 & Impulse & 2.5 Kyr & 40 pc & $0.28^\circ$\ Im3 & Impulse & 19 Kyr & 110 pc & $0.76^\circ$\ Im4 & Impulse & 100 Kyr & 250 pc & $1.7^\circ$\ Im5 & Impulse & 2 Myr & 1.13 Kpc & $7.8^\circ$\ C1 & Continuous & 7.5 Myr & 2.19 pc & $15^\circ$\ C2 & Continuous & $\gtrsim 1$ Gyr & $\infty$ & $\infty$\ In Figure \[fig:morpho\] we show the projected density of cosmic-ray protons for the putative impulsive and continuous sources listed in Table \[tab:sources\]. In particular, we show the evolution of a single impulsive source over the times from Table \[tab:sources\] as well as the continuous models C1 & C2 along with a representative superposition of impulsive sources ($\rm{Im4}+10\times\rm{Im5}$) which we will employ in what follows. The overall normalization is left arbitrary for the sake of illustration. It is crucial to note that this is the cosmic-ray proton density, and that it must be multiplied with the spatially varying target gas density in order to obtain spatial distribution of the $\gamma$-ray flux. As a guideline, we also show the prompt [$\gamma$-ray]{} flux for an annihilating dark matter candidate following an NFW profile of inner-slope $\gamma$ between 1.1 and 1.3 and scale-length $r_s=24$ kpc. These two values bracket the signal morphology resulting from the analyses of Ref. [@Abazajian:2014fta; @Daylan:2014rsa]. As we will demonstrate in the next section, the $\rm{Im4}+10\times\rm{Im5}$ and the C1 models have the correct proton densities to reproduce the GCE after convolution with the gas profile and are reshaped to closely match the $\gamma=1.3$ profile[^1]. In the plot, the shaded region indicates the angular region of interest, bounded at low angular scales ($\approx 0.25^\circ$) by the point-spread function of Fermi-LAT, and at the approximate angular scales ($\approx 12^\circ$) where statistical and systematic uncertainties currently render the excess invisible over backgrounds. It is important to note that the recent bursts (Im1, Im2, Im3 and Im4), or superposition thereof, provide highly concentrated populations of cosmic-ray protons in the Galactic center, possibly yielding a bright, centralized, and spherically symmetric $\gamma$-ray emission. ![The density of cosmic-ray protons, at an energy of 2 GeV, projected along the line of sight as a function of the angular distance from the Galactic center in the spherically symmetric analytic diffusion approximation. Shown in dotted blue lines is the evolution of an impulsive source after .5, 2.5, 19, 100, and 2000 Kyr from top to bottom. We also show our summed impulse model (thick black), a 7.5 Myr old continuously emitting source (thin black), and a stationary continuous source (black dashed). After a convolution with the gas density profile, the summed impulse and 7.5 Myr old continuous models have $\gamma$-ray flux profiles which approximately match that of an annihilating dark matter candidate following an NFW of inner slope $\gamma=1.3$ (shown in dashed red for several values of $\gamma$). The shaded region shows the angular scales which are both above the Fermi-LAT point-spread function (lower-bound $\approx 0.25^\circ$) and bright enough to be differentiated from the background (upper bound $\approx 10^\circ-15^\circ$).[]{data-label="fig:morpho"}](analytic_flux.pdf){width="0.9\columnwidth"} Note that the time-scales we employ in the present estimates are not accidental: for example, model Im5 is close to the age of the Fermi bubbles, as estimated e.g. by Ref. [@Guo:2011eg] and Ref. [@Yang:2012fy] to be around 1-3 Myr, while model Im4 is also close to another alternate age estimate for the bubbles, $4\times 10^5$ yr, obtained by Ref. [@Su:2010qj], as well as matching age estimates of $10^4-10^5$ yr for the supernovae remnant Sgr A East at the Galactic Center. Also notice that for the time-scales listed above we are never in the regime where $t_1\gg \tau_{\rm pp}$ with the exception of the stationary continuous source, where protons are replenished over the region of interest anyway. ### Numerical Simulations {#sec:nummorph} The hadronic $\gamma$-ray emission from $\pi^0$ decay traces both the density of cosmic-ray protons and the spatial distribution of the target interstellar gas. While the discussion above shows that with one or more burst injections, a variety of cosmic-ray density profiles can be obtained (including highly centrally concentrated ones), the present discussion must include the target density for hadronic inelastic processes. We note again that the template analyses of Refs. [@Daylan:2014rsa; @Abazajian:2014fta] are predicated on a uniform distribution of cosmic-ray protons, and therefore neglect any gradients introduced by sources and by a non-trivial cosmic-ray morphology in the region of interest such as those shown in Fig. \[fig:morpho\]. In order to simulate in detail the $\gamma$-ray emission from the region and to assess the role of the cosmic-ray distribution, we employ the code [Galprop v54.1.2423]{} [@galprop][^2] which provides a 3+1-dimensional numerical solution to cosmic-ray transport along with empirically calibrated semi-analytical models of atomic, molecular, & ionized hydrogen (HI, HII, H$^+$) gas in the Galaxy, in addition to a sophisticated treatment of pion production and decay. For simulations longer than 50 Kyr we employ a [Galprop]{} simulation consisting of a $10 \times 10$ kpc box centered on the Galactic plane with the x-axis defined by the Sun-GC line. The half-height along the z-direction is 4 kpc with a lattice spacing of 200 pc along each axis. For shorter simulations, the box-size is reduced to a sufficiently large cube of dimension 4 kpc with lattice spacing reduced to 50 pc. A source of cosmic-ray protons is then defined as a narrow sub-grid Gaussian localized at the Galactic center. In the case of impulsive source models, the [Galprop]{} code has been modified to inject protons in time following a $\delta$-function centered at $t=0$. Cosmic-ray transport is then solved forward in time with the [Galprop]{} code, using ‘explicit-time mode’ with step sizes of $\Delta t=10^2, 10^3$ yr for sources younger and older than 50 Kyr, respectively. As in the previous section, we assume an isotropic diffusion tensor with diagonal entries $D(E)=D_0(E/4~{\rm GeV})^{+0.33}$ and a diffusion constant $D_0=6.1 \times 10^{28}~{\rm cm^2 s^{-1}}$. For our morphological study of impulsive sources, we have explicitly verified that the diffusion constant and the diffusion time (the “age” of the source) are approximately degenerate for the quantity $D_0 t_{\rm diff}$ held constant. This is expected in the limiting case of Eq. (\[eq:imp\]) where the diffusion time is much shorter than the proton cooling timescale. In other words, holding the quantity $D_0 t_{\rm diff}$ constant will preserve the shape of the diffusion cloud, although the flux scales as $D_0^{-1}$. This implies that if the diffusion constant differs in the Galactic center our results will still hold, but diffusion timescales will change, as will the energetics in the case of a continuous source. The region of interest under consideration here extends to $\pm1.5~$kpc at 10 degrees, while the height of the diffusion zone is much larger and set to $\pm 4$ kpc. Unless this half-height is reduced to $h_{\rm dif}\lesssim 2$ kpc, variations in the height of the diffusion zone are also of negligible impact, and are thus not considered. Diffusive reacceleration is incorporated using a Kolmogorov spectrum for interstellar turbulence ($\delta_{\rm turb}=1/3$) and an [Alfvén ]{}velocity of 30 km/s. As mentioned at the beginning of this section, a crucial ingredient that a full cosmic-ray simulation allows us to test is the role of the interstellar gas distribution in predicting the morphology of the diffuse $\gamma$-ray emission. In our simulations, the interstellar gas consists of three components: molecular, atomic, and ionized hydrogen. In [Galprop]{}, the first two components are modeled as independent, cylindrically symmetric distributions of seven galactocentric rings derived from surveys of HI & CO, where the latter is used as a tracer of molecular hydrogen [@moskalenko:2002]. These surveys are then combined with distance information derived from the line-of-sight velocity distributions and Galactic rotation curves to assign a gas density to each ring as a function of height. Finally column densities from the analytic model are renormalized to agree with the survey gas column densities, breaking the cylindrical symmetry and reproducing the observed asymmetric gas structures. Using this gas model, [Galprop]{} propagates the cosmic-ray protons and convolves the resulting density with the gas model in order to produce a projected map of the $\gamma$-ray flux. The smallest resolvable scales are thus ultimately limited by the gas map resolution. In the case of HI and $H_2$, this amounts to an angular resolution of 0.5$^\circ$ and 0.25$^\circ$ respectively. Notably, the latter is approximately of the same characteristic size as the Fermi PSF above a few GeV. Within the Galactic plane, the mass fraction of ionized hydrogen is only a few percent when compared with the other two components. For the sake of comparison with the ‘inner-Galaxy analysis’ of Ref. [@Daylan:2014rsa], we focus on Galactic latitudes $\|b\|>1^\circ$ where the ionized Warm Interstellar Medium (WIM) makes up a significant portion of the diffuse $\gamma$-ray signal. In [Galprop]{}, the WIM is based on the commonly used NE2001 model of Cordes & Lazio [@cordes1; @cordes2] with scale-heights doubled to 2 kpc to ensure consistency with recent pulsar dispersion data as described in Gaensler et al 2008 [@Gaensler]. We emphasize that our gas model is nearly identical to that used to derive the hadronic component of the Fermi-LAT Collaboration’s Galactic Diffuse Model, although the Fermi diffuse model also includes inverse Compton scattering and bremsstrahlung contributions from high-energy electrons, which are not of interest in testing possible issues with the hadronic component of the diffuse emission. For a thorough description of the gas model we discussed above, see Ref. [@fermi_diffuse] and enclosed references. One difference of limited importance in our implementation of the scale-factor $X_{\rm CO}(R)$. This parameter captures the ratio between the survey-derived integrated CO line intensity and the $H_2$ column density. In contrast to the fixed value used by the Fermi-LAT team, we choose this ratio to increase as a function of Galactic radius, in accordance with the findings of Ref. [@Galprop_x_co]. As this function is nearly flat in the inner Galaxy, this change is not expected to play a significant role. The gas model and diffusion setup are now defined and we thus proceed to a morphological comparison between centralized proton sources and the measured Galactic center excess. In the analyses of Refs. [@Abazajian:2014fta; @Daylan:2014rsa], the basic features of the excess emission show an approximately spherical shape with flux approximately 3% of the brightness of the Fermi diffuse model in the central $5^\circ \times 5^\circ$ window [@Abazajian:2014fta] centered on the GC. We define three benchmark cases of interest: \(i) a continuously emitting central source of high-energy cosmic-ray protons, which has reached steady state over $\gtrsim 10^9$ year timescales, \(ii) a continuous source which was started injecting protons 7.5 Myr ago, a time-scale consistent with ages proposed for the Fermi-bubbles, and \(iii) a two-component impulsive source where protons were injected at ages of 19 Kyr, 100 Kyr, and 2 Myr, summed with free relative normalizations. In what follows, we calculate the $\gamma$-ray emission profile of our models as a function of the projected distance from the Galactic center. We then fit this profile to the GCE to determine statistical compatibility and study the remaining spatial properties. ![Projected flux density at 2 GeV as a function of from a proton source at the Galactic center. For non-dark matter lines, results are derived from a full [Galprop]{} simulation of diffusion and subsequent neutral pion decay averaged over the north + south regions with the Galactic plane ($\|b\|\pm 1^\circ$) masked out upon integration. In black we show radial flux profiles for our summed impulsive (thick), a 7.5 Myr old continuous source (thin), and a steady-state continuous source (dashed). In blue-dashed and blue-dotted we show the individual impulsive sources at 100 Kyr and 2 Myr. Finally, we show NFW profiles with inner slopes 1.3 and 1.1 in solid and dashed red. Data points are taken from Daylan et al (2014) [@Daylan:2014rsa].[]{data-label="fig:flux"}](flux_10Myr.pdf){width="\columnwidth"} In Figure \[fig:flux\] we plot the projected $\gamma$-ray flux, integrated along the line-of-sight and assuming a solar position of $r_\odot=$8.5 kpc, for each model as a function of radius from the Galactic center and compare against the ‘concentric ring’ analysis of Daylan et al (2014) [@Daylan:2014rsa] (black data-points). In practice, we use the same convention for this figure as in Ref. [@Daylan:2014rsa]: specifically, we average the line-of-sight integrated flux over circular annuli of increasing radius and a full-width of 1 degree, with the masked Galactic plane regions excluded. Also shown for comparison are NFW profiles of inner slopes 1.1 and 1.3, as suggested in Ref. [@Daylan:2014rsa]. In order to fit each model to the data, we choose normalizations using a (logarithmic) least-squares fit weighted by the (log) inverse variances of each of the nine points. We then calculate the chi-squared per 9-2 degrees of freedom. For the NFW models fit in Ref. [@Daylan:2014rsa], the normalization and slope were free parameters. For our proton source models, the normalization is allowed to vary and source ages were chosen by hand to provide a reasonable fit. Both of these parameters are included when counting degrees of freedom. In case (iii), i.e. the summed impulsive model, we do not include the 19 Kyr component since its contribution is negligible outside of the masked region (although it could be important to match the Galactic center analysis in the central few degrees, as we will show below). We then sacrifice an additional degree of freedom and allow the normalization of the 100 Kyr and 2 Myr components to float independently. Thus the summed model includes 2 ages and 2 normalizations. The energetics of the normalizations are assumed arbitrary at this point. We will explore how reasonable the resulting normalization values we infer actually are in section \[sec:SNR\] where a concrete astrophysical scenario is discussed. The profile slope of the continuous source in steady-state appears to be slightly too flat to match the observed emission and does not provide a particularly good fit to the data ($\chi^2/\rm{d.o.f.}=6.15$). However, if this emission were initiated at an age of $\mathcal{O}($5-10) Myr, the corresponding diffusion radius would be approximately 10 degrees. In this case, the resulting emission profile is significantly steepened, providing a very good fit – $\chi^2/\rm{d.o.f.}=1.31$ – compared to the $\alpha=1.3$ NFW profile where $\chi^2/\rm{d.o.f.}=1.14$. Our best fitting model is the 100 Kyr+2 Myr impulsive model with $\chi^2/\rm{d.o.f.}=1.50$. We find that for the summed impulse model, the best-fit injection luminosities have relative normalization 1:10, the larger corresponding to an event at 2 Myr. Although this precise ratio depends on the relative ages of the two components, this fact does indicate that two events with relatively comparable energetics provides good agreement with the observed excess and may indicate that events of similar nature and origin might have fueled the two cosmic-ray bursts needed to explain the observed morphology. ![image](flux_map_merged.pdf){width="2.1\columnwidth"} In Figure \[fig:skymap\] we investigate the overall spatial distribution of the emission from a new population of cosmic-ray protons injected in the Galactic center region. The Figure shows the $\gamma$-ray flux associated with a central proton source for the benchmark impulse times of 0.5, 2.5 and 19 Kyr (upper panels) and of 100 Kyr, 2 Myr and continuous (lower panels). We use a linear scale in the three upper panels to help the Reader visually compare our results with what shown e.g. in Fig. 9, right panels, of Ref. [@Daylan:2014rsa]. To the end of emphasizing the emission outside the Galactic plane, we instead employ a logarithmic scale for the older bursts and continuous sources in the lower panels. In each case, the fluxes are rescaled such that the maximum flux equals unity. The Galactic plane mask ($\|b\|<1^\circ$) is bounded by white lines (or is masked out) and reference reticles have been overlaid at radial increments of 2$^\circ$. The top three panels show that a recent (from a fraction of a Kyr to tens of Kyr) impulsive cosmic-ray proton injection event in the Galactic center region yields a highly spherically symmetric and concentrated source, with morphological properties very closely resembling and matching those found in the Galactic center analysis of Ref. [@Daylan:2014rsa] (see their Fig. 9, right panels), as well as in the GCE source residuals shown in the bottom panels of Fig. 1 in Ref. [@Abazajian:2014fta], and in the residual found in Ref. [@gordon_macias:2013] and shown in Fig. 3. As long as the injection episode is recent enough, the morphology primarily traces the distribution of cosmic-ray protons, and is relatively insensitive to the details of the target gas density distribution — the diametrically opposite regime from what assumed in the diffuse Galactic emission background models of Ref. [@Daylan:2014rsa; @Abazajian:2014fta; @gordon_macias:201]. It is evident that the sub-Myr simulations show a significant degree of spherical symmetry outside the masked regions. Also, an excess with the same morphological aspect as in in fig. 9, right panels, of Ref. [@Daylan:2014rsa] can be easily reproduced by young or very young sources, as shown in the three upper panels. As the diffusion time increases to to several Myr, the emission profile becomes more elongated and spherical symmetry is degraded. At higher latitudes ($\|b\|\gtrsim 2^\circ$), most of the spherical symmetry is, however, restored as the molecular and atomic gas distributions fall off, and the ionized component produces a more isotropic emission. In the template analyses of Refs. [@Abazajian:2014fta; @Daylan:2014rsa], a portion of this residual ridge emission may also be absorbed by the Fermi diffuse model, although it is difficult to exactly pinpoint this effect without repeating the full maximum likelihood analysis. It is also evident that gas structure is mostly washed out for recent impulsive sources, and that it becomes increasingly more prominent for older sources and for the continuous emission cases. Finally, we note that if a substantial portion of the inner excess is due to unresolved millisecond pulsars, much of the Galactic ridge would remain at a lower relative luminosity. Quantitatively examining the angular profile for each source at a variety of different radii shows that within $\pm 45^\circ$ of the north and south Galactic poles, there is a high degree of spherical symmetry with typical (positive) variations on the order of 20% with respect to the flux at Galactic north. At larger angles, however, the flux rapidly rises as one approaches the Galactic plane to values many times larger than the Galactic north flux. Although this does significantly illuminate the Galactic plane, it is unclear how important a role this plays in the analysis of Daylan et al [@Daylan:2014rsa], where spherical symmetry was tested by scanning the axis ratio of the (now ellipsoidal) dark matter template. Their analysis found a strong statistical preference in both the inner Galaxy and Galactic center analyses for an axis ratio of approximately $1:1\pm0.3$. While this template distortion does provide a simple test, its geometry is not physically motivated and does not correctly probe the bar+sphere shape expected from a central hadronic source. In Appendix C of Ref. [@Daylan:2014rsa], the authors examine the excess in two regions: north/south, defined by angles within the 45$^\circ$ of the poles, and east/west, defined as the complementary region dominated by the Galactic disk. While both regions exhibit an excess, the E/W template shows a significantly enhanced peak of the signal compared to a flatter N/S spectrum [@Daylan:2014rsa]. This seems to indicate that either the Fermi-bubbles template absorbs much of the excess N/S emission, or that the emission is, in fact, more extended along the disk, as is seen in our benchmark models with a central cosmic-ray proton source. In further testing the axis-ratio, Ref. [@Daylan:2014rsa], again, uses ellipsoidal projections of the NFW emission, this time allowing the template to rotate (there is still no test for a rectilinear disk component), finding a small statistical preference for an axis ratio of 1 to 1.3-1.4 elongated at an angle of $\approx 35^\circ$ counter-clockwise from the Galactic disk. It is possible that this component of the excess is in fact a component of an extended central molecular gas bulge, as advocated e.g. in Ref. [@gas_model], which is oriented at $\sim 14^\circ$ CCW and is not modeled by the cylindrically symmetric [Galprop]{} gas model and that, as a result, is therefore not included in Fermi Diffuse Galactic template. In Appendix 4 of Ref. [@Daylan:2014rsa] the hypothesis of an excess proton density is tested by adding an additional template based on the Schlegel-Finkbeiner-Davis dust map [@SFD]. The gas-correlated dust map is then spatially modulated so that the resulting template is given by $$\label{eq:modulation} \rm{Modulation} = SFD(\vec{r}) \times \frac{\int_{\rm l.o.s.} \rho_{\rm NFW}^2(\vec{r})}{g(\vec{r})}$$ where the NFW profile’s inner slope was scanned to maximally absorb the emission, preferring an inner slope $\gamma=1.1$. The functional form for $g(\|l\|,\|b\|)$ was then assumed to be the product of a latitudinal linear $\times$ exponential function and a longitudinal Gaussian. This function was then fit over $\|b\|<45^\circ$ and $\|l\|<70^\circ$ to also maximally absorb residuals. It was found that the modulated dust absorbed a significant component of the excess when an additional NFW template was omitted. However, when the NFW template was included in the analysis, it absorbed nearly the entire excess and the modulated dust map appears uncorrelated with the excess. It was concluded that gas-correlated emission does not provide a suitable description of the GCE. We disagree with this conclusion for the following reasons: 1. The morphology of the underlying population of cosmic-ray protons which reproduces the GCE is shown by the 7.5 Myr continuous source shown in Figure \[fig:morpho\] and is clearly very different from any of the NFW profiles shown. In the modulated dust template analysis, the functional forms chosen for $g(\vec{r})$ would need to be drastically different in order to reproduce distribution of protons matching that of Figure \[fig:morpho\]. In particular, any analysis must consider that the target gas density already falls off as one moves away from the Galactic center, and that the dust map should be initially modulated by the expected proton density, not proportionally to a projected NFW profile. For example, if one takes $g(r)=1$ in Eq. \[eq:modulation\], the resulting $\gamma$-ray template would fall off much faster than $r^{-3}$ when integrating over unmasked regions as was done for Fig. \[fig:flux\]. As can be seen in Fig. \[fig:morpho\], within the inner few degrees of the Galactic center, our 7.5 Myr continuous hadronic source would correspond approximately to a dust profile modulated by an NFW profile of inner slope $\gamma\approx 0.45$, which would then be required to steepen to more than $\gamma=1.6$ by $10^\circ$ in order to not severely overestimate the flux at large radii. 2. As seen in Figure \[fig:skymap\], the gas-correlated emission from cosmic-ray populations younger than a few hundred Kyr remains highly spherically symmetric at high latitudes. Only in substantially older sources ($\gtrsim 1~\rm{Myr}$) does the gas structure of the Galaxy become completely dominant in shaping the $\gamma$-ray morphology. In particular, this indicates that much of the dust structure lies at radii intermediate between the Earth and the Galactic center, whereas protons from a young cosmic-ray source have only reached the inner-most rings. Our simulations take this 3-dimensional structure into account using gas velocity measurements to construct a model of Galactic structure and indicate that a 2-dimensional map of the column-density simply cannot account for a non- uniform cosmic-ray density. 3. If astrophysical in nature, the residual is likely to be the result of several emission sources. A substantial emission component in the inner few degrees naturally needs to be attributed to unresolved MSPs [@Yuan:2014rca] which exhibit an approximately spherically symmetric, or slightly ellipsoidal profile (see however [@Hooper:2013nhl]). Such an addition would inevitably alter the preferred templates in the unmasked Galactic center analysis. To summarize this section, we have used the cosmic-ray propagation code [Galprop]{} to simulate the $\gamma$-ray emission associated with neutral pion decay as cosmic-ray protons from a central proton source diffuse and interact with interstellar gas. Using a gas model identical to that of the Fermi-LAT Galactic diffuse template, we studied a variety of continuous and impulsive proton injection histories. Under standard assumptions for the diffusion setup, it was shown that one can reasonably reproduce the spatial morphology of the observed Galactic center excess using source histories that are potentially correlated with past Galactic activity. Specifically, the radial flux profile can be very closely matched if a continuous proton source turned on within the past 5-10 Myr, or if two or more events of comparable energy occurred at ages of around 0.1 and 2 Myr, although these simple benchmarks only represent a few possible scenarios. The spatial distribution of these source’s $\gamma$-ray emission may be somewhat more extended along the Galactic plane compared to the observed GCE, although without repeating the full likelihood analysis, a direct comparison is difficult. Indeed, a repeated likelihood analysis using the hadronic templates derived here is key to helping rule out a hadronic origin for the GCE and will be studied in detail in follow-up work. The spatially concentrated excess found in the ‘Galactic center’ analysis of Ref. [@Daylan:2014rsa] is reproduced by young impulsive sources active from a fraction to a few Kyr ago in the center of the Galaxy, or perhaps even by efficient trapping of the 100 Kyr cosmic-ray population in sub-resolution molecular clouds at the GC. At Galactic latitudes above 2-3 degrees emission from the Galactic ridge becomes no longer dominant and at angles within $\approx \pm 45^\circ$ of the Galactic poles, our sources exhibit a very high degree of spherical symmetry while the projected gas structure is left largely unresolved relative to the steady-state Galactic diffuse model. Finally, we discussed possible correlations of the GCE with unmodeled gas components in the Galactic center as well as pointing out important issues with the modulated dust template analysis in Ref. [@Daylan:2014rsa]. In the next sections we turn to a study of the spectral characteristics of the GCE. Spectral Properties {#sec:spectrum} ------------------- Three independent recent analyses of the GCE have found spectra which share a characteristic peak near 2 GeV, with little excess emission over background either below a few hundred MeV or above $10$ GeV. Although the location of the spectral peak is relatively robust, the shape of the excess is very sensitive to the modeling of point sources in the field, with additional systematic uncertainties such as the Galactic diffuse emission, and with differing “regions of interest”, leading to a large variation in the reported low and high energy spectral slopes. While most models are relatively well fit by a hard exponentially cut-off power law for the [photon]{} spectrum (and, as a result, reasonably well fit by dark matter models), we show below that a power-law [proton]{} spectrum with a break at energies of $\approx10$ GeV also provides good fits to the excess spectrum. A crucial feature of the differential $\gamma$-ray spectrum produced through the inelastic scattering of astrophysical high-energy protons on interstellar gas, is a characteristic maximum flux at 100 MeV induced by the rapid downturn of the inclusive $\pi^0$ production cross-section below 1 GeV. Importantly, in the spectral energy distribution representation, $E_\gamma^2 dN/dE_{\gamma}$, this peak is shifted to $\approx 1 \rm{GeV}$ where both pulsar spectra and the GCE approximately peak. It is thus a remarkable and unfortunate coincidence that the claimed GCE spectrally peaks at $\approx 2 \rm{GeV}$, where the likelihood of confusion with astrophysical sources is maximal. Here, we consider three reference spectral models for the underlying cosmic-ray proton population, and thus for the resulting $\gamma$-ray spectrum. The first cosmic-ray spectrum we consider is a power-law with an exponential cutoff (PLExp) where the proton spectrum at momentum $p_{\rm p}$ is given by, $$n_{\rm p}(p_{\rm p})\sim\ p_{\rm p}^{-\Gamma}\ \exp[-p_{\rm p}/p_{\rm c}]. \label{eqn:plexpcut}$$ The second and third models have a broken power-law injection of protons of the following functional form: $$n_{\rm p}(p_{\rm p})\sim\ \left\{ \begin{array}{lr} (p_{\rm p}/p_{\rm br})^{-\Gamma_1} & : p_{\rm p}<p_{\rm br}\\ (p_{\rm p}/p_{\rm br})^{-\Gamma_2} & : p_{\rm p}>p_{\rm br}, \end{array} \right. \label{eqn:BPL}$$ where we allow the second index to be arbitrary in one case (BPL), and where we fix it to $\Gamma_2=\Gamma_1+1$ in the other (BPLFix). The BPLFix model will later be motivated by the possibility of proton acceleration by supernova remnants taking place inside dense and partially ionized molecular clouds. We then calculate the resulting $\gamma$-ray spectrum using one of [Galprop’s]{} newest models, employing the detailed low-energy parameterizations of Dermer (1986) [@dermer:1986] with interpolation to the Monte-Carlo studies of Kachelrieß & Ostapchenko (2013) which better fit available collider data at high energies [@Kachelriess:2012] (see App. \[app:pion\_decay\] for details). ![image](ML_spectra.pdf){width="2.1\columnwidth"} We then take data from the two analyses of Daylan et al (2014) [@Daylan:2014rsa], where different versions of the Fermi-LAT Galactic Diffuse Model were used to extract the GCE spectrum (using the template from the [P6v11]{} and [P7v6]{} releases, respectively), from Abazajian et al (2014) [@Abazajian:2014fta], and from Gordan and Mac[í]{}as (2013) [@gordon_macias:2013]. We perform a maximum likelihood fit for each of our three spectral models, and compute the reduced $\chi^2$ for $f=N-M$ degrees of freedom where $N$ is the number of data points and $M$=3 for PLExp and BPLFix and 4 for the general BPL. It is crucial to note that Daylan et al [@Daylan:2014rsa] do not provide an estimate of the systematic uncertainties (which are expected to be relatively large), nor do we attempt to include any such estimate. The error bars quoted in the analysis of Ref. [@Daylan:2014rsa] arise purely from counting statistics. Abazajian et al [@Abazajian:2014fta] do estimate the relative systematic error Galactic diffuse model based on variations in the spectral form chosen for the GCE, but they do not provide a specific number. Based on their Fig. 8, we estimate the error (conservatively small) as $1\times 10^{-8}$ GeV/cm$^{2}$/s, and combine this in quadrature with the statistical errors for each point. Gordan & Mac[í]{}as [@gordon_macias:2013] provide the most rigorous test of systematic uncertainties related to the Galactic diffuse model by looking at residuals as a window is scanned along the Galactic plane in regions with no contaminating point sources. This results in an estimated $\approx 11\%$ standard deviation from Fermi’s diffuse background model. However, if we combine their statistical and systematic errors in quadrature, the fit is very poorly constrained. We therefore use only systematic uncertainties (which are typically larger) for this case. Below we discuss the results of Figure \[fig:galprop\_spectra\], but one can already see from the substantial variations between the four extracted spectra that estimating the systematic uncertainties is a highly non-trivial issue. We thus urge caution when interpreting the reduced $\chi^2$ values we quote, which should be taken only as a rough indicator of fit quality. Figure \[fig:galprop\_spectra\] shows the best fits for each of the three spectral models. In the top-left panel is the Daylan et al analysis which uses the the non-reprocessed [P6v11]{} diffuse model. The excess is very well fit by the PLExp model which closely matches the prompt emission from a light dark matter candidate. The BPL model also provides an exceptionally good fit, although the pre-break index is unphysically steep, at $\Gamma_1=-0.7$ while the second index converges to a value $\Gamma_2\approx$17 with a relatively large break energy $E_{\rm br}=23.7$ GeV, effectively mimicking the PLExp model (the two lines are in fact hardly distinguishable in the figure). Of more interest is our BPLFix model, which provides a reasonable, though not optimal, fit to the data considering the underestimated error bars. The best-fit low-energy index $\Gamma_1$=2 is intriguingly equal to the canonical value $\Gamma \approx 2$ expected from the theory of linear diffusive shock acceleration (DSA) thought to drive supernovae and black-hole acceleration processes. [Note that there exist systematic uncertainties arising in the low and high-energy ranges from modeling of the inclusive $pp\to\pi^0+$ anything cross section, as is discussed in App. \[app:pion\_decay\] and Ref. [@Stecker1973a]. Such uncertainties can be as large as 15% below 1 GeV up to 40% above 100 GeV, and thus affect any conclusion of the precise values needed for the cosmic-ray proton injection spectrum.]{} The top-right panel shows the Daylan et al [P7v6]{} analysis, which includes a Fermi-LAT model of the bubbles in the Galactic diffuse template in addition to the independent Finkbeiner bubble template. Unlike the [P6v11]{} analysis, which used mismatched photon data from the [P7]{} release, this model is appropriately calibrated to the full [P7]{} event data. Compared to the [P6v11]{} analysis, this approach yields a substantial flattening of the spectrum, with all models providing equally good fits, with nearly identical $\gamma$-ray spectra. $\Gamma_1$ is found to vary between 1.65 and 2.13 and in both BPL models $\Gamma_2\approx 2.6$. The similarity between the BPL and BPLFix models is remarkable, given the significant difference in their initial spectral index. This indicates a weak spectral dependence on $\Gamma_1$ due to the natural ‘GeV-bump’ associated with pion decay. This is also observed for in the fits to the other analyses, where the initial index can have completely unphysical values $\gamma_1\gtrsim 15$ with only a very small change in the log-likelihood. Later we will show contour plots for the BPLFix model which indicate a strong covariance between the break momentum and the low-energy spectral index, and acceptable values of $\Gamma_1$ over the large range 1.25-2.5. In the bottom-left panel we show spectra taken from the full model of Abazajian et al (Figure 3), Ref. [@Abazajian:2014fta], with statistical errors added as discussed above. Even our conservative estimate of the systematic error leads to large uncertainties in the spectrum, and all of our models provide acceptable fits. Although the BPLFix model does not appear to fit the data particularly well, we encourage the reader to review Figure 8 of Ref. [@Abazajian:2014fta] where a range of GCE spectra are shown depending on the spectral model used in the likelihood fit. The data shown here is for the measured residual – as opposed to what results from a specific dark matter template – and corresponds approximately to the most strongly peaked model. The “mean model” of Fig. 8 in Ref. [@Abazajian:2014fta] has a significantly softer low-energy spectrum. The fit is also severely impacted by the asymmetrically small number of data points above the bump. Finally, in the lower-right panel we show data from Gordan & Mac[í]{}as (2013) which we found, again, to be well fit by all models, with a preference for a slightly hardened low-energy index of $\Gamma_1$=1.73 for the BPLFix model and a break energy of 13.7 GeV. Collectively, our results reveal two characteristic features: Firstly, in most cases there is a slight preference for the PLExp model; the BPL with free indices typically tend to converge towards a PLExp form. One exception is the [P7v6]{} fit from Daylan where the BPLFix model is actually preferred. The BPLFix models provide a reasonable fit throughout, with the exception of Daylan et al’s [P6v11]{} which, however, does not include any treatment of systematic uncertainties. Second, for a flat $p_{\rm p}^{-2}$ proton spectrum, the $\gamma$-radiation from $\pi^0$ decays naturally peaks at $\approx$1.25 GeV, slightly below the observed excess peak at $1.5-2$GeV. In order to shift the peak to these higher energies we prefer a slightly harder initial spectral index $\Gamma_1$ between approximately 1.6 to 2, although there is low sensitivity to this parameter. The placement of the spectral break is typically near $p_{\rm br}=10-50$ GeV and provides an effective control of the width of the spectral peak while the second index $\Gamma_2$ controls the cutoff rate as is expected from the nearly flat $\pi^0$ production cross-section above 1 GeV given in Eq. (\[eq:pion\_cross\]). The preference for a slightly hardened spectral index could arise naturally if the emission is a combination of e.g. SNR accelerated protons with index $\approx 2$ and MSP emission which can easily have Inverse-Compton spectra harder than 1.5. As an additional cautionary note, we reiterate that the theoretical predictions for the $\gamma$-ray spectra from proton-proton collisions are affected by significant systematic uncertainties associated to the modeling of the $pp\to\pi^0+$ anything production cross section. Such uncertainty feeds into the inferred spectral properties for the cosmic-ray populations associated with a given $\gamma$-ray emission. We discuss and evaluate quantitatively such uncertainties in the App. \[app:pion\_decay\]. For now, it is important to note that any conclusion on the nature of the GCE based on spectral considerations alone ought to include this source of systematic uncertainty as well. In addition to the ‘GeV bump’ feature of the pion-decay spectrum, we point out the discussion of Section 4.2.3 in Ref. [@aharonian], which describes the temporal evolution of the spectrum of a cosmic-rays which are accelerated inside a molecular cloud, where large gas densities and magnetic fields can trap low-energy protons on timescales of $10^5$ yr. For an impulsive accelerator and a cloud of very high density, high energy-protons can suffer substantial energy losses and propagate in a more rectilinear fashion, allowing escape while the low-energy protons remain inside. The cloud is thus illuminated with a spectral energy distribution peaked at a few GeV with a steepened high-energy falloff at ages greater than $10^4$ years. The low-energy index remains virtually unchanged unless the source is very young and brehmstrahlung from secondary electrons is contributing strongly. By $10^5$ yr the cloud’s peak flux decreases by 2 orders of magnitude and becomes part of the diffuse background. Although this produces gas-correlated emission that could potentially be resolved, very close to the Galactic center the spatial resolution of Fermi-LAT is limited to scales larger than about 30 pc, larger than most of the (many) molecular clumps orbiting in the central few parsecs. Such sources cannot thus be spatially differentiated from the central point source with $\gamma$-ray observations. If the escaping high-energy emission is already suppressed, as in our BPLFix model, this would appear as an additional spectral break at approximately the same energy. This very scenario may be realized at the Galactic center for the $\sim 10^4-10^5$ year old supernova remnant, Sgr A East, which we discuss in detail later. Almost certainly, molecular clouds are trapping protons at the Galactic center on scales unresolvable by Fermi-LAT and effectively reproducing the morphology of a younger source. In summary, we proposed three models for the spectrum of a new population of cosmic-ray protons which could explain the GCE: an exponentially cutoff power law, and two broken power laws with free and fixed ($\Delta \Gamma=1$) changes to the spectral index, respectively. We calculated the $\gamma$-ray spectra resulting from inelastic collisions of the protons on interstellar gas, noting that nearly all physically reasonable proton injection spectra exhibit a bump near $\approx 1$ GeV in the $\gamma$-ray $E^2 {\rm d}N/{\rm d}E$ distribution. For each model we performed a maximum likelihood fit to each of the four GCE residuals and found good fits in all cases over a broad range of parameter values. We concluded that [the core spectral features of the GCE]{} – namely a hard low-energy spectral index, a peak between 1-3 GeV, and a rapid decline above a few GeV – [can be naturally produced by an additional population of cosmic-ray protons in the inner Galaxy]{}. In the next section, we provide theoretical and phenomenological evidence that such a population is likely to exist in the Galactic center. Physical Models for the GC Excess ================================= In this section we demonstrate that the needed luminosity and spectral properties for the cosmic ray population we invoke to explain the GCE have sound physical motivations. In particular, we explain in Sec. \[sec:breaks\] how the spectral breaks in the cosmic-ray proton spectra we consider might have arisen in the Galactic center region, and related observational evidence; we then estimate in Sec. \[sec:SNR\] the energetics required by a cosmic-ray interpretation of the GCE, and argue that the time-scales and energy scales are plausible and in line with observations and theoretical expectations. A Mechanism and Evidence For GeV Spectral Breaks {#sec:breaks} ------------------------------------------------ For half a century, the bulk of Galactic cosmic rays has been thought to originate from supernova remnants (SNRs) which inject 3-30% of the total supernova energy ($\rm{E_{SN}} \approx 10^{51}$ erg) into protons and other light nuclei [@pionSNR]. A detailed theory of diffusive shock acceleration is still incomplete, but simplified linear models predict that supernova shocks propagating through an ionized gas precursor can accelerate protons and other nuclei up to $10^{15}$ eV with a resulting proton spectrum of $p_{\rm p}^{-2}$ at the source [@blandford1987]. When combined with sophisticated models of nuclear propagation through the Galaxy and solar system, this source spectrum successfully reproduces the locally measured spectrum of cosmic-ray nuclei. Direct confirmation of this acceleration model was provided only very recently (2013) by the Fermi-LAT collaboration following the detection of $\gamma$ radiation characteristic of $\pi^0$-decay in association with two known SNRs, IC443 and W44 [@pionSNR]. In order to postulate a viable astrophysical model for the Galactic center residual – i.e. without invoking new particle physics – we require either a substantial reduction in the $10^{15}$ eV high-energy cutoff, or a strong spectral break near $\approx$ 10 GeV which renders the signal invisible over that of the diffuse sea of background cosmic-rays where the $\gamma$ spectrum is roughly $\propto E_{\gamma}^{-2.7}$. In what follows, we describe recent proposals that modify the canonical theory of DSA in the presence of dense molecular clouds which surround the inner Galaxy, as well as actual realizations of this scenario as seen in recent Fermi SNR observations showing significant breaks at $\mathcal{O}(10$ GeV) in the underlying proton spectrum. It is thus possible to provide a natural explanation for the spectrum, energetics, and morphology of the GCE requiring only the assumption of an enhanced central supernova activity over the past few million years. In DSA, shock waves propagating through ionized interstellar medium compress the plasma and transfer kinetic energy downstream through either two-body collisions, or through collective electromagnetic effects if the collision cross section is very small. In the compressed zone preceding the shock front, resonant scattering of [Alfvén ]{}waves efficiently accelerates particles until their gyro-radius $r_g=cp/(eB)$ exceeds the width of the shock layer [@Drury1983]. While this test particle case assumes a fully ionized cosmic-ray precursor, the Galactic center is only partially ionized, with well over 80% of the gas content associated with neutral molecular hydrogen in the inner 200 pc, which completely engulfs the region of central starburst activity. Malkov, Diamond, and Sagdeev [@Malkov2005; @Malkov2011] demonstrated that when the upstream edge of supernovae shocks interact with molecular clouds, ion-neutral collisions effectively damp a range of otherwise resonant [Alfvén ]{}waves, severely deteriorating particle confinement within a slab of momentum space, and steepening the spectral index of protons by precisely one at an energy given in Ref. [@Malkov2005] as $$p_{\rm br}/m_{\rm p} c \approx 16 B_\mu^2 T_4^{-0.4}n_0^{-1}n_i^{-1/2}, \label{eqn:cutoff_energy}$$ where $B_\mu$ is the magnetic field strength in units of $\mu G$, $T_4$ is the temperature of the ionized precursor in units of $10^4~K$, and $n_0$, $n_i$ are the neutral and ionized gas density given in in units of ${\rm cm^{-3}}$, respectively. Similar developments in non-linear DSA have shown that over 1-10 GeV the spectrum can be as steep as $E_{\rm p}^{-4}$ depending on the shock speed and environment, flattening out again above a few TeV [@Blasi2012]. The mechanism described above successfully reproduces at least 6 of the 16 current Fermi-LAT observations of SNRs [@fermi_snr1; @fermi_snr2; @fermi_snr3; @fermi_snr4; @Dermer2013c; @fermi_snr5], although the uncertainties associated with estimating the relevant environmental parameters are considerable. The 10 remaining observations have not yet incorporated this model into the analysis. In Ref. [@fermi_snr3], several SNRs observed by Fermi were shown to be interacting with molecular clouds based on radio observations of 1720 MHz OH maser emission, providing a strong indication of shocked $H_2$. The spectra were then reproduced by fitting the underlying proton distribution according to an exponentially cutoff power-law, as we do above. SNRs interacting with highest density clouds were found to have low cutoff energies and hard proton spectra with \[$\Gamma ,E_{\rm c}]=$ \[1.7,160 GeV\] and \[1.7,80 GeV\] compared to the low-density cases, where \[2.4,1 TeV\] and \[2.45,1 TeV\]. For another SNR, W44, an independent analysis found that the $\gamma$-ray emission was well fit by a hard proton spectrum of index between 1.74 and 2 with a cutoff at $p_{\rm c} \approx$ 10 GeV/c [@fermi_snr2]. While these examples provide a representative sample of the expected range for the low-energy spectral index and cutoff energies, we do not necessarily expect a hardened spectrum to be correlated with high gas densities. These SNR spectra match the $\gamma$ radiation expected from an exponentially cutoff proton spectrum quite well, possibly indicating that the theory of Ref. [@Malkov2005] is underestimating the true breaking strength due to ion-neutral damping, or that an additional cutoff mechanism is at play. In either scenario, a more pointed spectral peak is predicted, and as a result the fit to the residual GCE spectrum in Section \[sec:spectrum\] is generally improved. The Galactic center hosts a zoo of high-energy astrophysical sources including several SNRs, resolved & unresolved pulsars, pulsar wind nebulae, and the central black hole Sgr A\. Most notably Sgr A East is a $\sim 10^4-10^5$ year old and 10 pc wide SNR rapidly expanding into the molecular cloud M–0.02–0.07, where a half-dozen sites show also show the 1720 MHz maser emission from shocked $H_2$ [@Yusef:1996]. This complex encompasses the central black hole with most of the structure residing within a few parsecs from Sgr A\ ($\lesssim 0.05^\circ$). This separation is too small to be spatially resolved by Fermi-LAT, which has a maximal angular resolution of about a quarter degree, hence it will appear as a point source, perhaps with minor spatial extension, whose spectrum cannot be differentiated from additional Galactic center sources[^3] An especially intriguing candidate for the recent injection of cosmic-ray protons in the inner Galaxy is Sgr A East. As an estimate of the expected flux from Sgr A East, we utilize a similar object, SNR W44. The latter is observed to have a differential flux of $\approx 1.25 \times 10^{-7}$ GeV/cm$^2$/s. Multiplying by the square of the distance ratio $d^2_{\rm W44}/d^2_{\rm GC}\approx(2.9~\rm{kpc/8.3~kpc})^2$ we obtain a flux of $5 \times 10^{-8}$ GeV/cm$^2$/s, precisely in line with the GCE residual and the Sgr A\* flux reported by Abazajian et al within a $1^\circ \times 1^\circ$ box centered on the GC [@Abazajian:2014fta]. (Note that the the two Daylan et al fluxes reported in Figure \[fig:galprop\_spectra\] are normalized by the solid angle of a thin annulus at $5^\circ$ from the GC). It remains to be assessed whether the spectral break energy near the Galactic center is compatible with the the results of Section \[sec:spectrum\], and whether a reasonable supernova rate is compatible with the observed flux. The environment of Sgr A East has been studied in detail at radio and X-ray wavelengths. Unfortunately, the complicated structure and rapid gradients in density, temperature, and magnetic field strength imply that there will be no single prediction for the spectral break energy predicted by Equation (\[eqn:cutoff\_energy\]), but, rather, a range of values dependent on the particular properties of the shocked region. Here we expect that nearly all of the supernova activity will take place very close to the Galactic center, with conditions not far removed from those of Sgr A East. The goal of the current study is to determine whether the conditions can plausibly reproduce the GCE, while a detailed environmental model and statistical treatment of uncertainties is reserved for future work. ![image](PLExp_contours.pdf){width="2.2\columnwidth"} The Central Molecular Zone (CMZ) is a large elliptical cloud with a gas mass fraction dominated by molecular hydrogen. It is thin and aligned with the Galactic disk, extending to a radius of approximately 150 pc from the Galactic center when projected along the line of sight[^4]. This cloud makes up 5-10% of the total Galactic molecular gas and is comprised of dense clumps of $H_2$ as well as of a lower density ambient component which completely fills the acceleration volume for any centralized SNR. In the inner 15 pc, typical densities can vary from the ambient value of $10^2~\rm{cm^{-3}}$ up to the dense molecular clouds at $10^5~\rm{cm}^{-3}$ [@Coil:2000; @gas_model], occasionally reaching even higher densities. The warm ionized hydrogen is significantly more extended and provides the precursor for shock acceleration. There is only weak power-law dependence of the break momentum on the density and temperature of the ionized component ($n_i^{-0.5}$ and $T^{-0.4}$). Both of these components are reasonably well measured in the Sgr A\* region using X-ray observations with ion densities near $10^3~\rm{cm^{-3}}$ and very hot plasma temperatures of $10^7$ K [@Galactic_center_environment]. The most important, and also the most uncertain factor in determining the break momentum, is the magnetic field strength in the shock propagation region. Zeeman splitting of OH molecules provides a measurement of the magnetic field strength along the line of sight, and indicates very strong fields in the large non-thermal radio filaments and possibly molecular clouds which can be as high as 1-4 mG [@Yusef:1996; @Ferriere2009] while Faraday rotation measurements indicate that the surrounding medium can be somewhat lower with a strength down to several hundred $\mu$G. For an extensive review of magnetic fields in the Galactic center, we point the Reader to Ref. [@Ferriere2009]. Efficient trapping of very low energy precursors in the very dense molecular clouds implies that these will be the primary acceleration sites for the resulting high energy cosmic-ray population, although a fraction will still originate from the surrounding lower density and lower magnetic field regions. In this case, the lower densities of the ionized and molecular components partially cancel the effect of the smaller magnetic field on the break momentum, but some broadening of the spectral peak may be expected toward lower energies. In order to estimate the range of break momenta achievable at the GC, we simply fix the least sensitive parameters to typical values, and set $n_i=10^3~\rm{cm^{-3}}$, $n_0=10^4~\rm{cm^{-3}}$, and $T=10^7$ K, while varying of $B$ between 0.5 mG and 4 mG. Doing this provides a break momentum between 0.79 and 51 GeV/c with a nominal value of 12.7 GeV/c for a 2 mG field strength. Without more accurate measurements and high-resolution 3-dimensional models of the Galactic center environment, it is extremely difficult to definitively compute the resulting cosmic-ray spectrum. If, in fact, these large magnetic fields are contained strictly to non-thermal radio filaments, or are much weaker then previously thought, as suggested in Ref. [@YusefZadeh:2012nh], the predicted momentum break would be significantly smaller, and the breaking mechanism would be disfavored as an explanation for the GCE. It is also very likely that current conditions at the Galactic center differ substantially from those of 1-10 Myr ago especially if the Fermi bubbles formed on comparable timescales. Compounded with uncertainties in non-linear DSA in the presence of ion-neutral damping, a conclusive statement is currently not possible. Nonetheless, the observation of break energies from ten to several hundred GeV in nearby SNR indicates that such scenarios are not uncommon, and provide evidence that the description advocated above is not unrealistic. In Figure \[fig:contours\] we show confidence intervals for the low-energy spectral index and break energy for the BPLFix and PLExp models of the proton spectrum as fitted to the two Daylan et al GCE residuals as well as that extracted by Gordon & Mac[í]{}as. We do not show the results of the fits to the Abazajian et al results due to the previously mentioned asymmetry in the number of points below and above the spectral peak which forces a very hard spectrum that clearly does not fit the rapid falloff above 2 GeV seen in the other datasets. While the residual found by Abazajian et al is indeed very hard at low energies, when an additional GCE template and spectral form is included as part of the fit, the low-energy index softens significantly becoming very similar to the other analyses. This behavior is clearly delineated in Fig. 8 of Ref. [@Abazajian:2014fta] and the enclosed discussion. In the left panel, the shaded regions along the x-axis show the range of the low-energy proton index which are compatible with Fermi-LAT observations of SNRs interacting with molecular clouds taken from Refs. [@fermi_snr2; @fermi_snr3], highlighting the canonical index $\Gamma_1=2$ predicted by linear DSA. In the shaded y-axis regions, we show expectations for the position of the spectral break in conditions typical of the very dense molecular clouds (dark cyan) and in the ambient lower density environment (darker+lighter cyan). It is promising that these contours are fully compatible with one-another when fitting to the BPLFix model. Clearly, if one assumes the BPLFix model, the parameter values are in line with those expected from SNR interacting with molecular clouds in the Galactic center. In the right panel we show similar regions shaded along the x-axis representing the range of the spectral indices compatible with Fermi-LAT observations where fitting the underlying proton spectrum used an exponentially cutoff power-law model [@fermi_snr1; @fermi_snr4; @fermi_snr5]. Although these studies also indicate GeV-TeV scale cutoff energies, it is unclear how such cutoff scales should change in the Galactic center environment without a theoretical understanding of the cutoff mechanism itself. In contrast to the BPLFix model, a PLExp spectrum reveals less compatibility among the three GCE residuals, with the main [P6v11]{} analysis of Daylan et al requiring an unphysically hard spectral index. Interestingly, two of the GCE datasets show a rapid upturn in the contour as the spectral index rises above $\Gamma=2$. In this region, the fit is almost completely insensitive to the cutoff energy up to at least $\approx 10$ TeV. Notably, a spectral index softer than 2 is commonly invoked when modeling radio and $\gamma$-ray emission from AGN in the context of hadronic injection. Although the relatively low momentum cutoff would still need to be explained, the insensitivity here could allow for a variety of possibilities, and warrants additional study. To summarize, we find that the occurrence of a break in the spectrum of cosmic-ray protons in the specific environment of the Galactic center is well-motivated. Observations of the $\gamma$-ray spectrum of several SNR with the Fermi LAT point to cosmic-ray proton spectral features aligning precisely with those needed to fit the spectrum of the GCE; the location of a spectral break in the accelerated cosmic-ray protons in the presence of dense molecular clouds in the inner Galaxy also falls squarely in the range that optimally fits the inferred $\gamma$-ray spectrum of the GCE. We thus conclude that the spectra we invoked to fit the GCE are well motivated by both theory and observation. SNe Rates and Starburst Histories {#sec:SNR} --------------------------------- In this section we explore the energetics required to produce the GCE with cosmic-ray protons injection at the center of the Galaxy. In the previous section, we showed that the flux measured from SNR W44 corresponds to the approximate luminosity needed to explain the GCE in the inner Galaxy. At radii larger than 1 degree, the GCE signal decays rapidly as shown in Fig. \[fig:flux\]. In Section \[fig:flux\] we showed that such a radial flux profile could be achieved rather naturally by the diffusion of protons injected at the Galactic center in several different episodes – for example, impulsive injection over 2-3 different epochs ($\approx 10^4, 10^5,\ \rm{and}\ 10^6$ yr) or continuously if the source was turned on around 7.5 Myr ago. Previously, we ignored the normalization of the flux and were only concerned with the relative normalization of the summed impulsive models This revealed that the 100 Kyr + 2 Myr summed model preferred relative normalizations of, respectively, 1:10. The energetics of these long-timescale events is more constrained than for more recent outbursts. We compute the $\gamma$-ray flux due to protons assuming a nuclear injection spectrum of index $\Gamma_1=2$ breaking to $\Gamma_2=3$ at 10 GeV. We find that the $100$ K and $10^6$ summed impulsive model requires a total injection of $\mathcal{O}(10^{52})$ erg into protons with energies above 100 MeV in order to produce flux compatible with the GCE consistent with the very recent findings of [@Yoast-Hull2014]. For continuous sources only a few million years old, the required energy is approximately $10^{38}$ erg/s, or a few $10^{48}$ erg/century, while continuous sources in steady-state are an order of magnitude less and comparable to the rates needed to maintain the current molecular gas temperatures near the Galactic center [@Yusef-Zadeh2012]. Stellar densities at the Galactic center are extremely high rising from a mass density of $10^4$ $M_\odot/{\rm pc}^{-3}$ at a radius of 10 pc to over $10^6$ $M_\odot {\rm pc}^{-3}$ in the central parsec (compared to the local density $\ll 1~M_\odot/{\rm pc}^3$). Measurements of the infrared luminosity near the Galactic center provide an indirect probe of the star formation rate. If this has not changed dramatically over short stellar evolution timescales ($10^8$ yr), the expected supernova rates are 0.01-0.1 per century [@Crocker2010] each injecting $\epsilon_{p} 10^{51}$ erg where $\epsilon_p$ is the fraction of the supernova energy channeled into proton acceleration, often taken to be near $0.1$ [@aharonian]. This implies an average continuous injection rate of $10^{48}-10^{49}$ erg/century, compatible with the observed excess signal. For impulsive sources, the same value of $\epsilon_p$ would require bursts of 10-100 supernovae to occur within a timescale relatively short – $10^4$ to $10^5$ yr – with respect to the diffusion timescale. While any realistic scenario would likely be an admixture of continuous and burst-like injections, the supernova rates required to reproduce the observed GCE flux in either case are well within the possible histories of the Galactic center Region. It is notable that the orbital time period for a typical molecular cloud at a radius of 1 pc is $10^5$ years providing ample opportunity for interactions with other clouds, or with the accretion disk surrounding the central black hole [@1993ApJ...408..496M]. Alternatively, this could be taken as possible evidence of intense supernovae or Sgr A\* activity several million years ago in which shocked molecular clouds became highly compressed, initiating star-formation. Supernovae bursts have also been proposed as a driver of the Fermi bubbles on Gyr timescales [@Crocker2010] and as a mechanism to explain the extremely hot plasma temperatures in the Galactic center where gas in excess of up to $10^8$ K are observed, hotter than the Galactic escape energy, implying extraordinary energy injection event(s) with total energy $10^{53}$ erg and a lifetime of order $10^5$ yr in order to remain contained near the Galactic center [@Galactic_center_environment]. Such extreme events have comparable timescales and energetics to produce the scenarios explored earlier. Another possibility which has been previously considered is the injection of protons directly from the central black hole [@2011ApJ...726...60C; @2012ApJ...753...41L; @2011PhRvD..84l3005H]. Our morphological analysis of Section \[sec:morphology\] is substantially blind to the spectrum over the very narrow energy range under consideration. Spectrally, the situation is more difficult as such low-energy cutoffs in the proton spectrum do not seem typical of active galaxies[^5]. It is possible that a yet unknown mechanism is responsible for producing a cutoff proton spectrum from Sgr A\. Such a scenario was in fact considered in Ref. [@2005ApJ...619..306A]. In this case, the black hole is taken to be in a quiescent state with a very hard proton spectrum $\Gamma=1$ exponentially cut off at 5 GeV. Secondary electrons produced in the hadronic interactions are of low enough energy to preserve their spectral shape and emit very hard infrared and millimeter synchrotron spectra, matching radio observations of Sgr. A\. In such a scenario, the soft protons could diffuse to large radii while the hard synchrotron emission would be confined to the confined to the ultra high magnetic fields in the immediate vicinity of the central black hole. A very recent result [@Yoast-Hull2014] examined the compatibility of radio and GeV/TeV $\gamma$-ray observations with predictions from two models of starburst galaxies based on the interactions of cosmic-rays (of supernova origin) in the Central Molecular Zone. Particularly careful attention was paid to the relevant astrophysical parameters, which are fully consistent with what we employed here. For each model, the average magnetic field strength, convective wind speed, ionized gas density, and free electron absorption fraction were allowed to vary in order to find optimized fits to data. The results strongly favor ion densities between 50 and 100 cm$^{-3}$ and magnetic fields between 100 and 350 $\mu$G throughout the entire ambient CMZ cloud. While radio and TeV observations are well fit, a GeV excess still persists. The addition of additional populations of either protons or electrons is then considered. In the case of protons, a soft spectral index $\Gamma\approx 3.1$ and a supernova rate enhanced by a factor $\sim 100$ are found to be consistent, but are dismissed from further analysis based on the required SNe rate. For electrons, the energetics are more compatible, but the spectral indices predicted for radio and $\gamma$-rays are found to be inconsistent with observations. We note that in our analysis, this is precisely the proton spectral index we predict above a $\sim 10$ GeV and that the required SNe rate is substantially reduced due to our much harder $\Gamma=2$ low-energy spectrum (which also matches the GeV excess in much greater detail than what considered in Ref. [@Yoast-Hull2014]). We find it remarkable that a completely independent analysis of the conditions required to fit starburst models to observations of the CMZ can naturally motivate an SNR explanation for the GCE at such a detailed level. To summarize, in this section we have presented observational and theoretical evidence for spectral breaks in the cosmic-ray spectrum when protons undergo diffusive shock acceleration by supernovae remnants which are inside or strongly interacting with partially-ionized molecular cloud complexes. This ion-neutral damping mechanism then predicts a break in the power-law index $\Gamma$ of precisely $\Delta\Gamma=$1 occurring at an energy parametrized by the local magnetic fields, ion/neutral number densities, and the temperature of the ionized precursor. We then discussed the conditions in the Galactic center environment needed to produce plausible break energies which were found to be of the correct order to explain the GCE if magnetic fields in the acceleration region are of approximately mG strength. Allowing the spectral index and break energy to float, we presented confidence contours for our fit to the Galactic center excess and showed that the preferred parameter space is spectrally compatible with an interpretation in terms of protons originating from GC SNR. Next, the energetics required to match the GCE were calculated, finding that each impulsive event requires tens to hundreds of supernovae (total energy $10^{52}$ erg) to occur on timescales somewhat smaller than the age of the outburst, or that quasi-continuous sources inject protons at a rate of order $10^{38}$ erg/s. Finally, we discussed evidence for sporadic increases in star-formation and supernovae rates in the Galactic center on the timescales relevant explain the extension of the GCE in terms of cosmic-ray diffusion and subsequent $\gamma$-rays of hadronic origin. Very large uncertainties plague each step of such an analysis and estimate. Nonetheless, the combination of spectral compatibility along with reasonable energetics and a plausible Galactic history provide a crucial background to any analysis of $\sim$GeV $\gamma$-ray data at the Galactic center. Discussion and Conclusions ========================== We presented a case for high-energy cosmic-ray protons injected in the Galactic center region as a plausible explanation to the reported Galactic center $\gamma$-ray excess over the expected diffuse background. Our study focused on whether such an explanation meets the required (i) morphology, (ii) spectrum and (iii) energetics. We demonstrated that cosmic rays injected on the order of a mega-year ago explain the observed spherical symmetry reported from the “inner Galaxy” analysis of Ref. [@Daylan:2014rsa], while a more recent (on the order of a few kilo-years old) episode would possess the same morphology obtained for the innermost portions of the Galaxy in the “Galactic center” analysis of Ref. [@Daylan:2014rsa] We showed that the $\gamma$-ray spectrum predicted by cosmic-ray proton energy distributions responsible for the emission observed from supernova remnants (such as broken power laws with specific spectral indexes, and exponentially suppressed power laws) provide excellent fits to the observed Galactic center excess. We pointed out that the preferred range for the break of the power law and for the spectral indexes inferred from the observed excess fall squarely in the ranges inferred from observations of supernova remnants, as well as in the general range expected from theory considerations. We also pointed out the importance of systematic effects in spectral reconstruction due to hadronic cross sections impacting the predictions for the $\gamma$-ray spectrum from inelastic proton-proton collisions. Finally, we inspected the time-scales, spectrum and energetics we invoked to reproduce the morphology and spectrum of the Galactic center excess in the context of one or more additional populations of cosmic-ray protons in the region. We demonstrated that the existence of such populations is motivated by a variety of observational and theoretical reasons, which we reviewed in detail. In conclusion, with the present study we gave proof of existence of a well-motivated alternative to dark matter annihilation or milli-second pulsars as an explanation to the reported Galactic center $\gamma$-ray excess. Our results indicate that conclusively claiming a signal of New Physics from $\gamma$-ray observations of the inner regions of the Galaxy must contend with a variety of additional astrophysical processes. In particular, we highlighted that one or more previously unaccounted-for populations of cosmic-ray protons in the Galactic center could potentially produce a $\gamma$-ray emission with a spectrum, morphology and intensity closely resembling those of the Galactic center $\gamma$-ray excess. Acknowledgments {#acknowledgments .unnumbered} =============== We thank Andy Strong, Amy Furniss, Roland Crocker, Gudlaugur J[ó]{}hannesson, and Tim Linden for very helpful discussions. EC is supported by a NASA Graduate Research Fellowship under NASA NESSF Grant No. NNX13AO63H. SP is partly supported by the US Department of Energy, Contract DE-FG02-04ER41268. Uncertainties on $\pi^0$ Emissivities {#app:pion_decay} ===================================== The $\gamma$-ray emissivity $q_\pi(E_\pi)$ of secondary neutral pions produced through inelastic scattering of cosmic-ray protons on interstellar hydrogen is given by the following expression: $$q_\gamma(E_\gamma)=2\int_{E_{\rm min}}^\infty\frac{q_\pi(E_\pi)}{\sqrt{E_\pi^2-m_\pi^2}}{\rm d}E_\pi,$$ where $E_{\rm min}=E_\gamma+m_\pi^2/(4E_\gamma)$ and the neutral pion production term,$q_{\pi}$, is defined by, $$q_\pi(E_\pi)=4 \pi n_{\rm H} \int_{m_{\rm p}}^\infty j_{\rm p}\left(\sqrt{E_{\rm p}^2-m_{\rm p}^2}~\right) \frac{\rm{d}\sigma_{{\rm pH}\to \pi^0}(E_{\rm p}, E_\pi)}{{\rm d} E_\pi}~{\rm d}E_{\rm p},$$ with $n_{\rm H}$ the target hydrogen gas density, $\sigma_{{\rm pp}\to \pi^0}$ the inclusive $\pi^0$ production cross section (p + p$\to \pi^0 +$ anything), and $j_{\rm p}(p_{\rm p})$ the cosmic-ray proton density as a function of the proton momentum, following recent results from Ref [@Dermer2012]. Note that many references use instead a proton spectrum following $E_{\rm tot}$ rather than $p_{\rm p}$ or kinetic energy $T_{\rm p}$. Although these asymptote to each other at $E\gg m_{\rm p}$, the assumption can have a non-negligible impact on the low-energy $\gamma$-ray spectrum for soft proton spectra $\Gamma \gtrsim 2.5$, where the low-energy protons contribute heavily. Since the cross-section falls off very rapidly below 1 GeV, this is negligible for the harder spectra of interest here. Remarkably, this cross-section is still not known to better than $\pm$10-20% near the pion production threshold of $T_{\rm p}=280$ MeV up to a few GeV, resulting in an important systematic uncertainty when using the $\gamma$-ray spectra to probe the underlying spectrum of nuclear cosmic-rays, or [vice versa]{} as is the case here. Until improved laboratory measurements are made available this remains a limiting factor in determining the global spectrum of diffuse Galactic protons using Fermi-LAT photon data [@Dermer2013; @Dermer2013a]. In this Appendix we demonstrate the systematic variations between four common models of the pion emissivity. The first model we consider is the simple $\delta$-function approximation for the cross section[@aharonian] as parametrized in Ref. [@Gaisser1990]; we then consider the three numerical models implemented in [Galprop]{}, which use cross-sections from Kamae et al (2006) [@Kamae2006], Dermer (1986) [@dermer:1986], and the model used throughout this paper: a combination of Dermer (1986) near threshold and interpolated to Kachelrieß & Ostapchenko (2013) at higher energies [@Kachelriess:2012], hereafter DKO. $$\begin{aligned} q_\pi(E_\pi) &= \frac{n_{\rm H}}{\kappa_\pi}\sigma_{\rm pp}^{\rm inel}\left(m_p+\frac{E_\pi}{\kappa_\pi}\right)\\ &\times j_{\rm p}\left( \sqrt{\left(m_p+\frac{E_\pi}{\kappa_\pi}\right)^2-m_{\rm p}^2}\right),\end{aligned}$$ $$\sigma_{\rm pp}^{\rm inel}(E_{\rm p})\approx (34.3 + 1.88 L +0.25 L^2) \left(1-\left(\frac{E_{\rm th}}{E_{\rm p}} \right)^4\right)^2, \label{eq:pion_cross}$$ ![Model variations in the $\gamma$-ray spectral energy distribution for cosmic-ray proton spectra following (in black) a broken power-law spectrum with $\Gamma_1=2, \Gamma_2=3$, and $p_{\rm br}=30$ GeV (see Eq. (\[eqn:BPL\])) and, in red, a flat power-law of index 2.82 representative of the ‘sea’ of Galactic protons. []{data-label="fig:pion_decay"}](pion_comparison.pdf){width="\columnwidth"} Below a few GeV, light hadronic states decaying through $\pi^0$’s provide the main contribution, primarily from the $\Delta (1232)$ resonance. As the proton energy increases, heavier resonances become more important as well as secondary photons from $\eta$ decays. The Dermer model includes the $\Delta(1232)$ using Stecker’s isobar model [@stecker1971cosmic] at low-energies with linear interpolation between 3 and 7 GeV to the scaling model of Stephens and Badhwar [@stephens]. At higher energies, however, this model violates the Feynman scaling hypothesis, where $E \rm{d}\sigma/\rm{d}^3p$ becomes independent of the center of mass energy $s$ for $s\gg m_{\rm p}^2$. Kamae et al [@Kamae2006] instead relies on parameterizations of Monte Carlo simulations in addition to corrections for the $\Delta(1232)$, the $N(1600)$ cluster of resonances, diffractive processes, non-scaling effects, and scaling violations which provides a better fit to high-energy observations than Dermer. The mixed DKO [@Kachelriess:2012] model used in this paper combines simulation/parametrization approaches by interpolating to results from event generator QGSJET-II at energies above $30$ GeV providing a better fit to available high-energy collider data. When fitting a proton spectrum to $\gamma$-ray data, Dermer provides the best fit below 1 GeV, but underestimates the higher-energy spectrum. Kamae et al has the opposite behavior, matching above 1 GeV, but overproducing photons below. The mixed model provides good fits in both regimes, and hence is the model of choice here. In the top panel of Figure \[fig:pion\_decay\] we show in black the $\gamma$-ray spectrum resulting from the fixed broken power-law (BPLFix) model of Eq. \[eqn:BPL\] with $\Gamma_1=2, \Gamma_2=3$, and $E_{\rm br}=30$ GeV as well as the background Galactic protons in red following a flat power law with index $\Gamma=2.82$ for each of the four models. Note that the relative normalization for each of the [Galprop]{} models is correct while the $\delta$-function case renormalized to match DKO at 2 GeV. In the lower panel we show the fractional variation in the spectral energy distributions of each model with respect to DKO. The two most important factors for an analysis of the Galactic center excess are the position and width of the spectral peak. The models which include the detailed low-energy characterization of Dermer produce the sharpest peak while that of Kamae et al is slightly broadened and peaks at 50% higher energy for the BPLFix model. This implies that the $\pi^0$ spectrum using Kamae et al requires slightly softer low and high-energy spectral indices than those of Dermer in order to match the GCE with a broken power law proton spectrum. Of general interest, but less importance to our analysis is the significant variation in the predicted Galactic background spectrum, where two distinctive peaks are seen in Dermer models compared to only one in the other two. [^1]: The other values of $\gamma$ shown will be used in a later discussion of dust template modulation. [^2]: Available at <http://sourceforge.net/projects/galprop/> [^3]: For reference, the template analysis of Daylan et al, which uses large photon statistics and an event selection which optimizes PSF, finds the most likely position for the GCE to be centered within about 3 arcmin of Sgr A\. The next generation of ground-based $\gamma$-ray telescopes is likely to resolve these structures at energies above 50 GeV. [^4]: Interestingly, the same gas model in Ref.[@gas_model] finds a large gas bulge extending to 450 pc which is rotated 13.5$^\circ$ CCW from the Galactic plane when projected along the line of sight with an axis ratio of 3:1. Daylan et al found a slightly preferred fit at roughly an angle of $35^\circ\pm$ CCW with an axis ratio of $1:1.4\pm .3$, possibly indicative of gas correlated emission. [^5]: Although many AGN spectra do have breaks in the $\gamma$-ray spectrum near 5 GeV, this results from absorption in the so-called ‘broad-line region’ within a few hundred Schwarzschild radii of the central black hole and does not provide a viable mechanism for extended* emission peaked in the GeV range.
--- abstract: \| We develop a link between degree estimates for rational sphere maps and compressed sensing. We provide several new ideas and many examples, both old and new, that amplify connections with linear programming. We close with a list of ten open problems. [AMS Classification Numbers]{}: 32H35, 32M99, 32V99, 90C05, 94A12. [Key Words]{}: rational sphere maps; CR complexity; proper holomorphic maps; compressed sensing; linear programming. In memory of Nick Hanges. address: - 'Dept. of Mathematics, Univ. of Illinois, 1409 W. Green St., Urbana IL 61801' - 'Dept. of Mathematics, Harvard ' - 'Dept. of Mathematics, Oklahoma State Univ.' author: - 'John P. D’Angelo' - Dusty Grundmeier - Jiri Lebl title: 'Rational sphere maps, linear programming, and compressed sensing' --- Introduction ============ This paper aims to make a surprising and powerful link between a basic problem in CR Geometry and the notion of compressed sensing from applied mathematics and statistics. First briefly consider the following question. Suppose that ${p \over q}$ is a rational mapping of degree $d$ that maps the unit sphere in the source ${\mathbb C}^n$ to the unit sphere in the target ${\mathbb C}^N$. Given $d$, what is the minimum possible value of $N$? Equivalently, given $N$, what is the maximum possible value of $d$? This problem remains open, but in the special case when $q=1$, ($p$ is a polynomial map), and the components of $p$ are orthogonal ($p$ is a monomial map), it has been completely solved. The method of solution involves combinatorial graph theory in addition to complex variable theory, and it does not seem to have been used outside of this problem. We modestly hope that this method will have applications in many other circumstances, so we now describe the link with compressed sensing. Consider an underdetermined linear system of equations $T{\bf x} = {\bf b}$ in finitely many unknowns. The general solution can be written ${\bf x}_0 + {\bf y}$, where $T({\bf x}_0) = {\bf b}$ and ${\bf y}$ is an arbitrary element of the null space of $T$. In [compressed sensing]{}, one seeks a solution ${\bf a} = (a_1,..., a_n)$ with as many components $a_j$ equal to $0$ as possible. In the language of Donoho (\[Do1\]), one wishes to minimize the $L^0$ norm $\\|{\bf a}\\|_0$ of the solution. While not a norm, this name for the number of non-zero components is clever, since $\\|{\bf a}\\|_0 = \sum \|a_j\|^0$ (if we put $0^0=0$.) The notion makes sense only after we have fixed a basis. In compressed sensing, several important results compare the $L^0$ norm with the $L^1$ norm $\sum \|a_j\|$. See \[Do2\]. We discuss this matter, in our context, in Section 8. As noted above, the study of rational sphere maps has led the authors to the same kind of question. The simplest version involves finding the largest possible degree $d$ of a monomial mapping sending the unit sphere in its source ${\mathbb C}^n$ to the unit sphere in its target ${\mathbb C}^N$, in terms of $n,N$. Equivalently, given the degree $d$, one seeks the minimum possible target dimension $N$. One obtains an underdetermined system of linear equations for the squared magnitudes of the coefficients of the distinct monomials. This problem has been solved as follows; for $n=1$, there is no bound on $d$. For $n=2$, the sharp bound is $d \le 2N-3$. See \[DKR\]. For $n\ge 3$, the sharp bound is $d \le {N-1 \over n-1}$. See \[LP\]. We can also regard these results as giving lower bounds on $N$ given the degree, thus making the connection to the previous paragraph. (Minimizing $N$ is equivalent to finding the [sparsest]{} solution, given that at least one of the coefficients of a monomial of maximum degree is not zero.) It seems that some of the techniques developed in the papers \[DKR\], \[G\], \[LP\] may be both useful and new when solving underdetermined linear systems in compressed sensing. Furthermore, in \[DX2\], a necessary and sufficient condition for a map $f$ to have minimum target dimension is that a certain group $H_f$ be trivial. Let $f= {p \over q}$ be a rational mapping in $n$ complex variables with values in ${\mathbb C}^N$ (and with no singularities on the closed unit ball). The condition that $f$ map the unit sphere in the source ${\mathbb C}^n$ to the unit sphere in the target ${\mathbb C}^N$ arises throughout CR geometry. We call $f$ a [rational sphere map]{}. When $f$ is not constant, $f$ is a proper holomorphic mapping between balls. By \[F\], if $n\ge 2$ and $f$ is a proper holomorphic mapping between balls with $N-n+1$ continuous derivatives at the sphere, then $f$ is a rational sphere map. Let $\\|z\\|^2$ denote the squared Euclidean norm of the vector $z \in {\mathbb C}^k$ for any $k$ and let $\langle z,w \rangle $ denote the Euclidean inner product, again in any dimension. In this notation we wish to find all polynomial solutions $p$ and $q$ to the equation $$\\|p(z)\\|^2 = \|q(z)\|^2 \eqno (1)$$ for $z$ satisfying $\\|z\\|^2=1$. We may assume, without loss of generality, that $p$ and $q$ have no common factors and that $q(0)=1$. We may further assume that $p(0)=0$, because the automorphism group of the target ball is transitive. If we fix the degrees of $p$ and $q$ to be at most $d$, then we can regard the coefficients of $p$ and $q$ as a finite number of unknowns. We obtain a [quadratic]{} system of equations by equating coefficients in (1) and using the condition $\\|z\\|^2=1$. See formula (9). When, however, we assume that the components of the map are monomials, we obtain a [linear]{} system in the squared magnitudes of the coefficients. The [degree estimate]{} $d \le 2N-3$ from \[DKR\] then tells us that the minimum $L^0$ norm $N$ of the solution is at least ${d+3 \over 2}$. Furthermore, this inequality is sharp, and polynomials satisfying this bound have many interesting properties. See Theorem 5.1. Similar conclusions apply in dimensions at least $3$, where we have $N \ge 1 + d(n-1)$. Again this result is sharp in the monomial case. See \[LP2\]. We regard this paper as an invitation to study the surprising connections between two beautiful but not obviously related topics: degree estimates for rational sphere maps and the compressed sensing techniques used in finding sparse solutions to linear systems. We therefore close the paper with a list of several open questions that will develop and expand these connections. Several of these questions also involve the linear programming problem of minimizing the $L^1$ norm given the degree of a monomial sphere map. Easy examples ============= We begin with two simple examples of the link we are emphasizing. Let $p$ be a quadratic monomial map in two variables with no constant term. Thus for complex constants $a,b,c,d,e$ we have $$p(z,w) = \left (a z, bw, c z^2, dzw, e w^2\right) .$$ The condition that $p(z,w)$ maps the sphere to the sphere becomes $$\|a\|^2 + \|c\|^2 = \|b\|^2 + \|e\|^2 = 1$$ $$\|a\|^2 + \|b\|^2 + \|d\|^2 = 2.$$ These equations are linear in the squared magnitudes, written ${\bf a} = (a_1,a_2,...,a_5)$. We obtain the linear system $$\begin{pmatrix} 1 && 0 && 1 && 0 && 0 \cr 0 && 1 && 0 && 0 && 1 \cr 1 && 1 && 0 && 1 && 0 \end{pmatrix} \ \begin{pmatrix} a_1 \cr a_2 \cr a_3 \cr a_4 \cr a_5 \end{pmatrix}= \begin{pmatrix} 1 \cr 1 \cr 2 \end{pmatrix}. \eqno (2)$$ The linear system (2) has general solution $$\begin{pmatrix} 1 \cr 1 \cr 0 \cr 0 &\cr 0\end{pmatrix} + a_4 \begin{pmatrix} -1 \cr 0 \cr 1 \cr 1 \cr 0 \end{pmatrix} + a_5 \begin{pmatrix}1 \cr 0 \cr -1 \cr 0 \cr 1 \end{pmatrix}. \eqno (3)$$ We seek solutions with nonnegative coefficients. There is one such solution with $L^0$ norm equal to $2$, and three such solutions with $L^0$ norm equal to $3$. We list these solutions, written in terms of the $a_j$, and then the corresponding maps. We have chosen the coefficients of the maps to be positive; they are actually determined only up to complex numbers of modulus $1$. $${\bf a} = (1,1,0,0,0) \ {\text {leads to}} \ (z,w, 0)$$ $${\bf a} = (1,0, 0,1,1) \ {\text {leads to}} \ (z,zw,w^2)$$ $${\bf a} = (0,1,1,1,0)\ {\text {leads to}} \ (w, zw, z^2)$$ $${\bf a} = (0,0,1,2,1) \ {\text {leads to}} \ (z^2, \sqrt{2}zw, w^2).$$ The first solution is first degree. There is a solution of degree $3$ to the analogous equations, but not for any degree larger than $3$. In other words, the degree of a polynomial (or even rational) map sending the unit sphere in ${\mathbb C}^2$ to the unit sphere in ${\mathbb C}^3$ can be of degree at most $3$. We next illustrate how the polynomial (or rational) case leads to a quadratic system. These quadratic equations are linear in the inner products of the various unknown vectors, and degree estimates can be regarded as a kind of quadratic compressed sensing. We also explain how to regard the rational case as a linear problem. The following simple example from \[D1\] illustrates the main idea. Assume that $p:{\mathbb C}^2 \to {\mathbb C}^5$ has degree at most two and $p(0)=0$. There are $5$ possible coefficient vectors; hence there is no loss of generality in starting with the target dimension $5$. We want to know how [small]{} it can be. We write $$p(z,w) = Az + Bw + C z^2 + Dzw + Ew^2. \eqno (4)$$ Here $A,B,C,D,E$ are elements of ${\mathbb C}^5$. The condition that $p$ map the sphere to the sphere becomes $$\\|Az + Bw + C z^2 + Dzw+ Ew^2\\|^2 = 1 \eqno (5)$$ when $\|z\|^2 + \|w\|^2 =1$. Expanding the squared norm on the left-hand side of (5) and then equating coefficients gives the following list of conditions on the unknowns: $$\langle A,B \rangle = - \langle D,E \rangle = - \langle C,D \rangle = \lambda \eqno (6.1)$$ $$\langle A,C \rangle = \langle A,D \rangle = \langle A,E \rangle = 0 \eqno (6.2)$$ $$\langle B,C \rangle = \langle B,D \rangle = \langle B,E \rangle = \langle C,E\rangle = 0 \eqno (6.3)$$ $$\\|A\\|^2 + \\|C\\|^2 =1 \eqno (6.4)$$ $$\\|B\\|^2 + \\|E\\|^2 = 1 \eqno (6.5)$$ $$\\|D\\|^2 = \\|C\\|^2 + \\|E\\|^2. \eqno (6.6)$$ There are $13$ linear equations in the $15$ variables (inner products including squared norms). The solution space to these equations involves three real parameters; we may choose them to be $\\|A\\|^2$, $\\|B\\|^2$, and $\langle A, B \rangle = \lambda$. We must have $ 0 \le \\|A\|^2 \le 1$, also that $0 \le \\|B\\|^2 \le 1$ and hence, by the Cauchy-Schwarz inequality, that the complex number $\lambda$ satisfies $\|\lambda\| \le 1$. The most succinct way to express this example is to write $L$ for the linear map with $L(z,w) = Az+Bw$. Then we have $$p(z) = Lz \oplus \left( (\sqrt{I - L^L}z ) \otimes z \right).$$ We summarize. The collection of polynomial mappings of degree at most two, with no constant term, and mapping the unit sphere in ${\mathbb C}^2$ to the unit sphere in ${\mathbb C}^5$, is parametrized by such linear maps $L$. When $L=0$, $p$ is a homogeneous quadratic map. When $L$ is unitary, $p$ is linear. In general $I - L^L$ must be non-negative definite. A good way for solving the system resulting from (5) is to homogenize. Replace the right-hand side of (5) with $(\|z\|^2 + \|w\|^2)^2$, expand the left-hand side, homogenize it, and finally equate coefficients. See Section 4. The rational case ================= Assume ${p \over q}$ is a rational mapping of degree at most $d$ that sends the unit sphere in its source to the unit sphere in its target. Let us write $$p(z) = \sum_{\|\alpha\| \le d} A_\alpha z^\alpha \eqno (7.1)$$ $$q(z) = \sum_{\|\beta\| \le d} b_\beta z^\beta. \eqno (7.2)$$ In (7.1) the $A_\alpha$ are vectors and in (7.2) the $b_\beta$ are scalars. The crucial condition that $\\|p(z)\\|^2 - \|q(z)\|^2 = 0$ on the unit sphere becomes $$\sum \left( \langle A_\alpha, A_ \gamma \rangle - b_\alpha {\overline {b_\gamma}}\right) z^\alpha {\overline z}^\gamma = 0 \eqno (8)$$ whenever $\sum_{j=1}^n \|z_j\|^2 = 1$. It takes some work to rewrite (8) as a linear system by eliminating the constraint. To do so, one first replaces $z$ by $e^{i\theta}z$ in (6) and equates Fourier coefficients. Here $\theta= (\theta_1,...,\theta_n)$ is a point on the $n$-torus. One obtains a finite number of independent equations. Then one homogenizes each equation. Equating coefficients yields, for each multi-index $\mu$, $$\sum \langle A_{\mu +\gamma}, A_{\gamma}\rangle z^\gamma {\overline z}^\gamma \\|z\\|^{2d - 2\|\gamma\|} = \sum b_{\mu +\gamma}{\overline b_{\gamma}}z^\gamma {\overline z}^\gamma \\|z\\|^{2d - 2\|\gamma\|}. \eqno (9)$$ We regard this system of equations as follows. . Assume that the denominator $q$ as in (7.2) is given. We want to find all $p$ as in (7.1) such that $\\|p(z)\\|^2 = \|q(z)\|^2$ on the unit sphere. The right-hand side of (9) provides the right-hand side of a linear system of the form ${\bf T}(u)= v$, where $u$ is a vector whose entries are the various inner products $\langle A_\alpha, A_\beta\rangle$. In the next definition we write ${\mathbb C}^n$ for the domain of a rational map $f$. To be precise, the domain must exclude the zero-set of the denominator $q$. The important issue is the target dimension. Let $f={p \over q}:{\mathbb C}^n \to {\mathbb C}^N$ be a rational sphere map. We say that $f$ is [target-minimal]{} if the image of $f$ lies in no affine subspace of ${\mathbb C}^N$ of lower dimension. Target minimality is equivalent to saying there is no smaller integer $k$ for which $\\|f\\|^2 = \\|g\\|^2$ for some rational map $g$ to ${\mathbb C}^k$. For example, the map $(z,w) \mapsto (z,w,0)$ is not target-minimal. Nor is the map $(z,w) \mapsto (\alpha z, \beta z, w)$ when $\|\alpha\|^2 + \|\beta\|^2 = 1$. Another way to express target-minimality is to expand $\\|f\\|^2$ in a power series about $0$; then $f$ is target-minimal if and only if the rank of the matrix of Taylor coefficients equals the target dimension of $f$. A third way to characterize target-minimality involves groups associated with mappings. See \[DX2\] and Section 11. The following non-trivial result (see \[D2\] and \[D3\]) is required in our subsequent discussion. This result is part of a program on Hermitian analogues of Hilbert’s $17$-th problem. Suppose $q:{\mathbb C}^n \to {\mathbb C}$ is a polynomial and $q(z) \ne 0$ for $z$ in the closed unit ball. Then there is an integer $N$ and a (non-constant) polynomial map $p:{\mathbb C}^n \to {\mathbb C}^N$ such that both statements hold: - $\\|p(z)\\|^2 = \|q\|^2 $ on the unit sphere; thus ${p \over q}$ is a rational sphere map. - ${p \over q}$ is reduced to lowest terms. The second condition prevents us from considering trivial examples such as $p=q$ or $p(z) = q(z) \left(z_1,..., z_n\right)$. We also note that the result is elementary when $n=1$. Thus given $q$, there is a smallest $N= N(q)$ for which the conclusion of the theorem applies. This number is the smallest $N$ for which there are vectors in ${\mathbb C}^N$ whose inner products satisfy the system of equations (9). The value of $N(q)$ depends on the coefficients of $q$; the following remark helps indicate the subtlety. Put $n=2$ and put $q(z_1,z_2) = 1 - \lambda z_1 z_2$. We must have $\|\lambda\| < 2$ for $q$ to be non-vanishing on the closed unit ball. Here $N(q)$ tends to infinity as $\lambda$ tends to $2$. Furthermore, the minimum possible degree of the numerator $p$ also tends to infinity as $\|\lambda\|$ tends to $2$. This fact is closely related to stabilization of Hermitian forms (see \[CD\]) and is explained in detail in \[D3\] for example. Theorem 3.1 suggests how to regard the problem in a linear framework. We think of the denominator $q$ as given. We try to find a numerator $p$ such that the conclusions of Theorem 3.1 hold. Doing so leads to a linear system in the inner products of the unknown (vector) coefficients of $p$. We seek the smallest dimension $N$ in which vectors exist for which this system of linear equations has a solution. There is no bound on the degree of $p$ nor on this dimension $N$ that depends only on $n$ and the degree of $q$. Thus the minimum $L^0$ norm depends on the right-hand side of the linear equation $T{\bf u} = {\bf v}$. The monomial case and homogenization ===================================== When discussing monomial maps $p$, we generally assume, with no loss of generality, that $p(0)=0$. The following simple idea has been used often by the authors in many papers. See for example \[D1\] and \[LP1\]. See \[W\] for a list (found using this idea) of all the monomial maps from the sphere in ${\mathbb C}^2$ to the sphere in ${\mathbb C}^5$. Assume $n\ge 2$. Suppose that $p:{\mathbb C}^n \to {\mathbb C}^N$ is a monomial map and $\\|p(z)\\|^2 =1 $ when $\\|z\\|^2 =1$. Put $x = (\|z_1\|^2,..., \|z_n\|^2)$. Write $s(x) = \sum_{j=1}^n x_j$. Write $$p(z) = ( ..., c_\alpha z^\alpha, ...).$$ There are several ways to create a linear system. One method is to eliminate a variable. A second method, using homogenization, is preferable. In the first method we write $(x_1,...,x_{n-1})= {\bf t}$. The squared norms of the $c_\alpha$, written $A_\alpha$, satisfy the equation $$\sum_{\alpha} A_\alpha x^\alpha = 1$$ on $s(x) = 1$. Write $\alpha = (\beta, \alpha_n)$. Replace $x_n$ by $1 - \sum_{j=1}^{n-1} x_j$ to obtain $$\sum_{\beta} A_\alpha {\bf t}^\beta (1 - \sum_{j=1}^{n-1} t_j)^{\alpha_n} = \sum_\mu C_\mu {\bf t}^\mu = 1. \eqno (10)$$ Equation (10) now holds for all ${\bf t}$ and we can equate coefficients. Thus (10) yields an underdetermined system of linear equations for the unknown coefficients $A_\mu$. The coefficient of the constant term equals unity; all the other coefficients vanish. Thus the right-hand side of the linear system is the column vector ${\bf v}$ with first coefficient $1$ and the remaining coefficients equal to $0$. We seek a solution to the equation $T({\bf a})= {\bf v}$, where the components of ${\bf a}$ are non-negative and where as many as possible vanish. It is also natural to fix the degree $d$ of $p$ and hence also assume that $c_\alpha \ne 0$ for some $\alpha$ of length $d$. The system arising from (10) is awkward. Homogenization provides an alternative and more symmetric way to approach this system of equations. Doing so illuminates the connection between compressed sensing and monomial sphere maps. Let $p:{\mathbb C}^n \to {\mathbb C}^N$ be a monomial sphere map; we have $ p(z) = (..., C_\alpha z^\alpha, ...)$ for complex numbers $C_\alpha$. Then $p$ maps the source sphere to the target sphere if $$\sum_\alpha \|C_\alpha\|^2 \|z\|^{2 \alpha}= 1 \eqno (11)$$ when $\sum \|z_j\|^2 =1$. Equation (11) depends only upon the $\|z_j\|^2$, and hence we obtain a real-variables problem by replacing $(\|z_1\|^2,..., \|z_n\|^2)$ by $(x_1,...,x_n)$. Put $s(x) = \sum_{j=1}^n x_j$. Write $c_\alpha = \|C_\alpha\|^2$. We get an equivalent equation to (11): $$\sum_\alpha c_\alpha x^\alpha = 1 \eqno (12)$$ on $s(x) =1$. Following ideas from \[D1\], we homogenize this equation. Assume that $p$ is of degree $d$. Homogenizing (12) yields $$\sum c_\alpha x^\alpha s(x)^{d - \|\alpha\|} = (s(x))^d = \sum_{\|\beta\|=d} { d \choose \beta} x^\beta. \eqno (13)$$ By homogeneity, (13) holds for all $x$. Equating coefficients yields a linear system for the unknown coefficients $c_\alpha$. Finding the minimum target dimension $N$ is precisely the same problem as solving this linear system with the smallest number of non-vanishing $c_\alpha$. Consider a sharp degree estimate $d \le \Phi(n,N)$. (Recall that $n \ge 2$.) We rewrite this inequality as $N \ge \Psi(d,n)$ and reinterpret it: [Interpretation in compressed sensing]{}. Of all solutions to (13), we seek those solutions for which the number of non-vanishing $c_\alpha$ is as small as possible, given that at least one of the coefficients of a largest degree term must be non-zero. We return to quadratic monomial maps in source dimension $2$, writing $(z,w)$ for the variables and $(x,y)$ for $(\|z\|^2, \|w\|^2)$. In this case $s(x,y)= x+y$. We assume, without loss of generality, that $p(0)=0$. Put $$p(z,w) = (Az,Bw, Cz^2,D zw, Ew^2).$$ Equation (13) becomes $$\|A\|^2 x(x+y) + \|B\|^2y(x+y) + \|C\|^2 x^2 + \|D\|^2 xy + \|E\|^2y^2 = x^2 + 2 xy + y^2.$$ Equating coefficients yields the system $$\begin{pmatrix} 1 & 0 & 1 & 0 & 0 \cr 1 & 1 & 0 & 1 & 0 \cr 0 & 1 & 0 & 0 & 1 \end{pmatrix} \ \begin{pmatrix} a \cr b \cr c \cr d \cr e \end{pmatrix} = \begin{pmatrix} 1 \cr 2 \cr 1\end{pmatrix}. \eqno (14)$$ The minimum number of non-vanishing coefficients is $2$. We have the solution where $a=b=1$ and the other coefficients are $0$. This map, however, is degree $1$. To make it degree $2$, we require that at least one of $c,d,e$ must be non-zero. Given this constraint, the minimum number becomes $3$. These solutions are given by $$(a,b,c,d,e) = (1,0,0,1,1)$$ $$(a,b,c,d,e) = (0,1,1,1,0)$$ $$(a,b,c,d,e) = (0,0,1,2,1).$$ The system of equations obtained by homogenization is not the same system as the one determined by elimination of a variable, although the corresponding maps are the same. In particular the right-hand sides of the equation differ. We prefer using the system obtained from homogenization. One advantage is that the coefficients are non-negative. The next proposition further illustrates the role of the monomial case. Recall that the [rank]{} of a holomorphic mapping $f:{\mathbb C}^n \to {\mathbb C}^N$ is the rank of the matrix of Taylor coefficients of $\\|f(z)\\|^2$. When $f$ is a monomial map, the matrix is diagonal, and the rank is the number of non-zero diagonal elements. In general the entries of the matrix are inner products of the unknown vector coefficients. Given $n, N, d$, the polynomial sphere map $p \colon {\mathbb{C}}^n \to {\mathbb{C}^N}$ of degree $d$ that minimizes the number of nonzero entries in the coefficient matrix of $\\|p(z)\\|^2$ is a monomial map. Consider a polynomial sphere map $p(z)$. Replace $z$ with $e^{i\theta} z$, where $\theta$ is in the $n$-torus. Let us average over the torus. Let $$r(z,\bar{z}) = \frac{1}{{(2\pi)}^n}\int_{{[0,2\pi]}^n} \\|p(e^{i\theta}z) \\|^2 d \theta .$$ Note that $r(z,\bar{z}) = 1$ whenever $\\|z\\| = 1$. Let us analyze the coefficient matrix of $r(z,\bar{z})$. Averaging a coefficient $a_{\alpha\beta} z^\alpha\bar{z}^\beta$ of $\\|p(z)\\|^2$ results in 0 unless $\alpha = \beta$. If $\alpha = \beta$, the coefficient is untouched. In other words, the coefficient matrix of $r(z,\bar{z})$ is zero off the diagonal, and the same as the coefficient matrix of $\\|p(z)\\|^2$ on the diagonal. In particular, $r(z,\bar{z})$ is a squared norm representing a monomial sphere map $m(z)$. As $p$ is of degree $d$, and the coefficient matrix of $\\|p(z)\\|^2$ is positive semidefinite, the diagonal, and therefore $r(z,\bar{z}) = \\| m(z) \\|^2$ has a nonzero term corresponding to a term of degree $d$ in $m(z)$. Furthermore, since they agree on the diagonals, the coefficient matrix of $\\| m(z) \\|^2$ has at most as many nonzero terms as the coefficient matrix of $\\|p(z)\\|^2$. The result follows. Source dimension two ==================== We begin with a remarkable collection of polynomials. Let $d$ be a positive integer. Define $p_d(x,y)$ by $$p_d(x,y) = \left({ x + \sqrt{x^2+4y} \over 2} \right)^d + \left({ x - \sqrt{x^2+4y} \over 2} \right)^d + (-1)^{d+1}y^d. \eqno (15)$$ This family of polynomials has many interesting properties. We mention just a few now, and say more later. See \[D1\], \[D3\], and their references. Define $p_d$ as in (15). Each $p_d$ is a polynomial of degree $d$ and 1. For each $d$, we have $p_d(x,y)=1 $ on $x+y=1$. 2. For each odd $d$, all the coefficients of $p_d$ are non-negative. 3. For each even $d$, all the coefficients of $p_d$ are non-negative except for the coefficient of $y^d$, which is $-1$. 4. $p_d(\eta x, \eta^2 y) = p_d(x,y)$ whenever $\eta$ is a $d$-th root of unity. 5. For each odd $d$, the polynomial $p_d$ has precisely ${d+3 \over 2}$ terms. 6. The polynomial $p_d$ is congruent to $x^d + y^d$ modulo $(d)$ if and only if $d=1$ or $d$ is prime. These polynomials exhibit striking algebraic combinatorial properties. To indicate why, instead of defining them as in (15), we could proceed as follows. Fix an odd integer $d=2r+1$ and assume that $p_d$ is of degree at most $d$. If, in addition, $p_d$ satisfies item (4) of Theorem 5.1, then there are constants $c_k$ and $c$ such that $$p_d(x,y) = \sum_{k=0}^r c_k x^{2r+1-2k}y^k + c y^{2r+1}.$$ The reason is that only invariant monomials can arise. Next make the crucial assumption that item (1) holds. Then $c=1$ and homogenization yields $$\sum_{k=0}^r c_k x^{2r+1-2k}y^k (x+y)^k+ y^{2r+1} = (x+y)^{2r+1}.$$ The corresponding linear system has a unique solution; in fact (see \[D1\]) $$c_k = \left({1 \over 4}\right)^{r-k} \sum_{j=k}^r {2r+1 \choose 2j} {j \choose k}.$$ The kind of sum arising here can be evaluated using the Wilf-Zeilberger methods (see \[PWZ\]). Considerable work yields the formula $$c_k = {(2r+1)(2r-k)! \over k! (2r+1-2k)!}.$$ Furthermore the proof of item (6) combines the method of homogenization with the analogous well-known primality property for the polynomials $(x+y)^d$. For the connection with sensing, the crucial point is that these polynomials have the fewest number of non-zero coefficients given their degree and that $p(x,y)=1$ on the line $x+y=1$. We state these results next. For each odd positive integer $2r+1$, there is a polynomial $p(x,y)$ of degree $2r+1$, with $r+2$ terms, such that - $p(x,y)- 1$ is divisible by $x+y-1$. - each coefficient of $p$ is non-negative. For each odd positive integer $d=2r+1$, there is a proper monomial map $f:{\mathbb B}_2 \to {\mathbb B}_{r+2}$ of degree $2r+1$. Thus $d=2N-3$. By this corollary there is an example where $d = 2N-3$. The next theorem shows that $d$ can be no larger if $N$ is fixed. \[DKR\] Let $p(x,y)$ be a polynomial such that $p(x,y)$ has all non-negative coefficients and $p(x,y)-1$ is divisible by $(x+y-1)$. Let $d$ denote the degree of $p$ and let $N$ denote the number of terms (distinct monomials) in $p$. Then $d \le 2N-3$, and (by Corollary 5.1) this result is sharp. We next recast these theorems in the language of compressed sensing. The first reformulation is direct. Using a lemma from \[LL\], we provide a considerably simpler second reformulation. Fix a positive integer $d$. Let $K= {d^2 + 3d \over 2}$ when $d$ is odd and ${d^2 + 3d+1 \over 2}$ when $d$ is even. We consider an unknown vector $\bf u$ in ${\mathbb R}^K$. We wish to solve a linear system $T{\bf u}= {\bf v}$, where $v$ is a specific vector in ${\mathbb R}^{d+1}$. We are given the standard orthonormal basis of these Euclidean spaces. In these coordinates, the components of $v$ are the binomial coefficients $(1,d,{d \choose 2},..., d,1)$. We call the last $d+1$ variables in the source [distinguished]{}. Our aim is to find ${\bf u}$ such that - $T{\bf u}= {\bf v}$ - At least one of the distinguished variables is not zero. - Each $u_j$ is non-negative. Let $N$ denote the minimum $L^0$ norm of all solutions. Then the above results state that $N= {d+3 \over 2}$ when $d$ is odd, and $N= {d \over 2} + 2$ when $d$ is even. Without the constraint that one of the distinguished variables is not $0$, then $N=2$. This solution corresponds to $u_1 = u_2 = 1$ and $u_j = 0$ for $j \ge 2$. The corresponding polynomial is $x+y$. For emphasis, we explain again why we have distinguished variables. We want the solution to correspond to a polynomial of degree precisely $d$, rather than to a polynomial of degree at most $d$. The number $K = {d^2 + 3 d \over 2}$ is the dimension of the space of polynomials of degree at most $d$ in two variables whose constant term is $0$. We have $$K = 2 + 3 + ... +(d+1) = {(d+1)(d+2) \over 2} - 1 = {d^2 + 3d \over 2}.$$ The number $d+1$ is the dimension of the space of homogeneous polynomials of degree $d$ in two variables. Suppose $f$ is a monomial map from two-space of degree at most $3$. There are $9$ possible monomials, hence $9$ unknown coefficients. Using the method of homogenization, the system of equations that must be solved becomes $$\begin{pmatrix} 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \cr 2 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \cr 1 & 2 & 0 & 1 & 1 & 0 & 0 & 1 & 0 \cr 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \end{pmatrix} \ \begin{pmatrix} a_1 \cr a_2 \cr a_3 \cr a_4 \cr a_5 \cr a_6 \cr a_7 \cr a_8 \cr a_9 \end{pmatrix} = \begin{pmatrix} 1 \cr 3 \cr 3 \cr 1 \end{pmatrix} \eqno (16)$$ This matrix has rank $4$, and there is a $5$-dimensional space of solutions. Of course we want non-negative solutions. If we want something of degree $3$, then one of the last four variables must be non-zero. It follows that at least [two]{} of these variables must be non-zero. The smallest target dimension possible for a map of degree $3$ is also $3$, obtained by the map $(\sqrt{3}zw, z^3, w^3)$. In the notation from (8), we have $a_4=3$, $a_6=1$, and $a_9 = 1$. This map is the special case of (15) when $r=1$. If $p$ is a proper rational mapping of degree $d$ between balls, then the rank of the terms of degree $d$ is at least the domain dimension. In particular, for a monomial map of degree $d$ in two variables, there must be at least $2$ terms of highest degree. See Lemma 5.1 and \[LL\] for various generalizations. We list the polynomials from (15) corresponding to $r=0,1,2,3,4$. $$x+y$$ $$x^3 + 3xy + y^3$$ $$x^5 + 5 x^3 y + 5 xy^2 + y^5$$ $$x^7 + 7 x^5 y + 14 x^3 y^2 + 7 x y^3 + y^7$$ $$x^9 + 9 x^7 y + 27 x^5 y^2 + 30 x^3 y^3 + 9 x y^4 + y^9.$$ A huge amount is known about these polynomials and various generalizations. See for example Theorem 5.1, \[D1\], \[D3\], \[G\], and their references. Assume $p$ has at most $5$ terms and $p(x,y)=1$ on $x+y=1$. By Theorem 5.2, the degree of $p$ is at most $7$. Put $p(x,y) = a_1 x + a_2 y + a_3 x^2 + ... + a_{35} y^7$. Homogenizing as before yields a system of $8$ equations in these $35$ unknowns. The sparsest solution is of course given by $x+y$; that is $a_1 = a_2 = 1$ and all the other variables equal $0$. If we assume, however, that there is at least one term of degree $7$, then the sparsest solution will have $5$ terms. There are four such examples: $$x^7 + y^7 + {7 \over 2} x^6 y + {7 \over 2} x y^6 + {7 \over 2} xy \eqno (17)$$ $$x^7 + y^7 + 7 x^3 y^3 + 7 x y^3 + 7 x^3 y \eqno (18)$$ $$x^7 + 7 x^5 y + 14 x^3 y^2 + 7 x y^3 + y^7 \eqno (19)$$ $$x^7 + 7 x y^5 + 14 x^2 y^3 + 7 x^3 y + y^7. \eqno (20)$$ Notice that (19) and (20) are examples of the form found in Theorem 5.1; (20) is obtained from (19) by interchanging $x$ and $y$. We note that (17) and (18) are symmetric in $x$ and $y$. Thus, in degree $7$, there are examples [not]{} of the form found in Theorem 5.1. The papers \[DL2\] and \[LL\] discuss this phenomenon in detail. See also Section 7. We repeat the main point: while the system of linear equations admits a solution with $L^0$ norm $N$ equal to $2$, if we assume that there is a term of degree $7$, then the minimum $L^0$ norm is $5$. In this reformulation we eliminate the notion of distinguished variables. We can do so because of a result proved about monomial proper maps with minimum target dimension $N$. We state Lemma 3.1 from \[LL\], in the language of this paper. Let $d$ be an odd integer and let $f(x,y)$ be a polynomial of degree $d$ such that the coefficients of $f$ are non-negative and $f(x,y) = 1$ on the line $x+y=1$. Suppose also that $f$ has $N$ terms where $N$ is minimal. (In other words, $N$ is the smallest $L^0$ norm of a solution to the problem.) Then $$f(x,y) = x^d + y^d + \ {\rm {lower \ order \ terms}} .$$ In solving the linear system, we can therefore set the coefficients of $x^d$ and $y^d$ equal to $1$ and set the rest of the distinguished variables equal to $0$. Doing so renders it unnecessary to discuss distinguished variables. We determine the values of $d+1$ of the variables automatically by using this lemma. We now get the following system. The variables $a_1,..., a_L$ are the coefficients of the monomials $x,y, x^2, xy, y^2, x^3,x^2y,xy^2, y^3,...$ of degree at most $d-1$. We homogenize as before, except that we set the result equal to $(x+y)^d - x^d + y^d$, obtaining two fewer equations in $d+1$ fewer variables. We illustrate by revisiting Example 5.3. We wish to solve $$(a_1 x + a_2 y) (x+y)^6 + (a_3 x^2 +a_4 xy + a_5 y^2 ) (x+y)^5 + ... (a_{21}x^6 + ... + a_{27} y^6)(x+y)$$ $$= (x+y)^7 - x^7 - y^7.$$ The lemma implies that $a_{28}=a_{35}=1$ and that $a_j = 0$ for $29 \le j \le 34$. The solutions are the same. Thus Lemma 5.1 determines the last $d+1$ variables; as claimed, the system has $d+1$ fewer unknowns and $2$ fewer equations. The paper \[LL\] has additional results enabling us to decrease the number of variables. For example, given that $f$ is as in Lemma 5.1, we may also assume that $a_1=a_2=0$. In fact the results of section 3 in \[LL\] imply the following restrictions of the solutions to the linear system. Let $d$ be an odd integer. Let $f(x,y)$ be a polynomial of degree $d$ such that the coefficients of $f$ are non-negative and $f(x,y) = 1$ on the line $x+y=1$. Assume that $f$ has $N$ terms for $N$ as small as possible. The following all hold: - The coefficient of $x^j$ is $0$ for $0 \le j \le d-1$. - The coefficient of $y^j$ is $0$ for $0 \le j \le d-1$. - The coefficient of $x^j y^k$ is $0$ for $j+k=d$ and neither $j$ nor $k$ zero. - The coefficient of $x^d$ is $1$ and the coefficient of $y^d$ is $1$. - If $d > 1$, then some coefficient of a monomial of degree $d-1$ is not zero. Using (all but the last item in) Proposition 5.1, we obtain a linear system with fewer unknowns. We write this new system - ${\bf T}{\bf u}= {\bf v}$. - Each $u_j$ is non-negative. We also note the following result, stated explicitly in (different language in) \[LL\]. The solutions to this system have rational components. sources and sinks ================= In this section we describe the main idea from \[DKR\], with the modest hope that it can be used more generally in finding the minimum $L^0$ norm of a solution to a linear system. Let $p(x,y)$ denote a polynomial of degree $d=2r+1$ with non-negative coefficients. Assume that $p(x,y)=1$ on the line $x+y=1$. Then the polynomial $p(x,y) - 1$ is divisible by $x+y-1$. We call the quotient $q(x,y)$. Then $q$ is of degree $d$. Let ${\bf G}_p$ denote the Newton diagram for $q$. Thus ${\bf G}_p$ is the collection of lattice points $(a,b)$ for which the coefficient of $x^a y^b$ in $q$ is not zero. If the coefficient is positive we label the lattice point with a ${\bf P}$, if it is negative we label it with a ${\bf N}$, and if zero with a ${\bf Z}$. When the point is labeled ${\bf P}$ we draw directed arrows upward and to the right from $(a,b)$ to $(a,b+1)$ and to $(a+1,b)$. When the point is labeled ${\bf N}$ we draw these arrows from $(a,b+1)$ and $(a+1,b)$ to $(a,b)$. We call a point in ${\bf G}_p$ a [source]{} if all arrows drawn at that point lead away from it, and we call it a [sink]{} if all arrows drawn at that point lead into it. It is easy to see that the origin is the only source. The first crucial observation is that the number of terms in $p$ is at least as large as the number of sinks in ${\bf G}_p$. The reason is that a sink at $(a,b)$ means that the coefficients of $x^{a+1}y^b$ and $x^a y^{b+1}$ in $p(x,y)$ must be positive, as no cancellation can occur. Next one observes that the graph ${\bf G}$ corresponding to $(x+y)^d$ has each lattice point $(a,b)$ labeled ${\bf P}$ when $a+b \le d-1$. The reason is simply that the geometric series gives $${ (x+y)^d - 1 \over x+y-1} = 1 + (x+y) + (x+y)^2 + ... + (x+y)^{d-1}.$$ The process of homogenization can be reversed to reach a given polynomial $p$. What happens to the number of sinks in the directed graphs when $p$ is a solution of degree $d$ with minimum $L^0$ norm? Assume first that $d=2r+1$ is odd. Then ${\bf G}$ has $2r+2$ sinks, the points $(a,b)$ where $a+b=d$. As we dehomogenenize, we change the picture. By \[DKR\] the following hold. The points $(d-1,0)$ and $(0,d-1)$ must have the labels ${\bf P}$ and hence there are sinks at $(d,0)$ and $(0,d)$. These correspond to Lemma 5.1. The remaining $2r$ sinks may coalesce down to $r$ sinks, but no further. We conclude that the total number of sinks is therefore at least $2 + r$, and hence the number of terms $N$ in $p$ is a least $2+r$. Thus $N \ge 2 + r$ and hence $2r+1 = d \le 2N-3$. The group invariant polynomials from Theorem 5.1 show that this bound is realized. We illustrate with the example $p(x,y)$ given by $$p(x,y) = x^5 + 5 x^3 y + 5 xy^2 + y^5.$$ We begin with $(x+y)^5$ and dehomogenize until we get to $p$. We use $\to$ to denote the effect of either rewriting terms or setting $x+y$ equal to $1$. $$(x+y)^5 = x^5 + 5 x^4 y + 10 x^3 y^2 + 10 x^2 y^3 + 5 x y^4 + y^5 \to$$ $$x^5 + 5x^3 y(x+y) + 5x^3 y^2 + 5 x^2 y^3 + 5 x^2 y^3 + 5 x y^4 + y^5 \to$$ $$x^5 + 5 x^3 y + 5 x^2y^2 (x+y) + 5 xy^3(x+y) + y^5 \to$$ $$x^5 + 5 x^3 y + 5 x^2 y^2 + 5 xy^3 + y^5 \to$$ $$x^5 + 5 x^3 y + 5 x y^2(x+y) + y^5 \to$$ $$x^5 + 5 x^3 y + 5 x y^2 + y^5.$$ The polynomial $q$ corresponding to $p$ is given by $$q(x,y) = 1 + x + x^2 + x^3 + x^4 + y + y^2 + y^3 + y^4 + 2xy - 2x y^2 - xy^3 + 3x^2 y + x^2 y^2- xy^3.$$ There are sinks at the points $(5,0)$, $(3,1)$, $(1,2)$, and $(0,5)$. Two of the original six sinks remain; the other four coalesced into two. The total number of sinks in the graph ${\bf G}_p$ is four, corresponding to the minimum $L^0$ norm of the linear system. When $d$ is even, it is easy to see that the minimum $L^0$ norm arises also from these examples. One simply multiplies either $x^{d-1}$ or $y^{d-1}$ by $x+y$ to create an example with one more term and of degree $d$. uniqueness ========== For certain odd integers $d$, the only polynomials $p(x,y)$ with $N$ terms, with positive coefficients, with $d=2N-3$, and with $p(x,y)=1$ on $x+y=1$ are those from Theorem 5.1. One can of course interchange the roles of $x$ and $y$. For other odd integers $d$, there exist additional polynomials with these properties. We say informally [uniqueness fails]{}. It is natural to ask for the precise set of $d$ for which these additional maps exist. The best results to date on this problem come from \[DL2\] and \[LL\]. All known examples where uniqueness fails come from the same procedure. It follows in all these examples, to be discussed below, that the explanation for the failure of uniqueness is that we can find a solution with a smaller $L^1$ norm. Suppose $p(x,y)$ is any polynomial with nonnegative coefficients. Then the sum of the coefficients of $p(x,y)$ equals $p(1,1)$. If we regard this sum as the $L^1$ norm of the solution to our linear system, then we have a formula for the $L^1$ norm. Consider for example the invariant polynomials $p_d$ given by (15) when $d=2r+1$ is odd. The sum of the coefficients is $$\left( {1 + \sqrt{5} \over 2 }\right)^{2r+1} + \left( {1 - \sqrt{5} \over 2 }\right)^{2r+1} + 1 = \varphi^{2r+1 } + \psi^{2r+1} + 1 \eqno (21)$$ and hence essentially a Fibonacci number. A standard formula (often erronenously attributed to Binet) for the $n$-th Fibonacci number is $$F_n = {1 \over \sqrt{5}} \left( \left( {1 + \sqrt{5} \over 2 }\right)^{n} - \left( {1 - \sqrt{5} \over 2 }\right)^{n}\right) = {1 \over \sqrt{5}} \left(\varphi^n - \psi^n\right). \eqno (22)$$ Since $\|\psi\| < 1$, for large $n=2r+1$, the $L^1$ norm (the sum of the coefficients) is close to $1+ \varphi^n$ which is close to $\sqrt{5}F_n$. All known sharp polynomials of degree $2d+1$ are found by a procedure which replaces terms in $p_{2r+1}$ using identities arising from using analogous polynomials $p_{2k}$, each of which has a single negative coefficient. The resulting effect always decreases the $L^1$ norm of the solution. We give a simple example. Begin with $p(x,y) = x^7 + 7 x^5 y +14 x^3 y^2 + 7 x y^3 + y^7$. We have $x^2+2y-y^2=1$ on the line $x+y=1$, and therefore $x^2 + 2y = 1 + y^2$ there. On the line we have $$x^7 + 7 x^5 y +14 x^3 y^2 + 7 x y^3 +y^7 = x^7 + 7 x^3 y(x^2 + 2y) + 7 xy^3 + y^7$$ $$\equiv x^7 + 7 x^3y(1+y^2) + 7 xy^3 + y^7 = x^7 + 7 x^3y + 7 x^3 y^3 + 7 xy^3 + y^7.$$ Here $\equiv$ denotes equality on the line $x+y=1$. We obtain a second polynomial of degree $7$, also with $5$ terms, also equal to $1$ on the line, and with non-negative coefficients. Thus the $L^0$ norm is the same, but the $L^1$ norm has decreased from $30$ to $23$. Looking back at formulas (17) through (20), we note that the corresponding $L^1$ norms are ${25/2}, 23, 30, 30$. Let us describe this procedure in more detail. Define $p_d$ by $$p_d(x,y) = \left({ x + \sqrt{x^2+4y} \over 2} \right)^d + \left({ x - \sqrt{x^2+4y} \over 2} \right)^d +(-1)^{d+1} y^d. \eqno (25)$$ Then $p_d(x,y)= 1$ on $x+y=1$, but $p$ has a negative coefficient when $d$ is even. Given an odd integer $d$ and an even integer $m$, the polynomial $$f_d(x,y) = p_d(x,y) - cx^a y^b(p_m(x,y) - 1) \eqno (26)$$ also equals $1$ on the line $x+y=1$. If we choose the monomial $cx^a y^b$ cleverly, then in passing from $p_d$ to $f_d$ we are replacing terms in $p_d$ with the same number of terms in $f_d$. In certain situations the polynomial $f_d$ will also have nonnegative coefficients. We can also iterate this procedure. It was proved in \[DL2\] that this procedure generates new sharp polynomials in various cases, including for example when - $d \ge 7$ and $d$ is congruent to $3$ modulo $4$. - $d \ge 7$ and $d$ is congruent to $1$ modulo $6$. It is important to emphasize two points. First of all, there exist numbers for which the only sharp polynomials are given by $p_d(x,y)$ and $p_d(y,x)$. In this case, one says [uniqueness holds]{}. Second of all, it is unknown whether there exist any sharp polynomials not constructed via this procedure. By the work in \[LL\], uniqueness is known to hold in degrees $1,3,5,9,17$. Later the third author verified it also for $d=21$; the proof was computer assisted and took a large amount of computer time. See \[L2\]. It is unknown whether the collection of numbers for which uniqueness holds is finite. sum of the coefficients and the $L^1$ norm ========================================== We write ${\bf S}_d$ for the set of polynomials $f(x,y)$ of degree $d$ with non-negative coefficients such that $f(x,y)=1$ when $x+y=1$. We saw in the last section that the sum of the coefficients, namely $f(1,1)$, carries some useful information. Given that the sum of the coefficients can be regarded as the $L^1$ norm of the solution to the linear system, it is natural to gather some information about this number. Let $m_d = \inf_{g \in {\bf S}_d} g(1,1)$. We begin with the following simple result. The infimum is not achieved because ${\bf S}_d$ is not closed under taking limits of the coefficients. For each degree $d$, we have $m_d =1$. For $d\ge 1$, the infimum is not attained. The result is trivial when $d=0$. For $d \ge 1$ and $0 \le \lambda \le 1$, put $g(x,y) = \lambda + (1 - \lambda )(x+y)^d$. Then $g(1,1) = \lambda + (1 -\lambda)2^d \ge 1$. Letting $\lambda$ tend to $1$ shows that the infimum is $1$. In general, if $g$ is not constant, the positivity of the constants forces $g(1,1) > 1$. If we restrict to polynomials with no constant term, then $m_d=2$. To see why, consider $g(x,y) = \lambda(x+y) + (1 - \lambda )(x+y)^d$. A similar situation arises by considering a polynomial with order of vanishing $\nu$ less than its degree $d$. On the other hand, if $f$ is homogeneous of degree $d$, then $f(x,y) = (x+y)^d$ and $f(1,1)$ achieves the [maximum]{} possible value of $2^d$. The next example illustrates a similar phenomenon. For $\lambda \in [0,3]$, put $$f_\lambda(x,y) = x^3 + y^3 + (3 -\lambda)xy + \lambda xy (x^3 + 3 xy + y^3).$$ Then $f_\lambda \in {\bf S}_5$ for $\lambda \in (0,3]$. The degree is $5$ when $\lambda \ne 0$. But when $\lambda=0$ the degree drops to $3$. Note that $f_\lambda(1,1) = 5 + 4 \lambda$. Hence, for $g$ of degree $5$ in this family of polynomials, the infimum of $g(1,1)$ is not achieved. Suppose $f \in {\bf S}_d$ and there are polynomials $g$ and $h$ with non-negative coefficients such that $f(x,y)=g(x,y) + (x+y) h(x,y)$ and $h$ is not the zero polynomial. (Thus $f$ is in the range of the partial tensor product operation. See \[D3\], for example, for this terminology.) Put $u = g+h$. Then $u(x,y) =1$ when $x+y=1$ and $u(1,1) < f(1,1)$. Thus we can decrease the $L^1$ norm when such $g,h$ exist. The following easy proposition offers a good reason for studying symmetric examples. In this result we consider a subset of ${\bf S}_d$ that is closed under interchanging the variables and averaging. In other words $g(x,y) \in S$ implies $g(y,x) \in S$ and $f,g \in S$ implies ${f+g \over 2} \in S$. Let $S \subset {\bf S}_d$ be closed under the operations of interchanging the variables and averaging. If $m= \inf_S g(1,1)$ is achieved, then there is a symmetric $h \in S$ with $h(1,1)=m$. Suppose $m= \inf_S g(1,1)$. Define a symmetric polynomial $h$ by $$h(x,y) = {g(x,y) + g(y,x) \over 2}.$$ Then $h(1,1) = g(1,1)$. Note that $h$ has non-negative coefficients, that $h(x,y) =1$ when $x+y=1$, and $h \in S$ by assumption. Put $g(x,y) = x^5 + y^5 + {10 \over 3} xy + {5 \over 3} (x^4 y + x y^4)$. Then $g$ is symmetric and $g(1,1) = {26 \over 3}$. Using linear programming, one can show that $g(1,1)$ is minimal among polynomials in ${\bf S}_7$, assuming that $g$ includes the monomials $x^7$ and $y^7$ each with coefficient $1$. Proposition 8.2 is important because it significantly decreases the number of unknowns in the linear system. Let us make a connection with uniqueness. At the $10$th workshop on Geometric Analysis of PDEs and Several Complex Variables in August 2019, the first author posed a new problem. For the integers $1,3,7,19$ there are sharp examples that are also symmetric in $x,y$. See formulas (17) and (18) above when the degree is $7$. Excluding the trivial case of degree $1$, the other cases provide examples where the group $\Gamma_f$ (defined in \[DX1\] and discussed in Section 11) is a semi-direct product of a torus and a group of order two. It is natural to ask for which integers there are sharp examples with this additional symmetry. One then considers a different compressed sensing problem. The allowed polynomials are $(xy)^a (x^b+y^b)$ and one proceeds in a similar fashion. In other words, one seeks constants $c[a,b]$ such that the following hold: 1. $ \sum_{a,b} c[a,b] (x y)^a (x^b + y^b) = 1$ on the line $x+y=1$. 2. $c[a,b] \ge 0$ for each $(a,b)$. 3. $2a + b \le d$ for each $(a,b)$ but $2a+b =d$ for some $(a,b)$. 4. The number of non-zero monomials is as small as possible. (When $b \ne 0$, a nonzero $c[a,b]$ contributes two monomials.) Proposition 5.1 allows us to decrease the number of variables. By Theorem 5.2, the smallest possible number of terms is ${d+3 \over 2}$. This value is achieved when $d=1,3,7, 19$. The third author has given a computer assisted verification that, up to degree $31$ and except for degrees $1,3,7,19$, the minimum number of terms [exceeds]{} ${d+3 \over 2}$. The first author hopes to discuss this problem in future work. Source dimension at least three =============================== When the source dimension is at least three, things are in some ways less interesting. The following results provide monomials for which $N$ is as small as possible given $d$. These polynomials are both easier to find and less interesting than the polynomials in Theorem 5.1. Assume $n \ge 3$. For $d \ge 1$, put $N= d(n-1)+1$. Then there is a monomial proper map $f: {\mathbb B}_n \to {\mathbb B}_N$ of degree $d = {N-1 \over n-1}$. Equivalently, there is a monomial $p$ in $n$ real variables, with non-negative coefficients, such that $p(x)=1$ on $s(x)=1$. Put $x = ({\bf t},u)$, where $u=x_n$ and ${\bf t} = (x_1,...,x_{n-1})$. Put $t= \sum_{j=1}^{n-1} x_j$. Then $s(x) = t+u$. For each positive integer $d$, we define a polynomial $p$ by $$p(t,u) = t(1+u+u^2 + ... + u^{d-1}) + u^d.$$ Then $p$ is of degree $d$, and all of its coefficients are non-negative. There are precisely $d(n-1)+1$ non-vanishing coefficients. On the set where $t+u=1$ we have $$p(t,u) = p(t,1-t) = t\left( {1- (1-t)^d \over 1-(1-t)}\right) + (1-t)^d = 1.$$ Replacing $(x_1,...,x_{n-1})$ by $(\|z_1\|^2,..., \|z_{n-1}\|^2)$ and $u$ by $\|z_n\|^2$ yields the squared norm of the desired map $f$. Maps as in the Proposition are called [Whitney maps]{}, because they generalize the map $$(x_1,...,x_n) \to (x_1,...,x_{n-1}, x_1 x_n,..., x_{n-1} x_n, x_n^2) \eqno (W)$$ studied by Whitney. The maps in (W) are proper maps from ${\mathbb R}^n$ to ${\mathbb R}^{2n-1}$. The following is one of the main results in \[LP2\]. Assume $n \ge 3$ and $f$ has degree $d$. Then $d \le {N-1 \over n-1}$. When $n\ge 4$, a complete list of the monomial sphere maps for which $d= {N-1 \over n-1}$ is known. We proceed analogously as the case $n=2$. Assume $n \ge 3$. Fix a positive integer $d$. Let $K$ denote the dimension of the vector space of polynomials of degree at most $d$ in $n$ variables with no constant term. Let $k$ denote the dimension of the space of homogeneous polynomials of degree $d$ in $n$ variables. We call the last $k$ variables in ${\mathbb R}^K$ distinguished. Let ${\bf v}$ denote the vector in ${\mathbb R}^k$ whose components are the multinomial coefficients. As before, the transformation $T$ arises via homogenization. We consider the linear system: - $T{\bf u}= {\bf v}$. - At least one (hence at least $n$) of the distinguished variables is not zero. - Each $u_j$ is non-negative. Let $N$ denote the minimum $L^0$ norm of the solution. Combining Proposition 9.1 and Theorem 9.2 tells us that $N = d(n-1) + 1$. The linear algebra problem again simplifies using homogenization techniques. Sparseness constraints ====================== The context of proper mappings between balls provides a new issue in understanding sparseness for these linear systems. Given a proper rational mapping in source dimension $n$, not every value of $N$ arises for [target-minimal maps]{}. We can achieve $N=1$ for non-constant rational sphere maps only when $n=1$. Thus, if $n\ge 2$, the value of $N$ cannot lie in the interval $1 < N < n$. This fact is easy to see using complex variable theory. It is harder to see, but still true, that the range $n < N < 2n-2$ cannot arise either. There is a general conjecture on the gaps that are possible. See \[HJX\] and \[HJY\] for work on the [gap conjecture]{}. We state and sketch the proof of a result from \[DL1\]. Fix the source dimension $n$. If $N \ge n^2 - 2n+2$, there is a target-minimal monomial example with $L^0$ norm equal to $N$. Put $T(n)= n^2-2n+2$. For each $N$ with $N \ge T(n)$, there is a target-minimal proper monomial map $f:{\mathbb B}_n \to {\mathbb B}_N$. The proof for $n=1$ is easy. We simply take an $N$-tuple of positive numbers $c_j^2$ whose sum is $1$. Use these numbers to define $f$ by $$f(z) = (c_1 z,c_2 z_2,...,c_N z^N).$$ For $n\ge 2$ we proceed as follows. For $x \in {\mathbb R}^m$, put $s(x)= \sum x_j$. Given a polynomial $p(x)$ with non-negative coefficients and with $p(x)=1$ on the line $s(x)=1$, and $c>0$, we may define new polynomials with the same properties by $$Wp(x) = p(x) - c x_n^d + c x_n^d s(x)$$ $$Vp(x) = p(x) - {c \over 2} x_n^d + {c \over 2} x_n^d s(x).$$ Regard $W$ and $V$ as operations which we may iterate, always applying them on the pure term of highest degree, we form $V^kW^js(x)$. One can then show for $N\ge T(n)$ that we obtain an example with the desired number of terms. See \[DL1\] for details. We mention several related results. Let $p$ be a rational sphere map of degree $d$ and source dimension $n$. Then the terms of degree exactly equal to $d$ must map into a subspace of dimension at least $n$. On the other hand, if $p$ is a [homogeneous]{} polynomial sphere map of degree $d$, then the target dimension of $p$ must be at least the binomial coefficient ${n+d-1 \choose d}$. The maps from Corollary 5.2 and Proposition 9.1 illustrate an interesting phenomenon. For these maps, which are not homogeneous for $d\ge 3$, the terms of highest degree map into a space of dimension at least (the source dimension) $n$, but no larger. For a typical rational sphere map, the terms of highest degree themselves already map into a space too large for the target dimension to be as small as possible. In other words, the solution of the compressed sensing problem must have enough terms of degree $d$ but not too many. Groups associated with mappings =============================== The papers \[DX1\] and \[DX2\] and the book \[D3\] associate groups with holomorphic mappings. These groups have many uses; we mention only those that directly bear on our discussion. Let ${\rm Aut}({\mathbb B}_n)$ denote the group of holomorphic automorphsims of the ball ${\mathbb B}_n$. Let $f:{\mathbb B}_n \to {\mathbb B}_N$ be a proper rational mapping. Then $A_f$ is the subgroup of ${\rm Aut}({\mathbb B}_n) \times {\rm Aut} ({\mathbb B_N})$ consisting of those pairs $(\gamma, \psi)$ for which $$f \circ \gamma = \psi \circ f. \eqno (29)$$ The Hermitian source group $\Gamma_f$ is the projection of $A_f$ onto its first factor; the Hermitian target group $T_f$ is the projection onto the second factor. It follows from the work in \[DX2\] that $f$ is target-minimal if and only if there is a group homomorphism for $\Gamma_f$ to $T_f$. When $f$ is target-minimal, and $\gamma \in \Gamma_f$, then the automorphism $\psi = \Phi(\gamma)$ from (29) is uniquely determined. The map $\Phi$ is easily seen to be a group homomorphism. When $f$ is not target-minimal, the automorphism $\psi$ is not uniquely determined. Therefore the groups provide information about the minimum target dimension $N$, the primary issue in this paper. The maps defined in (15) are all target-minimal. We compute this homomorphism in the degree $3$ case. The other cases are similar. Put $f(z,w) = (z^3, \sqrt{3}zw, w^3)$. Then $f:{\mathbb B}_2 \to {\mathbb B}_3$ is a proper holomorphic map. Here $\Gamma_f$ is the group generated by the diagonal unitary matrices and the element of order two interchanging the variables. Here $T_f$ is the group generated by $$\begin{pmatrix} e^{i 3 \alpha} & 0 & 0 \cr 0 & e^{i(\alpha+\beta)} & 0 \cr 0 & 0 & e^{i 3 \beta} \end{pmatrix}$$ $$\begin{pmatrix} 0 & 0 & 1 \cr 0 & 1 & 0 \cr 1 & 0 & 0 \end{pmatrix}.$$ The kernel of $\Phi$ is the cyclic group of order three generated by $$\begin{pmatrix} \eta & 0 \cr 0 & \eta^2 \end{pmatrix}$$ where $\eta$ is a primitive cube root of $1$. Here ${\rm kernel}(\Phi)$ is the set of $\gamma$ for which $f \circ \gamma = f$. These groups can be used to decide when a proper map between balls is equivalent to a monomial map. We have the following general results: Let $f:{\mathbb B}_n \to {\mathbb B}_N$ be a rational proper mapping. Then: - $\Gamma_f = {\rm Aut}({\mathbb B}_n)$ if and only if $f$ is a linear fractional transformation. - $\Gamma_f$ is noncompact if and only if $\Gamma_f = {\rm Aut}({\mathbb B}_n)$. Otherwise $\Gamma_f$ is contained in a conjugate of ${\bf U}(n)$. - $\Gamma_f$ is a conjugate of ${\bf U}(n)$ if and only if $f$ is equivalent to a juxtaposition of tensor powers. - $\Gamma_f = {\bf U}(n)$ if and only if $f$ is a juxtaposition of tensor powers. - $\Gamma_f$ contains an $n$-torus if and only if $f$ is equivalent to a monomial map. - Let $G$ be a finite subgroup of ${\rm Aut}({\mathbb B}_n)$. Then there is an $N$ and a proper rational map $f:{\mathbb B}_n \to {\mathbb B}_N$ such that $\Gamma_f = G$. - Let $G$ be a finite subgroup of ${\bf U}(n)$. Then there is an $N$ and a proper polynomial map $f:{\mathbb B}_n \to {\mathbb B}_N$ such that $\Gamma_f = G$. The last two parts of this theorem suggest additional questions of the sort considered throughout this paper. Given a finite subgroup $G$ of the unitary group, there is a map $f$ for which $\Gamma_f=G$. Relating either the smallest possible degree or the minimum possible target dimension to the group seems difficult. In \[D3\] it is noted that a map $f$ for which $\Gamma_f$ is trivial must be of degree at least $3$ and that $3$ is possible. Open problems ============= 1. Assume $n\ge 2$. Let $f:{\mathbb C}^n \to {\mathbb C}^N$ be a rational sphere map of degree $d$. When $n=2$, prove that $d \le 2N-3$. When $n\ge 3$, prove that $d \le {N-1 \over n-1}$. 2. Put $n=2$. For each odd degree $d$, find the minimum $N=N(d)$ for which there is a monomial sphere map $f:{\mathbb C}^2 \to {\mathbb C}^N$ of degree $d$ with $\\|f(z,w)\\|^2 = \\|f(w,z)\\|^2$. For example, when $d=1,3,7,19$ the answer is ${d+3 \over 2}$ but for other small odd $d$ this minimum cannot be achieved. Also, is the number of odd $d$ for which $N(d)= {d+3 \over 2}$ finite or infinite? See Remark 8.3 for more discussion. 3. Put $n=2$. Suppose that $p(x,y)$ is a sharp polynomial of degree $d$. What is the minimum value of the $L^1$ norm $p(1,1)$? 4. Let $S$ be the largest subset of ${\bf S}_d$ such that each $f\in S$ contains the monomials $x^d$ and $y^d$ with coefficient $1$. Find the minimum value of $f(1,1)$. See Example 8.2 and Remark 8.3. In other words, find the minimum $L^1$ norm of all solutions to the sensing problem. Stated otherwise, given the degree $d$, we wish to minimize a linear function (the sum) of the coefficients $c[a,b]$ subject to the first three constraints from Remark 8.3. Proposition 8.2 considerably reduces the search space. Can one characterize the polynomials realizing the minimum? 5. If the previous problems are too difficult, can one find asymptotic relations between the $L^1$ and $L^0$ norms as the degree increases? Compare with \[Do2\]. 6. Find all sharp monomial sphere maps when $n=3$; the answer is known for $n\ge 4$. See Proposition 9.1, Theorem 9.1, and \[LP\]. 7. Express the results from \[BEH\] and \[BH\] in the language of compressed sensing. 8. Extend the ideas of this paper to the hyperquadric setting. See for example \[Gr\], \[GLV\], \[LP1\], and \[LP2\]. 9. Relate the gap conjecture of Huang-Ji (See \[HJX\] and \[HJY\]) to compressed sensing. 10. Given a finite subgroup $G$ of ${\bf U}(n)$, what is the minimum possible degree of a rational sphere map $f$ with $\Gamma_f = G$? Can one relate this problem to known results in invariant theory? bibliography ============ \[BEH\] M. S. Baouendi, P. F. Ebenfelt, and X. Huang, Holomorphic mappings between hyperquadrics with small signature difference, [Amer. J. Math.]{} 133 (2011), no. 6, 1633-1661. \[BH\] M. S. Baouendi and X. Huang, Super-rigidity for holomorphic mappings between hyperquadrics with positive signature., [J. Differential Geom.]{} 69 (2005), no. 2, 379-98. \[CD\] D. W. Catlin and J. P. D’Angelo, A stabilization theorem for Hermitian forms and applications to holomorphic mappings, [Math. Res. Lett.]{} 3 (1996), no. 2, 149-166. \[D1\] J. D’Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces, CRC Press, Boca Raton, Fla., 1992. \[D2\] J. D’Angelo, Proper holomorphic mappings, positivity conditions, and isometric imbedding, [J. Korean Math Society]{}, May 2003, 1-30. \[D3\] J. D’Angelo, Hermitian Analysis: from Fourier series to CR Geometry, second edition, Springer, 2019. \[DKR\] J. D’Angelo, S. Kos, and E. Riehl, A sharp bound for the degree of proper monomial mappings between balls, [J. Geom. Anal.]{} 13 (2003), no. 4, 581-593. \[DL\] J. D’Angelo and J. Lebl, Homotopy equivalence for proper holomorphic mappings, [Adv. Math.]{} 286 (2016), 160-180. \[DL1\] J. D’Angelo and J. Lebl, On the complexity of proper mappings between balls, [Complex Variables and Elliptic Equations]{}, Volume 54, Issue 3, Holomorphic Mappings (2009), 187-204. J. D’Angelo and J. Lebl, Complexity results for CR mappings between spheres, [Int. J. of Math.]{}, Vol. 20, No. 2 (2009), 149-166. \[DX1\] J. D’Angelo and M. Xiao, Symmetries in CR complexity theory, [Adv. Math.]{} 313 (2017), 590-627. \[DX2\] J. D’Angelo and M. Xiao, Symmetries and regularity for holomorphic maps between balls, [Math. Res. Lett.]{}, Vol. 25, no. 5 (2018), 1389-1404. \[Do1\] D. Donoho, Compressed sensing, [IEEE Trans. Inform. Theory]{}, Vol. 52, No. 4 (2006), 1289-1306. \[Do2\] D. Donoho, For most large underdetermined systems of linear equations the minimal $L^1$-norm solution is also the sparsest solution, [Comm. Pure Appl. Math.]{}, Vol. 59, no. 6 (2006), 797-829. \[F\] F. Forstnerič, Extending proper holomorphic maps of positive codimension, [Inventiones Math.]{}, 95(1989), 31-62. \[Gr\] D. Grundmeier, Signature pairs for group-invariant Hermitian polynomials, [Internat. J. Math.]{} 22 (2011), no. 3, 311-343. \[GLV\] D. Grundmeier, J. Lebl, and L. 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Lebl, and H. Peters, Polynomials constant on a hyperplane and CR maps of spheres, [Illinois J. Math.]{} 56 (2012), no. 1, 155-175. \[PWZ\] M. Petkovsek, H. Wilf, and D. Zeilberger, $A=B$. A. K. Peters, Ltd., Wellesley, MA, 1996. \[W\] S. Wono, Proper Holomorphic Mappings in Several Complex Variables, PhD thesis, Univ. Illinois, 1993.
--- author: - 'Vitalii Ozvenchuk [^1]' - Elena Bratkovskaya - Olena Linnyk - Mark Gorenstein - Wolfgang Cassing title: 'Dynamical equilibration in strongly-interacting parton-hadron matter' --- Introduction {#intro} ============ Nucleus-nucleus collisions at ultra-relativistic energies are studied experimentally and theoretically to obtain information about the properties of hadrons at high density and/or temperature as well as about the phase transition to a new state of matter, the quark-gluon plasma (QGP). Whereas the early ‘big-bang’ of the universe most likely evolved through steps of kinetic and chemical equilibrium, the laboratory ‘tiny bangs’ proceed through phase-space configurations that initially are far from an equilibrium phase and then evolve by fast expansion. On the other hand, many observables from strongly-interacting systems are dominated by many-body phase space such that spectra and abundances look ‘thermal’. It is thus tempting to characterize the experimental observables by global thermodynamical quantities like ’temperature’, chemical potentials or entropy [@Ref1; @Ref2; @Ref3; @Ref4; @Ref5; @Ref6; @Ref7; @Ref8]. We note, that the use of macroscopic models like hydrodynamics [@Ref9; @Ref10; @Ref11; @Ref12] employs as basic assumption the concept of local thermal and chemical equilibrium. The crucial question, however, how and on what timescales a global thermodynamic equilibrium can be achieved, is presently a matter of debate. Thus nonequilibrium approaches have been used in the past to address the problem of timescales associated to global or local equilibration [@Ref13; @Ref14; @Ref15; @Ref16; @Ref17; @Ref18; @Ref19; @Ref20; @Ref21]. In view of the increasing ‘popularity’ of thermodynamic analyses a thorough microscopic study of the questions of thermalization and equilibration of confined and deconfined matter within a transport approach appears necessary. The model {#sec:model} ========= In this contribution, we study the kinetic and chemical equilibration in ‘infinite’ parton-hadron matter within the novel Parton-Hadron-String Dynamics (PHSD) transport approach [@Ref22; @Ref23], which is based on generalized transport equations on the basis of the off-shell Kadanoff-Baym equations [@Ref24; @Ref25] for Green’s functions in phase-space representation (in the first order gradient expansion, beyond the quasiparticle approximation). In the KB theory, the field quanta are described in terms of propagators with complex self-energies. Whereas the real part of the self-energies can be related to mean-field potentials, the imaginary parts provide information about the lifetime and/or reaction rates of time-like “particles”. The basis of the partonic phase description is the dynamical quasiparticle model (DQPM) [@Ref26; @Ref27] matched to reproduce lattice QCD results – including the partonic equation of state – in thermodynamic equilibrium [@Ref24]. In fact, the DQPM allows a simple and transparent interpretation for thermodynamic quantities as well as correlators – measured on the lattice – by means of effective strongly interacting partonic quasiparticles with broad spectral functions. The transition from partonic to hadronic degrees of freedom is described by covariant transition rates for fusion of quark-antiquark pairs or three quarks (antiquarks), obeying flavor current conservation, color neutrality as well as energy-momentum conservation. ![Snapshot of the spatial distribution of hadrons (blue) and partons (red) at an evolution time of 40 fm/c after the systems was initialized by solely partons at an energy density below critical.[]{data-label="fig1"}](1.eps){width="50.00000%"} The ‘infinite’ matter is simulated within a cubic box with periodic boundary conditions initialized at various values for baryon density (or chemical potential) and energy density. The size of the box is fixed to $9^3$ fm$^3$. The initialization is done by populating the box with light ($u,d,s$) quarks, antiquarks and gluons with random space positions and the momenta distributed according to the Fermi-Dirac distribution. The total numbers of the quarks and antiquarks are chosen so that the system with various desired values of the energy density $\varepsilon$ (the total energy of the particles divided by the size of the box) and baryon potential $\mu$ can be studied. ![Snapshot of the spatial distribution of light quarks and antiquarks (red), strange quarks and antiquarks (green) and gluons (blue) at a time of 40 fm/c. []{data-label="fig2"}](2.eps){width="50.00000%"} In the course of the subsequent transport evolution of the system by PHSD, the numbers of gluons, quarks and antiquarks are adjusted dynamically through the inelastic collisions to equilibrium values, while the elastic collisions lead to eventual thermalization of all the particle species (e.g. $u,d,s$ quarks and gluons, if the energy density in the system is above critical). Please note that if the energy density in a local cell drops below critical either due to the local fluctuations or because the system was initialized with a low enough number of partons, a transition from initial pure partonic matter to hadronic degrees of freedom occurs dynamically by interactions. Results {#sec:res} ======= ![The reaction rates for elastic parton scattering (blue), gluon splitting (green) and flavor neutral $q\bar{q}$ fusion (red) as a function of time. []{data-label="fig3"}](3.eps){width="50.00000%"} We present in the Fig. \[fig1\] a snapshot of the spatial distribution of hadrons (blue) and partons (red) at an evolution time of 40 fm/c after the systems was initialized by solely partons at an energy density of 0.4 GeV/fm$^3$. At this energy density – slightly below the critical energy density for the deconfinement phase transition – most of the partons have already formed hadrons at this point of the the evolution of the system, but a small fraction of partons have yet not hadronized. The remaining partons are approximately in the thermal equilibrium with the hadrons. ![The energy spectra for the off-shell u (red) and s quarks (green) and gluons (blue) in equilibrium for a system initialized at an energy density of 5.37 GeV/fm$^3$.[]{data-label="fig4"}](4.eps){width="50.00000%"} In Fig. \[fig2\], we show a snapshot of the systems that has been initialized at an energy density of 2.18 GeV/fm$^3$, which is clearly above the critical energy density. One can see in Fig. \[fig2\] the spatial distribution of light quarks and antiquarks (red), strange quarks and antiquarks (green) and gluons (blue) at a time of 40 fm/c. At this energy density no hadrons are seen. The different parton species are in thermal and chemical equilibrium. ![Abundances of the u,d s quarks+antiquarks and gluons as a function of time for a system initialized at an energy density of 9.43 GeV/fm$^3$. []{data-label="fig5"}](6.eps){width="50.00000%"} After a few fm/c the system – initialized at an energy density of 2.87 GeV/fm$^3$ – has achieved chemical and thermal equilibrium, since the reactions rates are practically constant and obey detailed balance for gluon splitting and $q\bar{q}$ fusion. This is shown in Fig. \[fig3\], where the reaction rates for elastic parton scattering (blue), gluon splitting (green) and flavor neutral $q\bar{q}$ fusion (red) are presented as a function of time. ![The average abundances of hadrons (blue) and partons (red) in equilibrium as functions of the energy density $\varepsilon$. []{data-label="fig6"}](5.eps){width="50.00000%"} Another indication that the system has achieved the thermal equilibrium is seen in the distribution of the kinetic energy of the particles. We show in Fig. \[fig4\] the energy spectra for the off-shell $u$ (red) and $s$ quarks (green) and gluons (blue) in equilibrium for a system initialized at an energy density of 5.37 GeV/fm$^3$. The spectra may well be described by a Boltzmann distribution with temperature T=240 MeV in the high energy regime. The deviations from the Boltzmann distribution at low energy E are due to the broad spectral functions of the partons. On the other hand, a sign of the chemical equilibration is the stabilization of the abundances of the different species. In Fig. \[fig5\], we show the particle abundances as a function of time for a system initialized at 9.43 GeV/fm$^3$. One can see that the chemical equilibration is reached after about 15 fm/c. It is interesting to observe in Fig. \[fig6\] the average abundances of hadrons (blue) and partons (red) in equilibrium as functions of the energy density $\varepsilon$. At energy densities below critical ($\approx$ 0.5 GeV/fm$^3$) the system evolves into a state, which is dominated by hadrons and has a very small fraction of partons due to rare fluctuations of local energy density to high values. At higher energy densities, the system is in a QGP final state, with a small hadron admixture. At high enough energy (above approx. 2 GeV/fm$^3$) we find that hadron fraction is negligible. In the regime of energy densities from 0.48 to 1.3 GeV/fm$^3$ the calculations have provided so far no stable equilibrium over time due to large fluctuations between hadronic and partonic configurations. Further studies are on the way. Conclusions {#sec:conclusions} =========== We have studied the kinetic and chemical equilibration in ‘infinite’ parton-hadron matter within the Parton-Hadron-String Dynamics transport approach (PHSD), which is based on a dynamical quasiparticle model for partons (DQPM) matched to reproduce lattice-QCD results – including the partonic equation of state – in thermodynamic equilibrium. The ‘infinite’ matter has been simulated within a cubic box with periodic boundary conditions initialized at different baryon density (or chemical potential) and energy density. Depending on the energy density, the system evolved into an ensemble of either hadrons or partons in chemical and thermal equilibrium. Abundances of the final particles depend on the energy density. The temperature of the degrees of freedom in the final state was measured by fitting the slopes of the Boltzmann-like tails of the thermal distributions of their kinetic energy. Acknowledgements {#acknowledgements .unnumbered} ================ V.O. is grateful for the financial support by the Helmholtz “Quark Matter" graduate school “H-QM". O.L., E.B. and M.G. acknowledge the support by the “HIC for FAIR" framework of the “LOEWE" program. [99]{} P.Braun-Munziger, J.Stachel, J.P.Wessels, N.Xu, Phys. Lett. B344, (1995) 43 P.Braun-Munziger, J.Stachel, J.P.Wessels, N.Xu, Phys. Lett. B365, (1996) 1 J.Stachel, Nucl. Phys. A654, (1999) 119c J.Cleymans, H.Satz, Z.Phyc. C57, (1993) 135 J.Sollfank, M.Gazdzicki, U.Heinz, J.Rafelski, Z. Phys. C61, (1994) 659 F.Becattini, M.Gazdzicki, J.Sollfank, Eur. Phys. J. C5, (1998) 143 C.Spieles, H.Stöcker, C.Greiner, Eur. Phys. J. C2, (1998) 351 J.Cleymans, H.Oeschler, K.Redlich, J.Phys. G25, (1999) 281 H.Stöcker, W.Greiner, Phys. Rep. 137, (1986) 277 U.Ornik et al., Phys. Rev. C54, (1996) 1381 S.Bernard et al., Nucl. Phys. A605, (1996) 566 J.Sollfank et al., Phys. Rev. C55, (1997) 392 P.Koch, B.Müller, J.Rafelski, Phys. 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--- bibliography: - 'auto\_generated.bib' title: 'Measurement of the production cross section in pp collisions at $\sqrt{s}=7$with lepton + jets final states' --- =1 $Revision: 170996 $ $HeadURL: svn+ssh://svn.cern.ch/reps/tdr2/papers/TOP-11-003/trunk/TOP-11-003.tex $ $Id: TOP-11-003.tex 170996 2013-02-11 18:19:52Z alverson $ Introduction ============ Since the discovery of the top quark at the Fermilab Tevatron collider [@top-discovery-cdf; @top-discovery-d0], considerable advances have been made in understanding its production rates and decay properties in $\Pp\Pap$ collisions. The advent of pp collisions at the Large Hadron Collider (LHC) [@lhc] has started a new phase of top quark physics, and the first measurement at the higher center-of-mass energy of 7was the top quark pair production cross section [@top-xsec-cms; @top-xsec-dil-cms; @top-xsec-atlas; @top-xsec-dil-atlas]. A precise measurement of the cross section provides constraints for QCD calculations presently available up to approximate next-to-next-to-leading order (NNLO) [@Aliev:2010zk; @Langenfeld:2009wd; @Kidonakis:2010dk; @Ahrens]. It is also important for probing new physics processes that can manifest themselves as an enhancement of the production rate. In this article, we present a precise measurement of the production cross section in pp collisions at $\sqrt{s}= 7\TeV$ utilizing a data set corresponding to an integrated luminosity of 2.3recorded by the Compact Muon Solenoid (CMS) experiment at the LHC. In the standard model (SM), top quarks are produced in pp collisions predominantly via the strong interaction as pairs, with each top quark decaying almost exclusively into a W boson and a bottom quark. In the analysis presented here, events are identified in final states in which one of the W bosons decays into a quark pair and the other into a charged lepton (electron or muon) and a neutrino, resulting in events that contain an electron or a muon, a neutrino, and four hadronic jets, two of which result from hadronization of the b and $\cPaqb$ quarks (b-jets). In order to improve the purity of the candidate event sample, we employ b-tagging algorithms, which are optimized for identification of b-jets. Decays of W bosons into $\tau$ leptons are not specifically selected in this analysis, albeit some events enter the event sample due to leptonic decays of the $\tau$. The technique for measuring the cross section from the candidate event sample consists of a simultaneous profile likelihood fit to the distribution of invariant masses of particles belonging to identified displaced vertices. These fits are performed as a function of the jet and b-tag multiplicities in the event. The method is similar to the one that was used in a previous CMS measurement [@top-xsec-cms], though a larger data sample is now studied. Several alternative methods have been employed. In one of these, we perform an inclusive measurement of production cross section without b-jet identification requirement, while others incorporate different b-tagging algorithms. The CMS Detector ================ The characteristic feature of the CMS detector is a superconducting solenoid of 6 in diameter, providing an axial magnetic field of 3.8. Charged particle trajectories are measured by the silicon pixel and strip subdetectors, covering $0 < \phi < 2\pi$ in azimuth and $\|\eta\| <~2.5$, where the pseudorapidity $\eta$ is defined as $\eta = - \ln[\tan\theta/2]$, with $\theta$ being the polar angle of the trajectory of the particle with respect to the counterclockwise-beam direction. Within the field volume, the silicon detectors are surrounded by a crystal electromagnetic calorimeter and a brass/scintillator hadron calorimeter that provide high resolution energy measurement of photons, electrons and hadronic jets. Muon detection systems are located outside of the solenoid and embedded in the steel return yoke. They provide muon detection in the range $\|\eta\| < 2.4$. A two-tier trigger system selects the most interesting pp collision events for use in physics analysis. A detailed description of the CMS detector can be found in Ref. [@cms]. Event selection \[sec:EventSelection\] ====================================== The sample of candidate events is collected using dedicated triggers, which require either a muon with transverse momentum ($\pt$) larger than 30or a high-$\pt$ electron. The criteria for the electron trigger evolved during the course of data-taking in order to maintain a reasonable trigger rate as the instantaneous luminosity of the LHC increased. For the initial data set, corresponding to an integrated luminosity 0.9, the threshold on the of electron candidates varied between 27 and 32. For the second part of the data set (1.4) the trigger required the presence of an electron with $\pt > 25$and at least three hadronic jets with $\pt > 30$. The recorded events are reconstructed using the CMS particle-flow algorithm [@ipartf], which categorizes observable particles into muons, electrons, photons, charged and neutral hadrons. Energy calibration is performed separately for each particle type. In the offline selection, muons are required to have a good-quality track with $\pt > 35\GeV$ and $\|\eta\| <2.1$, and the reconstructed tracks in the silicon tracker are consistent with the track information from the muon systems [@CMS_mu]. Electrons are identified using a combination of the shower shape information in electromagnetic calorimeter and track-cluster matching [@CMS_e], and are required to have $\pt > 35\GeV$ and $\|\eta\| < 2.5$. Electron candidates in the transition region between the barrel and forward electromagnetic calorimeters, $1.44 < \|\eta\| < 1.57$, are not used for the measurement. We also reject electrons coming from photon conversions [@CMS_e]. Since the lepton from a W decay is expected to be isolated from other activity in the event, we apply isolation requirements. The relative isolation is defined as $I_{\text{rel}} =\big(\sum \ET^{\text{charged}} + \sum \ET^{\text{photon}} + \sum \ET^{\text{neutral}}\big)/\pt$, where $\pt$ is the lepton transverse momentum, and $\ET^{\text{charged}}$, $\ET^{\text{photon}}$, and $\ET^{\text{neutral}}$ are transverse energies of the charged particles, the reconstructed photons, and the neutral particles not identified as photons. The sum of the transverse energies is computed in a cone of size $\Delta R =\sqrt{(\Delta\phi)^2+(\Delta\eta)^2} = 0.3$ around the lepton direction, excluding the lepton candidate itself. We require $I_{\text{rel}}$ to be less than 0.125 for muons and 0.10 for electrons. The signal events are required to have only one electron or muon whose origin is consistent with the reconstructed primary pp interaction vertex [@CMS_vertex], defined as the vertex with the largest value for the scalar sum of the $\pt$ of the associated tracks. Events with an additional electron or muon candidate that satisfies less strict lepton identification requirements are vetoed. Jets are reconstructed using the particle-flow algorithm and are clustered using the anti-jet technique [@ktalg] with a distance parameter of 0.5, as implemented in <span style="font-variant:small-caps;">FastJet</span> v2.4.2 [@fastjet1; @fastjet2]. In order to account for extra activity within a jet cone from multiple pp interactions per beam crossing, referred to as a pileup, jet energies are corrected for charged hadrons that originate from a vertex other than the primary one, and for the amount of pileup expected in the jet area from neutral jet constituents. Jet energies are also corrected for non-linearities due to different responses in the endcap and barrel calorimeters, and differences between true and simulated calorimeter responses [@jec_11]. Each jet is required to have a transverse momentum $\pt > 35\GeV$ and $\|\eta\| < 2.4$. We select events with at least one jet, or at least three jets for events collected with the electron + jets trigger. To reduce background processes, we require at least one of the jets to be identified as a b-jet by a displaced secondary vertex algorithm known as Simple Secondary Vertex High Efficiency [@CMS_btag] with a medium working point. The algorithm has a b-tag efficiency of 55% and a light parton (u, d, s, g) mistag rate of 1.5%. In addition, events are required to have a significant amount of missing transverse energy $(\ETslash)$ as evidence of a neutrino from the W boson decay. This is defined as the magnitude of the negative vector sum of the transverse momenta of all of the objects found by the particle-flow algorithm. We require $\ETslash > 20\GeV$ for both the electron + jets and muon + jets channels. Signal and Background Modeling\[sec:sigback\] ============================================= Pair production of top quarks is modeled using the <span style="font-variant:small-caps;">MadGraph v5.1.1</span> [@Alwall:2011] Monte Carlo (MC) event generator, assuming the mass of the top quark m$_\cPqt = 172.5$. The top quark pairs are generated with up to three additional hard jets using v6.424 with tune Z2 [@Sjostrand:2006za] to model parton-showering (PS), and the shower matching is performed using the -MLM prescription [@Alwall:2011]. The generated events are further passed through the full CMS detector simulation based on [@Geant4]. The presence of pileup is incorporated by simulating additional interactions with a multiplicity matching that observed in data. Leptonically decaying W + jets events constitute by far the largest background. These together with Z + jets events are also generated using with up to four jets subject to the matrix-element (ME) description. The W + jets events are generated inclusively with respect to jet flavor. Reconstructed jets are further matched to partons in the simulation, and the W + bottom quark and W + charm quark components are separated from the W + light-flavor (u, d, s, and gluon) component based on the parton flavor. Other backgrounds include single-top-quark production, simulated with <span style="font-variant:small-caps;">powheg v1.0</span> [@powheg1; @powheg2; @powheg3], QCD multijet simulated with , and photon + jet events, which constitute a background for the electron + jets channel, generated by . The set of parton distribution functions used by is CTEQ6L1 [@cteq_2010], while and use CTEQ6M [@cteq_2010]. The W and Drell–Yan production processes are normalized based on NNLO cross sections, determined using <span style="font-variant:small-caps;">fewz</span> [@fewz]. They correspond to $\sigma_{\PW\to\ell \nu} = 31.3 \pm 1.6\unit{nb}$ and $\sigma_{\cPZ/\gamma^\to\ell \ell} = 3048\pm 132\unit{pb}$, where for the Drell–Yan production the invariant mass of two leptons ($\ell = \Pe$ or $\mu$) is greater than 50. The single-top-quark $t$-channel production is normalized to the recent CMS measurement of $\sigma_{\cPqt} = 67.2 \pm 6.1\unit{pb}$ [@cms_SingleTop]. The single-top-quark associated production $(\cPqt\PW)$ is normalized to the approximate NNLO cross section $\sigma_{\cPqt\PW} = 15.7 \pm 1.2\unit{pb}$ [@Kidonakis:2010tW], and the $s$-channel is normalized to the next-to-next-to-leading-logarithm prediction of $\sigma_{\cPqt} = 4.6 \pm 0.2\unit{pb}$ [@NNLLtop]. The QCD multijet normalization is obtained by fitting SM contributions to the full $\ETslash$ distribution in data, though only the yield of QCD multijet events with $\ETslash > 20$enters the normalization. For the electron + jets channel, the QCD multijet background distributions are obtained from MC, and for the muon + jets channel, they are obtained from a background-enriched data sample defined as $I_{\text{rel}} > 0.125$ and $\ETslash <20\GeV$. Cross Section Measurement ========================= The cross section measurement is performed using a maximum profile likelihood fit to the number of reconstructed jets ($N_{\text{jet}}$), the number of b-tagged jets ($N_{\text{tag}}$), and the secondary vertex mass (SVM) distribution in the data. We consider ten event subsamples with $N_{\text{jet}}$ values of 1–4 and ${\ge}5 $, and $N_{\text{tag}}$ values of 1 and ${\ge}2$. The SVM is defined as the mass of the sum of four-vectors of the tracks associated to the secondary vertex with an assumption that all particles have the pion mass. For events with two b-tagged jets, SVM corresponds to the highest-$\pt$ b-tagged jet. The SVM distribution yields a good discrimination between the contributions from light- and heavy-flavor quark production [@top-xsec-cms]. The results are obtained by maximizing a binned Poisson likelihood that incorporates contributions from , W + jets, Z + jets, single-top-quark, and QCD multijet production processes. Performing a simultaneous fit across different jet and b-tag multiplicity bins, including regions dominated by background events, constrains the background contributions, resulting in a more precise measurement of the production cross section. The W + jets, Z + jets, and single-top-quark background processes are initially normalized to the expected event yields according to their theoretical cross sections. The QCD and photon + jets normalizations are evaluated as described above individually in each $N_{\text{jet}}$ and $N_{\text{tag}}$ subsample, for both channels. These background normalizations are the initial values that enter the profile likelihood fit. The cross section measurement is performed by fitting to the data to obtain corrections to these initial values. The W + jets backgrounds are split into W + b jets, W + c jets, and W + light-flavor (LF) sub-samples, with all three components free in the fit. During the likelihood maximization, the normalizations of each of these components are extracted. The normalizations of the and W + jets contributions are allowed to float freely. The contributions from small backgrounds, QCD multijet and Z + jets, are conservatively constrained with Gaussian uncertainties of 100% and 30% of their expected event yields, respectively. The single-top-quark contribution is constrained with an uncertainty of 10% [@cms_SingleTop]. The expected event yield for each background component, per $N_{\text{jet}}$ and $N_{\text{tag}}$, is also a function of other parameters, such as the jet energy scale (JES), the b-tagging efficiency and the mistag rate. In addition, the $N_{\text{jet}}$ spectrum is affected by the choice of the renormalization and factorization ($Q^2$) scales. For the W + jets simulation we use a dynamical mass scale of $(m_{\PW})^2 + (\sum \pt^{\text{jet}})^2$, where $m_{\PW}$ is the mass of the W and $\sum \pt^{\text{jet}}$ is the sum of the transverse momenta from the jets in the event. The magnitude of the scale is allowed to vary in the fit by incorporating an effective parameter $c_{Q^2}$ into the likelihood with initial value 1.0, and which is allowed to vary between 0.5 and 2.0. The profile likelihood maximization provides simultaneous measurements of each of these parameters, background contributions and the cross section. There are alternative control samples to estimate the JES, the b-tagging efficiency and the mistag rate. The JES uncertainty is measured in control samples to be approximately 3% [@jec_11], and this determines a Gaussian constraint on this parameter in the likelihood. To account for differences between simulation and data in the b-tagging efficiencies and the mistag rates, we weight the tagged jets in the simulation up or down by a data-to-simulation scale factor. The b-tagging efficiency and the mistag rate scale factors are constrained to be $1.0 \pm 0.1$ in the fit, where 10% is the uncertainty in the b-tagging efficiency and the mistag rate [@CMS_btag]. The systematic uncertainties related to JES, $Q^2$ scale, b-tagging and mistag scale factors are included as nuisance parameters in the profile likelihood fit. Other systematic uncertainties are not directly included in the profile likelihood and taken as additional systematic uncertainties outside of the fit result and are described below. The efficiencies for triggering, reconstructing, and identifying isolated leptons are determined using $\cPZ\to\Pe\Pe$ and $\cPZ\to\mu\mu$ samples of events, and found to be very similar in the data and simulation. We correct for small differences observed, and account for an additional systematic uncertainty of 3% on these values. The unclustered energy in the detector results in an additional resolution uncertainty of less than 1% on the $\ETslash$ scale. The difference in jet energy resolution determined in simulation and data results in an uncertainty of less than 1%. The theoretical uncertainties in modeling of production are evaluated from dedicated simulated event samples by varying the theoretical parameters of interest around their nominal values. Such variations are used to construct alternative distributions, from which simulated events can be generated. For each variation, 4000 pseudo-experiments are generated and fitted with the standard configuration. The mean bias of the fitted cross section is taken as the size of the systematic uncertainty due to the source under study. These include differences in the signal due to renormalization and factorization scales (4%), the scale for the ME partons to PS matching scheme (2%), pileup modeling in simulation (less than 1%), and the parton distribution function model (less than 1%). The total uncertainty for the modeling, when adding the above uncertainties in quadrature, is 5.0%. The systematic uncertainty on the SVM shape is also considered. We have studied several effects, which include pixel resolution and jet-track-association modeling, as well as pileup dependence. These have a negligible effect on the SVM shapes of , single-top-quark, W and Z + jets events. The uncertainty on the SVM shape from QCD multijet background is obtained as follows. For the electron + jets channel we generate pseudo-experiments based on the default and alternative QCD shapes obtained from simulation. To increase the statistical accuracy of the QCD multijet background, the default shape employed in the fit is taken from events with relaxed requirements on the electron isolation and identification, and no $\ETslash$ requirement imposed. The alternative shape is obtained from the region corresponding to the event selection used in the cross section measurement. For the muon + jets channel, the statistical fluctuations in the normalization for the $\ETslash$ distributions obtained from muon non-isolated ($I_{\text{rel}} > 0.125$) and isolated ($I_{\text{rel}} < 0.125$) regions are taken as the systematic uncertainty. The integrated luminosity of the event sample is determined with an uncertainty of 2.2% [@Lumi]. The list of systematic uncertainties is summarized in Table \[tab:sys\_summary\]. These include both the uncertainties related to the nuisance parameters in the likelihood fit and the additional uncertainties evaluated from alternative distributions as described above. The individual systematic uncertainties related to the nuisance parameters in the fit are shown for illustrative purposes only. These are obtained as follows. First, the total fit uncertainty is evaluated when the parameter of interest is fixed in the fit. Then this uncertainty is subtracted in quadrature from the total fit uncertainty when all parameters are varied in the fit. Since the treatment of the $Q^2$ uncertainty in the likelihood fit is dependent on parametrization, we also performed the cross-check with the $Q^2$ uncertainty treated outside of the fit, and obtained consistent results. The combined systematic uncertainty of the measurement is 6.5%, taking into account the correlations between the nuisance parameters. ------------------------------------------------- ---------- --------- ---------- Source Electron Muon Combined channel channel analysis Lepton ID/reconstruction/trigger efficiency 3.0 2.0 3.0 $\ETslash$ resolution due to unclustered energy 0.9 0.3 0.8 Jet energy resolution 0.5 0.5 0.6 + jets renorm./fact. scales 3.5 4.3 4.3 + jets ME to PS matching 2.2 1.8 2.2 Pileup 0.5 0.2 0.6 Parton distribution function choice 0.3 0.3 0.3 QCD multijet SVM distribution 1.4 0.4 0.6 Subtotal 5.5 5.1 5.9 Jet energy scale 3.9 3.0 2.4 b-tagging efficiency and mistag rate 3.5 2.8 2.1 W + jets renorm./fact. scale 1.6 1.5 1.6 Total systematic uncertainty 7.0 6.2 6.5 ------------------------------------------------- ---------- --------- ---------- The measurement is performed separately for the electron + jets and muon + jets channels, as well as simultaneously for both channels, yielding [Electron + jets]{} $$\sigma_{\ttbar} = 160.6 \pm 3.2\stat \pm 11.2\syst \pm 3.5\lum\unit{pb},$$ [Muon + jets]{} $$\sigma_{\ttbar} = 164.2 \pm 2.8\stat \pm 10.1\syst \pm 3.6\lum\unit{pb},\ \text{and}$$ [Combined*]{} $$\sigma_{\ttbar} = 158.1 \pm 2.1\stat \pm 10.2\syst \pm 3.5\lum\unit{pb}.$$ ![Results of the combined fit for the electron + jets channel, for single b-tag events (top panel), and for $\geq$2b-tag events (bottom panel). The distributions within each panel correspond to events with 1-, 2-, 3-, 4-, and $\ge$5-jets, respectively. The error bars indicate statistical uncertainties only.[]{data-label="fig:ele_tagged_2fb"}](legend_f18_ncol4_lessWhite "fig:"){width="\cmsFigWidth"} ![Results of the combined fit for the electron + jets channel, for single b-tag events (top panel), and for $\geq$2b-tag events (bottom panel). The distributions within each panel correspond to events with 1-, 2-, 3-, 4-, and $\ge$5-jets, respectively. The error bars indicate statistical uncertainties only.[]{data-label="fig:ele_tagged_2fb"}](121203_tagged_both_svm_1j_1t_el_svm_2j_1t_el_svm_3j_1t_el_svm_4j_1t_el_svm_5j_1t_el.pdf "fig:"){width="\cmsFigWidth"} ![Results of the combined fit for the electron + jets channel, for single b-tag events (top panel), and for $\geq$2b-tag events (bottom panel). The distributions within each panel correspond to events with 1-, 2-, 3-, 4-, and $\ge$5-jets, respectively. The error bars indicate statistical uncertainties only.[]{data-label="fig:ele_tagged_2fb"}](121203_tagged_both_svm_2j_2t_el_svm_3j_2t_el_svm_4j_2t_el_svm_5j_2t_el.pdf "fig:"){width="\cmsFigWidth"} ![ Results of the combined fit for the muon + jets channel, for single b-tag events (top panel), and for $\geq$2b-tag events (bottom panel). The distributions within each panel correspond to events with 1-, , 3-, 4-, and $\ge$5-jets, respectively. The error bars indicate statistical uncertainties only. \[fig:muo\_tagged\_2fb\]](legend_f18_ncol4_lessWhite "fig:"){width="\cmsFigWidth"} ![ Results of the combined fit for the muon + jets channel, for single b-tag events (top panel), and for $\geq$2b-tag events (bottom panel). The distributions within each panel correspond to events with 1-, , 3-, 4-, and $\ge$5-jets, respectively. The error bars indicate statistical uncertainties only. \[fig:muo\_tagged\_2fb\]](121203_tagged_both_svm_1j_1t_mu_svm_2j_1t_mu_svm_3j_1t_mu_svm_4j_1t_mu_svm_5j_1t_mu.pdf "fig:"){width="\cmsFigWidth"} ![ Results of the combined fit for the muon + jets channel, for single b-tag events (top panel), and for $\geq$2b-tag events (bottom panel). The distributions within each panel correspond to events with 1-, , 3-, 4-, and $\ge$5-jets, respectively. The error bars indicate statistical uncertainties only. \[fig:muo\_tagged\_2fb\]](121203_tagged_both_svm_2j_2t_mu_svm_3j_2t_mu_svm_4j_2t_mu_svm_5j_2t_mu.pdf "fig:"){width="\cmsFigWidth"} The comparison of the corresponding observed and fitted SVM distributions is shown in Figures \[fig:ele\_tagged\_2fb\] and \[fig:muo\_tagged\_2fb\]. As a by-product, the fit provides the size of contributions from the SM processes that are backgrounds to production, as well as in-situ evaluations of other parameters varied in the profile likelihood fit, such as the b-tagging efficiency and the JES correction factor (on top of the standard jet corrections). The results of the combined fit, as well as the results of the fits performed in the electron + jets and muon + jets samples separately, are listed in Table \[table:side\_by\_side\], with correlations among parameters shown in Table \[table:corr\_combined\_shyfts\_12\]. The cross section is given in pb, while the contributions from other standard model processes are quoted as scale factors with respect to their theoretical predictions described above. These measured scale factors do not account for a full treatment of the systematic uncertainties and hence are strictly valid only in the context of the fit presented in this paper. The b-tagging scale factor defined as the ratio of the b-tagging efficiencies in data and simulation is determined to be $96 \pm 1$%, consistent between the electron + jets and muon + jets channels. The JES correction factor is found to be $100.4 \pm 1.6$% and $98.1 \pm 1.2$% in the electron + jets and muon + jets channels, respectively, yielding $100.2 \pm 1.0$% in the combined fit. The W + c jets contribution in the data is found to be larger than SM predictions, both in the electron + jets and muon + jets channels. This contribution includes single charm and double charm production, which are both present in the selected events. The W + b jets contribution in the data is also found to be slightly higher than in the simulation. The W + LF jets scale factor in the electron channel is significantly lower than in the muon case. This is because of the presence of a much larger QCD multijet contribution in the electron sample, and its large correlation with the W + LF jets component. The combined W + LF jets/QCD multijet scale factors for muons and electrons are in agreement, being $0.84 \pm 0.09\%$ and $0.71 \pm 0.07\%$, respectively. Because of large correlated uncertainties and due to differences in the correlation matrix for the electron + jets and muon + jets channels, which result from different QCD multi jet contributions in the two channels, the result of the combined fit resides outside of the individual electron + jets and muon + jets measurements. The correlation matrix for the combined fit is given in Table \[table:corr\_combined\_shyfts\_12\]. Using 4000 alternative data sets constructed from the simulated events we determine that the combined cross section lies between the individual channel results only in 60% of the cases. For the combined fit we have seven out of ten parameters that are common to both channels, residing outside of the ${\pm}1 \sigma$ interval between individual electron + jets and muon + jets measurements. Using simulated events we determine this to occur in 10% of the cases. Fit parameters Electron + jets Muon + jets Combined ------------------------ --------------------- --------------------- ---------------------- $\sigma_{\ttbar}$ (pb) $ 160.6 \pm 6.6 $ $ 164.2 \pm 5.5 $ $ 158.1 \pm 4.1 $ Single top $ 1.05 \pm 0.10 $ $ 1.08 \pm 0.10 $ $ 1.17 \pm 0.10 $ W + b jets $ 1.19 \pm 0.35 $ $ 0.95 \pm 0.18 $ $ 1.28 \pm 0.16 $ W + c jets $ 1.54 \pm 0.15 $ $ 1.48 \pm 0.05 $ $ 1.55 \pm 0.04 $ W + LF jets $ 0.20 \pm 0.08 $ $ 0.57 \pm 0.07 $ $ 0.52 \pm 0.06 $ Z + jets $ 1.13 \pm 0.29 $ $ 1.08 \pm 0.29 $ $ 1.43 \pm 0.29 $ $c_{Q^2}$ $ 1.02 \pm 0.16 $ $ 0.94 \pm 0.06 $ $ 1.05 \pm 0.05 $ b-tag $ 0.95 \pm 0.01$ $ 0.97 \pm 0.01$ $ 0.96 \pm 0.01 $ JES $ 1.00 \pm 0.02$ $ 0.98 \pm 0.01 $ $ 1.00 \pm 0.01 $ Mistag $ 1.00 \pm 0.10 $ $ 1.00 \pm 0.10 $ $ 1.00 \pm 0.10 $ Single t W + b W + c W + LF Z + jets $c_{Q^2}$ b-tag JES Mistag ----------- ------------- ------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------ $ 1.00 $ $ -0.13 $ $ -0.48 $ $ 0.33 $ $ 0.03 $ $ 0.07 $ $ -0.07 $ $ -0.70 $ $ -0.81 $ $ 0.00 $ Single t $ -0.13 $ $ 1.00 $ $ -0.52 $ $ 0.04 $ $ 0.03 $ $ -0.03 $ $ 0.06 $ $ -0.08 $ $ 0.09 $ $ -0.00 $ W + b $ -0.48 $ $ -0.52 $ $ 1.00 $ $ 0.05 $ $ 0.13 $ $ -0.16 $ $ 0.27 $ $ 0.26 $ $ 0.42 $ $ -0.02 $ W + c $ 0.33 $ $ 0.04 $ $ 0.05 $ $ 1.00 $ $ 0.01 $ $ 0.15 $ $ 0.71 $ $ -0.38 $ $ -0.26 $ $ -0.02 $ W + LF $ 0.03 $ $ 0.03 $ $ 0.13 $ $ 0.01 $ $ 1.00 $ $ -0.19 $ $ 0.21 $ $ -0.03 $ $ -0.05 $ $ -0.83 $ Z + jets $ 0.07 $ $ -0.03 $ $ -0.16 $ $ 0.15 $ $ -0.19 $ $ 1.00 $ $ 0.23 $ $ -0.01 $ $ -0.10 $ $ 0.01 $ $c_{Q^2}$ $ -0.07 $ $ 0.06 $ $ 0.27 $ $ 0.71 $ $ 0.21 $ $ 0.23 $ $ 1.00 $ $ -0.02 $ $ 0.15 $ $ -0.02 $ b-tag $ -0.70 $ $ -0.08 $ $ 0.26 $ $ -0.38 $ $ -0.03 $ $ -0.01 $ $ -0.02 $ $ 1.00 $ $ 0.43 $ $ -0.02 $ JES $ -0.81 $ $ 0.09 $ $ 0.42 $ $ -0.26 $ $ -0.05 $ $ -0.10 $ $ 0.15 $ $ 0.43 $ $ 1.00 $ $ 0.01 $ Mistag $ 0.00 $ $ -0.00 $ $ -0.02 $ $ -0.02 $ $ -0.83 $ $ 0.01 $ $ -0.02 $ $ -0.02 $ $ 0.01 $ $ 1.00 $ The cross section is measured assuming a value of the top quark mass $m_{\cPqt} = 172.5$. The measured cross section of production has a dependence on $m_\cPqt$, which is evaluated using dedicated MC samples and can be parameterized in the range of 160–185as $$\sigma_{\ttbar} = 158.1\unit{pb} - (m_{\cPqt} -172.5\GeV ) \times (1.14 \pm 0.18\unit{pb}/\GeVns).$$ Alternative Analyses {#sec:crosscheck} ==================== In addition to the main result, we have performed several alternative analyses in the electron + jets and the muon + jets channels using different event selections and different methods to suppress background contributions and measure the cross section. One analysis does not rely on b-tagging, a second one makes use of the kinematical information from the top quark decays, and a third one relies on a data-based estimate of the dominant background. The analysis without relying on use of the b-tagging algorithms considers the data set corresponding to 4.6(4.9) in the electron (muon) + jets channel. The selected events are required to have an electron with $\pt > 35$or a muon with $\pt > 26$, and at least 4 jets with $\pt > 30$. No missing transverse energy requirement is imposed. The cross section is measured using a binned log-likelihood fit to the mass of the three-jet combination with the highest $\pt$ in the event ($M_3$). The , W/Z + jets, and QCD multijet components are unconstrained during the fit, with the QCD multijet contributions in the electron + jets and muon + jets channels treated independently. The single-top-quark normalization is constrained to within 30% of its theoretical value. The , single-top-quark, and W/Z + jets processes are modeled using the simulation, while the QCD multijet contribution is estimated from data using a side-band region with the relative isolation of the lepton greater than 0.25. Signal events, as well as W + jets events, are heavily suppressed by this selection, and subtracted based on simulation. The shape of the subtracted QCD multijet contribution is used for the fit in the signal region, since the $M_3$ distribution of QCD events does not depend on the relative isolation of the lepton in the event. ![ The mass of the three-jet combination with the highest transverse momenta.[]{data-label="fig:emu_data_mlfit_res_AN11_443"}](Combined_channel_fit_result.pdf){width="50.00000%"} The observed and the fitted $M_3$ distributions are shown in Figure \[fig:emu\_data\_mlfit\_res\_AN11\_443\]. The dominant sources of systematic uncertainty are JES, ME to PS matching, and the $\mathrm{Q^2}$ scale uncertainties. In the electron + jets channel the cross section measurement yields $$\sigma_{\ttbar} = 157.1 \pm 3.7\stat^{+17.2}_{-11.4}\syst\pm 3.5 \lum\unit{pb},$$ in the muon + jets channel the cross section is measured as $$\sigma_{\ttbar} = 161.6 \pm 3.5\stat^{+14.8}_{-22.1} \mathrm{~(syst.)} \pm 3.6 \lum\unit{pb},$$ The combined measurement in the electron + jets and muon + jets channels yields a cross section of $$\sigma_{\ttbar} = 159.7 \pm 2.6 \stat^{+13.1}_{-14.7}\syst\pm 3.5 \lum\unit{pb}.$$ Another measurement uses kinematic information from the leptonic top quark decay $\cPqt\to\cPqb\PW \to \cPqb\ell \nu_\ell$, namely the mass of the two-particle system consisting of a lepton and a jet associated with a b quark. The jet-to-parton assignment among the four leading jets is performed minimizing a least-squares residual based on the masses of the reconstructed W boson and hadronically decaying top quark in $\cPqt\to\cPqb\PW \to \cPqb\cPq\cPaq'$. The baseline event selection is similar to the reference analysis, complemented with the requirement that the jet assigned to the leptonic top quark decay is b-tagged using an algorithm based on measuring the significance of a track impact parameter [@CMS_btag]. The technique is applied to the muon + jets data sample, corresponding to an integrated luminosity 4.9. The result of this measurement is $\sigma_{\ttbar} = 162.4 \pm 5.4\stat^{+7.5}_{-11.0}\syst\pm 3.6\lum\unit{pb}$, where the dominant systematic uncertainty is due to the JES. Finally, the third method does not rely on MC simulation for the W + jets background, but exploits the W charge asymmetry [@cms_WchargeAsym] in W + jets production at the LHC. The shape of the lepton pseudo-rapidity distribution for the W + jets component is obtained from the data by subtracting the observed distribution for $\ell^-$ from the one corresponding to $\ell^+$. The cross section is measured by fitting a combination of signal and background components to the observed lepton $\|\eta\|$ spectrum using a data sample corresponding to an integrated luminosity of 0.9 (1.0)for electron (muon) + jets. The cross section is measured with a large expected uncertainty of 42% (23%) in the electron (muon) + jets channel, and agrees with the results of the other analyses. Summary ======= The production cross section measurement has been performed at $\sqrt{s}=7\TeV$ using the data collected with the CMS detector and corresponding to an integrated luminosity of 2.3. The cross section is measured using a profile likelihood fit to the number of reconstructed jets, the number of b-tagged jets, and the secondary vertex mass distribution. The measured cross section for an assumed top quark mass of 172.5is $$\sigma_{\ttbar} = 158.1 \pm 2.1\stat \pm 10.2\syst \pm 3.5\lum\unit{pb},$$ which is in agreement with the QCD predictions of $164^{+10}_{-13}\unit{pb}$ [@Aliev:2010zk; @Langenfeld:2009wd], $163^{+11}_{-10}\unit{pb}$ [@Kidonakis:2010dk] and $149\pm11\unit{pb}$ [@Ahrens] that are based on the full next-to-leading-order (NLO) matrix elements and the resummation of the leading and NLO soft logarithms. Acknowledgements {#acknowledgements .unnumbered} ================ We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centres and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: BMWF and FWF (Austria); FNRS and FWO (Belgium); CNPq, CAPES, FAPERJ, and FAPESP (Brazil); MEYS (Bulgaria); CERN; CAS, MoST, and NSFC (China); COLCIENCIAS (Colombia); MSES (Croatia); RPF (Cyprus); MoER, SF0690030s09 and ERDF (Estonia); Academy of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRT (Greece); OTKA and NKTH (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); NRF and WCU (Republic of Korea); LAS (Lithuania); CINVESTAV, CONACYT, SEP, and UASLP-FAI (Mexico); MSI (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Armenia, Belarus, Georgia, Ukraine, Uzbekistan); MON, RosAtom, RAS and RFBR (Russia); MSTD (Serbia); SEIDI and CPAN (Spain); Swiss Funding Agencies (Switzerland); NSC (Taipei); ThEPCenter, IPST and NSTDA (Thailand); TUBITAK and TAEK (Turkey); NASU (Ukraine); STFC (United Kingdom); DOE and NSF (USA). Individuals have received support from the Marie-Curie programme and the European Research Council (European Union); the Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the Ministry of Education, Youth and Sports (MEYS) of Czech Republic; the Council of Science and Industrial Research, India; the Compagnia di San Paolo (Torino); and the HOMING PLUS programme of Foundation for Polish Science, cofinanced from European Union, Regional Development Fund. The CMS Collaboration \[app:collab\] ==================================== =5000=500=5000
--- abstract: 'Recursive partitioning approaches producing tree-like models are a long standing staple of predictive modeling, in the last decade mostly as “sub-learners” within state of the art ensemble methods like Boosting and Random Forest. However, a fundamental flaw in the partitioning (or splitting) rule of commonly used tree building methods precludes them from treating different types of variables equally. This most clearly manifests in these methods’ inability to properly utilize categorical variables with a large number of categories, which are ubiquitous in the new age of big data. Such variables can often be very informative, but current tree methods essentially leave us a choice of either not using them, or exposing our models to severe overfitting. We propose a conceptual framework to splitting using leave-one-out (LOO) cross validation for selecting the splitting variable, then performing a regular split (in our case, following CART’s approach) for the selected variable. The most important consequence of our approach is that categorical variables with many categories can be safely used in tree building and are only chosen if they contribute to predictive power. We demonstrate in extensive simulation and real data analysis that our novel splitting approach significantly improves the performance of both single tree models and ensemble methods that utilize trees. Importantly, we design an algorithm for LOO splitting variable selection which under reasonable assumptions does not increase the overall computational complexity compared to CART for two-class classification. For regression tasks, our approach carries an increased computational burden, replacing a $O(log(n))$ factor in CART splitting rule search with an $O(n)$ term.' author: - \| Amichai Painsky amichaip@eng.tau.ac.il\ School of Mathematical Sciences\ Tel Aviv University\ Tel Aviv, Israel Saharon Rosset saharon@post.tau.ac.il\ School of Mathematical Sciences\ Tel Aviv University\ Tel Aviv, Israel bibliography: - 'sigproc.bib' title: 'Cross-Validated Variable Selection in Tree-Based Methods Improves Predictive Performance' --- Classification and regression trees, Random Forests, Gradient Boosting Introduction ============ The use of trees in predictive modeling has a long history, dating back to early versions like CHAID [@kass1980exploratory], and gaining importance in the 80s and 90s through the introduction of approaches like CART [@breiman1984classification] and C4.5/C5.0 [@quinlan2014c4; @quinlan2004data]. A tree describes a recursive partitioning of a feature space into rectangular regions intended to capture the relationships between a collection of explanatory variables (or features) and a response variable, as illustrated in Figure \[fig:tree\]. The advantages of trees that made them attractive include natural handling of different types of features (categorical/ordinal/numerical); a variety of approaches for dealing with missing feature values; a natural ability to capture non-linear and non-additive relationships; and a perceived interpretability and intuitive appeal. Over the years it has become widely understood that these advantages are sometimes overstated, and that tree models suffer from a significant drawback in the form of inferior predictive performance compared to modern alternatives (which is not surprising, given the greedy nature of their model building approaches). However the last 15 years have seen a resurgence in the interest in trees as “sub-learners” in ensemble learning approaches like Boosting [@friedman2001] and Random Forest [@breiman2001random]. These approaches take advantage of the favorable properties described above, and mitigate the low accuracy by averaging or adaptively adding together many trees. They are widely considered to be among the state of the art tools for predictive modeling [@hastie2009elements]. As mentioned, popular tree building algorithms like CART can handle both numerical and categorical features and build models for regression, two-class classification and multi-class classification. The splitting decisions in these algorithms are based on optimizing a splitting criterion over all possible splits on all variables. The split selection problems that arise present both computational and statistical challenges, in particular for categorical features with a large number of categories. Assume $K$ categories overall, then the space of possible binary splits includes $O(2^K)$ candidates, and a naive enumeration may not be feasible. However, as demonstrated in CART [@breiman1984classification], there is no need to enumerate over all of them in order to find the optimal one for regression or two-class classification, because it is enough to sort the categories by their mean response value and consider only splits along this sequence. The complexity of splitting is therefore $O(\max(K \log(K), n))$, where $n$ is the number of observations being split. However, if $K$ is large, the splitting still presents a major statistical (overfitting) concern. At the extreme, if $K=n$ (for example, if the categorical variable is in fact a unique identifier for each observation), then it is easy to see that a single split on this variable can perfectly separate a two-class classification training set into its two classes. Even if $K$ is smaller than $n$ but large, it is intuitively clear (and demonstrated below) that splitting on such a categorical variable can result in severe overfitting. This is exasperated by the fact that this overfitting in fact causes common tree building splitting criteria like Gini to preferably select such categorical variables and “enjoy” their overfitting. Some popular tree building software packages recognize this problem and limit the number of categories that can be used in model building. In [R]{}, many tree-based models limit the number of categories to $32$ (as [randomForest]{}, for example), while Matlab’s regression-tree and classification-tree routines ([fitrtree, fitctree]{}) leave this parameter for the users to define. In what follows we denote implementations which discard categorical features with a number of categories larger than $K$ as limited-$K$ while versions which do not apply this mechanism will be denoted as unlimited-$K$. The ad-hoc limited-$K$ approach is of course far from satisfactory, because the chosen number can be too small or too large, depending on the nature of the specific dataset and the categorical features it contains. It also fails to address the conceptual/theoretical problem: since features that are numerical or ordinal or have a small number of categories present a smaller number of unique splits than categorical features with many categories, there is a lack of uniformity in split criteria selection. Finally, some large datasets may well have categorical features with many categories that are still important for good prediction, and discarding them is counterproductive. In this work we amend these concerns through a simple modification: We propose to select the variable for splitting based on LOO scores rather than on training sample performance, an approach we term Adaptive LOO Feature selection (ALOOF). A naive implementation of ALOOF approach would call for building $n$ different best splits on each feature, each time leaving one observation out, and select the feature that scores best in LOO. As we demonstrate below, this implementation can be avoided in most cases. The result of our amendment is a “fair” comparison between features in selection, and consequently the ability to accommodate all categorical features in splitting, regardless of their number of categories, with no concern of major overfitting. Importantly, this means that truly useful categorical variables can always be taken advantage of. The preceding concepts are demonstrated in Figure \[fig:sim3\]. The simulation setting includes $n=300$ observations and an interaction between a numerical variable $x_1$ and a categorical variable $x_2$ with $K=50$ categories denoted $c_1,...,c_{50}$, with a parameter $\alpha$ controlling the strength of the interaction: $$\begin{aligned} \label{intro_model} y &=& \alpha \cdot \left( {\mathbb I}\left[x_1>0\right] {\mathbb I}\left[ x_2 \in \{c_1,...,c_{25}\} \right]\right. + \left. {\mathbb I}\left[x_1 \leq 0\right] {\mathbb I}\left[ x_2 \in \{c_{26},...,c_{50}\} \right] \right) + \epsilon, \end{aligned}$$ where $\epsilon \sim N(0,1)$ i.i.d. At small $\alpha$ the signal from the categorical variable is weak and less informative. In this case, the limited-$K$ approach is preferable (here, as in all subsequent analyses, we use a CART limited-$K$ Matlab implementation that eliminates categorical variables with $K>32$ values). As $\alpha$ increases, the categorical variable becomes more informative and unlimited-$K$ (which refers to a CART implementation on Matlab that does not discard any categorical variable) outperforms the limited-$K$ approach for the same $n$ and $K$. ALOOF (which is also implemented on a Matlab platform) successfully tracks the preferred approach in all situations, even improving slightly on the unlimited-$K$ approach for large $\alpha$ by selecting the categorical variable less often in the tree. The rest of this paper is organized as follows: In Section \[Sec2\] we formalize the LOO splitting approach in ALOOF and discuss its statistical properties, we then describe our algorithmic modifications for implementing it in Section \[Sec3\] and demonstrate that for two-class classification they lead to an approach whose computational complexity is comparable to CART splitting under reasonable assumptions. In Section \[Sec4\] we demonstrate via some toy simulations the striking effect that ALOOF can have on improving the performance of tree methods. We begin Section \[Sec5\] with a comprehensive case study, where large-K categorical variables are prevalent and important. We demonstrate the significant improvement in both interpretability and prediction from using ALOOF vs regular CART splitting. We then preset an extensive real data study in both regression and classification, using both single CART trees and ensemble methods, where ALOOF offers across-the-board improvement. Related work ------------ We focus here on two threads of related work, which relates to our approach from two different aspects. The Conditional Inference Trees (CIT) framework [@hothorn2006unbiased] proposes a method for unbiased variable selection through conditional permutation tests. At each node, the CIT procedure tests the global null hypothesis of complete independence between the response and each feature, and selects the feature with the most significant association (smallest p-value). It terminates once the global null can no longer be rejected. The advantage of this approach is that it proposes a unified framework for recursive partitioning which embeds tree-structured models into a well defined statistical theory of conditional inference procedures. However, practically it requires the computation of p-values under permutations tests, either by exact calculation, Monte Carlo simulation, or asymptotic approximation. These permutation tests are typically computationally expensive and necessitate impractical run-time as the number of observations grows (as demonstrated in our experiments). More critically, despite its well established theory, the statistical significance criterion the CIT procedure uses is not the relevant criterion for variable selection in tree-based modeling. Notice that in each node, the feature that should be most “informative" is the one that splits the observations so that the (test-set) generalization error is minimal. This criterion does not typically select the same variable that results with the smallest p-value under the null assumption on the train-set. As our experiments in Section \[Sec5\] show, our ALOOF method, based on the generalization error criterion, is superior the the CIT approach, where the CIT is computationally feasible. In a second line of work, Sabato and Shalev-Schwartz [@sabato2008ranking] suggest ranking categorical features according to their generalization error. However, their approach is limited to considering K-way splits (i.e., a leaf for every categorical value), as opposed to binary splits which group categories, which characterize CART and other tree methods commonly used in statistics. Their appraoch, which is also solely applicable for classification trees, offers strong theoretical results and low computational complexity (as it does not perform actual splitting), albeit in the K-way scenario which is significantly simpler theoretically than the binary split case we are considering, which requires optimizing over $O(2^K)$ possible splits. Thus our goals are similar to theirs, but since we are dealing with a more complex problem we cannot adopt either their computational approach or their theoretical results. In our classification examples we compare our approach to theirs, demonstrating the advantage of binary splitting for attaining better predictive performance in practice. The approach of [@sabato2008ranking] subsumes and improves on various previous lines of work in the machine learning literature on using cross validation to select variable in K-way splitting scenarios [@Frank1; @Frank2]. Formulation of ALOOF {#Sec2} ==================== We present here the standard CART splitting approach [@breiman1984classification] based on least squares for regression and Gini index (closely related to least squares) for classification, and then describe our modifications to adapt them to LOO splitting variable selection. We choose to concentrate on CART as arguably the most widely used tree implementation, but our LOO approach can be adapted to other algorithms as well. CART splitting rules -------------------- Assume we have $n$ observations denoted by their indexes $\{1,...,n\}$. Each observation $i$ is comprised of the pair $(x_i,y_i)$, where $x_i$ is a $p$ dimensional vector containing $p$ candidate variables to split on. We assume WLOG that the variables $x_{\cdot 1}...x_{\cdot q}$ are categorical with number of categories $K_1,...,K_q$ respectively, and the features $x_{\cdot q+1},...,x_{\cdot p}$ are numerical or ordinal (treated identically by CART). A split $s$ is a partition of the $n$ observations into two subsets $R(s),L(s)$ such that $R(s)\cap L(s)=\emptyset$ and $R(s)\cup L(s) = \{1,...,n\}$. For each variable $j$, denote the set of possible splits by $S_j$ and their number by $s_j = \|S_j\|$. Categorical variable $j$ has $s_j = 2^{K_j-1}-1$ possible binary splits, and each numerical/ordinal variable has $s_j \leq n$ possible splits between its unique sorted values. A specific split comprises a selection of a splitting variable $j$ and a specific split $s \in S_j$, and is evaluated by the minimizer of an “impurity” splitting criterion given the split $s$, denoted by ${\cal L}(s)$. In regression, CART uses the squared error loss impurity criterion: $${\cal L}(s) = \sum_{i \in L(s)} (y_i - \bar{y}_L)^2 + \sum_{i \in R(s)} (y_i - \bar{y}_R)^2,$$ where $\bar{y}_L, \bar{y}_R$ are the means of the response $y$ over the sets $L(s), R(s)$ resepectively. In two-class classification, it uses the Gini index of impurity: $${\cal L}(s) = n_L \hat{p}_L (1-\hat{p}_L) + n_R \hat{p}_R (1-\hat{p}_R),$$ where $n_L,n_R$ are the numbers of observations in $L(s),R(S)$ respectively and $\hat{p}_L,\hat{p}_R$ are the observed proportions of “class 1” in $L(s),R(S)$. The Gini index is easily seen to be closely related to the squared error loss criterion, as it can be viewed as the cross-term of a squared error loss criterion with 0-1 coding for the classes. The Gini criterion for multi-class classification is a straight forward generalization of this formula, but we omit it since we do not consider the multi-class case in the remainder of this paper. Given this notation, we can now formulate the CART splitting rule as: $$(j^, s^) = {\mathrm{arg}\displaystyle\min}_{\substack{j \in \{1, \dots, p \}\\ s\in S_j}} {\cal L}(s),$$ and the chosen pair $ (j^, s^)$ is the split that will be carried out in practice. A naive implementation of this approach would require considering $\sum_j s_j$ possible splits, and performing $O(n)$ work for each one. In practice, the best split for each numerical/ordinal variable can be found in $O(n \log(n))$ operations, and for each categorical variable in $O(\max(n, K_j \log(K_j)))$ operations, for both regression and classification problems [@breiman1984classification; @hastie2009elements]. Given these $p$ best splits, all that is left is to find their optimum. LOO splitting rules ------------------- For LOO splitting, our goal is to find the best variable to split on based on LOO scores for each variable under the given impurity criterion. The actual split performed is then the best regular CART split on the selected variable. The resulting LOO splitting algorithm is presented in Algorithm \[alg:high-level LOO approach\]. \[1\] $L(j)=0$ $s_{ij} = \arg\min_{s \in S_j} {\cal L}^{(-i)}(s)$ $L(j) = L(j) + R(s_{ij},i)$ $j* = \arg\min_{j \in \{1...,p\}} L(j)$ $s* = \arg\min_{s \in S_{j}} {\cal L} (s)$ \[alg:high-level LOO approach\] The result of Algorithm \[alg:high-level LOO approach\] is again a pair $(j^,s^)$ describing the chosen split. The additional notations used in this description are: 1. ${\cal L}^{(-i)}(s)$: the impurity criterion for candidate split $s$, when excluding the observation $i$. For example, if we assume WLOG that $i\in L(s)$ and the impurity criterion is the suaqred loss, we get $${\cal L}^{(-i)}(s) = \sum_{l \in L(s), l\neq i} (y_l - \bar{y}^{(-i)}_L)^2 + \sum_{l \in R(s)} (y_l - \bar{y}_R)^2,$$ where $\bar{y}^{(-i)}_L$ is the average calculated without the $i$th observation. If the criterion is Gini or the observation falls on the right side, the obvious modifications apply. Consequently, $s_{ij}$ is the best split for variable $j$ when excluding observation $i$. 2. $R(s,i)$: the LOO loss of the $i$th observation for the split $s$. In the above case (squared loss, left side) we would have: $$R(s,i) = (y_i - \bar{y}^{(-i)}_L)^2,$$ with obvious modifications for the other cases. $L(j)$ is simply the sum of $R(s_{ij},i)$ over $i$, i.e., the total LOO loss for the $j$th variable when the best LOO split $s_{ij}$ is chosen for each $i$. A naive implementation of this algorithm as described here would require performing $n$ times the work of regular CART splitting, limiting the usefulness of the LOO approach for large data sets. As we show below, for two-class classification we can almost completely eliminate the additional computational burden compared to regular CART splitting, by appropriately enumerating the LOO splits. For regression, the worst case performance of our approach is unfortunately close to being $n$ times worse than regular CART splitting. In practice, the added burden is much smaller. What is ALOOF estimating? {#unbias} ------------------------- The quantity $L(j)$ in our LOO approach directly estimates the generalization error of splitting on variable $j$ using a CART impurity criterion: for each observation $i$, the best split is chosen based on the other $n-1$ observations, and judged on the (left out) $i^{th}$ observation. Hence $L(j)$ is an unbiased estimate of generalization impurity error for a split on variable $j$ based on a random sample of $n-1$ observations. The selected splitting variable $j^$ is the one that gives the lowest unbiased estimate. Hence our goal of judging all variables in a fair manner, that identifies the variables that are truly useful for reducing impurity and not just over-fitting is attained. Since we eventually perform the best split on our complete data at the current node ($n$ observations), there is still a small gap remaining between the “$n-1$ observations splits” being judged and the “$n$ observations split” ultimately performed, but this difference is negligible in most practical situations (specifically, when $n$ is large). ALOOF stopping rules {#stop} -------------------- Tree methods like CART usually adopt a grow-then-prune approach, where a large tree is first grown, then pruned (cut down) to “optimal” size based on cross-validation or other criteria [@breiman1984classification]. Because cross-validation is built-in to the ALOOF splitting variable selection approach, it obviates the need to take this approach. For every variable, ALOOF generates an unbiased estimate of the generalization error (in Gini/squared loss) of splitting on this variable. If an additional split on any variable would be detrimental to generalization error performance, this would manifest in the ALOOF estimates (in expectation, at least) and the splitting would stop. This approach is not perfect, because ALOOF generates an almost-unbiased estimate for every variable, then takes the best variable, meaning it may still be over-optimistic and continue splitting when it is in fact slightly overfitting. However in our experiments below it is demonstrated that the combination of ALOOF’s variable selection with ALOOF’s stopping rules is superior to the CART approach. It should be noted that in Boosting or Random Forest approaches utilizing trees as a sub-routine, it is usually customary to avoid pruning and determine in advance the desirable tree size (smaller in Boosting, larger in Random Forest). Similarly, when we implement ALOOF within these methods, we also build the tree to the same desirable size, rather than utilize ALOOF stopping rules. Efficient implementation {#Sec3} ======================== As described above, the ALOOF algorithm utilizes a seemingly computationally expensive LOO cross-validation approach. However, we show that with a careful design, and under reasonable assumptions, we are able to maintain the same computational complexity of the CART algorithm for two-class classification modeling. In addition, we show that for regression tasks ALOOF carries some increase of the computational burden, replacing a $O(log(n))$ factor in CART splitting rules with an $O(n)$ term. We present our suggested implementation for both categorical and numerical/ordinal variables, in both two-class classification and regression settings. Two-class classification — categorical variables ------------------------------------------------ Assume a categorical variable with $K$ different categories. As described above, the CART algorithm performs a split selection by sorting the $K$ categories by their mean response value and consider only splits along this sequence. Therefore, the complexity of finding the best split for this variable is $O(\max(K \log(K), n))$, where $n$ is the number of observations being split. We now present our suggested implementation and compare its computational complexity with CART. First, we sort the observations according to the mean response value of the $K$ categories, just like CART does. Then, for each pair $(x_i,y_i)$ we leave out, we recalculate its category’s mean response (an $O(1)$ operation), and resort the categories by their new means. Notice that since only a single category changes its mean ($x_i$’s category), resorting the sequence takes only $O(\log(K))$. Once the sequence is sorted again we simply go over the $K$ splits along the sequence. Therefore, for each pair $(x_i,y_i)$ ALOOF necessitates $O(K)$ operations. It is important to notice that despite the fact we have $n$ observations, there are only two possible pairs for each category, as $y_i$ takes only two possible values in the classification setting. This means we practically need to repeat the LOO procedure only for $2K$ pairs. Hence, our total computational complexity is $O(\max(K^2, n))$ We further show that under different reasonable assumptions, our computational complexity may even be equal to the complexity of CART splitting. First, assuming the number of categories grows no faster than the square root of $n$, $K^2=O(n)$, it is immediate that both algorithms result with a complexity of $O(n)$. This setting is quite common for the majority of categorical variables in real-world datasets. Second, under the assumption that a single observation cannot dramatically change the order of the categories in the sorted categories sequence, then the worst case complexity is lower than $O(\max(K^2, n))$. More specifically, assuming a single observation can move the position of its category by at most $B$ positions in the sorted categories sequence, then the complexity we achieve is $O(\max(K\log(K), KB, n))$. This assumption is valid, for example, when all categories have about the same number of observations. To emphasize this point, consider the case where the number of observations in each category is exactly the same, so that each category consist of $\sfrac{n}{K}$ observations. In this case the effect of changing the value of a single observation (on the category’s mean) is proportional to ($\sfrac{K}{n}$). In other words, $\sfrac{K}{n}$ corresponds to the smallest possible difference between categories’ means. Moreover, when leaving out a single observation, the change in this category’s mean is bounded by this value. Hence, resorting the categories (following the LOO operation) is not more than an exchange of positions of adjacent categories in the sorted categories sequence ($B=1$). This leads to an overall complexity equal to CART’s. Two-class classification — numerical variables ---------------------------------------------- For numerical variables, CART performs a split selection by first sorting the $n$ pairs of observations according to their $x_i$ values, and then choosing a cut (from at most $n-1$ possible cuts) that minimizes the Gini criterion along both sides of the cut. By scanning along the list with sufficient statistics this takes $O(n)$ operations, leading to overall complexity $O(n\log(n))$ due to the sorting. Our suggested implementation again utilizes the fact that $y_i$ takes only two possible values to achieve a complexity identical to CART’s. Here we denote the values as $y_i \in \{-1,1\}$, we initialize our algorithm by sorting the observations in the same manner CART does. We then take out the observation $y_i=1$ with the lowest value of $x_i$, and find its best cut (by going over all $n-2$ possible cuts). We then place this observation back and take out the observations for which $y_i=1$, with the second lowest value of $x_i$. Notice that the only cuts whose Gini index values are affected by these operations are the cuts between the two pairs we replaced. Denote this group of cuts by $N_2$. This means that when exchanging the two observations we need to update the Gini values of at most $\|N_2\|$ cuts to find the best LOO cut. Continuing in the same manner leads to a total complexity of $O(\sum{\|N_j\|})=O(n)$ for finding the best cut of all observations with $y_i=1$. This process is then repeated with observations for which $y_i=-1$. Therefore, the overall complexity of our suggested implementation is $O(2n+n\log(n))=O(n\log(n))$, just like CART. We present pseudo-code of our suggested approach in Algorithm \[alg:algorithm\]. To avoid an overload of notation we only present the case of observations for which $y_i=1$. The key idea is presented in steps 20-21 of the algorithm, where it is emphasized that only the splits in $N_j$ have to be considered in every step. \[alg1\] \[1\] Observations $(x_1,y_1),\ldots,(x_n,y_n)$. $j=1$, $n_1=$number of observations for which $y_i=1$, $optimal$[\_]{}$cut=$ array of size $n_1$ holding the best LOO cut for each of the $y_i=1$ observations. Sort observations according to $x_i$ values. Set $A=\{(x_1,y_1),\ldots,(x_n,y_n)$ such that $y_i=1\}$. Set $A=$sort $A$ according to its $x_i$ values in ascending order. $A=A\setminus (x_1,y_1)$. $optimal$[\_]{}$cut_j=\infty$. $optimal$[\_]{}$cut$[\_]{}$Gini=\infty$. Set $cut_i=\frac{x_i+x_{i+1}}{2}$. Set $Gini_i=$ Gini index value with respect to the cut $cut_i$. $optimal$[\_]{}$cut_j=cut_i$. $optimal$[\_]{}$cut$[\_]{}$Gini=Gini_i$. $j=j+1$ Set $A=A\cup (x_{j-1},y_{j-1})$. Set $A=A\setminus(x_j,y_j)$. $optimal$[\_]{}$cut_j=optimal$[\_]{}$cut_{j-1}$. $N_j=\{(x_i,y_i) \in A$ such that $x_{j-1} <x_i<x_{j} \}$ . Set $cut_i=\frac{x_i+x_{i+1}}{2}$. Set $Gini_i=$ Gini index value with respect to the cut $cut_i$. $optimal$[\_]{}$cut_j=cut_i$. $optimal$[\_]{}$cut$[\_]{}$Gini=Gini_i$. $j=j+1$. $optimal$[\_]{}$cut$ \[alg:algorithm\] Regression — categorical variables ---------------------------------- The regression modeling analysis case is similar to the two-class classification in the presence of categorical variable, with a simple modification of the loss impurity criterion. Both CART and our suggested ALOOF method replace the Gini index with the squared error loss impurity criterion and follow the exact same procedures mentioned above. This leads to a overall complexity of $O(\max(K\log(K),n)$ for the CART algorithm and $O(\max(n,K^2))$ for our ALOOF method. As before, the computational complexity gap between the two methods may reduce under some reasonable assumptions. One simple example is when $K^2=O(n)$. Another example is in cases where we can assume that the effect of a single observation on the mean response value of each category is bounded. This happens, for instance, when the the range of the response values is bounded and there is approximately the same number of observations in each category. Regression — numerical variables -------------------------------- In this setting as well, we compare both algorithms’ implementation to those of the classification problem. The CART algorithm first sorts the $n$ observation pairs according to their $x_i$ values, and then chooses a cut that minimizes the squared error loss along both sides of the cut. As in the classification case, scanning along the list with sufficient statistics this takes $O(n)$ operations, leading to overall complexity $O(n\log(n))$ due to the sorting. As we examine the LOO method in this regression setting, we notice that unlike CART, it fails to simply generalize the two-class classification problem. Specifically, it is easy to show that for each observation pair $(x_i, y_i)$ that is left out, the best cut may move in a non-monotonic manner (along the values of the numerical variable $x$), depending on the specific values of the observations that were drawn. This unfortunate non-monotonic behavior leads to a straight-forward implementation, where we first sort the observations by their $x_i$ values and then find the best cut for each observation that is taken-out by exhaustive search. The overall complexity is therefore $O(n^2)$, which compared with the CART implementation, replaces the $O(\log(n))$ factor with $O(n)$. Illustrative simulations {#Sec4} ======================== We start with two synthetic data experiments which demonstrate the behavior of ALOOF compared with CART in the presence of a categorical variable with varying number of categories. In the first experiment we examine our ability to detect a categorical variable which is not informative with respect to the response variable. Our goal is to model and predict the response variable from two different explanatory variables, where the first is numerical and informative while the other is categorical with $K$ different categories, and independent of the response variable. We draw $1000$ observations which are split into $900$ observations for the train-set and $100$ for the test-set. We repeat this process multiple times and present averaged results. Figure \[fig:sim1\_1\] shows the test-set’s average mean square error (MSE) for different number of categories $K$, using three different methods. The upper curve corresponds to the classic CART model, which is extremely vulnerable to categorical variables with large number of categories, as discussed above. The full line is our suggest ALOOF method, which manages to identify the uselessness of the categorical variable and discards it. The line with the [x]{} symbols at the bottom is the CART model without the categorical variable at all, which constitutes a lower bound on the MSE we can achieve, using a tree-based model. In the second synthetic data experiment we revisit the model presented in the Introduction (\[intro\_model\]) as an example of a model with an informative categorical variable. We fix the strength of the interaction between the numerical and the categorical variables ($\alpha=15$) and repeat the experiment for different values of $K$. Figure \[fig:sim1\_2\] presents the results we achieve. As in the previous example ALOOF demonstrates superior performance to both its competitors as $K$ increases. This exemplifies the advantage of using ALOOF both when the categorical variable is informative or uninformative, for any number of categories $K$. We now turn to compare the modeling complexity of the ALOOF approach with that of CART. Since CART has a tree-depth parameter it may provide a variety of models which differ in their model complexity and the corresponding MSE they achieve. ALOOF, on the other hand, results in a single model, using the stopping policy described in Section \[stop\], with its corresponding MSE. Therefore, to have a fair comparison between the two methods we use degrees of freedom ($df$). We start with a brief review of the main ideas behind this concept. Following [@efron1986biased] and [@hastie2009elements], assume the values of the feature vectors $X=(x_1,...,x_n)$ are fixed (the fixed-x assumption), and that the model gets one vector of response variable $Y=(y_1,...,y_n)$ for training, drawn according to the conditional probability model $p(Y\|X)$ at the n data points. Denote by $Y^{new}$ another independent vector drawn according to the same distribution. $Y$ is used for training a model $\hat{f}(x)$ and generating predictions $\hat{y_i}=\hat{f}(x_i)$ at the $n$ data points. We define the training mean squared error as $$MRSS=\frac{1}{n}\\|Y-\hat{Y}\\|^2_2$$ and compare it to the expected error the same model incurs on the new, independent copy, denoted in [@hastie2009elements] as $ERR_{in}$, $$ERR_{in}=\frac{1}{n}\mathbb{E}_{Y^{new}}\\|Y^{new}-\hat{Y}\\|^2_2$$ The difference between the two is the optimism of the prediction. As Efron [@efron1986biased] and others have shown, the expected optimism in MRSS is $$\mathbb{E}_{Y,Y^{new}}(ERR_{in}-MRSS)=\frac{2}{n}\sum_{i}cov(y_i,\hat{y_i})\label{eq:ERR-MRSS}$$ For linear regression with homoskedastic errors with variance $\sigma^2$, it is easy to show that (\[eq:ERR-MRSS\]) is equal to $\frac{2}{n}d\sigma^2$ where $d$ is the number of regressors, hence the degrees of freedom. In nonparametric models (such as tree-based models), one usually cannot calculate the actual degrees of freedom of a modeling approach. However, in simulated examples it is possible to generate good estimates $\hat{df}$ of $df$ through repeated generation of $Y, Y^{new}$ samples to empirically evaluate (\[eq:ERR-MRSS\]). In the following experiment we compare the models’ degrees of freedom for CART and ALOOF. As explained above, CART generates a curve of values where the $df$ increases with the size of the tree (hence, the complexity of the model) and the MSE is the error it achieves on the test-set. Our ALOOF method provides a single value of MSE for the model we train. We draw $n=200$ observations from the same setup as in Figure \[fig:sim1\_1\] and achieve the results shown in Figure \[fig:sim2\] for three different values of $K$. The straight line at the bottom of each figure is simply to emphasize that the MSE ALOOF achieves is uniformly lower than CART’s curve, although it imposes more $df$ compared to the optimum of the CART curve. It is important to notice that this “CART optimum" is based on Oracle knowledge; in practice, the pruned CART model is selected by cross-validation. Typically, it fails to find the model which minimizes the MSE on the test-set. As we look at the results we achieve for different values of $K$ we notice that ALOOF’s superiority grows as the number of uninformative categories increases, as we expect. Real data study {#Sec5} =============== We now apply our ALOOF approach to real-world datasets and compare it with different commonly used tree-based methods. For obvious reasons, the data-sets we focus on are ones that include categorical features with a relatively large number of categories. All these dataset are collected from UCI repository [@Lichman:2013], CMU Statlib [^1] and Kaggle[^2] and are publicly available. Throughout this section we use ten-fold cross validation to achieve an averaged validation error on each dataset. In addition, we would like to statistically test the difference between ALOOF and its competitors and assert its statistical significance. However, the use of $K$-fold cross validation for model evaluation makes this task problematic, as the variance of cross validation scores cannot be estimated well [@bengio]. Some mitigations have been proposed to this problem in the literature, but we are not aware of a satisfactory solution. Therefore we limit our significance testing to policies that circumvent this problem: - For larger datasets ($>1000$ observations) we use a single 90-10 training-test division to test for significance separately from the cross validation scheme. - Combining the results of all datasets, we perform a sign test to verify the “overall” superiority of ALOOF. We start our demonstration with a comprehensive case study of the Melbourne grants dataset[^3] Case study: Melbourne Grants ---------------------------- The Melbourne Grants dataset was featured in a competition on the Kaggle website. The goal was to predict the success of grant applications based on a large number of variables, characterizing the applicant and the application. The full training dataset, on which we concentrate, included 8708 observations and 252 variables, with a binary response (success/failure). We only kept 26 variables (the first ones, excluding some identifiers), and eliminated observations that had missing values in these variables, leaving us with a dataset of $n=3650$ observations and $p=26$ variables. This dataset is interesting because some of the clearly relevant variables are categorical with many values, including: - The Field/Academic classification of the proposal (510 categories, encoded as codes from the Australian Standard Research Classification (ASRC)). - The Sponsor of the proposal (225 categories, an anonymized identifier). - The Department of the proposer (90 categories). - The Socio-economic code describing the potential impact of the proposed research (318 categories, encoded as codes from the Australian and New Zealand Standard Research Classification (ANZSRC)). Other clearly relevant variables include the history of success of failure by the current proposer, academic level, etc. In Figure \[fig:melb\] we present the top three level of three trees built on the entire dataset, using unlimited-K CART, limited-K CART and ALOOF. As expected, the unlimited-K version uses exclusively the largest-K categorical variables, the limited-K version is not exposed to them, so identifies the history of success as the most relevant non-categorical information, while ALOOF combines the use of one categorical variable (Sponsor), with the history of success. The ten-fold cross-validation error of the three complete trees, as shown in Table \[table:melb\], is $0.245, 0.260, 0.218$, respectively. As discussed above, the ability to accurately infer significance from $K$-fold cross validation is limited. Therefore, in addition to the cross validation experiments, we arbitrarily partition the dataset into $90\%$ train-set and $10\%$ test-set, and apply a $t$-test on results we achieve on the test-set. This way we conclude that the advantage of ALOOF is statistically significant at level $0.05$ on a single random fold of the ten-fold CV. The advantage of ALOOF over CART is preserved when they are used as sub-learners in Gradient Boosting (GB) and Random Forest (RF), also shown in Table \[table:melb\]. For GB we use $50$ trees and limit their complexity by defining the minimum number of observations in the trees’ terminal nodes to be $0.05\cdot n$. We train the model with a learning parameter of $\nu=0.1$ (see [@friedman2001] for further details). For the RF method we use [Matlab]{}’s procedure [treeBagger]{} with $500$ trees, where the rest of the parameters are maintained in their default values. Although the number of trees we utilize in both ensemble methods may seem relatively small, our experiments show that additional trees are not necessary as they do not significantly change the results we achieve. \[table:melb\] Approach Unlimited-K Limited-K ALOOF ------------------- ------------- ----------- ------------- Single tree $0.245$ $0.260$ $0.218^$ Gradient Boosting $0.186$ $0.192$ ${0.165^}$ Random Forest $0.172$ $0.185$ ${0.153^}$ : Misclassification results of two CART versions and ALOOF on the Melbourne Grant dataset. Results reported are from ten-fold CV. Results are shown for a single tree, Gradient Boosting trees and Random Forest (see text for details). The stars denote that the performance of ALOOF is statisticially significantly superior for all three versions. To gain further insight we can examine the variable importance measure commonly used in RF [@breiman2001random], which judges the performance based on permutation tests on out-of-bag (OOB) samples. In Table \[tab:RFimp\] we show the top five variables for each RF version. We can see that the lists of unlimited-K and limited-K CART have only one variable in common (Grant category code), while the ALOOF list combines the top three variables from the unlimited-K list and the top two from the limited-K list. This reiterates ALOOF’s ability to take advantage of both categorical and numerical variables in a balanced manner. \[tab:RFimp\] Rank Unlimited-K Limited-K ALOOF ------ --------------------- ---------------------------- ---------------------------- $1$ Sponsor No. of successful grants Sponsor $2$ Field No. of unsuccessful grants No. of successful grants $3$ Socio-economic code Grant category code No. of unsuccessful grants $4$ Department Faculty Field $5$ Grant category code Grant’s value Socio-economic code : Top variables according to Random Forest’s variable importance measure for each of the three tree-building approaches. As can be seen, ALOOF successfully identifies and combines the categorical and numeric variables that are most useful. Regression problems ------------------- We now turn to real-world regression problems. We first demonstrate our suggested approach with three classical small-size datasets. In the Automobile dataset, we model the price of a car given a set of relevant features. This set of features include the car’s brand which consists of $32$ categories. As previously discussed, the presence of such a categorical feature may cause overfitting as there are only $192$ observations. The Baseball experiment provides two designated datasets for hitters and catchers, where the goal is to predict the salary of each player in the preceding year. Here, the “problematic" categorical feature is the player’s team, consisting of $24$ different categories. Table \[table:real-world-small\] summarizes the results we achieve applying three different method. The CART column corresponds to a standard unlimited-K CART algorithm. CIT is the [R]{} implementation of the Conditional Inference Tree ([ctree]{}) using its default parameters of $10000$ Monte-Carlo replications to attain the distribution of the test statistic and a corresponding p-value threshold of $0.05$. ALOOF is our suggested algorithm. As our datasets are relatively small, we use ten-fold cross validation to estimate an averaged $MSE$. Ours results demonstrate the advantage that ALOOF’s LOO selection scheme has over both the overfitting approach of CART and significance-based approach of CIT in generating good prediction models. \[table:real-world-small\] [ M[3.5cm]{} M[1.9cm]{} M[1.2cm]{} M[0.9cm]{} M[1.1cm]{} N]{} Dataset (size) & ------------------- Categorical variables ($K_j$) ------------------- : Regression real-world data experiments on small datasets. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. CART is a standard CART implementation, CIT corresponds to the Conditional Inference Trees [@hothorn2006unbiased] and ALOOF is our suggested method. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation &CART &CIT &ALOOF&\ -------------- Auto $(192)$ -------------- : Regression real-world data experiments on small datasets. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. CART is a standard CART implementation, CIT corresponds to the Conditional Inference Trees [@hothorn2006unbiased] and ALOOF is our suggested method. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation & ------------------ Car brand $(32)$ ------------------ : Regression real-world data experiments on small datasets. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. CART is a standard CART implementation, CIT corresponds to the Conditional Inference Trees [@hothorn2006unbiased] and ALOOF is our suggested method. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation &$9.48$ &$13.24$ &${8.22}$&\ -------------------------- Baseball hitters $(264)$ -------------------------- : Regression real-world data experiments on small datasets. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. CART is a standard CART implementation, CIT corresponds to the Conditional Inference Trees [@hothorn2006unbiased] and ALOOF is our suggested method. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation & Team $(24)$ &$0.140$ &$0.136$ &$0.131$&\ --------------------------- Baseball catchers $(176)$ --------------------------- : Regression real-world data experiments on small datasets. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. CART is a standard CART implementation, CIT corresponds to the Conditional Inference Trees [@hothorn2006unbiased] and ALOOF is our suggested method. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation & Team $(24)$ &$0.076$ &$0.075$ &$0.062$&\ In addition to these illustrative small-size datasets we apply our suggested method to more modern and large-scale regression problems. Table \[table:real-world-large-reg\] presents the results we achieve in five different regression experiments. In Boston Housing we predict the value of each house, where the town’s name is the categorical variable which may cause overfitting. In the Strikes experiment we model volume of large strikes (days lost due to industrial disputes) over a total of more than $30$ years. The name of the country in which the strike took place is the categorical variable with relatively many categories. The Donations dataset is from KDD* Cup of $1998$ (excluding missing values) where participants were asked to model the amount of donations raised, given a list of variables. These include several categorical variables with large number of variables, as described in Table \[table:real-world-large-reg\]. The Internet usage dataset consists of two modeling problems. The first models the number of years a household is connected to an Internet provider while the second models the household’s income. The variables used for modeling include multiple socio-economical variables where the problematic categorical ones are the occupation, language and the residence country of each household. In all of these experiment we apply our ALOOF algorithm and compare it with a limited-$K$ and an unlimited-$K$ CART algorithms (as described above). Notice that in these large-scale experiments we cannot apply the CIT method [@hothorn2006unbiased] as it is computationally infeasible. In addition to these single tree models we also apply the ensemble methods Gradient Boosting (GB) and Random Forest (RF). We apply both of these methods using either unlimited-K CART trees or our suggested ALOOF method as sub-learners (limited-K is not competitive in relevant problems, where important categorical predictors are present, as demonstrated in the detailed Melbourne Grants case study). We apply GB and RF with the same set of parameters mentioned above. As in previous experiments we use ten-fold cross validation to achieve an averaged validation $MSE$. In addition, in order to attain statistically significant results we infer on a single random fold of the ten-fold CV for sufficiently large datasets (above $1000$ observations: the last three datasets in Table \[table:real-world-large-reg\]). A star indicates that the difference between ALOOF and its best competitor is statistically significant according to this $t$-test. The results we achieve demonstrate the advantage of using ALOOF both as a single tree and as a sub-learner in ensemble methods. Note that while the MSE ALOOF achieves in a single tree setting is consistently significantly lower than its competitors, it is not always the case with ensemble methods. This emphasizes the well-known advantage of using ensemble methods over a single tree, which can sometime mitigate the shortcomings of a single tree. However, it is still evident that ALOOF based ensemble methods are preferable (or at least equal) to the CART based methods. Moreover, it is notable that in some cases a single ALOOF tree achieves competitive results to ensemble based methods, such as in the Internet dataset. Classification problems ----------------------- In addition to the real-world regression experiments we also examine our suggested approach on a varity two-class classification problems. For each examined dataset we provide its number of observations and the portion of positive response values, as reference to the misclassification rate of the methods we apply. The Online Sales dataset describes the yearly online sales of different products. It is originally a regression problem which we converted to a two-class classification by comparing each response value to the mean of the response vector. Its variables’ names are confidential but there exist several categorical variables with relatively large number of categories, as described in Table \[table:real-world-large-class\]. The Melbourne Grants dataset is described in detail above. The Price Up dataset provides a list of products and specifies whether the prices were raised at different stores in different time slots. It provides a set of variables which consist of two “problematic" categorical variables: The name of the product and its brand. In the Salary Posting experiment our task is to determine the salary of a job posting given a set of relevant variables. As in the Online Sales, this dataset too is originally a regression problem which we converted to a two-class classification by comparing each response value with the mean of the response vector. Its “large-$K$" categorical variables are the location of the job, the company and the title of the offered position. Lastly, the Cabs Cancellations dataset provides a list of cab calls and specifies which ones were eventually canceled. The area code is the categorical variable which contains a large number of categories. As in the regression experiments, we apply different tree-based modeling methods to these datasets and compare the misclassification rate we achieve. In addition to the methods used in the regression problems we also apply the method of [@sabato2008ranking], which is designed for classification problems. We again perform a ten-fold cross validation to estimate the misclassification rate. As before, we also arbitrarily partition large datasets (the last four in Table \[table:real-world-large-reg\]) into $90\%$ train-set and $10\%$ test-set, and apply a paired test on the difference between the results we achieve using ALOOF and each of its competitors. A star indicates that the difference between ALOOF and its best competitor is statistically significant. As in the regression experiments, it is evident that our suggested ALOOF method is preferable both as a single tree and as sub-learner in ensemble methods. Note that in several experiments (such as Salary Postings) our advantage is even more remarkable compared with a limited-$K$ CART, which simply discards useful categorical variables only because they have too many categories. It is also notable that the single ALOOF tree is competitive with CART-based ensemble methods for several of the datasets. In particular, for the last three datasets the gain from replacing a single ALOOF tree with either GB or RF using CART is not statistically significant (whereas GB/RF + ALOOF does significantly improve on the CART ensemble counterparts). In addition to the misclassification rate, we also evaluated the performance of our suggested algorithm under the Area Under the Curve (AUC) criterion. The results we achieved were very simialr with those in Table \[table:real-world-large-reg\] and are omitted from this paper for brevity. In order to obtain a valid overall statistical inference on the results we achieve, we examined the null hypothesis that ALOOF performs equally well to each of its competitors, based on all the datasets together, using a one-sided sign test as suggested by [@demvsar2006statistical]. For every competitor, we count the number of datasets (out of ten in Tables \[table:real-world-large-reg\] and \[table:real-world-large-class\] combined) in which ALOOF wins and apply a one-sided sign test. ALOOF outperforms both versions of CART on all datasets examined, performs better in GB for nine of ten, and for RF in eight of ten, with corresponding one-sided p values of 0.001, 0.01 and 0.05, respectively, after continuity correction. Discussion and conclusion ========================= In this paper we have demonstrated that the simple cross-validation scheme underlying ALOOF can alleviate a major problems of tree-based methods that are not able to properly utilize categorical features with large number of categories in predictive modeling. By adopting a LOO framework for selecting the splitting variable we allow ALOOF to eliminate categorical features that do not improve prediction, and select the useful ones even if they have many categories. As our simulations and real data examples demonstrate, the effect of using the ALOOF approach for splitting feature selection can be dramatic in improving predictive performance, specifically when categorical features with many values are present and carry real information (i.e., cannot simply be discarded). A key aspect of our approach is the design of efficient algorithms for implementing ALOOF that minimize the extra computational work compared to simple CART splitting rules. As our results in Section \[Sec3\] indicate, ALOOF is more efficient in two-class classification than in regression. In practice, in our biggest regression examples in Section \[Sec5\] (about $10,000$ observations), building an ALOOF tree takes up to $15$ times longer than building a similar size CART tree (the exact ratio obviously depends on other parameters as well). This means that at this data size, regression tree with ALOOF is still a practical approach even with GB or RF. For larger data sizes, an obvious mitigating approach to the inefficiency of ALOOF in regression is to avoid using LOO in this case, instead using L-fold cross validation with $L<<n$. This would guarantee the computational complexity is no larger than $L$ times that of CART splitting even with a naive implementation. The price would be a potential reduction in accuracy, as the [almost unbiasedness]{} discussed in Section \[unbias\] relies on the LOO scheme. Acknowledgments =============== This research was partially supported by Israel Science Foundation grant 1487/12 and by a returning scientist fellowship from the Israeli Ministry of Immigration to Amichai Painsky. The authors thank Jerry Friedman and Liran Katzir for useful discussions. \[table:real-world-large-reg\] [M[1.6cm]{} M[2.6cm]{} M[1.5cm]{} M[1.75cm]{} M[1.1cm]{} M[1.2cm]{} M[1.2cm]{} M[1.2cm]{} M[1.2cm]{} N]{} Dataset (size) & ------------------- Categorical variables ($K_j$) ------------------- : Regression real-world data experiments. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. limited-$K$ CART is a CART implementation which discards categorical variables with $K_j>32$ while the unlimited-$K$ CART is an implementation which does not discard any variable. Gradient Boosting (GB) modeling with $50$ trees and Random Forest (RF) with $500$ are also provided, both with either CART or ALOOF as a sub-learner. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation. In addition, a star indicates that the difference between ALOOF its best competitor in the same category (single tree, GB, RF) is statistically significant on a single arbitrary partitioning to train and test sets (this test was only applied to the last three datasets, with more than 1000 observations). & ------------- Limited-$K$ CART ------------- : Regression real-world data experiments. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. limited-$K$ CART is a CART implementation which discards categorical variables with $K_j>32$ while the unlimited-$K$ CART is an implementation which does not discard any variable. Gradient Boosting (GB) modeling with $50$ trees and Random Forest (RF) with $500$ are also provided, both with either CART or ALOOF as a sub-learner. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation. In addition, a star indicates that the difference between ALOOF its best competitor in the same category (single tree, GB, RF) is statistically significant on a single arbitrary partitioning to train and test sets (this test was only applied to the last three datasets, with more than 1000 observations). & --------------- Unlimited-$K$ CART --------------- : Regression real-world data experiments. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. limited-$K$ CART is a CART implementation which discards categorical variables with $K_j>32$ while the unlimited-$K$ CART is an implementation which does not discard any variable. Gradient Boosting (GB) modeling with $50$ trees and Random Forest (RF) with $500$ are also provided, both with either CART or ALOOF as a sub-learner. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation. In addition, a star indicates that the difference between ALOOF its best competitor in the same category (single tree, GB, RF) is statistically significant on a single arbitrary partitioning to train and test sets (this test was only applied to the last three datasets, with more than 1000 observations). &ALOOF & -------- GB via CART -------- : Regression real-world data experiments. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. limited-$K$ CART is a CART implementation which discards categorical variables with $K_j>32$ while the unlimited-$K$ CART is an implementation which does not discard any variable. Gradient Boosting (GB) modeling with $50$ trees and Random Forest (RF) with $500$ are also provided, both with either CART or ALOOF as a sub-learner. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation. In addition, a star indicates that the difference between ALOOF its best competitor in the same category (single tree, GB, RF) is statistically significant on a single arbitrary partitioning to train and test sets (this test was only applied to the last three datasets, with more than 1000 observations). & -------- GB via ALOOF -------- : Regression real-world data experiments. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. limited-$K$ CART is a CART implementation which discards categorical variables with $K_j>32$ while the unlimited-$K$ CART is an implementation which does not discard any variable. Gradient Boosting (GB) modeling with $50$ trees and Random Forest (RF) with $500$ are also provided, both with either CART or ALOOF as a sub-learner. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation. In addition, a star indicates that the difference between ALOOF its best competitor in the same category (single tree, GB, RF) is statistically significant on a single arbitrary partitioning to train and test sets (this test was only applied to the last three datasets, with more than 1000 observations). & -------- RF via CART -------- : Regression real-world data experiments. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. limited-$K$ CART is a CART implementation which discards categorical variables with $K_j>32$ while the unlimited-$K$ CART is an implementation which does not discard any variable. Gradient Boosting (GB) modeling with $50$ trees and Random Forest (RF) with $500$ are also provided, both with either CART or ALOOF as a sub-learner. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation. In addition, a star indicates that the difference between ALOOF its best competitor in the same category (single tree, GB, RF) is statistically significant on a single arbitrary partitioning to train and test sets (this test was only applied to the last three datasets, with more than 1000 observations). & -------- RF via ALOOF -------- : Regression real-world data experiments. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. limited-$K$ CART is a CART implementation which discards categorical variables with $K_j>32$ while the unlimited-$K$ CART is an implementation which does not discard any variable. Gradient Boosting (GB) modeling with $50$ trees and Random Forest (RF) with $500$ are also provided, both with either CART or ALOOF as a sub-learner. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation. In addition, a star indicates that the difference between ALOOF its best competitor in the same category (single tree, GB, RF) is statistically significant on a single arbitrary partitioning to train and test sets (this test was only applied to the last three datasets, with more than 1000 observations). &\ --------- Boston Housing $(506)$ --------- : Regression real-world data experiments. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. limited-$K$ CART is a CART implementation which discards categorical variables with $K_j>32$ while the unlimited-$K$ CART is an implementation which does not discard any variable. Gradient Boosting (GB) modeling with $50$ trees and Random Forest (RF) with $500$ are also provided, both with either CART or ALOOF as a sub-learner. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation. In addition, a star indicates that the difference between ALOOF its best competitor in the same category (single tree, GB, RF) is statistically significant on a single arbitrary partitioning to train and test sets (this test was only applied to the last three datasets, with more than 1000 observations). &Town name $(88)$ &$24.35$ &$24.07$ &${20.83}$ &$9.24$ &${7.88}$ &$9.08$ &${8.99}$&\ --------- Strikes $(626)$ --------- : Regression real-world data experiments. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. limited-$K$ CART is a CART implementation which discards categorical variables with $K_j>32$ while the unlimited-$K$ CART is an implementation which does not discard any variable. Gradient Boosting (GB) modeling with $50$ trees and Random Forest (RF) with $500$ are also provided, both with either CART or ALOOF as a sub-learner. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation. In addition, a star indicates that the difference between ALOOF its best competitor in the same category (single tree, GB, RF) is statistically significant on a single arbitrary partitioning to train and test sets (this test was only applied to the last three datasets, with more than 1000 observations). &Country $(18)$ &$315.95$ &$315.95$ &${280.23}$ &$256.31$ &$257.12$ &$255.59$ &$256.06$&\ ----------- Donations $(6521)$ ----------- : Regression real-world data experiments. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. limited-$K$ CART is a CART implementation which discards categorical variables with $K_j>32$ while the unlimited-$K$ CART is an implementation which does not discard any variable. Gradient Boosting (GB) modeling with $50$ trees and Random Forest (RF) with $500$ are also provided, both with either CART or ALOOF as a sub-learner. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation. In addition, a star indicates that the difference between ALOOF its best competitor in the same category (single tree, GB, RF) is statistically significant on a single arbitrary partitioning to train and test sets (this test was only applied to the last three datasets, with more than 1000 observations). & ----------------------- Socio-cluster $(62)$, ADI code $(180)$, DMA code $(180)$, MSA code $(245)$, Mailing list $(348)$, Zip code $(3824)$ ----------------------- : Regression real-world data experiments. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. limited-$K$ CART is a CART implementation which discards categorical variables with $K_j>32$ while the unlimited-$K$ CART is an implementation which does not discard any variable. Gradient Boosting (GB) modeling with $50$ trees and Random Forest (RF) with $500$ are also provided, both with either CART or ALOOF as a sub-learner. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation. In addition, a star indicates that the difference between ALOOF its best competitor in the same category (single tree, GB, RF) is statistically significant on a single arbitrary partitioning to train and test sets (this test was only applied to the last three datasets, with more than 1000 observations). &$30.99$ &$39.30$ &${22.92^}$ &$23.34$ &${18.89^}$ &$21.39$ &${16.28^}$&\ -------------- Internet - years online $(10108)$ -------------- : Regression real-world data experiments. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. limited-$K$ CART is a CART implementation which discards categorical variables with $K_j>32$ while the unlimited-$K$ CART is an implementation which does not discard any variable. Gradient Boosting (GB) modeling with $50$ trees and Random Forest (RF) with $500$ are also provided, both with either CART or ALOOF as a sub-learner. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation. In addition, a star indicates that the difference between ALOOF its best competitor in the same category (single tree, GB, RF) is statistically significant on a single arbitrary partitioning to train and test sets (this test was only applied to the last three datasets, with more than 1000 observations). & -------------------- Occupation $(46)$, Language $(99)$, Country $(129)$ -------------------- : Regression real-world data experiments. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. limited-$K$ CART is a CART implementation which discards categorical variables with $K_j>32$ while the unlimited-$K$ CART is an implementation which does not discard any variable. Gradient Boosting (GB) modeling with $50$ trees and Random Forest (RF) with $500$ are also provided, both with either CART or ALOOF as a sub-learner. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation. In addition, a star indicates that the difference between ALOOF its best competitor in the same category (single tree, GB, RF) is statistically significant on a single arbitrary partitioning to train and test sets (this test was only applied to the last three datasets, with more than 1000 observations). &$0.727$ &$0.754$ &${0.703^}$ &$0.699$ &$0.698$ &$0.698$ &$0.698$&\ ------------ Internet - household income $(10108)$ ------------ : Regression real-world data experiments. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. limited-$K$ CART is a CART implementation which discards categorical variables with $K_j>32$ while the unlimited-$K$ CART is an implementation which does not discard any variable. Gradient Boosting (GB) modeling with $50$ trees and Random Forest (RF) with $500$ are also provided, both with either CART or ALOOF as a sub-learner. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation. In addition, a star indicates that the difference between ALOOF its best competitor in the same category (single tree, GB, RF) is statistically significant on a single arbitrary partitioning to train and test sets (this test was only applied to the last three datasets, with more than 1000 observations). & -------------------- Occupation $(46)$, Language $(99)$, Country $(129)$ -------------------- : Regression real-world data experiments. For each dataset we introduce its categorical variables whose number of categories $K_j$ is relatively large. limited-$K$ CART is a CART implementation which discards categorical variables with $K_j>32$ while the unlimited-$K$ CART is an implementation which does not discard any variable. Gradient Boosting (GB) modeling with $50$ trees and Random Forest (RF) with $500$ are also provided, both with either CART or ALOOF as a sub-learner. The performance of each method is measured by averaged $MSE$, via ten-fold cross validation. In addition, a star indicates that the difference between ALOOF its best competitor in the same category (single tree, GB, RF) is statistically significant on a single arbitrary partitioning to train and test sets (this test was only applied to the last three datasets, with more than 1000 observations). &$4.144$ &$4.308$ &${4.103^}$ &$4.099$ &${4.005^}$ &$3.991$ &$3.989$&\ \[table:real-world-large-class\] [ M[1.9cm]{} M[2.5cm]{} M[1.45cm]{} M[1.75cm]{} M[0.7cm]{} M[1.1cm]{} M[0.95cm]{} M[1.1cm]{} M[0.9cm]{} M[1.1cm]{} N]{} Dataset (size, portion of positives) & ------------------- Categorical variables ($K_j$) ------------------- & ------------- Limited-$K$ CART ------------- & --------------- Unlimited-$K$ CART --------------- &S&S &ALOOF & -------- GB via CART -------- & -------- GB via ALOOF -------- & -------- RF via CART -------- & -------- RF via ALOOF -------- &\ --------------- Online Sales $(725, 0.29)$ --------------- & ---------------------- Confidential $(70)$ Confidential $(273)$ Confidential $(406)$ ---------------------- &$0.198$ &$0.1903$ &$0.153$ &${0.151}$ &$0.145$ &${0.135}$ &$0.134$ &${0.121}$&\ ---------------- Melbourne Grants $(3650, 0.29)$ ---------------- & ------------------- Department $(99)$ Sponsor $(234)$ Field $(519)$ ------------------- &$0.260$ &$0.245$ &$0.240$ &${0.218^}$ &$0.186$ &${0.165^}$ &$0.172$ &${0.153^}$&\ ----------------- Price Up $(3980, 0.045)$ ----------------- & ----------------- Brand $(27)$ Product $(216)$ ----------------- &$0.035$ &$0.047$ &$0.036$ &${0.033^}$ &$0.039$ &${0.034^}$ &$0.034$ &$0.034$&\ ----------------- Salary Posting $(10000, 0.31)$ ----------------- & ------------------ Location $(390)$ Company $(390)$ Title $(8324)$ ------------------ &$0.330$ &$0.149$ &$0.148$ &${0.142^}$ &$0.142$ &$0.142$ &$0.140$ &${0.138^}$&\ ------------------ Cabs Cancellations $(34293, 0.096)$ ------------------ &Area code $(586)$ &$0.093$ &$0.095$ &$0.088$ &${0.083^}$ &$0.088$ &${0.083^*}$ &$0.083$ &$0.083$&\ [^1]: <http://lib.stat.cmu.edu/> [^2]: <http://www.kaggle.com/competitions> [^3]: <https://www.kaggle.com/c/unimelb>
--- abstract: 'We study the Quantum Electrodynamics of 2D and 3D Dirac semimetals by means of a self-consistent resolution of the Schwinger-Dyson equations, aiming to obtain the respective phase diagrams in terms of the relative strength of the Coulomb interaction and the number $N$ of Dirac fermions. In this framework, 2D Dirac semimetals have just a strong-coupling instability characterized by exciton condensation (and dynamical generation of mass) that we find at a critical coupling well above previous theoretical estimates, thus explaining the absence of that instability in free-standing graphene samples. On the other hand, we show that 3D Dirac semimetals have a richer phase diagram, with a strong-coupling instability leading to dynamical mass generation up to $N$ = 4 and a line of critical points for larger values of $N$ characterized by the vanishing of the electron quasiparticle weight in the low-energy limit. Such a critical behavior signals the transition to a strongly correlated liquid, characterized by noninteger scaling dimensions that imply the absence of a pole in the electron propagator and are the signature of non-Fermi liquid behavior with no stable electron quasiparticles.' address: 'Instituto de Estructura de la Materia, Consejo Superior de Investigaciones Científicas, Serrano 123, 28006 Madrid, Spain' author: - 'J. González' title: Phase diagram of the Quantum Electrodynamics of 2D and 3D Dirac semimetals --- Introduction ============ The discovery of graphene[@novo] has marked the beginning of a new chapter in condensed matter physics, with the appearance of new fundamental concepts and materials with unconventional properties. We have known about other genuine 2D materials like the transition metal dichalcogenides, and we have learned about the features hidden in the band structure of the topological insulators[@top1; @top2]. Lately, we have also seen the discovery of 3D Dirac semimetals which are the higher-dimensional analog of graphene[@liu; @neupane; @borisenko; @yu], and we have witnessed the ongoing search of Weyl semimetals with a built-in breakdown of parity and time-reversal invariance. In the above instances, most part of the unconventional features of the materials come from the peculiar geometrical and topological properties of the band structure. Moreover, these are electron systems that are prone to being placed in the strong coupling regime, with a large relative strength of the Coulomb interaction. Graphene should be a clear example of electron system with strong interaction, given the large ratio between the square of the electron charge and the Fermi velocity in the 2D material. However, graphene is not a prototype of strongly correlated system, even in the case of the free-standing material in vacuum, as experimental observations have shown no sign of electronic instability down to very low doping levels[@exp2]. On the other hand, most part of theoretical studies[@khves; @gus; @vafek; @khves2; @her; @jur; @drut1; @hands2; @gama; @fer; @ggg; @sabio; @prb] have estimated that 2D Dirac semimetals should have an excitonic instability at a critical point below the maximum interaction strength attained in graphene suspended in vacuum. The absence of any signature of a gap in the electronic spectrum, even below the meV scale, is certainly a puzzling evidence regarding the behavior of the 2D material. Given that the theoretical analyses have been mainly based on a ladder approximation to the electron self-energy corrections, the discrepancy between theory and experiment calls into question the use of such approximate methods in electron systems that are placed in the strong-coupling regime[@barnes]. In this paper, we apply a nonperturbative approach to the investigation of the effects of the Coulomb interaction in both 2D and 3D semimetals, with the aim of mapping more confidently the different phases that may appear in those electron systems. More precisely, we carry out the self-consistent resolution of the Schwinger-Dyson equations for the Quantum Electrodynamics (QED) of 2D and 3D Dirac semimetals, in which the scalar part of the electromagnetic potential is used to mediate in each case the long-range $e$-$e$ interaction. This approach has to be implemented in general with some kind of truncation to guarantee its practical feasibility. In this regard, we have relied on a formulation of the equations that amounts to including all kinds of diagrammatic contributions except those containing vertex corrections. Nevertheless, the solutions obtained in this way account for the renormalization of all the quasiparticle parameters, giving rise to frequency and momentum-dependent forms of the quasiparticle weight, the Fermi velocity and the dynamical Dirac fermion mass. In this framework, we will see that 2D Dirac semimetals have just a strong-coupling instability characterized by exciton condensation (and dynamical generation of mass) that we find at a critical coupling well above the estimates based on a ladder approximation, thus explaining the absence of that instability in free-standing graphene samples. On the other hand, we will show that 3D Dirac semimetals have a richer phase diagram, with a strong-coupling instability leading to dynamical mass generation up to $N$ = 4 and a line of critical points for larger values of $N$ characterized by the vanishing of the electron quasiparticle weight in the low-energy limit[@rc]. We will see that such a critical behavior marks the transition to a strongly correlated liquid, characterized by noninteger scaling dimensions that imply the absence of a pole in the electron propagator and are the signature of non-Fermi liquid behavior with no stable electron quasiparticles[@barnes2]. Quantum Electrodynamics of Dirac semimetals =========================================== We focus on the QED of Dirac semimetals, for which the Fermi velocity $v_F$ is much smaller than the speed of light. The dynamics of these systems can be then described by the interaction of a number $N$ of four-component Dirac spinor fields $\psi_i ({\bf r})$ representing the electron quasiparticles and the scalar part $\phi ({\bf r})$ of the electromagnetic potential. In principle, each spinor can represent the electronic states around a different Dirac point in momentum space, but we will not make more explicit such a discrimination as it does not play any role in the subsequent analysis. The hamiltonian can be written in general as $$H = i v_F \int d^D r \; \psi^{\dagger}_i ({\bf r}) \gamma_0 \mbox{\boldmath $\gamma $} \cdot \mbox{\boldmath $\nabla $} \psi_i ({\bf r}) + e \int d^D r \; \psi^{\dagger}_i ({\bf r}) \psi_i ({\bf r}) \; \phi ({\bf r}) \label{ham}$$ where $\{ \gamma_\alpha \}$ is a set of Dirac matrices satisfying $\{\gamma_\alpha , \gamma_\beta \} = 2\eta_{\alpha \beta}$ (with the Minkowski metric in $D+1$ dimensions $\eta = {\rm diag}(-1,1,\ldots 1)$). The expression (\[ham\]) holds equally well for dimension $D = 2$ and $3$, but the propagator of the $\phi $ field is very different in the two cases. The scalar field has to mediate the $e$-$e$ interaction with long-range Coulomb potential $V({\bf r}) = 1/\|{\bf r}\|$, irrespective of the spatial dimension. At $D = 2$, this leads to a bare propagator $D_0 ({\bf q},\omega )$ for the $\phi $ field $$\left. D_0 ({\bf q})\right\|_{D = 2} = \frac{1}{2 \|{\bf q}\|}$$ while in 3D space the bare propagator is instead $$\left. D_0 ({\bf q})\right\|_{D = 3} = \frac{1}{{\bf q}^2} \label{lr}$$ The effects of the interaction can be characterized through the corrections undergone by the scalar and the Dirac field propagators. The full Dirac propagator $G ({\bf k},\omega_k )$ has in general a representation of the form $$G ({\bf k},\omega_k )^{-1} = (\omega_k - v_F \gamma_0 \mbox{\boldmath $\gamma $} \cdot {\bf k} ) - \Sigma ({\bf k},\omega_k )$$ in terms of a self-energy correction $\Sigma ({\bf k},\omega_k )$ that contributes to renormalize the bare quasiparticle parameters. This object is given in turn by the equation $$i \Sigma ({\bf k},\omega_k ) = - e^2 \int \frac{d^D p}{(2\pi )^D} \frac{d\omega_p }{2\pi } D({\bf p},\omega_p ) G({\bf k} - {\bf p}, \omega_k - \omega_p) \Gamma ({\bf p},\omega_p;{\bf k},\omega_k) \label{si}$$ where $D({\bf p},\omega_p )$ stands for the full propagator of the scalar potential and $\Gamma ({\bf q},\omega_q;{\bf k},\omega_k)$ represents the irreducible three-point vertex. More precisely, this function is defined by the expectation value $$i e \Gamma ({\bf q},\omega_q;{\bf k},\omega_k) = \langle \phi ({\bf q},\omega_q) \psi_i ({\bf k}-{\bf q},\omega_k - \omega_q) \psi^{\dagger}_i ({\bf k},\omega_k) \rangle_{\rm 1PI} \label{gamma}$$ where 1PI means that we must take the irreducible part of the correlator. Furthermore, $D({\bf p},\omega_p )$ has also its own equation representing it in terms of the full propagators and the irreducible vertex. We can write $$D ({\bf q},\omega_q )^{-1} = D_0 ({\bf q})^{-1} - \Pi ({\bf q},\omega_q )$$ with the polarization $\Pi ({\bf q},\omega_q )$ being given by $$i\Pi ({\bf q},\omega_q ) = N e^2 \int \frac{d^D p}{(2\pi )^D} \frac{d\omega_p }{2\pi } {\rm Tr} \left[ G({\bf q} + {\bf p}, \omega_q + \omega_p) \Gamma ({\bf q},\omega_q;{\bf p},\omega_p) G({\bf p},\omega_p ) \right] \label{pi}$$ Moreover, the form of the irreducible vertex $\Gamma ({\bf q},\omega_q;{\bf k},\omega_k)$ is constrained by the Ward identity arising from the reduced gauge invariance of the model, admitting also a representation in terms of the full propagators[@bloch]. The expressions (\[si\]) and (\[pi\]) correspond to the Schwinger-Dyson equations of the model. Together with a suitable representation of the irreducible vertex, they may lead to valuable information about the form of the full fermion and interaction propagators. In general, however, one has to resort to some kind of truncation to achieve a self-consistent resolution of the integral equations. In what follows, we will apply a common procedure, the so-called bare vertex approximation, by which $\Gamma ({\bf q},\omega_q;{\bf k},\omega_k)$ is set equal to the unit matrix in the resolution of (\[si\]) and (\[pi\]). We note that this truncation does not satisfy the mentioned Ward identity, which relates the irreducible vertex to the derivative with respect to the frequency of the fermion self-energy. In this regard, the present work focuses on the investigation of dynamical effects in the renormalization of the quasiparticle parameters, which could be further improved by introducing a suitable ansatz for the vertex (in similar fashion as in the fully relativistic QED[@bloch]). Nevertheless, we note that the relevant features reported below within our approach are consistent with the results found by means of renormalization group methods[@rc], which supports the present formulation. The main advantage of adopting the mentioned truncation is that it leads to a very convenient implementation of the self-consistent approach, allowing us to attain easily convergence in the recursive resolution of the Schwinger-Dyson equations. Without the vertex corrections, (\[si\]) and (\[pi\]) lead indeed to closed self-consistent equations, shown diagrammatically in Fig. \[one\]. This representation allows us to establish a comparison with other standard approaches used to deal with many-body corrections. In particular, it becomes clear that the contributions accounted for by the diagrams in Fig. \[one\] have a much more comprehensive content than other approaches dealing with the RPA sum of bubble diagrams for the polarization. This makes the present computational scheme much more reliable to describe the electron system away from the weak-coupling regime, incorporating effects like the renormalization of the Fermi velocity and the quasiparticle weight which are essential to capture the different critical points of the Dirac semimetals. \ Self-consistent resolution of Schwinger-Dyson equations ======================================================= The integral equations represented in Fig. \[one\] can be solved numerically by applying a recursive procedure, after rotating first all the frequencies in the complex plane, $\overline{\omega} = -i \omega$, to make the passage to a Euclidean space in the variables $(\overline{\omega }, {\bf k})$. In practice, the integrals can be done numerically by discretizing the frequency and momentum variables. By choosing a set of frequencies $\overline{\omega} = \pi (2n+1)T$ with $n = 0, \pm 1, \pm 2, \ldots $, we can interpret such a discretization as the result of placing the theory at finite temperature $T$. On the other hand, computing with a grid in momentum space is equivalent to describing a system with finite spatial size. In this case, we can check the finite-size scaling of the results in order to extrapolate the behavior over large distances. For the self-consistent resolution, it becomes convenient to represent the fermion propagator in terms of renormalization factors $z_\psi({\bf k}, i\overline{\omega})$ for the electron wave-function and $z_v({\bf k}, i\overline{\omega})$ for the Fermi velocity, adding moreover another factor $z_m({\bf k}, i\overline{\omega})$ to allow for the dynamical generation of a mass for the Dirac fermions. Thus we write the full Dirac propagator in the form $$G({\bf k}, i\overline{\omega}) = \left( z_\psi({\bf k}, i\overline{\omega}) i \overline{\omega} - z_v({\bf k}, i\overline{\omega}) v_F \gamma_0 \mbox{\boldmath $\gamma $} \cdot {\bf k} - z_m({\bf k}, i\overline{\omega}) \gamma_0 \right)^{-1} \label{anstz}$$ In this way, the resolution consists in finding the functions $z_\psi({\bf k}, i\overline{\omega}), z_v({\bf k}, i\overline{\omega})$ and $z_m({\bf k}, i\overline{\omega})$ that attain the self-consistency in the Schwinger-Dyson equations. One more important detail is that the polarization $\Pi ({\bf q},\omega_q )$ may develop spurious divergences when computing the momentum integrals with a simple cutoff $\Lambda_k $. In general, only a gauge-invariant regularization scheme can produce results without non-physical power-law dependences on the cutoff[@np2]. These are anyhow additive contributions to the polarization, which makes possible to get rid of them by a suitable subtraction procedure. Thus, computing with the momentum cutoff, the polarization at $D = 2$ shows a contribution proportional to $\Lambda_k $, while the corresponding function at $D = 3$ has terms growing as large as $\Lambda_k^2 $. In our self-consistent resolution, we have carried out the frequency integrals first with a cutoff $\Lambda \gg v_F \Lambda_k$, implementing afterwards the subtraction procedure to remove the power-law dependences on $\Lambda_k$ from the polarization. In this way, we have ended up with expressions of $\Pi ({\bf q},\omega_q )$ that are functionals of the renormalization factors, displaying leading behaviors at small ${\bf q}$ proportional to $\|{\bf q}\|$ and ${\bf q}^2$, respectively, for $D = 2$ and $D = 3$. 2D Dirac semimetals ------------------- Solving the Schwinger-Dyson equations at $D = 2$, we find in general that the function $z_\psi({\bf k}, i\overline{\omega})$ giving the quasiparticle weight remains bounded, while $z_v({\bf k}, i\overline{\omega})$ diverges in the limit of small momentum ${\bf k} \rightarrow 0$. As long as the effective Fermi velocity depends on the momentum scale, it is convenient to define the bare value $v_B = z_v(\Lambda_k, 0) v_F$, which can be taken as a good measure of the Fermi velocity at the microscopic scale (it is always verified that $z_\psi(\Lambda_k,0) \approx 1$). We can then define the unrenormalized coupling giving the bare interaction strength as $$\alpha = e^2/4\pi v_B$$ which can take different values depending on the particular Fermi velocities of the 2D Dirac semimetals. As an illustration of the general behavior, Fig. \[two\] represents the solution obtained for $z_\psi({\bf k}, i\overline{\omega})$ and $z_v({\bf k},i \overline{\omega})$ for $N = 2$ and $\alpha = 2.2$, that is, for parameters that should be appropriate to describe graphene samples suspended in vacuum. The resolution has been carried out taking a discretization of the frequency variables such that $2\pi T \approx 0.01$ eV. In this case, the self-consistency in the equations is only attained when $z_m({\bf k}, i\overline{\omega})$ is set identically equal to zero. The behavior found for $z_\psi({\bf k}, i\overline{\omega})$ and $z_v({\bf k}, i\overline{\omega})$ is in agreement with the general trend obtained from renormalization group methods, which found the divergence of the Fermi velocity in the low-energy limit as a most relevant feature[@np2; @prbr]. For comparison, we have also represented in Fig. \[two\] the dependence on the energy scale $\varepsilon $ of the inverse of the quasiparticle weight $z (\varepsilon )$ and the renormalized Fermi velocity $v (\varepsilon )$ obtained from the renormalization group approach in the large-$N$ approximation[@prbr]. It can be observed anyhow that the plot of $z_v ({\bf k},0)/z_\psi ({\bf k},0)$ (full line in Fig. \[two\](b)) follows the experimental results of Ref. (Fig. 2(c) in that paper) much more accurately than the scale dependence of the Fermi velocity obtained with the renormalization group method in the large-$N$ approximation (dashed line in Fig. \[two\](b))[@footv]. ![Plot of the factors $z_\psi (0,i\omega)$ and $z_v ({\bf k},0)/z_\psi ({\bf k},0)$ (full lines in (a) and (b)) for a 2D Dirac semimetal with $N = 2$, bare coupling $\alpha = 2.2$, and $2\pi T \approx 0.01$ eV. The dashed lines represent the dependence on the energy scale $\varepsilon $ of the inverse of the quasiparticle weight $z (\varepsilon )$ (in (a)) and the renormalized Fermi velocity $v (\varepsilon )$ (in (b)) obtained with the renormalization group approach for the same bare coupling in the large-$N$ approximation.[]{data-label="two"}](zrealgrg.eps "fig:"){width="3.9cm"} ![Plot of the factors $z_\psi (0,i\omega)$ and $z_v ({\bf k},0)/z_\psi ({\bf k},0)$ (full lines in (a) and (b)) for a 2D Dirac semimetal with $N = 2$, bare coupling $\alpha = 2.2$, and $2\pi T \approx 0.01$ eV. The dashed lines represent the dependence on the energy scale $\varepsilon $ of the inverse of the quasiparticle weight $z (\varepsilon )$ (in (a)) and the renormalized Fermi velocity $v (\varepsilon )$ (in (b)) obtained with the renormalization group approach for the same bare coupling in the large-$N$ approximation.[]{data-label="two"}](vrealgrg.eps "fig:"){width="3.9cm"}\ (a) (b) ![Plot of $z_\psi (0,i\omega)$, $z_v ({\bf k},0)/z_\psi ({\bf k},0)$, and $z_m ({\bf k},0)/z_\psi ({\bf k},0)$ measured in eV (full lines in (a), (b) and (c)) for a 2D Dirac semimetal with $N = 2$, bare coupling $\alpha = 3.37$, and $2\pi T \approx 0.01$ eV. The dashed lines represent the dependence on the energy scale $\varepsilon $ of the inverse of the quasiparticle weight $z (\varepsilon )$ (in (a)) and the renormalized Fermi velocity $v (\varepsilon )$ (in (b)) obtained with the renormalization group approach for the same bare coupling in the large-$N$ approximation.[]{data-label="three"}](zabovegrg.eps "fig:"){width="3.9cm"} ![Plot of $z_\psi (0,i\omega)$, $z_v ({\bf k},0)/z_\psi ({\bf k},0)$, and $z_m ({\bf k},0)/z_\psi ({\bf k},0)$ measured in eV (full lines in (a), (b) and (c)) for a 2D Dirac semimetal with $N = 2$, bare coupling $\alpha = 3.37$, and $2\pi T \approx 0.01$ eV. The dashed lines represent the dependence on the energy scale $\varepsilon $ of the inverse of the quasiparticle weight $z (\varepsilon )$ (in (a)) and the renormalized Fermi velocity $v (\varepsilon )$ (in (b)) obtained with the renormalization group approach for the same bare coupling in the large-$N$ approximation.[]{data-label="three"}](vabovegrg.eps "fig:"){width="3.9cm"} ![Plot of $z_\psi (0,i\omega)$, $z_v ({\bf k},0)/z_\psi ({\bf k},0)$, and $z_m ({\bf k},0)/z_\psi ({\bf k},0)$ measured in eV (full lines in (a), (b) and (c)) for a 2D Dirac semimetal with $N = 2$, bare coupling $\alpha = 3.37$, and $2\pi T \approx 0.01$ eV. The dashed lines represent the dependence on the energy scale $\varepsilon $ of the inverse of the quasiparticle weight $z (\varepsilon )$ (in (a)) and the renormalized Fermi velocity $v (\varepsilon )$ (in (b)) obtained with the renormalization group approach for the same bare coupling in the large-$N$ approximation.[]{data-label="three"}](mabovegrate.eps "fig:"){width="4.4cm"}\ (a) (b) (c) With our nonperturbative approach, moreover, we can ask whether the tendency towards a noninteracting Fermi liquid, implied by the growth of the Fermi velocity, can be arrested by some instability as the bare coupling $\alpha $ is increased. The outcome of this search is that the other relevant feature of the 2D Dirac semimetals is the development of a nonvanishing mass $z_m({\bf k}, i\overline{\omega})$ at sufficiently large interaction strength, as illustrated in Fig. \[three\]. This corresponds to the onset of a phase with chiral symmetry breaking and dynamical generation of a gap for the Dirac fermions, as predicted by several other methods[@khves; @gus; @vafek; @khves2; @her; @jur; @drut1; @hands2; @gama; @fer; @ggg; @sabio; @prb]. We observe that the divergence of $z_v$ in the low-energy limit does not prevent the development of a nonvanishing dynamical mass $z_m$, while $z_\psi$ remains finite accross the transition. This latter fact implies that chiral symmetry breaking proceeds without an anomalous scaling of the Dirac fermion field, in agreement with field theory studies of that phenomenon[@nev]. In general, we also expect that the divergent growth of the Fermi velocity is arrested at the energy scale for which the quasiparticle dispersion becomes gapped (which is not appreciated in the case of Fig. \[three\](b) as the infrared cutoff set in that plot by finite-size effects is slightly above 1 meV). The present approach has the virtue of allowing an accurate determination of the critical coupling $\alpha_c$ for the transition to the broken symmetry phase. We have represented in Fig. \[four\] the plot of the critical coupling obtained as a function of $N$, which leads to a map of the two different phases in the QED of the 2D Dirac semimetals. In agreement with earlier analyses, we observe that $\alpha_c$ turns out to grow with $N$, though in the present resolution there seems to be no upper limit in the number of Dirac fermions for the development of the transition[@foot]. We have also checked that the approach to the critical coupling seems to be consistent with a transition of infinite order, since the dynamical mass exhibits an inflection point as a function of coupling constant above $\alpha_c$ which is the signature of that kind of transition under finite-size effects[@dress]. ![Phase diagram of the QED of 2D Dirac semimetals showing the region with dynamically generated fermion mass $z_m \neq 0$. The critical line has been obtained from the self-consistent resolution of the Schwinger-Dyson equations with $2\pi T \approx 0.01$ eV. The dashed line marks the nominal interaction strength $\alpha \approx 2.2$ corresponding to graphene suspended in vacuum.[]{data-label="four"}](phase2d2.eps){width="6.0cm"} In any case, the most relevant result regarding the phase diagram in Fig. \[four\] is that the point corresponding to real graphene samples suspended in vacuum falls in the region with no dynamical generation of mass. For the number $N = 2$ of Dirac fermions in graphene, the critical coupling obtained for chiral symmetry breaking turns out to be indeed well above most part of previous estimates relying on a restricted sum of many-body corrections. The present results explain therefore that no gap has been found in the electronic spectrum of graphene, even in experiments looking very close to the Dirac point[@exp2]. The reason for the unexpectedly large values of the critical coupling can be traced back to the combination of the slight suppression of the quasiparticle weight and the large growth of the Fermi velocity at low energies[@fv]. These two effects cooperate to reduce significantly the effective strength of the Coulomb interaction for the development of the gap, stressing the importance of a proper account of all the renormalization factors for the accurate determination of the transition to the broken symmetry phase. 3D Dirac semimetals ------------------- The 3D Dirac semimetals have in general a number of Dirac points that have attached (each of them) fermions with the two different chiralities. This is in particular the case of materials recently discovered like Na$_3$Bi and Cd$_3$As$_2$, as already clarified by their theoretical analysis[@chi1; @chi2]. Such a distinctive feature of the 3D Dirac semimetals is relevant in the present study, since it makes possible the dynamical generation of mass and opening of a gap in these systems from the hybridization of two chiralities at the same Dirac point. More precisely, the low-energy electronic states in both Na$_3$Bi and Cd$_3$As$_2$ can be naturally arranged into four-component spinors around each of two Dirac points, in such a way that the Dirac matrices appear in the chiral representation $$\begin{aligned} \gamma_0 \gamma_i = \eta_i \left(\begin{array}{cc} \sigma_i & 0 \\ 0 & - \sigma_i \end{array}\right)\end{aligned}$$ with $\eta_1 = \eta_2 = 1$ and $\eta_3 = \pm 1$ depending on the Dirac point. The dynamical mass generation that we impose with the ansatz (\[anstz\]) corresponds in this scheme to the mixing of the two chiralities, realized by the Dirac matrix $$\begin{aligned} \gamma_0 = \left(\begin{array}{cc} 0 & - \mathbbm{1} \\ - \mathbbm{1} & 0 \end{array}\right) \label{mix}\end{aligned}$$ A term proportional to (\[mix\]) has been identified in Refs. [@chi1] and [@chi2] as one of the possible perturbations of the Dirac hamiltonian in Na$_3$Bi and Cd$_3$As$_2$. The physical meaning of such a term has to be found in the breakdown of the threefold and fourfold rotational symmetry in each case, having an effect similar to that induced by strain in the crystal lattice. The dynamical mass generation we are discussing corresponds then to the development of an expectation value $$\langle \psi^{\dagger}_i \gamma_0 \psi_i \rangle \neq 0 \label{dii}$$ within each Dirac point $i$. We can imagine nevertheless the possibility of a symmetry breaking pattern with order parameter given by $$\langle \psi^{\dagger}_i M \psi_j \rangle \neq 0 \label{dij}$$ with a suitable matrix $M$ and a pair of Dirac points $i \neq j$ [@hut]. Such a condensation has the feature of involving a finite momentum ${\bf Q}$, needed to connect different Dirac points. This can be pictured diagrammatically, as the order parameters (\[dii\]) and (\[dij\]) can be characterized in terms of the respective three-point vertices in Fig. \[five\]. The point is that, as implied by the renormalization group approach in Ref. [@jhep] for the 2D case, one has to hybridize the two chiralities at the same Dirac point so that the long-range Coulomb interaction may induce the strongest scaling of the three-point vertex, leading eventually to the condensation signaled by (\[dii\]). The vertex in Fig. \[five\](b) is built from spinors at different Dirac points, which will not map in general onto each other upon a rigid shift by the momentum ${\bf Q}$ and may lead therefore to a weaker overlap. This means that, for a generic 3D Dirac semimetal with long-range Coulomb interaction, the maximum strength will be set by the vertex in Fig. \[five\](a), ensuring at least the condensation given by (\[dii\]) whenever symmetry breaking is to take place in the system. ![Diagrammatic representation of the three-point vertices involving the composite operators $\psi^{\dagger}_i \gamma_0 \psi_i$ and $\psi^{\dagger}_i M \psi_j$, with respective momentum transfer ${\bf q} = 0$ and ${\bf q} = {\bf Q}$ connecting Dirac points.[]{data-label="five"}](vert3.eps){width="6.5cm"} Apart from the phenomenon of dynamical mass generation, the 3D Dirac semimetals have a tendency to develop at strong coupling a drastic attenuation of the quasiparticle weight at low energies, with a much softer renormalization of the Fermi velocity in comparison to their 2D Dirac analogues[@ros]. This behavior has been already found in an analytic study of the 3D electron systems in the large-$N$ limit, where it has been possible to establish rigorously the existence of a critical coupling at which the quasiparticle weight vanishes in the low-energy limit[@rc]. In Fig. \[six\] we can see the effect of the critical behavior in the functions $z_\psi({\bf k}, i\overline{\omega})$ and $z_v ({\bf k}, i\overline{\omega})$, computed in the present approach for $N = 6$ close to the critical point and with a discretization such that $2\pi T \approx 0.02$ eV. For smaller values of $N$, we will see that there is however an interplay between that quasiparticle attenuation and the tendency to dynamical generation of mass. For $N \leq 4$, this latter effect becomes actually dominant, leading to a phase with chiral symmetry breaking that is the analog of the broken symmetry phase found in the 2D Dirac semimetals[@nomu]. ![Plot of the factors $z_\psi (0,i\omega)$ and $z_v ({\bf k},0)/z_\psi ({\bf k},0)$ for a 3D Dirac semimetal with $N = 6$, bare coupling $g = 36.8$, and $2\pi T \approx 0.02$ eV.[]{data-label="six"}](zmflN6L.eps "fig:"){width="3.9cm"} ![Plot of the factors $z_\psi (0,i\omega)$ and $z_v ({\bf k},0)/z_\psi ({\bf k},0)$ for a 3D Dirac semimetal with $N = 6$, bare coupling $g = 36.8$, and $2\pi T \approx 0.02$ eV.[]{data-label="six"}](vbelowgrate.eps "fig:"){width="3.9cm"}\ (a) (b) From the self-consistent resolution of the Schwinger-Dyson equations, we have determined for each value of $N$ the critical coupling at which the electron system becomes first unstable, either from the vanishing of the quasiparticle weight or from the dynamical generation of a gap. In the case of the 3D Dirac semimetals, one has to take special care to refer the parameters to a given scale, since quantities like the electron charge and the Fermi velocity are renormalized at low energies. In this respect, we have chosen to define the bare electron charge $e_B$ at the highest value of the momentum cutoff, according to the relation $e_B^2 = \Lambda_k^2 D(\Lambda_k, 0) e^2$. Then, we can take for the microscopic parameter $e_B$ the standard value of the electron charge. As in the case of the 2D Dirac semimetals, we have also defined the bare Fermi velocity by $v_B = z_v(\Lambda_k, 0) v_F$. Thus, we have computed all the critical couplings referred to the relative interaction strength at the microscopic scale, given by $$g = N e_B^2/4 \pi v_B \label{defg}$$ ![Phase diagram of the QED of 3D Dirac semimetals, showing the region with dynamical generation of mass ($z_m \neq 0$) and the region corresponding to non-Fermi liquid behavior (with ${\rm Re} (z_\psi (0, i \omega )) \propto \omega^\gamma , \; \gamma < 0$). The critical lines have been obtained from the self-consistent resolution of the Schwinger-Dyson equations with $2\pi T \approx 0.02$ eV.[]{data-label="seven"}](phase3dr.eps){width="6.0cm"} The results we have obtained are condensed in the phase diagram shown in Fig. \[seven\]. We observe that there is always a phase connected to weak coupling for all values of $N$, characterized by a gapless spectrum of quasiparticles whose parameters remain regular at low energies. This phase terminates for $N \leq 4$ in the dynamical generation of a fermion mass at sufficiently strong coupling, which in our approach is reflected in the onset of a nonvanishing $z_m ({\bf k}, i\overline{\omega})$. This is shown in Fig. \[eight\], where we have represented the different renormalization factors for $N = 2$ at a coupling above the critical point. For this value of $N$, we observe that the renormalization of the quasiparticle weight is a moderate effect as the gap opens up in the electronic spectrum. This soft behavior has been also observed in the studies of dynamical mass generation in the fully relativistic QED, carried out by means of the self-consistent resolution of the corresponding Schwinger-Dyson equations[@bloch]. ![Plot of the factors $z_\psi (0,i\omega)$, $z_v ({\bf k},0)/z_\psi ({\bf k},0)$ and $z_m ({\bf k},0)/z_\psi ({\bf k},0)$ (measured in eV) for a 3D Dirac semimetal with $N = 2$, bare coupling $g = 15.1$, and $2\pi T \approx 0.02$ eV.[]{data-label="eight"}](zaboveg3d.eps "fig:"){width="3.9cm"} ![Plot of the factors $z_\psi (0,i\omega)$, $z_v ({\bf k},0)/z_\psi ({\bf k},0)$ and $z_m ({\bf k},0)/z_\psi ({\bf k},0)$ (measured in eV) for a 3D Dirac semimetal with $N = 2$, bare coupling $g = 15.1$, and $2\pi T \approx 0.02$ eV.[]{data-label="eight"}](vaboveg3d.eps "fig:"){width="3.9cm"} ![Plot of the factors $z_\psi (0,i\omega)$, $z_v ({\bf k},0)/z_\psi ({\bf k},0)$ and $z_m ({\bf k},0)/z_\psi ({\bf k},0)$ (measured in eV) for a 3D Dirac semimetal with $N = 2$, bare coupling $g = 15.1$, and $2\pi T \approx 0.02$ eV.[]{data-label="eight"}](maboveg3d.eps "fig:"){width="4.2cm"}\ (a) (b) (c) The case of $N = 4$ is however specially interesting, since there is then an interplay between the dynamical generation of mass and the strong attenuation of the quasiparticle weight. This can be observed in Fig. \[nine\], where we have plotted $z_\psi({\bf k}, i\overline{\omega})$, $z_v ({\bf k}, i\overline{\omega})$ and $z_m ({\bf k}, i\overline{\omega})$ for different couplings below and above the point where the mass develops. We see that, while the breakdown of chiral symmetry takes place before the system is completely destabilized by the large growth of $z_\psi({\bf k}, i\overline{\omega})$, this latter effect may still have a large impact on the observation of the quasiparticles in the electron system. ![Plot of the factors $z_\psi (0,i\omega)$, $z_v ({\bf k},0)/z_\psi ({\bf k},0)$ and $z_m ({\bf k},0)/z_\psi ({\bf k},0)$ (in eV) for a 3D Dirac semimetal with $N = 4$, bare coupling $g = 30.1, 27.6$, $24.9$ (from top to bottom in (a) and (c), from bottom to top in (b)), and $2\pi T \approx 0.01$ eV.[]{data-label="nine"}](zN4long.eps "fig:"){width="3.9cm"} ![Plot of the factors $z_\psi (0,i\omega)$, $z_v ({\bf k},0)/z_\psi ({\bf k},0)$ and $z_m ({\bf k},0)/z_\psi ({\bf k},0)$ (in eV) for a 3D Dirac semimetal with $N = 4$, bare coupling $g = 30.1, 27.6$, $24.9$ (from top to bottom in (a) and (c), from bottom to top in (b)), and $2\pi T \approx 0.01$ eV.[]{data-label="nine"}](vN4long.eps "fig:"){width="3.9cm"} ![Plot of the factors $z_\psi (0,i\omega)$, $z_v ({\bf k},0)/z_\psi ({\bf k},0)$ and $z_m ({\bf k},0)/z_\psi ({\bf k},0)$ (in eV) for a 3D Dirac semimetal with $N = 4$, bare coupling $g = 30.1, 27.6$, $24.9$ (from top to bottom in (a) and (c), from bottom to top in (b)), and $2\pi T \approx 0.01$ eV.[]{data-label="nine"}](mN4long.eps "fig:"){width="4.2cm"}\ (a) (b) (c) On the other hand, we find that for $N > 4$ there is always a critical coupling at which the divergence of $z_\psi (0,i\omega)$ takes place before the dynamical generation of mass, according to the trend illustrated in Fig. \[six\]. The present approach allows us moreover to investigate the phase of the electron system above the critical point. The self-consistent resolution of the Schwinger-Dyson equations gives rise in that case to renormalization factors that get in general an imaginary part, as shown in Fig. \[ten\]. This has to be interpreted as the signature of a nonperturbative instability of the electron quasiparticles since, in the conventional perturbative approach, the self-energy corrections obtained after Wick rotation $\omega = i \overline{\omega }$ can only account for the renormalization of the real part of the quasiparticle parameters. In our approach, the divergence of the imaginary part of $z_\psi (0,i\overline{\omega })$ at $\overline{\omega } = 0$ points actually to the development of a strongly correlated liquid, which is confirmed by the concomitant suppression of the quasiparticle weight in the low-energy limit, observed in the plots of Fig. \[ten\]. ![Plot of the real and imaginary parts of the factors $z_\psi (0,i\omega)$ and $z_v ({\bf k},0)$ for a 3D Dirac semimetal with $N = 6$, bare coupling $g = 37.2, 37.4$, $37.6$ (from lower to higher absolute value in both sides of the plot), and $2\pi T \approx 0.02$ eV.[]{data-label="ten"}](zbyndN6.eps "fig:"){width="3.9cm"} ![Plot of the real and imaginary parts of the factors $z_\psi (0,i\omega)$ and $z_v ({\bf k},0)$ for a 3D Dirac semimetal with $N = 6$, bare coupling $g = 37.2, 37.4$, $37.6$ (from lower to higher absolute value in both sides of the plot), and $2\pi T \approx 0.02$ eV.[]{data-label="ten"}](vbyndN6.eps "fig:"){width="3.9cm"}\ (a) (b) A detailed analysis of the renormalization factor $z_\psi (0,i\overline{\omega })$ in Fig. \[ten\] reveals indeed that the real part of such a function follows accurately a power-law behavior as $\overline{\omega } \rightarrow 0$. This can be clearly seen in the plots of Fig. \[eleven\], where $\overline{\omega } \: {\rm Re} (z_\psi (0,i\overline{\omega }))$ is represented in linear and logarithmic scale. The fit to the power-law dependence $$\overline{\omega } \: {\rm Re} (z_\psi (0,i\overline{\omega })) \propto \overline{\omega }^\mu$$ gives an exponent $\mu \approx 0.7$, with little variation between the three curves for the different couplings. Equivalently, this corresponds to having a nonvanishing anomalous dimension of the Dirac fermion field with a value of $\approx 0.3$. The inspection of the renormalization of $v_F$, dictated by the function ${\rm Re} (z_v ({\bf k},0))/{\rm Re} (z_\psi ({\bf k},0))$, leads in this case to a picture very similar to that in Fig. \[six\](b). This shows that $\mu $ corresponds to the exponent governing both the frequency and momentum scaling of the electron propagator (implying therefore a dynamical critical exponent equal to 1). ![Plot of $\omega \: {\rm Re} (z_\psi (0,i\omega))$ for a 3D Dirac semimetal with the same parameters as in Fig. \[ten\], with (a) linear and (b) logarithmic scale (with the three different curves virtually collapsed onto the same line).[]{data-label="eleven"}](decayrN6.eps "fig:"){width="3.9cm"} ![Plot of $\omega \: {\rm Re} (z_\psi (0,i\omega))$ for a 3D Dirac semimetal with the same parameters as in Fig. \[ten\], with (a) linear and (b) logarithmic scale (with the three different curves virtually collapsed onto the same line).[]{data-label="eleven"}](decayrN6log.eps "fig:"){width="3.9cm"}\ (a) (b) From the physical point of view, the main consequence of the noninteger exponent $\mu $ is the absence of a pole in the electron propagator, which reflects the lack of low-energy fermion excitations. This is characteristic of correlated systems in low dimensions, where the interactions may drive into a non-Fermi liquid phase with nontrivial scaling exponents[@varma; @bares; @nayak; @hou; @cast]. Our system provides in this respect an example of such a behavior at $D = 3$, illustrating moreover the transition to the strongly correlated phase from the renormalized Fermi liquid, which is the phase of 3D Dirac semimetals at sufficiently weak coupling. Conclusion ========== We have seen that the behavior of 2D and 3D Dirac semimetals is governed by quite different effects in their respective strong-coupling regimes. In our approach to the 2D semimetals, there is in general a tendency of the Fermi velocity of quasiparticles to grow large in the low-energy limit, in agreement with the renormalization group studies carried out in the large-$N$ approximation[@prbr]. In the case of the 3D Dirac semimetals, we observe instead that the electron system is prone to develop an attenuation of the quasiparticle weight, with a less significant renormalization of the Fermi velocity. This tendency has been also identified in the large-$N$ limit of the 3D Dirac semimetals, which shows the existence of a critical coupling characterized by the divergence of the electron scaling dimension[@rc]. In the 2D Dirac semimetals, the divergence of the Fermi velocity in the low-energy limit is the dominant feature that explains for instance the absence of significant correlation effects in the graphene layer. Our nonperturbative solution of the Schwinger-Dyson equations incorporates naturally the scaling of the Fermi velocity, allowing us to reach very good agreement with the experimental measures from graphene samples at very low doping levels[@exp2]. As a result of such a renormalization, we have found that the interaction strength has to be set to relatively large values, above those attained in the suspended graphene samples, in order to open up a phase with exciton condensation and dynamical generation of a gap in the 2D Dirac semimetals. The picture changes into a richer phase diagram for the 3D Dirac semimetals, as a consequence of the interplay between the attenuation of the electron quasiparticles that prevails at large $N$ and the tendency to dynamical mass generation (analogous to the chiral symmetry breaking of the fully relativistic QED[@mas; @fom; @fuk; @mir; @gus2; @kon; @atk; @min]) that is dominant at small $N$. Both effects seem to coexist at the interface found for a number of Dirac fermions $N =4$. Most interestingly, our self-consistent resolution has also revealed the phase of the system above the large-$N$ critical point, allowing us to characterize the properties of a strongly correlated liquid that is reminiscent of other systems with suppression of electron quasiparticles making the transition from marginal Fermi liquid[@varma] to non-Fermi liquid behavior[@bares; @nayak; @hou; @cast]. We remark that our analysis of the 3D Dirac semimetals can be extended to map also the large-$N$ regime of 3D Weyl semimetals. These are a class of semimetals in which a number of Weyl points host fermions with a given chirality, represented in terms of two-component spinors. This means that a self-consistent resolution of the Schwinger-Dyson equations may be also carried out for these systems, writing now the full propagator of the Weyl fermions around a given Weyl point as $$G({\bf k}, i\overline{\omega}) = \left( z_\psi({\bf k}, i\overline{\omega}) i \overline{\omega} - z_v({\bf k}, i\overline{\omega}) v_F \mbox{\boldmath $\sigma $} \cdot {\bf k} \right)^{-1}$$ We may parallel the above approach to predict the existence of a phase with suppression of electron quasiparticles, similar to that found for the 3D Dirac semimetals and covering the right part of the phase diagram in Fig. \[seven\]. At small $N$ (taken now as the number of pairs of Weyl points), a strong-coupling phase corresponding to fermion condensation may also arise, with an order parameter mixing fermions at different Weyl points as in (\[dij\]). The strength of this instability cannot be assessed generically, however, since it may be highly dependent on the particular overlapping of spinors from different Weyl points. Nevertheless, we may expect a strong-coupling symmetry broken phase for 3D Weyl semimetals at small $N$, with a phase boundary determined by the particular form of the spinors in the material hosting the Weyl points. We finally comment on the feasibility to observe the strong-coupling phases of the 3D semimetals, according to the values of the Fermi velocity and number of Dirac or Weyl points found in different materials. We recall in this regard that the best known examples of 3D Dirac semimetals (Na$_3$Bi and Cd$_3$As$_2$) have a number $N = 2$ of Dirac fermions and Fermi velocities that have been measured with certain accuracy[@liu; @cd3]. The quasiparticle dispersion shows in both cases an anisotropy that is reflected in the values of $v_F$, which may be reduced by a factor of $\sim 4$ in one of the directions in momentum space[@nao]. We can make conservative estimates of the coupling defined in (\[defg\]) by taking the largest Fermi velocity in each case, with the result that $g \sim 10$ for Na$_3$Bi and $g \sim 3$ for Cd$_3$As$_2$. These couplings turn out to be below the critical coupling for dynamical mass generation, which corresponds to the critical point at $g^* \approx 14.7$ for $N = 2$ in the phase diagram of Fig. \[seven\]. This places the two mentioned 3D Dirac semimetals in the gapless phase, showing that the transition to the regime with dynamical mass generation would require a Fermi velocity at least about $50 \%$ smaller than the largest value in Na$_3$Bi. On the other hand, there should be good prospects to observe the strong-coupling phase at large $N$ in Weyl semimetals. The most promising candidates for this class are the pyrochlore iridates and TaAs, which have 12 pairs of Weyl points. With the value of the Fermi velocity measured for TaAs[@taas], the large value of $N$ already sets the effective coupling $g$ for this material well above the critical coupling for the non-Fermi liquid regime, which is at $g^* \approx 21.1$ for $N = 12$. This leads us to expect that such a material should not behave as a regular Fermi liquid when observed at filling levels sufficiently close to the Weyl points. In general, we conclude that the strong-coupling phases that we have studied in this paper are not beyond reach, and that they may be found in systems with reasonably low Fermi velocities, or with a sufficiently large number of Dirac or Weyl points already exhibited by several materials. We acknowledge the financial support from MICINN (Spain) through grant FIS2011-23713 and from MINECO (Spain) through grant FIS2014-57432-P. [24]{} K. S. Novoselov, A. K. 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--- abstract: \| In this paper we study posterior consistency for different topologies on the parameters for hidden Markov models with finite state space. We first obtain weak and strong posterior consistency for the marginal density function of finitely many consecutive observations. We deduce posterior consistency for the different components of the parameter. We finally apply our results to independent emission probabilities, translated emission probabilities and discrete HMMs, under various types of priors. [[Keywords:]{}]{} Bayesian nonparametrics, consistency, hidden Markov models. author: - \| Elodie Vernet\ elodie.vernet@math.u-psud.fr bibliography: - 'biblio2.bib' title: Posterior consistency for nonparametric hidden Markov models with finite state space --- Introduction {#Sect:intro} ============ Hidden Markov models (HMMs) have been widely used in diverse fields such as speech recognition, genomics, econometrics since their introduction in @BaPe66. The books @macdonald:zucchini:1997, @macdonald:zucchini:2009 and @CaMoRy05 provide several examples of applications of HMMs and give a recent (for the latter) state of the art in the statistical analysis of HMMs. Finite state space HMMs are stochastic processes $(X_t,Y_t)_{t\in \mathbb{N}}$ such that $(X_t)_{t\in \mathbb{N}}$ is a Markov chain taking values in a finite set, and conditionally to $(X_t)_{t\in \mathbb{N}}$, the random variables $Y_t$, $t\in \mathbb{N}$, are independent, the distribution of $Y_{t}$ depending only on $X_{t}$. The conditional distributions of $Y_{t}$ given $X_{t}$ for all possible values of $X_t$ are called emission distributions. The name “hidden Markov model” comes from the fact that the observations are the $Y_{t}$’s only, one cannot access to the states $(X_t)_t$ of the Markov chain. Finite state space HMMs can be used to model heterogeneous variables coming from different populations, the states of the (hidden) Markov chain defining the population the observed variable comes from. HMMs are very popular dynamical models especially because of their computational tractability since there exist efficient algorithms to compute the likelihood and to recover the posterior distribution of the hidden states given the observations. Frequentist asymptotic properties of estimators of HMMs parameters have been studied since the 1990s. Consistency and asymptotic normality of the maximum likelihood estimator have been established in the parametric case, see @DoMa01, @Do04 and references in @CaMoRy05, see also @DoMoOlHa11 for the most general consistency result up to now. As to Bayesian asymptotic results, there are only very few and recent results, see @GuSh08 when the number of hidden states is known, @GaRo12 when the number of hidden states is unknown. All these results concern parametric HMMs. Non parametric HMMs in the sense that the form of the emission distribution is not specified have only very recently been considered, since identifiability remained an open problem until @GaRo13 and @GaClRo13, who prove a general identifiability result. Because parametric modeling of emission distributions may lead to poor results in practice, in particular for clustering purposes, recent interest in using non parametric HMMs appeared in applications, see @YaPaRoHo11, @GaClRo13 and references therein. Theoretical results for estimation procedures in non parametric HMMs have also been obtained only very recently: @DuCo12 concerns regression models with hidden (markovian) regressors and unknown regression functions in Gaussian noise, and @GaRo13 is about translated emission distributions. In this paper, we obtain posterior consistency results for Bayesian procedures in finite state space non parametric HMMs. To our knowledge, this is the first result on posterior consistency in such models. In Section \[T\], we prove posterior consistency in terms of the weak topology and the $L_1$norm on marginal densities of consecutive observations. Our main result is obtained under assumptions on the emission densities and on the prior which are very similar to the ones in the i.i.d. case, see Theorem \[th1\]. This result relies on a new control of the Kullback-Leibler divergence for HMMs, see Lemma \[KL\]. In Section \[P\] we build upon the recent identifiability result to deduce from Theorem \[th1\] posterior consistency for each component of the parameters. We obtain in general posterior consistency for the transition matrix of the Markov chain and for the emission probability distribution in the weak topology, see Theorem \[th2\]. Finally, some examples of priors that fulfill the assumptions are studied in Section \[E\]. Particularly in Section \[D\] the discrete case is thoroughly studied with a Dirichlet process prior. Sufficient and almost necessary assumptions to apply Theorem \[th1\] are given in Proposition \[thD\]. Moreover in this framework, posterior consistency of the marginal smoothing distributions, used in segmentation or classification, is derived in Theorem \[FD\]. All proofs are given in . Settings and main Theorem ========================= Notations {#N} --------- We now precise the model and give some notations. Recall that finite state space HMMs are stochastic processes $(X_t,Y_t)_{t\in \mathbb{N}}$ such that $(X_t)_{t\in \mathbb{N}}$ is a Markov chain taking values in a finite set, and conditionally on $(X_t)_{t\in \mathbb{N}}$, the random variables $Y_t$, $t\in \mathbb{N}$, are independent. The distribution of $Y_{t}$ depending only on $X_{t}$ is called the emission distribution. The number $k$ of hidden states is known, so that the state space of the Markov chain is set to $\{1, \dots, k\}$. Throughout the paper, for any integer $n$, an $n$-uple $(x_{1},\ldots,x_{n})$ is denoted $x_{1:n}$. Let $\Delta_k = \{(x_1,\dots,x_k)~ : ~ x_i\geq 0 , ~ i=1,\dots k ~ ; ~ \sum_{i=1}^k x_i=1\}$ denote the $k-1$-dimensional simplex. Let $Q$ denote the $k\times k$ transition matrix of the Markov chain, so that identifying $Q$ as the $k$-uple of transition distributions (the lines of the matrix), we write $Q\in \Delta_k^k$. We denote $\mu\in\Delta_k$ the initial probability measure, that is the distribution of $X_1$. For $\underline{q}\geq 0$, we also define $$\Delta^k(\underline{q})=\{ Q \in \Delta_k^k ~ : ~ \min_{i,j\leq k } Q_{i,j} \geq \underline{q}\},$$ so that $\Delta^k(0)=\Delta_k^k $. We now recall some properties of Markov chains with transition matrix in $\Delta^k(\underline{q})$. Note that $\underline{q}$ needs to be less than $\frac{1}{k}$ for $\Delta^k(\underline{q})$ to be non empty. Then for all $Q$ in $\Delta^k(\underline{q})$, $\max_{i,j}Q_{i,j} \leq 1-(k-1)\underline{q}$. Also, if $Q\in\Delta^k(\underline{q})$, then for any $i \in \{1, \dots,k\}$ and $A \subset \{1, \dots,k\}$, $\sum_{j\in A}Q_{i,j}\geq k \underline{q} u(A) $, with $u$ the uniform probability on $\{1, \dots,k\}$. Besides if $Q \in \Delta^k(\underline{q})$ with $\underline{q}>0$, the chain is irreducible, positive recurrent and admits a unique stationary probability measure denoted $\mu^Q$ for which $\underline{q} \leq \mu^Q(i) \leq 1- (k-1) \underline{q}$, $1\leq i \leq k$. We assume that the observation space is $\mathbb{R}^{d}$ endowed with its Borel sigma field. Let $\mathcal{F}$ be the set of probability density functions with respect to a reference measure $\lambda$ on $\mathbb{R}^{d}$. $\mathcal{F}^k$ is the set of possible emission densities, that is for $f=(f_{1},\ldots,f_{k})\in\mathcal{F}^k$, the distribution of $Y_{t}$ conditionally to $X_{t}=i$ will be $f_{i}\lambda$, $i=1,\ldots,k$. See Figure \[graph\] for a visualization of the model. ![The model[]{data-label="graph"}](chaine_schema1.pdf){width="10cm"} Let $$\Theta=\{ \theta = (Q,f) ~ : ~ Q \in \Delta_k^k, f \in \mathcal{F}^k\}$$ and $$\Theta(\underline{q})=\{ \theta = (Q,f) ~ : ~ Q \in \Delta^k(\underline{q}), f \in \mathcal{F}^k\}.$$ Then $\mathbb{P}^\theta$ (resp. $\mathbb{P}^{\theta,\mu}$) denotes the probability distribution of $(X_t,Y_t)_{t\in \mathbb{N}}$ under $\theta$ and initial probability $\mu^\theta:=\mu^Q$ (respectively $\mu$). Let $p^\theta_l$ ($p^{\theta,\mu}_l$ resp.) denote the probability density of $Y_1, \dots, Y_l$ with respect to $\lambda^{\otimes l}$ under $\mathbb{P}^\theta$ (resp. $\mathbb{P}^{\theta,\mu}$). and $P_l^\theta$ ($P^{\theta,\mu}_l$ resp.) the marginal distribution of $Y_1, \dots, Y_l$ under $\mathbb{P}^\theta$ (resp. $\mathbb{P}^{\theta,\mu}$). So for any $\theta \in \Theta$, initial probability $\mu$, and measurable set $A$ of $\{1,\dots,k\}^l\times (\mathbb{R}^d)^l $: $$\begin{split} \mathbb{P}^{\theta,\mu} &((X_{1:l},Y_{1:l}) \in A) \\ &=\int \sum_{x_1, \dots, x_l=1}^k \mathds{1}_{(x_1, \dots,x_l,y_1,\dots,y_l)\in A} ~ \mu_{x_1} Q_{x_1,x_2} \dots Q_{x_{l-1},x_l} \\ &\qquad f_{x_1}(y_1) \dots f_{x_l}(y_l) \lambda(dy_1)\dots\lambda(dy_l) , \end{split}$$ $$\begin{split} & p^{\theta,\mu}_l (y_1, \dots,y_l) = \sum_{x_1, \dots, x_l=1}^k \mu_{x_1} Q_{x_1,x_2} \dots Q_{x_{l-1},x_l} f_{x_1}(y_1) \dots f_{x_l}(y_l), %\quad \lambda^{\otimes l} \text{ a.s.} \end{split}$$ and $\displaystyle{P_l^{\theta, \mu}= p_l^{\theta, \mu} \lambda^{\otimes l}.}$ We denote by $\delta_\mu \otimes \pi$ the prior on $\Delta_k \times \Theta$, where $\mu \in \Delta_k$ is an initial probability measure. We assume that $\pi$ is a product of probability measures on $\Theta$, $\pi= \pi_Q \otimes \pi_f$ such that $\pi_Q$ is a probability distribution on $\Delta_k^k$ and $\pi_f$ is a probability distribution on $\mathcal{F}^k$. We assume throughout the paper that the observations are distributed from $\mathbb{P}^{\theta^}$ so that their distribution is a stationary HMM. We are interested in posterior consistency, that is to prove that with $\mathbb{P}^{\theta^}$-probability one, for all neighborhood $U$ of $\theta^$ : $$\lim_{n \to +\infty} \pi(U\| Y_{1:n})= 1 .$$ The choice of a topology on the parameters arises here. For any distance or pseudometric $D$, we denote $N(\delta,A,D)$ the $\delta$-covering number of the set $A$ with respect to $D$, that is the minimum number $N$ of elements $a_1, \dots, a_N$ such that for all $a \in A$, there exists $n \leq N$ such that $D(a,a_n)\leq \delta$. For $k \times k$ matrices $M$, we use $$\lVert M \rVert= \max_{1 \leq i,j \leq k} \lvert M_{i,j} \rvert.$$ For vectors $v$ in $\mathbb{R}^k$, we denote $$\lVert v \rVert_1 = \sum_{1\leq i \leq k} \lvert v_i \rvert.$$ For probabilities $P_1$ and $P_2$, let $p_1$ and $p_2$ be their respective densities with respect to some dominated measure $\nu$. We use the total variation norm : $$\lVert P_1-P_2 \rVert_{TV}= \frac{1}{2} \int \lvert p_1-p_2 \rvert d\nu = \frac{1}{2} \lVert p_1-p_2 \rVert_{L_1(\nu)}$$ and the Kullback-Leibler divergence : $$\begin{split} KL(P_1,P_2) & =\left\{ \begin{array}{ll} \int p_1 \log(\frac{p_1}{p_2}) d\nu & \text{ if } P_1 << P_2, \\ + \infty & \text{ otherwise.} \end{array} \right.\\ \end{split}$$ We also denote $KL(p_1,p_2)$ for $KL(p_1\nu,p_2\nu)$. On $\mathcal{F}^k$ we use the distance $d(\cdot,\cdot)$ defined for all $g=(g_1, \dots,g_k)$, $\tilde{g}=(\tilde{g_1}, \dots,\tilde{g_k}) $ by $$d(g,\tilde{g}) = \max_{1 \leq j \leq k} \lVert g_j - \tilde{g_j} \rVert_{L_1(\lambda)}$$ On $\Theta(\underline{q})$, we use the following pseudometric for $l\geq3$, $l \in \mathbb{N}$, $$D_l(\theta, \theta')= \int \| p_l^\theta (y_1, \dots, y_l) - p_l^{\theta'} (y_1, \dots, y_l)\|\lambda(dy_1)\dots\lambda(dy_l) = \lVert p_l^\theta - p_l^{\theta'} \rVert_{L_1(\lambda^{\otimes l})}.$$ Then a $D_l$-neighborhood of $\theta$ is a set which contains a set $\{ \theta' ~ : ~ D_l(\theta,\theta')<\epsilon\}$ for some $\epsilon>0$. We also use the weak topology on marginal distributions $(P_l^\theta)_\theta$. We recall that in any neighborhood of $P_l^\theta $ in the weak topology on probability measures there is a subset which is a union of sets of the form $$\left\{P ~ : ~ \left\lvert \int h_j dP - \int h_j p_l^\theta d\lambda^{\otimes l} \right\rvert < \epsilon_j, ~ j=1,\dots,N \right\} ,$$ where for all $1 \leq j \leq N$, $\epsilon_j>0$ and $h_j$ is in the set $\mathcal{C}_b((\mathbb{R}^d)^l)$ of all bounded continuous functions from $(\mathbb{R}^d)^l$ to $\mathbb{R}$. We prove posterior consistency in this general nonparametric context using this weak topology on marginal distributions $(P_l^\theta)_\theta$ and the $D_l$-pseudometric in Section \[T\]. We study the posterior consistency for the transition matrix and the emission probabilities separately in Section \[P\]. Finally the sign $\lesssim$ is used for inequalities up to a multiplicative constant possibly depending on fixed parameters. Main Theorem {#T} ------------ In this section we state our general theorem on posterior consistency for nonparametric hidden Markov models in the weak topology on marginal distributions $(P_l^\theta)_\theta$ and the $D_l$-topology. We consider the following assumptions. Fix $l\geq 3$. (A1) For all $\epsilon>0$ small enough there exists a set $\Theta_\epsilon \subset \Theta(\underline{q})$ such that $\pi(\Theta_\epsilon)>0$ and for all $\theta=(Q,f) \in \Theta_\epsilon$, (A1a) $\displaystyle{\lVert Q - Q^ \rVert < \epsilon}$, (A1b) $\displaystyle{\max_{1\leq i\leq k} \int f_i^(y) \max_{1\leq j \leq k} \log \left(\frac{ f_j^(y)}{f_j(y)} \right) \lambda(dy) < \epsilon}, $ (A1c) for all $y \in \mathbb{R}^d$ such that $\displaystyle{\sum_{i=1}^k f^_i(y)>0}$, $\displaystyle{ \sum_{j=1}^k f_j(y) >0}$, (A1d) $\displaystyle{\sup_{y ~ : ~ \sum_{i=1}^k f^_i(y)>0} \max_{1 \leq j \leq k} f_j(y) < +\infty}$ (A1e) $\displaystyle{\sum_{i=1}^k \int f^_i(y) \left\lvert \log \left( \sum_{j=1}^k f_j(y)\right) \right\rvert \lambda(dy) < + \infty}$ (A2) For all $n>0$, for all $\delta>0$ there exists a set $\mathcal{F}_n \subset \mathcal{F}^k$ and a real number $r_1>0$ such that $\pi_f\big(({\mathcal{F}_n})^c\big)\lesssim e^{-nr_1}$ and such that $$\sum_{n>0} N\left(\frac{\delta}{36l}, \mathcal{F}_n , d(\cdot,\cdot)\right) \exp\left(-\frac{n \delta^2 k^2 \underline{q}^2}{32 l}\right)< + \infty.$$ \[th1\] Let $\underline{q}>0$. Assume that the support of the prior $\pi$ is included in $\Theta(\underline{q})$ and that for all $1\leq i\leq k$, $\mu_i \geq \underline{q}$. 1. \[a.\] If Assumption (A1) holds then for all weak neighborhood $U$ of $P_l^{\theta^}$, $$\mathbb{P}^{\theta^}\left(\lim_{n \to \infty} \pi(U\|Y_{1:n})=1 \right)=1.$$ 2. \[b.\] Moreover if Assumptions (A1) and (A2) hold then, for all $\epsilon>0$, $$\mathbb{P}^{\theta^}\left(\lim_{n \to \infty} \pi(~ \left\{ \theta: ~ D_l(\theta,\theta^)<\epsilon \right\} ~ \|Y_{1:n})=1 \right)=1.$$ We assume everywhere in the paper that the support of the prior is included in $\Theta(\underline{q})$. It means the results of this paper can only be applied to priors $\pi_Q$ on transition matrices which vanish close to the border of $\Delta_k^k$. This assumption is satisfied by a product of truncated Dirichlet distribution i.e. if the lines $Q_{i,\cdot}$ of $Q$ are independently distributed from a law proportional to: $$Q_{i,1}^{\alpha_1-1} \dots Q_{i,k}^{\alpha_k-1} \mathds{1}_{\left\{\underline{q} \leq Q_{i,j} \leq 1, ~ \forall 1\leq j \leq k\right\}} dQ_{i,1} \dots dQ_{i,k}$$ where $\alpha_1, \dots, \alpha_k >0$. In the case of density estimation with i.i.d. observations it is usual to control the Kullback-Leibler support of the prior to show weak posterior consistency and to control in addition a metric entropy to obtain strong consistency see Chapter 4 of @GhRa03. Assumptions (A1) and (A2) are similar in spirit. Assumption (A1) replaces the assumption on the true density function being in the Kullback-Leibler support of the prior in the i.i.d. case. (A1a) ensures that the transition matrices of $\Theta_\epsilon$ are in a ball of radius $\epsilon$ around the true transition matrix. Under (A1b) the emission densities are in an $\epsilon$ Kullback-Leibler ball around the true one. (A1c), (A1d) and (A1e) are assumptions under which the log-likelihood converges $\mathbb{P}^{\theta^}\text{\!\!\! -a.s.}$ and in $L_1(\mathbb{P}^{\theta^})$. (A2) is very similar to the assumptions of the metric entropy of Theorem 4.4.4 in @GhRa03. In Appendix \[Prth1\], the proof of Theorem \[th1\] relies on the method of @Ba88. It consists in controlling Kullback-Leibler neighborhoods and building tests. . \[KL\] Let $\theta^$ be in $\Theta(\underline{q})$. If (A1) holds then for all $0<\epsilon<1$, there exists $N \in \mathbb{N}$ such that for all $n\geq N$ and for all $\theta \in \Theta_\epsilon$: $$\frac{1}{n} KL(\mathbb{P}^{\theta^}_n,\mathbb{P}^{\theta,\mu}_n) \leq \frac{3}{\underline{q}} ~ \epsilon .$$ Consistency of each component of the parameter {#P} ---------------------------------------------- In this Section we look at the consequences of Theorem \[th1\] on posterior consistency for the transition matrix and the emission probabilities separately. Estimating consistently the components of the parameter is of great importance. First one may want to know the proportion of each population or the probability of moving from one population to another, i.e. the transition matrix. Secondly, these components are important to recover the smoothing distribution and then clustering the observations, see @CaMoRy05 and Theorem \[FD\]. The consistency of each component, i.e. the transition matrix and the emission distributions does not directly result from consistency of the marginal distribution of the observations, see @DuCo12. Obviously, identifiability seems to be necessary to obtain this implication yet it is not sufficient. We obtain posterior consistency for the components of the parameter thanks to the result of identifiability of @GaClRo13, an inequality linking the $D_l$ pseudometric to distances on each component of the parameter and an argument of compactness. We use a product topology on the set of parameters. In particular we study consistency in the topology associated with the sup norm on transition matrices $\lVert \cdot \rVert$ and the weak topology on probabilities for the emission probabilities up to label switching. To deal with label switching, we need the following definitions. Let $\mathcal{S}_k$ denote the symmetric group on $\{1, \dots, k\}$. Let $\sigma$ be a permutation in $\mathcal{S}_k$, for all matrices $Q \in \Delta^k_k$, we denote $\sigma Q$ the following matrix : for all $1 \leq i,j \leq k$, $$(\sigma Q)_{i,j}= Q_{\sigma(i),\sigma(j)}.$$If $(X_t,Y_t)_{t \in \mathbb{N}}$ is distributed from $P^{(Q,f)}$ and $\tilde{X}_t=\sigma^{-1}(X_t)$, for $\sigma \in \mathcal{S}_k$, then $(\tilde{X}_t,Y_t)_{t \in \mathbb{N}}$ is distributed from $P^{(\sigma Q, (f_{\sigma(1)}, \dots, f_{\sigma(k)}))}$, i.e the labels of the Markov chain have been switched. Under the assumptions of Theorem \[th1\] and of identifiability we prove that the posterior concentrates around $(Q^,f^)$ up to label switching, i.e. around $\{\sigma Q^ ,(f^_{\sigma(1)}, \dots, f^_{\sigma(k)})\}_{\sigma \in \mathcal{S}_k}$, in Theorem \[th2\] whose proof is given in Appendix \[P:th2\]. In other words we obtain posterior consistency considering neighborhoods of the form $$\left\{ \exists \sigma \in \mathcal{S}_k; ~ \sigma Q \in U_{Q^}, ~ f_{\sigma(i)} \in U_{f^_i}, i=1\dots k \right\}$$ where $U_{Q^}$ is a neighborhood of $Q^$ and for all $1 \leq i \leq k$, $U_{f^_i}$ is a weak neighborhood of $f^_i \lambda$. That is to say we consider the product of the sup norm topology on transition matrices and of the weak topology on the emission distributions up to label switching. \[th2\] Let $\theta^=(Q^,f^)$. Suppose $f^_1 \lambda, \dots,f^_k \lambda$ are linearly independent and $Q^$ has full rank. Let $\underline{q}>0$, assume that $\mu_i\geq\underline{q}$, that the support of the prior $\pi$ is included in $\Theta(\underline{q})$ and that (A1) and (A2) hold. Then for all weak neighborhood $U_{f^_i}$ of $f^_i \lambda$, for all $1\leq i \leq k$ and for all neighborhood $U_{Q^}$ of $Q^$, $$\label{sp} \mathbb{P}^{\theta^}\left(\lim_{n \to +\infty} \pi\bigg( \left\{ \exists \sigma \in \mathcal{S}_k; ~ \sigma Q \in U_{Q^}, ~ f_{\sigma(i)} \in U_{f^_i}, ~ i=1\dots k \right\} \bigg\| ~ Y_{1:n}\bigg) = 1 \right)=1.$$ In particular, Equation implies that for all $\epsilon >0$ $$\mathbb{P}^{\theta^}\left(\lim_{n \to +\infty} \pi\left( ~ \bigcup_{\sigma \in \mathcal{S}_k} \left\{Q : \lVert Q - \sigma Q^* \rVert < \epsilon \right\} ~ \bigg\| ~ Y_{1:n}\right) = 1 \right)=1 .$$ It means that under the assumptions of Theorem \[th2\], the posterior concentrates around $\{\sigma Q^, \sigma \in \mathcal{S}_k\}$. Equation also implies that for all $N \in \mathbb{N}$, for all $h_i \in \mathcal{C}_b(\mathbb{R}^d)$, for all $\epsilon_i>0,~ 1 \leq i\leq N,$ $$\mathbb{P}^{\theta^}\Bigg(\lim_{n \to +\infty} \pi\bigg( ~ \bigcup_{\sigma \in \mathcal{S}_k} \left\{P: \left\lvert \int h_i dP- \int h_i f_{\sigma(j)}^d\lambda \right\rvert < \epsilon_i \right\} \bigg\| ~ Y_{1:n}\bigg) =1 \Bigg)=1.$$ This last result is a weak result which allows to consistently recover smooth functionals of the emission distributions $(f^_j)_j$. We obtain stronger results in Sections \[TEP\] and \[D\]. Examples of priors on $f$ {#E} ========================= In this section we apply Theorems \[th1\] and \[th2\] for different types of priors and emission models. In Section \[MG\] we deal with emission probabilities which are independent mixtures of Gaussians. Translated emission probabilities are studied in Section \[TEP\]. Assumptions (A1) and (A2) are purposely designed to resemble the types of assumptions found in density estimation for i.i.d. observations. This allows us to use existing results on consistency in the case of i.i.d. observations. This is done in Sections \[MG\] and \[TEP\] following @To06. Contrariwise we develop a new method to deal with for the discrete case in Section \[D\]. Independent mixtures of Gaussians {#MG} --------------------------------- We consider the well known location-scale mixture of Gaussian distributions as prior model for each $f_i$, namely each density under the prior is written as $$\label{eqMG:normal} g(y)=\int_{\mathbb{R}\times (0,+ \infty)} \phi_\sigma(y-z)dP(z,\sigma)=: \phi P$$ where $\phi_\sigma$ is the Gaussian density with mean zero and variance $\sigma^2$ and $P$ is a probability measure on $\mathbb{R}\times (0,+\infty)$. In this part, $\lambda$ is the Lebesgue measure on $\mathbb{R}$. Let $\pi_P$ be a probability measure on the set of probability measures on $\mathbb{R}\times (0,+\infty)$. Denote $\pi_g$ the distribution of $g$ expressed as when $P \sim \pi_P$. Then we consider the prior distribution on $f=(f_1, \dots, f_k)$ defined by $\pi_f=\pi_g^{\otimes k}$. We need the following assumptions to apply Theorem \[th1\] and \[th2\]: (B1) $$\pi_P\left(P~ :~ \int \frac{1}{\sigma} dP(z,\sigma) < \infty \right) =1 ,$$ (B2) for all $1\leq j \leq k$, $f^_j$ is positive, continuous on $\mathbb{R}$ and bounded by $M<\infty$, (B3) for all $1\leq i \leq k$, $$\left\lvert \int_{\mathbb{R}} f^_i(y) \max_{ 1 \leq j \leq k} \log(f^_j(y)) \lambda(dy) \right\rvert <\infty$$ (B4) for all $1\leq i \leq k$, $1\leq j \leq k$, $$\int_{\mathbb{R}} f^_i(y) \log\left(\frac{f^_j(y)}{\psi_j(y)}\right) \lambda(dy) <\infty$$ where $\psi_j(y)= \inf_{t\in [y-1,y+1]} f^_j(t)$. (B5) for all $1\leq i \leq k$, there exists $\eta>0$ such that $$\int_{\mathbb{R}} \lvert y \rvert^{2(1+\eta)} f^_i(y) \lambda(dy) <\infty.$$ (B6) for all $\beta>0$, $\kappa>0$, there exist a real number $\beta_0>0$, two increasing and positive sequences $a_n$ and $u_n$ tending to $+\infty$ and a sequence $l_n$ decreasing to $0$ such that $$\pi_P\bigg(P~ :~P((-a_n,a_n] \times (l_n,u_n]) < 1- \kappa \bigg) \leq \exp(-n\beta_0),$$ $$\text{with }\qquad \frac{a_n}{l_n} \leq n\beta, \qquad \log\left(\frac{u_n}{l_n}\right) \leq n \beta$$ \[th:MG\] Let $\underline{q}>0$. Assume that the support of the prior $\pi$ is included in $\Theta(\underline{q})$ and that for all $1\leq i\leq k$, $\mu_i\geq\underline{q}$. Assume that $Q^$ is in the support of $\pi_Q$ and that the weak support of $\pi_P$ contains all probability measures that are compactly supported. Then - (B1), (B2), (B3), (B4), (B5) imply (A1) - and (B6) implies (A2). In particular in the case of the Dirichlet process mixture $DP(\alpha G_0)$ with base measure $\alpha G_0$, where $G_0$ is a probability measure on $\mathbb{R}\times (0,+\infty)$ and $\alpha>0$, Assumption (B1) holds as soon as $$\label{eqrk} \int_{\mathbb{R}\times (0,+\infty)} \frac{1}{\sigma} G_0(dz,d\sigma)<+\infty.$$ Indeed, $$\begin{split} \int \int \frac{1}{\sigma} P(dz,d\sigma) \pi_P(dP) & = \int \int \int_{\mathbb{[\sigma,+\infty)}} \frac{1}{t^2} \lambda(dt) P(dz,d\sigma) \pi_P(dP) \\ & = \int \frac{1}{\sigma} G_0(dz,d\sigma). \end{split}$$ Moreover Assumption (B6) easily holds as soon as for all $\beta>0$, there exist a real number $\beta_0>0$,two increasing and positive sequences $a_n$ and $u_n$ tending to $+\infty$ and a sequence $l_n$ decreasing to $0$ such that $$\label{eqrk2} \begin{split} G_0\left(\left(-a_n,a_n]\times (l_n,u_n]\right)^c \right)\leq \exp(-n \beta_0) \\ \frac{a_n}{l_n} \leq n\beta, \qquad \log\left(\frac{u_n}{l_n}\right) \leq n \beta \end{split}$$ are verified (see Remark 3.1 of @To06). Translated emission probabilities {#TEP} --------------------------------- In this section we consider the special case of translated emission distributions that is to say for all $1 \leq j \leq k$, $$f_j(\cdot)=g( \cdot - m_j)$$ where $g$ is a density function on $\mathbb{R}$ with respect to $\lambda$ and for all $1 \leq j \leq k$, $m_j$ is in $\mathbb{R}$. In this part, $\lambda$ is still the Lebesgue measure on $\mathbb{R}$ and $d=1$ This model has been in particular considered by @YaPaRoHo11 for the analysis of genomic copy number variation. First a corollary of Theorem \[th2\] is given. Then the particular case of location-scale mixture of Gaussians on $g$ is studied. Let $$\Gamma=\{ \gamma=(Q,m,g), Q \in\Delta^k_k, m \in \mathbb{R}^k,m_1=0<m_2< \dots< m_k, g\in \mathcal{F} \}$$ and $$\Gamma(\underline{q})=\{ \gamma=(Q,m,g) \in \Gamma, Q \in \Delta^k(\underline{q}) \}.$$ To $\gamma=(Q,m,g) \in \Gamma$ we associate $\theta=(Q,(g(\cdot-m_1),\dots,g(\cdot-m_k))) \in \Theta$. We then denote $\mathbb{P}^\gamma$ for $\mathbb{P}^\theta$. We assume that $\pi_f$ is a product of probability measure, $$\pi_f=\pi_m \otimes \pi_g$$ where $\pi_g$ is a distribution on $\mathcal{F}$ and $\pi_m$ is a probability measure on $\mathbb{R}^k$. Note that under $\Gamma$, the model is completely identifiable, see Theorem 2.1 of @GaRo13. The uncertainty we had until now because of the label switching is resolved here. In Corollary \[th3\] additionally to posterior consistency for the transition matrices, we obtain posterior consistency for the parameters of $m_j$ and for the weak convergence on the translated probability $g\lambda$. Under a stronger assumption, we get posterior consistency for the $L_1$-topology on the translated probability. Fix $l\geq 3$. The following assumption replaces (A2) in the context of translated emission probabilities: (C2) for all $n>0$, for all $\delta>0$ there exists a set $\mathcal{F}_n \subset \mathbb{R}^k \times \mathcal{F}$ and a real number $r_1>0$ such that $\pi_f\big(({\mathcal{F}_n})^c\big)\lesssim e^{-nr_1}$ $$\sum_{n>0} N\left(\frac{\delta}{36l}, \mathcal{F}_n , d(\cdot,\cdot)\right) \exp\left(-\frac{n \delta^2 k^2 \underline{q}^2}{32 l}\right)< + \infty.$$ \[th3\] Let $\gamma^=(Q^,m^,g^)$ be in $\Gamma(\underline{q})$. Suppose $m^_1=0<m^_2< \dots< m^_k $ and $Q^$ has full rank. Let $\underline{q}>0$, assume that $\mu_i\geq\underline{q}$, that the support of the prior $\pi$ is included in $\Gamma(\underline{q})$, that (A1) is verified with $f_j(\cdot)=g(\cdot-m_j), ~ 1\leq j \leq k$ and (C2) holds. Then for all $\epsilon>0$, $$\mathbb{P}^{\gamma^}\Big(\lim_{n \to +\infty} \pi(\left\{Q: \lVert Q- Q^* \rVert < \epsilon \right\} \big\| ~ Y_{1:n}) =1 \Big)=1,$$ $$\mathbb{P}^{\gamma^}\Big(\lim_{n \to +\infty} \pi(\left\{m: \forall 1 \leq j \leq k, ~ \lvert m_j- m^_j \rvert < \epsilon \right\} \big\| ~ Y_{1:n}) =1 \Big)=1,$$ and for all $N \in \mathbb{N}$, for all $h_i \in \mathcal{C}_b(\mathbb{R}^d)$, for all $\epsilon_i>0,~ 1 \leq i\leq N,$ $$\mathbb{P}^{\gamma^}\Bigg(\lim_{n \to +\infty} \pi\bigg(\left\{P: \left\lvert \int h_i dP- \int h_i g^d\lambda \right\rvert < \epsilon_i \right\} \bigg\| ~ Y_{1:n}\bigg) =1 \Bigg)=1.$$ If moreover $\max_{1 \leq j \leq k} \mu^_j>1/2$ and $g^$ is uniformly continuous, then for all $\epsilon>0$, $$\mathbb{P}^{\gamma^}\Big(\lim_{n \to +\infty} \pi\left(\left\{g: \lVert g- g^ \rVert_{L_1(\lambda)} < \epsilon \right\} \| Y_{1:n}\right) =1 \Big)=1.$$ The proof of Corollary \[th3\], in Appendix \[P:th3\], relies on the identifiability result of @GaRo13 and the technique of proof of Theorem \[th2\]. In the same way as in Section \[MG\], we propose to apply Theorem \[th1\] and Corollary \[th3\] to a prior based on location-scale mixtures of Gaussians. In this part we study a particular prior on the translated emission $g$ which is the location-scale mixture of Gaussians. $g$ is a sample drawn from $\pi_{g}$ if $$g(y)=\int_{\mathbb{R}\times (0,+\infty)} \phi_\sigma(y-z)dP(z,\sigma)$$ where $P$ is a sample drawn from $\pi_P$ and $\pi_P$ is a probability measure on probability measures on $\mathbb{R}\times(0, + \infty)$. The following assumption help in proving (C2): (D6) for all $\beta>0$, $\kappa>0$, there exist a real number $\beta_0>0$, three increasing sequences of positive numbers $m_n$, $a_n$ and $u_n$ tending to $+\infty$ and a sequence $l_n$ decreasing to $0$ such that $$\pi_P\bigg(P~ :~P((-a_n,a_n] \times (l_n,u_n]) < 1- \kappa \bigg) \leq \exp(-n\beta_0),$$ $$\pi_m\bigg( ([-m_n,m_n]^k)^c\bigg) \leq exp(-n \beta_0),$$ $$\frac{a_n}{l_n} \leq n\beta, \qquad \log\left(\frac{u_n}{l_n}\right) \leq n \beta, \qquad \log\left(\frac{m_n}{l_n}\right) \leq n\beta$$ \[th:tGM\] Let $\underline{q}>0$ and $\gamma^$ in $\Gamma(\underline{q})$. Assume that the support of the prior $\pi$ is included in $\Gamma(\underline{q})$ and that for all $1\leq i\leq k$, $\mu_i \geq \underline{q}$. Assume that $Q^$ is in the support of $\pi_Q$, that $m^$ is in the support of $\pi_m$ and that the weak support of $\pi_P$ contains all probability measures that are compactly supported. If (B1) is verified and (B2), (B3), (B4) and (B5) are verified with $f_j(\cdot)=\textcolor{black}{g}(\cdot-m_j), ~ 1 \leq j \leq k$ then (A1) holds. Moreover (D6) implies (C2). The proof of Proposition \[th:tGM\] is very similar to that of Proposition \[th:MG\] and is given in Appendix \[PtGM\]. Independent discrete emission distributions {#D} ------------------------------------------- Discrete emission probabilities, i.e. when the support of $\lambda$ is included in $\mathbb{N}$, have been successfully used, for instance in genomics in @GaClRo13. Note that for discrete emission probabilities, weak and $l_1$ convergences are the same so that weak posterior convergence implies $l_1$ posterior consistency. Thus Assumption (A2) becomes unnecessary in Theorems \[th1\] and \[th2\]. \[FD\] $$\begin{split} \lim_{n \to +\infty} \pi\bigg( \max_{1 \leq a_{1:m}\leq k} \lvert P^\theta(X_{1:m}=a_{1:m}~\|Y_{1:n}) \\ &\hspace{-4cm} - P^{\theta^}(X_{1:m}=a_{1:m} ~\|~Y_{1:n})\rvert <\epsilon \|Y_{1:n}\bigg) = 1 \text{ in } P^{\theta^}\text{-probability}. \end{split}$$ In the we apply Theorems \[th1\], \[th2\] and \[FD\] to a specific prior on the set of probability measures on $\mathbb{N}$ in the case of a HMM with discrete emission distributions. We consider a Dirichlet process $DP(\alpha G_0)$ with $\alpha$ a positive number and $G_0$ some probability measure on $\mathbb{N}$. We then consider a prior probability measure on $\Theta$ defined by $$\pi=\pi_Q \otimes DP(\alpha G_0)^{\otimes k}.$$ In Proposition \[thD\], we give sufficient and amost necessary conditions to obtain (A1). Proposition \[thD\] is proved in Appendix \[P:thD\]. \[thD\] Let $\underline{q}>0$. Assume that the support of the prior $\pi$ is included in $\Theta(\underline{q})$, that $Q^$ is in the support of $\pi_Q$ and that for all $1\leq i\leq k$, $\mu_i\geq\underline{q}$. If $$\text{ (E1) for all }1 \leq i \leq k, ~ \sum_{l \in \mathbb{N}} \frac{f^_i(l)}{G_0(l)} < + \infty$$ then (A1) holds. Moreover if $$\text{(T) for all } 1\leq i \leq k, \sum_{l \in \mathbb{N}} f^_i(l) (-\log f^_i(l)) <+ \infty .$$ then (A1b) implies (E1). Therefore (E1) is not only sufficient to prove (A1b) but up to the weak assumption (T) it is also necessary. We deduce from Proposition \[thD\] that $$\begin{split} \Bigg\{ g^: ~ &\mathbb{N} \to (0,1) \quad \text{such that} \quad \sum\limits_{l \in \mathbb{N}} g^(l) =1, \\ & \sum\limits_{l \in \mathbb{N}} g^(l) (-\log(g^(l))< + \infty \quad \text{and} \quad \sum\limits_{l \in \mathbb{N}} \frac{g^(l)}{G_0(l)} < + \infty \Bigg\} \end{split}$$ is a subset of the Kullback-Leibler support of the Dirichlet process $DP(\alpha G_0)$. Acknowledgements {#acknowledgements .unnumbered} ================ I want to thank Elisabeth Gassiat and Judith Rousseau for their valuable comments. I also want to thank the reviewer and the editor for their helpful comments. Proofs of key results {#kr} ===================== Proof of Lemma \[KL\] {#AKL .unnumbered} --------------------- For all $\theta, ~ \theta^* \in \Delta^k(\underline{q})$ the Kullback-Leibler divergence between $p^{\theta^}_n$ and $p_n^\theta$ verifies $$\begin{split} \frac{1}{n} & KL(p^{\theta^}_n,p^{\theta,\mu}_n) \\ &= \frac{1}{n} \mathbb{E}_{p^{\theta^}_n} \left( \log\left(\frac{p^{\theta^}_n(Y_{1:n})}{p^\theta_n(Y_{1:n})}\right)\right)\\ &= \frac{1}{n} \mathbb{E}_{p^{\theta^}_n} \left( \log\left(\frac{\sum_{i_1,\dots, i_n=1}^k \mu^_{i_1} Q^_{i_1,i_2} \dots Q^_{i_{n-1},i_n} f^_{i_1}(Y_1) \dots f^_{i_n}(Y_n)} {\sum_{i_1,\dots, i_n=1}^k \mu_{i_1} Q_{i_1,i_2} \dots Q_{i_{n-1},i_n} f_{i_1}(Y_1) \dots f_{i_n}(Y_n)}\right)\right)\\ &= \frac{1}{n} \mathbb{E}_{p^{\theta^}_n} \left( \log\left(\frac{\sum\limits_{i_1,\dots, i_n=1}^k \frac{\mu^_{i_1} Q^_{i_1,i_2} \dots Q^_{i_{n-1},i_n} f^_{i_1}(Y_1) \dots f^_{i_n}(Y_n)}{\mu_{i_1} Q_{i_1,i_2} \dots Q_{i_{n-1},i_n} f_{i_1}(Y_1) \dots f_{i_n}(Y_n)}\mu_{i_1} Q_{i_1,i_2} \dots Q_{i_{n-1},i_n} f_{i_1}(Y_1) \dots f_{i_n}(Y_n)} {\sum_{i_1,\dots, i_n=1}^k \mu_{i_1} Q_{i_1,i_2} \dots Q_{i_{n-1},i_n} f_{i_1}(Y_1) \dots f_{i_n}(Y_n)}\right)\right)\\ & \leq \frac{1}{n} \mathbb{E}_{p^{\theta^}_n} \left( \log\left(\max_{1 \leq i_1,\dots, i_n\leq k} \frac{\mu^_{i_1} Q^_{i_1,i_2} \dots Q^_{i_{n-1},i_n} f^_{i_1}(Y_1) \dots f^_{i_n}(Y_n)}{\mu_{i_1} Q_{i_1,i_2} \dots Q_{i_{n-1},i_n} f_{i_1}(Y_1) \dots f_{i_n}(Y_n)} \right)\right)\\ & \leq \frac{1}{n} \mathbb{E}_{p^{\theta^}_n} \left( \log\left(\max_{1 \leq i\leq k} \frac{\mu^_{i}}{\mu_i} \left(\max_{1 \leq i,j\leq k} \frac{Q^_{i,j}}{Q_{i,j}}\right)^{n-1} \max_{1 \leq i\leq k} \frac{f^_{i}(Y_1)}{f_{i}(Y_1)} \dots \max_{1 \leq i\leq k} \frac{f^_{i}(Y_n)}{f_{i}(Y_n)} \right)\right)\\ & \leq \frac{1}{n\underline{q}} \max_{1 \leq i\leq k} \left\lvert \mu_i - \mu^_i \right\rvert + \frac{n-1}{n\underline{q}} \max_{1 \leq i,j \leq k} \left\lvert Q_{i,j} - Q^_{i,j} \right\rvert + \max_{1 \leq i\leq k} \int f^_i(y) \max_{1 \leq j \leq k} \log \frac{f^_j(y)}{f_j(y)} \lambda(dy). \end{split}$$ The last inequality comes from the following assumption: $$\min\limits_{1\leq i,j\leq k} (\mu_i,\mu^_i, Q_{i,j}, Q^_{i,j})\geq \underline{q} .$$ Then for all $\epsilon>0$, for $n$ large enough, for all $\theta \in \Theta_\epsilon$, $$\frac{1}{n} KL(p^{\theta^}_n,p^{\theta,\mu}_n) \leq \frac{3}{\underline{q}} ~ \epsilon$$ Proof of Theorem \[th1\] {#Prth1 .unnumbered} ------------------------ This proof relies on Theorem 5 of @Ba88. We do not assume (A2) in the first part of the proof. First we prove that for all $a>0$, $$\label{eqmerge} \mathbb{P}^{\theta^}\left( \frac{\int_\Theta p^\theta_n (Y_1, \dots, Y_n) \pi(d\theta)}{p^{\theta^}_n(Y_1, \dots, Y_n)} \leq \exp(-a n) \text{ i.o.} \right)=0$$ that is to say $$p^{\theta^}_n (y_1, \dots, y_n) \lambda(dy_1) \dots \lambda(dy_n)$$ and $$\int_\Theta p^\theta_n (y_1, \dots, y_n) \lambda(dy_1) \dots \lambda(dy_n) \pi(d\theta)$$ merge with probability one. Let $\epsilon>0$. Note that Assumption (A1a) implies that $Q^ \in \Delta^k(\underline{q})$. Then by Lemma \[KL\], there exists a real $\tilde{\epsilon}>0$ such that for $n$ large enough, for all $\theta \in \Theta_{\tilde{\epsilon}}$, $$\label{kleq}\frac{1}{n}KL(p^{\theta^}_n,p^{\theta,\mu}_n) < \epsilon.$$ Moreover by Proposition 1 of @Do04, if $\theta \in \Theta(\underline{q})$ and if (A1c), (A1d) and (A1e) hold, $$\frac{1}{n} \log \left( \frac{p^{\theta^}_n(Y_{1:n})}{p^{\theta,\mu}_n(Y_{1:n})}\right)$$ converges $\mathbb{P}^{\theta^}$-almost surely and in $L^1(\mathbb{P}^{\theta^})$. Let $\bar{L}(\theta)$ denote this limit: $$\lim_{n \to \infty} \frac{1}{n} \log \left( \frac{p^{\theta^}_n(Y_{1:n})}{p^{\theta,\mu}_n(Y_{1:n})} \right) =: \bar{L}(\theta), ~ \mathbb{P}^{\theta^}\text{-a.s. and in } L^1(\mathbb{P}^{\theta^}).$$ Then for all $\theta \in \Theta_{\tilde{\epsilon}}$, $$\label{klimeq}\bar{L}(\theta) \leq \epsilon.$$ So for all $\epsilon>0$, there exists $\tilde{\epsilon}$ such that $$\pi\left(\theta : \bar{L}(\theta)<\epsilon\right) ~ \geq ~ \pi(\Theta_{\tilde{\epsilon}}) ~ >0.$$ By Lemma 10 of @Ba88, for all $a>0$, (\[eqmerge\]) is verified. We now have to build the tests described in Theorem 5 in @Ba88, to obtain posterior consistency first for the weak topology and secondly for the $D_l$-pseudometric. In the case of the weak topology, we follow the ideas of Section 4.4.1 in @GhRa03. Using page 142 of @GhRa03, it is sufficient to consider $$U = \left\{P ~ : ~ \int h dP - \int h p_l^{\theta^} d\lambda^{\otimes l} < \epsilon, \right\},$$ for all $\epsilon>0$ and $0 \leq h \leq 1$ in the set $\mathcal{C}_b((\mathbb{R}^d)^l)$. Choosing $\alpha$ and $\gamma$ as in page 128 of @GhRa03, if $$S^n=\left\{ y_1, \dots, y_n ~: ~ \frac{l}{n} \sum_{j=0}^{n/l-1} h(y_{jl+1}, \dots, y_{jl+ l}) > \frac{\alpha + \gamma}{2} \right\},$$ then $$\label{plip} \begin{split} P^{\theta^} (S^n) & = P^{\theta^}\left\{ \sum_{j=0}^{n/l-1} \left (h(y_{jl+1}, \dots, y_{jl+ l}) - \int h p_l^{\theta^} d\lambda^{\otimes l}\right)>\frac{n}{l} \frac{\gamma - \alpha }{2} \right\} \\ &\leq \exp \left(- \frac{n (\gamma - \alpha ) ^2 (\min_{i,j}Q^_{i,j})^2 }{2 l (2-k\min_{i,j}Q^_{i,j})^2 } \right) \end{split}$$ and for all $\theta \in \Theta(\underline{q})$ such that $\int h dP^\theta - \int h p_l^{\theta^} d\lambda^{\otimes l} \geq \epsilon$, $$\label{plop} \begin{split} P^\theta((S^n)^c) &\leq P^{\theta}\left\{ \sum_{j=0}^{n/l-1} \left (- h(y_{jl+1}, \dots, y_{jl+ l}) + \int h p_l^{\theta} d\lambda^{\otimes l}\right) \geq \frac{n}{l} \frac{\gamma - \alpha }{2} \right\} \\ & \leq \exp \left(- \frac{n (\gamma - \alpha ) ^2 (\min_{i,j}Q_{i,j})^2 }{2 l (2-k\min_{i,j}Q_{i,j})^2 } \right) \leq \exp \left(- \frac{n (\gamma - \alpha ) ^2 \underline{q}^2 }{2 l } \right), \end{split}$$ using the upper bound from the proof of Theorem 4 of @GaRo12 based on Corollary 1 in @Ri00. Using Theorem 5 of @Ba88 and combining Equations and , $$\begin{split} &P^{\theta^}\Bigg( \pi \bigg(\Big\{\theta : \int h dP^\theta - \int h p_l^{\theta^} d\lambda^{\otimes l} < \epsilon \Big\}^c ~ \bigg\| ~ Y_{1:n}\bigg)\geq e^{-nr} \text{, i.o. } \Bigg)=0 \end{split}$$ which implies that for all weak neighborhood $U$ of $P_l^{\theta^}$, $$P^{\theta^}(\left( \pi(U^c\|Y_{1:n})\geq \exp(-nr) \text{ i.o. } \right)=0,$$ so that $$\mathbb{P}^{\theta^}\left(\lim_{n \to \infty} \pi(U\|Y_{1:n})=1 \right)=1 .$$ We now assume (A2) and obtain consistency for the $D_l$-pseudometric. Let $\epsilon>0$ and let $$U=\left\{ \theta ~ : ~D_l(\theta,\theta^)< \frac{2 \epsilon}{k \underline{q}} \right\} \supset \left\{ \theta ~ : ~D_l(\theta,\theta^)< \epsilon \frac{2 - k \min_{1\leq i,j\leq k} Q_{i,j} }{k \min_{1\leq i,j\leq k} Q_{i,j} } \right\},$$ be a $D_l$-neighborhood of $\theta^$. Let $$B_n^c=\Delta^k(\underline{q}) \times \mathcal{F}_n,$$ so that $$\label{eqB} \pi(B_n)= \pi_f({\mathcal{F}_n}^c)\lesssim \exp(-n r_1).$$ In the proof of Theorem 4 of @GaRo12, it is proved that for all $n$ large enough, there exists a test $\psi_n$ such that $$\label{eqth1} \begin{split} \mathbb{E}^{\theta^}(\psi_n) &\leq N\left(\frac{\epsilon}{12}, \Delta^k(\underline{q}) \times \mathcal{F}_n, D_l\right) \exp\left(-\frac{n \epsilon^2}{8l} \frac{ k^2 (\min_{i,j}Q^_{i,j})^2}{(2-k\min_{i,j}Q^_{i,j})^2}\right)\\ &\leq N\left(\frac{\epsilon}{12}, \Delta^k(\underline{q}) \times \mathcal{F}_n, D_l\right) \exp\left(-\frac{n \epsilon^2 k^2 \underline{q}^2}{32 l}\right) \end{split}$$ $$\label{eqtest1} \sup_{\theta \in U^c \cap B_n^c} \mathbb{P}^{\theta,\mu} (1-\psi_n) \leq \exp\left( -\frac{n \epsilon^2}{32l} \right) .$$ Note that for all $\theta, \tilde{\theta}$ in $\Theta(\underline{q})$, $$D_l(\theta, \tilde{\theta}) \leq \lVert \mu^\theta - \mu^{\tilde{\theta}} \rVert_1 + k(l-1) \lVert Q-\tilde{Q} \rVert + l \max_{1 \leq j\leq k} \lVert f_j - \tilde{f_j} \rVert_{L_1(\lambda)}$$ The function $Q \to \mu^{Q}$ is continuous on the compact $\Delta^k(\underline{q})$ and thus is uniformly continuous: there exists $\alpha>0$ such that for all $\theta, \tilde{\theta}$ in $\Theta(\underline{q})$ such that $\lVert Q - \tilde{Q} \rVert <\alpha$ then $\lVert \mu^\theta -\mu^{\tilde{\theta}} \rVert_1 < \frac{\epsilon}{36}$. This implies that $$\label{eqth11} \begin{split} N & \left(\frac{\epsilon}{12}, \Delta^k(\underline{q}) \times \mathcal{F}_n, D_l\right) \\ & \leq N\left(\min\bigg(\frac{\epsilon}{36k(l-1)},\alpha\bigg), \Delta^k(\underline{q}) , \lVert \cdot \rVert \right) N\left(\frac{\epsilon}{36l}, \mathcal{F}_n, d(\cdot,\cdot) \right)\\ & \leq \left(\max \left(\frac{36k(l-1)}{\epsilon},\frac{1}{\alpha}\right) \right)^{k(k-1)} N\left(\frac{\epsilon}{36l}, \mathcal{F}_n, d(\cdot,\cdot) \right) \end{split}$$ Then combining Equations (\[eqB\]), (\[eqth1\]), (\[eqtest1\]), (\[eqth11\]) and using Theorem 5 of @Ba88, there exists $r>0$ such that $$\label{eqBa} \mathbb{P}^{\theta^}\Bigg( \pi \left(U^c\|Y_{1:n}\right) \geq \exp(-nr) \text{ i.o. } \Bigg) =0 .$$ And Equation (\[eqBa\]) implies that for all $\epsilon>0$, $$\mathbb{P}^{\theta^}\left(\lim_{n \to \infty} \pi(~ \left\{ \theta: ~ D_l(\theta,\theta^)< \epsilon \right\} ~ \| ~ Y_{1:n})=1 \right)=1.$$ Proof of Theorem \[th2\] {#P:th2 .unnumbered} ------------------------ Using Theorem \[th1\], it is sufficient to show that for all weak neighborhood $U_{f^}$ of $f^ \lambda$ and neighborhood $U_{Q^}$ of $Q^$, there exists a $D_3$-neighborhood $U_{\theta^}$ of $\theta^$ such that $$\label{eqsp} U_{\theta^} \subset \left\{ \exists \sigma \in \mathcal{S}_k; ~ \sigma Q \in U_{Q^}, ~ f_{\sigma(i)} \in U_{f^_i}, ~ i=1\dots k \right\} .$$ Following @GaClRo13, it is equivalent to show that for all sequences $\theta^n$ in $ \Theta(\underline{q})$ such that $D_3(\theta^n,\theta^) \to 0$, there exists a subsequence, that we denote again $\theta^n$, of $\theta^n$ and $\bar{\theta} \in \Theta$ such that $\lVert Q^n-\bar{Q}\rVert \to 0$, $f^n_i \lambda$ tends to $\bar{f}_i \lambda$ in the weak topology on probabilities for all $i\leq k$ and $p_3^{(Q^,f^)} = p_3^{(\bar{Q},\bar{f})}$. Let $\theta^n$ in $ \Theta(\underline{q})$ such that $D_3(\theta^n,\theta^) \to 0$. As $\Delta^k(\underline{q})$ is a compact set, there exists a subsequence of $Q^{n}$ that we denote again $Q^{n}$ which tends to $\bar{Q} \in \Delta^k(\underline{q})$. Writing $\mu^n$ the (sub)sequence of the stationary distribution associated to $Q_n$, then $\mu^{n} \to \bar{\mu}$ where $\bar{\mu}$ is the stationary distribution associated to $\bar{Q}$. Moreover, $$\begin{split} D_3( & \theta^{n}, \theta^)= \lVert p_3^{\theta_n} - p_3 ^{\theta^} \rVert_{L_1(\lambda^{\otimes 3})} \\ & \geq \int \Big\lvert \sum_{1 \leq i_1, i_2, i_3 \leq k }\mu_{i_1}^{n} Q_{i_1,i_2}^{n} Q_{i_2,i_3}^{n} f_{i_1}^{n}(y_1) f_{i_2}^{n}(y_2) f_{i_3}^{n}(y_3) - \\ & \quad \mu_{i_1}^ Q_{i_1,i_2}^* Q_{i_2,i_3}^* f_{i_1}^(y_1) f_{i_2}^(y_2) f_{i_3}^(y_3) \Big\rvert ~ \lambda(dy_1)\lambda(dy_2)\lambda(dy_3) \\ & \geq - \sum_{1 \leq i_1, i_2, i_3 \leq k }\left \lvert \mu_{i_1}^{n} Q_{i_1,i_2}^{n} Q_{i_2,i_3}^{n} - \bar{\mu}_{i_1} \bar{Q}_{i_1,i_2} \bar{Q}_{i_2,i_3} \right\rvert + \\ & \quad \int \Big\lvert \sum_{1 \leq i_1, i_2, i_3 \leq k } \bar{\mu}_{i_1} \bar{Q}_{i_1,i_2} \bar{Q}_{i_2,i_3} f_{i_1}^{n}(y_1) f_{i_2}^{n}(y_2) f_{i_3}^{n}(y_3)- \\ & \qquad \mu_{i_1}^ Q_{i_1,i_2}^* Q_{i_2,i_3}^* f_{i_1}^(y_1) f_{i_2}^(y_2) f_{i_3}^(y_3) \Big\rvert ~ \lambda(dy_1)\lambda(dy_2)\lambda(dy_3) \end{split}$$ Since $\displaystyle{ \sum_{1 \leq i_1, i_2, i_3 \leq k }\Big\lvert \mu_{i_1}^{n} Q_{i_1,i_2}^{n} Q_{i_2,i_3}^{n} - \bar{\mu}_{i_1} \bar{Q}_{i_1,i_2} \bar{Q}_{i_2,i_3} \Big\rvert}$ tends to zero, $$\label{semconv} \begin{split} \lim_n & \int \Big\lvert \sum_{1 \leq i_1, i_2, i_3 \leq k } \bar{\mu}_{i_1} \bar{Q}_{i_1,i_2} \bar{Q}_{i_2,i_3} f_{i_1}^{n}(y_1) f_{i_2}^{n}(y_2) f_{i_3}^{n}(y_3)- \\ & \qquad \mu_{i_1}^ Q_{i_1,i_2}^* Q_{i_2,i_3}^* f_{i_1}^(y_1) f_{i_2}^(y_2) f_{i_3}^(y_3) \Big\rvert ~ \lambda(dy_1)\lambda(dy_2)\lambda(dy_3) =0 \end{split}$$ Let $F^n_1, \dots, F^n_k$ be the probability distribution with respective densities $f^n_1, \dots, f^n_k$ with respect to $\lambda$. Since $$\sum_{i_1,i_2,i_3} \bar{\mu}_{i_1} \bar{Q}_{i_1,i_2} \bar{Q}_{i_2,i_3} F^n_{i_1} \otimes F^n_{i_2} \otimes F^n_{i_3}$$ converges in total variation, it is tight and for all $1 \leq i\leq k$, $(F_i^n)_n$ is tight. By Prohorov’s theorem, for all $1 \leq i \leq k$ there exists a subsequence denoted $F^n_i$ of $F^n_i$ which weakly converges to $\bar{F}_i$. This in turns implies that $$\sum_{i_1,i_2,i_3} \bar{\mu}_{i_1} \bar{Q}_{i_1,i_2} \bar{Q}_{i_2,i_3} F^n_{i_1} \otimes F^n_{i_2} \otimes F^n_{i_3}$$ weakly converges to $$\sum_{i_1,i_2,i_3} \bar{\mu}_{i_1} \bar{Q}_{i_1,i_2} \bar{Q}_{i_2,i_3} \bar{F}_{i_1} \otimes \bar{F}_{i_2} \otimes \bar{F}_{i_3},$$ which combined with , leads to $$\begin{split} \sum_{i_1,i_2,i_3} & \bar{\mu}_{i_1} \bar{Q}_{i_1,i_2} \bar{Q}_{i_2,i_3} \bar{F}_{i_1} \otimes \bar{F}_{i_2} \otimes \bar{F}_{i_3}\\ &= \sum_{ i_1, i_2, i_3 } \mu_{i_1}^ Q_{i_1,i_2}^* Q_{i_2,i_3}^* f_{i_1}^\lambda \otimes f_{i_2}^\lambda \otimes f_{i_3}^\lambda \end{split}$$ By @GaClRo13, $\bar{Q}=Q^$, so $\bar{\mu}=\mu^$ and $\bar{F}_i=f^_i \lambda$ up to a label swapping, that is there exists a permutation $\sigma \in \mathcal{S}_k$ such that $\sigma\bar{Q}= Q^$ and $\bar{F}_{\sigma(i)}= f^_{i} \lambda$ so that Equation holds. Proof of Theorem \[FD\] {#P:FD .unnumbered} ----------------------- To prove Theorem \[FD\] we need the following lemma: \[lem\] Let $\epsilon>0$, for all $0<\epsilon_1<1$, $N>0$, $1 \leq j < N$ and $c>0$ such that $$0<\frac{\epsilon_1 2^{2N}k^N}{c(c-\epsilon_1)} < \frac{\epsilon}{3} \text{ and } \frac{2 (1-\underline{q})^{N+1-j}}{\underline{q} + (1-\underline{q})^{N+1-j}} < \frac{\epsilon}{3}.$$ If $p_N^{\theta^}(Y_{1:N})>c$, then for all $1\leq l \leq k$ and for all $n>N$, $$\begin{split} \bigg\{ \theta \in \Delta^k(\underline{q}) : \lVert p_N^{\theta^} - p_N ^{\theta} \rVert_{l_1}< \epsilon_1 , ~ \exists \sigma \in \mathcal{S}_k, ~ \lvert\mu^\theta_{\sigma(i)}-\mu^_i\rvert < \epsilon_1 , ~ \lVert \sigma Q - Q^ \rVert < \epsilon_1 , \\ \max_{1\leq i \leq k}\lVert f_{\sigma(i)}-f^_i \rVert_{l_1}< \epsilon_1 \bigg\} \\ \subset \left\{ \theta \in \Delta^k(\underline{q}) : ~ \lvert P^{\theta^}(X_j=l ~ \| ~ Y_{1:n}) - P^{\theta}(X_j=l ~ \| ~ Y_{1:n}) \rvert ~ < ~ \epsilon \right\} \end{split}$$ Let $\theta \in \Delta^k(\underline{q})$ be such that $\lVert p_N^{\theta^} - p_N ^{\theta} \rVert_{l_1}< \epsilon_1$ and there exists $\sigma \in \mathcal{S}_k$ such that $\max_{1\leq i \leq k } \lvert \mu^\theta_{\sigma(i)}-\mu^_i\rvert < \epsilon_1$, $\lVert \sigma Q - Q^* \rVert < \epsilon_1$ and $\max_{1\leq i \leq k}\lVert f_{\sigma(i)}-f^_{i} \rVert_{l_1}< \epsilon_1 $. To bound $\lvert P^{\theta^}(X_j=l ~ \| ~ Y_{1:n}) - P^{\theta}(X_j=l ~ \| ~ Y_{1:n}) \rvert $, we now prove that it is sufficient to bound $\lvert P^{\theta^}(X_j=l ~ \| ~ Y_{1:N}) - P^{\theta}(X_j=l ~ \| ~ Y_{1:N}) \rvert $ with $N<n$ a well chosen fixed integer thanks to the exponential forgetting of the HMM. Let $1 \leq a \leq k$, $$\label{lem1} \begin{split} \lvert & P^{\theta^}(X_j=l ~ \| ~ Y_{1:n}) - P^{\theta}(X_j=l ~ \| ~ Y_{1:n}) \rvert \\ & \leq A_{\theta^} + \lvert P^{\theta^}(X_j=l ~ \| ~ Y_{1:N}) - P^{\theta}(X_j=l ~ \| ~ Y_{1:N}) \rvert + A_{\theta}, \end{split}$$ where for $\tilde{\theta}\in\{\theta, \theta^\}$, $$\begin{split} A_{\tilde{\theta}} = & \bigg\lvert \frac{P^{\tilde{\theta}}(Y_{1:N},X_j=l) \sum\limits_{1 \leq b \leq k} P^{\tilde{\theta}} (Y_{N+1:n}~ \| X_{N+1}=b) P^{\tilde{\theta}}( X_{N+1}=b \| X_j=l,Y_{j:N} )}{\sum\limits_{1\leq m \leq k} P^{\tilde{\theta}}(Y_{1:N},X_j=m) \sum\limits_{1 \leq b \leq k} P^{\tilde{\theta}} (Y_{N+1:n}~ \| X_{N+1}=b) P^{\tilde{\theta}}( X_{N+1}=b \| X_j=m , Y_{j:N})} - \\ & \frac{P^{\tilde{\theta}}(Y_{1:N},X_j=l) \sum\limits_{1 \leq b \leq k} P^{\tilde{\theta}} (Y_{N+1:n}~ \| X_{N+1}=b) P^{\tilde{\theta}}( X_{N+1}=b \| X_j=a, Y_{j:N} )}{\sum\limits_{1\leq m \leq k} P^{\tilde{\theta}}(Y_{1:N},X_j=m) \sum\limits_{1 \leq b \leq k} P^{\tilde{\theta}} (Y_{N+1:n}~ \| X_{N+1}=b) P^{\tilde{\theta}}( X_{N+1}=b \| X_j=a , Y_{j:N} )} \bigg\rvert. \end{split}$$ Using Corollary 1 of @Do04, i.e. the exponential forgetting of the HMM, we obtain for all $(\omega,m) \in \{1, \dots, k\}^2$, $$\begin{split} & \left \lvert P^{\tilde{\theta}}( X_{N+1}=b \| X_j=m , Y_{j:N} ) - P^{\tilde{\theta}}( X_{N+1}=b \| X_j=\omega , Y_{j:N} ) \right \rvert \\ & ~ \leq (1-\underline{q})^{N+1-j} \leq (1-\underline{q})^{N+1-j} \frac{ P^{\tilde{\theta}}( X_{N+1}=b \| X_j=\omega , Y_{j:N} )}{\underline{q}} \end{split}$$ so that for $\tilde{\theta}\in\{\theta, \theta^\}$, $$\label{lem2}A_{\tilde{\theta}} \leq \frac{2 (1-\underline{q})^{N+1-j}}{\underline{q} + (1-\underline{q})^{N+1-j}}.$$ Moreover, $$\begin{split} & P^{\theta^}(X_j=l ~ \| ~ Y_{1:N}) - P^{\theta}(X_j=l ~ \| ~ Y_{1:N}) \\ & = \frac{\sum\limits_{a_{1:j-1},a_{j+1:N}}\mu^_{a_1}Q^_{a_1,a_2} \dots Q^_{a_{j-1},l}Q^_{l,a_{j+1}} \dots Q^_{a_{N-1},a_N}f^_{a_1}(Y_{a_1})\dots f^_l(Y_j)\dots f^_{a_N}(Y_N)}{p^{\theta^}_N(Y_{1:N})} \\ & - \frac{\sum\limits_{a_{1:j-1},a_{j+1:N}}\mu^{}_{a_1}Q_{a_1,a_2} \dots Q_{a_{j-1},l}Q_{l,a_{j+1}} \dots Q_{a_{N-1},a_N}f_{a_1}(Y_{a_1})\dots f_l(Y_j)\dots f_{a_N}(Y_N)}{p^{\theta}_N(Y_{1:N})} \\ & \leq \frac{(1+\epsilon_1/c) \sum\limits_{a_{1:j-1},a_{j+1:N}}\mu^_{a_1} \dots f^_{a_N}(Y_N) -\sum\limits_{a_{1:j-1},a_{j+1:N}}\mu^{}_{a_1}\dots f_{a_N}(Y_N)}{(1+\epsilon_1/c)p_N^{\theta^}(Y_{1:N})}\\ & \leq \frac{(1+\epsilon_1/c) \sum\limits_{a_{1:j-1},a_{j+1:N}}\mu^_{a_1} \dots f^_{a_N}(Y_N) -\sum\limits_{a_{1:j-1},a_{j+1:N}}(\mu^{}_{a_1}-\epsilon_1)\dots (f_{a_N}(Y_N)-\epsilon_1)}{c+\epsilon_1} \\ & \leq \frac{\max(\epsilon_1,\epsilon_1/c) \sum\limits_{a_{1:j-1},a_{j+1:N}} 2^{2N} }{c+ \epsilon_1} \leq \frac{\epsilon_1 2^{2N}k^N}{c(c+\epsilon_1)}. \end{split}$$ Similarly $$P^{\theta}(X_j=l ~ \| ~ Y_{1:N}) - P^{\theta^}(X_j=l ~ \| ~ Y_{1:N}) \leq \frac{\epsilon_1 2^{2N}k^N}{c(c-\epsilon_1)}$$ so that $$\label{lem3} \big\lvert P^{\theta^}(X_j=l ~ \| ~ Y_{1:N}) - P^{\theta}(X_j=l ~ \| ~ Y_{1:N}) \big\rvert \leq \frac{\epsilon_1 2^{2N}k^N}{c(c-\epsilon_1)}.$$ Combining Equations (\[lem1\]), (\[lem2\]) and , we obtain $$\begin{split} \lvert &P^{\theta^}(X_j=l ~ \| ~ Y_{1:n}) - P^{\theta}(X_j=l ~ \| ~ Y_{1:n}) \rvert \\ &\leq 2 \frac{2 (1-\underline{q})^{N+1-j}}{\underline{q} + (1-\underline{q})^{N+1-j}} + \frac{\epsilon_1 2^{2N}k^N}{c(c-\epsilon_1)} < \epsilon. \end{split}$$ We prove Theorem \[FD\] for $m=1$, one may easily generalizes the proof. Let $\beta>0$, $j>0$ and $\epsilon>0$, we fix $N$ and $c>0$ such that $$\frac{2 (1-\underline{q})^{N+1-j}}{\underline{q} + (1-\underline{q})^{N+1-j}}<\frac{\epsilon}{3} \text{ and } P^{\theta^}\big(p_N^{\theta^}(Y_{1:N})>c\big)>\sqrt{1-\beta}$$ then we choose $\epsilon_1$ such that $$0< \frac{\epsilon_1 2^{2N}k^N}{c(c-\epsilon_1)} < \frac{\epsilon}{3}.$$ Posterior consistency for the marginal distribution in $l_1$ and for all components of the parameter i.e. Theorems \[th1\] and \[th2\] imply that there exists $M$ such that $P^{\theta^}$-a.s., for all $n \geq M$, $$\pi\left( \{ \theta ~: ~ D_N(\theta,\theta^) \}<\epsilon_1 ~ \big\| ~ Y_{1:n} \right)>\frac{\sqrt{1-\beta}+1}{2}$$ and $$\begin{split} \pi\bigg( \{ \theta ~: ~ \exists \sigma \in \mathcal{S}_k, ~ \max_{1\leq i\leq k} \lvert \mu_{\sigma(i)} &- \mu^_i \rvert < \epsilon_1, ~ \lVert \sigma Q - Q^\rVert<\epsilon_1, ~\\ &\max_{1 \leq i \leq k} \lVert f_{\sigma(i)} -f^_i \rVert_{l_1} \}<\epsilon_1 ~ \bigg\| ~ Y_{1:n} \bigg)>\frac{\sqrt{1-\beta}+1}{2} \end{split}$$ so that for all $n\geq \max(N,M)$, $$\begin{split} \mathbb{E}^{\theta^} & \left( \pi \bigg(\left\lvert P^\theta(X_j=l ~\|Y_{1:n}) - P^{\theta^}(X_j=l ~\|~Y_{1:n})\right\rvert <\epsilon \|Y_{1:n}\bigg) \right) \\ & \geq \mathbb{E}^{\theta^} \left( \mathds{1}_{p^{\theta^}_N(Y_{1:N})>c} \pi \bigg( \left\lvert P^\theta(X_j=l ~\|Y_{1:n}) - P^{\theta^}(X_j=l ~\|~Y_{1:n})\right\rvert <\epsilon \|Y_{1:n}\bigg) \right) \\ & \geq 1 -\beta. \end{split}$$ Then for all $\alpha>0$, $$\begin{split} P^{\theta^} &\left( \pi\bigg( \left\lvert P^\theta(X_j=l ~\|Y_{1:n}) - P^{\theta^}(X_j=l ~\|~Y_{1:n})\right\rvert <\epsilon \|Y_{1:n}\bigg) < 1 -\alpha \right) \\ & \leq \frac{1}{\alpha} \left( 1- \mathbb{E}^* \left( \pi\bigg( \left\lvert P^\theta(X_j=l ~\|Y_{1:n}) - P^{\theta^}(X_j=l ~\|~Y_{1:n})\right\rvert <\epsilon \|Y_{1:n}\bigg) \right)\right) \\ & \to 0. \end{split}$$ Proof of Proposition \[thD\] {#P:thD .unnumbered} ---------------------------- Note that for all $1 \leq i \leq k$, $$\begin{split} \int_{\mathcal{F}^k} & \sum_{l=1}^{+\infty} f^_i(l) \max_{1\leq j \leq k} \left( -\log(f_j(l)) \right) (DP(\alpha G_0))^{\otimes k}(df) \\ &\leq \sum_{l=1}^{+\infty} f^_i(l) \sum_{1\leq j \leq k} \int_{\mathcal{F}^k}\left( -\log(f_j(l)) \right) (DP(\alpha G_0))^{\otimes k}(df)\\ & \lesssim \sum _{l=1}^{+\infty} \frac{f^_i(l)}{\alpha G_0(l)} \end{split}$$ so that using Assumption (E1), $$\begin{split} \big(DP(\alpha G_0)\big)^{\otimes k} & \bigg(f_1, \dots ,f_k ~ : ~ \forall 1\leq i \leq k, \\ & \qquad \sum_{l=1}^{+\infty} f_i^(l)\max_{1 \leq j \leq k} (-\log(f_j(l)))<+\infty \bigg)=1. \end{split}$$ Note that for all $\epsilon>0$, $$\begin{split} & \left\{f_1, \dots ,f_k ~ : ~ \forall 1\leq i \leq k, ~ \sum_{l=1}^{+\infty} f_i^(l)\max_{1 \leq j \leq k} (-\log(f_j(l)))<+\infty \right\} \\ & \subset \bigcup_{N \in \mathbb{N}} \left\{ f_1, \dots ,f_k ~ : ~ \forall 1\leq i \leq k, ~ \sum_{l=N}^{+\infty} f_i^(l)\max_{1 \leq j \leq k} (-\log(f_j(l)))< \epsilon \right\}, \end{split}$$ thus arguing by contradiction, for all $\epsilon>0$, there exists $L_\epsilon$ such that $$\begin{split} \big(DP(\alpha G_0)\big)^{\otimes k} & \bigg(f_1, \dots ,f_k ~ : ~ \forall 1\leq i \leq k, \\ &\qquad \sum_{l>L_\epsilon} f_i^(l)\max_{1 \leq j \leq k} (-\log(f_j(l)))<\epsilon \bigg)>0. \end{split}$$ Using the tail free property of the Dirichlet process, for all $ 1 \leq j \leq k$, $$\sum_{l>L_\epsilon} f_i^(l)\max_{1 \leq j \leq k} (-\log(f_j(l)))<\epsilon$$ and $$\label{eqind} \left(\frac{f_j(1)}{f_j(l \leq L_\epsilon)}, \dots, \frac{f_j(L_\epsilon)}{f_j(l \leq L_\epsilon)} \right)$$ are independent given $f_j(l>L_\epsilon)$ and given $f_j(l>L_\epsilon)$ has a Dirichlet distribution with parameter $(\alpha G_0(1), \dots, \alpha G_0(L_\epsilon))$. Then for all $\epsilon>0$, there exists $L_\epsilon$ such that for all $\delta \in (0,1)$, $$\label{eqE4} \begin{split} \big(DP(\alpha G_0)\big)^{\otimes k} & \bigg(f_1, \dots ,f_k ~ : ~ \forall 1 \leq i \leq k, ~ \sum_{l>L_\epsilon} f_i^(l)\max_{1 \leq j \leq k} (-\log(f_j(l)))<\frac{\epsilon}{2},\\ & \qquad \forall l \leq L_\epsilon, ~ \lvert f_j(l)- f^_j(l) \rvert \leq c\delta \bigg) >0 \end{split}$$ where $ c = \min_{1 \leq i \leq k} \min_{l \leq L_\epsilon, f^_i(l)>0} f^_i(l)$. For all $f_1, \dots, f_k$ such that for all $1 \leq i \leq k$, $$\sum_{l>L_\epsilon} f_i^(l)\max_{1 \leq j \leq k} (-\log(f_j(l)))<\frac{\epsilon}{2}$$ and for all $l \leq L_\epsilon$, $ \lvert f_i(l)- f^_i(l) \rvert \leq c\delta $, (A1e) holds and $$\label{eqE3} \begin{split} \sum_{l \in \mathbb{N}} & f^_i(l) \max_{1\leq j \leq k}\log\left(\frac{f^_j(l)}{f_j(l)} \right) \\ & = \sum_{l\leq L_\epsilon} f^_i(l) \max_{1\leq j \leq k}\log\left(\frac{f^_j(l)}{f_j(l)} \right) + \sum_{l > L_\epsilon} f^_i(l) \max_{1\leq j \leq k}\log(f^_j(l)) \\ & \quad + \sum_{l > L_\epsilon} f^_i(l) \max_{1\leq j \leq k}(-\log(f_j(l)))\\ & \leq \frac{\delta}{1-\delta} + 0 + \frac{\epsilon}{2} \leq \epsilon \end{split}$$ for $\delta$ small enough. For such a $\delta$ denote $$\begin{split} \Theta_\epsilon = \{ Q ~ : ~ \lVert Q-Q^* \rVert \leq \epsilon \} \times \{f_1, \dots ,f_k ~ : ~ \sum_{l>L_\epsilon} f_i^(l)\max_{1 \leq j \leq k} (-\log(f_j(l)))<\frac{\epsilon}{2}, \\ \forall l \leq L_\epsilon, ~ \lvert f_j(l)- f^_j(l) \rvert \leq c\delta \} \end{split}$$ Using Equation (\[eqE3\]), (A1b) holds. Moreover $$\sum_{i=1}^k \sum_{l \in \mathbb{N}} f^_i(l) \left\lvert \log\left( \sum_{j=1}^k f_j(l)\right) \right\rvert \leq \sum_{i=1}^k \sum_{l\in \mathbb{N}} f^_i(l) \left(\log(k) - \log\left(\min_{1\leq j \leq k} f_j(l)\right)\right) <+\infty$$ so that (A1e) holds. Furthermore (A1d) and (A1c) are obviously checked. Using the assumption that $Q^$ is in the support of $\pi_Q$, (A1a) is checked. Then using Equation , (A1) holds and the first part of Proposition \[thD\] follows. We now prove the second part of Proposition \[thD\]. We first give a representation of a discrete Dirichlet process with independent Gamma distributed random variables. \[LDir\] Let $(Z_l)_{l \in \mathbb{N}}$ be independent random variables such that for all $ l \in \mathbb{N} $, $$Z_l \sim \Gamma(\alpha G_0(l),1),$$ then $\sum_{l=1}^L Z_l$ converges almost surely and its limit has a gamma distribution $\Gamma(\alpha,1)$. Moreover denote $$f : \left\{ \begin{array}{ll} \mathbb{N} &\to [0,1]\\ i &\to f(i)=Z_i/(\sum_{l=1}^{+\infty} Z_l) \end{array}\right. ,$$ then $f$ is distributed from a Dirichlet process $DP(\alpha G_0)$. First for all $t\in \mathbb{R}$ $$\begin{split} \lim_{L\to \infty} \mathbb{E}\left( \exp\left( i t \sum_{l=1}^L Z_l \right)\right) & = \lim_{L\to \infty} \prod_{l=1}^L \mathbb{E}(\exp(it Z_l)) = \lim_{L\to \infty} \prod_{l=1}^L (1-it)^{-\alpha G_0(l)} \\ & = (1-it)^{-\alpha}, \end{split}$$ thus $\sum_{l} Z_l$ converges in law and equivalently almost surely (see Section 9.7.1 in @Du) and is distributed from a gamma distribution $\Gamma(\alpha,1)$. Let $\{B_1, \dots, B_M\}$ be a partition of $\mathbb{N}$, $$\begin{split} (f(B_1),\dots, f(B_M)) &= \left(\frac{\sum_{l \in B_1} Z_l}{ \sum_{l\in \mathbb{N}} Z_l}, \dots ,\frac{ \sum_{l \in B_M}Z_l}{\sum_{l \in \mathbb{N}} Z_l}\right)\\ & \sim Dir((\alpha G_0(B_1), \dots, \alpha G_0(B_M))) \end{split}$$ since $\left(\sum_{l \in B_1} Z_l, \dots ,\sum_{l \in B_M}Z_l\right)$ are independent random variables and for all $1\leq i \leq M$, $$\sum_{l \in B_i} Z_l \sim \Gamma(\alpha G_0(B_i),1).$$ Finally $f$ is drawn from a Dirichlet process $DP(\alpha G_0)$. We assume (A1b) i.e. for all $\epsilon>0$, $$DP(\alpha G_0)^{\otimes k }\left(\left\{f \in \mathcal{F}^k , \forall i \in \{1, \dots, k\} ~ \sum_{l \in \mathbb{N}} f^_i(l)\max_{1\leq j \leq k} \log \frac{f_j^(l)}{f_j(l)}<\epsilon\right\}\right)>0.$$ Let $\epsilon>0$, define $\mathcal{F}_\epsilon$ as the set of $f=(f_1, \dots, f_k)$ such that for all $1\leq i \leq k$, for all $f\in \mathcal{F}_\epsilon$, $$\sum_{l \in \mathbb{N}} f^_i(l) \log\left( \frac{f^_i(l)}{f_i(l)}\right)< \epsilon.$$ Then $DP(\alpha G_0)^{\otimes k }(\mathcal{F}_\epsilon)>0$. Since $\sum_l f^_i(l) (-\log f^_i(l))$ converges, then $\sum_l f^_i(l) (-\log f_i(l))$ converges. Using Lemma \[LDir\], we can write $f_i$ with independent gamma distributed random variables $(Z_l)_{l\in \mathbb{N}}$: $$f_i(l)=\frac{Z_l}{\sum_{j\in \mathbb{N}} Z_j},$$ where $Z_l \sim \Gamma(\alpha G_0(l),1)$. Then $\sum_{l\in \mathbb{N}} f^_i(l) (-\log(Z_l))$ converges since $\sum_{j \in \mathbb{N}} Z_j$ is finite almost surely. Since $DP(\alpha G_0)^{\otimes k }(\mathcal{F}_\epsilon)>0$, for all $1\leq i \leq k$ with positive probability , $$\sum_{l\in \mathbb{N}} f^_i(l) (-\log(Z_l))$$ converges. Using the Kolmogorov $0$-$1$ law and the Three-Series Theorem (see Section 9.7.3 in @Du), $\sum_{l\in \mathbb{N}} f^_i(l) (-\log(Z_l))$ converges almost surely and $$\begin{aligned} {2} &\sum_{l\in \mathbb{N}} \mathbb{P}(\lvert f^_i(l) (-\log(Z_l)) \rvert >1) <+ \infty \label{seq1} \\ &\sum_{l\in \mathbb{N}} \mathbb{E}\big(f^_i(l) (-\log(Z_l)) \mathds{1}_{\lvert f^_i(l) (-\log(Z_l)) \rvert \leq 1}\big) <+ \infty \label{seq2} \\ &\sum_{l\in \mathbb{N}} \text{var}\big(f^_i(l) (-\log(Z_l)) \mathds{1}_{\lvert f^_i(l) (-\log(Z_l))\rvert \leq 1}\big) <+ \infty .\end{aligned}$$ Equation implies that $$\begin{split} + \infty & > \sum_{l\in \mathbb{N}} \mathbb{P}(\lvert f^_i(l) (-\log(Z_l)) \rvert >1) \\ &\geq \sum_{l\in \mathbb{N}} \frac{1}{\Gamma(\alpha G_0(l))} \int_0^{\exp(-1/f^_i(l))} x^{\alpha G_0(l)-1} e^{-x} dx \\ & \geq \sum_{l\in \mathbb{N}} \frac{1}{\alpha G_0(l) \Gamma(\alpha G_0(l))} \exp\left(-\exp\left(\frac{-1}{f^_i(l)}\right) -\frac{\alpha G_0(l) }{f^_i(l)}\right) \\ & \gtrsim \sum_{l\in \mathbb{N}} \exp\left(-\frac{\alpha G_0(l) }{f^_i(l)}\right). \end{split}$$ Then $$\lim_l \frac{f^_i(l)}{G_0(l)} = 0.$$ Moreover Equation implies that $$\begin{split} + \infty & > \sum_l \mathbb{E}\big(f^_i(l) (-\log(Z_l)) \mathds{1}_{\lvert f^_i(l) (-\log(Z_l)) \rvert\leq 1}\big) \\ & \geq \sum_l \bigg( \int_{\exp(-1/f^_i(l))}^{1} \frac{1}{\Gamma(\alpha G_0(l))} f^_i(l) (-\log(x)) x^{\alpha G_0(l)-1} e^{-x} dx \\ & \qquad \qquad + \int_{1}^{\exp(1/f^_i(l))} \frac{1}{\Gamma(\alpha G_0(l))} f^_i(l) (-\log(x)) x^{\alpha G_0(l)-1} e^{-x} dx \bigg) \\ & \geq \sum_l \bigg( \frac{e^{-1} f^_i(l)}{\Gamma(\alpha G_0(l)} \int_{\exp(-1/f^_i(l))}^{1} (-\log(x)) x^{\alpha G_0(l)-1} dx \\ & \qquad \qquad -\frac{1}{\Gamma(\alpha G_0(l))} \int_{1}^{\exp(1/f^_i(l))} e^{-x}dx \bigg) \\ &\gtrsim - \alpha + \sum_l \frac{e^{-1} f^_i(l)}{\alpha^2 G_0^2(l)\Gamma(\alpha G_0(l))} \\ & \qquad \qquad \left( 1 - \exp\left(- \frac{\alpha G_0(l) }{f^_i(l)}\right) - \frac{\alpha G_0(l) }{f^_i(l)} \exp\left(- \frac{\alpha G_0(l) }{f^_i(l)}\right) \right) \\ & \gtrsim - \alpha + \sum_l \frac{f^_i(l)}{G_0(l)} \end{split}$$ so that $\sum_l \frac{f^_i(l)}{G_0(l)} < + \infty$. Other proofs {#op} ============ Proof of Proposition \[th:MG\] {#proof-of-proposition-thmg .unnumbered} ------------------------------ The proof uses many ideas of @To06. We now prove that Assumptions (B1), (B2), (B3), (B4) and (B5) imply (A1). A reproduction of the proof of Theorem 3.2. and Lemma 3.1 of @To06 shows that Assumptions (B2), (B3), (B4) and (B5) imply that for all $\epsilon>0$, for all $1 \leq j \leq k$ there exists a weak neighborhood $V_j$ of a compactly supported probability $\tilde{P}_j$ such that for all $f_j=\phi P_j$, $P_j\in V_j$, $$\label{th51:KL} \int_{\mathbb{R}} f^_i(y) \max_{1\leq j \leq k} \log\left(\frac{f^_j(y)}{f_j(y)} \right)\lambda(dy) < \epsilon .$$ Let $0<\underline{\sigma}<\bar{\sigma}$ and $\zeta>0 $ be such that for all $1\leq j \leq k$ $$\tilde{P}_j([-\zeta,\zeta] \times [\underline{\sigma},\bar{\sigma}])=1.$$ Let $\delta=\underline{\sigma}/2$. For all $1\leq j\leq k$ define $$U_j=\{P ~ : ~ \left\lvert \int_{\mathbb{R} \times (0,+\infty)} \xi dP - \int_{\mathbb{R} \times(0,+\infty)} \xi d\tilde{P}_j \right\rvert <\epsilon \},$$ where $\xi: \mathbb{R} \times (0,+\infty) \to [0,1]$ is a piecewise affine continuous function such that $\xi(z,\sigma)=1$ for all $z \in [-\zeta,\zeta]$ and $\sigma \in [\underline{\sigma},\bar{\sigma}]$ and $\xi(z,\sigma)=0$ for all $z \in [-\zeta-\delta,\zeta+\delta]^c$ and $\sigma \in [\underline{\sigma} - \delta,\bar{\sigma} + \delta]^c$. For all $\epsilon>0$, define $$\Theta_{\epsilon}= \{ Q ~:~ \lVert Q - Q^* \rVert <\epsilon \} \times (V_1 \cap U_1) \times \dots \times (V_k \cap U_k) .$$ Then for all $(Q,\phiP_1, \dots, \phiP_k) \in \Theta_\epsilon$, (A1b) is true according to Equation (\[th51:KL\]). In addition, for all $y\in \mathbb{R}$, $$\label{th51:minor} \begin{split} f_j(y)& \geq \int_{[-\zeta- \delta,\zeta+ \delta] \times [\underline{\sigma}-\delta,\bar{\sigma}+\delta]} \phi_{\sigma}(y-z)P_j(dz,d\sigma)\\ & \geq \frac{1}{\bar{\sigma} + \delta} ~ \phi_{\underline{\sigma} - \delta} \big(\max(\lvert y-\xi-\delta \rvert,\lvert y+\xi+\delta \rvert)\big) ~ (1-\epsilon) \end{split}$$ which implies (A1c). Moreover using assumption (B1), $\Pi_P$-a.s. there exists $C>0$ such that for all $1\leq j \leq k$, $$f_j(y) \leq \int \frac{1}{\sigma} P_j(dz,d\sigma) \leq C.$$ Then $$\begin{split} &\left\lvert \log \left( \frac{1}{k} \sum_{j=1}^k f_j(y) \right) \right\rvert \\ & \leq \lvert \log(C) \rvert + \lvert \log(\bar{\sigma}+\delta) \rvert - \log(1 - \epsilon) + \frac{(\max( y-\xi-\delta , y+\xi+\delta ))^2}{2 (\underline{\sigma}-\delta)^2} \end{split}$$ which implies (A1e) under (B5). Furthermore (B1) implies (A1d). As $\Theta_\epsilon$ is a product of neighborhoods of elements in the support of their respective prior, $\pi(\Theta_\epsilon)>0$, so (A1) is checked. Now we prove that Assumption (B6) implies Assumption (A2). Let $\delta>0$. For all $a,l,u, \kappa >0$, such that $l<u$ denote $\mathcal{F}^{\kappa}_{a,l,u}= \{\phiP ~ : ~ P((-a,a]\times(l,u])>1- \kappa \} $. Using Section 4 of @To06, there exist $b_0,b_1,b_2$ only depending on $\kappa$ such that $$\begin{split} \log(N(3\kappa,(\mathcal{F}^{\kappa}_{a,l,u})^k,d))& \leq k \log(N(3\kappa,\mathcal{F}^{\kappa}_{a,l,u},\lVert \cdot \rVert_{L_1(\lambda)}))\\ &\leq k b_0 \left(b_1 \frac{a}{l} + b_2 \log\left(\frac{u}{l} \right) +1 \right) \end{split}$$ Choosing $\kappa=\frac{\delta}{336l} $ and $\beta<\frac{\delta^2 k \underline{q}^2}{32lb_0(b_1+b_2)}$, assumption (B6) shows that assumption (A2) holds. Proof of Corollary \[th3\] {#P:th3 .unnumbered} -------------------------- By repeating the proof of Theorem \[th2\] and using the result of identifiability of Theorem 2.1 of @GaRo13 , if $\lim_{n\to\infty} D_3(\gamma^n,\gamma^)=0$, there exists a subsequence of $\gamma_n$, which we also denote $\gamma_n$, such that $Q^n$ tends to $Q^$ and for all $1\leq j \leq k$, $g^n(\cdot-m^n_j)\lambda$ weakly tends to $g^(\cdot-m^_{j})\lambda$. Particularly $ g^n(\cdot)\lambda$ weakly tends to $g^(\cdot)\lambda$. These weak convergences imply the pointwise convergence of the characteristic functions. As for all $t\in \mathbb{R}$, $$\int e^{ity}g^n(y-m^n_j)d\lambda(y) = e^{itm_j^n} \int e^{ity} g^n(y) d\lambda(y)$$ then $\lim_{n\to \infty} e^{itm_j^n}= e^{itm^_{j}}$ for all $t$ such that $\int e^{ity}g^(y) d\lambda(y) \neq 0$. As any characteristic function is uniformly continuous and equal to $1$ at $0$, there exists $\alpha>0$ such that $\int e^{ity}g^(y) d\lambda(y) \neq 0$ for all $\lvert t \rvert<\alpha$. Thus for all $1\leq j \leq k$, $\lim_{n \to \infty} m^n_j=m^_j$. This implies the first part of Corollary \[th3\]. If moreover $\max_{1\leq j \leq k} \mu^_j > \frac{1}{2}$ and $g^$ is uniformly continuous, using the following inequality proved in the proof of Corrolary 1 in @GaRo13 $$\begin{split} \lVert D_1(\gamma^n,\gamma^) \rVert_{L_1} \geq & \left(2 \max_{1 \leq j \leq k} \mu_j^* -1\right) \lVert g^n - g^* \rVert_{L_1(\lambda)} \\ &-\max_{1\leq j \leq k} \lvert \mu^_j - \mu^n_i \rvert - \max_{1\leq j \leq k}\lVert g^(\cdot-m_j^n) - g^(\cdot-m_j^) \rVert_{L_1(\lambda)} \end{split}$$ we obtain that $\lim _{n \to \infty} \lVert g^n - g^* \rVert_{L_1(\lambda)}=0$ which implies the last part of Corollary \[th3\]. Proof of Proposition \[th:tGM\] {#PtGM .unnumbered} ------------------------------- As in the proof of Proposition \[th:MG\], many ideas come from @To06. We first prove (A1) assuming that (B1), (B2), (B3), (B4) and (B5) are verified with $f_j(\cdot)=g(\cdot-m_j),~ 1 \leq j \leq k$. With the same ideas of the proof of Theorem 3.2 in @To06, for all $\epsilon>0$ there exists a probability $\tilde{P}$ on $\mathbb{R}\times(0,+\infty)$ such that there exists $0<\underline{\sigma}<\bar{\sigma}$ and $a>0$ satisfying $$\tilde{P}((-a,a]\times(\underline{\sigma},\bar{\sigma}])=1$$ and $$\int g^(y-m^_i) \max_{1\leq j \leq k} \log \frac{g^(y-m^_j)}{\phi * \tilde{P} (y-m^_j)} \lambda(dy)\leq \frac{\epsilon}{3},$$ using Assumptions (B2), (B3), (B4) and (B5). Let $G=[-a,a]\times[\underline{\sigma},\bar{\sigma}]$. Using the proof of Lemma 3.1 in @To06 for all $C>\max_{1\leq j \leq k}\lvert m^_j \rvert + a + \bar{\sigma}$, for all $m_j \in [m^_j-a, m^_j +a]$, and for all $P$ such that $P(G)>\frac{\underline{\sigma}}{\bar{\sigma}}$, $$\label{eqplop} \begin{split} \int_{\lvert y \rvert > C} & g^(y-m^_i) \max_{1 \leq j \leq k} \log \frac{\phi\tilde{P}(y-m^_j)}{\phiP(y-m_j)}\lambda(dy) \\ & \leq \int_{\lvert y \rvert > C} g^(y-m_i) \max_{1 \leq j \leq k} \frac{1}{2} \left(\frac{\lvert y\rvert + \lvert m^_j \rvert +2a}{\underline{\sigma}} \right)^2 \lambda(dy) <\infty \end{split}$$ Using assumption (B5) and Equation , we fix $C$ such that $$\int_{\lvert y \rvert > C} g^(y-m^_i) \max_{1 \leq j \leq k} \log \frac{\phi\tilde{P}(y-m^_j)}{\phiP(y-m_j)}\lambda(dy)\leq \frac{\epsilon}{3}$$ Let $G_\delta=[-a-\delta,a+\delta]\times[\underline{\sigma}-\delta,\bar{\sigma}+\delta]$, with $\delta$ chosen in $(0,\min(\frac{\underline{\sigma}}{2},\frac{a}{2})]$. Let $\xi: \mathbb{R} \times (0,+\infty) \to [0,1]$ be a piecewise affine continuous function such that $\xi(z,\sigma)=1$ on $G$ and $\xi(z,\sigma)=0$ on $G_\delta^c$. Let $$c=\inf_{\scriptsize \begin{array}{c} \underline{\sigma} - \delta \leq \sigma \leq \bar{\sigma} + \delta, \\ \lvert y \rvert \leq C, \\ \lvert \theta \rvert \leq a +\max_{j} \lvert m^_j \rvert + \delta \end{array}} \phi_\sigma\left(y-\theta\right).$$ By Arzela-Ascoli theorem there exists $y_1, \dots, y_I$ such that for all $y \in [-C,C]$ and $1\leq j \leq k$, there exists $1 \leq i \leq I$ such that $$\sup_{(z, \sigma) \in G_\delta} \left\lvert \phi_\sigma\left(y-m^_j-z \right) - \phi_\sigma\left(y_i-m^_j-z \right) \right\rvert < c\delta$$ Let $$\begin{split} V_\delta= &\bigg\{ P ~ : ~ \Big \lvert \int \xi(z,\sigma) \phi_{\sigma}(y_i-m^_j-z)dP(z,\sigma) - \\ & \qquad \int \xi(z,\sigma) \phi_{\sigma}(y_i-m^_j-z)d\tilde{P}(z,\sigma) \Big\rvert < c\delta \bigg\}. \end{split}$$ For all $P \in V_\delta$, for all $m_j \in \left[m^_j - \frac{c \underline{\sigma} \delta \sqrt{2}}{\sqrt{\pi}},m^_j + \frac{c \underline{\sigma} \delta \sqrt{2}}{\sqrt{\pi}}\right]$ and for all $1\leq j \leq k$, we get $$\left\lvert \frac{\int \xi(z,\sigma) \phi_{\sigma}(y-m^_j-z) dP(z, \sigma)}{\int \xi(z,\sigma) \phi_{\sigma}(y-m_j-z) d\tilde{P}(z, \sigma)} - 1\right\rvert \leq 4 \delta$$ thus $$\begin{split} &\int_{\lvert y \rvert \leq C} g^(y-m^_i) \max_{1 \leq j \leq k} \log \frac{\phi\tilde{P}(y-m^_j)}{\phiP(y-m^_j)}\lambda(dy)\\ &\quad \leq \int_{\lvert y \rvert \leq C} g^(y-m^_i) \max_{1 \leq j \leq k} \log \frac{\int \xi(z,\sigma) \phi_\sigma(y-m^_j-z)d\tilde{P}(z,\sigma)}{\int\xi(z,\sigma)\phi_\sigma(y-m^_j-z)dP(z, \sigma)}\lambda(dy)\\ & \quad \leq \frac{4\delta}{1-4 \delta} \end{split}$$ Then for $\delta$ small enough, for all $g=\phiP$ such that $P \in V_\delta\cap \{P ~ : ~ P(G)>\frac{\underline{\sigma}}{\bar{\sigma}} \}=\tilde{V}_\delta$, for all $m_j \in \left[m^_j - \frac{c \underline{\sigma} \delta \sqrt{2}}{\sqrt{\pi}},m^_j + \frac{c \underline{\sigma} \delta \sqrt{2}}{\sqrt{\pi}}\right]=M_j^\delta$ and for all $1\leq i \leq k$, $$\label{EQ1} \max_{1\leq i\leq k} \int g^(y-m^_i) \max_{1\leq j \leq k} \log \left(\frac{ g^(y-m^_j)}{g(y-m_j)} \right) dy < \epsilon,$$ moreover, $$\label{EQ2} \begin{split} g(y-m_i) & \geq \int_{G} \phi_\sigma(y-m_i-z)P(dz,d\sigma) \\ & \geq \frac{\underline{\sigma}}{\bar{\sigma}}\phi_{\underline{\sigma}} (\max(\lvert y-m_i -a \rvert, \lvert y-m_i + a \rvert)) P(G)\\ & \geq \frac{\underline{\sigma}}{\bar{\sigma}} \phi_{\underline{\sigma}} (\max(\lvert y-m_i - a \rvert, \lvert y-m_i + a \rvert)) \frac{\underline{\sigma}}{\bar{\sigma}}>0. \end{split}$$ Using assumption (B1) , there exists $\tilde{C}<0$ such that $g\leq \tilde{C}$ thus for all $P\in \tilde{V}_\delta$ and $m_j \in M_j^\delta$ for all $1 \leq j\leq k$, $$\label{EQ3} \begin{split} \sum_{i=1}^k & \mu^_i \int g^(y-m^_i) \left\lvert \log \left( \frac{1}{k} \sum_{j=1}^k g(y-m_j)\right) \right\rvert dy \\ & \leq \sum_{i=1}^k \mu^_i \int g^(y-m^_i) \max_{1 \leq j\leq k} \bigg( \lvert \log(\tilde{C}) \rvert \\ & \qquad + 2\log(\frac{\underline{\sigma}}{\bar{\sigma}}) + \frac{(\max( y-m_j-a , y-m_j+a ))^2}{2 \underline{\sigma}^2} \bigg) \\ & < \infty \end{split}$$ Assumption (B1) ensures that (A1d) holds. Finally for all $\epsilon>0$, there exists $\delta>0$ such that (A1) holds with $\Theta_\epsilon = \{ Q ~ : ~ \lVert Q -Q^ \rVert < \min(\epsilon, \underline{q}/2) \} \times M_1^\delta \times \dots \times M_k^\delta \times \tilde{V}_\delta$ using Equations , and . We now prove (C2) thanks to Assumption (D6). Let $$\mathcal{F}_{a,l,u,\underline{m}} =[-\underline{m},\underline{m}]^k \times \mathcal{F}_{a,l,u},$$ where $\mathcal{F}_{a,l,u}=\mathcal{F}_{a,l,u}^{2} $ is defined in the proof of Proposition \[th:MG\]. Note that for all $(m,\phiP), (\tilde{m},\phi\tilde{P}) \in \mathcal{F}_{a,l,u,\underline{m}}$, for all $1\leq i \leq k$, $$\begin{split} &\lVert \phiP(\cdot -m_i)-\phi\tilde{P}(\cdot - \tilde{m}_i) \rVert_{L_1(\lambda)}\\ &\leq \lVert \phiP(\cdot -m_i)-\phiP(\cdot - \tilde{m}_i) \rVert_{L_1(\lambda)} +\lVert \phiP(\cdot)-\phi\tilde{P}(\cdot) \rVert_{L_1(\lambda)} \end{split}$$ The second term is dealt with in the proof of Proposition \[th:MG\]. As to the first part, we bound $$\lVert \phiP(\cdot -m_i)-\phiP(\cdot - \tilde{m}_i) \rVert_{L_1(\lambda)} \leq \frac{1}{l} \sqrt{\frac{2}{\pi}} \lvert m_i - \tilde{m}_i \rvert$$ Then for all $\kappa>0$, $a,l,u,\underline{m}>0$ such that $l<u$, $$N(3\kappa, \mathcal{F}_{a,l,u,\underline{m}}, d) \leq \left(\frac{2 \underline{m}}{l\kappa}+1\right)^k N(2\kappa, \mathcal{F}_{a,l,u}, \lVert \cdot \rVert_{L_1(\lambda)})$$ For all $\kappa>0$, let $$\mathcal{F}^\kappa_{a,l,u,\underline{m}} =[-\underline{m},\underline{m}]^k \times \mathcal{F}^{\kappa}_{a,l,u}.$$ Following the ideas of Lemmas 4.1 and 4.2 in @To06, there exist $c_0,c_1,c_2,c_3$ only depending on $\kappa$ such that $$\log\left( N(\kappa,\mathcal{F}^\kappa_{a,l,u,\underline{m}}), d \right) \leq c_0 \left(c_1 k \log\frac{\underline{m}}{l} + c_2 \frac{a}{l} + c_3 \log \frac{u}{l} +1 \right),$$ so that (D6) implies (C2) with suitable choices of $\kappa$ and $\beta$.
--- abstract: 'Biased (degree-dependent) percolation was recently shown to provide new strategies for turning robust networks fragile and vice versa. Here we present more detailed results for biased edge percolation on scale-free networks. We assume a network in which the probability for an edge between nodes $i$ and $j$ to be retained is proportional to $(k_ik_j)^{-\alpha}$ with $k_i$ and $k_j$ the degrees of the nodes. We discuss two methods of network reconstruction, sequential and simultaneous, and investigate their properties by analytical and numerical means. The system is examined away from the percolation transition, where the size of the giant cluster is obtained, and close to the transition, where nonuniversal critical exponents are extracted using the generating functions method. The theory is found to agree quite well with simulations. By introducing an extension of the Fortuin-Kasteleyn construction, we find that biased percolation is well described by the $q\rightarrow 1$ limit of the $q$-state Potts model with inhomogeneous couplings.' author: - \| Hans Hooyberghs$^{1}$, Bert Van Schaeybroeck$^{1}$, André A. Moreira$^{2}$,\ José S. Andrade, Jr.$^{2,3}$, Hans J. Herrmann$^{2,3}$ and Joseph O. Indekeu$^{1}$ date: '*' title: 'Biased Percolation on Scale-free Networks' --- Introduction ============ In recent years, much attention has been devoted to the study of real-life networks. Such networks may be modelled by points or nodes connected by edges. One feature is the scale-free topology, described by a probability distribution $P(k)$ for the number of edges $k$ of a node, which falls off as a power law $k^{-\gamma}$ for large values of $k$. Most of the investigated cases turn out to have a topological exponent, or “degree exponent", $\gamma$, in the range $2 < \gamma < 3.5$. For $\gamma > 2$ the mean degree $\langle k \rangle$ is finite, and for $\gamma > 3$ also the variance $\langle k^2\rangle$ is finite. Like fully random Poisson-distributed networks (with a typical scale), also scale-free networks are of “small-world" type. By now, many properties have been revealed and investigated thoroughly: these include degree-degree correlations, clustering and directedness of the edges in the network [@albert; @boccalettia; @dorogovtsev; @dorogovtsev2] . Another well-known property of scale-free networks is their resilience against random failure, a robustness caused by the presence of hubs (nodes with very high degree). On the other hand, these hubs may cause the network to be very vulnerable when a targeted attack is performed. In the limit of infinitely large networks, the network is said to be robust* when even after removing an arbitrary fraction of the edges, there is still a nonzero probability that two randomly chosen nodes are part of a connected cluster. On the other hand, when removing edges from a fragile network, a point will be reached when the giant cluster, the one with a size comparable to the network size, is destroyed; this very point is called the percolation threshold. The percolation transition is a genuine phase transition and is normally of second order so that critical exponents can be properly defined [@goltsev; @schwartz; @cohen3]. Since the first studies of percolation on scale-free networks [@cohen], a lot of work has been done on node percolation [@callaway; @cohen; @cohen2; @newman3; @gallos; @cohen3], bond percolation [@newman; @callaway; @allard; @dallasta], percolation on multitype networks [@allard; @newman; @newman2; @dallasta], clustered networks [@serrano2; @serrano3], correlated networks [@newman3; @goltsev; @serrano3; @serrano4], directed networks [@schwartz; @serrano4; @newman2], degree-dependent edge percolation [@newman; @dallasta; @wu] and degree-dependent node percolation [@gallos]. The percolation transition has many connections to real systems. For example, it can be related to disease propagation models [@newman; @kenah; @hastings], [^1]. In this analogy, the infection of an individual is represented by the activation of a node of the (social) network. When a giant cluster of active nodes emerges, an epidemic is established. Disease propagation on such networks can be efficiently suppressed by selective vaccination, depending for example on the connectedness of each node. An alternative interpretation of a network with a certain fraction of deactivated edges is in terms of a transport network in which the edges transmit data or deliverables between nodes with a certain transmission probability. This probability depends in general on the degrees of the connected nodes. For example, communication with the highly connected hubs on the internet is in general more efficient. It is, however, also possible that nodes with more edges are less robust. Indeed, in more social terms, friendships involving people which have many acquaintances are more likely to end than friendships between people with few connections. Or, as another example, traffic on a network induces high loads on highly connected nodes which in turn makes them more vulnerable to failure. Clearly, the resilience of an edge in a real network may depend strongly on the degrees of the nodes it connects. We study the properties of a network after biased or degree-dependent edge removal. More specifically, we consider networks in which the edge between nodes $i$ and $j$ is [retained]{} with a probability proportional to its weight $$\begin{aligned} \label{weights} w_{ij}=(k_ik_j)^{-\alpha},\end{aligned}$$ where $k_i$ and $k_j$ denote the degrees of nodes $i$ and $j$ respectively, and $\alpha$ is the “bias exponent”. By tuning $\alpha$, we can explore three qualitatively different regimes: random failure ($\alpha=0$), the attack of edges connected to hubs ($\alpha>0$), and the depreciation of edges between the least connected nodes ($\alpha<0$). Henceforth, we call the regime $\alpha>0$ “centrally biased” (CB). The converse regime, $\alpha<0$, is termed “peripherally biased” (PB). A degree dependence similar to that in Eq. has already been considered in Refs. and where Ising spin couplings $J_{ij}$ on scale-free networks were taken to be proportional to $w_{ij}$. The motivation for introducing degree-dependent couplings was the observation that for $\gamma \le 3$ the system is always “ordered" (critical temperature $T_c = \infty$) due to the dominance of the hubs. However, degree-dependent couplings make it possible to compensate high degree with weak interaction (assuming $\alpha >0$) so that the effect of the hubs can be neutralized. In doing so, it was discovered that a network with “interaction exponent" $\alpha$ and degree exponent $\gamma$ has the same critical behavior as a network with interaction exponent zero (uniform couplings $J$) and degree exponent $$\begin{aligned} \label{mappy} \overline{\gamma}=\frac{\gamma-\alpha}{1-\alpha}.\end{aligned}$$ In this way it was possible to “trade interactions for topology" and study the rich mean-field critical behaviour, with nonuniversal critical exponents depending on $\gamma$ [@dorogovtsev2], simply by varying $\alpha$ in a given network with fixed $\gamma$. The same exponent mapping will be recovered in this work in the following sense: at percolation the properties of a network with bias exponent $\alpha$ and degree exponent $\gamma$ are the same as those of a network with bias exponent zero and degree exponent $\overline{\gamma}$, [or]{} degree exponent $\gamma$, depending on conditions that will be specified. The significance and potential usefulness of biased depreciation of a network is now becoming more clear. Indeed, it has been shown that networks with $\gamma > 3$ are [fragile]{} under random failure, while networks with $\gamma < 3$ are [robust]{} under random removal of edges or nodes [@cohen]. If it should turn out, and under certain conditions this is what we find, that the depreciated network behaves as one in which $\gamma$ is replaced by $\overline{\gamma}$, it becomes possible to control the robustness or fragility of a network systematically by tuning the bias exponent $\alpha$. In other words, a network that is robust under random failure may turn out to be fragile under biased failure, and the other way round. Note that applying bias does not presuppose global knowledge about the network (location of the hubs, ...) but only requires local information on nodes and their degree. The exponent equality can be intuitively understood from the following heuristic argument, which is safe to use provided $\alpha > 0$ and $k$ is sufficiently large. Using Eq. , one can anticipate that after depreciation of the network, a node with degree $k$ will, on average, have a new degree $\overline{k}$ proportional to $k^{1-\alpha}$. Since all nodes remain in place during the depreciation process, the original degree distribution $P(k)$ changes into a new distribution $\overline{P}(\overline{k})$ after depreciation, the relation between them being: $$\begin{aligned} \label{mappy} P(k)dk=\overline{P}(\overline{k})d\overline{k}.\end{aligned}$$ Using $\overline{k}\propto k^{1-\alpha}$, one directly infers that indeed $\overline{P}(\overline{k})\propto \overline{k}^{-\overline{\gamma}}$ and the network after depreciation thus acquires degree exponent $\overline{\gamma}$ and the corresponding percolation properties. A more rigorous proof of this plausible expectation is given in the Appendix. The paper is organized as follows: In Secs. \[sequentialapproach\] and \[simultaneousapproach\] we introduce random scale-free networks and present two distinct approaches by means of which a network can be reconstructed in a degree-dependent manner. Based on these schemes, we focus in Sec. \[characteristics\] on the degree distribution and the degree-degree correlations of the network after (partial) reconstruction. The percolation threshold is then extracted from these degree characteristics in Sec. \[percolationtrigger\]. The theory of generating functions for degree-dependent percolation on random networks will be extensively presented in Sec. \[generousfunctions\]. In Sec. \[kastfort\] the equivalence of our model with the Potts model is elaborated and using this equivalence and finite-size scaling theory, we arrive at the critical exponents for the percolation transition in Sec. \[critexponents\]. Finally, our results are extensively compared to simulational results in Sec. \[comparison\]. Our conclusions are presented in Sect. \[conclusions\]. A summary of part of the results presented here has been reported in Ref. . Degree-Dependent Percolation On Random Graphs {#sec_intro} ============================================= This section concerns (maximally) random scale-free networks. These are networks generated with the so-called configuration model, which assumes that the degrees of the nodes in the network are distributed according to a probability $P(k)$ which is taken to be the power law: $$\begin{aligned} \label{nodedistrib} P(k) = C k^{-\gamma},\end{aligned}$$ for values of $k$ between the minimal and maximal degrees $m$ and $K$, respectively. $C$ is the normalization constant. In order to ensure a finite mean degree we take $\gamma > 2$. The graph is then completed by connecting the stubs emanating from all nodes. The probability $P_n(k)$ that a randomly chosen edge leads to a node of degree $k$ must therefore be: $$\begin{aligned} \label{nearestn} P_n(k) = \frac{kP(k)}{{\langle k \rangle}},\end{aligned}$$ where $\langle \cdot \rangle$ denotes the average over the nodes, obtained using probability distribution $P(k)$. The probability distribution $P_n$ is also called the nearest-neighbor degree distribution. Random networks are constructed by connecting the earlier mentioned stubs randomly. We do not allow self-connections nor multiple connections between nodes and use the method proposed in Ref. to avoid degree-degree correlations in the network. To quantify degree-degree correlations, let us introduce the probability $P(k,q)$ that nodes of degree $k$ and $q$ are connected. If no correlations are present, $P(k,q)$ reduces to: $$\begin{aligned} P(k,q)=P_n(k)P_n(q)=\frac{kqP(k)P(q)}{{\langle k \rangle}^2}.\end{aligned}$$ Below and close to the critical point, large and random networks can locally be treated as trees and loops are sparse so that their effect can, to a good approximation, be ignored. The local tree-like structure will be used in Sec. \[generousfunctions\] when the generating functions method is introduced. We continue with presenting two distinct depreciation methods to study degree-dependent percolation. The statistical edge properties are now being considered, which will allow us in Sec. \[characteristics\] to obtain the statistics of nodal properties. Sequential Approach {#sequentialapproach} ------------------- Our first method, which we call the sequential approach, starts from a random network with all $N$ node degrees distributed according to the degree distribution $P(k)$. Initially all edges are removed and we aim at reintroducing a fraction $f$ of the total number of edges $N_e=\langle k\rangle N/2$. This is achieved by activating one edge in each time step $t$. Consequently, the probability that the edge between nodes $i$ and $j$ is activated is $w_{ij}/Z_t$ where $Z_t$ is the sum of weights $w_{ij}$ of all non-activated edges after $t-1$ steps. Thus, the probability $\rho_{ij}(f)$ that an edge between nodes $i$ and $j$ is again present after the reinclusion of a fraction $f$ of the edges, is [@cohen4]: $$\begin{aligned} \label{john} \rho_{ij}(f) = 1 - \prod_{t=1}^{fN_e} \left(1 - \frac{w_{ij}}{Z_t} \right).\end{aligned}$$ For sufficiently large networks $w_{ij}/ Z_t$ is typically small compared to one and Eq. is well approximated by: $$\begin{aligned} \label{timothy} \rho_{ij}(f) \approx 1 - e^{-D_fw_{ij}},\end{aligned}$$ with the positive parameter $D_f=\sum_{t=1}^{fN_e} Z_t^{-1}$. It can be argued that for a sufficiently narrow distribution of the weights [^2]: $$\begin{aligned} \label{justlinear} Z_t=\langle w\rangle_e(N_e -t+1),\end{aligned}$$ where $\langle\cdot \rangle_e$ denotes the average over all edges. The following property is readily derived, $$\begin{aligned} \label{wproperty} \sqrt{\langle w\rangle_e} = \frac{\langle k^{1-\alpha}\rangle}{\langle k \rangle} = \frac{\gamma -2}{\gamma -2 + \alpha} m^{-\alpha}.\end{aligned}$$ Using Eq. , $D_f$ can be determined, such that for large $N_e$, $$\begin{aligned} \label{dfexplicit} D_f=-\ln\left[1-f\right]/\langle w\rangle_e\end{aligned}$$ and thus [@abramowitz]: $$\begin{aligned} \label{snoopy} \rho_{ij}(f) =1-\left[1-f\right]^{w_{ij}/\langle w\rangle_e}\end{aligned}$$ It is instructive to consider a few asymptotic regimes of Eq. . First, in the case $\alpha=0$, one recovers the expression for degree-independent percolation $\rho_{ij}=f$ as expected. Secondly, for arbitrary $\alpha$, we can distinguish the dilute limit and the dense limit in terms of $f$, and find: $$\begin{aligned} \rho_{ij}&\sim fw_{ij}/\langle w\rangle_e\text{ when }f \rightarrow 0,\label{tim}\\ \rho_{ij}& \sim 1\text{ when }f \rightarrow 1 .\end{aligned}$$ Thirdly, when $w_{ij}/\langle w\rangle_e\ll [-\ln\left(1-f\right)]^{-1}$: $$\begin{aligned} \label{beammeup} \rho_{ij}\sim -\ln\left(1-f\right)w_{ij}/\langle w\rangle_e.\end{aligned}$$ We proceed by defining the marginal distribution $\rho_k$ as the mean probability that an edge connected to a node with degree $k$ is present in the network after reconstruction. Thus $$\begin{aligned} \label{marginal} \rho_k= \sum_{q=m}^K P_n(q)\rho_{kq},\end{aligned}$$ where $\rho_{kq}=1-e^{-D_fw_{kq}}$. A good analytic approximation to $\rho_k$ can be obtained by substituting $\rho_{kq}$ into Eq. , considering an integral instead of a sum, taking the macroscopic limit $K \rightarrow \infty$, and expanding the exponential, $$\begin{aligned} \rho_k = 1 - \sum^{\infty}_{n=0}\frac{(-)^n (D_f \,m^{-\alpha} k^{-\alpha})^n}{(1+ n\alpha/(\gamma-2))\,n!}.\end{aligned}$$ Alternatively, this result follows straightforwardly from the fact that the marginal distribution involves the incomplete Gamma function [@abramowitz]. The usefulness of this explicit form can best be appreciated by first considering the range $0<\alpha \ll \gamma -2$, for which we obtain the simple analytic result $$\begin{aligned} \label{ghinzu} \rho_k &\sim & 1 - \exp\left\{- \frac{\gamma -2}{\gamma -2 + \alpha} D_f m^{-\alpha} k^{-\alpha}\right\}\nonumber \\ & =& 1 - \exp\left\{- D_f \sqrt{\langle w\rangle_e}\,k^{-\alpha}\right\}.\end{aligned}$$ Although this result is strictly only valid for the specified range of $\alpha$ specified above, numerical inspection shows that it is a rather good approximation to the integral representation of the sum Eq. for a wider range of $\alpha$, [including negative values]{}. In fact, the result is useful in the entire interval of our interest $\alpha \in [2-\gamma, 1]$. Using the previously obtained approximation to $D_f$, Eq. , it can be further simplified to $$\begin{aligned} \label{statement} \rho_k \approx 1 - [1-f]^{k^{-\alpha}/\sqrt{\langle w\rangle_e}}.\end{aligned}$$ Based on the asymptotic regimes of $\rho_{ij}$ it is also possible to extract the behavior of $\rho_k$. When $f\rightarrow 0$, we may use Eq. : $$\begin{aligned} \label{bob} \rho_{k}&\sim k^{-\alpha}f/\sqrt{\langle w\rangle_e}\end{aligned}$$ On the other hand, when $\alpha>0$ and $k\gg k_{\times}$, where the cross-over value for $k$ is given by $k_{\times} \equiv D_f^{1/\alpha}/m$, we get: $$\begin{aligned} \label{scalubl} \rho_{k}\sim k^{-\alpha}D_f \sqrt{\langle w\rangle_e}\approx -k^{-\alpha}\ln\left(1-f\right)/ \sqrt{\langle w\rangle_e}.\end{aligned}$$ Simultaneous Approach {#simultaneousapproach} --------------------- As an alternative to the sequential approach we introduce now the simultaneous approach. Again we start from a fully depreciated uncorrelated network with degree distribution $P(k)$. We then visit each edge (between nodes $i$ and $j$) once and activate this edge with probability $$\begin{aligned} \label{simultaneousexpression} \rho_{ij}=f w_{ij}/\langle w\rangle_e.\end{aligned}$$ In contrast to the sequential approach, $\rho_{ij}$ is now history-independent. Note also that $\langle \rho_{ij}\rangle_e=f$ as it must be. For the marginal distribution $\rho_{k}$ one finds: $$\begin{aligned} \label{marginalpaparel} \rho_{k}= k^{-\alpha}f/\sqrt{\langle w\rangle_e},\end{aligned}$$ which satisfies $f\rho_{kq}=\rho_{k}\rho_{q}$ and is the same as in the $f\rightarrow 0$ limit of the sequential approach (Eq. ). However, for each value of $k$ and $q$, the probability $\rho_{kq}$ must be less than or equal to one. This means that Eq. is only well-defined for values of $f$ for which $$\begin{aligned} \label{fu} f< f_u\equiv\left(\frac{\langle k^{1-\alpha}\rangle}{\langle k\rangle}\right)^2\times (\text{min}(m^{\alpha},K^{\alpha}))^2.\end{aligned}$$ It can be calculated that, in the macroscopic limit, the rhs of Eq. vanishes when $\alpha<0$ and therefore the simultaneous approach is only meaningful for positive $\alpha$ and provided $$\begin{aligned} \label{fff} f<\left(\frac{\gamma-2}{\gamma-2+\alpha}\right)^2.\end{aligned}$$ To reach fractions above this limit in the simulations, we [iterate]{} the simultaneous approach. The first iteration involves the usual simultaneous approach with $f=f_u$; the second iteration is initialized by considering a new network consisting of all edges that have not been reintroduced during the first sweep. For that network one calculates the probabilities $w_{ij}/Z_2$ and a new value of $f_u$, which is the minimum of the set $\{\langle w\rangle_{e}/w_{ij}\}$ where also the average is only over edges of the new network. One then applies the simultaneous approach until the new $f_u$ is reached, after which a third iteration can be initialized if necessary. Such iterations, however, introduce correlations and history dependence. Note that the sequential approach can be seen as an extreme case of an iterated simultaneous approach in which only one edge is reconstructed in each iteration. Degree Distribution and Correlations of the Reconstructed Network {#characteristics} ----------------------------------------------------------------- We now seek to obtain the degree distribution and characterize degree-degree correlations for the network after reconstruction. The following is valid for both the simultaneous and sequential approaches. Henceforth we adopt the convention that an overbar indicates quantities in the diluted, or depreciated, network. For the node degree distribution $\overline{P}(\overline{k})$ and the degree-degree correlations embodied in $\overline{P}(\overline{k},\overline{q})$ of the depreciated network we can write $$\begin{aligned} \overline{P}(\overline{k})&=\sum_{k=\overline{k}}^KP(A_{k}\wedge B_{k\rightarrow \overline{k}}),\\ \overline{P}(\overline{k},\overline{q})&=\sum_{k=\overline{k}}^K\sum_{q=\overline{q}}^KP(C_{qk}\wedge B_{q\rightarrow \overline{q}}\wedge B_{k\rightarrow \overline{k}}\wedge D)/f.\label{fritz}\end{aligned}$$ Here we introduced the notation, for events A-D, - $A_{k}$: a randomly chosen node of the original network has degree $k$. - $B_{k\rightarrow \overline{k}}$: the degree of a node goes from $k$ in the original to $\overline{k}$ in the depreciated network. - $C_{qk}$: the nodes connected by a randomly chosen edge of the original network have degrees $q$ and $k$. - $D$: the chosen edge has not been removed from the original network. For the node degree distribution, one readily finds $$\begin{aligned} \label{binomial} \overline{P}(\overline{k})&= \sum_{k=\overline{k}}^K P(k)\left( \begin{array}{c} k \\ \overline{k} \end{array} \right) \rho_{k}^{\overline{k}}(1-\rho_{k})^{k-\overline{k}}.\end{aligned}$$ For large values of $k$, i.e., $k \gg k_{\times}$, and $\alpha>0$, the probability of retaining a node of degree $k$ falls off as $\rho_k\propto k^{-\alpha}$; this is valid using the sequential approach (see Eq. ), as well as the simultaneous one (see Eq. ). Substituting this into Eq. and approximating the binomial distribution in Eq. by a normal distribution, one arrives at: $$\begin{aligned} \label{propto} \overline{P}(\overline{k})\propto \overline{k}^{-\overline{\gamma}} \text{ for } \overline{k}\rightarrow \infty.\end{aligned}$$ where $\overline{\gamma}$ is defined in Eq. . This result, which is proven in the Appendix, confirms the validity of the expectation raised in the Introduction. Note that in case $\alpha=0$, $\overline{P}(\overline{k})\propto \overline{k}^{-\gamma}$ as it must be. We introduce now averaging over nodes of the reconstructed network: $$\begin{aligned} \ll \cdot\gg=\sum_{\overline{k}=0}^K \overline{P}(\overline{k})\,\cdot.\end{aligned}$$ For further purposes, we calculate now the first and second moment of $\overline{P}(\overline{k})$ in terms of the moments of $P(k)$: \[newmoments\] $$\begin{aligned} \ll \overline{k}\gg&=\langle k\rho_k\rangle=f\langle k\rangle,\\ \ll \overline{k}^2\gg&=\langle k\rho_k(k\rho_k-\rho_k+1)\rangle.\end{aligned}$$ The degree-degree correlations are embodied in Eq. . This function can be further worked out to yield: $$\begin{aligned} \overline{P}(\overline{k},\overline{q})=&\sum_{k=\overline{k}}^K\sum_{q=\overline{q}}^KP_n(q)P_n(k)\rho_{kq}\nonumber\\ &\times\left( \begin{array}{c} q-1\\ \overline{q}-1 \end{array} \right) \rho_{q}^{\overline{q}-1}(1-\rho_{q})^{q-\overline{q}}\nonumber\\ &\times \left( \begin{array}{c} k-1\\ \overline{k}-1 \end{array} \right) \rho_{k}^{\overline{k}-1}(1-\rho_{k})^{k-\overline{k}}/f.\end{aligned}$$ This can be reduced to: $$\begin{aligned} \label{correlated} \overline{P}(\overline{k},\overline{q})=&\overline{k}\overline{q}\sum_{k=\overline{k}}^K\sum_{q= \overline{q}}^K\frac{P(q)P(k)}{(f\langle k\rangle)^2} P(B_{q\rightarrow \overline{q}})P(B_{k\rightarrow \overline{k}})\frac{f\rho_{kq}}{\rho_{k}\rho_{q}}.\end{aligned}$$ Eq. expresses the degree-degree correlations of a network after degree-dependent depreciation of a fully uncorrelated network. The question that can now be raised is when the depreciated network is also free of correlations, or, when is $ \overline{P}(\overline{k},\overline{q})=\overline{P}_n(\overline{k})\overline{P}_n(\overline{q})$? It is readily checked that this is true provided $$\begin{aligned} \label{correlationfreecriterion} f\rho_{kq}=\rho_{k}\rho_{q}.\end{aligned}$$ As Eq. is valid for the simultaneous approach (see Eqs. and ), no correlations appear in the reconstructed network (after a single iteration). For the sequential approach, on the other hand, Eq. is generally not satisfied and the reconstructed network will be correlated. The following limiting case of the sequential approach is interesting: take $\alpha$ positive and consider an edge between two nodes of large degrees $k$ and $q$ such that $k\gg k_{\times} $ and $q\gg k_{\times}$. We may then substitute Eqs. and into Eq. . One soon arrives at the result: $$\begin{aligned} \label{disassort} \frac{\overline{P}(\overline{k},\overline{q})}{\overline{P}_n(\overline{k})\overline{P}_n(\overline{q})}=-\frac{f}{\ln\left(1-f\right)}.\end{aligned}$$ Since the rhs is smaller than one, this demonstrates that the sequential approach causes disassortative mixing in the depreciated network when $\alpha>0$. In other words, nodes with large degrees tend to be connected to nodes with small degrees and vice versa. Using simulations, we will present evidence in Sect. \[comparison\] that such correlations are introduced. Finally, note also that Eq. reduces to the correct nearest-neighbor degree distribution upon summing over $\overline{q}$: $$\begin{aligned} \label{nearestneighbor} \overline{P}_n(\overline{k})=\frac{\overline{k}\,\overline{P}(\overline{k})}{f\langle k\rangle}.\end{aligned}$$ Percolation Threshold for central bias {#percolationtrigger} ====================================== Here we focus solely on centrally biased (CB) depreciation ($\alpha>0$) using the simultaneous approach. One may wonder what happens if centrally biased (CB) depreciation is applied to a robust network with $\gamma<3$ such that the edges between and emanating from hubs are preferentially removed. Since, in that case, $\overline{\gamma} > \gamma$, one may speculate that a robust network may turn fragile and that the threshold for this to occur is $\overline{\gamma}=3$ instead of the threshold $\gamma=3$ valid for degree-independent percolation. We will address this question further and conclude that it is indeed so. On the other hand, if we start from a fragile network ($\gamma >3$) and apply CB, it is logical that the net remains fragile. Upon removing edges linked to hubs with a larger probability, we are more likely to destroy the coherence of the network. The question can then still be posed how much the percolation threshold of the reconstructed network is shifted. Our first task now is to calculate the critical fraction at which the network becomes disconnected. According to Molloy and Reed [@molloy], the critical fraction of a random network can be found by looking at the average nearest-neighbor distribution. If, upon following a random edge, the attained node has more than two neighbors, the network is said to be percolating, that is, a giant cluster will be present in the network. Note that this criterion is exact for the simultaneous but not for the sequential approach, due to the appearance of degree-degree correlations. In the reconstructed network, the Molloy-Reed criterion reads: $$\begin{aligned} \label{molloyreed} 1=\frac{\ll \overline{k}(\overline{k}-1)\gg}{\ll \overline{k}\gg},\end{aligned}$$ or equivalently, using Eq. : $$\begin{aligned} \label{starsturntodust} 2\langle k\rho_k\rangle=\langle k\rho_k(k\rho_k-\rho_k+1)\rangle.\end{aligned}$$ Using Eq. for the simultaneous approach, we find the following expression for the critical fraction $f_c$ at percolation [@moreira]: $$\begin{aligned} \label{criterion584} f_c=\frac{\langle k^{1-\alpha}\rangle^2}{\langle k\rangle(\langle k^{2-2\alpha}\rangle-\langle k^{1-2\alpha}\rangle)}.\end{aligned}$$ For unbiased depreciation of the network ($\alpha=0$) this last expression reduces to the well-known formula of random percolation on random networks [@cohen]. Eq. allows us now to find out whether the network is robust, or in other words, whether $f_c\rightarrow 0$. This vanishing occurs, in the macroscopic limit, when the term $\langle k^{2-2\alpha}\rangle$ diverges and therefore: $$\begin{aligned} \overline{\gamma} \begin{cases} >3\text{: the network is fragile,} \\ <3\text{: the network is robust}. \end{cases}\end{aligned}$$ The scaling relation of $f_c$ as a function of the network size can be found when $\overline{\gamma} <3$; using $N\propto K^{\gamma-1}$, one finds: $$\begin{aligned} f_c \propto N^{\frac{\overline{\gamma}-3}{\overline{\gamma}-1}}. \label{fcN}\end{aligned}$$ Whether or not a network is robust for the degree-dependent attack is thus not solely a property of the network. Also the exponent $\alpha$ plays a crucial role in the arguments and its effect can be absorbed by using the exponent $\overline{\gamma}$ instead of the exponent before dilution, $\gamma$. In Sect. \[critexponents\] we will take a closer look at the regime around the percolation threshold and we will find that the same mapping from $\gamma$ to $\overline{\gamma}$ is valid. Generating Functions Approach {#generousfunctions} ============================= We now introduce the generating functions approach for degree-dependent percolation. By this method, certain properties of the finite clusters in the network are easily obtained; this in turn allows to draw conclusions about the giant cluster. The method is exact if loops in the network can be ignored; since in the macroscopic limit, the average loop sizes in the finite clusters diverge [@bianconi], the method turns out to be exact. This will be apparent in Sect. \[comparison\] when comparing the analytical results with simulations. Introduction {#randomnetworks} ------------ Generating functions are used in a wide branch of mathematical problems concerning series [@wilf]. A generating function of a series is the power series which has as coefficients the elements of the series. Applied to the context of percolation problems, this series is taken to be that of the discrete probability distributions characterizing the network under consideration [@essam; @newman]. We explain first the general formalism while closely following the approach of Newman [@newman], which we adapt for degree-dependent edge percolation [^3]. The most fundamental generating function is the one that generates the degree distribution of the network $$\begin{aligned} G_0(h) = \sum_{k=m}^K P(k) e^{-hk}.\label{G0}\end{aligned}$$ We also define the generating function for the distribution of residual edges of a node reached upon following a random edge: $$\begin{aligned} G_1(h) = \sum_{k=m}^K P_n(k)e^{-h(k-1)}. \label{G0}\end{aligned}$$ The exponent of $e^{-h}$ is $k-1$ because the edge which is used to reach the node is not counted. There is a threefold advantage in working with the generating functions instead of working with the degree distribution itself. First, moments can be obtained easily from the generating functions. For instance, the average degree is given by $$\begin{aligned} {\langle k \rangle}= \sum_{k=m}^K kP(k) = -G_0'(0),\end{aligned}$$ where $G_0'$ denotes the derivative with respect to $h$. Higher moments can be obtained with higher-order derivatives. Secondly, we can benefit from the so-called powers property of generating functions: if the distribution of a property $k$ of an object is generated by a function $G(h)$, then the generating function of the sum of $n$ independent realizations of $k$ is $G(h)^n$. For instance, if we randomly choose $n$ nodes in our network, the distribution of the sum of the degrees is generated by $G_0(h)^n$. Thirdly, the use of generating functions will allow us in Sect. \[kastfort\] to highlight the equivalence with the $q\rightarrow 1$ limit of the $q$-state Potts model where the parameter $h$ will play the role of the magnetic field. Self-consistent equations {#selfconsistent} ------------------------- We can now define the equivalents of $G_0(h)$ and $G_1(h)$ for the network after dilution as $F_0(h)$ and $F_1(h)$: \[genererendefunctiesrule\] $$\begin{aligned} F_0(h)&=\sum_{\overline{k}=0}^K \overline{P}(\overline{k}) e^{-h\overline{k}},\\ F_1(h)&=\sum_{\overline{k}=0}^K \overline{P}_n(\overline{k})e^{-h(\overline{k}-1)}.\end{aligned}$$ Note that the minimal degree in the network after dilution is zero instead of $m$. Eq. can be worked out further using Eqs. and : $$\begin{aligned} F_0(h) &= \sum_{k=m}^K P(k) (1-\rho_k + e^{-h}\rho_k)^k\label{F0},\\ F_1(h) &= \sum_{k=m}^K \frac{\rho_k P_n(k) }{f}(1-\rho_k + e^{-h}\rho_k)^{k-1} \label{F1}.\end{aligned}$$ The most interesting quantity for us is the size distribution of the finite clusters, the generating function of which can be readily derived using $F_0$ and $F_1$. Let $H_0$ denote the generating function for the probability that a randomly chosen node belongs to a connected cluster of a given (finite) size. Furthermore, let $H_1$ be the generating function for the probability that upon following a randomly chosen edge to one end, a cluster of a given (finite) size is reached. If the network can be treated as a tree, these generating functions satisfy the following self-consistency equations [^4]: \[H\] $$\begin{aligned} H_1(h) &=& e^{-h}F_1[H_1(h)],\label{H0}\\ H_0(h) &=& e^{-h}F_0[H_1(h)].\label{H1}\end{aligned}$$ Here the function $F_{0,1}[H_1(h)]$ denotes the function $F_{0,1}$ in which $e^{-h}$ is replaced by $H_1(h)$. The proof of these relations relies on the aforementioned powers property and is expounded in Ref. . The percolation threshold can now be derived with the aid of these functions. Several macroscopic quantities can be easily identified in the depreciated network [@dorogovtsev2]. For example, we define $\mathcal{P}_{\infty}$ as the probability that a node belongs to the giant cluster, $\mathcal{L}_{\infty}$ as the edge probability for being in the giant cluster and $\mathcal{S}$ as the average cluster size of finite clusters: \[richard\] $$\begin{aligned} &\mathcal{P}_{\infty}=1-H_0(0),\\ &\mathcal{L}_{\infty}=1-(H_1(0))^2,\\ &\mathcal{S}=-H_0'(0).\label{timothy2}\end{aligned}$$ Moreover, the degree distribution of nodes in the giant cluster varies as $$\begin{aligned} \overline{P}_{gc}(\overline{k})\propto (1-(H_1(0))^{\overline{k}} )\overline{P}(\overline{k}),\end{aligned}$$ from which it follows that the degree distribution of the finite clusters varies as $\overline{P}_{fc}(\overline{k})\propto (H_1(0))^{\overline{k}} \overline{P}(\overline{k})$. In case there are both finite clusters and a giant cluster, the degree distributions have the asymptotic behavior ($\overline{k}\rightarrow \infty$): $$\begin{aligned} &\overline{P}_{gc}(\overline{k})\sim \overline k\,^{-\overline{\gamma}},\\ &\overline{P}_{fc}(\overline{k})\sim e^{-\overline{k}/\lambda},\end{aligned}$$ with $\lambda=-\ln(H_1(0))$. In other words, in the presence of a giant cluster, only the degree distribution of the giant cluster falls off with a power law with exponent $\overline{\gamma}$. The average cluster size in the diluted network, on the other hand, can be further worked out by differentiating Eqs. with respect to $x$: $$\begin{aligned} \mathcal{S} = 1 + \frac{f\langle k\rangle}{1 + F_1'(0)}. \label{avcl}\end{aligned}$$ Hence the average cluster size diverges when $$\begin{aligned} 1 = -F_1'(0).\end{aligned}$$ This is yet another way of writing the Molloy-Reed criterium Eq. . Full Derivation of Self-consistent Equations {#fullderivation} -------------------------------------------- We prove now that the self-consistent Eqs. are only valid in case no correlations are introduced in the reconstructed network, or, when $f\rho_{kq}=\rho_k\rho_q$, as is valid for the simultaneous method only. Here we will give a precise derivation of the self-consistent Eqs. , thereby taking into account the degree-dependence of the functions $H_1$. Let us first look at the generating function $\hat{H}_1^{q\rightarrow k}(h)$ for the probability that an edge, which connects nodes of degree $q$ and $k$, branches out in a cluster of a given edge number along the node of degree $k$. It is readily derived that $\hat{H}_1^{q\rightarrow k}$ satisfies the equation: $$\begin{aligned} \label{hoeplala} \hat{H}_1^{q\rightarrow k}(h)=e^{-h} \left(1+\sum_{\hat{k}=m}^K P_n(\hat{k})\rho_{k\hat{k}}[\hat{H}_1^{k\rightarrow \hat{k}}(h)-1]\right)^{k-1}.\end{aligned}$$ We proceed by defining $H_1^q(h)=\sum_{k} P_n(k) \rho_{qk}\hat{H}_1^{q\rightarrow k}(h)/\rho_q$, such that we arrive at a set of self-consistent equations for each value of $q$: $$\begin{aligned} \label{exactH1} H_1^q(h)&=e^{-h}\sum_{k=m}^K\frac{P_n(k) \rho_{qk}}{\rho_q}\left(1+\rho_k[H_1^k(h)-1]\right)^{k-1}.\end{aligned}$$ Note that, as derived in Sec. \[characteristics\], no correlations are induced during depreciation when $f\rho_{kq}=\rho_k\rho_q$. In that case, this equation reduces to Eq. . After solving Eq. with respect to $H_1^k$ for all values of $k$, we can also calculate: $$\begin{aligned} H_0(h)&=\sum_{k=m}^K e^{-h}P(k)\left[1+\rho_k[H_1^k(h)-1]\right]^{k}.\end{aligned}$$ Again, this expression reduces to Eq. in case of correlation-free depreciation when $H_1^k$ is independent of $k$. Below the percolation transition, a trivial solution exists: $H_1^{k}(0)=1$. This solution, however, turns unstable at the percolation threshold. The threshold value may be derived by linearization of $H_1^{k}(0)$ around its equilibrium value: $H_1^{k}(0)=1-\varepsilon_{k}$ with $\varepsilon_{k}\ll 1$. The percolation criterion is then: $$\begin{aligned} \label{stelsel} \varepsilon_{q}&=\sum_{k=m}^K\frac{P_n(k) \rho_{qk}\rho_{k}}{\rho_{q}}(k-1)\varepsilon_{k}.\end{aligned}$$ Again, in absence of correlations in the network, that is, when the criterion $f\rho_{qk}=\rho_k\rho_q$ is satisfied, this reduces to the earlier encountered Molloy-Reed criterion of Eq. . Eq. is the criterion for the percolation threshold for correlated systems, such as the one created using the sequential method. However, solving Eq. to obtain $f_c$ constitutes a rather difficult task. Original and Depreciated Network {#upgrade} -------------------------------- We show now that another approach exists by which one easily derives the self-consistent equations characterizing the network. This method is closer to the one followed in other works. Let us call $R^{i\rightarrow j}$ the probability that an edge in the network does not lead to a vertex connected via the remaining edges to the giant component (infinite cluster) and $\rho_{ij}$ the probability that the edge between nodes $i$ and $j$ is active. Then, following the edge along node $j$, one finds: $$\begin{aligned} R^{i\rightarrow j}=1-\rho_{ij}+\rho_{ij}\prod_{z=1\ldots k_j-1} R^{j\rightarrow z}.\end{aligned}$$ This type of equation was already obtained for the $q\rightarrow 1$ limit of the $q$-state Potts model [@dorogovtsev4]. Using the tree approximation, we can rewrite everything as a function of the degrees of the nodes: $$\begin{aligned} \label{selfish} R^{q\rightarrow k}=1-\rho_{qk}+\rho_{qk}\left[\sum_{\hat{k}=m}^K P_n(\hat{k})R^{k\rightarrow \hat{k}}\right]^{k-1}.\end{aligned}$$ This self-consistent set of equations is equivalent to the ones that we obtained in Eq. . Indeed, after the transformation $R^{q\rightarrow k}-1=\rho_{qk}(\hat{R}^{q\rightarrow k}-1)$, we find: $$\begin{aligned} \label{selfish2} \hat{R}^{q\rightarrow k}=\left[1+\sum_{\hat{k}=m}^K P_n(\hat{k})\rho_{k \hat{k}}(\hat{R}^{k\rightarrow \hat{k}}-1)\right]^{k-1},\end{aligned}$$ which is exactly the same as Eq. for $h=0$. The purpose of this derivation is to show that our self-consistent equation is in agreement with Eq. . For degree-independent $R$ and $\rho$, an equation similar to Eq. appears frequently in the literature. The difference between Eq. and Eq. is that $\hat{H}_1$ is normalized with respect to the depreciated network, whereas $\hat{R}$ is normalized with respect to the original network. Equivalence with the Potts model {#kastfort} ================================ There exists an equivalence between edge percolation and the $q\rightarrow 1$ limit of the $q$-state Potts model. This connection was first worked out by Fortuin and Kasteleyn in Ref. . Although initially used for lattice models, the connection was very general and is valid for any network [@fortuin; @wu]. Moreover, their proof can easily be generalized to incorporate edge-dependent coupling constants and edge-dependent removal into the Potts model and the percolation model, respectively. We explain here in more detail this equivalence and reformulate our percolation problem as a spin-like problem which will allow us to derive critical exponents and compare them with the ones obtained for the Potts model. We will also find support for our simple scaling relation using exponent $\overline{\gamma}$ (Eq. ), as was already encountered in studies concerning degree-dependent Ising interactions on scale-free networks [@giuraniuc1; @giuraniuc2]. Note that the Ising model [@leone; @dorogovtsev3] and the Potts model [@igloi; @dorogovtsev4], together with their critical properties, were already studied on scale-free networks. The Potts model can be seen as a generalization of the Ising model in which each site $i$ has a spin $\sigma_i$. In the Potts model, these spins can take $q$ distinct values $0,\dots,q-1$ and the Potts Hamiltonian is: $$\begin{aligned} \mathcal{H} =-\sum_{<ij>}J_{ij} \delta_{\sigma_i,\sigma_j}- hk_BT\sum_i \delta_{\sigma_i,0}.\end{aligned}$$ Here $<ij>$ indicates nearest-neighbor sites $i$ and $j$, $J_{ij}$ is the coupling constant and $\delta$ the Kronecker delta function. Note that the Ising model corresponds to the $q=2$ Potts model. The Fortuin-Kasteleyn theorem states now that the free energy of the $q\rightarrow 1$ limit of the $q$-state Potts model is the same as the “free energy” of the percolating network where the latter is the generating function of the cluster size distribution function $n_s$: $$\begin{aligned} \mathcal{F}(f,h)&=\left\langle \sum_s n_s e^{-hs}\right\rangle.\end{aligned}$$ Here the average is performed over all networks in which the probability to retain the edge between nodes $i$ and $j$ is $\rho_{ij}$. The parameter $\rho_{ij}$ in the percolation problem corresponds in the following way to parameters of the Potts model [@herrmann] [^5]: $$\begin{aligned} \rho_{ij}&\quad\leftrightarrow\quad 1 - e^{-J_{ij}/k_BT}.\end{aligned}$$ From this relation, we can immediately identify the probability $\mathcal{P}_{\infty}$ for a node to be in the infinite cluster and the average cluster size $\mathcal{S}$, earlier introduced in Eq. . As we are interested in the behavior near criticality, we introduce $$\begin{aligned} \epsilon=f-f_c,\end{aligned}$$ and obtain $$\begin{aligned} \mathcal{P}_{\infty}(\epsilon)&=1+\left.\frac{\partial \mathcal{F}}{\partial h}\right\|_{h=0},\\ \mathcal{S}(\epsilon)& =\left.\frac{\partial^2 \mathcal{F}}{\partial h^2}\right\|_{h=0}.\end{aligned}$$ Note also that $\mathcal{F}(\epsilon,0)$ gives the total number of finite clusters. Scaling Theory and Critical Exponents {#critexponents} ===================================== In the following section, we introduce finite-size scaling in order to find critical exponents near the percolation transition. In order to solve the scaling relation, we use a Landau-like theory which we derive from the exact relations . We follow closely the approach presented in Refs. and for finite-size scaling in systems with dimensions above the upper critical dimension. However, we will find that the forms of the Landau-like theories of Refs. and were too limited for studying the percolation transition in case the distribution function has a very fat tail, that is when $2<\gamma<3$. According to finite-size scaling, the free energy $\mathcal{F}$ of a large but finite network with $N$ nodes close to criticality can be written in the general form [@stauffer]: $$\begin{aligned} \label{freeenergy} \mathcal{F}(\epsilon,h)=N^{-1}\mathbb{F}\left(\epsilon N^{1/\nu_{\epsilon}},hN^{1/\nu_{h}}\right),\end{aligned}$$ where $\mathbb{F}$ is a well-behaved function. The variable $\epsilon N^{1/\nu_{\epsilon}}$ originates from the existence of a “correlation number” $N_{\xi}$ (instead of a correlation length) which scales as $N_{\xi}\propto \epsilon^{-\nu_{\epsilon}}$ such that the first variable of $\mathbb{F}$ can be rewritten as $(N/N_{\xi})^{1/\nu_{\epsilon}}$ [@botet1; @botet2]. It is then obvious that close to criticality, the (singular part of the) free energy scales as: \[robert\] $$\begin{aligned} \mathcal{F}(\epsilon,0)&\propto\epsilon^{\nu_{\epsilon}},\\ \mathcal{F}(0,h)&\propto h^{\nu_{h}}.\end{aligned}$$ As a second scaling ansatz, we assume that, in the macroscopic limit, the scaling of the cluster size distribution $n_s$ can be written as [^6]: $$\begin{aligned} \label{clusterdistribution} n_s(\epsilon)=s^{-\tau}\mathbb{G}(\epsilon s^{\sigma}),\end{aligned}$$ where again $\mathbb{G}$ is a well-behaved function. The form of $n_s$ one usually has in mind is $n_s \propto s^{-\tau}e^{-s/s^}$ [@newman], which is essentially a damped power-law with cutoff $s^$, valid for large cluster sizes. At criticality very large clusters arise, caused by the diverging cutoff $s^$ according to $s^\propto \epsilon^{-1/\sigma}$ such that $n_s(0)\sim s^{-\tau}$. Using the analogy with the Potts model, we define the usual critical exponents $\alpha$, $\gamma_p$, $\beta$ and $\delta$ for the percolation problem as: $$\begin{aligned} &\mathcal{F}(\epsilon,0)\sim \epsilon^{2-\alpha},\\ &\mathcal{P}_{\infty}(\epsilon)\sim \epsilon^{\beta},\\ &\mathcal{S}(\epsilon)\sim \epsilon^{-\gamma_p},\\ &\left.\frac{\partial \mathcal{F}}{\partial h}\right\|_{\epsilon=0}+1 \sim h^{1/\delta}.\end{aligned}$$ Using the scaling forms of Eqs. and , standard techniques provide us with exponent relations by which all critical exponents can be related to $\nu_{h}$ and $\nu_{\epsilon}$. One arrives at [@stauffer] \[exponents\] $$\begin{aligned} \beta &=\nu_{\epsilon}(1-\nu_{h}^{-1}),\\ \gamma_p&=\nu_{\epsilon}(2\nu_{h}^{-1}-1),\\ \alpha &=2-2\beta-\gamma_p,\\ \sigma &=(\beta+\gamma_p)^{-1},\\ \tau &=2+\beta(\beta+\gamma_p)^{-1},\\ \delta &=(\beta+\gamma_p)/\beta,\\ \nu_f &=\beta+\gamma_p.\end{aligned}$$ The last exponent $\nu_f$ is related to the usual fractal dimension $d_f$ in the same way that $\nu_{\epsilon}$ is related to the usual dimension $d$ and quantifies how the cluster size $s$ scales with $\epsilon$ close to criticality, i.e., $s \propto \epsilon^{-\nu_f}$. Note that the cutoff size $s^$ scales like $s$, since $\nu_f = 1/\sigma$. The problem we are left with now is to find the scaling exponents $\nu_{h}$ and $\nu_{\epsilon}$ for percolation on scale-free networks. This can be done in an exact way since we know the equation of state from Eq. as a function of the order parameter: $$\begin{aligned} \psi(\epsilon,h)=1-H_1(\epsilon,h).\end{aligned}$$ As we are merely interested in the behavior near the transition where $\epsilon\ll 1$, $h\ll 1$ and $\psi\ll 1$, we can expand Eq. . For the case $\overline{\gamma}>3$, we find the form [@cohen3]: $$\begin{aligned} \label{eos} h =-c_1\epsilon \psi +c_2\psi^2+ ... + c_s\psi^{\overline{\gamma}-2}+\ldots,\end{aligned}$$ in which all $c_i$ as well as the coefficient of the singular term, $c_s$, are positive constants. This equation also follows from minimization of the free energy: $$\begin{aligned} \label{tristan} \mathcal{F}(\epsilon,h)\propto -h\psi-\overline{c}_1\epsilon \psi^2 +\overline{c}_2\psi^3+... +\overline{c}_s\psi^{\overline{\gamma}-1}+\ldots,\end{aligned}$$ with respect to the order parameter $\psi$. We distinguish two cases now. First, when $4<\overline{\gamma}$, we know that $f_c$ is finite and the relevant part of the equation of state for $\psi$ becomes: $$\begin{aligned} h=-c_1\epsilon \psi +c_2\psi^2.\end{aligned}$$ Solving for $\psi$, and substitution into Eq. one then simply finds that the free energy scales as: \[robert\] $$\begin{aligned} \mathcal{F}(\epsilon,0)&\propto \epsilon^{3},\\ \mathcal{F}(0,h)&\propto h^{3/2}.\end{aligned}$$ In other words, when $4<\overline{\gamma}$, we find that $\nu_{\epsilon}=3$ and $\nu_{h}=3/2$. From these two exponents, and using Eq. , we list all other exponents in the last column of Table $1$. Note that, as expected, these exponents agree with the usual mean-field results for percolation [@stauffer; @essam]. Secondly, when $3<\overline{\gamma}<4$, the relevant part of Eq. reduces to $$\begin{aligned} h =-c_1\epsilon \psi +c_s\psi^{\overline{\gamma}-2},\end{aligned}$$ from which follows that \[robert2\] $$\begin{aligned} \mathcal{F}(\epsilon,0)&\sim \epsilon^{\frac{\overline{\gamma}-1}{\overline{\gamma}-3}},\\ \mathcal{F}(0,h)&\sim h^{\frac{\overline{\gamma}-1}{\overline{\gamma}-2}}.\end{aligned}$$ Therefore, we come to: $$\begin{aligned} \nu_{\epsilon}=\frac{\overline{\gamma}-1}{\overline{\gamma}-3}\text{ and } \nu_{h}=\frac{\overline{\gamma}-1}{\overline{\gamma}-2},\end{aligned}$$ and we obtain all other exponents as given in the second column of Table $1$. $2<\overline{\gamma}<3$ $3<\overline{\gamma}<4$ $\overline{\gamma}>4$ ------------ ------------------------------------------------------ ----------------------------------------------------- ----------------------- $\beta$ $\dfrac{1}{3-\overline{\gamma}}$ $\dfrac{1}{\overline{\gamma}-3}$ $1$ $\tau$ $\dfrac{2\overline{\gamma}-3}{\overline{\gamma}-2}$ $\dfrac{2\overline{\gamma}-3}{\overline{\gamma}-2}$ $5/2$ $\sigma$ $\dfrac{3-\overline{\gamma}}{\overline{\gamma}-2}$ $\dfrac{\overline{\gamma}-3}{\overline{\gamma}-2}$ $1/2$ $\alpha$ $-\dfrac{3\overline{\gamma}-7}{3-\overline{\gamma}}$ $-\dfrac{5-\overline{\gamma}}{\overline{\gamma}-3}$ $-1$ $\gamma_p$ $-1$ $1$ $1$ $\delta$ $\overline{\gamma}-2$ $\overline{\gamma}-2$ $2$ $\nu_f$ $\dfrac{\overline{\gamma}-2}{3-\overline{\gamma}}$ $\dfrac{\overline{\gamma}-2}{\overline{\gamma}-3}$ $2$ : Critical Exponents\[table1\] \[table:nonlin\] Lastly, in case $2<\overline{\gamma}<3$, the critical fraction $f_c=0$. Again, we expand Eq. using the small parameters $\epsilon\ll 1$, $\|h\|\ll 1$ and $\psi\ll 1$. The equation of state for $\psi$ and the associated free energy become: $$\begin{aligned} h =&-c_s(\epsilon\psi)^{\overline{\gamma}-2}+\psi+\ldots,\\ \mathcal{F}(\epsilon,h)\propto &\,\epsilon\left(-h\psi+\overline{c}_s\epsilon^{\overline{\gamma}-2}\psi^{\overline{\gamma}-1}+\psi^2/2\right)+\ldots,\end{aligned}$$ where $c_s>0$. It follows that \[robert2\] $$\begin{aligned} \mathcal{F}(\epsilon,0)&\sim \epsilon^{\frac{\overline{\gamma}-1}{3-\overline{\gamma}}},\\ \left.\mathcal{F}(\epsilon,h)\right\|_{\epsilon\rightarrow 0}&\sim \|h\|^{\frac{\overline{\gamma}-1}{\overline{\gamma}-2}}.\end{aligned}$$ In the last expression, the limit $\epsilon\rightarrow 0$ is only taken in the free energy and $h$ is taken small and negative such that $\psi$ is still positive. Therefore, $$\begin{aligned} \label{nuuus} \nu_{\epsilon}=\frac{\overline{\gamma}-1}{3-\overline{\gamma}}\text{ and } \nu_{h}=\frac{\overline{\gamma}-1}{\overline{\gamma}-2}.\end{aligned}$$ The other exponents are listed in the first column of Table $1$. It must be noted here that in practice, the exponents of Eq. may be impossible to find with the configurational model in case we start from a robust network ($2<\gamma<3$). This stems from the fact that a structural cutoff for the maximally allowed degree $K$ must be introduced to obtain an uncorrelated network. Such cutoff can be of the form $K\sim N^{1/\omega}$ with $\omega\in [2,\infty[$. However, it is well-known that the cutoff affects the critical exponents [@castellano]. Indeed, performing the averages in Eq. with use of the cutoff, one readily obtains $f_c\propto N^{(1-\alpha)(\overline{\gamma}-3)/\omega}$ (cf. Eq. ) and therefore: $$\begin{aligned} \label{nuu} \nu_{\epsilon}=-\frac{\omega}{(1-\alpha)(\overline{\gamma}-3)}.\end{aligned}$$ In case we start from a fragile network ($\gamma>3$), $\omega$ equals $\gamma-1$ and the $\nu_{\epsilon}$ of Eq. is retrieved. Therefore, degree-dependent edge removal may be used as a tool to observe critical exponents in the delicate regime $2<\overline{\gamma}<3$, without* the use of a structural cut-off [@castellano]. In our simulations as presented in Sect. \[comparison\], we take $\omega=2$ in case $\gamma<3$. The exponents obtained for the case $\overline{\gamma}>3$ reduce to the ones for node percolation in the limit $\alpha = 0$ when $\overline{\gamma}=\gamma$ [@cohen3]. However, in the limit $\alpha=0$ the exponents in the first column do not coincide with those given in Refs. and . Close to the percolation transition, it is possible, with the use of Eqs. to calculate the average degree in the giant cluster. We can identify this as $$\begin{aligned} \lim_{f\rightarrow f_c}\frac{fN_e(1-(H_1)^2)}{N(1-H_0)}=\frac{(f_c+\epsilon)\langle k\rangle 2N }{N\langle k\rangle (f_c+\epsilon)}=2.\end{aligned}$$ This however, must not be confused with the Molloy-Reed Criterion, which states that the average degree of a neighboring site in the entire network has degree two. In sum, we have now calculated the most important critical exponents for a percolation process. Extra support for our critical exponents comes from scaling relations and the connection with the Potts model. There is one feature which appears in all the calculated exponents: the only dependence on $\alpha$ arises through the exponent $\overline{\gamma}$. Random percolation on a network with degree exponent $\overline{\gamma}$ gives the same critical exponents as percolation with bias exponent $\alpha$ on a network with degree exponent $\gamma$. This equivalence was found before for degree-dependent interactions on scale-free networks [@giuraniuc1; @giuraniuc2]. It is not surprising that the same behavior appears both for edge percolation and for degree-dependent interactions, since both can be linked with the Fortuin-Kasteleyn construction. Comparison with Numerical Results {#comparison} ================================= In this section we test the previously derived analytical results using simulations. The networks are generated using the uncorrelated configurational model which was introduced in Sect. \[sec\_intro\]. Each simulation involves three free parameters: the degree exponent $\gamma$ of the network, the minimal node degree $m$ and the number of vertices $N$. Unless mentioned otherwise, we set $m=1$. In the configuration model, the degrees of the nodes are determined initially from the discrete degree distribution [^7] and then the connections are assigned at random. To obtain an uncorrelated network, the maximal degree is set to $\sqrt{N}$ when $2<\gamma<3$ [@catanzaro]. In some simulations (see Figs. \[pcN\] and \[cross-overfig\]) no degree cutoff was imposed. If $\gamma \geq 3$, the maximal degree is simply $N-1$. Both the sequential and the simultaneous approach are implemented. Sequential Approach {#sequential-approach} ------------------- In this first subsection, we discuss simulation results concerning the sequential approach. Most of these results can also be found in Ref. . ### Scaling of the Critical Point First, we investigate the finite-size behavior of the critical fraction $f_c$ of nodes. The results for the sequential approach are shown in Fig. \[pcN\]. For networks with $\gamma=2.5$ submitted to CB with an effective value $\overline{\gamma}=4$ (continuous black line), we observe that the critical fraction $f_c$ converges to a finite value as $N$ grows, confirming the conjecture that a robust network may turn fragile under CB. In the opposite case, a network with $\gamma=4$ submitted to PB with an effective $\overline{\gamma}=2.5$ (dashed red line), has a critical fraction that decays with the vertex number $N$ as a power-law, $f_c\sim{N^{-1/\nu_{\epsilon}}}$. The best fit to the data in this case results in $1/\nu_{\epsilon}={0.35\pm{0.02}}$, consistent with the value $1/3$ expected from Eq. . This result shows that a fragile network under PB will behave in the same fashion as a robust network with a degree distribution controlled by $\overline{\gamma}$ under random failure [^8]. Note that this simulation result confirms Eq. , although this equation was derived for the simultaneous approach. Indeed, to deduce Eq. , correlations in the diluted network are neglected. These simulation results indicate that this is an acceptable approximation. ### Properties of the Diluted Network During our theoretical discussion of the sequential approach, certain properties of the diluted network became apparent, such as a cross-over behavior as a function of the degree $k$ for the mean edge preservation probability $\rho_k$. The probability $\rho_k$ can easily be inferred from our simulations by calculating the ratio of the new node degree $\overline{k}$ to the old node degree $k$ for each node and averaging over all nodes with the same degree. The result is shown in Fig. \[cross-overfig\] in which the expected crossover behaviors are marked by arrows and the continuous lines are fits to the data of Eq. . When CB is applied (inset), a cross-over between a regime with $\rho_k\approx 1$ to a decreasing power law is found. As a second characteristic of diluted networks, the emergence of correlations in the diluted network is discussed. The theoretical result of Eq. suggests disassortative mixing in the diluted network in case central bias is applied. To observe these correlations in the simulations, the mean nearest-neighbor degree is calculated as a function of the node degree. The result of our simulation is shown in Fig. \[correl4\]. As expected, no correlations are present in the original network (top curve in red, $f=1$). However, with 10% of its edges removed, the mean nearest-neighbor degree in the diluted net clearly decreases as the degree of the node increases (bottom curve in black). This is a clear indication of disassortative mixing after sequentially removing a certain fraction of edges and this confirms our theoretical prediction. . Simultaneous Approach {#simultaneous-approach} --------------------- ### Comparison with Theory and Sequential Approach This subsection deals with the iterated simultaneous approach as introduced at the end of Sect. \[simultaneousapproach\]. To examine the percolation transition, we search for the probability to belong to the largest cluster, $\mathcal{P}_{\infty}$, as a function of the fraction $f$ of included edges. Results are given in Figs. \[orderparam1\], \[orderparam2\] and \[orderparam3\]. For random edge removal it is shown in Fig. \[orderparam1\] that the simultaneous approach coincides with the sequential approach; in that case, only one iteration is necessary to attain $f=1$. When, on the other hand, $\alpha>0$, the simultaneous approach can only be used up to a certain value of $f_u$, smaller than one (See Eq. ). However, the iterative procedure can be used until all links are included. In general, we can include over 98% of the links with a relatively small number of weight-recalculations or iterations. For instance, in case a network with $\gamma = 2.5$ is subjected to percolation with $\alpha = 0.2$, one finds that $f_u = 0.51$ and 15 iterative steps are necessary to reach $f = 0.98$. The results of such iterative simultaneous approach can be found in figure \[orderparam2\]. It is immediately clear that the sequential and the simultaneous no longer coincide. This is not surprising since the definitions of edge retaining probabilities $\rho_{ij}$ are different for the sequential and the simultaneous approaches when $\alpha \neq 0$. Furthermore, the sequential approach introduces correlations which are absent in the simultaneous approach. Note, however, that the difference between the two approaches is not a mere consequence of the appearance of correlations in the sequential approach. Indeed, if only the (dissassortative) correlations were present, the sequential approach should have a larger critical fraction $f_c$ than the simultaneous approach [@goltsev]; yet, we find the inverse to be true as evidenced in Fig. \[orderparam3\]. At the point at which the weights are recalculated, the giant cluster probability $\mathcal{P}_{\infty}$ of the iterative simultaneous approach shows kinks as a function of $f$. For instance, a conspicuous kink appears at $f_u = 0.69$ in Fig. \[orderparam3\]. The iterative simultaneous and sequential methods approach each other as $f\rightarrow 1$ and coincide at $f=1$. Note that the critical fraction $f_c$ which can be extracted from Figs. \[orderparam1\], \[orderparam2\] and \[orderparam3\] is nonzero although it is expected to be zero for $2\leq \overline{\gamma}\leq 3$. This is a consequence of the finite-size effects which cause $f_c$ to scale with the system size according to Eq. (see also Fig. \[pcN\]). Although there are differences between the sequential and the simultaneous approach, the differences are clearly not very large. The largest relative deviations occur around $f_c$ while the largest absolute deviations are situated around the lowest $f_u$ and are typically not more than 10%. Although we find the critical fraction $f_c$ of the simultaneous approach to be always larger than the one of the sequential approach, both values deviate by less than 10%. We conclude that the simultaneous and the sequential approach do differ, but the differences are not large and both approaches are qualitatively similar. Analytical results for the probability of the largest cluster $\mathcal{P}_{\infty}$ in the regime $f<f_u$ can be obtained by solving Eq. numerically for $H_1(1)$ which is then introduced in Eq. . For random edge removal ($\alpha=0$), $f_u = 1$ and thus a theoretical result is available for all $f$-values. Moreover, also for other values of $\alpha$ the generating functions theory calculation (see black line in Fig. \[orderparam3\]) agrees well with simulation results throughout the entire reconstruction process. The theoretical model is thus justified by the simulations. ### Properties of the Diluted Network We end the section with an overview of the properties of the diluted network. The focus lies again on the appearance of correlations and on the cross-over. In the simultaneous approach (with $f<f_u$), no cross-over can appear for the node retaining probability $\rho_k$. Indeed, expression is exact and predicts a decreasing power law for $\rho_k$, that is when $f<f_u$. However, this expression is only valid as long as no recalculation of the weights is performed. As soon as we iterate the simultaneous approach, the results for the simultaneous and sequential methods start approaching each other. Since the sequential approach contains a cross-over, we expect the appearance of a cross-over in the iterative simultaneous approach. Analoguous arguments apply to the appearance of correlations in the diluted network. The emergence of a cross-over in the iterative simultaneous approach is indeed found and shown in Fig. \[crossoverpar\]. When 10% of the edges are included, no cross-over at all appears. This is consistent with our arguments since we can simply include this fraction of edges in one sweep. Also in the second sweep, no cross-over appears. However, after 10 recalculations of the weights, the cross-over is undoubtedly present. Once again, our theoretical arguments are verified. Furthermore, the appearance of correlations is confirmed by our simulation results as evidenced in Fig. \[correl5\]. Indeed, disassortative correlations are apparent only for large values of $f$, after several iterations have been performed. Note also that these correlations disappear only very slowly upon approach of the point $f=1$ where no correlations are present (see Fig. \[correl4\]). Conclusions =========== We have performed a detailed study of biased percolation on scale-free networks with degree exponent $\gamma$ (with $\gamma > 2$) and shown that it is possible to tune a robust network fragile and vice versa. Biased percolation involves degree-dependent removal of edges, more specifically, we assumed that the probability to retain an edge is proportional to $(k_ik_j)^{-\alpha}$ with $k_i$ and $k_j$ the degrees of the attached nodes. For $\alpha >0$ the bias is [central]{} since links between highly connected nodes are preferentially depreciated, while the converse, [peripheral bias]{}, corresponds to $\alpha < 0$. Our most important result is that, at percolation, the properties of a network with bias exponent $\alpha$ and degree exponent $\gamma$ are the same as those of a network with bias exponent zero and degree exponent $\overline{\gamma}=(\gamma-\alpha)/(1-\alpha)$, [or]{} degree exponent $\gamma$, depending on the sign and the range of $\alpha$. Let us first elaborate on this main result, in the light of the present work and recapitulating arguments presented in the preliminary report [@moreira]. For $\alpha >0$ (with the restriction $\alpha < 1$) the new degree exponent $\overline{\gamma} > \gamma$ governs the critical properties of the network that results when the percolation threshold is reached after biased depreciation. The exponent $\overline{\gamma}$ controls the large-degree behavior of the new degree distribution. This new $\overline P (\overline k)$ is not simply scale-free but [asymptotically]{} scale-free. There is a cross-over value $k_{\times}$, so that for $\overline{k} < k_{\times}$ the exponent $\gamma$ is dominant and for $ \overline{k} > k_{\times}$ the exponent $\overline{\gamma}$ takes over. For $\overline{\gamma} > 3$ the biased depreciation process will reach the percolation threshold at a finite fraction of retained edges. The network is then fragile under central bias, regardless of whether the network is fragile ($\gamma >3$) or robust ($\gamma < 3$) under random removal. For $\overline{\gamma} < 3$ the biased depreciation will (in an infinite system) not reach a percolation point since the critical fraction of retained edges, $f_c$, is zero. The network, which is robust for random removal, remains robust under centrally biased removal. However, for a finite system $f_c$ is small but finite and scales with system size in a manner governed by the exponent $\overline{\gamma}$, whereas the scaling properties of $f_c$ for random removal are governed by $\gamma < \overline \gamma$. We have shown, by analytic proof, that $\overline\gamma$ governs the percolation critical behavior for the case $\alpha >0$. Also our numerical results support this conclusion. For $\alpha < 0$ (with the restriction $2-\gamma < \alpha$), the critical behavior at percolation is more subtle. Peripherally biased removal is less destructive than random depreciation and it is possible that a network that is fragile under random failure becomes robust when peripheral bias is applied. Noting that $\overline\gamma < \gamma$ it is obvious that robustness is preserved for networks with $\gamma < 3$. Conversely, fragility persists for sure when $3 < \overline\gamma$. However, it is not obvious what to expect when $\overline\gamma < 3 < \gamma$. The behavior of the new degree distribution $\overline P (\overline k)$ for $\overline{k} > k_{\times}$ is, for $\alpha < 0$, controlled by the exponent $\gamma$, so it would seem that the properties of the network under random failure are simply not affected by peripheral bias. However, a finite-size scaling analysis at criticality reveals that the cross-over value $k_{\times}$ is larger than the maximal degree in the network, implying that the new degree distribution will be controlled by $\overline\gamma$ instead of $\gamma $, [provided]{} $2-\gamma < \alpha < 3 - \gamma$ (we assume $\gamma > 3$ since this discussion only makes sense for networks fragile under random failure). We conclude that sufficiently strong peripheral bias can turn a fragile network robust, and numerical evidence supports this conclusion. On the other hand, for $3 - \gamma < \alpha < 0$, it is not clear whether the network stays fragile under peripherally biased failure when $\overline \gamma $ drops below 3, which happens for $\alpha < (3-\gamma)/2$. This problem is still largely open to future investigation. Two distinct approaches by means of which a network can be reconstructed in a degree-dependent manner, the sequential and the simultaneous approach, have been introduced to perform the edge removal process. For the sequential approach, we obtained a very useful [analytic approximation]{} to the marginal distribution $\rho_k$, which is the mean probability that an edge connected to a node with degree $k$ is present in the network after reconstruction. This analytic form clearly features the cross-over value $k_{\times}$ which plays a crucial role in the network properties. The simultaneous approach, which is a simpler scheme useful for $\alpha > 0$ and for edge number fractions below a value dependent on $\gamma$ and $\alpha$, can be iterated so as to provide an alternative to the sequential approach (for $\alpha >0$). The iterations introduce a history-dependence and lead to the emergence of $k_{\times}$, rendering both reconstruction methods qualitatively similar. For both approaches the new degree distributions have been calculated and the degree-degree correlations emerging in the depreciated network have been characterized, by means of standard combinatorial methods. The main finding as regards the correlations is that the sequential approach causes [disassortative mixing]{} in the depreciated network when $\alpha > 0$. For the simultaneous reconstruction approach, the exact (since correlation-free) percolation threshold $f_c$ is derived for central bias ($\alpha >0$) as a function of (non-integer) moments of the degree distribution, for $\overline \gamma > 3$. On the other hand, for $\overline\gamma < 3$ the exact finite-size scaling law for the vanishing of $f_c$ is obtained. These results fully demonstrate the validity of our exponent mapping (\[mappy\]) for central bias. A [generating functions]{} approach is introduced for degree-dependent edge percolation, extending previous work on random percolation. This approach allows to obtain the size distribution of finite clusters close to the percolation transition as well as other critical properties. If the network can be treated as a tree, which is valid for all finite clusters, the generating functions satisfy self-consistency equations. We have derived the extensions of these equations for degree-dependent percolation, allowing for correlated networks, and have shown that they reduce to the original equations provided no correlations are present. We have also derived the criterion for the percolation threshold for degree-dependent percolation and have shown that it reduces to the familiar Molloy-Reed criterion when correlations are absent. Further, our self-consistency equations reduce, for random percolation, to equations frequently encountered in the literature. Our generating functions formalism is new in the sense that it extends known results on random percolation to biased percolation which may involve correlated networks. In the following, however, we draw further conclusions for the statistical properties (including critical exponents) of [uncorrelated]{} networks only. Using the equivalence between the $q \rightarrow 1$ limit of the Potts model and edge percolation, we have shown that critical exponents for our biased percolation problem can be obtained from the Potts model free energy by extending this equivalence to inhomogeneous (edge-dependent) couplings in the Potts model and edge-dependent removal probabilities in percolation. The generating functions approach has been combined with the extension of the Fortuin-Kasteleyn construction for the Potts model and with [finite-size scaling]{} in order to extract the critical exponents of the percolation transition, for uncorrelated networks. We have found that the critical exponents are functions of $\overline\gamma$, assuming that the degree distribution after depreciation is governed by degree exponent $\overline\gamma$, asymptotically for large degree. For $\overline\gamma$ we obtain critical exponents that reduce to literature values of random percolation simply by substituting $\overline\gamma \rightarrow \gamma$. However, in the more delicate regime $2 < \overline\gamma < 3$ this correspondence is not satisfied. A critical assessment of this discrepancy is not given here, but left to future scrutiny. We conclude that, in all cases, the only way in which the bias exponent $\alpha$ enters in the critical exponents of the percolation transition, is through the new degree exponent $\overline\gamma$. Furthermore, we have used numerical simulations to study the properties of the network after depreciation and near the percolation transition. We verified that robust networks can turn fragile under centrally biased failure and that fragile networks can turn robust under (sufficiently) strong peripherally biased failure, using the sequential approach. Although correlations are introduced in this approach, the results agree well with the predictions for uncorrelated networks. Also the cross-over behavior of the new degree distributions was tested and found to agree well with the analytical expectations. As regards correlations introduced by the sequential approach, we have been able to verify the occurrence of disassortative mixing predicted theoretically for $\alpha >0$. The critical properties at percolation were checked by simulations using the (iterated) simultaneous approach and also compared with results obtained by simulations using the sequential approach. Specifically, we have found that for biased percolation the sequential and the (iterated) simultaneous approach give rise to different results. In particular, the size of the giant cluster predicted by the generating functions theory agrees very well with the simulations for the (iterated) simultaneous approach. Nevertheless, the differences are often small and we may conclude that both methods are qualitatively similar. Finally, we have also provided evidence for the theoretically expected appearance of cross-over effects and degree-degree correlations for the (iterated) simultaneous approach. Overall, we conclude that good agreement has been found between simulations and theory. Appendix ======== There exists a more formal way to prove that the degree distribution in the diluted net satisfies $\overline{P}(\overline{k})\propto \overline{k}\,^{-\overline\gamma}$ for large degrees $\overline{k}$ when central bias is applied. We start from the degree distribution in the diluted network $\overline{P}(\overline{k})$ which was calculated in Eq. . For large values of $k$, i.e., $k \gg k_{\times}$, and $\alpha>0$, the probability of retaining a node of degree $k$ falls off as $\rho_k\propto k^{-\alpha}$; this is valid using the sequential approach (see Eq. ), as well as the simultaneous one (see Eq. ). If both $k\rho_k$ and $k (1- \rho_k)$ are large, the binomial distribution can be approximated by a normal distribution with mean $k\rho_k$ and variance $k\rho_k(1-\rho_k)$. The latter condition is always true if we apply CB, since then $1-\rho_k \approx 1$ for large $k$ as edges between the most connected nodes are almost certainly removed. Since $k \rho_k \propto k^{1 - \alpha}$, the first requirement holds only if $\alpha <1$. Inserting the normal distribution with variance $k\rho_k(1-\rho_k) \approx C_0k^{1-\alpha}$, with $C_0$ a constant, in Eq. and approximating the sum by an integral yields $$\begin{aligned} \overline{P}(\overline{k}) \propto \int_{\overline{k}}^{\infty} dk\:k^{-\gamma+\frac{\alpha - 1}{2}}\exp\left(-\frac{\left[\,\overline{k}-C_0k^{1-\alpha}\right]^2}{2C_0k^{1-\alpha}}\right).\end{aligned}$$ Now we introduce the auxiliary variable $u\equiv k/\overline{k}^{\,1/(1-\alpha)}$ and rewrite the integral as follows: $$\begin{aligned} \overline{P}(\overline{k})\propto \overline{k}^{\frac{1-\gamma}{(1-\alpha)}-\frac{1}{2}} &\int_{\overline{k}^{\frac{-\alpha}{1-\alpha}}}^{\infty}du\: u^{-\gamma + \frac{\alpha-1}{2}}\nonumber \\ &\times\exp\left(-\overline{k}\frac{\left[1-C_0u^{1-\alpha}\right]^2}{2C_0u^{1-\alpha}}\right).\end{aligned}$$ For $\overline{k}\rightarrow \infty$, the integrand has only non-vanishing values in a neighborhood $\Delta u \approx1/\sqrt{\overline{k}}$ around $u_c = C_0^{-1/(1-\alpha)}$. If $0<\alpha <1$, the lower bound of integration vanishes for large $\overline{k}$. Thus $u_c$ certainly lies in the domain of integration and the integral can simply be approximated by $C_1/\sqrt{\overline{k}}$ with $C_1$ a constant. After some trivial power counting, we arrive at $$\begin{aligned} \overline{P}(\overline{k}) \propto \overline{k}\,^{-\frac{\gamma-\alpha}{1-\alpha}}. \label{gammaac}\end{aligned}$$ Thus we obtain the anticipated behavior for CB. The exponent $\overline{\gamma}$ controls the decay of the degree distribution in the diluted network at large $\overline{k}$. The case $\alpha = 1$ can be studied using a Poisson distribution approximation for the binomial factors, which results in $\overline{P}(\overline{k})$ being a Poisson-type degree distribution $\overline{P}(\overline{k}) \sim \overline{k}^{\,1-\gamma}/\overline{k}!$ for large $k$. Thus $\alpha < 1$ is a natural restriction, because the scale-free behavior is destroyed if stronger CB is applied. 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[^4]: As explained later, these self-consistency equations are only valid in case the simultaneous approach is used, that is when no correlations are present in the network after depreciation. [^5]: This relation can be interpreted as follows: Consider that the average rate of spin-flipping between $i$ and $j$ is $J_{ij}/k_BT$, then, in the continuum limit, the probability $\rho_{ij}$ that the spin will be flipped after one time unit is equal to $$\rho_{ij}=1-\lim_{\delta t\rightarrow 0}(1-J_{ij}\delta t/k_BT)^{1/\delta t}=1-e^{-J_{ij}/k_BT}.$$ [^6]: Note that for $d$-dimensional lattices with $d>8$, the cluster size distribution can indeed be written in this form [@stauffer; @harris]. Since our considered random networks have essentially an infinite dimension, this scaling ansatz is justified. [^7]: Note that it is essential to take the distribution function to be defined on a [discrete]{} support ($k= 0,1,2, ...$) in order to match the simulations with our analytical work. [^8]: We remark that the choice of $\alpha$ in the PB case here is a little bit special, since $\alpha = 3-\gamma$, which means that this case is just the borderline case between the strong PB and the weak PB discussed in Ref. . The simulations indicate that this borderline case belongs to the strong PB regime.
--- author: - 'A.M. van Genderen' - 'A. Lobel' - 'H. Nieuwenhuijzen' - 'G.W. Henry' - 'C. de Jager' - 'E. Blown' - 'G. Di Scala' - 'E.J. van Ballegoij' date: 'Received..../Accepted....' title: 'Pulsations, eruptions, and evolution of four yellow hypergiants.' --- Introduction ============ In the present paper we explore large photometric data sets of four yellow hypergiants (YHGs) , and (the Big Three), and , the class of which is still undetermined (Arkhipova et al. 2009; Oudmaijer et al. 2009; Le Coroller et al. 2003; Ferguson & Ueta 2010; Sahin et al. 2016). This latter could be a YHG or a red supergiant (RSG), but also a post-asymptotic giant branch(AGB) star, thus a much lower-mass object and has been incorporated into this study due to its spectroscopic and photometric similarities with both classes. The three types of objects, YHGs, RSGs and post-AGBs, are almost entirely convective. Late-type supergiants showing the H$\alpha$ line with one or more broad emission components and exceptional broad absorption lines in their spectra are called hypergiants Ia$^{+}$. The approximate ranges for the M$_{\rm bol}$=-8.7–-9.6, for the luminosity logL/L$_{\rm \odot}$=5.3–5.7 and for the temperature T$_{\rm eff}$=4000K–6000K (excluding eruptive episodes). Typical for these stars are their very extended atmospheres, surface gravities around zero, high mass-loss rates, and strongly developed large-scale photospheric and atmospheric motion fields (de Jager 1980, 1998). The $\kappa$-mechanism of the two partially ionized He zones is likely responsible for the pulsations with a quasi-period of a few hundred days (e.g. Fadeyev 2011). These stars are also in the state of gravitational contraction of their He core (Meynet et al. 1994; Stothers & Chin 1996). On a timescale of one to a few decades, a pulsation develops into an atmospheric eruptive episode, lasting about one to two years, reaching a high temperature (6500K–7500K; we note that this is not an evolutionary temperature rise). Lobel (2001) presented non-LTE calculations, showing that in this temperature range, cool supergiants become dynamically unstable due to the decrease of the first adiabatic index stability-integral $\langle$$\Gamma_{1}$$\rangle$ over a major fraction of the extended atmosphere as a result of partial, thermal and photo-ionization of hydrogen. At these high T$_{\rm eff}$ large fractions of partially ionized H gas (Tgas$\simeq$8000K) can synchronously recombine under pulsation decompression and drive the fast expansion of the entire atmosphere in an outburst event with strong global cooling. An enormous amount of energy is released which blows away part of the upper atmospheric layers. Subsequently, a deep light minimum follows with T$\sim$4300K and the spectrum shows TiO absorption bands. The mass-loss rate as recorded during the eruption of $\rho$Cas in the year 2000 amounted to $\sim$ 3.10$^{-2}$M$_{\sun}$yr$^{-1}$ (Lobel et al. 2003). This amount is of the same order as that predicted by hydrodynamical models by Tuchman et al. (1978) and Stothers (1999). Figure1 depicts a schematic HR diagram. A cool portion of the HR diagram appeared to be almost void of stars (Humphreys 1978). Part of this void was theoretically studied and defined by de Jager (1998) and observationally checked by Nieuwenhuijzen & de Jager (2000). Very relevant for this study was also the comparison of the evolutionary tracks of HR8752 and other YHGs with the models of Meynet et al. (1994) by Nieuwenhuijzen et al. (2012) in their Fig.1 and Sect.1.2. In this latter analysis, IRC+10240 also appeared to be a YHG moving on an evolutionary track (Oudmaijer (1998). This area was divided into two voids: the Yellow Void and the Yellow-Blue Void. These represent critical phases in the evolution of YHGs, bounded by the three semi-vertical curves. The curve at lowest T$_{\rm eff}$$\sim$ 5200K is at photospheric values where H is close to beginning its ionization. The next curve at $\sim$7500K (dashed) is at photospheric values where H is close to being completely ionised and the curve at the highest T$_{\rm eff}$ between 8900K and 12500K marks the parameters where neutral He is close to being fully ionized. The extreme photospheric values of the four YHGs studied in this paper are given in TableA:Data summary of the Appendix. It should be noted that with the exception of HR8752, no Hipparcos or Gaia Data Processing and Analysis Consortium (DPAC) parallaxes were used for the distances. Together with other experts in this field we advise against using these latter for very big stars and stars brighter than six, as then the reliability of parallaxes is very uncertain (van Leeuwen, priv. comm. 2019). To prevent crowding in Fig.1, only one or two selected red or blue evolutionary loops (hereafter referred to as RL and BL, respectively) per YHG are schematically represented by lines (or dotted curves), and their observed evolutionary directions by an arrow. For HR8752: a RL and a BL (based on Nieuwenhuijzen et al. 2012) are connected by the curved dotted line (during which the 1973 eruption happened) to indicate the evolutionary direction (not the true track). How long this zigzag behaviour lasts is unknown; it is meant to lose enough mass (mainly during the eruptive episodes) in order to evolve further to the blue to eventually become an SDor variable (also referred to as a luminous blue variable (LBV)), or a Wolf-Rayet star (WR-star) (Oudmaijer et al. 2009; Oudmaijer & de Wit 2013). Critical comments by Sterken (2014, priv.comm.) on the supposition that HR5171A is a contact binary on account of a double-waved light curve constructed by Chesneau et al. (2014) contributed to our growing realization at the time (2014) that little was known about the photometric properties of the light curves of individual quasi-periodic pulsations and long-term light variations (LTVs) of YHGs. Therefore, our focus was to probe the large photometric databases, comprising 50 to 120yr of observations of the four selected YHGs to the bottom, otherwise pivotal developments will be absent. The most important aims of this study are to find out why spectroscopic temperatures are always higher than photometric ones; to decipher whether or not in addition to HR8752 (Nieuwenhuijzen et al. 2012), the three other selected YHGs RL and BL evolutions on timescales of decades, interrupted by eruptive episodes; to find out whether or not the observed LTVs represent these evolutionary tracks; and to find out whether the light variations of HR5171A are due to pulsations alone, and why it reddened stubbornly from 1960 to 1981. We also aim to find an explanation for the variable trends of quasi-periods. Sections1 and 2 are devoted to the $\sim$1yr stellar instabilities and the stellar properties. Section3 is devoted to the connection between LTVs and the evolutionary loops of YHGs on the HR diagram and the variable trends of quasi-periods. Section4 is devoted to a number of theoretical and speculative explanations for some observational highlights. Section5 summarises the most important novel findings. Miscellaneous photometric and physical details and pulsation properties. ========================================================================= Databases and published extinctions. ------------------------------------ Based on decades up to more than a century of visual, mono-, and multi-colour photometry, it was obvious that the four selected objects show more or less the same pattern of ’semi-regularity’ of pulsations, and appear coherent. The quasi-periods are a few hundred days in length and the pulsations are superimposed on the LTVs; we refer to AppendixB:2.1. for references and descriptions. Interstellar extinctions and reddening are required for our calculations of stellar properties: these are summarised in Table1. HR8752 $\rho$Cas HR5171A HD179821 ---------- ------------ ------------ ------------ ------------ A$_{v}$ 3.08$^{1}$ 1.44$^{2}$ 4.32$^{3}$ 2.24$^{3}$ R 4.4$^{1}$ 3.2 $^{3}$ 3.6$^{5}$ 3.2$^{3}$ E$(B-V)$ 0.70$^{1}$ 0.45$^{4}$ 1.2$^{3}$ 0.7$^{6}$ E$(U-B)$ 0.67 0.44 E$(V-R)$ 0.66 0.40 E$(V-I)$ 1.31 0.73 E$(R-I)$ 0.65 0.34 : Adopted interstellar extinctions, reddening, and the extinction laws R for the four objects used in this paper. The reddening values, dependant on the extinction law R, in the Johnson $UBVRI$ were retrieved from the tables of Steenman & Thé (1989).[]{data-label="table:1"} Pulsation properties. ---------------------- Figures2 and 3 show two typical time-series for the YHG cadence of pulsations. The first represents the very accurate $\rho$Cas photometry in $V$ and $B$ by Henry (1995, 1999) for the time interval 2003–2017 relative to the comparison star HD223173=HR9010 ($\triangle$ signifies variable minus comparison star). Henry’s observations concern the T2 $VRI$ (1986–2001) and the T3 $BV$ (2003–2018) Automatic Telescope Projects (ATP). For the relative magnitudes we used for the comparison star: (Johnson system) $V$=5.513, $B$=7.160, $R$=4.220, $I$=3.390. See stub-tables TablesM.1. and M.2. in the Appendix for Henry’s complete tables in electronic form available at the CDS. The mean lines represent the LTVs. Between 2007 and 2011 (JD2454168–JD2455819) Klochkova et al. (2014) derived ten spectral temperatures (T(Sp)) of between 5777K and 6744K, thus showing a range of $\sim$1000K. The two panels of Fig.3 show the cadence pattern of HR5171A (1989–1995 and 2009–2016, respectively), which is not significantly different from those of $\rho$Cas in Fig.2, of HR8752 (not depicted), and of HD179821 (e.g. Arkhipova et al. 2009). Both panels of Fig.3 also show a significant declining visual brightness of the LTV. In the upper panel of Fig.3 we matched professional photometry in $V$ based on several sources: van Genderen (1992, hereafter called PaperI); Chesneau et al. 2014) and visual observations by the American Association of Variable Star Observers (AAVSO) represented by the red curves which are based on the smoothed black dots. AAVSO data were shifted 04 upwards, the Hipparcos $H_p$ magnitudes shifted 01 downwards, and the $y$ ($uvby$) of the Strömgren system were shifted 01 upwards in order to match the $V$ of the Walraven $VBLUW$ system transformed into $V$ of the $UBV$ Johnson system (Paper I). ![image](img308.pdf){width="12cm"} No shift of the magnitude scale for the AAVSO data has been applied in the lower panel showing the difference with the six photo-electric $V$ magnitudes represented by plus signs (these $BV$ Johnson and $RI$ Cousins data are listed in TableI.1. of the Appendix). The systematic difference between the two observation techniques is common knowledge (Bailey 1978): the visual data are generally fainter by 02-04 than the photo-electric data (in the upper panel we applied a correction of 04). The vertical line in the lower panel near JD2456755 represent an observation made with the aid of the Astronomical Multi-BEam combineR (AMBER) instrument of the VLTI instrument at the ESO by Wittkowski et al. (2017a), indicating that T$_{\rm eff}$=4290K$\pm$760K. We estimate that the brightness was $V$$\sim$6.4. Our definition of the quasi-period, or duration $\triangle$D in days, is through timing measurements of light maxima. The amplitudes of the pulsations of the four objects vary between 02 and $\sim$05, and are variable from cycle to cycle. Eruptions are striking, among other things, by the relatively large depth of the minimum ($\sim$1$^{\rm m}$) after the eruption maximum, called max1, of which the amplitude is sometimes slightly higher than for normal pulsation maxima. The duration of the entire eruption episode is about twice the duration of an ordinary pulsation. Because of such markers we identified one new eruption in Fig.2 ($B$ panel): in 2013 (the maximum named no.8). This eruption has been independently confirmed by spectroscopic observations (Aret et al. 2016). However, it was of a much weaker caliber than those of for example 1986 and 2000. The pulsation in 2004, of which the maximum is labelled no.7 in Fig.2, showed more or less the same markers, but its light curve was even weaker than that of 2013, and it is definitely not an eruption (no spectroscopic indications). ![image](img019.pdf){width="12cm"} There is some ambiguity about the two high peaks near JD2456500 in the lower panel of Fig.3, but because of the large scatter of the observations we considered them tentatively as a single peak. Errors for visual magnitudes of very red objects like HR5171A, are larger than those for less-red stars because of the Purkinje effect. A quasi-period search in the literature is summarized in AppendixC:2.2. After inspection of the large body of recorded photometric time-series of $\rho$Cas, HR5171A, HR8752, and HD179821, our conclusion is that YHGs are subject to a coherent sequence of pulsations, but that each pulsation is unique. These pulsations are quasi-periodic. Additionally, the quasi-periodicity is uninterrupted without any cessation of pulsations (including the eruptive pulsation episodes lasting twice as long). Only light curves of great precision reveal bumps, plateaus, and depressions lasting for weeks to months. This incessant cyclic behaviour without any gaps in the sequence, which otherwise would signal one or more missing cycles (e.g. in case of one small-scale non-radial pulsation only happening accidentally on the invisible hemisphere), suggests that we are dealing with a more-or-less global instability pattern. In case of non-radial pulsations (Lobel et al. 1994; Fadeyev 2011), each hemisphere of the star should be well covered with a number of such oscillations moving up and down more or less in concert. The bumps and dips mentioned above could well be caused by this process. ### Semi-regular, weakly chaotic, or chaotic? The observations represented in Figs.2 and 3 (representative for the other two objects as well) show that YHGs are subject to coherent sequences of pulsations. One can easily identify minima and maxima, although showing a variable shape from cycle to cycle, without significant secondary dips and peaks. Periods and amplitudes of individual cycles vary by no more than a factor two or three (eruptive episodes excluded). Semi-regularity (semi=half) is an often-used but vague description for the variability of the different types of evolved stars. Below we summarise two of the most interesting studies dedicated to modeling the pulsation sequences of late-type stars and compare these latter with those of YHGs. Following suggestions by Ashbrook et al. (1954) and Deeming (1970), Wisse (1979) used a continuous second-order auto-regression process to derive model light curves of red giants. The latter author distinguished two groups, one corresponding to strongly damped systems generating irregular (chaotic) model curves, and the other one consisting of slightly damped and undamped systems, generating light curves showing striking similarities with the real ones of Lb, SRb, and SRa giants; for example (i) the uniqueness of each cycle, (ii) a trace of a quasi-periodicity, and (iii) the presence of humps and shoulders. Wisse tentatively concluded that red giants are likely subject to random processes being dominant over one regular process. Differently from their many predecessors, Icke et al. (1992) approached the study of such quasi-periodic behaviour with numerical studies of the dynamics of a driven one-zone stellar oscillation. These latter authors concentrated on the increase of the irregularity of the outer layers (the ‘mantle’) of evolved AGB stars, which are low-mass stars. The more AGB stars evolve, the more mass they lose, and the less massive the mantle becomes, the more the oscillations become irregular. We assume that the results of Wisse (1979) and Icke et al. (1992) are in general also valid for many other ‘semi-regular’ convective variables such as YHGs, although these other variables are likely to be much more massive. Yellow Hypergiants have lost at least half of their initial mass, and are still highly eruptive on a timescale of decades, or less. Additionally, we assume that the mantle of YHGs will also have a lower mass compared to the total mass, similar to AGBs. The calculations of the mantle motions by Icke et al. (which are assumed to be representative of the light curves), yield a number of global properties. A fine-looking series of oscillations may eventually, sometimes quite suddenly, switch to a different pattern. We noted such sudden switches also in the mean quasi-periods of YHGs: after a number of oscillations, for example about half a dozen, the mean jumps to a smaller (or larger) one. With respect to this type of intermittent behaviour, Icke et al.: state that if a stellar oscillator is found to be intermittent, an impractical amount of observing time is needed to establish the periodicity. This is exactly the case for YHGs (apart from the sudden switches of the mean quasi-periods mentioned above); each pulsation has a different duration. Additionally, we observe that also with respect to morphology, each pulsation is different as well. In short, each pulsation is unique, and predicting any property of the next pulsation is impossible. Icke et al. also concluded that stars with a large interior radius relative to that of the mantle likely show a more chaotic behaviour. For YHGs the opposite is likely the case: the outer layers are extremely extended, thus reducing the chaotic character. Noteworthy is that for the calculation of the model light curves of different kinds of evolved late-type stars, Wisse (1979) and Icke et al. (1992) did not take into account any potential long-term variation; Sect.4. In conclusion, we characterise the instability of the four objects studied here as weakly chaotic, that is, not as chaotic as the individual oscillations that are clearly separated from each other. In other words, the cadence of oscillations appears to be coherent. HR5171A: a contact binary? -------------------------- One of the important purposes of Figs.2 and 3 was to emphasise that despite the weakly chaotic character, the quasi-periodic sequences of YHGs show a coherent behaviour. Therefore, it was easy to conclude that HR5171A does not show any trace of a second variable source as proposed by Chesneau et al. (2014). The light curve of these latter authors consists of a double-peaked brightness oscillation with two unequal amplitudes, and with a supposed binary period of about 1300 d. This is unlikely, as if this were the case Fig.3 would not show any coherent succession of light oscillations, but a quite chaotic distribution of the observations. The resulting brightness fluctuations would vary from very small to twice the present size (Fig.3), by two oscillations with slightly different amplitudes, but significantly different periods (the one with half the binary period: about 660 d, and the other one by pulsation, at the time about 500 d). With respect to the morphology of the observed light curves observed in Figs.2 and 3, it happens to be that pulsations of $\rho$Cas and HR5171A showed minima with alternating depths for some time. The selection of cycles as done by Chesneau et al. and depicted in their Fig.8 was obviously sensitive to this bias, resulting in the double wave with two unequal minima. However, there are mitigating circumstances: at the time very little was known about this treacherous photometric property (hopefully, the current paper fills this gap). Another argument in favour of pulsations only is that the monitoring of the V$_{\rm rad}$ curve during two pulsations (no.16 and no. 17 in Fig.1 of Paper İ) by Lobel et al. (2015) showed a regular phase lag of about 0.4 with respect to the light curve. This is also typical for $\rho$Cas (Lobel et al. 2003). However, we do not deny the possible presence of some companion, albeit it perhaps only in the field of view of the Hipparcos satellite; see the photometric analysis by Eyer (1998) and footnote no.2 in van Genderen et al. (2015). Based on AMBER/VLTI observations, Chesneau et al. (2014) discovered a bright spot in front of the primary disk. The PIONIER/VLTI interferometric observations by Wittkovski et al. (2017b, c) obtained impressive near-infrared (NIR) $H$-band images. They claim that the contact companion, suggested by Chesneau et al. (2014) is tentatively supported by a few images of HR5171A showing no precise circular circumference. One image even showed a clear bubble in the SW direction, which according to these authors might be the companion. A tentative model for the orbit is proposed. Secondary features like bright patches are assigned to a convection cell. It can also be concluded that if a low-mass contact companion to HR5171A were present, any geometrical light variation due to deformation is obviously too small to be clearly detected. The diameters of those few images mentioned above of HR5171A look different from each other, obviously due to the pulsations (light curve at the time shown in Fig.15), and the circumference is far from perfectly circular. This is understandable considering the very low surface gravity. Therefore, we also conclude that the well-covered and clearly plotted pulsation sequences like those in Figs.2 and 3, are essential to understand the characteristic instabilities of YHGs and possibly to correct questionable interpretations. Inconsistencies between brightness and colour, and between spectroscopic and photometric temperatures. ------------------------------------------------------------------------------------------------------ The analyses presented in the following sections have not been performed previously. We embarked on the (tedious) traditional method of plotting many tens of individual pulsations per object on graphic paper, comprising thousands of data points (for light and colour curves). The intention was to get an idea about the degree of their variable morphological properties, whether quasi-periods and light amplitudes are different during hot and cool stages, and more importantly to try to understand why spectroscopic and photometric temperatures, T(Sp) and T(Phot), always differ. Pulsations during which a T(Sp) was obtained were particularly important for our purposes. Walker (1983), studying photometry and spectroscopy of HR8752, was probably the first to notice that a relationship between the photometric magnitudes (and thus the continuum radiation) and the spectrum of YHGs was not always straightforward. He was not the only one. For example, Arkhipova et al. (2009) noted ambiguous relations between brightness and colours of the HD179821 pulsations during 2000–2009, and attributed them to increasing optical depth during high mass-loss episodes. We also noticed that differences exist between the temperatures based on spectroscopy and photometry: large and small for high and low stellar temperatures, respectively. Such differences not only exist for $\rho$Cas, but also for the three other objects. The purpose of the following sections is to enhance our understanding of the photometric properties and the probable cause of the fact that T(Sp) is usually higher than T(Phot). Correlation diagram $V$/$(B-V)$: similarities and differences between hot and cool YHGs. ---------------------------------------------------------------------------------------- Studying the morphology of the series of detailed light and colour curves provides valuable information about the wavelength dependency of amplitudes, the variation of the duration of pulsations, the size of amplitudes, and the mean rate of variation of temperature and radius. Enhanced mass-loss episodes could well result in inconsistent behaviour between brightness and colour indexes, and consequently in unreliable stellar properties. Therefore, we were interested in the properties of the correlation diagrams $V$/$(B-V)$. Contrary to YHGs (including many RSGs), Pop. I Cepheids and RRLyrae stars are radially pulsating stars, and obey a number of fixed properties. Most of their light curves have steep ascending branches with respect to the descending ones. Maximum acceleration of the expansion velocity happens during this phase, and is responsible for a fast temperature rise (e.g. Pel 1978, Lub 1977, 1979). Therefore, at the same visual brightness $V$, $(B-V)$ is slightly bluer (T$_{\rm eff}$ is higher). Our analysis of the four objects reveals a greater amount of photometric diversity than what is seen for the Cepheids and RR Lyrae stars with respect to the correlation diagrams. We noted an eye-catching difference between the $V$/$(B-V)$ diagrams of the pulsations of the hot YHG HD179821 (T$_{\rm eff}$=6800K, Arkhipova et al. 2009) and those of the cooler HR5171A, $\rho$Cas, and HR8752 when the latter was also as cool as its two sisters: about 5000K (Nieuwenhuijzen et al. 2012). Intrinsic differences between the properties of light and colour curves between hot and cool stages of YHGs can be expected, which must be related to their evolutionary stage. Figure 1 shows that in the hot stage, HD179821 and HR8752 were crossing the Yellow Void (and HR8752 even evolved into the Yellow-Blue Void) along a BL evolution: the stars contracted and the mean density increased. On the contrary, the YHGs HR8752, $\rho$Cas, and HR5171A in the cool stage are moving along a RL just outside the Yellow Void (Fig.1). Indeed, we discovered that light curves of YHGs HD179821 and HR8752 in the hot stage after the 1973 eruption show smaller visual amplitudes (01–02) and much shorter quasi-periods (100$^{\rm d}$–200$^{\rm d}$) than the cooler YHGs. Furthermore, about 30% of the pulsations, best documented for HD179821, even showed shifted colour extremes with respect to the light extremes (thus, light and colour curves do not always run in phase). Sometimes the reddest colour is reached during the ascending branch, and the bluest colour during the descent, and vice versa. A very global conclusion is that we discovered that variable light curve properties like quasi-periods and amplitudes depend on whether the star is hot (periods are short, amplitudes are small), or cool (periods are longer, amplitudes are larger). We return to this subject in Sect.3. To look for the cause of the temperature inconsistency, we have to use another amplitude ratio; see the following section. Ratios Ampl$B$/Ampl$V$ and Ampl$V$/Ampl$(B-V)$: influence of enhanced mass loss episodes. ----------------------------------------------------------------------------------------- Other possibilities to characterise the variability of pulsating stars are the amplitude ratios Ampl$B$/Ampl$V$ (amplitudes in $B$ are always larger than in $V$) and Ampl$V$/Ampl$(B-V)$ (for ascending and descending branches, which are mainly used here). Both ratios are a measure of the temperature change during the pulsation. The size of the amplitudes is independent of extinction and reddening corrections, and therefore so are their ratios. –[$\rho$Cas]{}: It appeared that for $\rho$Cas the mean ratio Ampl$B$/Ampl$V$ for tens of ascending and descending branches of pulsations (or only parts of them if there is a lack of data points, and eruptions excluded) and based on the robotic data sets 2003–2017 (by G.W.H.) is 1.6$\pm$0.3 (st.dev.). The corresponding ratio Ampl$V$/Ampl$(B-V)$=1.7$\pm$0.5 (st.dev.) It should be noted that in the 1986–2000 time interval, only $VRI$ photometry was performed. $(V-R)_{0}$ can easily be transformed into $(B-V)_{0}$ by dividing it by 0.75; see (eq.1) of Sheffer & Lambert (1992). We call these ‘normal’ pulsations, as they are the majority. This ratio also applies to well-defined light curves of HR5171A and HR8752. We note that our designation ‘normal’ is relative, and it only serves as a reference point. ‘Abnormal’ pulsations also exist, with amplitude ratios significantly larger or smaller (rare) than the value of 1.7$\pm$0.5 above. Between the two massive eruptions of $\rho$Cas in 1986 and 2000, many pulsations with overly large amplitude ratios occurred. That could be established because the star was monitored photometrically (1986–2000, in $UBV$ and $VRI$) and spectroscopically (1993–2002, by Lobel et al. 2003). Figure1 of Lobel et al. 2003 shows the $V$ and visual light curves and the V$_{\rm rad}$ curves, while the dates of the spectra and the presence of emission lines are indicated. References used for this analysis are listed in Appendix D: 2.6. The fact that the ratios Ampl$V$/Ampl$(B-V)$ are higher than 2.2 up to about 5, and also the relatively large variation of these ratios within this group, is a warning that some irregular phenomenon has disturbed the intrinsic photometric parameters $B$ and $V$. The height of the ratios means that the denominator is significantly smaller than for the ‘normal’ ones, pointing to some selective absorption. Therefore, $(B-V)$ curves show overly small amplitudes, and the $(B-V)$ values are too red, causing the overly small temperature ranges. Two such pulsations, i.e., those in 1993 ($\sim$JD2449300) and 1998 ($\sim$JD2451065), with relatively high spectroscopic temperatures (T(Sp)=7250K), high maxima, and showing emission lines in the spectrum due to enhanced mass loss episodes, had ratios of 5.1 and 2.5, and 4.4 and 4.1, for ascending and descending branches, respectively (spectroscopy by Israelian et al. 1999; Lobel et al. 2003). The pulsation in 1997 ($\sim$JD2450575) however, that is in between the two above, without emission lines, and a with maximum rivaling the couple above, had abnormal ratios as well, 4.4 and 5.8, indicating that the absorption was still present. Lastly, the ratio for the ascent to the minimum ($\sim$JD2451320) preceding the 2000-eruption max.1 ($\sim$JD2451590) is also too high: 2.5, while no obvious emission was reported. The spectroscopic temperature rose to 7600K, and then the brightness declined to a deep minimum, indicating that $\rho$Cas expanded and cooled to about $\sim$4400K. De Jager (1998, Ch.8) suspected that pulsations during an enhanced mass-loss episode show larger light amplitudes. This appeared to be indeed the case for all eruptive maxima. However, ordinary pulsations sometimes show relatively large amplitudes during line-emission periods (e.g. 1993 and 1998), but not always like those observed in 1997 and 1998. –[HD179821]{}: Amplitude ratios were derived for most of the light curves of HD179821. A few detailed pulsations are shown for example in Fig.2 of Arkhipova et al. (2009) and in Figs.1, 3, and 4 of Le Coroller et al. (2003). Their light curves also include observations by ASAS, Hrivnak (2001), and Hipparcos (ESA, 1997). Just like in the case of $\rho$Cas, we also identified ‘normal’ and ‘abnormal’ pulsations for HD179821 in the two time intervals 1991–1999 (JD2448500–JD2451500) and 2000–2006 (JD2451700–JD2453700) on account of the ratios. Smaller ratios than 2.2 belong to the ‘normal’ ones, as they are the majority as well. The mean Ampl$B$/Ampl$V$for HD179821 is 1.6$\pm$0.3 (st.dev., n=14), thus similar to that of $\rho$Cas (1.6$\pm$0.3). One peculiar pulsation, was excluded: the one of $\sim$JD2452500. Consequently, the mean ratio Ampl$V$/Ampl$(B-V)$=1.7$\pm$0.5 for HD179821 is similar to the one for $\rho$Cas. At least six pulsations of HD179821 are abnormal (the cases with light and colour curves running out of phase were excluded, as mentioned in Sect.2.5., although they are abnormal as well). Indeed, often the H$\alpha$ profiles of HD179821 offered proof for enhanced mass-loss episodes (Tamura & Takeuti 1991; Zacs et al. 1996); see also Arkhipova et al. (2001). We would like to mention one concrete case with Ampl$V$/Ampl$(B-V)$=6.2, and coinciding with an enhanced mass-loss episode based on the H$\alpha$ profile, observed at JD2452793, by Sanchez Contreras et al. (2008). –[HR8752]{}: This object showed a connection between enhanced mass-loss episodes and abnormal amplitude ratios during its cool YHG stage, especially between 1976 and 1981. At the time, Smolinski et al. (1989) identified emission features in the spectrum (see Figs. 9 and 10 of de Jager 1998), while $BV$ photometry indicated overly small $(B-V)$ amplitudes (caution is called for: the number of data points was at times rather low). The photometric observations were conducted by Moffett & Barnes 1979; Percy & Welch 1981; Walker 1983: Arellano Ferro 1985; Zsoldos & Olah 1985). Only for one light curve of HR8752 in 1976 (thus, after the 1973 eruption) were the maximum ($\sim$JD2443100) and the following minimum ($\sim$JD2443400) properly defined by $B$ and $V$ data points, for which the amplitude ratio was relatively abnormal pointing indeed to an enhanced mass-loss episode (Smolinski et al. 1989): Ampl$V$/Ampl$(B-V)$=6.7. In conclusion, we discovered that for YHGs, overly high Ampl$V$/Ampl$(B-V)$ values signal instable atmospheric conditions, sometimes supported by the presence of emission lines in the spectrum. The overly low $(B-V)$ amplitudes are likely caused by some gas layer with an enhanced opacity, and showing variability during the oscillations. The absorption is obviously wavelength dependent: more for $B$ (a rough estimation: of the order of 01 up to one magnitude. The higher the temperature, the higher the absorption) than for $V$, but the extinction law remains unknown. Furthermore, such a layer seems to be able to survive some successive pulsations, considering the continuation of abnormal amplitude ratios for pulsations with no noticeable emission lines. This appears to be the case for the time interval 1986-2000 of $\rho$Cas and for HD179821, considering the occurrence of a number of pulsations with abnormal amplitude ratios, while no emission lines were present in the spectra. A sketch of a few fictitious pulsations in $B$ and $V$ is presented in Fig.D.1. of AppendixD:2.6.: to illustrate this cyclic behaviour with the aid of three sets of amplitudes and their ratios. More evidence for a difference between T(Sp) and T(Phot) is given in AppendixE:2.6. The temperature scales: calculations of stellar properties. ----------------------------------------------------------- If one would like to derive stellar properties from the photometry of many types of stars, such as the temperature (e.g. to compute M$_{\rm bol}$, the radius R and its variation), a short discussion is needed on a number of temperature calibrations for all stars based on broadband continuum photometry. The Schmidt-Kaler (SK) calibration (1982) uses the $(B-V)_{0}$ to derive the T$_{\rm eff}$ and the BC and is valid for stars up to supergiants of luminosity class Iab only, simply because at the time there was a lack of well-calibrated super- and hypergiants. If one uses the SK calibration of Iab stars to derive temperatures for hypergiants, they will be too high. Therefore one should use the de Jager-Nieuwenhuijzen (dJN) calibration (de Jager & Nieuwenhuijzen 1987); see AppendixF:2.7. $\rho$Cas pulsations and the definition for long-term variations. ----------------------------------------------------------------- Despite the fact that stellar parameters from YHGs derived from photometry are expected to be unreliable (Sects.2.4–2.6), to start with the temperature derived from $(B-V)$, it would be of most importance to start a photometric and spectroscopic monitoring program (daily) for individual pulsations. In this way the trend of both temperatures can be studied, the purpose being to learn more about the absorption law. It would be even more rewarding if the mass loss rate were monitored as well, so that the explanation of the ‘cyclic absorption variation’ (Appendix, FigD.1.) can be verified. As only high-quality photometry is required for such a program, like the multi-colour photometry of Henry (1995, 1999; see Sect.2.2), we selected two fine individual pulsations from the $\rho$Cas sequence in Fig.2, that is, Figs.4 and 5; see AppendixG: 2.8. for an explanation of why spectroscopic monitoring is also indispensable. Figure2 shows the 2003–2017 $BV$ sequence of quasi-periods (bright is up). The magnitudes $V$ and $B$ are relative to the comparison star HD223173 ($V$=5.513, $B$=7.160). Similar high-quality observations made recently by one of the current authors (EJvB) are represented by dots on the very right (see Appendix B: 2.1). Long-term variations are the mean curves sketched through the median brightness and colour indices of all pulsations. Although already known and discussed by many researchers for decades, we are the first to suggest this definition. The numbers placed above individual $B$ pulsations in Fig.2 refer to pulsations which are mentioned in this paper. The LTV in $V$ runs almost constant over time, contrary to the one in $B$; it shows a wavy trend, with an amplitude of $\sim$02. This means that the individual pulsations in the maxima of the LTV ($B$) are on average bluer and hotter than in the minima. Additionally, the LTV in $B$ also shows a significant dip (including pulsation no.2, depicted in Fig.4), lasting $\sim$300$^{\rm }$ d, but in $V$ it is hardly visible. A similar type of relatively fast variation, independent of the pulsations, was identified in the light variations of the other three YHGs. Figs.4 and 5 show the finely detailed $V$ and $B$ light curves relative to the comparison star, with the extremely small scatter of 001–002. One can make out the slow contraction phase to maximum light and then the expansion phase (only partly) to minimum light of a very low-gravity photosphere with a huge dimension of a few astronomical units. We selected Fig.4 because of two spectral temperature T(Sp) determinations, made by Klochkova et al. (2014) (JD2455409 and JD2455463). One coincides with a gap in the time-series (dashed curves) representing the minimum, and therefore interpolated magnitudes are used below, and the second one coincides with the plateau. Below, T(Sp) is compared with the calculated photometric temperatures based on the SK and dJN methods using the observed $(B-V)$ obtained at the same time. The ascending branch in Fig.4 shows a plateau and a peculiar symmetrically dome-shaped maximum lasting 37$^{\rm }$ d with amplitudes of 008 and 005 in $B$ and $V$, respectively. Subsequently, the branch shows a decline almost equal to the rise, and then a plateau. After a gap in the time-series, a new pulsation started, but brighter than the previous one according to Fig.2. The pulsation in Fig.5 (with no T(Sp) determinations), labelled no.3 in Fig.2, showed almost the same surprising light-curve morphology. The duration of the dome-shaped maximum lasts longer, namely 65$^{\rm }$ d, and the amplitudes are larger: 0115 and 0070 in $B$ and $V$, respectively. The ascending branch is slightly bumpy (which is intrinsic) with a timescale of weeks. A great variety of secondary features exist in other pulsations of $\rho$Cas, that is, no.5 (Fig.2), which has a plateau in the ascending branch lasting almost two months, and no.6 (Fig.2) shows a maximum that is regrettably interrupted by a gap in the time-series and is flat and lasts no less than four months. One can only speculate as to the causes of all these features. Table2 summarises the calculated stellar properties for the selected locations in Fig.4. As we assume that the plateaus represent constant temperature, radius, and so on, we have chosen the start and the end of the plateau, together with the minimum and the maximum. The physical properties are horizontally sorted. Below the double line, we show the differences in time, $\triangle$JD, and the $V_{0}$ and $(B-V)_{0}$, the T(Sp) and the two calculated temperatures for Iab stars based on the SK calibration and for Ia$^{\rm +}$ stars using the dJN calibration. Table3 summarises for Fig.4, in different temperature columns, the BC, M$_{\rm bol}$, and the radius R (in R$_{\rm \odot}$) derived from the formula: logR/R$_{\rm \odot}$=8.47 - 0.2M$_{\rm bol}$ - 2logT$_{\rm eff}$. The symbol $\triangle$R represents the decrease, or increase of R (- or +, respectively) with respect to the previous location, and the V$_{\rm rad, puls}$ is in kms$^{-1}$ (+ for contraction and - for expansion) between two successive locations (vertically sorted). We note that they represent the mean velocity between the two locations. We used the accepted distance of 3.1kpc. As YHGs are in general non-radial pulsating stars (Sect.2.2.), the $\triangle$R, and V$_{\rm rad,puls}$ are lower limits. Location min plateau plateau max --------------- ------- --------- --------- ------- JD– 5409 5464 5496 5515 2450000 $\triangle$JD – 55d – 19d $V_{0}$ 3.28 3.12 3.13 3.07 $(B-V)_{0}$ 0.79 0.73 0.72 0.70 T(Sp) 5777K 6044K T(SK) 5450K 5640K 5660K 5710K T(dJN) 4890K 4980K 5000K 5040K : Extinction free visual brightness, colour indices and three types of temperature determinations for four instances in time in the ascending branch (two for the plateau: the first for the beginning and the second for the end of the plateau) of a pulsation of $\rho$Cas shown in Fig.4. We note that T(SK) and T(dJN) are based on the $(B-V)_{0}$, therefore, they are photometric temperatures T(Phot).[]{data-label="table:2"} --------------------- -------------------- -------------------- -------------------- T(Sp) T(SK) T(dJN) [min]{} [min]{} [min]{} 5777K 5450K 4890K BC -0.13 -0.16 -0.20 M$_{\rm bol}$ -9.31 -9.35 -9.39 R 668R$_{\rm \odot}$ 779R$_{\rm \odot}$ 970R$_{\rm \odot}$ T(Sp) T(SK) T(dJN) [plateau]{} [plateau]{} [plateau]{} 6044K 5640K 4980K BC -0.10 -0.14 -0.19 M$_{\rm bol}$ -9.43 -9.48 -9.53 R 636R$_{\rm \odot}$ 747R$_{\rm \odot}$ 981R$_{\rm \odot}$ $\triangle$R -22R$_{\rm \odot}$ -7R$_{\rm \odot}$ +27R$_{\rm \odot}$ V$_{\rm rad, puls}$ +3.2kms$^{-1}$ +1.0kms$^{-1}$ -4.0kms$^{-1}$ T(Sp) T(SK) T(dJN) [plateau]{} [plateau]{} [plateau]{} 5660K 5000K BC -014 -0.18 M$_{\rm bol} $ -9.46 -9.51 R 735R$_{\rm \odot}$ 964R$_{\rm \odot}$ T(SK) T(dJN) [max]{} [max]{} 5710K 5040K BC -0.13 -0.17 M$_{\rm bol}$ -9.52 -9.56 R 742R$_{\rm \odot}$ 971R$_{\rm \odot}$ $\triangle$R +7R$_{\rm \odot}$ +7R$_{\rm \odot}$ V$_{\rm rad, puls}$ -3.0kms$^{-1}$ -3.0kms$^{-1}$ --------------------- -------------------- -------------------- -------------------- : BC, M$_{\rm bol}$, and radius R for four locations in the ascending branch (two for the plateau: the first for the beginning and the second for the end) of the pulsation of $\rho$Cas shown in Fig.4, the radius variation $\triangle$R and contraction/expansion velocity $V_{\rm rad, puls}$. The latter represents the mean velocity between the selected locations. We note that they represent pure radial velocities of the continuum radiating layer, and are independent of the space velocity. []{data-label="table:3"} The summary below lists the average differences between the calculated temperatures and radii based on different temperature scales. The average differences in T and R according to Tables2 and 3 and based on two temperature scales are: T(Sp)–T(dJN)$\sim$990K, R(Sp)–R(dJN)$\sim$-300R$_{\rm \odot}$. According to Klochkova (2014) and Klochkova et al. (2014), the uncertainties in T(Sp), based on ratios of selected spectral lines being sensitive temperature indicators, are in the range of 40–160K only. Considering the temperature differences we report, as shown in Tables2 and 3, these differences are negligible. Therefore, the differences between T(Sp) and T(Phot) (=T(dJN) and T(SK)) in Tables2 and 3 are due to the effect we discovered, and for which we offer an explanation in AppendixFig.D.1. Obviously, the observed $(B-V)$ is too red, and is therefore inappropriate for deriving accurate temperatures and all other stellar properties. For example, the trend of the calculated photometric radii at the four locations in Fig.4, especially for the YHG calibration, are incorrect. They clearly run opposite to radius variations of YHGs, and should decrease towards maximum brightness (=contraction, plus sign) instead of increasing. Its ratio Ampl$V$/Ampl$(B-V)$=2.3, meaning that the $B$ amplitude with respect to $V$ amplitude is indeed too small (Sect.2.6.). This suggests that absorption increases with temperature: at maximum light the absorption of $B$ increased with respect to the absorption in $V$. The V$_{\rm rad}$ value of 3.2kms$^{-1}$ (second column), which is based on spectroscopy and should be multiplied by 1.4 to correct for projection effects, and so on, could become of the right order if the overly low radius variation (for which the correction is unknown) is also taken into account: Lobel et al. 2003: 5–10kms$^{-1}$ and Klochkova et al. 2014: 7kms$^{-1}$. The above is in support of the existence of a cyclic absorption behaviour (AppendixFig.D.1.): as a consequence, the photometry of $\rho$Cas (and of all YHGs) does not represent a normal YHG photosphere. Stellar properties based on photometric data weakened by absorption with an unknown absorption law are unreliable. Again, we estimated that the absorption in $B$ is of the order of 01 up to a few times this value; in $V$ the absorption is presumably about a factor two smaller, but that is very uncertain. The higher the mass-loss rate and the higher the temperature, the higher the absorption. The photometric parameters of the pulsation in Fig.5 (no spectral temperatures known), of which the amplitude ratio is 1.2, and thus ‘normal’, are less inconsistent, yet the calculated temperatures (dJN) are still too low by hundreds of degrees with respect to the T(Sp) in Fig.4, while both pulsations do not differ much with respect to the medium brightness. Although the photometry of the other YHGs are less accurate, our conclusions were similar. Therefore, we suspect that a high-opacity layer is always present in the atmospheres of YHGs, and its absorption capacity depends on the density of that layer (higher by enhanced mass loss) and on the temperature (higher by a temperature rise), and vice versa. The magnitudes of $\rho$Cas, used in Tables2 and 3 offer a chance to calculate errors due to scatter as small as $\pm$001. The result is approximately $\pm$15K and $\pm$10R$_{\rm \odot}$. The relatively large size of these errors demonstrates the extreme sensitivity of the calculated stellar properties to ‘very small’ photometric errors. Our message is that this analysis underlines the importance of a long-lasting photometric and spectroscopic monitoring all over the world; see AppendixG: 2.8. Proposed distance reductions for $\rho$Cas and HR5171A. ------------------------------------------------------- Below we propose shorter (conventional) distances for $\rho$Cas and HR5171A. We ignore the distances based on parallaxes by the satellites Hipparcos and Gaia (Sect.1). The size of the radii derived from the spectroscopic and radial velocity monitoring campaign during the eruption in the year 2000 motivated us to question whether the accepted distance of $\rho$ Cas by Zsoldos & Percy (1991) of 3.1kpc$\pm$0.5kpc is too large. We note that the sizes of the radii obtained in this way were derived independently of the distance by Lobel et al. (2003). They estimated that the extreme dimensions of the stellar radius at the eruption maximum and at the deep minimum brightness were $\sim$400R$_{\rm \odot}$ and $\sim$1000R$_{\rm \odot}$, respectively. The calculated mean radius of an average pulsation with a normal amplitude in $V$ of for example 021, is $\sim$960R$_{\rm \odot}$ (dJN), which is too close to the radius for the deep eruption (photometric) minimum (when the star has reached its maximum size) to be plausible. Radii based on the SK calibration are about 200R$_{\rm \odot}$ smaller, but this is definitely only valid for SGs of type Iab. If de Jager & Nieuwenhuijzen (1987) were found to have assumed overly extreme properties for hypergiants relative to the Iab SGs, then a reduction of the radius difference above from 200R$_{\rm \odot}$ to say 100R$_{\rm \odot}$ would yield a radius of 860R$_{\rm \odot}$ (instead of the above 960$_{\rm \odot}$), which would still be too high to be credible. Therefore, the error on the dJN radius cannot be much more than that on the SK radius. Other options are to decrease the interstellar extinction from 144 (Table1) to 10 for example, and the reddening from 045 to about 030, but these values are not likely. Our choice of 2.5kpc$\pm$0.3kpc is rather arbitrary. The M$_{\rm bol}$ becomes about 045 less negative (luminosity lower), and the calculated radii are reduced by a factor of $\sim$1.2: $\sim$770R$_{\rm \odot}$, as well as the radius variations and the radial pulsation velocities. For us, these values are more acceptable with respect to the derived extreme dimensions above. This case serves as a reminder that our attempts to obtain more precise information on YHGs are seriously hampered as long as distances remain uncertain, not only for $\rho$Cas, but also for HR5171A below, and HD179821. The accepted distance of HR5171A is 3.6kpc, assuming that it belongs to Gum48d and that HR5171B, at a distance of 10" is a nearby companion: optically or physically (Humphreys et al. 1971; Schuster 2007; Schuster et al. 2006). However, subsequently M$_{\rm bol}$=-10 – -11 depending on the reddening (10–14) and reddening law used (3.1–3.5), creates a problem, as this is much too bright for a hypergiant. As far back as 2013 one of us (HN) emphasised that the accepted distance is too large. As a result, the radius would then also be much too large: between 6 and 12AU. We think that this is too big, until new outcomes are undisputable (van Genderen et al. 2015). A distance of between 1 and 2kpc for example, would entail a luminosity that is more in accordance with its Ia$^{+}$ hypergiant character, for example with M$_{\rm bol}$ lying between -8.7 and -9.6 (Sect.1) and a radius between 3 and 5AU. We do not think that HR5171A is a RSG as advocated by Wittkovski et al. (2017a); see Sect.2.2. HR5171A shows too many similarities with $\rho$Cas for example, like its eruptive activity, and above all, the spectra of the Big Three are almost identical. However, the H$\alpha$ line of HR8752 strongly differs with respect to the shape because of the permanent presence of a strong emission line. Thus, the physical properties of the winds of the Big Three are in many respects identical (Lobel et al. 2015); in any case, RSGs do not show eruptive events. A portion of the light curve between JD2447200 and JD2448800 (1988–1992) was spectroscopically monitored, resulting in a well-covered V$_{\rm rad}$ curve with an amplitude of 10kms$^{-1}$ (Lobel et al. 2015). This correlated very well with the $V$ magnitudes between the maxima of pulsations16 and 17 (numbered according Fig.1 in PaperI), indicating a steady contraction towards maximum brightness and the reverse to the brightness minimum. We note that at the time HR5171A had reached its faintest brightness: $V$$\sim$70 and reddest colour so far: $(B-V)$$\sim$2.6 (Figs.13 and 14). AppendixH.2.9. presents a description of a method which is in principle suitable for deriving the distance of a pulsating star with the aid of the calculated V$_{\rm rad puls}$ and the spectroscopically obtained V$_{\rm rad}$, but $B$ and $V$ magnitudes of the pulsation cycle should be undisturbed. Another argument favouring a much shorter distance than 3.6kpc is based on the fact that at this distance, the position of HR5171A would be below the Humphreys-Davidson limit on the theoretical HR diagram (shown in Fig.6 of PaperI) instead of far above it. At below 3.6 kpc, the distance with respect to the semi-empirical P=constant line for $\sim$500 d (the mean quasi-period of HR5171A between the 1960s and 1990s; van Leeuwen et al. 1998) would be more in accordance with the positions of the other variable supergiants. Further, the variability of the pulsation quantity Q=P$\rho^{-1/2}$ appeared to be compatible with stability studies of stellar models and was first described by Maeder & Rufener (1972). Based on that study, Burki (1978) derived a formula between the dependency of the quasi-period P on the mass M/M$_{\rm \odot}$, M$_{\rm bol}$ and on T$_{\rm eff}$. Applying in his Eq.(5) the approximate but plausible input parameters 25M/M$_{\rm \odot}$, -8.8, and 4300K, respectively, the quasi-period becomes P = 473$^{\rm d}$, that is, roughly of the same order as the observed mean quasi-periods (see Fig.13). By varying the input parameters by trial and error, say by $\pm$5M/M$_{\rm \odot}$, $\pm$02 and $\pm$200K, respectively, in either directions, results are of the same order. A third argument for a shorter distance refers to the energy budget of the Gum48d nebula. Schuster (2007) considered HR5171B as the single central engine of the associated HII region RCW80, just like Karr et al. (2009). Based on calculations on the present energy budget, the latter concluded that the nebula only needed one single ionizing O-type star, that is, the present B-type star HR5171B, only a few million years ago still on the main sequence, just like HR5171A at the same time. In other words, for the present energy budget of Gum48d, the presence of HR5171A as a member of Gum48a is superfluous. If HR5171A is indeed a foreground star, a new place of birth should be found. On the contrary, Humphreys et al. (1971) and Schuster (2007) offered a number of arguments based on various independent techniques and physical considerations favouring the larger distance. Therefore, the distance remains debatable for the time being. In conclusion, if our preferred distance of 1.5kpc$\pm$0.5kpc were found to be correct, it would dethrone HR5171A as one of the biggest stars known (Chesneau et al. 2014; Wittkovski et al. 2017c). This distance reduction would also reduce the flux of the reported Blue Luminescence by PAH molecules in the 1970s (van Genderen et al. 2015) by a factor $\sim$6. This is because the excesses were measured relative to the stellar flux in the $L$ channel and would stay unaffected for both scenarios (1) and (2): i.e. the source of the excess lies in Gum48d, or in the outer envelope of HR5171A, respectively. HR5171A possesses an optically thick extended molecular and dust envelope at about 1.5 stellar radii (Wittkovski et al. 2017a). According to studies by Gorlova et al. (2009) and Oudmaijer & de Wit (2013) a dense optically thick layer, formed by the wind just above the photosphere is able to shield the stellar radiation as well. Thus, there are ample possibilities to locate neutral PAH molecules to become excited to the upper electronic state by high energetic photons (3.5–5eV, A.N. Witt 2013, priv.comm). There after, these molecules recombine and emit within a near-UV band (coinciding with the $L$ band of the Walraven $VBLUW$ photometric system) radiation, called blue luminescence, discovered by Vijh et al. (2004) in the Red Rectangle nebula. The long-term variations ======================== Light and colour curves, and trends of light amplitudes and quasi-periods. -------------------------------------------------------------------------- In this section we outline the photometric properties of LTVs (on which the pulsations are superimposed and that was defined in Sect.2.8.) exhibited by the four selected objects. Our interest is also focussed on their possible influence on pulsations and the occurrence of eruptions which has never been explored before. Light and colour curves of LTVs often appear whimsical with timescales of hundreds up to a few thousand days. The evolutionary models usually reveal straight tracks on the HR diagram. Yet, we wondered whether LTVs represent in fact the evolutionary tracks, which once transformed into luminosity and temperature might not be straight tracks on the HR diagram. Whether straight or irregular, the subsequent zigzag movements of the YHGs on the HR diagram are supposed to be part of the need to loose enough mass for the final course to the blue (Sect.1, Fig.1). These LTVs have already been noted and discussed by numerous researchers and observers in the past for HR5171A (see for references AppendixI:3.1.), but generally they were reticent about a possible evolutionary origin, contrary to our present suspicions based on the thorough study of HR8752 from 1850 until 2005, including the eruptive episode from 1973 by Nieuwenhuijzen et al. (2012). These latter authors convincingly showed that the long-term variations of HR8752 shown in our Figs.6 and 7 are evolutionary tracks, or loops: a red one until the eruptive episode in 1973, and then a blue one (Figs.5, 6, 10, etc. in Nieuwenhuijzen et al. 2012). These evolutionary loops (RL and BL) represent the red and blue tracks (no.3 and no.4) of the models of Meynet et al. (1994), respectively. The most obvious conclusion from the study by Nieuwenhuijzen et al. is that the LTVs of the other three objects should have something to do with evolution as well. To prove that conjecture, we scrutinized the literature for scattered photometric observations of pulsations from which the LTV at the time could be derived. More or less complete light and colour curves were put together (thus curves sketched through the median brightness and colour). As this has never been done before, and because of the importance of these curves we show them in their entirety in Figs.6–14. In Tables4 and 5 of Sect.3.2 we tabulate the characteristic zigzag changes of brightness and colours, still referred to as ‘evolutionary modes’, and their timescales. Large empty intervals in time series, portions with bad sampling, and the presence of eruptive phases decrease the reliability of sketched LTVs, but generally they cannot be grossly in error. In the case of scattered historic visual or photographic observations of stars dating back to 1850, like in the case of HR8752 (Luck 1975; Arellano Ferro 1985; Zsoldos 1986a, b; Nieuwenhuijzen et al. 2012), we connected individual data points and the averages of small clusters of data points. Because of their importance these curves merit careful description. [–HR8752:]{} Figures6 and 7 depict the LTVs for HR8752 in $V$, $B,$ and $U$, and $(B-V)$ and $(U-B)$, respectively, for the time interval 1941–1994. Based on a collection of numerous historical data since 1840, and modern photometric data of HR8752, Nieuwenhuijzen et al. (2012) concluded that HR8752 evolved through an increasing reddening and cool episode in the 1960s and 1970s. It is noteworthy that the corresponding red minimum in our Fig.7 is also obvious in the logT$_{\rm eff}$/JD diagram of Fig.10 of Nieuwenhuijzen et al. (2012). These latter authors suggested that a massive eruption should have happened around 1973, although photometry was absent. Their assumption was based on spectroscopic observations made at the time. Luck (1975) derived from spectral scans a temperature of 4000K in August 1973, and $(B-V)$=1.78 in November 1974 but alas, by the scarcity of photometric $B$ and $V$ magnitudes, we could not calculate T(Phot), despite the expected amplitude of $\sim$10. This can be explained by the fact that the gaps in the time-series are much larger than the duration of an eruption: 400$^{\rm d}$–700$^{\rm d}$. Additionally, we found out that light amplitudes and quasi-periods of HR8752 since 1976 decreased linearly until 1993 to about 005–01 and 100$^{\rm d}$–150$^{\rm d}$, respectively. This is a consequence of the continuous contraction of the star and the increase of its atmospheric density. [–HD179821:]{} The first record of an LTV of HD179821 started in 1899 (JD2416000) with a photographic $B_{\rm pg}$ dataset until 1989 (collected by Arkhipova et al. 2001). Individual data points, or small concentrations of data points are connected by a line (Fig.8). This LTV is of course less reliable between 1899 and 1989, yet it gives a good impression of the instability of the star at the time and its cool phase. Unfortunately, no details of a possible eruption were observed, but we consider it not unlikely that one occurred somewhere between 1925 and 1960. Our assembled data show on average a gradual rise in light amplitude of $\sim$05. There is some overlap with modern observations by means of two data points which precisely match the modern $B$ magnitude sequence made photoelectrically until about 1989 ($\sim$JD2448000, Figs.9 and 10). Therefore, there is no significant reason to mistrust the reliability of the observations dating back to 1899. Thereafter, the photometric instability from the visual to the UV of the LTVs was very small: 01–02, until 2009 (JD2453000). It appears that the $B_{\rm pg}$ was 9.5 in 1899 ($\sim$JD2415000), declined to a deep minimum with $B_{\rm pg}$10 around 1925 ($\sim$JD2424000), and was at a second deep minimum around 1960 ($\sim$JD2437000). The $B$ in Fig.9 gradually rose to a magnitude of 9.4–9.3 in 2009 ($\sim$JD2455000), showing small fluctuations, at most by $\pm$01, just like in $V$. The modern portion of the LTV in $U$ (Fig.9) from JD2448000 until JD2452500 (1990–2003) shows that the star was still rising in brightness by 04. The gradual blueing of $(B-V)$ in Fig.10 between JD2448000 (1990) and JD2453000 (2004) and the subsequent reddening until 2009 is of crucial importance for the interpretation of the evolutionary state of HD179821 below. This reddening trend is supported by the new $UBV$ photometry until 2017 by Ikonnikova et al. (2018). If HD179821 is indeed a YHG an eruptive episode can be expected within a few decades. Noteworthy are two bumps and dips in the LTVs for $V$, $B,$ and $U$ in Fig.9 between JD2449000 and JD2450500. Sometimes $UB$ and $V$ run in opposite directions. The LTVs in $JK$ shown in Fig.6 of Arkhipova et al. (2009) mimic the one in $V$. The same is more or less the case for the bumps and dips. [–$\rho$Cas :]{} Figure11 shows the LTVs of $\rho$Cas for the $UBVRI$ data between 1965 and 2015. The arrows at the top indicate observed eruptive episodes. The mean quasi-periods with standard deviations and the number of cycles used (bracketed) are given in the second panel. The selected time intervals are indicated by the length of the bar. Figure12 shows the LTVs for the colour indices $(B-V)$, $(U-B)$, $(V-R),$ and $(V-I),$ the magnitude scale of which is about three times larger than in Fig.11. The LTVs are based on the observations by many observers (Sect.2.1.), but the majority are by Henry (1995, 1999). The preference of eruptions to occur when the star is relatively faint and red is obvious. The correlation between the trends of the quasi-periods and the $(B-V)$ is obvious and depends on the type of the evolutionary track: increasing and decreasing during RL and BL evolutions, respectively (Figs.11 and 12), just like HR8752. [–HR5171A :]{} For HR5171A we had the disposition of photographic $B$ magnitudes (Johnson system) to reconstruct its photometric history from 1900 until 1950 and between 1978 and 1989. They were obtained from Harvard plates of which the scanning was made within the framework of the DASCH scanning project. The measurements were kindly put at our disposal by Dr. Josh Grindlay, head of this project, before the official release of the data. The position of HR5171A was close to the edge of the plate. Probable errors are 02–03. It should be noted that the Harvard magnitudes include the nearby blue optical companion HR5171B (10$\arcsec$), which is likely not a physical companion, but a member of Gum48d, contrary to HR5171A, at least if its proposed shorter distance in Sect.2.9. is correct. Its contribution to the total brightness is, depending on the brightness of HR5171A, smaller than $\sim$ 025. According to Stickland & Harmer (1978), HR8752 also has a nearby B-type companion, but that one is a physical companion; see also Nieuwenhuijzen et al. (2012, Sect.1.1. The brightness limits ($B$=84–98) and the mean magnitude (91) of the data points in the 1900–1950 set which is not shown in the present paper appear to be similar to the modern ones after 1952 (Fig.1 in PaperI, and Fig. 3 in Chesneau et al. 2014). These papers refer to the original sources like those of Harvey 1972; Humphreys et al. 1971; Dean 1980; from ASAS-3 by Pojmanski 2002; from AAVSO amongst others by Otero; and from the Long-Term Photometry of Variables (LTPV) group: Sterken 1983, Manfroid et al. 1991, Sterken et al. 1993; from Hipparcos discussed by van Leeuwen et al. 1998. The Harvard $B$ data collected in the time interval 1978–1989 match the photoelectric data satisfactory. Our conclusion is that HR5171A did not alter its variability pattern during the last 120yrs. Figures13 and 14 represent the LTVs of HR5171A in $B$ and $V$, and in $(B-V)$, respectively, between 1953 and 2018: the black lines connecting the black dots. These latter figures are mainly based on the papers by van Genderen (1979 and 1992 Fig.1) and Chesneau et al. (2014) discussing the photometric history of HR5171A. The meaning of the other curves in Fig.14 is explained in Sect.3.4. Between 1994 and 2012 observations were made only by visual observers, but the means, represented by the dashed oscillating curves fit the sketched LTV in $V$ reasonably well. However, some systematic deviations can be attributed to the fact that for the construction of part of the LTV (between JD2450900 and JD2455500), we relied mostly on those made by Otero (see Chesneau et al. 2014) and on the All Sky Automated Survey-3 (ASAS) data (Pojmanski 2002; see Fig.3 in Chesneau et al. 2014). Systematic differences in the magnitude scales of different observers amounting to a few 01 are normal. The last set of $BV$ photometry (2013–2018) is listed in TableI.1 of the Appendix. Only three mean quasi-periods could be derived (top of Fig.13), as reliable pulsations between 1995 and 2012 were lacking. As a consequence of the above, and the fact that after the 1975 eruption the star became heavily obscured (discussed in Sect.3.4.1.) no reliable correlation between mean quasi-periods and colour could be determined. On the contrary, the third mean quasi-period on the very right has become significantly shorter which is in agreement with the observed blueing trend until the last observations of 2018. The 1975 eruption named after Dean (1980) was only observed during the brightness rise after a deep minimum. The second eruption of HR5171A, including the deep minimum, was well observed, but only in the visual: the Otero minimum of 2000. The maximum just before the deep decline is the eruption maximum. Alas, multi-colour photometry and spectroscopy were lacking. The arrow at JD2447300, 1988 (see Fig.13), is a local minimum of the LTV during the $VBLUW$ photometric time series (Fig.3 upper panel), containing three pulsation cycles nos.15–17 (in Fig.1 of PaperI). The deepest visual pulsation minimum is called the Jones-Williams minimum of 1995 (JD2449750, Fig.3 upper panel). The smoothed LTV in the $VBLUW$ system of Walraven is shown and discussed inFig.J.1 of the Appendix. Brightness and colour ranges of the red and blue ’evolutionary modes’. ---------------------------------------------------------------------- TableJ.1 of the Appendix presents a summary of the maximum ranges of the LTVs in brightness and colour index. These ranges serve a useful purpose, amongst others, giving a more precise determination of the actual evolutionary trends on the HR diagram in a later paper. Those of HD179821 are smallest. The largest ranges for the four objects occur in $U$ and $B$, that is, up to 12, and the smallest ones occur in $R$ and $I$. Those in $(U-B)$ are larger than in $(B-V)$. Another property is that the trends of the light variations in different pass bands often run in variable directions, and therefore in various ‘modes’. For example, $V$ and $B$ run in opposite directions and with different rates for a few years, then both decline in brightness with equal rates for another couple of years, after which $V$ declines much faster than the rise in $B$, and so on. We determined the evolutionary modes for HR8752, $\rho$Cas, and HD179821 and listed them chronologically in Table4, concentrating on those in $V$ and $B$ only, as $(B-V)$ is (apart from the reddening effect by the high-opacity layer) a temperature indicator. The red and blue evolutionary loops of HR8752 are shown in Fig.6 in Nieuwenhuijzen et al. 2012. The modes for HR5171A are collected in Table5, among which there are a few short-term modes. These tables list the type of modes with respect to the trends of $V$ and $B$ (brightness up, down, or constant), and of $(B-V)$ (to the red or to the blue, the latter in boldface), and the dates, JDs, and their durations. ----------- ---------------- -------------- ----------- ------------- ---------- Star Mode Trend Dates JD- Duration of $(B-V)$ 2400000 (d) HR8752 $B_{\rm pg}$-d 1899–1923 15000-23600 8600 $B_{\rm pg}$-u 1923–2001 23600-52000 28400 $V$-uu B-u red 1942–1950 30500–33500 3000 $V$-d $B$-d const 1950–1957 33500–36000 2509 $V$-d $B$-d red 1957–1976 36000–43000 7000 $V$-d $B$-u [blue]{} 1976–2005 43000–53500 10500 $\rho$Cas $V$-u $B$-d red 1965–1982 34000–45000 6000 $V$-d $B$-dd red 1982–1986 45000–46500 1500 $V$-u $B$-uu [blue]{} 1986–1991 46500–48500 2000 $V$-c $B$-d red 1991–1998 48500–51000 2500 $V$-d $B$-dd red 1998–2000 51000–51300 300 $V$-c $B$-uu [blue]{} 2001–2009 52000–55500 3500 $V$-c $B$-dd red 2009–2013 55000–56500 1500 $V$-c $B$-u [blue]{} 2013–2015 56500–58100 1600 HD179821 $V$-d $B$-u [blue]{} 1990–1993 48000–49200 1200 $V$-dd $B$-uu [blue]{} 1993–2004 49200–53200 4000 $V$-c $B$-d red 2004–2008 53200–54500 1300 ----------- ---------------- -------------- ----------- ------------- ---------- ---------------- -------------- ----------- ------------- ---------- Mode Trend Dates JD- Duration of $(B-V)$ 2400000 (d) $V$-c $B$-c const 1953–1958 34400–36400 2000 $V$-c $B$-d red 1958–1962 36400–37800 1400 $V$-d $B$-dd red 1962–1965 37800–38800 1000 $V$-d $B$-dd red 1965–1969 38800–40400 1600 $V$-d $B$-dd red 1969–1973 40400–42000 1600 ($V$-dd $B$-d) [blue]{} 1973–1975 42000–42500 500 $V$-uu $B$-u red 1975–1976 42500–43000 500 $V$-u $B$-uu [blue]{} 1976–1977 43000–43200 200 $V$-c $B$-d red 1977–1981 43200–44800 1600 $V$-c $B$-c const 1981–1985 44800–46100 1300 $V$-d $B$-d const 1985–1988 46100–47300 1200 $V$-uu $B$-u red 1988–1991 47300–48400 1100 $V$-dd $B$-d [blue]{} 1991–1995 48400–50000 1600 $V$-dd $B$-d [blue]{} 2013–2018 56330–58140 1810 ---------------- -------------- ----------- ------------- ---------- Red and blue loop evolutions: their $(B-V)$ ranges and quasi-period trends. ---------------------------------------------------------------------------- Below we derive approximate evolutionary tracks, and an important quantity: the range in $(B-V)$ during the evolution, and thus the variation of T$_{\rm eff}$, assuming that it is almost independent of the atmospheric absorption (Sects.2.4.–2.6.). Additionally, we try to establish the true nature of HD179821 and discuss its probable isolated position in space. The study of HR8752 by Nieuwenhuijzen et al. (2012) revealed that HR8752 evolved along a red loop evolution and after the 1973 eruption along a blue loop evolution (hereafter called RL and BL evolutions). Therefore, all red modes and all blue modes listed in Tables4 and 5 and preceding and following an eruption represent shorter pieces of one RL and one BL evolution, respectively. This probably means that the evolution tracks on the HR diagram are not always straight, but show irregularities. Below, we firstly discuss these RL and BL evolutions of HR8752, $\rho$Cas, and HD179821. [–HR8752:]{} The construction of a RL from 1850 (best documented from 1895) until the 1973 eruption and subsequently the BL until 2005, are based on a mix of different observational techniques (Nieuwenhuijzen et al. 2014, their Table4 and Figs.5 to 24). The $(B-V)$ ranges are 016 (1895–1963) and 064 (1976–2005). Between 1963 and 1976, no $BV$ photometry was available. Observed and model timescales (by Meynet et al. 1994), that is, a few decades, are of the same order. Multi-colour photometry has been absent since 1993, but spectral observations were made until between 2000 and 2005 from which T$_{\rm eff}$ and logg from Kurucz’s LTE models could be derived (Nieuwenhuijzen et al. 2012, their Table4 and Fig.2). During this BL evolution, temperatures and gravities fit the extrapolation line of the gradual rise of T$_{\rm eff}$ up to 8000K and logg from 1993 until 2005 very well. Nieuwenhuijzen et al. (2012) claimed that HR8752 left the first instability region on the HR diagram, crossed the Yellow Void (Fig.1), and then was on its way to stability further to the blue, and that during further evolution it must still go through the second potential unstable region, where He starts to ionize. However, as mentioned before, the recent $BV$ photometry (by EJvB, AppendixB:2.1.) indicates that in 2017/2019 the $(B-V)_{0}$ of the star stopped blueing, but is still in the Yellow-Blue Void (Fig.1), and shows some variability in brightness and colour owing to its very-low-amplitude pulsations. An additional analysis made by us of the AAVSO $V$ (Johnson) observations of HR8752 from 1993 until 2019 is presented in AppendixK:3.3. [–$\rho$Cas :]{} According to our collection of fragments of LTVs from numerous papers, $\rho$Cas has passed through three RL and two BL evolutions since the $BV$ photometry started in 1968; see the $(B-V)$ curve in Fig.12: a RL from 1968 to the 1986 eruption triggering a BL until about 1991, succeeded by a RL until the 2000 eruption, triggering the next BL until about 2008, then a RL until the end of the figure in 2018. Also, here the timescales are one to a few decades only. The ranges in $(B-V)$ of two RL and two BL evolutions lie between 020 and 045, meaning that the temperature ranges can be derived after correction for reddening (Table1). They lie in the range of 5200K to 4500K (excluding the extreme temperatures during the eruption maxima and deep minima of about 7500K and 4000K, respectively). Based on the existence of such a ‘sequence of events’, likely valid for all YHGs, we conclude that a third blue loop of $\rho$Cas must have taken place after the massive 1946 eruption (light curve in visual and photographic magnitudes by Gaposhkin 1949). Indeed, one to two decades later, in the 1960s and 1970s (in Figs.11 and 12 on the very left), the spectral type was F8, which is relatively blue (Humphreys 1978; de Jager et al.1988; Arellano Ferro & Mendoza 1993). The LTV at the time $(B-V)_{0}$=0.60, which corresponds with T$_{\rm eff}$$\sim$6000K. According to Fig.12, the blue loop after the 2000 eruption reached roughly the same $(B-V)_{0}$ and consequently the same temperature as the eruption of 1946 until about 1968 (the latter date on the very left of Fig.12). Whether this was also the case for the blue loop after the 1986 eruption is uncertain, because of some inconsistency between the $(B-V)$ based on $BV$ photometry (dots in Fig.12) and the $(B-V)$ based on a transformation from the $VRI$ photometry (circles in Fig.12). It appears that an inconsistency exists between the trends of dots and circles which is disappointing, but usually the transformation yields more satisfactory results. [–HD179821 :]{} One glance at the light curves of this object was enough to recognize a BL evolution. It is staggering how closely the past history of HD179821 imitated that of HR8752, even if one compares the pulsation properties until 2018 made with the aid of the AAVSO Light-Curve Generator. The amplitudes and timescales of the quasi-periods are smaller and shorter (100$^{\rm d}$–150$^{\rm d}$), respectively, as far as the visual observations of HR8752 allow the comparison with the $UBV$ photometry by Arkhipova et al (2001, 2009), Le Coroller et al. (2003) of HD179821. The pulsations of both objects also share some photometric morphological properties. Ikonnikova et al. (2018) reported that the quasi-periods of HD179821 became longer than 250d between 2010 and 2017, which can be expected when evolving along an RL evolution! The BL of HD179821 started somewhere in the time interval 1925–1960, because at the time the star was very faint in B$_{\rm pg}$, and therefore very cool; see Fig.8. The blue loop ended in 2004 (about JD2453000) when $(B-V)$ reddened again, marking a new RL evolution (Figs.9 and 10). The range of the colour $(B-V)$=022. Given the characteristic ‘cyclic sequence of events’ for HR8752 and HD179821, an eruption could have happened somewhere between 1925 and 1960, but like in the case of HR8752 (Sect.3.1.) stayed unnoticed. The timescale is uncertain, but is approximately five decades, and therefore not in disagreement with theory (the gaps in the time series of Fig.8 are in most cases much longer than the duration of eruptions: 400$^{\rm d}$–700$^{\rm d}$). The bluest colour (corrected for IS extinction) of HD179821 in 2004 was $(B-V)_{0}$$\sim$055, its reddest colour in 1990, 077. Hence, the maximum photometric temperature is T(Phot)=5290K (dJN), while at the time the mean spectroscopic temperature according to Arkhipova et al. (2009) was T(Sp)= 6800K$\pm$50K (a difference of 1500K, see Sects.2.4.– 2.6.). According to its position at the time in Fig.1, it stopped just before entering the Yellow-Blue Void and was heading back according to the available observations until 2009 (Fig.10). With the Straizhys (1982) temperature calibration T(Phot) a comparable result was obtained by Arkhipova et al. (2009): 5400–6000K. (As noted before new $UBV$ photometry by Ikonnikova et al. (2018), HD179821 continued its reddening trend until the end of the observations in 2017). All the similarities above suggests that HD179821 is not a RSG, or a post-AGB star as is often claimed, but presumably a YHG at a very large distance (M$_{\rm bol}$$<$-8), supporting conclusions by Gledhill et al. 2002, Gledhill & Takami 2001, Molster et al. 2002, Arkhipova et al. (2009), and Oudmaijer et al. (2009). However, as a consequence, the large z-distance to the galactic plane creates kinematical problems, which can only be solved by a reliable trigonometric parallax (Arkhipova et al. 2009). Its large distance and its isolated position with respect to its place of birth (a young stellar cluster) is not unusual: see Smith & Tombleson (2015: LBVs are ‘anti-social’). The isolated positions of many LBVs (the probable descendants of YHGs) are remarkable. Smith & Tombleson believe that LBVs are flung out of their former binary orbit by a SN explosion. See also the related discussions by Humphreys et al. (2016) and Aghakhanloo et al. (2017). In the case of the suggested distance reduction for HR5171A, as discussed in Sect.2.9., its position also becomes isolated, as it will no longer be a member of the young stellar cluster Gum48d. Our conclusions are that the observed $(B-V)$ ranges for the BL evolution lie roughly between 02 and 06, with T$_{\rm eff}$ roughly between 8000K and 4000K, roughly corrected for the too red $(B-V)$ values (TableA.1 of the Appendix). Other novelties are that HD179821 is likely a YHG, and that all four YHGs seem to be isolated objects. HR5171A - a special YHG ----------------------- ### The mysterious reddening episode 1960–1981. The most confusing property of HR5171A for a generation of researchers was the exceptional reddening episode by 07 in $(B-V)$ observed between about 1960 ($\sim$JD2436000) and 1981 ($\sim$JD2444000); see Figs.13 and 14. This happened before and after the 1975 eruption. After 1981 the reddening increase came to an end. Below we offer a plausible explanation with the aid of Fig.14. (Its caption explains the meaning of the other curves: only schematically sketched). Firstly, the long-lasting steep reddening episode starting around 1960 until 1981 cannot be caused by an increasing dust extinction, as the $B$ decline relative to that of $V$ happened much too quickly. Secondly, because of the quick succession of three short-lasting modes: 500$^{\rm d}$, 200$^{\rm d}$ and 500$^{\rm d}$), a blue, a red, and a blue one, respectively, between 1973 and 1977 (Table5). Obviously, something exceptional happened at the time, disturbing the brightening phase after the 1975 eruption, as well as the two colour variations of the following pulsations in 1977 and 1978 (numbered 7 and 8 in Fig.1 in PaperI). As observations were scarce at the time, no reliable details can be offered. However, as the three other YHGs highlight an alternating succession of a RL, an eruption, and a BL, with a timescale of a few decades only, this should also apply to HR5171A, but we are convinced that another event should have happened simultaneously. We propose the following plausible scenario. The sketched alternating sequence of red and blue curves in Fig.14 represents the simplified and inevitable representation of the RL and BL evolutions, bounded to the two eruptions. We believe that the first observed reddening episode 1960–1974 was in fact a RL evolution, represented by the first red dashed curve, as it has a $(B-V)$ range of 035 (falling within the range of the other three YHGs, Sect.3.3.). Additionally, the $B$ decline relative to the decline of $V$ 1960–1974 in Fig.13 is quite unlike an increasing extinction by more dust. The reddening in the second episode 1976–1981 started right after the 1975 eruption; it increased even faster than the one between 1960–1974, and stopped in 1981. This is quite awkward. It is plausible that the latter reddening episode was mainly due to a massive shell ejection during the 1975 eruption, schematically represented by the black dashed curve in Fig.14. The $B$ absorption was obviously much higher than in $V$. This event happened simultaneously with the first BL evolution, starting right after the 1975 eruption. The result of both events produced the observed LTV represented by the black dots connected by straight lines. A shell event happening right after the 1975 eruption is likely considering the existence of a massive optically thick extended molecular and dust envelope around HR5171A; see Sects.2.9 and 3.4.2. Obviously, such events happened in the past as well. This first BL evolution was succeeded by a new (second) RL evolution bounded by the impending eruption in 2000. Hence, the observed $(B-V)$ in Fig.14 became bluer as $V$ declined more than $B$ between 1990 and 1995; see Fig.13. We note that this probably explains the relatively large depth of the Jones–Williams pulsation minimum at JD249750 (1995); see Figs.3, and 13. The continuation of the dashed black curve to the right at the bottom of Fig.14 roughly indicates the presumable continuation of the intrinsic reddening by the shell, and its subsequent decrease. After the 2000 eruption, a second BL evolution then developed which is responsible for the blueing trend of $(B-V)$ after 2001, and was still going on early 2019 (Sect.3.4.3.). In conclusion, here we offer a new and very plausible explanation for the persistent reddening of HR5171A between 1960 and 1981. ### The 1977 excess of Balmer continuum radiation of HR5171A and the near-infrared $JHKL$ connection. Firstly, we relate the Balmer continuum excess of HR5171A observed in 1977 to its 1975 eruption episode and shell event with the aid of the $VBLUW$ photometric system. Subsequently, the effects of RL and BL evolutions and the 1975 eruption on the $JHKL$ light curves are discussed. One can only speculate as to any relation between the matter blown into space after the atmospheric explosion of 1975 and the anomalous high UV excess happening during the development of the very-high-amplitude pulsation of 1977 immediately following the eruption (labelled no.7 in Fig.1 of PaperI). The Balmer continuum radiation excess amounted to about 1$^{\rm m}$ (see Fig.1 in van Genderen et al.2015). No such excess was observed in 1971, 1973, or between 1980 and 1991 when also $VBLUW$ photometry was made. A glance at the $B_{\rm pg}$ light curve of the Harvard DASCH scanning project (not shown, only discussed in Sect.3.1.) indicates that such deep brightness declines could have happened in the past as well. (Maximum light was not observed during the 1975 eruptive episode, nor was the deep brightness decline to a deep minimum; only the steep brightness rise was seen thereafter: see Fig.13.). Nevertheless, after 1981 the observed $(B-V)$ curve until the end of our time series in 2019 remained overly red by approximately 06 compared to simplified RL and BL evolutions, of which the brightness level in Fig.14 should be more or less correct. A causal connection with the 1975eruption and the effects of its abundant mass-loss is plausible. It is not surprising that HR5171A is imbedded in an extended dust and molecular envelope (Chesneau et al. 2014; Wittkovski et al. 2017a, and Sect.2.9). We note that an overall long-term variation of the Balmer continuum radiation exists (AppendixJ:3.4.2. and Fig.J.1.). The NIR $JHKL$ photometric light and colour curves, obtained at the SAAO and presented and discussed by Chesneau et al. (2014) in their Figs.4 and 7, clearly show the same BL and RL evolutions after the 1975 eruption until 2012, just like the ones sketched in our Fig.14. The first BL evolution reached maximum brightness at 1990, just like in our sketch of Fig.14 at the bluest $(B-V)$. The 2000 eruption was reached after a RL with a brightness decline of 06 in $L$ and $K$, but less in $H$ and $J$. The NIR 1975 eruption minimum was at least 03 fainter on average than that of the 2000 eruption. It is a pity that at the time of the latter, no $B$ magnitudes were made, meaning that the $(B-V)$ trend until 2012 is unknown (Fig.14). The trend of the colour indices was discussed by Chesneau et al. (2014: Sect.4.1.). In conclusion: HR5171A was likely subject to a massive shell event after the 1975 eruption. Blue- and red-loop evolutions appear to also be detectable in the NIR photometry. ### Current BL evolution of HR5171A. In the autumn of 2015, our attention was aroused by the frequent reports of one of the present authors (EB) on the behaviour of HR5171A based on his monitoring of the visual brightness and supported by many AAVSO colleagues. They are Alexandre Amorim, Brazil; Giorgio Di Scala, Australia; Hiroshi Matsuyama, Japan; Andrew Pearce, Australia; Peter Reinhard, Austria; Eric Blown, New Zealand; Eduardo Goncalves, Brazil; Alan Plummer, Australia; Antonio Padilla Filho, Brazil; Peter Williams, Australia: the visual brightness decline of a pulsation to a minimum in the autumn of 2015 seemed to happen relatively quickly (called the Blown-minimum at JD2457300, October 2015); see red arrow. The pulsation amplitudes became suddenly smaller after 250 d (JD2457520), and the quasi-periods shorter until the end of our dataset at JD2458200 (March 2018), a time interval of about 2yr. At the same time $(B-V)$, based on $BV$ photometry shown in TableI.1 of the Appendix, became bluer – a similar behaviour to that of HR8752 and $\rho$Cas. It appeared that from the end of December 2015 to 2018, the light curve plotted by the AAVSO Light Curve Generator (JD2457100–JD2458300, March 2015–July 2018), shown in Fig.15, revealed a most astonishing pattern: a mean decrease of the visual brightness (see Fig.3 second panel of which Fig15 represents the continuation), smaller light amplitudes, and much shorter quasi-periods within a time span of only $\sim$250d. This happened between the maximum at JD2457400, right after the Blown-minimum, until JD2457650. During this pulsation, including the Blown-minimum and the maximum afterwards, some spectra were taken. An echelle spectrum was obtained, including H$\alpha$ by E. Budding (ed.budding@gmail.com), and a high resolution spectrum was obtained by Dr. N.I. Morrell (nmorrell@lco.cl), but there were no abnormal features; the spectra looked normal (private comm. 2015). The three bars at the bottom of Fig.15 mark the epochs for the three $H$-band images of HR5171A made by Wittkovski et al. (2017c). These images show a variable circumference, we presume not of the photosphere, but of its NIR $H$ band-radiating layer (although we expect the photosphere to be disfigured as well). In conclusion, thanks to the continuous visual observations, we detected a surprising feature: during the BL evolution in 2016, the pulsation amplitudes of HR5171A suddenly appeared much smaller (and probably also with shorter periods). This phenomenon has not been observed before. Radius variation and contraction rate of HD179821 during the BL evolution. --------------------------------------------------------------------------- Below we explore the effects of a BL evolution on the radius, the T(Phot), and the contraction rate, which has never been done before. As we are dealing with evolutionary tracks on the HR diagram, we take it for granted that the stellar property variations keep pace with the temperature. Hence, one expects that radial velocity curves also show LTVs. Indeed, radial velocity monitoring campaigns of $\rho$Cas (monitored by Lobel et al. 2003, Figs.1 and 4) reveal LTVs as well, and as far as we can judge these are more or less in concert with those of the visual brightness. As long-term radial velocities for HD179821 are lacking, we explored the effects of the BL evolution on its radii and photometric temperatures. We note that a calculated T(Phot) is always lower than a T(Sp). For example, an overly red $(B-V)$ is responsible for an overly low temperature, and according to the formula logR/R$_{\rm \odot}$=8.48–0.2M$_{\rm bol}$–2logT$_{\rm eff}$, the radius becomes larger for an almost constant M$_{\rm bol}$. Therefore, the results should not be taken too literally, as we are only interested in the general effects of a BL evolution on the stellar properties. The illuminating result is presented in Fig.16. The distance of HD179821 is a serious matter of debate: between $\sim$1kpc and $\sim$6kpc (e.g. Surendinanath et al. 2002; Arkhipova et al. 2001, 2009; Le Coroller et al. 2003; Nordhaus et al. 2008; Patel et al. 2008; Oudmaijer et al. 2009; Ferguson & Ueta 2010; Sahin et al. 2016), M$_{\rm bol}$$\sim$-4.3 and -8.2, respectively, but according to our conclusion in Sect.3.3., a large distance is favoured. We calculated the trend of R/R$_{\rm \odot}$ as a function of the $V$ and $(B-V)$ of the LTVs (here only a blue loop track, of which the photometry is depicted in Figs.9 and 10, respectively) for most of the pulsation maxima of HD179821, and for both distances. For the dJN method we then need to differentiate between an Ib–II supergiant and a hypergiant between the Ia$^{+}$ and Ia luminosity classes, for the short and long distances, respectively. Averages of the s-parameter for the two pairs of luminosity classes were used to derive T$_{\rm eff}$ and BC (AppendixF:2.7.). The extinction and reddening used are from Table1. On the left (dots) the trend of the radii for 6kpc on the right (crosses) for 1kpc. The scatter around the mean lines is largely intrinsic. This can be explained as follows. About 30% of the light ($V$) and colour ($(B-V)$) curves of HD179821 do not run in phase. As a consequence, at a fixed $V$ brightness, a great variety of $(B-V)$ colours exist, and therefore also a great variety of temperatures. According to the formula above, this causes a scatter as well in the computed radii, while the effect of the variation of M$_{\rm bol}$ is small. The radii at maximum light vary from 400 to 450R$_{\rm \odot}$ and 60 to 80R$_{\rm \odot}$ for 6 and 1 kpc, respectively. The trends in Fig.16 from the right to the left indicate a decrease of the radii for the distances d=6kpc and 1kpc, by about 60R$_{\rm \odot}$ (15%) and 11R$_{\rm \odot}$ (15%) by an LTV visual brightness decrease of 011 in $V$, and an LTV blueing of 015 in $(B-V)$, respectively. We derived the contraction rate during the BL evolution of HD179821 (roughly between JD2449000 and JD2453200) for a distance of 6kpc: the result is $\sim$2.0R$_{\rm \odot}$y$^{-1}$. Theoretical and speculative explanations for two observational highlights. ========================================================================== Below we discuss two speculative subjects. Firstly, the one raised in Sect.2.3. We believe that all convective stars, such as YHGs, are subject to a random process that may be identified as the convection dynamics acting on the periodic pressure waves by the $\kappa$-mechanism of the He-ionization zone (Fadeyev 2011). This principle has been used by Icke et al. (1992) for their numerical study of the instability of post-AGB stars, producing model light curves with some similar characteristics to those of YHGs. It should be noted that Wisse (1979) and Icke et al. (1992) did not include the LTVs (representing the BL and RL evolutions) in their calculations, on which the much shorter quasi-periods are superimposed. This means that the calculations should have been done with one more variable. Based on a comparison of the main characteristics of the observed time series of the four YHGs, and those of the calculated time series without LTVs, we conclude that LTVs have obviously little effect: the observed pulsations also show a weakly chaotic character just like the models. Now we discuss observations detailed in Sects.2.4.–2.6. and in AppendixD:2.6., namely indications for the presence of a variable high-opacity layer in the atmospheres of YHGs, responsible for a wavelength-dependent absorption. Deviating photometric and calculated stellar properties emerged from analyses of individual light and colour curves, for example by means of the $V$/$(B-V)$ and Ampl$V$/Ampl$(B-V)$ diagrams. As spectral temperatures are independent of continuum absorption, they always appear to be higher than photometric ones. The differences decrease from about 3000K to a few 100K from high (about 8000K) to low (about 4500K) temperatures, respectively. As a result of the abnormally high amplitude ratios Ampl$V$/Ampl$(B-V)$, the calculated temperatures for HR8752, for example, are about 200K too low (compared to the results of Nieuwenhuijzen et al. 2012: their Fig.2 based on MK spectral types and LTE effective temperatures), and the calculated stellar radii in the maximum are largest: 946R/R$_{\rm \odot}$ and in the minimum smallest: 866R/R$_{\rm \odot}$, respectively. The dimensions on the other hand should be the other way around (estimated relative and absolute errors are about 100R/R$_{\rm \odot}$, compared to the results of Nieuwenhuijzen et al. (2012) in their Fig.20). The photometric observations on which our calculations are based, are from Walker (1983); Ferland priv. comm. to Lambert & Luck (1978); Moffett & Barnes (1979). A discussion on variable transparency sources in stellar atmospheres is given by Lobel (1997, Ch.1.). For example, it is known that the opacity depends on the density of partially ionized hydrogen gas. The quasi-periodic pulsations of YHGs expand and compress their very extended atmospheres causing variable opacity and absorption of radiation that yield temporal changes of the stellar T$_{\rm eff}$ and mass-loss rate. During the pulsation-compressions the increased surface gravity raises the gas density of extended layers in the optically thin portions of the outer atmosphere (i.e. $\tau_{\rm R}$<$2/3$ where the local radiation temperature T$_{\rm rad}$ is not in equilibrium with T$_{\rm gas}$), consequently altering the selective absorption of incoming radiation and producing significant optical depth effects. Lobel (2001) discussed the possibility that the T$_{\rm eff}$ variations with pulsations can alter the temperature of the incoming radiation field (with T$_{\rm rad}$ variations proportional to the T$_{\rm eff}$ changes) inside these atmospheric layers, further shifting the local (LTE) thermal H ionization balance due to (non-LTE) photo-ionization. The deviations from an average T$_{\rm rad}$ occur because of variations in the local opacity sources at different atmospheric heights. The modified partial H ionization alters the layer opacity. But it also changes the overall compressibility (or bulk modulus) through $\Gamma_{1}$ and hence its local dynamic stability in the stellar atmosphere that may cause the unreliability of $(B-V)$ to derive an accurate temperature. Conclusions. ============ The most important novelties to come from the present study can be summarised as follows. 1\. Spectroscopic temperatures (T(Sp)) are always higher than photometric temperatures (T(phot)) independent of the temperature scale used. Differences (a few thousand K near 8000K) decrease towards lower temperatures (a few hundred K near 4200K), and must be due to the presence of a temperature-dependent high-opacity layer in the atmosphere. After all, T(Sp) is relatively insensitive to continuum absorption, contrary to T(Phot). As a result, $B$ and $V$ magnitudes are inconsistent with respect to each other; $B$ has suffered from more absorption than $V$. Obviously, the higher the temperature, the higher the opacity of that atmospheric layer. It goes without saying that this was caused by increased mass loss and subsequently by a higher gas density of that layer rendering a too low T(Phot) (Sects.2.4.–2.6.). 2\. The analysis of the LTVs of $\rho$Cas, HD179821, and HR5171A, and confirming the LTV of HR8752 by Nieuwenhuijzen et al. 2012, revealed that they mainly consist of RL and BL evolutions. These evolutions are responsible for the alternating T$_{\rm eff}$ variations of 1000K–4000K. The corresponding observed $(B-V)$ ranges are 02–06 (Sects.3.2. and 3.4.1.). 3\. The above suggests that HD179821 is likely a YHG, not a RSG, nor a post-AGB star. If correct, the star should be located at a large distance (probably larger than 5kpc), and far above the galactic plane, creating a kinematical problem (Sect.3.3.) as already mentioned by Arkhipova et al. (2009). 4\. In light of the results described in (2) above, a schematic sequence of RL and BL evolutions could be defined for HR5171A in Fig.14, separated by the 1975 and 2000 eruptions. The most plausible explanation for the persistent reddening episode in 1960–1981 of HR5171A is that it consists of an RL evolution between 1960 and the 1975 eruption. Thereafter, a severe absorption episode should have happened right after this eruption by a massive shell ejection, simultaneously with a BL evolution (Sect.3.4.1.). 5\. Quasi-periods, pulsation amplitudes, and stellar radii decrease and increase during BL and RL evolutions, respectively (Sects.2.5., 3.1., 3.3, 3.4.3., 3.5., Figs.11 and 13 and AppendixK:3.3.). 6\. Based on light-curve markers, a new but weak eruption of $\rho$Cas was identified which happened in 2013, and which was independently spectroscopically confirmed by Aret et al. (2016; Sect. 2.2. in the present paper). 7\. The light variations of HR5171A of the order of a year are due to pulsations alone. Any significant contribution to the light variation of a distorted primary by a possible much smaller contact companion is out of the question (Sect.2.3.; Chesneau et al. 2014, their Fig.8). However, we cannot exclude an unobserved companion. 8\. The photometric observations reveal that each pulsation is unique, any predictability on the morphology and duration of the next pulsation is impossible. These facts are in accordance with the light-curve models of Icke et al. (1992) and Wisse (1979) for evolved long-period variables (Sect.2.3.). Based on the model light curves of Icke et al. (1992), we characterize the pulsations of the four YHGs as ‘weakly chaotic’ rather than ‘chaotic’. For both types of model light curves, the incessant sequences of photometric pulsations appear coherent, which agrees with the observations (Figs.2 and 3). 9\. The membership of HR5171A to the stellar cluster Gum48d at a distance of 3.6kpc is doubtful. Its distance is probably of the order of 1.5kpc$\pm$0.5kpc (Sect.2.9.). If this supposition is correct, it would prove that HR5171A is not amongst the biggest stars known. For $\rho$Cas, a shorter distance is preferred as well: 2.5kpc$\pm$0.5kpc, instead of the accepted one of 3.1kpc$\pm$0.5kpc, (Sect.2.9.). 10\. We would like to finish this paper with the motto of the Royal Astronomical Society, London, 1820: ‘Quicquid nitet notandum’, or ‘Whatever shines should be observed’. See AppendixL: Message to observers. A.L. acknowledges support in part by the Belgian Federal Science Policy Office under contract No.BR/143/A2/BRASS and funding received from the European Union’s Framework Programme for Research and Innovation Horizon 2020 (2014–2020) under the Marie Sklodowska-Curie grant Agreement No.823734. G.W.H. thanks Lou Boyd for his support of Fairborn Observatory and acknowledges long-term support from Tennessee State University and the State of Tennessee through its Centers of Excellence Program. We acknowledge with thanks all the variable star observers worldwide from the AAVSO, especially S. Otero, VSX Team, for some invaluable comments on this paper and offering his historical data of HR5171A which were used in this paper, and which will be of great benefit for future research. We like to thank Mr. Roland Timmerman for making Fig.1. An anonymous referee is gratefully acknowledged for the invaluable comments and suggestions, which improved the presentation of this paper. We would like to thank the following colleagues for their stimulating and fruitful correspondence since 2013, on various subjects addressed in this paper: Prof.dr. R.M. Humphreys, Dr. R.D. Oudmaijer, Dr. F. van Leeuwen (Hipparcos), Dr. B. Mason (Washington Double Star Cat.), Dr. J. Grindlay (Harvard DASCH scanner), Prof. dr. V. Icke, Dr. J. Lub, Prof.dr. J.W. Pel, Dr. D. Terrell, Prof.dr. A. Witt (Blue Luminescence of PAH molecules), Mr. E. Budding and Dr. N. I. Morrell (spectrograms of HR5171A), Dr. C. Sterken, Dr. A. Meilland and Dr. E. Chapellier. This research made use of the excellent near-infrared (JHKL) data obtained at the SAAO, and discussed by Chesneau et al. (2014). 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Astron. 36, 65 Zsoldos, E., 1986a, The Obs., 106, 156 Zsoldos, E., 1986b, IBVS No.2913 Zsoldos, E., & Olah, K., 1985, IBVS No.2715 Zsoldos, E., & Percy, J.R., 1991, A& A 246, 441 1. Data summary. ================ See TableA.1. ----------- ----------- ------------------ ---------------------- --------- Star Time logT$_{\rm eff}$ logL/L$_{\rm \odot}$ $(B-V)$ RL or BL interval range range range HR8752 RL 1895–1963 3.72–3.70 5.25–5.38 016 BL 1976–2005 3.70–3,90 5.60–5.36 064 $\rho$Cas RL 1968–1998 3.72–3.66 5.48–5.54 040 HR5171A RL 1958–1974 3.70–3.65 5.40–5.30 035 HD179821 BL 1990–2004 3.69–3.83 5.20–5.10 022 ----------- ----------- ------------------ ---------------------- --------- 2.1. Databases. ================ Excellent and homogeneous time-series exist for $\rho$Cas initiated by Henry (1995, 1999; see footnoteNo.1): 1986–2001 ($VRI$) and 2003–2018 ($BV$), with a time resolution down to about 1$^{\rm }$ d and often with an accuracy of 002–001. The longest recorded time-series also including eruptions are those made for $\rho$Cas by visual observers since the 1960s, summarised by Beardsley (1961) for example. Others are Bailey (1978) and the AAVSO observers. A great amount of $(U)BV$ photometry was performed until the early 1990s for example by Brodskaya (1966), Fernie et al. (1972), Landolt (1968, 1973), Percy & Welch (1981), Leiker & Hoff (1987, 1990), Leiker et al. (1988, 1989, 1991), Halbedel (1985, 1986, 1988a, 1991b, 1993a), Henry (1995, see paragraph above), Percy et al. (1993, 2000), Zsoldos & Percy (1991), and by the Hipparcos satellite 1989-1993 (ESA 1997). A number of $BV$ observations were obtained between 2016 and 2018 by one of the current authors: (EJvB) and plotted in Fig.2. The observations were made with a 10" f/6.3, Meade LX200 telescope. See for the observations: https://www.aavso.org/data-access, and for the most recent ones: in window ‘Pick a star’ on www.aavso.org. HR8752 was rather intensively monitored during numerous observing campaigns in the eighties until 1993 in $BV$, but less in $U$ ($UBV$): Arellano Ferro (1985), Halbedel (1985, 1986, 1988a, 1991a, 1993b), Mantegazza et al. (1988), Moffett & Barnes (1979), Parsons & Montemayor (1982), Percy & Welch (1981), Walker (1983), Zsoldos & Olah (1985), and by the Hipparcos satellite 1989–1993 (ESA 1997). Lastly, a number of $BV$ observations were made between 2016 and 2019 by one of the present authors (EJvB, see above). Splendid time-series for HD179821 were obtained by: Arkhipova et al. (2001, 2009, $UBV$) between 1990 and 2008, who also retrieved photographic magnitudes for the time interval 1899-1989, by Le Coroller et al. (2003: $UBVRI$) and by Hrivnak et al. (2001: $V$), partly simultaneously with the previous groups. HR5171A was monitored in various multi-colour photometric systems: 1953-2016, collected from literature in PaperI and by Chesneau et al. 2014, by the Long-Term Photometry of Variables project at La Silla (LTPV) organised by Sterken 1983, see Manfroid et al. 1991, Sterken et al. 1993; by the Hipparcos satellite, ESA 1997, see van Leeuwen et al. 1998; and by one of the present authors (GDS: Appendix Table I:1.). A large body of splendid $JHKL$ data was obtained at the SAAO from 1975 until 2013, published and discussed by Chesneau et al. (2014). The AAVSO and ASAS observers appeared to be very productive for the Big Three ($\rho$Cas, HR8752 and HR5171A), especially in the last three decades; very valuable sequences of pulsations were recorded. 2.2. Quasi-period search. ========================= Particularly in the 1980s and 1990s, the type of the quasi-periodic pulsations, radial or non-radial, was heavily debated. Models indicated that they should be non-radial (Maeder 1980; de Jager 1993, 1998; Fadeyev 2011; Lobel 1997: Ch.2; Lobel et al. 1994. It is believed that they are probably due to gravity waves (de Jager et al. 1991) The many attempts to find a consistent quasi-period, and to decipher whether they are radial or non-radial pulsations, has not lead to cogent results (e.g. Arellano Ferro 1985; Halbedel 1991b; Zsoldos & Percy 1991; Percy et al. 2000; Arkhipova et al. 2001). Le Coroller et al. (2003) and Arkhipova et al. (2009) found a dominant bimodal behaviour for HD179821 of $\sim$200$^{\rm }$ d and $\sim$140$^{\rm }$ d for the time interval 1994–2000. Their ratio is 0.7. The authors suggest that they represent a fundamental tone and the first overtone, respectively, being indicative of a high stellar mass 15–20M$_{\rm \odot}$ and T$_{\rm eff}$=6300K according to theoretical models of Zalewski (1986). No reliable result could be obtained for the data between 2000 and 2008 (Arkhipova et al. 2009), while between 2009 and 2017 the quasi-periods increased to more than 250d according to Ikonnikova et al. (2018). 2.6. Ratios Ampl$V$/Ampl$(B-V)$, and the influence of enhanced mass loss episodes. ================================================================================== References of photometric $\rho$Cas data used for this analysis are: the multi-colour photometric observations made by Henry (1995, 1999); Leiker & Hoff (1987); Leiker et al. (1988, 1989, 1990, 1991); Halbedel (1988, 1991b, 1993a); Zsoldos & Percy (1991); Percy et al. (2000). Figure D.1. explains an abnormal amplitude ratio Ampl$V$/Ampl$(B-V)$ with the aid of three fictitious pulsations in $B$ and $V$. The first set represents a so-called normal pulsation (see below) as they are the majority, with Ampl$V$/Ampl$(B-V)$=1.7$\pm$0.5. The second set has become smaller according to the observations, mostly in $B$ and less so in $V$ by absorption; see the continuous curves. The corresponding ratio is 3.9. The dashed curves are the intrinsic ones. The third set of amplitudes refers to a ‘normal’ pulsation again without increased absorption, and for which the ratio is ‘normal’: 1.8. To offer a reliable estimate is hardly possible, the inconsistencies in $B$ are 01 up to perhaps 1$^{\rm m}$, and much less in $V$, depending on the optical thickness of the high-opacity layer. Caution is called for: we have indications that all pulsations, even ‘normal’ ones, suffer from the same type of absorption, albeit less. Obviously, a high-opacity layer with a variable density seems to be omnipresent. Estimated errors in T(Phot) and radius will be a few 100K (too low), and a few 100R$_{\rm \odot}$ (too large). 2.6. More evidences for a difference between T(Sp) and T(Phot). =============================================================== –[$\rho$Cas.]{} The $\rho$Cas pulsations 1969–1970 had a mean T(Sp)=7100K (Lobel et al. 1994). The observed mean $(B-V)$=1.1, after correction for extinction (Table1) $(B-V)_{0}$=0.65, which means a T(Phot)=5100K. This is 2000K cooler than T(Sp) mentioned above. Assuming that T(Sp) is correct, the corresponding $(B-V)_{0}$ should be about 0.15. Obviously, the colour $(B-V)$ of $\rho$Cas, although in a relatively hot stage, was too red by about 05 at the time. –[HD179821.]{} Arkhipova et al. (2009) list in their Table5 five spectral temperatures based on CaII and hydrogen P lines, made between 1994 and 2008, using the method of Mantegazza (1991): yielding the average T(Sp)=6800K$\pm$50K. This agrees with model calculations by Zacs et al. (1996), Reddy & Hrivnak (1999), and Kipper (2008), while a 7000K is suggested by the far-IR brightness (Nordhaus et al. 2008), and a 7350K$\pm$250K by Sahin et al. (2016). However, all these temperatures appeared to be higher than the photometric one for HD179821 based on the Straizhys (1982) temperature calibration: 5400K–6000K. This temperature range for HD179821 is of the same order as the one based on the calculated dJN temperatures in the previous sections. Using observations by Arkhipova et al. (2009), we determined the s-parameter from the $(B-V)_{0}$ according to the dJN calibration, and the photometric T$_{\rm eff}$. We summarise the results: T(Sp)–T(Phot)=1320K and 1210K, if the distance is 6kpc and 1kpc, respectively, with a standard error of the order of 65K. Hence, also here spectral temperatures are at least 1000K higher than the dJN temperatures. As the distance difference is extreme, we had to deal with two temperature calibrations, that is, for stars belonging to different luminosity classes: Iab–Ia and IB–II, respectively. –[HR8752.]{} Nieuwenhuijzen et al. (2012) did not report such differences for HR8752, but they do exist and are shown below. This is because these latter authors had the advantage of having of a list of precise MK classifications from which LTE effective temperatures could be derived and subsequently the s-parameter and $(B-V)_{0}$, which could be used to indirectly confirm the existence of that obstinate difference when their temperatures are compared with the ones based on the photometric observations. Two $BV$ observations from the cool YHG stage of HR8752, which were simultaneously observed with T(Sp) determinations: at JD2443737 (May 1979), T(Sp)=5602K and T(Phot)=4880K, difference=722K, and at JD2445898 (October 1984), T(Sp)=5343K and T(Phot)=5160K, the difference= 183K. See Tables1 and 4 and Fig.2 of Nieuwenhuijzen et al. (2012) for references and errors. Photometry by Walker (1983) and Zsoldos & Olah (1985). –[HR5171A.]{} AMBER/VLTI observations of HR5171A by Wittkowski et al. (2017a), yielded the temperature of T$_{\rm eff}$=4290K$\pm$760K at JD2456755=2014. This date point is indicated by a vertical line piece in the lower panel of Fig3, and is arbitrarily placed as the $V$ magnitude is unknown; it is situated within the series of $BV$ photometry (Johnson system) and $RI$ photometry (Cousins system) represented by plus symbols in Fig.3, of which the data are listed in TableI.1. One observation of TableI.1. (JD2456886) was made only 131 d after the temperature determination of Wittkovski et al. It appears that with the dJN method, the temperature of HR5171A is T$_{\rm eff}$=4210K, thus well within the error bar of the Wittkovski et al. temperature. Indeed, inconsistencies in temperature determinations become smaller for lower temperatures. In this context it is very relevant to mention that Chesneau et al. (2014) suggested the presence of a special veiling envelope rendering a decoupling of photometric and spectroscopic observations. For example, these latter authors showed that CO molecular lines were affected by this veiling, and could not be used for the spectral classification (their Fig.10). They also concluded that the strong veiling of the K-band spectrum of HR5171A was the reason why a spectral classification was impossible. However, here the circumstellar envelope is supposed to be the culprit. It can be concluded that the results in for example Sect.2.8. support the permanent presence of a high-opacity layer causing a variable selective absorption (Appendix Fig.D.1.). Its presence is almost undisputable. It is more optically thick for $B$ (a rough estimation: 01 up to one magnitude) than for $V$ (but the extinction law is uncertain and will be the focus of further research). Naturally the absorption has almost no effect on the spectroscopy, and as a result the differences T(Sp)–T(Phot) increase with temperature. It also appeared that the higher the T(Sp), the higher the gas density of that layer (the higher the selective absorption), meaning that a connection with increased mass loss is almost certain. See Sect.4 for a reference on transparency problems in stellar atmospheres, and for further discussion on the subject. 2.7. Temperature scales. ======================== A short discussion is needed on a number of temperature calibrations based on broadband photometry. A new temperature calibration was introduced by de Jager & Nieuwenhuijzen (1987) and is valid for all main sequence stars up to the hypergiant luminosity class Ia$^{+}$. This calibration is based on the continuous spectral variable ‘s’ derived from $(B-V)_{0}$ for all luminosity classes from V to Ia$^{+}$. This s-parameter determines T$_{\rm eff}$ and the BC. As a result, temperatures for Ia$^{+}$ stars turn out to be cooler, and stellar radii larger than for Iab supergiants, as expected. An additional novelty of the dJN calibration is, that it allows one to differentiate smoothly between the luminosity classes by means of the variable b-parameter, b=0.0 for Ia$^{+}$, 0.6 for Ia, and 1.0 for Iab, and so on, to 0.5 for class V. As the Big Three, namely HR8752, $\rho$Cas, and HR5171A (having similar spectra, Lobel et al. 2015), are without any doubt of class Ia$^{+}$ (with b=0), their bolometric magnitudes should vary during the pulsations, and considering their spectral types, be roughly between -8.7 and -9.6. If our calculations result in a slightly low luminosity, we neglect them and still consider the star as a Ia$^{+}$ YHG with b=0. Our consideration is that also uncertainties in distances and reddening are not negligible at all. We found and described in Sects.2.4.–2.6. and Appendix E:2.6. that photometric temperatures for YHGs are always lower than spectroscopic temperatures, and offer an explanation. We also tried to derive temperatures from theoretical spectral energy distributions (T(SED)), by comparing the observed SEDs based on the photometric fluxes between two channels (e.g. between $B$ and $V$, or $V$ and $R$, etc.), corrected for interstellar extinction, with the corresponding theoretical fluxes of atmospheric models (logg=0–0.5, v$_{\rm turb}$=2kms$^{-1}$, solar abundance) by Castelli & Kurucz (2003) and Castelli (2014). Our test case was $\rho$Cas using $BV$ and $VRI$ photometry by Henry (1995, 1999; see Sect.2.2.) coinciding 6 T(Sp) values from Klochkova et al. (2014) and 4 T(Sp) from Lobel et al. (2003). The extinction and reddening used are listed in Table1. It turned out that the differences $\triangle$T=T(Sp)–T(SED) for each set of $BV$, $VI,$ and $VR$ are linearly related with the observed brightness $V$ (the three running parallel), and are thus roughly related with the temperature: the brighter the star, the larger the difference between T(Sp) and T(SED), the latter being lower. For $BV$ the difference grows from $\sim$200K near $V$=472 (the coinciding T(Sp) determinations only cover a small brightness range), to $\sim$800K near $V$=45. For $VI$ these numbers are from $\sim$0K near $V$=51 to $\sim$2500K near $V$=42. For $VR$ these numbers are from $\sim$600K near $V$=51 to $\sim$2900K near $V$=42. These inconsistencies, increasing with temperature, point to an increasing flux excess of the observations with respect to the models in the sense that $V$ shows slightly more flux than $B$, $R$ more than $V$, and $R$ much more than $I$. These relative excesses transformed into magnitudes range from $\sim$01 to 07. We cannot offer a plausible explanation. It may be that the models for such extreme stars are not yet perfect. Therefore, we disregarded the T(SED) results. 2.8. Monitoring of V$_{\rm rad}$ is indispensable. ================================================== Individual pulsations are unique, each (at least most of them) can be considered more or lesl a global feature: the entire surface moves up and down, albeit not necessarily spherically symmetric as we are probably dealing with non-radial pulsations. This is supported by the radial velocity monitoring mentioned above of $\rho$Cas and HR5171A, revealing an oscillating trend in pace with the light variations without significant irregularities (as far as the accuracy of the current radial velocity curves allow such a statement), and showing a stable phase lag of about 0.4 (Lobel et al. 2003, 2015). Minimum radius happens without any doubt always at maximum brightness and vice versa (as far as the available measurements allow such an assertion). This in contrast with the calculated properties for many YHG pulsations and based on multi-colour photometry, which is deteriorated by atmospheric absorption. An explanation for the higher reliability of radial velocity measurements is as follows. They are mainly governed by the central part of the pulsating sphere, and are much less affected by the outer ring suffering from projection effects and limb darkening, or by a variable non-circular and irregularly shaped circumference (as seen from the Earth), see recent images of HR5171A by Wittkovsky et al. (2017c) discussed in Sect.3.4.3. of the present paper. Additionally, one expects little disturbing effects on the measurements by dynamically active surface features or variable absorbing effects by the atmosphere. On the contrary, multi-colour photometry is highly sensitive to most of these factors. Therefore, we would like to stress that long-lasting monitoring of V$_{\rm rad}$ curves, especially with a small time resolution, is indispensable for deeper studies, and was a regrettable deficiency of the present analysis. It should be noted that radial velocity curves also show a long-term variation more or less in accordance with the LTVs (Figs.2 and 3 in Lobel et al. 2003). 2.9. Distance of HR5171A ======================== We attempted to calculate the stellar properties of HR5171A in a similar way as we did for two pulsations of $\rho$Cas (Tables2 and 3), this time in order to compare V$_{\rm rad, puls}$ with $V_{\rm rad}$, the spectroscopic velocity. The goal was to find the distance. Our intention was to compare both radial velocity amplitudes, the one based on the geometrical velocities of the stellar surface V$_{\rm rad, puls}$, with aid of calculations similar to those for $\rho$Cas (Table3), and the other based on the atmospheric spectral lines (10kms$^{-1}$ mentioned in Sect.2.9.) and multiplied by $\sim$1.4 (to correct for projection and limb darkening), hence about 14kms$^{-1}$. The purpose was to determine the true distance of HR5171A. After all, radius, radius variation, and radial pulsation velocity are distance dependent. When the latter (in kms$^{-1}$) is plotted versus a number of assumed distances used for the calculations, a linear relation is obtained. The right distance should then be where V$_{\rm rad, puls}$=V$_{\rm rad}$ (corrected)=14kms$^{-1}$. For the calculations we used distances of 3.6kpc down to 1kpc. The attempt failed, most likely due to inconsistent $V$ and $(B-V)$ parameters (just like in the case of the pulsations of $\rho$Cas in Tables2 and 3, and of HR8752 and HD179821 discussed in Sect.2.4.–2.6.). Yet the ratio Ampl$V$/Ampl$(B-V)$ of the ascending branch of HR5171A above was not exceptionally high: 1.8, the one for the descent was high: 2.9. However, due to scarcity of data points, these numbers are not reliable. In any case, as a result of the photometric inconsistencies, calculated radial pulsation velocities appeared to be too small, or ran opposite to the observed V$_{\rm rad}$ values. Just to get some idea, we list below only the calculated temperature and radius ranges for 1.5kpc (temperature similar for 3.6kpc). We estimate that the values for T(Phot) are too faint by a few hundred degrees, and the radii are too small by about 100R$_{\sun}$ . Summary for 1.5kpc: dJN: T ranges 4020–4200K, and R ranges 1060–1160R$_{\rm \odot.}$ The best fit to the optical spectrum of 1992 (in the minimum after maximum no.17 (in PaperI Fig.1) by Lobel et al. (2015) is for 5000K and logg=0. As usual, the spectroscopic temperature is higher than the photometric one above. This type of distance determination will only be successful when the photometric parameters of a pulsation are consistent. 3.1. Long-term variations ========================= Long-term variations were noted and discussed by many observers: for HR5171A, for example, Harvey 1972, PaperI, Chesneau et al. (2014); for $\rho$Cas: for example, Arellano Ferro (1985), Percy & Keith (1985), Leiker et al. (1988, 1989, 1991), Halbedel (1988b, 1991b, 1993a, b) and Zsoldos & Percy (1991); for HD179821: Arkhipova et al. (2001, their Fig11) and Le Coroller et al. (2003). --------- ------ --------- ----------------- ----------------- ----------------- JD– 2450000 $V$ $(B-V)$ $(V-R)_{\rm c}$ $(V-I)_{\rm c}$ $(R-I)_{\rm c}$ 6324.03 6.53 2.54 6330.01 6.59 2.44 6331.06 6.58 2.46 6337.09 6.59 2.51 6886.50 6.60 2.46 1.38 2.73 1.35 7399.18 7.03 2.47 1.54 2.85 1.31 7426.07 6.84 2.41 1.49 2.62 1.13 7636.86 6.93 2.35 1.49 2.60 1.11 8132.95 6.81 2.49 1.46 2.74 1.28 8136.19 6.79 2.49 1.46 2.72 1.25 --------- ------ --------- ----------------- ----------------- ----------------- 3.4.2. Long-term variation of HR5171A in the $VBLUW$ system. ============================================================ FigureJ.1. shows the detailed smoothed LTVs for the four colour indices in (log intensity scale) of the Walraven $VBLUW$ system (described by Lub & Pel 1977; de Ruiter & Lub 1986) for the decade 1980-1991, and derived from Fig.2 of PaperI. They all show undulating declining reddening trends in $V-B$ (equivalent to the $(B-V)$ of the $UBV$ system), $B-L$ and $B-U$ (is roughly similar to the $(B-U)$ of the $UBV$ system, but due to the large differences in band widths it is impossible to derive transformation formulae). The average declines in especially $V-B$ and $B-U$, confirm the reddening trend of $(B-V)$ in Figs.13 and 14. The small bar on the right in the top panel indicates the local brightness minimum in the $VBLUW$ time series. Most interesting is the $U-W$ curve being dominated by two cycles of about 2000d mainly due to a $W$ ($\lambda$3235Å) brightness variation. The second cycle with a prominent maximum around (1989–1990), amplitude $\sim$0.12 log scale, or 03, points to an increase of the Balmer continuum radiation relative to the observations made in 1971, 1973, 1980 and 1981 (see the two-colour diagrams in Fig.5 in PaperI). It is also striking that some of the highest amplitudes of the pulsations in the near-UV pass bands $L$, $U,$ and $W$ (Fig.2 in PaperI) happen during these two $U-W$ maxima in Fig.J.1., roughly around JD2445200 and JD2447600. We presume that this undulating Balmer continuum radiation is somehow related to the circumstellar matter (Sect.2.9.). ----------- ------ ---------------- ----- ----- ----- ------- -------- ------- ------- ------- star $V$ $B$ $U$ $R$ $I$ $B-V$ $U-B$ $R-I$ $V-I$ $R-I$ dates ($B_{\rm pg}$) HR8752 1940–1994 0.6 0.8 1.2 0.9 $>$0.7 HR5171A 1900–1950 (1.3) 1953–2018 0.9 1.2 0.7 $\rho$Cas 1963–2018 0.5 0.5 0.8 0.3 0.2 0.4 0.6 0.2 0.4 0.2 HD179821 1899–1989 (0.5) 1990–2009 0.15 0.12 0.4 0.2 0.3 ----------- ------ ---------------- ----- ----- ----- ------- -------- ------- ------- ------- 3.3. Trends of brightness, quasi-periods, and amplitudes of HR8752 during the BL evolution 1993-2019. ===================================================================================================== The trends of amplitudes and quasi-periods of HR8752 were described in Sect.3.1. We explored the AAVSO website for further information, not only on the trends of amplitudes and quasi-periods, but also on the brightness. The scatter of the AAVSO mean light curves sometimes hampered the determination of a significant mean quasi-period (also due to the low intrinsic amplitudes of about 01). However, AAVSO photometric $V$ (Johnson) observations were made as well and were a great help. After 1993 ($V$$\sim$5.05), all $UBV$ monitoring stopped for some unknown reason. The photometric $V$ magnitudes were usually about 02–04 brighter than the visual estimates (as expected). The mean brightness $V$ and estimated amplitudes were as follows, interval 1993–1995: 5.15 and 01–02; 1995–1998: 5.2 and 01; 1998–2001: 5.25 and 005; 2002–2005: 5.3; 2005–2008: 5.3; 2008–2011: 5.3 and 01; 2011–2014: 5.25; 2012–2018: 5.3 and 01. The conclusion is that the brightness $V$ of HR8752 stayed almost constant from the early 1990s until 2019 (the finishing date of this paper). The light amplitudes (if real) were of the order of 01. 5. Message to observers ======================= Young professionals should start a new era of worldwide coordinated observing campaigns for YHGs (e.g. with the photometric and spectroscopic opportunities offered by the Las Cumbres Observatory, a global telescope network of robotic telescopes (headquarters in Goleta, CA, USA). While the AAVSO observers continued observing, most of the professionals stopped (or had to stop) observing HR5171A and HR8752 all of a sudden in the 1990s. Considering the unprecedented number of exciting and spectacular activities of YHGs, and not to forget all other types of variables, with respect to their physics, instability, and evolution, this halt 30 years ago was deplorable, supporting Sterken’s (2002: Sect.3.1.) concern on the same matter. Hopefully, our results herald coordinated campaigns for long-lasting spectroscopic monitoring of YHGs for obtaining detailed radial velocities, spectral characteristics, T(Sp), and so on. Naturally, such campaigns would go hand in hand with multi-colour photometry, preferably with a time resolution of a week or less. Only then can reliable information on for example the nature of the high-opacity layer and the absorption law be obtained, as well as information on the detailed development of eruptive episodes and on the subtle changes of pulsating and stellar properties along the BL and RL evolutions. Electronic tables for the photometric observations of $\rho$Cas. ================================================================ ----------------- --------- --------- --------- --------- --------- --------- Date Var $V$ Var $R$ Var $I$ Chk $V$ Chk $R$ Chk $I$ (HJD - 2400000) (mag) (mag) (mag) (mag) (mag) (mag) 46679.8629 -0.840 -0.731 -0.605 99.999 99.999 99.999 46679.9081 -0.838 -0.720 99.999 0.738 1.685 2.323 46680.7515 -0.839 -0.741 -0.605 0.728 1.666 2.318 46680.8244 -0.850 -0.722 -0.607 0.737 1.690 2.317 46680.8862 -0.845 -0.749 -0.626 0.746 1.660 2.339 ----------------- --------- --------- --------- --------- --------- --------- ---------------- --------- --------- --------- --------- Date Var $B$ Var $V$ Chk $B$ Chk $V$ (HJD - 2400000 (mag) (mag) (mag) (mag) 52800.9656 -1.261 -0.921 -0.402 0.840 52807.9611 -1.272 -0.925 -0.401 0.853 52828.9681 -1.282 -0.939 -0.402 0.848 52894.7499 -1.273 -0.950 -0.405 0.830 52895.6689 -1.286 -0.957 -0.409 99.999 ---------------- --------- --------- --------- ---------
--- abstract: 'The paper discusses some results concerning the multiplication of non-commutative random variables that are c-free with respect to a pair $( \Phi, \varphi) $, where $ \Phi $ is a linear map with values in some Banach or C$^\ast$-algebra and $ \varphi $ is scalar-valued. In particular, we construct a suitable analogue of the Voiculescu’s $ S $-transform for this framework.' address: - 'Department of Mathematics, University of Texas at San Antonio, One UTSA Circle San Antonio, Texas 78249, USA' - 'Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva 8410501, Israel' - 'Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E6, Canada' author: - Mihai Popa - Victor Vinnikov - 'Jiun-Chiau Wang' title: 'On the multiplication of operator-valued c-free random variables' --- Introduction ============ The terminology “c-free independence”(or c-freeness) was first used in the 1990’s by M. Bozejko, R. Speicher and M.Leinert (see [@bs], [@bls]) to denote a relation similar to D.-V. Voiculescu’s free independence, but in the framework of algebras endowed with two linear functionals (see Definition \[def:1\] below). The additive c-free convolution and the analytic characterization of the correspondent infinite divisibility was described in 1996 (see [@bls]); appropriate instruments for dealing with the multiplicative c-free convolution appeared a decade later, in [@mvp-jcw]. There, for $ X $ a non-commutative random variable, we define an analytic function ${{}^cT}_X(z) $, inspired by Voiculescu’s $ S $-transform, such that if $ X $ and $ Y $ are c-free , then ${{}^cT}_{XY}(z)= {{}^cT}_X(z) \cdot {{}^cT}_Y(z) $. Alternate proofs of this result were given in [@mp] and [@mp-proc]. The present work discusses the multiplicative c-free convolution in the framework of [@mlot], namely when one of the functionals is replaced by a linear map with values in a (not necessarily commutative) Banach algebra. In particular, we show that the combinatorial methods from [@mp] can be adapted to this more general framework. Notably, in the case of free independence over some Banach algebra, as showed in [@dykema], [@dykema2], the analogue of Voculescu’s $S$-transform satisfies a “twisted multiplicative relation”, namely $ T_{XY}(b)= T_X( T_Y(b)\cdot b \cdot T_Y(b)^{ - 1})\cdot T_Y(b) $. The main result of the present work, Theorem \[mainthm\], shows that the non-commutative ${{}^cT}$-transform satisfies the usual multiplicative relation: $ {{}^cT}_{XY}(z) = {{}^cT}_X(z) \cdot {{}^cT}_Y(z) $ for $ X, Y $ c-free non-commutative random variables. The paper is organized as follows. Section 2 presents some preliminary notions and results, mainly concerning the lattice of non-crossing linked partitions and its connection to free and c-free cumulants and to $t$- and ${}^ct$-coefficients. The main result of the section is Proposition \[vanishtcoeff\], the characterization of c-freeness in terms of ${}^ct$-coefficients. Section 3 restates the results on planar trees used in [@mp] and utilize them for the main result of the paper, Theorem \[mainthm\]. Section 4 discusses some aspects of infinite divisibility for the multiplicative c-free convolution in operator-valued framework. The results from the scalar case (see [@mvp-jcw]) can be easily extended to the framework of a commutative algebra of operators, but they are generally not valid in the non-commutative case. Framework and notations ======================= Non-crossing partitions and c-free cumulants -------------------------------------------- In the first part of this paper we will consider $ A $ and $ B $ to be two unital Banach algebras and $ \varphi:A {\longrightarrow}\mathbb{C} $, respectively $ \Phi: A {\longrightarrow}B $ to be two unital and linear maps. If $ A $ and $ B $ are C$^{\ast}$- or von Neumann algebras, then we will require that $ \varphi $, respectively $ \Phi $ to be positive, respectively completely positive. \[def:1\] $ A_1 $ and $ A_2 $, two unital subalgebras of $ A $, are said to be c-free with respect to $ ( \Phi, \varphi )$ if for all $ n $ and all $ a_1, a_2, \dots, a_n $ such that $\varphi(a_j)=0$ and $a_j\in A_{\varepsilon(j)}$ with $\varepsilon(j)\in\{1,2\}$ and $\varepsilon(j )\neq\varepsilon(j + 1 ) $ we have that 1. $ \varphi ( a_1 \cdots a_n ) = 0 $ 2. $ \Phi ( a_1 \cdots a_n )= \Phi(a_1 ) \cdots \Phi ( a_ n ) $. Two elements, $ X $ and $ Y, $ of $ A $ are said two be c-free with respect to $ ( \Phi, \varphi ) $ if the unital subalgebras of $ A $ generated by $ X $ and $ Y $ are c-free, as above. If $ A $ is a C$^\ast$- or a von Neumann algebra, then we will require that the unital C$^\ast$-, respectively von Neumann subalgebras of $ A $ generated by $ X $ and $ Y $ are c-free. If only condition [(i)]{} holds true, then the subalgebras $A_1, A_2 $ (respectively the elements $ X, Y $) are said to be free with respect to $\varphi$. As shown in [@bls], [@mp-commstoch], there is a convenient combinatorial characterization of c-freeness in terms of non-crossing partitions, that we will summarize below. A non-crossing partition $\gamma$ of the ordered set $\{1,2,\dots,n\}$ is a collection $C_1,\dots, C_k$ of mutually disjoint subsets of $\{1,2,\dots,n\}$, called blocks, such that their union is the entire set $ \{ 1, 2, \dots, n \} $ and there are no crossings, in the sense that there are no two blocks $C_l, C_s$ and $i<k<p<q$ such that $i,p\in C_l$ and $k,q\in C_s$. Example 1: Below is represented graphically the non-crossing partition\ $\pi=(1,5,6), (2,3), (4),(7,10),(8,9)$: $$\begin{picture}(10,8) \put(-16.5,0.5){1} \put(-15.9,3){\circle{1}} \put(-16,3){\line(0,1){5}} \put(-12.5,0.5){2} \put(-12,3){\circle{1}} \put(-12,3){\line(0,1){3}} \put(-8.5,0.5){3} \put(-8,3){\circle{1}}\put(-8,3){\line(0,1){3}} \put(-12,6){\line(1,0){4}} \put(-4.5,0.5){4} \put(-3.9,3){\circle{1}}\put(-4,3){\line(0,1){3}} \put(-0.5,0.5){5} \put(0,3){\circle{1}}\put(0,3){\line(0,1){5}} \put(3.5,0.5){6} \put(4,3){\circle{1}}\put(4,3){\line(0,1){5}} \put(-16,8){\line(1,0){20}} \put(7.5,0.5){7} \put(8.1,3){\circle{1}}\put(8,3){\line(0,1){5}} \put(11.5,0.5){8} \put(12.1,3){\circle{1}}\put(12,3){\line(0,1){3}} \put(15.5,0.5){9} \put(16.2,3){\circle{1}}\put(16,3){\line(0,1){3}} \put(12,6){\line(1,0){4}} \put(19,0.5){10} \put(20.2,3){\circle{1}}\put(20,3){\line(0,1){5}} \put(8,8){\line(1,0){12}} \end{picture}$$ The set of all non-crossing partitions on the set $ \{ 1, 2, \dots, n \} $ will be denoted by $NC(n)$. It has a lattice structure with respect to the reversed refinement order, with the biggest, respectively smallest element $\mathbbm{1}_{n}=(1,2,\dots,n)$, respectively $0_{n}=(1),\dots,(n)$. For $\pi,\sigma\in NC(n)$ we will denote by $\pi\bigvee\sigma$ their join (smallest common upper bound). For $\gamma\in NC(n)$, a block $B=(i_{1},\dots,i_{k})$ of $\gamma$ will be called interior if there exists another block $D\in\gamma$ and $i,j\in D$ such that $i<i_{1},i_{2},\dots,i_{k}<j$. A block will be called exterior if is not interior. The set of all interior, respectively exterior blocks of $\gamma$ will be denoted by $\text{Int}(\gamma)$, respectively $\text{Ext}(\gamma)$. The set $ \text{Ext} ( \gamma) $ is totally ordered by the value of the first element in each block. For $ X_1, \dots, X_n \in A $, we define the free, respectively c-free, cumulants $ \kappa_n (X_1, \dots, X_n ) $, respectively $ {}^c\kappa_n ( X_1, \dots, X_n ) $ as the multilinear maps from $ A^n $ to $ \mathbb{C} $, respectively $ B $, given by the recurrences below: $$\begin{aligned} \varphi( X_1 \cdots X_n ) & = \sum_{ \gamma \in {\textrm{NC}}( n ) } \prod_{\substack{ C = \text{block in} \gamma \\ C = (i_1, \dots, i_l ) } } \kappa_l (X_{ i_1}, \dots, X_{ i_l } )\\ \Phi ( X_1 \dots X_n ) & = \sum_{ \gamma \in {\textrm{NC}}( n ) } [ \prod_{\substack{ B \in \text{Ext} ( \gamma ) \\ B = ( j_1, \dots, j_l ) } } {{}^c\kappa}_l ( X_{ j_1 } \cdots X_{ j_l } ) ] \cdot [ \prod_{\substack{ D \in \text{Int} ( \gamma ) \\ D = (i_1, \dots, i_s ) } } \kappa_s (X_{ i_1}, \dots, X_{ i_s } ) ]\end{aligned}$$ with the convention that if $ \Lambda = \{ \alpha(1),\alpha(2), \dots, \alpha(n) \} $ is a totally ordered set and $ \{ X_{\lambda} \}_{\lambda \in \Lambda} $ is a collection of elements from $ B $, then $$\prod_{ \lambda \in \Lambda } X_\lambda = X_{ \alpha(1) } \cdot X_{ \alpha(2)}\cdots X_{ \alpha(n) }.$$ We will us the shorthand notations $ \kappa_n ( X ) $ for $ \kappa_n ( X, \dots, X ) $ and $ {{}^c\kappa}_n (X) $ for $ {{}^c\kappa}_n ( X, \dots, X ) $. As shown in [@ns], [@vdn], and in [@mp-commstoch], if $ X_1 $ and $ X_2 $ are c-free, then $$\begin{aligned} R_{ X_1 + X_2 }(z) & = R_{X_1} ( z ) + R_{X_2} ( z ) \\ {{}^cR}_{ X_1 + X_2} ( z ) & = {{}^cR}_{X_1} ( z ) + {{}^cR}_{X_2} ( z )\end{aligned}$$ where, for $ X \in A $, we let $ R_X ( z ) = \sum_{ n =1 }^\infty \kappa_n ( X ) z^n $ and $ {{}^cR}_X ( z ) = \sum_{ n = 1 }^\infty {{}^c\kappa}_n ( X ) z^n $. That is, the mixed free and c-free cumulants in $ X_1 $ and $ X_2 $ vanish. We will prove next a result analogous to Theorem 14.4 from [@ns], more precisely a lemma about c-free cumulants with products as entries, in fact an operator-valued version of Lemma 3.2 from [@mvp-jcw]. The Kreweras complementary $ \text{Kr}(\pi)$ of $ \pi \in NC(n)$ is defined as follows (see [@ns], [@kreweras]). Consider the symbols $\overline{1},\dots,\overline{n}$ such that $1<\overline{1}<2<\dots<n<\overline{n}$. Then $Kr(\pi)$ is the biggest element of $NC(\overline{1},\dots,\overline{n})\cong NC(n)$ such that $ \pi\cup \text{Kr}(\pi) $ is an element of $ NC(1,\overline{1},\dots,n,\overline{n}) $. The total number of blocks in $\gamma$ and ${\text{Kr}}(\gamma)$ is $n+1$. For $ \gamma \in NC(n) $ and $ p \in \{ 1, 2, \dots, n \} $, we will denote by $ \gamma [ p ] $ the block of $ \gamma $ that contains $ p $. \[lemma:1\] Suppose that $ X, Y $ are two c-free elements of $ A $. Then 1. $ \displaystyle \kappa_n (XY) = \sum_{ \gamma\in NC(n) } \prod_{ B \in \gamma } \kappa_{ \| B \| } (X) \cdot \prod_{ D \in \text{Kr}(\gamma) } \kappa_{ \| D \| } ( Y ) $ 2. $ \displaystyle {{}^c\kappa}_n (XY) = \sum_{ \gamma\in NC(n) } {{}^c\kappa}_{ \| \gamma [ 1 ] \| } ( X ) \cdot {{}^c\kappa}_{ \| \text{Kr} ( \gamma) [ \overline{n} ] \| } ( Y ) \cdot \prod_{ \substack{ B \in \gamma\\ B \neq \gamma[ 1 ] } } \kappa_{ \| B \| } (X) \cdot \prod_{\substack{ D \in \text{Kr}(\gamma) \\ D \neq \text{Kr}( \gamma)[ \overline{n}] } } \kappa_{ \| D \| } ( Y ) $ Part (i) is shown in [@ns], Theorem 14.4. We will show part (ii) by induction on $ n $. For $ n = 1 $, the statement is trivial, since $ {{}^c\kappa}_2 ( X, Y ) = 0 $ from the c-freeness of $ X $ and $ Y $, therefore $ {{}^c\kappa}_1 (XY) = \Phi ( X Y ) = {{}^c\kappa}_1 (X) {{}^c\kappa}_1 (Y ). $ For the inductive step, in order to simplify the writting, we will introduce several new notations. Let $ NC_S ( n )=\{ \pi \in NC( n):$ elements from the same block of $ \pi $ have the same parity $\}$. For $\sigma\in NC_{S}( n)$, denote $\sigma_{+}$, respectively $\sigma_{-}$ the restriction of $\sigma$ to the even, respectively odd, numbers and define $ NC_{0}(n)=\{\sigma:\sigma\in NC(n),\sigma_{+}=Kr(\sigma_{-})\}. $ Also, we will need to consider the mappings $$NC(n)\times NC(m)\ni(\pi,\sigma)\mapsto\pi\oplus\sigma\in NC(m+n),$$ the juxtaposition of partitions, and $$NC(n)\ni\sigma\mapsto\widehat{\sigma}\in NC(2n)$$ constructed by doubling the elements, that is if $ (i_1, i_2, \dots, i_s ) $ is a block of $ \sigma $, then $ ( 2 i_1 -1, 2 i_1 , 2 i_2 -1, 2 i_2, \dots, 2 i_s - 1, 2 i_s ) $ is a block of $ \widehat{\sigma} $. Then, for $ \pi \in NC (n ) $, we define $$\begin{aligned} \kappa_\pi[ X_1, \dots, X_n ] & = \prod_{\substack{ C \in \pi \\ C = (i_1, \dots, i_l ) } } \kappa_l (X_{ i_1}, \dots, X_{ i_l } )\\ {\mathcal{K}}_\pi[ X_1, \dots, X_n ] & = \prod_{\substack{ B \in \text{Ext} ( \gamma ) \\ B = ( j_1, \dots, j_l ) } } {{}^c\kappa}_l ( X_{ j_1 } \cdots X_{ j_l } ) ] \cdot [ \prod_{\substack{ D \in \text{Int} ( \gamma ) \\ D = (i_1, \dots, i_s ) } } \kappa_s (X_{ i_1}, \dots, X_{ i_s } )\end{aligned}$$ Remark that $ \kappa_{ \pi \oplus \sigma } = \kappa_\pi \cdot \kappa_\sigma $ and $ {\mathcal{K}}_{ \pi \oplus \sigma } = {\mathcal{K}}_\pi \cdot {\mathcal{K}}_\sigma $. Also, if $ ( i_1, i_2, \dots, i_s )$ is an exterior block of $\pi$, then $$\pi_{ \| \{ i_1 + 1, \dots, i_s - 1 \} } \cup {\text{Kr}}( \pi_{ \| \{ i_1 + 1, \dots, i_s - 1 \} } ) = \pi \cup {\text{Kr}}(\pi)_ { \| \{ i_1 + 1, \dots, i_s - 1 \} },$$ so Lemma \[lemma:1\] is equivalent to $$\begin{aligned} \kappa_{\pi}[XY,\dots,XY]& =\sum_{\substack{\sigma\in NC_{S}(2n)\\ \sigma\bigvee\widehat{0_{n}}=\widehat{\pi}} }\kappa_{\sigma}[X,Y,\dots,X,Y] \\ {\mathcal{K}}_{\pi}[XY,\dots,XY] &= \sum_{\substack{\sigma\in NC_{S}(2n)\\ \sigma\bigvee\widehat{0_{n}} =\widehat{\pi}} }{\mathcal{K}}_{\sigma}[X,Y,\dots,X,Y],\end{aligned}$$ therefore$ \phi\left((XY)^{n}\right) = {{}^c\kappa}_n (XY)+\sum_{\substack{\pi\in NC(n)\\ \pi\neq\mathbbm{1}_{n}} }{\mathcal{K}}_{\pi}[XY,\dots,XY]. $ On the other hand, $$\phi\left((XY)^{n}\right)=\phi(X\cdot Y\cdots X\cdot Y)=\sum_{\sigma\in NC(2n)}{\mathcal{K}}_{\sigma}[X,Y,\dots,X,Y].$$ Since the mixed cumulants vanish, the equation above becomes $$\begin{aligned} \phi\left((XY)^{n}\right) & = & \sum_{\sigma\in NC_{S}(2n)}{\mathcal{K}}_{\sigma}[X,Y,\dots,X,Y]\\ & = & \sum_{\sigma\in NC_{0}(2n)}{\mathcal{K}}_{\sigma}[X,Y,\dots,X,Y]+\sum_{\substack{\sigma\in NC_{S}(2n)\\ \sigma\notin NC_{0}(2n)} }{\mathcal{K}}_{\sigma}[X,Y,\dots,X,Y] \end{aligned}$$ But $NC_{S}(2n)=\bigcup_{\pi\in NC(n)}\{\sigma:\ \sigma\in NC_{S}(2n),\sigma\bigvee\widehat{0_{n}}=\widehat{\pi}\}$, and, for $\sigma\in NC_{s}(2n)$, one has that $\sigma\in NC_{0}(2n)$ if and only if $\sigma\bigvee\widehat{0_{n}}=\mathbbm{1}_{2n}$, Therefore: $$NC_{S}(2n)\setminus NC_{0}(2n)=\bigcup_{\substack{\pi\in NC(n)\\ \pi\neq\mathbbm{1}_{n}} }\{\sigma:\ \sigma\in NC_{S}(2n),\sigma\bigvee\widehat{0_{n}}=\widehat{\pi}\},$$ henceforth $$\begin{aligned} \sum_{\substack{\sigma\in NC_{S}(2n)\\ \sigma\notin NC_{0}(2n)} }{\mathcal{K}}_{\sigma}[X,Y,\dots,X,Y] & = & \sum_{\substack{\pi\in NC(n)\\ \pi\neq\mathbbm{1}_{n}} }\sum_{\substack{\sigma\in NC_{S}(2n)\\ \sigma\bigvee\widehat{0_{n}}=\widehat{\pi}} }{\mathcal{K}}_{\sigma}[X,Y,\dots,X,Y]\\ & = & \sum_{\substack{\pi\in NC(n)\\ \pi\neq\mathbbm{1}_{n}} }{\mathcal{K}}[XY,\dots,XY]. \end{aligned}$$ so the proof is now complete. Non-crossing linked partitions and $t$-coefficients --------------------------------------------------- ${}$\ By a non-crossing linked partition $\pi$ of the ordered set $\{1,2,\dots,n\}$ we will understand a collection $B_1,\dots, B_k$ of subsets of $\{1,2,\dots,n\}$, called blocks, with the following properties: 1. $\displaystyle{\bigcup_{l=1}^kB_l=\{1,\dots,n\}}$ 2. $B_1,\dots,B_k$ are non-crossing. 3. for any $1\leq l,s\leq k$, the intersection $B_l\bigcap B_s$ is either void or contains only one element. If $\{j\}=B_i\bigcap B_s$, then $\|B_s\|, \|B_l\|\geq 2$ and $j$ is the minimal element of only one of the blocks $B_l$ and $B_s$. For $ \pi $ as above we define $s(\pi)$ to be the set of all $1\leq k\leq n$ such that there are no blocks of $\pi$ whose minimal element is $k$. A block $B=i_1<i_2<\dots <i_p$ of $\pi$ will be called exterior if there is no other block $D$ of $\pi$ containing two elements $l,s$ such that $l\leq i_1<i_p<s$. The set of all non-crossing linked partitions on $\{1,\dots, n\}$ will be denoted by $NCL(n)$. Example 2: Below is represented graphically the non-crossing linked partition $\pi=(1,4,6,9), (2,3), (4,5),(6,7,8),(10,11), (11,12)$ from $ NCL(12)$. Its exterior blocks are $(1, 4, 6, 9)$ and $(10, 11)$. $$\begin{picture}(10,8) \put(-16,3){\circle{1}} \put(-12,3){\circle{1}} \put(-12,3){\line(0,1){3}} \put(-8,3){\circle{1}}\put(-8,3){\line(0,1){3}} \put(-4,3){\circle{1}} \put(0,3){\circle{1}}\put(0,3){\line(0,1){3}} \put(4,3){\circle{1}} \put(8,3){\circle{1}}\put(8,3){\line(0,1){3}} \put(12,3){\circle{1}}\put(12,3){\line(0,1){3}} \put(16,3){\circle{1}} \put(20,3){\circle{1}} \put(24,3){\circle{1}} \put(28,3){\circle{1}}\put(28,3){\line(0,1){3}} \put(-12,6){\line(1,0){4}} \put(8,6){\line(1,0){4}} \put(-4,3){\line(4,3){4}} \put(4,3){\line(4,3){4}} \put(24,3){\line(4,3){4}} \put(-16.5,0.2){1} \put(-12.5,0.2){2} \put(-8.5,0.2){3} \put(-4.5,0.2){4} \put(-0.5,0.2){5} \put(3.5,0.2){6} \put(7.5,0.2){7} \put(11.5,0.2){8} \put(15.5,0.2){9} \put(19,0.2){10} \put(23,0.2){11} \put(27,0.2){12} \linethickness{.55mm} \put(-16,3){\line(0,1){5}} \put(-16,8){\line(1,0){32}} \put(20,8){\line(1,0){4}} \put(20,3){\line(0,1){5.1}} \put(24,3){\line(0,1){5.1}} \put(16,3){\line(0,1){5}} \put(4,3){\line(0,1){5.1}} \put(-4,3){\line(0,1){5}} \end{picture}$$ Similarly to [@dykema] and [@mp], we define the $ t $-coefficients, respectively $ {{}^ct}$-coefficients, as follows. Take $ A^{\circ} = A \setminus \ker \varphi $. Then, for $ n $ a positive integer, the maps $ t_n : A \times ( A^\circ )^n {\longrightarrow}\mathbb{C} $ and $ {}^c t_n : A \times (A^\circ)^n {\longrightarrow}B $ are given by the following recurrences: $$\label{eq:31} \varphi( X_1\cdots X_n ) = \sum_{ \pi\in {\textrm{NCL}}( n ) } [ \prod_{\substack{ B \in \pi \\ B = ( i_1, \dots, i_l ) } } t_{ l-1 } ( X_{ i_1} \, \dots, X_{ i_l } ) \cdot \prod_{ p \in s( \pi ) } t_0 ( p ) ]\\$$ respectively $$\begin{aligned} \label{eq:32} \Phi ( X_1 \cdots X_n ) & = & \sum_{ \pi \in {\textrm{NCL}}(n) } [ \prod_{ \substack{ B \in \text{Ext}( \pi ) \\ B = ( i_1, \dots, i_l ) } } {{}^ct}_{ l-1 } ( X_{ i_1} \, \dots, X_{ i_l } ) \label{eq:41}\\ && \hspace{1cm}\cdot \prod_{ \substack{ D \in \text{Int} ( \pi ) \\ D = ( j_1, \dots , j_s ) } } t_{ s-1 } ( X_{ j_1} , \dots, X_{ j_s } ) \cdot \prod_{ p \in s(\pi ) } t_0 ( X_p ). \nonumber\end{aligned}$$ To simplify the writing we will use the shorthand notations $ t_\pi[X_1,\dots, X_n] $, respectively ${{}^ct}_\pi[X_1,\dots, X_n] $ for the summing term of the right-hand side of (\[eq:31\]), respectively (\[eq:41\]); also, if $ X_1 = \dots = X_n= X$, we will use the shorter notations $ t_\pi[X] $, $ {{}^ct}_\pi[X] $ respectively $t_n(X)$, $ {{}^ct}_n(X)$. Note that all the factors in $ t_\pi $ are $ t $-coefficients, but the development of $ {{}^ct}_\pi $ contains both $ {{}^ct}$- and $ t $-coefficients. The lattice $NCL(n)$ and c-freeness in terms on $ t $-coefficients ------------------------------------------------------------------ On the set $NCL(n)$ we define a order relation by saying that $\pi{\succeq}\sigma$ if for any block $B$ of $\pi$ there exist $D_1,\dots, D_s$ blocks of $\sigma$ such that $\displaystyle {B=D_1\cup\dots\cup D_s}$. With respect to the order relation ${\succeq}$, the set $NCL(n)$ is a lattice that contains $NC(n)$. We say that $i$ and $j$ are connected in $\pi\in NCL(n)$ if there exist $B_1,\dots,B_s$ blocks of $\pi$ such that $i\in B_1$, $j\in B_s$ and $B_k\cap B_{k+1}\neq \varnothing$, $1\leq k\leq s-1$. For every $\pi\in NCL(n)$ there exist a unique partition $c(\pi)\in NC(n)$ defined as follows: $i$ and $j$ are in the same block of $c(\pi)$ if and only if they are connected in $\pi$. We will use the notation $$[c(\pi)]=\{\sigma\in NCL(n): c(\sigma)=c(\pi)\}.$$ In Example 2 from above, we have $c(\pi)=(1, 4, 5, 6, 7, 8, 9), (2, 3), (10, 11, 12)$ for $ \pi = (1, 4, 6, 9), (2, 3), (4, 5), (6, 7, 8), (10, 11), (11, 12) $. For every $\gamma \in NC(n)$, the set $[\gamma]$ is a sublattice of $NCL(n)$ with maximal element $\gamma$. Moreover, if $\gamma$ has the blocks $B_1,\dots, B_s$, each $B_l$ of cardinality $k_l$, then we have the following ordered set isomorphism: $$\label{factorization} [c(\pi)] \simeq [\mathbbm{1}_{k_1}]\times\cdots\times[\mathbbm{1}_{k_s}]$$ \[cum-tcoeff\] For any positive integer $n$ and any $X_1,\dots, X_n\in A $ we have that $$\begin{aligned} \kappa_n(X_1,\dots,X_n) & =\sum_{\pi\in[\mathbbm{1}_n]}t_\pi[X_1,\dots, X_n]\\ {{}^c\kappa}_n(X_1,\dots,X_n) & =\sum_{\pi\in[\mathbbm{1}_n]}{{}^ct}_\pi[X_1,\dots, X_n]\end{aligned}$$ First relation is shown in Proposition 1.4 from [@mp]. The second relation is trivial for $ n = 1 $. For $ n > 1 $, note that $$\sum_{\pi\in NCL(n)} {{}^ct}_\pi[X_1,\dots,X_n] =\sum_{\gamma\in NC(n)}\sum_{\pi\in[\gamma]} {{}^ct}_\gamma[X_1,\dots,X_n]$$ and the similar relation for $ t_\pi $. Suppose that $ \pi \in NCL(n) $ and $ \gamma \in NC(n) $ are such that $ \pi \in [ \gamma ] $. From the definition of $ {{}^ct}$- and $ t $-coefficients we have that $$\label{eq:24} {{}^ct}_\pi [ X_1, \cdots, X_n ] = \prod_{ \substack{ B \in \text{Ext} ( \pi )\\ B = ( i_1, \dots, i_s ) } } {{}^ct}_{ \pi_{ \| B } } [ X_{ i_1 }, \dots, X_{ i_s } ] \cdot \prod_{ \substack{ D \in \text{Int} ( \pi )\\ D = ( j_1, \dots, j_t ) } } t_{ \pi_{ \| D } } [ X_{ j_1 }, \dots, X_{ j_t } ]$$ and $$\label{eq:25} t_\pi[X_1,\dots,X_n] = \prod_{ \substack{ B \in \pi )\\ B = ( i_1, \dots, i_s ) } } {{}^ct}_{ \pi_{ \| B } } [ X_{ i_1 }, \dots, X_{ i_s } ] .$$ Therefore, the equation (\[eq:32\]) becomes $$\begin{aligned} \Phi( X_1 \cdots X_n ) = & \sum_{ \gamma \in NC(n) } [ \prod_{ \substack { B \in \text{Ext}(\pi ) \\ B = ( i_1, \dots, i_s ) } } ( \sum_{ \substack{ \pi \in NCL(n) \\ \pi \in [ \gamma ] } } {{}^ct}_{ \pi_{ \| B } } [ X_{ i_1 }, \dots, X_{ i_s } ] ) \\ & \cdot \prod_{ \substack { D \in \text{Int}(\pi ) \\ D = ( j_1, \dots, j_l ) } } ( \sum_{ \substack{ \pi \in NCL(n) \\ \pi \in [ \gamma ] } } t_{ \pi_{ \| D } } [ X_{ j_1 }, \dots, X_{ j_l } ] ) ]\end{aligned}$$ and the factorization (\[factorization\]) gives: $$\begin{aligned} \Phi( X_1 \cdots X_n ) = & \sum_{ \gamma \in NC(n) } [ \prod_{ \substack { B \in \text{Ext}(\pi ) \\ B = ( i_1, \dots, i_s ) } } ( \sum_{\sigma\in[\mathbbm{1}_s] } {{}^ct}_{ \sigma } [ X_{ i_1 }, \dots, X_{ i_s } ] ) \\ & \cdot \prod_{ \substack { D \in \text{Int}(\pi ) \\ D = ( j_1, \dots, j_l ) } } ( \sum_{ \sigma\in[\mathbbm{1}_l] } t_{ \sigma } [ X_{ j_1 }, \dots, X_{ j_l } ] ) ]\end{aligned}$$ The conclusion follows now utilizing the moment-cumulant recurrence and induction on $n$. For $ X \in A^\circ $, we define the formal power series $ T_X ( z ) = \sum_{ n = 0 }^\infty t_n ( X ) z^n $ and $ {}^cT_X ( z ) = \sum_{ n = 0 }^\infty {{}^ct}_n ( X ) z^n $. Also, we consider the moment series of $ X $, namely $ M_X(z)= \sum_{n=1}^\infty \Phi(X^n) z^n $, and $ m_X(z) = \sum_{ n =1}^\infty \varphi(X^n) z^n $. As shown in [@mvp-jcw], Lemma 7.1(i), the recurrence \[eq:31\] gives that $$\label{eq:38} T_X(m_X(z)) \cdot ( 1 + m_X(z)) =\frac{1}{z} m_X(z ).$$ The proposition below gives an analogous relation for the series $ {{}^cT}_X(z) $ (i. e. a non-commutative analogue of Lemma 7.1(ii) from [@mvp-jcw]). \[analytic-ct\] With the notations above, we have that $$\label{eq:39} {{}^cT}_X (m_X(z) ) \cdot ( 1 + M_X( z ) ) = \frac{1}{z} M_X(z).$$ Let $ NCL(1, p) = \{ \pi \in NCP(p) : \pi \ \text{has only one exterior block} \}$. Note that for each $ \tau \in NCL(n) $, there exists a unique triple $ p \leq n $, $ \pi \in NCL(1, p ) $ and $ \sigma \in NCL(n-p) $ such that $ \tau = \pi \oplus \sigma $. Indeed, taking $ p $ to be the maximal element of the block of $ c(\tau) $ containing 1, it follows that $$\tau = \tau_{ \| \{ 1, 2, \dots, p \}} \oplus \tau_{ \| \{ p+1, \dots, n \}}$$ and that $ \tau_{ \| \{ 1, 2, \dots, p \}} \in NCL(1, p) $. Conversely, each triple $ p, \pi, \sigma $ as above determine a unique $ \pi \oplus \sigma \in NCL(n) $, hence $$NCL(n) = \bigcup_{ p \leq n } \{ \pi \oplus \sigma : \ \pi \in NCL(1, p), \sigma \in NCL( n - p ) \}$$ therefore the recurrence \[eq:32\] gives $$\begin{aligned} \Phi(X^n) &= \sum_{ \pi \in NCL(n)} {{}^ct}_pi(X) \nonumber\\ & = \sum_{ p \leq n } [ \sum_{ \pi \in NCL(p)} {{}^ct}_\pi(X) \cdot ( \sum_{ \sigma \in NCL(n-p)} {{}^ct}_\sigma(X))] \nonumber \\ & = \sum_{ p \leq n } [ \sum_{ \pi \in NCL(p)} {{}^ct}_\pi(X) \cdot \Phi(X^{ n - p}) ]\label{new:01}. \end{aligned}$$ Let $ NCL (1, q, p ) = \{ \pi \in NCL(1, p) : \ \text{ the exterior block of $ \pi $ has exactly $ q $ elements} \} $. Fix $ \pi \in NCL(1, q, p) $ and let $ (1, i_1, i_2, \dots, i_{q-1} )$ be the exterior block of $ \pi $. Define $ \widetilde{\pi} \in NCL(p-1)$ such that, with the notations following the recurrences (\[eq:31\])–(\[eq:32\]), $ {{}^ct}_\pi(X)= {{}^ct}_{ q-1}\cdot t_{ \widetilde{\pi}}(X) $, as follows: 1. if $ (j_1, j_2, \dots, j_s ) $ is an interior block of $ \pi $, then $(j_1 -1, j_2-1, \dots, j_s-1 ) $ is a block in $ \widetilde{\pi} $; 2. if $ j >1 $ and the only block of $ \pi $ containing $ j $ is exterior, then $ ( j-1) $ is a block of $ \widetilde{\pi} $ i.e. $ \widetilde{\pi} $ is obtaining by “ deleting ” the 1 and the exterior block of $ \pi $. For each $ i_l $ in the exterior block $(1, i_1, i_2, \dots, i_{q-1})$ of $ \pi $ we define $ i_l^\prime $ to be the maxinal element connected to $ i_l $ in $ \pi $, and let $ i_0^\prime =0$. Then each set $ S(l) = \{ i_{l-1}^\prime +1, i_{l-1}^\prime +2, \dots, i_l^\prime\} $ is nonvoid and we have the decomposition $ \widetilde{\pi } = \widetilde{\pi }_{ \| S(1)} \oplus \widetilde{\pi }_{ \| S(2)} \oplus \dots \oplus \widetilde{\pi }_{ \| S(q-1)} .$ Example 3. If $ \pi = \{(1, 4, 6, 9), (2, 3), (4, 5), (6, 7, 8)\} $, then $$\widetilde{\pi}=\{ (1, 2), (3, 4), (5, 6, 7), (8) \} = \{ (1, 2), (3, 4)\} \oplus \{ (1, 2, 3)\} \oplus \{ (1)\},$$ see the diagram below: $$\begin{picture}(24,4) \put(20,0){\circle{1}} \put(20,0){\line(0,1){3}} \put(24,0){\circle{1}} \put(24,0){\line(0,1){3}} \put(20,3){\line(1,0){4}} \put(28,0){\circle{1}} \put(28,0){\line(0,1){3}} \put(28, 3){\line(1,0){4}} \put(32,0){\circle{1}} \put(32,0){\line(0,1){3}} \put(33, 1){ $\oplus$} \put(38,0){\circle{1}} \put(38,0){\line(0,1){3}} \put(38,3){\line(1,0){8}} \put(42,0){\circle{1}} \put(42,0){\line(0,1){3}} \put(46,0){\circle{1}} \put(46,0){\line(0,1){3}} \put(47,1){ $\oplus$} \put(52,0){\circle{1}} \put(52,0){\line(0,1){3}} \put(4,3){\line(1,0){4}} \put(-8,0){\line(4,3){4}} \put(0,0){\line(4,3){4}} \put(-20,0){\circle{1}} \put(-16,0){\circle{1}} \put(-16,0){\line(0,1){3}} \put(-12,0){\circle{1}} \put(-12,0){\line(0,1){3}} \put(-8,0){\circle{1}} \put(-4,0){\circle{1}} \put(-4,0){\line(0,1){3}} \put(0,0){\circle{1}} \put(4,0){\circle{1}} \put(4,0){\line(0,1){3}} \put(8,0){\circle{1}} \put(8,0){\line(0,1){3}} \put(12,0){\circle{1}} \put(-16,3){\line(1,0){4}} \put(4,3){\line(1,0){4}} \put(-8,0){\line(4,3){4}} \put(0,0){\line(4,3){4}} \linethickness{.55mm} \put(-20,0){\line(0,1){5}} \put(-20,5){\line(1,0){32}} \put(12,0){\line(0,1){5}} \put(0,0){\line(0,1){5}} \put(-8,0){\line(0,1){5}} \put(14,1 ){$\longrightarrow$} \end{picture}$$ Using the equality $ {{}^ct}_\pi(X)= {{}^ct}_{ q-1}\cdot t_{ \widetilde{\pi}}(X) $, we obtain $$\begin{aligned} \sum_{ \pi \in NCL(1, q, p)} {{}^ct}_\pi(X) &= {{}^ct}_{ q-1}(X) \cdot \sum_{ r_1 + \dots r_{ q-1}=p } ( \prod_{k=1}^{q-1} \sum_{ \sigma\in NCL(r_k)} t_{\sigma}(X) )\nonumber\\ &= {{}^ct}_{ q-1}(X)\sum_{ r_1 + \dots r_{ q-1}=p } ( \prod_{k=1}^{q-1} \varphi(X^{r_k}). )\label{new:02} \end{aligned}$$ Since $ \displaystyle NCL(1, p) = \bigcup_{q=1}^p NCL(1, q, p) $, the equations (\[new:01\]) and (\[new:02\]) give that $$\Phi(X^n) = \sum_{p=1}^n ( \sum_{ q=1}^p {{}^ct}_{q-1}(X) \cdot\sum_{ r_1 + \dots r_{ q-1}=p } ( \prod_{k=1}^{q-1} \varphi(X^{r_k}),$$ which is the relation of the left hand side and right hand side coefficients of $ z^{n-1} $ in equation (\[eq:39\]). We conclude this section with the following result. \[vanishtcoeff\](Characterization of c-freeness in terms of ${}^ct$-coefficients)\ Two elements $ X, Y $ from $ A^\circ $ are c-free if and only if all their mixed $ t $- and $ {{}^ct}$-coefficients vanish, that is for all $ n $ and all $ a_1, \dots, a_n \in \{ X, Y \} $ such that $ a_k = X $ and $ a_l = Y $ for some $ k , l $ we have that $${{}^ct}_{ n -1}( a_1, \dots, a_n ) = t_{n-1} ( a_1, \dots, a_n ) = 0.$$ We will show by induction on $ n $ the equivalence between vanishing of mixed free and c-free cumulants of order $ n $ in $ X $ and $ Y $ and vanishing of mixed $ t $- and $ {{}^ct}$-coefficients of order up to $ n -1 $. For $ n = 2 $ the result is trivial, since $ k_2 ( X, Y ) = t_1 ( X, Y ) $ and $ {{}^c\kappa}_2 ( X, Y ) = {{}^ct}_1 ( X, Y ) $. For the inductive step suppose that $ a_1, \dots, a_n $ are not all $ X $ nor all $ Y $. Proposition \[cum-tcoeff\] gives $$\begin{aligned} \kappa_n ( a_1, \dots, a_n ) =& t_{n-1}(a_1,\dots, a_n)+ \sum_{\substack{\pi\in[\mathbbm{1}_n]\\ \pi\neq \mathbbm{1}_n}} t_\pi [a_1,\dots, a_n] \\ {{}^c\kappa}_n ( a_1, \dots, a_n ) =& {{}^ct}_{n-1}(a_1,\dots, a_n)+ \sum_{\substack{\pi\in[\mathbbm{1}_n]\\ \pi\neq \mathbbm{1}_n}} {{}^ct}_{ \pi } [ a_1, \dots, a_n ].\end{aligned}$$ Fix $ \pi \in [\mathbbm{1}_n] $, $ \pi \neq \mathbbm{ 1 }_n $. Since $ \pi $ is connected, there is $ ( i_1, \dots, i_s ) $, a block of $ \pi $ with $ s < n $ such that $ a_{ i_1 }, \dots, a_{ i_2 } $ are not all $ X $ not all $ Y $, so equations (\[eq:24\]) and (\[eq:25\]) and the induction hypothesis imply that $ t_\pi (a_1, \dots, a_n ) ={{}^ct}_{\pi }( a_1, \dots, a_n ) = 0 $, hence we have that $ \kappa_n ( a_1, \dots, a_n ) = t_{n-1}(a_1,\dots, a_n) $ and $ {{}^c\kappa}_n ( a_1, \dots, a_n ) = {{}^ct}_{n-1}(a_1,\dots, a_n) $ so q.e.d.. planar trees and the multiplicative property of the $T$-transform ================================================================= In this section we will use the combinatorial arguments from [@mp] to show that whenever $ X $ and $ Y $ are two c-free elements from $ A^\circ $, we have that $ T_{ XY} (z ) = T_{ X } ( z ) \cdot T_{ Y } ( z ) $ and $ {{}^cT}_{ XY} (z ) = {{}^cT}_{ X} (z) \cdot {{}^cT}_{ Y } (z ) $ where, for $ Z \in A^\circ $, we define $ T_Z ( z ) = \sum_{ n = 0 }^\infty t_n ( Z ) z^n $, respectively $ {{}^cT}_Z ( z ) = \sum_{ n = 0 }^\infty {}^ct_n ( Z ) z^n $ Planar trees ------------ ${}$ We will start with a review of the notations and results from [@mp]. By an elementary planar tree* we will denote a graph with $m\geq 1$ vertices, $v_1,v_2,\dots, v_m$, and $m-1$ (possibly 0) edges, or branches, connecting the vertex $v_1$ (that we will call root) to the vertices $v_2, \dots, v_m$ (that we will call offsprings). By a planar tree we will understand a graph consisting in a finite number of levels, such that: 1. first level consists in a single elementary planar tree, whose root will be considered the root of the planar tree; 2. the $k$-th level will consist in a set of elementary planar trees such that their roots are offsprings from the $(k-1)$-th level. Below are represented graphically the elementary planar tree $ C_1 $ and the 2-level planar tree $ C_2 $: $$\begin{picture}(22,18) \put(-22,8){\circle{1}} \put(-22,8){\line(2,3){4}} \put(-18,14){\circle{1}} \put(-18,8){\circle{1}}\put(-18,8){\line(0,1){6}} \put(-14,8){\circle{1}} \put(-14,8){\line(-2,3){4}} \put(-19,4){$ C_1$} \put(8,16){\circle{1}} \put(5,11){\circle{1}} \put(5,11){\line(3,5){3}} \put(11,11){\circle{1}} \put(11,11){\line(-3,5){3}} \put(8,6){\circle{1}} \put(8,6){\line(3,5){3}} \put(14,6){\circle{1}} \put(14,6){\line(-3,5){3}} \put(5,2){$ C_2$} \multiput(17,6)(4,0){3}{\line(1,0){2}} \multiput(16,11)(4,0){3}{\line(1,0){2}} \multiput(15,16)(4,0){3}{\line(1,0){2}} \multiput(17,2)(4,0){3}{\line(1,0){2}} \put(26,3){level 3} \put(26,8){level 2} \put(26,13){level 1} \end{picture}$$ The set of all planar trees with $ n $ vertices will be denoted by $ \mathfrak{T}(n) $. If $ C $ is a planar tree, the set of elementary trees composing it will be denoted by $ E( C ) $, the elementary tree containing the root of $ C $ will be denoted by $ \mathfrak{r}(C) $, and we define $ \mathfrak{b}(C) = E( C) \setminus \{ \mathfrak{r}(C) \} $. On the set of vertices of a planar tree we consider the “left depth first” order from [@aep], given by: 1. roots are less than their offsprings; 2. offsprings of the same root are ordered from left to right; 3. if $v$ is less that $w$, then all the offsprings of $v$ are smaller than any offspring of $w$. ( I.e. the order in which the vertices are passed by walking along the branches from the root to the right-most vertex, not counting vertices passed more than one time, see the example below). Example 4: $$\begin{picture}(18,18) \put(8,18){1} \put(8,16){\circle{1.5}} \put(3,11){\circle{1.5}} \put(3,11){\line(1,1){5}} \put(8,16){\line(2,-1){10}} \put(18,11){\circle{1.5}} \put(18,11){\line(-3,-5){3}} \put(18,11){\line(3,-5){3}} \put(21,6){\circle{1.5}} \put(15,6){\circle{1.5}} \put(8,16){\line(0,-1){5}} \put(8,11){\circle{1.5}} \put(7.5,8){4} \put(3,6){\circle{1.5}}\put(3,6){\line(0,1){5}} \put(0,11){2} \put(1.5,3 ){3} \put(19.1,11.3){5} \put(14,3.2){6} \put(22,3.2){7} \end{picture}$$ Next, consider, as in [@mp], the map $\Theta: [\mathbbm{1}_n]{\longrightarrow}\mathfrak{T}(n)$ such that $\Theta(\pi)$ is the unique planar tree with the property that if $(i_1, \dots, i_s)$ is a block of $\pi$, then the vertices of $\Theta(\pi) $ form an elementary tree from $ E (\Theta(\pi))$. More precisely, if $(1,2,i_1,\dots, i_s)$ is the block of $\pi$ containing 1, then the first level of $\Theta(\pi)$ is the elementary planar tree of root numbered 1 and $ s + 1 $ offsprings numbered $(2,i_1,\dots, i_s)$. The second level of $\Theta(\pi)$ consists on the elementary trees representing the blocks (if any) having $2,i_1,\dots, i_s$ as first elements etc (see Example 5 below). As shown in [@mp], the map $ \Theta $ is well-defined and bijective. Example 5: ![image](first.pdf){width="3in" height="1.1cm"} For $ X \in A^\circ $, define the maps $ {\mathcal{E}}_{ X} $, respectively $ {\widetilde{\mathcal{E} } }_{ X } $ from $ \bigcup_{ n \in \mathbb{N } } \mathfrak{T} ( n ) $ to $ \mathbb{C} $, respectively to $ B $, as follows. If $ C $ is an elementary planar tree with $ n$ vertices, then let $ {\mathcal{E}}_X ( C ) = t_{ n-1} ( X ) $ and $ {\widetilde{\mathcal{E} } }_X (C) = {{}^ct}_{ n - 1 } ( X ) $; for $ W \in \mathfrak{T}(n ) $, let $$\begin{aligned} {\mathcal{E}}_X( W ) & = \prod_{ C \in E( W ) } {\mathcal{E}}_X ( C ) \\ {\widetilde{\mathcal{E} } }_X ( W ) & = {\widetilde{\mathcal{E} } }_X ( \mathfrak{r} ( W ) ) \cdot \prod_{ C \in \mathfrak{b}(W) } {\mathcal{E}}_X ( C ).\end{aligned}$$ As in [@mp], Proposition \[cum-tcoeff\] and the bijectivity $\Theta $ give that $$\label{eq:10} \kappa_n ( X ) = \sum_{ C \in \mathfrak{T}(n) } {\mathcal{E}}_X ( C ) \ \ \text{and} \ \ {{}^c\kappa}_n ( X ) = \sum_{ C \in \mathfrak{T}(n) } {\widetilde{\mathcal{E} } }_X ( C ).$$ Bicolor planar trees and the Kreweras complement ------------------------------------------------ By a bicolor elementary planar tree we will understand an elementary tree together with a mapping from its offsprings to $\{0, 1\}$ such that the offsprings whose image is 1 are smaller than (with respect to the order relation considered above, i.e. at right of) the offsprings with image 0. Branches toward offsprings of color 0, respectively 1, will be also said to be of color 0, respectively 1. The set of all bicolor planar trees with $n$ vertices will be denoted by $\mathfrak{EB}(n)$. Following [@mp], branches of color 1 will be represented by solid lines and branches of color 0 by dashed lines. Example 6: The graphical representation of $\mathfrak{EB}(4)$: ![image](cb4.pdf){width="3in" height=".8cm"} A planar tree whose constituent elementary trees are all bicolor will be called bicolor planar tree ; the set of all bicolor planar trees will be denoted by $\mathfrak{B}(n)$. Let $ NCL_S (2n) = \{ \pi\in NCL(2n) : $ elements from the same block of $ \pi $ have the same parity $\}$. We will say that the blocks of a partition from $ NCL_S(2n) $ with odd elements are of color 1 and the ones with even elements are of color 0; blocks of color 1 will be graphically represented by solid lines and blocks of color 0 by dashed lines. Example 7: Representation of $ (1, 7), (2, 6), (3, 5), (4), (8, 12), (9, 11), (10) $: ![image](kreweras.pdf){width="3in" height=".9cm"} As shown in [@mp], there exist a bijection $ \Lambda : NCL_S ( 2n ) {\longrightarrow}\mathfrak{B}(n) $, constructed as follows (see also Example 8 below): 1. If $(i_1,\dots, i_s)$ and $(j_1,\dots, j_p )$ are the two exterior blocks of $\pi$, where $ j_1 =i_s + 1 $, then the first level of $\Lambda(\pi)$ is the elementary tree with $s-1+p-1$ offsprings, the first $s-1$ of color 1, representing $(i_2,\dots, i_s)$, in this order, and the last $p-1$ of color 0, representing $(j_2,\dots, i_p)$, in this order. 2. Suppose that $i_1$ and $i_2$ are consecutive elements in a block $B$ of $\pi$ already represented in some tree of $\Lambda(\pi)$. Let $ t_2 = i_2 -1 $ and $ t_1 = j_s + 1 $ if $ i_1 $ is the minimal element of some block $ (j_1, j_2, \dots, j_s )$ of $ \pi $ other than $ B $ and $ t_1 = i_1 + 1 $ otherwise. Let $ B_1, \dots , B_s $ be, in this order, the exterior blocks with more than one element of $ \pi_{\| t_1, t_1 + 1 \dots, t_2 } $ such that each $ B _k $ has $p(k) $ elements. Finally, if $i_2 $ is the minimal elements of some block $ ( i_2, l_1, \dots, l_{p(s+1)} ) $ of $ \pi $, then let $ D =( i_2, l_1, \dots, l_{p(s+1)} ) $ (otherwise let $ p(s+1) = 0 $). Then we represent $ B_1, \dots, B_s, D $ by an elementary tree with root the vertex representing $ i_2 $ and of $ p(1) + \dots p(s+ 1) - s $ offsprings, where the first $ p(1) - 1 $ are representing, and keeping the color of, the vertices of $ B_1 $ except for the minimal one, the next $p(2) - 1 $ representing elements of $ B_2 $ except for the minimal one etc. Example 8: ![image](lambda.pdf){width="8cm" height="2cm"} For $ X , Y \in A^\circ $, we define the maps $\displaystyle \omega_{XY}$ and $ {\widetilde{\omega}}_{XY} $ from $\cup_{ n\in \mathbb{N} } \mathfrak{B}( n ) $ to $\mathbb{C} $, respectively $ B$ as follows. If $ C_0\in\mathfrak{EB}( n )$ has $ k $ offsprings of color 1 and $ n - ( k +1 ) $ offsprings of color 0, then we define $$\begin{aligned} \omega_{X,Y}(C_0) & =t_k(X)t_{n-k-1}(Y) \\ {\widetilde{\omega}}_{X,Y}(C_0) & ={{}^ct}_k(X) {{}^ct}_{ n - k -1 } ( Y ). \end{aligned}$$ For $ W \in\mathfrak{B} ( n ) $, define $$\begin{aligned} \omega_{ X , Y }( W ) & = \prod_{ D \in E ( W ) } \omega_{X,Y}( D )\\ {\widetilde{\omega}}_{ X, Y } ( W ) & = {\widetilde{\omega}}_{ X, Y } ( \mathfrak{r} ( W ) ) \cdot \prod_{ D \in \mathfrak{b} ( W ) } \omega_{X,Y}( D ).\end{aligned}$$ Remark that, for $ \pi \in NC_S ( 2 n ) $, the definitions of $ \Lambda $ and $ \omega_{ X, Y } $, $ {\widetilde{\omega}}_{ X , Y } $ give $$\label{eq:11} \omega_{X,Y}(\Lambda(\pi)) = \kappa_{\pi_-}[X]\kappa_{\pi_+}[Y],$$ respectively $$\label{eq:12} {\widetilde{\omega}}_{X,Y}(\Lambda(\pi)) = {{}^c\kappa}_{ \| \pi_{-} [ 1 ] \| } ( X ) \cdot {{}^c\kappa}_{ \| \pi_+ [ 2n ] \| } ( Y )\cdot \prod_{ \substack{ B \in \pi_{ - }\\ B \neq \pi_{ - } [ 1 ] } } \kappa_{ \| B \| } (X) \cdot \prod_{\substack{ D \in \pi_{ + } \\ D \neq \pi_{ + } [ 2n] } } \kappa_{ \| D \| } ( Y )$$ The multiplicative property of the $ {{}^cT}$-transform ------------------------------------------------------- \[mainthm\] If $ X,Y$ are c-free elements from $A^\circ$, then $T_{XY}(z)=T_X(z)T_Y(z)$ and $ {{}^cT}_{ XY}(z) ={{}^cT}_X (z)\cdot {{}^cT}_Y(z) $. We need to show that, for all $m\geq 0$ $$\label{final} t_m(XY)=\sum_{k=0}^mt_k(X)t_{m-k}(Y))\ \ \text{and} \ {{}^ct}_m(XY)=\sum_{k=0}^m {{}^ct}_k(X) {{}^ct}_{m-k}(Y))$$ If $ C_n $ denotes the elementary planar tree with $ n $ vertices, with the notations from the previous two sections, the equations (\[final\]) are equivalent to $$\label{finaltree} {\mathcal{E}}_{XY}(A_n)=\sum_{B\in\mathfrak{EB}(n)}\omega_{X,Y}(B) \ \ \text{and} \ {\widetilde{\mathcal{E} } }_{XY}(A_n)=\sum_{B\in\mathfrak{EB}(n)}{\widetilde{\omega}}_{X,Y}(B)$$ i.e. for example, $$\begin{picture}(28,7) \put(-21,4){${\mathcal{E}}_{XY}($} \put(-14,3){\circle{1}} \put(-12,3){\circle{1}} \put(-10,3){\circle{1}} \put(-12,7){\circle{1}} \put(-14,3){\line(1,2){2}} \put(-12,3){\line(0,1){4}} \put(-10,3){\line(-1,2){2}} \put(-9,4){$)$} \put(-7,4){$=$} \put(-5,4){$\omega_{XY}($} \put(2,3){\circle{1}} \put(4,3){\circle{1}} \put(6,3){\circle{1}} \put(4,7){\circle{1}} \put(2,3){\line(1,2){2}} \put(4,3){\line(0,1){4}} \put(6,3){\line(-1,2){2}} \put(7,4){$)$} \put(9,4){$+$} \put(11,4){$\omega_{XY}($} \put(18,3){\circle{1}} \put(20,3){\circle{1}} \put(22,3){\circle{1}} \put(20,7){\circle{1}} \put(18,3){\line(1,2){2}} \put(20,3){\line(0,1){4}} \multiput(20,7)(.25,-.5){8}{\line(0,1){.2}} \put(23,4){$)$} \put(25,4){$+$} \put(27,4){$\omega_{XY}($} \put(34,3){\circle{1}} \put(36,3){\circle{1}} \put(38,3){\circle{1}} \put(36,7){\circle{1}} \put(34,3){\line(1,2){2}} \multiput(36,3)(0,.5){8}{\line(0,1){.2}} \multiput(36,7)(.25,-.5){8}{\line(0,1){.2}} \put(39,4){$)$} \put(41,4){$+$} \put(43,4){$\omega_{XY}($} \put(50,3){\circle{1}} \put(52,3){\circle{1}} \put(54,3){\circle{1}} \put(52,7){\circle{1}} \multiput(52,7)(-.25,-.5){8}{\line(0,1){.2}} \multiput(52,3)(0,.5){8}{\line(0,1){.2}} \multiput(52,7)(.25,-.5){8}{\line(0,1){.2}} \put(55,4){$)$} \end{picture}$$ We will prove (\[finaltree\]) by induction on $ n $. For $ n =0$, the result is trivial. Suppose (\[finaltree\]) true for $m\leq n-1$. Relation (\[eq:10\]) and Lemma \[lemma:1\] give $$\begin{aligned} \sum_{C \in\mathfrak{T}(n)}{\widetilde{\mathcal{E} } }_{XY}( C ) &= {{}^c\kappa}_n( X Y )\\ =& \sum_{ \pi \in NC_S( 2n ) } {{}^c\kappa}_{ \| \pi_{-} [ 1 ] \| } ( X ) \cdot {{}^c\kappa}_{ \| \pi_+ [ 2n ] \| } ( Y ) \cdot \prod_{ \substack{ B \in \pi_{ - }\\ B \neq \pi_{ - } [ 1 ] } } \kappa_{ \| B \| } (X) \cdot \prod_{\substack{ D \in \pi_{ + } \\ D \neq \pi_{ + } [ 2n] } } \kappa_{ \| D \| } ( Y )\end{aligned}$$ and equation (\[eq:12\]) and the bijectivity of $ \Lambda $ give: $$\label{eq1:7} \sum_{C \in\mathfrak{T}(n)}{\widetilde{\mathcal{E} } }_{XY}( C ) =\sum_{B\in\mathfrak{B}(n)}{\widetilde{\omega}}_{X,Y}(B).$$ Similarly, we have that $$\label{eq1:8} \sum_{C \in\mathfrak{T}(n)}{\mathcal{E}}_{XY}( C ) =\sum_{B\in\mathfrak{B}(n)}\omega_{X,Y}(B).$$ All non-elementary trees from $\mathfrak{T}(n)$ are composed of elementary trees with less than $ n $ vertices. The relations (\[eq1:7\]) and (\[eq1:8\]) imply that the image under ${\widetilde{\mathcal{E} } }_{XY}$, respectively $ {\mathcal{E}}_{ XY } $ of any such tree is the sum of the images under $ {\widetilde{\omega}}_{ XY } $, respectively under $\omega_{XY}$ of its colored versions. Hence $$\sum_{\substack{C\in\mathfrak{T}(n)\\C\neq C_n}}{\mathcal{E}}_{XY}(C) = \sum_{\substack{B\in\mathfrak{B}(n)\\B\notin\mathfrak{EB}(n)}} \omega_{X,Y}(B) \ \ \ \text{and} \ \sum_{\substack{C\in\mathfrak{T}(n)\\C\neq C_n}}{\widetilde{\mathcal{E} } }_{XY}(C) = \sum_{\substack{B\in\mathfrak{B}(n)\\B\notin\mathfrak{EB}(n)}} {\widetilde{\omega}}_{X,Y}(B).$$ Finally, (\[eq1:7\]), (\[eq1:8\]) and the two equations above give that $${\mathcal{E}}_{XY}(A_n)=\sum_{B\in\mathfrak{B}(n)}\omega_{X,Y}(B) \ \ \ \text{and} \ \ {\widetilde{\mathcal{E} } }_{XY}(A_n)=\sum_{B\in\mathfrak{B}(n)}{\widetilde{\omega}}_{X,Y}(B)$$ which imply (\[finaltree\]). Infinite Divisibility ===================== Fix a unital $C^{}$-subalgebra $B$ of $L(H)$, the $C^{}$-algebra of bounded linear operators on a Hilbert space $H$. In this section, we study the infinite divisibility relative to the c-freeness. A natural framework for such a discussion is in a c-free probability space $(A,\Phi,\varphi)$, that is, the algebras $A$ is also a concrete $C^{}$-algebra acting on some Hilbert space $K$, the linear map $\Phi:A\rightarrow B$ is a unital completely positive map, and the expectation functional $\varphi$ is a state on $L(K)$. Note that we have the norm $\|\|\Phi\|\|=\|\|\Phi(1)\|\|=1$. The distribution* of a unitary $u \in (A,\Phi,\varphi)$, written as the spectral integral $$u=\int_{\mathbb{T}}\xi\,dE_u (\xi),$$ is the pair $(\mu, \nu)$, where $\nu=\varphi \circ E_{u}$ is a positive Borel probability measure on the circle $\mathbb{T}=\{\|\xi\|=1\}$, and $\mu$ is a linear map from $\mathbb{C}[\xi, 1/\xi]$, the ring of Laurent polynomials, into the $C^{}$-algebra $B$ such that $$\mu(f)=\Phi(f(u, u^{})), \quad f\in \mathbb{C}[\xi,1/\xi].$$ Of course, the positivity of $\Phi$ and the Stinespring theorem (see [@paulsen], Theorem 3.11) imply that the map $\mu$ extends to a completely positive map on $C(\mathbb{T})$, the $C^{}$-algebra of continuous functions on $\mathbb{T}$. More generally, given any sequence $\{A_n\}_{ n\in\mathbb{Z}} \subset L(H)$, it is known (see [@paulsen]) that the operator-valued trigonometric moment sequence* $$A_n=\mu(\xi^n),\quad n\in \mathbb{Z},$$ extends linearly to a completely positive map $\mu:C(\mathbb{T})\rightarrow L(H)$ if and only if the operator-valued power series $F(z)=A_0/2+\sum_{k=1}^{\infty}z^kA_k$ converges on the open unit disk $\mathbb{D}$ and satisfies $F(z)+F(z)^{}\geq 0$ for $z\in\mathbb{D}$. In particular, for the unitary $u$ this implies that its moment generating series $$M_{u}(z)=\Phi(zu(1-zu)^{-1})=\sum_{k=1}^{\infty}z^k\mu(\xi^k)$$ and $$\label{eq:18} m_{u}(z)=\varphi(zu(1-zu)^{-1})=\int_{\mathbb{T}}\frac{z\xi}{1-z\xi}\,d\nu(\xi)$$ satisfy the properties: $$I+M_{u}(z)+M_{u}(z)^{} \geq 0\quad \text{and}\quad 1+m_{u}(z)+\overline{m_{u}(z)} \geq 0$$ for $\|z\|<1$. Thus, the formula $$\label{B-trans} B_{u}(z)=\frac{1}{z}M_{u}(z)(I+M_{u}(z))^{-1},\quad z\in \mathbb{D},$$ defines an analytic function from the disk $\mathbb{D}$ to the algebra $B$, with the norm $\|\|B_{u}(z)\|\|<1$ for $z\in\mathbb{D}$. Analogously, the function $$b_{u}(z)=\frac{m_{u}(z)}{z+zm_{u}(z)},\quad z\in\mathbb{D},$$ will be an analytic self-map of the disk $\mathbb{D}$. Notice that we have $B_{u}(0)=\Phi(u)$ and $b_{u}(0)=\varphi(u)$. Conversely, suppose we are given two analytic maps $B:\mathbb{D}\rightarrow B$ and $b:\mathbb{D}\rightarrow \mathbb{D}$ satisfying $\|\|B(z)\|\|<1$ for $\|z\|<1$. Then the maps $M(z)=zB(z)(I-zB(z))^{-1}$ and $m(z)=zb(z)/(1-zb(z))$ are well-defined in $\mathbb{D}$, and they can be written as the convergent power series: $$M(z)=\sum_{k=1}^{\infty}z^kA_k\quad \text{and} \quad m(z)=\sum_{k=1}^{\infty}a_kz^k,\quad z\in \mathbb{D},$$ where the operators $A_k\in B$ and the coefficients $a_k \in \mathbb{D}$. Since $I-\|z\|^2B(z)B(z)^{} \geq 0$, we have $$I+M_{u}(z)+M_{u}(z)^{}=(I-zB(z))^{-1}(I-\|z\|^2B(z)B(z)^{})[(I-zB(z))^{-1}]^{} \geq 0$$ for every $z \in \mathbb{D}$. Therefore, the solution of the operator-valued trigonometric moment sequence problem implies that the map $$\mu(\xi^{n})=\begin{cases} A_{n}, & n>0;\\ I, & n=0;\\ A_{n}^{}, & n<0. \end{cases}$$ extends linearly to a completely positive map from $C(\mathbb{T})$ into $B$. In the case of $m(z)$, we obtain a Borel probability measure $\nu$ on $\mathbb{T}$ satisfying . The pair $(\mu,\nu)$ is uniquely determined by the analytic maps $B$ and $b$. It is now easy to construct a c-free probability space $(A,\Phi,\varphi)$ and a unitary random variable $u\in A$ so that the distribution of $u$ is precisely the pair $(\mu,\nu)$. Indeed, we simply let $A=C(\mathbb{T})$, whose members are viewed as the multiplication operators acting on the Hilbert space $L^{2}\left(\mathbb{T};\nu\right)$, $\Phi=\mu$, and the variable $u$ can be defined as $$(uf)(\xi)=\xi f(\xi),\quad \xi \in \mathbb{T},\quad f\in L^{2}\left(\mathbb{T};\nu\right).$$ In summary, we have identified the distribution of $u$ with the pair $(B_u,b_u)$ of contractive analytic functions. A unitary $u\in A$ is said to be c-free infinitely divisible* if for every positive integer $n$, there exists identically distributed c-free unitaries $u_1,u_2,\cdots,u_n$ in $A$ such that $u$ and the product $u_1u_2\cdots u_n$ have the same distribution. It follows from the definition of the c-freeness that if a unitary $u \in A$, with the distribution $(\mu,\nu)$, is c-free infinitely divisible, then the law $\nu$ must be infinitely divisible with respective to the free multiplicative convolution $\boxtimes$, that is, to each $n \geq 1$ there exists a probability measure $\nu_n$ on $\mathbb{T}$ such that $$\nu=\nu_n \boxtimes \nu_n \boxtimes \cdots \boxtimes \nu_n \quad (n\,\,\text{times}).$$ The theory of $\boxtimes$-infinite divisibility is well-understood, see \[5\], and we shall focus on the c-free infinitely divisible distribution $\mu$, or equivalently, on the function $B_u$. From Equation (\[B-trans\]) and Proposition \[analytic-ct\], we have that the ${{}^cT}$-transform of $u$ satisfies $${{}^cT}_{u}\left(m_{u}(z)\right)=B_{u}(z).\quad$$ Therefore, Theorem 2.1 yields immediately the following characterization of c-freely infinite divisibility. A unitary $u \in (A,\Phi,\varphi)$ with distribution $(\mu,\nu)$ is c-free infinitely divisible if and only if $\nu$ is $\boxtimes$-infinitely divisible and the function $B_u$ is infinitely divisible in the sense that to each $n \geq 1$, there exists an analytic map $B_n:\mathbb{D}\rightarrow B$ such that $\|\|B_n(z)\|\|<1\quad \text{and} \quad B_u(z)=\left[B_n(z)\right]^n,\quad z\in \mathbb{D}.$ It was proved in \[5\] that a $\boxtimes$-infinitely divisible law $\nu$ is the Haar measure $d\theta/2\pi$ on the circle group $\mathbb{T}=\{\exp(i\theta):\theta\in (-\pi,\pi]\}$ if and only if $\nu$ has zero first moment. We now show the c-free analogue of this result. Let $u \in (A,\Phi,\varphi)$ be a c-free infinitely divisible unitary with $\Phi(u)=0$. If $\varphi(u)=0$, then one has $\Phi(u^n)=0$ for all integers $n \neq 0$. Denote by $(\mu,\nu)$ the distribution of $u$. Assume first that $\varphi(u)=0$, hence the law $\nu$ equals $d\theta/2\pi$. The c-free infinitely divisibility of $u$ shows that there exist c-free and identically distributed unitaries $u_1$ and $u_2$ in $A$ such that $\varphi(u_1)=0=\varphi(u_2)$ and $u=u_1u_2$ in distribution. Therefore, for $n>1$, we have $$\begin{aligned} \Phi(u^n) & = & \Phi( \underbrace{(u_1u_2)(u_1u_2)\cdots(u_1u_2)}_{n\:\text{times}})\\ & = & \Phi(u_1)\Phi(u_2)\cdots\Phi(u_1)\Phi(u_2)\\&=& \Phi(u_1u_2)\Phi(u_1u_2)\cdots\Phi(u_1u_2)=\Phi(u)^n=0. \end{aligned}$$ The case of $n<0$ follows from the identity $\Phi(u^{n})=\Phi(u^{-n})^{}$. An interesting case for c-freely infinite divisibility arises from the commutative situation. To illustrate, suppose $B=C(X)$, the algebra of continuous complex-valued functions defined on a Hausdorff compact set $X\subset\mathbb{C}$ equipped with the usual supremum norm. Denote by $\mathcal{M}$ the family of all Borel finite (positive) measures on $\mathbb{T}$, equipped with the weak\-topology from duality with continuous functions on $\mathbb{T}$. Then we shall have the following Let $\nu$ be a $\boxtimes$-infinitely divisible law on $\mathbb{T}$ and $\nu \neq d\theta/2\pi$. A unitary $u \in (A,\Phi,\varphi)$ is c-free infinitely divisible if and only if its ${{}^cT}$-transform admits the following Lévy-Hinčin type representation: $${{}^cT}_{u}\left(\frac{z}{1-z}\right)(x)=\gamma_{x} \exp \left(\int_{\xi \in \mathbb{T}}\frac{\xi z +1}{\xi z -1}\,d\sigma_{x}(\xi)\right), \quad x\in X, \quad z \in \mathbb{D},$$ where the map $x \mapsto \gamma_{x}$ is a continuous function from $X$ to the circle $\mathbb{T}$ and the map $x \mapsto \sigma_{x}$ is weak\-continuous from $X$ to $\mathcal{M}$. The integral representation follows directly from the characterization of c-free infinite divisibility in the scalar-valued case [@mvp-jcw]. To conclude, we need to show the continuity of the functions $\sigma_{x}$ and $\gamma_x$. To this purpose, observe that $$\left\|{{}^cT}_{u}\left(\frac{z}{1-z}\right)(x)\right\|=\exp \left(-\int_{\xi \in \mathbb{T}}\frac{1-\|z\|^2}{\|z - \xi \|^2}\,d\sigma_{x}(1/\xi)\right).$$ In particular, we have $$\exp \left(-\sigma_{x}(\mathbb{T})\right)=\|{{}^cT}_{u}(0)(x)\|.$$ Thus, if $\{x_\alpha\}$ is a net converging to a point $x \in X$, then the family $\{\sigma_{x_\alpha}(\mathbb{T})\}$ is bounded, and for every $z \in \mathbb{D}$ we have $$\int_{\xi \in \mathbb{T}}\frac{1-\|z\|^2}{\|z - \xi \|^2}\,d\sigma_{x_\alpha}(1/\xi)\rightarrow \int_{\xi \in \mathbb{T}}\frac{1-\|z\|^2}{\|z - \xi \|^2}\,d\sigma_{x}(1/\xi)$$ as $x_\alpha \rightarrow x$. Since bounded sets in $\mathcal{M}$ are compact in the weak-star topology, the above convergence of Poisson integrals determines uniquely the limit $\sigma_x$ of the net $\{\sigma_{x_\alpha}\}$. We deduce that the measures $\{\sigma_{x_\alpha}\}$ converge weakly to $\sigma_{x}$, and therefore the function $\sigma_x$ is continuous. This and the identity $${{}^cT}_{u}\left(0\right)(x)=\gamma_{x} \exp\left(-\sigma_{x}(\mathbb{T})\right)$$ imply further the continuity of the function $\gamma_x$. We end the paper with the remark that if the algebra $B$ is non-commutative, a $B$-valued contractive map $B(z)$ does not always have contractive analytic $n$-th roots for each $n\geq 1$, even when $B(z)$ is invertible for every $z\in \mathbb{D}$. To illustrate, consider the constant map $$B(z)=\left[\begin{array}{cc} \lambda^{2} & 2\lambda\\ 0 & \lambda^{2} \end{array}\right],\quad z \in \mathbb{D}.$$ Then $B(z)$ is an invertible contraction if the modulus $\|\lambda\|$ is sufficiently small, and yet its square roots $$\left[\begin{array}{cc} \pm \lambda & 1\\ 0 & \pm \lambda \end{array}\right]$$ have the same norm $\sqrt{1+\|\lambda\|^2}$. Acknowledgements {#acknowledgements .unnumbered} ---------------- This work was partially supported by a grant of the Romanian National Authority for Scientific Research, CNCS – UEFISCDI, project number PN-II-ID-PCE-2011-3-0119. [HD]{} M. Anshelevich, E. G. Effros, M. Popa, Zimmermann type cancellation in the free Faa di Bruno algebra. J. Funct. Anal. 237(2006), no. 1, 76–104 S. Belinschi, H. Bercovici Partially defined semigroups relative to multiplicative free convolution,* S. Belinschi, H. Bercovici Hinčin’s theorem for multiplicative free convolution, H. Bercovici, V. Pata Stable laws and domains of attraction in free probability theory H. Bercovici and D. Voiculescu Lévy-Hinčin type theorems for multiplicative and additive free convolution, H. Bercovici, J.-C. Wang Limit theorems for free multiplicative convolutions Trans. Amer. Math. Soc. 360 (2008), no. 11, 6089–6102 F. Boca Free products of completely positive maps and spectral sets M. Bożejko and R. 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Multilinear function series in conditionally free probability with amalgamation. M. Popa. A new proof for the multiplicative property of the boolean cumulants with applications to operator-valued case M. Popa. A Fock space model for addition and multiplication of c-free random variables M. Popa, J.-C. Wang. On multiplicative conditionally free convolution. R. Speicher. Combinatorial Theory of the Free Product with amalgamation and Operator- Valued Free Probability Theory. Mem. AMS, Vol 132, No 627 (1998) D.V. Voiculescu, K. Dykema, A. Nica. Free random variables.
--- abstract: 'In a companion paper [@SVP1], we gave a detailed account of the well-posedness theory for singular vortex patches. Here, we discuss the long-time dynamics of some of the classes of vortex patches we showed to be globally well-posed in [@SVP1]. In particular, we give examples of time-periodic behavior, cusp formation in infinite time at an exponential rate, and spiral formation in infinite time.' author: - 'Tarek M. Elgindi^1^' - 'In-Jee Jeong^2^' bibliography: - 'thesis.bib' title: 'On Singular Vortex Patches, II: Long-time dynamics' --- Introduction ============ A vortex patch is a solution to the incompressible Euler equation: $$\partial_t\omega+u\cdot\nabla\omega=0$$ $$u=\nabla^\perp \Delta^{-1}\omega$$ which is of the form $\omega(x,t)={\bf 1}_{\Omega(t)}(x)$ with $\Omega(t)\subset\mathbb{R}^2$. Such solutions play an important role in the rigorous analysis of general solutions to the incompressible Euler equation as well as an important role in modeling, from hurricanes to Jupiter’s Great Red Spot ([@Hur1],[@Mar],[@PD]). Consequently, the dynamical behavior of planar vortex patches has been considered by many authors for well over a century. The simplest example of a planar vortex patch is the “Rankine vortex” which is the stationary solution $\omega(x,t)={\bf 1}_{B_1(0)}(x)$. This solution was shown to be dynamically stable in certain topologies, while asymptotic stability is known to fail as we shall now observe. Indeed, a remarkable observation of Kirchoff is that if $\Omega(0)$ is the interior of an ellipse, then ${\bf 1}_{\Omega(t)}$ solves the incompressible Euler equation with $\Omega(t)=R_{\alpha(0)t}\Omega(0)$ where $R_{\theta}$ is just the linear transformation which rotates vectors counterclockwise by angle $\theta$ and $\alpha(0)$ is some constant depending on the initial ellipse $\Omega(0)$. In fact, as has been shown in works of Burbea [@Bu] and then many others ([@HMV; @HM; @HM2; @CCG3]), there is a rich family of time-periodic vortex patch solutions near the base circular solution. We refer the interested readers to a recent development [@GSPSY; @HM3] and references therein. Such vortex patches have a very regular behavior, but it is unclear whether such behavior is generic even in an asymptotic sense. In a different direction, it was observed in several numerical experiments and some theoretical works ([@CT; @Maj; @Dr; @DrMc; @CM1; @CM2; @Drit88; @PuM]) that small deformations of the initial state ${\bf 1}_{B_1(0)}$ often lead to the boundary of the vortex patch forming filaments which spiral around the center of the patch. Even more drastic behavior was observed in other numerical simulations ([@But]) which indicated self intersection in finite time. Finite time self-intersection or pinching was then ruled out in important work of Chemin [@C] and then in works of Bertozzi-Constantin [@BeCo] and Serfati [@Ser1] in the case when the boundary of $\Omega(0)$ is initially smooth. These works, however, do not preclude very rapid formation of small scales. In other words, while a vortex patch cannot self intersect in finite time, it is still possible that a vortex patch self intersects or develops high oscillations in infinite time at a very rapid rate as $t\rightarrow\infty$ as the early numerical simulations seem to suggest. To our knowledge, the infinite-time phenomena described above (spiral formation and pinching) have not been rigorously confirmed, though some results have been obtained in spatial domains with solid boundaries when the vortex patch is allowed to touch the boundary. The purpose of this work is to rigorously establish some of the phenomena described above and others. Specifically, our goal is to give examples of vortex patches that exhibit non-trivial dynamical behavior as $t\rightarrow\infty$ even though they remain “globally regular.” Our first result concerns existence of purely rotating patches which are given by a union of infinite sectors. We prove some classification results on rotating sectors, and obtain in particular existence of purely rotating solutions with zero mean. Note that the existence of rotating patches with compact support which have corners at the origin has been numerically observed in [@LF1]. Our second result is the construction of a compactly supported vortex patch in $\mathbb{R}^2$ consisting of eight flower petals coming out of a center (asymptotically near ${\bf 0}$, $\Omega(0)$ looks like eight sectors emanating from the origin). Such a patch can be placed into a very natural global well-posedness class so that for all $t>0$, the local picture at the origin remains asymptotically like eight flower petals and there is propagation of regularity for all time. As $t\rightarrow\infty$, however, we see four of the eight petals being ejected from the origin at an exponential rate in time. That is, each of those pieces of the vortex patch cusp exponentially fast as $t\rightarrow\infty$. While we do not give any global information on the dynamics of the full patch, we are able to control the dynamics exactly at the origin, which reduces to an ODE system, to establish the cusp formation in infinite time. This “localization” phenomenon also happens in the case of 3D vortex patches with 3D corners, and based on this observation one can establish finite-time blow up of $\V \omega(t)\V_{L^\infty}$ in time ([@EJ3]). We mention that this type of blow-up result is similar to the one proved in [@EJB; @EJE]. Our third result is of a more global nature. We take now four identical flower petals intersecting at the origin so that $\Omega(0)$ is 4-fold symmetric. Once more, we have shown in [@SVP1] that such initial data can be placed in a natural global well-posedness class. Again, the dynamics exactly at the corner is governed precisely by a predetermined system of ODE; in this case, however, the ODE just dictates that the four corners rotate at a fixed speed depending only on the size of the corners. To make a meaningful statement on the long-time behavior of the vortex patch we need another, global, piece of information. Now we also stipulate that the vortex patch be very close to a disk (the symmetric difference should have small measure). Due to the (infinite-time) stability of the circular vortex patch which has been known for some time ([@WP]), we now also have that the velocity field of the vortex patch is close in a certain topology to that of the circular vortex patch. This implies that “most” points of the vortex patch (those away from the origin) are rotating counterclockwise with speed $1$. Now if one can ensure that there is a discrepancy between the rotation of the corner (whose speed can be determined a-priori) and most of the points, for all time, one would believe that the boundary of the vortex patch must develop a spiral in infinite time. Carrying this out rigorously requires us to employ a few analytic tools but also some basic topological methods, but the basic idea is to use both the local stability of the corner dynamics and the global stability of the disk to give spiral formation. While a spiral forms in this case, it remains an open problem whether the perimeter of the boundary of the patch actually becomes unbounded. We emphasize that vortex patches utilized in the current work involve corners meeting at the origin in a symmetric fashion. The boundary of the patch can be infinitely smooth away from the origin. The work of Danchin [@Da] has established that if the patch boundary is smooth away from a closed set, this property propagates globally in time. Global persistence of certain cusp structures were obtained in [@Da2]. In the case of patches with a single isolated corner, we have shown in [@SVP1] that, roughly speaking, the corner structure cannot propagate continuously in time in general. We refer the interested readers to numerical simulations [@CS; @CD]. On the other hand, if the corners meet at the origin with rotational symmetry (e.g. Figure \[fig:petal\]), then we have global well-posedness; see the statements of Theorem \[mainthm:wellposedness\] below for details. Outline of the paper {#outline-of-the-paper .unnumbered} -------------------- The rest of this paper is organized as follows. First in Section \[sec:prelim\], we fix some notations and precisely state the central well-posedness result from the companion paper [@SVP1]. In particular, the ODE system satisfied by the corner angles will be the basis of all the main results in this work. Then in Sections \[sec:rot\], \[sec:cusp\], and \[sec:spiral\], we construct patch solutions which rotate with constant speed, cusp, and spiral, respectively. Acknowledgments {#acknowledgments .unnumbered} --------------- T.M. Elgindi was partially supported by NSF DMS-1817134. I.-J. Jeong has been supported by the POSCO Science Fellowship of POSCO TJ Park Foundation and the National Research Foundation of Korea(NRF) grant (No. 2019R1F1A1058486). Preliminaries {#sec:prelim} ============= In this section, we collect the notations, conventions, and recall the relevant main results of the companion work. Notations and definitions {#notations-and-definitions .unnumbered} ------------------------- - For $\theta \in [0,2\pi)$, we let $R_{\theta}$ be the matrix of counterclockwise rotation around the origin by the angle $\theta$. We say that a set $\Omega \subset \mathbb{R}^2$ is $m$-fold (rotationally) symmetric if $R_{2\pi/m}(\Omega) = \Omega$. Similarly, a scalar function $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ is $m$-fold symmetric if $f(x) = f(R_{2\pi/m}x)$ for any $x \in \mathbb{R}^2$. On the other hand, a vector field $v : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is $m$-fold symmetric if $v(R_{2\pi/m}x) = R_{2\pi/m} v(x)$. - Given two angles $0 \le \theta_1 < \theta_2 < 2\pi$, we define the sector $$\begin{split} S_{\theta_1,\theta_2} = \{ (r,\theta) : \theta_1 < \theta < \theta_2 \}. \end{split}$$ - The classical Hölder spaces are defined as follows: for $0 < \alpha \le 1$ and an open set $U \subset \mathbb{R}^2$, $$\begin{split} \V f \V_{C^\alpha(\overline{U})} &= \V f\V_{L^\infty(U)} + \V f\V_{{C}_^\alpha(\overline{U})} \\ &= \sup_{x \in U} \|f(x)\| + \sup_{x \ne x'} \frac{\|f(x) - f(x')\|}{\|x-x'\|^\alpha}. \end{split}$$ - By a $C^{1,\alpha}$-diffeomorphism, we mean a uniformly bi-Lipschitz map $\Psi : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ satisfying $\nabla\Psi, \nabla(\Psi^{-1}) \in C^{\alpha}(\mathbb{R}^2)$. - We reserve the letter $K$ for the Biot-Savart kernel $$\begin{split} K(x) = \frac{1}{2\pi} \frac{x^\perp}{\|x\|^2}. \end{split}$$ Convolution against $\nabla K$ is defined in the sense of principal value integration. - A point in $\mathbb{R}^2$ is denoted by $x = (x_1,x_2)$ or by $y = (y_1,y_2)$. Often we slightly abuse notation and consider polar coordinates $(r,\theta)$, where $r = \|x\|$ and $\theta = \arctan(x_2/x_1)$. - Given $x \in \mathbb{R}^2$ and $r > 0$, we define $B_x(r) = \{ y \in \mathbb{R}^2 : \|x-y\| < r \}$. In the following we shall fix some value of $0<\alpha<1$ (a specific choice of $\alpha$ does not make any differences) and suppress the dependence of constants on $\alpha$. Moreover, with the exception of Section \[sec:rot\], the strength of the vorticity will be normalized to be 1. Main well-posedness results from [@SVP1] {#main-well-posedness-results-from .unnumbered} ---------------------------------------- The main result of [@SVP1] states that for a patch with “corners” meeting symmetrically at a point (which can be taken to be the origin without loss of generality), the uniform Hölder regularity up to the corner propagates globally in time. Moreover, the angles of the corners and the region between corners satisfy a closed system of ODEs. We shall encode the uniform regularity with a $C^{1,\alpha}$-diffeomorphism $\Psi : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ satisfying $\Psi(\bf{0}) = \bf{0}$ and $\nabla\Psi\|_{\bf{0}} = \mathrm{Id}$. These properties will be always assumed whenever we use the term “$C^{1,\alpha}$-diffeomorphism”. By a corner, we mean a region which can be (locally) mapped from the exact sector by a $C^{1,\alpha}$-diffeomorphism. [A]{}\[mainthm:wellposedness\] Consider $\omega_0 = {\bf{1}}_{\Omega_0}$ where $\Omega_0$ satisfies the following properties: $\Omega_0$ is $m$-fold rotationally symmetric around the origin with $m \ge 3$; the boundary $\partial\Omega_0$ is $C^{1,\alpha}$-smooth away from the origin; and there exists some $r_0>0$ and a $C^{1,\alpha}$-diffeomorphism $\Psi_0$ such that $$\label{eq:local_expression} \begin{split} & \Psi_0(\Omega_0) \cap B_{\bf{0}}(r_0) = \left(\bigcup_{k = 0}^{m-1} \bigcup_{i = 1}^N S_{ \beta_{i,0} + \frac{2\pi k}{m}, \beta_{i,0} + \zeta_{i,0} + \frac{2\pi k}{m} } \right)\cap B_{\bf{0}}(r_0) \end{split}$$ where the sectors $S_{ \beta_{i,0} + \frac{2\pi k}{m}, \beta_{i,0} + \zeta_{i,0} + \frac{2\pi k}{m} }$ do not intersect with each other for $1 \le i \le N$ and $0 \le k \le m-1$. Then, the corresponding patch solution $\Omega(t)$ enjoys the same properties for all $t > 0$, with some $C^{1,\alpha}$-diffeomorphism $ \Psi(t)$ and $r(t) > 0$; that is, $$\begin{split} & \Psi(t)(\Omega(t)) \cap B_{\bf{0}}(r(t)) = \left(\bigcup_{k = 0}^{m-1} \bigcup_{i = 1}^N S_{ \beta_{i}(t) + \frac{2\pi k}{m}, \beta_{i}(t) + \zeta_{i}(t) + \frac{2\pi k}{m} } \right)\cap B_{\bf{0}}(r(t)) \end{split}$$ for some non-overlapping sectors $S_{ \beta_{i}(t) + \frac{2\pi k}{m}, \beta_{i}(t) + \zeta_{i}(t) + \frac{2\pi k}{m} }$. Moreover, the set of angles $\{ \beta_i,\zeta_i \}_{i=1}^N$ is determined for all $t > 0$ completely by the initial conditions $\{\beta_{i,0},\zeta_{i,0}\}_{i=1}^N$: introducing for convenience $\gamma_{i+\frac{1}{2}}(t):= \beta_{i+1}(t) - \beta_i(t) - \zeta_i(t) $ (separation angles), we have the following system: $$\label{eq:ode0} \begin{split} \frac{d \beta_1}{dt} =C_m' \sum_{i=1}^N \sin(\frac{m}{4}(2\beta_i + \zeta_i)) \sin(\frac{m}{4}\zeta_i ) - C_m'' \sum_{i=1}^N \zeta_i , \end{split}$$ $$\label{eq:ode1} \begin{split} \frac{d\zeta_j(t)}{dt} & = C_m \sin\left(\frac{m}{4}\zeta_j\right)\sum_{l=1}^{N} \mathrm{sgn}(j-l) \sin\left(\frac{m}{4}\zeta_l\right) \cos\left( \frac{m}{4}\left( 2(\beta_j -\beta_l) + (\zeta_j - \zeta_l) \right) \right) \end{split}$$ and $$\label{eq:ode2} \begin{split} \frac{d\gamma_{j+ \frac{1}{2}}(t) }{dt}& = C_m \sin\left(\frac{m}{4}\gamma_{j+\frac{1}{2}}\right) \sum_{l=1}^{N}\mathrm{sgn}(j+\frac{1}{2}-l) \sin\left(\frac{m}{4}\zeta_l\right) \cos\left( \frac{m}{4}\left( (\beta_{j+1} -\beta_l) + (\beta_j -\beta_l) + (\zeta_j - \zeta_l) \right) \right) \end{split}$$ for some constants $C_m, C_m' > 0$ and $C_m'' \ge 0$ depending only on $m$. The last condition on $\Omega_0$ in the above says that up to a diffeomorphism, $\Omega_0$ is a union of non-intersecting sectors meeting at the origin locally; for a nice example, see Figure \[fig:petal\] where $m = 3$. The proof of this result spans Sections 2–4 from [@SVP1], and here we just use it as a black box to deduce interesting dynamics of vortex patches. ![A petal domain[]{data-label="fig:petal"}](petal.pdf) A few remarks are in order. - Specifying $2N$ variables $\{ \beta_i, \zeta_i \}_{i=1}^N$ is equivalent to specifying $\beta_1, \zeta_1,\cdots, \zeta_N, \gamma_{1+\frac{1}{2}}, \cdots, \gamma_{N-\frac{1}{2}}$, via the relations $$\begin{split} \beta_j - \beta_l = \left(\gamma_{j-\frac{1}{2}} + \cdots + \gamma_{l+\frac{1}{2}}\right) + \left( \zeta_{j-1} + \cdots + \zeta_l \right), \qquad j > l . \end{split}$$ - The absolute constants $C_m, C_m',$ and $C_m''$ can be determined as follows: when the vorticity exactly has the form $$\begin{split} \omega(r,\theta) = h(\theta) \end{split}$$ in polar coordinates with some profile $h$ depending only on the angle satisfying $h(\theta) = h(\theta + \frac{2\pi}{m})$ for some $m \ge 3$, the angular part of the corresponding velocity is simply $2rH(\theta)$ in polar coordinates, where $$\begin{split} 4H + H'' = h. \end{split}$$ The kernel expression for this elliptic problem can be found: $$\begin{split} H(\theta) = \int_{S^1} (c_m \sin(\frac{m}{2}\|\theta - \theta'\|) - c_m' )h(\theta')d\theta' \end{split}$$ for some constants $c_m,c_m' $. - It follows from that the patch boundary $\partial\Omega(t)$ remains uniformly $C^{1,\alpha}$ up to the origin for all times. Indeed, in the proof of Theorem \[mainthm:wellposedness\], one has to propagate the following important pieces of information: for any finite $T>0$, $u(t) = K {\bf{1}}_{\Omega(t)}$ satisfy $$\begin{split} & \V \nabla u(t) \V_{L^\infty([0,T];L^\infty(\mathbb{R}^2))} < C(T) \end{split}$$ and $$\begin{split} & \V \nabla u(t) \V_{L^\infty([0,T];C^\alpha(\overline{\Omega(t)}))} < C(T) \end{split}$$ for some constant $C(T)$. That is, the velocity field is globally uniformly Lipschitz and $C^{1,\alpha}$ in the interior of the patch, uniformly up to the boundary. - We emphasize that the angles are determined for all $t > 0$ not just up to an overall constant (this fact is expressed in ). - It is not difficult to see from that the sum $\zeta_1 + \cdots + \zeta_N$ is constant in time. This can be also derived from the conservation of circulation along a small loop containing the origin. - In the simplest case of $N = 1$ in , the corners rotate with a constant angular speed for all time, which is determined only by the initial angle and $m$. - While strictly speaking the above result does not allow some (or all) angles to be zero, an analogous global well-posedness result can be shown even in such a setting; see [@SVP1 Section 4.5]. Of course, in that case, angles which are initially zero (i.e. cusp) must remain so for all times. Rotating patches {#sec:rot} ================ We demonstrate in this section that the system – gives rise to a large set of periodic (rotating) patch solutions to the 2D Euler equation. To be clear, we say that a solution $\omega(t)$ to 2D Euler is rotating if there exists some nonzero constant $c $ such that $\omega(t,r,\theta) = \omega_0(r, \theta - ct)$. In particular it is a time-periodic solution. This is achieved by considering patches supported on multiple sectors with different strengths. For the simplicity of computations we shall focus on the case $m = 4$. Under this assumption, the sectors can be identified with intervals on $[-\frac{\pi}{4},\frac{\pi}{4})$ with endpoints identified by quotienting out the symmetry. Assume further that the $i$-th interval has strength $A_i$, which is an invariant of the 2D Euler equation. Then, the system of motion takes the following form, modulo a multiplicative constant: $$\label{eq:ode1-4} \begin{split} \frac{d\zeta_j(t)}{dt} & = \sin\left(\zeta_j\right)\sum_{l=1}^{N} \mathrm{sgn}(j-l) A_l \sin\left(\zeta_l\right) \cos\left( 2(\beta_j -\beta_l) + (\zeta_j - \zeta_l) \right) \end{split}$$ and $$\label{eq:ode2-4} \begin{split} \frac{d\gamma_{j+ \frac{1}{2}}(t) }{dt}& = \sin(\gamma_{j+\frac{1}{2}}) \sum_{l=1}^{N}\mathrm{sgn}(j+\frac{1}{2}-l) A_l \sin\left(\zeta_l\right) \cos\left( (\beta_{j+1} -\beta_l) + (\beta_j -\beta_l) + (\zeta_j - \zeta_l) \right) . \end{split}$$ Further restricting to the case $N = 2$, and by assuming that the angles $\zeta_1, \zeta_2$, and $\gamma_{1+\frac{1}{2}}$ are stationary, we obtain $$\label{eq:ode-2-stat} \left\{ \begin{aligned} 0 & = -A_2 \sin(\zeta_1)\sin(\zeta_2) \cos(2\gamma + \zeta_1 + \zeta_2) \\ 0 & = A_1 \sin(\zeta_1)\sin(\zeta_2) \cos(2\gamma + \zeta_1 + \zeta_2) \\ 0 & = \sin(\gamma)\left( A_1 \sin(\zeta_1)\cos(\gamma+\zeta_1) - A_2 \sin(\zeta_2) \cos(\gamma+\zeta_2) \right) \end{aligned} \right.$$ Assuming that $0 < \gamma, \zeta_1, \zeta_2 < \frac{\pi}{2}$ (we also have $\gamma+\zeta_1+\zeta_2<\frac{\pi}{2}$), it is straightforward to see that holds if and only if $$\label{eq:sol} \begin{split} & 2\gamma + \zeta_1+\zeta_2 = \frac{\pi}{2}, \quad A_1 \sin(\zeta_1)\cos(\gamma+\zeta_1) = A_2 \sin(\zeta_2) \cos(\gamma+\zeta_2) \end{split}$$ Therefore we have a three-dimensional solution set in the five-dimensional phase space. We have arrived at the following proposition. In the case $N = 2$, $\omega(r,\theta) = \sum_{i=1}^2 \sum_{k=0}^{3} A_i {\bf 1}_{ \beta_i + \frac{\pi k}{4} }$ defines a rotating solution to 2D Euler if and only if holds. We have particular solutions $$\begin{split} &\zeta_1 = \frac{\pi}{8} + \xi, \quad \zeta_2 = \frac{\pi}{8} - \xi, \quad \gamma = \frac{\pi}{8},\quad A_1 = \sin(\frac{\pi}{8} - \xi)\cos(\frac{\pi}{4}-\xi), \quad A_2 = \sin(\frac{\pi}{8}+\xi)\cos(\frac{\pi}{4}+\xi) \end{split}$$ for any $0 \le \xi \le \frac{\pi}{8}$. In particular, there exist rotating patches with zero mean: $$\begin{split} & \int_{S^1} \omega \, d\theta = 0. \end{split}$$ The fact that there is a constant speed of rotation with zero mean sounds counter-intuitive. However, one should keep in mind that individual “fluid particles” are rotating at their own speed (constant in time), which altogether averages out to zero. It seems difficult to completely classify the set of periodic patches especially when the coefficients $A_i$ are allowed and $N $ is large. Still we have the following result which says that the support of a rotating solution cannot be too localized. We recall that (see [@EJ1; @SVP1]), the solution of $$4H + H''= h$$ is simply given by $$\label{eq:sol-H} \begin{split} H(\theta) = \frac{\pi}{8}\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \left\|\sin(2(\theta-\theta'))\right\| h(\theta')d\theta'. \end{split}$$ and that $2H$ is the speed of rotation. (Of course, the exact constant $\frac{\pi}{8}$ does not make a crucial difference.) Let $\mathcal{I}$ be a disjoint union of intervals contained in $[-\frac{\pi}{8},\frac{\pi}{8}]$ and assume that $\omega_0 = {\bf 1}_\mathcal{I} $ defines a rotating solution. Then $\omega_0 = {\bf 1}_{[b,a]}$ for some $-\frac{\pi}{8} \le b \le a \le \frac{\pi}{8}$. Consider the maximal value of $0 \le a \le \frac{\pi}{8}$ where $\mathcal{I} \subset [-a,a]$. Without loss of generality, we may assume that $\bar{\mathcal{I}}$ contains the point $a$, and let $I = [b,a]$ be the connected component of $\mathcal{I}$ containing $a$. Then, from we obtain that $$\begin{split} H(a) - H(b) = \frac{\pi}{8} \int_{\mathcal{I} \backslash I} \sin(2(a-\theta)) - \sin(2(b-\theta)) d\theta \end{split}$$ and note that $\sin(2x)$ is strictly increasing for $x \in [0,\frac{\pi}{4}]$. From the assumption on $\mathcal{I}$, $0 < a-\theta < \frac{\pi}{4} $ for all $\theta \in \mathcal{I}\backslash I$, and therefore $H(a) - H(b) = 0$ if and only if $\mathcal{I} = I$. The support condition is sharp; when the support is allowed to lie on $[-\frac{\pi}{8}-\epsilon, \frac{\pi}{8}+\epsilon]$ (say), then $\omega = {\bf 1}_{[-\frac{\pi}{8}-\epsilon,-\frac{\pi}{8}+\epsilon]} + {\bf 1}_{[\frac{\pi}{8}-\epsilon,\frac{\pi}{8}+\epsilon]}$ defines a rotating solution. The above argument covers the case when $\mathcal{I} = \cup_{i=1}^N I_i$ and $\omega = \sum_{i=1}^N A_i{\bf 1}_{I_i} $ with $A_i > 0$. Moreover, one can show along the same lines that there is no time-periodic solution (up to rotation) whose support (modulo 4-fold symmetry) is contained in an interval of size $\frac{\pi}{4}$ for all time. In the next section, we investigate the dynamics under the same assumption that $m = 4$ and $N = 2$. It turns out that any initial data converges to a rotating solution. As we shall see, in the special case $A_1 = A_2$, two angles generically converge to a single angle as $t \rightarrow +\infty$ which implies cusp formation. Cusp formation in infinite time {#sec:cusp} =============================== To demonstrate that there is some non-trivial long time dynamics of sectors, we consider the case when there are two sectors in a fundamental domain, assuming 4-fold rotational symmetry. Let us define $\Omega_1 \subset [0,\frac{\pi}{2})$ be the set obtained from $\Omega$ by quotienting out the rotational symmetry, and further write $\Omega_1 = \Omega_1^1 \cup \Omega_1^2$ (see Figure \[fig:two\_corners\]), and in this case, up to a rotation of $\mathbb{R}^2$, we only need to specify three angles; angles of each $\Omega_1^j$ and the angle in between. We denote them by $\zeta_1$, $\zeta_2$, and $\gamma$, respectively (see Figure \[fig:two\_corners\]). Then, the systems of equations , reduce to, up to a multiplicative constant which we neglect, $$\label{eq:ode_reduced} \left\{ \begin{aligned} \dot{\zeta_1} &= - \sin(\zeta_1) \sin(\zeta_2) \cos(2\gamma + \zeta_1 + \zeta_2), \\ \dot{\zeta_2} &= \sin(\zeta_2)\sin(\zeta_1) \cos(2\gamma + \zeta_1 + \zeta_2), \\ \dot{\gamma} &= \sin(\gamma) \sin(\zeta_1 - \zeta_2) \cos(\gamma + \zeta_1 + \zeta_2). \end{aligned} \right.$$ (Observe that $\zeta_1 + \zeta_2$ is an invariant of motion, as it should be). ![Evolution of two angles[]{data-label="fig:two_corners"}](two_corners.pdf) It can be easily shown that for exact infinite sectors, the whole patch $\Omega$ defines a purely rotating state if and only if $\zeta_2 = \zeta_1$ and $\gamma = \frac{\pi}{4} - \zeta_1$ (in which case $\Omega$ is indeed 8-fold symmetric) or one of the sectors is degenerate, that is, either $\zeta_2 = 0$ or $\zeta_1 = 0$. In this restricted setting, it can be established that any state converges as $ t\rightarrow +\infty$ to such a purely rotating space, and generically to a state where one of the angles become zero. Indeed, fix $\zeta_1(0) + \zeta_2(0) = \frac{\pi}{4}$, and moreover $\gamma(0) = \zeta_2(0)$. Then using the system one sees that $\gamma = \zeta_2 $ for all time. Alternatively, assuming such an initial data, subtracting $-\frac{1}{2}$ from vorticity everywhere in the plane gives an odd configuration with respect to the line separating $\zeta_2$ and $\gamma$ (see Figure \[fig:two\_corners\]). Since the Euler equations preserve odd symmetries of vorticity, it follows that $\gamma= \zeta_2$ for all time. Therefore, the one-dimensional system we get is: $$\label{eq:ode_1D} \begin{split} \dot{\gamma} = \sin(\gamma)\sin(\frac{\pi}{4} - 2\gamma) \cos(\gamma + \frac{\pi}{4}). \end{split}$$ On the other hand, if we keep the assumption $\zeta_1(0)+ \zeta_2(0) = \frac{\pi}{4}$ but now take $\gamma(0) = \zeta_1(0)$, we instead obtain $$\label{eq:ode_1D_prime} \begin{split} \dot{\gamma} = -\sin(\gamma)\sin(\frac{\pi}{4} - 2\gamma) \cos(\gamma + \frac{\pi}{4}). \end{split}$$ Then, one sees that for any initial value $0 < \gamma(0) < \frac{\pi}{4}$, it always converges to $\pi/8$ for $t \rightarrow + \infty$ in the former case, and either $0$ to $\frac{\pi}{4}$ in the latter. Actually, under the constraint $\zeta_1 + \zeta_2 = \frac{\pi}{4}$, the former case is the only situation where the forward asymptotic state is 8-fold symmetric, not just 4. See a plot of the phase portrait in Figure \[fig:phase\_portrait\]. ![The phase portrait of the system under the assumption $\zeta_1 = \frac{\pi}{4} - \zeta_2$. The axis correspond to variables $\zeta_2$ and $\gamma$ taking values in $[0,\frac{\pi}{4}]$, which play symmetric roles.[]{data-label="fig:phase_portrait"}](phase_portrait.pdf) Thanks to Theorem \[mainthm:wellposedness\], we have shown the following \[thm:cusping\] Assume that $\Omega_0$ is a 4-fold symmetric $C^{1,\alpha}$-patch with two angles $0 < \zeta_1(0)$ and $0 < \zeta_2(0)$ separated by an angle $0 < \gamma$, in a fundamental domain. Further assume that $\zeta_1(0) + \zeta_2(0) = \frac{\pi}{4}$ and $\gamma(0) \ne \zeta_2(0)$. Then, depending on whether $\zeta_2(0) > \gamma(0)$ or $\gamma(0) > \zeta_2(0)$ holds, we have $\zeta_1(t) \rightarrow 0$ or $\zeta_2(t) \rightarrow 0$, respectively, as $t \rightarrow + \infty$. That is, one of the two components of $\Omega_t$ in each fundamental domain cusps in infinite time. The speed of cusp formation is exponential in time. We only need to show that the angle collapses with an exponential rate. Without loss of generality, assume that $\zeta_1 \rightarrow 0$ as $t \rightarrow +\infty$. Then, we have $\gamma \rightarrow 0$ as well, and the equation becomes $$\begin{split} \dot{\zeta_1} \approx -C\zeta_1 \end{split}$$ for some positive constant $C > 0$. This finishes the proof. It is likely that for any finite number of sectors, with constant vorticity, there is asymptotic convergence of a purely rotating state. Spiral formation in infinite time {#sec:spiral} ================================= In this subsection, we construct a patch solution which develops an infinite spiral as time goes to infinity. To be more precise, the winding number of the boundary of our patch solution around the origin increases at least linearly with time. This will be achieved by combining the well-known stability result for the circular vortex patch with the persistence of the corner angle (Theorem \[mainthm:wellposedness\] in the case $N = 1$). The idea is very simple: one may perturb the circular patch near the center so that the patch locally looks like a symmetric union of sectors. Then, an explicit computation shows that the rotation speed at the corner is different from the bulk rotation speed, which forces a spiral to form linearly in time. In the following we make this idea precise. We begin with the statement of our result. \[thm:spiral\] There exists a vortex patch $\Omega_0$ of compact support, with boundary $C^\infty$ away from the origin, whose solution $\Omega(t)$ spirals linearly in time as $t \rightarrow +\infty$. More precisely, for each $t \ge 0$, there is an injective curve $\gamma(t) \subset \partial\Omega(t)$ whose winding number around the origin is at least $ct$ with some constant $c > 0$ depending only on $\Omega_0$. In particular, for any line $L$ passing through the origin, we have $$\begin{split} \forall t \ge 0, \quad \|\{ x \in \mathbb{R}^2 : x \in \partial\Omega(t) \cap L \} \| \ge ct - c' \end{split}$$ for some $c' > 0$. In addition, at least one of the following options hold: (i) the perimeter of the patch goes to infinity: $$\begin{split} \limsup_{t \rightarrow \infty} \|\partial\Omega(t)\| = + \infty, \end{split}$$ (ii) the turns accumulate at the origin: $$\begin{split} \forall r > 0, \quad \lim_{t \rightarrow \infty} \|\{ \|x\| \le r : x \in \partial\Omega(t) \cap L \} \| = +\infty \end{split}$$ for any line $L$ containing the origin. When the domain of the fluid has boundary and a non-trivial fundamental group, spiral formation in infinite time is very easy to achieve. One can simply take the annulus $\mathbb{A} = \{ 1 \le r \le 2 \}$ to be the fluid domain and take a patch $\Omega_0 \subset \mathbb{A}$ which extends to both components of $\partial\mathbb{A}$. It is not difficult to arrange that the velocities on two component of $\partial\mathbb{A}$ are bounded away from each other for all times. This implies that the pieces of $\Omega(t)$ lying on different components of $\partial\mathbb{A}$ will rotate with different angular speeds for all times, which means linear in time spiral formation. Note that in this example the length of the patch boundary trivially goes to infinity as time goes to infinity. This example is essentially due to Nadirashvili [@Nad] who observed in this setup infinite in time growth of the vorticity gradient for smooth vorticities (instead of patches). Even in the case of the trivial fundamental group, when boundary is present, the boundary of a large chunk of vorticity could play the role of an essential curve. For concreteness take the square domain $[0,1]^2$ and place a patch $U_0$ which has area almost equal to 1 and does not touch the boundary of the square. Then now the region $[0,1]^2\backslash U_0$ is homeomorphic to an annulus. Take a patch $\Omega_0 \subset [0,1]^2\backslash U_0 $ whose boundary intersects $\partial([0,1]^2)$ and $\partial U_0$. We demand that the intersection with $\partial([0,1]^2)$ is contained in the segment $[0,1]\times \{ 0\}$ (such an example of $\Omega_0$ is depicted in Figure \[fig:spiral\_square\_initial\]). One may add rotated images of $\Omega_0$ around the center $(\frac{1}{2},\frac{1}{2})$ to make the configuration 4-fold symmetric, which adds extra stability of the scenario. One can show that the central patch $U(t) = \Phi(t,U_0)$ (where $\Phi(t,\cdot)$ is the particle trajectory map associated with the initial patch data) rotates around the center infinitely many times as $t \rightarrow \infty$ (cf. Kiselev-Sverak [@KS]). On the other hand, the part of the patch boundary touching $[0,1]\times\{0\}$ simply converges to the corner $(1,0)$. This guarantees spiral formation in infinite time. The length of the boundary of the patch again goes to infinity in this example. ![Spiral formation in the case of the square[]{data-label="fig:spiral_square_initial"}](spiral_square.pdf) In view of the examples above, the main point of Theorem \[thm:spiral\] is that we can achieve spiral formation in the absence of the boundary of the fluid domain. \[ex:spiral\] Here we give an explicit example of $\Omega_0$ with which the conclusion of Theorem \[thm:spiral\] holds. The patch $\Omega_0$ will be given by the 4-fold symmetrization of a patch $\Omega_0^1$ which lies strictly inside the region $\{ \|x_2\| < x_1 \}$ except at the origin. For simplicity we use polar coordinates to define $\Omega_0^1$. For sufficiently small $\delta, \nu > 0$ and any $0 < \theta_0 < \frac{\pi}{4} - \nu$, take the points $$\begin{split} A^{\pm} = (\delta, \pm \theta_0), \quad B^{\pm} = (1, \pm(\frac{\pi}{4} - \nu) ) \end{split}$$ in the $(r,\theta)$-coordinates. Then, draw straight lines between the origin $O$ and $A^+$, and between $A^+$ and $B^+$. Similarly connect $O$ and $A^-$, and $A^-$ and $B^-$ by straight lines. Finally, connect $B^+$ with $B^-$ by an arc which belongs to the unit circle centered at $O$. This gives a closed piecewise smooth curve which is depicted in Figure \[fig:spiral\_initial\]. Then we may smooth out the patch boundary locally near the points $A^\pm$, $B^\pm$ so that in the ball $B(0,\delta/2)$, $\Omega_0^1$ is still a sector and the patch boundary is $C^\infty$-smooth except at $O$. Then, simply set $\Omega_0 = \cup_{j = 0}^3 R_{j\pi/2} \Omega_0^1$. Note that by taking $\delta, \nu \rightarrow 0^+$, the area of $\Omega_0$ converges to that of the unit circle. We shall use this example in the proof of Theorem \[thm:spiral\]. ![Initial data for the result of Theorem \[thm:spiral\] (before smoothing)[]{data-label="fig:spiral_initial"}](spiral.pdf) Before we give a proof of Theorem \[thm:spiral\], let us briefly review the $L^1$-stability theorem for the circular vortex patch first proved by Wan and Pulvirenti [@WP].[^1] Here we state the version given later by Sideris and Vega [@SiVe]. We also mention classical (but weaker) stability results for the circular patch by Dritschel [@Dr2] and Saffman [@Saff]. Let ${\bf 1}_B$ be the characteristic function supported on the unit ball inside $\mathbb{R}^2$. For any bounded open set $\Omega_0 \subset \mathbb{R}^2$, we have $$\begin{split} \V {\bf 1}_{\Omega(t)} - {\bf 1}_B \V_{L^1}^2 \le 4\pi \sup_{ \Omega_0 \triangle B } \|1 - \|x\|^2\| \V {\bf 1}_{\Omega_0} - {\bf 1}_B\V_{L^1} \end{split}$$ for all $t \ge 0$, where $\Omega(t)$ is the patch solution in $\mathbb{R}^2$ associated with initial data $\Omega_0$. We shall now define the notion of winding number for curves satisfying $\gamma : [0,a] \rightarrow \mathbb{R}^2$, $0\notin \gamma((0,a])$, and $\gamma(0) = 0$ with well-defined tangent vector at $0$. Then, we may consider the continuous map $h_\gamma : (0,a] \rightarrow S^1$ defined for $b \in (0,a]$ by taking the angle of $\gamma(b) \in \mathbb{R}^2$ in polar coordinates. Here, $S^1$ denotes the interval $[0,2\pi]$ with the endpoints identified, so that there is a natural projection map $\pi : \mathbb{R} \rightarrow S^1$. The map $h_\gamma$ has a unique extension (up to an additive constant) to a continuous map $\tilde{h}_\gamma : [0,a] \rightarrow \mathbb{R}$ satisfying $\pi \circ \tilde{h}_\gamma = h_\gamma$. Here, we may define $\tilde{h}_\gamma$ continuously at 0 using the assumption that $\gamma$ has a well-defined tangent vector at $0$. Then, we define the winding number of $\gamma$ by $N[\gamma] = (\tilde{h}_\gamma(a) - \tilde{h}_\gamma(0))/2\pi$. We shall recall the following elementary \[lem:winding\] Let $\gamma : [0,a] \rightarrow \mathbb{R}\backslash \{ 0 \}$ be a continuous curve not touching the origin except at $0$ and has a well-defined tangent at 0. Then, for any line $L$ passing through the origin, $$\begin{split} \left\| \{ x : x \in L \cap \gamma([0,a]) \} \right\| \ge \lfloor N[\gamma]\rfloor \end{split}$$ where $ \lfloor N \rfloor$ denotes the largest integer not exceeding $N$. We omit the proof, which is a simple application of the intermediate value theorem (cf. Figure \[fig:winding\]). ![The winding number for $\gamma(t)$ is defined by the $\theta$-coordinate of the right endpoint in $(r,\theta)$ coordinates, where $\theta$ is now varying on $\mathbb{R}$ instead of $S^1$. A lower bound on $u^\theta(t,\cdot)$ at the right endpoint of $\gamma(t)$ guarantees increase of the winding number with $t$. Moreover, a large winding number guarantees a large number of intersections with any line in $\mathbb{R}^2$ passing through the origin.[]{data-label="fig:winding"}](winding.pdf) We are ready to give a proof of Theorem \[thm:spiral\]. The idea is simply to take $\gamma$ to be a curve on $\partial\Omega_0$ starting at the origin and ending at a point in $\partial\Omega_0 \cap \partial B$ where $B$ is the unit disc. Then both endpoints rotate with different angular speed (strictly speaking the origin is fixed but points on $\partial\Omega(t)$ arbitrarily close to the origin rotates at a constant angular speed by persistence of the corner angle), causing the winding number to grow (linearly) with time. A slight complication arises since the initial endpoint of such a curve may slowly move towards the origin as the patch evolves, which then may not rotate around the origin with the desired angular speed. We fix some $0 < \theta_0 < \frac{\pi}{4}$ and for any given $\epsilon > 0$, we may take the 4-fold symmetric patch $\Omega_0$ as in Example \[ex:spiral\] with $\delta, \nu > 0$ sufficiently small, so that the stability theorem gives $$\begin{split} \V {\bf 1}_{\Omega(t)} -{\bf 1}_B \V_{L^1} \le \epsilon^2\quad\forall t \ge 0 \end{split}$$ where $\Omega(t)$ is the patch solution with initial data $\Omega_0$. Below we shall choose $\epsilon > 0$ to be sufficiently small with respect to several parameters depending only on $\theta_0$. Assume for a moment that, for any $t \ge 0$, there is an injective curve $\gamma(t) \subset \partial\Omega(t)$ which has winding number greater than $ct$ for some constant $c = c(\theta_0) > 0$. Using this, the rest of the statements of Theorem \[thm:spiral\] follows immediately. The statement regarding the number of intersections follows from Lemma \[lem:winding\]. To see that the second statement holds, assume that the option (ii) does not hold; that is, there exists some $r > 0$, $$\begin{split} \lim_{ t \rightarrow \infty} \| \{ \|x\| < r : x \in \partial\Omega(t) \cap L \} \| \ne +\infty. \end{split}$$ Then there exists a sequence of time moments $t_k \rightarrow +\infty $ such that $$\begin{split} \| \{ \|x\| < r : x \in \partial\Omega(t_k) \cap L \} \| \le M \end{split}$$ for some $M > 0$. For each $t_k$, we deduce from $N[\gamma(t_k)] \ge ct_k$ and the above that whenever $k$ is sufficiently large, $\partial\Omega(t_k) \ge c'rt_k$ for some constant $c' > 0$. This establishes (i). Returning to the proof of the above claim, let $u(t)$ and $u_B$ be the velocities associated with patches ${\bf 1}_{\Omega(t)}$ and ${\bf 1}_B$, respectively. Then, from $$\begin{split} u(t,x) - u_B(x) = \frac{1}{2\pi} \int_{\mathbb{R}^2} \frac{(x-y)^\perp}{\|x-y\|^2} \left( {\bf 1}_{\Omega(t)}(y) - {\bf 1}_{B}(y) \right) dy \end{split}$$ we compute that, after splitting $\mathbb{R}^2$ into regions $\{ \|x-y\| < r \}$ and $\{ \|x-y\| \ge r\}$ and then choosing $r$ to make two terms on the right hand side equal, $$\begin{split} \|u(t,x) - u_B(x)\| & \le \frac{1}{2\pi}\left( r \V {\bf 1}_{\Omega(t)}- {\bf 1}_{B}\V_{L^1} + r^{-1} \V {\bf 1}_{\Omega(t)} - {\bf 1}_{B} \V_{L^\infty} \right) \\ & \le \frac{1}{\pi} \V {\bf 1}_{\Omega(t)}- {\bf 1}_{B}\V_{L^1}^{\frac{1}{2}} \V {\bf 1}_{\Omega(t)}- {\bf 1}_{B}\V_{L^\infty}^{\frac{1}{2}} \end{split}$$ so that we conclude $$\label{eq:vel_L_infty} \begin{split} \V u(t) - u_B \V_{L^\infty(\mathbb{R}^2)} \le \epsilon \end{split}$$ for all $t \ge 0$. We now compute explicitly the angular velocity of the patch boundary at the corner, which remains the same for all times. At the initial time, we may write $$\begin{split} u_{\Omega_0} = u_S + \tilde{u}, \end{split}$$ where $u_{\Omega_0}$ and $u_S$ are velocities associated with patches $\Omega_0$ and $S = \cup_{j=0}^3 \{ (r,\theta) : -\theta_0 +j\pi/2 < r < \theta_0 + j\pi/2 \}$. Since the patches $\Omega_0$ and $S$ coincide in a ball centered at 0, $\|\tilde{u}(x)\| \ll \|x\|$ as $\|x\| \rightarrow 0$. We then compute $$\begin{split} \lim_{r \rightarrow 0}\frac{(u_{\Omega_0}\cdot e^\theta)(r,\theta_0)}{r} = \frac{(u_{S}\cdot e^\theta)(r,\theta_0)}{r} &= \int_0^{2\pi} \|\sin(2(\theta_0 - \theta'))\| {\bf 1}_{[-\theta_0,\theta_0]} d\theta' \\ &= \frac{1}{4}(1 - \cos(4\theta_0)). \end{split}$$ From the ODE system of \[mainthm:wellposedness\], $$\begin{split} \forall t > 0, \quad \lim_{r \rightarrow 0}\frac{1}{r}(u_{\Omega(t)}\cdot e^\theta)(r,\theta_0 +\frac{t}{4}(1 - \cos(4\theta_0)) ) = \frac{1}{4}(1 - \cos(4\theta_0)). \end{split}$$ Note that $$\begin{split} 0 < \frac{1}{4}(1 - \cos(4\theta_0)) < \frac{1}{2}, \end{split}$$ whereas $$\frac{u_B\cdot e^\theta}{r} = \frac{1}{2}$$ for $r \le 1$. For simplicity, let us set $c_0 = \frac{1}{4}(1 - \cos(4\theta_0))$ and work in a reference frame which rotates in the clockwise direction (note that the patches are rotating in the counter-clockwise direction since $e^\theta = (e^r)^\perp$) with angular speed $c_0$ around the origin, so that the tangent vectors at the corner are stationary for all times. In this frame, the unit disc now rotates with angular speed $c_1 := \frac{1}{2} - c_0$. Claim. For any $T > 0$ there exists a point $x(T) \in \Omega(T)^1 := \Phi(T,\Omega_0^1)$ such that the curve $\Phi(T,\gamma(T))$ has winding number at least $cT$ for some constant $c > 0$, where $\gamma(T)$ is any injective curve belonging to the initial patch $\Omega_0^1$ and connecting the origin with the point $\Phi_T^{-1}(x(T)) =: \tilde{x}(T)$. To show this, we observe that by taking $\epsilon > 0$ sufficiently small, an arbitrarily high portion of the points inside the patch rotates with an angular speed comparable to that of the unit disc. That is, for given small $r_0 > 0$, if $\rho \ge r_0$, then we have from that $$\begin{split} \V u(t)\cdot e^\theta - u_B \cdot e^\theta \V_{L^\infty} \le \epsilon, \end{split}$$ and evaluating it at a point $x$ with distance to the origin $\rho$, since $u_B\cdot e^\theta = c_1\rho$, we obtain $$\begin{split} \|u^\theta(t,x) - c_1\| \le \frac{\epsilon}{\rho} \le \frac{\epsilon}{r_0}. \end{split}$$ That is, given $r_0 > 0$, we may take $\epsilon > 0$ smaller if necessary to guarantee that $$\label{eq:lowerbound_av} \begin{split} u^\theta(t,x) \ge \frac{c_1}{2},\quad \forall t\ge 0, \|x\| \ge r_0. \end{split}$$ Assume that a point $\tilde{x} \in \Omega_0^1 $ has the property that $$\begin{split} \frac{1}{T}\left\| A \right\| := \frac{1}{T}\left\| \{ 0 \le t \le T : \|\Phi(t,\tilde{x})\| \ge r_0 \} \right\| \ge \eta , \end{split}$$ where $\eta>0$ is sufficiently close to 1. Then, taking $\gamma(T)\subset \overline{\Omega_0^1}$ to be an injective curve connecting the origin to $\tilde{x}$, we have that the winding number of the image $\Phi(T,\gamma(T))$ satisfies (cf. Figure \[fig:winding\]) $$\begin{split} N[\Phi(T,\gamma(T))] = \int_0^T u^\theta(t, \Phi(t, \tilde{x}) ) dt &= \int_A u^\theta(t, \Phi(t, \tilde{x}) ) dt + \int_{[0,T]\backslash A} u^\theta(t, \Phi(t, \tilde{x}) ) dt \\ &\ge \eta T \frac{c_1}{2} - C(1-\eta)T, \end{split}$$ where $C > 0$ is an absolute constant in the estimate $$\begin{split} \|u^\theta(t,x)\| := \frac{\|u(t,x)\cdot e^\theta\|}{\|x\|} \le C\V \omega(t)\V_{L^\infty} = C \end{split}$$ which holds under the 4-fold symmetry assumption on $\omega$. Here, we have used that $$\begin{split} \frac{d}{dt} N[\Phi(t,\gamma(t))] = u^\theta(t,\Phi(t,\tilde{x})), \end{split}$$ which is a direct consequence of the definition of the winding number. Once we have chosen $\eta$ sufficiently close to 1 that $$\begin{split} \frac{\eta c_1}{2} > \frac{C(1-\eta)}{2} \end{split}$$ holds, we conclude that $$\begin{split} N[\Phi(T,\gamma(T))] \ge \frac{\eta c_1}{4} T \end{split}$$ which finishes the proof of the Claim with $x(T) := \Phi(T, \tilde{x})$. We shall now show existence of such a point. Assume towards contradiction that there exists some $0 < \eta < 1$ and $T > 0$ for which every point in $\Omega_0^1$ spends less than $\eta$-fraction of time outside the ball $B_0(r_0)$ during the time interval $[0,T]$. The total area of points from $\Omega(t)^1$ which can belong to $B_0(r_0)$ at any moment of time $0 \le t \le T$ is clearly bounded by $ \pi r_0^2 / 4$. Integrating over time, we get an upper bound of $\pi r_0^2T/4$. On the other hand, since every point of $\Omega_0^1$ is forced to spend at least $\eta$-fraction of time inside $B_0(r_0)$, we get a lower bound of $\eta T \|\Omega_0^1\|$ for the total area of points from $\Omega(t)^1$ lying in $B_0(r_0)$, integrated over the time interval $[0,T]$. We get a contradiction once $$\begin{split} \eta T \|\Omega_0^1\| > \frac{\pi r_0^2 T}{4}, \end{split}$$ which is easy to arrange by taking $\|\Omega_0^1\|$ close to $\frac{\pi}{4}$ and $r_0 > 0$ small. We are now in a position to finish the proof, by a simple continuity argument. Fix some $T > 0$ and take the point $\tilde{x}(T)$ satisfying the property described in the above Claim. There is a curve $\Phi(T,\gamma(T)) \subset \Omega(T)^1$ with winding number at least $cT$. Using this it is easy to find, with a continuity argument, an injective curve along the boundary $\partial(\Omega(T)^1)$, starting at the origin, with winding number exceeding that of $\Phi(T,\gamma(T))$. To see this we consider the curve $\tilde{\gamma}$ lying on $[0,\infty) \times S^1$ defined by expressing the image of $\gamma(T)$ in $\mathbb{R}^2$ with polar coordinates. Then using the canonical projection map $\mathbb{R} \rightarrow S^1$, we can lift this curve (its image, to be precise) to lie on $[0,\infty) \times \mathbb{R}$. Then, by definition of the winding number, $\tilde{\gamma}$ has a point in its image with second coordinate at least $cT$. Repeating this procedure for the boundary $\partial(\Omega(T)^1)$, viewed as a closed and injective curve starting and ending at the origin, we see that the corresponding image in $[0,\infty) \times \mathbb{R}$ should contain the image of $\tilde{\gamma}$. In particular, there is a point on this image with its second coordinate strictly larger than $cT$. The proof is now complete. [^1]: Strictly speaking, the result in this paper is stated for patches contained inside a ball but the authors mention that the domain can be replaced by $\mathbb{R}^2$.
--- abstract: 'The ATLAS detector is used to search for excited leptons in the electromagnetic radiative decay channel $\ell^\rightarrow \ell\gamma$. Results are presented based on the analysis of $pp$ collisions at a center-of-mass energy of 7 TeV corresponding to an integrated luminosity of 2.05 . No evidence for excited leptons is found, and limits are set on the compositeness scale $\Lambda$ as a function of the excited lepton mass $m_{\lstar}$. In the special case where $\Lambda = m_{\lstar}$, excited electron and muon masses below 1.87 TeV and 1.75 TeV are excluded at 95% C.L., respectively.' author: - The ATLAS Collaboration bibliography: - 'lstar\_prd2011.bib' title: \| Search for excited leptons in proton–proton collisions\ at $\sqrt{s}=7$ TeV with the ATLAS detector --- Introduction ============ The Standard Model (SM) of particle physics is an extremely successful effective theory which has been extensively tested over the past forty years. However, a number of fundamental questions are left unanswered. In particular, the SM does not provide an explanation for the source of the mass hierarchy and the generational structure of quarks and leptons. Compositeness models address these questions by proposing that quarks and leptons are composed of hypothetical constituents named preons [@patisalam]. In these models, quarks and leptons are the lowest-energy bound states of these hypothetical particles. New interactions among quarks and leptons should then be visible at the scale of the constituents’ binding energies, and give rise to excited states. At the Large Hadron Collider (LHC), excited lepton $\lstar$ production via four-fermion contact interactions can be described by the effective Lagrangian [@BaurSpiraZerwas] $${\cal L}_{\rm contact} =\frac{g_\ast^2}{2\Lambda^2} j^\mu j_\mu,$$ where $g_\ast^2$ is the coupling constant, $\Lambda$ is the compositeness scale, and $j_\mu$ is the fermion current $$j_\mu = \eta_L\overline{f}_L\gamma_\mu f_L + \eta'_L \overline{f}_L^\ast\gamma_\mu f_L^\ast + \eta_L''\overline{f}_L^\ast\gamma_\mu f_L + h.c. + (L\rightarrow R).$$ For simplicity and consistency with recent searches, the following prescription is used: $g_\ast^2=4\pi$, $\eta_L = \eta'_L=\eta''_L=1$, and $\eta_R = \eta'_R=\eta''_R=0$ such that chiral symmetry is conserved [@eichten][@contactI]. The above ansatz ignores underlying preon dynamics and is valid as long as the mass of the excited leptons is below the scale $\Lambda$. In the well-studied case of the homodoublet-type [@hagiwara; @cabbibo; @BaurSpiraZerwas], the relevant gauge-mediated Lagrangian describing transitions between excited and ground-state leptons is $${\cal L}_{\rm GM} = \frac{1}{2\Lambda} \overline{\ell}_R^\ast \sigma^{\mu\nu}\left[g f \frac{\tau^a}{2}W_{\mu\nu}^a+g'f'\frac{Y}{2}B_{\mu\nu} \right] \ell_L + h.c., \label{eqn:lagrangian_trans}$$ where $\ell_L$ is the lepton field, $W_{\mu\nu}$ and $B_{\mu\nu}$ are the $SU(2)_L$ and $U(1)_Y$ field strength tensors, $g$ and $g'$ are the respective electroweak couplings, and $f$ and $f'$ are phenomenological constants chosen to be equal to 1. The ${\cal L}_{\rm GM} $ term allows the decay of excited leptons via the electromagnetic radiative mode ${\ell^\ast}^\pm \to \ell^\pm \gamma$, a very clean signature which is exploited in this search. For a fixed value of $\Lambda$, the branching ratio $B({\ell^\ast}^\pm \to \ell^\pm \gamma)$ decreases rapidly with increasing $\lstar$ mass. For $\Lambda = 2$ , $B({\ell^\ast}^\pm \to \ell^\pm \gamma)$ is 30% for $m_\lstar = 0.2$ and decreases exponentially to about 2.3% for $m_\lstar = 2$ . Previous searches at LEP [@LEPresults], HERA [@HERAresults], and the Tevatron [@TEVATRONresults] have found no evidence for such excited leptons. For the case where $\Lambda = m_{\lstar}$, the CMS experiment has excluded masses below 1.07 for $e^$ and 1.09 for $\mu^$ at the 95% credibility level (C.L.) [@Chatrchyan:2011pg]. Analysis strategy ================= This article reports on searches for excited electrons and muons in the $\ell^\rightarrow \ell \gamma$ channel based on 2.05 of 7 TeV $pp$ collision data recorded in 2011 with the ATLAS detector [@atlas:detector]. The benchmark signal model considered is based upon theoretical calculations from Ref. [@BaurSpiraZerwas]. In this model, excited leptons may be produced singly via $q\overline{q}\rightarrow \ell^\overline{\ell}$ or in pairs via $q\overline{q}\rightarrow \ell^\overline{\ell}^$, due to contact interactions. As the cross section for pair production is much less than for single production, the search for excited leptons is based on the search for events with $\ell\overline{\ell}\gamma$ in the final state: three very energetic particles, isolated, and well separated from one another. For both the $e^$ and $\mu^$ searches, the dominant background arises from Drell-Yan (DY) processes accompanied by either a prompt photon from initial or final state radiation () or by a jet misidentified as a photon (). The dominant irreducible background results in the same final state as the signal, whereas background can be highly suppressed by imposing stringent requirements on the quality of the reconstructed photon candidate. Small contributions from and diboson ($WW$, $WZ$ and $ZZ$) production are also present in both channels. events, as well as semileptonic decays of heavy flavor hadrons, and multijet events can be heavily suppressed by requiring the leptons and photons to be isolated and thus have a negligible contribution to the total background. The signature for excited leptons can present itself as a peak in the invariant mass of the $\ell + \gamma$ system because the width of the $\ell^$ is predicted to be narrower than the detector mass resolution for excited lepton masses $m_{\lstar} \lesssim 0.5 \Lambda$. This peak could be easily resolved from the background. However, it is difficult to identify which of the two leading leptons in the event comes from the decay. To avoid this ambiguity, one can search for an excess in the $\ell\overline{\ell}\gamma$ invariant mass ($m_{\ell\ell\gamma}$) spectrum. This approach is effective for the whole $m_{\lstar} - \Lambda$ parameter space probed as one can search for an excess of events with $m_{\ell\ell\gamma} > 350$ , which defines a nearly background-free signal region. Optimization studies demonstrate that the observable $m_{\ell\ell\gamma}$ provides better signal sensitivity than $m_{\ell\gamma}$, particularly for lower masses. The analysis strategy therefore relies on $m_{\ell\ell\gamma}$ for the statistical interpretation of the results. ATLAS detector ============== ATLAS is a multi-purpose detector with a forward-backward symmetric cylindrical geometry and nearly 4$\pi$ coverage in solid angle. It consists of an inner tracking detector immersed in a 2 T solenoidal field, electromagnetic and hadronic calorimeters, and a muon spectrometer. Charged particle tracks and vertices are reconstructed in silicon-based pixel and microstrip tracking detectors that cover $\|\eta\| <$ 2.5 and transition radiation detectors extending to $\|\eta\| <$ 2.0 [@coordinates]. A hermetic calorimetry system, which covers $\|\eta\|$ $<$ 4.9, surrounds the superconducting solenoid. The liquid-argon electromagnetic calorimeter, which plays an important role in electron and photon identification and measurement, is finely segmented. It has a readout granularity varying by layer and cells as small as $0.025 \times 0.025$ in $\eta \times \phi$, and extends to $\|\eta\| < 2.5$ to provide excellent energy and position resolution. Hadron calorimetry is provided by an iron-scintillator tile calorimeter in the central rapidity range $\|\eta\| < 1.7$ and a liquid-argon calorimeter with copper and tungsten as absorber material in the rapidity range $1.5 < \|\eta\| < 4.9$. Outside the calorimeter, there is a muon spectrometer which is designed to identify muons and measure their momenta with high precision. The muon spectrometer comprises three toroidal air-core magnet systems: one for the barrel and one per endcap, each composed of eight coils. Three layers of drift tube chambers and/or cathode strip chambers provide precision ($\eta$) coordinates for momentum measurement in the region $\|\eta \|< 2.7$. A muon trigger system consisting of resistive plate chambers in the barrel and thin-gap chambers for $\|\eta\| > 1$ provides triggering capability up to $\|\eta\| = 2.4$ and measurements of the $\phi$ coordinate. Simulated samples ================= The excited lepton signal samples are generated based on calculations from Ref. [@BaurSpiraZerwas] at leading order (LO) with 4.5.1 [@comphep] interfaced with 6.421 to handle parton showers and hadronization [@CompHep_Pythia; @pythia], using MRST2007 LO\* [@mrst] parton distribution functions (PDFs). Only single production of excited leptons is simulated, with the decaying exclusively via the electromagnetic channel. The sample is generated with 1.2.3 [@SherpaManual] using CTEQ6.6 [@cteq] PDFs, requiring the dilepton mass to be above 40 . To avoid phase-space regions where matrix elements diverge, the angular separation between the photon and leptons is required to be $R(\ell,\gamma) = \sqrt{(\Delta \eta)^2 + (\Delta \phi)^2}> 0.5$ and the transverse momentum () of the photon is required to be $\ptgam>10$ GeV. To ensure adequate statistics at large $m_{\ell\ell\gamma}$, an additional sample is generated with $\ptgam>40$ GeV, and is equivalent to $\sim 300$ of data. The background is generated with 2.13 [@Mangano:2002ea], while the background is produced with 3.41 [@mcatnlo]. In both cases, 4.31 [@jimmy] is used to describe multiple parton interactions and 6.510 [@herwig] is used to simulate the remaining underlying event and parton showers and hadronization. CTEQ6.6 PDFs are used for both backgrounds. To remove overlaps between the and the samples, events with prompt energetic photons are rejected if the photon-lepton separation is such that $R(\ell,\gamma) > 0.5$. The diboson processes are generated with using MRST2007 LO\* PDFs. For all samples, final-state photon radiation is handled via [@fsr_ref]. The generated samples are then processed through a detailed detector simulation [@atlas:sim] based on 4 [@geant] to propagate the particles and account for the detector response. A large sample of MC minimum bias events is then mixed with the signal and background MC events to simulate pile-up from additional $pp$ collisions. Simulations are normalized on an event-by-event basis such that the distribution of the number of interactions per event agrees with the spectrum observed in data. Although includes higher-order QCD contributions beyond the Born amplitude, such as the real emission of partons in the initial state, it omits virtual corrections. For this reason, the cross section is calculated at next-to-leading order ($\sigma_{\rm NLO}$) using MCFM [@MCFM] with MSTW 2008 NLO PDFs [@mstw]. The theoretical precision of the $\sigma_{\rm NLO}$ estimate is $\sim 6\%$, and the ratio $\sigma_{\rm NLO} / \sigma_{\rm SHERPA}$ is used to determine a correction factor as a function of $m_{\ell\ell\gamma}$. The cross section is initially normalized to predictions calculated at next-to-next-to-leading order (NNLO) in perturbative QCD as determined by the FEWZ [@fewz] program using MSTW 2008 NNLO PDFs. Since the misidentification of jets as photons is not well modeled, the prediction is adjusted at the analysis level using data-driven techniques described below. Cross sections for diboson processes are known at NLO with an uncertainty of 5%, while the cross section is predicted at approximate-NNLO, with better than 10% uncertainty [@Moch:2008qy; @Langenfeld:2009tc]. Data and preselection ===================== The data, which correspond to a total integrated luminosity of 2.05 , were collected in 2011 during stable beam periods of 7 TeV $pp$ collisions. For the $e^$ search, events are required to pass the lowest unprescaled single electron trigger available. For the first half of the data this corresponds to a threshold of 20 , and a threshold of 22 for the later runs. For the $\mu^$ search, a single muon trigger with matching tracks in the muon spectrometer and inner detector with combined $\ptmu > 22$ is used to select events. In addition, events with a muon with $\ptmu>40$ in the muon spectrometer are also kept. Collision candidates are then identified by requiring a primary vertex with a $z$ position along the beam line of $\|z\|<200$ mm and at least three associated charged particle tracks with $\pt > 0.4$ . The lepton selection consists of the same requirements used in the ATLAS search for new heavy resonances decaying to dileptons [@Zprime2011]. Electron candidates are formed from clusters of cells in the electromagnetic calorimeter associated with a charged particle track in the inner detector. For the $e^$ search, two electron candidates with $\pte > 25$ and $\|\eta\| < 2.47$ are required. Electrons within the transition region $1.37 < \|\eta\| < 1.52$ between the barrel and the endcap calorimeters are rejected. The electron identification criteria [@egamma_id] on the transverse shower shape, the longitudinal leakage into the hadronic calorimeter, and the association with an inner detector track are applied to the cluster. The electron’s reconstructed energy is obtained from the calorimeter measurement and its direction from the associated track. A hit in the first active pixel layer is required to suppress background from photon conversions. To further suppress background from jets, the leading electron is required to be isolated by demanding that the sum of the transverse energies in the cells around the electron direction in a cone of radius $R<0.2$ be less than 7 . The core of the electron energy deposition is excluded and the sum is corrected for transverse shower leakage and pile-up from additional $pp$ collisions to make the isolation variable essentially independent of [@ambientenergy]. In cases where more than two electrons are found to satisfy the above requirements, the pair with the largest invariant mass is chosen. To minimize the impact of possible charge misidentification, the electrons are not required to have opposite electric charges. Muon tracks are reconstructed independently in both the inner detector and muon spectrometer, and their momenta are determined from a combined fit to these two measurements. For the $\mu^$ search, two muons with $\ptmu > 25$ are required. To optimize the momentum resolution, each muon candidate is required to have a minimum number of hits in the inner detector and to have at least three hits in each of the inner, middle, and outer layers of the muon spectrometer. This requirement results in a muon fiducial acceptance of $\|\eta\|<2.5$. Muons with hits in the barrel-endcap overlap regions of the muon spectrometer are discarded because of large residual misalignments. The effects of misalignments and intrinsic position resolution are otherwise included in the simulation. The resolution at 1 TeV ranges from 13% to 20%. To suppress background from cosmic rays, the muon tracks are required to have transverse and longitudinal impact parameters $\|d_0\|<0.2$ mm and $\|z_0\| < 1$ mm with respect to the primary vertex. To reduce background from heavy flavor hadrons, each muon is required to be isolated such that $\Sigma\pt(R<0.3)/\ptmu <0.05$, where only inner detector tracks with $\pt >1$ enter the sum. Muons are required to have opposite electric charges. In cases where more than two muons are found to satisfy the above requirements, the pair of muons with the largest invariant mass is considered. The dielectron and dimuon distributions are inspected for consistency with background predictions to ensure that the resolution and efficiency corrections were adjusted properly in the simulation. Excellent agreement is found around the mass of the $Z$, in terms of both the peak position and width of the dilepton invariant mass distributions. For the mass range $70 < m_{\ell\ell} < 110$ , the number of events observed in data agrees to within 1% of the background predictions for both the electron and muon channels. Furthermore, the tails of the and distributions in the simulation are found to closely match the data. The presence of at least one photon candidate with $\ptgam > 20$ and pseudorapidity $\|\eta \| < 2.37$ is then necessary for the events to be kept. Photons within the transition region between the barrel and the endcap calorimeters are excluded. Photon candidates are formed from clusters of cells in the electromagnetic calorimeter. They include unconverted photons, with no associated track, and photons that converted to electron-positron pairs, associated to one or two tracks. All photon candidates are required to satisfy the photon definition [@ATLASphotons]. This selection includes constraints on the energy leakage into the hadronic calorimeter as well as stringent requirements on the energy distribution in the first sampling layer of the electromagnetic calorimeter, and on the shower width in the second sampling layer. The photon definition is designed to increase the purity of the photon selection sample by rejecting most of the jet background, including jets with a leading neutral hadron (usually a $\pi^0$) that decays to a pair of collimated photons. To further reduce background from misidentified jets, photon candidates are required to be isolated by demanding that the sum of the transverse energies of the cells within a cone $R < 0.4$ of the photon be less than 10 . As for the electron isolation, the core of the photon energy deposition is excluded and the sum is corrected for transverse shower leakage and pile-up. Because no background predictions are simulated for $R(\ell,\gamma) < 0.5$, photons are required to be well separated from the leptons with $R(\ell,\gamma) > 0.7$. This requirement has a negligible impact on signal efficiency. Finally, if more than one photon in an event satisfies the above requirements, the one with the largest is used in the search. For the above selection criteria, the total signal acceptance times efficiency ($A\times \epsilon$) is $\sim 56\%$ in the $e^$ channel for masses $m_{e^} > 600$ . This value includes the acceptance of all selection cuts and the reconstruction efficiencies, and reflects the lepton and photon angular distributions. In comparison, $A\times \epsilon$ is $\sim 32\%$ for $m_{\mu^} > 600$ . The lower acceptance in the $\mu^$ channel is due to the stringent selection on the muon spectrometer hits used to maximize the resolution, in particular the limited geometrical coverage of the muon spectrometer with three layers of precision chambers. Background determination ======================== All background predictions are evaluated with simulated samples. These include the dominant and irreducible background, as well as events where a jet is misidentified as a photon. The rate of jet misidentification is overestimated in the simulation so the predictions are adjusted to data as described below. Small contributions from and diboson production are also present at low $m_{\ell\ell\gamma}$. Background from multijet events and semileptonic decays of heavy flavor hadrons are heavily suppressed by the isolation requirements and are negligible in the signal region. The estimates are adjusted to data in a control region defined by $m_{\ell\ell\gamma} < 300$ . This region represents less than 1% of the signal parameter-space for $m_\lstar \ge 200$ . The nominal strategy consists of counting the number of events in data in this control region and comparing it to the MC background predictions. The excess of background events found in the simulation is attributed to the mis-modeling of the rate of jets misidentified as photons, and the number of events is scaled down accordingly. As a result, the number of events in the control region is the same in the MC simulations as in data as shown in Table \[table-control\]. The estimates are validated using various data-driven methods, notably by using misidentification rates evaluated in jet-enriched samples, and applying these rates to data samples using an approach similar to the one described in Ref. [@ATLASphotons]. The main reason for the overestimation of the jet misidentification rate in the simulation is due to the mis-modeling of the jet shower shapes. A enriched sample was used to correct the shower shapes of jets in the simulations, such that the efficiency for jets to pass the photon requirement in the MC simulation is comparable to the rate measured in data. This correction depends strongly on the generator used (e.g. vs ) and results in a 15% uncertainty in the background estimate. The largest difference between the nominal background determination and the alternative estimates is assigned as a systematic uncertainty and dominates the total error in the estimates presented in Table \[table-control\]. The corresponding scaling factors applied to the simulation are $0.51\pm 0.14$ and $0.61\pm 0.21$ for the $e^$ and $\mu^$ channels, respectively, i.e. within uncertainties of one another. Furthermore, the ratio of the number of events outside the control region to the number of events inside is found to be the same in the MC simulations as in the data-driven techniques: 0.06 for both the $e^$ and $\mu^$ channels. This finding indicates that the jet misidentification rate as a function of the jet is modeled properly. Region $[$$]$ diboson data -------------------------- ------------- --------------- --------------- -------------- ------ $m_{ee\gamma} < 300$ $306 \pm 8$ $138 \pm 38$ $8.3 \pm 0.8$ $2.4\pm 0.5$ 455 $m_{ee\gamma} > 300$ $25 \pm 2$ $8.1 \pm 1.6$ $0.8 \pm 0.2$ $0.5\pm 0.2$ 29 $m_{\mu\mu\gamma} < 300$ $255 \pm 8$ $89 \pm 31$ $4.9 \pm 0.6$ $0.9\pm 0.3$ 350 $m_{\mu\mu\gamma} > 300$ $14 \pm 1$ $5.4 \pm 1.4$ $0.9 \pm 0.3$ $0.1\pm 0.1$ 19 : Data yields and background expectations inside ($m_{\ell\ell\gamma} < 300$ ) and outside the $m_{\ell\ell\gamma}$ control region after adjusting the background. The uncertainties shown are purely statistical, except for the background for which the total uncertainty is dominated by systematic uncertainties. \[table-control\] Comparisons between data and the resulting background expectations for the , , $m_{\ell\gamma}$ and $m_{\ell\ell\gamma}$ distributions are shown in Figs. \[fig:lpt\] to \[fig:llg\]. No significant discrepancies are observed between data and the simulations. In particular, the background prediction for the photon shape matches the data for both the $e^$ and $\mu^$ searches, which suggests that the tuning of the jet misidentification rate for the background is adequate. ![Lepton distributions for the $e^$ (top) and $\mu^$ (bottom) channels. The expected background uncertainties shown correspond to the sum in quadrature of the statistical uncertainties as well as the uncertainty in the normalization measured in the control region.[]{data-label="fig:lpt"}](fig_01a.eps "fig:"){width="3.5truein"} ![Lepton distributions for the $e^$ (top) and $\mu^$ (bottom) channels. The expected background uncertainties shown correspond to the sum in quadrature of the statistical uncertainties as well as the uncertainty in the normalization measured in the control region.[]{data-label="fig:lpt"}](fig_01b.eps "fig:"){width="3.5truein"} ![Photon distributions for the $e^$ (top) and $\mu^$ (bottom) channels. The expected background uncertainties shown correspond to the sum in quadrature of the statistical uncertainties as well as the uncertainty in the normalization measured in the control region.[]{data-label="fig:gpt"}](fig_02a.eps "fig:"){width="3.5truein"} ![Photon distributions for the $e^$ (top) and $\mu^$ (bottom) channels. The expected background uncertainties shown correspond to the sum in quadrature of the statistical uncertainties as well as the uncertainty in the normalization measured in the control region.[]{data-label="fig:gpt"}](fig_02b.eps "fig:"){width="3.5truein"} ![Distributions of the invariant mass of the $\ell\gamma$ systems for the $e^$ (top) and $\mu^$ (bottom) channels. Combinations with both the leading and subleading leptons are shown. The expected background uncertainties shown correspond to the sum in quadrature of the statistical uncertainties as well as the uncertainty in the normalization measured in the control region. For both channels, one event lies outside the mass range shown.[]{data-label="fig:lg"}](fig_03a.eps "fig:"){width="3.5truein"} ![Distributions of the invariant mass of the $\ell\gamma$ systems for the $e^$ (top) and $\mu^$ (bottom) channels. Combinations with both the leading and subleading leptons are shown. The expected background uncertainties shown correspond to the sum in quadrature of the statistical uncertainties as well as the uncertainty in the normalization measured in the control region. For both channels, one event lies outside the mass range shown.[]{data-label="fig:lg"}](fig_03b.eps "fig:"){width="3.5truein"} ![Distributions of the invariant mass for the $\ell\ell\gamma$ system for the $e^$ (top) and $\mu^$ (bottom) channels. The expected background uncertainties shown correspond to the sum in quadrature of the statistical uncertainties as well as the uncertainty in the normalization measured in the control region. For both channels, one event lies outside the mass range shown.[]{data-label="fig:llg"}](fig_04a.eps "fig:"){width="3.5truein"} ![Distributions of the invariant mass for the $\ell\ell\gamma$ system for the $e^$ (top) and $\mu^$ (bottom) channels. The expected background uncertainties shown correspond to the sum in quadrature of the statistical uncertainties as well as the uncertainty in the normalization measured in the control region. For both channels, one event lies outside the mass range shown.[]{data-label="fig:llg"}](fig_04b.eps "fig:"){width="3.5truein"} Signal region optimization ========================== The signal search region is optimized as a function of $m_\lstar$ using simulated events by determining the lower bound on $m_{\ell\ell\gamma}$ that maximizes the significance defined as $$S_L = \sqrt{ 2 \ln \left[ \left( 1 + S/B \right)^{S+B} e^{-S} \right] },$$ where $S$ and $B$ are the number of signal and background events, respectively. The optimum threshold value is found to be $m_{\ell\ell\gamma} = m_\lstar + 150$ . Additionally, to improve the sensitivity particularly at low $m_\lstar$, background contributions from DY processes are suppressed further by requiring events to satisfy $m_{\ell\ell} > 110$ . The signal efficiency for these two additional requirements is $>99$% for $m_\lstar \ge 200$ . Because few events survive the complete set of requirements, the shape of the and backgrounds are individually fitted using an exponential function $\exp(P_0 + P_1 \times m_{\ell\ell\gamma})$ over the mass range $250 \gev < m_{\ell\ell\gamma} < 950$ . The sum of these two fits is then used to obtain the total background prediction for $m_{\ell\ell\gamma} > 350$ GeV. The resulting background estimates and data yields are shown in Table \[tab:eyield\] for the $e^$ and $\mu^$ searches, as well as in Figs. \[fig:lg-final\] and \[fig:llg-final\]. ![Distributions of the invariant mass of the $\ell\gamma$ systems for the $e^$ (top) and $\mu^$ (bottom) channels after requiring $m_{\ell\ell} > 110$ . Combinations with both the leading and subleading leptons are shown. The expected background uncertainties shown correspond to the sum in quadrature of the statistical uncertainties as well as the uncertainty in the normalization measured in the control region. Note that the last bin contains the sum of all entries with $m_{\ell\gamma}> 950$ .[]{data-label="fig:lg-final"}](fig_05a.eps "fig:"){width="3.5truein"} ![Distributions of the invariant mass of the $\ell\gamma$ systems for the $e^$ (top) and $\mu^$ (bottom) channels after requiring $m_{\ell\ell} > 110$ . Combinations with both the leading and subleading leptons are shown. The expected background uncertainties shown correspond to the sum in quadrature of the statistical uncertainties as well as the uncertainty in the normalization measured in the control region. Note that the last bin contains the sum of all entries with $m_{\ell\gamma}> 950$ .[]{data-label="fig:lg-final"}](fig_05b.eps "fig:"){width="3.5truein"} ![Distributions of the invariant mass for the $\ell\ell\gamma$ system for the $e^$ (top) and $\mu^$ (bottom) searches after requiring $m_{\ell\ell} > 110$ . The and backgrounds were fitted, and the total uncertainties from the fit as well as the uncertainty in the normalization measured in the control region are displayed as the shaded area. Note that the last bin contains the sum of all events with $m_{\ell\ell\gamma}> 1450$ .[]{data-label="fig:llg-final"}](fig_06a.eps "fig:"){width="3.5truein"} ![Distributions of the invariant mass for the $\ell\ell\gamma$ system for the $e^$ (top) and $\mu^$ (bottom) searches after requiring $m_{\ell\ell} > 110$ . The and backgrounds were fitted, and the total uncertainties from the fit as well as the uncertainty in the normalization measured in the control region are displayed as the shaded area. Note that the last bin contains the sum of all events with $m_{\ell\ell\gamma}> 1450$ .[]{data-label="fig:llg-final"}](fig_06b.eps "fig:"){width="3.5truein"} Systematic uncertainties ======================== In this section, the dominant systematic uncertainties in the and background predictions are first described, followed by a description of the experimental systematic uncertainties that affect both the background and signal yields, and by a discussion of the theoretical uncertainties which affect both the $e^$ and $\mu^$. The dominant systematic uncertainty in the irreducible background comes from the fit of its background shape and normalization due to the limited number of events with $m_{\ell\ell} > 110$ . This uncertainty increases with $m_\lstar$ from about 20% at 200 to 100% for $m_\lstar > 800$ . The second largest uncertainty in the background is of theoretical nature and arises from the NLO computations. This uncertainty is obtained by varying the renormalization and factorization scales by factors of two around their nominal values and combining with uncertainties arising from the PDFs and values of the strong coupling constant $\alpha_s$. For $m_\lstar = 200$ ($m_\lstar>800$ ), the resulting theoretical uncertainty in the number of background events in our signal region is 7% (10%) for both channels. The uncertainty in the normalization is determined to be 38% (35%) for the $e^$ ($\mu^$) channel, which covers the range of values obtained by the different estimates as well as their uncertainties in the $m_{\ell\ell\gamma}<300$ GeV control region. Uncertainties in the prediction from the shape of the fitted distribution are added in quadrature to the normalization uncertainty. Experimental systematic uncertainties that affect both signal and background yields include the uncertainty from the luminosity measurement of 3.7% [@lumi2011], and uncertainties in particle reconstruction and identification as described below. A 3% systematic uncertainty is assigned to the photon efficiency. This value is obtained by comparing the signal efficiency with and without photon shower shape corrections (2%), by studying the impact of material mis-modeling in the inner detector (1%) and the by determining the reconstruction efficiency for various pile-up conditions (1%) [@diphotonExo]. The electron trigger and reconstruction efficiency is evaluated in data and in MC simulations in several $\eta \times \phi$ bins to high precision. Correction factors are applied to the simulations accordingly and have negligible uncertainties. A 1% systematic uncertainty in the electron efficiency at high is assigned. This uncertainty is estimated by studying the electron efficiency as a function of the calorimeter isolation criteria. The calorimeter energy resolution is dominated at high by a constant term which is 1.1% in the barrel and 1.8% in the endcaps. The simulation is adjusted to reproduce this resolution at high energy, and the uncertainty in this correction has a negligible effect on and . The calorimeter energy scale is corrected by studying $J/\psi\rightarrow ee$ and $Z\rightarrow ee$ events. Calibration constants are obtained for several regions and deviate at most by 1.5% of unity, and have small uncertainties. Thus, uncertainties on the calorimeter energy scale and resolution result in negligible uncertainties in the background and signal yields. The combined uncertainty in yields arising from the trigger and reconstruction efficiency for muons is estimated to increase linearly as a function of $\ptmu$ to about 1.5% at 1 . This uncertainty is dominated by a conservative estimate of the impact of large energy loss from muon bremsstrahlung in the calorimeter which can affect reconstruction in the muon spectrometer. The uncertainty from the resolution due to residual misalignments in the muon spectrometer propagates to a change in the number of events passing the $m_{\mu\mu\gamma}$ cut, and affects the sensitivity of the search. The muon momentum scale is calibrated with a statistical precision of 0.1% using the $Z\to\mu\mu$ mass peak. Thus, uncertainties on the muon momentum scale and resolution result in negligible uncertainties in the background and signal yields. An additional 1% systematic uncertainty is assigned to the $e^$ and $\mu^$ signal efficiencies to account for the fact that the dependence on $\Lambda$ is neglected in this analysis. This uncertainty is obtained by studying the signal $A\times \epsilon$ for various excited lepton masses and compositeness scales. Theoretical uncertainties from renormalization and factorization scales and PDFs have negligible impact on the signal efficiency and are not included in the results presented below. Results ======= A summary of the data yields and background expectations as a function of a lower bound on $m_{\ell\ell\gamma}$ is shown in Table \[tab:eyield\] for the $e^$ and $\mu^$ searches. The uncertainties displayed correspond to the sum in quadrature of the statistical and systematic uncertainties. The significance of a potential excited lepton signal is estimated by a $p$-value, the probability of observing an outcome at least as signal-like as the one observed in data, assuming that a signal is absent. The lowest $p$-values obtained are 3% in the $e^$ channel (for $m_{ee\gamma} > 950$ ), and 17% in the $\mu^$ channel (for $m_{\mu\mu\gamma} > 850$ ), which indicates that the data are consistent with the background hypothesis. Given the absence of a signal, an upper limit on the cross section times branching ratio $\sigma B$ is determined at the 95% C.L. using a Bayesian approach [@bayesianMethod] with a flat, positive prior on $\sigma B$. Systematic uncertainties are incorporated in the limit calculation as nuisance parameters. The limits are translated into bounds on the compositeness scale as a function of the mass of the excited leptons by comparing them with theoretical predictions of $\sigma B$ for various values of $\Lambda$. The expected exclusion limits are determined using simulated pseudo-experiments (PE) containing only SM processes, by evaluating the 95% C.L. upper limits for each PE for each fixed value of $m_\lstar$. The median of the distribution of limits represents the expected limit. The ensemble of limits is used to find the $1\sigma$ and $2\sigma$ envelopes of the expected limits as a function of $m_\lstar$. Figure \[fig:limits1d\] shows the 95% C.L. expected and observed limits on $\sigma B(\lstar \rightarrow \ell\gamma)$ for the $e^{}$ and $\mu^{}$ searches. For $m_{\lstar}> 0.9$ , the observed and expected limits on $\sigma B$ are 2.3 fb and 4.5 fb for the $e^{}$ and $\mu^{}$, respectively. The green and yellow bands show the expected $1\sigma$ and $2\sigma$ contours of the expected limits. When the expected number of background events is zero, there is an effective quantization of the expected limits obtained from the PE, and no downward fluctuation of the background is possible. These effects explain the behavior of the $1\sigma$ and $2\sigma$ contours of the expected limits for large masses. Theoretical predictions of $\sigma B$ for three different values of $\Lambda$ are also displayed in Fig. \[fig:limits1d\], as well as the theoretical uncertainties from renormalization and factorization scales and PDFs for $\Lambda = 2$ TeV. These uncertainties are shown for illustrative purposes only and are not included in determining mass limits. The mass limits obtained for various $\Lambda$ values are used to produce exclusion limits on the $m_{\lstar}-\Lambda$ plane as shown in Fig. \[fig:limits2d\]. In the special case where $\Lambda = m_{\lstar}$, masses below 1.87 TeV and 1.75 TeV are excluded for excited electrons and muons, respectively. ----------------------------- ----------------- ----------------- ------ ----------- ----------------- ----------------- ------ ----------- $m_{\ell\ell\gamma}$ region $[$$]$ total bkg data $p$-value total bkg data $p$-value $> 0.35$ $10.1 \pm 1.9$ $11.5 \pm 2.2$ 8 0.92 $5.2 \pm 1.4 $ $6.0 \pm 1.6 $ 6 0.40 $> 0.45$ $4.6 \pm 1.0$ $5.1 \pm 1.2$ 2 0.83 $3.1 \pm 0.8 $ $3.4 \pm 0.9 $ 3 0.42 $> 0.55$ $2.1 \pm 0.7$ $2.3 \pm 0.8$ 1 0.80 $1.8 \pm 0.6 $ $2.0 \pm 0.7 $ 1 0.72 $> 0.65$ $0.98 \pm 0.47$ $1.02 \pm 0.49$ 1 0.32 $1.09 \pm 0.49$ $1.14 \pm 0.51$ 1 0.72 $> 0.75$ $0.45 \pm 0.29$ $0.46 \pm 0.30$ 1 0.16 $0.65 \pm 0.39$ $0.67 \pm 0.39$ 1 0.28 $> 0.85$ $0.20 \pm 0.16$ $0.21 \pm 0.17$ 1 0.11 $0.39 \pm 0.29$ $0.39 \pm 0.29$ 1 0.17 $> 0.95$ $0.09 \pm 0.09$ $0.10 \pm 0.09$ 1 0.03 $0.23 \pm 0.21$ $0.23 \pm 0.21$ 0 0.78 $> 1.05$ $0.05 \pm 0.05$ $0.05 \pm 0.05$ 0 0.81 $0.14 \pm 0.14$ $0.14 \pm 0.14$ 0 0.92 ----------------------------- ----------------- ----------------- ------ ----------- ----------------- ----------------- ------ ----------- : Data yields and background expectation as a function of a lower bound on $m_{\ell\ell\gamma} = m_\lstar + 150$ . The uncertainties represent the sum in quadrature of the statistical and systematic uncertainties. The probability for the background only hypothesis ($p$-value) is also provided. \[tab:eyield\] ![Cross section $\times$ branching ratio limits at 95% C.L. as a function of $e^$ and of $\mu^$ mass. Theoretical predictions for excited leptons produced for three different compositeness scales are shown, as well as the theoretical uncertainties from renormalization and factorization scales and PDFs for $\Lambda = 2$ TeV. For $m_{\lstar} > 0.9$ , the observed limit on $\sigma B$ is 2.3 fb (4.5 fb) for $e^{}$ ($\mu^{}$). \[fig:limits1d\]](fig_07a.eps "fig:"){width="3.5truein"} ![Cross section $\times$ branching ratio limits at 95% C.L. as a function of $e^$ and of $\mu^$ mass. Theoretical predictions for excited leptons produced for three different compositeness scales are shown, as well as the theoretical uncertainties from renormalization and factorization scales and PDFs for $\Lambda = 2$ TeV. For $m_{\lstar} > 0.9$ , the observed limit on $\sigma B$ is 2.3 fb (4.5 fb) for $e^{}$ ($\mu^{}$). \[fig:limits1d\]](fig_07b.eps "fig:"){width="3.5truein"} ![Exclusion limits in the $m_{\lstar} - \Lambda$ parameter space for $e^$ and $\mu^$. Regions to the left of the experimental limits are excluded at 95% C.L. No limits are set for the hashed region as the approximations made in the effective contact interaction model do not hold for $m_{\lstar} > \Lambda$. The best limits from the Tevatron experiments as well as from the CMS experiment based on 36 pb$^{-1}$ are also shown. \[fig:limits2d\]](fig_08a.eps "fig:"){width="3.5truein"} ![Exclusion limits in the $m_{\lstar} - \Lambda$ parameter space for $e^$ and $\mu^$. Regions to the left of the experimental limits are excluded at 95% C.L. No limits are set for the hashed region as the approximations made in the effective contact interaction model do not hold for $m_{\lstar} > \Lambda$. The best limits from the Tevatron experiments as well as from the CMS experiment based on 36 pb$^{-1}$ are also shown. \[fig:limits2d\]](fig_08b.eps "fig:"){width="3.5truein"} Conclusions =========== The results of a search for excited electrons and muons with the ATLAS detector are reported, using a sample of $\sqrt{s}=7$ $pp$ collisions corresponding to an integrated luminosity of 2.05 . The observed invariant mass spectra are consistent with SM background expectations. Limits are set on the cross section times branching ratio $\sigma B(\lstar \rightarrow \ell\gamma)$ at 95% C.L. For $m_{\lstar} > 0.9$ , the observed upper limits on $\sigma B$ are 2.3 fb and 4.5 fb for the $e^{}$ and $\mu^{}$ channels, respectively. The limits are translated into bounds on the compositeness scale $\Lambda$ as a function of the mass of the excited leptons. In the special case where $\Lambda = m_{\lstar}$, masses below 1.87 TeV and 1.75 TeV are excluded for $e^$ and $\mu^$, respectively. These limits are the most stringent bounds to date on excited leptons for the parameter-space region with $m_{\lstar} \ge 200$ . Acknowledgements ================ We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently. We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC, Australia; BMWF, Austria; ANAS, Azerbaijan; SSTC, Belarus; CNPq and FAPESP, Brazil; NSERC, NRC and CFI, Canada; CERN; CONICYT, Chile; CAS, MOST and NSFC, China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic; DNRF, DNSRC and Lundbeck Foundation, Denmark; ARTEMIS, European Union; IN2P3-CNRS, CEA-DSM/IRFU, France; GNAS, Georgia; BMBF, DFG, HGF, MPG and AvH Foundation, Germany; GSRT, Greece; ISF, MINERVA, GIF, DIP and Benoziyo Center, Israel; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; FOM and NWO, Netherlands; RCN, Norway; MNiSW, Poland; GRICES and FCT, Portugal; MERYS (MECTS), Romania; MES of Russia and ROSATOM, Russian Federation; JINR; MSTD, Serbia; MSSR, Slovakia; ARRS and MVZT, Slovenia; DST/NRF, South Africa; MICINN, Spain; SRC and Wallenberg Foundation, Sweden; SER, SNSF and Cantons of Bern and Geneva, Switzerland; NSC, Taiwan; TAEK, Turkey; STFC, the Royal Society and Leverhulme Trust, United Kingdom; DOE and NSF, United States of America. The crucial computing support from all WLCG partners is acknowledged gratefully, in particular from CERN and the ATLAS Tier-1 facilities at TRIUMF (Canada), NDGF (Denmark, Norway, Sweden), CC-IN2P3 (France), KIT/GridKA (Germany), INFN-CNAF (Italy), NL-T1 (Netherlands), PIC (Spain), ASGC (Taiwan), RAL (UK) and BNL (USA) and in the Tier-2 facilities worldwide.
--- abstract: 'In this article, I calculate the contributions of the nuclear matter induced condensates up to dimension 5, take into account the next-to-leading order contributions of the nuclear matter induced quark condensate, study the properties of the scalar, pseudoscalar, vector and axialvector heavy mesons in the nuclear matter with the QCD sum rules in a systematic way, and obtain the shifts of the masses and decay constants. Furthermore, I study the heavy-meson-nucleon scattering lengths as a byproduct, and obtain the conclusion qualitatively about the possible existence of heavy-meson-nucleon bound states.' --- \ Zhi-Gang Wang [^1]\ Department of Physics, North China Electric Power University, Baoding 071003, P. R. China PACS numbers: 12.38.Lg; 14.40.Lb; 14.40.Nd [[Key Words:]{}]{} Nuclear matter, QCD sum rules Introduction ============ The suppression of $J/\psi$ production in relativistic heavy ion collisions is considered as an important signature to identify the quark-gluon plasma [@Matsui86]. The dissociation of $J/\psi$ in the quark-gluon plasma due to color screening can result in a reduction of its production. The interpretation of suppression requires the detailed knowledge of the expected suppression due to the $J/\psi$ dissociation in the hadronic environment. The in-medium hadron properties can affect the productions of the open-charmed mesons and the $J/\psi$ in the relativistic heavy ion collisions, the higher charmonium states are considered as the major source of the $J/\psi$ [@Jpsi-Source]. For example, the higher charmonium states can decay to the $D\bar{D}$, $D^\bar{D}^$ pairs instead of decaying to the lowest state $J/\psi$ in case of the mass reductions of the $D$, $D^$, $\bar{D}$, $\bar{D}^$ mesons are large enough. We have to disentangle the color screening versus the recombination of off-diagonal $\bar{c}c$ (or $\bar{b}b$) pairs in the hot dense medium versus cold nuclear matter effects, such as nuclear absorption, shadowing and anti-shadowing, so as to draw a definite conclusion on appearance of the quark-gluon plasma [@Recombine-cc; @CNM]. The upcoming FAIR (Facility for Antiproton and Ion Research) project at GSI (Institute for Heavy Ion Research) in Darmstadt (Germany) provides the opportunity to study the in-medium properties of the charmoniums or charmed hadrons for the first time. The CBM (Compressed Baryonic Matter) collaboration intends to study the properties of the hadrons in the nuclear matter [@CBM], while the $\rm \bar{P}ANDA$ (anti-Proton Annihilation at Darmstadt) collaboration will focus on the charm spectroscopy, and mass and width modifications of the charmed hadrons in the nuclear matter [@PANDA]. However, the in-medium mass modifications are not easy to access experimentally despite the interesting physics involved, and they require more detailed theoretical studies. On the other hand, the bottomonium states are also sensitive to the color screening, the $\Upsilon$ suppression in high energy heavy ion collisions can also be taken as a signature to identify the quark-gluon plasma [@QGP-rev]. The suppressions on the $\Upsilon$ production in ultra-relativistic heavy ion collisions will be studied in details at the Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC). Extensive theoretical and experimental studies are required to explore the hadron properties in nuclear matter. The connection between the condensates and the nuclear density dependence of the in-medium hadron masses is not straightforward. The QCD sum rules provides a powerful theoretical tool in studying the in-medium hadronic properties [@SVZ79; @PRT85], and has been applied extensively to study the light-flavor hadrons and charmonium states in the nuclear matter [@C-parameter; @Drukarev1991; @Jpsi-etac]. The works on the heavy mesons and heavy baryons are few, only the $D$, $B$, $D_0$, $B_0$, $D^$, $B^$, $D_1$, $B_1$, $\Lambda_Q$, $\Sigma_Q$, $\Xi_{QQ}$ and $\Omega_{QQ}$ are studied with the QCD sum rules [@Hay; @WangHuang; @Azi; @Hil; @Hilger2; @WangH]. The heavy mesons (heavy baryons) contain a heavy quark and a light quark (two light quarks), the existence of a light quark (two light quarks) in the heavy mesons (heavy baryons) leads to large difference between the mass-shifts of the heavy mesons (heavy baryons) and heavy quarkonia in the nuclear matter. The former have large contributions from the light-quark condensates in the nuclear matter and the modifications of the masses originate mainly from the modifications of the quark condensates, while the latter are dominated by the gluon condensates, and the mass modifications are mild [@Jpsi-etac; @Hay; @WangHuang; @Azi; @Hil; @Hilger2; @WangH]. In previous works [@Hay; @WangHuang; @Azi; @Hil; @Hilger2], the properties of the heavy mesons in the nuclear matter are studied with the QCD sum rules by taking the leading order approximation for the contributions of the quark condensates. In this article, I take into account the next-to-leading order contributions of the quark condensates, and study the properties of the scalar, pseudoscalar, vector and axialvector heavy mesons in the nuclear matter with the QCD sum rules in a systematic way, and make predictions for the modifications of the masses and decay constants of the heavy mesons in the nuclear matter. Measuring those effects is a long term physics goal based on further theoretical studies on the reaction dynamics and on the exploration of the experimental ability to identify more complicated processes [@CBM; @PANDA]. Furthermore, I study the heavy-meson-nucleon scattering lengths as a byproduct. From the negative or positive sign of the scattering lengths, I can obtain the conclusion qualitatively that the interactions are attractive or repulsive, which favor or disfavor the formations of the heavy-meson-nucleon bound states. For example, the $\Sigma_c(2800)$ and $\Lambda_c(2940)$ can be assigned to be the $S$-wave $DN$ state with $J^P ={\frac{1}{2}}^-$ and the $S$-wave $D^N$ state with $J^P ={\frac{3}{2}}^-$ respectively based on the QCD sum rules [@ZhangDN]. The article is arranged as follows: I study in-medium properties of the heavy mesons with the QCD sum rules in Sec.2; in Sec.3, I present the numerical results and discussions; and Sec.4 is reserved for my conclusions. The properties of the heavy mesons in the nuclear matter with QCD sum rules =========================================================================== I study the scalar, pseudoscalar, vector and axialvector heavy mesons in the nuclear matter with the two-point correlation functions $\Pi(q)$ and $\Pi_{\mu\nu}(q)$, respectively. In the Fermi gas approximation for the nuclear matter, I divide the $\Pi(q)$ and $\Pi_{\mu\nu}(q)$ into the vacuum part $\Pi^0(q)$ and $\Pi^0_{\mu\nu}(q)$ and the static one-nucleon part $\Pi_N(q)$ and $\Pi^N_{\mu\nu}(q)$, and expand the $\Pi_N(q)$ and $\Pi^N_{\mu\nu}(q)$ up to the order ${\mathcal{O}}(\rho_N)$ at relatively low nuclear density [@Drukarev1991; @Hay], $$\begin{aligned} \Pi(q) &=& i\int d^{4}x\ e^{iq \cdot x} \langle T\left\{J(x)J^{\dag}(0)\right\} \rangle_{\rho_N}\nonumber \\ & =& \Pi_{0}(q)+ \frac{\rho_N}{2m_N}T_{N}(q)\, , \nonumber \\ \Pi_{\mu\nu}(q) &=& i\int d^{4}x\ e^{iq \cdot x} \langle T\left\{J_\mu(x)J_\nu^{\dag}(0)\right\} \rangle_{\rho_N} \nonumber\\ &=&\Pi^{0}_{\mu\nu}(q)+ \frac{\rho_N}{2m_N}T^{N}_{\mu\nu}(q)\, , \end{aligned}$$ where the $\rho_N$ is the density of the nuclear matter, and the forward scattering amplitudes $T_{N}(q)$ and $T^{N}_{\mu\nu}(q)$ are defined as $$\begin{aligned} T_{N}(\omega,\mbox{\boldmath $q$}\,) &=&i\int d^{4}x e^{iq\cdot x}\langle N(p)\| T\left\{J(x)J^{\dag}(0)\right\} \|N(p) \rangle\, , \nonumber\\ T^{N}_{\mu\nu}(\omega,\mbox{\boldmath $q$}\,) &=&i\int d^{4}x e^{iq\cdot x}\langle N(p)\| T\left\{J_\mu(x)J_\nu^{\dag}(0)\right\} \|N(p) \rangle\, ,\end{aligned}$$ where the $J(x)$ and $J_\mu(x)$ denote the isospin averaged currents $\eta(x)$, $\eta_5(x)$, $\eta_\mu(x)$ and $\eta_{5\mu}(x)$, respectively, $$\begin{aligned} \eta(x) &=&\eta^\dag(x) =\frac{\bar{c}(x) q(x)+\bar{q}(x) c(x)}{2}\, , \nonumber\\ \eta_{5}(x) &=&\eta_{5}^\dag(x) =\frac{\bar{c}(x)i \gamma_5q(x)+\bar{q}(x)i\gamma_5 c(x)}{2}\,,\nonumber\\ \eta_\mu(x) &=&\eta_\mu^\dag(x) =\frac{\bar{c}(x)\gamma_\mu q(x)+\bar{q}(x)\gamma_\mu c(x)}{2}\, , \nonumber\\ \eta_{5\mu}(x) &=&\eta_{5\mu}^\dag(x) =\frac{\bar{c}(x)\gamma_\mu \gamma_5q(x)+\bar{q}(x)\gamma_\mu\gamma_5 c(x)}{2}\,,\end{aligned}$$ which interpolate the scalar, pseudoscalar, vector and axialvector mesons $D_0$, $D$, $D^$ and $D_1$, respectively. I choose the isospin averaged currents since the $D_0$, $D$, $D^$ and $D_1$ mesons are produced in pairs in the antiproton-nucleon annihilation processes. The $q$ denotes the $u$ or $d$ quark, the $q^{\mu}=(\omega,\mbox{\boldmath $q$}\,)$ is the four-momentum carried by the currents $J(x)$ and $J_\mu(x)$, the $\|N(p)\rangle$ denotes the isospin and spin averaged static nucleon state with the four-momentum $p = (m_N,0)$, and $\langle N(\mbox{\boldmath $p$})\|N(\mbox{\boldmath $p$}')\rangle = (2\pi)^{3} 2p_{0}\delta^{3}(\mbox{\boldmath $p$}-\mbox{\boldmath $p$}')$ [@Hay]. I can decompose the correlation functions $T^{N}_{\mu\nu}(\omega,\mbox{\boldmath $q$}\,)$ as $$\begin{aligned} T^{N}_{\mu\nu}(\omega,\mbox{\boldmath $q$}\,) &=&T_{N}(\omega,\mbox{\boldmath $q$}\,)\left(-g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2}\right)+T^0_N(\omega,\mbox{\boldmath $q$}\,) q_\mu q_\nu\nonumber\\ &&+T^1_N(\omega,\mbox{\boldmath $q$}\,) \left(q_\mu u_\nu+q_\nu u_\mu \right)+ T^2_N(\omega,\mbox{\boldmath $q$}\,) u_\mu u_\nu\, ,\end{aligned}$$ according to Lorentz covariance, where the $T_{N}(\omega,\mbox{\boldmath $q$}\,)$ denotes the contributions of the vector and axialvector charmed mesons, and the $T^{0/1/2}_{N}(\omega,\mbox{\boldmath $q$}\,)$ are irrelevant in the present analysis. In the limit $\mbox{\boldmath $q$}\rightarrow {\bf 0}$, the forward scattering amplitude $T_{N}(\omega,\mbox{\boldmath $q$}\,)$ can be related to the $DN$ ($D_0N$, $D^N$ and $D_1N$) scattering $T$-matrix, $$\begin{aligned} {{\cal T}_{D/D_0/D^/D_1\,N}}(m_{D/D_0/D^/D_1},0) = 8\pi(m_N+m_{D/D_0/D^/D_1})a_{D/D_0/D^/D_1} \, ,\end{aligned}$$ where the $a_{D/D_0/D^/D_1}$ are the $D/D_0/D^/D_1\,N$ scattering lengths. I can parameterize the phenomenological spectral densities $\rho(\omega,0)$ with three unknown parameters $a,\,b$ and $c$ near the pole positions of the charmed mesons $D$, $D_0$, $D^$ and $D_1$ according to Ref.[@Hay], $$\begin{aligned} \rho(\omega,0) &=& -\frac{1}{\pi} \mbox{Im} \left[\frac{{{\cal T}_{D/D_0N}}(\omega,{\bf 0})}{\left(\omega^{2}- m_{D/D_0}^2+i\varepsilon\right)^{2}} \right]\frac{f_{D/D_0}^2m_{D/D_0}^4}{m_c^2}+ \cdots \,, \nonumber \\ &=& a\,\frac{d}{d\omega^2}\delta\left(\omega^{2}-m_{D/D_0}^2\right) + b\,\delta\left(\omega^{2}-m_{D/D_0}^2\right) + c\,\delta\left(\omega^{2}-s_{0}\right)\, ,\end{aligned}$$ for the pseudoscalar and scalar currents $\eta_5(x)$ and $\eta(x)$, $$\begin{aligned} \rho(\omega,0) &=& -\frac{1}{\pi} \mbox{Im} \left[\frac{{{\cal T}_{D^/D_1N}}(\omega,{\bf 0})}{\left(\omega^{2}- m_{D^/D_1}^2+i\varepsilon\right)^{2}} \right]f_{D^/D_1}^2m_{D^/D_1}^2+ \cdots \,, \nonumber\\ &=& a\,\frac{d}{d\omega^2}\delta\left(\omega^{2}-m_{D^/D_1}^2\right) + b\,\delta\left(\omega^{2}-m_{D^/D_1}^2\right) + c\,\delta\left(\omega^{2}-s_{0}\right)\, ,\end{aligned}$$ for the vector and axialvector currents $\eta_\mu(x)$ and $\eta_{5\mu}(x)$. Now the hadronic correlation functions $\Pi(\omega,0)$ and $\Pi_{\mu\nu}(\omega,0)$ at the phenomenological side can be written as $$\begin{aligned} \Pi(\omega,0)&=&\frac{\left( f_{D/D_0}+\delta f_{D/D_0}\right)^2\left( m_{D/D_0}+\delta m_{D/D_0}\right)^4}{m_c^2}\frac{1}{\left(m_{D/D_0}+\delta m_{D/D_0}\right)^2-\omega^2}+\cdots \nonumber\\ &=&\frac{ f_{D/D_0}^2m_{D/D_0}^4}{m_c^2}\frac{1}{m_{D/D_0}^2-\omega^2}+\cdots\nonumber\\ &&+\frac{\rho_N}{2m_N}\left[\frac{a}{\left(m_{D/D_0}^2-\omega^2\right)^2}+\frac{b}{m_{D/D_0}^2-\omega^2} +\cdots \right]\, ,\end{aligned}$$ $$\begin{aligned} \Pi_{\mu\nu}(\omega,0)&=&\left( f_{D^/D_1}+\delta f_{D^/D_1}\right)^2\left( m_{D^/D_1}+\delta m_{D^/D_1}\right)^2\frac{1}{\left(m_{D^/D_1}+\delta m_{D^/D_1}\right)^2-\omega^2}\nonumber\\ &&\left( -g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2}\right) +\cdots \, ,\nonumber\\ &=&f_{D^/D_1}^2m_{D^/D_1}^2\frac{1}{m_{D^/D_1}^2-\omega^2}\left( -g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2}\right)+\cdots\nonumber\\ &&+\frac{\rho_N}{2m_N}\left[\left(\frac{a}{\left(m_{D^/D_1}^2-\omega^2\right)^2}+\frac{b}{m_{D^/D_1}^2-\omega^2} +\cdots\right)\left( -g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2}\right)+\cdots \right] \, .\nonumber\\\end{aligned}$$ In Eqs.(6-7), the first term denotes the double-pole term, and corresponds to the on-shell effect of the $T$-matrix, $$\begin{aligned} a&=&-8\pi(m_N+m_{D/D_0})a_{D/D_0}\frac{f_{D/D_0}^2m_{D/D_0}^4}{m_c^2}\, ,\end{aligned}$$ for the currents $\eta_5(x)$ and $\eta(x)$ and $$\begin{aligned} a&=&-8\pi(m_N+m_{D^/D_1})a_{D^/D_1}f_{D^/D_1}^2m_{D^/D_1}^2\, ,\end{aligned}$$ for the currents $\eta_\mu(x)$ and $\eta_{5\mu}(x)$; the second term denotes the single-pole term, and corresponds to the off-shell effect of the $T$-matrix; the third term denotes the continuum term or the remaining effects, where the $s_{0}$ is the continuum threshold parameter. In general, the continuum contributions are approximated by $\rho_{QCD}(\omega,0)\theta(\omega^2-s_0)$, where the $\rho_{QCD}(\omega,0)$ are the perturbative QCD spectral densities, and $\theta(x)=1$ for $x\geq 0$, else $\theta(x)=0$. In this article, the QCD spectral densities are of the type $\delta(\omega^2-m_Q^2)$, which include both the ground state and continuum state contributions, I have attributed the excited state contributions to the continuum state contributions, so the collective continuum state contributions can be approximated as $c\,\delta(\omega^2-s_0)$, then I obtain the result $c/\left(s_0-\omega^2\right)$ in the hadronic representation, see Eq.(15). The doublet $\left(D(2550), D(2600)\right)$ or $\left(D_J(2580), D^_J (2650)\right)$ is assigned to be the first radial excited state of the doublet $(D,D^)$ [@WangHQET]. The single-pole contributions come from the doublet $\left(D(2550), D(2600)\right)$ or $\left(D_J(2580), D^_J (2650)\right)$ are of the form $1/\left(m_{D(2550)/D(2600)}^2-\omega^2\right)$, so the approximation $c/\left(s_0-\omega^2\right)$ is reasonable. Then the shifts of the masses and decay constants of the charmed-mesons can be approximated as $$\begin{aligned} \delta m_{D/D_0/D^/D_1} &=&2\pi\frac{m_{N}+m_{D/D_0/D^/D_1}}{m_Nm_{D/D_0/D^/D_1}}\rho_N a_{D/D_0/D^/D_1}\, ,\end{aligned}$$ $$\begin{aligned} \delta f_{D/D_0}&=&\frac{m_c^2}{2f_{D/D_0}m_{D/D_0}^4}\left(\frac{b\rho_N}{2m_N}-\frac{4f_{D/D_0}^2m_{D/D_0}^3\delta m_{D/D_0}}{m_c^2} \right) \, , \nonumber\\ \delta f_{D^/D_1}&=&\frac{1}{2f_{D^/D_1}m_{D^/D_1}^2}\left(\frac{b\rho_N}{2m_N}-2f_{D^/D_1}^2m_{D^/D_1}\delta m_{D^/D_1} \right) \, .\end{aligned}$$ In calculations, I have used the following definitions for the decay constants of the heavy mesons, $$\begin{aligned} \langle 0\|\eta(0)\|D_0+\bar{D}_0\rangle &=&\frac{f_{D_0}m^2_{D_0}}{m_c} \,,\nonumber\\ \langle 0\|\eta_5(0)\|D+\bar{D}\rangle &=&\frac{f_{D}m^2_{D}}{m_c} \,,\nonumber\\ \langle 0\|\eta_\mu(0)\|D^+\bar{D}^\rangle &=&f_{D^}m_{D^}\epsilon_\mu\,,\nonumber\\ \langle 0\|\eta_{5\mu}(0)\|D_1+\bar{D}_1\rangle &=&f_{D_1}m_{D_1}\epsilon_{\mu}\,,\end{aligned}$$ with summations of the polarization vectors $\sum_\lambda \epsilon_\mu(\lambda,q)\epsilon^_\nu(\lambda,q)=-g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2}$. In the low energy limit $\omega\rightarrow 0$, the $T_{N}(\omega,{\bf 0})$ is equivalent to the Born term $T_{N}^{\rm Born}(\omega,{\bf 0})$. Now I take into account the Born terms at the phenomenological side, $$\begin{aligned} T_{N}(\omega^2)&=&T_{N}^{\rm Born}(\omega^2)+\frac{a}{\left(m_{D/D_0/D^/D_1}^2-\omega^2\right)^2}+\frac{b}{m_{D/D_0/D^/D_1}^2-\omega^2}+\frac{c}{s_0-\omega^2} \, ,\end{aligned}$$ with the constraint $$\begin{aligned} \frac{a}{m_{D/D_0/D^/D_1}^4}+\frac{b}{m_{D/D_0/D^/D_1}^2}+\frac{c}{s_0}&=&0 \, .\end{aligned}$$ The contributions from the intermediate spin-$\frac{3}{2}$ charmed baryon states are zero in the soft-limit $q_\mu \to 0$ [@Wangzg], and I only take into account the intermediate spin-$\frac{1}{2}$ charmed baryon states in calculating the Born terms, $$\begin{aligned} \left(D/D_0/D^/D_1\right)^0(c\bar{u})+p(uud)\ \mbox{or}\ n(udd)&\longrightarrow& \Lambda_c^+,\Sigma_c^+(cud)\ \mbox{or}\ \Sigma_c^0(cdd) \, ,\nonumber\\ \left(D/D_0/D^/D_1\right)^+(c\bar{d})+p(uud)\ \mbox{or}\ n(udd) &\longrightarrow& \Sigma_c^{++}(cuu)\ \mbox{or}\ \Lambda_c^+,\Sigma_c^+(cud)\, ,\end{aligned}$$ where $M_{\Lambda_c}=2.286\,\rm{GeV}$ and $M_{\Sigma_c}=2.454\,\rm{GeV}$ [@PDG]. I can take $M_H\approx 2.4\,\rm{GeV}$ as the average value, where the $H$ means either $\Lambda_c^+$, $\Sigma_c^+$, $\Sigma_c^{++}$ or $\Sigma_c^0$. In the case of the bottom baryons, I take the approximation $M_H=\frac{M_{\Sigma_b}+M_{\Lambda_b}}{2}\approx 5.7\,\rm{GeV}$ [@PDG]. I write down the Feynman diagrams and calculate the Born terms directly, and obtain the results, $$\begin{aligned} T_{N}^{\rm Born}(\omega,{\bf0})&=&\frac{2m_N(m_H+m_N)} {\left[\omega^2-(m_H+m_N)^2\right]\left[\omega^2-m_{D}^2\right]^2}\left(\frac{f_{D}m_{D}^2g_{DNH}}{m_c}\right)^2\,,\end{aligned}$$ for the current $\eta_5(x)$, $$\begin{aligned} T_{N}^{\rm Born}(\omega,{\bf0})&\to&T_{N}^{\rm Born}(\omega,{\bf0})\,\left( {\rm with}\,\,\, m_N \to -m_N\, , \,\,\, D\to D_0 \right)\,,\end{aligned}$$ for the current $\eta(x)$, $$\begin{aligned} T_{N}^{\rm Born}(\omega,{\bf0})&=&\frac{2m_N(m_H+m_N)} {\left[\omega^2-(m_H+m_N)^2\right]\left[\omega^2-m_{D^}^2\right]^2}\left(f_{D^}m_{D^}g_{D^NH}\right)^2\,,\end{aligned}$$ for the current $\eta_\mu(x)$, $$\begin{aligned} T_{N}^{\rm Born}(\omega,{\bf0})&\to&T_{N}^{\rm Born}(\omega,{\bf0})\,\left( {\rm with}\,\,\, m_N \to -m_N\, , \,\,\, D^\to D_1 \right)\,,\end{aligned}$$ for the current $\eta_{5\mu}(x)$, where the $g_{D/D_0/D^/D_1NH}$ denote the strong coupling constants $g_{D/D_0/D^/D_1N\Lambda_c}$ and $g_{D/D_0/D^/D_1N\Sigma_c}$. On the other hand, there are no inelastic channels for the $(\bar{D}/\bar{D}_0/\bar{D}^/\bar{D}_1)^0 N$ and $(\bar{D}/\bar{D}_0/\bar{D}^/\bar{D}_1)^- N$ interactions, and $T_{N}^{\rm Born}(0)=0$. In calculations, I have used the following definitions for the hadronic coupling constants, $$\begin{aligned} \langle\Lambda_c/\Sigma_c(p-q)\|D(-q)N(p)\rangle &=&g_{\Lambda_c/\Sigma_cDN}\overline{U}_{\Lambda_c/\Sigma_c}(p-q) i\gamma_5 U_N(p)\,,\nonumber\\ \langle\Lambda_c/\Sigma_c(p-q)\|D_0(-q)N(p)\rangle &=& g_{\Lambda_c/\Sigma_cD_0N}\overline{U}_{\Lambda_c/\Sigma_c}(p-q) U_N(p)\,,\nonumber\\ \langle\Lambda_c/\Sigma_c(p-q)\|D^(-q)N(p)\rangle &=&\overline{U}_{\Lambda_c/\Sigma_c}(p-q)\left( g_{\Lambda_c/\Sigma_cD^N}\!\not\!{\epsilon}+i\frac{g^T_{\Lambda_c/\Sigma_cD^N}}{M_N+M_{\Lambda_c/\Sigma_c}}\sigma^{\alpha\beta}\epsilon_\alpha q_\beta\right)U_N(p)\,,\nonumber\\ \langle\Lambda_c/\Sigma_c(p-q)\|D_1(-q)N(p)\rangle &=&\overline{U}_{\Lambda_c/\Sigma_c}(p-q)\left( g_{\Lambda_c/\Sigma_cD_1N}\!\not\!{\epsilon}+i\frac{g^T_{\Lambda_c/\Sigma_cD_1N}}{M_N+M_{\Lambda_c/\Sigma_c}}\sigma^{\alpha\beta}\epsilon_\alpha q_\beta\right)\gamma_5U_N(p)\,,\nonumber\\ \end{aligned}$$ where the $U_N$ and $\overline{U}_{\Lambda_c/\Sigma_c}$ are the Dirac spinors of the nucleon and the charmed baryons $\Lambda_c/\Sigma_c$, respectively. In the limit $q_\mu \to 0$, the strong coupling constants $g_{\Lambda_c/\Sigma_cD^N}^T$ and $g_{\Lambda_c/\Sigma_cD_1N}^T$ have no contributions. For example, near the thresholds, the $D^N$ can translate to the $DN$, $D^N$, $\pi\Sigma_c$, $\eta\Lambda_c$, etc, we can take into account the intermediate baryon-meson loops or the re-scattering effects with the Bethe-Salpeter equation to obtain the full $D^N \to D^N$ scattering amplitude, and generate higher baryon states dynamically [@DN-BSE-Negative]. We can saturate the full $D^N\to D^N$ scattering amplitude with the tree-level Feynman diagrams describing the exchanges of the higher resonances $\Lambda_c(2595)$, $\Sigma_c({\frac{1}{2}}^-)$, etc. While in other coupled-channels analysis, the $\Lambda_c(2595)$ emerges as a $DN$ quasi-bound state rather than a $D^N$ quasi-bound state [@DN-BSE-Negative]. The translations $D^N $ to the ground states $\Lambda_c $ and $\Sigma_c$ are favored in the phase-space, as the $\Lambda_c(2595)$ and $\Sigma_c({\frac{1}{2}}^-)$ with $J^P={\frac{1}{2}}^-$ have the average mass $m_{H'}\approx 2.7\,\rm{GeV}$ [@PDG; @Wang-Negative]. In fact, $m_{H'}^2> s_0$, I can absorb the high resonances into the continuum states in case the high resonances do not dominate the QCD sum rules. In calculations, I observe that the mass-shift $\delta m_{D^}$ does not sensitive to contributions of the ground states $\Lambda_c $ and $\Sigma_c$, the contributions from the spin-$\frac{1}{2}$ higher resonances maybe even smaller. In this article, I neglect the intermediate baryon-meson loops, their effects are absorbed into continuum contributions. At the low nuclear density, the condensates $\langle{\cal {O}}\rangle_{\rho_N}$ in the nuclear matter can be approximated as $$\begin{aligned} \langle{\cal{O}}\rangle_{\rho_N} &=&\langle{\cal{O}}\rangle+\frac{\rho_N}{2m_N}\langle {\cal{O}}\rangle_N \, ,\end{aligned}$$ based on the Fermi gas model, where the $\langle{\cal{O}}\rangle$ and $\langle {\cal{O}}\rangle_N$ denote the vacuum condensates and nuclear matter induced condensates, respectively [@Drukarev1991]. I neglect the terms proportional to $p_F^4$, $p_F^5$, $p_F^6$, $\cdots$ at the normal nuclear matter with the saturation density $\rho_N=\rho_0=\frac{2p_F^3}{3\pi^2}$, as the Fermi momentum $p_F=0.27\,\rm{GeV}$ is a small quantity [@Drukarev1991]. I carry out the operator product expansion to the nuclear matter induced condensates $\frac{\rho_N}{2m_N}\langle {\cal{O}}\rangle_N$ up to dimension-5 at the large space-like region in the nuclear matter, and take into account the one-loop corrections to the quark condensate $\langle \bar{q}q\rangle_N$. I insert the following term $$\begin{aligned} \frac{1}{2!}\,\, ig_s \int d^D y \bar{\psi}(y)\gamma^\mu \psi(y)\frac{\lambda^a}{2}G^a_\mu(y)\,\, ig_s \int d^D z \bar{\psi}(z)\gamma^\nu \psi(z)\frac{\lambda^b}{2}G^b_\nu(z) \, ,\end{aligned}$$ with the dimension $D=4-2\epsilon$, into the correlation functions $T_N(q)$ and $T^N_{\mu\nu}(q)$ firstly, where the $\psi$ denotes the quark fields, the $G^a_\mu$ denotes the gluon field, the $\lambda^a$ denotes the Gell-Mann matrix, then contract the quark fields with Wick theorem, and extract the quark condensate $\langle\bar{q}{q}\rangle_N$ according to the formula $\langle N\|q_\alpha^i q_\beta^j\|N\rangle=-\frac{1}{12}\langle \bar{q}q\rangle_N\delta_{ij}\delta_{\alpha\beta}$ to obtain the perturbative corrections $\alpha_s\langle\bar{q}{q}\rangle_N$, where the $i$ and $j$ are color indexes and the $\alpha$ and $\beta$ are Dirac spinor indexes. There are six Feynman diagrams make contributions, see Fig.1. Now I calculate the first diagram explicitly for the current $\eta_5(x)$ in Fig.1, $$\begin{aligned} 2T^{(\alpha_s,1)}_N(q^2) &=&-\frac{{\rm Tr}(\frac{\lambda^a}{2} \frac{\lambda^b}{2})\langle \bar{q}q\rangle_Ng_s^2 \mu^{2\epsilon}}{12} \frac{i}{(2\pi)^D}\int d^Dk{\rm Tr}\left\{ i\gamma_5 \frac{i}{\!\not\! {k}}\gamma^\alpha \gamma^\beta \frac{i}{\!\not\! {k}} i \gamma_5 \frac{i}{\!\not\! {q}+\!\not\! {k}-m_c} \frac{-i\delta_{ab}g_{\alpha\beta}}{k^2}\right\} \nonumber\\ &=&-\frac{4Dm_c\langle \bar{q}q\rangle_Ng_s^2 \mu^{2\epsilon}}{3(2\pi)^D}i\frac{\partial}{\partial t}\frac{(-2\pi i)^2}{2\pi i} \int_{m_c^2}^{\infty} ds \frac{\int d^Dk \delta \left( k^2-t\right)\delta\left( (k+q)^2-m_c^2\right)}{s-q^2}\mid_{t=0} \nonumber\\ &=& -\frac{Dm_c\langle \bar{q}q\rangle_Ng_s^2 \mu^{2\epsilon}\left[1+\epsilon(\log4\pi-\gamma_E) \right]}{12\pi^2} \int_{m_c^2}^{\infty} ds \frac{1}{s-q^2} \frac{s+m_c^2}{s^{1-\epsilon}(s-m_c^2)^{1+2\epsilon}} \, , \end{aligned}$$ where I have used Cutkosky’s rule to obtain the QCD spectral density. There exists infrared divergence at the end point $s=m_c^2$. It is difficult to carry out the integral over $s$, I can perform the Borel transform $B_{M^2}$ firstly, then carry out the integral over $s$, $$\begin{aligned} B_{M^2}2T^{(\alpha_s,1)}_N(q^2)&=& -\frac{Dm_c\langle \bar{q}q\rangle_Ng_s^2 \mu^{2\epsilon}\left[1+\epsilon(\log4\pi-\gamma_E) \right]}{12\pi^2M^2} \int_{m_c^2}^{\infty} ds \frac{s+m_c^2}{s^{1-\epsilon}(s-m_c^2)^{1+2\epsilon}}\exp\left( -\frac{s}{M^2}\right) \nonumber\\ &=&\frac{m_c\langle \bar{q}q\rangle_Ng_s^2}{3\pi^2M^2}\exp\left(-\frac{m_c^2}{M^2} \right)\left( \frac{1}{\epsilon}-\log4\pi+\gamma_E\right)+\frac{m_c\langle \bar{q}q\rangle_Ng_s^2}{3\pi^2M^2}\Gamma\left(0,\frac{m_c^2}{M^2} \right) \nonumber\\ &&-\frac{m_c\langle \bar{q}q\rangle_Ng_s^2}{6\pi^2M^2}\exp\left(-\frac{m_c^2}{M^2} \right)+\frac{m_c\langle \bar{q}q\rangle_Ng_s^2}{3\pi^2M^2}\exp\left(-\frac{m_c^2}{M^2} \right)\log\frac{m_c^2\mu^2}{M^4} \, , \end{aligned}$$ where $$\begin{aligned} \Gamma(0,x)&=&e^{-x}\int_0^\infty dt \frac{1}{t+x}e^{-t} \, .\end{aligned}$$ Other diagrams are calculated analogously, I regularize the divergences in $D=4-2\epsilon$ dimension, then remove the ultraviolet divergences through renormalization and absorb the infrared divergences into the quark condensate $\langle \bar{q}q\rangle_N$. ![The perturbative $\mathcal{O}(\alpha_s)$ corrections to the quark condensate $\langle\bar{q}q\rangle_N$. ](qq-aphs.eps){width="14cm"} I calculate the contributions of other condensates at the tree level, the calculations are straightforward and cumbersome. In calculations, I use the following formulas, $$\begin{aligned} \langle q_{\alpha}(x)\bar{q}_{\beta}(0)\rangle_N&=&-\frac{1}{4}\left[\left(\langle\bar{q}q\rangle_N+x^{\mu}\langle\bar{q}D_{\mu}q\rangle_N +\frac{1}{2}x^{\mu}x^{\nu}\langle\bar{q}D_{\mu}D_{\nu}q\rangle_N +\cdots\right)\delta_{\alpha\beta}\right.\nonumber \\ &&\left.+\left(\langle\bar{q}\gamma_{\lambda}q\rangle_N+x^{\mu}\langle\bar{q} \gamma_{\lambda}D_{\mu} q\rangle_N +\frac{1}{2}x^{\mu}x^{\nu}\langle\bar{q}\gamma_{\lambda}D_{\mu}D_{\nu} q\rangle_N +\cdots\right)\gamma^{\lambda}_{\alpha\beta} \right] \, ,\end{aligned}$$ and $$\begin{aligned} \langle g_{s}q^i_{\alpha}\bar{q}^j_{\beta}G_{\mu\nu}^{a}\rangle_N&=&-\frac{1}{96} \frac{\lambda^a_{ij}}{2}\left\{\langle g_{s}\bar{q}\sigma Gq\rangle_N\left[\sigma_{\mu\nu}+i(u_{\mu}\gamma_{\nu}-u_{\nu}\gamma_{\mu }) \!\not\! {u}\right]_{\alpha\beta} +\langle g_{s}\bar{q}\!\not\! {u}\sigma Gq\rangle_N\right.\nonumber\\ &&\left.\left[\sigma_{\mu\nu}\!\not\! {u}+i(u_{\mu}\gamma_{\nu}-u_{\nu}\gamma_{\mu} )\right]_{\alpha\beta} -4\langle\bar{q}u D u D q\rangle_N\left[\sigma_{\mu\nu}+2i(u_{\mu}\gamma_{\nu}-u_{\nu}\gamma_{\mu} )\!\not\! {u}\right]_{\alpha\beta}\right\} \, , \nonumber\\\end{aligned}$$ where $D_\mu=\partial_\mu-ig_s\frac{\lambda^a}{2}G^a_\mu$, $$\begin{aligned} \label{ } \langle\bar{q}\gamma_{\mu}q\rangle_N&=&\langle\bar{q}\!\not\!{u}q\rangle_N u_{\mu} \, , \nonumber \\ \langle\bar{q}D_{\mu}q\rangle_N&=&\langle\bar{q}u D q\rangle_N u_{\mu}=0\, , \nonumber \\ \langle\bar{q}\gamma_{\mu}D_{\nu}q\rangle_N&=&\frac{4}{3}\langle\bar{q} \!\not\! {u}u D q\rangle_N\left(u_{\mu}u_{\nu}-\frac{1}{4}g_{\mu\nu}\right) \, , \nonumber \\ \langle\bar{q}D_{\mu}D_{\nu}q\rangle_N&=&\frac{4}{3}\langle\bar{q} u D u D q\rangle_N\left(u_{\mu}u_{\nu}-\frac{1}{4}g_{\mu\nu}\right) -\frac{1}{6} \langle g_{s}\bar{q}\sigma Gq\rangle_N\left(u_{\mu}u_{\nu}-g_{\mu\nu}\right) \, , \nonumber \\ \langle\bar{q}\gamma_{\lambda}D_{\mu}D_{\nu}q\rangle_N&=&2\langle\bar{q} \!\not\! {u}u D u D q\rangle_N\left[u_{\lambda}u_{\mu}u_{\nu} -\frac{1}{6} \left(u_{\lambda}g_{\mu\nu}+u_{\mu}g_{\lambda\nu}+u_{\nu}g_{\lambda\mu}\right)\right] \nonumber\\ &&-\frac{1}{6} \langle g_{s}\bar{q}\!\not\! {u}\sigma Gq\rangle_N(u_{\lambda}u_{\mu}u_{\nu}-u_{\lambda}g_{\mu\nu}) \, ,\end{aligned}$$ and $$\langle G_{\alpha\beta}^{a}G_{\mu\nu}^{b}\rangle_N=\frac{\delta^{ab}}{96} \langle GG\rangle_N\left(g_{\alpha\mu}g_{\beta\nu}-g_{\alpha\nu}g_{\beta\mu}\right)+O\left(\langle \textbf{E}^{2}+\textbf{B}^{2}\rangle_N\right).$$ Once analytical results at the level of quark-gluon degree’s of freedom are obtained, then I set $\omega^2=q^2$, and take the quark-hadron duality below the continuum threshold $s_0$, and perform the Borel transform with respect to the variable $Q^2=-\omega^2$, finally obtain the following QCD sum rules: $$\begin{aligned} a\, C_a+b\, C_b &=&C_f \, ,\end{aligned}$$ $$\begin{aligned} C_a &=&\frac{1}{M^2}\exp\left(-\frac{m_{D}^2}{M^2}\right)-\frac{s_0}{m_{D}^4}\exp\left(-\frac{s_0}{M^2}\right) \, ,\nonumber\\ C_b&=&\exp\left(-\frac{m_{D}^2}{M^2}\right)-\frac{s_0}{m_{D}^2}\exp\left(-\frac{s_0}{M^2}\right) \, ,\end{aligned}$$ $$\begin{aligned} C_f&=& \frac{2m_N(m_H+m_N)}{(m_H+m_N)^2-m_{D}^2}\left(\frac{f_{D}m_{D}^2g_{DNH}}{m_c}\right)^2\left\{ \left[\frac{1}{M^2}-\frac{1}{m_{D}^2-(m_H+m_N)^2}\right] \exp\left(-\frac{m_{D}^2}{M^2}\right)\right.\nonumber\\ &&\left.+\frac{1}{(m_H+m_N)^2-m_{D}^2}\exp\left(-\frac{(m_H+m_N)^2}{M^2}\right)\right\}-\frac{m_c\langle\bar{q}q\rangle_N}{2}\left\{1+\frac{\alpha_s}{\pi} \left[ 6-\frac{4m_c^2}{3M^2} \right.\right.\nonumber\\ &&\left.\left.-\frac{2}{3}\left( 1-\frac{m_c^2}{M^2}\right)\log\frac{m_c^2}{\mu^2}-2\Gamma\left(0,\frac{m_c^2}{M^2}\right)\exp\left( \frac{m_c^2}{M^2}\right) \right]\right\}\exp\left(- \frac{m_c^2}{M^2}\right) \nonumber\\ &&+\frac{1}{2}\left\{-2\left(1-\frac{m_c^2}{M^2}\right)\langle q^\dag i D_0q\rangle_N +\frac{4m_c }{M^2}\left(1-\frac{m_c^2}{2M^2}\right)\langle \bar{q} i D_0 i D_0q\rangle_N+\frac{1}{12}\langle\frac{\alpha_sGG}{\pi}\rangle_N\right\} \nonumber\\ &&\exp\left(- \frac{m_c^2}{M^2}\right)\, ,\end{aligned}$$ for the current $\eta_5(x)$, $$\begin{aligned} C_i &\to& C_i\left( {\rm with}\,\,\, m_N \to -m_N\, , \,\,\,m_c \to -m_c\, , \,\,\,D \to D_0\right)\, , \end{aligned}$$ for the current $\eta(x)$, $$\begin{aligned} C_a &=&\frac{1}{M^2}\exp\left(-\frac{m_{D^}^2}{M^2}\right)-\frac{s_0}{m_{D^}^4}\exp\left(-\frac{s_0}{M^2}\right) \, ,\nonumber\\ C_b&=&\exp\left(-\frac{m_{D^}^2}{M^2}\right)-\frac{s_0}{m_{D^}^2}\exp\left(-\frac{s_0}{M^2}\right) \, ,\end{aligned}$$ $$\begin{aligned} C_f&=& \frac{2m_N(m_H+m_N)}{(m_H+m_N)^2-m_{D^}^2}\left(f_{D^}m_{D^}g_{D^NH}\right)^2\left\{ \left[\frac{1}{M^2}-\frac{1}{m_{D^}^2-(m_H+m_N)^2}\right] \exp\left(-\frac{m_{D^}^2}{M^2}\right)\right.\nonumber\\ &&\left.+\frac{1}{(m_H+m_N)^2-m_{D^}^2}\exp\left(-\frac{(m_H+m_N)^2}{M^2}\right)\right\}-\frac{m_c\langle\bar{q}q\rangle_N}{2}\left\{1+\frac{\alpha_s}{\pi} \left[ \frac{8}{3}-\frac{4m_c^2}{3M^2} \right.\right.\nonumber\\ &&\left.\left.+\frac{2}{3}\left( 2+\frac{m_c^2}{M^2}\right)\log\frac{m_c^2}{\mu^2}-\frac{2m_c^2}{3M^2}\Gamma\left(0,\frac{m_c^2}{M^2}\right)\exp\left( \frac{m_c^2}{M^2}\right) \right]\right\}\exp\left(- \frac{m_c^2}{M^2}\right) \nonumber\\ &&+\frac{1}{2}\left\{-\frac{4\langle q^\dag i D_0q\rangle_N}{3} +\frac{2m_c^2\langle q^\dag i D_0q\rangle_N}{M^2}+\frac{2m_c\langle\bar{q}g_s\sigma Gq\rangle_N}{3M^2}+\frac{16m_c\langle \bar{q} i D_0 i D_0q\rangle_N}{3M^2}\right.\nonumber\\ &&\left.-\frac{2m_c^3\langle \bar{q} i D_0 i D_0q\rangle_N}{M^4}-\frac{1}{12}\langle\frac{\alpha_sGG}{\pi}\rangle_N\right\}\exp\left(- \frac{m_c^2}{M^2}\right)\, ,\end{aligned}$$ for the current $\eta_\mu(x)$, $$\begin{aligned} C_i &\to& C_i\left( {\rm with}\,\,\, m_N \to -m_N\, , \,\,\,m_c \to -m_c\, , \,\,\,D^* \to D_1\right)\, , \end{aligned}$$ for the current $\eta_{5\mu}(x)$, where $i=a,b,f$. In this article, I neglect the contributions from the heavy quark condensates $\langle \bar{Q}Q\rangle$, $\langle \bar{Q}Q\rangle=-\frac{1}{12\pi m_Q}\langle \frac{\alpha_s GG}{\pi} \rangle$ up to the order $\mathcal{O}(\alpha_s)$ (here I count the condensate $ \langle \frac{\alpha_s GG}{\pi} \rangle$ as of the order $\mathcal{O}(\alpha_s)$), the heavy quark condensates have practically no effect on the polarization functions, for detailed discussions about this subject, one can consult Ref.[@PRT85]. In Ref.[@QQcond], Buchheim, Hilger and Kampfer study the contributions of the condensates involve the heavy quarks in details, the results indicate that those condensates are either suppressed by the heavy quark mass $m_Q$ or by the additional factor $\frac{\alpha_s}{4\pi}$ (or $g_s^2/(4\pi)^2$). Neglecting the in-medium effects on the heavy quark condensates cannot affect the predictions remarkably, as the main contributions come from the terms $\langle \bar{q}q\rangle_N$. Differentiate above equation with respect to $\tau=\frac{1}{M^2}$, then eliminate the parameter $b$ ($a$), I can obtain the QCD sum rules for the parameter $a$ ($b$), $$\begin{aligned} a&=&\frac{C_f\left(-\frac{d}{d\tau}\right)C_b-C_b\left(-\frac{d}{d\tau}\right)C_f}{C_a\left(-\frac{d}{d\tau}\right)C_b-C_b\left(-\frac{d}{d\tau}\right)C_a}\, , \nonumber\\ b&=&\frac{C_f\left(-\frac{d}{d\tau}\right)C_a-C_a\left(-\frac{d}{d\tau}\right)C_f}{C_b\left(-\frac{d}{d\tau}\right)C_a-C_a\left(-\frac{d}{d\tau}\right)C_b}\, . \end{aligned}$$ With the simple replacements $m_c \to m_b$, $D/D_0/D^/D_1 \to B/B_0/B^/B_1$, $\Lambda_c \to \Lambda_b$ and $\Sigma_c \to \Sigma_b$, I can obtain the corresponding the QCD sum rules for the bottom mesons in the nuclear matter. Numerical results and discussions ================================= At the normal nuclear matter with the saturation density $\rho_N=\rho_0=\frac{2p_F^3}{3\pi^2}$, where the Fermi momentum $p_F=0.27\,\rm{GeV}$ is a small quantity, the condensates $\langle{\cal {O}}\rangle_{\rho_N}$ in the nuclear matter can be approximated as $\langle{\cal{O}}\rangle_{\rho_N} =\langle{\cal{O}}\rangle+\frac{\rho_N}{2m_N}\langle {\cal{O}}\rangle_N $, the terms proportional to $p_F^4$, $p_F^5$, $p_F^6$, $\cdots$ can be neglected safely, where the $\langle{\cal{O}}\rangle=\langle0\|{\cal{O}}\|0\rangle$ and $\langle {\cal{O}}\rangle_N=\langle N\| {\cal{O}}\|N\rangle$ denote the vacuum condensates and nuclear matter induced condensates, respectively [@Drukarev1991]. The input parameters at the QCD side are taken as $\rho_N=(0.11\,\rm{GeV})^3$, $\langle\bar{q} q\rangle_N={\sigma_N \over m_u+m_d } (2m_N)$, $\langle\frac{\alpha_sGG}{\pi}\rangle_N= - 0.65 \,{\rm {GeV}} (2m_N)$, $\sigma_N=45\,\rm{MeV}$, $m_u+m_d=12\,\rm{MeV}$, $\langle q^\dagger iD_0 q\rangle_N=0.18 \,{\rm{GeV}}(2m_N)$, $\langle\bar{q}g_s\sigma G q\rangle_N=3.0\,{\rm GeV}^2(2m_N) $, $\langle \bar{q} iD_0iD_0 q\rangle_N+{1\over8}\langle\bar{q}g_s\sigma G q\rangle_N=0.3\,{\rm{GeV}}^2(2m_N)$, $m_N=0.94\,\rm{GeV}$ [@C-parameter], $m_c=(1.3\pm0.1)\,\rm{GeV}$, $m_b=(4.7\pm0.1)\,\rm{GeV}$, $\alpha_s=0.45$ and $\mu=1\,\rm{GeV}$. If we take the normalization $\langle N(\mbox{\boldmath $p$})\|N(\mbox{\boldmath $p$}')\rangle = (2\pi)^{3} \delta^{3}(\mbox{\boldmath $p$}-\mbox{\boldmath $p$}')$, then $\langle{\cal{O}}\rangle_{\rho_N} =\langle{\cal{O}}\rangle+\rho_N\langle {\cal{O}}\rangle_N $, the unit $2m_N$ in the brackets in the values of the condensates $\langle\bar{q} q\rangle_N$, $\langle\frac{\alpha_sGG}{\pi}\rangle_N$, $\cdots$ disappears. I choose the values of the nuclear matter induced condensates determined in Ref.[@C-parameter], which are still widely used in the literatures. Although the values of some condensates are updated, those condensates are irrelevant to the present work. The updates focus on the four-quark condensate [@update-4q]. In this article, I take into account the condensates up to dimension-5, the four-quark condensates have no contributions, the dominant contributions come from the nuclear matter induced condensate $\langle\bar{q} q\rangle_N$, $\langle\bar{q} q\rangle_N={\sigma_N \over m_u+m_d } (2m_N)$. The value $m_u+m_d=12\,\rm{MeV}$ is obtained from the famous Gell-Mann-Oakes-Renner relation at the energy scale $\mu=1\,\rm{GeV}$, while the value $\sigma_N=45\,\rm{MeV}$ is still widely used [@update-4q]. The parameters at the hadronic side are taken as $m_D=1.870\,\rm{GeV}$, $m_B=5.280\,\rm{GeV}$, $m_{D_0}=2.355\,\rm{GeV}$, $m_{B_0}=5.740\,\rm{}GeV$, $m_{D^}=2.010\,\rm{GeV}$, $m_{B^}=5.325\,\rm{GeV}$, $m_{D_1}=2.420\,\rm{GeV}$, $m_{B_1}=5.750\,\rm{}GeV$, $f_{D}=0.210\,\rm{GeV}$, $f_B=0.190\,\rm{GeV}$, $f_{D_0}=0.334 \frac{m_c}{m_{D_0}}\,\rm{GeV}$, $f_{B_0}=0.280\frac{m_b}{m_{B_0}}\,\rm{GeV}$, $f_{D^}=0.270\,\rm{GeV}$, $f_{B^}=0.195\,\rm{GeV}$, $f_{D_1}=0.305\,\rm{GeV}$, $f_{B_1}=0.255\,\rm{GeV}$, $s^0_{D}=(6.2\pm0.5)\,\rm{GeV}^2$, $s^0_{B}=(33.5\pm1.0)\,\rm{GeV}^2$, $s^0_{D^}=(6.5\pm0.5)\,\rm{GeV}^2$, $s^0_{B^}=(35.0\pm1.0)\,\rm{GeV}^2$, $s^0_{D_0}=(8.0\pm0.5)\,\rm{GeV}^2$, $s^0_{B_0}=(39.0\pm1.0)\,\rm{GeV}^2$, $s^0_{D_1}=(8.5\pm0.5)\,\rm{GeV}^2$ and $s^0_{B_1}=(39.0\pm1.0)\,\rm{GeV}^2$, which are determined by the conventional two-point correlation functions using the QCD sum rules [@WangHuang; @WangJHEP]. I neglect the uncertainties of the decay constants to avoid double counting as the main uncertainties of the decay constants originate from the uncertainties of the continuum threshold parameters $s_0$. The value of the strong coupling constant $g_{DN\Lambda_c}$ is $g_{\Lambda_c DN}=6.74$ from the QCD sum rules [@Nielsen98], while the average value of the strong coupling constants $g_{\Lambda_cDN}$ and $g_{\Sigma_cDN}$ from the light-cone QCD sum rules is $\frac{g_{\Lambda_cDN}+g_{\Sigma_cDN}}{2}=6.775$ [@Khodjamirian1108], those values are consistent with each other. The average value of the strong coupling constants $g_{\Lambda_c D^N}$ and $g_{\Sigma_c D^N}$ from the light-cone QCD sum rules is $\frac{g_{\Lambda_c D^N}+g_{\Sigma_cD^N}}{2}=3.86$ [@Khodjamirian1108]. In this article, I take the approximation $g_{DN\Lambda_c}\approx g_{DN\Sigma_c}\approx g_{BN\Lambda_b}\approx g_{BN\Sigma_b}\approx g_{D_0N\Lambda_c}\approx g_{D_0N\Sigma_c}\approx g_{B_0N\Lambda_b}\approx g_{B_0N\Sigma_b}\approx6.74$ and $g_{\Lambda_cD^N}\approx g_{\Sigma_cD^N} \approx g_{\Lambda_cD_1N}\approx g_{\Sigma_cD_1N}\approx g_{\Lambda_bB^N}\approx g_{\Sigma_bB^N} \approx g_{\Lambda_bB_1N}\approx g_{\Sigma_bB_1N}\approx3.86$. ![(Color online) The shifts of the masses of the heavy mesons in the nuclear matter with variations of the Borel parameter $M^2$, the I (II) denotes contributions up to the next-to-leading order (leading order) are included. ](massPD.EPS "fig:"){width="7cm"} ![(Color online) The shifts of the masses of the heavy mesons in the nuclear matter with variations of the Borel parameter $M^2$, the I (II) denotes contributions up to the next-to-leading order (leading order) are included. ](massVD.EPS "fig:"){width="7cm"} ![(Color online) The shifts of the masses of the heavy mesons in the nuclear matter with variations of the Borel parameter $M^2$, the I (II) denotes contributions up to the next-to-leading order (leading order) are included. ](massSD.EPS "fig:"){width="7cm"} ![(Color online) The shifts of the masses of the heavy mesons in the nuclear matter with variations of the Borel parameter $M^2$, the I (II) denotes contributions up to the next-to-leading order (leading order) are included. ](massAD.EPS "fig:"){width="7cm"} ![(Color online) The shifts of the masses of the heavy mesons in the nuclear matter with variations of the Borel parameter $M^2$, the I (II) denotes contributions up to the next-to-leading order (leading order) are included. ](massPB.EPS "fig:"){width="7cm"} ![(Color online) The shifts of the masses of the heavy mesons in the nuclear matter with variations of the Borel parameter $M^2$, the I (II) denotes contributions up to the next-to-leading order (leading order) are included. ](massVB.EPS "fig:"){width="7cm"} ![(Color online) The shifts of the masses of the heavy mesons in the nuclear matter with variations of the Borel parameter $M^2$, the I (II) denotes contributions up to the next-to-leading order (leading order) are included. ](massSB.EPS "fig:"){width="7cm"} ![(Color online) The shifts of the masses of the heavy mesons in the nuclear matter with variations of the Borel parameter $M^2$, the I (II) denotes contributions up to the next-to-leading order (leading order) are included. ](massAB.EPS "fig:"){width="7cm"} ![(Color online) The shifts of the decay constants of the heavy mesons in the nuclear matter with variations of the Borel parameter $M^2$, the I (II) denotes contributions up to the next-to-leading order (leading order) are included. ](decayPD.EPS "fig:"){width="7cm"} ![(Color online) The shifts of the decay constants of the heavy mesons in the nuclear matter with variations of the Borel parameter $M^2$, the I (II) denotes contributions up to the next-to-leading order (leading order) are included. ](decayVD.EPS "fig:"){width="7cm"} ![(Color online) The shifts of the decay constants of the heavy mesons in the nuclear matter with variations of the Borel parameter $M^2$, the I (II) denotes contributions up to the next-to-leading order (leading order) are included. ](decaySD.EPS "fig:"){width="7cm"} ![(Color online) The shifts of the decay constants of the heavy mesons in the nuclear matter with variations of the Borel parameter $M^2$, the I (II) denotes contributions up to the next-to-leading order (leading order) are included. ](decayAD.EPS "fig:"){width="7cm"} ![(Color online) The shifts of the decay constants of the heavy mesons in the nuclear matter with variations of the Borel parameter $M^2$, the I (II) denotes contributions up to the next-to-leading order (leading order) are included. ](decayPB.EPS "fig:"){width="7cm"} ![(Color online) The shifts of the decay constants of the heavy mesons in the nuclear matter with variations of the Borel parameter $M^2$, the I (II) denotes contributions up to the next-to-leading order (leading order) are included. ](decayVB.EPS "fig:"){width="7cm"} ![(Color online) The shifts of the decay constants of the heavy mesons in the nuclear matter with variations of the Borel parameter $M^2$, the I (II) denotes contributions up to the next-to-leading order (leading order) are included. ](decaySB.EPS "fig:"){width="7cm"} ![(Color online) The shifts of the decay constants of the heavy mesons in the nuclear matter with variations of the Borel parameter $M^2$, the I (II) denotes contributions up to the next-to-leading order (leading order) are included. ](decayAB.EPS "fig:"){width="7cm"} In Figs.2-3, I plot the shifts of the masses and decay constants of the heavy mesons in the nuclear matter with variations of the Borel parameter $M^2$, respectively. From the figures, I can see that there appear platforms. In this article, I choose the Borel parameters $M^2$ according to the criterion that the uncertainties originate from the Borel parameters $M^2$ are negligible. The values of the Borel parameters $M^2$ are shown explicitly in Table 1. From Figs.2-3 and Table 1, I can see that the Borel parameters $M^2$ in the QCD sum rules for the mass-shift $\delta m$ and decay-constant-shift $\delta f$ of the same meson are different. It is not un-acceptable, as the mass-shift $\delta m$ and decay-constant-shift $\delta f$ come from different QCD sum rules, not coupled QCD sum rules, see Eq.(39), the platforms maybe appear in different places in different QCD sum rules. I can obtain the shifts of the masses and decay constants of the heavy mesons in the nuclear matter in the Borel windows, which are shown explicitly in Table 2. From the Table 2, I can obtain the fractions of the shifts $\frac{\delta m_{D/D^/D_0/D_1}}{m_{D/D^/D_0/D_1}}\leq 5\%$, $\frac{\delta f_{D/D^/D_0/D_1}}{f_{D/D^/D_0/D_1}}\leq 10\%$, $\frac{\delta m_{B/B^/B_0/B_1}}{m_{B/B^/B_0/B_1}}= (5-15)\%$ and $\frac{\delta f_{B/B^/B_0/B_1}}{f_{B/B^/B_0/B_1}}= (25-55)\%$, which are shown explicitly in Table 3. In calculations, I observe that the main contributions come from the terms $m_c\langle\bar{q}q\rangle_N$ and $m_b\langle\bar{q}q\rangle_N$. From Table 3, I can see that the next-to-leading order corrections $\alpha_s \langle\bar{q}q\rangle_N$ are important. In the case of the shifts $\frac{\delta m_{B^/B_1}}{m_{B^/B_1}}$ and $\frac{\delta f_{B^/B_1}}{f_{B^/B_1}}$, the next-to-leading order contributions $\alpha_s \langle\bar{q}q\rangle_N$ and the leading order contributions $\langle\bar{q}q\rangle_N$ are almost equivalent. In this article, I choose the special energy scale $\mu=1\,\rm{GeV}$. The logarithm $\log \frac{m_b^2}{\mu^2}$ in the next-to-leading contributions is very large and enhances the next-to-leading contributions greatly. Although the nuclear matter induced condensates evolve with the renormalization group equation, their evolving behaviors with the energy scales are not well known, as this subject has not been studied in details yet at the present time. A larger energy scale $\mu$ can lead to smaller logarithm $\log \frac{m_b^2}{\mu^2}$ therefore more reasonable predictions. In Table 4, I present the main uncertainties, which originate from the uncertainties of the heavy quark masses and the continuum threshold parameters. $\delta m_{D}$ $\delta m_{D^}$ $\delta m_{D_0}$ $\delta m_{D_1}$ $\delta m_{B}$ $\delta m_{B^}$ $\delta m_{B_0}$ $\delta m_{B_1}$ ------- ---------------- ------------------ ------------------ ------------------ ---------------- ------------------ ------------------ ------------------ -- $M^2$ $4.4-5.4$ $4.6-5.6$ $6.0-7.0$ $6.6-7.6$ $29-33$ $30-34$ $32-36$ $32-36$ $\delta f_{D}$ $\delta f_{D^}$ $\delta f_{D_0}$ $\delta f_{D_1}$ $\delta f_{B}$ $\delta f_{B^}$ $\delta f_{B_0}$ $\delta f_{B_1}$ $M^2$ $1.9-2.9$ $3.5-4.5$ $4.3-5.3$ $5.3-6.3$ $25-29$ $27-31$ $30-34$ $31-35$ : The Borel parameters in the QCD sum rules for the shifts of the masses and decay constants of the heavy mesons in the nuclear matter, the unit is $\rm{GeV}^2$. $\delta m_{D}$ $\delta m_{D^}$ $\delta m_{D_0}$ $\delta m_{D_1}$ $\delta m_{B}$ $\delta m_{B^}$ $\delta m_{B_0}$ $\delta m_{B_1}$ --------- ---------------- ------------------ ------------------ ------------------ ---------------- ------------------ ------------------ ------------------ -- [NLO]{} $-72$ $-102$ $80$ $97$ $-473$ $-687$ $295$ $522$ [LO]{} $-47$ $-70$ $54$ $66$ $-329$ $-340$ $209$ $260$ [@Hay] $-48$ [@Hil] $+45$ $+60$ [@Azi] $-46$ $-242$ $\delta f_{D}$ $\delta f_{D^}$ $\delta f_{D_0}$ $\delta f_{D_1}$ $\delta f_{B}$ $\delta f_{B^}$ $\delta f_{B_0}$ $\delta f_{B_1}$ [NLO]{} $-6$ $-26$ $11$ $31$ $-71$ $-111$ $56$ $134$ [LO]{} $-4$ $-18$ $7$ $21$ $-48$ $-55$ $39$ $67$ [@Azi] $-2$ $-23$ : The shifts of the masses and decay constants of the heavy mesons in the nuclear matter, where the NLO (LO) denotes contributions up to the next-to-leading order (leading order) are included, the unit is MeV. $\frac{\delta m_{D}}{m_{D}}$ $\frac{\delta m_{D^}}{m_{D^}}$ $\frac{\delta m_{D_0}}{m_{D_0}}$ $\frac{\delta m_{D_1}}{m_{D_1}}$ $\frac{\delta m_{B}}{ m_{B}}$ $\frac{\delta m_{B^}}{m_{B^}}$ $\frac{\delta m_{B_0}}{m_{B_0}}$ $\frac{\delta m_{B_1}}{m_{B_1}}$ --------- ------------------------------ ---------------------------------- ---------------------------------- ---------------------------------- ------------------------------- ---------------------------------- ---------------------------------- ---------------------------------- -- [NLO]{} $-4\%$ $-5\%$ $3\%$ $4\%$ $-9\%$ $-13\%$ $5\%$ $9\%$ [LO]{} $-3\%$ $-3\%$ $2\%$ $3\%$ $-6\%$ $-6\%$ $4\%$ $5\%$ $\frac{\delta f_{D}}{f_{D}}$ $\frac{\delta f_{D^}}{f_{D^}}$ $\frac{\delta f_{D_0}}{f_{D_0}}$ $\frac{\delta f_{D_1}}{f_{D_1}}$ $\frac{\delta f_{B}}{f_{B}}$ $\frac{\delta f_{B^}}{f_{B^}}$ $\frac{\delta f_{B_0}}{f_{B_0}}$ $\frac{\delta f_{B_1}}{f_{B_1}}$ [NLO]{} $-3\%$ $-10\%$ $6\%$ $10\%$ $-37\%$ $-57\%$ $24\%$ $53\%$ [LO]{} $-2\%$ $-7\%$ $4\%$ $7\%$ $-25\%$ $-28\%$ $17\%$ $26\%$ : The fractions of the shifts of the masses and decay constants of the heavy mesons in the nuclear matter, where the NLO (LO) denotes contributions up to the next-to-leading order (leading order) are included. $\delta (\delta m_{D})$ $\delta(\delta m_{D^})$ $\delta(\delta m_{D_0})$ $\delta(\delta m_{D_1})$ $\delta(\delta m_{B})$ $\delta(\delta m_{B^})$ $\delta(\delta m_{B_0})$ $\delta(\delta m_{B_1})$ -------------- ------------------------- --------------------------- --------------------------- --------------------------- ------------------------- --------------------------- --------------------------- --------------------------- -- $\delta m_Q$ $\pm14$ $\pm4$ $\pm26$ $\pm6$ $\pm18$ $\pm1$ $\pm25$ $\pm1$ $\delta s_0$ $\pm9$ $\pm14$ $\pm12$ $\pm13$ $\pm65$ $\pm80$ $\pm39$ $\pm70$ $\delta (\delta f_{D}$) $\delta (\delta f_{D^})$ $\delta (\delta f_{D_0})$ $\delta (\delta f_{D_1})$ $\delta (\delta f_{B})$ $\delta (\delta f_{B^})$ $\delta (\delta f_{B_0})$ $\delta (\delta f_{B_1})$ $\delta m_Q$ $\pm1$ $\pm1$ $\pm3$ $\pm2$ $\pm2$ $\pm1$ $\pm3$ $\pm0$ $\delta s_0$ $\pm1$ $\pm7$ $\pm4$ $\pm8$ $\pm21$ $\pm25$ $\pm15$ $\pm34$ : The uncertainties of the shifts of the masses and decay constants of the heavy mesons in the nuclear matter originate from the uncertainties of the heavy quark masses and continuum threshold parameters, where the unit is MeV. The mass-shifts of the negative (positive) parity mesons are negative (positive), the decays of the high charmonium states to the $D\bar{D}$ and $D^\bar{D}^$ ($D_0\bar{D}_0$ and $D_1\bar{D}_1$) pairs are enhanced (suppressed) in the phase space, and we should take into account those effects carefully in studying the production of the $J/\psi$ so as to identifying the quark-gluon plasmas. The currents $\bar{Q} q$ and $\bar{Q}i\gamma_5 q$ (also $\bar{Q}\gamma_\mu q$ and $\bar{Q}\gamma_\mu\gamma_5 q$) are mixed with each other under the chiral transformation $q\to e^{i\alpha\gamma_5}q$, the currents $\bar{Q} q$, $\bar{Q}i\gamma_5 q$, $\bar{Q}\gamma_\mu q$, $\bar{Q}\gamma_\mu\gamma_5 q$ are not conserved in the limit $m_q \to 0$, it is better to take the doublets $(D,D_0)$ and $(D^,D_1)$ as the parity-doublets rather than the chiral-doublets. The quark condensate $\langle\bar{q}q\rangle_{\rho_N}$ serves as the order parameter, and undergoes reduction in the nuclear matter, the chiral symmetry is partially restored; however, there appear new medium-induced condensates, which also break the chiral symmetry. In this article, the $\langle {\cal{O}}\rangle_N$ are companied by the heavy quark masses $m_Q$, $m_Q^2$ or $m_Q^3$, the net effects cannot warrant that the chiral symmetry is monotonously restored with the increase of the $\rho_N$. When the $\rho_N$ is large enough, the order parameter $\langle\bar{q}q\rangle_{\rho_N} \to 0$, the chiral symmetry is restored, the Fermi gas approximation for the nuclear matter breaks down, and the parity-doublets maybe have degenerated masses approximately. In this article, I study the parity-doublets at the low $\rho_N$, the mass breaking effects of the parity-doublets maybe even larger, see Table 2. We expect that smaller mass splitting of the parity-doublets at the high nuclear density is favored, however, larger mass splitting of the parity-doublets at the lower nuclear density cannot be excluded. In Refs.[@Hil; @Hilger2], the mass center $\overline{m}_P$ of the pseudoscalar mesons increases in the nuclear matter while the mass center $\overline{m}_S$ of the scalar mesons decreases in the nuclear matter, the mass breaking effect $\overline{m}_S-\overline{m}_P$ of the parity-doublets is smaller than that in the vacuum. In Table 5, I show the scattering lengths $a_{D}$, $a_{D^}$, $a_{D_0}$, $a_{D_1}$, $a_{B}$, $a_{B^}$, $a_{B_0}$, $a_{B_1}$ explicitly, the $a_{D}$, $a_{D^}$, $a_{B}$, $a_{B^}$ are negative, which indicate the interactions $DN$, $D^N$, $BN$, $B^N$ are attractive, the $a_{D_0}$, $a_{D_1}$, $a_{B_0}$, $a_{B_1}$ are positive, which indicate the interactions $D_0N$, $D_1N$, $B_0N$, $B_0N$ are repulsive. It is difficult (possible) to form the $D_0N$, $D_1N$, $B_0N$, $B_0N$ ($DN$, $D^N$, $BN$, $B^N$) bound states. In Ref.[@ZhangDN], Zhang studies the $S$-wave $DN$ and $D^N$ bound states with the QCD sum rules, the numerical results indicate that the $\Sigma_c(2800)$ and $\Lambda_c(2940)$ can be assigned to be the $S$-wave $DN$ state with $J^P ={\frac{1}{2}}^-$ and the $S$-wave $D^N$ state with $J^P ={\frac{3}{2}}^-$, respectively. $a_{D}$ $a_{D^}$ $a_{D_0}$ $a_{D_1}$ $a_{B}$ $a_{B^}$ $a_{B_0}$ $a_{B_1}$ --------- --------- ----------- ----------- ----------- --------- ----------- ----------- ----------- -- [NLO]{} $-1.1$ $-1.5$ $1.3$ $1.6$ $-8.9$ $-12.9$ $5.6$ $9.9$ [LO]{} $-0.7$ $-1.1$ $0.9$ $1.1$ $-6.2$ $-6.4$ $4.0$ $5.0$ : The heavy-meson-nucleon scattering lengths, where the NLO (LO) denotes contributions up to the next-to-leading order (the leading order) are included, the unit is fm. ![(Color online) The contributions of the perturbative term (I) and quark condensate term (II) in the QCD sum rules for the $D$ mesons in the vacuum. Furthermore, I show the mass $m_D$ (III) and decay constant $f_D$ (IV) explicitly, which are normalized to be 1 at the value $M^2=1.5\rm{GeV}^2$. ](Vacuum-PD.EPS){width="8cm"} In the present work and Refs.[@Hay; @WangHuang; @Azi], the correlation functions are divided into a vacuum part and a static one-nucleon part, and the nuclear matter induced effects are extracted explicitly; while in Refs.[@Hil; @Hilger2], the pole terms of (or ground state contributions to) the hadronic spectral densities of the whole correlation functions are parameterized as $\Delta\Pi(\omega)= F_{+}\delta(\omega-m_{+})- F_{-}\delta(\omega+m_{-})$, where $m_{\pm}=m\pm\Delta m$ and $F_{\pm}=F\pm\Delta F$, and QCD sum rules for the mass center $\overline{m}$ and the mass splitting $\Delta m$ are obtained. In the leading order approximation, the present predictions of the $\delta m_{D}$ and $\delta m_{B}$ are compatible with that of Refs.[@Hay; @Azi] and differ greatly from that of Refs.[@Hil; @Hilger2], see Table 2. The values obtained from the QCD sum rules depend heavily on the Borel windows, the values extracted from different Borel windows especially in different QCD sum rules maybe differ from each other greatly. In Refs.[@Hil; @Hilger2], the authors study the masses of the heavy mesons in the nuclear matter directly by including both the vacuum part and the static one-nucleon part in the QCD sum rules, then the continuum contributions are well approximated by $\rho_{QCD}(\omega^2)\theta\left(\omega^2-{\omega_0^{\pm}}^2\right)$, where the ${\omega_0^{\pm}}^2$ are the continuum threshold parameters, it is one of the advantages of Refs.[@Hil; @Hilger2]. However, they define the moments $S_n(M^2)$ to study the mass-shifts, $$\begin{aligned} S_n(M^2)&=&\int_{\omega_0^-}^{\omega_0^+}d\omega \omega^n \Delta\Pi(\omega) \exp\left( -\frac{\omega^2}{M^2}\right)\, ,\end{aligned}$$ the odd moment $o=S_0(M^2)$ and the even moment $e=S_1(M^2)$, then obtain $\frac{do}{d(1/M^2)}=-S_2(M^2)$ and $\frac{de}{d(1/M^2)}=-S_3(M^2)$ by assuming the $F_{\pm}$ and $m_{\pm}$ are independent on the Borel parameters at the phenomenological side. In fact, $\frac{do}{d(1/M^2)}\neq-S_2(M^2)$ and $\frac{de}{d(1/M^2)}\neq-S_3(M^2)$ at the operator product expansion side according to the QCD spectral densities $\Delta\Pi(\omega)$, which depend on the Borel parameters explicitly, the approximations $\frac{do}{d(1/M^2)}=-S_2(M^2)$ and $\frac{de}{d(1/M^2)}=-S_3(M^2)$ lead to undetermined uncertainties. In Refs.[@Hil; @Hilger2], the perturbative $\mathcal{O}(\alpha_s)$ corrections to the perturbative terms are taken into account. In the QCD sum rules for the pseudoscalar $D$ mesons in the vacuum, if we take into account the perturbative $\mathcal{O}(\alpha_s)$ corrections to the perturbative term and vacuum condensate term, the two criteria (pole dominance and convergence of the operator product expansion) of the QCD sum rules leads to the Borel window $M^2=(1.2-1.8)\,\rm{GeV}^2$, the resulting predictions of the mass $m_{D}$ and decay constant $f_{D}$ are consistent with the experimental data. In Fig.4, I plot the contributions of the perturbative term and quark condensate term in the operator product expansion. From the figure, I can see that the main contributions come from the perturbative term, the quark condensate $\langle \bar{q}q\rangle$ plays a less important role. The modifications of the condensates in the nuclear matter are mild, for example, $\langle\bar{q}q\rangle_{\rho_N}\approx 0.64 \langle\bar{q}q\rangle $, while the perturbative contributions are not modified (or modified slightly by introducing a minor splitting $\Delta s_0$, ${\omega_0^{\pm}}^2=s_0\pm\Delta s_0$) by the nuclear matter. If we turn on the in-medium effects, the contributions of the quark condensate are even smaller, the Borel windows are determined dominantly by the perturbative terms [@Hil; @Hilger2]. If the perturbative $\mathcal{O}(\alpha_s^2)$ corrections to the perturbative terms are also included, the contributions of the perturbative are even larger [@WangJHEP], the QCD sum rules are dominated by the perturbative terms, which are not (or slightly) affected by the nuclear matter. It is not favored to extract the mass-shifts in the nuclear matter, and impairs the predictive ability. In the present work and Refs.[@Hay; @WangHuang; @Azi], the correlation functions are divided into the vacuum part and the static one-nucleon part, which are of the orders ${\mathcal{O}}(0)$ and ${\mathcal{O}}(\rho_N)$, respectively. We can obtain independent QCD sum rules from the two parts respectively. The QCD sum rules correspond to the orders ${\mathcal{O}}(0)$ and ${\mathcal{O}}(\rho_N)$ respectively can have quite different Borel parameters. In this article, I separate the nuclear matter induced effects unambiguously, study the QCD sum rules correspond to the order ${\mathcal{O}}(\rho_N)$, and determine the Borel parameters by the criteria of the QCD sum rules. In the conventional QCD sum rules, we usually choose the Borel parameters $M^2$ to satisfy the following three criteria: $\bf{1_\cdot}$ Pole dominance at the phenomenological side; $\bf{2_\cdot}$ Convergence of the operator product expansion; $\bf{3_\cdot}$ Appearance of the Borel platforms. In the present work and Refs.[@Hay; @WangHuang; @Azi], the nuclear matter induced effects are extracted explicitly, the resulting QCD sum rules are not contaminated by the contributions of the vacuum part, the Borel windows are determined completely by the nuclear matter induced effects, it is the advantage. As the QCD spectral densities are of the form $\delta(\omega^2-m_Q^2)$, we have to take the hadronic spectral densities to be the form $\delta(\omega^2-m_{H}^2)$ and model the continuum contributions with the function $\delta(\omega^2-s_0)$, and determine the $s_0$ by some constraints, see Eq.(16), where the $H$ denotes the ground state and excited state heavy mesons. In this article, I attribute the higher excited states to the continuum contributions, the $\delta$-type hadronic spectral densities make sense. So the pole dominance at the phenomenological side can be released as the continuum contributions are already taken into account. Furthermore, I expect that the couplings of the interpolating currents to the excited states are more weak than that to the ground states, the uncertainties originate from continuum contributions are very small. For example, the decay constants of the pseudoscalar mesons $\pi(140)$ and $\pi(1300)$ have the hierarchy $f_{\pi(1300)}\ll f_{\pi(140)}$ from the Dyson-Schwinger equation [@CDRoberts], the lattice QCD [@Latt-pion], the QCD sum rules [@QCDSR-pion], etc, or from the experimental data [@pion-exp]. In the present work and Refs.[@Hay; @WangHuang; @Azi], large Borel parameters are chosen to warrant the convergence of the operator product expansion and to obtain the Borel platforms, and small Borel parameters cannot lead to platforms. In the Borel windows, where the platforms appear, the main contributions come from terms $\langle \bar{q}q\rangle_N$, the operator product expansion is well convergent. The criteria $\bf{2}$ and $\bf{3}$ can be satisfied. The continuum contributions are not suppressed efficiently for large Borel parameters compared to that for small Borel parameters. In calculations, I observe that the predictions are insensitive to the $s_0$, the uncertainties originate from the continuum threshold parameters $s_0$ are very small in almost all cases, the large Borel parameters make sense. Furthermore, the continuum contributions are already taken into account. On the other hand, from Eqs.(8-9) and Eqs.(12-13), we can see that the mass-shifts $\delta m_{D/D_0/D^/D_1}$ and decay constant shifts $\delta f_{D/D_0/D^/D_1}$ reduce to zero in the limit $\rho_N \to 0$, the QCD sum rules correspond to the nuclear matter induced effects decouple, their Borel parameters (irrespective of large or small) are also irrelevant to the ones in the QCD sum rules for the vacuum part of the correlation functions. So the present predictions are sensible. The predictions depend on the in-medium hadronic spectral functions [@Kwon-2008], for example, there are two generic prototypes of the in-medium spectral functions for the $\rho$ meson, they differ in details at the low mass end of the spectrum. The Klingl-Kaise-Weise spectral function emphasizes the role of chiral in-medium $\pi\pi$ interactions [@KKW-97], while the Rapp-Wambach spectral function focuses on the role of nucleon-hole, $\Delta(1232)$-hole and $N^(1520)$-hole excitations [@RW-99]. Both of the spectral functions account quite well for the low-mass enhancements observed in dilepton spectra from high-energy nuclear collisions. However, the QCD sum rules analysis of the lowest spectral moments reveals qualitative differences with respect to their Brown-Rho scaling properties [@Kwon-2008]. If the simple spectral densities $F\delta(\omega^2-M_{P/V}^2)$ analogous to the ones in Refs.[@Hil; @Hilger2] are taken, where the $P$ denotes the pseudoscalar mesons $\pi$, $\eta_c$, the $V$ denotes the vector mesons $\rho$, $\omega$, $\phi$, $J/\psi$, the $F$ denotes the constant pole residues, the in-medium mass-shifts $\delta M_{P/V}$ are smaller than zero qualitatively [@Lee-Vector]. I expect that the S-wave mesons $q^{\prime}\bar{q}$, $c\bar{q}$, $c\bar{c}$ with the spin-parity $J^P=0^-$ (or $1^-$) have analogous in-medium mass-shifts, at least qualitatively. Further studies based on more sophisticated hadronic spectral densities are needed. In fact, there are controversies about the mass-shifts of the $D$ and $B$ mesons in the nuclear matter, some theoretical approaches indicate negative mass-shifts [@DN-BSE-Negative], while others indicate positive mass-shifts [@Positive-BD]. The different predictions originate mainly from whether or not the heavy pseudoscalar and heavy vector mesons are treated on equal footing in the coupled-channel approaches. If we obtain the meson-baryon interaction kernel by treating the heavy pseudoscalar and heavy vector mesons on equal footing as required by heavy quark symmetry, the mass-shift $\delta M_{D}$ is negative [@DN-BSE-Negative], which is consistent with the present work; furthermore, the attractive D-nucleus interaction can lead to the formation of $D$-nucleus bound states, which can be confronted to the experimental data in the future directly [@D-Nuclei]. The upcoming FAIR project at GSI provides the opportunity to study the in-medium properties of the charmoniums or charmed hadrons for the first time, however, the high mass of charmed hadrons requires a high momentum in the antiproton beam to produce them, the conditions for observing in-medium effects seem unfavorable, as the hadrons sensitive to the in-medium effects are either at rest or have a small momentum relative to the nuclear medium. We have to find processes that would slow down the charmed hadrons inside the nuclear matter, but this requires more detailed theoretical studies. Further theoretical studies on the reaction dynamics and on the exploration of the experimental ability to identify more complicated processes are still needed. Conclusion ========== In this article, I divide the two-point correlation functions of the scalar, pseudoscalar, vector and axialvector currents in the nuclear matter into two parts, i.e. the vacuum part and the static one-nucleon part, then study the in-medium modifications of the masses and decay constants by deriving QCD sum rules from the static one-nucleon part of the two-point correlation functions. In the operator product expansion, I calculate the contributions of the nuclear matter induced condensates up to dimension 5, especially I calculate the next-to-leading order contributions of the in-medium quark condensate and obtain concise expressions, which also have applications in studying the mesons properties in the vacuum. In calculation, I observe that the next-to-leading order contributions of the in-medium quark condensate are very large and should be taken into account. All in all, I study the properties of the scalar, pseudoscalar, vector and axialvector heavy mesons with the QCD sum rules in a systematic way, and obtain the shifts of the masses and decay constants in the nuclear matter. The numerical results indicate that the mass-shifts of the negative parity and positive parity heavy mesons are negative and positive, respectively. For the pseudoscalar meson $D$, I obtain the prediction $\delta M_{D}<0$, which is in contrast to the prediction in Refs.[@Hil; @Hilger2], where the mass-shift is positive $\delta M_{D}>0$. In Refs.[@Hil; @Hilger2], the authors study the masses of the heavy mesons in the nuclear matter directly by including both the vacuum part and static one-nucleon part with the QCD sum rules, and parameterize the spectral density of the whole correlation functions by a simple function $\Delta\Pi(\omega)= F_{+}\delta(\omega-m_{+})- F_{-}\delta(\omega+m_{-})$. I discuss the differences between the QCD sum rules in the present work and that in Refs.[@Hil; @Hilger2] in details, and show why I prefer the present predictions. In the present work and Refs.[@Hay; @WangHuang; @Azi; @Hil; @Hilger2], the finite widths of the mesons in the nuclear matter are neglected, further studies based on the more sophisticated hadronic spectral densities by including the finite widths are needed. As the masses of the heavy meson paries, such as the $D\bar{D}$, $D^\bar{D}^$, $D_0 \bar{D}_0$, $D_1 \bar{D}_1$ are modified in the nuclear environment, we should take into account those effects carefully in studying the production of the $J/\psi$ (and $\Upsilon$) so as to identifying the quark-gluon plasmas. Furthermore, I study the heavy-meson-nucleon scattering lengths as a byproduct, and obtain the conclusion qualitatively about the possible existence of the heavy-meson-nucleon bound states. 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--- abstract: 'Let $G$ be a finite permutation group of degree $n$ and let ${\rm ifix}(G)$ be the involution fixity of $G$, which is the maximum number of fixed points of an involution. In this paper we study the involution fixity of almost simple primitive groups whose socle $T$ is an alternating or sporadic group; our main result classifies the groups of this form with ${\rm ifix}(T) {\leqslant}n^{4/9}$. This builds on earlier work of Burness and Thomas, who studied the case where $T$ is an exceptional group of Lie type, and it strengthens the bound ${\rm ifix}(T) > n^{1/6}$ (with prescribed exceptions), which was proved by Liebeck and Shalev in 2015. A similar result for classical groups will be established in a sequel.' address: - 'T.C. Burness, School of Mathematics, University of Bristol, Bristol BS8 1UG, UK' - 'E. Covato, Bristol, UK' author: - 'Timothy C. Burness' - Elisa Covato title: On the involution fixity of simple groups --- Introduction {#s:intro} ============ Let $G {\leqslant}{\rm Sym}({\Omega})$ be a permutation group on a finite set ${\Omega}$. Let ${\rm fix}(g)$ be the number of elements in ${\Omega}$ fixed by $g \in G$ and set $${\rm fpr}(g,{\Omega}) = \frac{{\rm fix}(g)}{\|{\Omega}\|},$$ which is called the fixed point ratio of $g$. This is a classical concept in permutation group theory and bounds on fixed point ratios find a wide range of applications, especially in the context of primitive groups. For instance, we refer the reader to the recent survey article [@Bur] for a discussion of some powerful applications concerning bases for permutation groups, the random generation of simple groups and the structure of monodromy groups of coverings of the Riemann sphere. In this paper we study ${\rm fix}(g)$ in the setting where $G$ is an almost simple primitive permutation group and $g \in G$ is an involution. We call $${\rm ifix}(G) = \max\{{\rm fix}(g) \,:\, \mbox{$g \in G$ is an involution}\}$$ the involution fixity of $G$ and we are interested in comparing ${\rm ifix}(G)$ with the degree of $G$. This is closely related to the more general concept of fixity, which is defined to be the maximal number of points fixed by a non-identity element. The latter notion was originally introduced by Ronse [@Ronse] in 1980 and there are more recent papers by Liebeck, Saxl and Shalev [@LSh; @SS] on the fixity of primitive groups (also see [@MW], where the transitive groups with fixity at most $2$ are studied). Let us also highlight work of Bender [@Bender] from the early 1970s, where the finite transitive groups $G$ with ${\rm ifix}(G) = 1$ are determined. Our main motivation stems from [@LSh], where Liebeck and Shalev use the O’Nan-Scott theorem to investigate the structure of the primitive groups of degree $n$ with fixity at most $n^{1/6}$. Their main result for an almost simple group $G$ with socle $T$ shows that ${\rm ifix}(T) > n^{1/6}$, with specified exceptions (see [@LSh Theorem 4]). With a view towards applications, it is desirable to strengthen this lower bound (at the expense of some additional exceptions). The first step in this direction was taken by Burness and Thomas in [@BThomas], where the almost simple groups with socle an exceptional group of Lie type $T$ and ${\rm ifix}(T) {\leqslant}n^{4/9}$ are determined. In this paper, we extend the analysis in [@BThomas] to the almost simple groups with socle an alternating or sporadic group. The remaining classical groups will be handled in a sequel, which will complete our study of involution fixity for almost simple primitive groups. Our main result is the following. In the statement, $\mathcal{S}$ denotes the set of finite simple groups that are either alternating or sporadic. \[t:main\] Let $G {\leqslant}{\rm Sym}({\Omega})$ be an almost simple primitive permutation group of degree $n$ with socle $T \in \mathcal{S}$ and point stabilizer $H$. Set $H_0 = H \cap T$. Then one of the following holds: - ${\rm ifix}(T) > n^{4/9}$. - $H_0$ has odd order and ${\rm ifix}(T) = 0$. - $(T,n) = (A_5,5)$ and ${\rm ifix}(T) =1$. - $n^{{\alpha}} {\leqslant}{\rm ifix}(T) {\leqslant}n^{4/9}$ and $(T,H_0,{\rm ifix}(T),n,{\alpha})$ is recorded in Table \[tab:main\]. $$\begin{array}{llllll}\hline T & H_0 & {\rm ifix}(T) & n & {\alpha}& \mbox{Conditions} \\ \hline A_5 & S_3 & 2 & 10 & 0.301 & \\ & D_{10} & 2 & 6 & 0.386 & \\ A_6 & 3^2{:}4 & 2 & 10 & 0.301 & \\ & A_5 & 2 & 6 & 0.386 & \mbox{$G = A_6$ or $S_6$} \\ & D_{10} & 4 & 36 & 0.386 & \mbox{$G = {\rm M}_{10}$, ${\rm PGL}_{2}(9)$ or $A_6.2^2$} \\ & S_4 & 3 & 15 & 0.405 & \mbox{$G = A_6$ or $S_6$} \\ & D_8 & 5 & 45 & 0.422 & \mbox{$G = {\rm M}_{10}$, ${\rm PGL}_{2}(9)$ or $A_6.2^2$} \\ A_7 & {\rm L}_{2}(7) & 3 & 15 & 0.405 & G = A_7 \\ A_9 & 3^2{:}{\rm SL}_{2}(3) & 8 & 840 & 0.308 & \\ & {\rm L}_{2}(8){:}3 & 8 & 120 & 0.434 & G = A_9 \\ A_{10} & {\rm M}_{10} & 24 & 2520 & 0.405 & \\ A_{11} & {\rm M}_{11} & 24 & 2520 & 0.405 & G = A_{11} \\ {\rm J}_{1} & 2^3{:}7{:}3 & 5 & 1045 & 0.231 & \\ & 11{:}10 & 12 & 1596 & 0.336 & \\ & 7{:}6 & 20 & 4180 & 0.359 & \\ & 19{:}6 & 20 & 1540 & 0.408 & \\ & {\rm L}_{2}(11) & 10 & 266 & 0.412 & \\ {\rm J}_{2} & A_5 & 60 & 10080 & 0.444 & \\ {\rm J}_{3} & 2^4{:}(3 \times A_5) & 50 & 17442 & 0.400 & \\ & 2^{2+4}{:}(3 \times S_3) & 85 & 43605 & 0.415 & \\ & 3^2.3^{1+2}{:}8 & 80 & 25840 & 0.431 & \\ {\rm McL} & 3^{1+4}{:}2S_5 & 56 & 15400 & 0.417 & \\ {\rm He} & 7^2{:}2.{\rm L}_{2}(7) & 64 & 244800 & 0.335 & \\ {\rm O'N} & 3^4{:}2^{1+4}D_{10} & 1064 & 17778376 & 0.417 & \\ {\rm Co}_{1} & 5^2{:}2A_5 & 3244032 & 1385925602181120 & 0.430 & \\ {\rm HN} & {\rm U}_{3}(8){:}3 & 800 & 16500000 & 0.402 & \\ {\rm Th} & 2^5.{\rm L}_{5}(2) & 2169 & 283599225 & 0.394 & \\ & 7^2{:}(3 \times 2S_4) & 645120 & 12860819712000 & 0.443 & \\ \hline \end{array}$$ \[r:main\] Let us make some comments on the statement of Theorem \[t:main\]. - The groups arising in part (ii) with $\|H_0\|$ odd are determined in [@LSa Theorem 2] (also see [@LSh Lemma 2.1]). The possibilities are as follows: $$\begin{array}{lll} \hline T & H_0 & \mbox{Conditions} \\ \hline A_p & {\rm AGL}_{1}(p) \cap T & \mbox{$p$ prime, $p \equiv 3 {\allowbreak\mkern4mu({\operator@font mod}\,\,4)}$} \\ & & \mbox{$G = S_p$ if $p=7,11,23$} \\ \mbox{${\rm J}_{3}$, ${\rm O'N}$} & \mbox{$19{:}9$, $31{:}15$ (resp.)} & G = T.2 \\ \mbox{${\rm M}_{23}$, ${\rm Th}$, $\mathbb{B}$} & \mbox{$23{:}11$, $31{:}15$, $47{:}23$ (resp.)} & \\ \hline \end{array}$$ - The number ${\alpha}$ recorded in the fifth column of Table \[tab:main\] is equal to $\log {\rm ifix}(T) / \log n$, expressed to $3$ significant figures. - The theorem reveals that there are only finitely many groups of the given form with $1 {\leqslant}{\rm ifix}(T) {\leqslant}n^{4/9}$. However, it is straightforward to show that there are infinitely many with $1 {\leqslant}{\rm ifix}(T) {\leqslant}n^{1/2}$. For example, we can take $T = A_p$ and $H = {\rm AGL}_{1}(p) \cap G$, where $p$ is any prime with $p \equiv 1 {\allowbreak\mkern4mu({\operator@font mod}\,\,4)}$ (see Remark \[r:aff2\]). - Theorem \[t:main\] already has an application in finite geometry. Indeed, we refer the reader to [@BPP Section 6], where the result is used to study point-primitive generalized quadrangles. By combining Theorem \[t:main\] with [@BThomas Theorem 1], we get the following corollary. \[c:main\] Let $G {\leqslant}{\rm Sym}({\Omega})$ be an almost simple primitive permutation group of degree $n$ with socle $T$ and point stabilizer $H$. Set $H_0 = H \cap T$ and assume $\|H_0\|$ is even and $T$ is not isomorphic to a classical group. Then one of the following holds: - ${\rm ifix}(T) > n^{1/3}$. - $(T,n) = ({}^2B_2(q),q^2+1)$ and ${\rm ifix}(T) =1$. - $(T,H_0,{\rm ifix}(T),n)=(A_9, 3^2{:}{\rm SL}_{2}(3), 8, 840)$ or $({\rm J}_{1}, 2^3{:}7^3{:}3, 5, 1045)$. The proof of Theorem \[t:main\] is presented in Sections \[s:alt\] and \[s:spor\], where we handle the groups with an alternating and sporadic socle, respectively. We freely employ computational methods, using [@GAP] and [Magma]{} [@magma], when it is feasible to do so. In particular, the argument for sporadic groups in Section \[s:spor\] makes extensive use of the character tables (and associated fusion maps) that are available in the Character Table Library [@GAPCTL]. As one might expect, the O’Nan-Scott theorem provides a framework for our proof when the socle $T$ is an alternating group. Indeed, this key result divides the possibilities for the point stabilizer $H$ into several families and we proceed by considering each family in turn. The notation we use in this paper is fairly standard. We will write $C_n$, or just $n$, for a cyclic group of order $n$ and $G^n$ denotes the direct product of $n$ copies of $G$. An unspecified extension of $G$ by a group $H$ will be denoted by $G.H$; if the extension splits then we write $G{:}H$. We adopt the standard notation for simple groups of Lie type from [@KL], which differs slightly from the notation in [@Atlas]. All logarithms are base two, unless stated otherwise. Symmetric and alternating groups {#s:alt} ================================ Let $G {\leqslant}{\rm Sym}({\Omega})$ be an almost simple primitive permutation group of degree $n$ with socle $T$ and point stabilizer $H$. Set $H_0 = H \cap T$ and note that $H$ is a maximal subgroup of $G$ such that $G = HT$. Then $n = \|T:H_0\|$ and $$\label{e:fix} {\rm fix}(t) = \frac{\|t^T \cap H_0\|}{\|t^T\|} \cdot n$$ for all $t \in T$, where $t^T$ denotes the conjugacy class of $t$ in $T$. We will adopt this notation for the remainder of the paper. In this section, we prove Theorem \[t:main\] for the groups with socle $T=A_m$. Recall that if $t \in T$ is an involution with cycle-shape $(2^k,1^{m-2k})$, then $$\|t^T\| = \frac{m!}{k!(m-2k)!2^k}.$$ We begin by handling the cases with $m {\leqslant}25$. \[p:alt\] The conclusion to Theorem \[t:main\] holds if $m {\leqslant}25$. This is a straightforward [Magma]{} [@magma] computation. First assume $G = A_m$ or $S_m$. Working in the natural permutation representation of degree $m$, we use the function `MaximalSubgroups` to construct a set of representatives of the conjugacy classes of maximal subgroups $H$ of $G$. Given an involution $t \in T$, we can then compute $\|t^T \cap H_0\|$ and $\|t^T\|$, which gives ${\rm fix}(t)$ via . We then obtain ${\rm ifix}(T)$ by taking the maximum over a set of representatives of the conjugacy classes of involutions in $T$ and the desired result quickly follows. Finally, if $T = A_6$ and $G$ is one of ${\rm PGL}_{2}(9)$, ${\rm M}_{10}$ or $A_6.2^2$ then we can proceed in an entirely similar manner, working with a permutation representation of $G$ of degree $10$. For the remainder of this section, we may assume $G = A_m$ or $S_m$ with $m > 25$. Our aim is to establish the bound ${\rm ifix}(T)>n^{4/9}$. The possibilities for $H$ are described by the O’Nan-Scott theorem (see [@LPS], for example), which divides the maximal subgroups of $G$ into the following families (in parts (e) and (f), $S$ denotes a non-abelian finite simple group): - Intransitive: $H = (S_k \times S_{m-k}) \cap G$, $1 {\leqslant}k < m/2$. - Imprimitive: $H = (S_k \wr S_r) \cap G$, $m=kr$, $1 < k < m$. - Affine: $H = {\rm AGL}_{d}(p) \cap G$, $m = p^d$, $p$ prime, $d {\geqslant}1$. - Product-type: $H = (S_k \wr S_r) \cap G$, $m=k^r$, $k {\geqslant}5$, $r {\geqslant}2$. - Diagonal-type: $H = (S^k.({\rm Out}(S) \times S_k)) \cap G$, $m = \|S\|^{k-1}$, $k {\geqslant}2$. - Almost simple: $S {\leqslant}H {\leqslant}{\rm Aut}(S)$, $m = \|H:K\|$ for some maximal subgroup $K$ of $H$. We will consider each family of subgroups in turn. Before we begin the analysis of case (a), let us record some useful preliminary lemmas. \[l:basic\] Suppose $\|H_0\|$ is even, $\|H_0\| {\leqslant}\|T\|^{{\alpha}}$ and $\|t^T\| {\leqslant}\|T\|^{{\beta}}$ for every involution $t \in H_0$. Then ${\rm ifix}(T) > n^{4/9}$ if $5-5{\alpha}-9{\beta}>0$. Let $t \in H_0$ be an involution. Then $\|t^T \cap H_0\| {\geqslant}1$ and $\|t^T\| {\leqslant}\|T\|^{{\beta}}$, so ${\rm fix}(t) {\geqslant}n\|T\|^{-{\beta}}$ and thus ${\rm ifix}(T) > n^{4/9}$ if $n>\|T\|^{9{\beta}/5}$. The result now follows since $n = \|T:H_0\| {\geqslant}\|T\|^{1-{\alpha}}$. \[l:inv\] If $T = A_m$ and $m > 20$, then $\|t^T\| < \|T\|^{11/20}$ for every involution $t \in T$. The groups with $m {\leqslant}54$ can be checked using [Magma]{}, so let us assume $m {\geqslant}55$. Recall that if $G$ is a finite group and $\mathcal{I}(G)$ is the set of involutions in $G$, then $\|\mathcal{I}(G)\|^2 < k(G)\cdot \|G\|$, where $k(G)$ is the number of conjugacy classes of $G$ (see [@Isaacs Chapter 4], for example). As a special case, we deduce that $$\|\mathcal{I}(S_m)\|^2 < m!p(m),$$ where $p(m)$ is the partition function, and thus it suffices to show that $$\label{e:part} 2^{11}p(m)^{10} < m!.$$ Indeed, if this inequality holds then $\|\mathcal{I}(S_m)\| < \|T\|^{11/20}$ and the desired bound follows. By the main theorem of [@Prib] we have $p(m) < m^{-3/4}e^{c\sqrt{m}}$, where $c = \pi\sqrt{2/3}$, so holds if $f(m)>1$, where $$f(m):=\frac{m^{15/2}m!}{2^{11}e^{10c\sqrt{m}}}.$$ It is easy to check that $f$ is an increasing function and $f(55)>1$. The result follows. \[l:cor\] Let $T = A_m$ with $m>20$. If $\|H_0\|$ is even and $\|H_0\|^{100}<\|T\|$, then ${\rm ifix}(T)>n^{4/9}$. This follows by combining Lemmas \[l:basic\] and \[l:inv\]. Intransitive subgroups {#ss:intran} ---------------------- In this section we will assume $H = (S_k \times S_{m-k}) \cap G$ is a maximal intransitive subgroup of $G$, where $1 {\leqslant}k < m/2$. We may identify ${\Omega}$ with the set of $k$-element subsets of $\{1, \ldots, m\}$. In particular, $n = \binom{m}{k}$. \[p:intrans\] If $m {\geqslant}7$, then ${\rm ifix}(T) > n^{1/2}$. We claim that ${\rm fix}(t)>n^{1/2}$, where $t = (1,2)(3,4) \in T$. If $k = 1$ then $n=m$, ${\rm fix}(t) = m-4$ and the result follows. Now assume $k {\geqslant}2$. Clearly, $t$ fixes a $k$-set $\Gamma$ if and only if $\Gamma \cap \{1,2,3,4\}$ is either empty, or one of $\{1,2\}$, $\{3,4\}$ or $\{1,2,3,4\}$. Therefore, $${\rm fix}(t) = \binom{m-4}{k} + 2\binom{m-4}{k-2} + \binom{m-4}{k-4}$$ where the final term is $0$ if $k=2$ or $3$. The cases with $m<10$ can be checked directly, so let us assume $m {\geqslant}10$. We claim that $$\label{e:bd1} \binom{m-4}{k} + 2\binom{m-4}{k-2} > \binom{m}{k}^{\frac{1}{2}},$$ which implies that ${\rm fix}(t) > n^{1/2}$. To see this, we first express the binomial coefficients $\binom{m-4}{k}$ and $\binom{m-4}{k-2}$ in terms of $\binom{m}{k}$ and we deduce that it suffices to show that $$\binom{m}{k}^{\frac{1}{2}}\left(\frac{f(k)g(k)}{m(m-1)(m-2)(m-3)}\right)>1,$$ where $f(k) = (m-k)(m-k-1)$ and $g(k) = 2k(k-1)+(m-k-2)(m-k-3)$. Since $k {\leqslant}\frac{1}{2}(m-1)$, we calculate that $f(k) {\geqslant}\frac{1}{4}(m^2-1)$ and $g(k) {\geqslant}\frac{2}{3}m^2-4m+\frac{21}{4}$. In addition, we have $\binom{m}{k} {\geqslant}\binom{m}{2}$ and thus holds if $h(m)>1$, where $$h(m) := \frac{\binom{m}{2}^{\frac{1}{2}}(m+1)\left(\frac{2}{3}m^2-4m+\frac{21}{4}\right)}{4m(m-2)(m-3)}.$$ One checks that $h(m)$ is an increasing function and $h(10)>1$. The result follows. Imprimitive subgroups {#ss:imprim} --------------------- Next we turn to the imprimtive subgroups of the form $H = (S_k \wr S_r) \cap G$, where $m=kr$ and $1<k<m$. We identify ${\Omega}$ with the set of partitions of $\{1, \ldots, m\}$ into $r$ subsets of size $k$. Note that $$n = \|{\Omega}\| = \frac{(kr)!}{k!^rr!} =: f(k,r).$$ \[p:imprim\] If $m {\geqslant}9$, then ${\rm ifix}(T) > n^{1/2}$. We claim that ${\rm fix}(t) > n^{1/2}$ for $t = (1,2)(3,4) \in T$. First assume $k=2$, so $r {\geqslant}5$. Clearly, $t$ stabilizes a partition in ${\Omega}$ if and only if the partition contains $\{1,2\}$ and $\{3,4\}$, or $\{1,3\}$ and $\{2,4\}$, or $\{1,4\}$ and $\{2,3\}$. Therefore, ${\rm fix}(t) = 3f(2,r-2)$ and it suffices to show that $g(r)>1$, where $$g(r):=\frac{9f(2,r-2)^2}{f(2,r)}.$$ It is easy to check that this is an increasing function with $g(5)>1$. Now assume $k {\geqslant}3$. A partition in ${\Omega}$ is fixed by $t$ if and only if it has a part containing $\{1,2\}$ and another containing $\{3,4\}$, or $k {\geqslant}4$ and it has a part containing $\{1,2,3,4\}$. Therefore, $${\rm fix}(t) = \binom{m-4}{k-2}\binom{m-k-2}{k-2}f(k,r-2) + \binom{m-4}{k-4}f(k,r-1)$$ and it suffices to show that $$g(k,r):= \binom{kr-4}{k-2}^2\binom{kr-k-2}{k-2}^2\frac{f(k,r-2)^2}{f(k,r)} >1.$$ Now, if $k$ is fixed then $g(k,r)$ is increasing as a function of $r$. Therefore, for $k {\geqslant}4$ we have $$g(k,r) {\geqslant}g(k,2) = \binom{2k-4}{k-2}^2\frac{1}{f(k,2)},$$ which is an increasing function in $k$ and one checks that $g(4,2)>1$. Similarly, if $k=3$ then $r {\geqslant}3$ and $g(3,r) {\geqslant}g(3,3)>1$. The result follows. Affine subgroups {#ss:affine} ---------------- In this section we assume $H = {\rm AGL}_{d}(p) \cap G$ and $m = p^d$, where $p$ is a prime and $d {\geqslant}1$. Note that $$n = \|{\Omega}\| {\geqslant}\frac{\|T\|}{\|{\rm AGL}_{d}(p)\|} = \frac{(p^d-1)!}{2\|{\rm GL}_{d}(p)\|}.$$ Write ${\rm AGL}_{d}(p) = V{:}L$, where $V = (\mathbb{F}_p)^d$ and $L = {\rm GL}(V)$. Now ${\rm AGL}_{d}(p)$ acts faithfully on $V$ by affine transformations $(v,x): u \mapsto v+u^x$ and this embeds ${\rm AGL}_{d}(p)$ in $S_m$. Note that if $t = (v,x) \in {\rm AGL}_{d}(p)$ then $t^2=1$ if and only if $v^x = -v$ and $x^2=1$. \[d:tk\] Fix a basis $\{e_1, \ldots, e_d\}$ for $V$. With respect to this basis, let us define $x_k = [-I_k,I_{d-k}]$ if $p \ne 2$ and $x_k = [A^k,I_{d-2k}]$ if $p=2$, where $A = \left(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right)$. In particular, if $p=2$ then $1 {\leqslant}k {\leqslant}d/2$ and $x_k$ is a block-diagonal matrix with $k$ blocks equal to $A$. For $k {\geqslant}1$ set $t_k = (v,x_k) \in {\rm AGL}_{d}(p)$, where $v=e_1$ if $p \ne 2$, otherwise $v = 0$. Note that $t$ is an involution. \[l:aff\] Let $t = t_k \in {\rm AGL}_{d}(p)$. Then $t$ has cycle-shape $(2^{p^{d-k}(p^k-1)/2},1^{p^{d-k}})$ as an element of $S_m$ and we have $\|C_{{\rm AGL}_{d}(p)}(t)\| = p^{d-k}\|C_{{\rm GL}_{d}(p)}(x_k)\|$. First consider the cycle-shape of $t$. Since $t$ is an involution, it suffices to show that it fixes exactly $p^{d-k}$ vectors in $V$. Suppose $w = \sum_{i}a_ie_i \in V$ is fixed by $t$. First assume $p \ne 2$. Here $w = w^t = w^{x_k}+e_1$ and thus $$\sum_{i=1}^{d}a_ie_i = (-a_1+1)e_1 + \sum_{i=2}^{k}(-a_i)e_i + \sum_{i=k+1}^da_ie_i,$$ so $a_1 = \frac{1}{2}$ and $a_i = 0$ for $2 {\leqslant}i {\leqslant}k$. There are no conditions on the coefficients $a_i$ for $i > k$, so $t$ fixes precisely $p^{d-k}$ vectors and the result follows. Similarly, if $p = 2$ then $w = w^t = w^{x_k}$ and $$\sum_{i=1}^{d}a_ie_i = \sum_{i=1}^{k}(a_{2i}e_{2i-1}+a_{2i-1}e_{2i}) + \sum_{i=2k+1}^{d}a_ie_i,$$ which implies that $a_{2i-1}=a_{2i}$ for $1 {\leqslant}i {\leqslant}k$. Therefore $t$ fixes $2^{k}2^{d-2k}=2^{d-k}$ vectors as claimed. Now let us consider the centralizer of $t$. Suppose $p \ne 2$ and $(u,y) \in {\rm AGL}_{d}(p)$. Then $(u,y)$ centralizes $t$ if and only if $y \in C_{{\rm GL}_{d}(p)}(x_k) = {\rm GL}_{k}(p) \times {\rm GL}_{d-k}(p)$ and $u+e_1^y = e_1+u^{x_k}$. Given $y \in C_{{\rm GL}_{d}(p)}(x_k)$, a straightforward calculation shows that there are $p^{d-k}$ vectors $u \in V$ such that $u+e_1^y = e_1+u^{x_k}$ and thus $$\|C_{{\rm AGL}_{d}(p)}(t)\| = p^{d-k}\|C_{{\rm GL}_{d}(p)}(x_k)\| = p^{d-k}\|{\rm GL}_{k}(p)\|\|{\rm GL}_{d-k}(p)\|.$$ Similarly, if $p=2$ then $(u,y) \in {\rm AGL}_{d}(2)$ centralizes $t$ if and only if $y \in C_{{\rm GL}_{d}(2)}(x_k)$ and $u^{x_k} = u$. Since the $1$-eigenspace of $x_k$ on $V$ is $(d-k)$-dimensional, we get $$\|C_{{\rm AGL}_{d}(2)}(t)\| = 2^{d-k}\|C_{{\rm GL}_{d}(2)}(x_k)\| = 2^{d-k+2dk-3k^2}\|{\rm GL}_{k}(2)\|\|{\rm GL}_{d-2k}(2)\|$$ and the result follows. \[p:aff1\] If $d=1$ then one of the following holds: - $p \equiv 3 {\allowbreak\mkern4mu({\operator@font mod}\,\,4)}$ and ${\rm ifix}(T) = 0$. - $p=5$, $n=6$ and ${\rm ifix}(T) = 2$. - $p \equiv 1 {\allowbreak\mkern4mu({\operator@font mod}\,\,4)}$, $p {\geqslant}13$ and ${\rm ifix}(T)> n^{4/9}$. First observe that $H_0 = p{:}\frac{1}{2}(p-1)$ and $n = (p-2)!$. In particular, if $p \equiv 3 {\allowbreak\mkern4mu({\operator@font mod}\,\,4)}$ then $\|H_0\|$ is odd and thus ${\rm ifix}(T) = 0$ as claimed. Now assume $p \equiv 1 {\allowbreak\mkern4mu({\operator@font mod}\,\,4)}$. If $p=5$ then $H_0 = D_{10}$ and ${\rm ifix}(T) = 2$, so let us assume $p {\geqslant}13$. Let $t \in H_0$ be an involution. By applying Lemma \[l:aff\], noting that $H_0$ has a unique conjugacy class of involutions, we deduce that $$\|t^T \cap H_0\| = p,\;\; \|t^T\| = \frac{p!}{2^{(p-1)/2}\left(\frac{1}{2}(p-1)\right)!}$$ and thus gives $$\label{e:eq1} {\rm ifix}(T) = {\rm fix}(t) = \frac{2^{(p-1)/2}\left(\frac{1}{2}(p-1)\right)!}{p-1}.$$ It follows that ${\rm ifix}(T)>n^{4/9}$ if and only if $f(p)>1$, where $$f(p):=\frac{2^{(p-1)/2}\left(\frac{1}{2}(p-1)\right)!}{(p-1)(p-2)!^{4/9}}.$$ One can check that $f(p+2)>f(p)$ and $f(13)>1$, whence ${\rm ifix}(T)>n^{4/9}$ as claimed. \[r:aff2\] The proof of Proposition \[p:aff1\] reveals that there are infinitely many groups $G$ as in Theorem \[t:main\] with $1 {\leqslant}{\rm ifix}(T) {\leqslant}n^{1/2}$. Indeed, if we take $T = A_p$ and $H = {\rm AGL}_{1}(p) \cap G$, where $p$ is a prime such that $p \equiv 1 {\allowbreak\mkern4mu({\operator@font mod}\,\,4)}$, then ${\rm ifix}(T)$ is given in and we deduce that ${\rm ifix}(T) > n^{1/2}$ if and only if $g(p) > 1$, where $$g(p):=\frac{2^{(p-1)/2}\left(\frac{1}{2}(p-1)\right)!}{(p-1)(p-2)!^{1/2}}.$$ But $g(p+2) < g(p)$ and $g(5)<1$, so ${\rm ifix}(T) {\leqslant}n^{1/2}$ for all primes $p$ with $p \equiv 1 {\allowbreak\mkern4mu({\operator@font mod}\,\,4)}$. \[p:aff2\] If $d {\geqslant}2$ then either ${\rm ifix}(T) > n^{4/9}$ or $(d,p,{\rm ifix}(T),n) = (2,3,8,840)$. First assume $d=2$, so $m = p^2$ and $p$ is odd. If $p = 3$ or $5$ then the result follows from Proposition \[p:alt\], so let us assume $p {\geqslant}7$. As in Definition \[d:tk\], set $t = t_2 = (e_1,x_2) \in H_0$. By applying Lemma \[l:aff\] we deduce that $$\|t^T\cap H_0\| {\geqslant}\|t^{H_0}\| = p^2,\;\; \|t^T\| = \frac{(p^2)!}{2^{(p^2-1)/2}(\frac{1}{2}(p^2-1))!}$$ and thus ${\rm ifix}(T)>n^{4/9}$ if $f(p)>1$, where $$f(p):= \frac{2^{(p^2-1)/2}p^{1/3}\left(\frac{1}{2}(p^2-1)\right)!}{(p-1)^{5/9}(p^2-1)^{5/9}(p^2)!^{4/9}}.$$ Here $f(p+2)>f(p)$ and $f(7)>1$, so ${\rm ifix}(T)>n^{4/9}$ as required. Now assume $d {\geqslant}3$. If $p=2$ and $d {\leqslant}6$, then a [Magma]{} calculation gives ${\rm ifix}(T)>n^{4/9}$. Similarly, one can check that the same conclusion holds if $p=3$ and $d {\leqslant}4$. In order to establish the desired bound in the remaining cases, set $t = t_2 \in H_0$ and note that $t$ has cycle-shape $(2^{p^{d-2}(p^2-1)/2},1^{p^{d-2}})$ by Lemma \[l:aff\]. Now $\|t^T \cap H_0\| {\geqslant}1$ and $\|{\rm GL}_{d}(p)\|<p^{d^2}$, so $n > (p^d)!p^{-d(d+1)}$ and it follows that ${\rm ifix}(t)>n^{4/9}$ if $g(d,p)>1$, where $$g(d,p) := \frac{\left(\frac{1}{2}p^{d-2}(p^2-1)\right)!\left(p^{d-2}\right)!2^{p^{d-2}(p^2-1)/2}}{p^{5d(d+1)/9}\left(p^d\right)!^{4/9}}.$$ If $d$ is fixed, then $g(d,p)$ is an increasing function in $p$. Similarly, one checks that if $p$ is fixed, then $g(d,p)$ is increasing as a function of $d$ (here we are assuming, as we may by the above remarks, that $d {\geqslant}7$ if $p=2$ and $d {\geqslant}5$ if $p=3$). Therefore, if $p {\geqslant}5$ then $g(d,p) {\geqslant}g(3,5)>1$. Similarly, if $p=3$ and $d {\geqslant}5$ then $g(d,p) {\geqslant}g(5,3)>1$ and for $p=2$ with $d {\geqslant}7$ we get $g(d,p) {\geqslant}g(7,2)>1$. We conclude that ${\rm ifix}(T)> n^{4/9}$ if $d {\geqslant}3$ and the proof of the proposition is complete. Product-type subgroups {#ss:prod} ---------------------- Now assume $H$ is a product-type subgroup of $G$, so $H = (S_k \wr S_r) \cap G$ and $m = k^r$, where $k {\geqslant}5$ and $r {\geqslant}2$. Set $\Gamma = \{1, \ldots, k\}$ and note that the embedding of $H$ in $G$ arises from the product action of $H$ on the Cartesian product $\Gamma^r$. That is, for every $(x_1,\ldots,x_r){\sigma}\in H$ and $(\gamma_1, \ldots, \gamma_r) \in \Gamma^r$ we have $$(\gamma_1, \ldots, \gamma_r)^{(x_1,\ldots,x_r){\sigma}} = \left(\gamma_1^{x_1}, \ldots, \gamma_r^{x_r}\right)^{{\sigma}} = \left(\gamma_{1^{{\sigma}^{-1}}}^{x_{1^{{\sigma}^{-1}}}},\ldots, \gamma_{r^{{\sigma}^{-1}}}^{x_{r^{{\sigma}^{-1}}}}\right).$$ In particular, let us observe that $$n {\geqslant}\frac{(k^r)!}{2(k!)^rr!}.$$ \[p:prod\] If $H$ is a product-type subgroup of $G$, then ${\rm ifix}(T)>n^{4/9}$. Fix the involution $t = (t_1, 1, \ldots, 1) \in (A_k)^r < H_0$, where $t_1 = (1,2)(3,4) \in A_k$. By considering the action of $H$ on $\Gamma^r$, it is easy to see that $t$ has exactly $(k-4)k^{r-1}$ fixed points and so it has cycle-shape $(2^{2k^{r-1}},1^{(k-4)k^{r-1}})$ as an element of $T$. Therefore, $$\|t^T\| = \frac{(k^r)!}{2^{2k^{r-1}}(2k^{r-1})!((k-4)k^{r-1})!}$$ and using the trivial bound $\|t^T \cap H_0\| {\geqslant}1$ we deduce that ${\rm fix}(t)>n^{4/9}$ if $f(k,r)>1$, where $$f(k,r):=\frac{2^{18k^{r-1}-5}(2k^{r-1})!^9((k-4)k^{r-1})!^9}{(k^r)!^4(k!)^{5r}(r!)^5}.$$ It is straightforward to check that $f(k,r)$ is an increasing function in both $k$ and $r$, which quickly implies that $f(k,r)>1$ unless $r=2$ and $k=5$ or $6$. If $(k,r) = (5,2)$ then $m=25$ and so this case was handled in Proposition \[p:alt\]. Similarly, if $(k,r) = (6,2)$ then a routine [Magma]{} computation shows that ${\rm ifix}(T)>n^{4/9}$. Diagonal-type subgroups {#ss:diag} ----------------------- Here $H = (S^k.({\rm Out}(S) \times S_k)) \cap G$ and $m = \|S\|^{k-1}$, where $k {\geqslant}2$ and $S$ is a non-abelian finite simple group. The embedding of $H$ in $G$ is afforded by a natural (faithful) action of $H$ on the set of cosets of the diagonal subgroup $\{(s, \ldots, s) \, : \, s \in S\}$ of $S^k$. \[p:diag\] If $H$ is a diagonal-type subgroup of $G$, then ${\rm ifix}(T)>n^{4/9}$. First assume $m<200$, so $k=2$ and $S$ is isomorphic to $A_5$ or ${\rm L}_{2}(7)$. In both cases, we can use the database of primitive groups in [Magma]{} to construct $H$ as a subgroup of $S_m$ and then it is a routine computation to check that $5-5{\alpha}-9{\beta}>0$ for constants ${\alpha}$ and ${\beta}$ such that $\|H_0\| {\leqslant}\|T\|^{{\alpha}}$ and $\|t^T\| {\leqslant}\|T\|^{{\beta}}$ for every involution $t \in H_0$. Therefore, ${\rm ifix}(T)>n^{4/9}$ by Lemma \[l:basic\]. For the remainder, we may assume $m {\geqslant}200$. We claim that $\|H_0\|^{100}<\|T\|$ and thus ${\rm ifix}(T)>n^{4/9}$ by Lemma \[l:cor\]. To see this, let us first observe that $\|{\rm Out}(S)\| {\leqslant}\|S\|/30$ by [@Quick Lemma 2.2], so $\|H_0\| {\leqslant}\frac{1}{30}\ell^{k+1}k!$ where $\ell = \|S\|$. It follows that $\|H_0\|^{100}<\|T\|$ if $f(k,\ell)>1$, where $$f(k,\ell):= \frac{1}{2}\left(\frac{30}{\ell^{k+1}k!}\right)^{100}(\ell^{k-1})!$$ If $k=2$ then $m = \ell$ and the condition $m {\geqslant}200$ implies that $\ell {\geqslant}360$ since $A_6$ is the smallest non-abelian simple group with order at least $200$ (up to isomorphism). For $\ell {\geqslant}360$, it is easy to check that $f(2,\ell)$ is increasing as a function of $\ell$ and we have $f(2,360)>1$. Similarly, if $k {\geqslant}3$ then $\ell {\geqslant}60$ and $f(k,\ell)$ is an increasing function in both $k$ and $\ell$. The result now follows since $f(3,60)>1$. Almost simple subgroups {#ss:as} ----------------------- To complete the proof of Theorem \[t:main\] for symmetric and alternating groups, we may assume that $T = A_m$ with $m>25$ and $H$ is an almost simple subgroup acting primitively on $\Gamma = \{1, \ldots, m\}$. We will write $S$ to denote the socle of $H$ (note that $S \ne T$ since $H$ is a core-free subgroup of $G$). First we handle the low degree groups with $m {\leqslant}600$. \[p:as1\] If $25<m {\leqslant}600$ then ${\rm ifix}(T)>n^{4/9}$. To construct $H$ as a subgroup of $G$ we use the database of primitive groups in [Magma]{}, via the command $$\texttt{PrimitiveGroups([26..600] : Filter:="AlmostSimple")}.$$ Once we have removed the groups with $S = T$, we are left with $766$ cases to consider. Define $${\alpha}(J) = \max\left\{\frac{\|t^{J}\|}{\|t^T\|} \,:\, \mbox{$t \in J$ is an involution}\right\}$$ for each subgroup $J$ of $H_0$. Given a specific subgroup $J$, we can compute ${\alpha}(J)$ by finding a set of representatives for the conjugacy classes of involutions in $J$ and then for each representative $t$ we compute the number of fixed points of $t$ on $\{1, \ldots, m\}$, which allows us to calculate $\|t^T\|$. Note that ${\rm ifix}(T) {\geqslant}{\alpha}(J)n$ For $m {\leqslant}60$ it is easy to check that ${\alpha}(H_0)>n^{-5/9}$ and thus ${\rm ifix}(T) > n^{4/9}$. Similarly, if $60<m{\leqslant}600$ and $P$ is a Sylow $2$-subgroup of $H_0$, then ${\alpha}(P)>n^{-5/9}$ and the result follows (this approach avoids the problem of computing a set of conjugacy class representatives in $H_0$, which can be expensive in terms of time and memory). For the remainder, we may assume $m>600$. Our basic aim is to establish the bound $$\label{e:T} \|H_0\|^{100}<\|T\|$$ whenever possible, noting that this gives ${\rm ifix}(T)>n^{4/9}$ via Lemma \[l:cor\]. To do this, it will be convenient to make a distinction between the cases where $H$ is standard or non-standard, according to the following definition. \[d:std\] Let $H {\leqslant}{\rm Sym}(\Gamma)$ be an almost simple primitive group with socle $S$ and point stabilizer $K$. Then $H$ is standard if one of the following holds: - $S=A_{k}$ is an alternating group and $\Gamma$ is a set of subsets or partitions of $\{1,\ldots, k\}$. - $S$ is a classical group with natural module $V$ and $\Gamma$ is a set of subspaces (or pairs of subspaces) of $V$. - $S = {\rm Sp}_{2d}(q)$, $q$ is even and $K \cap S = {\rm O}_{2d}^{\pm}(q)$. In all other cases, $H$ is non-standard. This definition facilitates the statement of the following key result of Liebeck and Saxl (see [@LSa2 Proposition 2]). \[p:bd\] Let $H {\leqslant}{\rm Sym}(\Gamma)$ be a non-standard almost simple primitive group of degree $m {\geqslant}25$. Then $\|H\|<m^5$. With this proposition in hand, we can very quickly reduce the problem to the cases where $H$ is standard. \[p:as2\] If $m>600$ and $H$ is non-standard, then ${\rm ifix}(T)>n^{4/9}$. Here $\|H\|<m^5$ by Proposition \[p:bd\] and one can check that $2m^{500} < m!$ (since $m>600$). Therefore holds and the result follows. \[p:as3\] If $m>600$, $H$ is standard and $S$ is alternating, then ${\rm ifix}(T)>n^{4/9}$. Write $S=A_k$ and first assume that the embedding of $H$ in $G$ is afforded by the action of $H$ on the set of $\ell$-element subsets of $\{1, \ldots, k\}$, so $m = \binom{k}{\ell}$ and $2 {\leqslant}\ell < k/2$. Note that $k {\geqslant}12$ since $m>600$. Now $m {\geqslant}\binom{k}{2} = \frac{1}{2}k(k-1)$ and it is straightforward to check that $$\|H_0\|^{100} {\leqslant}(k!)^{100} < \frac{1}{2}\left(\frac{1}{2}k(k-1)\right)! {\leqslant}\|T\|$$ for all $k {\geqslant}98$. Similarly, if $k {\leqslant}97$ and $\ell {\geqslant}3$ then $m {\geqslant}\binom{k}{3}$ and one checks that holds, so we may assume that $\ell=2$, $36 {\leqslant}k {\leqslant}97$ and $m = \frac{1}{2}k(k-1)$. Here we compute $$\label{e:al} {\alpha}= \frac{\log k!}{\log \|T\|},\;\; {\beta}= \frac{\log \gamma}{\log \|T\|},$$ where $$\gamma = \max\left\{\frac{m!}{2^{2j}(2j)!(m-4j)!}\,:\, 1 {\leqslant}j {\leqslant}m/4\right\}$$ is the size of the largest conjugacy class of involutions in $T$. One checks that $5-5{\alpha}-9{\beta}>0$ in each case, whence ${\rm ifix}(T)>n^{4/9}$ by Lemma \[l:basic\]. Now assume that the embedding of $H$ corresponds to the action on the set of partitions of $\{1, \ldots, k\}$ into $r$ subsets of size $\ell$, where $1<\ell<k$. Here $m = \frac{k!}{(\ell!)^rr!}$ and the condition $m>600$ implies that $k {\geqslant}10$. It is easy to check that $m {\geqslant}\binom{k}{4}$ and by arguing as in the previous paragraph we deduce that holds if $k {\geqslant}12$. The same bound also holds when $k=10$ since $r=5$, $\ell=2$ and $m=945$. In order to complete the proof of Theorem \[t:main\] for $T = A_m$ we may assume that $m>600$ and $H$ is an almost simple classical group over $\mathbb{F}_q$ with socle $S$. Let $V$ be the natural module for $S$ and set $\ell = \dim V$. In view of Proposition \[p:as2\], we may also assume that $H {\leqslant}{\rm Sym}(\Gamma)$ is a standard group, which means that $\Gamma$ is either a set of subspaces (or pairs of subspaces) of $V$, or $S = {\rm Sp}_{\ell}(q)$, $q$ is even and $\Gamma$ is the set of cosets of a subgroup ${\rm O}_{\ell}^{\pm}(q)$ of $S$ (see Definition \[d:std\]). Let $K$ be a point stabilizer for the action of $H$ on $\Gamma$, so $m = \|H:K\|$. \[r:cond\] Due to the existence of a number of exceptional isomorphisms among the low dimensional classical groups, we may assume that $S$ is one of the following: $${\rm L}_{\ell}(q),\, \ell {\geqslant}2; \; {\rm U}_{\ell}(q), \, \ell {\geqslant}3; \; {\rm PSp}_{\ell}(q), \, \ell {\geqslant}4;\; {\rm P{\Omega}}_{\ell}^{{\epsilon}}(q),\, \ell {\geqslant}7.$$ In addition, in view of the isomorphisms $${\rm L}_{2}(4) \cong {\rm L}_{2}(5) \cong A_5, \; {\rm L}_{2}(9) \cong {\rm PSp}_{4}(2)' \cong A_6,\; {\rm L}_{3}(2) \cong {\rm L}_{2}(7),\; {\rm L}_{4}(2) \cong A_8,\; {\rm PSp}_{4}(3) \cong {\rm U}_{4}(2)$$ (see [@KL Proposition 2.9.1]), we may assume that $$S \ne {\rm L}_{2}(4), \, {\rm L}_{2}(5), \, {\rm L}_{2}(9), \, {\rm L}_{3}(2), \, {\rm L}_{4}(2), \, {\rm PSp}_{4}(2)', \, {\rm PSp}_{4}(3).$$ \[p:as4\] If $m>600$, $H$ is standard and $S$ is classical, then ${\rm ifix}(T)>n^{4/9}$. We adopt the set-up introduced above, including the conditions on $S$ presented in Remark \[r:cond\]. Write $q=p^f$, where $p$ is a prime. We will prove that holds unless $(H,m) = ({\rm L}_{10}(2), 2^{10}-1)$. Since $m = \|S:S \cap K\|$ it follows that $m {\geqslant}P(S)$, where $P(S)$ is the minimal degree of a nontrivial permutation representation of $S$. The minimal degrees are presented in [@GMPS Table 4] (which corrects a couple of slight errors in [@KL Table 5.2.A]) and by inspection we deduce that $m > q^{\ell-2}$. Similarly, the order of ${\rm Aut}(S)$ is recorded in [@KL Table 5.1.A] and it is easy to see that $\|H\| {\leqslant}\|{\rm Aut}(S)\|<2q^{\ell^2}$. If $S = {\rm L}_{2}(q)$ then $m > \max\{600,q\}$, $\|H\| {\leqslant}q(q^2-1) \log_pq$ and it is routine to verify the bound in . Similarly, if $\ell=3$ then $S = {\rm L}_{3}^{{\epsilon}}(q)$, $m > \max\{600,q^2+q\}$, $$\|H\| {\leqslant}\|{\rm Aut}({\rm U}_{3}(q))\| = 2q^3(q^2-1)(q^3+1)\log_pq$$ and we quickly deduce that holds. Now assume $\ell {\geqslant}4$. If $q {\geqslant}31$ then one checks that $$\label{e:ell} 2^{101}q^{100\ell^2} < (q^{\ell-2})!$$ and this establishes the bound in . More precisely, if $\ell {\geqslant}5$ then the inequality in is satisfied unless $(\ell,q)$ is one of the following: $$\label{e:cases} \ell = 5, \, q {\leqslant}9; \; \ell=6, \, q {\leqslant}5; \; \ell=7, \, q {\leqslant}4; \; \ell=8, \, q {\leqslant}3; \; \ell=9,10,11,12, \, q = 2.$$ Suppose $\ell=4$ and $q {\leqslant}29$. If $S = {\rm PSp}_{4}(q)$ then $q {\geqslant}4$ (see Remark \[r:cond\]), $m > \max\{600,q^2\}$ and $$\|H\| {\leqslant}\|{\rm Aut}({\rm PSp}_{4}(q))\| {\leqslant}2q^4(q^2-1)(q^4-1)\log_pq,$$ which implies that holds. Now assume $S = {\rm L}_{4}^{{\epsilon}}(q)$, so $$\|H\| {\leqslant}\|{\rm Aut}({\rm U}_{4}(q))\| {\leqslant}2q^6(q^2-1)(q^3+1)(q^4-1)\log_pq.$$ If $m> \max\{600,q^4\}$ then holds, so let us assume $600 < m {\leqslant}q^4$. By inspecting [@BG Table 4.1.2], which records the degree of every standard classical group, we deduce that $S = {\rm L}_{4}(q)$ and $m=(q^4-1)/(q-1)$ is the only possibility, so $q {\geqslant}9$ and we get $\|{\rm Aut}(S)\|^{100}<\|T\|$. Very similar reasoning establishes the bound in for all the remaining cases in with $\ell {\leqslant}9$, so to complete the proof we may assume that $\ell \in \{10,11,12\}$ and $q=2$. If $\ell \in \{11,12\}$ and $S = {\rm L}_{\ell}^{{\epsilon}}(2)$ then $m {\geqslant}2^{\ell}-1$ and we deduce that holds. Similarly, if $\ell=12$ and $S \ne {\rm L}_{12}^{{\epsilon}}(2)$ then the bound $m > 2^{10}$ is sufficient. Finally, let us assume $\ell=10$. If $S \ne {\rm L}_{10}^{{\epsilon}}(2)$ then $\|H\| {\leqslant}\|{\rm Sp}_{10}(2)\|$ and one checks that the condition $m>600$ implies that $m {\geqslant}2^{10}-1$, which allows us to verify the bound in . Now assume $S = {\rm L}_{10}^{{\epsilon}}(2)$, so $\|H\| {\leqslant}2\|{\rm U}_{10}(2)\|$. If $m>2^{11}$ then $\|H\|^{100}<\|T\|$, so let us assume $m {\leqslant}2^{11}$, in which case $H = {\rm L}_{10}(2)$ and $m = 2^{10}-1$ (so $\Gamma$ is the set $1$-dimensional subspaces of $V$). Here $\|H\|^{100} > \|T\|$, but if we define ${\alpha}= \log \|H\| /\log \|T\|$ and ${\beta}$ as in , then it is easy to check that $5-5{\alpha}-9{\beta}>0$ and thus ${\rm ifix}(T)>n^{4/9}$ by Lemma \[l:basic\]. Sporadic groups {#s:spor} =============== In this final section we complete the proof of Theorem \[t:main\] by handling the groups where the socle $T$ is a sporadic simple group. Our first result quickly reduces the problem to the Baby Monster and Monster (denoted by $\mathbb{B}$ and $\mathbb{M}$, respectively). \[l:spor1\] The conclusion to Theorem \[t:main\] holds if $T \ne \mathbb{B}, \mathbb{M}$ is a sporadic group. This is an easy computation using the Character Table Library [@GAPCTL]. In each case, the character table of $G$ is available in [@GAPCTL] and we use the `Maxes` function to access the character table of the maximal subgroup $H$. In addition, [@GAPCTL] stores the fusion map from $H$-classes to $G$-classes, which allows us to compute ${\rm fix}(t)$ via for all $t \in G$. In particular, we can compute ${\rm ifix}(T)$ precisely and the result follows. To complete the proof of Theorem \[t:main\], we may assume that $T = \mathbb{B}$ or $\mathbb{M}$. In both cases, we claim that ${\rm ifix}(T)>n^{4/9}$. \[l:spor2\] The conclusion to Theorem \[t:main\] holds if $T = \mathbb{B}$. Here $G = T$ is the Baby Monster and we proceed as in the proof of the previous proposition, noting that the character tables of $G$ and $H$ are available in [@GAPCTL] (as before, we use the `Maxes` function to access the character table of $H$). In addition, in all but one case, the fusion map from $H$-classes to $G$-classes is also stored and this reduces the analysis to the case $H = (2^2 \times F_4(2)).2$. Here we use the function `PossibleClassFusions` to determine a set of candidate fusion maps (there are $64$ such maps in total) and for each possibility one checks that ${\rm ifix}(T) = 1609085288448 > n^{4/9}$. \[l:spor3\] The conclusion to Theorem \[t:main\] holds if $T = \mathbb{M}$. Let $G = T = \mathbb{M}$ be the Monster. By inspecting the <span style="font-variant:small-caps;">Atlas</span> [@Atlas], we see that $G$ has two conjugacy classes of involutions, labelled `2A` and `2B`, where $$\|\texttt{2A}\| = 97239461142009186000,\;\; \|\texttt{2B}\| = 5791748068511982636944259375.$$ As discussed in [@Wil], $G$ has $44$ known conjugacy classes of maximal subgroups and any additional maximal subgroup is almost simple with socle one of ${\rm L}_{2}(8)$, ${\rm L}_{2}(13)$, ${\rm L}_{2}(16)$ or ${\rm U}_{3}(4)$. Let us define the following three collections of known maximal subgroups of $G$: $$\begin{aligned} \mathcal{A} & = \{2^{10+16}.{\Omega}_{10}^{+}(2), 2^{2+11+22}.({\rm M}_{24} \times S_3), 2^{5+10+20}.(S_3 \times {\rm L}_{5}(2)), 2^{3+6+12+18}.({\rm L}_{3}(2) \times 3S_6) \} \\ \mathcal{B} & = \{ 3^8.{\rm P{\Omega}}_{8}^{-}(3).2, (3^2{:}2 \times {\rm P{\Omega}}_{8}^{+}(3)).S_4, 3^{2+5+10}.({\rm M}_{11} \times 2S_4), 3^{3+2+6+6}.({\rm L}_{3}(3) \times {\rm SD}_{16})\} \\ \mathcal{C} & = \{({\rm L}_{2}(11) \times {\rm L}_{2}(11)){:}4, 11^2{:}(5 \times 2A_5), 7^2{:}{\rm SL}_{2}(7), {\rm L}_{2}(29){:}2, {\rm L}_{2}(19){:}2\}\end{aligned}$$ First assume $H$ belongs to one of the $44$ known classes of maximal subgroups. If $H$ is not contained in $\mathcal{A}$, $\mathcal{B}$ or $\mathcal{C}$, then we use the function `NamesOfFusionSources` to access the character table of $H$ in and in each case we can work with the stored fusion map from $H$-classes to $G$-classes. This allows us to compute ${\rm ifix}(T)$ as in the proof of Proposition \[l:spor1\] and it is straightforward to verify the desired bound. Now assume $H$ is one of the subgroups in $\mathcal{C}$. Here the character table of $H$ is available in , but the fusion map is not stored. So in these cases we proceed as in the proof of Proposition \[l:spor2\], using the function `PossibleClassFusions`. In each case, we find that ${\rm ifix}(T)$ is independent of the choice of fusion map and we calculate that ${\rm ifix}(T)>n^{4/9}$. Next suppose $H \in \mathcal{A} \cup \mathcal{B}$. If $H \in \mathcal{A}$ then [@BOW Proposition 3.9] gives $\|t^G \cap H\|$, where $t$ is contained in the `2A` class of involutions in $G$. This allows us to compute ${\rm fix}(t)$ precisely and we deduce that ${\rm ifix}(T)>n^{4/9}$. Now assume $H \in \mathcal{B}$ and let ${\alpha}$ be the size of the largest conjugacy class of involutions in $H$. We use [Magma]{} to compute ${\alpha}$, working with a representation of $H$ given in [@WebAt]. For $H = 3^8.{\rm P{\Omega}}_{8}^{-}(3).2$, this is a matrix representation of dimension $204$ over $\mathbb{F}_3$ and we use `LMGClasses` to compute a set of conjugacy class representatives; in the remaining cases, we work with a permutation representation of degree less than $10^5$. Now, if ${\beta}$ is the size of the `2B` class of involutions in $G$ (see above), then ${\rm ifix}(T) {\geqslant}n{\alpha}{\beta}^{-1}$ and in each case it is easy to check that that this lower bound is greater than $n^{4/9}$. For example, if $H = 3^8.{\rm P{\Omega}}_{8}^{-}(3).2$ then ${\alpha}= 1982806371$ and the above bound yields ${\rm ifix}(T) > n^{4/9}$. To complete the proof of the proposition, we may assume $H$ is an almost simple maximal subgroup with socle $S = {\rm L}_{2}(8)$, ${\rm L}_{2}(13)$, ${\rm L}_{2}(16)$ or ${\rm U}_{3}(4)$. Let ${\alpha}$ be the size of the largest class of involutions in $S$ and define ${\beta}= \|\texttt{2B}\|$ as above. Then one checks that $$\left(\frac{\|T\|}{\|{\rm Aut}(S)\|}\right)^{5/9} > \frac{{\beta}}{{\alpha}}$$ and since $H {\leqslant}{\rm Aut}(S)$, we immediately deduce that ${\rm ifix}(T)>n^{4/9}$ as required. This completes the proof of Theorem \[t:main\]. [999]{} J. Bamberg, T. Popiel and C.E. Praeger, Simple groups, product actions, and generalized quadrangles, Nagoya Math. J. 234 (2019), 87–126. H. Bender, Transitive Gruppen gerader Ordnung, in denen jede Involution genau einen Punkt festlässt, J. Algebra 17 (1971), 527–554. W. Bosma, J. Cannon and C. Playoust, The [<span style="font-variant:small-caps;">Magma</span>]{} algebra system I: The user language, J. Symb. Comput. 24 (1997), 235–265. T. Breuer, The haracter [T]{}able [L]{}ibrary, Version 1.2.1, package,\ `http://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib`, 2012. T.C. 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--- abstract: 'The Free-Electron Laser Laboratory at the University of Hawai‘i has constructed and tested a scanning wire beam position monitor to aid the alignment and optimization of a high spectral brightness inverse-Compton scattering x-ray source. X-rays are produced by colliding the electron beam from a pulsed S-band linac with infrared laser pulses from a mode-locked free-electron laser driven by the same electron beam. The electron and laser beams are focused to diameters at the interaction point to achieve high scattering efficiency. This wire-scanner allows for high resolution measurements of the size and position of both the laser and electron beams at the interaction point to verify spatial coincidence. Time resolved measurements of secondary emission current allow us to monitor the transverse spatial evolution of the e-beam throughout the duration of a macro-pulse while the laser is simultaneously profiled by pyrometer measurement of the occulted infrared beam. Using this apparatus we have demonstrated that the electron and laser beams can be co-aligned with a precision better than as required to maximize x-ray yield.' author: - \| M. R. Hadmack[^1] and E. B. Szarmes\ University of Hawai‘i Free-Electron Laser Laboratory, Honolulu, HI 96822, USA title: 'SCANNING WIRE BEAM POSITION MONITOR FOR ALIGNMENT OF A HIGH BRIGHTNESS INVERSE-COMPTON X-RAY SOURCE[^2]' --- INTRODUCTION ============ A compact high brightness x-ray source is currently under development at the University of Hawai‘i Free-Electron Laser Laboratory, based on inverse-Compton scattering of electron bunches with synchronous laser pulses from an infrared free-electron laser (FEL)[@hadmack2012phd; @madey2013oce]. One of the more challenging aspects of realizing a Compton backscatter x-ray source is co-alignment of the electron and laser beams. With high intensities, and spot sizes as small as , it is not possible to align these beams without special diagnostic tools. The resolution of available beam position monitors (BPMs) and optical transition radiation (OTR) screens is limited to about by the sampling electronics and video cameras used. Wire scanners are commonly employed on accelerator beam-lines as alignment aides. The “flying wire” type scanners, such as those used at CERN, are too large for use in the space allocated on the Mk V beam-line at UH and are incompatible with the bunch structure of our accelerator. The x-ray interaction point is shared by two other insertable diagnostic devices in a crowded vacuum chamber, also housing the x-ray interaction point laser optics. The wire scanner described here is based on the designs used at NBS-LANL[@cutler1987performance] and the SLC[@ross1991wire] and adapted to the constraints of our beamline configuration. This system also includes the capability to resolve the temporal evolution of the electron beam profile over the macropulse duration (approximately ). HARDWARE ======== The wire scanner head shown in Fig. \[fig:swbpm\_fork\_photo\] consists of two -diameter graphite fibers stretched across the gap in an aluminum fork. The wires are oriented such that when the scanner insertion axis is inclined above the beam plane, the two wires are oriented horizontally and vertically. In this way a single axis of motion allows the beam to be scanned in both axes. ![The wire scanner fork electrically isolates the carbon fiber from the grounded fork. Fibers are soldered to the signal lead and clamped at the other end.[]{data-label="fig:swbpm_fork_photo"}](swbpm_fork_photo.jpg){width="75mm"} The secondary emission current from the wire is conducted via the scanner shaft to a vacuum feedthrough on the assembly shown in Fig. \[fig:swbpm\_assembly\_photo\]. The fork itself is grounded to avoid charge accumulation from the beam halo. ![Beam profiler drive assembly with motor and LVDT with an early prototype fork. The support rod conducts the signal to the vacuum feedthrough on the far end.[]{data-label="fig:swbpm_assembly_photo"}](swbpm_assembly_photo.jpg){width="80mm"} ![The wire scanner installed with other diagnostic devices in the x-ray scattering chamber.[]{data-label="fig:chamberinstalled"}](chamberinstalled.jpg){width="85mm"} Most wire scanners in operation today utilize bremsstrahlung radiation detectors to measure beam interception of the wire. However, on a linear machine it is more difficult to position a PIN diode detector close to the source without substantial background radiation. Moving the detector to a suitably shielded location results in a large bremsstrahlung beam diameter, making efficient detection difficult and introducing errors due to diffraction. Figure \[fig:chamberinstalled\] shows the assembly integrated in the x-ray scattering chamber installed at the interaction point. The wire scanner assembly consists of a precision vacuum translation stage, a linear-variable-differential-transformer (LVDT) position sensor, and a DC motor drive. The motor speed is controlled by software to achieve high resolution at the beam repetition rate. Position is measured with an LVDT attached to the translation stage; its resolution is limited by 12 bit readout electronics to steps over a range. The LVDT read-back is calibrated against the actual translation stage motion with calipers. Wire scans are typically performed with the actuator speed set to so that the position changes by twice the LVDT limiting step size each macropulse event, thus ensuring monotonic position data. A full scan using both the horizontal and vertical wire takes approximately . For each accelerator macropulse trigger three quantities are measured: the current from the wire, the laser pulse transmission, and the position. The intercepted electron beam current is inferred by the current resulting from secondary electrons ejected from the wire. The wire current signal is terminated in and sampled with a digital oscilloscope. A pyroelectric detector viewing the transmitted beam measures the occlusion of the laser beam by the wire. The pyrometer’s response time is considerably slower than that of the wire current. The pulse peak voltage is sampled with a boxcar integrator and digitized with 12 bit precision. The data acquisition software is implemented in Python with a graphical user interface (GUI) built using wxPython. The software acquisition is triggered using control lines on an RS232 serial port to monitor the accelerator’s TTL trigger. When a trigger event is detected, data is read from the oscilloscope and boxcar integrator, both of which are synchronously triggered. Data is also acquired asynchronously from the LVDT controller via a serial connection. Figure \[fig:swbpm\_blockdiagram\] illustrates the data acquisition system. The GUI shown in Fig. \[fig:swbpm\_gui\] provides the operator with a live stripchart of both the laser and current measurements throughout a scan. The GUI allows for configuration and control of automated scans and data storage. Data is stored in a custom binary format and includes full oscilloscope waveforms for every position step. ![A PC acquires beam current, laser intensity, and scan position data and controls the drive motor.[]{data-label="fig:swbpm_blockdiagram"}](swbpm_blockdiagram.eps){width="65mm"} ![The GUI displays the electron beam (yellow) and laser (purple) beam profiles in real time during a scan. Figure shows background data, not actual scan results.[]{data-label="fig:swbpm_gui"}](swbpm_gui.png){width="80mm"} The GUI stripchart is only used as a rough guide for scan operation while detailed data analysis is performed using an offline tool, also implemented in Python. Figure \[fig:scan1\] shows a sample wire scan analysis. The graphic in the upper part of the figure is a representation of the evolution of the beam current spatial distribution over a macropulse. The vertical columns in the image are individual oscilloscope waveforms for each position along the horizontal axis; interpolation is applied to account for non-uniformly spaced positions. The lower plot shows the transmitted laser pulse energy compared to the wire current integrated over a particular duration of interest within the pulse. The integration region is typically chosen to overlap the laser pulse in the last of the macropulse. ![Secondary emission current as a function of position and time within the macropulse. The lower plot compares the integrated current from with the laser signal (axis inverted). This scan includes both the horizontal (right) and vertical (left) profiles.[]{data-label="fig:scan1"}](130719_154103.png){width="85mm"} RESULTS ======= Preliminary experiments have been conducted to measure the sizes, positions, and stability of the laser and electron beams. The wire scanner was initially commissioned with a tungsten wire. This wire was operated successfully for several months with large beam diameters and limited resolution; however, a microfocused (sub diameter) beam quickly severed the wire. Next, a tungsten wire was chosen to reduce the absorption volume and to enhance the spatial resolution of the scans by a factor of eight. Again, however, a microfocused beam destroyed the wire on the first pass. Ultimately, -diameter carbon monofilament from Specialty Materials, Inc. has proven robust enough to endure the highly focused electron beam and several of infrared laser exposure. Since the carbon filament could not be wrapped in the same manner as the more flexible tungsten, it was necessary to modify the wire scanner fork. Figure \[fig:swbpm\_fork\_photo\] illustrates how the carbon fibers are clamped on one end between Vespel plastic discs while the other ends are soldered to a copper tab attached to the signal lead. The scan data in Fig. \[fig:scan1\] shows an electron beam focused to $w_x,w_y = \SI{350}{\um},\SI{115}{\um}$, where the left feature is the evolution of the vertical beam profile and the right represents the horizontal. The total charge intercepted on each wire is the same, so the area under the beam profile curves is constant, resulting in lower peak signal for the horizontal scan. In this scan the laser is well aligned to the electron beam resulting in suppression of laser operation while the beam is intercepted by the wire. ![Horizontal axis wire scan showing the laser displaced from the e-beam position. Interception of the e-beam by the wire inhibits lasing and results in a second co-aligned peak.[]{data-label="fig:scan2"}](130718_161013.png){width="85mm"} ![image](vault_layout.eps){width="160mm"} Figure \[fig:scan2\] shows an example where the laser is misaligned from the electron beam. This horizontal scan (vertical wire only) gives an electron beam width of and a laser beam width of with a beam separation of , correctable with a motorized mirror. The laser was intentionally defocused at the interaction point for this scan to preclude wire damage during these early experiments. It is interesting to note that while the laser size and position are measurable by transmission to the pyroelectric detector (red trace), a laser signal also appears distinctly in the wire current. We believe that this laser induced wire current is the result of thermionic emission from the wire due to heating from laser exposure. This hypothesis is supported by the observation that the laser induced current extends several hundred beyond both the end of the electron beam and laser macropulses while in fact the laser beam exposure of the wire begins a microsecond earlier and is therefore presumed to be a thermal artifact. The photon energy of the laser is not sufficient to generate photoelectrons. This feature provides a useful way to measure both laser and electron beams with a single sensor signal. Figure \[fig:scan2\] also illustrates the time resolved nature of this wire scanner system. The trajectory in this image indicates that the beam’s horizontal position slews nearly a over the macropulse. The large slew in the first microsecond is an inevitable consequence of beam-loading in the linear accelerator and is typically ignored for experimental purposes. The remainder of the macropulse, however, also shows a position slew that is the consequence energy slew in the beam. The diagnostic chicane shown in Fig. \[fig:markv\] contains a number dipole bend magnets upstream of the interaction point that produce energy dependent deflections in the beam. Characterization and mitigation of this energy/position slew in the beam are of critical importance to operation of the free-electron laser and for beam alignment of the inverse-Compton scattering interaction. A naïve integration of the wire current or a transition radiation image would significantly overestimate the instantaneous beam size as well as produce an ambiguity in the centroid position during the time interval of interest for scattering. Even with a significant transverse evolution in the beam, the instantaneous size and position can be precisely measured. The “flying wire” type beam profilers employed on many large accelerators and storage rings employ a high velocity wire capable of scanning many stored bunches in a single sweep[@igarashi2002aa]. While this technique enables much faster profile acquisition, the position becomes correlated with a particular time within the bunch train. Thus, these systems are not capable of revealing the temporal structure of the macropulse in the manner described above and can overestimate the beam size. Typically this is not a problem for a storage ring that is filled with nearly identical bunches. However, the transient beam loading experienced in a linac with a thermionic gun results in beam evolution that must be considered. Wire scan repeatability was measured from the analysis of a scan sequence with the same e-beam configuration. The beam centroid can be measured with an uncertainty of $\sigma_{x,y} = \SI{9}{\um}$ and a beam width uncertainty of $\sigma_w = \SI{4}{\um}$. Scans are always performed in the same direction to eliminate hysteresis due to a backlash in the translator lead-screw. CONCLUSION ========== A scanning wire beam position monitor has been successfully constructed and operated at the University of Hawaii Free-Electron Laser Laboratory. This custom design satisfies the tight space restrictions imposed by the need to share access to the interaction point of the inverse-Compton x-ray source with other diagnostic devices and laser optics. The use of a commercially available linear vacuum translator significantly reduces the engineering time and cost of the system. carbon fiber has been selected as a suitable material for scans of a sub- microfocused electron beam operated at with average current over macropulses. Further studies of the carbon filament damage threshold for both electron beam and laser exposure are necessary for effective optimization of the inverse-Compton scattering interaction point focus. The laser can easily be attenuated to avoid wire damage once this is known. However, the electron beam current cannot be varied and the damage threshold will impose a limit to the average current density allowed on the wire. In principle, the width resolution is sufficient to achieve the focal spot specification of the UH x-ray source. Alignment of the beams can be verified with a scan repeatability of better than when the hysteresis is accounted for. Combining time-resolved wire scans with quadrupole magnet scans will give us the capability to perform time-resolved emittance measurements of the e-beam. This will be a vital capability for our continued efforts to improve extended pulse length thermionic electron gun technology[@kowalczyk2013las]. ACKNOWLEDGMENT ============== We acknowledge [Specialty Materials, Inc.](http://specmaterials.com/) for providing the carbon monofilament and John M. J. Madey for his advice and operational support on this project. [9]{} M.R. Hadmack. Ph.D. thesis, University of Hawai‘i, 2012. J. M. J. Madey et al. , 2013. R.I. Cutler, J. Owen, and J. Whittaker. , p. 625, 1987. M.C. Ross et al. , 1991. S. Igarashi et al. , 482(1–2):32, 2002. J. M. D. Kowalczyk, M. R. Hadmack, and J. M. J. Madey. , 2013. [^1]: hadmack@hawaii.edu [^2]: Work supported by the US Department of Homeland Security DNDO ARI program GRANT NO. 2010-DN-077-ARI045-02
--- abstract: 'We consider a diffusion model with limit cycle reaction functions. In an unbounded domain, diffusion spreads pattern outwards from the source. Convection adds instability to the reaction-diffusion system. The result of this instability is a readiness to create pattern. We choose the Lambda-Omega reaction functions for their simple limit cycle. We carry out a transformation of the dependent variables into polar form. From this we consider the initiation of pattern to approximate a travelling wave. We carry out numerical experiments to test our analysis. These confirm the premise of the analysis, that the initiation can be modelled by a travelling wave. Furthermore, the analysis produces a good estimate of the numerical results. Most significantly, we confirm that the pattern consists of two different types.' author: - 'E. H. Flach' - 'S. Schnell' - 'J. Norbury' title: \| Limit cycles in the presence of convection,\ a travelling wave analysis --- Introduction ============ Morphological patterning, such as animal coat markings, may be caused by a chemical field [@Turing1952The-Chemical-Ba; @Murray1981A-Pre-pattern-f; @Murray1981On-Pattern-Form]. The Turing model now has strong experimental support [@Maini2006The-Turing-Mode for a review], [@Sick2006WNT-and-DKK-Det; @Jung1998Local-inhibitor]. This mechanism has some limitations, and so we continue to investigate variations of the model [@Maini1996Spatial-and-spa]. We consider the standard reaction-diffusion model with the addition of convection [@Bamforth2001Flow-distribute; @Bamforth2000Modelling-flow-; @Kuznetsov1997Absolute-and-co]. The more general form is a system with advection, as in [@Andresen1999Stationary-spac]. Our motivation is theoretical, to see the effect of convection on the robustness of the pattern formation. We consider that a weak amount of convection, or a similar effect, may be present in vivo. This may soften the standard requirements for the formation of pattern, increasing the applicability of the model. There may be a direct biological application, such as the formation of the vertebral precursors (somitogenesis). The organism growth could produce such a convective effect [@Kaern2001Chemical-waves-]. However, convection-induced patterning is not generally supposed for somitogenesis [@Schnell2002Models-for-patt]. Our system does not fit the standard Turing analysis since our reaction functions are already unstable. However, we have shown previously that an unstable function may produce a Turing pattern [@Flach2007Turing-pattern-]. Furthermore, we have shown that, in the presence of convection, we can see similar behaviour irrespective of the stability of the fixed point [@Flach2007Turing-pattern-]. Experimentally, convection is introduced from the boundaries [@Kaern1999Flow-distribute; @Miguez2006Robustness-and-]. The effect of boundary conditions is also likely to be relevant in vivo. The boundary can have a significant effect on the pattern formed [@Bamforth2000Modelling-flow-]. However, we follow the suggestion of Cross and Hohenberg to examine the system in a boundary-free environment first [@Cross1993Pattern-formati]. In numerical simulations we make form an initial point disturbance. Pattern is formed, spreading outwards from its initiation point. The disturbance is oscillatory and complex. Our focus here is the way in which the pattern propagates. The speed of propagation has been given for a two-species system with no convection [@Bricmont1994Stability-of-mo] and one with equal convection [@Bamforth2000Modelling-flow-]. A related theoretical study considers the relationship between the onset of the instability and the longer-term behaviour of the non-equilibrium state [@Sherratt1998Invading-wave-f]. Here the simplest reaction-diffusion system with oscillatory kinetics was considered, and complex behaviours were found. Limit cycles are inherent in these oscillatory models, since an unstable spiral at the steady state must be bounded [@Schnakenberg1979Simple-chemical]. In this paper, we seek to exploit the limit cycle to aid our analysis. To this end, we select the Lambda-Omega reaction functions. These are chosen to produce the simplest limit cycle possible: the unit circle. The reaction functions chosen in a full model correspond to actual reaction mechanisms. Examples of these include the Belousov-Zhabotinsky reaction, and Brusselator-type systems such as the chlorine dioxide-iodine-malonic acid reaction. The Schnakenberg model can be considered an intermediate step towards these functions. For an initial investigation, we have chosen simple, related functions. The intention is to extend the analysis to more realistic models. Secondarily, the results in this simple case may form a useful basis for comparison to more complex systems. We transform the problem so that the oscillations are removed, or at least reduced. This brings the onset of the pattern into much sharper relief. The two aspects – the pattern, and the initiation of the pattern – are clearly distinguished. Thus this onset of pattern is effectively a phase transition. The transformation is to convert the dependent variables into polar form. We then consider that the formation of the limit cycle approximates a travelling wave, and so we employ a Fisher solution to the problem [@Wang1988Exact-and-expli]. The relative sizes of the parameters affects our analysis: we consider various situations. We carry out numerical experiments to discover which of these estimates are valid. Limit cycles and the $\lambda$-$\omega$ function ================================================ We suppose that some form of chemical mechanism is the underlying basis for biological pattern formation. If the chemicals are well-mixed, then the law of mass action valid, and an ODE is an appropriate model: $$\begin{aligned} u' &=& f \ , \nonumber\\ v' &=& g \ . \label{eq:ODE}\end{aligned}$$ ![Phase space for the limit-cycle reaction , , given by numerical solution. The phase curves spiral out from the steady state (the origin) to meet the limit cycle (the unit circle). The trajectories starting far away from the steady state spiral into the limit cycle. Solid curves starting with circles are the trajectories.[]{data-label="fig:ODE"}](ef05fig1.eps){width="4.2in"} We consider reaction functions chosen for their simplicity in the form of the limit cycle produced:$$\begin{aligned} f &=& -v +u(1-u^2-v^2) \ , \nonumber\\ g &=& u +v(1-u^2-v^2) \ . \label{eq:lambdaomegafunctions}\end{aligned}$$ This is a simple form of the $\lambda$-$\omega$ class of functions. Here the steady state is the origin and the limit cycle is the unit circle. There is no parameter to determine the stability: the steady state is unstable and the limit cycle stable. This behaviour is clear in the $(u,v)$ phase plane (see [Figure]{} \[fig:ODE\]). The circled points on the diagram show the start of the phase plane trajectories. The phase curves spiral out from the steady state to join the limit cycle. The trajectories starting far away from the steady state spiral into the limit cycle. These functions are chemically unrealistic, as they stand. However, we can relate these functions to the Schnakenberg reactions, chosen to be the simplest chemical form which can produce a limit cycle. This mechanism was proposed theoretically, but has been used as a model for actual reaction mechanisms [@Epstein1998An-Introduction; @Gray1994Chemical-Oscill]. The key reaction there is the cubic autocatalytic one, $U + 2V \rightarrow 3V$. Using the law of mass action, this produces the term $uv^2$. The reverse step gives $v^3$. These types of terms form the core of the $\lambda$-$\omega$ functions, the cubic terms. We consider a new coordinate system for the dependent variables ${u, v}$ as a polar form ${r, \theta}$ as follows:$$\begin{aligned} r^2 &=& u^2 +v^2 \ , \nonumber\\ \tan\theta &=& \frac{v}{u} \ . \end{aligned}$$ By differentiating these identities, we transform the first order ODE into:$$\begin{aligned} \dot{r} &=& r(1-r^2) \ , \nonumber\\ \dot{\theta} &=& 1 \ , \label{eq:polaridentities}\end{aligned}$$ The $\theta$ equation clearly decouples and is resolvable. The remaining single-species ODE in $r$ has steady states at $-1, 0, 1$. The negative state is unrealistic because $r$ must be positive, zero is unstable and one is stable. The point $r=1$ is our limit cycle: the unit circle. Furthermore, $\theta \approx t$, time and so progression around the limit cycle is constant. This type of analysis is often used in similar cases, such as the analysis of travelling wave trains [@Murray1989Mathematical-Bi]. Pattern formation {#sec:pattern} ================= Biological pattern formation is by definition spatially differentiated. There are many models for this. One is that chemicals diffuse within an organism, then cells respond differently dependent on the concentration of one of these chemicals. If the concentration of the chemicals has formed a pattern, then this is reflected by the cells. For an initial analysis, we choose the simplest case: one spatial dimension and some diffusion. By diffusion we refer to the averaged gross effect of random motion of the chemicals; with passive movement this is equivalent to Brownian motion. In the case of general reaction functions, this system is the one proposed by Turing [@Turing1952The-Chemical-Ba]. We add a convective term to the system. This can be considered a disturbance to the system, as an investigation of stability. For a small amount of convection, this could correspond to axial growth [@Kaern2001Chemical-waves-]. However, for convection to be significant in axial growth we would require extremely slow diffusion of the chemicals to fit the model. The general system is then as follows: $$\begin{aligned} u_t &=& \varepsilon_1 u_{\xi\xi} -p u_\xi+ f \ , \nonumber\\ v_t &=& \varepsilon_2 v_{\xi\xi} -q v_\xi + g \ . \label{eq:original}\end{aligned}$$ We consider different convection on each species ($p \neq q$). This effect could be created in a chemical flow reactor, with one of the reactants held fixed in a packed bed [@Kaern2001Chemical-waves-]. For biological applications, one of the chemicals may be held within a cell, while the other flows freely. If both chemicals flow freely, it is possible that their movement may be hampered by obstacles such as the extra-cellular matrix. In this case, a larger molecule may be affected more strongly than a smaller one, and different convection speeds may result for each chemical. This concept parallels that of different diffusion rates, as in the Turing model. We consider that $p>0$. If this is not the case, we make the transformation $(p, q, \xi) \rightarrow (-p, -q, -\xi)$, giving a positive value for $p$ with no change to the form of . We can remove one of the convective terms by a simple change of coordinates $x = \xi - pt$: $$\begin{aligned} u_t &=& \varepsilon_1 u_{xx} + f \ , \nonumber\\ v_t &=& \varepsilon_2 v_{xx} - \gamma v_x + g \ , \label{eq:full}\end{aligned}$$ with $f$ and $g$ as in and $\varepsilon_1$, $\varepsilon_2$ and $\gamma$ positive constants. From the original system we have $\gamma = q-p$. If $\gamma$ turns out to be negative, we can reverse the sign as we did for $p$ and $q$. Equivalently, we choose the coordinate change $x = pt -\xi$ previously, and set $\gamma = p-q$. In the case where the convection is the same on both species ($p=q$), then both convection terms are removed ($\gamma=0$). This recovers the basic Turing model. We have reaction functions that have limit-cycle behaviour and convection, which is known to drive instability. The appearance of pattern is then to be expected, although the form might be more difficult to predict. Satnoianu, Merkin and Scott [@Satnoianu1998Spatio-temporal] studied a similar system to previously, with Schnakenberg reaction functions in place of the $\lambda$-$\omega$ ones. They found that periodic behaviour is emergent in the system over a broad parameter range. ![Pattern found for a diffusion system with convection and limit-cycle reaction kinetics . The initial disturbance propagates and becomes pronounced, forming a regular pattern with aligned oscillations. The propagation is linear, forming a V-shape. The convective effect is slight: the pattern is skewed slightly to the right. In this case, the direction of the internal oscillations is distinct from the angle of propagation of the pattern. This is a numerical solution of using NAG D03PCF, plotting species $u$ with $\gamma=\varepsilon_1=\varepsilon_2=1$. The reactants are initially at steady state: $\left(u,v\right)=\left(0, 0 \right)$, with a small disturbance at $x=0$. The boundaries are held at zero derivative: $u_x=0$, $v_x=0$.[]{data-label="fig:pattern"}](ef05fig2.eps){width="4.2in"} In the numerical experiment, we start at the steady state $(0, 0)$, except for a small disturbance at $x=0$. We try to simulate a boundless environment – to this end we find zero derivative boundary conditions the most effective. The initial disturbance propagates and becomes pronounced, forming a regular pattern with aligned oscillations. The propagation is linear, forming a V-shape. The convective effect is to skew the pattern to the right ([Figure]{} \[fig:pattern\]). Given the parameters in the figure ($\gamma=\varepsilon_1=\varepsilon_2=1$), the direction of the internal oscillations is distinct from the angle of propagation of the pattern. The emergence of this pattern is the main focus of our study. Travelling wave analysis ======================== Before we start any specific analysis, we rescale the spatial variable $x$ to remove one of the parameters. We choose $x=\sqrt{\varepsilon_1}y$. This yields the system: $$\begin{aligned} u_t &=& u_{yy} + f \ , \nonumber\\ v_t &=& v_{yy} + \bar{\varepsilon} v_{yy} - \bar{\gamma} v_y + g \ , \label{eq:rescaled}\end{aligned}$$ where $\bar{\gamma} = \gamma/\sqrt{\varepsilon_1}$. Following suite, we expect $\bar{\varepsilon_2} = \varepsilon_2/\varepsilon_1$, but we go one step further in defining $\bar{\varepsilon} = \bar{\varepsilon_2} - 1$. This produces a symmetric first spatial derivative on both equations, with $\bar{\varepsilon}$ the difference between the two diffusion rates. We wish to again convert ${u, v}$ into the polar form $(r, \theta)$. During the conversion, we see that there is grouping of the terms parameterised by $\bar{\varepsilon}$ and $\bar{\gamma}$: $$\begin{aligned} r_t &=& r_{yy} - r\theta_y^2 + \frac{v}{r}[\bar{\varepsilon}v_{yy} - \bar{\gamma}v_y] + r(1-r^2) \ , \nonumber\\ \theta_t &=& \theta_{yy} + 2\frac{r_y}{r}\theta_y + \frac{u}{r^2}[\bar{\varepsilon}v_{yy} - \bar{\gamma}v_y] + 1 \ . \end{aligned}$$ We complete the transformation: $$\begin{aligned} r_t &=& \left(1+\bar{\varepsilon}\sin^2\theta\right)r_{yy} \nonumber\\ && + \left(2\bar{\varepsilon}\sin\theta\cos\theta.\theta_y - \bar{\gamma}\sin^2\theta\right)r_y \nonumber\\ && + \left(1 -r^2 -\theta_y^2 +\bar{\varepsilon}\sin\theta\cos\theta.\theta_{yy} -\bar{\varepsilon}\sin^2\theta.\theta_y^2 -\bar{\gamma}\sin\theta\cos\theta.\theta_y\right)r \ , \nonumber\\ \theta_t &=& \left(1 +\bar{\varepsilon}\cos^2\theta\right)\theta_{yy} \nonumber\\ && +\left(2\frac{r_y}{r} +2\bar{\varepsilon}\frac{r_y}{r}\cos^2\theta -\bar{\varepsilon}\sin\theta\cos\theta.\theta_y -\bar{\gamma}\cos^2\theta \right)\theta_y \nonumber\\ && +1 +\frac{\bar{\varepsilon}r_{yy} -\bar{\gamma}r_y}{r}\sin\theta\cos\theta \ . \end{aligned}$$ ![Polar form of the pattern. The new coordinate $r$ transitions sharply from $0$ to $1$. This is a travelling wave, propagating outwards. There are minor ripples in the established solution: the limit cycle settled on by the PDE is not the unit circle, but is close. The oscillation is at an angle to the travelling wave front. There is a difference in behaviour between the left and right sides of the pattern: the angle of alignment and the frequency of the pattern is different on either side. There is also a clear centre to the propagation, roughly at $x= \gamma/2 t$. This is a numerical solution of using NAG D03PCF, plotting species $u$ with $\varepsilon_1=\varepsilon_2=1$, $\gamma=2$. The reactants are initially at steady state: $\left(u,v\right)=\left(0, 0 \right)$, with a small disturbance at $x=0$. The boundaries are held at zero derivative: $u_x=0$, $v_x=0$.[]{data-label="fig:polar"}](ef05fig3.eps){width="4.2in"} If we translate our numerical results into this polar form, the onset of the pattern becomes very clearly demarked: the new coordinate $r$ transitions sharply from $0$ to $1$. The primary behaviour is that of a travelling wave, propagating outwards. The internal behaviour of the pattern, the steady oscillation, is reduced to a secondary effect. There are minor ripples in the established solution: the limit cycle settled on by the PDE is not the unit circle, but is close. There is a difference in behaviour between the left and right sides of the pattern: the angle of alignment and the frequency of the pattern is different on either side ([Figure]{} \[fig:polar\]). Next we examine this travelling wave analytically. Simple version -------------- We consider the very simplest situation: equal diffusion on both chemical species ($\bar{\varepsilon}=0$) combined with no convection of $v$ ($\bar{\gamma}$=0). We make a simplifying assumption for an initial analysis: $\theta_y\approx 0$, $\theta_{yy} \approx 0$. The problem reduces to $$\begin{aligned} r_t &=& r_{yy} + r(1-r^2) \ , \nonumber\\ \theta_t &=& 1 \ . \end{aligned}$$ Then $\theta$ decouples, as for the ODE. The solution for $\theta$ is $\theta = t$, to within an arbitrary constant. The remaining equation in $r$ is of the form similar to the Fisher equation and should therefore yield a propagating wave solution. This analysis is covered in greater detail in [@Murray1989Mathematical-Bi]. We look for a solution of the form $$\begin{aligned} r(y,t) = R(z) \ , && z=y-\bar{c}t \ . \end{aligned}$$ which gives $$\begin{aligned} R'' + \bar{c} R' + R(1-R^2) = 0 \ . \end{aligned}$$ We carry out a phase plane analysis in the $(R, R')$ phase plane. We find that $\bar{c}>0$ gives a stable point at $(R, R') = (0,0)$, which suggests the solution we are looking for. The point $R=1$, which corresponds to our limit cycle, is always a saddle point. We are looking for a phase plane trajectory that leaves this saddle point and goes to the zero steady state: this will be our travelling wave solution. Small wave speed, $\bar{c} < 2$, gives a spiral in the phase plane, which would imply $r < 0$ at some point on the trajectory. From the definition of $r$ we know this to be impossible and thus this is unrealistic for a travelling wave solution. For $\bar{c} \geq 2$ we have a node and the trajectory discussed above, leaving from the saddle point will head directly to the node, remaining in the fourth quadrant of the phase plane and therefore realistic. This trajectory equates to the travelling wave that we are looking for. We expect that the least speed wave will be achieved so we predict a wave of speed $\bar{c}=2$. Converting this back into our unscaled system, we have $c_R = 2\sqrt{\varepsilon_1}$ to the right. There is also a solution to the left: $c_L = -2\sqrt{\varepsilon_1}$. In the original system, we have $\textit{wavespeed} = p \pm 2\sqrt{\varepsilon_1}$. Full system ----------- We now apply the method for the full system. Here we may have different diffusion on the two species ($\bar{\varepsilon} \ne 0$) or some convection ($\bar{\gamma} > 0$). We again make the assumption $\theta' \approx 0$, $\theta'' \approx 0$. We introduce the travelling wave coordinate $z= y-ct$ and look for a solution $R(z) = r(y, t)$ . We also linearise the system, dropping the $R^3$ term: $$\begin{aligned} \left(1+\bar{\varepsilon}\sin^2\theta\right)R'' +\left(c - \bar{\gamma}\sin^2\theta\right)R' + R &=& 0 \ , \nonumber\\ \frac{\bar{\varepsilon}R'' - \bar{\gamma}R'}{R}\sin\theta\cos\theta +1 &=& 0 \ . \end{aligned}$$ Using the same approach as before, we find $\bar{c} = \bar{\gamma}\sin^2\theta \pm 2\sqrt{1+\bar{\varepsilon}\sin^2\theta}$. Converting this back to the unscaled coordinate system we have $c= \gamma\sin^2\theta \pm 2\sqrt{\varepsilon_1\cos^2\theta+\varepsilon_2\sin^2\theta}$. We see that $\gamma$ and $\varepsilon_2$ remain linked by the $\sin^2$ term. These parameters occur on the $v$ differential equation, and $v = r\sin\theta$. In the original system we have $\textit{wavespeed} = p\cos^2\theta + q\sin^2\theta \pm 2\sqrt{\varepsilon_1\cos^2\theta+\varepsilon_2\sin^2\theta}$. Again, the parameters for each species $u$ and $v$ are linked by the functions $\cos^2$ and $\sin^2$ respectively. This general result for the wavespeeds gives a dependence on the angle variable $\theta$. We consider some general numerical experiments to assess the behaviour of the system. What we find is two distinct types of behaviour. We refine our wavespeed estimates in light of these particular cases. Small parameter behaviour ------------------------- For parameters close to those of the simple system, namely small convection ($\gamma \ll 1$) and near-equal diffusion ($\varepsilon_1 \approx \varepsilon_2$), we expect the behaviour to remain similar to that of the simple system. Previously we have seen ODE-like and Turing-type behaviour in a convection-free system [@Flach2007Turing-pattern-]. In both cases, we see that the internal angle of the pattern (the group direction) is not in line with the wavefront. As such, the oscillation is perceptible at the wavefront. This suggests that an average value approximation of $\theta$ may be an appropriate estimate. In this case we approximate $\sin^2 \theta$ with its average value: $1/2$. This yields $\bar{c} \approx \bar{\gamma}/2 \pm 2\sqrt{1+\bar{\varepsilon}/2}$. In the unscaled coordinate system we have $c \approx \gamma/2 \pm\sqrt{2}\sqrt{\varepsilon_1+\varepsilon_2}$. If we consider the limit as the amount of diffusion tends to zero, we see that the centreline of the propagation is at $\gamma/2$. In this situation, the convection does not increase the separation of the left and right wavefronts; the spread of the propagation is solely due to diffusion. In the original system the incident fronts are at $\textit{wavespeed} \approx \cfrac{p+q}{2} \pm\sqrt{2}\sqrt{\varepsilon_1+\varepsilon_2}$. For dimensional reasons, it is appropriate to write $\varepsilon_i = \hat{\varepsilon_1}^2$. Then the diffusive contribution to the wavespeed becomes $\sqrt{2}\sqrt{\hat{\varepsilon_1}^2+\hat{\varepsilon_2}^2}$, which is reminiscent of Pythagorean form. More simply, the term is additive and the wavespeed is proportional to the diffusion of each species. Large parameter behaviour ------------------------- As the strength of convection $\bar{\gamma}$ increases, we see an alignment of the incident wave with the internal group direction. This means that the value of $\theta$ is approximately constant at the onset of the wave. We investigate the effect of this on the wavespeed estimate. We consider the range of values possible for the right wavespeed with a fixed value for $\theta$. Since $\sin^2$ has a range of $[0, 1]$ then $\bar{c}$ must range at least between $2$ and $\bar{\gamma} + 2\sqrt{1 + \bar{\varepsilon}}$. This second value is not necessarily greater than the first, for example $\bar{\gamma} = 0$ and $\bar{\varepsilon} = -1/2$ gives $\sqrt{2} \approx 1.4$, which is less than $2$. However, once $\bar{\gamma} > 2$ the second value is greater than the first, so we take this as an estimate for the transition to large parameter behaviour. Thus the range of possible values for the right wavespeed is $\bar{c}_R \in [2, \bar{\gamma} + 2\sqrt{1 + \bar{\varepsilon}}]$. On the left we have a similar result: $\bar{c}_L \in [-2, \bar{\gamma} - 2\sqrt{1 + \bar{\varepsilon}}]$ is the range once $\bar{\gamma}>2$. We note that for $\sin^2=0$, $\theta = 0 \pmod{\pi}$ and for $\sin^2=1$, $\theta=\pi/2 \pmod{\pi}$. We propose that the maximum spread of the instability is achieved. That is, the wavespeeds are maximised within their given possible ranges. For this, the left wavespeed must reach its lower limit, and the right its highest. This gives $\bar{c}_L \approx -2$ and $\bar{c}_R \approx \bar{\gamma} + 2\sqrt{1+\bar{\varepsilon}}$. Then the onset angle to the left is expected to be approximately $0 \pmod{\pi}$, and to the right of the right wave the angle should be roughly $\pi/2 \pmod{\pi}$. ![Large parameter behaviour of $\theta$. The oscillations are aligned with the travelling waves. To the left of the left travelling wave (the light, solid line), the angle is roughly $0$, and to the right of the right wave (the dark, dashed line) the angle is approximately $\pi/2$. There is a smooth transition between the left and the right of the pattern. The oscillations at the centre line ($\gamma/2$; medium, dot-dashed line) are roughly perpendicular to the centre line itself. This is a numerical solution of using NAG D03PCF, plotting species $u$ with $\gamma=5$, $\varepsilon_1=\varepsilon_2=1$. The reactants are initially at steady state: $\left(u,v\right)=\left(0, 0 \right)$, with a small disturbance at $x=0$. The boundaries are held at zero derivative: $u_x=0$, $v_x=0$.[]{data-label="fig:polar_angle"}](ef05fig4.eps){width="4.2in"} We see this is the case in [Figure]{} \[fig:polar\_angle\]. There is a smooth transition between the left and the right of the pattern. The oscillations at the centre line (approximately $\gamma/2$) are roughly perpendicular to the centre line itself. In the unscaled coordinate system, for strong convection, we have $c_L \approx -2\sqrt{\varepsilon_1}$ and $c_R \approx \gamma + 2\sqrt{\varepsilon_2}$. In the original system, this is $\textit{left wavespeed} \approx p - 2\sqrt{\varepsilon_1}$ and $\textit{right wavespeed} = q + 2\sqrt{\varepsilon_2}$ for $q > p$. In the other case $p > q$, there is a reversal and $\textit{left wavespeed} \approx q - 2\sqrt{\varepsilon_2}$ and $\textit{right wavespeed} = p + 2\sqrt{\varepsilon_1}$. This analytical prediction was made from observing that the internal waves become aligned with the onset of the instability. However, when we make the assumption that the spread of the propagation is maximised, it follows that the onset angle $\theta$ become constant. Furthermore, the suggestion of $\bar{\gamma}=2$ as a transition value between behaviours occurs naturally from the algebra. Numerical experiments ===================== Having carried out an analysis together with some initial numerical investigation, we now conduct a more comprehensive experiment. We vary our parameters over a wide range and measure the incident wavespeeds in each case. The results confirm our analyses, as we can see in [Figure]{} \[fig:ef05fig5\]. For the left wavespeed the difference between the low and high parameter estimate is pronounced, so the transitional behaviour is noticable. There is less difference between the estimates on the right, and the transition is close to the intersection of the estimates. This seems quite remarkable, that the behavioural transition occurs where the estimates intersect. As a result, there is no discernable area of transitional behaviour. In both cases the low parameter behaviour endures longer for $\varepsilon$ high, compared to $\varepsilon$ low. This suggests that the transition, caused primarily by the convection, $\gamma$, is held in check more strongly by the diffusion on $v$, where the convection is applied. Absolute versus convective instability ====================================== There is a concept of stability in systems such as these, which considers whether the system at a particular spatial point, once destabilised, will return to the steady state. If so, the system is classified as convectively unstable. If the chemical species do not return to stasis then the system is deemed absolutely unstable. In line with our analysis, we define absolute instability as $\operatorname{sign}\left(\textit{left wavespeed} \right) \ne \operatorname{sign}\left(\textit{right wavespeed} \right)$, with equivalence giving convective instability. Combining this with our results, we see that for small parameters we require $\gamma^2 > 8 \left( \varepsilon_1+\varepsilon_2 \right)$ for convective instability in the unscaled system. This inequality is unlikely to be satisfied whilst keeping within the small parameter regime. However, if we consider the original system, the condition is $\left(p+q\right)^2 > 8 \left( \varepsilon_1+\varepsilon_2 \right)$, which is quite possible to satisfy while still remaining in the ‘small parameter’ regime). For example, $p=q=2$ gives $\gamma=0$, and then $\varepsilon_1= \varepsilon_2 = 1$ satisfies the condition. For large parameters we always have absolute instability in the unscaled system. Referring to the observed data ([Figure]{} \[fig:ef05fig5\]), we see the left and right wavespeeds always have different signs. Thus, we can classify this system as only exhibiting absolutely unstable behaviour. However, if we consider the original system we find that $q>p>2\sqrt{\varepsilon_1}$ or $p>q>2\sqrt{\varepsilon_2}$ will give convective instability rather than absolute instability. We see that, in the original system, we can always shift the stability from absolute to convective with sufficient convection applied to the system. In the flow coordinate system, the stability is essentially absolute. However, when we convert to this system we make an arbitrary choice to remove the parameter p rather than, for example, q. Then the distinction between absolute and convective instability becomes similarly arbitrary. An alternative perspective ========================== At the start of our analysis, in section \[sec:pattern\], we chose a new coordinate system for the purpose of eliminating one of the convective terms. This reduction in the number of parameters proved useful in the ensuing work. However, the choice of coordinates, as we observed previously, was arbitrary. Now we consider an alternative coordinate system, one centred on the convection. From we choose $x= \xi -\cfrac{p+q}{2} t$. This gives the system as: $$\begin{aligned} u_t &=& \varepsilon_1 u_{xx} + \hat{\gamma} u_x + f \ , \nonumber\\ v_t &=& \varepsilon_2 v_{xx} - \hat{\gamma} v_x + g \ , \label{eq:flow}\end{aligned}$$ where $\hat{\gamma} = \cfrac{q-p}{2}$. We can recalculate the apparent wavespeeds from this perspective. For our ‘small parameter’ regime we have $\hat{c} = \pm \sqrt{2}\sqrt{\varepsilon_1 + \varepsilon_2}$. Here we see that we have aligned our system centrally with the flow. In our large parameter case, we first consider $q>p$. This ensures $\hat{\gamma} > 0$. Now the left wavespeed is $\hat{c}_L = -\left(\hat{\gamma} + 2\sqrt{\varepsilon_1}\right)$, and the right wavespeed is $\hat{c}_R = \hat{\gamma} + 2\sqrt{\varepsilon_2}$. In the other case $p<q$, we define $\tilde{\gamma} = \cfrac{p-q}{2} > 0$:$$\begin{aligned} u_t &=& \varepsilon_1 u_{xx} - \tilde{\gamma} u_x + f \ , \nonumber\\ v_t &=& \varepsilon_2 v_{xx} + \tilde{\gamma} v_x + g \ . \label{eq:flow2}\end{aligned}$$ Now the wavespeeds are $\tilde{c}_L = -\left(\tilde{\gamma} + 2\sqrt{\varepsilon_2}\right)$, and the right wavespeed is $\tilde{c}_R = \tilde{\gamma} + 2\sqrt{\varepsilon_1}$. Consider the right wavespeed. In each case it is a function of only one of the diffusion parameters. We examine the equation for the relevant species. In both cases, we find a negative convection term for that species. This suggests that negative convection drives the onset of instability. For the left wavespeed the connection is with the positive convection. However, in this direction $x$ is decreasing. From this perspective, the signs of the convective terms change. In this sense then, the connection is the same. If we understand that the negative convection drives the instability wavefront, then we can consider only the associated species. For example, looking for the right wavespeed in , we see that $u$ is the relevant species. We make the assumption that the other species $v$ is much smaller than $u$, and so disregard it. Then we can carry out a travelling wave analysis for this single species. The result is exactly the same as we found in our polar-form analysis, with a much simpler approach. We can confirm this behaviour by observing that the initiation angles $0$ and $\pi/2$ correspond to $u$ and $v$ only, respectively. We can go one step further in unravelling the puzzle. If we simply separate the species – perhaps we could call this a super-linearisation – we can produce two wavespeeds, one from each species. In our example , for the right wavespeed we have the possibility of $\tilde{\gamma}+2\sqrt{\varepsilon_1}$ from the $u$ equation and $-\tilde{\gamma}+2\sqrt{\varepsilon_2}$ from $v$. Then we apply the ‘maximisation’ principle suggested earlier. For large $\tilde{\gamma}$, we can be sure that the wavespeed given from the $u$ equation will maximise the propagation. Thus we can find the wavespeeds by applying two simple concepts to the problem. The behaviour for small convection relies on some more complex way of combining the two species. The wavespeed in this case is dependent on the strength of diffusion of both the species. Therefore the polar-form approach given previously retains some merit. Discussion ========== The addition of convection to the reaction-diffusion model has produced many patterns [@Bamforth2000Modelling-flow-; @Satnoianu2003Coexistence-of-]. Advection, being the broader definition, has potential to produce even more. Aspects of the pattern have been observed, such as the different behaviour to the right and the left of the free-forming pattern [@Satnoianu1998Spatio-temporal]. Previously analyses have considered a variety of models, most restricted in different ways. No movement for one species [@Satnoianu1998Interaction-bet], equal diffusion and equal convection [@Bamforth2000Modelling-flow-] and fixed diffusion-convection ratio [@Satnoianu2000Non-Turing-stat] have all been considered. Here, we have considered the most general diffusion and convection possible, as in more recent work [@Satnoianu2003Coexistence-of-]. Much previous work was devoted to the discussion of ‘flow and diffusion distributed structures’ [@Satnoianu2001Parameter-space]. This is spatial pattern with no variation in time. Our reduction of the system to two parameters allows for a clear and concise analysis of the system. However, after our change to convective coordinates, the meaning of ‘time-constant’ is lost. In fact, in our reduced system, a time-constant solution is only possible for the trivial case, $\gamma=0$. The standard analysis to date has been a linear existence analysis for pattern, involving dispersion relations. This follows the Turing derivation, but is difficult for this more complex model. In our analysis we have employed a different approach. We have not directly considered the internal behaviour of the pattern, but instead looked for approximate descriptions of the propagation speed of the pattern. This information has already been found for other versions of this system (less general diffusion or convection, more complex reactions) [@Satnoianu1998Interaction-bet; @Bamforth2000Modelling-flow-]. To understand the understand a system well, it is best to isolate influences [@Cross1993Pattern-formati]. This is why we consider the pattern forming away from the spatial boundaries. In this instance we also have symmetrical, parameterless reactions, giving the most elementary form possible. Previous work has shown boundary effects mixed in with other behaviour [@Satnoianu2003Coexistence-of-], and that the boundary can control the existence of pattern [@Bamforth2000Modelling-flow-]. Our analysis is the natural one given the system and its behaviour. We have combined two simple techniques to reduce a complex problem to a more manageable one. We consider only the (apparently) linear aspect. Our analysis predicted the speed of the incident travelling waves over the full parameter range. In combination with the numerical investigation, we demonstrate two different behaviours of the system. There is a distinct transition between them, with the transition region indicated by the analysis. Our work also confirms the validity of the Fisher-type analysis in this two-species system. Furthermore, we give some insight to the different behaviour to right and left of the pattern. Having successfully analysed this simple $\lambda$-$\omega$ system we must attempt a similar analysis on a more complex system. The next step is to consider a more realistic set of reaction functions, such as the Schnakenberg ones. In this case, we already know that first-order effects will come into play [@Flach2006Limit-cycles-in]. This should lead to a clearer understanding of the Turing phenomenon. Acknowledgements {#acknowledgements .unnumbered} ================ We would like to acknowledge support from NIH grant number R01GM076692. Any opinions, findings, conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the NIH or the United States Government. The authors are grateful to two anonymous reviewers for their helpful comments. [10]{} A. M. Turing, Philos. Trans. R. Soc. London, Ser. B [237]{}, 37 (1952). J. D. 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--- abstract: '[We report on the generation of continuous-wave squeezed vacuum states of light at the telecommunication wavelength of 1550nm. The squeezed vacuum states were produced by typeI optical parametric amplification (OPA) in a standing-wave cavity built around a periodically poled potassium titanyl phosphate (PPKTP) crystal. A non-classical noise reduction of 5.3dB below the shot noise was observed by means of balanced homodyne detection.]{}' author: - Moritz Mehmet - Sebastian Steinlechner - Tobias Eberle - Henning Vahlbruch - André Thüring - Karsten Danzmann - Roman Schnabel title: 'Observation of continuous-wave squeezed light at 1550nm' --- Squeezed states of light were proposed to improve the sensitivity of laser interferometers for the detection of gravitational waves (GW) [@Cav81], and to establish quantum communication channels [@YHa86], e.g. for quantum key distribution [@Ralph99; @Hillery00]. For any application of squeezed states of light, a low decoherence level is required, i.e. optical loss and thermally driven noise sources need to be minimized. In this respect the laser wavelength of 1550nm has emerged as a very interesting topic. Firstly, at this wavelength conventional silica based telecom glass fibers show low optical loss and can be used for the transmission of squeezed states. Losses of as low as 0.2dB/km were already measured in the late 70’s [@Miya79], and ultra low loss (ULL) fibers with an attenuation of 0[.]{}17-0[.]{}18dB/km are commercially available today [@Li08]. Secondly, at this wavelength, crystalline silicon constitutes an excellent test mass material for interferometric applications with low optical loss and high mechanical quality [@McGuigan]. GW detectors require the generation of squeezed states in a single spatio-temporal mode of continuous-wave light, whereas quantum channels can also be established in the pulsed laser regime. In the past years, squeezed states at wavelengths beyond 1.5$\mu$m were mainly generated in the latter regime. Noise powers of 6.8dB below vacuum noise at 1.5$\mu$m[@Dong08], 3.2dB at 1.535$\mu$m[@ETZH07], and 1.7 dB at 1.55$\mu$m[@NSHMYG02] were observed. Very recently, continuous-wave squeezed vacuum states at 1560nm were generated by an optical parametric oscillator based on periodically poled LiNbO$_3$ (PPLN), and a nonclassical noise suppression of 2.3dB was observed [@Feng08]. Here, we report on the generation of continuous-wave squeezed vacuum states at a wavelength of 1550nm based on periodically poled potassium titanyl phosphate (PPKTP). Squeezing of 5.3dB was observed by balanced homodyne detection. The visibility of the mode-matching between the squeezed field and a spatially filtered local oscillator beam was measured to be 99%, thereby proving high spatial mode quality of the squeezed states. The light source in our setup, as schematically depicted in Fig. \[setup\], was a high power erbium micro fiber laser providing about 1.6W of continuous-wave radiation at 1550nm. The laser beam was first sent through a ring mode cleaner (MC) cavity with a finesse of 350 and a line width of 1[.]{}2MHz for p-polarized light. Thus reducing mode distortions of the laser’s TEM$_{00}$ spatial mode profile as well as its phase and amplitude fluctuations at frequencies above the MC linewidth. ![Schematic of the setup. After being sent through a mode cleaner (MC) cavity, one part of the light is used as a control beam for the OPA and the local oscillator for balanced homodyne detection. The other part is frequency doubled in a SHG cavity to provide the 775nm field to pump the OPA. The squeezed field leaves the OPA in the counter direction to the pump, and is measured with the homodyne detector. PBS: polarizing beam splitter; DBS: dichroic beam splitter; HBS: homodyne beam splitter; MC: mode cleaner cavity; PD: photo diode; EOM: electro-optical modulator.[]{data-label="setup"}](Fig1.eps) Approximately 10mW of the transmitted light served as a local oscillator (LO) for balanced homodyne detection, while the remaining power of about one 1W was used for second harmonic generation (SHG) to provide the frequency doubled pump field for the OPA. Both, SHG and OPA were realized as single-ended standing-wave cavities formed by two mirrors and the non-linear crystal in between. In both cavities we employed a PPKTP crystal of dimension $10\times$2$\times$1mm$^3$ with flat, anti-reflection (AR) coated front and end faces. Inside a polyoxymethylene (POM) housing, each crystal is embedded in a copper fixture mounted on a Peltier element. Together with an integrated thermistor this enabled us to actively fine-tune the crystal temperature for efficient nonlinear coupling. A highly reflective (HR) mirror with a power reflectivity r$>$99[.]{}98% for both the fundamental and second harmonic field faces one AR-side of the crystal and a piezo-driven out-coupling mirror was mounted on the opposite side. The OPA out-coupling mirror had 90% and 20% power reflectivity for 1550nm and 775nm, respectively. For the SHG we also used 90% reflectivity for the fundamental but only a marginal reflectivity for the second harmonic. The mirrors and the ring-piezo were mounted inside aluminum blocks that were rigidly attached to the POM housing. Considering the refractive index of $n$=1.816 for PPKTP at 1550nm and the spacing of 20mm between crystal end faces and mirrors, the cavity waist size $w_0$, free spectral range FSR, and line width (FWHM) were calculated to be $\omega_0$=60$\mu$m, FSR=2.6GHz, and FWHM= 43MHz, respectively. When the SHG cavity was locked on resonance it produced up to 800mW at 775nm, which was separated from the fundamental by a dichroic beam splitter (DBS). The harmonic beam passed a combination of a half waveplate and a polarizing beam splitter for pump power adjustment, a Faraday isolator to prevent the SHG from retro-reflected light, and an electro optical modulator (EOM), and was mode matched to the TEM$_{00}$-mode of another MC cavity (MC$_{775}$) with characteristics equal to those of MC$_{1550}$. The transmitted beam was then carefully aligned to match the OPA-cavity TEM$_{00}$ mode. The length control of the cavities in our setup was accomplished by means of a modulation/demodulation (Pound-Drever-Hall, PDH) scheme utilizing custom made EOMs and matched photo detectors. Details on the particular implementation can be found in Fig. \[setup\]. The squeezed states left the OPA in the counter direction to the second-harmonic pump, where another DBS separated the two of them. The measurement of field quadratures variances was accomplished by means of balanced homodyne detection, for which the squeezed field was subsequently made to interfere with the LO on a 50/50-beam splitter. A piezo-actuated steering mirror was employed to shift the LO phase relative to the squeezed field. To adjust the visibility we injected a control beam through the HR back side of the OPA. This control beam was matched to o the OPA TEM$_{00}$ mode. The light that was transmitted propagated congruent to the mode to be squeezed, and, by locking the OPA cavity length, could be used to overlap with the LO on the homodyne beam splitter (HBS). We reached a fringe visibility of 99[.]{}0%. The two outputs of the 50/50-beam splitter were each focused down and detected by a pair of Epitaxx ETX-500 photodiodes. The difference current was fed to a spectrum analyzer. To verify our detector’s linearity we took measurements of the vacuum noise power against the incident LO power at a sideband frequency of 5MHz as depicted in Fig. \[fig1\]. Changing the LO power by a factor of two, entailed a 3dB shift of the corresponding noise trace, showing that the detector was quantum noise limited and operated linearly in the measurement regime. ![Noise power levels of the homodyne detector were measured at different LO powers at a centre frequency of 5MHz with the signal port blocked. Box sizes indicate the standard deviation of the fit and an estimated $\pm$5% uncertainty of the power meter used. The graph shows that our homodyne detector was quantum noise limited and operated linearly within our measurement regime.[]{data-label="fig1"}](Fig2.eps) We found the optimum pump power for our OPA to be 300mW, yielding a noise reduction of 5.3dB in the squeezed quadrature. This entailed an increase of 9.8dB in the anti-squeezed quadrature. To switch between the two, a piezo-actuated mirror was used to phase shift the LO with respect to the squeezed field . The measured noise curves are depicted in Fig. \[fig2\]. Trace (a) is the measured vacuum noise when the signal port of the HBS is blocked. The associated power of the incident LO was approximately 4 mW. Upon opening the signal port and injecting the squeezed field of the resonant OPA, trace (d) was recorded by linearly sweeping the LO-phase, thereby changing the measured quadrature from anti-squeezed to squeezed values. By holding the homodyne angle fixed, continuous traces of the squeezing (b) and anti-squeezing (c) were recorded. All traces were recorded at a sideband frequency of 5MHz and are, apart from (d), averaged twice. The contribution of electronic dark noise of our detector was negligible (18dB below the shot noise) and was not subtracted from the measured data. ![Noise powers of the squeezed light emitted by the OPA at a sideband frequency of 5MHz normalized to the shot-noise level (trace (a)). All traces were recorded with a resolution bandwidth of 300kHz and a video bandwidth of 300Hz. Squeezing (b) and anti-squeezing (c) curves were averaged twice. Curve (d) was recorded by linearly sweeping the LO-phase which continuously rotated the measured quadrature from anti-squeezing to squeezing.[]{data-label="fig2"}](Fig3.eps) The observed squeezed noise power was 5.3dB below shot noise, however, the observed anti-squeezing was about 10dB above shot noise, revealing an uncertainty product of about a factor of three above the minimum uncertainty. With an increased pump power we observed further increased anti-squeezing, but a constant squeezing level. Following the argumentation in [@Vahlb08] this observation implies that our measurement was not limited by phase noise [@Taken07; @Franzen06] but by optical losses. With 0.25% residual reflectance of our crystal AR coatings and 0.1%/cm absorption loss within the crystal we estimate the escape efficiency of the OPA cavity to be 90%. Together with a propagation loss of approximately 3%, we estimate the quantum efficiency of our photo detectors to be 90%$\pm4$%. We therefore expect that higher levels of squeezing from PPKTP could be observed in future utilizing better photo diodes and an OPA optimized for better escape efficiency. We note that PPKTP has already been successfully applied for the generation of squeezed and entangled states at wavelengths between 532nm and 1064nm [@Hetet07; @Aoki06; @Taken07; @Goda08NatPhys; @Gross08] with the maximum squeezing strength of 9dB observed at 860nm in [@Taken07]. The strongest squeezing to date was reported in [@Vahlb08] where a MgO:LiNbO$_{3}$ crystal enabled the observation of a noise reduction of 10dB below shot noise at 1064nm. However, at 1550nm the phase matching condition of this material is uncomfortably high and temperature gradients would significantly complicate the stable operation of a squeezed light source. This makes PPKTP the preferable material for the generation of squeezed light at 1550nm. In conclusion, we have demonstrated strong squeezing at the telecommunication wavelength of 1550nm. Our experiment proved that PPKTP is an effective material for the generation of squeezed states at this wavelength. The spatio-temporal mode of the squeezed field had a high purity ensuring the compatibility with quantum memories and quantum repeaters. By implementing a control scheme according to [@Vahlb06] squeezing in the detection band of current GW detectors can be realized. 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--- abstract: 'We introduce a $k$-fold boosted version of our Boostrapped Average Hierarchical Clustering cleaning procedure for correlation and covariance matrices. We then apply this method to global minimum variance portfolios for various values of $k$ and compare their performance with other state-of-the-art methods. Generally, we find that our method yields better Sharpe ratios after transaction costs than competing filtering methods, despite requiring a larger turnover.' author: - bibliography: - 'report\_KBAHC.bib' title: 'Reactive Global Minimum Variance Portfolios with $k-$BAHC covariance cleaning' --- covariance matrix cleaning; portfolio optimization, global minimum variance portfolios, realized risk. Introduction ============ Portfolio optimization works best when the asset covariance matrix is optimally cleaned. The necessity to filter covariance matrices was recognized a long time ago in this context [@michaud1989markowitz]. Cleaning may be optimal in two respects: first, estimation noise has to be filtered out when the number of data points $t$ is comparable to the number of assets $n$ (the so-called curse of dimensionality). This is often the case as the non-stationary nature of the dependence between asset price returns dictates to take as small a $t$ as possible [@bongiorno2020nonparametric]. Many filtering methods have been proposed, either for covariance or correlation matrices themselves (see [@bun2017cleaning] for a review) or for the portfolio optimization methods objectives [@markowitz1959portfolio; @black1990asset; @duffie1997overview; @hull1998value; @krokhmal2002portfolio; @roncalli2013introduction; @meucci2015risk]. Secondly, a good filtering method should also be able to retain the most stable structure of dependence matrices. As a consequence, what filtering method is optimal may depend on asset classes and market conditions. For these reasons, using a flexible yet robust method brings a more consistent performance. A third ingredient to improve portfolio optimization is to account for stochastic volatility itself, i.e., to use both asset-level volatility model and covariance matrix cleaning, as in [@engle2019largecov]. Since this work is devoted to the influence of covariance cleaning itself, we will not use this ingredient. Here we shall focus on covariance cleaning; we refer the reader to the extensive review of [@bun2017cleaning]. There are two main ways to clean covariance matrices: either to filter the eigenvalues of the corresponding correlation matrix and or to make assumptions on the structure of correlation matrices, i.e., to use an ansatz. Eigenvalue filtering rests on the spectral decomposition of the covariance or correlation matrix into a sum of outer product of eigenvectors and eigenvalues. Remarkable recent progresses lead to the proof that provided if $n<t$ and if the system is stationary, the Rotationally Invariant Estimator (RIE) [@bun2016rotational] converges to the oracle estimator (which knows the realized correlation matrix) at fixed ratio $q=t/n$ and in the large system limit $n$ and $t\to\infty$. In practice, computing RIE is far from trivial for finite $n$, i.e., for sparse eigenvalue densities; several numerical methods address this problem, such as QuEST [@ledoit2012nonlinear], Inverse Wishart regularisation [@bun2017cleaning], or the cross-validated approach (CV hereafter) [@bartz2016cross]. Note that these methods only modify the eigenvalues, and keep the empirical eigenvectors intact. The structure-based approach requires a well-chosen ansatz that needs to be suitable for the system under study. For example, linear shrinkage uses a target covariance (or correlation) matrix, and also has an eigenvalue filtering interpretation [@potters2005financial]. Factor models belong to the structure-based approach. A particular case is hierarchical factor models, which have been shown to yield remarkably good Global Minimum Variance (GMV henceafter) portfolios [@tumminello2007hierarchically; @pantaleo2011improved; @tumminello2007kullback]. A problem of the hierarchical ansatz is its sensitivity to the bootstrapping of the original data, which does not yield many statistically validated clusters in correlation matrices of equity returns [@bongiorno2019nested]. Very recently, we have leveraged this sensitivity to build a more flexible estimator, which consists in averaging filtered hierarchical clustering correlation or covariance matrices of bootstrapped data (BAHC) [@bongiorno2020covariance]. BAHC not only allows an imperfect hierarchical structure, i.e., a moderate overlapping among clusters, but also a probabilistic superposition of quite distinct hierarchical structures. When applied to GMV portfolios, BAHC yields similar or better realized risk than the optimal eigenvalues filtering methods but for a much smaller $t$ than its competitors, which gives portfolios that are much more reactive to changing market conditions. It can be further improved, as shown below. This paper proposes to extend BAHC to account for the structure of the correlation matrix that is not described by BAHC, i.e., the residuals. The rationale is that the latter may also contain a structure that persists in the out-of-sample window, hence that they should not be erased by the filtering method. The idea is to apply to filter recursively the residuals and to average the filtered matrices of bootstrapped data. The order of recursion, denoted by $k$, is a parameter of the method, which we proposed to call $k-$BAHC. This new method is equivalent to BAHC when $k=1$ by convention. The higher $k$, the finer the details kept by $k-$BAHC, which, as shown below, improves the out-of-sample GMV portfolios up to a point. When $k$ tends to infinity, the filtered correlation matrix converges to the unfiltered correlation matrix averaged over many bootstrap copies. This matrix is almost surely positive definite in the high-dimensional regime $t<n$, despite the fact that the empirical unfiltered correlation matrix is not positive definite [@bongiorno2020bootstraps]. As shown below, the optimal average $k$ depends on the size of the in-sample window for a data set of US equities. It is generally an increasing function of the in-sample window length $t_{in}$: for small $t_{in}$, most of the variations of the empirical correlation matrices are due to estimation noise, which is best filtered by a small $k$; as $t_{in}$ increases, the relative importance of estimation noise decreases and thus a higher $k$ should be preferred. Methods ======= Let us start with some notations of standard quantities: let ${\bf R}$ be a $n \times t$ matrix of price returns. Its $n\times n$ covariance matrix, denoted by ${\bf \Sigma}$, has elements $\sigma_{ij}$, where $$\sigma_{ij} =\frac{1}{t} \sum_{h=1}^t \left( r_{ih} - \bar{r}_i \right) \left(r_{jh} - \bar{r}_j \right)$$ and where $\bar{r}_i = \sum_{h=1}^t r_{ih}/t$ is the sample mean of vector $r_i$. The Pearson correlation matrix ${\bf C}$ has elements $$c_{ij} = \frac{\sigma_{ij}}{\sqrt{\sigma_{ii}\,\sigma_{jj}}}$$ As $k-$BAHC is an extension of BAHC, itself is a bootstrapped version of the strictly hierarchical filtering method of [@tumminello2007hierarchically], let us start with the hierarchical clustering. Hierarchical Clustering ----------------------- Hierarchical clustering agglomerates groups of objects recursively according to a distance matrix taken here as $\textbf{D}=1-\textbf{C}$ with elements $d_{ij}$; $\textbf{D}$ respects all the axioms of a proper distance. Accordingly, the distance between clusters $p$ and $q$, denoted by $\rho_{pq}$ is defined as the average distance between their elements $$\rho_{pq} = \frac{\sum_{i \in \mathfrak{C}_p} \sum_{j \in \mathfrak{C}_q} d_{ij}}{n_q\,n_p},\label{eq:rho_pq}$$ where $\mathfrak{C}_p$ and $\mathfrak{C}_q$ denote the $n_p$, respectively $n_q$ elements of clusters $p$ and $q$ respectively. Hierarchical agglomeration works as follows: one starts by giving each element its own cluster. Then, the two clusters $(p,q)$ with the smallest distance $\rho_{pq}$ are merged into a new cluster $s$ which contains the elements $\mathfrak{C}_s = \mathfrak{C}_p \cup \mathfrak{C}_q$. This algorithm is applied until all nodes form a single unique cluster. This defines a tree, called a dendrogram, which uniquely identifies the genealogy of cluster merges, denoted by $\mathfrak{G}$. Hierarchical Clustering Average Linkage Filtering (HCAL) {#sec:ALCA} -------------------------------------------------------- Defining a merging tree is not enough to clean correlation matrices. [@tumminello2007hierarchically] propose to average all the elements of the sub-correlation matrix defined from the indices $\mathfrak{F}_{pq} = \{ (i,j)\,:\, i \in \mathfrak{C}_p,\, j \in \mathfrak{C}_q\}$, i.e., to replace $c_{ij}$ by $$c_{ij}^< = c_{ji}^<=1 - \rho_{pq} \;\; \mbox{where} \;\;\; (p,q ) \in \mathfrak{G},\, (i,j) \in \mathfrak{F}_{pq},$$ where $\rho_{pq}$ is the average distance between clusters $p$ and $q$ (see Eq. ), with $c_{ii}^<$ set to 1. This defines the HCAL-cleaned correlation matrix ${\bf C}^<$, which corresponds to a hierarchical factor model [@tumminello2007hierarchically]. HCAL-filtered matrices have two interesting properties: by construction, ${\bf C}^<$ is positive definite when the correlation matrix is dominated by a global mode, i.e., when the average correlation is large, as in equity correlation matrices [@tumminello2007hierarchically]. Secondly, ${\bf C}^<$ is the simplest matrix that has the same dendrogram as the empirical correlation matrix ${\bf C}$; this means that by applying the HCAL to both ${\bf C}$ and ${\bf C}^<$, the resulting dendrograms will be identical. This, however, is also one of the main limitations of this approach as it prevents any overlap among clusters; in addition, the dendrogram of ${\bf C}$ may not be the true one. $k-$BAHC -------- The method we propose rests on two ingredients: a recursive HCAL filtering of the residuals of filtered correlation matrices, and bootstrapping the return matrix (time-wise) in order to make the method more flexible, as in the BAHC method. ### $k-$HCAL Filtering {#sec:HOf} Let us define the filtered matrix of order $k=0$ as ${\bf C}^<_{(0)}={\bf 0}$. The residue matrix of order $k$ is then $$\label{eq:res} {\bf E}_{(k)}= {\bf C} - {\bf C}^<_{(k)}.$$ When $k=0$, ${\bf E}_{(0)}={\bf C}$; For any value of $k\in\mathbb{N}_+$, we can apply the filtering procedure of sec. \[sec:ALCA\] to the residue matrix ${\bf E}_{(k)}$ to obtain a filtered residue matrix ${\bf E}^<_{(k)}$. Then the $k+1-$HCAL filtered matrix is obtained with $$\label{eq:Ck} {\bf C}^<_{(k+1)} = {\bf C}^<_{(k)} + {\bf E}^<_{(k)}.$$ For example, $k=1$ correspond to HCAL-filtered matrix. The recursive application of Eqs. and allows us to compute the filtered matrix at any order $k$. It is worth noticing that by iterating Eqs. and $$\label{eq:kinf} \lim_{k \to \infty} {\bf C}^<_{(k)} = {\bf C}$$ since the residue become smaller and smaller. It is important to point out that, ${\bf C}^<_{(k)}$ is not in general a semi-positive definite matrix for $k>1$, and in most cases, some small negative eigenvalues have been observed in our numerical experiments. These eigenvalues, according to Eq., shrink to non-negative values when $k$ approach infinity. For any order $k>1$, we set the possibly negative eigenvalues to 0. ### Bootstrap-based regularization {#sec:BAHC} In the spirit of the BAHC method [@bongiorno2020covariance], our recipe prescribes to create a set of $m$ bootstrap copies of the data matrix ${\bf R}$, denoted by $\{{\bf R}^{(1)},\, {\bf R}^{(2)},\, \cdots, {\bf R}^{(m)}\}$. A single bootstrap copy of the data matrix ${\bf R}^{(b)} \in \mathbb{R}^{n \times t}$ is defined entry-wise as $r^{(b)}_{ij} = r_{i {\bf s}^{(b)}_j}$, where ${\bf{s}}^{(b)}$ is a vector of dimension $t$ obtained with random sampling by replacement of the elements of the vector $\{1, 2, \cdots, t\}$. The vector ${\bf s}^{(b)}$, $b=1,\cdots,m$ are sampled independently. We compute the Pearson correlation matrix ${\bf C}^{(b)}$ of each bootstrap ${\bf R}^{(b)}$ of the data matrix, from which we derive the $k-$HCAL-filtered matrix ${\bf C}^{(b)<}_{(k)}$. Finally, the filtered Pearson correlation matrix ${\bf C}^{k\textrm{-BAHC}}$ is defined as the average over the $m$ filtered bootstrap copies, i.e., $$\label{eq:corfilt} {\bf C}^{k-\textrm{BAHC}} = \sum_{b=1}^m \frac{{\bf C}_{(k)}^{(b)<}}{m}$$ While ${\bf C}^{(b)<}_{(k)}$ is a semi-positive definite matrix, the average of these filtered rapidly becomes positive-definite, as shown in [@bongiorno2020bootstraps]. This convergence is fast, and it is guaranteed almost surely if $m \geq n$, but in most of the cases is reached for $m$ much smaller then $n$. Finally, $k$-BAHC filtered covariance is obtained from the sample univariate variance according to $$\label{eq:covfilt} \sigma_{ij}^{k\textrm{-BAHC}} = c_{ij}^{k\textrm{-BAHC}} \sigma_{ii} \sigma_{jj}$$ The main advantage of $k-$BAHC over $k-$HCAL is not to force ${\bf C}^{k-\textrm{BAHC}}$ to be embedded in a purely recursive hierarchical structure. Results ======= Data ---- We consider the daily close-to-close returns from 1999-01-04 to 2020-03-31 of US equities, adjusted for dividends, splits, and other corporate events. More precisely, the data set consists of 1295 assets taken from the union of all the components of the Russell 1000 from 2010-06 to 2020-03. The number of stocks with data varies over time: it ranges from 497 in 1999-02-18 to 1172 in 2018-01-17. Spectral Properties ------------------- One of the reasons why the original BAHC filtering achieves a similar or better realized variance than its competitors that focus on filtering the eigenvalues of the correlation matrix only, that the resulting eigenvectors have a larger overlap with the out-of-sample eigenvectors than the unfiltered empirical eigenvectors while still filtering eigenvalues nearly as well as the optimal methods [@bongiorno2020covariance]. This sub-section is devoted to investigate how the eigenvector components change as $k$ is increased. It turnouts that the localization of eigenvectors is crucial in understanding the role of $k$. To understand why localization matters to portfolio optimization, it is worth recalling that Global Minimum Portfolios correspond to the optimal weights $$\label{eq:minrisk} {\bf w}^* = \frac{{\bf \Sigma}^{-1} \mathbb{1}}{\mathbb{1} {\bf \Sigma}^{-1} \mathbb{1}'}$$ that is a sum by rows (or columns) of the inverted covariance matrix, then normalized to one. The inverted covariance matrix can be expressed in terms of spectral decomposition of ${\bf \Sigma}$ as $${\bf \Sigma}^{-1} = \sum_{i=1}^n \frac{1}{\lambda_i} {\bf v}_i {\bf v}_i',$$ where $\lambda_i$ and ${\bf v}_i$ are respectively the $i-$th eigenvalue and and its associated eigenvector of ${\bf \Sigma}$. This means that the composition of the eigenvectors related to the highest eigenvalues is irrelevant and the portfolio allocation is dominated by the eigenvectors of the smallest eigenvalues. Let us assume that the eigenvalue are ordered, i.e., that $\lambda_1>\lambda_2>\cdots>\lambda_n$. If the smallest eigenvalue is much smaller than all the others ones, i.e., $\lambda_n \ll \lambda_{n-1}$, the largest part of investment will be on the $j$-stocks such that $\|v_{nj}\|\gg 0$. Therefore, the localization of the non-zero elements of the eigenvectors is crucial to understand the portfolio allocation. The statistical characteristics of the eigenvectors localization is typically described in term of Inverse Participation Ratio (IPR), defined as $$\mbox{IPR}_i =\frac{1}{ \sum_{j=1}^{n} v_{ij}^4}$$ where the index $i$ refers the $i$-th eigenvalue. The smaller the value of $\mbox{IPR}_i$, the more localized its associated eigenvector, the most localized case corresponding to IPR$=1$. Figure \[fig:IPRcum\] reports the cumulative distribution function of $\mbox{IPR}$ of the eigenvectors for different recursion orders $k$. The dependence on $k$ is obvious: $1-$BACH has the most localized eigenvectors and the larger the value of $k$, the less localized the components of the eigenvectors. In the limit $k \to \infty$, one recovers the empirical, unfiltered, covariance matrix. In addition, the IPR of the latter two are hardly different from the random matrix null expectation obtained by shuffling price returns asset by asset in the data matrix. Figure \[fig:IPRscat\] gives more details about the IPRs associated with the eigenvalues. It makes it obvious that IPRs are different for small eigenvalues, while no clear pattern emerges for the outliers $\lambda_i\gtrsim 10^{-3}$. Since the lowest eigenvalues are the ones that affect mainly GVM portfolio optimization, a filtering procedure that modifies the IPR of such eigenvalues will produce a substantial difference in the portfolio allocation. Global Minimum Variance portfolios ---------------------------------- This part explores how the realized risk of GMV portfolios depends on the recursion order $k$ and compares it with the performance obtained from sample covariance and the Cross-Validated (CV) eigenvalue shrinkage [@bartz2016cross], which is a strong contender for the best realized risk [@bongiorno2020covariance]. Two types of tests are carried out: because our data covers many different market regimes and a variable number of assets, we first ask what is the average realized risk of each covariance cleaning method for a random collection of assets in a randomly chosen period of fixed length. This allows us to assess the performance of each cleaning scheme in a fair way and to control the effect of the calibration window length. In the second part, we compare the performance of these optimal portfolios with all available stocks at any given time. ### Random assets, random periods The experiments of this part are carried out in the following way: for each calibration window length $\Delta t_{in} \in [20, 2000]$ we randomly choose a time $t$ between 2000-01-03 and 2020-03-30 that defines a calibration widow $[t-\Delta t_{in}, t[$, and a test window $[t, t+ \Delta t_{out}[$ with $\Delta t_{out}=21$ days. We then sample $n=100$ stocks over the available assets within the calibration and test windows. Finally, we compute the GMV portfolios with and without short positions using $k-$BAHC, the state-of-art Cross-Validated (CV) eigenvalue shrinkage [@bartz2016cross] and the unfiltered empirical covariance matrix. Figure \[fig:VOL\] shows the realized risk of GMV portfolios obtained with the chosen filtering schemes and with the empirical (sample) covariance matrix. The $k-$BACH estimators outperform both CV and the sample covariance estimators for $\Delta t_{in}<300$ in the long-short case and for every $\Delta t_{in}$ for the long-only case (Figures \[fig:VOLlongshort\] and \[fig:VOLlong\]). The highest performance of CV is obtained for $\Delta t_{in}\approx 350$; however, the highest absolute minimum is obtained for $k-$BAHC with $\Delta t_{in}\approx 200$, which requires much shorter calibration times and thus yields more reactive portfolios. What values to take for $k$ (and for this data set) depends on $\Delta t_{in}$. In the high-dimensional regime ($q>1$), i.e. for $\Delta t_{in}<90$, the best results are obtained is for $k=1$; however, when $\Delta t_{in}$ increases the performance of $k=1$ becomes even worse than the sample covariance. From this analysis is clear that the larger the calibration window size, the larger the approximation order $k$ must be. Figures \[fig:Klongshort\] and \[fig:Klong\] show the average optimal $k^*$ that minimizes the realized risk as a function of $\Delta t_{in}$ for the long-short and long-only case. They confirm that a longer calibration window requires a higher approximation order both for the long-short and long-only cases; however, whereas for the long-short the increment seems linear with $\Delta t_{in}$, this dependence for the long-only case is sub-linear (and much noisier). It is worth remarking that the fits of the right plots of Figure \[fig:VOL\] are obtained with $k\le20$: larger values of $k$ might further improve the performance for larger $\Delta t_{in}$; however, they would require a comparatively greater computational effort. ### Full-universe, full period backtest In this section we performed a set of portfolio optimizations with monthly computations of new portolio weights (and relabancing) over the full time-period \[2000-01-02, 2020-03-31\] for all the considered covariance estimators and equally weighted portfolios (EQ hence after), and for different in-sample window lengths. The backtests include transaction costs set to 2 bps. A slight complication comes from the variable number of available assets. Thus, as any time, $q=n/t$ varies and generally increases with time at fixed calibration window length. In any case, it is worth keeping in mind that $n$ is relatively large, i.e., between 497 and 1172. In particular, at each rebalancing time-step we considered all the available stocks listed in both the in-sample and out-of-sample periods. The present work investigates in detail short calibration windows: we chose a sequence of $\Delta t_{in} \in [21,252]$ days by steps of $21$ days, i.e., about $1,2, \cdots, 12$ months. In addition, for sake of completeness we also included longer calibration windows $\Delta t_{in} = 300, 350, 400, 500, 750, 1000, 1500, 2000$. For each method and calibration window, we computed the realized annualized volatility, the realized annualized return, the Sharpe ratio, the gross-leverage, the concentration of the portfolio, and the average turnover. Figure \[fig:profit\] shows the behavior of equally weighted portfolios and GMV portfolios obtained by $11-$BAHC, the globally best value for the data set that we used. As expected, GMV reduce the realized risk with respect to EW portfolios, and cleaning the covariance matrix is clearly beneficial as well. Let us start with realized risk, the focus of this paper. The realized risk of EQ portfolios is much larger than that resulting from the other methods, which is hardly surprising as the latter account for the covariance matrix (Tables \[tab:SDlongshort\] and \[tab:SDlong\]); $k-$BACH achieves the smallest realized risk, and the best value for $k$ increases as $\Delta t_{in}$ increases. Although GMV does not guarantees a positive return, we also report the Sharpe ratios of the various filtering methods. Because computing Sharpe ratios with moments is not efficient for heavy-tailed variables, we use the efficient and unbiased moment-free SR estimator introduced in [@challet2017sharper] and implemented in [@packagesharperRatio]. Sharpe ratios paint a picture similar to realized risk (see Tables \[tab:SRlongshort\] and \[tab:SRlong\]): $k-$BAHC outperforms all the other methods for medium to large values of $k$ for almost every $\Delta t_{in}$ both in the long-only and long-short cases, especially when the calibration window is smaller than a year, which corresponds to $q=n/t\in[2,4.5]$, i.e., quite deep in the high-dimensional regime. In particular, in the long-short case, the highest SR equals $1.25$ for $k= 7, 11, 18$ in the remarkably short calibration window $\Delta t_{in}=105$ days (about 5 months), and is significantly higher than the best performance of CV (SR$=1.18$) obtained for a much higher calibration window $\Delta t_{in}=400$. This shows that reactive portfolio optimization is invaluable. In the long-only case, the improvement of $k-$BAHC is smaller: the best SR (1.13) is that of $18-$BAHC, whereas the highest SR of CV is $1.07$. However, as shown from the cumulative performances in Figure \[fig:profit\], the relative performance changes over time. To overcome this limitation, we evaluated the SR every year, and we performed a dense ranking of all the methods after having rounded the related SRs to the second decimal (to ensure some equal ranks). Finally we associated a score $\langle \mbox{rank} \rangle$ to every method defined as the average dense rank over the years. The results for the long-short and long-only cases are summarized in Table \[tab:rank\]. It is worth noting that different medium to large values of $k$ of $k-$BACH outperform all the other methods, and in particular the optimal performances are achieved with a calibration window length shorter than for CV by a factor of about four. \[tab:rank\] We checked that the portfolios obtained with $k-$BAHC are more concentrated than the other ones, which is consistent with the fact that the IPR of the relevant eigenvectors is smaller. The concentration of a portfolio can be measured with $$n_{\textrm{eff}} = \frac{1}{\sum_{i=1}^n w_i^2}$$ as proposed in [@bouchaud2003theory]; However, as noticed in [@pantaleo2011improved], this quantity does not have a clear interpretation when short selling is allowed. To overcome this issue, [@pantaleo2011improved] introduced the $n_{90}$ metrics which measures the smallest number of stocks that amount for at least 90% of the invested capital. Accordingly, we used $n_{90}$ is for the long-short case and $n_{\textrm{eff}}$ for the long-only one. Looking at Tables \[tab:N90longshort\] and \[tab:Nefflong\], the number of stocks selected is systematically smaller for every $k$ and calibration window for $k-$BAHC for both long-only and long-short portfolios. That said, $k-$BAHC has two drawbacks. First, the gross leverage is generally larger than for CV in the long-short case (see Table \[tab:Grosslongshort\]). However, if we compare the values of gross leverage corresponding to the larger SR for CV and $k-$BAHC for $\Delta t_{in}$ within one year, they differ only by $0.52$ ($2.47$ for CV and $3.00$ for $k-$BAHC). On the other hand, without constraining the calibration window, the highest SR for CV is achieved for $\Delta t_{in}=400$ and the gross leverage reaches $3.31$, which is larger than for other methods. The other drawback of $k-$BAHC is that it requires a larger turnover for long-short portfolios. A natural turnover metrics, denoted by $\gamma$, was defined in [@reigneron2019case] as $$\gamma = \frac{1}{\tau} \sum_{h=0}^{\tau-1} \sum_{i=1}^n \left\| w_i(t_0+h \Delta t_{out} ) -w_i(t_0+(h+1)\Delta t_{out}) \right\|,$$ where $\tau$ is the number of rebalancing operations and $t_0$ is the initial time. $\gamma$ measures the average changes in the portfolio allocation between two consecutive portfolio allocations. Table \[tab:Gammalongshort\] shows that $k-$BAHC has a $\gamma$ typically twice as large as CV, except for $k=1$ for large $\Delta t_{in}$ for long-short portfolios. For the long-only case (Table \[tab:Gammalong\]) CV still outperforms $k-$BAHC in that respect, although not by much.All performance measures take into account into account the rebalancing costs. Note that the larger turnover comes from the fact that portfolios are more concentrated, i.e., select fewer assets. It is therefore more likely that the set of selected changes at every weight updates. Discussion ========== By combining recursive hierarchical clustering average linkage and bootstrapping of the data matrix yields a globally better way to filtering asset price covariance matrices. We have shown that this method filters the eigenvectors associated with small eigenvalues of the covariance matrix by making them more concentrated, which in turn yields portfolios with fewer assets. Because $k-$ BACH captures more of the persistent structure of covariance matrices with shorter calibration windows, it leads to better realized variance of Global Minimum Variance portfolios than even the best method that optimally filters the eigenvalues of the correlation matrix. Finally, it is able to achieve its best performance for significantly smaller calibration window lengths, which makes $k-$BAHC portfolios more reactive to changing market conditions. The main drawback is that it requires a larger turnover. This is due, in part, to the fact that resulting portfolios are more concentrated, hence that the fraction of capital in which to invest change more rapidly than less specific methods. Whether this reflects a genuine change of market structure or a by-product of the specific assumptions of $k-$BACH is an interesting open question. Future work will investigate how $k-$BAHC may improve other kinds of portfolio optimization schemes and other financial applications of covariance matrices. Acknowledgement(s) {#acknowledgements .unnumbered} ================== This work was performed using HPC resources from the “Mésocentre” computing center of CentraleSupélec and École Normale Supérieure Paris-Saclay supported by CNRS and Région Île-de-France (<http://mesocentre.centralesupelec.fr/>) Funding {#funding .unnumbered} ======= This publication stems from a partnership between CentraleSupélec and BNP Paribas. \[tab:SDlongshort\] \[tab:SDlong\] \[tab:SRlongshort\] \[tab:SRlong\] \[tab:N90longshort\] \[tab:Nefflong\] \[tab:Grosslongshort\] \[tab:Gammalongshort\] \[tab:Gammalong\]
--- abstract: 'In this paper, we study the uniform Hölder continuity of the generalized Riemann function $R_{\alpha,\beta}$ (with $\alpha>1$ and $\beta>0$) defined by $$R_{\alpha,\beta}(x)=\sum_{n=1}^{+\infty}\frac{\sin(\pi n^\beta x)}{n^\alpha},\quad x\in\operatorname{\mathbb{R}},$$ using its continuous wavelet transform. In particular, we show that the exponent we find is optimal. We also analyse the behaviour of $R_{\alpha,\beta}$ as $\beta$ tends to infinity.' author: - 'F. Bastin' - 'S. Nicolay' - 'L. Simons' title: \| About the Uniform Hölder Continuity of\ Generalized Riemann Function --- Keywords: Hölder continuity, Continuous wavelet transform, Riemann function\ 2010 Mathematical Subject Classification: 26A16, 42C40, 30B50 Introduction ============ In the $\text{19}^\text{th}$ century, Riemann introduced the function $R$ defined by $$R(x)=\sum_{n=1}^{+\infty}\frac{\sin(\pi n^2 x)}{n^2},\quad x\in\operatorname{\mathbb{R}},$$ in order to construct a continuous but nowhere differentiable function (see [@DBR] for some historical informations). The regularity of this function has been extensively studied by many authors. In 1916, Hardy [@H] showed that $R$ is not differentiable at irrational numbers and at some rational numbers. In the seventies, Gerver [@G1] and other people [@I; @M; @Q; @S; @HT] proved that $R$ is only differentiable at the rational numbers $(2p+1)/(2q+1)$ (with $p\in\operatorname{\mathbb{Z}}$ and $q\in\operatorname{\mathbb{N}}$) with a derivative equals to $-1/2$. The Hölder spaces allow to define a notion of smoothness or regularity for a function. In some way, it is an “intermediate level” between continuity and differentiability. Following [@Ja; @JMR; @Da; @T], we adopt the next definition for Hölder spaces. Let $\alpha\in[0,1)$, $f\in L^{\infty}(\operatorname{\mathbb{R}})$ and $x_0\in\operatorname{\mathbb{R}}$. (1) The function $f$ belongs to $C^\alpha(x_0)$ if there exists $C>0$ and $\varepsilon>0$ such that $$\|f(x)-f(x_0)\|\leq C \|x-x_0\|^{\alpha}$$ for all $x\in(x_0-\varepsilon,x_0+\varepsilon)$. In this case, we say that $f$ is [Hölder continuous with exponent $\alpha$ at $x_0$]{}. (2) The function $f$ belongs to $C^{\alpha}(\operatorname{\mathbb{R}})$ if there exists $C>0$ such that $$\|f(x)-f(y)\|\leq C \|x-y\|^{\alpha}$$ for all $x,y\in\operatorname{\mathbb{R}}$. In this case, we say that $f$ is [uniformly Hölder continuous with exponent $\alpha$ (on $\operatorname{\mathbb{R}}$)]{}. The spaces defined above are embedded: if $\alpha<\beta$ for $\alpha,\beta\in[0,1)$, then $C^{\beta}(x_0)\subset C^\alpha(x_0)$ for any $x_0\in\operatorname{\mathbb{R}}$ and $C^{\beta}(\operatorname{\mathbb{R}})\subset C^\alpha(\operatorname{\mathbb{R}})$. This property allows to define a notion of regularity, known as Hölder exponent. Let $f\in L^\infty(\operatorname{\mathbb{R}})$ and let $x_0\in\operatorname{\mathbb{R}}$. (1) The [Hölder exponent of $f$ at $x_0$]{} is $$H_f(x_0)=\sup\left\{\alpha\in[0,1):f\in C^{\alpha}(x_0)\right\}.$$ (2) The [uniform Hölder exponent of $f$ (on $\operatorname{\mathbb{R}}$)]{} is $$H_f(\operatorname{\mathbb{R}})=\sup\left\{\alpha\in[0,1):f\in C^{\alpha}(\operatorname{\mathbb{R}})\right\}.$$ Following this definition, if $f$ is differentiable, then $H_f(\operatorname{\mathbb{R}})=1$. Moreover, $H_f(\operatorname{\mathbb{R}})<1$ implies that $f$ is not differentiable. However, there exist non-differentiable functions with a uniform Hölder exponent equal to $1$; the Takagi function (see [@Ta; @SS]) is a famous example. Based on a work with Littlewood [@HL], Hardy [@H] showed that $R$ is not Hölder continuous with exponent $3/4$ at irrational numbers and at some rational numbers. Using the continuous wavelet transform (of $R$), Holschneider and Tchamitchian [@HT] established that $R$ is uniformly Hölder continuous with exponent $1/2$ and gave some results about its Hölder continuity at some particular points. With some similar techniques, Jaffard and Meyer [@Ja; @JM] determined the Hölder exponent of $R$ at each point and proved that $R$ is a multifractal function, i.e. that the function $x\mapsto H_R(x)$ is not constant. A generalization of $R$ is given by the function $R_{\alpha,\beta}$ defined by $$\label{Rgen} R_{\alpha,\beta}(x)=\sum_{n=1}^{+\infty} \frac{\sin(\pi n^\beta x)}{n^\alpha},\quad x\in\operatorname{\mathbb{R}},$$ with $\alpha>1$ and $\beta>0$. Other generalizations of $R$ are possible; for example, one can replace the element $n^\beta$ in the definition of $R_{\alpha,\beta}$ by a polynomial with entire coefficients (see [@CU; @Q]). The function $R_{\alpha,\beta}$ defined in is clearly continuous and bounded on $\operatorname{\mathbb{R}}$. If $\beta\in(0,\alpha-1)$, it is easy to check that $R_{\alpha,\beta}$ is continuously differentiable on $\operatorname{\mathbb{R}}$ (because the series of derivatives uniformly converge on $\operatorname{\mathbb{R}}$). If $\beta\geq\alpha+1$, Luther [@L] proved that $R_{\alpha,\beta}$ is nowhere differentiable. If $\beta\in[\alpha-1,\alpha+1)$, several partial results about the differentiability of $R_{\alpha,\beta}$ are known (see [@Q; @L]). Moreover, some results are also known for the cases $\beta=2$ (see [@H; @Ja]), $\beta=3$ (see [@G]) and $\beta\in\operatorname{\mathbb{N}}\setminus\{0\}$ (see [@CU1]). Concerning the Hölder continuity and also the Hölder exponent of $R_{\alpha,\beta}$, several particular cases have been studied (see [@B; @Ja; @JM; @JJ; @CU1; @U]). In this paper, we study the uniform Hölder continuity of $R_{\alpha,\beta}$ with $\beta\geq\alpha-1$ in order to complete and generalize a result of Johnsen [@JJ] in 2010 which claims that, if $\beta>\alpha-1$, then $R_{\alpha,\beta}$ is uniformly Hölder continuous with an exponent superior or equal to $(\alpha-1)/\beta$. To achieve this, we use some techniques different from the ones of Johnsen. Our approach is based on the continuous wavelet transform of $R_{\alpha,\beta}$ related to the Lusin wavelet, and is similar to the ones used to obtain the Hölder continuity of $R$ in [@JMR; @Ja; @HT]. This method has two advantages: we can consider both the cases $\beta=\alpha-1$ and $\beta>\alpha-1$ to study the uniform Hölder continuity of $R_{\alpha,\beta}$ and then show the optimality of the so obtained exponent. In other words, we calculate the uniform Hölder exponent of $R_{\alpha,\beta}$ for $\beta\geq\alpha-1$. These results are summarized in the following theorem. \[Main\] We have $$H_{R_{\alpha,\beta}}(\operatorname{\mathbb{R}})=\left\{ \begin{array}{ll}\vspace{1.5ex} 1&\text{if }\beta=\alpha-1\\ \displaystyle\frac{\alpha-1}{\beta}&\text{if }\beta>\alpha-1 \end{array}\right..$$ If we fix $\alpha>1$, the uniform Hölder exponent of $R_{\alpha,\beta}$ decreases to $0$ as $\beta$ increases to infinity. In order to illustrate this phenomenon, we give the graphical representation of $R_{\alpha,\beta}$ for some $\beta$. For $\beta$ large enough, we can observe that $R_{\alpha,\beta}$ seems to be the function $x\mapsto \sin(\pi x)$ with some noise or fluctuations all around. In fact, this function is simply the first term of the series defining $R_{\alpha,\beta}$. We show that $R_{\alpha,\beta}$ can be, on average, compared to the function $x\mapsto\sin(\pi x)$ and we measure the amplitude of these fluctuations. The paper is organized as follows. In Section \[CWT\], we recall some helpful properties about the continuous wavelet transform and the tool that it provides to study the Hölder continuity of a function. We will extensively take advantage of the properties of the Lusin wavelet. The proof of Theorem \[Main\] is given in Section \[ProofMain\]. We analyse in Section \[Graphic\] the behaviour of $R_{\alpha,\beta}$ as $\beta$ increases. We present the graphical representation of $R_{2,\beta}$ for some particular values of $\beta$. In section \[FinalRem\], we give some additional comments about the more general case of nonharmonic Fourier series. We also show the limitations of the Lusin wavelet to investigate the research of the maximal possible Hölder exponent of $R_{\alpha,\beta}$ at a point. Hölder continuity and continuous wavelet transform {#CWT} ================================================== Let us recall some notions about the continuous wavelet transform and the Hölder continuity of a function (see [@Da; @Ja; @JMR; @T; @Ho; @HT]). The natural space associated to the continuous wavelet transform is the Hilbert space $L^2(\operatorname{\mathbb{R}})$. Such a setting is of no interest for the function $R_{\alpha,\beta}$, since it does not belong to $L^2(\operatorname{\mathbb{R}})$. As $R_{\alpha,\beta}$ is a continuous and bounded function on $\operatorname{\mathbb{R}}$, the continuous wavelet transform of a function of $L^\infty(\operatorname{\mathbb{R}})$ is more appropriate. The function $\psi$ is a [wavelet]{} if $\psi\in L^1(\operatorname{\mathbb{R}})\cap L^2(\operatorname{\mathbb{R}})$ and $\hat{\psi}(0)=0$, where $\hat{\psi}$ denotes the Fourier transform of $\psi$: $$\hat{\psi}(\xi)=\int_{\operatorname{\mathbb{R}}}e^{-ix\xi}\psi(x)\,dx,\quad\xi\in\operatorname{\mathbb{R}}.$$ Using the wavelet $\psi$, the [continuous wavelet transform]{} of a function $f\in L^\infty(\operatorname{\mathbb{R}})$ is the function $\mathcal{W}_\psi f$ defined by $$\mathcal{W}_\psi f(a,b)=\int_{\operatorname{\mathbb{R}}} f(x)\,\frac{1}{a}\,\overline{\psi}\left(\frac{x-b}{a}\right)\,dx,\quad a>0,\,b\in\operatorname{\mathbb{R}},$$ where $\overline{\psi}$ denotes the complex conjugate of $\psi$. In order to study the uniform Hölder continuity of $R_{\alpha,\beta}$, we will use a peculiar wavelet, known as the Lusin wavelet: $$\label{Lusin} \psi(x)=\frac{1}{\pi(x+i)^2},\quad x\in\operatorname{\mathbb{R}}.$$ Since $$\hat{\psi}(\xi)=\left\{\begin{array}{ll} -2\xi e^{-\xi}&\text{if }\xi\geq 0\\ 0&\text{if }\xi<0 \end{array}\right.,$$ this wavelet belongs to the second Hardy space $$H^2(\operatorname{\mathbb{R}})=\left\{f\in L^2(\operatorname{\mathbb{R}}): \hat{f}=0\,\text{ a.e. on }(-\infty,0)\right\}.$$ Such a property will be useful to obtain a simple explicit expression of $\mathcal{W}_\psi R_{\alpha,\beta}$ (in comparison with the derivatives of a gaussian function for example). An exact reconstruction formula exists in such a situation: if $\psi$ belongs to $H^2(\operatorname{\mathbb{R}})$ and if $f$ belongs to a certain class of continuous and bounded functions on $\operatorname{\mathbb{R}}$, we can recover $f$ from $\mathcal{W}_\psi f$ using a second wavelet satisfying some additional properties. This result is strongly inspired by Proposition 2.4.2 in [@Da] and Theorem 2.2 in [@HT]. For the sake of completeness, we give in the appendix a proof based on the ideas of [@Da; @HT; @Ho] and adapted to our case. \[LusinRecons\] Let $\psi$ be a wavelet which belongs to $H^2(\operatorname{\mathbb{R}})$. Let $\varphi$ be a differentiable wavelet such that $x\mapsto x\varphi(x)$ is integrable on $\operatorname{\mathbb{R}}$, such that $D\varphi$ is square integrable on $\operatorname{\mathbb{R}}$ and such that $$\label{CondAdm} \int_0^{+\infty}\overline{\hat{\psi}}(\xi)\hat{\varphi}(\xi)\,\frac{d\xi}{\xi}=1.$$ If $f$ is a continuous and bounded function on $\operatorname{\mathbb{R}}$ and is weakly oscillating around the origin, i.e. such that $$\lim_{r\to +\infty}\; \sup_{x\in\operatorname{\mathbb{R}}}\left\|\frac{1}{2r}\int_{x-r}^{x+r} f(t)\,dt\right\|=0,$$ then we have $$f(x)=\lim_{\substack{\varepsilon\to 0^+\\ r\to+\infty}} 2\int_\varepsilon^r\left(\int_{-\infty}^{+\infty}\mathcal{W}_\psi f(a,b)\,\frac{1}{a}\varphi\left(\frac{x-b}{a}\right)\,db\right)\,\frac{da}{a}$$ for all $x\in\operatorname{\mathbb{R}}$. Thanks to this reconstruction formula, the Hölder continuity of a function can be characterized with its continuous wavelet transform, provided that the wavelet satisfies some additional conditions. We will use the following result to study the Hölder continuity of the generalized Riemann function (see [@JMR; @Ja; @HT]). \[CaractUnif\] Let $\alpha\in(0,1)$, let $\psi$ be a wavelet such that $x\mapsto x^\alpha\psi(x)$ is integrable on $\operatorname{\mathbb{R}}$ and let $f$ be a function as in Theorem \[LusinRecons\]. (1) We have $f\in C^{\alpha}(\operatorname{\mathbb{R}})$ if and only if there exists $C>0$ such that $$\|\mathcal{W}_\psi f(a,b)\|\leq C\,a^{\alpha}$$ for all $a>0$ and $b\in\operatorname{\mathbb{R}}$. (2) Let $x_0\in\operatorname{\mathbb{R}}$. If $f\in C^{\alpha}(x_0)$, then there exist $C>0$ and $\eta>0$ such that $$\|\mathcal{W}_\psi f(a,b)\|\leq C\,a^{\alpha}\left(1+\left(\frac{\|b-x_0\|}{a}\right)^\alpha\right)$$ for all $a\in(0,\eta)$ and $b\in(x_0-\eta,x_0+\eta)$. Conversely, if there exist $\alpha'\in(0,\alpha)$, $C>0$ and $\eta>0$ such that $$\|\mathcal{W}_\psi f(a,b)\|\leq C\,a^{\alpha}\left(1+\left(\frac{\|b-x_0\|}{a}\right)^{\alpha'}\right)$$ for all $a\in(0,\eta)$ and $b\in(x_0-\eta,x_0+\eta)$, then $f\in C^\alpha(x_0)$.$\square$ Let us note that the necessary conditions in Theorem \[CaractUnif\] do not need all the hypotheses on the function $f$: the continuity and the weak oscillation around the origin of $f$ are not useful for these implications. The generalized Riemann function and the Lusin wavelet satisfy the conditions of the two previous theorems. Indeed, we know that $R_{\alpha,\beta}$ is continuous and bounded and that the Lusin wavelet $\psi$ belongs to $H^2(\operatorname{\mathbb{R}})$. Moreover, $R_{\alpha,\beta}$ is weakly oscillating around the origin because $$\left\|\frac{1}{2r}\int_{x-r}^{x+r} R_{\alpha,\beta}(t)\,dt\right\| \leq\left\|\frac{1}{2r}\sum_{n=1}^{+\infty}\frac{\cos((x-r)\pi n^\beta)-\cos((x+r)\pi n^\beta)}{\pi n^{\alpha+\beta}}\right\| \leq\frac{\zeta(\alpha+\beta)}{\pi r}$$ for all $x\in\operatorname{\mathbb{R}}$ and $r>0$, and $x\mapsto x^\alpha\psi(x)$ is clearly integrable for $\alpha\in(0,1)$. Besides, it is easy to find a differentiable wavelet $\varphi$ such that $x\mapsto x\varphi(x)$ is integrable on $\operatorname{\mathbb{R}}$, such that $D\varphi$ is square integrable on $\operatorname{\mathbb{R}}$ and such that $$\int_0^{+\infty}\hat{\varphi}(\xi)e^{-\xi}\,d\xi=-\frac{1}{2}.$$ In the following, $\psi$ will systematically denote the Lusin wavelet (see ). Hölder continuity of generalized Riemann function {#ProofMain} ================================================= Since we know that the function $R_{\alpha,\beta}$ is continuously differentiable on $\operatorname{\mathbb{R}}$ if $\alpha>1$ and $\beta\in(0,\alpha-1)$, we may assume $\beta\geq\alpha-1$ in the study of the uniform Hölder continuity of $R_{\alpha,\beta}$. To prove Theorem \[Main\], we first need to determine the continuous wavelet transform of $R_{\alpha,\beta}$ related to the Lusin wavelet, as in [@JMR; @Ja; @HT] where the case $\alpha=\beta=2$ is treated. We have $$\begin{aligned} \label{Wab} \mathcal{W}_\psi R_{\alpha,\beta}(a,b)=ia\pi\sum_{n=1}^{+\infty}\frac{e^{i\pi n^\beta(b+ia)}}{n^{\alpha-\beta}}\end{aligned}$$ for all $a>0$ and $b\in\operatorname{\mathbb{R}}$. We can write $$R_{\alpha,\beta}(x)=\frac{1}{2}\left(T_{\alpha,\beta}(x)-\widetilde{T}_{\alpha,\beta}(x)\right)$$ for $x\in\operatorname{\mathbb{R}}$ with $$T_{\alpha,\beta}(x)=-i\sum_{n=1}^{+\infty}\frac{e^{i\pi n^{\beta}x}}{n^\alpha}\qquad\text{and}\qquad\widetilde{T}_{\alpha,\beta}(x)=T_{\alpha,\beta}(-x).$$ In other words, $R_{\alpha,\beta}$ is the odd part of $T_{\alpha,\beta}$. Let us fix $a>0$ and $b\in\operatorname{\mathbb{R}}$. We have $$\mathcal{W}_\psi T_{\alpha,\beta}(a,b) =\int_{\operatorname{\mathbb{R}}}T_{\alpha,\beta}(x)\,\frac{1}{a}\overline{\psi}\left(\frac{x-b}{a}\right)\,dx =\frac{a}{\pi}\int_{\operatorname{\mathbb{R}}}\frac{T_{\alpha,\beta}(x)}{(x-(b+ia))^2}\,dx.$$ For $\eta>0$ and $r>0$, let us denote by $\gamma_{\eta,r}$ the closed path formed by the juxtaposition of the two following ones: the first path describes the segment $[-r+i\eta,r+i\eta]$ and the second one the half-circle of center $i\eta$ and radius $r$ included in $H=\{z\in\operatorname{\mathbb{C}}:\Im z>0\}$. The function $T_{\alpha,\beta}$ is holomorphic on $H$ because the series uniformly converges on every compact set of $H$. As the point $b+ia$ is situated inside the curve described by $\gamma_{\eta,r}$ for $\eta\in(0,a)$ and $r>a$, we obtain $$\begin{aligned} \mathcal{W}_\psi T_{\alpha,\beta}(a,b) &=&\frac{a}{\pi}\lim_{r\to+\infty}\lim_{\eta\to 0^+}\int_{\gamma_{\eta,r}}\frac{T_{\alpha,\beta}(z)}{(z-(b+ia))^2}\,dz\\ &=&2ia\,(DT_{\alpha,\beta})(b+ia)\\ &=&2ia\pi\sum_{n=1}^{+\infty}\frac{e^{i\pi n^\beta(b+ia)}}{n^{\alpha-\beta}},\end{aligned}$$ thanks to Cauchy’s integral formula. Similarly, the continuous wavelet transform of $\widetilde{T}_{\alpha,\beta}$ is given by $$\mathcal{W}_\psi\widetilde{T}_{\alpha,\beta}(a,b) =\int_{\operatorname{\mathbb{R}}}T_{\alpha,\beta}(-x)\,\frac{1}{a}\overline{\psi}\left(\frac{x-b}{a}\right)\,dx =\frac{a}{\pi}\lim_{r\to+\infty}\lim_{\eta\to 0^+}\int_{\gamma_{\eta,r}}\frac{T_{\alpha,\beta}(z)}{(z-(-b-ia))^2}\,dz=0$$ by homotopy invariance, because the point $-b-ia$ does not belong to $H$. We thus have the conclusion. Let us now analyse $\mathcal{W}_\psi R_{\alpha,\beta}$ in order to study the uniform Hölder continuity of $R_{\alpha,\beta}$ with Theorem \[CaractUnif\]. We have $$\label{WabMaj} \|\mathcal{W}_\psi R_{\alpha,\beta}(a,b)\|\leq a\pi\sum_{n=1}^{+\infty}\frac{e^{-a\pi n^\beta}}{n^{\alpha-\beta}} =\|\mathcal{W}_\psi R_{\alpha,\beta}(a,0)\|$$ for $a>0$ and $b\in\operatorname{\mathbb{R}}$. The function $f_{\alpha,\beta}:x\mapsto x^{\beta-\alpha}\,e^{-a\pi x^\beta}$ is differentiable on $(0,+\infty)$ and $$Df_{\alpha,\beta}(x)=e^{-a\pi x^\beta}\,x^{\beta-\alpha-1}\,\left((\beta-\alpha)-a\pi\beta x^\beta\right),\quad x>0.$$ Then, $f_{\alpha,\beta}$ is decreasing on $(0,+\infty)$ if $\beta\in[\alpha-1,\alpha)$ and on $(((\beta-\alpha)/a\pi\beta)^{1/\beta},+\infty)$ if $\beta\geq\alpha$. The next developments are mainly based on the classical comparison principle between series and integral (when the general term is decreasing), which we recall in the following lemma. Let $N\in\operatorname{\mathbb{N}}$ and let $f$ be a decreasing and positive function defined on $[N,+\infty)$. The series $\sum_{n=N+1}^{+\infty}f(n)$ converges if and only if $f$ is integrable on $[N,+\infty)$; in this case we have $$\int_{N+1}^{+\infty}f(x)\,dx\leq\sum_{n=N+1}^{+\infty}f(n)\leq\int_{N}^{+\infty}f(x)\,dx.$$ $\square$ We note that $f_{\alpha,\beta}$ is integrable on $(0,+\infty)$ only if $\beta>\alpha-1$. We therefore split the study of the uniform Hölder continuity and the calculus of the uniform Hölder exponent of $R_{\alpha,\beta}$ into two cases: $\beta>\alpha-1$ and $\beta=\alpha-1$. \[ExpH\] If $\beta>\alpha-1$, then $$H_{R_{\alpha,\beta}}(\operatorname{\mathbb{R}})=\frac{\alpha-1}{\beta}.$$ 1\. Let us first consider the case $\beta\in(\alpha-1,\alpha)$. The function $f_{\alpha,\beta}$ is decreasing on $[1,+\infty)$ and we have $$\|\mathcal{W}_\psi R_{\alpha,\beta}(a,b)\| \leq a\pi\left(e^{-a\pi}+\sum_{n=2}^{+\infty}\frac{e^{-a\pi n^\beta}}{n^{\alpha-\beta}}\right) \leq a\pi\left(e^{-a\pi}+\int_1^{+\infty}\frac{e^{-a\pi x^\beta}}{x^{\alpha-\beta}}\,dx\right)$$ for $a>0$ and $b\in\operatorname{\mathbb{R}}$. For the second term of the right hand side of the last inequality, we obtain $$\label{GammaMaj} \int_1^{+\infty}\frac{e^{-a\pi x^\beta}}{x^{\alpha-\beta}}\,dx \leq \int_0^{+\infty}\frac{e^{-a\pi x^\beta}}{x^{\alpha-\beta}}\,dx =\frac{1}{\beta}\pi^{\frac{\alpha-1}{\beta}}\,\Gamma\left(\frac{1+\beta-\alpha}{\beta}\right)\,a^{\frac{\alpha-1}{\beta}-1}$$ for $a>0$, where $\Gamma$ is defined by $$\Gamma(x)=\int_0^{+\infty}e^{-t}\,t^{x-1}\,dt,\quad x>0,$$ as usual. For the first term, we note that the function $a\mapsto e^{-a\pi} a^{1-\frac{\alpha-1}{\beta}}$ is bounded on $(0,+\infty)$ because $\alpha-1<\beta$. Then, there exists $C_{\alpha,\beta}>0$ such that $$\|\mathcal{W}_\psi R_{\alpha,\beta}(a,b)\|\leq C_{\alpha,\beta}\,a^{\frac{\alpha-1}{\beta}}$$ for all $a>0$ and $b\in\operatorname{\mathbb{R}}$, which implies $R_{\alpha,\beta}\in C^{\frac{\alpha-1}{\beta}}(\operatorname{\mathbb{R}})$ using Theorem \[CaractUnif\]. Let us show the optimality of this exponent $(\alpha-1)/\beta$ related to the uniform Hölder continuity. Let $C>0$ and $\eta>0$; we have $$\|\mathcal{W}_\psi R_{\alpha,\beta}(a,0)\|=a\pi\sum_{n=1}^{+\infty}\frac{e^{-\pi n^\beta a}}{n^{\alpha-\beta}} \geq a\pi\int_1^{+\infty}\frac{e^{-a\pi x^\beta}}{x^{\alpha-\beta}}\,dx =\frac{1}{\beta}\,(a\pi)^{\frac{\alpha-1}{\beta}}\,\Gamma\left(\frac{\beta-\alpha+1}{\beta},a\pi\right)$$ for $a>0$, where $\Gamma$ is the incomplete Gamma function defined by $$\Gamma(x,y)=\int_y^{+\infty}e^{-t}t^{x-1}\,dt,\quad (x,y)\in(0,+\infty)\times [0,+\infty).$$ Let us recall that $\Gamma(x,0)=\Gamma(x)$ and $\Gamma(x,y)$ converges to $\Gamma(x)$ as $y\to 0^+$ for all $x>0$. Since $\Gamma((\beta-\alpha+1)/\beta,a\pi)\to\Gamma((\beta-\alpha+1)/\beta)$ and $a^\eta\to 0$ as $a\to 0^+$, there exists $A>0$ such that, for all $a\in(0,A)$, we have $$\|\mathcal{W}_\psi R_{\alpha,\beta}(a,0)\| >C\,a^{\frac{\alpha-1}{\beta}+\eta}.$$ Hence the conclusion using Theorem \[CaractUnif\]. 2\. Let us now consider the case $\beta\geq\alpha$ and let us write $N_a=\lfloor((\beta-\alpha)/a\pi\beta)^{1/\beta}\rfloor+1$, where $\lfloor x\rfloor$ denotes the largest integer smaller than or equal to the real $x$. If $a>1$, then $N_a=1$ and we can proceed as in the previous case. Let us therefore suppose that $a\in(0,1]$. We have $$\begin{aligned} \|\mathcal{W}_\psi R_{\alpha,\beta}(a,b)\|&\leq & a\pi\left(\sum_{n=1}^{N_a}\frac{e^{-a\pi n^\beta}}{n^{\alpha-\beta}}+\sum_{n=N_a+1}^{+\infty}\frac{e^{-a\pi n^\beta}}{n^{\alpha-\beta}}\right)\\ &\leq & a\pi\left(N_a\,N_a^{\beta-\alpha}+\int_{N_a}^{+\infty}\frac{e^{-a\pi x^\beta}}{x^{\alpha-\beta}}\,dx\right)\\ &\leq & a\pi\left(\left(\left(\frac{\beta-\alpha}{\pi\beta}\right)^{\frac{1}{\beta}}+a^{\frac{1}{\beta}}\right)^{\beta-\alpha+1}a^{\frac{\alpha-1}{\beta}-1}+\int_{0}^{+\infty}\frac{e^{-a\pi x^\beta}}{x^{\alpha-\beta}}\,dx\right)\\ & \leq & a^{\frac{\alpha-1}{\beta}}\pi\left(\left(\left(\frac{\beta-\alpha}{\pi\beta}\right)^{\frac{1}{\beta}}+1\right)^{\beta-\alpha+1}+\frac{1}{\beta}\pi^{\frac{\alpha-1}{\beta}}\,\Gamma\left(\frac{1+\beta-\alpha}{\beta}\right)\right),\end{aligned}$$ where we have used relation to obtain the last inequality. We then have $R_{\alpha,\beta}\in C^{\frac{\alpha-1}{\beta}}(\operatorname{\mathbb{R}})$ using Theorem \[CaractUnif\]. Let us show the optimality of the exponent related to the uniform Hölder continuity. Let $C>0$ and $\eta>0$; we have $$\begin{aligned} \sum_{n=1}^{+\infty}\frac{e^{-\pi n^\beta a}}{n^{\alpha-\beta}}&\geq & \sum_{n=N_a}^{+\infty}\frac{e^{-\pi n^\beta a}}{n^{\alpha-\beta}}\\ &\geq & \int_{N_a}^{+\infty}\frac{e^{-a\pi x^\beta}}{x^{\alpha-\beta}}\,dx\\ &=&\frac{1}{\beta}\,(a\pi)^{\frac{\alpha-1}{\beta}-1}\int_{a\pi N_a^\beta}^{+\infty}e^{-u}\,u^{\frac{\beta-\alpha+1}{\beta}-1}\,du\\ &\geq & \frac{1}{\beta}\,(a\pi)^{\frac{\alpha-1}{\beta}-1}\,\Gamma\left(\frac{\beta-\alpha+1}{\beta},\left(\left(\frac{\beta-\alpha}{\beta}\right)^{1/\beta}+(a\pi)^{1/\beta}\right)^\beta\right)\end{aligned}$$ for $a>0$. As in the case $\beta\in(\alpha-1,\alpha)$, there exists $A>0$ such that, for all $a\in(0,A)$, we have $$\|\mathcal{W}_\psi R_{\alpha,\beta}(a,0)\| >C\,a^{\frac{\alpha-1}{\beta}+\eta},$$ hence the conclusion using once again Theorem \[CaractUnif\]. In fact, by taking $b=2k$ with $k\in\operatorname{\mathbb{Z}}$, we can show that $R_{\alpha,\beta}\in C^{\frac{\alpha-1}{\beta}}(2k)$ and that the exponent cannot be improved because $\mathcal{W}_\psi R_{\alpha,\beta}(a,2k)=\mathcal{W}_\psi R_{\alpha,\beta}(a,0)$ for all $a>0$. In other words, we have $$H_{R_{\alpha,\beta}}(2k)=\frac{\alpha-1}{\beta}.$$ Since this quantity is strictly smaller than $1$, $R_{\alpha,\beta}$ is consequently not differentiable at $2k$. We have $H_{R_{\alpha,\alpha-1}}(\operatorname{\mathbb{R}})=1$. We have $$\|\mathcal{W}_\psi R_{\alpha,\alpha-1}(a,b)\|\leq a\pi\left(e^{-a\pi}+\int_1^{+\infty}\frac{e^{-a\pi x^{\alpha-1}}}{x}\,dx\right) =a\pi\left(e^{-a\pi}+\frac{1}{\alpha-1}E_1(a\pi)\right)$$ for $a>0$ and $b\in\operatorname{\mathbb{R}}$, where $E_1$ is the exponential integral defined by $$E_1(x)=\int_1^{+\infty}\frac{e^{-xt}}{t}\,dt,\quad x>0.$$ Since we have $$\label{E1} \frac{1}{2}\,e^{-x}\,\ln\left(1+\frac{2}{x}\right)<E_1(x)<e^{-x}\ln\left(1+\frac{1}{x}\right)$$ for all $x>0$ (see [@AS] p. 229), we obtain $$\|\mathcal{W}_\psi R_{\alpha,\alpha-1}(a,b)\|\leq a\pi\,e^{-a\pi}\left(1+\frac{1}{\alpha-1}\,\ln\left(1+\frac{1}{a\pi}\right)\right)$$ for $a>0$ and $b\in\operatorname{\mathbb{R}}$. Let us fix $\delta\in(0,1)$. There exists $A>0$ such that, for all $a\in(0,A)$, we have $$\frac{1}{\alpha-1}\,\frac{\ln\left(1+\frac{1}{a\pi}\right)}{\left(1+\frac{1}{a\pi}\right)^{\delta}}<1$$ and then $$\|\mathcal{W}_\psi R_{\alpha,\alpha-1}(a,b)\| \leq a\pi\,e^{-a\pi}\left(1+\left(1+\frac{1}{a\pi}\right)^\delta\right) \leq a\pi\left(1+2^\delta \left(1+\left(\frac{1}{a\pi}\right)^\delta \right)\right).$$ There also exists $A'\in(0,A)$ such that, for all $a\in(0,A')$, we have $$\|\mathcal{W}_\psi R_{\alpha,\alpha-1}(a,b)\|\leq C_{\delta}' a^{1-\delta},$$ where $C_\delta '$ is a positive constant (depending only on $\delta$). Since the function $$a\mapsto a^\delta e^{-a\pi}\left(1+\frac{1}{\alpha-1}\,\ln\left(1+\frac{1}{a\pi}\right)\right)$$ is bounded on $[A',+\infty)$, we also have $$\|\mathcal{W}_\psi R_{\alpha,\alpha-1}(a,b)\|\leq C_{\delta}''a^{1-\delta}$$ for $a\in[A',+\infty)$, where $C_\delta''$ is a positive constant. We thus obtain $$\|\mathcal{W}_\psi R_{\alpha,\alpha-1}(a,b)\|\leq C_\delta\,a^{1-\delta}$$ for all $a>0$ and $b\in\operatorname{\mathbb{R}}$ where $C_\delta=\max\{C_\delta',C_\delta''\}$, which implies $R_{\alpha,\alpha-1}\in C^{1-\delta}(\operatorname{\mathbb{R}})$ using Theorem \[CaractUnif\]. Let us now show that this exponent of uniform Hölder continuity is optimal. Let $C>0$; we have $$\|\mathcal{W}_\psi R_{\alpha,\alpha-1}(a,0)\|\geq a\pi\int_1^{+\infty}\frac{e^{-a\pi x^{\alpha-1}}}{x}\,dx=\frac{a\pi}{\alpha-1} E_1(a\pi)\geq a\,\frac{\pi}{2(\alpha-1)}\,e^{-a\pi}\ln\left(1+\frac{2}{a\pi}\right)$$ for all $a>0$ thanks to and so, there exists $A>0$ such that, for all $a\in(0,A)$, we have $$\|\mathcal{W}_\psi R_{\alpha,\alpha-1}(a,0)\|>Ca,$$ hence the conclusion using one last time Theorem \[CaractUnif\]. Behaviour of $R_{\alpha,\beta}$ as $\beta$ increases {#Graphic} ==================================================== If we fix $\alpha>1$, we know that the uniform Hölder exponent of $R_{\alpha,\beta}$ decreases as $\beta$ increases, thanks to Theorem \[Main\]. Moreover, we know that this exponent is exactly the Hölder exponent of $R_{\alpha,\beta}$ at the origin. This phenomenon is clearly illustrated in Figure \[Graphique\] in the case $\alpha=2$. ![\[Graphique\] Graphical representation of $R_{2,1}$, $R_{2,3/2}$, $R_{2,2}$, $R_{2,4}$ and $R_{2,10}$](2-1.eps "fig:"){width="7.5cm"} ![\[Graphique\] Graphical representation of $R_{2,1}$, $R_{2,3/2}$, $R_{2,2}$, $R_{2,4}$ and $R_{2,10}$](2-1.5.eps "fig:"){width="7.5cm"}\ ![\[Graphique\] Graphical representation of $R_{2,1}$, $R_{2,3/2}$, $R_{2,2}$, $R_{2,4}$ and $R_{2,10}$](2-2.eps "fig:"){width="7.5cm"} ![\[Graphique\] Graphical representation of $R_{2,1}$, $R_{2,3/2}$, $R_{2,2}$, $R_{2,4}$ and $R_{2,10}$](2-4.eps "fig:"){width="7.5cm"}\ ![\[Graphique\] Graphical representation of $R_{2,1}$, $R_{2,3/2}$, $R_{2,2}$, $R_{2,4}$ and $R_{2,10}$](2-10.eps "fig:"){width="7.5cm"} As $\beta$ tends to infinity, we note that the graphical representation of $R_{\alpha,\beta}$ looks like to the one of the function $s:x\mapsto\sin(\pi x)$ (in a certain sense to establish), with some noise or fluctuations all around. In the next two propositions, we give a convergence result and show that the fluctuations have a constant amplitude (i.e. independent from $\beta$). To do so, let us recall the usual definition of the mean of an integrable function over a bounded interval. Let $a,b\in\operatorname{\mathbb{R}}$ be such that $a<b$ and let $f$ be an integrable function on $(a,b)$. The [mean of the function $f$ over the inverval $(a,b)$]{} is defined by $$m_f^{a,b}=\frac{1}{b-a}\int_a^b f(x)\,dx.$$ \[Mean\] Let $\alpha>1$. For all $a,b\in\operatorname{\mathbb{R}}$ such that $a<b$, we have $$\lim_{\beta\to+\infty}m_{R_{\alpha,\beta}}^{a,b}=m_s^{a,b}.$$ We have $$\left\|\int_a^b(R_{\alpha,\beta}(x)-\sin(\pi x))\,dx\right\| =\left\|\sum_{n=2}^{+\infty}\frac{\cos(\pi n^\beta a)-\cos(\pi n^\beta b)}{\pi n^{\alpha+\beta}}\right\|\leq\frac{2}{\pi}(\zeta(\alpha+\beta)-1)$$ and we know that $\zeta(x)\to 1$ as $x\to+\infty$, hence the conclusion. \[Fluctuations\] Let $\alpha>1$ and let $\beta\in\operatorname{\mathbb{N}}\setminus\{0\}$. The function $R_{\alpha,\beta}$ is periodic of period $2$ and we have $$\int_{-1}^{1}\left(R_{\alpha,\beta}(x)-\sin(\pi x)\right)^2\,dx=\zeta(2\alpha)-1.$$ The periodicity of $R_{\alpha,\beta}$ is easy to check. Let us calculate the integral. By developing $x\mapsto R_{\alpha,\beta}(x)-\sin(\pi x)$ in Fourier series, we have $$R_{\alpha,\beta}(x)-\sin(\pi x)=\frac{a_0}{2}+\sum_{m=1}^{+\infty}\left(a_m\cos(\pi m x)+b_m\sin(\pi m x)\right)$$ in $L^2([-1,1])$ where $a_0=a_m=0$ and $$\begin{aligned} b_m&=&2\int_0^1(R_{\alpha,\beta}(x)-\sin(\pi x))\,dx\\ &=&\sum_{n=2}^{+\infty}\frac{1}{n^\alpha}\int_0^1\left(\cos(x\pi(n^\beta-m))-\cos(x\pi(n^\beta+m))\right)\,dx\\ &=&\left\{ \begin{array}{ll}\vspace{1.5ex} \displaystyle\frac{1}{m^{\alpha/\beta}}&\text{if $m=k^\beta$ for one $k\in\operatorname{\mathbb{N}}\setminus\{0,1\}$}\\ 0&\text{otherwise} \end{array} \right.\end{aligned}$$ for all $m\in\operatorname{\mathbb{N}}\setminus\{0\}$. Consequently, by Parseval formula, we obtain $$\int_{-1}^{1}\left(R_{\alpha,\beta}(x)-\sin(\pi x)\right)^2\,dx =\sum_{m=1}^{+\infty} b_m^2=\sum_{k=2}^{+\infty}\frac{1}{k^{2\alpha}}=\zeta(2\alpha)-1.$$ The two previous propositions are illustrated in Figure \[GraphiqueBeta\]. Let us end this section with a simple remark about the behaviour of $R_{\alpha,\beta}$ as $\alpha$ tends to infinity. ![\[GraphiqueBeta\] Mean value and amplitude of fluctuations of $x\mapsto R_{2,10}(x)-\sin(\pi x)$](rieman-sin.eps "fig:"){width="7.5cm"} ![\[GraphiqueBeta\] Mean value and amplitude of fluctuations of $x\mapsto R_{2,10}(x)-\sin(\pi x)$](rieman-sin-2.eps "fig:"){width="7.5cm"}\ Proposition \[Mean\] is also “satisfied” for $\alpha$: we have $$\lim_{\alpha\to +\infty}m_{R_{\alpha,\beta}}^{a,b}=m_s^{a,b}$$ for all $\beta>0$ and all $a,b\in\operatorname{\mathbb{R}}$ such that $a<b$. Moreover, by Proposition \[Fluctuations\], we have $$\lim_{\alpha\to+\infty}\int_{-1}^1\left(R_{\alpha,\beta}(x)-\sin(\pi x)\right)^2\,dx=0$$ for all $\beta\in\operatorname{\mathbb{N}}\setminus\{0\}$. In fact, a stronger result holds: for any fixed $\beta>0$, $R_{\alpha,\beta}$ uniformly converges on $\operatorname{\mathbb{R}}$ to $s$ as $\alpha$ tends to infinity because we have $$\left\|R_{\alpha,\beta}(x)-\sin(\pi x)\right\|\leq\sum_{n=2}^{+\infty}\frac{1}{n^\alpha}=\zeta(\alpha)-1$$ for all $x\in\operatorname{\mathbb{R}}$. Final remarks {#FinalRem} ============= About nonharmonic Fourier series -------------------------------- A part of Theorem \[Main\] can be adapted for particular nonharmonic Fourier series. Let us first recall the notion of nonharmonic Fourier series (see [@L; @Y; @Ja1]). Let $\boldsymbol{a}=(a_n)_{n\in\operatorname{\mathbb{N}}\setminus\{0\}}$ be a sequence of complex numbers and let $\boldsymbol{\lambda}=(\lambda_n)_{n\in\operatorname{\mathbb{N}}\setminus\{0\}}$ be an increasing sequence of positive numbers which converges to infinity. A [nonharmonic Fourier series]{} (related to the sequences $\boldsymbol{a}$ and $\boldsymbol{\lambda}$) is a function $S$ defined by $$S(x)=\sum_{n=1}^{+\infty}a_n\,e^{i\lambda_n x},\quad x\in\operatorname{\mathbb{R}},$$ if the series converges. If the series $\sum_{n=1}^{+\infty}a_n$ is absolutely convergent, then the above series (related to $S$) uniformly converges on $\operatorname{\mathbb{R}}$. We will assume that this is the case in the remainder of this discussion. Such a function $S$ is then continuous and bounded on $\operatorname{\mathbb{R}}$. As for $R_{\alpha,\beta}$, we can calculate the continuous wavelet transform of $S$ (related to the Lusin wavelet). Since $\lambda_n>0$ for all $n\in\operatorname{\mathbb{N}}\setminus\{0\}$, $S$ is a holomorphic function on $H$ and we have $$\mathcal{W}_\psi S(a,b)=-2a\sum_{n=1}^{+\infty} a_n \lambda_n\,e^{i\lambda_n (b+ia)}$$ for $a>0$ and $b\in\operatorname{\mathbb{R}}$, similarly to . If we assume that there exist positive constants $C_1$, $C_2$ and $C_3$, $\alpha>1$ and $\beta>0$ such that $$\|a_n\|\leq \frac{C_1}{n^\alpha}\quad\text{and}\quad C_2 n^\beta\leq \lambda_n\leq C_3 n^\beta$$ for all $n\in\operatorname{\mathbb{N}}\setminus\{0\}$, we then obtain $$\|\mathcal{W}_\psi S(a,b)\|\leq 2a C_1 C_3\sum_{n=1}^{+\infty} \frac{e^{-C_2 a n^\beta}}{n^{\alpha-\beta}}$$ for $a>0$ and $b\in\operatorname{\mathbb{R}}$ and we recover an expression similar to the one obtained for $\|\mathcal{W}_\psi R_{\alpha,\beta}(a,b)\|$ in . Using the same reasoning as in the study of the uniform Hölder continuity of $R_{\alpha,\beta}$ with $\alpha>1$ and $\beta\geq\alpha-1$, we can formulate the following result. With the previous assumptions on $\boldsymbol{a}$ and $\boldsymbol{\lambda}$, we have $S\in C^{\frac{\alpha-1}{\beta}}(\operatorname{\mathbb{R}})$ if $\beta>\alpha-1$ and $S\in C^{1-\delta}(\operatorname{\mathbb{R}})$ for all $\delta\in(0,1)$ if $\beta=\alpha-1$. About the Lusin wavelet ----------------------- If $\alpha=\beta=2$, we know that the largest Hölder exponent of $R=R_{2,2}$ at a point is $3/2$ and that it is attained at the rational numbers $(2p+1)/(2q+1)$ with $p\in\operatorname{\mathbb{Z}}$ and $q\in\operatorname{\mathbb{N}}$ (see [@JM]). The continuous wavelet transform related to the Lusin wavelet of $R$ does not allow to find this exponent. Indeed, for $a>0$, we have $$\mathcal{W}_\psi R(a,1)=ia\pi\sum_{n=1}^{+\infty}e^{i\pi n^2(1+ia)}=ia\pi\sum_{n=1}^{+\infty}(-1)^n e^{-a\pi n^2} =\frac{ia\pi}{2}\left(\sum_{n\in\operatorname{\mathbb{Z}}} e^{i\pi n}\, e^{-a\pi n^2} -1\right)$$ and, by the Poisson summation formula, $$\|\mathcal{W}_\psi R(a,1)\|=\frac{a\pi}{2}\left\|\sum_{n\in\operatorname{\mathbb{Z}}} \frac{1}{\sqrt{a}}e^{-\frac{(\pi+n)^2}{4a\pi}}-1\right\|=\frac{a\pi}{2}\left\|\frac{e^{-\frac{\pi}{4a}}}{\sqrt{a}}\left(1+2\sum_{n=1}^{+\infty} e^{-\frac{n^2}{4a\pi}}\cosh\left(\frac{n}{2a}\right)\right)-1\right\|.$$ Let $C>0$ and $\eta>0$. We have $$\lim_{a\to 0^+}\frac{e^{-\frac{\pi}{4a}}}{\sqrt{a}}\left(1+2\sum_{n=1}^{+\infty} e^{-\frac{n^2}{4a\pi}}\cosh\left(\frac{n}{2a}\right)\right)=0$$ since we have $$2\sum_{n=8}^{+\infty} e^{-\frac{n^2}{4a\pi}}\cosh\left(\frac{n}{2a}\right) \leq\int_{7}^{+\infty} e^{-\frac{x^2}{4a\pi}}\left(1+e^{\frac{x}{2a}}\right)\,dx \leq \pi\sqrt{a}+\int_{7}^{+\infty}e^{-\frac{1}{2}(\frac{x^2}{2\pi}-x)}\,dx$$ for all $a\in(0,1)$. The sum begins with the term related to $n=8$ for two reasons. On the one hand, the function $g:x\mapsto e^{-\frac{x^2}{4a\pi}}\cosh\left(\frac{x}{2a}\right)$ is differentiable on $\operatorname{\mathbb{R}}$ and $$Dg(x)=\frac{e^{-\frac{x^2}{4a\pi}}}{2a}\left(-\frac{x}{\pi}\cosh\left(\frac{x}{2a}\right)+\sinh\left(\frac{x}{2a}\right)\right)\leq0\quad\Leftrightarrow\quad x\geq\pi\tanh\left(\frac{x}{2a}\right),$$ which implies that $g$ is decreasing on $[4,+\infty)$. On the other hand, the function $x\mapsto \frac{x^2}{2\pi}-x$ is positive on $[7,+\infty)$. Consequently, there exists $A\in(0,1)$ such that, for all $a\in(0,A)$, we have $$\frac{\pi}{2C}\left\|\sum_{n\in\operatorname{\mathbb{Z}}} \frac{1}{\sqrt{a}}e^{-\frac{(\pi+n)^2}{4a\pi}}-1\right\|>a^{\eta}$$ and then $$\label{Lusin32} \|\mathcal{W}_\psi R(a,1)\|>Ca^{1+\eta}.$$ In fact, the Lusin wavelet has only one vanishing moment since $\hat{\psi}(0)=0$ and $(D\hat{\psi})(0)\neq 0$, because the function $x\mapsto x\psi(x)$ is not integrable on $\operatorname{\mathbb{R}}$. Inequality thus shows that the second vanishing moment is essential for the study of the Hölder continuity of $R$ when the exponent is (strictly) greater than $1$. We could otherwise find $D>0$ and $\delta>0$ such that $$\|\mathcal{W}_\psi R(a,b)\|\leq D\,a^{3/2}\left(1+\left(\frac{\|b-1\|}{a}\right)^{3/2}\right)$$ for all $a\in(0,\delta)$ and $b\in(1-\delta,1+\delta)$ and then $\|\mathcal{W}_\psi R(a,1)\|\leq D\,a^{3/2}$ for all $a\in(0,\delta)$, which is in contradiction with by taking $C=D$, $\eta=1/2$ and $a\in(0,\min\{\delta,A\})$. Appendix ======== Let us give a proof of Theorem \[LusinRecons\]. It is based on the ideas of [@Da; @HT; @Ho] and adapted to the case of the continuous wavelet transform related to a wavelet $\psi$ which belongs to $H^2(\operatorname{\mathbb{R}})$. Let us fix $x\in\operatorname{\mathbb{R}}$ and $r>\varepsilon>0$. We write $$f_{\varepsilon,r}(x)=\int_\varepsilon^r\left(\int_{-\infty}^{+\infty}\mathcal{W}_\psi f(a,b)\,\frac{1}{a}\varphi\left(\frac{x-b}{a}\right)\,db\right)\frac{1}{a}\,da$$ and we have $$f_{\varepsilon,r}(x)=(M_{\varepsilon,r}\star f)(x)$$ by Fubini’s theorem, where $M_{\varepsilon,r}$ is defined by $$M_{\varepsilon,r}(t)=\int_\varepsilon^r\left(\int_{-\infty}^{+\infty}\overline{\psi}\left(-\frac{b}{a}\right)\varphi\left(\frac{t-b}{a}\right)\,db\right)\frac{1}{a^3}\,da,\quad t\in\operatorname{\mathbb{R}}.$$ Since $M_{\varepsilon,r}\in L^1(\operatorname{\mathbb{R}})$ and the support of $\hat{\psi}$ is included in $(0,+\infty)$, we have $$\hat{M}_{\varepsilon,r}(\xi) =\int_\varepsilon^r \overline{\hat{\psi}}(a\xi)\hat{\varphi}(a\xi)\frac{1}{a}\,da =\left\{ \begin{array}{ll} 0&\text{if}\;\xi\leq 0\\ \displaystyle\int_{\varepsilon\xi}^{r\xi}\overline{\hat{\psi}}(a)\hat{\varphi}(a)\frac{1}{a}\,da&\text{if}\;\xi>0 \end{array} \right..$$ Moreover, we have $$\label{M} \hat{M}_{\varepsilon,r}(\xi) =m(\varepsilon\xi)-m(r\xi)$$ for all $\xi\in\operatorname{\mathbb{R}}$, where $m$ is defined by $$m(\xi) =\left\{\begin{array}{ll} \displaystyle\int_\xi^{+\infty}\overline{\hat{\psi}}(a)\hat{\varphi}(a)\frac{1}{a}\,da&\text{if}\;\xi\geq 0\\ \displaystyle\int_{-\xi}^{+\infty}\overline{\hat{\psi}}(-a)\hat{\varphi}(-a)\frac{1}{a}\,da&\text{if}\;\xi< 0 \end{array}\right..$$ It is easy to check that $m(0)=1$, $m=0$ on $(-\infty,0)$ and that $m$ is continuous only on $\operatorname{\mathbb{R}}\setminus\{0\}$. Since we have the three following properties: $\hat{\psi}$ is bounded, $\varphi$ is differentiable and $D\varphi\in L^2(\operatorname{\mathbb{R}})$, we obtain $$\|m(\xi)\| \leq \left(\int_0^{+\infty}\|a\hat{\varphi}(a)\|^2da\right)^{1/2}\left(\int_\xi^{+\infty}\frac{\|\hat{\psi}(a)\|^2}{a^4}\,da\right)^{1/2} \leq \frac{C'}{\xi^{3/2}}$$ for all $\xi>0$, by Cauchy-Schwarz inequality, where $C'$ is a positive constant. Then, $m$ is bounded and there exists $C>0$ such that $$\|m(\xi)\|\leq\frac{C}{(1+\|\xi\|)^{3/2}}$$ for all $\xi\in\operatorname{\mathbb{R}}$. So $m\in L^1(\operatorname{\mathbb{R}})\cap L^2(\operatorname{\mathbb{R}})$ and we can define $M$ by $M(\xi)=\hat{m}(-\xi)/\pi$ for all $\xi\in\operatorname{\mathbb{R}}$. By definition, $M$ is continuous and bounded on $\operatorname{\mathbb{R}}$. Moreover, $m$ is differentiable on $\operatorname{\mathbb{R}}\setminus\{0\}$ and $$Dm(\xi)=\left\{ \begin{array}{ll} 0&\text{if}\;\xi<0\\ \displaystyle-\overline{\hat{\psi}}(\xi)\hat{\varphi}(\xi)\frac{1}{\xi}&\text{if}\;\xi>0 \end{array} \right..$$ Since $\hat{\varphi}(0)=0$ and $x\mapsto x\varphi(x)$ is integrable on $\operatorname{\mathbb{R}}$, we have $$\hat{\varphi}(\xi) =\left\|\int_{\operatorname{\mathbb{R}}}\varphi(x)\left(e^{-ix\xi}-1\right)dx\right\| =\left\|\int_{\operatorname{\mathbb{R}}}x\varphi(x)\left(\int_0^\xi -ie^{-ixt}\,dt\right)dx\right\| \leq C''\|\xi\|$$ for all $\xi\in\operatorname{\mathbb{R}}$, where $C''$ is a positive constant. Consequently, $Dm\in L^2(\operatorname{\mathbb{R}})$ because $\psi\in L^2(\operatorname{\mathbb{R}})$. So $M\in L^1(\operatorname{\mathbb{R}})$ since we can write $M$ as the product of two square integrable functions: for all $x\in\operatorname{\mathbb{R}}$, we have $$M(x)=\frac{1}{\sqrt{1+x^2}}\left(\sqrt{1+x^2}\,M(x)\right),$$ where the second factor is square integrable, because $m$ and $Dm$ are square integrable on $\operatorname{\mathbb{R}}$. Moreover, by the Dirichlet condition for Fourier inversion theorem (since $m$ and $Dm$ are piecewise continuous), we have $$\int_{\operatorname{\mathbb{R}}} M(x)\,dx=\hat{M}(0)=m(0^+)+m(0^-)=1$$ using where $m(0^\pm)=\lim_{\xi\to 0^{\pm}}m(\xi)$. By definition of $M$ and by Fourier inversion theorem in , we have $$M_{\varepsilon,r}(t)=\frac{1}{2}\left(\frac{1}{\varepsilon}M\left(\frac{t}{\varepsilon}\right)-\frac{1}{r}M\left(\frac{t}{r}\right)\right)$$ for all $t\in\operatorname{\mathbb{R}}$ and we then obtain $$f_{\varepsilon,r}(x)=\frac{1}{2}\left(\int_{\operatorname{\mathbb{R}}}\frac{1}{\varepsilon}M\left(\frac{x-t}{\varepsilon}\right)f(t)\,dt-\int_{\operatorname{\mathbb{R}}}\frac{1}{r}M\left(\frac{x-t}{r}\right)f(t)\,dt\right).$$ The first integral converges to $f(x)$ as $\varepsilon$ tends to $0^+$ by Lebesgue theorem. The second integral converges to $0$ as $r$ tends to $+\infty$ thanks to Lemma 6.3.3 in [@Ho], because $f$ is bounded and weakly oscillating around on the origin, and $M\in L^1(\operatorname{\mathbb{R}})$ is of integral equal to $1$. The conclusion follows. [99]{} M. Abramovitz, I. 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Luther, [The Differentiability of Fourier Gap Series and “Riemann’s Example” of a Continuous, Nondifferentiable Function]{}, Journal Approximation Theory [48]{}, 303-321, 1986. E. Mohr, [Wo ist die Riemannsche Funktion nicht differenzierbar?]{}, Annali di Matematica Pura ed Applicata [123]{} (1), 93-104, 1980. H. Queffelec, [Dérivabilité de certaines sommes de séries de Fourier lacunaires]{}, Comptes rendus de l’Académie des sciences de Paris [273]{}, 291-293, 1971. A. Shidfar, K. Sabetfakhri, [On the Continuity of van der Waerden’s Function in the Hölder Sense]{}, American Mathematical Monthly [93]{}, 375-376, 1986. A. Smith, [The Differentiability of Riemann’s Function]{}, Proceedings of the American Mathematical Society [34]{} (2), 463-468, 1972. T. Takagi, [A Simple Example of the Continuous Function without Derivative]{}, Proceedings of the Physico-Mathematical Society of Japan [1]{}, 176-177, 1903. B. Torrésani, [Analyse continue par ondelettes]{}, CNRS Editions, Paris, 1995. A. 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--- abstract: 'We investigate the circular motion of charged test particles in the gravitational field of a charged mass described by the Reissner-Nordström (RN) spacetime. We study in detail all the spatial regions where circular motion is allowed around either black holes or naked singularities. The effects of repulsive gravity are discussed by finding all the circles at which a particle can have vanishing angular momentum. We show that the geometric structure of stable accretion disks, made of only test particles moving along circular orbits around the central body, allows us to clearly distinguish between black holes and naked singularities.' author: - 'Daniela Pugliese$^{1,2}$, Hernando Quevedo$^{1,3}$, and Remo Ruffini$^1$' title: 'Motion of charged test particles in Reissner–Nordström spacetime' --- Introduction {#xanes} ============ Let us consider the background of a static gravitational source of mass $M$ and charge $Q$, described by the Reissner–Nordström (RN) line element in standard spherical coordinates $$\label{11metrica} ds^2=-\frac{\Delta}{r^2}dt^2+\frac{r^2}{\Delta}dr^2 +r^2\left(d\theta^2+\sin^2\theta d\phi^2\right)\ ,$$ where $\Delta = (r-r_+)(r-r_-)$ and $r_\pm = M\pm\sqrt{M^2 -Q^2}$ are the radii of the outer and inner horizon, respectively. Furthermore, the associated electromagnetic potential and field are $$\label{EMF} A=\frac{Q}{r}dt,\quad F=dA=-\frac{Q}{r^2}dt\wedge dr \ ,$$ respectively. The motion of a test particle of charge $q$ and mass $\mu$ moving in a RN background (\[11metrica\]) is described by the following Lagrangian density: $$\label{LagrangianaRN} \mathcal{L}=\frac{1}{2} g_{\alpha\beta}\dot{x}^{\alpha}\dot{x}^{\beta}+\epsilon A_\alpha x^\alpha,$$ where $A_{\alpha}$ are the components of the electromagnetic 4–potential, the dot represents differentiation with respect to the proper time, and the parameter $\epsilon = q/\mu$ is the specific charge of the test particle. The equations of motion of the test particle can be derived from Eq.[ ]{}(\[LagrangianaRN\]) by using the Euler–Lagrange equation. Then, $$\label{COv} \dot{x}^{\alpha}\nabla_{\alpha}\dot{x}^{\beta}=\epsilon F^{\beta}_{\ \gamma}\dot{x}^{\gamma},$$ where $F_{\alpha\beta}\equiv A_{\alpha,\beta}-A_{\beta,\alpha}$. Since the Lagrangian density (\[LagrangianaRN\]) does not depend explicitly on the variables $t$ and $\phi$, the following two conserved quantities exist $$\begin{aligned} \label{10000} p_t&\equiv& \frac{\partial\mathcal{L}}{\partial \dot{t}}=-\left(\frac{\Delta}{r^2}\dot{t}+\frac{\epsilon Q}{r}\right)=-\frac{E}{\mu},\\ \label{100000} p_{\phi}&=&\frac{\partial\mathcal{L}}{\partial \dot{\phi}}=r^2\sin^2\theta \dot{\phi}=\frac{L}{\mu},\end{aligned}$$ where $L$ and $E$ are respectively the angular momentum and energy of the particle as measured by an observer at rest at infinity. Moreover, to study the motion of charged test particles in the RN spacetime it is convenient to use the fact if the initial position and the tangent vector of the trajectory of the particle lie on a plane that contains the center of the body, then the entire trajectory must lie on this plane. Without loss of generality we may therefore restrict ourselves to the study of equatorial trajectories with $\theta =\pi/2$. On the equatorial plane $\theta=\pi/2$, the motion equations can be reduced to the form $\dot r^2 + V^2 = E^2/\mu^2$ which describes the motion inside an effective potential $V$. Then, we define the potential $$\label{9} V_{\pm}=\frac{E^{\pm}}{\mu}=\frac{\epsilon Q}{r}\pm \sqrt{\left(1+\frac{L^2}{\mu^2r^2}\right)\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)}$$ as the value of $E/\mu$ that makes $r$ into a “turning point” $(V=E/\mu)$; in other words, the value of $E/\mu$ at which the (radial) kinetic energy of the particle vanishes [@RuRR; @Chandra; @Levin:2008mq; @Bilic:2006bh]. The effective potential with positive (negative) sign corresponds to the solution with $$\lim_{r\rightarrow\infty} E^{+}=+\mu;\quad \left(\lim_{r\rightarrow\infty} E^{-}=-\mu\right),$$ where E\^[+]{}(L,,r)E\^[-]{}(L,,r), and the following relation holds: $$\label{IR} E^{+}(L,\epsilon,r)=-E^{-}(L,-\epsilon,r).$$ The behavior of the effective potential strongly depends on the sign of $\epsilon Q$; in particular in the case of $\epsilon Q<0$, negative energy states for the solution $E^{+}$ can exist (see also [@Grunau:2010gd; @[11]; @[12]; @[13]; @[14]; @Belbruno:2011nn; @Barack:2011ed]). The problem of finding exact solutions of the motion equations of test particles moving in a RN spacetime has been widely studied in literature in many contexts and ways. For a recent discussion we mention the works [@Grunau:2010gd; @[11]; @[12]; @[13]; @[14]; @Belbruno:2011nn; @Barack:2011ed]. In particular, in a recent paper [@Grunau:2010gd] the full set of analytical solutions of the motion equations for electrically and magnetically charged test particles is discussed in terms of the Weierstrass ($\gamma$, $\sigma$ and $\zeta$) functions. The general structure of the geodesics was discussed and a classification of their types was proposed. Remarkably, analytical solutions are found in the case of a central RN source not only with constant electric charge, but also with constant magnetic charge. It is interesting to notice that if either the test particle or the central body possesses both types of charge, it turns out that the motion is no longer confined to a plane. In the present work, we consider only equatorial circular orbits around a central RN source with constant electric charge. Instead of solving directly the equations of motion, we explore the properties of the effective potential function associated to the motion. Thus, we discuss and propose a classification of the equatorial orbits in terms of the two constants of motion: the energy $E/\mu$ and the orbital angular momentum $L/(\mu M)$. In fact, we focus our attention on some peculiar features of the circular motion and the physics around black holes and naked singularities. In particular, we are interested in exploring the possibility of distinguishing between black holes and naked singularities by studying the motion of circular test particles. In this sense, the present work complements and is different from previous studies [@Grunau:2010gd; @[11]; @[12]; @[13]; @[14]; @Belbruno:2011nn; @Barack:2011ed]. In a previous work [@Pugliese:2010ps; @Pugliese:2010he], we analyzed the dynamics of the RN spacetime by studying the motion of neutral test particles for which the effective potential turns out to coincide with $V_+$ as given in Eq.(\[9\]) with $\epsilon=0$. We will see that in the case of charged test particles the term $\epsilon Q/r$ drastically changes the behavior of the effective potential, and leads to several possibilities which must analyzed in the case of black holes and naked singularities. In particular, we will show that for particles moving along circular orbits there exist stability regions whose geometric structure clearly distinguishes naked singularities from black holes (see also [@Virbhadra:2002ju; @Virbhadra:2007kw] and [@do1; @Dabrowski:2002iw]). The plan of this paper is the following: In Sec. \[sec:veff\] we investigate the behavior of the effective potential and the conditions for the motion of positive and negative charged test particles moving on circular orbits around the central charged mass. This section also contains a brief analysis of the Coulomb approximation of the effective potential. In Sec. \[BHBHTR\], we will consider the black hole case while in Sec. \[NSNSTRE\] we shall focus on the motion around naked singularities. The conclusions are in Sec. \[milka\]. Circular motion {#sec:veff} =============== The circular motion of charged test particles is governed by the behavior of the effective potential (\[9\]). In this work, we will mainly consider the special case of a positive solution $V_+$ for the potential in order to be able to compare our results with those obtained in the case of neutral test particles analyzed in [@Pugliese:2010ps; @Pugliese:2010he]. Thus, the radius of circular orbits and the corresponding values of the energy $E$ and the angular momentum $L$ are given by the extrema of the function $V_{+}$. Therefore, the conditions for the occurrence of circular orbits are: $$\label{fg2} \frac{d V_{+}}{d r}=0, \quad V_+=\frac{E^+}{\mu}.$$ When possible, to simplify the notation we will drop the subindex $(+)$ so that, for example, $V=E/\mu$ will denote the positive effective potential solution. Solving Eq.[ ]{}(\[fg2\]) with respect to $L$, we find the specific angular momentum = , \[ECHEEUNAL\] where $\Sigma\equiv r^2-3Mr+2Q^2$, of the test particle on a circular orbit of radius $r$. The corresponding energy can be obtained by introducing the expression for the angular momentum into Eq.[ ]{}(\[9\]). Then, we obtain $$\label{Lagesp} \frac{E^{\pm}}{\mu}=\frac{\epsilon Q}{r} +\frac{\Delta\sqrt{2\Sigma+\epsilon^2Q^2\pm Q\sqrt{\epsilon^2(4\Sigma+\epsilon^2Q^2)}}} {\sqrt{2}r\|\Sigma\|}\ .$$ The sign in front of the square root should be chosen in accordance with the physical situation. This point will be clarified below by using the formalism of orthonormal frames. An interesting particular orbit is the one in which the particle is located at rest as seen by an observer at infinity, i.e., $L=0$. These “orbits” are therefore characterized by the following conditions \[Nolia\] L=0,=0. [@Interessantissimo]. Solving Eq.[ ]{}(\[Nolia\]) for $Q\neq0$ and $\epsilon\neq0$, we find the following radius r\_[s]{}\^\[rE\] . Table[ ]{}\[TabL0\] shows the explicit values of all possible radii for different values of the ratio $Q/M$. A particle located at $r=r_{s}$ with angular momentum $L=0$ will have the energy (see also [@CONS; @Bonnor; @Interessantissimo; @Bini:2006dp; @Bini:2008zza; @Bini:2006pk; @Bini:2005xg]) (+) . The minimum radius for a stable circular orbit occurs at the inflection points of the effective potential function; thus, we must solve the equation $$\label{Lagespep} \frac{d^2 V}{d r^2}=0,$$ for the orbit radius $r$, using the expression (\[ECHEEUNAL\]) for the angular momentum $L$. From Eq.[ ]{}(\[fg2\]) and Eq.[ ]{}(\[Lagespep\]) we find that the radius of the last stable circular orbit and the angular momentum of this orbit are related by the following equations (L\^2+Q\^2-1) r\^6-6 L\^2 r\^5+6 L\^2(1+ Q\^2) r\^4-2L\^2(2 L\^2+5Q\^2) r\^3\ + L\^2(3 L\^2+3 L\^2 Q\^2+3 Q\^4) r\^2-6 L\^4 Q\^2 r+2 L\^4 Q\^4=0 ,\[nube\] and \[nuvola\] Q\^2 r\^2-r\^3+L\^2 (2 Q\^2-3 r+r\^2)+Q r\^3 =0 , where in order to simplify the notation we introduced the normalized quantities $L\rightarrow L/(M/\mu)$, $r\rightarrow r/M$, and $Q\rightarrow Q/M$. Equation (\[nube\]) depends on the test particle specific charge $\epsilon$ via the function $L$ as given in Eq.[ ]{}(\[ECHEEUNAL\]). It is possible to solve Eq.[ ]{}(\[nube\]) for the last stable circular orbit radius as a function of the free parameter $L$. We find the expression &=&\ for the angular momentum of last stable circular orbit. Eq.[ ]{}(\[Carmara\]) can be substituted in Eq.[ ]{}(\[nuvola\]) to find the radius of the last stable circular orbit. Coulomb potential approximation ------------------------------- Consider the case of a charged particle moving in the Coulomb potential $$U(r)=\frac{ Q}{r} \ .$$ This means that we are considering the motion described by the following effective potential $$\label{vecchiaRR} V_+= \frac{E^{+}}{\mu}=\frac{\epsilon Q}{r} +\sqrt{1+\frac{L^2}{\mu^2 r^2}}\ ,$$ where $\epsilon Q<0$. The Coulomb approximation is interesting for our further analysis because it corresponds to the limiting case for large values of the radial coordinate $r$ \[cf. Eq.(\[9\])\]. Circular orbits are therefore situated at $r=r_c$ with $$\label{mai più} r_c = \sqrt{\frac{L^2}{\mu^{2}}\left(\frac{L^2}{\epsilon^2 Q^{2}}-1\right)}\quad\mbox{and}\quad \frac{L^2}{\mu^2}\geq \epsilon^2 Q^2,$$ and in the case $\epsilon=0$ with $Q>0$, circular orbits exist in all $r>0$ for $L=0$. We conclude that in this approximation circular orbits always exist with orbital radius $r_c$ and angular momentum satisfying the condition $\|L\|/\mu \geq \|\epsilon Q\|$. For the last stable circular orbit situated at $r=r_{{\mbox{\tiny{lsco}}}}$ we find $$\label{mascalzone} r_{{\mbox{\tiny{lsco}}}}=0\quad\mbox{with}\quad \frac{E^+(r_{{\mbox{\tiny{lsco}}}})}{\mu}= 0 \quad\mbox{and}\quad \frac{\|L\|}{\mu}=\|\epsilon Q\|\ .$$ This means that, in the approximation of the Coulomb potential, all the circular orbits are stable, including the limiting case of a particle at rest on the origin of coordinates. Furthermore, Eqs.[ ]{}(\[mai più\]–\[mascalzone\]) show that, in contrast with the general RN case, for a charged particle moving in a Coulomb potential only positive or null energy solutions can exist. See Fig. \[Plotfxue\] where the potential (\[vecchiaRR\]) is plotted as a function of the orbital radius for different values of the angular momentum. --------------------------------------------------- ![[]{data-label="Plotfxue"}](Plotfxue.eps "fig:") --------------------------------------------------- Black holes {#BHBHTR} =========== In the case of a black hole $(M^2>Q^2)$ the two roots $V_{\pm}$ of the effective potential are plotted as a function of the ratio $r/M$ in Fig. [ ]{}\[Pcomparativa2u\] for a fixed value of the charge–to–mass ratio of the test particle and different values of the angular momentum $L/(M \mu)$ (see also [@Pradhan:2010ws; @Gladush:2011cz; @Olivares:2011xb; @Zaslavskii:2010aw; @Gad; @Dotti:2010uc]). ------------------------------------------------------- ![ []{data-label="Pcomparativa2u"}](Plot1.eps "fig:") ------------------------------------------------------- Notice the presence of negative energy states for the positive solution $V_+=E^+/\mu$ of the effective potential function. Negative energy states for $V_+$ are possible only in the case $\epsilon Q<0$. In particular, the largest region in which the $V_{+}$ solution has negative energy states is $$\label{assurdo!} M+\sqrt{M^2-Q^2}<r\leq M+\sqrt{M^2-Q^2\left(1-\epsilon^2\right)}$$ and corresponds to the limiting case of vanishing angular momentum ($L=0$). For $L\neq 0$ this region becomes smaller and decreases as $L$ increases. For a given value of the orbit radius, say $r_0$, such that $r_0<M+\sqrt{M^2-Q^2\left(1-\epsilon^2\right)} $, the angular momentum of the test particle must be chosen within the interval 0<<[r\_0\^2]{} (-1) for a region with negative energy states to exist. This behavior is illustrated in Fig. \[Pcomparativa2u\]. Fig. [ ]{}\[Plot0501\] shows the positive solution $V_+$ of the effective potential for different values of the momentum and for positive and negative charged particles. In particular, we note that, at fixed $Q/M$ for a particle with $\|\epsilon\|<1$, in the case $\epsilon Q>0$ the stable orbit radius is larger than in the case of attractive electromagnetic interaction, i. e., $\epsilon Q<0$. -------------------------------------------------------------------------------------------------------- -- ![ []{data-label="Plot0501"}](Plot0501.eps "fig:") ![ []{data-label="Plot0501"}](Plot05m01.eps "fig:") -------------------------------------------------------------------------------------------------------- -- In Fig. [ ]{}\[Pcomparativa2a\], the potential $V_+$ of an extreme black hole is plotted for different, positive and negative values of the test particle with charge–to–mass ratio $\epsilon$. In this case, it is clear that the magnitude of the energy increases as the magnitude of the specific charge of the particle $\epsilon$ increases. -------------------------------------------------------------- ![ []{data-label="Pcomparativa2a"}](Normalizzato.eps "fig:") -------------------------------------------------------------- As mentioned in Sec. \[sec:veff\], in the case of the positive solution for the effective potential the conditions for the existence of circular orbits =0, V=, =0. lead to Eqs.(\[ECHEEUNAL\]) and (\[Lagesp\]) in which the selection of the $(\pm)$ sign inside the square root should be done properly. To clarify this point we consider explicitly the equation of motion for a charged particle in the gravitational field of a RN black hole. $$\label{100} a(U)^{\alpha}=\epsilon F^{\alpha}_{\phantom\ \beta}U^{\beta},$$ where $a(U) = \nabla_{U} U$ is the particle’s 4–acceleration. Introducing the orthonormal frame e\_= \_[t]{},e\_=\_[r]{},e\_=\_,e\_=\_ , with dual \^=dt,\^=dr,\^=rd,\^=rd , the tangent to a (timelike) spatially circular orbit $u^{\alpha}$ can be expressed as $$u=\Gamma(\partial_t+\zeta\partial_{\phi})=\gamma\left(e_{\hat{t}}+\nu e_{\hat{\phi}}\right)\ ,$$ where $\Gamma$ and $\gamma$ are normalization factors $$\Gamma^2=(-g_{tt}-\zeta^2g_{\phi\phi})^{-1}\quad\mbox{and}\quad\gamma^2=(1-\nu^2)^{-1},$$ which guarantees that $u_{\alpha}u^{\alpha} = -1$. Here $\zeta$ is the angular velocity with respect to infinity and $\nu$ is the “local proper linear velocity" as measured by an observer associated with the orthonormal frame. The angular velocity $\zeta$ is related to the local proper linear velocity by $$\zeta=\sqrt{-\frac{g_{tt}}{g_{\phi\phi}}}\nu\ .$$ Since only the radial component of the 4–velocity is non-vanishing, Eq.[ ]{}(\[100\]) can be written explicitly as $$\label{11} 0=\gamma (\nu^{2}-\nu_{g} ^{2})+\frac{\nu_{g} }{\zeta_{g}}\frac{\epsilon Q}{r^{2}}\ ,$$ where $$\label{RR} \zeta_{g}= \pm\frac{\sqrt{Mr-Q^2}}{r^2}\ ,\quad\nu_{g} =\sqrt{\frac{Mr-Q^2}{\Delta}}\ .$$ This equation gives the values of the particle linear velocity $\nu=\pm \nu_{\epsilon}^{\pm}$ which are compatible with a given value of $\epsilon Q$ on a circular orbit of radius $r$, i. e., $$\label{12} \nu_{\epsilon}^{\pm}=\nu_g \sqrt{1-\frac{Q^{2} \epsilon ^{2}}{2 r^4 \zeta_g^{2}}\pm \frac{Q }{r^{2} \zeta_g \nu_g } \sqrt{\frac{\epsilon ^{2}}{\gamma_g^{2}}+\frac{Q^{2} \epsilon ^4 \nu_g ^{2}}{4 r^4 \zeta_g^{2}}}},$$ where $$\gamma_{g}=\left(\frac{\Delta}{r^2-3Mr+2Q^2}\right)^{1/2},$$ and \_\^=(1-\_\^\^[2]{}) \^[-1/2]{}. In the limiting case of a neutral particle ($\epsilon =0)$, Eq.(\[11\]) implies that the linear velocity of the particle is $\nu_g $. We introduce the limiting value of the parameter $\epsilon$ corresponding to a particle at rest, $\nu=0$, in Eq.[ ]{}(\[11\]), i. e., $$\label{13} \epsilon_{0}=\nu_{g} \zeta_g \frac{r^{2}}{Q}=\frac{M r-Q^{2}}{Q \sqrt{\Delta}}.$$ By introducing this quantity into Eq.[ ]{}(\[11\]), one gets the following equivalent relation $$\label{14} \frac{\epsilon}{\epsilon_{0}}=\gamma \left(1-\frac{\nu^{2}}{\nu_{g} ^{2}}\right),$$ whose solution (\[12\]) can be conveniently rewritten as $$\label{15} \nu_{\epsilon}^{\pm}=\nu_{g} \left[\Lambda\pm\sqrt{\Lambda^{2}-1+(\epsilon/\epsilon_0)^{2}}\right]^{1/2}\ ,$$ where $$\label{16} \Lambda=1-\frac{\nu_{g} ^{2}}{2}\left(\frac{\epsilon}{\epsilon_0}\right)^{2}.$$ Moreover, from Eq.[ ]{}(\[14\]) it follows that $\epsilon<0$ implies that $\nu^{2}> \nu_{g} ^{2}$ (because $\epsilon_{0}$ is always positive for $r > r_+$), so that the allowed solutions for $\nu$ can exist only for $r \geq r^+_\gamma$, where r\_\^+ (3M+) , the equality corresponding to $\nu_{g} = 1$. In this case, the solutions of Eq.[ ]{}(\[11\]) are given by $\nu=\pm\nu_{\epsilon}^{+}$. For $ \epsilon> 0$, instead, solutions can exist also for $r_+ < r < r^+_\gamma$. The situation strongly depends on the considered range of values of $\epsilon$ and is summarized below. Equation [ ]{}(\[15\]) gives the following conditions for the existence of velocities $$\begin{aligned} \label{170} \Lambda^{2}-1+(\epsilon/\epsilon_{0})^{2} &\geq& 0\ , \\ \label{180} \Lambda\pm\sqrt{\Lambda^{2}-1+(\epsilon/\epsilon_{0})^{2}}&\geq& 0\ . $$ The second condition, Eq.[ ]{}(\[180\]), is satisfied by $$\label{19} r\geq r_l \equiv\frac{3M}{2}+\frac{1}{2}\sqrt{9M^{2}-8Q^{2}-\epsilon^{2}Q^{2}}.$$ Moreover for $Q=M$ and $\epsilon=1$ it is $\Lambda+\sqrt{\Lambda^{2}-1+(\epsilon/\epsilon_{0})^{2}}\geq0$ when $M<r<(3/2)M$. However it is also possible to show that condition Eq.[ ]{}(\[fg2\]) is satisfied for $0<Q<M$ and $\epsilon>0$ only in the range $r\geq r_l$. Requiring that the argument of the square root be nonnegative implies $$\label{20} \epsilon\le\epsilon _{l}\equiv \frac{\sqrt{9M^{2}-8Q^{2}}}{Q}.$$ The condition (\[180\]) will be discussed later. From the equation of motion (\[14\]) it follows that the velocity vanishes for $\epsilon/\epsilon_0 = 1$, i. e., for \[cf. Eq.(\[rE\])\] $$\label{21} r=r_{s}\equiv \frac{Q^{2}}{\epsilon^{2}Q^{2}-M^{2}} \left[M(\epsilon^{2}-1)+\sqrt{\epsilon^2(\epsilon^2-1)(M^{2}-Q^{2})}\right]\ ,$$ which exists only for $\epsilon> M/Q$. We thus have that > 1 r > r\_[s]{}, whereas < 1 r\_+ < r < r\_[s]{} . On the other hand, the condition $\nu = 0$ in Eq.[ ]{}(\[15\]) implies that $$\begin{aligned} \label{22} \left[\Lambda\pm\sqrt{\Lambda^{2}-1+(\epsilon/\epsilon_{0})^{2}}\right]_{\epsilon/\epsilon_0=1}&=& 0, \\ \label{230}\mbox{i. e.} \quad \left[\Lambda\pm\sqrt{\Lambda^{2}-1}\right]_{r=r_{s}}&=& 0, $$ thus $\nu_{\epsilon}^{-}$ is identically zero whereas $\nu_{\epsilon}^{+}= {2}\Lambda(r_{s})=0$ only for $$\label{24} \epsilon=\tilde{\epsilon}\equiv \frac{1}{\sqrt{2}Q}\sqrt{5M^{2}-4Q^{2}+\sqrt{25M^{2}-24Q^{2}}} .$$ Finally, the lightlike condition $\nu=1$ is reached only at $r = r_\gamma^+$, where $\nu_g = 1 = \nu$. The behavior of charged test particles depends very strongly on their location with respect to the special radii $r_+$, $r_l$, $r_\gamma^+$, and $r_{s}$. In Sec. \[sec:stb\] the behavior of these radii will be analyzed in connection with the problem of stability of circular orbits. On the other hand, the particle’s 4–momentum is given by $P = m U-qA$. Then, the conserved quantities associated with the temporal and azimuthal Killing vectors $\xi=\partial_{t}$ and $\eta=\partial_{\phi}$ are respectively \[250\] P &=&--=-,\ P &=&=, where $E/\mu$ and $L/\mu$ are the particle’s energy and angular momentum per unit mass, respectively (see also Eqs.(\[ECHEEUNAL\]) and (\[Lagesp\]) ). Let us summarize the results. ### Case $\epsilon<0$ The solutions are the geodesic velocities $\nu=\pm\nu_{\epsilon}^+$ in the range $r\geq r_\gamma^+$ as illustrated in Fig. [ ]{}\[Figure1a\]. Orbits with radius $r= r_\gamma^+$ are lightlike. We can also compare the velocity of charged test particles with the geodesic velocity $\nu_g $ for neutral particles. For $r> r_\gamma^+$ we see that $\nu_{\epsilon}^+>\nu_g $ always. This means that, at fixed orbital radius, charged test particles acquire a larger orbital velocity compared to that of neutral test particles in the same orbit. -------------------------------------------------- ![[]{data-label="Figure1a"}](Figure1.eps "fig:") -------------------------------------------------- As it is possible to see from Eq.[ ]{}(\[12\]) and also in Fig. [ ]{}\[Figure1an\], an increase in the particle charge $\epsilon<0$ corresponds to an increase in the velocity $\nu_{\epsilon}^+$. ---------------------------------------------------- ![[]{data-label="Figure1an"}](Figure1n.eps "fig:") ---------------------------------------------------- As the orbital radius decreases, the velocity increases until it reaches the limiting value $\nu_{\epsilon}^+=1$ which corresponds to the velocity of a photon. This fact can be seen also in Fig. [ ]{}\[Figure1ab\], where the energy and angular momentum for circular orbits are plotted in terms of the distance $r$. Clearly, this graphic shows that to reach the photon orbit at $r=r_\gamma^+$, the particles must acquire and infinity amount of energy and angular momentum. ----------------------------------------------------- -- ![[]{data-label="Figure1ab"}](FigureEL1.eps "fig:") ----------------------------------------------------- -- In Fig. [ ]{}\[Figure1absil\] we analyze the behavior of the particle’s energy and angular momentum in terms of the specific charge $\epsilon$. It follows that both quantities decrease as the value of $\|\epsilon\|$ decreases. ----------------------------------------------------------- -- ![[]{data-label="Figure1absil"}](FigureEL1sil.eps "fig:") ----------------------------------------------------------- -- ### Case $\epsilon=0$ The solutions are the geodesic velocities $\nu=\pm\nu_g $ in the range $r\geq r_\gamma^+$. This case has been studied in detail in [@Pugliese:2010ps]. ### Case $\epsilon>0$ Depending on the explicit values of the parameters $Q$ and $\epsilon$ and the radial coordinate $r$, it is necessary to analyze several subcases. a) : $\epsilon < M/Q$ and $r \geq r_l$. There are two different branches for both signs of the linear velocity: $\nu=\pm\nu_{\epsilon}^+$ in the range $r_l \leq r\leq r_\gamma^+$, and $\nu=\pm\nu_{\epsilon}^-$ in the whole range $r\geq r_l$. The two branches join at $r=r_l$, where $\nu^+_{\epsilon} =\nu^{-}_{\epsilon}=\nu_{g} \sqrt{\Lambda}$, as shown in Fig. [ ]{}\[Figure2abcd\]. First we note that in this case for $r> r_\gamma^+$ it always holds that $\nu_{\epsilon}^{-}<\nu_g $. This means that, at fixed orbital radius, charged test particles possess a smaller orbital velocity than that of neutral test particles in the same orbit. This is in accordance to the fact that in this case, a black hole with $\epsilon Q>0$, the attractive gravitational force is balanced by the repulsive electromagnetic force. In the region $r>r_\gamma^+$, the orbital velocity increases as the radius approaches the value $r_{\gamma}^+$ (see Fig. [ ]{}\[Figure2abcd\]). The interval $r_l \leq r\leq r_\gamma^+$ presents a much more complex dynamical structure. First we note that, due to the Coulomb repulsive force, charged particle orbits are allowed in a region which is forbidden for neutral test particles. This is an interesting result leading to the possibility of accretion disks in which the innermost part forms a ring of charged particles only. Indeed, suppose that an accretion disk around a RN black hole is made of neutral and charged test particles. Then, the accretion disk can exist only in the region $r\geq r_l$ with a ring of charged particles in the interval $[r_l,r_\gamma^+)$. Outside the exterior radius of the ring $(r>r_\gamma^+)$, the disk can be composed of neutral and charged particles. ------------------------------------------------------------------------------------------------------------ -- ![[]{data-label="Figure2abcd"}](Figure2a.eps "fig:") ![[]{data-label="Figure2abcd"}](Figure2az.eps "fig:") ------------------------------------------------------------------------------------------------------------ -- This situation can also be read from Fig. [ ]{}\[Figure2a\] where the energy and the angular momentum are plotted as functions of the radial distance $r/M$. ----------------------------------------------------- ------------------------------------------------------ ![[]{data-label="Figure2a"}](Figure2aEl.eps "fig:") ![[]{data-label="Figure2a"}](Figure2aElb.eps "fig:") ----------------------------------------------------- ------------------------------------------------------ \ b) : $M/Q <\epsilon < \tilde{\epsilon}$ and $r_l \leq r\leq r_{s}$. Since $r < r_{s}$, one has that $\epsilon/\epsilon_0 < 1$, implying that both solutions $\nu^{+}_{\epsilon}$ and $\nu^{-}_{\epsilon}$ can exist. There are two different branches for both signs: $\nu=\pm\nu^+_{\epsilon}$ in the range $r_l \leq r\leq r_{\gamma}^{+}$, and $\nu^{-}_{\epsilon}$ in the entire range $r_l\leq r\leq r_{s}$. The two branches join at $r= r_l$. Note that for increasing values of $\epsilon$, the radius $r_{s}$ decreases and approaches $r_l$, reaching it at $\epsilon =\tilde{\epsilon}$, and as $\epsilon$ tends to infinity $r_s$ tends to the outer horizon $r_+$ (see Fig. [ ]{}\[PlotRsRt\]). --------------------------------------------------- ![[]{data-label="PlotRsRt"}](PlotRsRt.eps "fig:") --------------------------------------------------- --------------------------------------------------------------------------------------------------------- -- ![[]{data-label="Figure2bc"}](Figure2b.eps "fig:") ![[]{data-label="Figure2bc"}](Figure2bEL.eps "fig:") --------------------------------------------------------------------------------------------------------- -- In particular, the interaction between the attractive gravitational force and the Coulomb force generates a zone $r_l \leq r\leq r_\gamma^+$ in which only charged test particles can move along circular trajectories while neutral particles are allowed in the region $r>r_\gamma^+$ (see Fig. \[Figure2bc\]). This result again could be used to construct around black holes accretion disks with rings made of charged particles.\ c) : $\tilde{\epsilon}<\epsilon <\epsilon_{l} $ and $r_{s} < r < r_\gamma^{+}$. The solution $\nu^{-}_{\epsilon}$ for the linear velocity is not allowed whereas the solution $\nu^+_{\epsilon}$ is valid in the entire range. In fact, the condition $r > r_{s}$ implies that $\epsilon/\epsilon_{0} > 1$, and therefore $\Lambda^{2}-1+(\epsilon/\epsilon_{0})^{2} > \Lambda^{2}$, so that the condition (\[180\]) for the existence of velocities is satisfied for the plus sign only. Therefore, the solutions are given by $\nu=\pm\nu^+_{\epsilon}$ in the entire range as shown in Fig. [ ]{}\[Figure2abcdz\]. At the radius orbit $r=r_{s}$, the angular momentum and the velocity of the test particle vanish, indicating that the particle remains at rest with respect to static observers located at infinity. In the region $r_{s} < r < r_\gamma^{+}$ only charged particles can move along circular trajectories. --------------------------------------------------------------------------------------------------------------- -- ![[]{data-label="Figure2abcdz"}](Figure2c.eps "fig:") ![[]{data-label="Figure2abcdz"}](Figure2cEL.eps "fig:") --------------------------------------------------------------------------------------------------------------- -- \ d) : $ \epsilon > \epsilon_l$ and $r_{s}<r<r^{+}_\gamma$. In this case the radius $r_l$ does not exist. The solutions are the velocities $\nu=\pm\nu^+_{\epsilon}$ in the entire range. Note that for $\epsilon\rightarrow\infty$ one has that $r_{s}\rightarrow r_+$. Also in this case we note that neutral particles can stay in circular orbits with a velocity $\nu_g $ only in the region $r>r^{+}_\gamma$ whereas charged test particles are allowed within the interval $r_{s}<r<r^{+}_\gamma$, as shown in Fig. [ ]{}\[Figure2abcdx\]. Clearly, for charged and neutral test particles the circular orbit at $r=r^{+}_\gamma$ corresponds to a limiting orbit. --------------------------------------------------------------------------------------------------------------- -- ![[]{data-label="Figure2abcdx"}](Figure2d.eps "fig:") ![[]{data-label="Figure2abcdx"}](Figure2dEL.eps "fig:") --------------------------------------------------------------------------------------------------------------- -- Stability {#sec:stb} --------- To analyze the stability of circular orbits for charged test particles in a RN black hole we must consider the condition (\[Lagespep\]) which leads to the Eqs.(\[nube\]), (\[nuvola\]), and (\[Carmara\]). So the stability of circular orbits strongly depends on the sign of $(\epsilon Q)$. The case $\epsilon Q\leq0$ is illustrated in Fig. [ ]{}\[ultimaepsilonEpm02\] where the radius of the last stable circular orbit $r_{{\mbox{\tiny{lsco}}}}$ is plotted for two different values of $\epsilon$ as a function of $Q/M$. It can be seen that the energy and angular momentum of the particles decrease as the value of $Q/M$ increases. These graphics also include the radius of the outer horizon $r_+$ and the radius $r_\gamma^+$ which determines the last (unstable) circular orbit of neutral particles. In Sec. \[BHBHTR\], we found that circular orbits for charged particles are allowed also inside the radius $r_\gamma^+$ for certain values of the parameters; however, since $r_\gamma^+< r_{{\mbox{\tiny{lsco}}}}$, we conclude that all those orbits must be unstable. ----------------------------------------------------------------------------------------------------------------------- -- ![[]{data-label="ultimaepsilonEpm02"}](PEpm02.eps "fig:") ![[]{data-label="ultimaepsilonEpm02"}](Plotm15e.eps "fig:") ----------------------------------------------------------------------------------------------------------------------- -- From Fig. [ ]{}\[ultimaepsilon\] we see that for $Q=0$ and $\epsilon=0$, the well-known result for the Schwarzschild case, $r_{{\mbox{\tiny{lsco}}}}= 6M$, is recovered. Also in the limiting case $Q=M$ and $\epsilon=0$, we recover the value of $r_{{\mbox{\tiny{lsco}}}}= 4M$ for neutral particles moving along circular orbits in an extreme BN black hole. ------------------------------------------------------------- ![[]{data-label="ultimaepsilon"}](ultimaepsilon.eps "fig:") ------------------------------------------------------------- In general, as the value of $\|\epsilon\|$ increases we see that the value of $ r_{{\mbox{\tiny{lsco}}}}$ increases as well. This behavior resembles the case of the radius of the last stable orbit for neutral test particles [@Pugliese:2010ps; @Pugliese:2010he]. Indeed, in the case $\epsilon Q<0$ the attractive Coulomb force reinforces the attractive gravitational force so that the general structure remains unchanged. We also can expect that an increase in the charge of the particle $\|\epsilon\|$ produces an increase in the velocity of the stable circular orbits. In fact, this can be seen explicitly from Eq.[ ]{}(\[12\]) and Fig. [ ]{}\[Figure1an\]. It then follows that the energy and angular momentum of the charged test particle increases as the value of $\|\epsilon\|$ increases. The case of $\epsilon Q>0$ is illustrated in Figs.[ ]{}\[BHP7\] and \[BHP05\]. The situation is very different from the case of neutral particles or charged particles with $\epsilon Q<0$. Indeed, in this case the Coulomb force is repulsive and leads to a non trivial interaction with the attractive gravitational force, see also [@Joshi; @Belinski:2008zz; @Luongo:2010we; @Pizzi:2008ti; @Belinski:2008bn; @Pizzi:2008zz; @Paolino:2008qi; @Manko:2007hi; @Alekseev:2007re; @Preti:2008zz]. It is necessary to analyze two different subcases. The first subcase for $\epsilon>1$ is illustrated in Fig. [ ]{}\[BHP7\] while the second one for $0<\epsilon<1$ is depicted in Fig. [ ]{}\[BHP05\]. ----------------------------------------------- -- ![[]{data-label="BHP7"}](PlotBHp7.eps "fig:") ----------------------------------------------- -- ------------------------------------------------- -- ![[]{data-label="BHP05"}](PlotBHp05.eps "fig:") ------------------------------------------------- -- We can see that in the case $0<\epsilon<1$ the stability regions are similar to those found in the case $\epsilon<0$ (cf. Figs.[ ]{}\[ultimaepsilonEpm02\] and \[BHP05\]). This means that for weakly–charged test particles, $0<\epsilon <1$, it always exists a stable circular orbit and $r_{{\mbox{\tiny{lsco}}}}\geq 4M$, where the equality holds for an extreme black hole. On the contrary, in the case $\epsilon >1$ there are regions of $Q$ and $\epsilon$ in which stable circular orbits cannot exist at all. As can be seen from Fig. \[BHP7\], charged particles moving along circular orbits with radii located within the region $r<r_\gamma^+$ or $r<r_{s}$ must be unstable. We conclude that the ring structure of the hypothetical accretion disks around a RN black hole mentioned in Sec. \[BHBHTR\] must be unstable. Naked singularities {#NSNSTRE} =================== The effective potential $V_{\pm}$ given in Eq.[ ]{}([9]{}) in the case of naked singularities $(M^2<Q^2)$ is plotted in Figs.[ ]{}(\[NKS\]–\[Plot15501\]) in terms of the radial coordinate $r/M$ for selected values of the ratio $\epsilon$ and the angular momentum $L/(M \mu)$ of the test particle, see also [@dfl; @062; @Cohen:1979zzb; @primorg; @Patil:2010nt; @Patil:2011aw; @Pradhan:2010ws; @F1]. -------------------------------------------------------------------------------------------- -- ![ []{data-label="NKS"}](Plot1sp1.eps "fig:") ![ []{data-label="NKS"}](Plot1sm.eps "fig:") ![ []{data-label="NKS"}](Plot1sp.eps "fig:") -------------------------------------------------------------------------------------------- -- The effective potential profile strongly depends on the sign of $\epsilon Q$. Moreover, the cases with $\|\epsilon\|\leq1$ and with $\|\epsilon\|>1$ must be explored separately. Fig. [ ]{}\[11:00M\] shows the effective potential for a particle of charge–to–mass $\epsilon$ in the range $[-10,-1]$. The presence of minima (stable circular orbits) in the effective potential with negative energy states is evident. Moreover, we note that the minimum of each potential decreases as $\|\epsilon\|$ increases. This fact is due to the attractive and repulsive effects of the gravitational and electric forces [@Joshi; @Belinski:2008zz; @Luongo:2010we; @Pizzi:2008ti; @Belinski:2008bn; @Pizzi:2008zz; @Paolino:2008qi; @Manko:2007hi; @Alekseev:2007re; @Preti:2008zz] . ----------------------------------------------- ![[]{data-label="11:00M"}](Plotfm.eps "fig:") ----------------------------------------------- In Fig. [ ]{}\[NKSepsilon\] the effective potential is plotted for negative and positive values of the charge–to–mass ratio $\epsilon$. We see that for a fixed value of the radial coordinate and the angular momentum of the particle, the value of the potential $V$ increases as the value of $\epsilon$ increases. ---------------------------------------------------------- -- ![[]{data-label="NKSepsilon"}](Normalizzato2.eps "fig:") ---------------------------------------------------------- -- In the Fig. [ ]{}\[Plot15501\] we plot the effective potential for a fixed $Q/M$ as function of the radial coordinate and the angular momentum for two different cases, $\epsilon=0.1$ and $\epsilon=-0.1$. We can see that in the first case the presence of a repulsive Coulomb force reduces the value of the radius of the last stable circular orbit for a fixed angular momentum. ------------------------------------------------------------------------------------------------------------ -- ![ []{data-label="Plot15501"}](Plot15501.eps "fig:") ![ []{data-label="Plot15501"}](Plot155m01.eps "fig:") ------------------------------------------------------------------------------------------------------------ -- We note the existence of stable “circular" orbits with $L=0$ at which the particle is at rest with respect to static observers located at infinity. Negative energy states are possible only in the case $\epsilon Q<0$. The region in which the solution $V_{+}$ has negative energy states is $$\label{assurdonk!} 0<r<M+\sqrt{M^2-Q^2 \left(1-\epsilon^2\right)} \quad \mbox{for}\quad \epsilon \leq-1 \ ,$$ and $$\label{assurdonk!w} 0<r<r_l^+ \quad \mbox{for} \quad 0\leq L<L_q\ , \quad \epsilon \leq-1 \ ,$$ $$\label{assurdonk!3} r_l^-<r<r_l^+ \quad \mbox{for}\quad 0\leq L<L_q\ , \quad -1<\epsilon \leq-\sqrt{1-\frac{M^2}{Q^2}} \ ,$$ where . In general, for a particle in circular motion with radius $r_0$ and charge–to–mass ratio $\epsilon$, around a RN naked singularity with charge $Q$ and mass $M$, the corresponding angular momentum must be chosen as <[r\_0\^2]{} ( -1), in order for negative energy states to exist. The conditions for circular motion around a RN naked singularity are determined by Eq.[ ]{}(\[fg2\]) which can be used to find the energy and angular momentum of the test particle. Indeed, Eqs.[ ]{}(\[ECHEEUNAL\]) and (\[Lagesp\]) define the angular momentum $L^\pm$ and the energy $E^{\pm}$, respectively, in terms of $r/M$, $Q/M$, and $\epsilon$. The explicit dependence of these parameters makes it necessary to investigate several intervals of values. To this end, it is useful to introduce the following notation $$\begin{aligned} r^{\pm}_l&\equiv&\frac{3 M}{2}\pm\frac{1}{2} \sqrt{9 M^{2}-8 Q^{2}-Q^{2} \epsilon ^{2}}\ ,\\ \tilde{\epsilon}_{\pm}&\equiv& \frac{1}{\sqrt{2}Q}\sqrt{5M^{2}\pm4Q^{2}+\sqrt{25M^{2}-24Q^{2}}}\ ,\end{aligned}$$ and \[r1novo\] \_&& . We note that \_[0]{}r\_[s]{}\^=r\_[\]{}= , which corresponds to the classical radius of a mass $M$ with charge $Q$, see for example [@Qadir:2006va; @Belgiorno:2000gq], and \_[0]{}r\_[l]{}\^=r\_\^= , which represents the limiting radius at which neutral particles can be in circular motion around a RN naked singularity [@Pugliese:2010ps]. The behavior of the charge parameters defined above is depicted in Fig. [ ]{}\[Plotettmp\] in terms of the ratio $Q/M>1$. ----------------------------------------------------- -- ![[]{data-label="Plotettmp"}](Plotettmp.eps "fig:") ----------------------------------------------------- -- It follows from Fig. [ ]{}\[Plotettmp\] that it is necessary to consider the following intervals: Q/M&&(1,5/(2)\],\ Q/M&&( 5/(2),(3)/7\],\ Q/M&&( (3)/7,\],\ Q/M&&\[, ). Our approach consists in analyzing the conditions for the existence of circular orbits by using the expressions for the angular momentum, Eq.[ ]{}(\[ECHEEUNAL\]), and the energy, Eq.[ ]{}(\[Lagesp\]), of the particle together with the expressions for the velocity obtained in Sec. \[BHBHTR\]. We consider separately the case $\epsilon>0$ in Secs.[ ]{}\[eme1\] and \[eme1a\], and $\epsilon<0$ in Secs. \[bicioc\] and \[bicioca\]. In the Appendix \[noten\] we present equivalent results by using the alternative method of the proper linear velocity of test particles in an orthonormal frame as formulated in Sec. \[BHBHTR\]. Case $\epsilon>1$ {#eme1} ----------------- For $\epsilon>0$ the condition (\[14\]) implies in general that $r>r_{}\equiv Q^{2}/M$. Imposing this constraint on Eqs.(\[ECHEEUNAL\]) and (\[Lagesp\]), we obtain the following results for timelike orbits. For $\epsilon>1$ and $M<Q<\sqrt{9/8}M$ circular orbits exist with angular momentum $L=L^{+}$ in the interval $r_\gamma^-<r<r_\gamma^+ $, while for $Q\geq\sqrt{9/8}M$ no circular orbits exist (see Fig. [ ]{}\[pelHa\]). Clearly, the energy and angular momentum of circular orbits diverge as $r$ approaches the limiting orbits at $r_\gamma^\pm $. This means that charged test particles located in the region $r_\gamma^-<r<r_\gamma^+ $ need to acquire an infinite amount of energy to reach the orbits at $r_\gamma^\pm $. The energy of the states is always positive. A hypothetical accretion disk would consist in this case of a charged ring of inner radius $r_\gamma^-$ and outer radius $r_\gamma^+$, surrounded by a disk of neutral particles. The boundary $r=r_\gamma^+$ in this case would be a lightlike hypersurface. --------------------------------------------- -- ![[]{data-label="pelHa"}](pelHa.eps "fig:") --------------------------------------------- -- Since for $\epsilon Q>0$ the Coulomb interaction is repulsive, the situation characterized by the values for $Q\geq\sqrt{9/8}M$ and $\epsilon>1$ corresponds to a repulsive electromagnetic effect that cannot be balanced by the attractive gravitational interaction. We note that the case $Q\geq\sqrt{9/8}M$ and $\epsilon>1$ could be associated to the realistic configuration of a positive ion or a positron in the background of a RN naked singularity. Case $0<\epsilon<1$ {#eme1a} ------------------- It turns out that in this case it is necessary to consider separately each of the four different regions for the ratio $Q/M$ that follow from Fig. [ ]{}\[Plotettmp\]. Moreover, in each region of $Q/M$ it is also necessary to consider the value of $\epsilon$ for each of the zones determined by the charge parameters $\epsilon_l$, $\tilde{\epsilon}_\pm$, and $\tilde{\tilde{\epsilon}}_\pm$, as shown in Fig. [ ]{}\[Plotettmp\]. We analyzed all the resulting cases in detail and found the values of the energy and angular momentum of charged test particles in all the intervals where circular motion is allowed. We summarize the results as follows. There is always a minimum radius $r_{min}$ at which circular motion is allowed. We found that either $r_{min}=r_{s}^+$ or $r_{min}=r_\gamma^-$. Usually, at the radius $r_{s}^+$ the test particle acquires a zero angular momentum so that a static observer at infinity would consider the particle as being at rest. Furthermore, at the radius $r_\gamma^-$ the energy of the test particle diverges, indicating that the hypersurface $r=r_\gamma^-$ is lightlike. In the simplest case, circular orbits are allowed in the infinite interval $[r_{min},\infty)$ so that, at any given radius greater than $r_{min}$, it is always possible to have a charged test particle moving on a circular trajectory. Sometimes, inside the infinite interval $[r_{min},\infty)$, there exists a lightlike hypersurface situated at $r_\gamma^+>r_{min}$. Another possible structure is that of a finite region filled with charged particles within the spatial interval $(r_{min}=r_\gamma^-,r_{max}=r_\gamma^+)$. This region is usually surrounded by an empty finite region in which no motion is allowed. Outside the empty region, we find a zone of allowed circular motion in which either only neutral particles or neutral and charged particles can exist in circular motion. Clearly, this spatial configuration formed by two separated regions in which circular motion is allowed, could be used to build with test particles an accretion disk of disconnected rings. A particular example of this case is illustrated in Fig. \[Mercedes\] ---------------------------------------------------- ---------------------------------------------------- ![[]{data-label="Mercedes"}](Mercedes.eps "fig:") ![[]{data-label="Mercedes"}](Mercedes1.eps "fig:") ![[]{data-label="Mercedes"}](Mercedes2.eps "fig:") ![[]{data-label="Mercedes"}](Mercedes3.eps "fig:") ---------------------------------------------------- ---------------------------------------------------- Case $\epsilon<-1$ {#bicioc} ------------------ The contribution of the electromagnetic interaction in this case is always attractive. Hence, the only repulsive force to balance the attractive effects of the gravitational and Coulomb interactions can be generated only by the RN naked singularity. This case therefore can be compared with the neutral test particle motion as studied in [@Pugliese:2010ps; @Pugliese:2010he]. Then, it is convenient, as in the case of a neutral test particle, to consider the two regions $Q>\sqrt{9/8}M$ and $M<Q\leq\sqrt{9/8}M$ separately. For $\epsilon<-1$ and for $Q>\sqrt{9/8}M$ circular orbits with $L=L^{+}$ always exist for $r>0$ (in fact, however, one has to consider also the limit $r>r_$ for the existence of timelike trajectories). This case is illustrated in Fig. [ ]{}\[Faria\] where the presence of orbits with negative energy states is evident. --------------------------------------------- -- ![[]{data-label="Faria"}](Faria.eps "fig:") --------------------------------------------- -- For $M<Q\leq\sqrt{9/8}M$ circular orbits exist with $L=L^{+}$ in $0<r<r_\gamma^-$ and $r>r_\gamma^+ $ (see Fig. [ ]{}\[Antonioc\]). ------------------------------------------------------------------------------------------------------ -- ![[]{data-label="Antonioc"}](Antonioc.eps "fig:") ![[]{data-label="Antonioc"}](Antonioca.eps "fig:") ------------------------------------------------------------------------------------------------------ -- We note that for neutral test particles in the region $M<Q\leq\sqrt{9/8}M$, (stable) circular orbits are possible for $r>r_{}=Q^2/M$. At $r=r_$, the angular momentum of the particle vanishes [@Pugliese:2010ps]. On the contrary, charged test particles with $\epsilon<-1$ can move along circular orbits also in the region $(0,r_{}]$. The value of the energy on circular orbits increases as $r$ approaches $r=0$. However, the angular momentum, as seen by an observer located at infinity, decreases as the radius of the orbit decreases. In the region $M<Q\leq\sqrt{9/8}M$, two limiting orbits appear at $r_\gamma^\pm $, as in the neutral particle case [@Pugliese:2010ps]. Case $-1<\epsilon<0$ {#bicioca} -------------------- For this range of the ratio $\epsilon$, it is also convenient to analyze separately the two cases $Q>\sqrt{9/8}M$ and $M<Q\leq\sqrt{9/8}M$. In each case it is necessary to analyze the explicit value of $\epsilon$ with respect to the ratio $M/Q$. Several cases arise in which we must find the regions where circular motion is allowed and the value of the angular momentum and energy of the rotating charged test particles. We summarize the results in the following manner. There are two different configurations for the regions in which circular motion of charged test particles is allowed. The first one arises in the case $Q>\sqrt{9/8}M$, and consists in a continuous region that extends from a minimum radius $r_{min}$ to infinity, in principle. The explicit value of the minimum radius depends on the value of $\epsilon$ and can be either $r_s^-$, $r_s^+$, or $r_{min}=Q^2/(2M)$. In general, we find that particles standing on the minimum radius are characterized by $L=0$, i. e., they are static with respect to a non-rotating observer located at infinity. The second configuration appears for $M<Q\leq\sqrt{9/8}M$. It also extends from $r_{min}$ to infinity, but inside it there is a forbidden region delimited by the radii $r_\gamma^-$ and $r_\gamma^+$. The configuration is therefore composed of two disconnected regions. At the minimum radius, test particles are characterized by $L=0$. On the boundaries ($r_\gamma^\pm$) of the interior forbidden region only photons can stand on circular orbits. A particular example of this case is presented in Fig. \[Grotta\]. ---------------------------------------------------------------------------------------------- -- ![[]{data-label="Grotta"}](Grotta.eps "fig:") ![[]{data-label="Grotta"}](Grotta1.eps "fig:") ---------------------------------------------------------------------------------------------- -- Stability {#stability} --------- To explore the stability properties of the circular motion of charged test particles in a RN naked singularity, it is necessary to investigate the equation (\[Lagespep\]) or, equivalently, Eqs.(\[nube\]), (\[nuvola\]), and (\[Carmara\]), considering the different values for $\epsilon$ and $Q/M>1$. We can distinguish two different cases, $\|\epsilon\|>1$ and $0<\|\epsilon\|<1$. Let us consider the case $\|\epsilon\|>1$. In particular, as it was shown in Sec. \[eme1\], for $\epsilon>1$ and $M<Q<\sqrt{9/8}M$ circular orbits exist with $L=L^{+}$ in the interval $r_\gamma^-<r<r_\gamma^+ $ whereas no circular orbits exist for $\epsilon>1$ and $Q>\sqrt{9/8}M$. For this particular case, a numerical analysis of condition (\[Lagespep\]) leads to the conclusion that a circular orbit is stable only if its radius $r_0$ satisfies the condition $r_0> r_{{\mbox{\tiny{lsco}}}}$, where $r_{{\mbox{\tiny{lsco}}}}$ is depicted in Fig. [ ]{}\[PlotVns11\]. We see that in general the radius of the last stable circular orbit is located inside the interval $(r_\gamma^-,r_\gamma^+)$. It then follows that the only stable region is determined by the interval $r_{{\mbox{\tiny{lsco}}}}<r<r_\gamma^+$. ---------------------------------------------------- -- ![[]{data-label="PlotVns11"}](PlotNSp7.eps "fig:") ---------------------------------------------------- -- Consider now the case $\epsilon<-1$. The numerical investigation of the condition (\[Lagespep\]) for the last stable circular orbit shows that in this case there are two solutions $r_{{\mbox{\tiny{lsco}}}}^\pm$ such that $r_{{\mbox{\tiny{lsco}}}}^-\leq r_{{\mbox{\tiny{lsco}}}}^+$, where the equality is valid for $Q/M\approx 1.72$. Moreover, for $Q/M=\sqrt{9/8}$ we obtain that $r_{{\mbox{\tiny{lsco}}}}^-=r_\gamma^- = r_\gamma^+$. This situation is illustrated in Fig. [ ]{}\[NSm7\]. Stable orbits corresponds to points located outside the region delimited by the curves $r=r_{{\mbox{\tiny{lsco}}}}^+$, $r=r_{{\mbox{\tiny{lsco}}}}^-$, and the axis $Q/M=1$. On the other hand, we found in Sec. \[bicioc\] that for $\epsilon<-1$ and $1<Q/M\leq\sqrt{9/8}$ circular orbits exist in the interval $0<r<r_\gamma^-$ and $r>r_\gamma^+ $. It then follows that the region of stability corresponds in this case to two disconnected zones determined by $0<r<r_\gamma^-$ and $r>r_{{\mbox{\tiny{lsco}}}}^+$. Moreover, we established in Sec. \[bicioc\] that for $\epsilon<-1$ and $\sqrt{9/8}<Q/M$ circular orbits always exist for $r>0$. Consequently, in the interval $\sqrt{9/8}<Q/M\lesssim 1.72$, the stable circular orbits are located in the two disconnected regions defined by $0<r<r_{{\mbox{\tiny{lsco}}}}^-$ and $r>r_{{\mbox{\tiny{lsco}}}}^+$. Finally, for $Q/M\gtrsim 1.72$ all the circular orbits are stable (see Fig. [ ]{}\[NSm7\]). ---------------------------------------------------------------------------------------------- -- ![[]{data-label="NSm7"}](PlotNSm7.eps "fig:") ![[]{data-label="NSm7"}](PlotNSm7z.eps "fig:") ---------------------------------------------------------------------------------------------- -- The case $0<\|\epsilon\|<1$ is much more complex, and needs to be described for different subcases following the classification of orbital regions traced in Sec. [ ]{}\[eme1a\] for the case $0<\epsilon<1$, and in Sec. [ ]{}\[bicioca\] for the case $-1<\epsilon<0$. The results for the specific ratio $\epsilon=0.5$ are given in Fig. [ ]{}\[NSp05\] and for $\epsilon=-0.5$ in Fig. [ ]{}\[NSm05\]. In general, we find that the results are similar to those obtained for the case $\epsilon<-1$. Indeed, the zone of stability consists of either one connected region or two disconnected regions. The explicit value of the radii that determine the boundaries of the stability regions depend on the particular values of the ratio $Q/M$. ------------------------------------------------- -- ![[]{data-label="NSp05"}](PlotNSp05.eps "fig:") ------------------------------------------------- -- ------------------------------------------------- ![[]{data-label="NSm05"}](PlotNSm05.eps "fig:") ------------------------------------------------- Conclusions {#milka} =========== In this work, we explored the motion of charged test particles along circular orbits in the spacetime described by the Reissner–Nordström (RN) metric. We performed a very detailed discussion of all the regions of the spacetime where circular orbits are allowed, using as parameters the charge–to–mass ratio $Q/M$ of the source of gravity and the charge–to–mass ratio $\epsilon=q/\mu$ of the test particle. Depending on the value of $Q/M$, two major cases must be considered: The black hole case, $\|Q/M\|\leq 1$, and the naked singularity case, $\|Q/M\|> 1$. Moreover, we found out that the two cases $\|\epsilon\|\leq 1$ and $\|\epsilon\|>1$ must also be investigated separately. Whereas the investigation of the motion of charged test particles with $\|\epsilon\|>1$ can be carried out in a relatively simple manner, the case with $\|\epsilon\|\leq 1$ is much more complex, because it is necessary to consider various subcases which depend on the explicit value of $\epsilon$ in this interval. To perform the analysis of circular motion of charged test particles in this gravitational field we use two different methods. The first one consists in using constants of motion to reduce the equations of motion to a single first–order differential equation for a particle moving in an effective potential. The properties of this effective potential are then used to find the conditions under which circular motion is possible. The second approach uses a local orthonormal frame to introduce a “local proper linear velocity" for the test particle. The conditions for this velocity to be timelike are then used to determine the regions of space where circular orbits are allowed. The results of both methods are equivalent and, in fact, for the sake of simplicity it is sometimes convenient to use a combination of both approaches. In this work, we analyzed in detail the conditions for the existence of circular orbits and found all the solutions for all the regions of space in the case of black holes and naked singularities. To formulate the main results of this work in a plausible manner, let us suppose that an accretion disk around a RN gravitational source can be made of test particles moving along circular orbits [@Kovacs:2010xm]. Then, in the case of black hole we find two different types of accretion disks made of charged test particles. The first type consists of a disk that begins at a minimum radius $R$ and can extend to infinity, in principle. In the second possible configuration, we find a circular ring of charged particles with radii $(r_{int},r_{ext})$, surrounded by the disk, i. e., with $r_{ext}<R$. For certain choices of the parameter $\epsilon$ the exterior disk might be composed only of neutral particles. A study of the stability of circular orbits shows that the second structure of a ring plus a disk is highly unstable. This means that test particles in stable circular motion around RN black holes can be put together to form only a single disk that can, in principle, extend to infinity. In the case of RN naked singularities we find the same two types of accretion disks. The explicit values of the radii $r_{min}$, $r_{ext}$, and $R$ depend on the values of the ratios $\epsilon$ and $Q/M$, and differ significantly from the case of black holes. In fact, we find that the case of naked singularities offers a much richer combination of values of the charge–to–mass ratios for which it is possible to find a structure composed of an interior ring plus an exterior disk. A study of the stability of this specific situation shows that for certain quite general combinations of the parameters the configuration is stable. This result implies that test particles in stable circular motion around RN naked singularities can be put together to form either a single disk that can extend, in principle, to infinity or a configuration of an interior ring with an exterior disk. This is the main difference between black holes and naked singularities from the viewpoint of these hypothetical accretion disks made of test particles. The question arises whether it is possible to generalize these results to the case of more realistic accretion disks around more general gravitational sources, taking into account, for instance, the rotation of the central body, [@LoraClavijo:2010ih; @Vogt:2004qx]. It seems reasonable to expect that in the case of Kerr and Kerr-Newman naked singularities, regions can be found where stable circular motion is not allowed so that an accretion disk around such an object would exhibit a discontinuous structure. Indeed, some preliminary calculations of circular geodesics in the field of rotating compact objects support this expectation. Thus, we can conjecture that the discontinuities in the accretion disks around naked singularities are a consequence of the intensity of the repulsive gravity effects that characterize these speculative objects. Furthermore, it was recently proposed that static compact objects with quadrupole moment can be interpreted as describing the exterior gravitational field of naked singularities [@quev11a; @quev11b]. It would be interesting to test the above conjecture in this relatively simple case in which rotation is absent. If the conjecture turns out to be true, it would give us the possibility of distinguishing between black holes and naked singularities by observing their accretion disks. Acknowledgments {#acknowledgments .unnumbered} =============== Daniela Pugliese and Hernando Quevedo would like to thank the ICRANet for support. We would like to thank Andrea Geralico for helpful comments and discussions. One of us (DP) gratefully acknowledges financial support from the A. Della Riccia Foundation. This work was supported in part by DGAPA-UNAM, grant No. IN106110. Velocity of test particles in a RN naked singularity {#noten} ==================================================== In this Appendix we explore charged test particles in circular motion in a RN naked singularity by using the tetrad formalism, as developed in Sec. \[BHBHTR\] for the black hole analysis. In Sec. \[NSNSTRE\], we studied the timelike circular motion in the naked singularity case by analyzing directly the existence conditions for the energy, Eq.[ ]{}(\[Lagesp\]), and the angular momentum, Eq.[ ]{}(\[ECHEEUNAL\]). Here we use the formalism of “local proper linear velocity" as measured by an observer attached to an orthonormal frame. The results are equivalent to those obtained by using the expressions for the energy and angular momentum. In Sec. [ ]{}\[BHBHTR\], we showed that the linear velocity of a test particle in a RN spacetime can be written as $$\label{vel} \nu_{\epsilon}^{\pm}=\nu_{g} \left[\Lambda\pm\sqrt{\Lambda^{2}-1+(\epsilon/\epsilon_0)^{2}}\right]^{1/2}\ ,$$ where $$\label{notvel} \Lambda=1-\frac{\nu_{g} ^{2}}{2}\left(\frac{\epsilon}{\epsilon_0}\right)^{2}\ , \quad \nu_g= \sqrt{\frac{Mr-Q^2}{\Delta}}\ , \quad \epsilon_0 = \frac{Mr-Q^2}{Q\sqrt{\Delta}} \ .$$ Then, the conditions for the existence of timelike velocities are $$\begin{aligned} \label{cond1} \Lambda^{2}-1+(\epsilon/\epsilon_{0})^{2} &\geq& 0 \ ,\\ \label{cond2} \Lambda\pm\sqrt{\Lambda^{2}-1+(\epsilon/\epsilon_{0})^{2}}&\geq& 0 \ ,\\ \label{cond3} (\nu_{\epsilon}^{\pm})^{2}&<&1\ .\end{aligned}$$ We first note that, in the case of a naked singularity, these conditions can be satisfied only for $r\geq Q^{2}/M$. For $\epsilon>1$ and $\epsilon<-1$ the solutions are the geodesic velocities $\nu=\pm\nu_{\epsilon}^{+}$. In fact, in this case, condition (\[cond2\]) with the minus sign is no more satisfied. On the other hand, conditions (\[cond1\]), (\[cond2\]), and (\[cond3\]) imply that circular timelike orbits exist for $Q/M>\sqrt{9/8}$ in the entire range $r>Q^{2}/M$. For $1<Q/M<\sqrt{9/8}$ circular orbits are possible in $r>Q^{2}/M$ and $r\neq r_\gamma^\pm \equiv[3M\pm\sqrt{9M^{2}-8Q^{2}}]/2$. Finally, for $Q/M=\sqrt{9/8}$ timelike circular orbits exist for all $r>Q^{2}/M$, except at $r=(3/2) M$. Moreover, the radii $r= r_\gamma^\pm $ correspond to photon orbits in the RN spacetime (see Fig. \[Sax11b\]). Consider now the case $\|\epsilon\|<1$. It is useful to introduce here the following notations: $$\begin{aligned} r^{\pm}_l&\equiv&\frac{3 M}{2}\pm\frac{1}{2} \sqrt{9 M^{2}-8 Q^{2}-Q^{2} \epsilon ^{2}},\\ \tilde{\epsilon}_{\pm}&\equiv& \frac{1}{\sqrt{2}Q}\sqrt{5M^{2}\pm4Q^{2}+\sqrt{25M^{2}-24Q^{2}}},\end{aligned}$$ and \[r1\] r\_[s]{}\^&& . First, consider the case $0<\epsilon<1$. For $\epsilon>0$ condition (\[14\]) implies that $r>Q^{2}/M$. Applying this constraint on conditions (\[cond1\]) and (\[cond2\]), we obtain the following results for timelike geodesics. 1. For $1<Q/M\leq5/(2\sqrt{6})$ the following subcases occur:\ a) : $0<\epsilon<\tilde{\epsilon}_{-}$ : Fig. [ ]{}\[Sax4\]a The velocity $\nu=\pm\nu_{\epsilon}^{+}$ exists in the range $Q^{2}/M<r\leq r_{l}^{-}$ and $r\geq r_{l}^{+}$ with $r\neq r_{\gamma}^{\pm}$, $\nu=\pm\nu_{\epsilon}^{-}$ exists in the range $r_{s}^+<r\leq r_{l}^{-}$ and $r\geq r_{l}^{+}$.\ b) : $\tilde{\epsilon}_{-}\le\epsilon \leq\tilde{\epsilon}_{+}$ : Fig. [ ]{}\[Sax4\]b The velocity $\nu=\pm\nu_{\epsilon}^{+}$ exists in the range $Q^{2}/M<r< r_{l}^{-}$ and $r\geq r_{l}^{+}$ with $r\neq r_{\gamma}^{\pm}$, $\nu=\pm\nu_{\epsilon}^{-}$ exists in the range $r\geq r_{l}^{+}$.\ c) : $\tilde{\epsilon}_{+}<\epsilon<\epsilon_{l}$, : Fig. [ ]{}\[Sax4\]. The velocity $\nu=\pm\nu_{\epsilon}^{+}$ exists in the range $Q^{2}/M<r\leq r_{l}^{-}$ and $r\geq r_{l}^{+}$ with $r\neq r_{\gamma}^{\pm}$, $\nu=\pm\nu_{\epsilon}^{-}$ exists in the range $r_{s}^{+}<r\leq r_{l}^{-}$ and $r\geq r_{l}^{+}$.\ d) : $\epsilon_{l} \le\epsilon <1$ : Fig. [ ]{}\[Sax4\]d The solutions are the geodesic velocities $\nu=\pm\nu_{\epsilon}^{+}$ in the range $r>Q^{2}/M$ with $r\neq r_{\gamma}^{\pm}$. The solution $\nu=\pm\nu_{\epsilon}^{-}$ exists for $\epsilon_{l} \le\epsilon <M/Q$ in the range $r>r_{s}^{+}$. 2. For $5/(2\sqrt{6})<Q/M<\sqrt{9/8}$ the following subcases occur: a) : $0<\epsilon<\epsilon_{l}$ : Fig. [ ]{}\[Sax4\]b The velocity $\nu=\pm\nu_{\epsilon}^{+}$ exists in the range $Q^{2}/M<r\leq r_{l}^{-}$ and $r> r_{l}^{+}$ with $r\neq r_{\gamma}^{\pm}$, $\nu=\pm\nu_{\epsilon}^{-}$ exists in the range $r_{s}^{+}<r\leq r_{l}^{-}$ and $r\geq r_{l}^{+}$.\ b) : $\epsilon_{l}\le\epsilon <1$ : Fig. [ ]{}\[Sax4\]a The velocity $\nu=\pm\nu_{\epsilon}^{+}$ exists in the range $r>Q^{2}/M$, $\nu=\pm\nu_{\epsilon}^{-}$ exists in the range $r> r_{s}^{+}$. 3. $Q/M\geq\sqrt{9/8}$ : Figs.[ ]{}\[Sax2\] and \[Sax2bc\] The velocity $\nu=\pm\nu_{\epsilon}^{+}$ exists in the range $r>Q^{2}/M$ for $Q/M>\sqrt{9/8}$ whereas for $Q/M=\sqrt{9/8}$ this is a solution in $r/M> 9/8$ with $r/M\neq 3/2$, $\nu=\pm\nu_{\epsilon}^{-}$ exists for $0<\epsilon<M/Q$ in the range $r>r_{s}^{+}$. ----------------------------------------------- ----------------------------------------------- ![ []{data-label="Sax11b"}](Sax1.eps "fig:") ![ []{data-label="Sax11b"}](Sax1b.eps "fig:") (a) (b) ![ []{data-label="Sax11b"}](Sax1c.eps "fig:") ![ []{data-label="Sax11b"}](Sax1d.eps "fig:") (c) (d) ----------------------------------------------- ----------------------------------------------- --------------------------------------------- --------------------------------------------- ![ []{data-label="Sax4"}](Sax4.eps "fig:") ![ []{data-label="Sax4"}](Sax4a.eps "fig:") (a) (b) ![ []{data-label="Sax4"}](Sax4b.eps "fig:") ![ []{data-label="Sax4"}](Sax4c.eps "fig:") (c) (d) --------------------------------------------- --------------------------------------------- -------------------------------------------- --------------------------------------------- ![ []{data-label="Sax5"}](Sax5.eps "fig:") ![ []{data-label="Sax5"}](Sax5a.eps "fig:") (a) (b) -------------------------------------------- --------------------------------------------- -------------------------------------------- --------------------------------------------- ![ []{data-label="Sax2"}](Sax2.eps "fig:") ![ []{data-label="Sax2"}](Sax2a.eps "fig:") (a) (b) -------------------------------------------- --------------------------------------------- ----------------------------------------------- ----------------------------------------------- ![ []{data-label="Sax2bc"}](Sax2b.eps "fig:") ![ []{data-label="Sax2bc"}](Sax2c.eps "fig:") (a) (b) ----------------------------------------------- ----------------------------------------------- The results for $-1<\epsilon<0$ are summarized below. 1. For $1<Q/M\leq5/(2\sqrt{6})$ the following subcases occur:\ a) : For $-1<\epsilon\leq-\epsilon_{l}$, the velocity $\nu=\pm\nu_{\epsilon}^{+}$ exists in the range $r>Q^{2}/M$ with $r\neq r_{\gamma}^{\pm}$, $\nu=\pm\nu_{\epsilon}^{-}$ exists for $-(M/Q)<\epsilon\leq-\epsilon_{l}$ in the range $r>r_{s}^{+}$ (see Fig. [ ]{}\[Sax4n\]a).\ b) : For $-\epsilon_{l}<\epsilon<-\tilde{\epsilon}_{+}$, the solution is $\nu=\pm\nu_{\epsilon}^{+}$ in the range $Q^{2}/M<r\leq r_{l}^{-}$ and $r\geq r_{l}^{+}$ with $r\neq r_{\gamma}^{\pm}$, $\nu=\pm\nu_{\epsilon}^{-}$ exists in the range $r_{s}^{+}<r\leq r_{l}^{-}$ and $r\geq r_{l}^{+}$ (see Fig. [ ]{}\[Sax4n\]b).\ c) : For $-\tilde{\epsilon}_{+}\le\epsilon \leq-\tilde{\epsilon}_{-}$, the velocity $\nu=\pm\nu_{\epsilon}^{-}$ exists in the range $r\geq r_{l}^{+}$. $\nu=\pm\nu_{\epsilon}^{+}$ exists for $-\tilde{\epsilon}_{+}<\epsilon<-\tilde{\epsilon}_{-}$ in the range $(Q^{2}/M)<r<r_{s}^{+}$, and $r\geq r_{l}^{+}$ with $r\neq r_{\gamma}^{\pm}$, and for $\epsilon=-\tilde{\epsilon}^{\pm}$ the velocity $\nu_{\epsilon}^{+}$ exists for $Q^{2}/M<r<r_{l}^{-}$ and $r\geq r_{l}^{+}$ with $r\neq r_{\gamma}^{\pm}$. Finally, for $Q=5/(2\sqrt{6})M$ and $\epsilon=-\tilde{\epsilon}_{+}$, $\nu_{\epsilon}^{+}$ exists for $(Q^{2}/M)<r<r_{l}^{-}$, and $r\geq r_{l}^{+}$ (see Fig. [ ]{}\[Sax4n\]c).\ d) : For $-\tilde{\epsilon}_{-} <\epsilon<0$, the solutions are the geodesic velocities $\nu=\pm\nu_{\epsilon}^{+}$ in the range $ (Q^{2}/M)<r\leq r_{l}^{-}$ and $r\geq r_{l}^{+}$ with $r\neq r_{\gamma}^{\pm}$. The solution $\nu=\pm\nu_{\epsilon}^{-}$ exists in $ r_{s}^{+}<r\leq r_{l}^{-}$ and $r\geq r_{l}^{+}$ (see Fig. [ ]{}\[Sax4n\]d). 2. For $5/(2\sqrt{6})<Q/M<\sqrt{9/8}$ the following subcases occur: a) : For $-1<\epsilon\leq-\epsilon_{l}$, the velocity $\nu=\pm\nu_{\epsilon}^{+}$ exists in the range $r>Q^{2}/M$ with $r\neq r_{\gamma}^{\pm}$, $\nu=\pm\nu_{\epsilon}^{-}$ exists for $-(M/Q)<\epsilon\leq-\epsilon_{l}$ in the range $r>r_{s}^{+}$ (see Fig. [ ]{}\[Sax4n\]b).\ b) : For $-\epsilon_{l}\le\epsilon <0$, the velocity $\nu=\pm\nu_{\epsilon}^{+}$ exists in the range $Q^{2}/M<r\leq r_{l}^{-}$ and $r\geq r_{l}^{+}$, $r\neq r_{\gamma}^{\pm}$, $\nu=\pm\nu_{\epsilon}^{-}$ exists in the range $ r_{s}^{+}<r\leq r_{l}^{-}$ and $r\geq r_{l}^{+}$ (see Fig. [ ]{}\[Sax4n\]a). 3. For $Q/M\geq\sqrt{9/8}$ the velocity $\nu=\pm\nu_{\epsilon}^{-}$ exists for $-(M/Q)<\epsilon<0$ in the range $r>r_{s}^{+}$. $\nu=\pm\nu_{\epsilon}^{+}$ is a solution for $Q/M>\sqrt{9/8}$ and $-1<\epsilon<0$ in $r>Q^{2}/M$ whereas for $Q/M=\sqrt{9/8}$ this is a solution in $r/M> 9/8$ with $r/M\neq 3/2$ (see Figs.[ ]{}\[Sax2n\] and \[Sax2bcn\]). ----------------------------------------------- ----------------------------------------------- ![ []{data-label="Sax4n"}](Sax4cn.eps "fig:") ![ []{data-label="Sax4n"}](Sax4bn.eps "fig:") (a) (b) ![ []{data-label="Sax4n"}](Sax4an.eps "fig:") ![ []{data-label="Sax4n"}](Sax4n.eps "fig:") (c) (d) ----------------------------------------------- ----------------------------------------------- ----------------------------------------------- ---------------------------------------------- ![ []{data-label="Sax5n"}](Sax5an.eps "fig:") ![ []{data-label="Sax5n"}](Sax5n.eps "fig:") (a) (b) ----------------------------------------------- ---------------------------------------------- ----------------------------------------------- ---------------------------------------------- ![ []{data-label="Sax2n"}](Sax2an.eps "fig:") ![ []{data-label="Sax2n"}](Sax2n.eps "fig:") (a) (b) ----------------------------------------------- ---------------------------------------------- ------------------------------------------------- ------------------------------------------------- ![ []{data-label="Sax2bcn"}](Sax2cn.eps "fig:") ![ []{data-label="Sax2bcn"}](Sax2bn.eps "fig:") (a) (b) ------------------------------------------------- ------------------------------------------------- [99]{} R. 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Introduction ============ A new paradigm of heavy ion phenomenology that the quark gluon plasma (QGP) is a strongly interacting liquid [@Bugaev:Ref1n] proved to be very successful not only in describing some of its properties measured by lattice quantum chromodynamics (QCD), but also in explaining some experimental observables that cannot be reproduced otherwise. Probably, the two most striking conclusions obtained within the new paradigm are as follows: first, at the cross-over temperature, where the string tension of color tube is almost vanishing, the potential energy of color charge is of the order of a few GeV [@Bugaev:Ref2n], i.e. it is 10 times larger than its kinetic energy, and, second, the QGP, so far, is the most perfect fluid since its shear viscosity in units of the entropy density is found to be the smallest one [@Bugaev:Ref3n; @Bugaev:Ref4n]. The first of these conclusions tells us that at the cross-over region there is no color charge separation [@Bugaev:Ref5n], whereas the second one naturally explains the great success of ideal hydrodynamics when applied to relativistic heavy ion collisions. Here we would like to discuss the recent progress achieved in our understanding of both the confinement phenomenon [@Bugaev:Ref6n; @Bugaev:Ref7n] and the physical origin of the cross-over [@Bugaev:Ref8n; @Bugaev:Ref9n; @Bugaev:Ref10n; @Bugaev:Ref11n; @Bugaev:Ref12n]. As we demonstrate below such a progress was made possible after realizing a principal role played by negative values of the surface tension coefficient of large QGP bags [@Bugaev:Ref6n; @Bugaev:Ref8n]. Also here we argue that the negative values of the surface tension, that are responsible for an existence of the cross-over transition to QGP at low baryonic densities, play the same role in ordinary liquids. Moreover, in this work we would like to draw an attention to the problem of the temperature dependence of surface tension coefficient in liquids by clearly showing that for many liquids the well known Guggenheim relation (see Eq. (\[EqBugaevX\])) is not so well established experimentally as it is usually believed. The work is organized as follows. Section 2 is devoted to the confining tube model, in which the Fisher topological term of the QGP bag free energy is accounted for. In section 3 we show that at the cross-over region the surface tension of QGP bags is necessarily negative and argue that this is the case for ordinary liquids as well. The maximum of the tube entropy observed in lattice QCD is explained in section 4, where the model of gas of free tubes is also developed. The conclusions are given in the last section. Color confining tube and sQGP ============================= A color confinement, i.e. an absence of free color charges, is usually described by the free energy of heavy (static) quark-antiquark pair [$F_{q\bar q} (T, L) = \sigma_{str} \cdot L$]{}. In the lattice QCD the functional dependence of $F_{q\bar q} (T, L)$ on the temperature $T$ and the separation distance $L$ can be extracted from the Polyakov line correlation in a color singlet channel. Then it is customary to define - [confinement:]{} the case of non-zero string tension, i.e. [$\sigma_{str} >0$]{}; - [deconfinement:]{} the case of vanishing string tension [$\sigma_{str} \rightarrow 0$]{} at $T \rightarrow T_{co}$, but one should remember that there is no color charge separation up to $T \ge 1.3 \,T_{co}$ values of the cross-over temperature $T_{co}$ [@Bugaev:Ref1n; @Bugaev:Ref5n]. The explanation of the latter is as follows: although at large distances $L \rightarrow \infty$ the potential energy of static $q\bar q$ pair is finite $U_{q\bar q} (T, L) = F_{q\bar q} - T \frac{\partial F_{q\bar q}}{\partial T} = F_{q\bar q} + T S_{q\bar q}$ near $T_{co}$, the values of $U (T, \infty)$ are very large (see Fig. \[Bugaev:fig:1\]). From Fig. \[Bugaev:fig:1\] one can conclude that near $T_{co}$ region QGP is a strongly interacting plasma (sQGP) which is similar to a liquid, since the ratio of the quark potential energy to its kinetic energy, the so called plasma parameter, $\frac{U (T, \infty)}{3\,T} \in 1-10$ has the range of values that is typical for ordinary liquids [@Bugaev:Ref1n]. ![The internal energy $U_\infty$ (left) and entropy $S_\infty$ (right) of confining tube connecting two static color charges found by lattice QCD simulations for infinite separation distance between charges [@Bugaev:Ref2n]. The internal energy is shown for 2 quark flavors.[]{data-label="Bugaev:fig:1"}](Bugaev_Uinf_T.pdf "fig:"){width="42.00000%"}![The internal energy $U_\infty$ (left) and entropy $S_\infty$ (right) of confining tube connecting two static color charges found by lattice QCD simulations for infinite separation distance between charges [@Bugaev:Ref2n]. The internal energy is shown for 2 quark flavors.[]{data-label="Bugaev:fig:1"}](Bugaev_Sinf_T.pdf "fig:"){width="42.00000%"} The second striking feature of the confining tube can be seen in the right panel of Fig. \[Bugaev:fig:1\] which clearly demonstrates that at $T=T_{co}$ the entropy of static $q\bar q$ pair is very large $S_{q\bar q} (T_{co}, \infty) \approx 20$. Such a value signals that really a [huge number of degrees of freedom $\sim \exp(20)$]{} is involved, but the origin of large energy $U_{q\bar q} (T, \infty)$ and entropy $S_{q\bar q} (T, \infty)$ values near $T_{co}$ for awhile remained [mysterious]{} [@Bugaev:Ref1n] despite many attempts to explain it. Another problem of principal importance for phenomenological models of deconfinement phase transition [@Bugaev:Ref8n; @Bugaev:Ref9n; @Bugaev:Ref10n; @Bugaev:Ref11n; @Bugaev:Ref12n; @Bugaev:Ref13o; @Bugaev:Ref14o; @Bugaev:Ref15o] is the value of the surface tension coefficient $\sigma_{surf}$ of QGP bags. There are several estimates for the surface tension coefficient $\sigma_{surf}$ of QGP bags [@Bugaev:Ref13n], but the question is whether can we determine $\sigma_{surf}$ from lattice QCD? Therefore, in this section we consider an approach that allows us to determine the surface tension coefficient of QGP bags directly from the lattice QCD. As it will be shown in the section 4 such an approach naturally explains an existence of the ‘mysterious maximum’ [@Bugaev:Ref1n] of the confining tube entropy. In order to estimate the surface tension of QGP bags let us consider the static quark-antiquark pair connected by the unbreakable color tube of length $L$ and radius $R \ll L$. In the limit of large $L$ the free energy of the color tube is $F_{str} \rightarrow \sigma_{str} L$. Now we consider the same tube as an elongated cylinder of the same radius and length [@Bugaev:Ref6n]. In this case we neglect the free energy of the regions around the color charges, but for our treatment of large separation distances $L \gg R$ this is sufficient. For the cylinder free energy we use the standard parameterization [@Bugaev:Ref8n; @Bugaev:Ref9n; @Bugaev:Ref10n; @Bugaev:Ref11n; @Bugaev:Ref12n] $$\begin{aligned} \label{EqBugaevI} &&\hspace{-0.5cm}F_{cyl} (T, L, R) = - p_v(T) \pi R^2 L + \sigma_{surf} (T) 2 \pi R L + T \tau \ln\left[ \frac{\pi R^2 L }{V_0} \right] \,, $$ where $p_v (T)$ is the bulk pressure inside a bag, $\sigma_{surf} (T)$ is the temperature dependent surface tension coefficient, while the last term on the right hand side above is the Fisher topological term [@Bugaev:Ref14n] which is proportional to the Fisher exponent $\tau = const > 1 $ [@Bugaev:Ref8n; @Bugaev:Ref9n; @Bugaev:Ref10n; @Bugaev:Ref11n; @Bugaev:Ref12n; @Bugaev:Ref15o] and $V_0 \approx 1$ fm$^3$ is a normalization constant. Since we consider the same object then its free energies calculated as the color tube and as the cylindrical bag should be equal to each other. Then for large separating distances $L \gg R$ one finds the following relation $$\label{EqBugaevII} \sigma_{str} (T) = \sigma_{surf} (T)\, 2 \pi R~ - ~p _v (T) \pi R^2 + \frac{T \tau}{L} \ln\left[ \frac{\pi R^2 L }{V_0} \right] \,. $$ In doing so, in fact, we match an ensemble of all string shapes of fixed $L$ to a mean elongated cylinder, which according to the original Fisher idea [@Bugaev:Ref14n] and the results of the Hills and Dales Model (HDM) [@Bugaev:Ref15n; @Bugaev:Ref16n] represents a sum of all surface deformations of such a bag. The last equation allows one to determine the $T$-dependence of bag surface tension as $$\label{EqBugaevIII} \sigma_{surf} (T) = \frac{\sigma_{str} (T)}{ 2 \pi R} ~ + ~ \frac{1}{2} \, p_v (T) R ~ - ~ \frac{T \tau}{2 \pi R L} \ln\left[ \frac{\pi R^2 L }{V_0} \right] \,, $$ if $R(T)$, $\sigma_{str} (T)$ and $p_v (T)$ are known. This relation opens a principal possibility to determine the bag surface tension directly from the lattice QCD simulations for any $T$. Also it allows us to estimate the surface tension at $T=0$. Thus, taking the typical value of the bag model pressure which is used in hadronic spectroscopy $p_v (T=0) = - (0.25)^4$ GeV$^4$ and inserting into (\[EqBugaevIII\]) the lattice QCD values $R=0.5$ fm and $\sigma_{str} (T=0) = (0.42)^2$ GeV$^2$ [@Bugaev:Ref17n], one finds $\sigma_{surf} (T=0) = (0.2229~ {\rm GeV})^3 + 0.5\, p_v\, R\approx (0.183~{\rm GeV})^3 \approx 157.4$ MeV fm$^{-2}$ [@Bugaev:Ref6n]. The last term in (\[EqBugaevIII\]) does not modify our above estimate at $T=0$, but, in contrast to [@Bugaev:Ref6n; @Bugaev:Ref7n], we keep it in order to demonstrate its importance for the confining tube with free color charges. The found value of the bag surface tension at zero temperature is very important for the phenomenological equations of state of strongly interacting matter in two respects. Firstly, according to HDM the obtained value defines the temperature at which the bag surface tension coefficient changes the sign [@Bugaev:Ref15n; @Bugaev:Ref16n; @Bugaev:Ref7n] $$\label{EqBugaevIV} T_\sigma = \sigma_{surf} (T=0)\, V_0^\frac{2}{3} \cdot \lambda^{-1}~ \in ~[148.4;~ 157.4] ~{\rm MeV} \, , $$ where the constant $\lambda = 1$ for the Fisher parameterization of the $T$-dependent surface tension coefficient [@Bugaev:Ref14n] or $\lambda \approx 1.06009$, if we use the parameterization derived within the HDM for surface deformations [@Bugaev:Ref15n; @Bugaev:Ref16n; @Bugaev:Ref7n]. Secondly, according to one of the most successful models of liquid-gas phase transition, i.e. the Fisher droplet model (FDM) [@Bugaev:Ref14n] the surface tension coefficient linearly depends on temperature. This conclusion is well supported by HDM and by microscopic models of vapor-liquid interfaces [@Bugaev:Ref21n]. Therefore, the temperature $T_\sigma$ in (\[EqBugaevIV\]), at which the surface tension coefficient vanishes, is also the temperature of the (tri)critical endpoint $T_{cep}$ of the liquid-gas phase diagram. On the basis of these arguments in Ref. [@Bugaev:Ref7n] we concluded that the value of QCD critical endpoint temperature is $T_{cep}= T_\sigma = 152.9 \pm 4.5 $ MeV. Hopefully, the latter can be verified by the lattice QCD simulations using Eq. (\[EqBugaevIII\]). Now the question is what is the surface tension coefficient above $T_{cep}$, i.e. at supercritical temperatures. There are no experimental data on usual liquids in this region. In FDM and in the other well known model of liquid-gas phase transition, the statistical multifragmentation model (SMM) [@Bugaev:Ref18n; @Bugaev:Ref19n; @Bugaev:Ref20n], the surface tension at supercritical temperatures is assumed to be zero, while in other models such a question is usually not discussed. The only exceptions known to us are the exactly solvable statistical models of quark gluon bags with surface tension [@Bugaev:Ref8n; @Bugaev:Ref9n], their extension which includes the finite widths of large/heavy QGP bags [@Bugaev:Ref10n; @Bugaev:Ref11n; @Bugaev:Ref12n] and recently formulated generalization of the SMM [@Bugaev:Ref125n]. For all these models it was demonstrated that the negative surface tension is the only physical reason of degeneration of the 1-st order phase transition into cross-over at supercritical temperatures. The question is whether the above suggested formalism can support such a conclusion. Surface tension coefficient at the cross-over temperature ========================================================= The above results, indeed, allow us to tune the interrelation with the color tube model and to study the bag surface tension near the cross-over to QGP. Consider first the vanishing baryonic densities. The lattice QCD data indicate that at large $R$ the string tension behaves as $$\label{EqBugaevV} \sigma_{str} ~ =~ \frac{\ln\left( L/L_0 \right)}{R^2 } g_0 \,, $$ where $L_0 > 0 $ and $g_0 > 0$ are some positive constants. Assuming the validity of Eq. (\[EqBugaevV\]) in the infinite available volume, one finds that for $\sigma_{str} (T)\rightarrow + 0$ the string radius diverges, i.e. $R \rightarrow \infty$. Using Eqs. (\[EqBugaevI\]) and (\[EqBugaevIII\]) we can write the total pressure $p_{tot}$ of the cylinder as follows $$\begin{aligned} p_{tot} (L, R, T)& =& p_v (T) - \frac{ \sigma_{surf} (T)}{R} - \frac{T \tau}{\pi R^2 L} \equiv \frac{ \sigma_{surf} (T)}{R} - \frac{\sigma_{str}}{\pi R^2} + \frac{T \tau}{\pi R^2 L} \left[ \ln \left( \frac{\pi R^2 L}{V_0} \right) - 1 \right] \nonumber \\ \label{EqBugaevVI} & = & \frac{ \sigma_{surf} (T)}{R} - \frac{g_0 \ln\left( L/L_0 \right)}{\pi R^4} + \frac{T \tau}{\pi R^2 L} \left[ \ln \left( \frac{\pi R^2 L}{V_0} \right) - 1 \right] \, . $$ This equation shows that for fixed separation distance $L$ in the limit $\sigma_{str} (T)\rightarrow + 0$ the leading term is given by the surface tension contribution, while the next to leading term corresponds to the contribution of the Fisher topological term, whereas the second term on the right hand side of (\[EqBugaevVI\]) is the smallest one. Therefore, it is evident that for small values of string tension (and large $R$) the main contribution to the total pressure and to its temperature derivative is given by the first term on the right hand side of (\[EqBugaevVI\]). To calculate the total entropy density $s_{tot}$ of the cylinder let us parameterize the string tension as $$\label{EqBugaevVII} \sigma_{str}= \sigma_{str}^0\, t^\nu$$ where $t \equiv \frac{T_{co} -T}{T_{co}} \rightarrow +0$ and $\nu =const >0$. From (\[EqBugaevVII\]) it follows $R = \left[ \frac{ g_0 \ln(L/L_0)}{ \sigma_{str}^0 t^\nu} \right]^\frac{1}{2}$ and then for $t \rightarrow ~0$ the entropy density $s_{tot}$ can be found from (\[EqBugaevVI\]) and (\[EqBugaevVII\]) as $$\begin{aligned} \label{EqBugaevVIII} s_{tot} & = & \left(\frac{\partial ~p_{tot}}{\partial~T} \right)_\mu \rightarrow \underbrace{- \frac{\nu}{2 \,R \, T_{co}} \, \frac{\sigma_{surf}}{t} }_{dominant~~since~~ t \rightarrow ~0 } \,+\, \frac{1}{R} \frac{\partial~\sigma_{surf} }{\partial ~T} \rightarrow - ~ \frac{\nu}{2 \, T_{co}} \left[ \frac{\sigma^0_{str}}{ g_0 \ln\left( L/L_0 \right)} \right]^\frac{1}{2} \, \frac{\sigma_{surf}}{t^{1 - \nu/2}} ~> ~0 \, .\end{aligned}$$ This equation shows that at $T= T_{co}$ the entropy density diverges for $\nu < 2$ and also that at the cross-over region the surface tension coefficient must be negative otherwise the system would be thermodynamically unstable since its entropy density would be negative. ![Surface tension of normal paraffins as a function of temperature from the triple point to the critical point. The filled circles indicate the experimental data [@Bugaev:Ref29n]. The lines are the different theoretical parameterizations [@Bugaev:Ref27n].[]{data-label="Bugaev:fig:2"}](Bugaev_Sigma_dat1a.pdf "fig:"){width="84.00000%"}\ Clearly, the results (\[EqBugaevI\])–(\[EqBugaevVIII\]) are valid for nonzero baryonic chemical potential $\mu$ up to the (tri)critical endpoint. The main modification in (\[EqBugaevI\])–(\[EqBugaevVIII\]) is an appearance of $\mu$-dependences of $p_v (T, \mu)$ and $T_{co} (\mu)$ [@Bugaev:Ref6n]. In the (tri)critical endpoint vicinity the behavior of $p_{tot}$ and $s_{tot}$ is defined by the $T$-dependence of the surface tension coefficient. We stress that there is nothing wrong or unphysical with the negative values of surface tension coefficient, since $ \sigma_{surf}\, 2 \pi R L$ in (\[EqBugaevI\]) is [the surface free energy]{} and, hence, as any free energy, it contains the energy part $E_{surf}$ and the entropy part $S_{surf}$ multiplied by temperature $T$, i.e. $F_{surf}= E_{surf} - T S_{surf} $ [@Bugaev:Ref15n; @Bugaev:Ref16n]. Therefore, at low temperatures the energy part dominates and the surface free energy is positive, whereas at high temperatures the number of configurations of a cylinder with large surface drastically increases and the surface free energy becomes negative since $S_{surf} > \frac{E_{surf}}{T}$. Moreover, the exactly solvable models with phase transition and cross-over [@Bugaev:Ref8n; @Bugaev:Ref9n; @Bugaev:Ref10n] have region of negative surface tension coefficient and they clearly show that the only reason why the 1-st order deconfinement phase transition degenerates into a cross-over at low baryonic densities is the negative values of $ \sigma_{surf}$ at this region and the above results independently prove this fact. We believe that the same is true for many ordinary liquids otherwise one has to search for an alternative explanation for the disappearance of the 1-st order liquid-gas phase transition at the supercritical temperatures. Of course, the experimental data in this region do not exists, but, nevertheless, there is indirect evidence for an existence of negative values of the surface tension coefficient at the supercritical temperatures. To demonstrate the validity of this statement we have to remind that the modern experimental data on the temperature dependence of the surface tension do not allow one to definitely conclude what is $T$-dependence at the vicinity of critical temperature $T_c$. In fact there are two alternative prescriptions [@Bugaev:Ref26n; @Bugaev:Ref26b] $$\begin{aligned} \label{EqBugaevIX} {\rm E\ddot{o}tv\ddot{o}s~~rule:} \quad \quad \frac{ \sigma_{surf}}{\rho_l^\frac{2}{3}} &=& a_E (T_c - T) \, , \\ {\rm Guggenheim~~rule:} \quad \quad \frac{ \sigma_{surf}}{\rho_l^\frac{2}{3}} &=& a_G (T_c - T)^n\, \quad {\rm with} \quad n \approx \frac{11}{9}\, , \label{EqBugaevX} $$ where $\rho_l$ is the temperature dependent particle density of the liquid phase. ![ The surface tension $\Gamma$ (in $N/m$) in terms of cluster surfaces (first and third rows) and the surface tension $\Gamma / \rho_l^{2/3}$ (in $(N/m)/(mole/l)^{2/3}$) in terms of cluster number (second and fourth rows)) as a function of $\varepsilon =(T_c-T)/T_c$ for: ÒquantumÓ fluids (hydrogen and helium), noble gases (krypton and xenon) and more complex fluids (methane and water). The thin solid lines show data points [@Bugaev:Ref33n] and the heavy dashed-dotted lines over the thin line show fits to $\Gamma = \Gamma_0\, \rho_l^{2/3} \varepsilon^{2 n} $ and $\Gamma / \rho_l^{2/3} = \Gamma_0\, \varepsilon$ according to Eqs. (\[EqBugaevX\]) and (\[EqBugaevIX\]), respectively. []{data-label="Bugaev:fig:3"}](Bugaev_Sigma_NIST.pdf "fig:"){width="100.00000%"}\ After the Guggenheim work [@Bugaev:Ref26n] the prescription (\[EqBugaevX\]) became a dominant one [@Bugaev:Ref27n]. Sometimes there appeared even confusions. Thus, in [@Bugaev:Ref28n] the authors determined the surface tension of water from the triple point to critical point and parametrized it by the polynomial of 9-th power $ \sigma_{surf} (T) = \sum\limits_{l=1}^9 a_l \, (T_c - T)^l$, but then the same authors refitted it to the prescription (\[EqBugaevX\]) [@Bugaev:Ref29n]. Here in Fig. \[Bugaev:fig:2\] which is taken from [@Bugaev:Ref27n] we show the temperature dependence of some paraffins. As one can see for n-Pentane and n-Heptane the data on temperature dependence of surface tension near the critical point, indeed, may show the nonlinear behavior similar to (\[EqBugaevX\]), but for the n-Hexane and n-Octane one can see the linear $T$-dependence of Eq. (\[EqBugaevIX\])! Therefore, in order to clarify this issue a few years ago a thorough analysis [@Bugaev:Ref32n] of the high quality NIST data [@Bugaev:Ref33n] was performed. Some of the results are shown in Fig. \[Bugaev:fig:3\] which is taken from Ref. [@Bugaev:Ref32n]. As one can see from Fig. \[Bugaev:fig:3\] for most of the analyzed liquids the linear prescription (\[EqBugaevIX\]) provides an essentially better fit with the only exceptions of xenon and methane. Therefore, our first conclusion is that for many liquids the rule (\[EqBugaevIX\]) better describes the data than the rule (\[EqBugaevX\]). The second conclusion one can draw from this discussion is that naive extrapolation of the linear $T$-dependence (\[EqBugaevIX\]) of the surface tension coefficient $\sigma_{surf}$ to supercritical temperatures $T > T_c$ would lead to the negative values of the surface tension coefficient. Of course, one may think that $\sigma_{surf} \equiv 0$ for $T > T_c$ like in the FDM [@Bugaev:Ref14n] and SMM [@Bugaev:Ref18n; @Bugaev:Ref19n; @Bugaev:Ref20n], but in this case one has to explain the reason why the $T$-derivative of $\sigma_{surf}$ has a discontinuity at $T = T_c$ while the pressure and all its first and second derivatives are continuous functions of it arguments in this region. The mysterious maximum of the lattice entropy and the gas of free tubes ======================================================================= The considered configuration of the unbroken confining tube is only one of many other configurations accounted by the lattice QCD thermodynamics. However, in order to explain a mysterious maximum of the lattice entropy (see Fig \[Bugaev:fig:1\]) it is sufficient to assume that the probability of the unbroken confining tube among other configurations measured by lattice QCD is $W (L) \sim [L \, g_0 \ln (L/L_0)]^{-1}$, i.e. in the limit $L \rightarrow \infty$ it is negligible for any $\nu \neq 0$. Then the contribution of the unbroken confining tube into the lattice free energy is small, since $W (L) F_{str} (L) \sim R^{-2} $ for $t \rightarrow +0$ and $R \rightarrow R_{lat}-0$ ($R_{lat}$ denotes the lattice size), but its contribution to the tube entropy $$\label{EqBugaevXI} W(L) S_{str} = - W \frac{d F_{str}}{d T} = W L \frac{\sigma_{str}^0 \nu}{T_{co}} t^{\nu-1} \rightarrow W L \frac{ \nu}{T_{co}} \left[ \frac{ \sigma_{str}^0}{[g_0 \ln (L/L_0)]^{1- \nu}} \right]^\frac{1}{\nu} R^\frac{2(1-\nu)}{\nu} \sim R^\frac{2(1-\nu)}{\nu} $$ is an increasing function of the tube radius $R$ for $\nu <1$. Clearly, if the available size of the lattice $R_{lat}$ would be infinite then the contribution of the unbroken tube would diverge, but for finite lattice size one should observe a fast increase at $T \rightarrow T_{co}$. The physical origin of a singular behavior of the tube entropy (\[EqBugaevXI\]) encoded in $\nu < 1$ is rooted in the formation of fractal surfaces of the confining tube in the cross-over temperature vicinity [@Bugaev:Ref6n]. This can be clearly seen from the power $\frac{2 (1-\nu)}{\nu}$ of $R$ on the right hand side of (\[EqBugaevXI\]) which is fractal for any $\nu \neq \frac{2}{2+n} $ where $n = 1, 2, 3, ...$ Moreover, the appearance of fractal structures at $T=T_{co}$ can be easily understood within our model, if one recalls that only at this temperature the fractal surfaces can emerge at almost no energy costs due to almost zero total pressure (\[EqBugaevVII\]). An explanation of the tube entropy decrease for $t < 0$ is similar [@Bugaev:Ref6n; @Bugaev:Ref7n]. It means that the fractal surfaces gradually disappear since for $T > T_{co}$ the tube gradually occupies the whole available lattice volume. Here we also would like to consider a toy model based on the total pressure (\[EqBugaevVI\]) of the confining tube, but for the non-static (or free) quark-antiquark pair. In this case the parameter $L$ should be considered as a free parameter which has to be determined from the maximum of the total pressure (\[EqBugaevVI\]). Finding from this condition the radius dependent separation distance $L_w (R)$ which corresponds to the most probable and the stable state of the free confining tube one has to substitute it into expression for the pressure (\[EqBugaevVI\]) and find the corresponding radius of the tube from the equation $p=p_{tot}(L_w (R), R, T)$ for the set of given external pressure $p$ and temperature $T$. Clearly the determination of an extremum of the total pressure (\[EqBugaevVI\]) with respect to $R$ with subsequent finding of the separation distance $L$ should give the same result, but the first way is technically easier. Instead of the quantitative analysis of the resulting pressure $p_{tot}(L_w (R), R, T)$ which requires the knowledge of values of all constants, i.e. $L_0$, $g_0$, $\tau$ e.t.c., we prefer to discuss some qualitative properties of the model and show that it has two phases. Also we have to stress that such a model cannot be applied to high pressures (or high densities) directly because in this case the pressure of the system should account for the short range repulsion between the tubes. Therefore in what follows it is assumed that the gas of tubes has some low particle density $\rho$ and, hence, one can consider this gas as an ideal gas with the pressure $p = T \rho$. To determine the density $\rho$ one has to maximize the pressure $p_{tot}$ first. From the vanishing derivative condition $$\begin{aligned} \label{EqBugaevXII} \frac{\delta ~p_{tot} (L, R, T)}{\delta~L} & = & - \frac{g_0 }{\pi R^4 L} - \frac{T \tau}{\pi R^2 L^2} \left[ \ln \left( \frac{\pi R^2 L}{V_0} \right) - 1 \right] + \frac{T \tau}{\pi R^2 L^2} = 0 \, $$ one can find the following equation for $L_w(R)$ $$\begin{aligned} \label{EqBugaevXIII} L_w & = & \frac{T \tau R^2}{g_0 } \left[2 - \ln \left( \frac{\pi R^2 L_w}{V_0} \right) \right] \, $$ and show that it corresponds to a maximum of pressure. From Eq. (\[EqBugaevXII\]) it is clearly seen that, in contrast to previous findings [@Bugaev:Ref6n; @Bugaev:Ref7n], the role of the Fisher topological term is a decisive one for an establishing Eq. (\[EqBugaevXIII\]). Consider a few limiting cases of Eq. (\[EqBugaevXIII\]). In the limit $T\rightarrow 0$ and finite $R$ one gets $$\label{EqBugaevXIV} L_w^0 ~\approx ~ - \frac{T \tau R^2}{g_0 } \ln \left( \frac{\pi R^4 \tau T}{V_0\, g_0 \, e^2} \right) \rightarrow~0 \,,$$ which can be interpreted as a confinement of color charges. The same solution is true for the case of $R \rightarrow 0$ and finite $T$. In the other extreme $T \rightarrow \infty$ (or $R \rightarrow \infty$) one obtains a different solution of Eq. (\[EqBugaevXIII\]) $$\label{EqBugaevXV} L_w^\infty~ \approx ~\frac{ V_0 e^2}{\pi R^2} \,,$$ which again shows that for large $R$ values the separation distance is small. This situation resembles what is observed in lattice QCD for the non-static color charges: the long tubes that connect such charges simply break up at some separation distances [@Bugaev:Ref1n]. From Eq. (\[EqBugaevXIII\]) one can show that the solution $L_w$ is a monotonically increasing function of $T$, while for $T \neq 0$ it always has a maximum as the function of $R$. Searching for the maximum of $L_w(R)$ (\[EqBugaevXIII\]) one can find the corresponding value of the radius $R_{max}$ and $L_w (R_{max})$ $$\begin{aligned} \label{EqBugaevXVI} R_{max} & = & \left[ \frac{g_0 \, V_0 \, e}{\pi \, \tau\,T} \right]^\frac{1}{4} \, ,\\ \label{EqBugaevXVII} L_w (R_{max}) & = & \left[ \frac{ V_0 \, e \, \tau \,T }{\pi \, g_0 } \right]^\frac{1}{2} \, , $$ which, evidently, obey the condition $\pi R_{max}^2 L_w (R_{max}) = V_0\, e $. The presence of the maximum of function $L_w (R)$ leads to an existence of two different radii for the same value of separation distance $L$, or in other words, there are two solutions of the equation $L = L_w(R)$. Clearly, the parameters of the maximum $R_{max}$ and $L_w (R_{max})$ given, respectively, by Eqs. (\[EqBugaevXVI\]) and (\[EqBugaevXVII\]), separate the regions of these solutions. Evidently, the latter correspond to two phases of the gas of tubes which have different tube radius for the same separation distance L. The analysis of these solutions shows that there are many possibilities which strongly depend on the values of the involved constants $L_0$, $g_0$, $\tau$ and $V_0$, whereas for low but non-vanishing temperatures one can show that the higher pressure corresponds to the phase of the tubes with smaller radius. This is clear because in case of low temperatures the leading contribution to the total pressure (\[EqBugaevVI\]) is given by the surface tension term $\frac{\sigma_{surf}}{R}$. Conclusions =========== In this work we discuss the most general relation between the tension of the color tube connecting the static quark-antiquark pair and the surface tension of the corresponding cylindrical bag. Such a relation allows us to determine the surface tension of the QGP bags at zero temperature and, under the plausible assumptions that are typical for ordinary liquids, to estimate the temperature of vanishing surface tension coefficient of QGP bags at zero baryonic charge density as $T_\sigma = 152.9 \pm 4.5 $ MeV. Using the Fisher conjecture [@Bugaev:Ref14n] and the exact results found for the temperature dependence of surface tension coefficient from the partition of surface deformations [@Bugaev:Ref15n; @Bugaev:Ref16n; @Bugaev:Ref7n], we conclude that the same temperature range corresponds to the value of QCD (tri)critical endpoint temperature, i.e. $T_{cep}= T_\sigma = 152.9 \pm 4.5 $ MeV. Then requiring the positive values for the confining tube entropy density we demonstrate that at the cross-over region the surface tension coefficient of the QGP bags is unavoidably negative. 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--- abstract: 'Fully Convolutional Networks have been achieving remarkable results in image semantic segmentation, while being efficient. Such efficiency results from the capability of segmenting several voxels in a single forward pass. So, there is a direct spatial correspondence between a unit in a feature map and the voxel in the same location. In a convolutional layer, the kernel spans over all channels and extracts information from them. We observe that linear recombination of feature maps by increasing the number of channels followed by compression may enhance their discriminative power. Moreover, not all feature maps have the same relevance for the classes being predicted. In order to learn the inter-channel relationships and recalibrate the channels to suppress the less relevant ones, Squeeze and Excitation blocks were proposed in the context of image classification with Convolutional Neural Networks. However, this is not well adapted for segmentation with Fully Convolutional Networks since they segment several objects simultaneously, hence a feature map may contain relevant information only in some locations. In this paper, we propose recombination of features and a spatially adaptive recalibration block that is adapted for semantic segmentation with Fully Convolutional Networks — the SegSE block. Feature maps are recalibrated by considering the cross-channel information together with spatial relevance. Experimental results indicate that Recombination and Recalibration improve the results of a competitive baseline, and generalize across three different problems: brain tumor segmentation, stroke penumbra estimation, and ischemic stroke lesion outcome prediction. The obtained results are competitive or outperform the state of the art in the three applications.' author: - 'Sérgio Pereira, Adriano Pinto, Joana Amorim, Alexandrine Ribeiro, Victor Alves, and Carlos A. Silva[^1] [^2][^3]' bibliography: - 'references.bib' title: Adaptive feature recombination and recalibration for semantic segmentation with Fully Convolutional Networks --- Segmentation, Deep Learning, Fully Convolutional Network, Recalibration, Recombination, Adaptive. Introduction ============ Medical image segmentation is often part of medical image analysis pipelines, being crucial for diagnostic, treatment planning, and follow-up [@litjens2017survey; @menze2015multimodal; @maier2017isles]. However, when manually done, segmentation is time demanding and prone to inter- and intra-rater variability [@menze2015multimodal; @maier2017isles]. Automatic segmentation can mitigate those issues, thus improving the efficiency and quality of medical care, allowing the experts to focus on other tasks [@magoulas1999machine]. In recent years, several approaches have been based on machine learning methods. Among them, many rely on Deep Learning using Convolutional Neural Networks (CNNs) [@litjens2017survey]. Convolutional Neural Networks are composed by convolutional layers that learn features directly from the data. Each of these layers consists of a set of learnable kernels that are convolved over the input data to generate a stack of feature maps. These models gathered a lot of interest after the results achieved by Krizhevsky et al. [@krizhevsky2012imagenet] in object recognition. Since then, CNN-based approaches have reached state-of-the-art results across many applications in medical image analysis [@pereira2016brain; @ronneberger2015u; @kamnitsas2017efficient]. In recent years, active research is being conducted regarding architectural design in deep neural networks. Several studies focused on enabling training of deeper networks, such as VGGNet [@simonyan2014very] that uses small kernels to reduce the number of parameters, or residual learning, elegantly implemented through short skip connections [@7780459]. However, other dimensions of CNNs, besides depth, are being studied, such as the cardinality [@xie2017aggregated], or the cross-channels relationships in a stack of feature maps [@hu2017squeeze]. The relationships among the channels may be used to enhance the representational power of the features. For instance, convolutional layers with $1 \times 1$ kernels were used as parametric pooling [@lin2013network], or as bottlenecks in ResNets [@7780459]. Interestingly, in the latter, these layers were also used to increase the number of feature maps after the previous bottleneck. Since layers with $1 \times 1$ kernels do not take a neighborhood into account, they work solely by recombining the channels of the feature maps. Differently, Hu et al. [@hu2017squeeze] explored the relationships among channels and proposed recalibration of feature maps with Squeeze-and-Excitation (SE) blocks. In this case, the end goal is to adaptively suppress the less relevant feature maps. Many advances in CNN‘s design for image applications are often proposed in the context of object recognition [@litjens2017survey]. In this problem, a single class label is inferred for the whole image. Similar approaches can be used for segmentation. For instance, Pereira et al. [@pereira2016brain] used a conventional CNN-based method that classifies the central voxel of a patch. So, in this case, it is straightforward to introduce general CNN blocks. However, recently, the more efficient Fully Convolutional Networks (FCNs) [@ronneberger2015u; @long2015fully] are being preferred for image segmentation over conventional CNNs. In these architectures, the fully-connected layers of CNNs are replaced by convolutional layers. In this way, it is possible to segment a full patch of voxels in just one forward pass. While in conventional CNNs the whole feature maps characterize the class of just one voxel, in FCNs there is a direct correspondence between the features in a given location of the feature maps and the voxel being classified in the same spatial location. Therefore, some architectural designs may not be well suited for FCNs, such as the SE block. The reason is that this block weights whole feature maps, while we may be interested in suppressing regions of the feature maps that are irrelevant when predicting voxels in the same spatial location. Besides learning channel-wise relationships to enhance the discriminative power of the feature maps, the SE block [@hu2017squeeze] may be interpreted as a channel-wise attention mechanism [@oktay2018attention]. Different kinds of attention have been recently proposed in the context of medical image segmentation using FCNs. Qin et al. [@qin2018autofocus] proposed a scale attention scheme that adaptively chooses the receptive field. This is achieved by processing several parallel branches with convolutional layers with different dilation rates. However, this increases the memory and computational requirements. Oktay et al. [@oktay2018attention] utilized attention as a soft mask to enhance the region of interest. However, this approach scales the same regions across all feature maps. Roy et al. [@roy] learn attention at the channel and spatial levels, but a single spatial attention map is inferred for all feature maps. Motivation and Contributions ---------------------------- The cross-channel relationships among feature maps were shown to encode relevant information [@hu2017squeeze]. Therefore, in this work, we study the recombination and recalibration of feature maps. Regarding recombination, convolutional layers with $1 \times 1$ kernels were previously used to decrease the number of feature maps [@lin2013network; @7780459], but also to increase their number afterwards [@7780459] to boost their representational power. We propose recombination, where we linearly expand the number of feature maps before compressing again, allowing the network to learn how to mix the information to generate more discriminative features. In the case of recalibration, the SE block has the desirable property of learning cross-channel relationships and suppressing the less relevant feature maps. However, it recalibrates whole feature maps, which makes it not well adapted for semantic segmentation with FCNs where it may be better to have spatially adaptive recalibration. So, in this work we also propose a recalibration block (the SegSE block) that is able to spatially recalibrate feature maps, while still considering the cross-channel relationships. This can be alternatively interpreted as a spatially adapted attention mechanism for each feature map; unlike [@oktay2018attention], where a single attention map is generated. Related to our work is the block proposed by Roy et al. [@roy]. But, channel and spatial recalibration are considered separately, while we learn them jointly. Also, a single spatial recalibration map is estimated for all channels without taking context into account, while we infer channel-specific maps. An early version of this work was presented at a conference [@segse]. This paper extends the previous work with further validation across several datasets and applications. Additionally, we provide more detailed descriptions on the method and discussion, and insights on the working process of the proposed approach. We observe that it indeed learns how to suppress or enhance features accordingly to the structures being segmented. The contributions in this paper can be summarized as follows. 1) We propose feature recombination by means of linear expansion and compression of the number of feature maps. 2) We propose a novel feature recalibration approach for FCNs – the SegSE block. This block may be interpreted as an attention mechanism for each channel of the stack of feature maps. 3) We validate our methods in several publicly available datasets for brain tumor segmentation, stroke penumbra estimation, and ischemic stroke lesion outcome prediction. Finally, 4) we inspect the attention maps to verify that they enhance specific features, while the original SE block suppresses whole feature maps, including potentially important local features. The remaining sections of this paper are organized as follows. The proposed methods are presented in Section \[sec:methods\]. The experimental setup is described in Section \[sec:exp\_set\]. Then, in Section \[sec:results\], we present the results and discussion. Finally, the main conclusions are presented in Section \[sec:conclusion\]. Methods {#sec:methods} ======= In this work, we approach semantic segmentation using 2D FCNs operating over patches. The architecture of our network is inspired in U-Net [@ronneberger2015u], as depicted in Fig. \[subfig:multi\_seg\] (note that rectangles represent a layer or set of layers). This type of FCN is well-established in medical image segmentation tasks [@litjens2017survey]. U-Net belongs to a broader class of architectures called encoder-decoder. As input we have an image patch, which may have several stacked channels. The encoder path encompasses different levels of abstraction, which are responsible for learning higher order features. Features computed by higher (deeper) convolutional layers are more abstract. However, these features may lack the fine details that are important for segmentation, which are better captured by the lower layers. Since the feature maps are down-sampled, we need to map the lower resolution feature maps back to the input patch resolution. This is gradually done by up-sampling in the decoder path. As we up-sample feature maps, we sum them with the feature maps of equivalent size of lower layers of the encoder path, through long skip connections. Further convolutional layers fuse the lower and higher level features. Finally, the last layer is a convolutional layer with $1\times 1$ kernels followed by the softmax activation function that infers the probability of each voxel belonging to each class [@ronneberger2015u]. We may think of a layer in a FCN as implementing a transformation function ($\mathrm{F^{Tr}}$) that maps the input feature maps $\bm{X}$ into some output feature maps $\bm{U}$. Therefore, it can be defined as $\mathrm{F^{Tr}}: \bm{X} \rightarrow \bm{U^{Tr}}$, with $\bm{X} \in \mathbb{R}^{H'\times W'\times C'}$ and $\bm{U}^{Tr} \in \mathbb{R}^{H\times W\times C}$, where $H$ and $W$ represent the height and width of the transformed feature maps, respectively. $H'$ and $W'$ represent the same measures but in relation to the input feature maps $\bm{X}$. $C$ and $C'$ are the number of feature maps, such that $\bm{X}=\big[\bm{x}_1, \bm{x}_2, \cdots, \bm{x}_{C'}\big]$, and $\bm{U^{Tr}}=\big[\bm{u}_1, \bm{u}_2, \cdots, \bm{u}_{C}\big]$. So, a convolutional layer defines a function $\mathrm{F^{conv}}$, such that $U^{conv}=\mathrm{F^{conv}}\left( \ \cdot \ ; \ k, \ d, \ n \right)$, where $k$, $d$, and $n$ represent the kernel size, the dilation rate, and the number of kernels, respectively. In this case, each output channel $c$ is computed as $u_c = \bm{v}^c * \bm{X} = \sum_{l=1}^{C'} \bm{v}_l^c * \bm{X}_l$, where $\bm{V} = \big[\bm{v}^1, \bm{v}^2, \cdots, \bm{v}^C\big]$ is the set of learnable kernels (the bias term is not represented for the sake of simplicity), and $$ is the convolution operation symbol. Recombination {#sec:recomb} ------------- In recombination we are interested in increasing the representational power of the features by mixing them linearly. To this end, we employ convolutional layers with $1\times 1$ kernels. First, we expand the number of features maps ($\mathrm{F^{exp}}$), before compressing again to the original number ($\mathrm{F^{comp}}$). So, recombination is defined as $\mathrm{F^{recomb}}: \bm{X} \rightarrow \bm{U}^{recomb}$, with $\bm{U}^{recomb} \in \mathbb{R}^{H' \times W' \times C'}$. This translates into the following operation: $$\begin{multlined} \mathrm{F^{recomb}}\left( \bm{X} \right) = \mathrm{F^{comp}}\left( \mathrm{F^{exp}} \left( \bm{X}; \ mC' \right); \ C' \right) = \\ \mathrm{F^{conv}}\left( \mathrm{F^{conv}} \left( \bm{X}; \ 1, \ 1, \ mC' \right); \ 1, \ 1, \ C' \right), \end{multlined}$$ where $m$ is the expansion factor. These operations are depicted in Fig. \[subfig:recombination\]. Recalibration ------------- Recalibration consists in learning the relationship among the feature maps of a layer, and suppressing the less relevant features. Originally, Hu et al. [@hu2017squeeze] proposed to recalibrate whole feature maps, which we refer as channel SE blocks (c.f. Fig. \[subfig:se\_diagram\]). However, in this work, we propose spatially adaptive recalibration (c.f. Fig. \[subfig:segse\_diagram\]). ### Channels Squeeze-and-Excitation Let us consider a function $\mathrm{F^{rec}}: \bm{X} \rightarrow \bm{U}^{rec}$, with $\bm{U}^{rec} \in \mathbb{R}^{H \times W \times C}$, that transforms the input feature maps into their recalibrated form. This is achieved by two sequential operations: squeeze and excitation [@hu2017squeeze]. The spatial squeeze operation consists in summarizing each channel into a scalar descriptor, such that we obtain a descriptor vector $\bm{z} = \big[ z_1, z_2, \cdots , z_{C'} \big]$, with $\bm{z} \in \mathbb{R}^{1 \times 1 \times C'}$. This operation can be performed by global average pooling, where a given $c$ channel $\bm{x}_c$ is squeezed as, $$z_c = F^{sq}\left( \bm{x}_c \right) = \frac{1}{H' \times W'} \sum_{i=1}^{H'} \sum_{j=1}^{W'} \bm{x}_c \left( i, j \right).$$ In the excitation operation, the network learns the dependencies among the channels in order to adaptively estimate the excitation or scaling factors $\bm{s}$. This can be understood as a gating mechanism that provides channel-wise attention. To this end, two fully-connected layers[^4] are employed, where the first one acts as bottleneck and the second one restores the dimension of the vector. These operations are described as, $$\bm{s} = F^{exc} \left( \bm{z} \right) = \sigma\left( \bm{W}_2 \delta \left( \bm{W}_1 \bm{z} \right) \right),$$ with $\bm{W}_1 \in \mathbb{R}^{C' \times \frac{C'}{r}}$, and $\bm{W}_2 \in \mathbb{R}^{\frac{C'}{r} \times C'}$. $\delta$ denotes the ReLU activation function, $\sigma$ denotes the sigmoid function, and $r$ denotes the reduction factor. Therefore, we can define the function $\mathrm{F^{SE}} \left( \bm{X} \right) = \mathrm{F^{exc}} \left( \mathrm{F^{sq}} \left( \bm{X} \right) \right)$. These operations are visually described in the SE block of Fig. \[subfig:recalibration\]. Finally, the recalibration, or scaling, of the feature maps is done by simple multiplication of the original channels by the corresponding recalibration factor. So, it may be defined as $\mathrm{F^{rec}} \left( \bm{X} \right) = \mathrm{F^{scale}} \left( \bm{X}, \ \mathrm{F^{SE}} \left( \bm{X} \right) \right)$. In this way, the recalibration of a feature map $c$ is defined as, $$\bm{u}_c^{rec} = \bm{x}_c \cdot s_c.$$ A visual depiction of channel SE can be found in Fig. \[subfig:se\_diagram\]. ### Spatially adaptive Squeeze-and-Excitation In semantic segmentation with FCNs there is a spatial correspondence between the units in the features maps and the pixels/voxels being segmented in the same locations. Hence, a given feature map may have relevant features for some voxels, whereas in other locations it may not be so important. Therefore, due to its global squeeze operation, channel SE blocks may end up suppressing whole feature maps that may contain important regions. For this reason, we argue that spatially adaptive SE blocks, as depicted in Fig. \[subfig:segse\_diagram\], are more appropriate for semantic segmentation, at least with FCNs. In this subsection, we start by presenting our simultaneous spatial and channel Squeeze-and-Excitation proposal. Next, we describe two variants that were studied in this work. #### Spatially adaptive Squeeze-and-Excitation for segmentation — SegSE The proposed block can be observed in Fig \[subfig:recalibration\] — SegSE block. In order to preserve the spatial structure and correspondence of the feature maps, we first replace the squeeze operation through global average pooling in the SE block by a convolutional layer with $3 \times 3$ kernels. The motivation for the squeeze operation is to aggregate contextual information, which in this case is captured by the kernel operating over neighboring voxels. Hence, we employ convolutional layers with dilated kernels [@dilated] to capture a larger context than simple $3 \times 3$ kernels, but without increasing the number of parameters over those kernels. In this way, instead of having a descriptor vector $\bm{z}$, as in the case of the channels SE block, we obtain feature maps as, $$\bm{Z}^{segSE} = \gamma \left( \mathrm{F^{conv}} \left( \bm{X}; \ k^{segSE}, \ d, \ n^{segSE} \right) \right),$$ where $n^{segSE} = \frac{C'}{r}$, $k^{segSE} = 3$, and $\gamma$ represents batch normalization followed by the ReLU activation function. The dilation factor $d$ may be chosen according to the scale of the layer it is operating. Layers that already take a large field of view and context into account may require less dilation. In general, the field of view increases, for instance, with the number of convolutional, or pooling layers. After these layers, each unit of the feature maps represents a larger region of the input space. Having $\bm{Z}^{segSE}$, the feature maps with recalibration factors are obtained by a convolutional layer with $1 \times 1$ kernels, followed by the sigmoid activation function, as $$\bm{S} = \sigma \left( \mathrm{F^{conv}} \left( \bm{Z}^{segSE}; \ k, \ d, \ n \right) \right),$$ with $k=1$, $d=1$, and $n=C'$. Therefore, we combine the squeeze and excitation procedures, since the convolutional layer with dilation also includes the bottleneck by decreasing the number of feature maps. Finally, the recalibrated feature maps are obtained by element-wise multiplication ($\odot$) of the input with $\bm{S}$. So, having a feature map $c$, it is recalibrated as $$\bm{u}_c^{segSE} = \bm{x}_c \odot \bm{s}_c, \label{eq:recal}$$ In this way, our SegSE block may be described by the function $\mathrm{F^{segSE}} : \bm{X} \rightarrow \bm{U}^{segSE}$. #### Variant 1 - No context A simpler approach consists in removing the global average pooling in the SE block, and replacing the fully-connected layers by convolutional layers with $1 \times 1$ kernels (Fig. \[subfig:recalibration\] — variant 1). Taking the previous description into account, this would translate into setting $k^{segSE} = 1$ and $d=1$. The issue with this approach is that contextual information is not considered, which we previously obtained through $3 \times 3$ convolutional kernels with dilation. #### Variant 2 - Pooling-based context Contextual information may be obtained by average pooling. Contrasting with global pooling, this acts over a kernel instead of the complete channel of the feature maps. This can be implemented through a pooling layer defined as $\mathrm{F^{avg. pool.}} : \bm{X} \rightarrow \bm{U}^{avg. pool.}$, such that $\bm{U}^{avg. pool.} = \mathrm{F^{avg. pool.}} \left( \bm{X}; \ p \right)$, where $p$ is the size of the pooling kernel and stride, which we assume to be an even number. Therefore, we obtain $\bm{U}^{avg. pool.} \in \mathbb{R}^{\frac{H'}{p} \times \frac{W'}{p} \times C'}$. Then, a convolutional layer with $1 \times 1$ kernels combines the feature maps and learns their relationship. Average pooling results in a down-sampled feature map. Hence, it is necessary to restore it to the original shape. We accomplish this through transposed convolutional layers [@dumoulin2016guide]. We employ these layers to double the size of the feature maps each time. So, if $p > 2$, it is necessary to employ more than one layer of transposed convolution. Therefore, the result of this block is $U^{convTblock} = \mathrm{F^{convTblock}} \left( \bm{X}; \ k_T = 3, \ s_T = 2, n_T=\frac{C'}{r}, \ l_T = \frac{p}{2} \right)$, where $k_T$ is the kernel size, $s_T$ is the stride, $n_T$ is the number of convolutional kernels, and $l_T$ is the number of transposed convolution layers. Each $l_T$ operation consists of $ \theta \left( \mathrm{F^{convT}} \left( \cdot ; k_T, \ s_T, \ n_T \right) \right) $, with $\theta$ being batch normalization. Having described these operations, this variant 2* is defined by the following procedures (and in Fig. \[subfig:recalibration\] — variant 2), $$\bm{Z}^{var 2} = \gamma \left( \mathrm{F^{conv}} \left( \mathrm{F^{avg. pool}} \left( \bm{X}; p \right); 1, \ 1, \frac{C'}{r} \right) \right),$$ $$\begin{multlined} \bm{S}^{var 2} = \sigma \left( \mathrm{F^{conv}} \left( \bm{Z}^{var 2}; \ k_T, \ s_T, \ n_T, \ l_T \right); 1 , 1 , C' \right), \end{multlined}$$ where, $k_T = 3$, $s_T = 2$, $n_T=\frac{C'}{r}$, and $l_T = \frac{p}{2}$. Finally, recalibration is obtained in a similar way as in equation \[eq:recal\]. Recombination and Recalibration ------------------------------- We combine both recombination and recalibration (RR) into the same block, as illustrated in Fig. \[subfig:recalibration\]. As in recombination, we first expand the number of feature maps, then we recalibrate them, and, finally, we compress the number of feature maps into the original number. This can be defined as, $$\bm{U}^{RR} = \mathrm{F^{RR}} \left( \bm{X} \right) = \mathrm{F^{comp}} \left( \mathrm{F^{rec}} \left( \mathrm{F^{exp}} \left( \bm{X}; \ mC' \right); \ k, \ d, \ n \right); \ C' \right)$$ Experimental Setup {#sec:exp_set} ================== We evaluate the proposed blocks in three medical applications: brain tumor segmentation, penumbra estimation in acute ischemic stroke, and ischemic stroke lesion outcome prediction. Although all the applications use Magnetic Resonance Imaging (MRI), they are quite diverse in terms of acquisitions. Brain tumor segmentation only takes structural MRI sequences into account. Penumbra estimation has a mixture of low resolution structural, diffusion, and perfusion MRI acquisitions. Finally, in ischemic stroke lesion outcome prediction we deal with perfusion and diffusion acquisitions only. Besides the differences in imaging data, these applications are distinct, too. Brain tumor represents a multi-class classification problem, while the others are binary problems. Additionally, in ischemic stroke lesion outcome prediction we are interested in predicting the status of a stroke lesion three months after the image acquisition and intervention. This differs from a pure segmentation problem, where we segment the objects that are visible at the moment. The proposed blocks were studied in the brain tumor segmentation task. Then, the best hyperparameters and network architecture were evaluated in the other applications for further validation. Some hyperparameters were kept constant across experiments. The expansion factor $m$ for recombination was set to 4. In the case of recalibration, the dilation factors $d$ of our SegSE blocks were defined according to the scale of the feature maps (c.f. Fig. \[subfig:multi\_seg\]) as $ \{ RR^1 , RR^2 , RR^3 \} = \{ 3, 2, 1 \}$. Similarly, the kernel sizes and strides $p$ of the average pooling of Variant 2 were defined as $ \{ RR^1 , RR^2 , RR^3 \} = \{ 4, 2, 2 \}$. Finally, we set the reduction factor $r$ to $10$. These hyperparameters were tuned using the validation set. During training, the cross entropy loss function was minimized using the Adam optimizer [@kingma2014adam] with learning rate of . Furthermore, we employed spatial dropout with probability of 0.05, and weight decay of . Artificial data augmentation consisted of random sagittal flipping and random rotations of $\{0^\circ, 90^\circ, 180^\circ, 270^\circ \}$. Patches were extracted from the axial plane of the MRI images. The FCNs were implemented using Keras with Theano backend. The proposed blocks are available in an online repository[^5]. Brain tumor segmentation {#sec:brain_tumor_train} ------------------------ For the task of brain tumor segmentation we used the Brain Tumor Segmentation Challenge (BRATS) 2017 and 2013 datasets [@menze2015multimodal; @bakas2017advancing]. BRATS 2017 has two publicly available sets: Training (285 subjects) and Leaderboard (46 subjects). BRATS 2013 encompasses three sets: Training (30 subjects), Leaderboard (25 subjects), and Challenge (10 subjects). For each subject, there are four MRI sequences available: T1-weighted (T1), post-contrast T1-weighted (T1c), T2-weighted (T2), and Fluid-Attenuated Inversion Recovery (FLAIR). All images are already interpolated to $1 \ mm$ isotropic resolution, skull stripped, and aligned. We further pre-processed them by correcting the bias field [@tustison2010n4itk], and standardizing the intensity histogram of each MRI sequence [@nyul2000new]. Only the Training sets contain manual segmentations publicly available. In BRATS 2017 it distinguishes three tumor regions: edema, necrotic/non-enhancing tumor core, and enhancing tumor. In BRATS 2013 the manual segmentations have necrosis and non-enhancing tumor separately, although we fuse these labels to be similar to BRATS 2017. Hence, the last layer of the FCN has 4 feature maps, three for tumor classes and one for normal tissue. Evaluation is performed for the whole tumor (all regions combined), tumor core (all, excluding edema), and enhancing tumor. Since annotations are not publicly available for 2017 Leaderboard, 2013 Leaderboard, and 2013 Challenge, metrics are computed by the CBICA IPP[^6] and SMIR[^7] online platforms. The development of the RR block was conducted in the larger BRATS 2017 Training set, which was randomly divided into training (60%), validation (20%), and test (20%); the identification of the subjects in each set is also available in the online repository. All the hyperparameters were found using the validation set, before evaluation in the test set. Afterwards, the test set was added to the training and the FCN was fine-tuned before evaluation in the Leaderboard set. However, networks tested in BRATS 2013 were trained in the 2013 Training set. Given the problem of data imbalance between the tumor tissues and normal tissue, and following recent developments in brain tumor segmentation [@pereira2017hierarchical; @pinto2018hierarchical; @wang2017automatic], we employ a hierarchical FCN-based brain tumor segmentation approach, as described in [@segse]. First, we detect the whole tumor as a binary segmentation problem. Then, we use this information to identify the multi-class segmentation object representing the tumor tissues in the region of interest. The binary brain tumor segmentation FCN is 3D and possesses a large field of view. These two properties are important to reduce the number of false positive detections. The architecture is depicted in Fig. \[fig:grade\], and the pipeline for brain tumor segmentation can be found in the Supplementary Material. Note that the 3D binary whole tumor detection network does not contain the RR block and is kept constant across all experiments. Therefore, the variants of the RR block in the more challenging multi-class FCN are the only source of variation in results, allowing us to better compare them and evaluate the benefits of our proposal. ![image](FIG3.pdf){width="90.00000%"} Stroke penumbra estimation -------------------------- In the case of penumbra estimation in acute ischemic stroke, we used the Stroke Perfusion Estimation (SPES) dataset of the MICCAI Ischemic Stroke Lesion Segmentation (ISLES) Challenge [@maier2017isles]. This is a binary classification problem, so, the last layer of the FCN outputs two feature maps. The dataset contains two sets: Training and Challenge. Training has 30 subjects with publicly available annotations. The Challenge set does not have publicly available annotations of its 20 patients, so, evaluation is performed by an online platform[^8]. Each patient contains 7 sequences: T1c, T2, Diffusion Weighted Imaging (DWI), Cerebral Blood Flow (CBF), Cerebral Blood Volume (CBV), Time-to-Peak (TTP), and Time-to-Max (Tmax). The images are already registered to the T1c, interpolated to a resolution of $2 \times 2 \times 2$ mm resolution, and skull stripped. Further pre-processing included the bias field correction [@tustison2010n4itk] and histogram standardization [@nyul2000new] of the structural sequences, the clipping of the Tmax values over 60 ($Tmax > 6$ s), and linear scaling of all sequences to the $\big[0, \ 255 \big]$ intensity range. Ischemic stroke lesion outcome prediction ----------------------------------------- In this experiment, we used the dataset from ISLES 2017 Challenge [@winzeck2018isles]. Similarly to penumbra estimation, this is a binary problem. There are two sets available: Training (43 patients), and Challenge (32 patients). While the former has publicly available annotations, in the latter the evaluation is conducted by an online platform[^9]. For each patient, there are available 5 perfusion maps (CBV, CBF, Mean Transit Time (MTT), TTP, and Tmax), and 1 diffusion map (ADC). The images were already aligned and skull stripped. Further pre-processing was similar to [@pinto2018enhancing], and consisted in resizing the images to $256 \times 256 \times 32$, clipping of the Tmax values to $[0, 20s]$ and the ADC values to $[0, 2600] \times 10^{-6} \ mm^2/s$, and linear scaling to $[0, 255]$. Finally, and following [@pinto2018enhancing], we employed the Dice loss function during training. Evaluation metrics ------------------ For quantitative evaluation we follow the metrics used in each challenge associated with each dataset. Therefore, for BRATS 2017 Leaderboard we use the Dice Coefficient (DC) and the 95^th^ percentile of the Hausdorff Distance (HD$_{95}$). However, BRATS 2013 employs the DC together with the Sensitivity and Positive Predictive Value (PPV) metrics. In the case of SPES, we report DC and Average Symmetric Surface Distance (ASSD). Regarding ISLES 2017, results are reported in terms of DC, PPV, and Sensitivity. The DC is a measure of overlap, but it is sensitive to the size of the lesions, and does not provide information regarding over- and under-segmentation. Nonetheless, such behavior can be inferred from Sensitivity and PPV. The distance metrics provide insights regarding the correctness of the contour of the segmentations [@menze2015multimodal; @maier2017isles; @winzeck2018isles]. Results and Discussion {#sec:results} ====================== In this section we evaluate the proposed recombination and recalibration blocks, and we observe experimentally the improvements obtained by the SegSE block. Then, we evaluate the baseline model and the best model in the independent sets of BRATS, SPES, and ISLES 2017. Finally, we compare with the state of the art. Evaluation of Recombination and Recalibration --------------------------------------------- We evaluate the effect of the variants of recombination and recalibration of feature maps in the brain tumor segmentation task. To that end, we use the controlled test set that consists of 20% of the BRATS 2017 Training set. Table \[tab:res\_test\] shows the quantitative results, while qualitative results can be found in Fig. \[fig:seg\] as segmentation examples. So, we start from a competitive baseline (as will be shown in Sections \[sec:brats\], \[sec:spes\], and \[sec:isles\]) consisting of a FCN similar to the one depicted in Fig. \[subfig:multi\_seg\], but where the RR block is absent. Then, we incrementally evaluate the proposed blocks. These results can be found in Table \[tab:res\_test\]. It is possible to observe that the addition of recombination (Subsection \[sec:recomb\]) through linear expansion followed by compression, leads to better DC and Sensitivity in all the tumor regions. This is especially expressive in the core region, where it achieves the highest DC. However, the HD$_{95}$ increased, which may be due to over-segmentation, since the sensitivity scores are higher than the ones obtained by the baseline. \[tab:res\_test\] ![Segmentation examples obtained by the baseline architecture and each of the evaluated blocks. The meaning of the colors in the segmentation is: blue — non-enhancing tumor core, red — enhancing tumor, and green — edema. The subject can be found in BRATS 2017 with ID Brats17\_TCIA\_430\_1.[]{data-label="fig:seg"}](FIG4.pdf){width="1.0\columnwidth"} In Table \[tab:res\_test\] we can also observe the results obtained by recalibration of the feature maps with recombination. Considering DC, the original SE block [@hu2017squeeze] yields worse scores in all tumor classes when compared with the recombination alone. Indeed, it manages to improve over the Baseline only in the tumor core region. The enhancing region is the region that suffers the most, since its DC deteriorates to values even lower than the baseline, which is due to a large decrease in sensitivity. Hence, it may be in accordance with the intuition that the SE block may end up suppressing features that are important for thinner and smaller structures, as observed in Fig. \[fig:recal\_senet\]. This block behaves in this way because it squeezes whole feature maps through their average and recalibrates them as a whole. Since the sensitivity of the enhancing region decreased, we may conclude that it is under-segmenting this region. Moreover, when we observe the results obtained by the different proposed attention mechanisms, DC and Sensitivity of the enhancing region are always better than with the SE block. Note that all the other settings recalibrate the feature maps in a spatially adaptive way. So, we conclude that the SE block, acting as whole feature map recalibration is not adapted for segmentation with FCNs. ![Recalibration of a potentially relevant feature map for a small tumor core that is suppressed by the SE block [@hu2017squeeze]. From left to right: the feature map before and after recalibration, and the manual segmentation.[]{data-label="fig:recal_senet"}](FIG5.pdf){width="1.0\columnwidth"} Considering the RR block with spatially adaptive recalibration, we may conclude that variants 1 and 2 also achieve worse DC when comparing with the baseline with recombination (Baseline + Recomb.), although in general they perform better in terms of HD$_{95}$. Still, they recover some enhancing tumor that the original SE (Baseline + RR SE) is unable to detect. However, the proposed RR block with SegSE achieves the best results. The DC of the tumor core and the whole tumor are similar to the ones obtained by recombination alone. Nevertheless, we observe an improvement in the enhancing region that achieved the highest score, being the tumor region with the finest details. This was a result of balanced PPV and sensitivity scores that denotes a good quality segmentation, since the HD$_{95}$ was simultaneously the lowest among all the evaluated settings. Additionally, it achieved the best HD$_{95}$ for the whole tumor, and the second for the core region. Qualitatively examining Fig. \[fig:seg\], the proposed RR block with SegSE appears to result in better segmentations. The reasons why RR with SegSE performs better than RR with variants 1 and 2 may be because variant 1 does not consider any context, while variant 2 suffers from the checkerboard effect introduced by pooling and up-sampling. In Supplementary Materials it can be found results obtained with 3D versions of the baseline networks and the proposed RR with SegSE. We observe that the proposed blocks also improve the performance in a 3D setting. A drawback of the SegSE block is that it increases the number of parameters. Therefore, to evaluate if its performance is due to the extra capacity, we proportionally increased the width of the baseline, such that the number of parameters becomes similar to the baseline + RR SegSE network. The results obtained with this larger network can be found in Table \[tab:res\_test\] as Wide Baseline. It is possible to observe from DC and HD$_{95}$ that it generally performed worse than the proposed RR SegSE block. Indeed, the Wide Baseline was able to achieve better scores only in the whole tumor and tumor core classes in the PPV metric. So, the improvements of the proposed block are due to learning better features, and not directly to the higher number of parameters. To further inspect the effect of the spatially adaptive recalibration procedure with the SegSE block, we can observe a feature map in Fig. \[fig:recal\]. In the left side it is represented a feature map before recalibration, where we can observe a high response for the tumor core, but also substantial response in the surrounding structures. After recalibration, however, we note how the feature map enhances the tumor core, while suppressing most of the surroundings. Hence, we may conclude that the SegSE block is able to selectively and adaptively recalibrate regions of interest in the feature maps, thus acting as an intra-channel attention mechanism. ![image](FIG6.pdf){width="85.00000%"} Brain tumor segmentation {#sec:brats} ------------------------ Table \[tab:res\_leaderboard\] presents the results obtained in BRATS 2017 Leaderboard with the baseline FCN, the RR with SE block, and the RR with SegSE block[^10]. It is possible to observe that the FCN with the proposed RR SegSE block outperforms the baseline in both DC and HD$_{95}$ in every tumor regions. This is especially noticeable in the tumor core where the DC improves from 0.758 to 0.798, and the HD$_{95}$ decreases from 11.1 mm to 8.947 mm. Additionally, with the RR SegSE block the FCN was able to segment the enhancing tumor better, which is a detailed and difficult region [@menze2015multimodal]. This can also be observed when comparing with the RR SE block, indicating that the proposed SegSE block is more suited for segmenting detailed regions. Moreover, the proposed SegSE block outperforms the SE block in terms of HD$_{95}$, which suggests that the contours of the obtained segmentations are more detailed and closer to the annotated ones. It is also possible to compare with the state of the art in BRATS 2017 Leaderboard in Table \[tab:res\_leaderboard\], where all the methods are CNN-based. Additionally, we evaluate the block proposed by Roy et al. [@roy] in our multi-class FCN. The authors implement an alternative channel and spatial attention mechanism, where a single spatial attention map is inferred for all channels. We evaluate this alternative using the parameters proposed by Roy, but in the same conditions as our RR SegSE block, i.e., after fine-tuning and in a hierarchical approach using the same binary segmentation FCN. We separate single prediction approaches from ensembles[^11]. The reason is that ensembles have a competitive advantage, since it is known that it is a way of significantly improving the performance if the models make different mistakes, as it alleviates the effect of the possible high variance of each of its models [@goodfellow2016deep; @kamnitsas_ensemble]. In the considered methods, ensembles resulted from training a variety of FCN architectures with different settings [@kamnitsas_ensemble], from training a FCN in each of the MRI planes (axial, coronal, and sagittal) [@wang2017automatic; @zhao20173d; @jungo2017towards], or from using several models trained previously for k-fold cross-validation in the Training set [@isensee2017brain]. \[tab:res\_leaderboard\] Comparing the single prediction approaches, it is possible to observe that our Baseline is competitive. This is relevant in the sense that it is hard to improve over a competitive approach, which sustains the added value of the RR SegSE block for semantic segmentation. In fact, our FCN with this block outperforms the other single prediction approaches in DC of the enhancing and the tumor core, as well as in HD$_{95}$ of the enhancing tumor, being very close to Jesson et al. [@jesson] in DC of the whole tumor. However, Jesson used a FCN with multiple prediction layers and loss functions in different scales. Additionally, the authors employed a learning curriculum to deal with class imbalance. Comparing with Roy et al. [@roy], we observe that their block obtains lower, but competitive, DC of complete and core, but the proposed RR SegSE block enables us to achieve better scores in DC of enhancing tumor and HD$_{95}$ of all tumor regions. The block proposed by Roy includes a branch where the original SE block is used to learn to recalibrate at the channels level. However, according to our observations with the Baseline + RR SE variant in Table \[tab:res\_test\] and Fig. \[fig:recal\_senet\], it may not be adequate for segmenting fine structures such as enhancing tumor, which may be a reason why the proposed RR SegSE block performs better. When we compare with ensemble methods, we observe that incorporating the RR SegSE block allowed our approach to achieve competitive results, especially in DC. The HD$_{95}$ metric suffers more from the presence of false positive detections, especially if they are far away from the object, and ensembles may effectively tackle this problem since different models do different mistakes. However, our results in DC are similar to the ones obtained by Kamnitsas et al. [@kamnitsas_ensemble], who won BRATS 2017 Challenge edition by generalizing well to the Challenge set, despite the fact that Wang et al. [@wang2017automatic] held the best Leaderboard set results. In Table \[tab:res\_2013\] we present the results on BRATS 2013 Leaderboard and Challenge sets. Even though these datasets are older and smaller, they allow us to compare with a larger variety of methods, such as CNN-based [@pereira2016brain; @shen; @zhao2018deep; @havaei2017brain], Random Forest and handcrafted features-based methods [@tustison2015optimal; @pinto2018hierarchical], and multi-atlas patch-based methods [@cordier2016patch]. Comparing our Baseline with the FCN with RR SegSE block, we can observe that the latter achieves an overall better performance in terms of DC in both the Leaderboard and Challenge set. Still, it is worth noting the competitiveness of the Baseline FCN. We also observe that the proposed RR SegSE block obtains better Dice and Sensitivity than the RR SE block, but lower PPV, except in the core region in the Challenge set. This may be a hint that the SE block may fall into undersegmentation by suppressing the tumor regions. Comparing with the state of the art, we verify that in the Challenge set our proposal obtains higher DC and Sensitivity than the other approaches. In the case of Leaderboard, the results are competitive with Zhao et al. [@zhao2018deep]. However, their approach is based on an ensemble of three CNNs (one for each MRI plane), followed by a Conditional Random Field formulated as a Recurrent Neural Network and extensive post-processing. To our knowledge, these are new state-of-the-art results in BRATS 2013 dataset. \[tab:res\_2013\] As previously mentioned, the hierarchical segmentation approach for brain tumor segmentation is helpful when dealing with class imbalance. A possible limitation of this strategy is that if the first stage is poor in a given challenging subject, it will affect the quality of the second, multi-class, stage. Nevertheless, the results in Table \[tab:res\_leaderboard\] and Table \[tab:res\_2013\] are competitive with the state of the art, which suggests that the proposed method does not fail to segment more tumors than the other methods, at least in these datasets. In summary, in both BRATS 2017 Leaderboard and BRATS 2013 Leaderboard and Challenge sets we can draw two observations. 1) The variant with the RR SegSE block improves over our competitive Baseline FCN. 2) Although our baseline FCN is simple, our results with the proposed block are competitive, or superior, when comparing with the state of the art. Stroke penumbra estimation {#sec:spes} -------------------------- Table \[tab:res:spes\] depicts the results obtained in stroke penumbra estimation in the SPES dataset with the proposed approach, as well as other state-of-the-art methods. We can observe that the Baseline FCN is competitive when comparing with the other methods. Nevertheless, the Baseline + RR SegSE block was able to improve results further in terms of both DC and ASSD. Therefore, the benefits of the proposed RR SegSE block appear to generalize in this problem as well. Moreover, the proposed SegSE block outperforms the SE block, both in DC and ASSD. In fact, the SE has a negative effect over the Baseline FCN in SPES. Although in Table \[tab:res:spes\] we report the metrics used in [@maier2017isles], the Baseline + RR SE obtained PPV and Sensitivity of $0.86 \pm 0.09$ and $0.77 \pm 0.17$, respectively, while Baseline + RR SegSE scored $0.81 \pm 0.11$ and $0.84 \pm 0.13$ in terms of PPV and Sensitivity, respectively. Hence, we conclude that in SPES the SE block may suppress large portions of the lesions, since it suppresses complete feature maps. In contrast, the SegSE block is spatially adaptive, thus adaptively suppressing regions of the feature maps. In this way, the proposed blocks appear to be able to balance PPV and Sensitivity. \[tab:res:spes\] Among the compared methods, only CA-Usher [@maier2017isles], Cl[è]{}rigues et al. [@clerigues2018sunet], and Pereira et al. [@pereira2018enhancing] are based on Representation Learning approaches. While CA-Usher and Cl[è]{}rigues employ CNNs, Pereira learned features from data using Restricted Boltzmann Machines. Observing Table \[tab:res:spes\], we verify that our approach achieves better scores than CA-Usher and Pereira et al. [@pereira2018enhancing], which may be due to those models being shallow. When we compare with the more recent FCN-based approach by Cl[è]{}rigues et al. [@clerigues2018sunet], we verify that both approaches obtained similar DC. The results of the participants in the SPES challenge are officially reported in [@maier2017isles] in terms of DC and ASSD. However, Cl[è]{}rigues et al. [@clerigues2018sunet] opted to use HD instead of ASSD. In this way, the authors report a HD score of $23.9 \pm 13.5 \ mm$, while the proposed Baseline + RR SegSE obtained $22.67 \pm 11.47 \ mm$. Therefore, although our approach performs similar to Cl[è]{}rigues et al. [@clerigues2018sunet] in DC, it is slightly better in HD, which may indicate a better delineation of the lesions. Two other top performing methods among the compared ones are CH-Insel and DE-Uzl, being both based on machine learning approaches with handcrafted features. The method proposed by the CH-Insel team employed the Random Forest classifier with texture features, and a bootstrapping scheme during training at the subject and voxel level. DE-Uzl is also based on the Random Forest classifier, but using intensity, hemispheric difference, local histograms, and center distances as features. Note that while CH-Insel obtained better DC than DE-Uzl, in terms of ASSD it was the other way around. When we compare the proposed Baseline + RR SegSE with those methods, it is possible to observe that it achieves the same DC as CH-Insel and better ASSD than both of those teams. Hence, the proposed approach appears to be competitive or outperform the state of the art in the SPES dataset. Ischemic stroke lesion outcome prediction {#sec:isles} ----------------------------------------- Table \[tab:res\_isles\] presents the results obtained in ISLES 2017 Challenge set regarding ischemic stroke lesion outcome prediction. We observe that most of the top performing methods obtain higher sensitivity than PPV scores. This may be due to the existence of very small lesions with just a few voxels. Therefore, the models may tend towards over-estimation to avoid missing any lesion. Taking the Baseline FCN into consideration, we note that it is competitive with the state of the art. Indeed, it achieves a DC score of 0.31, placing it only below HKU-1. However, the proposed SegSE block is able to improve the DC score by 0.03, achieving a score of 0.34. This results from an increase of the Sensitivity metric, while PPV decreases only 0.01. Hence, we conclude that the proposed blocks contribute for predicting the lesions better. This is maybe due to their ability to suppress irrelevant regions, thus allowing the model to focus on the important parts of the image. The SegSE block also outperforms the SE block in this application, although the latter is also able to improve the DC of the Baseline FCN to 0.32. The RR SE block results in a large gap between Sensitivity and PPV, which may result in a poorer prediction than the one obtained by the RR SegSE block. Indeed, this is confirmed by the HD and ASSD of $29.09 \pm 14.88 \ mm$ and $5.17 \pm 3.25 \ mm$, respectively, obtained by our RR SegSE block, against the HD and ASSD of $35.58 \pm 15.58 \ mm$ and $6.32 \pm 4.33 \ mm$, respectively, of the RR SE block. \[tab:res\_isles\] The methods in Table \[tab:res\_isles\] represent the top-10 methods that participated in ISLES 2017 challenge, together with Pinto et al. [@pinto2018enhancing], which, to our knowledge, was the first approach to incorporate information from the perfusion Dynamic Susceptibility Contrast-enhanced MRI sequence. All of these methods are based on CNNs, mostly FCN-based. Moreover, SNU-1, SNU-2, HKU-1, and MIPT are ensembles of different FCN architectures and training settings. As expected, these ensemble-based approaches are among the highest ranked ones. SNU-2 and HKU-1 obtained DC of 0.31 and 0.32, respectively. However, the proposed Baseline + RR SegSE was able to outperform both methods by achieving a DC of 0.34. Moreover, the standard deviation was also smaller. Regarding PPV, both our method and SNU-2 performed similarly, with a 0.36 score, while the proposed method obtained sensitivity of 0.55, which is higher than both SNU-2 and HKU-1. Ischemic stroke lesion outcome prediction differs from segmentation in the sense that we are predicting the final infarct core at a three month follow-up acquisition. In this scenario, the proposed blocks were able to improve an already competitive baseline FCN. This may be due to the capabilities of the proposed blocks to suppress irrelevant regions of the features maps, by taking context into account. To the best of our knowledge, the results obtained in ISLES 2017 are state-of-the-art. Conclusion {#sec:conclusion} ========== In summary, channel recalibration of feature maps consists in learning the dependencies among channels, and use it to suppress the less relevant ones. Although desirable, this approach is not well suited for semantic segmentation with FCNs, where several voxels in a patch are segmented simultaneously. In this case, a feature map may contain regions that are relevant for certain voxels, but are less important for others. Also, convolutional layers with $1\times 1$ kernels were used before as bottlenecks in a way to decrease the computational complexity. In this work, we propose to use layers with $1 \times 1$ kernels to recombine features, by an expansion followed by compression of the feature maps. Additionally, we propose a spatially adaptive recalibration block. With this block, we are able to suppress only the less relevant regions of the feature maps, while maintaining the important parts, behaving as an intra-channel attention mechanism. The proposed recalibration block (SegSE) employs dilated convolution for aggregating context. Experimentally, we show that the proposed RR with SegSE block leads to improvements over a competitive baseline. This behavior was observed in all three applications: brain tumor segmentation, stroke penumbra estimation, and ischemic stroke lesion outcome prediction. Our baseline FCN is a simple encoder-decoder FCN, and in this work we aimed at studying spatially adaptive recalibration. However, when we added the RR SegSE block we were able to achieve competitive or state-of-the-art results. Finally, the proposed block is general and it should be possible to add it to other FCN architectures for semantic segmentation. Acknowledgments {#acknowledgments .unnumbered} =============== Sérgio Pereira was supported by a scholarship from the Fundação para a Ciência e Tecnologia (FCT), Portugal (scholarship number PD/BD/105803/2014). This work is supported by FCT with the reference project UID/EEA/04436/2013, COMPETE 2020 with the code POCI-01-0145-FEDER-006941 and by COMPETE: POCI-01-0145-FEDER-007043 and FCT within the Project Scope:UID/CEC/00319/2013. [^1]: Copyright (c) 2019 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. [^2]: S. Pereira, A. Pinto, J. Amorim, A. Ribeiro, and C. A. Silva are with CMEMS-UMinho Research Unit, University of Minho, Guimarães, Portugal (e-mail: id5692@alunos.uminho.pt (S. Pereira), csilva@dei.uminho.pt (C. Silva)) [^3]: S. Pereira, A. Pinto, and V. Alves are with Centro Algoritmi, University of Minho, Braga, Portugal (e-mail: valves@di.uminho.pt (V. Alves)) [^4]: Equivalently implemented as convolutional layers with $1 \times 1$ kernels. [^5]: <https://github.com/sergiormpereira/rr_segse> [^6]: <https://ipp.cbica.upenn.edu/> [^7]: <https://www.smir.ch/BRATS/Start2013> [^8]: <https://www.smir.ch/ISLES/Start2015> [^9]: <https://www.smir.ch/ISLES/Start2017> [^10]: In the earlier version of this work [@segse] the fine-tuning stage described in Subsection \[sec:brain\_tumor\_train\] was not performed. Hence, the results in the Leaderboard set reported in [@segse] were obtained after training the models with 60% of the subjects in BRATS 2017 Training set. This explains the improved results in Table \[tab:res\_leaderboard\]. [^11]: In single prediction approaches the final segmentation results from the predictions of a model, instead of the combination of several predictions, as in ensembles.
--- abstract: 'In this paper we define a new invariant of the incomplete hyperbolic structures on a $1$-cusped finite volume hyperbolic $3$-manifold $M$, called the ortholength invariant. We show that away from a (possibly empty) subvariety of excluded values this invariant both locally parameterises equivalence classes of hyperbolic structures and is a complete invariant of the Dehn fillings of $M$ which admit a hyperbolic structure. We also give an explicit formula for the ortholength invariant in terms of the traces of the holonomies of certain loops in $M$. Conjecturally this new invariant is intimately related to the boundary of the hyperbolic Dehn surgery space of $M$.' address: \| Department of Mathematics, University of Melbourne\ Parkville, 3052, Australia author: - 'James G. Dowty' title: A new invariant on hyperbolic Dehn surgery space --- Introduction {#S:Intro} ============ Let $M$ be an orientable 3-manifold which admits a complete, finite-volume hyperbolic structure with a single cusp. Thurston’s hyperbolic Dehn surgery theorem applied to $M$ says that all but a finite number of topological Dehn fillings on $M$ have hyperbolic structures. To prove this, Thurston introduced the deformation space $\defm$ of incomplete hyperbolic structures on $M$ whose completions have ‘Dehn surgery-type’ singularities (see [@ThurstonNotes]). This space includes any hyperbolic cone-manifold whose set of non-singular points is diffeomorphic to $M$. The space $\defm$ is of continuing interest because it offers a possible approach (due to Thurston) to the Geometrization Conjecture. In particular, non-hyperbolic geometric structures and topological decompositions along incompressible spheres and tori can often be produced by understanding the ways that hyperbolic structures can degenerate near the boundary of $\defm$ (see Thurston [@ThurstonNotes Chapter 4], Kerckhoff [@Kerckhoff] and Kojima [@KojSurvey]). This approach has recently yielded a proof of the Orbifold Theorem (see Cooper-Hodgson-Kerckhoff [@OrbifoldNotes Chapter 7] or Boileau-Porti [@BPorti]). In this paper we introduce a $\mathbb{C}^n$-valued invariant $\orth$ of each incomplete hyperbolic structure in $\defm$, called the [ortholength invariant]{}. This invariant is defined in terms of a topological ideal triangulation $K$ of $M$, but it is actually independent of $K$ in the sense that for ‘generic’ $K$, $\orth$ determines the ortholength invariant corresponding to any other ideal triangulation (see Theorem \[T:Param\]). The main result of this paper says that away from a (possibly empty) subvariety of values, the ortholength invariant locally parameterises $\defm$ and it is a complete invariant of the (topological) Dehn fillings of $M$ which admit a hyperbolic structure (see Theorem \[T:CP\]). For a hyperbolic structure whose metric completion is a cone-manifold, the ortholength invariant is essentially a list of hyperbolic cosines of the complex distances[^1] from the cone-manifold’s singular set to itself along the edges[^2] of $K$. Hence there is a close connection between the ortholength invariant and the [tube radius]{} of a hyperbolic Dehn filling, i.e. the supremum of the radii of the embedded hyperbolic tubes in $M$ about the singular set of the Dehn filling. The importance of the tube radius has emerged recently from the work of Kojima [@KojPi] and Hodgson-Kerckhoff [@Kerckhoff], [@HK]. In these papers, the condition that the tube radius stays bounded away from zero as cone-angles are varied has emerged as the key to ensuring that the hyperbolic structures do not degenerate. (By contrast, it is possible for hyperbolic cone-manifolds to degenerate while their volume remains bounded above zero, e.g. see [@KojPi], especially Theorem 7.1.2 or the example of §7.2.) This suggests a relationship between the boundary of $\defm$ and the ortholength invariant which deserves further study (see Conjecture \[C:Conjecture\]). The terminology ‘ortholength invariant’ follows [@Meyerhoff], where Meyerhoff defines the complex ortholength spectrum of a closed hyperbolic 3-manifold as the set of complex distances between pairs of the manifold’s simple closed geodesics. He shows that the ortholength spectrum plus some combinatorial data determines the manifold up to isometry. Hence the fact that $\orth$ is a complete invariant of the topological Dehn fillings of $M$ which admit a hyperbolic structure can be interpreted as a strengthened version of Meyerhoff’s result, namely that a finite (and computable) subset of the ortholength spectrum plus slightly stronger combinatorial information than assumed in [@Meyerhoff] determines the filled manifold up to isometry. The rest of this paper is set out as follows. In Section \[S:Lines\] we show that it is possible to parameterise a configuration of lines in $\hyp$ in terms of the complex distances between them. In Section \[S:OrthInv\] we define the ortholength invariant and give a formula for it in terms of the traces of the holonomies of certain loops in $M$, thereby showing that $\orth$ is a rational map from the $\isom$-character variety $\Char$ of $M$ into $\mathbb{C}^n$. The main section of this paper is Section \[S:Param\], where we prove that the ortholength invariant locally parameterises incomplete hyperbolic structures on $M$ and is a complete invariant when restricted to the (topological) Dehn fillings of $M$ which admit a hyperbolic structure (for those hyperbolic structures whose ortholength invariant does not lie in a certain subvariety). We also prove (under a weak technical assumption) that the ortholength invariant is a birational equivalence from certain interesting irreducible components of $\Char$ to certain irreducible components of a complex affine algebraic variety $\param \subseteq \mathbb{C}^n$. In Section \[S:Eg\] we calculate the ortholength invariant when $M$ is the figure-8 knot complement. Section \[S:Conc\] concludes this paper with a conjecture and a discussion of some applications for the ortholength invariant, including the construction of incomplete hyperbolic structures on $M$. This paper is based on my doctoral thesis and I would like to acknowledge and thank my advisor Craig D. Hodgson for his guidance throughout the work presented here. Configurations of lines in hyperbolic 3-space {#S:Lines} ============================================= We say that $n$ oriented lines $l_1, \ldots, l_n$ in hyperbolic 3-space $\hyp$ [realise]{} a set of complex numbers $x_{ij} \in \mathbb{C}$ ($i,j = 1, \ldots ,n$) if $\cosh$ of the complex distance[^3] between each pair of lines $l_i$ and $l_j$ is equal to $x_{ij}$. This section is motivated by the following question. \[Q:InfCh1\] When is it possible to realise a given set of complex numbers $x_{ij} \in \mathbb{C}$ ($i,j = 1, \ldots ,n$) by an arrangement of lines and how unique is such an arrangement? A complete answer to Problem \[Q:InfCh1\] is given in Theorem \[T:SolnCh1\]. A corollary to this theorem (Corollary \[C:N=4\]) says that there exists an arrangement of four lines which realises a set of complex numbers $x_{ij} \in \mathbb{C}$ ($i,j = 1, \ldots ,4$) if and only if these complex numbers satisfy a ‘hextet’ equation. This corollary is the key result which allows us to locally parameterise hyperbolic structures on a $3$-manifold by ‘ortholengths’ (see Section \[S:Param\]). The results of this section prior to Problem \[Prob:Ch1\] are simply re-statements of some of the ideas Fenchel presented in [@Fenchel]. For related material see Thurston [@ThurstonBook §§2.3-2.6]. The end-points of a line in $\hyp$ are two distinct points on the sphere at infinity $\si$, and conversely any two such points determine a line. We identify the set of [oriented lines]{} in $\hyp$ with $\endpts$ where $\Delta = \{ (x,x) \mid x \in \si\}$, and we refer to an element of $\endpts$ as the [ordered end-points]{} of an oriented line in $\hyp$. A simple calculation shows that every traceless matrix $A \in \mbox{SL}_2\mathbb{C}$ has eigenvalues $\pm i$. The eigenvectors of $A$ correspond to points on the sphere at infinity $\si$ which are fixed by the action[^4] of $A$. Hence there is an oriented line corresponding to $A$ whose ordered end-points are $(p,q) \in \endpts$, where $p$ corresponds to eigenvalue $-i$ and $q$ corresponds to eigenvalue $i$. It is not hard to check that the map $A \mapsto (p,q)$ is a homeomorphism from $$\OrLines \defeq \{ l \in \mbox{SL}_2\mathbb{C} \mid \mbox{tr}~ l = 0 \}$$ to the space of oriented lines in $\hyp$. We identify these two spaces and from now on we will use the same symbol to denote both an oriented line and the corresponding element of $\OrLines$. Note that as a consequence of this, $-l \in \OrLines$ denotes the same line in $\hyp$ as $l \in \OrLines$ but with the opposite orientation. Now, $\isom$ acts by orientation-preserving isometries on the set of oriented lines in $\mathbb{H}^3$. Under the correspondence between oriented lines and $\OrLines$, this gives us a corresponding action of $\isom$ on $\OrLines$, given by $g \cdot l = \tilde{g} l \tilde{g}^{-1}$ for any $g \in \isom$ and $l \in \OrLines$, where $\tilde{g} \in \mbox{SL}_2\mathbb{C}$ is either of the matrices covering $g$. Hence studying the geometric properties of arrangements of lines in $\hyp$ is the same as studying the properties of subsets of $\OrLines$ which are invariant under the conjugacy action of $\mbox{SL}_2\mathbb{C}$. Given two oriented lines $l,m \in \OrLines$, an obvious conjugacy invariant of the pair is $\mbox{tr} (lm)$. Lemma \[L:InnerProduct\] (below) says that this invariant is essentially $\cosh$ of the complex distance between $l$ and $m$. However, before stating this lemma we introduce some notation[^5]. The complex vector space $\Lines$ of [line matrices]{} consists of all $2 \times 2$ complex matrices with zero trace. This space is endowed with a symmetric bilinear form $\lacute \cdot,\cdot \racute$ given by $$\lacute l,m \racute = - \frac{1}{2} \mbox{tr} (lm)$$ for any $l,m \in \Lines$. It is not hard to check that the bilinear form $\lacute \cdot,\cdot \racute$ is non-degenerate on $\Lines$. Also, each non-singular line matrix $l \in \Lines$ acts on $\mathbb{H}^3$ as a half-turn about some line so there is an (unoriented) line associated with each non-singular line matrix. This is where the terminology ‘line matrices’ comes from. \[L:InnerProduct\] For any $l, m \in \OrLines$, $$\begin{aligned} \label{E:FundLink} \lacute l,m \racute = \cosh ( \dist (l,m)) \end{aligned}$$ where $\dist (l,m)$ is the complex distance between $l$ and $m$. See Fenchel [@Fenchel §V.3]. Alternatively, the reader can take Lemma \[L:InnerProduct\] as the definition of the complex distance between two oriented lines in $\hyp$. Note that $\lacute l,l \racute = \det l$ for any $l \in \Lines$ so $\OrLines = \{l\in \Lines \mid \lacute l,l \racute = 1 \}$ is the set of normalised line matrices. Then by Lemma \[L:InnerProduct\] we can rephrase Problem \[Q:InfCh1\] in terms of linear algebra as follows. \[Prob:Ch1\] Given some complex numbers $x_{ij} \in \mathbb{C}$ ($i, j = 1, \ldots , n$) so that $x_{ij} = x_{ji}$ and $x_{ii} = 1$, do there exist oriented lines $l_1, \ldots, l_n \in \OrLines$ so that $$\lacute l_i, l_j \racute = x_{ij}?$$ To what extent do these conditions determine the lines $l_1, \ldots, l_n$ if they exist? Now, the action of $\isom$ on $\OrLines$ extends to an action on $\Lines$ in an obvious way[^6] and clearly $\lacute \cdot,\cdot \racute$ is invariant under this action. Also note that $\lacute \cdot,\cdot \racute$ is invariant under the map $\Lines \to \Lines$ given by $l \mapsto -l$. The following lemma says that these are the only isomorphisms of $(\Lines, \lacute \cdot,\cdot \racute)$. \[L:IsomL\] For each linear map $\phi: \Lines \to \Lines$ which preserves $\lacute \cdot,\cdot \racute$ there is some element $g \in \isom$ and some choice $\pm 1$ of sign so that $$\pm \phi (l) = g \cdot l$$ for any $l \in \Lines$. Consider three lines $l_1,l_2,l_3 \in \OrLines$ for which $\lacute l_i,l_j\racute = 0$ for each $i\not= j$. By Lemma \[L:InnerProduct\], these lines all meet at a common point of $\hyp$ where they are mutually perpendicular. We assume that the orientations of the $l_i$ have been chosen so that they define a right-handed frame at this common point (see Figure \[F:CoordSystem\]). Now, if $\phi: \Lines \to \Lines$ preserves $\lacute \cdot,\cdot \racute$ then $\lacute \phi(l_i), \phi(l_j) \racute = 0$ for each $i\not= j$ and so by Lemma \[L:InnerProduct\], $\phi(l_1), \phi(l_2), \phi(l_3)$ are also mutually perpendicular lines which meet at a common point. By replacing $\phi$ by $- \phi$ if need be we can assume $\phi(l_1), \phi(l_2), \phi(l_3)$ define a right-handed frame. Since $\isom$ acts transitively on the bundle of right-handed orthogonal frames of $\hyp$ (see [@ThurstonBook §2.2]) there is an isometry $g\in \isom$ which takes $l_i$ to $\phi(l_i)$ for each $i = 1,2,3$. Hence $\phi(l_i) = g \cdot l_i$ for each $i = 1,2,3$. Since the $l_i$ form a basis for $\Lines$ (linear dependence would imply that $\lacute l_i,l_i\racute = 0$ for some $i$) this proves the lemma. In fact, from this proof it is clear that the action of $\isom \times \{\pm1\}$ on $\Lines$ induces an isomorphism from $\isom \times \{\pm1\}$ to the group $\mbox{O}_3(\mathbb{C})$ of isomorphisms of $(\Lines,\lacute \cdot,\cdot \racute)$ (see also Footnote \[FootnoteRef\], above). Note that $-1$ acts on $\Lines$ by simultaneously reversing the orientations of all lines and is not related to the orientation-reversing isometries of $\hyp$. Let $(V,g)$ denote a finite-dimensional complex vector space $V$ equipped with a symmetric, bilinear form $g$. Then associated to $g$ there is a map $V \to V^$ from $V$ to its dual given by $v \mapsto g(v,\cdot)$. The [rank]{} of $g$ (denoted $\rk(g)$) is the dimension of the image of this map and $g$ is [non-degenerate]{} if this map is an isomorphism. To solve Problem \[Prob:Ch1\] we will use the following lemma from linear algebra. \[L:BiForm\] Let $(V,g)$ and $(W,h)$ be two finite-dimensional complex vector spaces equipped with symmetric, bilinear forms $g$ and $h$. If $\rk(g) \leq \rk(h)$ then there exists a linear transformation $\phi : V \to W$ so that $$\phi^h = g$$ i.e. so that $g(x,y) = h(\phi x, \phi y)$ for any $x,y \in V$. Furthermore, if $h$ is non-degenerate and $\rk(g) = \rk(h)$ or $\rk(g) = \rk(h)-1$ then $\phi$ is unique up to composing it on the left with an isomorphism of $(W,h)$. This is an elementary consequence of the fact that $(V,g)$ is isomorphic to $(\mathbb{C}^n, E_r)$ for $n = \mbox{dim}(V)$ and $r = \rk(g)$, where $E_r$ is the bilinear form defined by $E_r(u,v) = u\transpose Gv$ for any $u,v \in \mathbb{C}^n$ and $G$ is the $n \times n$ diagonal matrix $\mbox{diag}(1, \ldots, 1, 0, \ldots, 0)$ with $r$ non-zero entries. Armed with Lemmas \[L:IsomL\] and \[L:BiForm\] we can now completely solve Problem \[Prob:Ch1\]. \[T:SolnCh1\] For $i,j = 1,\ldots , n$ let $x_{ij} \in \mathbb{C}$ be given complex numbers so that $x_{ij} = x_{ji}$ and $x_{ii} =1$. Then there exist $l_1, \ldots, l_n \in \OrLines$ for which $\lacute l_i, l_j \racute = x_{ij}$ if and only if $\rk(X) \leq 3$, where $X$ is the $n \times n$ matrix whose $(i,j)^{th}$ entry is $x_{ij}$. If some $x_{ij} \not= \pm 1$ then the arrangement of lines $l_1, \ldots, l_n$ is unique up to the action of $\isom \times \{\pm 1\}$, i.e. unique up to orientation-preserving isometry and simultaneous reversal of orientations. If all $x_{ij} = \pm 1$ then there may be non-isometric arrangements of lines which realise the $x_{ij}$. Let $x_{ij} \in \mathbb{C}$ be given complex numbers and suppose that there exist lines $l_1, \ldots, l_n \in \OrLines$ so that $x_{ij} = \lacute l_i, l_j \racute$. Then since $\Lines$ is 3-dimensional, any four of the lines must be linearly dependent. From this it follows that any four rows of $X$ are linearly dependent. For example, there exist non-zero constants $\alpha_1, \ldots, \alpha_4 \in \mathbb{C}$ so that $\alpha_1 l_1 + \ldots + \alpha_4 l_4 = 0$ and hence $$\alpha_1 \lacute l_1, l_i \racute +\ldots+ \alpha_4 \lacute l_4, l_i \racute= 0$$ (i.e. $\alpha_1 x_{1i}+\ldots+\alpha_4 x_{4i} =0$) for each $i = 1, \ldots, n$. Hence $\rk(X) \leq 3$. So now suppose that for $i,j = 1, \ldots, n$ we have $x_{ij} \in \mathbb{C}$ so that $x_{ij} = x_{ji}$ and $x_{ii}=1$ and that $\rk(X) \leq 3$ where $X = [x_{ij}]$ is the matrix defined in the statement. Define a symmetric, bilinear form $g$ on $\mathbb{C}^n$ by $$g(x, y) = x \transpose X y$$ for any $x, y \in \mathbb{C}^n$. Then $\rk(g) \leq 3 = \rk (\lacute \cdot, \cdot \racute )$ so setting $(V,g) = (\mathbb{C}^n,g)$ and $(W,h) = (\Lines,\lacute \cdot,\cdot\racute)$ in Lemma \[L:BiForm\] gives us a linear map $\phi : \mathbb{C}^n \to \Lines$ so that $\phi^* \lacute \cdot, \cdot \racute = g$. For each $i = 1, \ldots, n$ let $l_i \defeq \phi e_i$ where $e_1, \ldots, e_n$ is the standard basis for $\mathbb{C}^n$. Then $$x_{ij} = e_i\transpose X e_j = g(e_i, e_j) = \lacute \phi e_i, \phi e_j \racute = \lacute l_i, l_j \racute$$ for any $i,j = 1, \ldots, n$. Note that $\lacute l_i, l_i \racute = x_{ii} = 1$ for each $i = 1, \ldots n$ so $l_i \in \OrLines$, i.e. each $l_i$ corresponds to an oriented line in $\hyp$. Now, if some $x_{ij} \not= \pm 1$ then $\rk(g) = 2$ or $3$ (this uses the fact that $X$ is symmetric and $x_{ii} = 1$). Hence by Lemma \[L:BiForm\], $\phi$ (and hence the arrangement of lines $l_1, \ldots, l_n$) is unique up to composing $\phi$ by an isomorphism of $(\Lines,\lacute \cdot,\cdot\racute)$. But by Lemma \[L:IsomL\] the isomorphisms of $(\Lines,\lacute \cdot,\cdot\racute)$ are exactly given by the action of $\isom \times \{\pm 1\}$ on $\Lines$. Hence the arrangement of lines $l_1, \ldots, l_n$ is unique up to orientation-preserving isometry and simultaneous reversal of orientations. The special cases of Theorem \[T:SolnCh1\] when $n= 3$ and $n=4$ are of particular interest to us. If three lines $l_1,l_2,l_3 \in \OrLines$ all have distinct end-points on $\si$ then any two of these lines has a common-perpendicular. Adding these three perpendiculars and truncating appropriately gives an arrangement of six line segments in $\hyp$ which meet at right-angles, i.e. a right-angled hexagon (see Figure \[F:RightHexagon\]). Motivated by this generic case we will simply say that any three oriented lines define a right-angled hexagon, without making the assumption that the lines have distinct end-points. Then Theorem \[T:SolnCh1\] applied to $l_1,l_2,l_3$ gives the well-known result that the complex distances along alternating edges of a right-angled hexagon determine the hexagon up to isometry (see [@Fenchel §VI.4]). Similarly, any four lines $l_1,\ldots,l_4 \in \OrLines$ which all have distinct end-points on $\si$ have six pair-wise common-perpendiculars. Adding these perpendiculars and truncating all lines appropriately gives an arrangement of lines in $\hyp$ loosely resembling a tetrahedron (see Figure \[F:HexTet\]). Since this arrangement is like a tetrahedron whose vertices have been stretched into the lines $l_1,\ldots,l_4$, turning its faces into right-angled hexagons, we call such an arrangement a [hextet]{}. As with right-angled hexagons, we will drop the requirement that the end-points of $l_1,\ldots,l_4$ be distinct and simply say that any four oriented lines in $\hyp$ define a hextet. Substituting $n=4$ into Theorem \[T:SolnCh1\] gives the following corollary. \[C:N=4\] Let $x_{ij} \in \mathbb{C}$ be given complex numbers for $i,j = 1,\ldots , 4$ so that $x_{ij} = x_{ji}$, $x_{ii} =1$. Then there exist $l_1, \ldots, l_4 \in \OrLines$ so that $\lacute l_i, l_j \racute = x_{ij}$ if and only if the $x_{ij}$ satisfy the [hextet equation]{} $$\begin{aligned} 0 &=& \det \left[ \matrix { 1 & x_{12} & x_{13} & x_{14} \cr x_{21} & 1 & x_{23} & x_{24} \cr x_{31} & x_{32} & 1 & x_{34} \cr x_{41} & x_{42} & x_{43} & 1 \cr } \right].\end{aligned}$$ If some $x_{ij} \not= \pm 1$ then these lines $l_1, \ldots, l_4$ are unique up to orientation-preserving isometry and simultaneous reversal of each line’s orientation. It was known to Fenchel (see [@Fenchel §V.3]) that if four lines $l_1, \ldots, l_4 \in \OrLines$ are given then the complex numbers $\lacute l_i, l_j \racute$ satisfy the above hextet equation. We finish this section with a brief discussion of degenerate arrangements of lines in $\hyp$. \[D:HexDegen\] An arrangement of lines $l_1, \ldots , l_n \in \OrLines$ is [ non-degenerate]{} if $l_1, \ldots , l_n$ spans $\Lines$ and the arrangement is [degenerate]{} otherwise. The following lemma gives us three more characterisations of degeneracy. \[L:Degen\] Let $l_1, \ldots, l_n \in \OrLines$ be given, let $x_{ij} = \lacute l_i,l_j \racute$ for each $i,j = 1, \ldots, n$ and let $X$ be the $n \times n$ matrix whose $(i,j)^{th}$ entry is $x_{ij}$. If some $x_{ij} \not= \pm 1$ then the following are equivalent: - $l_1, \ldots, l_n$ is a degenerate arrangement of lines. - $l_1, \ldots, l_n$ have a common perpendicular line. - There exists an orientation-preserving isometry taking the arrangement $l_1, \ldots, l_n$ to the arrangement $-l_1, \ldots, -l_n$. - $\rk(X) = 2$. Suppose that $l_1, \ldots, l_n$ are linearly dependent. Some $x_{ij} \not= \pm 1$ so without loss of generality we assume $x_{12} \not= \pm 1$ and hence that $l_1$ and $l_2$ have no end-points in common. Therefore $l_1$ and $l_2$ are linearly independent and they also have a common perpendicular $n \in \OrLines$ (whose orientation is not unique). Then for each $i = 3, \ldots, n$ there exist $\alpha_1, \alpha_2 \in \mathbb{C}$ so that $l_i = \alpha_1 l_1 + \alpha_2 l_2$. But $\lacute l_1, n \racute = \lacute l_2, n \racute = 0$ and so $\lacute l_i, n \racute = 0$ and hence $l_i$ is also perpendicular to $n$ (by Lemma \[L:InnerProduct\]). [(2) $\Rightarrow$ (3)]{}If $l_i \in \OrLines$ is perpendicular to $n \in \OrLines$ then the half-turn about $n$ takes $l_i$ to $-l_i$. [(3) $\Rightarrow$ (1)]{}Suppose there is some $g \in \isom$ so that $g \cdot l_i = -l_i$ for each $i = 1,\ldots, n$. If $l_1, \ldots, l_n$ span $\Lines$ then $g$ is an isometry which takes every oriented line to the same line but with the opposite orientation, which is absurd. [(1) $\Rightarrow$ (4)]{}If we assume that any three of the lines $l_1, \ldots, l_n$ are linearly dependent then it follows that any three of the rows of $X$ are linearly dependent, too, and hence that $\rk(X) <3$. For example, if $\alpha_1 l_1 + \alpha_2 l_2 + \alpha_3 l_3 = 0$ then $$\begin{aligned} 0 &=& \alpha_1 \lacute l_1,l_i \racute + \alpha_2 \lacute l_2,l_i \racute + \alpha_3 \lacute l_3,l_i \racute \cr &=& \alpha_1 x_{1i}+\alpha_2 x_{2i} +\alpha_3 x_{3i}\end{aligned}$$ for each $i = 1, \ldots, n$. But since some $x_{ij} \not= \pm 1$, $\rk(X) \not = 1$ and so $\rk(X) = 2$. [(4) $\Rightarrow$ (1)]{}Conversely, suppose that $\rk(X) = 2$ and assume (in order to derive a contradiction) that the $l_1, \ldots, l_n$ span $\Lines$. Without loss of generality assume that $l_1, l_2, l_3$ form a basis for $\Lines$. Then since $\rk(X) = 2$, the first three rows of $X$ are linearly dependent so there exist $\alpha_1, \alpha_2, \alpha_3 \in \mathbb{C}$ so that $$\begin{aligned} 0 &=& \alpha_1 x_{1i}+\alpha_2 x_{2i} +\alpha_3 x_{3i} \cr &=& \lacute \alpha_1 l_1 + \alpha_2 l_2 + \alpha_3 l_3, l_i \racute\end{aligned}$$ for each $i = 1, \ldots, n$. But since $\lacute \cdot, \cdot \racute$ is non-degenerate and (by assumption) $l_1, \ldots, l_n$ span $\Lines$, this implies $\alpha_1 l_1 + \alpha_2 l_2 + \alpha_3 l_3 = 0$ which is a contradiction. The ortholength invariant {#S:OrthInv} ========================= In this section we define the ortholength invariant of each incomplete hyperbolic structure in the hyperbolic Dehn surgery space $\defm$ of $M$. This invariant is given purely in terms of holonomy representations so its definition naturally extends to a map $\orth:\Char \dashrightarrow \mathbb{C}^n$ from the $\isom$-character variety $\Char$ of $M$ to $\mathbb{C}^n$. We show that $\orth$ is a rational map whose image lies inside a variety $\param \subseteq \mathbb{C}^n$. For examples of $\orth$, $\param$ and $\Char$, see Section \[S:Eg\]. Let $M$ be an oriented, finite volume, $1$-cusped hyperbolic $3$-manifold (as in Section \[S:Intro\]) and let $N$ be an embedded horoball neighbourhood of the cusp. Then $N$ is diffeomorphic to the product of a $2$-torus $T^2$ with the half-open interval $[0,1)$ and furthermore the complement in $M$ of the interior of $N$ is a compact $3$-manifold with boundary $\partial N \cong T^2$ (e.g. see Thurston [@ThurstonBook §4.5]). Let $K$ be a (topological) ideal triangulation of $M$ (see Benedetti-Petronio [@BP §E.5-i]) which meets $N$ ‘nicely’, i.e. so that inside any tetrahedron $\triangle$ of $K$, $N$ has four connected components, each being a punctured neighbourhood of one of the vertices of $\triangle$. Let $$ be a base-point for $M$ which lies in $\partial N$. Now, let $\rho: \pi_1(M,) \to \isom$ be a homomorphism which satisfies the condition that $\rho(\pi_1(\partial N,))$ fixes exactly two points on the sphere at infinity $\si$ (e.g. $\rho$ could be the holonomy representation of an incomplete hyperbolic structure of $\defm$). A simple investigation of the fixed-points of the Abelian group $\rho(\pi_1(\partial N,))$ shows that this condition is equivalent to the requirement that $\rho(\pi_1(\partial N,))$ is a non-trivial group of non-parabolic[^7] isometries which is not isomorphic to $\mathbb{Z}/2 \oplus \mathbb{Z}/2$ (where $\mathbb{Z}/2$ is the group with two elements). These conditions in turn are equivalent to the algebraic conditions that $$\begin{aligned} \label{E:OrthDomain} \mbox{neither\quad} \tr^2\rho(m) = \tr^2\rho(l) = 4 \cr \mbox{nor\quad} \tr^2\rho(m) = \tr^2\rho(l) = \tr^2\rho(ml) = 0, \end{aligned}$$ where $m$ and $l$ are any pair of generators of $\pi_1(\partial N,) \cong \mathbb{Z} \oplus \mathbb{Z}$. Now, by our assumption that $\rho$ satisfies the conditions (\[E:OrthDomain\]), it follows that the group $\rho(\pi_1(\partial N,))$ fixes a unique line in $\hyp$. Choose an orientation and call the resulting oriented line $\sigma$. Let $\pi: \tM \to M$ be the universal cover of $M$ and choose some base-point $\tilde{} \in \pi^{-1}()$. This choice allows us to identify the deck-transformations of $\pi: \tM \to M$ with $\pi_1(M,)$. Let $\tN$ be the connected component of $\pi^{-1}(N)$ which contains $\tilde{}$. Let $e_1, \ldots, e_n$ be the edges of $K$ and for each $i = 1, \ldots, n$ choose a lift $\tilde{e}_i$ of edge $e_i$ in $\tM$. To each end of $\tilde{e}_i$ there is a corresponding connected component of $\pi^{-1}(N)$. Denote these connected components by $\tN_1$ and $\tN_2$ (note that it is possible that $\tN_1 =\tN_2$ if $e_i$ is homotopically trivial). Then since $\tN_1$ and $\tN_2$ both cover $N$, there exist deck-transformations $\gamma_1$ and $\gamma_2$ for which $\gamma_i(\tN) = \tN_i$ (for each $i = 1,2$). We define the [ortholength]{} $d_i$ corresponding to edge $e_i$ to be $$d_i \defeq \dist(\rho(\gamma_1) \cdot \sigma, \rho(\gamma_2) \cdot \sigma)$$ i.e. define $d_i$ to be the complex distance between $\rho(\gamma_1) \cdot \sigma$ and $\rho(\gamma_2) \cdot \sigma$. While $\gamma_1$ and $\gamma_2$ are not unique, $\rho(\gamma_1) \cdot \sigma$ and $\rho(\gamma_2) \cdot \sigma$ do not depend on their arbitrariness and so are well-defined oriented lines in $\hyp$. Also, the definition of $d_i$ doesn’t depend on our choice of lift $\tilde{e_i}$ of edge $e_i$. This is because any other lift is of the form $\alpha \cdot \tilde{e_i}$ for some deck-transformation $\alpha$. In the above prescription this has the effect of replacing $\tN_i$ by $\alpha(\tN_i)$, i.e. replacing $\gamma_i$ by $\alpha \gamma_i$ ($i = 1,2$). But since $\dist(\rho(\alpha) \rho(\gamma_1) \cdot \sigma, \rho(\alpha) \rho(\gamma_2) \cdot \sigma) =\dist(\rho(\gamma_1) \cdot \sigma, \rho(\gamma_2) \cdot \sigma)$, $d_i$ is not affected by this change. Also, conjugating $\rho$ by some $g \in \isom$ has the effect of replacing $\sigma$ by $g \cdot \sigma$ and each $\rho(\gamma_1)$ by $g\rho(\gamma_1) g^{-1}$, which clearly leaves $d_i$ unchanged. This shows that $d_i$ is independent of our choice of $\tilde{}$ (which we used to identify $\pi_1(M,)$ with the deck-transformations of $\widetilde{M} \to M$) and also that $d_i$ only depends on the conjugacy class of $\rho$. We define the ortholength invariant $\orth(\rho)$ with respect to the ideal triangulation $K$ evaluated at $\rho$ to be $$\orth(\rho) \defeq (\cosh d_1, \ldots, \cosh d_n) \in \mathbb{C}^n.$$ We think of $\orth(\cdot)$ as the function which associates $\cosh$ of the ortholength $d_i$ to edge $e_i$ of $K$ for each $i = 1, \ldots, n$. Now, as noted in Footnote \[FootnoteRef\] (and see also [@BZ]) $\isom$ is naturally isomorphic to $\mbox{SO}_3\mathbb{C}$ and so the space of representations $\mathcal{R}(M)$ of $\pi_1(M,)$ into $\isom$ is a complex algebraic variety. The $\isom$-character variety $\Char$ of $M$ is the ‘quotient’ (in the sense of algebraic geometry, see [@BZ §§3,4]) of $\mathcal{R}(M)$ by the conjugacy action of $\isom$. This space $\Char$ has a natural algebraic structure which makes it an affine algebraic variety whose co-ordinate ring is the ring of regular functions on $\mathcal{R}(M)$ which are invariant under the $\isom$-conjugacy action. As noted above, $\orth(\cdot)$ is defined on all representations satisfying the conditions (\[E:OrthDomain\]) and $\orth(\cdot)$ is constant on conjugacy classes. Hence $\orth(\cdot)$ descends[^8] to a function defined on the complement of a proper[^9] sub-variety of $\Char$. We denote this function by $\orth$ as well, and we write $\orth:\Char \dashrightarrow \mathbb{C}^n$ to indicate that $\orth$ is not necessarily defined[^10] as a set-theoretic function at all points of $\Char$. Now, given a tetrahedron $\triangle$ of $K$, choose some tetrahedron $\widetilde{\triangle}$ in $\tM$ which covers it. Each corner of $\widetilde{\triangle}$ meets a unique connected component of $\pi^{-1}(N)$ and these connected components determine four (not necessarily distinct) oriented lines $\rho(\gamma_1) \cdot \sigma, \ldots, \rho(\gamma_4) \cdot \sigma$ in $\hyp$, as above. The complex distance between any pair of these lines is equal to the ortholength corresponding to one of the edges of $\triangle$. Hence the four oriented lines $\rho(\gamma_1) \cdot \sigma, \ldots, \rho(\gamma_4) \cdot \sigma$ determine a hextet which realises the complex distances associated to the edges of $\triangle$. By Corollary \[C:N=4\] this implies that the hyperbolic cosines of these ortholengths satisfy a certain algebraic equation for each tetrahedron of $K$. Hence the image of $\orth$ lies in the following (not necessarily irreducible) complex algebraic variety $\param \subseteq \mathbb{C}^n$. \[D:param\] The [ortholength space]{} $\param\! \subseteq \mathbb{C}^n$ corresponding to a (topological) ideal triangulation $K$ of $M$ with $n$ edges is the complex affine algebraic variety consisting of those points of $\mathbb{C}^n$ which satisfy the [hextet equations]{} of all the tetrahedra of $K$. Here the hextet equation of a tetrahedron $\triangle$ of $K$ is $$0 = \det \left[ \matrix { 1 & x_{12} & x_{13} & x_{14} \cr x_{21} & 1 & x_{23} & x_{24} \cr x_{31} & x_{32} & 1 & x_{34} \cr x_{41} & x_{42} & x_{43} & 1 \cr } \right]$$ where the vertices of $\triangle$ have been numbered arbitrarily from $1$ to $4$ and where we denote $\cosh$ of the ortholength associated to the edge of $\triangle$ between vertices $i$ and $j$ by $x_{ij}= x_{ji}$. Note that the hextet equation of $\triangle$ doesn’t depend on the arbitrary numbering of its vertices. Although $\param$ lies in an $n$-dimensional space (one dimension for each edge of $K$) and is defined by $n$ hextet equations (one for each tetrahedron of $K$) for ‘generic’ $K$, $\param$ has an irreducible component which is a complex curve. This follows from Theorem \[T:Param\] (below) and from the fact that Dehn surgery space is diffeomorphic to $\mathbb{C}$ in a neighbourhood of the complete structure (see Thurston [@ThurstonNotes §5.5]). Conversely, it should be possible to prove that the irreducible component of $\Char$ which contains the complete hyperbolic structure is one-dimensional over $\mathbb{C}$ via Theorem \[T:Param\] and a lemma about the dimension of $\param$. We finish this section with an explicit formula for the ortholength invariant. This formula is given in terms of a presentation for $\pi_1(M,)$ based on the ideal triangulation $K$ of $M$. The generators for this presentation consist of generators for $\pi_1(\partial N,)$ plus loops $\alpha_i \not\in \pi_1(\partial N,)$ which lie in $\partial N \cup e_i$, where $e_i$ is the $i^{th}$ edge of $K$ ($i = 1, \ldots, n$). \[P:orth\] Let $\rho:\pi_1(M,) \to \isom$ be a representation on which $\orth$ is defined (i.e. for which the conditions (\[E:OrthDomain\]) hold) and for $i = 1, \ldots,n$ let $\tilde{h}, \tilde{g}_i \in \mbox{SL}_2\mathbb{C}$ be matrices which cover $\rho(\beta), \rho(\alpha_i) \in \isom$, for $\alpha_i$ as above and some non-trivial $\beta \in \pi_1(\partial N,)$. Then the $i^{th}$ co-ordinate $\cosh d_i$ of $\orth(\rho) \in \param \subseteq \mathbb{C}^n$ is $$\cosh d_i = 2 \frac{ \mbox{tr}( \tilde{h}\tilde{g}_i) \mbox{tr} (\tilde{h}^{-1}\tilde{g}_i) - \mbox{tr}^2 \tilde{g}_i } { \mbox{tr}^2 \tilde{h} - 4 } -1$$ where $i = 1, \ldots, n$. Note that $\mbox{tr}( \tilde{h}\tilde{g}_i) \mbox{tr} (\tilde{h}^{-1}\tilde{g}_i)$, $\mbox{tr}^2 \tilde{g}_i$ and $\mbox{tr}^2 \tilde{h}$ are independent of the choice of matrices $\tilde{h}$ and $\tilde{g}_i$ covering $\rho(m)$ and $\rho(\alpha_i)$ and so these three functions define elements of the co-ordinate ring of $\Char$ (see [@BZ]). Hence from the above formula it is clear that $\orth:\Char \dashrightarrow \param$ is a rational map. Suppose we have the set-up as in the statement, and let $\rho(\pi_1(\partial N,))$ fix a geodesic $\sigma$ of $\mathbb{H}^3$, which we give an arbitrary orientation. Then $\cosh d_i$ is equal to $\cosh(\dist(\sigma, g_i \cdot \sigma))$, where $g_i = \rho(\alpha_i) \in \isom$. Since $\cosh(\dist(\sigma, g_i \cdot\sigma))$ is invariant under conjugating $\rho$ by an orientation-preserving isometry, we can assume $$\begin{aligned} \tilde{h} = \left[ \matrix{ e^{x/2} & 0 \cr 0 & e^{-x/2} \cr} \right] & \tilde{g}_i = \left[ \matrix{ a & b \cr c & d \cr} \right],\end{aligned}$$ (where $ad -bc = 1$) for the purposes of calculating $\cosh(\dist(\sigma, g_i \cdot \sigma))$. We’ll then express our answer in terms which are invariant under conjugacy, and then our formula will hold true for general $\tilde{h}$ and $\tilde{g}_i$. The axis of $\tilde{h}$ has end-points $0$ and $\infty$ on the sphere at infinity, so the line matrix of $\sigma$ is $\pm$[$\left[ \matrix{ i & 0 \cr 0 & -i \cr} \right]$]{}$ \in \mbox{SL}_2\mathbb{C}$. Then by Lemma \[L:InnerProduct\] and the fact that the line matrix of $g_i \cdot \sigma$ is $\pm \tilde{g}_i $[$\left[ \matrix{ i & 0 \cr 0 & -i \cr} \right] $]{}$ \tilde{g}_i^ {-1}$, we have $$\begin{aligned} \cosh(\dist(\sigma, g_i \cdot \sigma)) &=& - \mbox{$\frac{1}{2}$ tr} \left[ \matrix{ i & 0 \cr 0 & -i \cr} \right] \left[ \matrix{ a & b \cr c & d \cr} \right] \left[ \matrix{ i & 0 \cr 0 & -i \cr} \right] \left[ \matrix{ d & -b \cr -c & a \cr} \right] \\ &=& ad + bc \\ &=& 2ad -1.\end{aligned}$$ So our task now is to express $ad$ invariantly. But $$\begin{aligned} \mbox{tr}(\tilde{h}\tilde{g}_i) = e^{x/2}a + e^{-x/2}d & \mbox{ and } & \mbox{tr}(\tilde{h}^{-1}\tilde{g}_i) = e^{-x/2}a + e^{x/2}d\end{aligned}$$ so $$\begin{aligned} \mbox{tr} (\tilde{h}\tilde{g}_i) \mbox{tr} (\tilde{h}^{-1}\tilde{g}_i) &=& a^2 + d^2 + ad(e^{x/2} + e^{-x/2}) \\ &=& (a+d)^2 - 2ad + 2ad \cosh x \\ &=& \mbox{tr}^2 \tilde{g}_i + 2ad ( \cosh x - 1).\end{aligned}$$ Combining this with $$\cosh x - 1 = 2 \cosh^2 (x/2) - 2 = (\mbox{tr}^2 \tilde{h} - 4)/2$$ gives us the required formula for $\cosh d_i$. Parameterising hyperbolic structures with ortholengths {#S:Param} ====================================================== In this section we prove that the ortholength invariant locally parameterises the deformation space $\defm$ and is a complete invariant when restricted to the (topological) Dehn fillings of $M$ which admit a hyperbolic structure (as long as the hyperbolic structure in question has ortholength invariant not lying in a certain (possibly-empty) subvariety). We also prove (under a weak technical assumption) that the ortholength invariant is a birational equivalence from certain interesting irreducible components of $\Char$ to certain irreducible components of $\param$. Fix a (topological) ideal triangulation $K$ of $M$ with $n$ edges and let a set of ortholength parameters $p = (p_1, \ldots, p_n) \in \param$ be given. Then a [realisation]{} of $p$ is a set of $n$ hextets (see Figure \[F:HexTet\]) each of which realises the ortholength parameters associated to the edges of a tetrahedron of $K$. More precisely, a realisation is an association of an oriented line to the four corners of each tetrahedron $\triangle$ of $K$ so that $\cosh$ of the complex distance between any pair of these four lines is equal to the ortholength parameter of the edge of $\triangle$ which lies between the corresponding pair of vertices of $\triangle$. The broad aim of this section is to construct an inverse to $\orth:\Char \dashrightarrow \param$. To each point $p \in \param$, Corollary \[C:N=4\] guarantees the existence of a realisation of $p$ by hextets. Our strategy for building a representation $\pi_1(M,) \to \isom$ is essentially to glue copies of these hextets together to form a type of skeletal developing map for $M$ whose rigidity then gives us a holonomy representation for free (see the proof of Lemma \[L:ConRep\], below). However, this approach is complicated slightly by the fact that the ortholength parameters $p = (p_1, \ldots, p_n)$ don’t quite determine the hextets up to isometry (see Corollary \[C:N=4\]), even if we assume no $p_i = \pm 1$. This forces us to make Definition \[D:coherence\], below. First note that each face $F$ of $K$ is contained in exactly two tetrahedra of $K$ and that these two inclusions composed with the realisation of $p$ by hextets gives two right-angled hexagons corresponding to $F$. Here we think of a right-angled hexagon corresponding to $F$ as an association of an oriented line to each of the three corners of $F$, and by an isometry between such hexagons we mean an orientation-preserving isometry which respects this association. \[D:coherence\] Let $p = (p_1, \ldots, p_n) \in \param$ be such that no $p_i = \pm 1$. Then a realisation of $p$ by hextets is [coherent]{} if the pair of right-angled hexagons corresponding to each face of $K$ are isometric. If a coherent realisation of $p$ exists then we say that $p$ is [coherent]{}. Note that any pair of right-angled hexagons corresponding to a face of $K$ have the same ortholengths and so by Theorem \[T:SolnCh1\] must be either isometric or isometric after reversing the orientations of the lines in one of the hexagons. Also note (from the definition of ortholength invariant, see Section \[S:OrthInv\]) that any point in the set-theoretical image of $\orth$ is coherent. Now, since the rigidity of hextets described in Corollary \[C:N=4\] may fail if all of the ortholength parameters are $\pm 1$, we will usually restrict our attention to $p \in \param \setminus \mathcal{T}$, where $$\mathcal{T} = \{(p_1, \ldots, p_n) \in \param \mid p_i = \pm 1 \mbox{ for some } i = 1, \ldots, n \}.$$ Also, let $F_1, \ldots, F_{2n}$ denote the faces of $K$ and for each $m = 1, \ldots, 2n$ define $$\mathcal{S}_m \defeq \{ p \in \param \mid \det \left[ \matrix {1 & p_{i} & p_{j} \cr p_{i} & 1 & p_{k} \cr p_{j} & p_{k} & 1} \right] = 0 \}$$ where $p_i,p_j,p_k$ are the co-ordinates of $p = (p_1, \ldots, p_n)$ corresponding to the edges of $F_m$. By Lemma \[L:Degen\], $\mathcal{S}_m$ is the set of ortholengths $p\in \param$ so that any realisation of $p$ by hextets has a degenerate hexagon corresponding to face $F_m$. Define $$\begin{aligned} \label{E:calS} \mathcal{S} \defeq \bigcup_{i \not= j} \mathcal{S}_i \cap \mathcal{S}_j\end{aligned}$$ i.e. a point of $\param$ lies in $\mathcal{S}$ only if it has multiple degeneracies. Note that if $\param$ is one complex-dimensional (which Theorem \[T:Param\] indicates whenever $M$ has a single cusp) then dimensional considerations suggest $\mathcal{S}$ will be empty. The following lemma shows how the existence and rigidity of hextets (Corollary \[C:N=4\]) allows us to construct holonomy representations from coherent ortholength parameters. \[L:ConRep\] To each [coherent]{} realisation of $p \in \param \setminus \mathcal{T}$ by hextets there exists a corresponding representation $\rho: \pi_1(M,) \to \isom$ so that $\orth(\rho) = p$. Furthermore, up to conjugacy there are at most a finite number of representations $\rho$ for which $\orth(\rho) = p$ and there is only one if $p \not\in \mathcal{S}$. Hence this lemma implies that, in $\param \setminus \mathcal{T}$, the image of $\orth :\Char \dashrightarrow \param$ is exactly the set of coherent ortholength parameters. Before proving Lemma \[L:ConRep\] we pause briefly to describe part of the relationship between Thurston and [SnapPea’s]{} parameterisation of $\Char$ via ideal hyperbolic tetrahedra (see [@ThurstonNotes], [@SnapPea], [@NZ]) and the parameterisation of $\Char$ described in Lemma \[L:ConRep\] via ortholengths $p \in \param$. To each hextet there is an associated ideal hyperbolic tetrahedron whose vertices are the second end-points[^11] of the four oriented lines comprising the hextet. So given a coherent realisation of $p \in \param$, we can associate a hyperbolic ideal tetrahedron to each (topological) ideal tetrahedron of $K$. Then [SnapPea’s]{} edge-consistency conditions are automatically satisfied for these geometric tetrahedra. To see this, consider all the tetrahedra of $K$ surrounding an edge $e_i$ of $K$. In a hextet which realises one of these tetrahedra, two of the four lines comprising the hextet correspond to edge $e_i$ (and hence $\cosh$ of the complex distance between these two lines is $p_i$). By acting by (orientation-preserving) isometries if need be we can assume that these two lines are the same for each hextet. But then the pair of right-angled hexagons corresponding to a face $F$ of $K$ containing edge $e_i$ must also coincide (since they are isometric by the assumption of coherence and they share two distinct lines). Associating hyperbolic ideal tetrahedra (as above) to each of these hextets therefore gives geometrical tetrahedra whose faces coincide and which fit together smoothly about their shared edge. The shape of a hyperbolic ideal tetrahedron is determined by a complex shape parameter (see [@ThurstonNotes]). The shape parameter $z\in \mathbb{C}$ of the hyperbolic tetrahedron associated (by the above procedure) to a given hextet satisfies an equation in $\cosh$ of the complex distances between the lines of the hextet. This equation is quite large but it is clearly quadratic in $z$, reflecting the fact that a set of ortholengths only determine a corresponding hextet up to orientation. Let $p \in \param\setminus \mathcal{T}$ be coherent and let $H_1, \ldots, H_n$ be a coherent realisation of $p$ by hextets, where $H_j$ corresponds to the $j^{th}$ tetrahedron $\triangle_j$ of $K$. We can pull the ideal triangulation $K$ back to an ideal triangulation $\widetilde{K}$ of the universal cover $\widetilde{M}$ of $M$ via the covering map $\widetilde{M} \to M$. Then every tetrahedron, face and edge of $\widetilde{K}$ covers one of the tetrahedra, faces or edges (respectively) of $K$. Let $\widetilde{K}^{(3)}$ denote the set of tetrahedra of $\widetilde{K}$. We will consider a function $H:\widetilde{K}^{(3)} \to (\OrLines)^4$ which associates a hextet to each tetrahedron of $\widetilde{K}$. This is as an association of an oriented line to each of the four corners of each tetrahedron of $\widetilde{K}$. We require that $H$ satisfies two conditions: (1) if $\widetilde{\triangle} \in \widetilde{K}^{(3)}$ covers the $j^{th}$ tetrahedron $\triangle_j$ of $K$ then the hextet $H(\widetilde{\triangle})$ is isometric to $H_j$ and (2) if two tetrahedra $\widetilde{\triangle}, \widetilde{\triangle}^\prime \in \widetilde{K}^{(3)}$ share a face $T$ of $\widetilde{K}$ then the right-angled hexagons corresponding to $T$ in $H(\widetilde{\triangle})$ and $H(\widetilde{\triangle}^\prime)$ are identical. We will show that to each coherent realisation of $p$ by hextets there is a corresponding function $H:\widetilde{K}^{(3)} \to (\OrLines)^4$ which satisfies the above two conditions, and that this function is unique up to (orientation-preserving) isometry. Before proving this we first show how the existence and rigidity of such a ‘skeletal developing map’ $H$ proves the lemma. Given such an $H$, by its uniqueness up to isometry we know that for each deck-transformation $\alpha$ of the covering $\widetilde{M} \to M$ there is a unique isometry $\rho(\alpha) \in \isom$ which makes the following diagram $$\begin{array}{ccc} \widetilde{K}^{(3)} & \stackrel{\alpha}{\to} & \widetilde{K}^{(3)} \\ \downarrow H & & \downarrow H \\ (\OrLines)^4 & \stackrel{{\rho(\alpha)}}{\rightarrow} & (\OrLines)^4 \end{array}$$ commute. Choosing a base-point $\tilde{} \in \widetilde{M}$ covering $ \in M$ allows us to identify the deck transformations of the covering $\widetilde{M} \to M$ with $\pi_1(M,)$, and hence $\rho$ becomes a function $\rho:\pi_1(M,) \to \isom$. It follows easily from the definition that $\rho:\pi_1(M,) \to \isom$ is actually a homomorphism, and it is well-defined up to conjugacy since using a different $H$ or a different base-point $\tilde{}$ simply has the effect of conjugating $\rho$ by an isometry. By construction, translates under $\rho(\pi_1(M,))$ of the line which is fixed by $\rho(\pi_1(N,))$ are simply the lines occurring in the image of $H$, so $\orth(\rho) = p$. Up to isometry there are at most $2^n$ different realisations of $p$ and so there are at most this many representations $\rho$ with $\orth(\rho) = p$ (up to conjugacy). Also, if $p \not\in \mathcal{S}$ then up to isometry there are at most two coherent realisations of $p$, and these are the same except that they have opposite orientations. (This is because the orientation of one hextet determines the orientations of all other hextets in a coherent realisation of $p$ whenever all or all but one of the hexagonal faces are non-degenerate, see point (3) of Lemma \[L:Degen\].) These two realisations therefore give rise to functions $H:\widetilde{K}^{(3)} \to (\OrLines)^4$ which are identical (up to isometry) except that all of their lines are of opposite orientation. Clearly the holonomy representations corresponding to such $H$ are identical. The rest of the proof is devoted to showing that, corresponding to a coherent realisation $H_1, \ldots, H_n$ of $p$, there is a function $H:\widetilde{K}^{(3)} \to (\OrLines)^4$ which satisfies the two conditions listed in the first paragraph of this proof and furthermore that such an $H$ is unique up to (orientation-preserving) isometry. Dual to $\widetilde{K}$ in $\widetilde{M}$ there is a $2$-dimensional CW-complex $C$ whose underlying space $\|C\|$ is a deformation retract of $\widetilde{M}$ and so in particular $\|C\|$ is simply connected. Let $\gamma$ denote a finite sequence $\gamma_1, \ldots, \gamma_m$ of oriented $1$-cells in $C$ so that $\gamma_i(1) = \gamma_{i+1}(0)$ for each $i = 1, \ldots, m-1$, where $\gamma_i(0)$ denotes the start of the $1$-cell $\gamma_i$ and $\gamma_i(1)$ denotes the end of it. Let $x_0 = \gamma_1(0)$ and let $x_i = \gamma_i(1)$ for each $i = 1, \ldots, m$. We think of $\gamma$ as a path in the $1$-skeleton of $C$ from $x_0$ to $x_m$. For each $i = 1, \ldots, m$, the $0$-cell $x_i$ is dual to a tetrahedron $\widetilde{\triangle}_i$ (say) of $\widetilde{K}$, and each tetrahedron $\widetilde{\triangle}_i$ covers a tetrahedron $\triangle_{j(i)}$ of $K$ (for some $j(i) \in \{1, \ldots, n \}$). Given a choice of hextet $H^\prime_0$ isometric to $H_{j(0)}$, we define a sequence $H^\prime_0, \ldots, H^\prime_m$ of hextets with each $H^\prime_i$ isometric to $H_{j(i)}$ as follows. Assume that $H^\prime_{i-1}$ is defined. Then the $1$-cell $\gamma_i$ is dual to some $2$-simplex $\widetilde{T}$ of $\widetilde{K}$ which is contained in both $\widetilde{\triangle}_{i-1}$ and $\widetilde{\triangle}_i$. Let $T$ be the $2$-simplex of $K$ covered by $\widetilde{T}$. By the coherence of $p$ we know that the right-angled hexagons in $H_{j({i-1})}$ and $H_{j(i)}$ corresponding to $T$ are isometric. Since $H^\prime_{i-1}$ is isometric to $H_{j(i-1)}$ we can define $H^\prime_{i}$ as the hextet which is isometric to $H_{j(i)}$ and for which the right-angled hexagons in $H^\prime_{i-1}$ and $H^\prime_i$ corresponding to $\widetilde{T}$ are identical. This uniquely determines $H^\prime_i$ because there is no non-trivial (orientation-preserving) isometry fixing the right-angled hexagon corresponding to $\widetilde{T}$, since no $p_i = \pm 1$ and so the hexagon has at least three end-points on $\si$. Hence to any such path $\gamma$ and any choice $H^\prime_0$ we have a corresponding sequence $H^\prime_0, \ldots, H^\prime_m$ of hextets. Now, choose some $\widetilde{\triangle} \in \widetilde{K}^{(3)}$ (which covers the $j^{th}$ tetrahedron $\triangle_j$ of $K$, say) and choose a hextet which is isometric to $H_j$ and denote it by $H(\widetilde{\triangle})$. Let $$ be the $0$-cell of $C$ dual to $\widetilde{\triangle}$. Given any other $\widetilde{\triangle}^\prime \in \widetilde{K}^{(3)}$, dual to some $0$-cell $x$ of $C$, choose a path $\gamma$ as above with $x_0 = $ and $x_m = x$. Then form the corresponding sequence $H^\prime_0, \ldots, H^\prime_m$ of hextets as defined above with $H^\prime_0 = H(\widetilde{\triangle})$ and define $H(\widetilde{\triangle}^\prime)$ to be $H^\prime_m$. To see that this definition is independent of the path from $$ to $x$, note that any two such paths are homotopic in $\|C\|$. It therefore suffices to show that the holonomy around the boundary of any of the $2$-cells of $C$ is trivial, i.e. that if $\gamma$ is a path tracing once around a $2$-cell of $C$ then in the corresponding sequence $H^\prime_0, \ldots, H^\prime_m$ of hextets defined above, $H^\prime_0 = H^\prime_m$. But for such a $\gamma$, all of the tetrahedra $\widetilde{\triangle}_0, \ldots, \widetilde{\triangle}_m$ share an edge in $\widetilde{K}$ and so by construction all of the hextets $H^\prime_0, \ldots, H^\prime_m$ have two lines in common. So the hextets $H^\prime_0$ and $H^\prime_m$ share two lines and are isometric (since $\widetilde{\triangle}_0 = \widetilde{\triangle}_m$). Then since there are no non-trivial orientation-preserving isometries fixing two distinct[^12] lines in $\hyp$, $H^\prime_0 = H^\prime_m$ as required. It is clear that $H:\widetilde{K}^{(3)} \to (\OrLines)^4$ as constructed satisfies the two defining conditions given in the first paragraph of this proof, and conversely any such $H$ can be constructed in this way. Hence the existence and rigidity of $H$ is proved. Non-coherent ortholengths correspond to representations into $\isom$ of the fundamental groups of finite-sheeted covers or branched covers of $M$, branched over the edges of $K$. Hence non-coherent ortholengths seem to be more related to the ideal triangulation $K$ than to intrinsic properties of the $3$-manifold $M$. We next show that the property of coherence is locally constant on the complement of a certain subvariety of $\param$. This proof is a deformation argument in which we make use of the fact that in some sense there exists a continuous local parameterisation of hextets by their ortholengths. We pause now briefly to give a precise description of this local parameterisation. Define the set of [reduced hextets]{} $\redh$ to be $$\redh \defeq \{ (l_1, \ldots, l_4) \in (\OrLines)^4 \mid l_1\! = \left[ \matrix {i & 0\cr 0 & -i\cr} \right]\! , l_2\! = \left[ \matrix {a & i-a\cr i+a & -a\cr} \right]\!, a \not= \pm i \in \mathbb{C} \},$$ i.e. the set of hextets whose lines $l_1, \ldots, l_4$ are in standard position so that $l_1$ has ordered end-points $(0,\infty)$ and $l_2$ has ordered end-points $(b,1)$ for some $b \in \si$ not equal to $0$, $1$ or $\infty$. Note that any hextet $l_1, \ldots, l_4$ with $\lacute l_1,l_2 \racute \not= \pm 1$ is isometric to exactly one of the hextets in $\redh$. Now, let $$\Pd = \{ (x_{12},x_{13},x_{14},x_{23},x_{24},x_{34}) \in \mathbb{C}^6 \mid x_{12} \not= \pm1 \mbox{ and }\det X = 0 \}$$ where $$\begin{aligned} \label{E:4X} X \defeq \left[ \matrix {1 & x_{12} & x_{13}&x_{14} \cr x_{12} & 1 & x_{23} & x_{24} \cr x_{13} & x_{23} & 1 & x_{34} \cr x_{14} & x_{24} & x_{34} & 1 \cr } \right]\end{aligned}$$ and let $\Od:\redh \to \Pd$ be the quadratic map given in the above co-ordinates by $x_{ij} = \lacute l_i, l_j \racute$. By Lemma \[L:InnerProduct\], $\Od$ takes a hextet and gives us $\cosh$ of the ortholengths between the hextet’s four lines. Now, by Corollary \[C:N=4\] and Lemma \[L:Degen\], $\Od$ is onto and it is two-to-one everywhere except on the set of degenerate hextets where $\Od$ is instead one-to-one. By Lemma \[L:Degen\] again, the degenerate hextets are the pre-image of those elements of $\Pd$ for which $\rk (X) = 2$. The restriction of $\Od$ to the non-degenerate hextets is a double cover onto the elements of $\Pd$ for which $\rk (X) = 3$. The deck-transformation for this cover is the involution $\sigma: \redh \to \redh$ given by $\sigma :(l_1,l_2,l_3, l_4) \mapsto (l_1, l_2, f_a(l_3), f_a(l_4))$ where $$f_a( \left[ \matrix {u & v\cr w & -u\cr} \right]) = \left[ \matrix {u & \frac{i-a}{i+a}w\cr \frac{i+a}{i-a}v & -u\cr} \right]$$ and $a \not= \pm i$ is determined by $l_2 = \left[ \matrix {a & i-a\cr i+a & -a\cr} \right]$. Similarly, we define the set of reduced right-angled hexagons to be $$\rhex \defeq \{ (l_1, l_2, l_3) \in (\OrLines)^3 \mid l_1 = \left[ \matrix {i & 0\cr 0 & -i\cr} \right] , l_2 = \left[ \matrix {a & i-a\cr i+a & -a\cr} \right], a \not= \pm i \in \mathbb{C} \},$$ and define $$\mathcal{P}(F) = \{ (x_{12},x_{13},x_{23}) \in \mathbb{C}^3 \mid x_{12} \not= \pm1 \}$$ and let $\OrthF:\rhex \to \mathcal{P}(F)$ be the map given in the above co-ordinates by $x_{ij} = \lacute l_i, l_j \racute$. As above, $\OrthF$ restricts to a double covering from the set of non-degenerate right-angled hexagons to the subset of $\mathcal{P}(F)$ for which $$\det \left[ \matrix {1 & x_{12} & x_{13} \cr x_{12} & 1 & x_{23} \cr x_{13} & x_{23} & 1} \right] \not= 0.$$ \[L:Coher1\] The property of coherence is locally constant on $\param \setminus (\mathcal{S} \cup \mathcal{T}).$ Let $a$ be a point of $\param \setminus (\mathcal{S} \cup \mathcal{T})$ and for each $i = 1, \ldots, n$, choose a numbering from $1$ to $4$ for the vertices of the $i^{th}$ tetrahedron $\triangle_i$ of $K$. Then we have a projection $\pi_i : \param \to \mathcal{P}(\triangle)$ essentially given by ignoring the edge parameters which do not appear on the edges of $\triangle_i$. We know that $\pi_i(a)$ is a smooth point of the hypersurface $\mathcal{P}(\triangle) \subseteq \mathbb{C}^6$, since $\Od$ is a local diffeomorphism onto a neighbourhood of $\pi_i(a)$ (since $\pi_i(a)$ corresponds to a non-degenerate hextet) and $\redh$ is easily seen to be smooth everywhere. Let $V_i\subseteq \mathcal{P}(\triangle)$ be a neighbourhood of $\pi_i(a)$ which is diffeomorphic to $\mathbb{C}^{5}$ and which is disjoint from the hypersurface given by the equation $\rk(X) = 2$, where $X$ is as defined in (\[E:4X\]). (This is possible since $a \not\in \mathcal{S}$ and hence $\pi_i(a)$ corresponds to a non-degenerate hextet.) Let $U$ be a connected neighbourhood of $a$ in $\param \setminus (\mathcal{S} \cup \mathcal{T})$ which is contained in $\pi_i^{-1}(V_i)$ for each $i = 1, \ldots, n$. Now, suppose that there exists some $p\in U$ which is coherent, say with a coherent realisation $H^p_1, \ldots, H^p_n \in \redh$, and let $q$ be any other point of $U$. For each $i = 1, \ldots, n$, the image of $U$ under $\pi_i : \param \to \mathcal{P}(\triangle)$ is contained in the path-connected, locally path-connected and simply connected set $V_i$. Also, the restriction of $\Od:\redh \to \mathcal{P}(\triangle)$ to $\Od^{-1}(V_i)$ is a covering projection. Hence by the lifting property of coverings (see [@Bredon]) there is a unique lift $\tilde{\pi}_i: U \to \redh$ of the restriction $\pi_i\|U$ of $\pi_i$ to $U$ for which $\tilde{\pi}_i(p) = H^p_i$. We define $H^q_i$ to be the hextet $\tilde{\pi}_i(q)$. Now, let $F$ be any face of $K$ and suppose that $F$ is contained in tetrahedra $\triangle_i$ and $\triangle_j$ of $K$. If we choose a numbering from $1$ to $3$ for the vertices of $F$ then we can define several projection maps. Firstly a projection $\pi_F:\param \to \mathcal{P}(F)$ essentially given by forgetting all ortholength parameters except those associated to the edges of $F$. Also, a map $\tau_i: \redh \to \rhex$ which first forgets the line of $H^q_i\in \redh$ corresponding to the vertex of $\triangle_i$ not contained in $F$ and which then acts by an (orientation-preserving) isometry to bring the resulting right-angled hexagon into standard position. Similarly we have a map $\tau_j: \redh \to \rhex$. Now, if the right-angled hexagons corresponding to $F$ in $H^q_i$ and $H^q_j$ are degenerate then they are isometric by Lemma \[L:Degen\]. Hence we can restrict our attention to the set $U^\prime$ consisting of $q \in U$ so that $H^q_i$ and $H^q_j$ are non-degenerate. Then $\tau_i \circ \tilde{\pi}_i: U^\prime \to \rhex$ and $\tau_j \circ \tilde{\pi}_j: U^\prime \to \rhex$ are both lifts of the restriction $\pi_F\|U^\prime$ of $\pi_F$ to $ U^\prime $ and they agree at $p$. Hence by the uniqueness of lifts to covering spaces (see [@Bredon]), $\tau_i \circ \tilde{\pi}_i = \tau_j \circ \tilde{\pi}_j$ and so the right-angled hexagons corresponding to $F$ in $H^q_i$ and $H^q_j$ are again isometric. \[L:Coher2\] The limit of a sequence of coherent points in $\param$ is coherent. This lemma follows from the fact that the inverse images of compact sets in $\Pd$ under $\Od : \redh \to \Pd$ are compact. A rational map between two irreducible complex varieties which is generically[^13] defined, injective and onto has a rational inverse and is called a birational equivalence (see [@Harris p.77]). Hence Lemmas \[L:ConRep\], \[L:Coher1\] and \[L:Coher2\] combine to give the following theorem. \[T:Param\] Let $C$ be an irreducible component of $\Char$ on which $\orth$ is generically defined and suppose that the set-theoretic image of $C$ under the rational map $\orth:\Char \dashrightarrow \param$ contains some point $p = (p_1, \ldots, p_n)\in \param$ for which $p \not\in \mathcal{S}$ (see equation (\[E:calS\])) and no $p_i = \pm 1$. Then $\orth$ restricts to a birational equivalence between $C$ and an irreducible component of $\param$. Let $\hol : \defm \to \Char$ be the map which takes a hyperbolic structure and returns its holonomy representation modulo conjugacy. Then as in Section \[S:OrthInv\], the ortholength invariant of an incomplete hyperbolic structure $M_0 \in \defm$ is $\orth \circ \hol (M_0)$. Recall also that we say that an incomplete hyperbolic structure on $M$ [corresponds to a (topological) Dehn filling]{} if the metric completion of $M$ is a closed manifold (homeomorphic to a topological Dehn filling of $M$) with a smooth hyperbolic structure. \[T:CP\] Let $M_0 \in \defm$ be an incomplete hyperbolic structure whose ortholength invariant $p = (p_1, \ldots, p_n) \in \param$ is not contained in $\mathcal{S}$ (see equation (\[E:calS\])) and which has no $p_i = \pm 1$. Then: - If $M_0$ corresponds to a (topological) Dehn filling of $M$ then $M_0$ is uniquely determined by $p$. - The ortholength invariant $\orth : \Char \dashrightarrow \param$ restricts to a diffeomorphism from a neighbourhood of $\hol(M_0)$ in $\Char$ to a neighbourhood of $p$ in $\param$. The holonomy map $\hol$ is a local homeomorphism (e.g. see [@ThurstonNotes §5.2] or [@Goldman]) so we can endow $\defm$ with a smooth structure coming from $\Char$. Then the second point of this theorem says that (generically) the ortholength invariant smoothly locally parameterises incomplete hyperbolic structures on $M$. A hyperbolic structure which corresponds to a topological Dehn filling of $M$ is determined by its holonomy representation. This is a ‘folk-lore’ result which is true essentially because, in this case, the image of the holonomy representation is a discrete Kleinian group, and its action on $\hyp$ has a quotient isometric to the metric completion of $M$. But by Lemma \[L:ConRep\], the ortholength invariant $p$ determines the holonomy representation whenever $p \not\in \mathcal{S}$ and no $p_i = \pm 1$. This proves the first point of the corollary. Now, $\orth$ is defined on every representation which arises as the holonomy of one of the hyperbolic structures of $\defm$ (see Section \[S:OrthInv\]). Then since the conditions (\[E:OrthDomain\]) are open, $\orth$ is defined in a neighbourhood of $\hol(M_0)$. From the formula of Proposition \[P:orth\] it is clear that $\orth$ is smooth in this neighbourhood. On the other hand, $p$ is coherent by the construction used to define the ortholength invariant in Section \[S:OrthInv\]. By Lemma \[L:Coher1\] this implies that all ortholength parameters in a neighbourhood of $p$ are also coherent. Hence by Lemma \[L:ConRep\] there is a local inverse to $\orth$ defined in a neighbourhood of $p$. This inverse is smooth by the construction given in Lemma \[L:ConRep\], since the representation given there is determined by a finite portion of the skeletal developing map $H$ and the hextets used to build this map all vary smoothly with $p$. This last result follows from the discussion preceding Lemma \[L:Coher1\] and the fact that if $p \not\in \mathcal{S}$ then the hextets in a realisation of $p$ are all non-degenerate. Theorems \[T:Param\] and \[T:CP\] are vacuous unless there exists some incomplete hyperbolic structure with ortholength invariant $p$ such that $p \not\in \mathcal{S}$ and no $p_i = \pm 1$. This will probably be the case whenever the edges of $K$ are homotopically non-trivial, however until such a general result can be established we have the following lemma. \[L:EP\] If the ideal triangulation $K$ is Epstein-Penner’s [@EP] ideal cell-decomposition of $M$ then there exists an incomplete hyperbolic structure with ortholength invariant $p$ such that $p \not\in \mathcal{S}$ and no $p_i = \pm 1$. The full proof of this lemma relies on the notion of a ‘tube domain’ (see Section \[S:Conc\] and [@Dowty Lemma 4.7]) so we content ourselves here with a sketch. Corresponding to the complete hyper-bolic structure on $M$ there is a 2-complex in $M$ which is the image of the boundary of the Ford domain of $M$ under the face-pairing identifications. This [Ford 2-complex]{} is dual to $K$ (see [@EP]). Let $\mbar$ be a (topological) Dehn filling of $M$ with very large Dehn surgery co-ordinates and let $\Sigma$ be the added ‘core’ geodesic of $\mbar$, hence $\mbar$ is hyperbolic and $M = \mbar \setminus \Sigma$. The cut-locus of $\Sigma$ in $\mbar$ is a 2-complex which (for very large Dehn fillings) is a small perturbation of the Ford 2-complex, and so in particular this cut-locus is also topologically dual to $K$ (see [@Dowty]). Now, the pre-image of $\Sigma$ under a (fixed) covering isometry $\hyp \to \mbar$ is a collection of disjoint lines in $\hyp$. If $F_j$ is a 2-simplex of $K$ and $\widetilde{F}_j$ is a lift to $\hyp$ then each corner of $\widetilde{F}_j$ determines one of these lines. The complex distances between these three distinguished lines are given by the three ortholength parameters $p_i$ attached to the corresponding edges of $F_j$, where $p = (p_1, \ldots, p_n)$ is the ortholength invariant of the hyperbolic structure on $M$ induced from $M \hookrightarrow \mbar$. Then the set of points of $\hyp$ which are equidistant from all three lines simultaneously is non-empty, since this set contains a lift of a certain part of the cut-locus of $\Sigma$, namely the 1-cell dual to $F_j$. But if $p$ lies in $\mathcal{S}_j$ then this is a contradiction. In this case, the three lines corresponding to the corners of $\widetilde{F}_j$ have a common perpendicular line $n$ (by Lemma \[L:Degen\]). Since $\mbar$ is a large Dehn filling, all of its ortholengths will be very large[^14] and so the distances between these three lines are also large (and no $p_i = \pm 1$). This implies that the locus of points equidistant from any pair of lines is close to the plane which is perpendicular to $n$ and lies mid-way between the two lines. Hence there can be no point of $\hyp$ which is equidistant from all three lines simultaneously. Examples {#S:Eg} ======== In this section we describe the ortholength invariant $\orth: \Char \dashrightarrow \param$ for the figure-8 knot complement $M$. ![The figure-8 knot[]{data-label="F:Fig8"}](Fig8.eps){height="3cm"} Let $M$ be the complement in $\mathbb{S}^3$ of the figure-8 knot (see Figure \[F:Fig8\]). A result of Epstein and Penner (see [@EP]) says that every 1-cusped hyperbolic 3-manifold has a canonical ideal cell-decomposition. For the figure-8 knot complement $M$, this cell-decomposition is the ideal triangulation $K$ shown[^15] in Figure \[F:tetn\_FP\] (see [@ThurstonBook]). This ideal triangulation for the figure-8 knot complement was first described by Thurston [@ThurstonNotes]. The ideal triangulation $K$ has two edge classes, $e_0$ and $e_1$. Let $p_0$ be the ortholength parameter corresponding to edge class $e_0$ and let $p_1$ correspond to $e_1$. Then the hextet equation (see Definition \[D:param\]) of tetrahedron $0$ is $$\begin{aligned} 0 &=& \det \left[ \matrix {1 & p_0 & p_1 & p_1 \cr p_0 & 1 & p_1 & p_0 \cr p_1 & p_1 & 1 & p_0 \cr p_1 & p_0 & p_0 & 1 } \right] \nonumber\\ &=& 1-3p_0^2 -3 p_1^2 + 4 p_0 p_1^2 + 4 p_0^2 p_1 + p_0^4 - 2p_0^3 p_1 - p_0^2 p_1^2 - 2p_0 p_1^3 + p_1^4 \nonumber \\ &=& \mbox{$\frac{1}{4}$} (p_0^2 + p_1^2 + p_0 p_1 - p_0 -p_1 - 1) (2p_1 - 3 p_0+1+\sqrt{5} (p_0-1)) \nonumber \\ && ~~\times (2p_1 - 3p_0 +1-\sqrt{5} (p_0-1)) \label{E:ParamEqn}.\end{aligned}$$ The hextet equation of tetrahedron $1$ is identical, so $\param \subseteq \mathbb{C}^2$ is the plane algebraic curve given by equation (\[E:ParamEqn\]). From (\[E:ParamEqn\]) we see that $\param$ is the union of a conic (whose projective completion is topologically a smooth sphere) and two lines. Following the notation of Section \[S:OrthInv\] we take $N \subseteq M$ to be a closed tubular neighbourhood of the figure-8 knot minus the knot itself. Then the generators of the presentation of $\pi_1(M,)$ based on $K$ (see the discussion preceding Proposition \[P:orth\]) are as shown in Figure \[F:Fig8\_alphan\]. Now, let $t_1 = l \alpha_1^{-1} m$ and $t_2 = m$. These are the generators of a presentation $$\begin{aligned} \pi_1(M,) = \lacute t_1, t_2 \| [t_2^{-1}, t_1] t_2^{-1} = t_1^{-1} [t_2^{-1}, t_1] \racute \label{E:WirtPres}\end{aligned}$$ which is derived from a Wirtinger presentation for $\pi_1(M,)$. We define two complex algebraic functions $X$ and $Y$ on the space of representations of $\pi_1(\!M,\!)$ into $\mbox{SL}_2\mathbb{C}$ by $$X(\tilde{\rho}) = \mbox{tr}(\tilde{\rho} (t_1)) = \mbox{tr}(\tilde{\rho}(t_2)) \mbox{ and } Y(\tilde{\rho}) = \mbox{tr}(\tilde{\rho} (t_1 t_2))$$ for any representation $\tilde{\rho}: \pi_1(M,) \to \mbox{SL}_2 \mathbb{C}$. These functions are clearly conjugacy invariant, so they descend to functions (also denoted $X$ and $Y$) on the $\mbox{SL}_2\mathbb{C}$-character variety[^16]. In fact, $X$ and $Y$ are the co-ordinate functions of an embedding of the $\mbox{SL}_2 \mathbb{C}$-character variety in $\mathbb{C}^2$ as the complex curve given by $$\begin{aligned} \label{E:SL2CEqn} 0 = (X^2 - Y -2) (X^2 Y - 2 X^2 -Y^2 +Y +1 )\end{aligned}$$ (see González-Acuña and Montesinos-Amilibia’s paper [@Montesinos]). Now, from the presentation (\[E:WirtPres\]) above it is clear that each representation $\rho: \pi_1(M,) \to \isom$ lifts to exactly two representation $\pm \tilde{\rho}: \pi_1(M,) \to \mbox{SL}_2 \mathbb{C}$. Hence there are well-defined algebraic functions $U$ and $V$ defined on the$\isom$-character variety $\Char$ by $$U(\rho) = X^2(\tilde{\rho}) \mbox{ and } V(\rho) = X^2(\tilde{\rho}) -Y(\tilde{\rho}),$$ i.e. $U$ and $V$ are elements of the co-ordinate ring of $\Char$. By (\[E:SL2CEqn\]), these functions define an embedding of $\Char$ into $\mathbb{C}^2$ as the complex curve given by the equation $$\begin{aligned} 0 = (V-2)(V^2 - UV + V + U -1). \label{E:CharUVEqn}\end{aligned}$$ Hence $\Char$ is composed of two irreducible components, one being the line $V = 2$ and the other being the rational curve $$\begin{aligned} U = \frac{V^2 + V -1}{V - 1}. \label{E:CharParam}\end{aligned}$$ The line $V=2$ corresponds to reducible representations (i.e. those which fix a point on $\si$) and the rational curve (\[E:CharParam\]) corresponds to conjugacy classes of irreducible representations (see [@Montesinos]). We can now calculate $\orth:\Char \dashrightarrow \param$ using the formula given in Proposition \[P:orth\]. The conditions (\[E:OrthDomain\]) are satisfied at all points of $\Char$ except the points with $(U,V)$ equal to $(4,2)$ or $(4,1 - \zeta)$, where $\zeta \in \mathbb{C}$ is a non-trivial cube root of unity. (These points correspond to the trivial representation and to the two orientation-preserving conjugacy classes of discrete and faithful representations.) Since the line $V=2$ corresponds to reducible representations (see [@Montesinos]) $\orth$ takes the whole line to the point $(1,1)$. On the curve (\[E:CharParam\]), Proposition \[P:orth\] implies that $$\begin{aligned} \orth(U,V) = \frac{1}{V^2 - 3V +3}(-V^2 + 3V -1, V^2-V -1). \label{E:lambda}\end{aligned}$$ From this expression it is easy to check that the image of $\orth$ lies inside the conic component of $\param$. From (\[E:lambda\]) and Figure \[F:tetn\_FP\], the ortholength invariant $\orth(U,V)$ lies in $\mathcal{S}$ whenever $$0 = (V-2)^2(V^2 +V-1) \mbox{ and } 0=(V -2)^2(V-1)^2(V^2+V-1),$$ where $(U,V)\in \Char$ belongs to the curve (\[E:CharParam\]), i.e. $\orth(U,V) \in \mathcal{S}$ when $V$ equals $2$ or $(-1 \pm \sqrt 5)/2$. Also, a simple calculation shows that $\orth(U,V) \in \param \subseteq \mathbb{C}^2$ has one or both co-ordinates equal to $\pm 1$ if and only if $V$ is $1$ or $2$. Hence by Theorem \[T:Param\], the image of $\orth$ is dense in the conic component of $\param$ and $\orth$ restricts to a birational equivalence between this component and the curve (\[E:CharParam\]). A simple calculation shows that the inverse of $\orth$ on this curve is given by $(p_0,p_1) \mapsto (U,V)$, where $$V = \frac{2p_0 + p_1 +1}{p_0+1}$$ and $U$ is given by the equation (\[E:CharParam\]). Future applications {#S:Conc} =================== The work presented in this paper arose out of an attempt to construct families of incomplete hyperbolic structures on a cusped $3$-manifold $M$ and to estimate the boundary of the deformation space $\defm$ in Dehn surgery co-ordinates. The connection between the ortholength invariant and these two problems is the concept of a [tube domain]{} (see [@Dowty]) which comes from collaborative work with Craig D. Hodgson. A tube domain can be defined for any (incomplete) hyperbolic structure in $\defm$, but the description is simplest for those hyperbolic structures which correspond to topological Dehn fillings of $M$. If $\mbar$ is a hyperbolic Dehn filling of $M$ then there is a covering isometry $\pi:\hyp \to \mbar$ and a distinguished simple closed geodesic $\Sigma$ in $\mbar$ so that $\mbar \setminus \Sigma$ is diffeomorphic to $M$. We define the [tube domain]{} $\dom$ of the Dehn filling to be (the closure of) the set $\widetilde{\dom}$ of points of $\hyp$ which are closer to a line $\widetilde{\Sigma} \subseteq \pi^{-1}(\Sigma)$ than to any other line of $\pi^{-1}(\Sigma)$, modulo the deck-transformations which preserve $\widetilde{\Sigma}$. There is a surjective map $\dom \to \mbar$ which makes the diagram $$\begin{array}{ccc} \widetilde{\dom} & \hookrightarrow & \hyp \\ \downarrow & & ~\downarrow \pi \\ \dom & \rightarrow & \mbar \end{array}$$ commute. The tube domain $\dom$ is diffeomorphic to a solid torus and its boundary is broken into (non-planar) faces which the map $\dom \to \mbar$ isometrically identifies in pairs. On the other hand, the ortholength invariant of the hyperbolic structure $\mbar \setminus \Sigma$ on $M$ determines the set of lines $\pi^{-1}(\Sigma)$ up to isometry (as detailed in the proof of Lemma \[L:ConRep\]). These in turn determine the corresponding tube domain $\dom$ and its face-pairing isometries, and so give us back the original hyperbolic structure $\mbar \setminus \Sigma$ on $M$. This gives us a method of constructing hyperbolic structures via tube domains which can be generalised to calculate hyperbolic structures with more general Dehn surgery-type singularities, including hyperbolic cone-manifolds whose singular sets are simple closed geodesics. This approach has been automated by Goodman-Hodgson in a computer program [@tube] called [Tube]{}. [Tube]{} naturally lends itself to calculating the tube radius and it has been used to calculate hyperbolic structures which [SnapPea]{} [@SnapPea] fails to compute. For instance, [Tube]{} has calculated a hyperbolic structure for m004(4,0) (the hyperbolic cone-manifold with cone-angles $\pi/2$ along the figure-8 knot) and for m007(3,1) (the manifold of third lowest volume on [SnapPea’s]{} closed census) while [SnapPea]{} fails to compute a hyperbolic structure for m004(4,0) and there is no known positively oriented tetrahedral decomposition of m007(3,1). Also, the cone-manifolds over the figure-8 knot complement with cone angle strictly less than $2\pi/3$ all have hyperbolic structures (see [@HLM] and also [@HodgsonThesis §29]) but for cone angles bigger than $2\pi/4.767 \ldots$ [SnapPea’s]{} tetrahedral construction fails. In fact, Choi [@Young] has shown that there cannot exist a positively-oriented hyperbolic ideal triangulation for the cone-manifold over the figure-8 knot complement when cone-angles are $2 \pi / 4.767 \ldots$ (i.e. when one of the tetrahedra of the canonical ideal triangulation flattens out). The second problem which motivates the study of the ortholength invariant is the estimation of the boundary of $\defm$. The ortholength invariant seems especially suited to this task because of its close relationship with the tube radius and because of some suggestive connections between the tube radius and the degeneration of hyperbolic structures on $M$ (e.g. see Kojima [@KojPi] and Hodgson-Kerckhoff [@Kerckhoff], [@HK]). For a given (incomplete) hyperbolic structure $M_0 \in \defm$, there is a (topological) ideal triangulation $K_0$ so that the tube radius of $M_0$ is given in terms of $\orthz(M_0)$. In particular, the tube radius of $M_0$ will be non-zero whenever all co-ordinates of $\orthz(M_0)$ lie in $\mathbb C \setminus [-1, 1]$. Each edge of this special ideal triangulation $K_0$ corresponds to a pair of faces of the tube domain of $M_0$. The combinatorics of the tube domains and hence of the corresponding ideal triangulations $K_0$ are locally constant in $\defm$ and also at the complete structure (see [@Dowty §4.5]). Since $\defm$ is non-compact near its boundary, it is conceivable that infinitely many inequivalent triangulations $K_0$ may arise as $M_0$ varies over $\defm$. However, we conjecture that this in not the case, i.e. for any given $M$ only a finite number $K_1, \ldots, K_m$ of (topological) ideal triangulations are needed to compute the tube radii of the incomplete hyperbolic structures in $\defm$. For our irreducible manifold $M$, the conjecture of [@Kerckhoff] reduces to the assertion that a family of hyperbolic structures degenerates if and only if their tube radii approach zero. This motivates the following conjecture. \[C:Conjecture\] Let $M$ be the underlying smooth manifold of an orientable, $1$-cusped hyperbolic 3-manifold of finite volume. Then there are a finite number of (topological) ideal triangulations $K_1, \ldots, K_m$ of $M$ so that a sequence of (incomplete) hyperbolic structures in $\defm$ has a limit in $\defm$ if and only if the corresponding $m$ sequences of ortholength invariants with respect to the ideal triangulations $K_1, \ldots, K_m$ all have limits with co-ordinates $\cosh d_i$ in $\mathbb{C} \setminus [-1, 1]$. For instance, when $M$ is the figure-8 knot complement, computational evidence suggests that this conjecture is true with $m = 1$ and $K_1$ the ideal triangulation of Section \[S:Eg\]. More specifically, in Dehn surgery co-ordinates the zero-volume set (which in this case is conjectured to be the boundary of $\defm$, see [@ThurstonNotes], [@HodgsonThesis]) appears to coincide with the frontier of the set of hyperbolic structures with $p_0, p_1 \not\in [-1, 1]\subseteq \mathbb{C}$, where $(p_0, p_1)$ is the ortholength invariant of Section \[S:Eg\]. Practical conditions to determine when the tube domain construction fails may provide rigorous estimates about the boundary of $\defm$, especially in special cases such as the figure-8 knot complement. Analogies between the construction (in practice) of Dirichlet domains and the construction of tube domains suggest possible necessary conditions for the failure of the tube domain construction though no useful estimates have emerged from this approach so far. We can also attempt to use [ortho-angles]{} (which are dual to the ortholengths) to parameterise $\defm$. The ortho-angles are more directly related to the Dehn surgery co-ordinates of $\defm$ than the ortholengths and they are very well-behaved near the complete structure. This makes the ortho-angle parameterisation of $\defm$ quite a promising line of research. Finally we note that even though we have restricted our attention in this paper to single-cusped $3$-manifolds, the main definitions and proofs[^17] trivially extend to $3$-manifolds with multiple cusps. [B]{} R. Benedetti and C. Petronio, [Lectures on hyperbolic geometry]{}, Springer, Berlin, 1992. M. Boileau and J. Porti, [Geometrization of 3-orbifolds of cyclic type]{}, [Asteŕisque]{}, [272]{}, 2001. S. Boyer and X. Zhang, [On Culler-Shalen seminorms and Dehn filling]{}, Ann. of Math. (2) [148]{} (1998), no. 3, 737–801. G. E. Bredon, [Topology and Geometry]{}, GTM 139, Springer, New York, 1993. Young-Eun Choi, [Positively oriented ideal triangulations on hyperbolic 3-manifolds]{}, Ph.D. thesis, Stanford, 2000. D. Cooper, C. Hodgson and S. Kerckhoff, [Three-dimensional Orbifolds and Cone-Manifolds]{}, Memoirs of the Mathematical Society of Japan, Vol 5, 2000. M. Culler and P. B. Shalen, [Varieties of group representations and splittings of $3$-manifolds]{}, Ann. of Math. [117]{} (1983), 109–146. J. G. Dowty, [Ortholengths and Hyperbolic Dehn Surgery]{}, Ph.D. thesis, Univ. of Melbourne, Australia, 2000. D. B. A. Epstein and R. C. Penner, [Euclidean decompositions of noncompact hyperbolic manifolds]{}, J. Differential Geom. [27]{} (1988), no. 1, 67–80. W. Fenchel, [Elementary geometry in hyperbolic space]{}, de Gruyter, Berlin, 1989. W. M. Goldman, [Geometric structures on manifolds and varieties of representations]{} in [Geometry of group representations (Boulder, CO, 1987)]{}, 169–198, Contemp. Math. 74, Amer. Math. Soc., Providence, RI, 1988. F. González-Acuña and J. M. Montesinos-Amilibia, [On the character variety of group representations in ${\rm SL}(2,\mathbb{C})$ and ${\rm PSL}(2,\mathbb{C})$]{}, Math. Z. [214]{} (1993), no. 4, 627–652. O. A. Goodman and C. D. Hodgson, [tube]{}, 1999. This program is available from: [http://www.ms.unimelb.edu.au/’176snap]{} J. Harris, [Algebraic Geometry: A First Course]{}, GTM 133, Springer, New York, 1992. H. Hilden, M. T. Lozano and J. M. Montesinos-Amilibia, [On a remarkable polyhedron geometrizing the figure eight knot cone manifolds]{}, J. Math. Sci. Univ. Tokyo [2]{} (1995), no. 3, 501–561. C. D. Hodgson, [Degeneration and regeneration of geometric structures on $3$-manifolds]{}, Ph. D. thesis, Princeton Univ., 1986. C. D. Hodgson and S. Kerckhoff, [Universal Bounds for Hyperbolic Dehn Surgery]{}, in preparation. S. P. Kerckhoff, [Deformations of hyperbolic cone-manifolds]{} in [Topology and Teichmüller spaces (Katinkulta, 1995)]{}, 101–114, World Sci. Publishing, River Edge, NJ, 1996. S. Kojima, [Deformations of hyperbolic $3$-cone-manifolds]{}, J. Differential Geom. [49]{} (1998), no. 3, 469–516. S. Kojima, [Hyperbolic $3$-manifolds singular along knots]{}, Chaos Solitons Fractals [9]{} (1998), no. 4-5, 765–777. G. R. Meyerhoff, [The ortho-length spectrum for hyperbolic $3$-manifolds]{}, Quart. J. Math. Oxford Ser. (2) [47]{} (1996), no. 187, 349–359. W. D. Neumann and D. Zagier, [Volumes of hyperbolic three-manifolds]{}, Topology [24]{} (1985), no. 3, 307–332. C. Petronio and J. Porti, [Negatively oriented ideal triangulations and a proof of Thurston’s hyperbolic Dehn filling theorem]{}, Expo. Math. [18]{} (2000), no. 1, 1–35. D. Rolfsen, [Knots and Links]{}, Publish or Perish, 1976. W. P. Thurston, [The geometry and topology of 3-manifolds]{}, lecture notes, Princeton Univ., 1978. These notes are available from: W. P. Thurston, [Three-dimensional manifolds, Kleinian groups and hyperbolic geometry,]{} [ Bull. Amer. Math. Soc. 6]{} (1982), 357–381. W. P. Thurston, [Three-dimensional geometry and topology. Vol. 1]{}, Princeton Univ. Press, Princeton, NJ, 1997. J. Weeks, [SnapPea, the computer program]{}, available from: [^1]: The complex distance between two lines is the hyperbolic distance between them plus $i$ times an angle of twist. [^2]: Not all homotopy classes of paths from the singular set to itself contain a distance-realising geodesic, however the ortholength invariant is defined purely in terms of the holonomy representation and so is well-defined in any case. [^3]: The complex distance between two lines is the hyperbolic distance between them plus $i$ times an angle of twist—see Fenchel [@Fenchel §V.3]. [^4]: $A$ acts as a projective transformation on $\si = \mathbb{CP}^1$. [^5]: \[FootnoteRef\]I am grateful to the referee for drawing my attention to the fact that $\Lines$ is the Lie algebra of $\mbox{SL}_2\mathbb{C}$, the corresponding action of $\mbox{SL}_2\mathbb{C}$ on $\Lines$ is the adjoint action, the form $\lacute \cdot,\cdot \racute$ (below) is a multiple of the Killing form and Lemma \[L:IsomL\] (below) is essentially the (known) result that $\isom$ is isomorphic to $\mbox{SO}_3\mathbb{C}$. [^6]: For $g \in \isom$ and $l\in\Lines$, $g\cdot l = \tilde{g} l \tilde{g}^{-1}$ where $\tilde{g} \in \mbox{SL}_2\mathbb{C}$ is one of the two matrices covering $g$. [^7]: An orientation-preserving isometry $g \in \isom$ is [parabolic]{} if $g \not= 1$ and $\tr^2g = 4$. [^8]: Since $\Char$ is not simply the quotient of the representation variety $\rep$ of $M$ by the conjugacy action of $\isom$ this is not immediately obvious. We need two more facts: (1) $\orth(\rho) = (1, \ldots, 1) \in \mathbb{C}^n$ whenever $\rho$ is reducible (i.e. whenever $\rho(\pi_1(M,))$ fixes a point on the sphere at infinity $\si$) and (2) if $\rho$ is irreducible and $\rho, \rho^\prime \in \rep$ project to the same point under the algebro-geometric quotient map $\rep \to \Char$ then $\rho$ and $\rho^\prime$ are conjugate (see [@BZ p. 753]). [^9]: The subvariety is proper since it doesn’t contain (the characters of) the holonomy representations of the (incomplete) hyperbolic structures of $\defm$. [^10]: Warning: Since $\Char$ is not necessarily irreducible it is possible that $\orth$ is not defined at all on some components of $\Char$. [^11]: The [second end-point]{} of an oriented line $(p,q) \in \endpts$ is $q$. [^12]: Since no $p_i = \pm 1$, the two lines have more than three end-points on $\si$. [^13]: A property is [generically]{} satisfied on a variety if it is true on the complement of a proper subvariety. [^14]: For large Dehn fillings the tube radius is large, see [@ThurstonNotes]. [^15]: The faces of the tetrahedra shown in Figure \[F:tetn\_FP\] are identified in pairs giving face-classes A,B,C and D. The class of each face is written at the vertex opposite the face. [^16]: Similarly to the $\isom$-character variety, the $\mbox{SL}_2\mathbb{C}$-character variety of $M$ is defined to be the algebro-geometric quotient of the space of representations $\pi_1(M,) \to \mbox{SL}_2 \mathbb{C}$ by the conjugacy action of $\mbox{SL}_2\mathbb{C}$, as defined in Culler-Shalen [@CS]. [^17]: In particular, Theorems \[T:Param\] and \[T:CP\] are true in the case of multiple cusps.
--- author: - \| M A Jivulescu$^{1,2}$, A Napoli$^1$, A Messina$^1$\ ${}^1$ MIUR, CNISM and Dipartimento di Scienze Fisiche ed Astronomiche,\ Università di Palermo, via Archirafi 36, 90123 Palermo, Italy\ ${}^2$ Department of Mathematics, “ Politehnica” University of Timişoara,\ P-ta Victoriei Nr. 2, 300006 Timişoara, Romania\ \ title: 'General solution of a second order non-homogenous linear difference equation with noncommutative coefficients ' --- SUMMARY The detailed construction of the general solution of a second order non-homogenous linear operator-difference equation is presented. The wide applicability of such an equation as well as the usefulness of its resolutive formula is shown by studying some applications belonging to different mathematical contexts. [Keywords:]{} difference equation, companion matrix, generating functions, noncommutativity. INTRODUCTION ============ In this paper we report the explicit representation of the general solution of the second order non-homogenous linear operator-difference equation $$\label{NHE} Y_{p+2}=\mathcal{L}_0Y_p+\mathcal{L}_1Y_{p+1}+\phi_{p+1},$$ where the unknown $\{Y_p\}_{p\in\mathbb{N}}$ as well as the non-homogenous term $\{\phi_p\}_{p\in \mathbb{N}}$ are sequences from a vectorial space $V$, and the coefficients $\mathcal{L}_0,\mathcal{L}_1$, are linear noncommutative operators mapping $V$ on itself, independent from the discrete variable $p\in \mathbb{N}$. This equation encompasses interesting problems arising in very different scenarios. If, for instance, the reference space $V$ is the complex Euclidean space $\mathbb{C}^n$, that is $Y_p$ and $\phi_{p+1}$ are $n$-dimensional vectors, $\mathcal{L}_0$ and $\mathcal{L}_1$ $n\times n$ complex matrices, then eq. (\[NHE\]) is the vectorial representation of a system of second-order linear non-homogenous difference equations. As another example, let’s identify $V$ as the vectorial space of all linear operators defined on a given Hilbert space. Now, the operators $\mathcal{L}_0$ and $\mathcal{L}_1$ act upon operators and for this reason are called superoperators. The master equations appearing in the theory of open quantum systems provide examples of equations belonging to this class[@Breuer]. It is of relevance to emphasize from the very beginning that the ingredients $Y_p$, $\phi_p$, $\mathcal{L}_0$ and $\mathcal{L}_1$ of eq. may be also interpreted as elements of an assigned algebra $V$. Let’s consider, for example, $V$ as the noncommutative algebra of all square matrices of order $n$, that is $M_n[\mathbb{C}]$. Then, eq. (\[NHE\]) defines a second order non-homogenous linear matrix-difference equation, where $Y_p, \phi_p, \mathcal{L}_0$ and $\mathcal{L}_1$ belong to $M_n(\mathbb{C})$. We wish further emphasis that if $V$ is the vectorial space of the smooth functions over an interval $I$, that is $C^\infty(I)$, then eq. (\[NHE\]) represents a wide class of functional-difference equations[@Kolmanovskii], including difference-differential equations or integro-difference equations[@Pinney; @Bellman; @Driver]. These few examples motivate the interest toward the search of techniques for solving the operator eq.(\[NHE\]), with $\mathcal{L}_0,\mathcal{L}_1$ noncommutative coefficients. In this paper we cope with such a problem and succeed in giving its explicit solution leaving unspecified the abstract “support space” wherein eq. (\[NHE\]) is formulated. This means that we do not choose from the very beginning the mathematical nature of its ingredients, rather we only require that all the symbols and operations appearing in eq. (\[NHE\]) are meaningful. Accordingly, vectors $Y_p$ may be added, this operation being commutative and, at the same time, may be acted upon by $\mathcal{L}_0$ or $\mathcal{L}_1$ ( hereafter called operators) transforming themselves into other vectorsof $V$. The symbol $Y_0=0$ simply denotes, as usual, the neutral element of the underlying space. Finally we put $(\mathcal{L}_a\mathcal{L}_b)Y\equiv \mathcal{L}_a(\mathcal{L}_bY)\equiv \mathcal{L}_a\mathcal{L}_bY$ with $a$ or $b=0,1$ and define addition between operators through linearity. The paper is organized as follows.\ The first section presents the solution of an arbitrary Cauchy problem associated with eq. . Some interesting consequences of such a result are derived in the subsequent section. The practical usefulness of our resolutive formula is shown in the third section where we solve some nontrivial functional-difference and integral-difference equations. Some concluding remarks are presented in the last section. EXPLICIT CONSTRUCTION OF THE RESOLUTIVE FORMULA OF EQ. ====================================================== Let’s begin by recalling that if $\{Y_p^{}\}_{p\in\mathbb{N}}$ and $\{Y_p\}_{p\in\mathbb{N}}$ are solutions of eq.(\[NHE\]), then $\{Y_p^{H}\}_{p\in\mathbb{N}}$ defined as $Y_p-Y_p^\equiv Y_p^{H}$ is a solution of the associated homogenous equation $$\label{HEQ} Y_{p+2}=\mathcal{L}_0Y_p+\mathcal{L}_1Y_{p+1}$$ Thus, as for the linear differential equations, and independently from the noncommutative nature of $\mathcal{L}_0$ and $\mathcal{L}_1$, solving eq. amounts at being able to construct the general integral of eq. and to find out a particular solution of eq. . To this end, we start with the following theorem which extends a recently published result[@Jivulescu] concerning the exact resolution of the following Cauchy problem$$\label{CP}\left\{\begin{array}{rl} Y_{p+2}=\mathcal{L}_0Y_p+\mathcal{L}_1Y_{p+1}, \\ Y_0=0,\quad Y_1=B\end{array}\right..$$ The solution of the Cauchy problem $$\label{HE}\left\{\begin{array}{rl} Y_{p+2}=\mathcal{L}_0Y_p+\mathcal{L}_1Y_{p+1}\\ Y_0=A,\quad Y_1=B\end{array}\right.,$$ can be written as $$\label{solH}Y_p^{(H)}=\alpha_pA+\beta_pB,$$ where the operators $\alpha_p$ and $\beta_p$ have the following form $$\label{alfa} \alpha_p=\left\{\begin{array}{ll} \sum\limits_{t=0}^{[\frac{p-2}{2}]}\{\mathcal{L}_0^{(t)}\mathcal{L}_1^{(p-2-2t)}\}\mathcal{L}_0&\quad if \quad p\geq2 \\ 0&\quad if \quad p=1\\ E&\quad if \quad p=0 \end{array}\right.,$$ $$\label{beta} \beta_p=\left\{\begin{array}{ll}\sum\limits_{t=0}^{[\frac{p-1}{2}]}\{\mathcal{L}_0^{(t)}\mathcal{L}_1^{(p-1-2t)}\} & \quad if \quad p\geq 2\\E&\quad if \quad p=1\\ 0& \quad if \quad p=0 \end{array}\right.,$$ We recall that the mathematical symbol $\{\mathcal{L}_0 ^{(u)}\mathcal{L}_1^{(v)}\}$, in accordance with ref [@Jivulescu], denotes the sum of all possible distinct permutations of $u$ factors $\mathcal{L}_0$ and $v$ factors $\mathcal{L}_1$, while $0$, $E:V\rightarrow V$ define the null and the identity operator in $V$, respectively. We omit the proof of this theorem since it is practically coincident with that given in ref [@Jivulescu]. Here instead we demonstrate the following Eq. (\[NHE\]) admits the particular solution $$\label{SOLP} Y_p^=\left\{\begin{array}{ll} \sum\limits_{r=1}^{p-1}\beta_{p-r}\phi_r, &\quad if \quad p\geq2 \\ 0,&\quad if \quad p=0,1\\ \end{array}\right.$$ It is immediate to verify, by direct substitution, that the sequence given by eq. (\[SOLP\]) satisfies eq. (\[NHE\]) written for $p=0$ and $p=1$. To this end, it is enough to exploit eqs. and getting $Y_2^=\beta_1\phi_1=\phi_1$ and $Y_3^=\beta_2\phi_1+\beta_1\phi_2=\mathcal{L}_1\phi_1+\phi_2$.\ For a generic $p\geq 2$, introducing $Y_p^$ in the right hand of eq. (\[NHE\]) yields $$\begin{aligned} \label{10}\mathcal{L}_0\sum\limits_{r=1}^{p-1}\beta_{p-r}\phi_r+\mathcal{L}_1\sum\limits_{r=1}^{p}\beta_{p+1-r}\phi_r+\phi_{p+1}\nonumber\\ =\label{OPT}\sum\limits_{r=1}^{p-1}(\mathcal{L}_0\beta_{p-r}+\mathcal{L}_1\beta_{p+1-r})\phi_r+\mathcal{L}_1\beta_1\phi_{p}+\phi_{p+1}\end{aligned}$$ Applying theorem (1) to the Cauchy problem expressed by eq. , we easily deduce that for $p\geq 2$ and $r=1,2,\dots, p-1$ the following operator identity $$\mathcal{L}_0\beta_{p-r}+\mathcal{L}_1\beta_{p+1-r}=\beta_{p+2-r},\quad$$holds. Thus, the expression given by eq. may be cast as follows $$\begin{aligned} \label{11} \sum\limits_{r=1}^{p-1}\beta_{p+2-r}\phi_r+\beta_{p+2-(p)}\phi_{p}+\beta_{p+2-(p+1)}\phi_{p+1}=\sum\limits_{r=1}^{p+1}\beta_{p+2-r}\phi_r\end{aligned}$$ where we have exploited the identity $\beta_2=\mathcal{L}_1\beta_1$ based on eq. (\[beta\]). Since the right hand of eq. coincides with $Y_{p+2}^$ as given by eq. (\[SOLP\]), we may conclude that $\{Y_p^\}_{p\in \mathbb{N}}$, expressed by eq. , provides a particular solution of eq. . On the basis of theorem (1) and (2) we hence may state our main result, that is The solution of the Cauchy problem $$\label{CPPNH}\left\{\begin{array}{rl} Y_{p+2}=\mathcal{L}_0Y_p+\mathcal{L}_1Y_{p+1}+\phi_{p+1}\\ Y_0=A,\quad Y_1=B\end{array}\right.,$$ is $$\label{solNH}Y_p=\alpha_pA+\beta_pB+\sum\limits_{r=1}^{p-1}\beta_{p-r}\phi_r$$ where $A$ and $B$ are generic admissible initial conditions and $\alpha_p$ and $\beta_p$ are defined by eqs. and , respectively. We emphasize that eq. furnishes a recipe to solve explicitly, that is in terms of its ingredients $\mathcal{L}_0, \mathcal{L}_1$ and $\{\phi_{p+1}\}_{p\in\mathbb{N}}$, the general Cauchy problem expressed by eq. . In the subsequent sections we will highlight that our result is effectively exploitable, providing indeed a useful approach to solve problems belonging to very different mathematical contexts. This circumstance adds a further robust motivation to investigate eq. and its consequences. We conclude this section looking for the structural form assumed by eq. solely relaxing the noncommutativity between the two operator coefficients $\mathcal{L}_0$ and $\mathcal{L}_1$. To this end, it is useful to recall the definition of the Chebyshev polynomials of the second kind $\mathcal{U}_p(x), x\in \mathbb{C}$[@SCH] $$\label{Cby}\mathcal{U}_p(x)=\sum\limits_{m=0}^{[p/2]}(-1)^m\frac{(p-m)!}{m!(p-2m)!}(2x)^{p-2m}$$ Indeed, taking into consideration that the number of all the different terms appearing in the operator symbol $\{\mathcal{L}_0 ^{(u)}\mathcal{L}_1^{(v)}\}$ coincides with the binomial coefficient $ \left(\begin{array}{c} u+v\\ m \end{array} \right)$, with $m=min(u,v)$ as well as assuming the existence of the operator $(-\mathcal{L}_0)^{-\frac{1}{2}}$, then the operators $\alpha_p$ and $\beta_p$ for $p\geq 2$ may be cast as follows $$\label{C11}\alpha_p=-(-\mathcal{L}_0)^{\frac{p}{2}}\mathcal{U}_{p-2}\left(\frac{1}{2}\mathcal{L}_1(-\mathcal{L}_0)^{-\frac{1}{2}}\right)$$ and $$\label{C22}\beta_p=(-\mathcal{L}_0)^{\frac{p-1}{2}}\mathcal{U}_{p-1}\left(\frac{1}{2}\mathcal{L}_1(-\mathcal{L}_0)^{-\frac{1}{2}}\right)$$ where $\mathcal{U}_p\left(\frac{1}{2}\mathcal{L}_1(-\mathcal{L}_0)^{-\frac{1}{2}}\right)$ means the operator value of the polynomial $\mathcal{U}_p$ defined in accordance with eq. for $x=\left(\frac{1}{2}\mathcal{L}_1(-\mathcal{L}_0)^{-\frac{1}{2}}\right)$. Thus, the solution of Cauchy problem may be rewritten, for $p\geq 2$, as $$\begin{aligned} \label{POWC}Y_p=(-\mathcal{L}_0)^{\frac{p-1}{2}}\mathcal{U}_{p-1}\left(\frac{1}{2}\mathcal{L}_1(-\mathcal{L}_0)^{-\frac{1}{2}}\right)B-(-\mathcal{L}_0)^{\frac{p}{2}}\mathcal{U}_{p-2}\left(\frac{1}{2}\mathcal{L}_1(-\mathcal{L}_0)^{-\frac{1}{2}}\right)A+\nonumber\\+\sum\limits_{r=1}^{p-1}(-\mathcal{L}_0)^{\frac{p-r-1}{2}}\mathcal{U}_{p-r-1}\left(\frac{1}{2}\mathcal{L}_1(-\mathcal{L}_0)^{-\frac{1}{2}}\right)\phi_{r}\end{aligned}$$ where $Y_0=A$ and $Y_1=B$ are the prescribed initial conditions. SOME CONSEQUENCES OF THE RESOLUTIVE FORMULA =========================================== The mathematical literature offers several ways of solving linear second difference equations such as the matrix method or the generating function method. In the following we will heuristically generalized these methods to the operator case. The novelty of our method enables to deduce, by comparison with these approaches, some interesting consequent identities. Indeed, the second-order operator difference equation may be traced back to the first-order vectorial representation $$\label{ME}\mathbf{Y}_{p+1}=C_1\mathbf{Y}_p+\Phi_{p+1}$$ where $\mathbf{Y}_p=\left(\begin{array}{c} Y_{p}\\Y_{p+1} \end{array} \right)$, $C_1=\left(\begin{array}{cc}0&E\\ \mathcal{L}_0&\mathcal{L}_1 \end{array} \right)$, $\Phi_{p+1}=\left(\begin{array}{c} 0\\\phi_{p+1} \end{array} \right)$, $\mathbf{Y}_0=\left(\begin{array}{c} A\\B \end{array} \right)$. Successive iterations easily lead us to the formal solution $$\label{c2}\mathbf{Y}_p=C_1^{p}\mathbf{Y}_0+\sum\limits_{r=1}^{p}C_1^{p-r}\Phi_{r}$$ On this basis, the solution of eq. may be written as [@Gohberg] $$Y_p=P_1C_1^{p}\mathbf{Y}_0+P_1\sum\limits_{r=1}^{p}C_1^{p-r}R_1\Phi_{r}$$ where $P_1=(E \quad 0)$ and $R_1=\left(\begin{array}{c} 0\\E \end{array} \right)$. This solution is of practical use only if we are able to evaluate the general integer power of the companion matrix $C_1$. Exploiting our procedure of writing the solution of eq. , the vector $\mathbf{Y}_p$ may be expressed, accordingly with eq. (\[solNH\]), in terms of operator sequences $\alpha_p$ and $\beta_p$ like $$\label{c1}\mathbf{Y}_p=\left(\begin{array}{c} \alpha_{p}A+\beta_{p}B+\sum\limits_{r=1}^{p-1}\beta_{p-r}\phi_r\\\alpha_{p+1}A+\beta_{p+1}B+\sum\limits_{r=1}^{p}\beta_{p+1-r}\phi_r \end{array} \right)=\left(\begin{array}{cc} \alpha_{p}&\beta_{p}\\\alpha_{p+1}&\beta_{p+1} \end{array} \right)\left(\begin{array}{c} A\\B \end{array} \right)+\left(\begin{array}{c} \sum\limits_{r=1}^{p-1}\beta_{p-r}\phi_r\\\sum\limits_{r=1}^{p}\beta_{p+1-r}\phi_r \end{array} \right)$$ Confining ourselves to the homogenous version of eq. , that is putting $\phi_{p+1}=0$ into eqs. and , we get the formula for the $p$-th power of the companion matrix $C_1$ as follows $$\label{PowC1}C_1^p=\left(\begin{array}{cc} 0&E\\ \mathcal{L}_0&\mathcal{L}_1 \end{array} \right)^p=\left(\begin{array}{cc} \alpha_{p}&\beta_{p} \\\alpha_{p+1}&\beta_{p+1} \end{array} \right)$$ Another possible way of treating eq. is via the generating functions method[@Goldberg; @Hildebrand]. We recall that, given the sequence $\{Y_p\}_{p\in \mathbb{N}}$, the associated generating function $Y(s), s\in \mathbb{C}$ is defined as $$\label{Gen}\mathcal{G}Y_p\equiv Y(s)\equiv\sum\limits_{p=0}^{\infty}Y_ps^p$$ under the assumption that the series converges when $\|s\|\leq \xi$, for some positive number $\xi$. The advantage of this method consists in the systematical possibility of transforming a difference equation in an algebraic one in the unknown $Y(s)$. In order to apply such approach to the operator-difference equation given by , we stipulate that $\mathcal{G}[\mathcal{L}_iY_p]$, $\mathcal{L}_i(\mathcal{G}Y_p)$ are both defined and that $\mathcal{G}[\mathcal{L}_iY_p]=\mathcal{L}_i(\mathcal{G}Y_p), i=0,1$. Accordingly, heuristically, we transform both sides of eq. getting $$\begin{aligned} \nonumber \frac{Y(s)-A-Bs}{s^2}=\mathcal{L}_1\frac{Y(s)-A}{s}+\mathcal{L}_0Y(s)+\Phi(s)\end{aligned}$$ Thus, assuming the existence of $(E-\mathcal{L}_1s-\mathcal{L}_0s^2)^{-1}$ within the convergence disk, we have $$Y(s)=(E-\mathcal{L}_1s-\mathcal{L}_0s^2)^{-1}A+(E-\mathcal{L}_1s-\mathcal{L}_0s^2)^{-1}(B-\mathcal{L}_1A)s+(E-\mathcal{L}_1s-\mathcal{L}_0s^2)^{-1}\Phi(s)s^2$$ or equivalently $$\begin{aligned} Y(s)=(E-\mathcal{L}_1s-\mathcal{L}_0s^2)^{-1}[E-\mathcal{L}_1 s]A+(E-\mathcal{L}_1s-\mathcal{L}_0s^2)^{-1}Bs+(E-\mathcal{L}_1s-\mathcal{L}_0s^2)^{-1}\Phi(s) s^2\end{aligned}$$ On the other hand, accordingly with eqs. and it holds that $$Y(s)=\sum\limits_{p=0}^{\infty}(\alpha_pA+\beta_pB+\sum\limits_{r=1}^{p-1}\beta_{p-r}\phi_r)s^p$$ Thus, one notes that imposing $\phi_{p+1}\equiv 0$ and $B=0$, respectively $A=0$, we heuristically find the generating function of the operator sequences $\alpha_p$ and $\beta_p$ in the closed form as $$\label{GenA}\mathcal{G}\alpha_p\equiv\sum\limits_{p=0}^{\infty}\alpha_ps^p:=(E-\mathcal{L}_1s-\mathcal{L}_0s^2)^{-1}[E-\mathcal{L}_1s]$$ respectively $$\label{GenB}\mathcal{G}\beta_p\equiv\sum\limits_{p=0}^{\infty}\beta_ps^p:=(E-\mathcal{L}_1s-\mathcal{L}_0s^2)^{-1}s$$ The particular case $\mathcal{L}_0=-E$ reproduces the generating functions of the Chebyshev polynomials of second kind. Extracting indeed for the sake of convenience the first two terms of the series, that is writing $\nonumber \sum\limits_{p=0}^{\infty}\alpha_ps^p=\alpha_0+\alpha_1s+\sum\limits_{p=2}^{\infty}\alpha_ps^p$ with the help of eq. and $\alpha_0=E, \alpha_1=0$ we get $$\begin{aligned} \label{last}(E-\mathcal{L}_1s+s^2)^{-1}[E-\mathcal{L}_1s]=E-\sum\limits_{p=2}^{\infty}\mathcal{U}_{p-2}[\frac{\mathcal{L}_1}{2}]s^p\end{aligned}$$ The eq. easily determines the generating function of the sequence $\{U_p[\mathcal{L}_1/2]\}_{p\in \mathbb{N}}$ in the form $$\begin{aligned} \nonumber\sum\limits_{p=2}^{\infty}\mathcal{U}_{p-2}[\frac{\mathcal{L}_1}{2}]s^p=(E-\mathcal{L}_1s+s^2)^{-1}[E-\mathcal{L}_1s+s^2-E+\mathcal{L}_1s]\\ \Leftrightarrow \sum\limits_{p=0}^{\infty}\mathcal{U}_{p}[\frac{\mathcal{L}_1}{2}]s^p=(E-\mathcal{L}_1s+s^2)^{-1}E\end{aligned}$$ The novel results obtained in this paper exploiting our resolutive formula ( eqs. , , ) clearly evidence that our recipe to manage eq. successfully integrate with other resolutive methods. Thus, we may claim that our resolutive formula do not possess a formal character only, since it helps to provide new interesting identities. APPLICATIONS OF OUR RESOLUTIVE FORMULA ====================================== An example of matrix-difference equation coped with our formula\ Let us consider the second order matrix-difference eqution $$\begin{aligned} Y_{p+2}=M_0Y_p+M_1Y_{p+1}+\Phi_{p+1}\end{aligned}$$ where $M_0,M_1\in \mathcal{M}_n(\mathbb{C})$ are noncommutative nilpotent matrices of index $2$, that is $M_i^2=0, \quad i=0,1$. Prescribing the initial conditions $Y_0=A, Y_1=B$, then the solution of this equation is given by the eq. (\[solNH\]). The analysis of the matrix term $\{M_0^{(u)}M_1^{(v)}\}$ which appears in the composition of the matrix-operator $\alpha_p$ and $\beta_p$ brings to light interesting peculiarities due to the specific nature of the coefficients $M_0$ and $M_1$. By definition, the term $\{M_0^{(u)}M_1^{(v)}\}$ represents the sum of all possible terms of $u$ factor $M_0$ and $v$ factors $M_1$. Thus, it is quite simple to deduce that now the matrix-term of the form $M_0^{\nu_1}M_1^{\nu_2}M_0^{\nu_3}M_1^{\nu_4}\dots$ is equal with zero, if $\nu_i>1, (\forall) i$. Hence, we deduce that the operator $\{M_0^{(u)}M_1^{(v)}\}$ survives solely when $u=v$ or $v=u\pm 1$. Indeed, when $u=v, u\geq2$, then in the sum $\{M_0^{(u)}M_1^{(u)}\}$ survive only the terms $\underbrace{[M_0M_1][M_0M_1]\dots [M_0M_1]}_{u-times}$ and $\underbrace{[M_1M_0][M_1M_0]\dots [M_1M_0]}_{u-times}$. Further, the only nonvanishing matrix-terms $\{M_0^{(u)}M_1^{(u+1)}\}$ are $M_1\underbrace{[M_0M_1][M_0M_1]\dots[M_0M_1]}_{u-times}$, as well as from $\{M_0^{(u)}M_1^{(u-1)}\}$ the terms $\underbrace{[M_0M_1][M_0M_1]\dots[M_0M_1]}_{(u-1)-times}M_0$, respectively. Exploiting the above results for the matrix term $\{M_0^{(t)}M_1^{(p-1-2t)}\}$ we establish that $$\begin{aligned} \beta_p=\left\{\begin{array}{rl} \{M_0^{(\frac{p-1}{3})}M_1^{(\frac{p-1}{3})}\}=\underbrace{[M_0M_1]\dots[M_0M_1]}_{k-times}+\underbrace{[M_1M_0]\dots[M_1M_0]}_{k-times}, \quad p=3k+1 \\\vspace{0.5cm} \{M_0^{(\frac{p-2}{3})}M_1^{(\frac{p+1}{3})}\}=M_1\underbrace{[M_0M_1]\dots [M_0M_1]}_{k-times},\quad p=3k+2 \\\vspace{0.5cm} \{M_0^{(\frac{p}{3})}M_1^{(\frac{p-3}{3})}\}=\underbrace{[M_0M_1]\dots [M_0M_1]}_{(k-1)-times}M_0,\quad p=3k, \end{array}\right.\end{aligned}$$ where $ k=1,2,\dots$. Similarly, we have that $$\begin{aligned} \alpha_p=\left\{\begin{array}{rl}\vspace{0.2cm} \{M_0^{(\frac{p-2}{3})}M_1^{(\frac{p-2}{3})}\}M_0=\underbrace{[M_0M_1]\dots[M_0M_1]}_{k-times}M_0,\quad p=3k+2 \\\vspace{0.2cm} \{M_0^{(\frac{p-3}{3})}M_1^{(\frac{p}{3})}\}M_0=M_1\underbrace{[M_0M_1]\dots[M_0M_1]}_{(k)-times}M_0,\quad p=3k+3\\\vspace{0.2cm} \{M_0^{(\frac{p-1}{3})}M_1^{(\frac{p-4}{3})}\}M_0=(\underbrace{[M_0M_1]\dots [M_0M_1]}_{(k-1)-times}M_0)M_0=0,\quad p=3k+1\end{array}\right.\end{aligned}$$ Hence, we may write the solution into a closed form $$\begin{aligned} \nonumber Y_p=\left[\delta_{\frac{p-2}{3},\left[\frac{p-2}{3}\right]}(M_0M_1)^{[\frac{p-2}{3}]}M_0+\delta_{\frac{p-3}{3},\left[\frac{p-3}{3}\right]}M_1(M_0M_1)^{[\frac{p-3}{3}]}M_0\right]A+\end{aligned}$$$$\begin{aligned} [\delta_{\frac{p-1}{3},\left[\frac{p-1}{3}\right]}(M_0M_1)^{[\frac{p-1}{3}]}+\delta_{\frac{p-1}{3},\left[\frac{p-1}{3}\right]}(M_1M_0)^ {[\frac{p-1}{3}]}+\delta_{\frac{p-2}{3},\left[\frac{p-2}{3}\right]}M_1(M_0M_1)^{[\frac{p-2}{3}]}+\delta_{\frac{p-3}{3},\left[\frac{p-3}{3}\right]}(M_0M_1) ^{[\frac{p-3}{3}]}M_0]B\nonumber\end{aligned}$$ $$\begin{aligned} +\sum\limits_{r=1}^{p-1}[\delta_{\frac{p-r-1}{3},\left[\frac{p-r-1}{3}\right]}\left((M_0M_1)^{[\frac{p-r-1}{3}]}+(M_1M_0)^{[\frac{p-r-1}{3}]}\right)+ \nonumber \\\delta_{\frac{p-r-2}{3},\left[\frac{p-r-2}{3}\right]}M_1(M_0M_1)^{[\frac{p-r-2}{3}]}+\delta_{\frac{p-r-3}{3},\left[\frac{p-r-3}{3}\right]}(M_0M_1)^ {[\frac{p-r-3}{3}]}M_0]\Phi_r\end{aligned}$$ An example of functional-difference equation coped with our method\ The three-term recurrence relation $$\label{ee}f_{p+2}(t)=-f_p(t-\tau_0)+f_{p+1}(t+\tau_1)$$ with the initial conditions $A=f_0(t)$ and $B=f_1(t)$ is an example of functional difference equation, traceable back to eq. . It is indeed well-known that if $f(t)$ is a function of class $C^{\infty}$, then the translation of its independent variable from $t$ to $t+\tau$ can be represented as the effect on the same function of the operator $exp[\tau \frac{d\cdot}{dt}]=\sum\limits_{k=0}^{\infty}\frac{1}{k!}[\tau \frac{d\cdot}{dt}]^k$. This operator appears in a natural way when one studies problems characterized by translational invariance in a physical context[@Sakurai]. Thus, by putting $\mathcal{L}_{i}=(-1)^{i+1}exp[(-1)^{i+1}\tau_i\frac{d.}{dt}],i=0,1$ the commutativity property of the two operator coefficients $\mathcal{L}_0$ and $\mathcal{L}_1$ allows us to write down the solution of eq. as follows $$\begin{aligned} \nonumber f_p(t)=\exp[-(\frac{p-1}{2})\tau_0\frac{d}{dt}]\mathcal{U}_{p-1}\left[\frac{1}{2}\exp[(\tau_1+\frac{\tau_0}{2})\frac{d}{dt}]\right]f_1(t)-\\ \exp[-(\frac{p}{2})\tau_0\frac{d}{dt}]\mathcal{U}_{p-2}\left[\frac{1}{2}\exp[(\tau_1+\frac{\tau_0}{2})\frac{d}{dt}]\right]f_0(t)\end{aligned}$$ Exploiting eq. we may write down that $$\begin{aligned} f_p(t)=\sum\limits_{k=0}^{[p-1/2]}(-1)^k \left(\begin{array}{c} p-1-k\\k \end{array} \right)f_1\left(t+(p-1-2k)\tau_1-k\tau_0\right)-\\ \sum\limits_{k=0}^{[p-2/2]}(-1)^k \left(\begin{array}{c} p-2-k\\k \end{array} \right)f_0\left(t+(p-2-2k)\tau_1-(k+1)\tau_0\right)\end{aligned}$$ Imposing, for example, the following initial conditions $f_0(t)=e^{-t}$ and $f_1(t)=e^{t}$ we get $$\begin{aligned} f_p(t)= \exp[-(\frac{p-1}{2})\tau_0]\mathcal{U}_{p-1}\left[\frac{1}{2}\exp[(\tau_1+\frac{\tau_0}{2})]\right]e^t-\\ \exp[( \frac{p}{2})\tau_0]\mathcal{U}_{p-2}\left[\frac{1}{2}\exp[-(\tau_1+\frac{\tau_0}{2})]\right]e^{-t}\end{aligned}$$\ An example of integro-difference equation coped with our method\ Consider the difference-differential equation $$\begin{aligned} f_{p+2}'(t)=\beta f_{p+1}(t)+\alpha f_{p}(t),\quad \alpha, \beta\in\mathbb{R}, \quad p=0,1,\dots\end{aligned}$$ where $f_p(t)$ is a $C^{\infty}(I)$ function with $f_0(t)$, $f_1(t)$ and $\{f_p(0),\quad p=0,1,\dots\}$ prescribed functions.\ The above equation may be rewritten in the equivalent form $$\begin{aligned} \label{41}f_{p+2}(t)=\mathcal{L}_1f_{p+1}(t)+\mathcal{L}_0f_p(t)+f_{p+2}(0)\end{aligned}$$ where $\mathcal{L}_0=\alpha \mathcal{L}$, $\mathcal{L}_1=\beta \mathcal{L}$ and $\mathcal{L}(\cdot)=\int\limits_0^t\cdot d\tau$. Eq. is a particular case of eq..\ The explicit solution of this equation requires the knowledge of the operator terms $\alpha_p$ and $\beta_p$. One remarks that $\beta_p$ is the sum of $\left[\frac{p-1}{2}\right]+1$ operator terms of the form $\{\mathcal{L}_0^t\mathcal{L}_1^{p-1-2t}\}$. Because $\mathcal{L}_0=\alpha \mathcal{L}$ and $\mathcal{L}_1=\beta \mathcal{L}$ then, for a finite $p$, holds $$\begin{aligned} \{\mathcal{L}_0^k\mathcal{L}_1^{p-1-2k}\}=\alpha^k\beta^{p-1-2k} \left(\begin{array}{c} p-1-k\\ k \end{array} \right)\mathcal{L}^{p-1-k}\end{aligned}$$ Therefore, by direct substitution into eq. it follows that $$\begin{aligned} \beta_p=\sum\limits_{k=0}^{\left[\frac{p-1}{2}\right]}\alpha^k\beta^{p-1-2k} \left(\begin{array}{c} p-1-k\\ k \end{array} \right)\mathcal{L}^{p-1-k}\end{aligned}$$ Similarly, we have that $$\begin{aligned} \alpha_p=\sum\limits_{k=0}^{\left[\frac{p-2}{2}\right]}\alpha^{k+1}\beta^{p-2-2k} \left(\begin{array}{c} p-2-k\\ k \end{array} \right)\mathcal{L}^{p-1-k}\end{aligned}$$ The solution of the corresponding homogenous equation in accordance with the prescribed initial conditions is then $$\begin{aligned} \nonumber f_p^{(H)}(t)=\sum\limits_{k=0}^{\left[\frac{p-2}{2}\right]}\alpha^{k+1}\beta^{p-2-2k} \left(\begin{array}{c} p-2-k\\ k \end{array} \right)\mathcal{L}^{p-1-k}\left(f_0(t)\right)+\\\sum\limits_{k=0}^{\left[\frac{p-1}{2}\right]}\alpha^k\beta^{p-1-2k} \left(\begin{array}{c} p-1-k\\ k \end{array} \right)\mathcal{L}^{p-1-k}\left(f_1(t)\right)\end{aligned}$$ Exploiting our central theorem (2), we may claim that $$\begin{aligned} f_p^=\sum\limits_{m=1}^{p-1}\beta_{p-m}f_{m+1}(0)= \sum\limits_{m=1}^{p-2}\beta_{p-m}f_{m+1}(0)+f_p(0)\end{aligned}$$ is the particular solution of the nonhomogenous equation for which $f_0=f_1=0, (\forall) t$. Equivalently, we have that $$\begin{aligned} f_p^= \sum\limits_{m=1}^{p-2}\sum\limits_{k=0}^{\left[\frac{p-m-1}{2}\right]}\alpha^k\beta^{p-m-1-2k} \left(\begin{array}{c} p-m-1-k\\ k \end{array} \right)\mathcal{L}^{p-m-1-k}\left(f_{m+1}(0)\right)+f_p(0) \end{aligned}$$ But, as shown in the Appendix, we may prove that $$\begin{aligned} \mathcal{L}^n(f(t))=\int\limits_0^tdt_n\int\limits_0^{t_n}dt_{n-1}\int\limits_0^{t_{n-1}}dt_{n-2}\dots\int\limits_0^{t_3}dt_2\int\limits_0^{t_2}f(t_1)dt_1=\frac{1}{(n-1)!}\int\limits_0^t(t-\tau)^{n-1}f(\tau)d\tau,\end{aligned}$$ so that we may write down that $$\begin{aligned} f_p^{(H)}(t)=\sum\limits_{k=0}^{\left[\frac{p-2}{2}\right]}\alpha^{k+1}\beta^{p-2-2k} \left(\begin{array}{c} p-2-k\\ k \end{array} \right)\frac{1}{(p-2-k)!}\int\limits_0^t(t-\tau)^{p-2-k}f_0(\tau)d\tau+\nonumber\\\sum\limits_{k=0}^{\left[\frac{p-1}{2}\right]}\alpha^k\beta^{p-1-2k} \left(\begin{array}{c} p-1-k\\ k \end{array} \right)\frac{1}{(p-2-k)!}\int\limits_0^t(t-\tau)^{p-2-r}f_1(\tau)d\tau \end{aligned}$$and $$\begin{aligned} \nonumber f_p^=\sum\limits_{m=1}^{p-2}\sum\limits_{k=0}^{\left[\frac{p-m-1}{2}\right]}\alpha^k\beta^{p-m-1-2k} \left(\begin{array}{c} p-m-1-k\\ k \end{array} \right)\frac{1}{(p-m-2-k)!}\int\limits_0^t(t-\tau)^{p-m-2-k}f_{m+1}(0)d\tau+f_p(0)\end{aligned}$$ Hence, the general solution of the proposed integral-difference equation is $$\begin{aligned} f_p(t)=\sum\limits_{k=0}^{\left[\frac{p-2}{2}\right]}\alpha^{k+1}\beta^{p-2-2k} \left(\begin{array}{c} p-2-k\\ k \end{array} \right)\frac{1}{(p-2-k)!}\int\limits_0^t(t-\tau)^{p-2-k}f_0(\tau)d\tau+\nonumber\\\sum\limits_{k=0}^{\left[\frac{p-1}{2}\right]}\alpha^k\beta^{p-1-2k} \left(\begin{array}{c} p-1-k\\ k \end{array} \right)\frac{1}{(p-2-k)!}\int\limits_0^t(t-\tau)^{p-2-k}f_1(\tau)d\tau+\\ \sum\limits_{m=1}^{p-2}\sum\limits_{k=0}^{\left[\frac{p-m-1}{2}\right]}\alpha^k\beta^{p-m-1-2k} \left(\begin{array}{c} p-m-1-k\\ k \end{array} \right)\frac{t^{p-m-1-k}}{(p-m-2-k)!(p-m-1-k)!}f_{m+1}(0)+f_p(0) \end{aligned}$$ CONCLUSIVE REMARKS ================== The novel and mean theoretical result of this paper is expressed by theorem (2) with which we demonstrate that eq. provides a particular solution of eq. . This result together with theorem (1) completes the resolution of this equation enabling us to write down formula for its general solution. The operator character of eq. and, as a consequence, the presence of generally noncommuting coefficients is the key to understand why such an equation may represent the canonical form of equations seemingly not related each other. The consequences of eq. and the applications reported in this paper, besides being interesting in their own, demonstrate indeed both the wide applicability of eq. as well as the practical usefulness of its resolutive formula. APPENDIX ======== For the sake of completeness we here report a proof of the well-known follwing identity $$\begin{aligned} \label{formula}\int\limits_0^tdt_n\int\limits_0^{t_n}dt_{n-1}\int\limits_0^{t_{n-1}}dt_{n-2}\dots\int\limits_0^{t_3}dt_2\int\limits_0^{t_2}f(t_1)dt_1=\frac{1}{(n-1)!}\int\limits_0^t(t-\tau)^{n-1}f(\tau)d\tau,\end{aligned}$$ where $f(t)$ is a $C^{\infty}$-function. The mathematical induction procedure will be exploited.\ For $n=1$ the above formula becomes the identity $$\begin{aligned} \int\limits_0^tf(t_1)dt_1=\int\limits_0^tf(\tau)d\tau\end{aligned}$$ Let’s suppose that the formula \[formula\] holds for any $r\leq n$ and we prove it validity for $n+1$. To proceed, it is convenient to define $$\begin{aligned} F_r(t)=\int\limits_0^tdt_r\int\limits_0^{t_r}dt_{r-1}\int\limits_0^{t_{r-1}}dt_{r-2}\dots\int\limits_0^{t_3}dt_2\int\limits_0^{t_2}f(t_1)dt_1\end{aligned}$$ writing down eq. as follows $$F_n(t)=\frac{1}{(n-1)!}\int\limits_0^t(t-\tau)^{n-1}f(\tau)d\tau$$ It is obviously that by definition $$\begin{aligned} F_{r+1}(t)=\int\limits_0^tF_r(t_{r+1})dt_{r+1}, \quad r\geq 1\end{aligned}$$ By the induction hypothesis we have that $$\begin{aligned} F_{n+1}'=F_n(t)=\frac{1}{(n-1)!}\int\limits_0^t(t-\tau)^{n-1}f(\tau)d\tau\end{aligned}$$ Our problem becomes the resolution of the following Cauchy problem $$\begin{aligned} \label{A} \left\{\begin{array}{rl} F_{n+1}'(t)=\frac{1}{(n-1)!}\int\limits_0^t(t-\tau)^{n-1}f(\tau)d\tau\\ F_{n+1}(0)=0\end{array}\right.\end{aligned}$$ From the well-known Leibnitz identity $$\begin{aligned} \frac{d}{dx}\int\limits_{a(x)}^{b(x)}f(x,y)dy=\int\limits_{a(x)}^{b(x)}\frac{\partial f(x,y)}{\partial x}dy+b'(x)f(x,b(x))-a'(x)f(x,a(x))\end{aligned}$$ we have that $$\begin{aligned} \frac{1}{(n-1)!}\int\limits_0^t(t-\tau)^{n-1}f(\tau)d\tau=\frac{1}{n(n-1)!}\int\limits_0^t\frac{\partial \left[(t-\tau)^{n}f(\tau)\right]}{\partial t}d\tau =\frac{d}{dt}\left[\frac{1}{n!}\int\limits_0^t(t-\tau)^nf(\tau)d\tau\right]\end{aligned}$$ Hence, the differential equation for $F_{n+1}(t)$ may be rewritten as $$\begin{aligned} F_{n+1}'(t)=\frac{d}{dt}\left[\frac{1}{n!}\int\limits_0^t(t-\tau)^nf(\tau)d\tau\right]\end{aligned}$$ such that the Cauchy problem has the solution $$\begin{aligned} F_{n+1}(t)=\frac{1}{n!}\int\limits_0^t(t-\tau)^nf(\tau)d\tau\end{aligned}$$ [111]{} Kolmanovskii V., Myshkis A. Applied Theory of Functional Differential equations, Kluwer Academic Publisher, 1992. Pinney E. Ordinary Difference-Differential Equations* , University of California Press, Berkely and Los Angeles, 1958. Bellman R., Cooke Kenneth Differential-Difference Equations , Academic Press, New York and London, 1963. Driver R. D., Cooke Kenneth Ordinary and Delay Differential Equations , Springer-Verlag, New York Heidelberg Berlin 1977. Breuer H.P., Petruccione F. The Theory of Open Quantum Systems Oxford University Press Inc., New York, 2002. Le Bellac M. Quantum Physics Cambridge University Press, Cambridge, 2006. Jivulescu M.A., Messina A., Napoli A., Petruccione F. Exact treatment of linear difference equations with noncommutative coefficients Mathematical Methods in Applied Science 2007; 30: 2147-2153. Murray Spiegel Schaum’s Mathematical Handbook of Formulas and Tables McGraw-Hill, 1998. Hildebrand Finite-Difference Equations and Simulations Prentice-Hall, Inc, Englewood Cliffs, N J 1968. Goldberg S. Difference Equations, John Wiley and Sons, Inc, New-York, 1958. Gohberg I., Lancaster P., Rodman L. Matrix polynomials, Academic Press, Inc, New-York, 1982. Sakurai J. J. Advance quantum mechanics, Addison Wesley, 1967.
--- abstract: 'Within the framework of geodetic brane gravity, the Universe is described as a 4-dimensional extended object evolving geodetically in a higher dimensional flat background. In this paper, by introducing a new pair of canonical fields $\{\lambda, P_{\lambda}\}$, we derive the quadratic Hamiltonian for such a brane Universe; the inclusion of matter then resembles minimal coupling. Second class constraints enter the game, invoking the Dirac bracket formalism. The algebra of the first class constraints is calculated, and the BRST generator of the brane Universe turns out to be rank-$1$. At the quantum level, the road is open for canonical and/or functional integral quantization. The main advantages of geodetic brane gravity are: (i) It introduces an intrinsic, geometrically originated, ’dark matter’ component, (ii) It offers, owing to the Lorentzian bulk time coordinate, a novel solution to the ’problem of time’, and (iii) It enables calculation of meaningful probabilities within quantum cosmology without any auxiliary scalar field. Intriguingly, the general relativity limit is associated with $\lambda$ being a vanishing (degenerate) eigenvalue.' address: \| Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel\ ([karasik@bgumail.bgu.ac.il, davidson@bgumail.bgu.ac.il]{}) author: - David Karasik and Aharon Davidson date: '14.7.2002' title: Geodetic Brane Gravity --- Introduction ============ Geodetic Brane Gravity (GBG) treats the universe as an extended object (brane) evolving geodetically in some flat background. This idea has been proposed more than twenty years ago by Regge and Teitelboim (’General Relativity a la String’) [@RT], with the motivation that the first principles which govern the evolution of the entire universe cannot be too different from those which determine the world-line behavior of a point particle or the world-sheet behavior of a string. Geometrically speaking, the $4$-dimensional curved space-time is a hypersurface embedded within a higher dimensional flat manifold. Following the isometric embedding theorems [@Cartan], at most $N=\frac{1}{2}n(n+1)$ background flat dimensions are required to locally embed a general $n$-metric. In particular, for $n=4$, one needs at most a $10$ dimensional flat background. This number can be reduced, however, if the $n$-metric admits some Killing-vector fields. In Regge and Teitelboim (RT) model, the external manifold (the bulk) is flat and empty, it contains neither a gravitational field nor matter fields. Other models were suggested, where the external manifold is more complicated [@RS; @Dvali; @Ida; @carter; @Stealth], it may be curved and contain bulk fields which may interact with the brane. RT action, therefore, does not contain bulk integrals, it is only an integral over the brane manifold, which may include the scalar curvature (${\cal R}^{n}$), a constant ($\Lambda$), and some matter Lagrangian (${\cal L}_{matter}$) [^1]. $$S = \int\left(\frac{1}{16\pi G^{n}}{\cal R}^{n}+\Lambda + {\cal L}_{matter}\right)\sqrt{-g^{n}}\,d^{n-1}x\,d\tau \label{action}$$ The Geodetic Brane has two parents: 1. General relativity gave the Einstein-Hilbert action, which makes the geodetic brane a gravitational theory. 2. Particle/String theory gave the embedding coordinates $y^{A}(x)$ [^2] as canonical fields, and this will lead to geodetic evolution. The $4$ dimensional metric is not a canonical field, it is just being induced by the embedding $g_{\mu\nu}(x)=\eta_{AB}y^{A}_{\,,\mu}(x)y^{B}_{\,,\nu}(x)$. Due to the fact that the Lagrangian (\[action\]) does not depend explicitly on $y^{A}$, but solely on the derivatives through the metric, the geodetic brane equations of motion are actually a set of conservation laws $$\left[({\cal R}^{\mu\nu}-\frac{1}{2}g^{\mu\nu}{\cal R} -8\pi GT^{\mu\nu})y^{A}_{\,;\mu}\right]_{;\nu}=0. \label{RT1}$$ Eq.(\[RT1\]) splits into two parts, the first is proportional to $y^{A}_{\,,\mu}$ and the second to $y^{A}_{\,;\mu\nu}$. Since the $4$-dimensional covariant derivative of the metric vanishes $g_{\mu\nu;\lambda}=0$, one faces the embedding identity $\eta_{AB}y^{A}_{\,;\lambda}y^{B}_{\,;\mu\nu}=0$. Therefore, the first and second covariant derivatives of $y^{A}$, viewed as vectors in the external manifold, are orthogonal, and each part of Eq.(\[RT1\]) should vanish separately. The part proportional to $y^{A}_{\,,\mu}$ implies that $T^{\mu\nu}_{\,;\nu}=0$. The second part is the geodetic brane equation [^3] $$\left({\cal R}^{\mu\nu} - \frac{1}{2}g^{\mu\nu} {\cal R} - 8\pi GT^{\mu\nu}\right)y^{A}_{\,;\mu\nu} = 0 ~. \label{RTeq}$$ - The matter fields equations remain intact, since the matter Lagrangian depends only on the metric. - Energy momentum is conserved. This is a crucial result, especially when the Einstein equations are not at our disposal. - Clearly, every solution of Einstein equations is automatically a solution of the corresponding geodetic brane equations. But the geodetic brane equations allow for different solutions [@Deser]. A general solution of eq.(\[RTeq\]) may look like $$\begin{aligned} & & {\cal R}^{\mu\nu} - \frac{1}{2}g^{\mu\nu} {\cal R} - 8\pi GT^{\mu\nu}=D^{\mu\nu} \label{RT2} \\ & & D^{\mu\nu}y^{A}_{\,;\mu\nu} = 0 \;\;\; D^{\mu\nu}\neq 0\end{aligned}$$ The non vanishing right hand side of eq.(\[RT2\]) will be interpreted by an [Einstein Physicist]{} as additional matter, and since it is not the ordinary $T^{\mu\nu}$ it may labeled [Dark Matter]{} [@DKL]. It has been speculated, relying on the structural similarity to string/membrane theory, that quantum geodetic brane gravity may be a somewhat easier task to achieve than quantum general relativity (GR). The trouble is, however, that the parent Regge-Teitelboim [@RT] Hamiltonian has never been derived! In this paper, by adding a new non-dynamical canonical field $\lambda$ we derive the quadratic Hamiltonian density of a gravitating brane-universe $${\cal H}=N^{k}y_{\|k}\cdot P - N\frac{8\pi G}{2\sqrt{h}}\left[ \left(\frac{\sqrt{h}}{8\pi G}\right)^{2}(\lambda + {\cal R}^{(3)}) + P\Theta(\Psi - \lambda I)^{-1}\Theta P \right] ~.$$ The derivation of the Geodetic Brane Hamiltonian is done here in a pedagogical way. In section \[embedding\] we translate the relevant geometric objects to the language of embedding. Each object is characterized by its tensorial properties with respect to both the embedding manifold and the brane manifold. We embed the ADM formalism [@ADM] in a higher dimensional Minkowski background, the $4$ dimensional spacetime manifold ($V_{4}$)is artificially separated into a $3$-dimensional space-like manifold ($V_{3}$) and a time direction characterized by the time-like unit vector orthogonal to $V_{3}$. For simplicity we restrict ourselves to $3$-dimensional space-like manifolds with no boundary (either compact or infinite), while the appropriate surface terms should be added when boundaries are present [@boundary]. Section \[hamiltonian\] is the main part of this Paper, where we derive the Hamiltonian. We first look at an empty universe with no matter fields, we present the gravitational Lagrangian density as a functional of the embedding vector $y^{A}(x)$, and derive the conjugate momenta $P_{A}(x)$. Reparametrization invariance causes the canonical Hamiltonian to vanish, (in a similar way to the ADM-Hamiltonian and string theory), and the total Hamiltonian is a sum of constraints. We introduce a new pair of canonical fields $\lambda,P_{\lambda}$ and make the Hamiltonian quadratic in the momenta. Following Dirac’s procedure [@Dirac] we separate the constraints into $4$ first-class constraints (reflecting reparametrization invariance), and $2$ second-class constraints (caused by the $2$ extra fields). We define the Dirac-Brackets and eliminate the second-class constraints. The final algebra of the constraints takes the familiar form of a relativistic theory, such as: The relativistic particle, string or membrane. In section \[matter\] we discuss the inclusion of arbitrary matter fields confined to the four dimensional brane. The algebra of the constraints remains unchanged, while the Hamiltonian is simply the sum of the gravitational Hamiltonian and the matter Hamiltonian. In section \[einstein\] the necessary conditions for classical Einstein gravity are formulated, they are - $\lambda$ must vanish. - The total (bulk) momentum of the brane vanishes. Section \[quantization\] deals with quantization schemes. We can use canonical quantization by setting the Dirac Brackets to be commutators $\displaystyle{\left\{,\right\}_{D} \longrightarrow i\hbar \left[,\right]}$. The wave-functional of a brane-like Universe [@HH] is subject to a Virasoro-type momentum constraint equation followed by a Wheeler-deWitt-like equation (first class constraints), the operators are not free, but are constrained by the second class constraints as operator identities. Another quantization scheme is the functional integral formalism, where we use the BFV [@BFV] formulation. The BRST generator [@BRST] is calculated, and the theory turns out to be rank $1$. This resembles ordinary gravity and string theory as oppose to membrane theory, where the rank is the dimension of the underlying space manifold. Section \[mini\] Geodetic Brane Quantum Cosmology is demonstrated. We apply the path integral quantization to the homogeneous and isotropic geodetic brane, within the minisuperspace model. A possible solution to the problem of time arises when one notices that while in GR the only dynamical degree of freedom is the scale factor of the universe, GBQC offers one extra dynamical degree of freedom (the bulk time) that may serve as time coordinate. Definitions, notations and some lengthy calculations were removed from the main stream of this work and were put in the appendix section. The Geometry of Embedding {#embedding} ========================= In this section we will formulate the relevant geometrical objects of the $V_{4}$ and $V_{3}$ manifolds in the language of embedding. Let our starting point be a flat $m$-dimensional manifold ${\cal M}$, with the corresponding line-element being $$ds^{2}=\eta_{AB}dy^{A}dy^{B} ~.$$ - An embedding function $y^{A}(x^{\mu}) \; (\mu=0,1,2,3)$ defines the $4$ dimensional hypersurface $V_{4}$ parameterized by the $4$ coordinates $x^{\mu}$. The $V_{4}$ tangent space is spanned by the vectors $y^{A}_{\,,\mu}$. (The $V_{3}$ hypersurface and tangent space are defined in a similar way). The induced $4$-dimensional metric is the projection of $\eta_{AB}$ onto the $V_{4}$ manifold: $g_{\mu\nu}=\eta_{AB}y^{A}_{\,,\mu}y^{B}_{\,,\nu}$. Choosing a time direction $t$ and space coordinates $x^{i} \; (i=1,2,3)$, the induced $4$-dimensional line-element takes the form $$ds^{2} = \eta_{AB}(y^{A}_{\, ,i}dx^{i}+ \dot{y}^{A}dt)(y^{B}_{\,,j}dx^{j}+ \dot{y}^{B}dt) ~,$$ The various projections of the metric $\eta_{AB}$ onto the space and time directions are denoted as the $3$-metric $h_{ij}$, the shift vector $N_{i}$, and the lapse function $N$ $$\begin{aligned} \eta_{AB}y^{A}_{\,,i}y^{B}_{\,,j} & = & h_{ij} ~, \label{ADMa} \\ \eta_{AB}y^{A}_{\,,i}\dot{y}^{B} & = & N_{i} ~, \label{ADMb} \\ \eta_{AB}\dot{y}^{A}\dot{y}^{B} & = & N_{i}N^{i} - N^{2} ~. \label{ADMc}\end{aligned}$$ These are not independent fields (as in Einstein’s gravity), but are functions of the embedding vector $y^{A}$. Nevertheless, it is a matter of convenience to write down the induced $4$-dimensional line-element in the familiar Arnowitt-Deser-Misner [@ADM] (ADM) form $$ds^{2}= -N^{2}dt^{2}+ h_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt) ~.$$ The vectors $(\dot{y}^{A},y^{A}_{\,,i})$ span the $4$-dimensional tangent-space of the $V_{4}$ spacetime manifold, while $y^{A}_{\,,i}$ span the $3$-dimensional tangent-space of the $V_{3}$ manifold. Using $h^{ij}$ as the inverse of the $3$-metric $h^{ij}h_{jk}=\delta^{i}_{k}$, one can introduce projections orthogonal to the $V_{3}$ manifold with the operator $$\begin{aligned} \Theta^{A}_{B} & = & \delta^{A}_{B}- y^{A}_{\,,a}h^{ab} y_{B,b} ~.\\ \Theta^{A}_{C}\Theta^{C}_{B} & = & \Theta^{A}_{B} ~. \label{Theta} \end{aligned}$$ Now, any vector $v^{A}$ can be separated into the projections tangent and orthogonal to the $V_{3}$ space $$v^{A}= v^{A}_{\parallel} + v^{A}_{\perp} = v^{B}y_{B,b}h^{ab}y^{A}_{\,,a} + v^{B}(\delta^{A}_{B}- y^{A}_{\,,a}h^{ab} y_{B,b}) ~.$$ An important role is played by the time-like unit vector orthogonal to $V_{3}$-space yet tangent to $V_{4}$-spacetime, $$\begin{aligned} n^{A} & \equiv & \frac{1}{N}\left(\dot{y}^{A}- N^{i}y^{A}_{\,,i}\right) = \frac{1}{N}\dot{y}^{B}\Theta^{A}_{B} ~, \label{n} \\ \eta_{AB}y^{A}_{\,,i}n^{B} & = & 0 ~, \\ \eta_{AB}n^{A}n^{B} & = & -1 ~.\end{aligned}$$ The tangent space of the embedding manifold ${\cal M}$ is spanned by the vectors: $y^{A}_{\,,i}$, $n^{A}$ and $L^{A}_{p}$ ($i=1,2,3\;p=1,..,m-4$). The vectors $L^{A}_{p}$ are chosen to be orthogonal to $y^{A}_{\,,i}$, $n^{A}$ and to each other. - The connections on the underlying $V_{3}$ are $\Gamma^{k}_{ij}=\eta_{AB}y^{A}_{\,,ij}y^{B}_{\,,l}h^{kl}$, this way, the covariant derivative of the $3$-metric vanishes $h_{ij\|k}=0$ (the stroke denotes 3-dimensional covariant derivative). As a result, one faces the powerful embedding identity $$\eta_{AB}y^{A}_{\,\|ij}y^{B}_{\,,k} \equiv 0 ~. \label{RT}$$ The vectors $y^{A}_{\,\|ij}$ are orthogonal to the $V_{3}$ tangent space and may be written as a combination of $n^{A}$ and $L^{A}_{p}$ [@exact]. $$y^{A}_{\,\|ij}=n^{A}K_{ij} + L^{A}_{p}\Omega^{p}_{ij} ~.$$ The projection of $y^{A}_{\,\|ij}$ in the $n^{A}$ direction is the extrinsic curvature of the $V_{3}$ hypersurface embedded in $V_{4}$ $$K_{ij} \equiv -\frac{1}{2N}\left( N_{i\|j}+N_{j\|i} -\frac{\partial h_{ij}}{\partial t}\right)= -\eta_{AB}y^{A}_{\,\|ij}n^{B} ~. \label{excurv}$$ The coefficient $\Omega^{p}_{ij}$ is the extrinsic curvature of $V_{3}$ with respect to the corresponding normal vector $L^{A}_{p}$. The intrinsic curvature of the $V_{3}$ manifold is also related to the second derivative of the embedding functions $y^{A}_{\,\|ij}$. The $3$-dimensional Riemann tensor is $${\cal R}^{(3)}_{iljk} \equiv \eta_{AB}(y^{A}_{\,\|ij}y^{B}_{\,\|kl} - y^{A}_{\,\|ik}y^{B}_{\,\|jl}) ~. \label{Rim}$$ For convenience we define the $\dot{y}^{A}$-independent symmetric tensor $$\Psi^{AB} \equiv (h^{ij}h^{ab}-h^{ia}h^{jb}) y^{A}_{\,\|ij}y^{B}_{\,\|ab} ~. \label{Psi}$$ Checking the indices, $\Psi^{AB}$ is a tensor in the embedding manifold, but a scalar in $V_{3}$ space. The trace of $\Psi^{A}_{\,B}$ is simply the $3$-dimensional Ricci scalar ${\cal R}^{(3)} = \eta_{AB}\Psi^{AB}$. Looking at eq.(\[RT\]), one can easily check that $$\Psi^{A}_{\,B}y^{B}_{\,,i} = 0 ~, \label{Psiy}$$ and $\Psi$ as an operator has at least $3$ eigenvectors with vanishing eigenvalue. Using definitions (\[Psi\],\[excurv\]), the contraction of $\Psi$ twice with $n^{A}$ is related to the extrinsic curvature $$K^{i}_{\,i}K^{j}_{\,j} - K_{ij}K^{ij} = \Psi_{AB}n^{A}n^{B} = \frac{1}{N^{2}} \Psi_{AB}\dot{y}^{A}\dot{y}^{B} ~.$$ Deriving the Hamiltonian {#hamiltonian} ======================== The gravitational Lagrangian density is the standard one $${\cal L} = \frac{1}{16\pi G}\sqrt{-g}{\cal R}^{(4)} ~.$$ Up to a surface term, it can be written in the form $${\cal L} = \frac{1}{16\pi G}N\sqrt{h}\left[ {\cal R}^{(3)} - ( K^{i}_{\,i}K^{j}_{\,j} - K_{ij}K^{ij} ) \right] ~. \label{lang}$$ Here, ${\cal R}^{(3)}$ denotes the 3-dimensional Ricci scalar, constructed by means of the 3-metric $h_{ij}$ (\[ADMa\]), whereas $K_{ij}$ (\[excurv\]) is the extrinsic curvature of $V_{3}$ embedded in $V_{4}$. Using the tensor $\Psi^{AB}$ (\[Psi\]) one can put the Lagrangian density (\[lang\]) in the form $${\cal L} = \frac{\sqrt{h}}{16\pi G}\left[ N{\cal R}^{(3)} - \frac{1}{N}\Psi_{AB}\dot{y}^{A} \dot{y}^{B} \right] ~. \label{lang2}$$ As one can see, the Lagrangian (\[lang2\]) does not involve mixed derivative $\dot{y}^{A}_{\, ,i}$ or second time derivative $\ddot{y}^{A}$. The first derivative $\dot{y}^{A}$ appears either explicitly or within $N$. Therefore the Lagrangian $$\fbox{${\cal L}(y,\dot{y},y_{\|i},y_{\|ij})$}$$ is ripe for the Hamiltonian formalism. The momenta $P_{A}$ conjugate to $y^{A}$ is simply $$P_{A}(x) \equiv \frac{\delta L}{\delta \dot{y}^{A}(x)} = \frac{\sqrt{h}}{16\pi G}\left\{ \left[{\cal R}^{(3)} + \frac{1}{N^{2}}\Psi_{BC} \dot{y}^{B}\dot{y}^{C}\right]\frac{\partial N} {\partial \dot{y}^{A}}-\frac{2}{N} \Psi_{AB}\dot{y}^{B}\right\} ~. \label{P1}$$ Using eq.(\[ADMb\],\[ADMc\]) to get $\frac{\partial N}{\partial \dot{y}^{A}} = -n_{A}$, while eq.(\[Psiy\]) tells us that $\frac{1}{N}\Psi_{AB}\dot{y}^{B} = \Psi_{AB}n^{B}$, the momentum (\[P1\]) becomes $$P^{A} = -\frac{\sqrt{h}}{16\pi G}\left\{\left[ {\cal R}^{(3)} + n_{B}\Psi^{BC}n_{C} \right] n^{A} + 2\Psi^{A}_{\,B}n^{B} \right\} ~. \label{P2}$$ The next step should be :“Solve eq.(\[P2\]) for $\dot{y}^{A}(y,P,y_{\|i},y_{\|ij})$”. But eq.(\[P2\]) involves only $n^{A}$, so one would like to solve eq.(\[P2\]) for $n^{A}(P,y,y_{\|i},y_{\|ij})$ first, and then to solve eq.(\[n\]) for $\dot{y}^{A}$ $$\dot{y}^{A} = Nn^{A} + N^{i}y^{A}_{\,,i} ~.$$ This looks innocent but even if one is able to solve eq.(\[P2\]) for $n^{A}$, any attempt to solve eq.(\[ADMb\]) for $N^{i}(n,y,y_{\|i})$ and eq.(\[ADMc\]) for $N(n,y,y_{\|i})$ will lead to a cyclic redefinition of $N^{i}$ and $N$. This situation is similar to other reparametrization invariant theories (such as the relativistic free particle, string theory etc.) and simply means that we have here $4 \times V_{3}$ primary constraints $$\begin{aligned} & & \eta_{AB}n^{A}n^{B} + 1 = 0 ~, \label{nn} \\ & & \eta_{AB}y^{A}_{\,,i}n^{B} = 0 ~. \label{ny}\end{aligned}$$ The constraints should be written in terms of canonical fields $(y^{A},P_{A})$. So one should solve eq.(\[P2\]) for $n^{A}(P)$, and then substitute in the above constraints. Any naive attempt to solve eq.(\[P2\]) for $n^{A}(y,P)$ falls short. The cubic equation involved rarely admits simple solutions. To ’linearize’ the problem we define a new quantity $\lambda$, such that $$P^{A}=-\frac{\sqrt{h}}{8\pi G} \left(\Psi - \lambda I\right)^{A}_{B}n^{B} ~. \label{P(n)}$$ - Comparing eq.(\[P2\]) with eq.(\[P(n)\]), the definition of $\lambda$ is actually another constraint $$n_{A}\Psi^{A}_{\,B}n^{B} + {\cal R}^{(3)} + 2\lambda = 0 ~. \label{lambda}$$ - An independent $\lambda$ comes along with its conjugate momentum $P_{\lambda}$. $\lambda$ is not a dynamical field therefore one faces another constraint $$P_{\lambda} = 0 ~. \label{P_lambda}$$ Assuming $\lambda$ is *not* an eigenvalue of $\Psi^{A}_{\,B}$, we solve (\[P(n)\]) for $n^{A}(\sqrt{h} ,\Psi ,P ,\lambda)$ and find $$n^{A}= -\frac{8\pi G}{\sqrt{h}}\left[\left(\Psi-\lambda I\right)^{-1}\right]^{A}_{\,B}P^{B} ~. \label{n(p)}$$ At this point we have $6 \times V_{3}$ primary constraints (\[nn\],\[ny\],\[lambda\],\[P\_lambda\]). We will follow Dirac’s way [@Dirac] to treat the constrained field theory we have in hand. First we will write down the various constraints in term of the canonical fields $\left( y^{A}(x), P_{A}(x), \lambda(x), P_{\lambda}(x)\right)$ : $$\begin{aligned} \phi_{0} & = & \frac{8\pi G}{2\sqrt{h}}\left[ \left(\frac{\sqrt{h}}{8\pi G}\right)^{2}(\lambda + {\cal R}^{(3)}) + P\Theta(\Psi - \lambda I)^{-1}\Theta P \right]\approx 0 ~, \label{phi0} \\ \phi_{k} & = & y_{\,\|k}\cdot P \approx 0 ~, \label{phik} \\ \phi_{4} & = & P_{\lambda} \approx 0 ~, \label{phi4} \\ \phi_{5} & = & \frac{8\pi G}{2\sqrt{h}}\left[ \left(\frac{\sqrt{h}}{8\pi G}\right)^{2} + P\Theta(\Psi - \lambda I)^{-2}\Theta P \right]\approx 0 ~. \label{phi5}\end{aligned}$$ \[phis\] - We use shorthanded notation to simplify the detailed expressions, $F \cdot G \equiv F^{A}G_{A}$ where $F$ and $G$ are vectors in the embedding space, and $P(\Psi - \lambda I)^{-2}P \equiv P_{A}\left[(\Psi - \lambda I)^{-2}\right]^{AB}P_{B}$ - We adopt Dirac’s notation $\phi \approx 0$ for weakly vanishing terms. - The embedding functions $y^{A}(x)$ and $\lambda(x)$ are scalars in the $V_{3}$ manifold. Their conjugate momenta $P_{A}(x),P_{\lambda}(x)$ are scalar densities of weight $1$. For convenience we normalize all constraints to be scalars in the embedding space, and scalar/vector densities of weight $1$ in $V_{3}$. This way, the Lagrange multipliers are of weight $0$. - $\phi_{k}$ is based on the constraint (\[ny\]) but it takes into account the embedding identity (\[Psiy\]) $$\phi_{k} = y_{\,\|k}\cdot P = -\frac{\sqrt{h}}{8\pi G }y_{\,\|k}(\Psi - \lambda I)n = \frac{\lambda \sqrt{h}}{8\pi G }y_{\,\|k}\cdot n \approx 0 ~.$$ - $\phi_{5}$ is based on the constraint (\[nn\]), but we added the projection operator $\Theta$ (\[Theta\]) in front of $P$. This step simplifies the final algebra of the constraints, and brings it to the familiar form of a relativistic theory. Inserting $\Theta$ in front of $P$, is equivalent to adding terms proportional to $\phi_{k}$ (\[phik\]), since $$\Theta^{A}_{B}P^{B}=(\delta^{A}_{B} -y^{A}_{\,,a}h^{ab}y_{B,b})P^{B} =P^{A}-y^{A}_{\,,a}h^{ab}\phi_{b}~.$$ - $\phi_{0}$ is also a combination of the constraints (\[lambda\]),(\[ny\]) and (\[nn\]), chosen such that $$\frac{\partial\phi_{0}}{\partial\lambda}= \phi_{5} \approx 0 ~.$$ - See appendix \[app FD\] for the definitions of functional derivatives and Poisson brackets. In a similar way to other parameterized theories, the canonical Hamiltonian density vanishes $${\cal H}_{c} = \dot{y}^{A}P_{A}-{\cal L} \approx 0 ~.$$ This means that the total Hamiltonian is a sum of constraints $$H=\int d^{3}x\,u^{m}(x)\phi_{m}(x) ~. \label{H1}$$ The constraints (\[phis\]) should vanish for all times, therefore their PB with the Hamiltonian should vanish (at least weakly). This imposes a set of consistency conditions for the functions $u^{m}(x)$ $$\begin{aligned} \dot\phi_{n}(x)=\left\{\phi_{n}(x),H\right\} & = & \left\{\phi_{n}(x),\int d^{3}z\,u^{m}(z) \phi_{m}(z)\right\} \nonumber \\ & \approx & \int d^{3}z \,u^{m}(z)\left\{\phi_{n}(x),\phi_{m}(z)\right\} \approx 0 ~. \label{consis}\end{aligned}$$ The basic Poisson brackets between the constraints are calculated in appendix \[app DB\], and in general has the form $$\begin{array}{\|c\|c\|c\|c\|c\|} \hline \displaystyle{\left\{\,,\,\right\}\approx} & \phi_{0}(z) & \phi_{l}(z) & \phi_{4}(z) & \phi_{5}(z) \\ \hline \phi_{0}(x) & 0 & 0 & 0 & \frac{8\pi G}{2\sqrt{h}}\,\alpha(x,z) \\ \hline \phi_{k}(x) & 0 & 0 & 0 & \phi_{5,\lambda}\lambda_{\|k}\delta(x-z) \\ \hline \phi_{4}(x) & 0 & 0 & 0 & -\phi_{5,\lambda}\delta(x-z) \\ \hline \phi_{5}(x) & -\frac{8\pi G}{2\sqrt{h}}\,\alpha(z,x) & -\phi_{5,\lambda}\lambda_{\|k}\delta(x-z) & \phi_{5,\lambda}\delta(x-z) & [F^{i}(x)+F^{i}(z)]\delta_{\|i}(x-z) \\ \hline \end{array} \label{PBarray}$$ The exact expressions for $\alpha$ and $F^{i}$ appears in appendix \[app DB\]. Now, insert the PB between the constraints (\[PBarray\]) into the consistency conditions (\[consis\]) to determine $u^{m}(x)$ $$\begin{aligned} \left\{\phi_{4}(x),H\right\} & \approx & \frac{\partial\phi_{5}}{\partial\lambda}(x)u^{5}(x) \approx 0 \Rightarrow u^{5}(x)=0 ~, \\ \left\{\phi_{0}(x),H\right\} & \approx & 0 \Rightarrow u^{0}(x) = -N(x) \; \emph{arbitrary} ~, \\ \left\{\phi_{k}(x),H\right\} & \approx & 0 \Rightarrow u^{k}(x) = N^{k}(x) \; \emph{arbitrary} ~, \\ \left\{\phi_{5}(x),H\right\} & \approx & \int d^{3}z \left[\frac{8\pi G}{2\sqrt{h}}\,\alpha(z,x)N(z) -\phi_{5,\lambda}\lambda_{\|k}\delta(x-z)N^{k}(z) +\phi_{5,\lambda}\delta(x-z)u^{4}(z)\right] \nonumber \\ & & \Rightarrow u^{4}(x) = N^{k}\lambda_{\|k}(x)- \phi_{5,\lambda}^{-1}(x) \int d^{3}z\, {\text{ $\frac{8\pi G}{2\sqrt{h}}$}} \,\alpha(z,x)N(z) ~.\end{aligned}$$ The first class Hamiltonian is then $$\begin{aligned} H & = & \int d^{3}x\left\{\frac{}{}N^{k} \left[y_{\|k}\cdot P + \lambda_{\|k}P_{\lambda}\right] \right.\nonumber \\ & -& N\frac{8\pi G}{2\sqrt{h}}\left[ (\frac{\sqrt{h}}{8\pi G})^{2}(\lambda + {\cal R}^{(3)}) + P\Theta(\Psi - \lambda I)^{-1}\Theta P \right. \nonumber \\ & & ~~~ \left. \left. +\int d^{3}z\,\alpha(x,z) \phi_{5,\lambda}^{-1}(z)P_{\lambda}(z) \right] \right\} \label{HT}\end{aligned}$$ As one can see, at this stage we have in the Hamiltonian $4$ arbitrary functions $N,N^{k}$ (Lagrange multipliers). This means we have $4$ first class constraints reflecting the reparametrization invariance ($4$-dimensional general coordinate transformation) $$\begin{aligned} \varphi_{0} & = & \frac{8\pi G}{2\sqrt{h}}\left[ (\frac{\sqrt{h}}{8\pi G})^{2}(\lambda + {\cal R}^{(3)}) + P\Theta(\Psi - \lambda I)^{-1}\Theta P \right. \nonumber \\ & & + \left. \int d^{3}z\,\alpha(x,z) \phi_{5,\lambda}^{-1}(z)P_{\lambda}(z) \right] \approx 0 ~, \label{varphi0} \\ \varphi_{k} & = & y_{\,\|k}P + \lambda_{\|k}P_{\lambda} \approx 0 ~. \label{varphik}\end{aligned}$$ \[varphi\] And we are left with $2$ second class constraints, reflecting the fact that we expanded our phase space with two extra fields $\lambda$ and $P_{\lambda}$ $$\begin{aligned} \theta_{1}=\phi_{4} & = & P_{\lambda} \approx 0 ~, \label{theta1} \\ \theta_{2}=\phi_{5} & = & \frac{8\pi G}{2\sqrt{h}}\left[ \left(\frac{\sqrt{h}}{8\pi G}\right)^{2} + P\Theta(\Psi - \lambda I)^{-2}\Theta P \right] \approx 0 ~. \label{theta2}\end{aligned}$$ Using the classical equation of motion for $y^{A}(x)$, $$\dot{y}^{A}(x) = \left\{y^{A}(x),H\right\} \approx N^{k}y_{\|k} - N\frac{8\pi G}{\sqrt{h}} (\Psi - \lambda I)^{-1}P ~,$$ one can identify the lapse function (\[ADMc\]) and the shift vector (\[ADMb\]) with $N,N^{k}$ respectively. Thus, recover the nature of the lapse function and the shift vector as Lagrange multipliers only at the stage of the solution to the equation of motion, not as an a priori definition. We would like to continue along Dirac’s path [@Dirac], and use [[Dirac Brackets]{}]{} (DB) instead of Poisson Brackets (PB). The DB are designed in a way such that the DB of a first class constraint with anything is weakly the same as the corresponding PB, while the DB of a second class constraint with anything vanish identically. Using DB, we actually eliminate the second class constraints (the extra degrees of freedom). The DB are defined as $$\left\{A,B\right\}_{D} \equiv \left\{A,B\right\}_{P} - \int d^{3}x\int d^{3}z \left\{A,\theta_{m}(x)\right\}_{P} C^{-1}_{mn}(x,z)\left\{\theta_{n}(z),B\right\}_{P} \label{db}$$ Where $C^{-1}_{mn}(x,z)$ is the inverse of the second class constraints PB matrix $$C_{mn}(x,z) \equiv \left\{\theta_{m}(x),\theta_{n}(z)\right\}.$$ In our case, $C_{mn}(x,z)$ is simply the $2\times 2$ bottom right corner of (\[PBarray\]) $$C_{mn}(x,z) = \left( \begin{array}{c\|c} 0 & \displaystyle{-\frac{\partial\phi_{5}} {\partial\lambda}(x)\delta(x-z)} \\ \hline \displaystyle{\frac{\partial\phi_{5}} {\partial\lambda}(x)\delta(x-z)} & \displaystyle{\left[F^{i}(x)+F^{i}(z)\right]\delta_{\|i}(x-z)} \end{array} \right) ~~ m,n = 1,2 \label{Cmn}$$ When dealing with field theory, the matrix $C_{mn}$ is generally a differential operator, and the inverse matrix is not unique unless one specifies the boundary conditions. We choose ’no-boundary’ as our boundary condition, therefore, integration by parts can be done freely, and the inverse matrix is $$C^{-1}_{mn}(x,z) = \left( \begin{array}{c\|c} \displaystyle{\left((\frac{\partial\phi_{5}} {\partial\lambda})^{-2}F^{i}(x) + (\frac{\partial\phi_{5}}{\partial\lambda})^{-2} F^{i}(z)\right)\delta_{\|i}(x-z)} & \displaystyle{\left(\frac{\partial\phi_{5}} {\partial\lambda}\right)^{-1}(x)\delta(x-z)} \\ \hline \displaystyle{-\left(\frac{\partial\phi_{5}} {\partial\lambda}\right)^{-1}(x)\delta(x-z)} & 0 \end{array} \right) \label{C-1}$$ The resulting DB are $$\begin{aligned} \left\{A,B\right\}_{D} & = & \left\{A,B\right\}_{P} + \int d^{3}x\left(\frac{\partial\phi_{5}} {\partial\lambda}\right)^{-2}F^{i}(x)\left[\frac{\delta A} {\delta\lambda(x)}(\frac{\delta B}{\delta\lambda(x)})_{\|i} - (\frac{\delta A}{\delta\lambda(x)})_{\|i} \frac{\delta B}{\delta\lambda(x)} \right] \nonumber \\ & & - \int d^{3}x\left(\frac{\partial\phi_{5}} {\partial\lambda}\right)^{-1}(x)\left[\frac{\delta A} {\delta\lambda(x)}\left\{\phi_{5}(x),B\right\} +\left\{A,\phi_{5}(x)\right\}\frac{\delta B}{\delta\lambda(x)} \right] \label{DB}\end{aligned}$$ This way, from now on, one should work with DB instead of PB and take the second class constraints to vanish strongly. This will omit the parts proportional to $P_{\lambda}$ from the first class constraints (\[varphi0\],\[varphik\]) and recover the original form (\[phi0\],\[phik\]). The algebra of the first class constraints takes the familiar form [@Dirac] of a relativistic theory $$\begin{aligned} \{\phi_{0}(x),\phi_{0}(z)\}_{D} & = & [h^{ij}\phi_{i}(x)+h^{ij}\phi_{i}(z)]\delta_{\|j}(x-z) \\ \{\phi_{0}(x),\phi_{k}(z)\}_{D} & = & \phi_{0}(z)\delta_{\|k}(x-z) \\ \{\phi_{k}(x),\phi_{l}(z)\}_{D} & = & \phi_{l}(x)\delta_{\|k}(x-z) + \phi_{k}(z)\delta_{\|l}(x-z)\end{aligned}$$ \[fDB\] The final first class Hamiltonian of a bubble universe is $$\fbox{$\displaystyle{ H = \int d^{3}x\left\{N^{k}y_{\|k}\cdot P -N\frac{8\pi G}{2\sqrt{h}}\left[ \left(\frac{\sqrt{h}}{8\pi G}\right)^{2}(\lambda + {\cal R}^{(3)}) + P\Theta(\Psi - \lambda I)^{-1}\Theta P \right]\right\} }$} \label{Hamiltonian}$$ At this stage, we have a first class Hamiltonian composed of four first class constraints, and accompanied with two second class constraints. The algebra of the first class constraints is the familiar algebra of other relativistic theories. Before moving on to quantization schemes we would like to study two more classical aspects: what happens if the action includes brane matter fields, and what is the relation between Einstein’s solutions to the geodetic brane solutions. Inclusion of Matter {#matter} =================== The inclusion of matter is done by adding the action of the matter fields to the gravitational action $$S = \int d^{4}x\left[\sqrt{-g} \frac{1}{16\pi G}{\cal R}^{(4)}+{\cal L}_{m}\right] ~.$$ The matter Lagrangian density depends in general on some matter fields, but also on the $4$-dimensional metric $g_{\mu\nu}$. The dynamics of the matter fields is actually not affected by the exchange of the canonical fields from $g_{\mu\nu}$ to $y^{A}$, and one expects the same equations of motion or the same ’matter’ Hamiltonian density. On the other hand the momenta $P_{A}$ gets a contribution from the matter Lagrangian $$\Delta P_{A}=\frac{\delta{\cal L}_{matter}} {\delta \dot{y}^{A}}=\sqrt{h}\left[ T_{nn}n^{A} - h^{ij}T_{ni}y^{A}_{\,,i} \right] ~.$$ This contribution depends on the various projections of the energy-momentum tensor $$T^{\mu\nu}\equiv\frac{2}{\sqrt{-g}} \frac{\delta{\cal L}_{matter}}{\delta g_{\mu\nu}} ~.$$ $T_{nn}$ is the matter energy density, or the projection of the energy-momentum tensor twice onto the $n^{A}$ direction $ T_{nn} \equiv \left(T^{\mu\nu}y^{A}_{\,,\mu}y^{B}_ {\,,\nu}\right)n_{A}n_{B} $. While in $T_{ni}$ the energy-momentum tensor is projected once onto the $n^{A}$ direction and once onto the $V_{3}$ tangent space. $ T_{ni} \equiv \left(T^{\mu\nu}y^{A}_{\,,\mu} y^{B}_{\,,\nu}\right)n_{A}y_{B,i}$. See appendix \[app MH\] for some examples of matter Lagrangians, Hamiltonians and the corresponding energy-momentum tensor projections. The momenta $P_{A}$ (\[P2\]) is now changed to $$P^{A} = -\frac{\sqrt{h}}{16\pi G}\left\{\left[ {\cal R}^{(3)} + n_{B}\Psi^{BC}n_{C} -16\pi G T_{nn} \right] n^{A} + 2\Psi^{A}_{\,B}n^{B} +16\pi G T_{ni}h^{ij}y^{A}_{\,,i}\right\} \label{PT}$$ Following the same logic that lead us from Eq.(\[P2\]) to the introduction of $\lambda$ (\[lambda\]), we will define $\lambda$ as $$n_{A}\Psi^{A}_{\,B}n^{B} + {\cal R}^{(3)} -16\pi G T_{nn} + 2\lambda = 0 ~. \label{lambdaT}$$ The effects of matter are thus $\lambda\rightarrow\lambda+8\pi G T_{nn}$, $P^{A}\rightarrow P^{A}-\sqrt{h} T_{ni}h^{ij}y^{A}_{\,,i}$ but $\Theta P$ is unchanged. The constraints are modified as follows $$\begin{aligned} \phi_{0} & \longrightarrow & \phi_{0} - \sqrt{h}T_{nn} \\ \phi_{k} & \longrightarrow & \phi_{k} + \sqrt{h}T_{nk} \label{tphik}\end{aligned}$$ Thus the Hamiltonian is changed to $$H_{G} \longrightarrow H_{G} + \int d^{3}x\sqrt{h} [N^{k}T_{nk}+NT_{nn}] = H_{G} + H_{m} ~, \label{tH}$$ Where $H_{m}$ is the matter Hamiltonian, calculated in terms of the matter fields alone as shown in appendix \[app MH\]. The algebra of the constraints (\[fDB\]) remains unchanged under the inclusion of matter, where the PB now include the derivatives with respect to matter fields as well. The Einstein Limit {#einstein} ================== In some manner Regge-Teitelboim gravity is a generalization of Einstein gravity. Any solution to Einstein equations is also a solution to RT equations (\[RTeq\]). We will derive here the necessary conditions for a RT-solution to be an Einstein-solution. - First, we use a purely geometric relation $$2G_{nn} = {\cal R}^{(3)} + n^{B}\Psi_{BC}n^{C} ~, \label{Gnn}$$ where $G_{nn}$ is the Einstein tensor twice projected onto the $n^{A}$-direction. The constraint associated with the introduction of $\lambda$ (\[lambdaT\]) is $$-2\lambda = {\cal R}^{(3)} + n^{B}\Psi_{BC}n^{C} -16\pi G T_{nn} = 2(G_{nn}-8\pi G T_{nn})~.$$ The Einstein solution of the equation is therefore associated with $$\fbox{$\displaystyle{\lambda = 0}$} ~.$$ As was shown in Eq.(\[Psiy\]), $\Psi$ has a degenerate vanishing eigenvalue. Therefore Einstein case with $\lambda=0$, will not allow for the essential $(\Psi-\lambda I)^{-1}$. One can not impose $\lambda=0$ as an additional constraint (as was proposed by RT [@RT]), but only look at it as a limiting case. - Second, we use the projection of the Einstein tensor once onto the $n^{A}$-direction and once onto the $V_{3}$ tangent space $G_{ni}$ $$G_{ni}h^{ij}y^{A}_{\,,j} = -\Psi^{A}_{\,B}n^{B} -\left(y^{A}_{\,,j}(Kh^{ij}-K^{ij})\right)_{\|i} ~, \label{Gni}$$ in eq.(\[PT\]) and put the momentum $P^{A}$ in the form $$P^{A} = -\frac{\sqrt{h}}{8\pi G}\left[ (G_{nn}-8\pi GT_{nn})n^{A} - (G_{ni}-8\pi GT_{ni}) h^{ij}y^{A}_{\,,j} + \left(y^{A}_{\,,j} (Kh^{ij}-K^{ij})\right)_{\|i}\right] ~. \label{P5}$$ It is clear that if Einstein equations $G_{nn}=8\pi GT_{nn}$ and $G_{ni}=8\pi GT_{ni}$ are both satisfied, the momentum $P_{A}$ makes a total derivative such that $$\oint d^{3}xP_{A} = 0 ~. \label{Ein2}$$ The total momentum $\oint d^{3}xP_{A}$ is a conserved Noether charge since the original Lagrangian does not depend explicitly on $y^{A}$ $$\mu^{A} \equiv \oint d^{3}xP^{A} = $ const.$~. \label{mu}$$ The universe, as an extended object, is characterized by the total momentum $\mu^{A}$. The necessary condition for an Einstein-solution is a vanishing $\mu^{A}$. $$\fbox{$\displaystyle{ \mu^{A} \equiv \oint d^{3}xP^{A} = 0 }$}~. \label{mu0}$$ - The condition (\[mu0\]) simply tells us that the total ’bulk’ momentum of the universe vanishes. This motivates us to use a new coordinate system for the embedding, namely, the ’center of mass frame’ $+$ ’relative coordinates’. As relative coordinates we will use the derivatives $y^{A}_{\,,i}\,$. This has a direct relation to the metric and therefore, we expect the equation of motion to resemble Einstein’s equations. The new system and the calculations appear in appendix \[app DM\]. Quantization ============ The treatment so far was classical, but the derivation of the Hamiltonian and the construction of the various constraints are the ingredients one needs for quantization. In the following sections we will describe two quantization schemes, canonical quantization and functional integral quantization. Canonical Quantization ---------------------- Dirac’s procedure leads us towards the canonical quantization of our constrained system. The following recipe was constructed by Dirac [@Dirac] for quantizing a constrained system within the Schrodinger picture - Represent the system with a state vector (wave functional) . - Replace all observables with operators. - Replace DB with commutators, $\displaystyle{\left\{,\right\}_{D} \longrightarrow i\hbar \left[,\right]}$ - First class constraints annihilate the state vector. - Second class constraints represent operator identities. - Since the commutator is ill defined for fields at the same space point, one must place all momenta to the right of the constraint. - First class constraints must commute with each other. This ensures consistency, and may call for operator ordering within the constraint. In our case, we can use the coordinate representation. The state vector is represented by a wave functional $\Phi[y]$. The DB (commutator) between $y^{A}$ and $P_{B}$ is canonical, therefore, these operators can be represented in a canonical way $$\begin{aligned} & & \hat{y}^{A}(x) \Rightarrow y^{A}(x) \nonumber \\ & & \hat{P}_{A}(x) \Rightarrow -i\hbar\frac{\delta} {\delta y^{A}(x)} \nonumber\end{aligned}$$ The operator $\hat{P_{\lambda}}$ vanishes identically. The DB of $\lambda$ with $y^{A}$,$P_{B}$ are not canonical, therefore the operator $\hat{\lambda}$ must be expressed as a function of $\hat{y^{A}}$,$\hat{P_{B}}$ . This can be done with the aid of the second class constraint (\[theta2\]). The first class constraints as operators must annihilate the wave functional. These constraints are recognized as 1. The momentum constraint (\[varphik\]) $$-i\hbar y^{A}_{\,\|k}\frac{\delta\Phi} {\delta y^{A}} = 0 ~, \label{momcons}$$ which simply means that the wave functional is a $V_{3}$ scalar and does not change its value under reparametrization of the space coordinates. This can be shown if one takes an infinitesimal coordinate transformation $$\begin{aligned} x^{k} & \longrightarrow & x^{k}+\epsilon^{k} ~,\nonumber \\ y^{A}(x) & \longrightarrow & y^{A}(x)+ \epsilon^{k}y^{A}_{\,\|k}(x) ~,\nonumber \\ \Phi[y] & \longrightarrow & \Phi[y] + \epsilon^{k}y^{A}_{\,\|k}\frac{\delta\Phi[y]} {\delta y^{A}} ~.\nonumber\end{aligned}$$ The wave functional is unchanged if and only if the momentum constraint holds. 2. The other constraint is the Hamiltonian constraint, and up to order ambiguities, the equation is the analog to the Wheeler de-Witt equation $$\frac{8\pi G}{2\sqrt{h}}\left[ \left(\frac{\sqrt{h}}{8\pi G}\right)^{2} (\hat{\lambda} + {\cal R}^{(3)})(x) -\hbar^{2}\left((\Psi - \hat{\lambda}I)^{-1}\right)^{AB}(x) \frac{\delta^{2}} {\delta y^{A}(x)\delta y^{B}(x)}\right]\Phi[y] = 0 ~. \label{WdW}$$ It is accompanied however, with the operator identity $$\frac{8\pi G}{2\sqrt{h}}\left[ \left(\frac{\sqrt{h}}{8\pi G}\right)^{2} + \hat{P}\Theta(\Psi - \hat{\lambda} I)^{-2}\Theta \hat{P}\right] =0$$ Functional Integral Quantization -------------------------------- Calculating functional integrals for a constrained system is not new. This was done for first class constraints by BFV [@BFV], And was generalized for second class constraints by Fradkin and Fradkina [@FF]. The first step is actually a classical calculation, that is, calculating the BRST generator [@BRST]. For this calculation we will adopt the following notations: - The set of canonical fields will include the Lagrange multipliers $N^{\mu}=(N,N^{i})$ that is $Q^{A}=\left( y^{A}, \lambda, N^{\mu}\right)^{T}$, and the corresponding conjugate momenta $\Pi_{A}=( P_{A}, P_{\lambda}, \pi_{\mu})$. The Lagrange multipliers are not dynamical, therefore, the conjugate momenta must vanish. This doubles the number of first class constraints $G_{a}=(\pi_{\mu}, \phi_{\nu})$. - For each constraint we introduce a pair of fermionic fields $\eta^{a}=\left( \rho^{\mu}, c^{\mu}\right)^{T}$, and the conjugate momenta ${\cal P}_{a}=(\bar{c}_{\nu},\bar{\rho}_{\nu})$. (In our case, all constraints are bosonic, therefore the ghost fields are fermions). - Each index actually represent a discrete index and a continuous index, for example, $y^{A}\equiv y^{A}(x)$. The summation convention is then generalized to sum over the continuous index as well $$N^{\mu}\phi_{\mu}\equiv \int d^{3}x N^{\mu}(x)\phi_{\mu}(x)~.$$ - We use Dirac Brackets as in (\[DB\]), but the Poisson Brackets are generalized to include bosonic and fermionic degrees of freedom $$\left\{L,R\right\} = \frac{\partial^{r} L}{\partial q^{A}} \frac{\partial^{l} R}{\partial p_{A}} -(-1)^{n_{L}n_{R}}\frac{\partial^{r} R}{\partial q^{A}} \frac{\partial^{l} L}{\partial p_{A}} ~.$$ Where $(q,p)$ is the set of canonical fields including the fermionic fields. $r,l$ denote right and left derivatives $$dR=\frac{\partial^{r} R}{\partial q}dq =dq\frac{\partial^{l} R}{\partial q}~.$$ And the fermionic index is $$n_{R}=\left\{\begin{array}{l l} 0 & $if R is a boson$ \\ 1 & $if R is a fermion$ \end{array} \right.$$ Let us now calculate the structure functions of the theory. The first order structure functions are defined by the algebra of the constraints $\{G_{a},G_{b}\}_{D}=G_{c}U^{c}_{ab}$. It is only the original constraints, (not the multipliers momenta), that have non vanishing structure functions (\[fDB\]). $$\left\{\left(\begin{array}{c}\pi_{\mu}(x)\\ \phi_{\mu}(x) \end{array}\right) ,\left(\pi_{\nu}(z), \phi_{\nu}(z)\right)\right\}_{D} = \left(\begin{array}{c c} 0 & 0 \\ 0 & \int d^{3}w\,\phi_{\lambda}(w) U^{\lambda}_{\mu\nu}(x,z,w) \end{array}\right) ,$$ and the relevant first order structure functions are $$U^{\lambda}_{\mu\nu}(x,z,w) = \left[\delta^{0}_{\mu}\delta^{0}_{\nu}h^{\lambda k} \left(\delta(w-x)+\delta(w-z)\right) +\delta^{\lambda}_{\mu}\delta^{k}_{\nu}\delta(w-z) +\delta^{k}_{\mu}\delta^{\lambda}_{\nu}\delta(w-x) \right]\delta_{,k}(x-z) \label{fosf}$$ (Generally, one should also look at $\{H_{0},G_{a}\}_{D}=G_{b}V^{b}_{a}$, but here $H_{0}=0$). The second order structure functions are defined by the Jacobi identity $\protect{{\cal A}(\{\{G_{a},G_{b}\}_{D},G_{c}\}_{D})=0}$, where ${\cal A}$ means antisymmetrization. Using the first order functions (\[fosf\]) one gets $\protect{{\cal A}(G_{d}[\{U^{d}_{ab},G_{c}\}_{D}+ U^{d}_{ec}U^{e}_{ab}])=0}$. This equation is satisfied if and only if the expression in the square brackets is again a sum of constraints $${\cal A}(\{U^{d}_{ab},G_{c}\}_{D}+ U^{d}_{ec}U^{e}_{ab}) = G_{f}U^{fd}_{abc} \label{sosf}.$$ The second order structure functions $U^{fd}_{abc}$ are antisymmetric on both sets of indices. In our case, the second order structure functions vanish, and the theory is of rank $1$. This resembles ordinary gravity and string theory as oppose to membrane theory, where the rank is the dimension of the underlying space manifold. The BRST generator of a rank $1$ theory is given by $\Omega = G_{a}\eta^{a} + \frac{1}{2}{\cal P}_{c} U^{c}_{ab}\eta^{b}\eta^{a}$. Here it is $$\Omega = \int d^{3}x[\pi_{\mu}\rho^{\mu}+ \phi_{\mu}c^{\mu}+ h^{kl}\bar{\rho}_{k} c^{0}_{\,,l}c^{0} +\bar{\rho}_{\mu} c^{\mu}_{\,,k}c^{k}](x). \label{Omega}$$ The main theorem of BFV [@BFV] is that the following functional integral does not depend on the choice of the gauge fixing Fermi function $\Psi$ : $$Z_{\Psi} = \int {\cal D}Q^{A}{\cal D}\Pi_{A} {\cal D}\eta^{a}{\cal D}{\cal P}_{a}\,M\, \exp[i \int dt(\Pi_{A}\dot{Q}^{A} + {\cal P}_{a}\dot{\eta}^{a} - H_{\Psi})]~. \label{Z1}$$ Where $M=\delta(\theta_{1})\delta(\theta_{2})(\det C_{mn})^{1/2}$ is taking care of the second class constraints, and, since the canonical Hamiltonian vanishes, $H_{\Psi}=-\{\Psi,\Omega\}_{D}$. The determinant of $C_{mn}$ for compact space manifolds, is calculated in a simple way in appendix \[app detC\]. An Example: Geodetic Brane Quantum Cosmology {#mini} ============================================ In the following example we would like to implement GBG to cosmology, and in particular to quantum cosmology. Detailed examples and calculations can be found in [@RTcos; @minipath], here we will just focus on global characteristics of the Feynman propagator for a geodetic brane within the minisuperspace model. Attention will be given to the differences between ’Geodetic Brane Quantum Cosmology’ and the standard ’Quantum Cosmology’. The standard and simple way to describe the cosmological evolution of the universe is to assume that on large scales the universe is homogeneous and isotropic. The geometry of such a universe is described by the Friedman-Robertson-Walker (FRW) metric $$ds^{2}=-N^{2}(t)dt^{2}+a^{2}(t)d\Omega_{3}^{2}, \label{FRW}$$ where $N(t)$ is the lapse function, $a(t)$ is the scale factor of the universe, and $$d\Omega_{3}^{2}=d\psi^{2}+\chi^{2}(\psi)d\Omega_{2}^{2} \label{dOmega3}$$ is the line element of the $3$ dimensional spacelike hypersurface which is assumed to be homogeneous and isotropic. $d\Omega_{2}^{2}$ is the usual line element on a $2$ sphere, and if the $3$ space is closed, flat or open respectively. In General Relativity, the components of the metric are the dynamical fields, the lapse function $N(t)$ is actually a Lagrange multiplier, and the only dynamical variable is the scale factor $a(t)$. This model is called minisuperspace, since, the infinite number of degrees of freedom in the metric is reduced to a finite number. The remnant of general coordinate transformation invariance, is time reparameterization invariance, that is, the arbitrariness in choosing $N(t)$. The usual and most convinient gauge is $N=1$. In GBG the situation is quite different. First, one has to embed the FRW metric (\[FRW\]) in a flat manifold. The minimal embedding of a FRW metric calls for one extra dimension. We will work here, for simplicity, with the closed universe $\chi=\sin\psi$. The embedding in a flat Minkowski spacetime with the signature $(-,+,+,+,+)$, is given by [@rosen] $$y^{A}=\left(\begin{array}{l}T(t) \\a(t)z^{I}(x)\end{array}\right) \;\;\; z^{I}=\left(\begin{array}{l}\sin\psi\sin\theta\cos\phi\\ \sin\psi\sin\theta\sin\phi\\ \sin\psi\cos\theta\\ \cos\psi \end{array}\right)~. \label{5embed}$$ The lapse function is given by $N(t)=\sqrt{\dot{T}^{2}-\dot{a}^{2}}$, it is [not]{} a Lagrange multiplier, but it depends on the two dynamical variables: the scale factor $a(t)$ and the external timelike coordinate $T(t)$. Time reparameterization invariance is, naturally, an intrinsic feature of , but, no gauge fixing is allowed here, since both $T(t)$ and $a(t)$ are dynamical. The gravitational Lagrangian (\[lang2\]), after integrating over the spatial manifold is $$L=\sigma\left(3Na-\frac{3a\dot{a}^{2}}{N}\right). \label{minilang}$$ $\sigma=\frac{2\pi^{2}}{8\pi G}$ is a scaling factor, for convinience we will set $\sigma=1$. The key for quantization is of course the Hamiltonian. One can derive the Hamiltonian directly from the Langrangian (\[minilang\]), or, to use the ready made Hamiltonian (\[Hamiltonian\]) and just insert the ’minimized’ expressions for the embedding vector, and the conjugate momenta. Minisuperspace Hamiltonian -------------------------- The first step is to introduce the coordinates and conjugate momenta. The general embedding vector $y^{A}$ is replaced by the dynamical degrees of freedom $a(t)$ and $T(t)$, while the spatial dependence is forced by the expression (\[5embed\]). It is expected that the conjugate momenta will have two degrees of freedom $P_{a}(t),P_{T}(t)$, the delicate issue is the spatial dependence of the momenta. Our choice is $$P_{A}=\left(\begin{array}{l}P_{T}(t) \\P_{a}(t)z^{I}(x)\end{array}\right)\cdot \frac{\sin^{2}\psi\sin\theta}{8\pi G} ~, \label{5momenta}$$ the factor $\sin^{2}\psi\sin\theta$ was inserted in order to keep the momenta a $3$-dimensional vector density. The spatial dependence is through $z^{I}(x)$ such that the momentum constraint (\[phik\]) vanishes strongly. And, the normalization is . In addition, we set $\lambda=\lambda(t)$ and $P_{\lambda}=P_{\lambda}(t)\frac{\sin^{2} \psi\sin\theta}{8\pi G}$. Inserting these expressions into the constraints (\[phis\]) and integrating over spatial coordinates, one is left with one first class constraint $$\varphi=\frac{1}{2}\left(6a+a^{3}\lambda +\frac{P_{T}^{2}}{a^{3}\lambda} +\frac{P_{a}^{2}}{6a-a^{3}\lambda} +\alpha P_{\lambda}\right)\approx 0 ~, \label{mini1cons}$$ and two second class constraints $$\begin{aligned} \theta_{1}&=&P_{\lambda}\approx 0 \nonumber\\ \theta_{2}&=&\frac{1}{2}\left(a^{3} -\frac{P_{T}^{2}}{a^{3}\lambda^{2}} +\frac{a^{3}P_{a}^{2}}{(6a-a^{3}\lambda)^{2}}\right)\approx 0~. \label{mini2cons}\end{aligned}$$ The Dirac brackets (\[DB\]) are defined as $$\begin{aligned} \left\{A,B\right\}_{D} & = & \left\{A,B\right\}_{P} -\left(\frac{P_{T}^{2}}{a^{6}\lambda^{3}} +\frac{a^{6}P_{a}^{2}}{(6a-a^{3}\lambda)^{3}}\right)^{-1} \left[\frac{\partial A}{\partial\lambda}\{\theta_{2},B\} +\{A,\theta_{2}\}\frac{\partial B}{\partial\lambda}\right]\end{aligned}$$ and the minisuperspace Hamiltonian is $$H=\frac{-N}{2}\left(6a+a^{3}\lambda +\frac{P_{T}^{2}}{a^{3}\lambda} +\frac{P_{a}^{2}}{6a-a^{3}\lambda} +\alpha P_{\lambda}\right) ~, \label{minihamiltonian}$$ We would like to focus on the Feynman propagator [@Feynman] $K(a_{f},T_{f},t_{f};a_{i},T_{i},t_{i})$ for the empty geodetic brane universe. Although the empty universe is a non-realistic model for our universe, the calculation of the propagator is simple and it demonstrates some of the main features and advantages of Geodetic Brane Quantum Cosmology over the standard quantum cosmology models. This propagator is the probability amplitude that the universe is in $(a_{f},T_{f})$ at time $t_{f}$, and it was in $(a_{i},T_{i})$ at time $t_{i}$. We will use a modified version of BFV integral offered by Senjanovic [@senj], where the ghosts and multipliers were integrated out. $$\begin{aligned} K(a_{f},T_{f},t_{f};a_{i},T_{i},t_{i})&=&\int d\mu\, \exp\left[2\pi i\int_{t_{i}}^{t_{f}}dt(\dot{a}P_{a}+\dot{T}P_{T} +\dot{\lambda}P_{\lambda})\right] \nonumber \\ d\mu&=& da\,dP_{a}\,dT\,dP_{T}\,d\lambda\,dP_{\lambda}\, \delta(\varphi)\,\delta(\chi)\,\|\{\chi,\varphi\}\|\, \delta(\theta_{1})\,\delta(\theta_{2})\, \|\det(\{\theta_{m},\theta_{n}\})\|^{1/2} \label{prop1}\end{aligned}$$ This propagator is calculated in phase space, where the measure is the Liuville measure $dx\,dp$. In addition, the measure $d\mu$ enforces the constraints (first and second class) by delta functions, it includes an arbitrary gauge fixing function $\chi$, the determinants of the Poisson brackets between first class constraints and the gauge fixing function and the determinants of the Poisson brackets between second class constraints. Attention should be given to the following issues: - The canonical Hamiltonian vanishes, therefore it is absent in the action. - The boundary conditions for the propagator determine the values of $a_{f},T_{f},a_{i},T_{i}$, but not the value of $\lambda$ nor the values of the momenta. Therefore, the momenta and $\lambda$ must be integrated over at the initial point. - The gauge fixing function $\chi$, although arbitrary, must be chosen such that it does not violate the boundary conditions nor the constraints. In addition, the Poisson brackets $\{\chi,\varphi\}$ must not vanish. - The determinant of the second class constraints Poisson brackets is simply $$\|\det(\{\theta_{m},\theta_{n}\})\|^{1/2} =\left\|\frac{\partial\theta_{2}}{\partial\lambda}\right\| =\left\|\frac{P_{T}^{2}}{a^{3}\lambda^{3}} +\frac{a^{6}P_{a}^{2}}{(6a-a^{3}\lambda)^{3}}\right\| \label{theta2,lambda}$$ - Our convention here is $\sigma=1$ and Planck constant $h=1$ ($\hbar=\frac{1}{2\pi}$). - In cases where matter is included, the inclusion of matter will affect in a few places. The action will include terms like $\dot{\phi}\pi$, an integration over matter fields and momenta will be added, and the first class constraint will have a contribution which is simply the matter Hamiltonian All other constraints remain intact. The calculation of the propagator (\[prop1\]) is carried out in a simple way following Halliwell [@halliwell], and the final propagator takes the form $$K_{\pm}(a_{f},T_{f};a_{i},T_{i}) =\int d\omega\, \exp\left[2\pi i\omega(T_{f}-T_{i}) \mp2\pi i\omega^{2}\left(F(x_{f})-F(x_{i})\right)\right] \label{prop3}$$ The index of $K_{\pm}$ and the $\mp$ in the exponent refers to the expanding/contracting scale factor. $\omega$ is the conserved bulk energy (the momentum conjugate to the bulk time coordinate $T$). Since the value of $\omega$ is not fixed at the initial condition, one must integrate over $\omega$. One should notice according to eq.(\[mu0\]) that the Einstein solution is assiciated with $\omega=0$. The function $F(x)$ is given by $$F(x)=\left\{\begin{array}{ll} \frac{1}{12}\left[3\text{Arcsin} x +\sqrt{1-x^{2}}(4x^{5}+2x^{3}-3x)\right] & \|x\|\leq 1 \\ \text{sgn}(x)\frac{\pi}{8}-\frac{i}{12}\left[3\text{sgn}(x) \text{Arccosh}\|x\|-\sqrt{x^{2}-1}(4x^{5}+2x^{3}-3x)\right] & 1<\|x\| \end{array}\right. \label{F}$$ Where $x=(\frac{3a}{\omega})^{1/3}$. Let us now examine the properties of the propagator (\[prop3\]). Actually, the propagator is independent of the internal time parameter $t$ (a common character of all parameterized theories), and depends exclusively on the value of $a$ and $T$ at the boundaries. - The most basic characteristic of a propagator is the possibility to propagate from an initial state to a final state through an intermidiate state. For example, the propagator for a non-relativistic particle is At the intemidiate time $t_{2}$, one must integrate over $x_{2}$. It is clear that there is no integration over $t_{2}$, $t$ is the evolution parameter, it must be monotonic $t_{3}>t_{2}>t_{1}$, and integration over $t_{2}$ makes no sense. Another characteristic of the propagator is . The situation with parameterized theories is quite different. The propagator is independent of the internal time, and integration over all dynamical variable diverges. The solution is, usually, to let one of the dynamical variables as ’time’, and integrate only over the other variables. The question is: How does the propagator (\[prop3\]) behaves at the intermidiate point ? What is the relevant evolution parameter and what integrations should be made ? One can check that if $a$ is taken to be the monotonic evolution parameter and integration over $T$ at the intermidiate point is done, then the propagator (\[prop3\]) is well behaved. $$\begin{aligned} K(a_{3},T_{3};a_{1},T_{1})&=& \int dT_{2}\int d\omega\,e^{2\pi i[\omega(T_{3}-T_{2}) -\omega^{2}(F(x_{3})-F(x_{2}))]}\int d\bar{\omega}\, e^{2\pi i[\bar{\omega}(T_{2}-T_{1}) -\bar{\omega}^{2}(F(\bar{x}_{2})-F(\bar{x}_{1}))]} \nonumber \\ &=& \int d\omega\,e^{2\pi i[\omega(T_{3}-T_{1}) -\omega^{2}(F(x_{3})-F(x_{1}))]} \\ K(a_{1},T_{2};a_{1},T_{1})&=& \int d\omega\,e^{2\pi i\omega(T_{2}-T_{1})}=\delta(T_{2}-T_{1}) \label{K32K21}\end{aligned}$$ This cannot be done within the standard quantum cosmology models, since there, the only dynamical variable is the scale factor $a$. Such a propagator of only one variable contains no information, it can tell that the varible is monotonic. The common solution in standard quantum cosmology is to add another dynamical variable such as a scalar field and to use one of them as the evolution parameter. Here we see one of the main advantages of Geodetic Brane Quantum Cosmology over the standard models, the problem of time has an intrinsic solution as we have one extra degree of freedom which serves as ’time’. - The most general wave function that can be generated using the propagator (\[prop3\]) is $$\Psi(a,T)=\int d\omega\,e^{2\pi i\omega T}\left[ A(\omega)e^{-2\pi i \omega^{2}F(x)} +B(\omega)e^{2\pi i \omega^{2}F(x)}\right]~. \label{wavefunction}$$ One can verify that the wave function (\[wavefunction\]) (and the propagator (\[prop3\])) satisfy the corresponding WDW equation $$\hbar^{2}\left[-\xi(x)\frac{\partial}{\partial a} \left(\frac{1}{\xi(x)}\frac{\partial}{\partial a}\right) +\xi^{2}(x)\frac{\partial^{2}}{\partial T^{2}}\right]\Psi(a,T)=0 \label{miniWDW}$$ Where $\xi(x)=(1+2x^{2})\sqrt{1-x^{2}}$, and Putting $-i\hbar\frac{\partial}{\partial T}=\omega$ and neglecting the term proportional to the first derivative $\frac{\partial \Psi}{\partial a}$, the equation (\[miniWDW\]) looks like a zero energy Schrodinger equation $$\left[-\hbar^{2}\frac{\partial^{2}}{\partial a^{2}} +V_{\omega}(a)\right]\Psi_{\omega}(a)=0~, \label{minischr}$$ with the potential $$V_{\omega}(a)=-\omega^{2}\left[1-\left(\frac{3a} {\omega}\right)^{2/3}\right] \left[1+2\left(\frac{3a}{\omega}\right)^{2/3}\right]^{2} =36a^{2}-3\omega^{4/3}(3a)^{2/3}-\omega^{2} \label{minipotential}$$ ![The potential $V_{\omega}(a)$[]{data-label="fig.potential"}](potential.eps) The classical turning point is $a=\omega/3$, and the empty brane universe can not expand classically byeond this point. The empty universe model is non-realistic, a more realistic model may include some matter fields, or at least a cosmological constant. Analysis of the cosmological constant universe can be found in [@RTcos]. - In accordance with section \[einstein\], one of the necessary conditions for an Einstein solution is Eq.(\[Ein2\]) $\int d^{3}x\,P_{A}=0$. Within our minisuperspace model, integrating the momenta (\[5momenta\]) over the spatial manifold one gets $\int d^{3}x\,P_{A}=(P_{T},0,0,0,0)^{T}$, thus the Einstein case is associated with $\omega=0$, and the only classcal regime is $a=0$. - The still open question is that of the boundary conditions. In particular $\Psi(a=0,T)$ and $\Psi(a\rightarrow\infty,T)$. One possibility is: $\Psi$ vanishes at the big bang ($a=0$) and $\Psi$ is bounded at $a\rightarrow\infty$. This will lead to $\omega$ quantization $\omega_{n}^{2}=8\hbar(n+1/4)$ where $n$ is a positive integer. Clearly, the Einstein case $\omega=0$ is excluded by such quatization condition. summary {#summary .unnumbered} ======= 1. In the present model of Geodetic Brane Gravity, the $4$ dimensional universe floats as an extended object within a flat $m$ dimensional manifold. It can be generalized however, to include fields in the surrounding manifold (bulk), this is done by adding the bulk action integral to the action of the brane. The brane will feel those bulk fields as forces influencing its motion [@carter]. The bulk fields may include matter fields or the bulk gravity [@RS; @Dvali; @Ida; @Stealth]. 2. In this paper we have derived the quadratic Hamiltonian of a brane universe. The Hamiltonian is a sum of $4$ first class constraints, while $2$ additional second class constraints are present. We used Dirac Brackets and found the algebra of first class constraints to be the familiar one from other relativistic theories (such as string, membrane or general relativity). The BRST generator turns out to be of rank $1$. 3. Geodetic brane gravity modifies general relativity, and introduces in a natural way [dark matter]{} components. Dark matter in inflationary models which accompanies ordinary matter to govern the evolution of the universe can be found in [@DKL]. 4. We have formulated the conditions for a solution to be that of general relativity, and showed that the Einstein case can be achieved only as a limiting case. 5. Canonical quantization is possible with the aid of Dirac brackets. The resulting Wheeler de-Witt equation includes operators which are not free, but are constrained by the second class constraints as operator identities. 6. The ground is ready for functional integral quantization, the BRST generator is of rank $1$, and the determinant of second class constraints has been brought to a simple form. 7. A simple application of geodetic brane gravity to cosmology is possible within the framework of a minisuperspace. Classical cosmological models appear in [@davidson; @higgs]. Canonical quantization appears in [@RTcos], and the complementary functional integral quantization in [@minipath]. 8. Another significant advantage of GBG over GR is the solution to the problem of time. While a homogeneous and isotropic metric is characterized by only one dynamical variable (the scale factor of the universe), the embedding vector contains two dynamical variables (the scale factor and the bulk time). Thus, taking the embedding vector to be the canonical variables, will enhance the theory with one extra variable that may be intepreted as a time coordinate. Functional Derivatives {#app FD} ====================== $\bullet$ Let $F[y]$ be a functional of $y(x)$ such that $\delta F = \int d^{3}x f(x)\delta y(x)$ then the functional derivative is $\displaystyle{\frac{\delta F}{\delta y(x)} \equiv f(x)}$. The chain rule holds for functional derivatives $\displaystyle{\frac{\delta F(G[y])}{\delta y(x)} =\frac{\partial F}{\partial G}\frac{\delta G[y]}{\delta y(x)}}$ $\bullet$ The delta distribution is a scalar density of weight $1$ such that for a $3$-scalar $f(x)$ $$f(x)=\int d^{3}z f(z)\delta^{3}(x-z)$$ The covariant derivative of the delta function $\delta^{3}_{\,\|i}(x-z)$ is defined for a $3$-vector $g^{i}(x)$ $$\int d^{3}x g^{i}(x)\delta^{3}_{\,\|i}(x-z) = -g^{i}_{\,\|i}(z).$$ $\bullet$ The delta function is symmetric with its two arguments $$\delta(x-z) = \delta(z-x) ~.$$ The first covariant derivative of the delta function is antisymmetric with its arguments $$\delta_{\,\|i}(x-z)\equiv \nabla_{x^{i}}\delta(x-z) = -\nabla_{z^{i}}\delta(z-x) \equiv -\delta_{\,\|i}(z-x) ~.$$ While the second covariant derivative is again symmetric. $\bullet$ The basic functional derivatives are $$\begin{aligned} \frac{\delta y^{A}(x)}{\delta y^{B}(z)} &=& \delta^{A}_{B}\delta(x-z) \\ \frac{\delta y^{A}_{\,\|i}(x)}{\delta y^{B}(z)} &=& \delta^{A}_{B}\delta_{\|i}(x-z) \\ \frac{\delta y^{A}_{\,\|ij}(x)}{\delta y^{B}(z)} &=& \left(\delta^{A}_{B}-y^{A}_{\,\|a}h^{ab}y_{B\|b}\right) \delta_{\|ij}(x-z) - y_{B\|ij}y^{A}_{\,\|a}h^{ak}\delta_{\|k}(x-z)\end{aligned}$$ For a general expression $\Phi(x,y,y_{\|i},y_{\|ij})$ the functional derivative is $$\begin{aligned} \frac{\delta \Phi(x)}{\delta y^{A}(z)} & = & \frac{\partial \Phi}{\partial y^{A}}(x)\delta(x-z) + \frac{\partial \Phi}{\partial y^{A}_{\,\|i}}(x) \delta_{\|i}(x-z) \nonumber \\ & + & \frac{\partial \Phi}{\partial y^{B}_{\,\|ij}}(x) \left[\left(\delta^{B}_{A}-y^{B}_{\,\|b}h^{ab}y_{A\|a}\right) \delta_{\|ij}(x-z) - y_{A\|ij}y^{B}_{\,\|b}h^{bk}\delta_{\|k}(x-z)\right]\end{aligned}$$ Another nontrivial example is the $3$-dimensional Christofel symbols $\Gamma^{i}_{kl}=h^{ij}y^{A}_{\,,j}y_{A,kl}$ $$\frac{\delta \Gamma^{i}_{kl}(x)}{\delta y^{A}(z)} = h^{ij}y_{A\|kl}(x)\delta_{\|j}(x-z) + h^{ij}y_{A\|j}(x)\delta_{\|kl}(x-z) ~.$$ $\bullet$ The Poisson brackets are defined in the usual way $$\left\{F,G\right\} = \int d^{3}x \left(\frac{\delta F}{\delta y^{A}(x)} \frac{\delta G}{\delta P_{A}(x)} - \frac{\delta F}{\delta P_{A}(x)} \frac{\delta G}{\delta y^{A}(x)} \right) ~. \label{PB}$$ Poisson Brackets of constraints {#app DB} =============================== We will start with the constraints (\[phis\]) $$\begin{aligned} \phi_{0} & = & \frac{8\pi G}{2\sqrt{h}}\left[ \left(\frac{\sqrt{h}}{8\pi G}\right)^{2}(\lambda + {\cal R}^{(3)}) + P\Theta(\Psi - \lambda I)^{-1}\Theta P %+ \frac{1}{\lambda}P\cdot y_{\,\|k} % h^{kl}y_{\,\|l}\cdot P \right]\approx 0 ~, \\ \phi_{k} & = & y_{\,\|k}\cdot P \approx 0 ~, \\ \phi_{4} & = & P_{\lambda} \approx 0 ~, \\ \phi_{5} & = & \frac{8\pi G}{2\sqrt{h}}\left[ \left(\frac{\sqrt{h}}{8\pi G}\right)^{2} + P\Theta(\Psi - \lambda I)^{-2}\Theta P %- \frac{1}{\lambda^{2}}P\cdot y_{\,\|k}h^{kl}y_{\,\|l}\cdot P \right]\approx 0 ~.\end{aligned}$$ The PB of these constraints are listed below $$\begin{aligned} \left\{\phi_{0}(x),\phi_{0}(z)\right\} & = & \left[Q^{i}(x)+Q^{i}(z)\right]\delta_{\|i}(x-z) \approx 0 ~. \\ \left\{\phi_{0}(x),\phi_{l}(z)\right\} & = & \phi_{0}(z)\delta_{\|l}(x-z)-\phi_{5}\lambda_{,l}(z) \delta(x-z)\approx 0 ~. \\ \left\{\phi_{0}(x),\phi_{4}(z)\right\} & = & \phi_{5}(z)\delta(x-z) \approx 0 ~. \\ \left\{\phi_{0}(x),\phi_{5}(z)\right\} & = & \left[B^{i}(x)+B^{i}(z)\right]\delta_{\|i}(x-z) +M(z)\delta(x-z) ~. \\ \left\{\phi_{k}(x),\phi_{l}(z)\right\} & = & \phi_{l}(x)\delta_{\|k}(x-z) + \phi_{k}(z)\delta_{\|l}(x-z) \approx 0 ~. \\ \left\{\phi_{k}(x),\phi_{4}(z)\right\} & = & 0 ~. \\ \left\{\phi_{k}(x),\phi_{5}(z)\right\} & = & \phi_{5}(x)\delta_{\|k}(x-z) -\frac{\partial\phi_{5}}{\partial\lambda}\lambda_{\|k}\delta(x-z) ~. \\ \left\{\phi_{4}(x),\phi_{4}(z)\right\} & = & 0 ~. \\ \left\{\phi_{4}(x),\phi_{5}(z)\right\} & = & -\frac{\partial\phi_{5}}{\partial\lambda}\delta(x-z) ~. \\ \left\{\phi_{5}(x),\phi_{5}(z)\right\} & = & \left[F^{i}(x)+F^{i}(z)\right]\delta_{\|i}(x-z) ~. \label{ConsPB}\end{aligned}$$ Where the shorthanded expressions are $$\begin{aligned} \frac{\partial\phi_{5}}{\partial\lambda} & = & \frac{8\pi G}{\sqrt{h}}\left[P\Theta(\Psi - \lambda I)^{-3}\Theta P %+ \frac{1}{\lambda^{3}}h^{ab}\phi_{a}\phi_{b} \right] \label{phi5,lambda} \\ K_{ij} & = & -\frac{8\pi G}{\sqrt{h}}P(\Psi-\lambda I)^{-1}y_{\|ij} \\ Q^{i} & = & h^{ij}\phi_{j} + 2\left[(Kh^{ij}-K^{ij})_{\|j} -\frac{8\pi G}{\sqrt{h}}h^{ij}\phi_{j}\right]\phi_{5} \approx 0 \\ B^{i} & = & \left[(Kh^{ij}-K^{ij})_{\|j} -\frac{8\pi G}{\sqrt{h}} h^{ij}\phi_{j}\right]\frac{\partial\phi_{5}}{\partial\lambda} + \left[\frac{\partial(Kh^{ij}-K^{ij})}{\partial\lambda}\right]_{\|j}\phi_{5} \nonumber \\ & \approx & \left[(Kh^{ij}-K^{ij})_{\|j}\right] \frac{\partial\phi_{5}}{\partial\lambda} \\ M & \approx & \frac{\sqrt{h}}{8\pi G}\left[\lambda\frac{\partial K}{\partial \lambda} - K +(R_{ij}-2K_{il}K_{j}^{l})\frac{\partial}{\partial\lambda}(Kh^{ij}-K^{ij})\right] \nonumber \\ & & + (Kh^{ij}-K^{ij})_{\|j}\left[ \left(\frac{\partial\phi_{5}}{\partial\lambda}\right)_{\|i} -2\frac{8\pi G}{\sqrt{h}}[P(\Psi-\lambda I)^{-1}]_{\|i}(\Psi-\lambda I)^{-2}P \right] \nonumber \\ & & - \frac{8\pi G}{\sqrt{h}} P(\Psi-\lambda I)^{-1}[(\Psi-\lambda I)^{-1}P]_{\|ij} \frac{\partial}{\partial\lambda}(Kh^{ij}-K^{ij}) \\ F^{i} & \approx & \frac{1}{3}\frac{\partial^{2}\phi_{5}} {\partial \lambda^{2}}\left(Kh^{ij}-K^{ij}\right)_{\|j} \nonumber\\ & & - \left(\frac{\partial\phi_{5}}{\partial\lambda}\right)^{2} \left[\left(\frac{\partial\phi_{5}}{\partial\lambda}\right)^{-1} \frac{\partial}{\partial\lambda}(Kh^{ij}-K^{ij})\right]_{\|j} \nonumber \\ & & - 2\frac{8\pi G}{\sqrt{h}} P(\Psi-\lambda I)^{-2}\left[(\Psi - \lambda I)^{-1}P \right]_{\|j}\frac{\partial}{\partial\lambda}(Kh^{ij}-K^{ij}) \label{F^i}\end{aligned}$$ Matter Hamiltonians {#app MH} =================== Consider here a few simple matter Lagrangians and Hamiltonians, $\bullet$ For a cosmological constant , $$\begin{aligned} {\cal L}_{matter} & = & -\sqrt{-g}2\Lambda ~, \\ T^{\alpha\beta} & = & -2\Lambda g^{\alpha\beta} ~.\end{aligned}$$ The corresponding energy/momentum projections are $$\begin{aligned} T_{nn} & = & 2\Lambda ~, \\ T_{ni} & = & 0 ~.\end{aligned}$$ The Hamiltonian is simply $${\cal H}_{matter} = -{\cal L}_{matter} = N\sqrt{h}2\Lambda = N\sqrt{h}T_{nn} ~.$$ $\bullet$ For a scalar field $\Phi(x)$, $$\begin{aligned} {\cal L}_{matter} & = & -\sqrt{-g}\left( \frac{1}{2}g^{\mu\nu}\partial_{\mu}\Phi\partial _{\nu}\Phi + V(\Phi)\right) ~, \\ T^{\alpha\beta} & = & \left(g^{\alpha\mu}g^{\beta\nu}- \frac{1}{2}g^{\alpha\beta}g^{\mu\nu}\right)\partial_{\mu} \Phi\partial_{\nu}\Phi-g^{\alpha\beta}V(\Phi) ~.\end{aligned}$$ The momentum $\Pi$ conjugate to $\Phi$ is given by $$\Pi = \frac{\delta{\cal L}}{\delta \dot{\Phi}}= \sqrt{h}\frac{1}{N}(\dot{\Phi}-N^{i}\Phi_{,i}) ~, \label{Pi}$$ and the corresponding energy/momentum projections are $$\begin{aligned} T_{nn} & = & \frac{1}{2}\left(\frac{1}{h}\Pi^{2}+ h^{ij}\Phi_{,i}\Phi_{,j}\right) + V ~, \\ T_{ni} & = & \frac{1}{\sqrt{h}}\Pi\Phi_{,i} ~.\end{aligned}$$ The matter Hamiltonian is $${\cal H}_{matter} = N\sqrt{h}(\frac{1}{2h}\Pi^{2}+ \frac{1}{2}h^{ij}\Phi_{,i}\Phi_{,j} + V) + N^{i}\Pi \Phi_{,i} = N\sqrt{h}T_{nn} + N^{i}\sqrt{h}T_{ni} ~.$$ $\bullet$ For a vector field $A_{\mu}(x)$, $$\begin{aligned} {\cal L}_{matter} & = & -\frac{1}{16\pi}\sqrt{-g} g^{\mu\lambda}g^{\nu\sigma}F_{\mu\nu} F_{\lambda\sigma} ~, \\ T^{\alpha\beta} & = & \frac{1}{4\pi}\left(g^{\alpha\mu} g^{\beta\nu}-\frac{1}{4}g^{\alpha\beta}g^{\mu\nu}\right) g^{\lambda\sigma}F_{\mu\lambda}F_{\nu\sigma} ~.\end{aligned}$$ The momentum $\Pi^{\mu}$ conjugate to $A_{\mu}$ is given by $$\begin{aligned} \Pi^{0} & = & 0 ~, \\ \Pi^{i} & = & \frac{\sqrt{h}}{4\pi N}h^{ij}\left( \dot{A}_{j}-A_{0,j}-N^{k}F_{kj} ~, \right)\end{aligned}$$ and the corresponding energy/momentum projections are $$\begin{aligned} T_{nn} & = & \frac{2\pi}{h}h^{ij}\Pi_{i}\Pi_{j}+ \frac{1}{16\pi}h^{ij}h^{kl}F_{ik}F_{jl} ~, \\ T_{ni} & = & \frac{1}{\sqrt{h}}h^{kl}\Pi_{k}F_{il} ~.\end{aligned}$$ The Hamiltonian is $${\cal H} = N\sqrt{h}(\frac{2\pi}{h}h^{ij}\Pi_{i} \Pi_{j} + \frac{1}{16\pi}h^{ij}h^{kl}F_{ik}F_{jl}) + N^{i}\Pi^{j}F_{ij} - A_{0}\Pi^{i}_{\,,i} = N\sqrt{h}T_{nn}+N^{i}\sqrt{h}T_{ni}-A_{0}\Pi^{i}_{\,,i} ~.$$ In this case the Hamiltonian picks up another Lagrange multiplier $A_{0}$, and an additional constraint $$-\Pi^{i}_{\,,i} = \frac{1}{4\pi}\sqrt{-g} F^{0\nu}_{\hspace{5pt};\nu} = 0 ~.$$ The center of mass and relative coordinates {#app DM} =========================================== We will try to make a canonical transformation to the new system. We will use a global pair ${\bf Y}^{A}(t),{\bf P}_{A}(t)$ to describe the total momentum and its conjugate coordinate. And, as relative coordinates we will use the directional derivatives $z^{A}_{\,i}(x)=y^{A}_{\,,i}(x)$ of the field $y^{A}(x)$. (This is the analog to a discrete system, where the relative coordinates are differences between the coordinates of the various particles involved). The variation of the Action with respect to $y^{A}_{\,,i}(x)$ is going to be very similar to the variation with respect to $h_{ij}$, and therefore will resemble Einstein’s equations. The new set of canonical ’coordinates $+$ fields’ ${\bf Y}^{A},{\bf P}_{A},z^{A}_{\,i}(x),\pi_{A}^{\,i}(x)$, must obey the canonical PB $$\begin{aligned} \left\{{\bf Y}^{A},{\bf P}_{B}\right\} & = & \delta^{A}_{B} ~,\label{YP} \\ \left\{{\bf Y}^{A},\pi_{B}^{\,i}(x)\right\} & = & 0 ~,\label{Ypi} \\ \left\{z^{A}_{\,i}(x),{\bf P}_{B}\right\} & = & 0 ~,\label{zP} \\ \left\{z^{A}_{\,i}(x),\pi_{B}^{\,j}(\bar{x})\right\} & = & \delta^{A}_{B}\delta_{i}^{\,j}\delta(x-\bar{x}) ~.\label{zpi}\end{aligned}$$ \[canPB\] We will write the transformation from the old set of fields to the new set as $$\begin{aligned} {\bf Y}^{A}(t) & = & \int d^{3}x\,f(x)y^{A}(t,x) \\ {\bf P}_{A}(t) & = & \int d^{3}x\,P_{A}(t,x) \\ z^{A}_{\,i}(t,x) & = & y^{A}_{\,,i}(t,x) \\ \pi_{A}^{\,i}(t,x) & = & \int d^{3}\bar{x}\,P_{A} (t,\bar{x})J^{i}(x,\bar{x})\end{aligned}$$ While the inverse transformation is $$\begin{aligned} y^{A}(t,x) & = & {\bf Y}^{A}(t)+ \int d^{3}\bar{x}\, z^{A}_{\,i}(t,\bar{x})J^{i}(\bar{x},x) \\ P_{A}(t,x) & = & {\bf P}_{A}(t)f(x) - \pi^{i}_{A\;,i}(t,x)\end{aligned}$$ The functions $f(x),J^{i}(x,\bar{x})$ are distributions over the $V_{3}$ manifold, they do not depend on the canonical fields, and in particular are independent of the $3$-metric. The solution to Eq.(\[canPB\]) put some restrictions on $f(x),J^{i}(x,\bar{x})$, and they must fulfill the following relations $$\begin{aligned} \int d^{3}x\,f(x) & = & 1 ~, \\ \int d^{3}\bar{x}\,f(\bar{x})J^{i}(x,\bar{x}) & = & 0 ~,\\ \frac{\partial J^{i}(x,\bar{x}) }{\partial \bar{x}^{j}} & = & \delta^{i}_{j}\delta(x-\bar{x}) ~,\\ \frac{\partial J^{i}(x,\bar{x}) }{\partial x^{i}} & = & f(x) - \delta(x-\bar{x})~. \label{Jf}\end{aligned}$$ We assume one can find such distributions and we move on to the dynamics. We will start with the Hilbert action (\[action\]) and do the variation with respect to the new variables $$\begin{aligned} \delta S & = & \frac{-1}{16\pi G}\int d^{4}x\,\sqrt{-g} (G^{\mu\nu}-8\pi G T^{\mu\nu}) \delta g_{\mu\nu} \nonumber \\ & = & \frac{-2}{16\pi G}\int d^{4}x\,\sqrt{-g} (G^{\mu\nu}-8\pi G T^{\mu\nu}) y_{A,\mu}\delta y^{A}_{\,,\nu} \nonumber \\ & = & \frac{2}{16\pi G}\int d^{4}x\,\left\{ \left[\sqrt{-g}(G^{\mu 0}-8\pi G T^{\mu 0}) y_{A,\mu}\right]_{,0}\left[\delta {\bf Y}^{A} + \int d^{3}\bar{x}\,\delta z^{A}_{\,i}(\bar{x}) J^{i}(\bar{x},x)\right]\right. \nonumber \\ & - & \frac{2}{16\pi G}\int d^{4}x\, \sqrt{-g}(G^{\mu i}-8\pi G T^{\mu i}) y_{A,\mu}\delta z^{A}_{\,i}(x) ~, \label{daction}\end{aligned}$$ The variation with respect to ${\bf Y}^{A}$ will lead to the conservation of the total momentum $$\frac{-2}{16\pi G}\int d^{3}x\, \left[\sqrt{-g}(G^{\mu 0}-8\pi G T^{\mu 0}) y_{A,\mu}\right] = \mu_{A} = \text{const.} \label{mu2}$$ The variation with respect to $z^{A}_{\,i}(x)$ will lead to an equation similar to Einstein’s equations, but, the right hand side does not vanish $$\sqrt{-g}\,y_{A,\alpha}[G^{\alpha i}-8\pi G T^{\alpha i}](x) = \int d^{3}\bar{x} J^{i}(x,\bar{x}) \left[\sqrt{-g}(G^{\alpha 0}-8\pi G T^{\alpha 0}) y_{A,\alpha}(\bar{x})\right]_{,0} ~. \label{dark2}$$ Multiply Eq.(\[dark2\]) by $\frac{1}{\sqrt{-g}}g^{\mu\nu}y^{A}_{\,,\nu}$ to get $$G^{\mu i}-8\pi G T^{\mu i}(x) = D^{\mu i}(x) = \frac{1}{\sqrt{-g}}g^{\mu\nu}y^{A}_{\,,\nu}(x) \int d^{3}\bar{x} J^{i}(x,\bar{x}) \left[\sqrt{-g}(G^{\alpha 0}-8\pi G T^{\alpha 0}) y_{A,\alpha}(\bar{x})\right]_{,0} ~. \label{dark3}$$ An [[Einstein physicist]{}]{} will interpret Eq.(\[dark3\]) as if there is some additional matter in the universe, and may call it [[dark matter]{}]{}. It is easy to reveal Eq.(\[RTeq\]) if one takes the derivative of Eq.(\[dark2\]) with respect to $x^{i}$ and use (\[Jf\]). Determinant of second class constraints PB {#app detC} ========================================== We would like to calculate the determinant of $C_{mn}(x,z)$ (\[Cmn\]). First we will find the eigenvalues of $C$. Take $v(x)$ to be a two components scalar function $$v(x)= \left(\begin{array}{c} g(x)\\ f(x) \end{array} \right) ~.$$ The eigenvalue equation of $C$ is $$\int d^{3}z\,C(x,z)v(z)= \alpha v(x) ~. \label{CEV}$$ Inserting Eq.(\[Cmn\]) into Eq.(\[CEV\]) one can see that the components of $v(x)$ are proportional, and must obey a differential equation $$\begin{aligned} & & g = -\frac{1}{\alpha}\frac{\partial\phi_{5}} {\partial\lambda}f ~, \\ & & 2F^{i}f_{\|i}+F^{i}_{\|i}f -\frac{1}{\alpha}\left(\frac{\partial\phi_{5}} {\partial\lambda}\right)^{2}f = \alpha f ~. \label{diff1}\end{aligned}$$ Multiplying (\[diff1\]) by $f$ one gets $$\left(F^{i}f^{2}\right)_{\|i} = \left[\alpha+\frac{1}{\alpha} \left(\frac{\partial\phi_{5}} {\partial\lambda}\right)^{2}\right]f^{2} ~. \label{diff2}$$ Eigenvalues of a differential operator are determined by the boundary conditions. Our boundary conditions are actually the fact that the $3$-manifold has no boundary. Thus integrating Eq.(\[diff2\]) over $V_{3}$ gives us $$\int d^{3}x\left[\alpha+\frac{1}{\alpha} \left(\frac{\partial\phi_{5}} {\partial\lambda}\right)^{2}\right]f^{2}(x) =0 ~. \label{alpha1}$$ Arranging Eq.(\[alpha1\]) one gets $$\alpha^{2}= -\frac{\int d^{3}x \left(\frac{\partial\phi_{5}} {\partial\lambda}\right)^{2}f^{2}(x)} {\int d^{3}xf^{2}(x)} ~. \label{alpha2}$$ - $C_{mn}(x,z)$ is a PB matrix and therefore antihermitian, this causes the eigenvalues of $C$ to be purely imaginary. - One can see that the eigenvalues of $C$ are affected only by the off diagonal terms $\frac{\partial\phi_{5}}{\partial\lambda}$, not by $F^{i}$. The structure of $\alpha^{2}$ is very simple. Define the probability density $$\bar{f}(x) \equiv \frac{f^{2}(x)} {\int d^{3}xf^{2}(x)} ~,$$ one sees that any eigenvalue of $C$ is simply the expectation value of $\left(\frac{\partial\phi_{5}} {\partial\lambda}\right)^{2}$ with respect to some probability distribution $\bar{f}$ $$\alpha^{2}_{\bar{f}}= -\left\langle\left(\frac{\partial\phi_{5}} {\partial\lambda}\right)^{2}\right\rangle_{\bar{f}} ~. \label{alpha3}$$ For each $\bar{f}$ one finds $2$ complex conjugate purely imaginary eigenvalues. The determinant of $C$ is therefore the multiplication of all eigenvalues $$\det C =\prod_{\bar{f}}\left\langle\left(\frac{\partial\phi_{5}} {\partial\lambda}\right)^{2}\right\rangle_{\bar{f}} \label{detC}$$ The probability density over a compact manifold can be parameterized by the appropriate harmonics, and the product is countable. See for example [@structure; @S3] for the compact $S_{3}$ harmonics. T. Regge and C. Teitelboim, in Proc. Marcel Grossman, p.77 (Trieste, 1975). S. 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--- abstract: 'Csiszár’s channel coding theorem for multiple codebooks is generalized allowing the codeword lenghts differ across codebooks. Also in this case, for each codebook an error exponent can be achieved that equals the random coding exponent for this codebook alone, in addition, erasure detection failure probability tends to $0$. This is proved even for sender and receiver not knowing the channel. As a corollary, a substantial improvement is obtained when the sender knows the channel.' author: - 'Lóránt Farkas and Tamás Kói [^1]' title: 'Universal Random Access Error Exponents for Codebooks of Different Word-Lengths' --- error exponent, variable length, asynchronous, random access, erasure Introduction ============ The discrete memoryless channel (DMC) coding theorem of Csiszár [@Csiszar] analyzes the performance of a codebook library of several constant composition codebooks consisting of codewords of the same length. The rate and the type of the codewords may be different for each codebook. The number of codebooks is subexponential in the codeword length. It is shown that simultaneously for each codebook the same error exponent can be achieved as the random coding exponent of this codebook alone. In other words, for transmitting messages that may be of different kinds, with specified rates: with the sender using different codebooks for different kinds of messages, the same reliability can be guaranteed for each message kind as if it were known that only messages of this kinds occur, with the given rates. Note that this theorem is used in [@Csiszar] to the engineeringly different problem of joint source-channel coding. As noted in [@Csiszar] the result is also relevant in unequal protection of messages: for better protection, important messages may be encoded via “more reliable” (smaller) codebooks, see for example Borade, Nakiboglu and Zheng [@Borade], Weinberger and Merhav [@Merhav] and Shkel, Tan and Draper [@Yanina] for more recent results. Luo and Ephremides in [@Ephremides] analyze a similar model in the context of random access communication for multiple access channel (MAC) which brings classical information theory closer to packet based random access communications models. Not using their concepts of standard communication rate and generalized random coding, the model of [@Ephremides] can be summarized as follows. Each user employs a random codebook partitioned into classes corresponding to different rate options. If the vector of the senders’ actual rate choices belongs to a preselected operation rate region, the decoder should reliably decode the messages sent, otherwise it should report collision. Wang and Luo in [@Jie-Luo] derive Gallager type error exponents for this model. In a slightly modified model Farkas and Kói in [@isit2013] give error exponents employing a mutual information based universal decoder, with application to joint source-channel coding for MAC. This paper generalizes the mentioned result of [@Csiszar] in a different direction, not addressing MACs. As in [@Csiszar], the sender is assumed to have a codebook library of several codebooks, each consisting of codewords of the same length and type. Before each message transmission, the sender chooses the codebook he will use, the receiver is unaware of this choice. As a new feature compared to [@Csiszar], here not only the rate and type but also the codeword length may vary across codebooks, thus a model in between fixed and variable length coding is addressed. This model appears natural, e.g., for communication situations where a channel is used alternatingly for transmitting messages of different kinds such as audio, data, video etc. We believe that this paper, though of theoretical nature providing asymptotic achievability results, may contribute to a better understanding of such communication situations. For channels with positive zero-error capacity, the above model does not provide mathematical challenges. Indeed, in that case (as noted also in [@Csiszar]) prior to each message transmission the sender can communicate his codebook choice over the channel without error, using codewords of length $o(n)$. This reduces the introduced model to the standard case of a single codebook. In the more common case of zero error capacity equal to $0$, no such simple strategy is available, and the fact that codewords of different length are used causes a certain asynchronism at the receiver, who should also estimate the boundaries of the codewords and avoid error propagation. To meet these challenges we introduce a mutual information based two-stage decoder. It is not obvious what to mean by decoding error in our model. By the definition we adopt, the $j$’th message is correctly decoded if the decoder correctly assigns this message to the time slot where the corresponding codeword is sent, including correct identification of the codeword boundaries. The receiver is not required to learn that this message has been sent as the $j$’th one (taking care of the possibility that at previous instances erroneously less or more messages have been decoded than actually sent). Our main result extends the result in [@Csiszar] to the above scenario, showing that simultaneously for each codebook choice the same error exponent can be achieved as the random coding error exponent for the chosen codebook alone. This is proved under the technical assumption that all codeword length ratios are between $D$ and $\frac{1}{D}$ for some $D \in (0,1]$ and the number of codebooks is subexponential in length-bound $n$. Recall that even in the standard case of a single codebook, a positive error exponent is achievable only for rate less than the mutual information over the channel with input distribution equal to the type of the codewords, and under this condition the random coding exponent is positive. It is desirable that when this condition fails, the decoder can report that reliable decoding is not possible. This feature is present in [@Ephremides] and [@isit2013] (but not in [@Csiszar]). In [@Ephremides] and [@isit2013], addressing MACs, the term collision detection is used, in our one-sender context we will use the term erasure detection. As part of main result, our universal decoder is shown suitable also for erasure detection: When the chosen codebook has random coding exponent $0$, an erasure is reported with probability approaching $1$, though here we do not obtain exponential speed of convergence (for more on this see Remark \[etan\]). This has been achieved with a completely universal construction: Neither the design of the codebook library nor the decoder depends on the channel. A corollary of the main theorem improves the result when the sender knows the channel while maintaining the universality of the decoder. The improvement leads to exponent also for erasure declaration failure probability, and shows that for each message kind the maximum of the random coding error exponent over the possible input distributions is achievable. Even the special case of this corollary for transmitting messages of a single kind is of interest, yielding a universal coding result for this classical problem that, to our knowledge, does not appear in the literature, see Remark \[singlecodebook\]. The proofs rely on the subtype technique of Farkas and Kói [@Hawaii] and [@Hongkong]. The hardest kind of error to deal with has been that of detecting the right codeword in a wrong position, partially overlapping with the correct one. This obstacle has been overcome employing a new concept of $\gamma$-independent sequences, and also second order types. We are aware of only one prior work extending results in [@Csiszar] in a direction like here, by Balakirsky [@Balakirsky] on joint source-channel coding error exponent for variable length codes. Channel coding with multiple codebooks is not explicitly mentioned in [@Balakirsky] but some ideas in our paper are similar to those there, due to the close mathematical relationship of these problems. Note that the topic of the paper is also connected (see the Discussion for details) to the area of strong asynchronism, see Tchamkerten, Chandar and Wornell [@Tchamkarten] and Polyanskiy [@Polyanskiy], and even more to Yıldırım, Martinez and Fàbregas [@HongkongML] concerning error exponents. Notation ======== The notation follows [@Csiszar], [@Nazari] and [@Hawaii] whenever possible. All alphabets are finite and $\log$ denotes logarithm to the base $2$. The set $\{1,2,\dots,M\}$ is denoted by $[M]$. The notation $subexp(n)$ denotes a quantity growing subexponentially as $n \rightarrow \infty$ (i.e. $\frac{1}{n}\log(subexp(n))\rightarrow 0$), that could be given explicitly. For some subexpontial sequences individual notations are used and the parameters on which these sequences depend will be indicated in parantheses. Random variables $X$, $Y$, etc., with alphabets $\mathcal{X}$, $\mathcal{Y}$, etc., will be assigned several different (joint) distributions. These will be denoted by $P^{X}$, $P^{XY}$, etc. or $V^{X}$, $V^{XY}$, etc. The first notation will typically refer to a distinguished (joint) distribution, the second one refers to distributions introduced for technical purposes such as representing joint types. The family of all distributions on $\mathcal{X} \times \mathcal{Y}$, say, is denoted by $\mathcal{P} (\mathcal{X} \times \mathcal{Y})$. If a multivariate distribution, say $V^{\hat{X}XY} \in \mathcal{P} (\mathcal{X} \times \mathcal{X} \times \mathcal{Y})$ is given then $V^{X}$, $V^{\hat{X}X}$, $V^{XY}$, $V^{Y\|X}$ etc. will denote the associated marginal or conditional distributions. The type of an $n$-length sequence ${\mathbf x}=x_1 x_2 \dots x_n \in \mathcal{X}^n$ is the distribution $P_{{\mathbf x}} \in \mathcal{P} (\mathcal{X})$ where $P_{{\mathbf x}}(x)$ is the relative frequency of the symbol $x$ in ${\mathbf x}$. The joint type of two or more $n$-length sequences is defined similarly and, for $({\mathbf x},{\mathbf y}) \in \mathcal{X}^n \times \mathcal{Y}^n$, say, it is denoted by $P_{({\mathbf x},{\mathbf y})}$. The family of all possible types of sequences ${\mathbf x}\in \mathcal{X}^n$ is denoted by $\mathcal{P}^n (\mathcal{X})$, and for $P \in \mathcal{P}^n (\mathcal{X})$ the set of all ${\mathbf x}\in \mathcal{X}^n$ of type $P_{{\mathbf x}}=P$ is denoted by $T^n_{P}$. Denote $\operatorname{H}_V(X)$, $\operatorname{H}_V(Y\|X)$, $\operatorname{I}_V(\hat{X}X \wedge Y)$ etc. the entropy, conditional entropy and mutual information etc. when the random variables $X$, $\hat{X}$, $Y$ have joint distribution $V=V^{\hat{X}XY}$. Furthermore, the empirical mutual information $\operatorname{I}({\mathbf x}\wedge {\mathbf y})$ of two sequences ${\mathbf x}$ and ${\mathbf y}$ (of equal length) is defined as $\operatorname{I}_V(X \wedge Y)$ with $V^{XY}=P_{({\mathbf x},{\mathbf y})}$. Given a DMC $W: \mathcal{X} \rightarrow \mathcal{Y}$ and $P \in \mathcal{P}(\mathcal{X})$ let $I(P,W)$ be equal to $\operatorname{I}_{V}(X\wedge Y)$ where $V^{X}=P$ and $V^{Y\|X}=W$. The maximum of $I(P,W)$ over all $P \in \mathcal{P}(\mathcal{X})$ is the capacity of the DMC $W$. The following elementary facts will be used (see, e.g., [@Csiszar2]): $$\begin{aligned} &\|\iP^n(\iX)\|\leq (n+1)^{\|\iX\|}, \label{basicfact1}\\ &\frac{2^{n\operatorname{H}(P)}}{(n+1)^{\|\iX\|}} \le \|T^n_{P}\|\le 2^{n\operatorname{H}(P)} \textnormal{ if }\, P \in \iP^n(\iX), \label{basicfact2}\\ &W^n({\mathbf y}\|{\mathbf x})=2^{-n\left(\operatorname{D}(V^{Y\|X}\\|W\|P_{\mathbf x})+\operatorname{H}_{V}(Y\|X)\right)} \textnormal{ where $V^{XY}=P_{({\mathbf x},{\mathbf y})}$.} \label{basicfact4}\end{aligned}$$ The concatenation of an $n_1$-type $V_1\in \iP^{n_1}(\iX)$ and an $n_2$-type $V_2\in \iP^{n_2}(\iX)$ is the $(n_1+n_2)$-type $V_1\oplus V_2\in \iP^{n_1+n_2}(\iX)$ with $$\begin{aligned} \left( V_1\oplus V_2 \right)(x)=\frac{n_1}{n_1+n_2}V_1(x)+\frac{n_2}{n_1+n_2}V_2 (x). \end{aligned}$$ The concatenation of joint types, say, $V_1 \in \mathcal{P}^{n_1}(\mathcal{X} \times \mathcal{Y})$ and $V_2 \in \mathcal{P}^{n_2}(\mathcal{X} \times \mathcal{Y})$ is defined similarly. If $V_1$, $V_2$, …, $V_k$ are $n_1$, $n_2$, …, $n_k$-types, respectively, let $$\operatorname{J}(V_1,V_2, \dots, V_k)=\operatorname{H}(V_1 \oplus \dots \oplus V_k)-\sum_{i=1}^{k}\frac{n_i}{n_1+\dots+n_k}\operatorname{H}(V_i). \label{F-def1}$$ The nonegative quantity in (\[F-def1\]) is a Jensen-Shannon divergence if $k=2$, and a generalized Jensen-Shannon divergence otherwise, in the sense of [@BURBEA] and [@Lin]. The second order type of a sequence $\mathbf{x}=x_1 \dots x_n \in \mathcal{X}^n$ is $P_{\mathbf{x}}^2 \in \mathcal{P}^{n-1}(\mathcal{X}\times \mathcal{X})$ defined by $$P_{\mathbf{x}}^2 (a,b) = \frac{1}{n-1}\|i: x_i = a, x_{i+1}=b\|.$$ In other words, $P_{\mathbf{x}}^2$ is the joint type of $\mathbf{x}'=x_1 \dots x_{n-1}$ and $\mathbf{x}''=x_2 \dots x_{n}$. Let $T^{n,2}_{V,a}$ denote the second order type class $\{\mathbf{x}: \mathbf{x}\in \mathcal{X}^n, P_{\mathbf{x}}^2=V,x_1=a \}$. We cite from [@Csiszar3] that $$\|T^{n,2}_{V,a}\|\le 2^{n\operatorname{H}_{V}(\hat{X}\|X)}. \label{markovrabecsles}$$ The next combinatorial construction will be substantially used in our proofs. Let a $(g+1)$-length sequence of positive integers $\mathbf{L}=(\hat{l},l^1,l^2,\dots, l^g)$, a non-negative integer $q$ and a collection of sequences $(\mathbf{\hat{x}},\mathbf{x}_1, \dots, \mathbf{x}_g)$ with $\mathbf{\hat{x}} \in \mathcal{X}^{\hat{l}}$, $\mathbf{x}_1 \in \mathcal{X}^{l^1}$, $\mathbf{x}_2 \in \mathcal{X}^{l^2}$, $\dots$, $\mathbf{x}_g \in \mathcal{X}^{l^g}$ be given. The sequences $\mathbf{\hat{x}},\mathbf{x}_1, \dots, \mathbf{x}_g$ are arranged in a two-row array as in Figure \[notionalakul\], i.e., $\mathbf{\hat{x}}$ is placed in the first row and $\mathbf{x}_1, \dots, \mathbf{x}_g$ are placed consecutively in the second row so that the second row ends by $q$ symbols after the first one; either row may start before the other one, depending on $\mathbf{L}$ and $q$. This configuration is referred to as $(\mathbf{L},q)$-array in the sequel. It will be always assumed that $\mathbf{\hat{x}}$ has a nonempty overlap with both $\mathbf{x}_1$ and $\mathbf{x}_g$, equivalently that $$q < l^g, \text{ and } \sum_{i=2}^g l^i -q < \hat{l}. \label{szimmetrikusfeltetel}$$ Note that the second inequality in (\[szimmetrikusfeltetel\]) trivially holds if $g=1$. ![Illustration for understanding some notations[]{data-label="notionalakul"}](notationalakul2mod) An $(\mathbf{L},q)$-array is divided into subblocks according to the starting and ending positions of the sequences $\mathbf{\hat{x}},\mathbf{x}_1, \dots, \mathbf{x}_g$ (see Fig. \[notionalakul\]). For technical reasons, we assume artificially that in the degenerate case of $q=0$ there is a $0$-length last block and in case of $\hat{l}=\sum_{i=1}^{g} l^i-q$ there is a $0$-length first block. Then the number of the subblocks is always equal to $g+2$. Their lengths, determined by $\mathbf{L}$ and $q$, will be denoted by $n_1, \dots, n_{g+2}$. Note that $q=n_{g+2}$. For $2 \le i \le g+1$, the $i$’th subblock consists of parts both in the first and the second row, let $V_i \in \mathcal{P}^{n_i}(\mathcal{X} \times \mathcal{X})$ denote their joint type. The first and last subblocks are contained in one row, their types are $V_1 \in \mathcal{P}^{n_1}(\mathcal{X})$ and $V_{g+2} \in \mathcal{P}^{n_1}(\mathcal{X})$. The subblock types $V_i$ will be often represented via dummy random variables, with $\hat{X}$ referring to the first and $X$ to the second row. When $V_i=V_i^{\hat{X}X}$, $V_i^{\hat{X}}$ and $V_i^X$ are the types of the parts of the $i$’th subblock in the first resp. second row. In the degenerate case when $n_1=0$ or $n_{g+2}=0$, let $V_1$ resp. $V_{g+2}$ be a dummy symbol regarded as the type of the empty sequence. Given $\mathbf{L}=(\hat{l},l^1,l^2,\dots, l^g)$ and $q$ satisfying (\[szimmetrikusfeltetel\]), each sequence $\mathbf{V}=(V_1,\dots,V_{g+2})$ of types $V_1 \in \mathcal{P}^{n_1} (\mathcal{X})$, $V_i \in \mathcal{P}^{n_1} (\mathcal{X} \times \mathcal{X})$, $i=2,\dots,g+1$, and $V_{g+2} \in \mathcal{P}^{n_{g+2}} (\mathcal{X})$, where $n_1, \dots, n_{g+2}$ are the subblock lengths determined by $(\mathbf{L},q)$, will be called a subtype sequence compatible with $(\mathbf{L},q)$. For $\mathbf{V}$ compatible with $(\mathbf{L},q)$, and sequences $\mathbf{\hat{x}} \in \mathcal{X}^{\hat{l}}$, $\mathbf{x}_i \in \mathcal{X}^{l^i}$, $i=2,\dots,g$ let $\mathds 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{\hat{x}};\mathbf{x}_1, \dots, \mathbf{x}_g)$ denote the indicator function equal to $1$ if $\mathbf{\hat{x}},\mathbf{x}_1, \dots, \mathbf{x}_g$ arranged in $(\mathbf{L},q)$ array has subtype sequence $\mathbf{V}$, and otherwise $0$. The set of collections of sequences $(\mathbf{\hat{x}},\mathbf{x}_1, \dots, \mathbf{x}_g)$ with $\mathds 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{\hat{x}};\mathbf{x}_1, \dots, \mathbf{x}_g)=1$, i.e., for which the corresponding $(\mathbf{L},q)$-array has subtype sequence $\mathbf{V}$, will be denoted by $\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}}$. In the sequel the following generalization of $\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}}$ is also needed. Let $\mathcal{I}$ be a set of prescribed equalities of form $\mathbf{x}_i=\mathbf{x}_j$ with $i,j \in [g]$, or $\mathbf{\hat{x}}=\mathbf{x}_1$ or $\mathbf{\hat{x}}=\mathbf{x}_g$ (the possibility of $\mathbf{\hat{x}}=\mathbf{x}_i$ for $1<i<g$ is excluded since $\mathbf{\hat{x}}$ has nonempty overlap with both $\mathbf{x}_1$ and $\mathbf{x}_g$). The set of those collections $(\mathbf{\hat{x}},\mathbf{x}_1, \dots, \mathbf{x}_g) \in \mathcal{T}^{\mathbf{L},q}_{\mathbf{V}}$ for which the equalities in $\mathcal{I}$ hold will be denoted by $\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}, \mathcal{I}}$. Of course, $\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}, \mathcal{I}}=\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}}$ if $\mathcal{I}$ is empty. Note that Section \[expurgation\] provides an introductory example of application of these notations, in the special case of $\mathbf{L}=(l,l)$ and $\hat{\mathbf{x}}=\mathbf{x}_1=\mathbf{x}$. Expurgation =========== A sequence $\mathbf{x} \in \mathcal{X}^{l}$ will be called $\gamma$-independent if its initial and final parts of length $r$ have empirical mutual information less than $\gamma$, for each $(\log l)^2 \le r \le \frac{l}{2}$. The subset of $\mathcal{T}^{l}_{P}$ consisting of $\gamma$-independent sequences is denoted by $\mathcal{T}^{l}_{P} (\gamma)$. \[expurgationlemma\] For each $P \in \mathcal{P}^{l} (\mathcal{X})$ and $\gamma>0$ $$\|\mathcal{T}_P^l \setminus \mathcal{T}_P^l(\gamma)\| \le p(l) 2^{- (\log l)^2 \gamma} \|\mathcal{T}_P^l\|,$$ where $p(l)$ denotes a polynomial factor not depending on $\gamma$. ![$(\mathbf{L},q)$ array with $\mathbf{L}=(l,l)$, $q=l-r$ and $(\hat{\mathbf{x}},\mathbf{x}_1)=(\mathbf{x},\mathbf{x})$.[]{data-label="expurgationcikk2"}](expurgationcikk2){width="70mm"} We present a proof using the concept of $(\mathbf{L},q)$-array introduced in Section \[notation\] as a simple example of the counting technique crucial for this paper. For a sequence $\mathbf{x}=x_1 \dots x_l$, the empirical mutual information of its initial and final parts of length $r$ is $\operatorname{I}_{V_2}(X \wedge \hat{X})$, where $(V_1,V_2,V_3)$ is the subtype sequence of the $(\mathbf{L},q)$-array with $\mathbf{L}=(l,l)$, $q=l-r$ and $\hat{\mathbf{x}}=\mathbf{x}_1=\mathbf{x}$, see Fig. \[expurgationcikk2\]. For a fixed $(\log l)^2\le r \le \frac{l}{2}$ let $\mathcal{V}_{\gamma}^{l,r}$ be the set of subtype sequences $\mathbf{V}=(V_1,V_2,V_3)$ compatible with $(\mathbf{L},l-r)$ for which there exists $\mathbf{x} \in \mathcal{T}^{l}_{P}$ with $\mathds 1^{\mathbf{L},l-r}_{\mathbf{V}}(\mathbf{x};\mathbf{x})=1$ and $\operatorname{I}_{V_2}(X \wedge \hat{X})\ge \gamma$. Then $$\label{expurgeq1} \|\mathcal{T}_P^l \setminus \mathcal{T}_P^l(\gamma)\| \le \sum_{r=(\log l)^2}^{\frac{l}{2}} \sum_{\mathbf{V} \in \mathcal{V}_{\gamma}^{l,r}} \|\mathcal{T}^{\mathbf{L},l-r}_{\mathbf{V}, \{\hat{\mathbf{x}}=\mathbf{x}_1\}}\|,$$ where $\mathcal{T}^{\mathbf{L},l-r}_{\mathbf{V}, \{\hat{\mathbf{x}}=\mathbf{x}_1\}}$ is defined as in the end of Section \[notation\] with $\mathcal{I}$ consisting of the equality $\{\hat{\mathbf{x}}=\mathbf{x}_1\}$. We divide the first and last subblocks into two pieces according to Fig. \[expurgationcikk2b\]. Formally for each $\mathbf{V}=(V_1,V_2,V_3)\in \mathcal{V}_{\gamma}^{l,r}$ we define $$\label{expurgset} \mathcal{V}_{\gamma, \mathbf{V}}^{l,r} \triangleq \left\{\hspace{-3pt} \begin{array}{l} (V_{11}^{\hat{X}},V_{12}^{\hat{X}},V_{31}^{X},V_{32}^{X}): \\ V_{11}^{\hat{X}} \in \mathcal{P}^{r}(\mathcal{X}), V_{12}^{\hat{X}} \in \mathcal{P}^{l-2r}(\mathcal{X}), V_{11}^{\hat{X}}\oplus V_{12}^{\hat{X}}= V_{1}^{\hat{X}} \\ V_{31}^{X} \in \mathcal{P}^{l-2r}(\mathcal{X}), V_{32}^{X} \in \mathcal{P}^{r}(\mathcal{X}), V_{31}^{X}\oplus V_{32}^{X}= V_{3}^{X} \\ V_{11}^{\hat{X}}=V_2^{X}, V_{12}^{\hat{X}}=V_{31}^{X}, V_{32}^{X}= V_{2}^{\hat{X}} \end{array}\hspace{-3pt}\right\}$$ Then using (\[basicfact1\]) and (\[basicfact2\]) $$\begin{aligned} &\|\mathcal{T}^{\mathbf{L},l-r}_{\mathbf{V}, \{\hat{\mathbf{x}}=\mathbf{x}_1\}}\|\le \sum_{(V_{11}^{\hat{X}},V_{12}^{\hat{X}},V_{31}^{X},V_{32}^{X}) \in \mathcal{V}_{\gamma, \mathbf{V}}^{l,r}} 2^{r \operatorname{H}_{V_{2}} (\hat{X}X)} 2^{(l-2r)\operatorname{H}_{V_{12}}(\hat{X})} \label{tovabboszt} \\ &=p'(l) 2^{r (\operatorname{H}_{V_{2}} (\hat{X}X)-\operatorname{H}_{V_2}(X)-\operatorname{H}_{V_2}(\hat{X}))} 2^{(l-2r)\operatorname{H}_{V_{12}}(\hat{X})+ r \operatorname{H}_{V_{11}}(\hat{X})+r \operatorname{H}_{V_{2}}(\hat{X})-l \operatorname{H}_{P}(X)} 2^{l \operatorname{H}_{P}(X)} \label{expuralgebra0} \\ &= p'(l)2^{-r\operatorname{I}_{V_2}(X \wedge \hat{X})} 2^{-l\operatorname{J}(V^{\hat{X}}_{11},V^{\hat{X}}_{12}, V^{\hat{X}}_{2})} 2^{l \operatorname{H}_{P}(X)}, \label{expuralgebra2}\end{aligned}$$ where $p'(l)$ denotes a polynomial factor not depending on $\gamma$. In (\[expuralgebra0\]) we used that $V_{11}^{\hat{X}}=V_2^{X}$. Substituting (\[expuralgebra2\]) into (\[expurgeq1\]), the positivity of $J(V^{\hat{X}}_{11},V^{\hat{X}}_{12}, V^{\hat{X}}_{2})$, (\[basicfact1\]) and the fact that $\operatorname{I}_{V_2}(X \wedge \hat{X})\ge \gamma$ prove the lemma. ![Further division of subblocks[]{data-label="expurgationcikk2b"}](expurgationcikk2b){width="70mm"} We will need the following consequence of Lemma \[expurgationlemma\]: for any positive numbers $\gamma_l$ with $\gamma_l \log l \rightarrow \infty$ as $l \rightarrow \infty$, for $l$ large enough $$\label{expurgsize} \frac{1}{2} \|\mathcal{T}^{l}_{P}\| \le \|\mathcal{T}^{l}_{P} (\gamma_l)\| \le \|\mathcal{T}^{l}_{P}\|$$ The model {#Modell} ========= The transmitter has a codebook library with multiple constant composition codebooks. The codewords’ length and type are fixed within codebooks, but can vary from codebook to codebook, subject to a bound on permissible codeword length ratios. \[constantcomposition\] Let $D \in (0,1]$, positive integers $n$, $M$, $l^1,l^2,\dots,l^M$ with $D n \le l^i\le n$ for all $i \in [M]$, distributions $\{ P^{i} \in \iP^{l^i} (\iX), {i}\in [M] \}$ and rates $\{ R^{i} , i \in [M]\}$ be given parameters. A codebook library with the above parameters, denoted by $\mathcal{A}$, consists of constant composition codebooks $(A^{1},\dots,A^{M})$ such that $A^{i}=\{{\mathbf x}^{i}_1,{\mathbf x}^{i}_2, \dots {\mathbf x}^{i}_{N^i} \} $ with ${\mathbf x}^{i}_{a} \in \mathcal{T}^{l^i}_{P^{i}}$, ${i} \in [M]$, $N^i = \left\lfloor 2^{l^{i} R^i} \right\rfloor$, $a \in [N^i]$. In the sequel, $n$ will be referred to as length-bound. The parameters in Definition \[constantcomposition\] will depend on $n$, except for the constant $D$, but this dependence will be suppressed for brevity. Actually the proof of Theorem \[maintheorem\] works if $D=D(n)$ goes to $0$ appropriately slowly. Note, however, that the appropriate speed of its convergence to $0$ would depend on the number of codebooks $M(n)$. For this reason and for the sake of simplicity we have chosen to fix $D$. ![Outline of the model[]{data-label="librarybolmodellcikk"}](librarybolmodellcikkmod){width="70mm"} The transmitter continuously sends messages to the receiver through a DMC $W: \mathcal{X} \rightarrow \mathcal{Y}$ that may be unknown to the sender and receiver. Before sending a message, the transmitter arbitrarily chooses one codebook of the library. This choice is not known to the receiver, who is cognisant only of the codebook library. His choices are described by an infinite codebook index sequence $\mathbf{h}=(h_1,h_2, h_3, \dots, h_j, \dots)$ where $h_j \in [M]$. In the sequel, $\mathbf{h}$ will be referred to as codebook schedule. To each fixed codebook schedule $\mathbf{h}$ there corresponds a sequence $B_1$, $B_2$, …of mutually independent random messages, where $B_j$ is uniformly distributed on $[N^{h_j}]$. To transmit $B_j=b$, the encoder assigns to it the $b$’th codeword of the codebook of index $h_j$. The transmission of this message starts at instance $s_j=\sum_{i=1}^{j-1} l^{h_i}+1$, depending on the codebook schedule but not on the actual messages. All formal statements in this paper refer to a fixed codebook schedule. Still, Theorem \[maintheorem\] below covers also scenarios where the schedule $\mathbf{h}$ is random (random process of any kind), providing the messages $B_j$ are conditionally independent with uniform conditional distributions given $\mathbf{h}$, as the bounds (\[felsobecsles\]) and (\[rcb\]) do not involve $\mathbf{h}$. A decoder is defined as a mapping of infinite channel output sequences $\mathbf{y}=y_1,y_2,\dots$ into decoder output sequences $\mathbf{o}=o_1,o_2, \dots$, where each $o_t$ either equals a pair $(h,b)$ with $h \in [M]$, $b \in [N^{l^{\hat{h}}}]$, or the space symbol “-”, or the string “erasure”. By correct decoding of message $B_j=b$ we mean that $o_t$ is equal to $(B_j,b)$ at the starting instance $t=s_j$ and to space at the remaining $l^{h_j}-1$ instances of the transmission of this message. Accordingly, the average decoding error probability for the $j$’th transmission is $$\label{errordecoding} Err_j^{\mathbf{h}} \triangleq Pr\left( O_{s_j} \ne (h_j,B_j) \text{ or } \exists \text{ } i \in \{ s_j+1,\dots,s_{j}+l^{h_j}-1 \} \text{ with } O_{i} \ne "-" \right).$$ As correct decoding with small error probability is not possible for codebooks with type $P$ and rate $R\ge I(P,W)$, when such codebook is used a good decoder should declare erasure. We define average erasure detection failure probability for the $j$’th message as $$\label{errorth} Edf_{j}^{\mathbf{h}} \triangleq Pr\left( \exists \text{ } i \in \{ s_j,\dots,s_{j}+l^{h_j}-1 \} \text{ with } O_{i} \ne \text{ "erasure" } \right).$$ It is required to be small when $R^{h_j}\ge I(P^{h_j},W)$. Note that in (\[errordecoding\]) and (\[errorth\]) the probabilities are calculated over the random choice of the messages, and the channel transitions. Capital letters are used to indicate randomness. The universal decoder in the proof of Theorem \[maintheorem\] does not use the whole channel output sequence to determine $o_t$ but only $y_t$ and the preceding $2\cdot l^{max}-2$ and the subsequent $2\cdot l^{max}-2$ output symbols, where $l^{max}$ is the maximal codeword length in the codebook library, i.e., $l^{max}=\max_{i \in [M]} l^i \le n$. This can be seen to imply that the error events in (\[errordecoding\]) (or in (\[errorth\])) corresponding to message transmission indices $j_1$, $j_2$ are independent if $\|j_1 - j_2\|$ exceeds a constant times $\frac{1}{D}$. The fact that there is only one sender raises the question whether it is possible to substitute average error terms (\[errordecoding\]) and (\[errorth\]) for maximal ones in Theorem \[maintheorem\] below, as for example in [@Csiszar]. Unfortunately, as the error events depend simultaneously on several codebooks, this does not seem possible. The standard argument for upgrading average error results to maximal error gives only that the statement of Theorem \[maintheorem\] also holds for the following error terms: $$\begin{aligned} \label{maxerror} &Errm_{j}^{\mathbf{h}} \triangleq \max_{b \in [N^{h_j}]} Pr\left( O_{s_j} \ne (h_j,B_j) \text{ or } \exists \text{ } i \in \{ s_j+1,\dots,s_{j}+l^{h_j}-1 \} \text{ with } O_{i} \ne "-" \| B_j=b \right),\\ &Edfm_{j}^{\mathbf{h}} \triangleq \max_{b \in [N^{h_j}]} Pr\left( \exists i \text{ } \in \{ s_j,\dots,s_{j}+l^{h_j}-1 \} \text{ with } O_{i} \ne \text{ "erasure" } \| B_j=b \right). \label{maxerrorc}\end{aligned}$$ Here the maximum is taken relative to the $j$’th transmission, while still averaging relative to the other transmissions. Main theorem {#maintheoremsection} ============ \[maintheorem\] For each $n$ let codebook library parameters as in Definition \[constantcomposition\] be given with $D$ fixed and $\frac{1}{n}log M \rightarrow 0$ as $n \rightarrow \infty$. Then there exist a sequence $\nu_n(\|\mathcal{X}\|,\|\mathcal{Y}\|,\{M\}_{n=1}^{\infty},D)$ with $\frac{1}{n}\log \nu_n \rightarrow 0$ and for each $n$ a codebook library $\mathcal{A}$ with the given parameters, and decoder mappings such that for all codebook schedule $\mathbf{h}$ and index $j$ (i) $$\label{felsobecsles} Err_j^{\mathbf{h}} \le \nu_n \cdot 2^{- l^{h_j} \mathcal{E}_{r}(R^{h_j},P^{h_j},W) },$$ where $$\label{rcb} \mathcal{E}_{r}(R,P,W) =\min_{{\genfrac{}{}{0pt}{}{V\in \mathcal{P}(\mathcal{X} \times \mathcal{Y})}{V_{X}=P}}} \operatorname{D}(V_{Y\|X}\|\|W\|P)+\|\operatorname{I}_V (X \wedge Y)-R\|^{+}$$ is the random coding error exponent function. (ii) If $R^{h_j}\ge I[W,P^{h_j}]$ then $$\label{felsobecsles2} Edf_{j}^{\mathbf{h}} \le \frac{1}{n}\log \nu_n.$$ As the random coding exponent function $\mathcal{E}_{r}(R,P,W)$ is positive if and only if $R<I(P,W)$ , the bound (\[felsobecsles\]) can be useful only if $R^{h_j}<I(P^{h_j},W)$. Recall that all parameters in Theorem \[maintheorem\] depend on length-bound $n$. Even if $R^{h_j}<I(P^{h_j},W)$, the first factor in (\[felsobecsles\]) may override the second one, but this does not happen for large $n$ if $I(P^{h_j},W)-R^{h_j}$ is bounded away from $0$. Then (\[felsobecsles\]) guarantees exponentially small error probability, with exponent $\mathcal{E}_{r}(R^{h_j},P^{h_j},W)$ relative to codeword length $l^{h_j}$. This result is the best possible for codebooks whose rates $R^{h_j}$ are sufficiently close to $I(P^{h_j},W)$, since even for a single codebook with codeword type $P$ the random coding error exponent is tight for rates less than $I(P,W)$ but larger than a critical rate $\tilde{R}(P,W)$. Possible improvements for codebooks of small rates are beyond the scope of this paper. For the standard mentioned properties of the function $\mathcal{E}_{r}(R,P,W)$ see for example [@Csiszar2]. \[etan\] For erasure declaration failure probability, Theorem \[maintheorem\] asserts only convergence to $0$, perhaps not exponentially fast. An argument similar to [@isit2013], Appendix C suggests that this may not be a shortcoming due to loose calculation, upper bound (\[hibabindikator2th\]) in the proof of Theorem \[maintheorem\] is not exponentially small, under reasonable assumptions. An exponentially small erasure declaration failure probability could be achieved by modifying the decoder used in Theorem \[maintheorem\], replacing the threshold $\eta_n \rightarrow 0$ in (\[dekodolomukth\]) by a positive contant, but at the expense of decreasing the decoding error probability exponents and perhaps declaring erasure also when decoding would be possible. As shown in Section \[improvements\], this problem, however, can be easily overcome if the sender knows the channel. The next packing lemma provides the appropriate codebook library for Theorem \[maintheorem\]. We emphasize that the constructed codebook library works simultaneously for all codebook schedules $\mathbf{h}$. ![Notations used in Lemma \[Rc-Packing-lemma\][]{data-label="packingtenycikk"}](packingtenycikkmod) Given codebook library parameters as in Def. \[constantcomposition\], for any sequence $\mathbf{k}=(\hat{k}, k_1,\dots, k_g)$ consisting of codebook indices (integers in $[M]$) let $\mathbf{L}(\mathbf{k})$ denote the sequence $(l^{\hat{k}},l^{k_1},\dots,l^{k_g})$. Further, given also a nonnegative integer $q$ satisfying (\[windowcondition1\]) and (\[windowcondition2\]) below, denote by $\mathcal{V}^{\mathbf{k},q,n}$ the family of those subtype sequences $\mathbf{V}=(V_1,... V_{g+1})$ compatible with $(\mathbf{L}(\mathbf{k}),q)$ for which the sequences $\hat{\mathbf{x}}$ and $\mathbf{x}_1,\dots,\mathbf{x}_g$ that form $(\mathbf{L}(\mathbf{k}),q)$-arrays with subtype sequence $\mathbf{V}$ have types $P^{\hat{k}}$ and $P^{k_i}$, ${i\in [g]}$ respectively. \[Rc-Packing-lemma\] Let a sequence of codebook library parameters be given as in Theorem \[maintheorem\]. Then there exist a sequence $\nu^{'}_n (\|\mathcal{X}\|,\{M\}_{n=1}^{\infty},D)$ with $\frac{1}{n} \log \nu'_n \rightarrow 0$ and for each $n$ a codebook-library $\mathcal{A}$ with the given parameters such that each codeword is $\gamma_n$-independent, i.e., ${\mathbf x}^{i}_{a} \in \mathcal{T}^{l^i}_{P^{i}}(\gamma_n)$, ${i} \in [M]$, $a \in [N^i]$ where $\gamma_n=(\log n)^{-\frac{1}{2}}$, and for each sequence $\mathbf{k}=(\hat{k}, k_1,\dots, k_g)$ consisting of codebook indices, non-negative integer $q$ with $$\begin{aligned} &q < l^{k_g}, \text{ } \sum_{i=2}^{g} l^{k_i}-q < l^{\hat{k}}, \label{windowcondition1}\\ &l^{\hat{k}}\le \sum_{i=1}^{g} l^{k_i}-q, \label{windowcondition2}\end{aligned}$$ and subtype sequence $\mathbf{V}=(V_1,V_2,\dots, V_{g+2}) \in \mathcal{V}^{\mathbf{k},q,n}$ $$\begin{aligned} &K^{\mathbf{k},q}[\mathbf{V}] \triangleq \sideset{}{'}\sum_{{\genfrac{}{}{0pt}{}{a_i \in [N^{k_i}], i \in [g]}{\hat{a} \in [N^{\hat{k}}]}}}\hspace{-11pt}\mathds 1^{\mathbf{L}(\mathbf{k}), q}_{\mathbf{V}}(\mathbf{x}_{\hat{a}}^{\hat{k}};{\mathbf x}_{a_1}^{k_1}, \dots, {\mathbf x}_{a_g}^{k_g}) \label{packingegyutt}\\ &\le \nu^{'}_n \cdot 2^{-\sum_{i=2}^{g+1} n_i \operatorname{I}_{V_i}(X \wedge \hat X) + \sum_{i \in [g]} l^{k_i}R^{k_i}+l^{\hat{k}}R^{\hat{k}}-l^{\hat{k}} \operatorname{J}(V_2^{\hat{X}}, \dots, V_{g+1}^{\hat{X}}) }, \notag\end{aligned}$$ where in (\[packingegyutt\]) the summation sign with the comma denotes standard summation except in the case of $g=1$,$\hat{k}=k_1$,$q=0$ when it is restricted for $\hat{a}$ in $[N^{\hat{k}}]=[N^{k_1}]$ to $\hat{a}\ne a_1$. Note that (\[windowcondition1\]) corresponds to (\[szimmetrikusfeltetel\]) ensuring that in an $(\mathbf{L}(\mathbf{k}),q)$-array $\hat{\mathbf{x}}$ has nonempty overlap with both $\mathbf{x}_1$ and $\mathbf{x}_g$ while (\[windowcondition2\]) ensures that, in addition, the first row is completely covered by the second row (see Fig. \[packingtenycikk\]). Moreover, the condition (\[windowcondition1\]) implies that $g \le \frac{2}{D}+1$. Choose the codebook library $\mathcal{A}$ at random, i. e., for all $i \in [M]$ the codewords of $\mathcal{A}^i$ are chosen independently and uniformly from $\mathcal{T}^{l^i}_{P^i} (\gamma_n)$. Fix arbitrarily a sequence $\mathbf{k}=(\hat{k}, k_1,\dots, k_g)$ consisting of codebook indices, a non-negative integer $q$ fulfilling (\[windowcondition1\]) and (\[windowcondition2\]), a subtype sequence $\mathbf{V}=(V_1,V_2,\dots, V_{g+2}) \in \mathcal{V}^{\mathbf{k},q,n}$ and codeword indices $\hat{a}\in [N^{\hat{k}}]$, $a_1 \in [N^{k_1}], \dots, a_g \in [N^{k_g}]$ such that if if $g=1$, $\hat{k}=k_1$, $q=0$ then $\hat{a}\ne a_1$. Then, as shown in Appendix \[bizonyitasvarhatoertek\], the following inequality holds $$\label{egysegesbecsles} \mathds{E}\left( 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right)\le \nu^{''}_n 2^{-\sum_{i=2}^{g+1} n_i \operatorname{I}_{V_i}(X \wedge \hat X) -l^{\hat{k}} \operatorname{J}(V_2^{\hat{X}}, \dots V_{g+1}^{\hat{X}})},$$ where $\nu^{''}_n$ is a subexponential function of $n$ that depends only on $D$ and the alphabet size $\|\mathcal{X}\|$. Let $E^{\mathbf{k},r}[\mathbf{V}]$ be the exponent in upper-bound (\[packingegyutt\]), i.e., $$E^{\mathbf{k},q}[\mathbf{V}]=-\sum_{i=2}^{g+1} n_i \operatorname{I}_{V_i}(X \wedge \hat X) -l^{\hat{k}} \operatorname{J}(V_2^{\hat{X}}, \dots, V_{g+1}^{\hat{X}})+ \sum_{i \in [g]} l^{k_i}R^{k_i}+l^{\hat{k}}R^{\hat{k}}.$$ It follows from (\[egysegesbecsles\]) that under this random selection the expected value of the expression $$\label{eztbecsultuk} K^{\mathbf{k},q}[\mathbf{V}]2^{-E^{\mathbf{k},q}[\mathbf{V}]}$$ is upper-bounded by $\nu^{''}_n$. Denote by $S$ the sum of (\[eztbecsultuk\]) for all possible $\mathbf{k}=(\hat{k}, k_1,\dots, k_g)$, $q$ and subtype sequences $\mathbf{V}$. As $M$ grows at most subexponentially and the number of types is polynomial, it follows that $\mathds{E}(S) \le \nu'_{n}$ for suitable $\nu^{'}_n (\|\mathcal{X}\|,\{M\}_{n=1}^{\infty},D)$ with $\frac{1}{n} \log \nu'_n \rightarrow 0$. Then there exists a realization of the codebook library with $S \le \nu'_{n}$. Hence, the lemma is proved if (\[egysegesbecsles\]) is proved. We would like to emphasize some interesting features of the rather technical proof of (\[egysegesbecsles\]) in Appendix \[bizonyitasvarhatoertek\]: in subcases 1b and 1d it exploits the $\gamma_n$-independence property of the codewords and in subcases 2c and 2d second order types are employed. Lemma \[Rc-Packing-lemma\] provides the appropriate codebook library $\mathcal{A}$. We define the following sequential decoder. Assume that decoding related to symbols $y_1,\dots, y_{t-1}$ is already performed (i.e., $o_1,o_2, \dots,o_{t-1}$ are already defined) and now instance $t$ is analyzed. In the first stage of decoding the decoder tries to find indices $\tilde{h}, \tilde{b}$ which uniquely maximize $$\label{dekodolomuk} l^h\left( \operatorname{I}(\mathbf{x}_{b}^{h} \wedge y_t y_{t+1} \dots y_{t+l^h -1})-R^{l^h} \right).$$ If the decoder successfully finds a unique maximizer $\tilde{h}, \tilde{b}$, the second stage of decoding starts. Let $\eta_n=\eta_n(\mathcal{X},\mathcal{Y},\{M\}_{n=1}^{\infty},D)$ be a sequence with $\eta_n \rightarrow 0$ as $n \rightarrow \infty$. In the second stage if $$\label{dekodolomukth} \left( \operatorname{I}(\mathbf{x}_{\tilde{b}}^{\tilde{h}} \wedge y_t y_{t+1} \dots y_{t+l^{\tilde{h}} -1})-R^{l^{\tilde{h}}} \right) > \eta_n$$ and for all $h, b$ and $d \in \{t-l^h+1, \dots, t-1 \} \cup \{t+1, \dots, t+l^{\tilde{h}}-1 \}$ the maximum of (\[dekodolomuk\]) is strictly larger than $$\label{dekodolomuk2} l^h\left( \operatorname{I}(\mathbf{x}_{b}^{h} \wedge y_{d} y_{d+1}\dots y_{d+l^h -1})-R^{l^h} \right),$$ the decoder decodes $\mathbf{x}_{\tilde{b}}^{\tilde{h}}$ as the codeword sent in the window $[t,t+l^{\tilde{h}}-1]$, i.e., $o_{t}$ becomes equal to $(\tilde{h}, \tilde{b})$, and $o_{t+i}$ becomes equal to $\text{"-"}$, $i \in [l^{\tilde{h}}-1]$. Then the decoder jumps to the instance $t+l^{\tilde{h}}$, where the same but shifted procedure is performed. If in the first stage the maximum is not unique or in the second stage at least one of the required inequalities is not fulfilled, the decoder reports erasure in instance $t$, i.e., $o_t$ becomes equal to $\text{"erasure"}$, and the decoder goes to instance $t+1$. See Fig. \[stage2\]. ![Universal two-stage decoder[]{data-label="stage2"}](dekodolcikkmod){width="95mm"} We prove that the codebook library $\mathcal{A}$ provided by Lemma \[Rc-Packing-lemma\] with the decoder specified above with appropriately chosen $\eta_n(\|\mathcal{X}\|,\|\mathcal{Y}\|, \{M\}_{n=1}^{\infty},D)$ fulfills Theorem \[maintheorem\]. Let a codebook schedule $\mathbf{h}$ and an index $j$ be given. Let $Y_{s_j}, Y_{s_j+1},\dots, Y_{s_j+l^{h_j}-1}$ denote the random output symbols affected by the $j$-th message $B_j$. Proof of part (i) of Theorem \[maintheorem\]: Let $\mathcal{E}_j^{\mathbf{h}}(\text{TH})$ be the following event corresponding to threshold criterion (\[dekodolomukth\]) $$\left\{ \left( \operatorname{I}(\mathbf{x}_{B_j}^{h_j} \wedge Y_{s_j}Y_{s_j+1} \dots Y_{s_j+l^{h_j}-1})-R^{l^{h_j}} \right)\le \eta_n \right\}, \label{errorpatternth}$$ and for $\hat{k} \in [M]$ and $d \in \{s_j-l^{\hat{k}}+1,s_j-l^{\hat{k}}+2, \dots, s_j+l^{h_j}-1 \}$ let $\mathcal{E}_j^{\mathbf{h}}(\hat{k},d)$ be the event $$\left\{\hspace{-3pt}\begin{array}{l} l^{\hat{k}}\left( \operatorname{I}(\mathbf{x}_{\hat{a}}^{\hat{k}} \wedge Y_{d} Y_{d+1}\dots Y_{d+l^{\hat{k}} -1})-R^{l^{\hat{k}}} \right) \\ \ge l^{h_j}\left( \operatorname{I}(\mathbf{x}_{B_j}^{h_j} \wedge Y_{s_j} Y_{s_j+1}\dots Y_{s_j+l^{h_j}-1})-R^{l^{h_j}} \right), \\ \text{for some } \hat{a} \in [N^{\hat{k}}] \text{ }(\hat{a}\ne B_j \text{ if } \hat{k}=h_j \text{ and } d=s_j) \end{array}\hspace{-3pt}\right\}. \label{errorpattern}$$ The mutual informations in (\[errorpatternth\]) and (\[errorpattern\]) are empirical ones as in (\[dekodolomukth\]), (\[dekodolomuk2\]), though involving random sequences as the capital letters indicate. Denote by $Err_j^{\mathbf{h}}(\text{TH})$ and $Err_j^{\mathbf{h}}(\hat{k},d)$ the probabilities of these events, respectively. Then $$\label{uniokorlatahibara} Err_j^{\mathbf{h}} \le Err_j^{\mathbf{h}}(\text{TH}) + \sum_{(\hat{k},d)} Err_j^{\mathbf{h}}(\hat{k},d).$$ Note that the first term in (\[uniokorlatahibara\]) is the probability that the threshold criterion is not fulfilled by the sent codeword while the sum of terms $Err_j^{\mathbf{h}}(\hat{k},d)$ provide upper-bound to the probability of the event that the sent codeword is outperformed in terms of (\[dekodolomuk\]) either in Stage 1 or 2. The event that the decoder skips time index $s_j$ due to an erroneous previous decoding, i.e., $O_{s_j}=\text{"-"}$, is contained in the latter event. By standard argument it follows from (\[basicfact1\])-(\[basicfact4\]) that $$Err_j^{\mathbf{h}}(\text{TH}) \le subexp(n) \cdot 2^{- l^{h_j} \mathcal{E}^n_{TH}(R^{h_j},P^{h_j},W)} \label{bizTH},$$ where $$\label{rcbth} \mathcal{E}^n_{TH}(R^{h_j},P^{h_j},W) \triangleq \min_{{\genfrac{}{}{0pt}{}{V\in \mathcal{P}(\mathcal{X} \times \mathcal{Y})}{V_{X}=P^{h_j}, \operatorname{I}_V(X \wedge Y)-R^{h_j}\le \eta_n }}} \operatorname{D}(V_{Y\|X}\|\|W \|P^{h_j}).$$ It follows from (\[rcb\]) that $\mathcal{E}_{r}(R^{h_j},P^{h_j},W) \le \mathcal{E}^n_{TH}(R^{h_j},P^{h_j},W)+\eta_n$. Hence, as $\eta_n \rightarrow 0$ we get $$Err_j^{\mathbf{h}}(\text{TH}) \le subexp(n) \cdot 2^{- l^{h_j} \mathcal{E}_{r}(R^{h_j},P^{h_j},W)}. \label{biz1}$$ For $\hat{k}=h_j$ and $d=s_j$ by standard argument $$Err_j^{\mathbf{h}}(\hat{k},d) \le subexp(n) \cdot 2^{- l^{h_j} \mathcal{E}_{r}(R^{h_j},P^{h_j},W)} \label{biz1b}$$ ![Messages and assigned codewords affecting output sequence $\mathbf{Y}=(Y_{d} Y_{d+1}\dots Y_{s_j + l^{h_j}-1})$ along with explanation of notation $q$.[]{data-label="hibabecslesujcikkmod1"}](hibabecslesujcikkmod1) To prove part (i) of the theorem it is enough to show that this upper-bound also applies to $Err_j^{\mathbf{h}}(\hat{k},d)$ for all $\hat{k} \in [M]$ and $d \in \{s_j-l^{\hat{k}}+1,s_j-l^{\hat{k}}+2, \dots, s_j+l^{h_j}-1 \}$ (the number of pairs $(\hat{k},d)$ is subexponential). Fix a pair $(\hat{k},d) \ne (h_j,s_j)$. Assume that $d \le s_j$ (the analysis of the case $d>s_j$ is similar). The codebook schedule $\mathbf{h}$ determines the number $g$ of messages which affect outputs $Y_{d}, Y_{d+1},\dots, Y_{s_j + l^{h_j}-1}$ (see Fig. \[hibabecslesujcikkmod1\]). Let $$N \triangleq \prod_{i=1}^{g} N^{h_{j-g+i}} = \prod_{i=1}^{g}2^{l^{h_{j-g+i}} R_{h_{j-g+i}}}.$$ Then $$Err_j^{\mathbf{h}}(\hat{k},d) = N^{-1} \sum_{a_i \in [N^{h_{j-g+i}}], i \in [g]} {\textnormal{Pr}}\{ \mathcal{E}_j^{\mathbf{h}}(\hat{k},d) \| B_{j-g+i}=a_i, i \in [g]\}. \label{teljesval}$$ Let $k_i$ denote $h_{j-g+i}$, $i \in [g]$, let $\mathbf{k}=(\hat{k},k_1, \dots,k_g)$ and $\mathbf{L}(\mathbf{k})=(l^{\hat{k}}, l^{k_1}, \dots, k^{k_g})$. Furthermore, let $q= s_{j} + l^{h_j} - (d + \hat{l})$ and $l=\sum_{i=1}^{g} l^{h_j - g +i}$ (see Fig. \[hibabecslesujcikkmod1\]). This time we consider an array that includes the codewords ${\mathbf x}_{\hat{a}}^{\hat{k}}, {\mathbf x}_{a_1}^{k_1},\dots,{\mathbf x}_{a_g}^{k_g}$ and also the output sequence ${\mathbf y}\in \mathcal{Y}^l$, namely, we arrange $({\mathbf x}_{\hat{a}}^{\hat{k}}; {\mathbf x}_{a_1}^{k_1},\dots,{\mathbf x}_{a_g}^{k_g})$ into $(\mathbf{L}(\mathbf{k}),q)$-array as in Section \[notation\] and we put the sequence $\mathbf{y}$ in a third row, the starting and ending position of $\mathbf{y}$ coincide with the starting position of ${\mathbf x}_{a_1}^{k_1}$ and the ending position of ${\mathbf x}_{a_g}^{k_g}$ respecticely. This 3-row array is also divided into subblocks according to the starting and ending positions of codewords ${\mathbf x}_{\hat{a}}^{\hat{k}}, {\mathbf x}_{a_1}^{k_1},\dots,{\mathbf x}_{a_g}^{k_g}$. We apply the notations and conventions of Section \[notation\], for example, the lengths of the subblocks are denoted by $n_1,n_2,\dots,n_{g+2}$ as before. See Fig. \[hibabecslesujcikkmod2\]. Let $\mathcal{VM}^{\mathbf{k},q,n}$ be the following family of 3-row array subtype sequences $$\label{hibatipus} \left\{\hspace{-3pt} \begin{array}{l} \mathbf{V}=(V_1,V_2,\dots,V_{g+2}): V_1=V^{XY}_1 \in \mathcal{P}^{n_1}(\mathcal{X} \times \mathcal{Y}), V_{g+2}=V^{XY}_{g+2} \in \mathcal{P}^{n_{g+2}}(\mathcal{X} \times \mathcal{Y})\\ V_i= V^{\hat{X}XY}_i \in \mathcal{P}^{n_i}(\mathcal{X} \times \mathcal{X} \times \mathcal{Y}), 2\le i \le g+1 \\ (V^{X}_1,V^{\hat{X}X}_2,\dots, V^{\hat{X}X}_{g+1},V^{X}_{g+2}) \in \mathcal{V}^{\mathbf{k},q,n}\\ l^{k_{g}} (\operatorname{I}_{V_{g+1} \oplus V_{g+2}}(X \wedge Y)-R^{k_{g}}) \le l^{\hat{k}} ( \operatorname{I}_{V_2 \oplus V_3 \oplus \dots \oplus V_{g+1}}(\hat{X} \wedge Y)-R^{\hat{k}}) \end{array}\hspace{-3pt}\right\},$$ where $\mathcal{V}^{\mathbf{k},q,n}$ is defined immediately prior to Lemma \[Rc-Packing-lemma\]. ![The 3-row array of codewords $\hat{\mathbf{x}}, \mathbf{x}_1, \dots, \mathbf{x}_g$ and output $\mathbf{y}$.[]{data-label="hibabecslesujcikkmod2"}](hibabecslesujcikkmod2) Then, using (\[basicfact4\]), the upper-bound (\[teljesval\]) can be further upper-bounded by $$\begin{aligned} &\sum_{\mathbf{V} \in \mathcal{VM}^{\mathbf{k},q,n}} \hspace{-4pt} N^{-1} \hspace{-4pt} \sum_{a_i \in [N^{k_i}], i \in [g]} \hspace{-1pt} \prod_{i=1}^{g+2} \hspace{-1pt} \big( 2^{-n_i (\operatorname{D}(V_i^{Y\|X}\|\|W\|V_i^{X})+\operatorname{H}_{V_i}(Y\|X))} \notag\\ &\cdot \big\| \{\mathbf{y} \in \mathcal{Y}^{l}:\mathds{1}^{\mathbf{L}(\mathbf{k}),q}_{\mathbf{V}} ( {\mathbf x}_{\hat{a}}^{\hat{k}}; {\mathbf x}_{a_1}^{k_1},\dots,{\mathbf x}_{a_g}^{k_g};{\mathbf y}) =1 \text{ for some } \hat{a} \in [N^{\hat{k}}] \big\|, \label{hibabindikator}\end{aligned}$$ where the indicator function $\mathds{1}^{\mathbf{L}(\mathbf{k}),q}_{\mathbf{V}} ( {\mathbf x}_{\hat{a}}^{\hat{k}}; {\mathbf x}_{a_1}^{k_1},\dots,{\mathbf x}_{a_g}^{k_g};{\mathbf y})$ equals $1$ (otherwise $0$) if placing ${\mathbf x}_{\hat{a}}^{\hat{k}}, {\mathbf x}_{a_1}^{k_1},\dots,{\mathbf x}_{a_g}^{k_g}$ and ${\mathbf y}$ into a 3-row array as above, the type of the $i$’th subblock is $V_i$ for each $i \in [g+2]$ with $n_i>0$. We can upper-bound the set size in (\[hibabindikator\]) two different ways. The first bound is $2^{\sum_{i=1}^{g+2} n_i \operatorname{H}_{V_i} (Y\|X)}$. The second bound is: $$\sum_{\hat{a} \in [N^{\hat{k}}]} 2^{n_1 \operatorname{H}_{V_1} (Y\|X) + \sum_{i=2}^{g+1} n_i \operatorname{H}_{V_i} (Y\|\hat{X}X)+n_{g+2} \operatorname{H}_{V_{g+2}} (Y\|X)} \mathds{1}^{\mathbf{L}(\mathbf{k}),q}_{\mathbf{V}'} ( {\mathbf x}_{\hat{a}}^{\hat{k}}; {\mathbf x}_{a_1}^{k_1},\dots,{\mathbf x}_{a_g}^{k_g} ),$$ where $\mathbf{V}' =(V_1^{X}, V_2^{\hat{X}X},\dots, V_{g+1}^{\hat{X}X},V_{g+2}^{X})$. Substituting these bounds into (\[hibabindikator\]) and using (\[packingegyutt\]) we get that: $$\begin{aligned} &Err_j^{\mathbf{h}}(\hat{k},d) \le \sup_{\mathbf{V} \in \mathcal{VM}^{\mathbf{k},q,n}} subexp(n) \hspace{-1pt} 2^{- \sum_{i=1}^{g+2} n_i \operatorname{D}(V_i^{Y\|X}\|\|W\|V_i^{X})} 2^{-\|\sum_{i=2}^{g+1} n_i \operatorname{I}_{V_i}(\hat{X}\wedge X Y)+l^{\hat{k}}J(V_2^{\hat{X}}, \dots, V_{f+1}^{\hat{X}})-l^{\hat{k}}R^{\hat{k}}\|^{+}} \label{hibabindikatorutolag} \\ &\le \sup_{\mathbf{V} \in \mathcal{VM}^{\mathbf{k},q,n}} subexp(n) \hspace{-1pt} 2^{- \sum_{i=1}^{g+2} n_i \operatorname{D}(V_i^{Y\|X}\|\|W\|V_i^{X})} 2^{-\|\sum_{i=2}^{g+1} n_i \operatorname{I}_{V_i}(\hat{X}\wedge Y)+l^{\hat{k}}J(V_2^{\hat{X}}, \dots, V_{f+1}^{\hat{X}})-l^{\hat{k}}R^{\hat{k}}\|^{+}}. \label{hibabindikator2}\end{aligned}$$ The term inside $\|\|^{+}$ is equal to $$\begin{aligned} -\sum_{i=2}^{g+1} n_i \operatorname{H}_{V_i}(\hat{X} \| Y) + l^{\hat{k}} \operatorname{H}(P^{\hat{k}}) -l^{\hat{k}}R^{\hat{k}} \label{lassanuccso}\end{aligned}$$ By convexity, (\[lassanuccso\]) can be lower-bounded by $$\begin{aligned} &-l^{\hat{k}} \operatorname{H}_{V_2 \oplus \dots \oplus V_{g+1}}(\hat{X} \| Y) + l^{\hat{k}}\operatorname{H}(P^{\hat{k}}) -l^{\hat{k}}R^{\hat{k}}=l^{\hat{k}} \operatorname{I}_{V_2 \oplus \dots \oplus V_{g+1}}(\hat{X}\wedge Y) -l^{\hat{k}}R^{\hat{k}}. \label{lassanuccso2}\end{aligned}$$ Substituting (\[lassanuccso2\]) into (\[hibabindikator2\]) and using (\[hibatipus\]) we get $$\begin{aligned} &Err_j^{\mathbf{h}}(\hat{k},d) \le \sup_{\mathbf{V} \in \mathcal{VM}^{\mathbf{k},q,n}} subexp(n) \cdot 2^{-\sum\limits_{i=1}^{g+2} n_i \operatorname{D}(V_i^{Y\|X}\|\|W\|V_i^{X})-\|l^{k_g} \operatorname{I}_{V_{g+1} \oplus V_{g+2}}(X \wedge Y) -l^{k_{g}} R^{k_g}\|^{+}} \label{lassanuccso3} \\ &\le \sup_{\mathbf{V} \in \mathcal{VM}^{\mathbf{k},q,n}} subexp(n) \cdot 2^{-\sum\limits_{i=g+1}^{g+2} n_i \operatorname{D}(V_i^{Y\|X}\|\|W\|V_i^{X})-\|l^{k_g} \operatorname{I}_{V_{g+1} \oplus V_{g+2}}(X \wedge Y) -l^{k_{g}} R^{k_g}\|^{+}} \label{lassanuccso4}\end{aligned}$$ Hence, the convexity of the divergence proves $$Err_j^{\mathbf{h}}(\hat{k},d) \le subexp(n) \cdot 2^{- l^{h_j} \mathcal{E}_{r}(R^{h_j},P^{h_j},W)}.$$ Note that in this part $\eta_n$ can be arbitrary positive sequence which goes to $0$ as $n\rightarrow \infty$. However, it will turn out from the proof of part (ii) that the sequence $\eta_n$ has to converge to $0$ sufficiently slowly. Proof of part (ii) of Theorem \[maintheorem\]: For $\hat{k} \in [M]$ and $d \in \{s_j-l^{\hat{k}}+1,s_j-l^{\hat{k}}+2, \dots, s_j+l^{h_j}-1 \}$ let $\mathcal{E}df_j^{\mathbf{h}}(\hat{k},d)$ be the event $$\left\{\hspace{-3pt}\begin{array}{l} \left( \operatorname{I}(\mathbf{x}_{\hat{a}}^{\hat{k}} \wedge Y_{d} Y_{d+1}\dots Y_{d+l^{\hat{k}} -1})-R^{l^{\hat{k}}} \right) > \eta_n \\ \text{for some } \hat{a} \in [N^{\hat{k}}] \text{ }(\hat{a}\ne B_j \text{ if } \hat{k}=h_j \text{ and } d=s_j) \end{array}\hspace{-3pt}\right\}, \label{errorpatternii}$$ and let $Edf_{j}^{\mathbf{h}}(\hat{k},d)$ denote its probability. Introduce also the notation $Edf_{j}^{\mathbf{h}}(\text{TH})=1-Err_j^{\mathbf{h}}(\text{TH})$. Then $$\label{uniokorlatahibaraii} Edf_{j}^{\mathbf{h}} \le Edf_{j}^{\mathbf{h}}(\text{TH}) + \sum_{(\hat{k},d)} Edf_{j}^{\mathbf{h}}(\hat{k},d).$$ By standard argument it follows from (\[basicfact2\]) and (\[basicfact4\]) that $$Edf_{j}^{\mathbf{h}}(\text{TH}) \le subexp(n) \cdot 2^{- l^{h_j} \mathcal{E}df_{TH}^n (R^{h_j},P^{h_j},W)} \label{bizTHii},$$ where $$\label{rcbthii} \mathcal{E}df_{TH}^n (R^{h_j},P^{h_j},W) \triangleq \min_{{\genfrac{}{}{0pt}{}{V\in \mathcal{P}(\mathcal{X} \times \mathcal{Y})}{V_{X}=P^{h_j}, \operatorname{I}_V(X \wedge Y)-R^{h_j} > \eta_n }}} \operatorname{D}(V_{Y\|X}\|\|W\|P^{h_j})=\min_{{\genfrac{}{}{0pt}{}{V\in \mathcal{P}(\mathcal{X} \times \mathcal{Y})}{V_{X}=P^{h_j}, \operatorname{I}_V(X \wedge Y)-R^{h_j} > \eta_n }}} \operatorname{D}(V\|\|P^{h_j}W).$$ In part (ii) it is assumed that $R^{h_j}\ge I[W,P^{h_j}]$. It follows that for each possible $V$ in (\[rcbthii\]) $$\label{rcbthii2} \operatorname{I}_{V}(X \wedge Y)-I[W,P^{h_j}] > \eta_n.$$ Hence using Lemma 2.7 of [@Csiszar2] it follows that there exists $\zeta_n (\eta_n, \|\mathcal{X}\|, \|\mathcal{Y}\|)$>0 such that for each possible $V$ in (\[rcbthii\]) its variational distance from $P^{h_j}W$ is at least $\zeta_n$. Then (\[bizTHii\]),(\[rcbthii\]) and Pinsker inequality show that $Edf_{j}^{\mathbf{h}}(\text{TH})$ goes to $0$ if $\eta_n(\|\mathcal{X}\|,\|\mathcal{Y}\|, \{M\}_{n=1}^{\infty},D)$ converges to $0$ sufficiently slowly. To prove part (ii) of the theorem it remains to show that also $ \sum_{(\hat{k},d)} Edf_{j}^{\mathbf{h}}(\hat{k},d)$ goes to $0$ if $\eta_n$ converges to $0$ sufficiently slowly. Fix a pair $(\hat{k},d)$. Assume that $d \le s_j$ (the analysis of the case $d>s_j$ is similar). Replicating the notations and arguments of the proof of part (i) of the theorem the analogue of (\[hibabindikator2\]) with (\[lassanuccso\]) and (\[lassanuccso2\]) gives that $$\begin{aligned} &Edf_{j}^{\mathbf{h}}(\hat{k},d) \le \sup_{\mathbf{V} \in \mathcal{VM}_{edf}^{\mathbf{k},q,n}} subexp(n) \prod_{i=1}^{g+2} \hspace{-1pt} 2^{-n_i \operatorname{D}(V_i^{Y\|X}\|\|W\|V_i^{X})} 2^{-\|l^{\hat{k}} \operatorname{I}_{V_2 \oplus \dots \oplus V_{g+1}}(\hat{X}\wedge Y) -l^{\hat{k}}R^{\hat{k}}\|^{+}}, \label{hibabindikator2th}\end{aligned}$$ where the family of subtype sequences $\mathcal{VM}_{edf}^{\mathbf{k},r,n}$ is equal to $$\label{hibatipusth} \left\{\hspace{-3pt} \begin{array}{l} \mathbf{V}=(V_1,V_2,\dots,V_{g+2}): V_1=V^{XY}_1 \in \mathcal{P}^{n_1}(\mathcal{X} \times \mathcal{Y}), V_{g+2}=V^{XY}_{g+2} \in \mathcal{P}^{n_{g+2}}(\mathcal{X} \times \mathcal{Y})\\ V_i= V^{\hat{X}XY}_i \in \mathcal{P}^{n_i}(\mathcal{X} \times \mathcal{X} \times \mathcal{Y}), 2\le i \le g+1 \\ (V^{X}_1,V^{\hat{X}X}_2,\dots, V^{\hat{X}X}_{g+1},V^{X}_{g+2}) \in \mathcal{V}^{\mathbf{k},q,n}, \operatorname{I}_{V_2 \oplus V_3 \oplus \dots \oplus V_{g+1}}(\hat{X} \wedge Y)-R^{\hat{k}} > \eta_n. \end{array}\hspace{-3pt}\right\}.$$ From (\[hibabindikator2th\]) and (\[hibatipusth\]) it follows that $\sum_{(\hat{k},d)} Edf_{j}^{\mathbf{h}}(\hat{k},d)$ can be upper-bounded by $subexp(n) 2^{-l^{\hat{k}} \eta_n}$ which goes to $0$ if $\eta_n(\|\mathcal{X}\|,\|\mathcal{Y}\|, \{M\}_{n=1}^{\infty},D)$ converges to $0$ sufficiently slowly. Improvement when the channel is known by the sender {#improvements} =================================================== Theorem \[maintheorem\] provides a universal result: Neither the design of the codebook library nor the decoder depends on the channel matrix $W$. In this section we outline a substantial improvement when the channel is known to the sender (it remains unknown to the receiver). Intuitively, supposing there are $M$ kinds of messages, with given rate and codeword length for each of them, the improvement will be based on an extended codebook library that contains several codebooks with different types for each message kind. From the codebooks available for a given message kind, the sender will chose the one whose type maximizes the random coding exponent for the actual channel, or if that maximum is $0$, the sender will use a one-codeword codebook just indicating the message kind. Formally, let a sequence of codebook library parameters be given as in Theorem \[maintheorem\], except for the prescribed types, i.e., let $D \in (0,1]$ be fixed, and for each $n$ let $M$ with $\frac{1}{n} \log M \rightarrow 0$, $l^1,l^2,\dots,l^M$ with $D n \le l^i\le n$ for all $i \in [M]$ and rates $\{ R^{i} , i \in [M]\}$ be given. We construct a sequence of codebook library parameters as follows. $D$ remains unchanged. Instead of $i \in [M]$, the codebook indices will be triplets $(i,P,s)$, where $i \in [M]$, $P \in P^{l^i}(\mathcal{X})$ and $s \in \{0,1\}$. Let the length, type and rate of the codebook indexed by triplet $(i,P,s)$ be equal to $l^{(i,P,s)}=l^i$, $P^{(i,P,s)}=P$ and $R^{(i,P,s)}=R^i \cdot s$, respectively. As the number of triplets $(i,P,s)$ remains subexponential in $n$ we can apply Theorem \[maintheorem\] with this modified sequence of codebook library parameters. The codebook library provided by Theorem \[maintheorem\] contains two codebooks corresponding to each pair $(l^i,R^i)$ and type $P$, one with the rate $R^i$ and one with rate $0$, i.e., consisting of only one codeword. This codebook library will be referred to as extended codebook library. An infinite message kind schedule $\mathbf{k}=(k_1,\dots,k_j \dots)$, $k_j \in [M]$, specifies for each $j$ the kind of message $k_j$ to be transmitted at time $j$. Relying on the channel knowledge, the sender constructs a codebook schedule $\mathbf{h}(\mathbf{k},W)=(h_1(k_1,W),\dots,h_j(k_j,W),\dots)$ as follows. For each $i \in [M]$ define $$\mathcal{E}_r^i (R,W) \triangleq \max_{P \in P^{l^{i}}(\mathcal{X})} \mathcal{E}_r (P,R,W),$$ where $\mathcal{E}_r (P,R,W)$ is defined in (\[rcb\]). Note that maximization for all $P \in \mathcal{P} (\mathcal {X})$, rather than only for $l_i$-types, gives the standard random coding exponent $E_r(R,W)$ of the channel. Thus, when $n$ and hence $l_i > Dn$ is large, $E_r^i (R,W)$ differs only negligibly from $E_r( R,W)$. Let $P_1$ and $P_0$ be types for which $\mathcal{E}_r^{k_j} (R,W)=\mathcal{E}_r (P_1,R^{k_j},W)$ and $\mathcal{E}_r^{k_j} (0,W)=\mathcal{E}_r (P_0,0,W)$ respectively. Let $h_j(k_j,W)$ be equal to $(k_j,P_1,1)$ if $E_r^{k_j} (R^{k_j},W) > 0$ and $(k_j,P_0,0)$ otherwise. This means that in each transmission $j$, the sender uses the optimal input distribution with the given rate. Moreover, when reliable message detection is not possible with the given rate, a $0$-rate (and hence reliable) codebook is used. In accordance with this codebook schedule construction, we modify the decoder used in the proof of Theorem \[maintheorem\]. If the output of the decoder is the only codeword in the codebook indexed by $(i,P,0)$ for some $i \in [M]$ and $P \in P^{l^i}(\mathcal{X})$ then the decoder reports “erasure” and supplements it with declaring that the receiver wanted to send $i$’th message kind but the channel is not supported it. Altogether, the next corollary follows from Theorem \[maintheorem\] (i) (part (ii) of Theorem \[maintheorem\] is not used). \[cor1\] Let $\nu_n$ be the sequence specified by Theorem \[maintheorem\]. For each infinite message kind schedule $\mathbf{k}=(k_1,\dots,k_j \dots)$, $k_j \in [M]$, and index $j$, using the extended codebook library with codebook schedule $\mathbf{h}(\mathbf{k},W)=(h_1(k_1,W),\dots,h_j(k_j,W),\dots)$ and the decoder specified above the followings hold. (i) If $\mathcal{E}_r^{k^j} (R^{k^j},W)>0$, the probability of incorrectly decoding the $j$’th message is less than $\nu_n \cdot 2^{- l^{k_j} \mathcal{E}_{r}^{k_j}(R^{k_j},W) }$. (ii) If $\mathcal{E}_r^{k^j} (R^{k^j},W)=0$, the decoder reports “erasure” and declares that the kind of the erased message is $k_j$, with probability at least $1-\nu_n \cdot 2^{- l^{k_j} \mathcal{E}_{r}^{k_j}(0,W) }$. The construction of $\mathbf{h}(\mathbf{k},W)=(h_1(k_1,W),\dots,h_j(k_j,W),\dots)$ above is very specific. Actually, for each transmission $j$ the sender can decide whether the exponent $\mathcal{E}_r^{k_j} (R^{k_j},W)$ is sufficient or not. If it is not sufficient for his purposes, he can choose to use the corresponding $0$-rate codebook instead of actual message transmission. \[singlecodebook\] Even the special case of Corollary \[cor1\] for the classical situation of transmitting messages of a single kind is of interest: If the sender but not the receiver knows the channel the random coding exponent of the actual channel (i.e., the random coding exponent maximized over input distribution) is achievable. In the literature, this fact is usually stated only when both sender and receiver know the channel. Note that this special case of the corollary also follows from [@Csiszar], though not explicitly stated there. Discussion ========== A generalization of the DMC coding theorem of [@Csiszar] has been studied allowing not just the rate and the type but also the length of the codewords to vary across codebooks. This generalization could provide a theoretical background for practical scenarios when different coding strategies are used for sending different kind of messages (e.g. audio, data, video). It has been shown that in this scenario simultaneously for each codebook choice of each transmission, the same error exponent can be achieved as the random coding error exponent for the chosen codebook alone, supplemented with non-exponential erasure detection. This has been achieved with a completely universal construction: Neither the design of the codebook library nor the decoder depends on the channel. When the channel is known to the sender, an improvement is given while maintaining the universality of the decoder. The improvement leads to exponent also for erasure declaration failure probability and shows that the maximum of the random coding error exponent over the possible input distributions is achievable for each message kind. The possible improvement via relaxing the universality of the decoder is not addressed in this paper. However, we note that in the model with equal codeword lengths, [@Csiszar4] shows that maximum likelihood decoding admits to achieve, individually for each codebook, also the expurgated error exponents that for small rates exceeds the random coding exponent. A similar result likely holds also for the model in this paper. The difficulty in the analysis of the model in this paper comes from the fact that the different codeword lengths cause a certain asynchronism at the receiver, who should also estimate the boundaries of the codewords and avoid error propagation. The asynchronous nature of this model gives a natural connection to “strong asynchronism” in [@Tchamkarten]. In that model the sender has only one codebook, sends a message only once in an exponentially large (in the codeword length) time window, when the sender is idle a special dummy symbol denoted by $$ is transmitted. The time of the message transmission is not known to the receiver. Tradeoff between the rate of the codebook and the exponent of the time window is investigated. The detailed investigation of the relation of this model to ours is beyond the scope of this paper. However, in order to arouse the reader’s attention we note the following. 1. It is a natural idea to try to employ Theorem \[maintheorem\] in its current form to the model of strong asynchronism via artificially introducing one-codeword codebooks consisting of dummy symbols $$. One problem with this idea is that the random coding error exponent of these artificial codebooks is $0$. A better option is to employ artificial one-codeword codebooks (similarly as in Section \[improvements\]) with positive random coding error exponent to model the idle periods of the sender. Then the exponentially small error probability in each transmission ensures that the decoder fails only with small probability even in an exponential large time window. Hence, this application of Theorem \[maintheorem\] would lead to a meaningful model, nevertheless, it would differ from the original model of strong asynchronism. 2. Theorem 3 of [@Polyanskiy] shows that the achievable pairs (rate, time window exponent) can be also achieved with a universal decoder. This is, however, shown only under an error criterion which does not require exact synchronization. Hence, the event of detecting the right codeword in a wrong position, partially overlapping with the correct one is omitted in the error analysis. [@HongkongML] provides error exponent in the model of strong asynchronism using maximum likelihood decoder. Here, exact synchronization is required but no simple single-letter expression is obtained for the exponent. As our paper does handle the event of partial overlap, we think that the tools used here can be used to strengthen Theorem 3 of [@Polyanskiy] and the error exponent analysis in the model of strong asynchronism. Proof of inequality (\[egysegesbecsles\]) {#bizonyitasvarhatoertek} ========================================= In this section we suppose that a sequence of codebook library parameters as in Lemma \[Rc-Packing-lemma\] is given. Let $\gamma_n=(\log n)^{-\frac{1}{2}}$. Choose the codebook library $\mathcal{A}$ at random, i. e., for all $i \in [M]$ the codewords of $\mathcal{A}^i$ are chosen independently and uniformly from $\mathcal{T}^{l^i}_{P^i} (\gamma_n)$. We prove rigorously that for arbitrarily sequence $\mathbf{k}=(\hat{k}, k_1,\dots, k_g)$ consisting of codebook indices, non-negative integer $q$ fulfilling (\[windowcondition1\]) and (\[windowcondition2\]), subtype sequence $\mathbf{V}=(V_1,V_2,\dots, V_{g+2}) \in \mathcal{V}^{\mathbf{k},q,n}$, and indices $\hat{a}\in [N^{\hat{k}}]$, $a_1 \in [N^{k_1}], \dots, a_g \in [N^{k_g}]$ supposing $\hat{a}\ne a_1 \text{ if $g=1$, $\hat{k}=k_1$ and $q=0$}$, the following inequality holds $$\label{egysegesbecsles2} \mathds{E}\left( 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right)\le \nu^{''}_n 2^{-\sum_{i=2}^{g+1} n_i \operatorname{I}_{V_i}(X \wedge \hat X) -l^{\hat{k}} \operatorname{J}(V_2^{\hat{X}}, \dots V_{g+1}^{\hat{X}})},$$ where $\nu^{''}_n$ is a subexponential function of $n$ that depends only on $D$ and the alphabet size $\mathcal{X}$. The proof relies strongly on the notations introduced in Section \[notation\]. We define a set of equalities $\mathcal{I}$ as follows: for all $i \in [g]$, equality $\hat{\mathbf{x}}=\mathbf{x_i}$ is in $\mathcal{I}$ iff $(\hat{k},\hat{a})=(k_i,a_i)$ and for all $i,j \in [g]$ equality $\mathbf{x_i}=\mathbf{x}_j$ is in $\mathcal{I}$ iff $(k_i,a_i)=(k_j,a_j)$. For notational convenience we also define a set $\mathcal{I}^{}$ consisting of the positive integers $j \in [g]$ such that $(k_i,a_i) \ne (k_j,a_j)$ for all $i<j$. Note that $\mathcal{I}$ determines $\mathcal{I}^{}$ but the reverse is not true. To prove (\[egysegesbecsles2\]) we separately investigate two cases: $n_2$ and $n_{g+1}$ are both less than or equal to $n-(\log n)^2$ (case 1) or at least one of them is larger than $n-(\log n)^2$ (case 2). Both can be larger then $n-(\log n)^2$ only in the case $g=1$ when $n_2=n_{g+1}$; in this case always $\mathcal{I}=\emptyset$, and the proof of subcase 1a below works. From now on, we assume that $g \ge 2$. CASE 1: By symmetry it can be assumed that $n_2 \le n_{g+1}$ (this assumption ensures that the counting from left to right works in all subcases). We have to separately investigate four subcases: $\{ \hat{\mathbf{x}}=\mathbf{x}_1 \} \notin \mathcal{I}$ , $\{ \hat{\mathbf{x}} =\mathbf{x}_g \} \notin \mathcal{I}$ (subcase 1a), $\{ \hat{\mathbf{x}}=\mathbf{x}_1 \} \in \mathcal{I}$ , $\{ \hat{\mathbf{x}} =\mathbf{x}_g \} \notin \mathcal{I}$ (subcase 1b) $\{ \hat{\mathbf{x}}=\mathbf{x}_1 \} \notin \mathcal{I}$ , $\{ \hat{\mathbf{x}} =\mathbf{x}_g \} \in \mathcal{I}$ (subcase 1c) and $\{ \hat{\mathbf{x}}=\mathbf{x}_1 \} \in \mathcal{I}$ , $\{ \hat{\mathbf{x}} =\mathbf{x}_g \} \in \mathcal{I}$ (subcase 1d). SUBCASE 1a: For the sake of clarity first we assume that not only equalities $\hat{\mathbf{x}}=\mathbf{x}_1 , \hat{\mathbf{x}}=\mathbf{x}_g$ are not in $\mathcal{I}$ but $\mathcal{I}$ is empty. This assumption is relaxed in the second part of the discussion of this subcase. Then using (\[basicfact2\]) and (\[expurgsize\]) it follows that $$\label{becslesalap} \mathds{E}\left( 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right) \le subexp(n)\cdot 2^{-l^{\hat{k}}\operatorname{H}(P^{\hat{k}})-\sum_{i=1}^{g} l^{k_i} \operatorname{H}(P^{k_i})}\cdot \|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}}\|$$ To upper-bound $\|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}}\|$ we perform counting from left to right (see Fig. \[packingtenycikk\]). This specific counting allows a uniform handling of cases. In this simple subcase any counting would work. $$\label{cardialitybound} \|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}}\|\le 2^{n_1 \operatorname{H}_{V_1}(X)+\sum_{i=2}^{g} n_i\operatorname{H}_{V_i}(\hat{X}X)+ n_{g+1} \operatorname{H}_{V_{g+1}}(\hat{X}X) + n_{g+2} \operatorname{H}_{V_{g+2}}(X) } \\$$ Substituting (\[cardialitybound\]) into (\[becslesalap\]), the fact that $\operatorname{H}_{V_i}(X)=\operatorname{H}(P^{k_{i-1}})$, $3 \le i \le g$, and some algebraic rearrangement give: $$\begin{aligned} &\mathds{E}\left( 1^{\mathbf{L},r}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right) \le subexp(n) 2^{\sum_{i=2}^{g+1} n_i (\operatorname{H}_{V_i}(\hat{X}X)-\operatorname{H}_{V_i}(X)-\operatorname{H}_{V_i}(\hat{X})) } 2^{-l^{k_1} \operatorname{H}(P^{k_1}) +n_1\operatorname{H}_{V_1}(X) + n_2 \operatorname{H}_{V_2}(X) }\notag\\ &\cdot 2^{-l^{k_g} \operatorname{H}(P^{k_g}) +n_{g+1}\operatorname{H}_{V_{g+1}}(X) + n_{g+2} \operatorname{H}_{V_{g+2}}(X) } 2^{-l^{\hat{k}}\operatorname{H}(P^{\hat{k}}) + \sum_{i=2}^{g+1} n_i \operatorname{H}_{V_i}(\hat{X}) }\label{becslesreszelejecase1a}\\ &=subexp(n) 2^{-\sum\limits_{i=2}^{g+1} n_i\operatorname{I}_{V_i}(X \wedge \hat X)} 2^{- l^{k_1}J(V_1^{X},V_2^{X})} 2^{-l^{k_g}J(V_{g+1}^{X}, V_{g+2}^{X})} 2^{- l^{\hat{k}}J(V_2^{\hat{X}},\dots,V_{g+1}^{\hat{X}})} \label{becsleskozepecase1a}\\ &\le subexp(n) 2^{-\sum\limits_{i=2}^{g+1} n_i\operatorname{I}_{V_i}(X \wedge \hat X)-l^{\hat{k}}J(V_2^{\hat{X}},\dots,V_{g+1}^{\hat{X}})}.\label{becslesvegsocase1a}\end{aligned}$$ Here in (\[becslesvegsocase1a\]) the positivity of the Jensen-Shannon divergence is used. Inequality (\[becslesvegsocase1a\]) implies (\[egysegesbecsles2\]) in this subcase, under the supplementary assumption that $\mathcal{I}$ is empty. Next we prove (\[egysegesbecsles2\]) in the general scenario of subcase 1a. Heuristically we can summarize the formal proof below that if $j \notin \mathcal{I}^$ then in (\[becslesalap\]) the term $l^{k_j}\operatorname{H}(P^{k_j})$ is missing and in (\[cardialitybound\]) instead of $\operatorname{H}_{V_{j}}(\hat{X}X)$ the term $\operatorname{H}_{V_{j}}(\hat{X}\|X)$ occurs, thus the same upper-bound is obtained since $$\operatorname{H}_{V_i}(\hat{X}X)-\operatorname{H}_{V_i}(X)-\operatorname{H}_{V_i}(\hat{X})= \operatorname{H}_{V_i}(\hat{X}\|X)-\operatorname{H}_{V_i}(\hat{X})=-\operatorname{I}_{V_i}(X \wedge \hat X). \label{mutualinfo}$$ Formally, assume first that $g \in \mathcal{I}^$. The analogues of (\[becslesalap\]) and (\[cardialitybound\]) are respectively $$\begin{aligned} &\mathds{E}\left( 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right) \le subexp(n)\cdot 2^{-l^{\hat{k}}\operatorname{H}(P^{\hat{k}})-\sum_{i \in \mathcal{I}^}l^{k_i}\operatorname{H}(P^{k_i})}\cdot \|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}, \mathcal{I}}\| \label{becslesalapism}\\ &\|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}, \mathcal{I}}\|\le 2^{n_1 \operatorname{H}_{V_1}(X)+\sum_{i: i-1 \in \mathcal{I}^{} \setminus \{g\} } n_{i} \operatorname{H}_{V_i}(\hat{X}X) +\sum_{i: i-1 \in [g] \setminus \mathcal{I}^{}} n_{i} \operatorname{H}_{V_i}(\hat{X}\|X)+ n_{g+1} \operatorname{H}_{V_{g+1}}(\hat{X}X)+ n_{g+2} \operatorname{H}_{V_{g+2}}(X) } \label{cardialitybound2ism}\end{aligned}$$ Substituting (\[cardialitybound2ism\]) into (\[becslesalapism\]), (\[mutualinfo\]) and the same algebraic rearrangement as in (\[becslesreszelejecase1a\]) give identical upper-bound to the one in (\[becsleskozepecase1a\]). In case of $g \notin \mathcal{I}^$ the analogues of (\[becslesalap\]) and (\[cardialitybound\]) are respectively $$\begin{aligned} &\mathds{E}\left( 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right) \le subexp(n)\cdot 2^{-l^{\hat{k}}\operatorname{H}(P^{\hat{k}})-\sum_{i \in \mathcal{I}^}l^{k_i}\operatorname{H}(P^{k_i})}\cdot \|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}, \mathcal{I}}\| \label{becslesalapismm}\\ &\|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}, \mathcal{I}}\|\le 2^{n_1 \operatorname{H}_{V_1}(X)+\sum_{i: i-1 \in \mathcal{I}^{}} n_{i} \operatorname{H}_{V_i}(\hat{X}X) +\sum_{i: i-1 \in [g-1] \setminus \mathcal{I}^{}} n_{i} \operatorname{H}_{V_i}(\hat{X}\|X)+ n_{g+1} \operatorname{H}_{V_{g+1}}(\hat{X}\|X) } \label{cardialitybound2ismm}\end{aligned}$$ Now substituting (\[cardialitybound2ismm\]) into (\[becslesalapismm\]) and proceeding similarly as before give upper-bound (\[becsleskozepecase1a\]) without $2^{-l^{k_g}J(V_{g+1}^{X}, V_{g+2}^{X})}$ which is omitted in the next step. This completes the proof of (\[egysegesbecsles2\]) in subcase 1a. Note that the argument which allowed the proof without the supplementary assumption works also in other subcases. Hence, from now on we assume that no equality $x_i=x_j$, $i,j \in [g]$, is in $\mathcal{I}$ except the equality $x_1=x_g$ in subcase 1d. SUBCASE 1b: According to the last paragraph of subcase 1a, it can be assumed that $\mathcal{I}=\{\hat{\mathbf{x}}=\mathbf{x}_1\}$. We can assume also that there exists a collection of sequences $(\hat{\mathbf{x}},\mathbf{x}_{1}, \dots \mathbf{x}_{g})$ with $\hat{\mathbf{x}} \in \mathcal{T}_{P^{\hat{k}}}^{l^{\hat{k}}}(\gamma_n)$ and $\mathbf{x}_{i} \in \mathcal{T}_{P^{k_i}}^{l^{k_i}}(\gamma_n)$ for all $i \in [g]$ with $\mathds 1_{\mathbf{V}, \mathcal{I}}^{\mathbf{L},q} (\hat{\mathbf{x}};\mathbf{x}_{1},\dots, \mathbf{x}_{g})=1$, otherwise $\mathds{E}\left( 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right)$ is equal to $0$. This assumption implies that $$\label{expurgalaseredmenye} 2^{n_2 \operatorname{I}_{V_2}(X \wedge \hat{X})} < subexp(n)$$ because if $n_2 < (\log n)^2$ then (\[expurgalaseredmenye\]) is immediate, while otherwise $\operatorname{I}_{V_2}(X \wedge \hat{X})< \gamma_n$ which also implies (\[expurgalaseredmenye\]). Then using (\[basicfact2\]) and (\[expurgsize\]) we get that $$\label{becslesalap1b} \mathds{E}\left( 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right) \le subexp(n)\cdot 2^{-l^{\hat{k}}\operatorname{H}(P^{\hat{k}})-\sum_{i=2}^{g} l^{k_i} \operatorname{H}(P^{k_i})}\cdot \|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}, \mathcal{I}}\|$$ To upper-bound $\|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}, \mathcal{I}}\|$ we perform counting again from left to right (see Fig. \[packingtenycikk\]) but we skip the first block, and write $n_2 \operatorname{H}_{V_2}(\hat{X})$ instead of $n_2 \operatorname{H}_{V_2}(\hat{X}X)$ related to the second block. $$\label{cardialitybound1b} \|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}, \mathcal{I}}\|\le 2^{n_2 \operatorname{H}_{V_2}(\hat{X})+\sum_{i=3}^{g} n_i\operatorname{H}_{V_i}(\hat{X}X)+ n_{g+1} \operatorname{H}_{V_{g+1}}(\hat{X}X) + n_{g+2} \operatorname{H}_{V_{g+2}}(X) } \\$$ Substituting (\[cardialitybound1b\]) into (\[becslesalap1b\]), the fact that $\operatorname{H}_{V_i}(X)=\operatorname{H}(P^{k_{i-1}})$, $3 \le i \le g$, and performing the same algebraic rearrangement as before give: $$\begin{aligned} &\mathds{E}\left( 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right) \le subexp(n) 2^{-\sum\limits_{i=3}^{g+1} n_i\operatorname{I}_{V_i}(X \wedge \hat X)} 2^{-l^{k_g}J(V_{g+1}^{X}, V_{g+2}^{X})} 2^{- l^{\hat{k}}J(V_2^{\hat{X}},\dots,V_{g+1}^{\hat{X}})} \label{becsleskozepecase1b}\\ &\le subexp(n) 2^{-\sum\limits_{i=2}^{g+1} n_i\operatorname{I}_{V_i}(X \wedge \hat X)-l^{\hat{k}}J(V_2^{\hat{X}},\dots,V_{g+1}^{\hat{X}})}.\label{becslesvegsocase1b}\end{aligned}$$ Here in (\[becslesvegsocase1b\]) the positivity of the Jensen-Shannon divergence and (\[expurgalaseredmenye\]) are used. Inequality (\[becslesvegsocase1b\]) implies (\[egysegesbecsles2\]). SUBCASE 1c: According to the last paragraph of subcase 1a it can be assumed that $\mathcal{I}=\{\hat{\mathbf{x}}=\mathbf{x}_g\}$. Now using again (\[basicfact2\]) and (\[expurgsize\]) we get that $$\label{becslesalap1c} \mathds{E}\left( 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right) \le subexp(n)\cdot 2^{-l^{\hat{k}}\operatorname{H}(P^{\hat{k}})-\sum_{i=1}^{g-1}l^{k_i}\operatorname{H}(P^{k_i})}\cdot \|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}, \mathcal{I}}\|$$ ![Further division of the $(g+1)$’th subblock in subcases (1c) and (1d)[]{data-label="packingtenycikkmodtovabb"}](packingtenycikkmodtovabb) To upper-bound $\|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}, \mathcal{I}}\|$ we divide the subblock of index $g+1$ into consecutive subblocks of length $n_{g+1,i}=(\log n)^2$ except perhaps for the last subblock that has length $n_{g+1,s}\le (\log n)^2$ (see Fig. \[packingtenycikkmodtovabb\]). Again we perform the counting from left to right. $$\begin{aligned} \label{cardialitybound1c} &\|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V},\mathcal{I}}\|\le \sum_{{\genfrac{}{}{0pt}{}{V_{g+1,i}\in \mathcal{P}^{n_i}(\mathcal{X}\times \mathcal{X}), i \in [s]}{V_{g+1,1} \oplus \dots \oplus V_{g+1,s}=V_{g+1}}}} 2^{n_1 \operatorname{H}_{V_1}(X)+\sum_{i=2}^{g} n_i\operatorname{H}_{V_i}(\hat{X}X)+\sum_{i=1}^{s} n_{g+1,i} \operatorname{H}_{V_{g+1,i}}(\hat{X}\|X) } \\ &\le subexp(n) 2^{n_1 \operatorname{H}_{V_1}(X)+\sum_{i=2}^{g} n_{i} \operatorname{H}_{V_i}(\hat{X}X) + n_{g+1} \operatorname{H}_{V_{g+1}}(\hat{X}\|X) } \label{cardialitybound21c}\end{aligned}$$ Here in (\[cardialitybound1c\]) the sum is over subtype sequences corresponding to the division in Fig. \[packingtenycikkmodtovabb\], where $V_{g+1,1}\oplus \dots \oplus V_{g+1,s}=V_{g+1}$. In (\[cardialitybound21c\]) we used the concavity of entropy and the fact that the number of subtype sequences in the sum is subexponential. Substituting (\[cardialitybound21c\]) into (\[becslesalap1c\]), the fact that $\operatorname{H}_{V_i}(X)=\operatorname{H}(P^{k_{i-1}})$, $3 \le i \le g$, and the same algebraic rearrangement as before give: $$\begin{aligned} &\mathds{E}\left( 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right) \le subexp(n) 2^{\sum_{i=2}^{g+1} n_i (\operatorname{H}_{V_i}(\hat{X}X)-\operatorname{H}_{V_i}(X)-\operatorname{H}_{V_i}(\hat{X})) } 2^{-l^{k_1} \operatorname{H}(P^{k_1}) +n_1\operatorname{H}_{V_1}(X) + n_2 \operatorname{H}_{V_2}(X) }\notag\\ &\cdot 2^{-l^{\hat{k}}\operatorname{H}(P^{\hat{k}}) + \sum_{i=2}^{g+1} n_i \operatorname{H}_{V_i}(\hat{X}) }\label{becslesreszelejecase1c}\\ &=subexp(n) 2^{-\sum\limits_{i=2}^{g+1} n_i\operatorname{I}_{V_i}(X \wedge \hat X)} 2^{- l^{k_1}J(V_1^{X},V_2^{X})} 2^{- l^{\hat{k}}J(V_2^{\hat{X}},\dots,V_{g+1}^{\hat{X}})} \label{becsleskozepecase1c}\\ &\le subexp(n) 2^{-\sum\limits_{i=2}^{g+1} n_i\operatorname{I}_{V_i}(X \wedge \hat X)-l^{\hat{k}}J(V_2^{\hat{X}},\dots,V_{g+1}^{\hat{X}})}.\label{becslesvegsocase1c}\end{aligned}$$ Here in (\[becslesvegsocase1c\]) the positivity of the Jensen-Shannon divergence is used. Inequality (\[becslesvegsocase1c\]) implies (\[egysegesbecsles2\]). SUBCASE 1d: In this subcase we have to combine the methods of previous subcases. According to the last paragraph of subcase 1a it can be assumed that $\mathcal{I}=\{\hat{\mathbf{x}}=\mathbf{x}_1,\hat{\mathbf{x}}=\mathbf{x}_g, \mathbf{x}_1=\mathbf{x}_g\}$. We can assume also that there exists a collection of sequences $(\hat{\mathbf{x}},\mathbf{x}_{1}, \dots \mathbf{x}_{g})$ with $\hat{\mathbf{x}} \in \mathcal{T}_{P^{\hat{k}}}^{l^{\hat{k}}}(\gamma_n)$ and $\mathbf{x}_{i} \in \mathcal{T}_{P^{k_i}}^{l^{k_i}}(\gamma_n)$ for all $i \in [g]$ with $\mathds 1_{\mathbf{V}, \mathcal{I}}^{\mathbf{L},q} (\hat{\mathbf{x}};\mathbf{x}_{1},\dots, \mathbf{x}_{g})=1$, otherwise $\mathds{E}\left( 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right)$ is equal to $0$. This assumption implies as in subcase 1b that $$\label{expurgalaseredmenye1d} 2^{n_2 \operatorname{I}_{V_2}(X \wedge \hat{X})} < subexp(n).$$ Now using again (\[basicfact2\]) we get that $$\label{becslesalap1d} \mathds{E}\left( 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right) \le subexp(n)\cdot 2^{-l^{\hat{k}}\operatorname{H}(P^{\hat{k}})-\sum_{i=2}^{g-1}l^{k_i}\operatorname{H}(P^{k_i})}\cdot \|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}, \mathcal{I}}\|$$ To upper-bound $\|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}, \mathcal{I}}\|$ we divide the subblock corresponding to $V_{g+1}$ into consecutive subblocks of length $(\log n)^2$ as in subcase 1c (see Fig. \[packingtenycikkmodtovabb\]) and as in subcase (1b) we perform the counting from left to right but we skip the first block and write $n_2 \operatorname{H}_{V_2}(\hat{X})$ instead of $n_2 \operatorname{H}_{V_2}(\hat{X}X)$ related to the second block. Using the same arguments as in subcases 1b and 1c we get $$\begin{aligned} \label{cardialitybound1d} &\|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V},\mathcal{I}}\|\le \sum_{{\genfrac{}{}{0pt}{}{V_{g+1,i}\in \mathcal{P}^{n_i}(\mathcal{X}\times \mathcal{X}), i \in [s]}{V_{g+1,1} \oplus \dots \oplus V_{g+1,s}=V_{g+1}}}} 2^{n_2 \operatorname{H}_{V_2}(\hat{X})+\sum_{i=3}^{g} n_i\operatorname{H}_{V_i}(\hat{X}X)+\sum_{i=1}^{s} n_{g+1,i} \operatorname{H}_{V_{g+1,i}}(\hat{X}\|X) } \\ &\le subexp(n) 2^{n_2 \operatorname{H}_{V_2}(\hat{X})+\sum_{i=3}^{g} n_{i} \operatorname{H}_{V_i}(\hat{X}X) + n_{g+1} \operatorname{H}_{V_{g+1}}(\hat{X}\|X) } \label{cardialitybound21d} \\ &\mathds{E}\left( 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right) \le subexp(n) 2^{-\sum\limits_{i=3}^{g+1} n_i\operatorname{I}_{V_i}(X \wedge \hat X)} 2^{- l^{\hat{k}}J(V_2^{\hat{X}},\dots,V_{g+1}^{\hat{X}})} \label{becsleskozepecase1d}\\ &\le subexp(n) 2^{-\sum\limits_{i=2}^{g+1} n_i\operatorname{I}_{V_i}(X \wedge \hat X)-l^{\hat{k}}J(V_2^{\hat{X}},\dots,V_{g+1}^{\hat{X}})}.\label{becslesvegsocase1d}\end{aligned}$$ Inequality (\[becslesvegsocase1d\]) implies (\[egysegesbecsles2\]). ![Illustration for case (2)[]{data-label="kepmarkov1"}](packingtenycikkmodtovabbmarkov) CASE 2: In this case $g = 2$ holds if $n$ is large enough. Moreover, by symmetry it can be assumed that $n_2 \le (\log n)^2$ and $n_3 \ge n-(\log n)^2$ (See Fig. \[kepmarkov1\]). Here also we separately investigate the same four subcases. SUBCASES 2a and 2b: The proofs are identical to the proofs of subcases 1a and 1b respectively. SUBCASE 2c: Here $\mathcal{I}=\{\hat{\mathbf{x}}=\mathbf{x}_1\}$. Using again (\[basicfact2\]) we get that $$\label{becslesalap2c} \mathds{E}\left( 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right) \le subexp(n)\cdot 2^{-l^{\hat{k}}\operatorname{H}(P^{\hat{k}})-l^{k_1}\operatorname{H}(P^{k_1})}\cdot \|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}, \mathcal{I}}\|$$ ![Further division of the third subblock in subcases 2c and 2d[]{data-label="kepmarkov2"}](packingtenycikkmodtovabbmarkov2mod) To upper-bound $\|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}, \mathcal{I}}\|$ we divide the third block into $n_2$ “virtual subblocks” consisting of non-consecutive elements: the first virtual subblock corresponds to indices $(1,1+n_2,\dots)$, the second one corresponds to indices $(2,2+n_2,\dots)$, $\dots$, the last corresponds to indices $(n_2,2n_2,\dots)$ (see Fig. \[kepmarkov2\]). Let $n_{3,1}$, …, $n_{3,s}$ denote the lengths of these “subblocks” (note that $\|n_{3,i}-n_{3,j}\|\le 1$ for all $i,j \in [s]$). Then: $$\begin{aligned} \label{cardialitybound2c} &\|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V},\mathcal{I}}\|\le \sum_{{\genfrac{}{}{0pt}{}{V_{3,i}\in \mathcal{P}^{n_{3,i}}(\mathcal{X}\times \mathcal{X}), i \in [s]}{V_{3,1} \oplus \dots \oplus V_{3,s}=V_{3}}}} 2^{n_1 \operatorname{H}_{V_1}(X)+ n_2\operatorname{H}_{V_2}(\hat{X}X)+\sum_{i=1}^{s} n_{3,i} \operatorname{H}_{V_{3,i}}(\hat{X}\|X) } \\ &\le subexp(n) 2^{n_1 \operatorname{H}_{V_1}(X)+ n_{2} \operatorname{H}_{V_2}(\hat{X}X) + n_{3} \operatorname{H}_{V_{3}}(\hat{X}\|X) } \label{cardialitybound22c}\end{aligned}$$ Here in (\[cardialitybound2c\]) inequality (\[markovrabecsles\]) is used and the sum is over type sequences corresponding to the division in Fig. \[kepmarkov2\]. These $V_{3,1},\dots, V_{3,s}$ have convex combination $V_{3}$. In (\[cardialitybound22c\]) again the concavity of the entropy and the fact that the number of subtype sequences in the sum is subexponential in $n$ are used. Substituting (\[cardialitybound22c\]) into (\[becslesalap2c\]) and the same algebraic rearrangement as before give: $$\begin{aligned} &\mathds{E}\left( 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right) \le subexp(n) 2^{-\sum\limits_{i=2}^{3} n_i\operatorname{I}_{V_i}(X \wedge \hat X)} 2^{- l^{k_1}J(V_1^{X},V_2^{X})} 2^{- l^{\hat{k}}J(V_2^{\hat{X}},V_{3}^{\hat{X}})} \label{becsleskozepecase2c}\\ &\le subexp(n) 2^{-\sum\limits_{i=2}^{3} n_i\operatorname{I}_{V_i}(X \wedge \hat X)-l^{\hat{k}}J(V_2^{\hat{X}},V_{3}^{\hat{X}})}.\label{becslesvegsocase2c}\end{aligned}$$ Here in (\[becslesvegsocase2c\]) the positivity of the Jensen-Shannon divergence is used again. Inequality (\[becslesvegsocase2c\]) implies (\[egysegesbecsles2\]). SUBCASE 2d: Here $\mathcal{I}=\{\hat{\mathbf{x}}=\mathbf{x}_1, \hat{\mathbf{x}}=\mathbf{x}_2,\mathbf{x}_1=\mathbf{x}_2 \}$. Note that (\[expurgalaseredmenye1d\]) trivially holds in this case. Using again (\[basicfact2\]) we get that $$\label{becslesalap2d} \mathds{E}\left( 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right) \le subexp(n)\cdot 2^{-l^{\hat{k}}\operatorname{H}(P^{\hat{k}})}\cdot \|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}, \mathcal{I}}\|$$ To upper-bound $\|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V}, \mathcal{I}}\|$ we divide the third block into $n_2$ “subblocks” consisting of non-consecutive elements as in subcase (2c) (see Fig. \[kepmarkov2\]) and we skip the first block and write $n_2\operatorname{H}_{V_2}(\hat{X})$ instead of $n_2\operatorname{H}_{V_2}(\hat{X}X)$ related to the second block. Using the same argument as in subcases 1d and 2c we get: $$\begin{aligned} \label{cardialitybound2d} &\|\mathcal{T}^{\mathbf{L},q}_{\mathbf{V},\mathcal{I}}\|\le \sum_{{\genfrac{}{}{0pt}{}{V_{3,i}\in \mathcal{P}^{n_{3,i}}(\mathcal{X}\times \mathcal{X}), i \in [s]}{V_{3,1} \oplus \dots \oplus V_{3,s}=V_{3}}}} 2^{n_2\operatorname{H}_{V_2}(\hat{X})+\sum_{i=1}^{s} n_{3,i} \operatorname{H}_{V_{3,i}}(\hat{X}\|X) } \\ &\le subexp(n) 2^{n_{2} \operatorname{H}_{V_2}(\hat{X}) + n_{3} \operatorname{H}_{V_{3}}(\hat{X}\|X) } \label{cardialitybound22d} \\ &\mathds{E}\left( 1^{\mathbf{L},q}_{\mathbf{V}}(\mathbf{X}_{\hat{a}}^{\hat{k}};\mathbf{X}_{a_1}^{k_1}, \dots, \mathbf{X}_{a_g}^{k_g})\right) \le subexp(n) 2^{- n_3\operatorname{I}_{V_3}(X \wedge \hat X)} 2^{- l^{\hat{k}}J(V_2^{\hat{X}},V_{3}^{\hat{X}})} \label{becsleskozepecase2d}\\ &\le subexp(n) 2^{-\sum\limits_{i=2}^{3} n_i\operatorname{I}_{V_i}(X \wedge \hat X)-l^{\hat{k}}J(V_2^{\hat{X}},V_{3}^{\hat{X}})}.\label{becslesvegsocase2d}\end{aligned}$$ Inequality (\[becslesvegsocase2d\]) implies (\[egysegesbecsles2\]). In the proof above the different divisions shown on Fig. \[packingtenycikkmodtovabb\] and Fig. \[kepmarkov2\] ensure that the numbers of terms in the corresponding sums are subexponential. Acknowledgment {#acknowledgment .unnumbered} ============== We would like to thank Prof. Imre Csiszár for his help and advice. We also thank the support of the Hungarian National Research Development and Innovation Office Grant K105840 and the MTA-BME Stochastics Research Group. [70]{} V. B. Balakirsky “Joint Source-Channel Coding Using Variable-Length Codes,” Problems of Information Transmission, vol. 37, No. 1, 2001, pp. 12-27. S. Borade, B. Nakiboglu and L. Zheng “Unequal Error Protection: An Information-Theoretic Perspective,” IEEE Trans. Inf. Theory, vol. 55, pp. 5511-5539, Dec. 2009. J. Burbea and, C. D. Rao “On the Convexity of Some Divergence Measures Based on Entropy Functions ,” IEEE Trans. Inf. Theory, vol. 28, pp. 489-495, May 1982. JACOB BURBEA AND C. RADHAKRISHNA RAO I. Csisz[á]{}r, J. 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Draper, “Unequal Message Protection: Asymptotic and Non-Asymptotic Tradeoffs,” IEEE Trans. Inf. Theory vol. 61, pp. 5396-5416 Oct. 2015. N. Weinberger and N. Merhav, “Codeword or Noise? Exact Random Coding Exponents for Joint Detection and Decoding,” IEEE Trans. Inf. Theory vol. 60, pp. 5077-5094 Sept. 2014. S. A. Yıldırım, A. Martinez and A. G. Fàbregas, “Achievable rates and exponents for asynchronous communication with ML decoding,” ISIT, Jun. 2015. [^1]: This paper has been presented in part at the recent result poster session of ISIT 2016, Barcelona. Lóránt Farkas is with the Department of Analysis, Budapest University of Technology and Economics, e-mail: lfarkas@math.bme.hu. Tamás Kói is with the Department of Stochastics, Budapest University of Technology and Economics, e-mail: koitomi@math.bme.hu. The work of the authors was supported by the Hungarian National Research Development and Innovation Office Grant K105840.
--- abstract: \| With the advent of Software Defined Networks (SDN), Network Function Virtualisation (NFV) or Service Function Chaining (SFC), operators expect networks to support flexible services beyond the mere forwarding of packets. The network programmability framework which is being developed within the IETF by leveraging IPv6 Segment Routing enables the realisation of in-network functions. In this paper, we demonstrate that this vision of in-network programmability can be realised. By leveraging the eBPF support in the Linux kernel, we implement a flexible framework that allows network operators to encode their own network functions as eBPF code that is automatically executed while processing specific packets. Our lab measurements indicate that the overhead of calling such eBPF functions remains acceptable. Thanks to eBPF, operators can implement a variety of network functions. We describe the architecture of our implementation in the Linux kernel. This extension has been released with Linux 4.18. We illustrate the flexibility of our approach with three different use cases: delay measurements, hybrid networks and network discovery. Our lab measurements also indicate that the performance penalty of running eBPF network functions on Linux routers does not incur a significant overhead. author: - 'Mathieu Xhonneux, Fabien Duchene, Olivier Bonaventure' bibliography: - 'paper.bib' title: Leveraging eBPF for programmable network functions with IPv6 Segment Routing --- <ccs2012> <concept> <concept\_id>10003033.10003034.10003038</concept\_id> <concept\_desc>Networks Programming interfaces</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003033.10003058.10003063</concept\_id> <concept\_desc>Networks Middle boxes / network appliances</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003033.10003099.10003102</concept\_id> <concept\_desc>Networks Programmable networks</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003033.10003099.10003103</concept\_id> <concept\_desc>Networks In-network processing</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012> Acknowledgements {#acknowledgements .unnumbered} ================ This work was partially supported by a Cisco URP grant and by CFWB within the ARC-SDN project.
--- abstract: 'Communication-based train control (CBTC) is gradually adopted in urban rail transit systems, as it can significantly enhance railway network efficiency, safety and capacity. Since CBTC systems are mostly deployed in underground tunnels and trains move in high speed, building a train-ground wireless communication system for CBTC is a challenging task. Modeling the tunnel channels is very important to design and evaluate the performance of CBTC systems. Most of existing works on channel modeling do not consider the unique characteristics in CBTC systems, such as high mobility speed, deterministic moving direction, and accurate train location information. In this paper, we develop a finite state Markov channel (FSMC) model for tunnel channels in CBTC systems. The proposed FSMC model is based on real field CBTC channel measurements obtained from a business operating subway line. Unlike most existing channel models, which are not related to specific locations, the proposed FSMC channel model takes train locations into account to have a more accurate channel model. The distance between the transmitter and the receiver is divided into intervals, and an FSMC model is applied in each interval. The accuracy of the proposed FSMC model is illustrated by the simulation results generated from the model and the real field measurement results.' author: - \| \ $^{\dag}$Department of Systems and Computer Engineering, Carleton University, Ottawa, ON, Canada\ $^{\ddag}$State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, P.R. China bibliography: - 'ref.bib' title: 'Finite-State Markov Modeling of Tunnel Channels in Communication-based Train Control (CBTC) Systems' --- Instruction =========== Urban rail transit systems are developing rapidly around the world. Duo to the huge urban traffic pressure, improving the efficiency and capacity of urban rail transit systems is increasingly in demand. Being a key subsystem in rail transit systems, communications-based train control (CBTC) is an automated train control system using bidirectional train-ground communications to ensure the safe operation of rail vehicles [@What_is_communication-based_train_control]. It can enhance the level of safety and service offered to customers and improve the utilization of railway network infrastructure. CBTC is a modern successor of a traditional railway signaling system using interlockings, track circuits, and signals [@CBTC_Standard]. Building a train-ground wireless communication system for CBTC is a challenging task. As urban rail transit systems are mostly deployed in underground tunnels, there are a large amount of reflections, scattering and barriers that severely affect the propagation performance of wireless communications. Moreover, due to the available commercial-off-the-shelf equipments, wireless local area networks (WLANs) are often adopted as the main method of train ground communications for CBTC systems [@ZHULI_FSMC]. However, most of the current IEEE 802.11 WLAN standards are not originally designed for the high speed environment in tunnels [@ZHULI_FSMC; @ZFBT12]. Furthermore, the fast movement of trains will cause frequent handoffs between WLAN access points (APs), which could affect CBTC performance severely. Modeling the channels of urban rail transit systems in tunnels is very important to design and evaluate the performance of CBTC systems. There are some previous works on radio wave propagation in tunnels [@ZHANGYP_Enhancement_of_rectangular_tunnel_waveguide_model], [@ZhangYP_Novel_Model]. A path loss model of tunnels is given in [@ZhangYP_Novel_Model], which describes the characteristics of the large scale fading. Authors of [@KEGUAN] illustrate a measurement method of $2.4GHz$ in a subway tunnel, and the research object is a distributed antenna system, which is not often applied in CBTC systems. Due to the good balance between accuracy and complexity, finite state Markov channel (FSMC) model has been successfully used in different channels, including Rayleigh fading [@Finite_state_Markov_channel_a_useful_model_for_radio_communication_channels], Ricean fading [@Finite-state_Markov_modeling_of_correlated_Rician-fading_channels] and Nakagami fading [@Fast_simulation_of_diversity_Nakagami_fading_channels_using_finite-state_Markov_models]. Although some excellent works have been done on modeling channels, most of them do not consider the unique characteristics in CBTC systems, such as high mobility speed, deterministic moving direction, and accurate train location information. In this paper, we develop a finite state Markov channel model for tunnel channels in CBTC systems. Some distinct features of the proposed channel model are as follows. - The proposed FSMC model is based on real field CBTC channel measurements obtained from business operating Beijing Subway Changping line. - Unlike most existing channel models, which are not related to specific locations, the proposed FSMC channel model takes train locations into account to have a more accurate channel model. - The distance between the transmitter and the receiver is divided into intervals, and an FSMC model is applied in each interval. - Lloyd-Max technique [@Least_squares_quantization_in_PCM] is used to determine the SNR level boundaries in the proposed FSMC model. - The accuracy of the proposed FSMC model is illustrated by the simulation results generated from the model and the real field measurement results. The effects of different parameters are also discussed. The rest of this paper is organized as follows. Section \[OverviewCBTC\] describes an overview of CBTC systems. In Section \[Sec\_Measurement\], the real field measurement configuration and scenario are described. In Section \[Sec\_FSMC\], the FSMC model is introduced. Then, Section \[Sec\_TestandDiscussion\] presents the real field measurement results and discussions. Finally, the paper is concluded in Section \[Sec\_CandF\] with future work. Overview of Communication-Based Train Control {#OverviewCBTC} ============================================= In CBTC systems, continuous bidirectional wireless communications between each mobile station (MS) on the train and the wayside access point (AP) are adopted instead of the traditional fixed-block track circuit. The railway line is usually divided into areas or regions. Each area is under the control of a zone controller (ZC) and has its own radio transmission system. Each train transmits its identity, location, direction and speed to the ZC. The radio link between each train and the ZC should be continuous so that the ZC knows the locations of all the trains in its area at all the time in order to guarantee train operation safety and efficiency. Wireless channels in CBTC systems are different from those in other wireless systems, since most CBTC systems are deployed in underground tunnels, where there are a large amount of reflections, scattering and barriers that severely affect the propagation performance of wireless communications. In order to design and evaluate the performance of CBTC systems, modeling of tunnel channels in CBTC systems should be carefully studied. Real Field CBTC Channel Measurements {#Sec_Measurement} ==================================== The objective of the real field CBTC channel measurements is to get the real field data of WLAN propagation in tunnels under real conditions of the subway line, which can be used to build an FSMC model. Measurement Equipment --------------------- Two sets of Cisco 3200 are used, and one is set as AP while the other one is set as the mobile station (MS). Both of them are set to work at the frequency of 2.412$GHz$, which is also called channel $1$. The output power of the AP is set as 30$dBm$. The AP is located on the wall of the tunnel, while the MS is located on the measurement vehicle. The transmitting antenna is a Yagi antenna connected with the AP, which is directional and vertically polarized. The half power beam width (HPBW) is $30 ^{\circ}$ and the gain of Yagi antenna is 13.5$dBi$. In addition, the Shark-fin antenna is applied as the receiving antenna connected with the MS, which is also directional and vertically polarized. The HPBW is $40 ^{\circ}$ and the gain of Shark-fin antenna is 10$dBi$.The measurement configuration settings are shown in Table \[Mconfig\]. Frequency ---------------------------- ------------------------ ------------------- Transmitting Power \[Transmitting Antenna]{} Type Yagi Antenna Polarization Direction Vertical Gain $13.5 dBi$ HPBW $30 ^{\circ}$ \[Receiving Antenna ]{} Type Shark-fin Antenna Polarization Direction Vertical Gain $10 dBi$ HPBW $40 ^{\circ}$ : Measurement Configuration[]{data-label="Mconfig"} The location of the train is obtained through a velocity sensor installed on the wheel of the measurement vehicle, which can detect the realtime velocity, and the resolution of position is millimeter per second. When the measurement vehicle is moving, the velocity sensor gets the speed transmitted to the singlechip computer immediately through a serial port. At the same time, the MS captures the signal strength and SNR at the current position, and the signal information together with the integrated displacement data by the singlechip computer can be stored in the laptop. Therefore, the signal strength and SNR mapping with the location of receiver can be obtained, which is useful to build an FSMC model depending on the distance between the transmitter and the receiver. Measurement Scenario -------------------- The measurement was performed in the straight section of tunnels in Beijing Subway Changping Line, and the cross section of tunnel is rectangular. The height of the tunnel is 4.91m and the width is 4.4m. The transmitting antenna is located 0.15m below the tunnel roof, which is 4.76m. The receiving antenna is set on the top of an iron bar, which is 3.8m and also the height of the top of the train . As the threshold of the receiver is $-90dBm$, the coverage of one AP is about $0m$-$500m$, which is also the experimental zone in our measurements. The tunnel where we performed the measurement is a section of straight tunnel, and Fig. \[RT\] shows the cross section of tunnel in Changping Subway Line, the Shark-fin antenna, the Yagi antennas and the AP set on the wall. ![(a) The tunnel where we performed the measurements in Beijing Changping Subway Line. (b) The Shark-fin antenna. (c) The Yagi antenna. (d) The AP set on the wall.[]{data-label="RT"}](Real_Picture.eps){width="45.00000%"} The Finite-State Markov Chain Channel Model {#Sec_FSMC} =========================================== To capture the characteristics of tunnel channels in CBTC systems, we define channel states according to the different received SNR levels, and use an FSMC to track the state variation. In this section, we first describe the FSMC model, followed by the determination of key model parameters, including SNR levels and SNR distribution. The Finite State Markov Channel Model ------------------------------------- Let $\Gamma$ denote the SNR of the received signal, whose range can be obtained from the experimental data. The range of SNR is partitioned into $N$ non-overlapping levels with thresholds $\{\Gamma_{n}, n = 1, 2, 3, ..., N+1\}$. Let $\textbf{S}=\{s_{1}, s_{2}, ..., s_{n}\}$ denote the finite channel states, and the channel state is $s_{n}$ when the SNR of the received signal belongs to the range $(\Gamma_{n}, \Gamma_{n+1})$. Then $\{\textbf{S}_{n}\}$ is a Markov process and the transition probability $p_{n,j}$ can be shown as follows, which is independent of the index $n$. $$p_{n,j}=P_{r}\{\textbf{S}_{k+1}=s_{n} \mid \textbf{S}_{k}=s_{j}\}, \label{STP}$$ where $k = 1, 2, 3, ..., $ and $n, j \in \{1 ,2, ..., N+1\}$. According to the property of first-order Markov chain, we assume that each state can only transit to the adjacent states, which means $p_{n,j}=0$, if $\mid{n-j}\mid>1$. With the definition, we can define a $K\times K$ state transition probability matrix $\textbf{P}$ with elements $p_{n,j}$. Due to the effect of large scale fading, the amplitude of SNR depends on the distance between the transmitter and the receiver. It is obvious that the SNR is usually higher when the receiver is close to the transmitter; while it is lower when the receiver is far away from the transmitter. As a result, the transition probability from the high received SNR state to the low received SNR state is different when the receiver is near or far away from the transmitter, which means that the Markov state transition probability is related to the location of the receiver. Therefore, only one state transition probability matrix, which is independent of the location of the receiver, may not generate accurate enough models to describe the tunnel channels. Thus, we divide the tunnel into $L$ intervals and one state transition probability matrix is generated for each interval. Specifically, $\textbf{P}^{l}, l \in \{1, 2, ..., L\}$, is the state transition probability matrix corresponding to the $l$th interval, and the relationship between the transition probability and the location of the receiver can be built. Then, $p_{n, j}^{l}$ is the state transition probability from state $s_{n}$ to state $s_{j}$ in the $l$th interval. Based on the measurement results, we need to determine the value of the state probability $p_{n}^{l}$ and the state transition probability $p_{n,j}^{l}$. Determine the SNR Level Thresholds of the FSMC Model ---------------------------------------------------- As mentioned above, getting the thresholds of SNR levels is the key factor that affects the accuracy of the FSMC model. There are many methods to select the SNR level boundaries, and the equiprobable partition method is frequently used in previous works [@Finite_state_Markov_channel_a_useful_model_for_radio_communication_channels; @Finite-state_Markov_modeling_of_correlated_Rician-fading_channels; @Fast_simulation_of_diversity_Nakagami_fading_channels_using_finite-state_Markov_models]. As nonuniform amplitude partitioning may be useful to obtain more accurate estimates of system performance measures [@Finite-state_Markov_modeling_of_fading_channels_a_survey_of_principles_and_applications], we choose the Lloyd-Max technique [@Least_squares_quantization_in_PCM] instead of the equiprobable method to partition the amplitude of SNR in this paper. Lloyd-Max is an optimized quantizer, which can decrease the distortion of scalar quantization. Lloyd-Max can realize uniform scalar quantization and non-uniform scalar quantization, and the latter one is used in this paper to divide the amplitude range of SNR. Firstly, a distortion function $D$ is defined as follows. $$D=\sum^{N+1}_{k=2}\int ^{x_{k}}_{x_{k-1}}f(\tilde{x}_{k}-x)p(x)dx, \label{LLM1}$$ where $x_{k}$ is the threshold of the $k$th SNR level, $f(x)$ is the error criterion function, and $p(x)$ is the probability distribution function of SNR. The error criterion function $f(x)$ is often taken as $x^{2}$ [@Digital_communications]. As a result, Then, the necessary conditions for minimum distortion are obtained by differentiating $D$ with respect to ${x_{k}}$ and ${\tilde{x_{k}}}$ as follows. $$\begin{aligned} \label{LLM4} &x_{k}=\frac{\tilde{x}_{k}+\tilde{x}_{k+1}}{2}.\\ \label{LLM5} &\int^{x_{k}}_{x_{k-1}}(\tilde{x}_{k}-x)p(x)dx=0.\end{aligned}$$ Therefore, all elements of ${\Gamma_n}$ can be obtained according to . Combined with , the value of ${\Gamma_n}$ can be updated until the value of $D$ is the minimum, and the optimal thresholds of the SNR levels can be got. As $p(x)$ is still not determined, we should discuss the distribution of SNR according to the experimental sampling data, which is the last step to obtain the thresholds of SNR regions. Determine the Distribution of SNR --------------------------------- Deriving the distribution of SNR is the crucial step of partitioning the levels of SNR. In fact, there are some classic models to describe the distribution of signal strength, such as Rice, Rayleigh and Nakagami, and then the corresponding models of SNR can also be obtained [@Digital_Communication_over_Fading_Channels]. We firstly obtain the distribution of the signal strength in order to determine the model of SNR. The Akaike information criterion (AIC) is adopted in this paper to get the approximate distribution model of the signal strength. The AIC is a measure of the relative goodness of fit of a statistical model. The general case of AIC is [@Model_selection_and_multimodel_inference] $$AIC=-2\ln{L}+2\eta,$$ where $\eta$ is the number of parameters in the statistical model, and $L$ is the maximized value of the likelihood function for the estimated model. In fact, according to the relationship of $\eta$ and the number of samples $n$, AIC needs to be changed to Akaike information criterion with a correction (AICc) when ${n}/\eta<40$ [@Model_selection_and_multimodel_inference]. $$AICc=AIC+\frac{2\eta(\eta+1)}{n-\eta-1}. \label{F_AICc}$$ AICc is adopted to estimate the model of the signal strength distribution instead of the classic AIC in the paper. In practice, one can compute AICc for each of the candidate models and select the model with the smallest value of AICc. The candidate models include Rice, Rayleigh, and Nakagami in the paper. Our model is related to the distance between the transmitter and receiver, and the tunnel should be divided into intervals. Thus, as mentioned above, we should divide the amplitude of SNR into several levels and firstly calculate the value of AICc for each model to determine the distribution function for each interval. Now we assume there are $L$ intervals, and then we select the most appropriate model based on the frequency of the minimum AICc value of different candidate models. In order to obtain enough data for each interval, we set the length of each interval as $40$ wavelengths of WLANs [@A_statistical_model_for_indoor_office_wireless_sensor_channels], and then there are $100$ intervals. Fig. \[AICc\] shows the frequencies of AICc of different distributions. From Fig. \[AICc\], we can observe that the Nakagami distribution provides the best fit in a majority of the cases. As a result, we can define $p(x)$ as the Nakagami distribution. ![Frequencies of AICc selecting a candidate distribution.[]{data-label="AICc"}](AICc.eps){width="45.00000%"} According to [@Digital_Communication_over_Fading_Channels], we can obtain the distribution of SNR, after the distribution of the signal strength is obtained. $$p(x)=\frac{m^{m}x^{m-1}}{\bar{x}^{m}\Gamma{m}}exp(-\frac{mx}{\bar{x}}), \label{SNRfunc}$$ where $x$ is the SNR data, $\bar{x}$ is the mean of SNR, $m$ is the fading factor of Nakagami distribution, and $\Gamma(.)$ is the gamma function. In fact, $m$ can be calculated when applying AICc through the maximum likelihood estimator for each interval. Based on , and , the thresholds $\{\Gamma_{n},n=1,2,...,N+1\}$ of SNR in each distance interval can be derived. Table \[Thresholds\_4\] and Table \[Thresholds\_8\] demonstrate the thresholds of the SNR levels at the location of $100m$ for different intervals, where we divide SNR into four and eight levels. As the distance intervals are different, the range of SNR is different and it brings different thresholds, which can provide one more accurate model. 5m 10m 20m 50m 100m ------------ ------------ ------------ ------------ ----------- -- $[95~100]$ $[90~100]$ $[80~100]$ $[50~100]$ $[0~100]$ 24 22 22 22 22 27.98 26.90 27.73 29.53 33.22 32.03 31.44 32.89 35.39 44.01 36.31 36.02 38.14 41.22 57.00 41 41 44 48 78 : Thresholds of SNR levels ($8$ levels) at the location of $100m$ for different intervals.[]{data-label="Thresholds_8"} 5m 10m 20m 50m 100m ------------ ------------ ------------ ------------ ----------- $[95~100]$ $[90~100]$ $[80~10m]$ $[50~100]$ $[0~100]$ 24 22 22 22 22 25.99 24.53 25.00 26.16 27.87 27.98 26.90 27.73 29.50 33.22 29.99 29.18 30.33 32.50 38.50 32.03 31.43 32.89 35.38 44.01 34.13 33.69 35.47 38.25 50.04 36.31 36.01 38.14 41.22 57.00 38.59 38.43 40.95 44.41 65.68 41 41 44 48 78 : Thresholds of SNR levels ($8$ levels) at the location of $100m$ for different intervals.[]{data-label="Thresholds_8"} Real Field Measurement Results and Discussions {#Sec_TestandDiscussion} ============================================== In this section, we compare our FSMC model with real field test results to illustrate the accuracy of the model. The effects of different parameters in the proposed model are discussed. The number of states in our model is first set as $4$. We also use $8$ states to study the effects of the number of states on the accuracy of the proposed model. In order to obtain the effects of distance intervals on the model, we choose the intervals as $5m$, $10m$, $20m$, $50m$, $100m$. We perform the measurements in the tunnels of Beijing Subway Changping Line many times so that enough data can be captured. We verify the accuracy of the FSMC model through another set of measurement data. First of all, we get the statistical state transition probabilities. Table \[STPM\_8\] illustrates the state transition probabilities of the FSMC model and the measurement data at the same location $(35m-40m)$ when there are eight states, and the distance interval is $5$m. Fig. \[5m\] shows the simulation results generated from our FSMC model and the experimental results from real field measurements. We can observe that there is greater agreement between them when the distance is $5m$ than that with the $100m$ distance interval. Next, we derive the Mean Square Error (MSE) to measure the degrees of approximation, shown in Fig. \[MSE\]. With the distance interval increasing, the MSE does also increase, which means the accuracy decreases. Moreover, it is obvious that the MSE of the FSMC model with $4$ states is larger than that with $8$ states. The number of states in the FSMC model plays a key role in the accuracy. Nevertheless, when the distance interval is $5m$, the difference of MSE is small for $4$ states FSMC model and $8$ states FSMC model. From this figure, we can see that the FSMC model with 4 states and $5m$ distance interval can provide an accurate enough channel model for tunnel channels in CBTC systems. ----- ------------- ----------- ------------- ------------- ----------- ------------- $p_{k,k-1}$ $p_{k,k}$ $p_{k,k+1}$ $p_{k,k-1}$ $p_{k,k}$ $p_{k,k+1}$ k=1 - 0.75 0.25 - 0.78 0.22 k=2 0.25 0.5 0.25 0.269 0.47 0.26 k=3 0.25 0.5 0.25 0.23 0.5 0.26 k=4 0.22 0.66 0.11 0.22 0.65 0.12 k=5 0.125 0.5 0.25 0.126 0.63 0.24 k=6 0.095 0.81 0.048 0.089 0.86 0.049 k=7 0.13 0.6 0.27 0.12 0.61 0.26 k=8 0.013 0.98 - 0.013 0.98 - ----- ------------- ----------- ------------- ------------- ----------- ------------- : The values of transition probabilities for the FSMC model with $8$ states and $5m$ interval[]{data-label="STPM_8"} ![The MSE between the FSMC model and the experimental data with $4$ states and $8$ states.[]{data-label="MSE"}](interval_5m_100m.eps){width="45.00000%"} ![The MSE between the FSMC model and the experimental data with $4$ states and $8$ states.[]{data-label="MSE"}](MSE_FSMC.eps){width="45.00000%"} Conclusions and Future Work {#Sec_CandF} =========================== Modeling the tunnel wireless channels of urban rail transit systems is important in designing and evaluating the performance of CBTC systems. In this paper, we have proposed an FSMC model for tunnel channels in CBTC systems. Since the train location is known in CBTC systems, the proposed FSMC channel model takes train locations into account to have a more accurate channel model. The distance between the transmitter and the receiver is divided into intervals, and an FSMC model is designed in each interval. The accuracy of the proposed model has been illustrated by the simulation results generated from the proposed model and the real field measurement. In addition, we have shown that the number of states and the distance interval have impacts on the accuracy of the proposed FSMC model. Future work is in progress to study the effects of wireless channels on the control performance of CBTC systems based on the proposed channel model. Acknowledgement {#acknowledgement .unnumbered} =============== This paper was supported by grants from the National Natural Science Foundation of China (No.61132003), the National High Technology Research and Development Program of China (863 Program) (2011AA110502), and projects (No. RCS2011ZZ007, RCS2012ZQ002, 2013JBM124, 2011JBZ014, RCS2010ZZ003, RCS2012K010).
--- abstract: \| A graph $G=(V,E)$ is called $(k,\ell)$-full if $G$ contains a subgraph $H=(V,F)$ of $k\|V\|-\ell$ edges such that, for any non-empty $F' \subseteq F$, $\|F'\| \leq k\|V(F')\| - \ell$ holds. Here, $V(F')$ denotes the set of vertices incident to $F'$. It is known that the family of edge sets of $(k,\ell)$-full graphs forms a family of matroid, known as the sparsity matroid of $G$. In this paper, we give a constant-time approximation algorithm for the rank of the sparsity matroid of a degree-bounded undirected graph. This leads to a constant-time tester for $(k,\ell)$-fullness in the bounded-degree model, (i.e., we can decide with high probability whether an input graph satisfies a property $P$ or far from $P$). Depending on the values of $k$ and $\ell$, it can test various properties of a graph such as connectivity, rigidity, and how many spanning trees can be packed. Based on this result, we also propose a constant-time tester for $(k,\ell)$-edge-connected-orientability in the bounded-degree model, where an undirected graph $G$ is called $(k,\ell)$-edge-connected-orientable if there exists an orientation $\vec{G}$ of $G$ with a vertex $r \in V$ such that $\vec{G}$ contains $k$ arc-disjoint dipaths from $r$ to each vertex $v \in V$ and $\ell$ arc-disjoint dipaths from each vertex $v \in V$ to $r$. A tester is called a one-sided error tester for $P$ if it always accepts a graph satisfying $P$. We show, for $k \geq 2$ and (proper) $\ell \geq 0$, any one-sided error tester for $(k,\ell)$-fullness and $(k,\ell)$-edge-connected-orientability requires $\Omega(n)$ queries. author: - 'Hiro Ito[^1]' - 'Shin-ichi Tanigawa[^2]' - 'Yuichi Yoshida[^3]' bibliography: - 'testing\_rigidity.bib' title: 'Constant-Time Algorithms for Sparsity Matroids' --- Introduction {#sec:intro} ============ Property testing is a relaxation of decision. In property testing, given an instance $I$, we are to distinguish the case in which $I$ satisfies a predetermined property $P$ from the case in which $I$ is “far” from satisfying $P$. The farness depends on each model. The main objective of property testing is to develop efficient algorithms running even in constant time, which is independent of sizes of instances. In this paper, we study about testing algorithms for two strongly related properties of undirected graphs, $(k, \ell)$-sparsity and $(k,\ell)$-edge-connected-orientability. A graph $G = (V, E)$ is called $(k, \ell)$-sparse if $\|F\| \leq k\|V(F)\| - \ell$ for any $F \subseteq E, \|F\| \geq 1$, where $V(F)$ denotes the set of vertices incident to edges in $F$. We note that $(k,\ell)$-sparsity becomes meaningful only when $2k - \ell \geq 1$. If otherwise, any non-empty graph cannot be $(k,\ell)$-sparse since just an edge violates the condition. A graph $G$ is called $(k, \ell)$-tight if $G$ is $(k, \ell)$-sparse and $\|E\| = kn - \ell$, where $n$ is the number of vertices in $G$. A graph $G$ is called $(k, \ell)$-full if $G$ contains a $(k,\ell)$-tight subgraph with $n$ vertices. Checking whether a given graph is $(k,\ell)$-full or not is one of main topics in this paper. Another topic studied in this paper is an orientability of undirected graphs. A (di)graph is called $k$-edge-connected (resp., $k$-vertex-connected) if deletion of any $k-1$ edges (resp., vertices) leaves the graph connected. By Menger’s theorem, this is equivalent to asking $k$ edge-disjoint (resp., $k$ openly-disjoint) paths between any pair of vertices. A digraph $D=(V,A)$ is called [$(k,\ell)$-edge-connected]{} with a root $r\in V$ if, for each $v\in V\setminus \{r\}$, $D$ has $k$ arc-disjoint dipaths from $r$ to $v$ and $\ell$ arc-disjoint dipaths from $v$ to $r$. An undirected graph $G=(V,E)$ is called [$(k,\ell)$-edge-connected-orientable]{} ($(k,\ell)$-ec-orientable, in short) if one can assign an orientation to each edge so that the resulting digraph is $(k,\ell)$-edge-connected with some root $r\in V$. Note that the choice of $r$ is actually not important, and we may specify any vertex as $r$. Nash-Williams’ graph-orientation theorem [@nash1960] implies that a graph $G$ admits an orientation such that the resulting digraph is $k$-edge-connected if and only if $G$ is $2k$-edge-connected. This implies that $(k,k)$-ec-orientability of a graph is equivalent to $2k$-edge-connectivity. Another famous result of Nash-Williams [@nash1964] for the forest-partition problem shows that an undirected graph $G$ contains $k$ edge-disjoint spanning trees if and only if $G$ is $(k,k)$-full. This theorem, combined with Edmonds’ arc-disjoint branching theorem [@edmonds1972], implies that $G$ is $(k,0)$-ec-orientable if and only if $G$ is $(k,k)$-full. In this sense, $(k,\ell)$-ec-orientability can be considered as an unified concept of the sparsity and the conventional edge-connectivity. In this paper, we give constant-time testers for $(k,\ell)$-fullness and $(k,\ell)$-ec-orientability in the bounded-degree model. In the bounded-degree model with a degree bound $d$ [@GR02], we only consider graphs with maximum degree at most $d$. A graph $G = (V, E)$ is represented by an oracle ${\mathcal{O}}_G$. Given a vertex $v$ and an index $i \in \{1,\ldots,d\}$, ${\mathcal{O}}_G$ returns the $i$-th edges incident to $v$. If there is no such vertex, ${\mathcal{O}}_G$ returns a special character $\bot$. It can be seen that ${\mathcal{O}}_G$ represents the incidence list of $G$, and we can see one entry of the incidence list by one query to ${\mathcal{O}}_G$. A graph is called $\epsilon$-far from a property $P$ if we must modify at least $\frac{\epsilon dn}{2}$ edges. In other words, we must modify at least $\epsilon$-fraction of the incidence list to make $G$ satisfy $P$. The query complexity of an algorithm is the number of accesses to ${\mathcal{O}}_G$. For a property $P$, an algorithm is called a tester for a property $P$ if it accepts graphs satisfying $P$ with probability at least $\frac{2}{3}$ and rejects graphs $\epsilon$-far from $P$ with probability at least $\frac{2}{3}$. Our main results are summarized as follows. \[thr:test-k-l-fullness\] Let $k \geq 1, \ell \geq 0$ be integers with $2k-\ell \geq 1$. In the bounded-degree model with a degree bound $d$, there is a testing algorithm for the $(k,\ell)$-fullness of a graph with query complexity $(k+d)^{O(1/\epsilon'^2)}(\frac{1}{\epsilon'})^{O(1/\epsilon')}$, where $\epsilon'=\frac{\epsilon}{k+d\ell}$. \[thr:test-k-l-orientability\] Let $k \geq 1, \ell \geq 0$ be integers. In the bounded-degree model with a degree bound $d$, there is a testing algorithm for the $(k,\ell)$-ec-orientability of a graph with query complexity $(k+d)^{O(1/\epsilon'^2)}(\frac{1}{\epsilon'})^{O(1/\epsilon')}$, where $\epsilon'=\max(\frac{\epsilon}{dk},\frac{d\epsilon }{\ell})$. The second result resolves an open problem raised by Orenstein [@Ore10], which asks the existence of a constant-time tester for $(k,\ell)$-ec-orientability. As mentioned below, the first result has numerous applications to both theoretical and practical problems. An algorithm is called a $(1,\beta)$-approximation algorithm for a value $x^$ if, with probability $\frac{2}{3}$, it outputs $x$ such that $x^ - \beta \leq x \leq x^$. For a graph $G=(V,E)$, it is known that the family of edge sets of $(k,\ell)$-sparse subgraphs forms a family of independent sets of a matroid on $E$. This matroid is called the [$(k,\ell)$-sparsity matroid]{} of $G$, denoted by ${\cal M}_{k,\ell}(G)$, and the rank function by $\rho_{k,\ell}:2^E\rightarrow \mathbb{Z}$. Although the detailed property will be discussed in the next section, we should note that $G$ is $(k,\ell)$-full if and only if $\rho_{k,\ell}(E)=kn-\ell$. To test $(k,\ell)$-fullness, we actually develop a constant-time $(1,\epsilon n)$-approximation algorithm for $\rho_{k,\ell}(E)$. For a property $P$, a tester is called a one-sided error tester* for $P$ if it always accepts graphs satisfying $P$. A general tester is sometimes called a two-sided error tester for comparison. Our testers for $(k,\ell)$-fullness and $(k,\ell)$-ec-orientability are two-sided error testers. On the contrary, we give the following lower bounds for one-sided error testers. \[thr:lower-k-l-fullness\] Let $k \geq 2, \ell \geq 0$ be integers with $2k - \ell \geq 1$. In the bounded-degree model, any one-sided error tester for $(k,\ell)$-fullness requires $\Omega(n)$ queries where $n$ is the number of vertices in an input graph. \[thr:lower-k-l-ec-orientability\] Let $k \geq 2, \ell \geq 0$ be integers with $k > \ell$. In the bounded-degree model, any one-sided error tester for $(k,\ell)$-ec-orientability requires $\Omega(n)$ queries where $n$ is the number of vertices in an input graph. It is not hard to show that there are one-sided error testers for $(1,\ell)$-fullness and $(1,\ell)$-ec-orientability. Also, we have a one-sided error tester for $(k,\ell)$-ec-orientability when $\ell \geq k$. We briefly mention why we use the bounded-degree model. Another famous model for graphs is the adjacency matrix model, in which a graph is represented by an oracle ${\mathcal{O}}_G$ such that, given two vertices $u$ and $v$, ${\mathcal{O}}_G$ answers whether there is an edge between $u$ and $v$. A graph $G$ is called $\epsilon$-far from $P$ in this model if we must modify $\frac{\epsilon n^2}{2}$ edges to make $G$ satisfy $P$. We show that testing $(k,\ell)$-fullness is trivial in this model. Note that we can make any graph $(k,\ell)$-full by adding $kn - \ell$ edges. Thus, any graph is at most $O(\frac{1}{n})$-far. Thus, for any $\epsilon > 0$, when $n = \Omega(\frac{1}{\epsilon})$, we can safely accept graphs without any computation. When $n = O(\frac{1}{\epsilon})$, we can test $(k,\ell)$-fullness using a standard polynomial-time algorithm. We have the same issue also for $(k,\ell)$-ec-orientability. #### Related works In the bounded-degree model, many testers are known for several fundamental graph properties (see e.g.,[@goldreich2010]). The most relevant works are testers for connectivity. For undirected graphs, Goldreich and Ron [@GR02] gave constant-time testers for $k$-edge-connectivity ($k \geq 1$), $2$-vertex-connectivity, and $3$-vertex-connectivity. Yoshida and Ito [@YI08] extended the result by showing constant-time testers for $k$-vertex-connectivity ($k \geq 1$). For digraphs, constant-time testers for $k$-edge-connectivity ($k \geq 1$) are given in [@YI10]. Recently, Orenstein [@Ore10] simplified those results, and he also gave constant-time testers for $k$-vertex-connectivity of digraphs ($k \geq 1$). We stress that the idea behind all the algorithms above is to detect a small evidence that a graph does not satisfy the property we are concerned with. However, as we discuss later, for $(k,\ell)$-sparsity and $(k,\ell)$-ec-orientability, there may not be any such small evidence. This fact makes our testers more involved. Regarding exact and deterministic algorithms for checking the $(k,\ell)$-fullness of a graph $G$ with $n$ vertices and $m$ edges, Imai [@Ima83] proposed an algorithm for computing the rank of ${\cal M}_{k,k}(G)$ in $O(n^2)$ time and that of ${\cal M}_{k,\ell}(G)$ in $O(nm)$ time for general $\ell$. Improved algorithms were proposed by Gabow and Westermann [@gabow:1992], which run in $O(n\sqrt{m+n\log n})$ time for $k=\ell$ and in $O(n^2)$ time for $k=2$ and $\ell=3$. Also, they proposed an $O(n\sqrt{n\log n})$-time algorithm for checking the $(2,3)$-tightness (but not fullness). An efficient and practical algorithm for computing the rank of ${\cal M}_{k,\ell}(G)$ for general $k$ and $\ell$ is the so-called pebble algorithm by Lee and Streinu [@lee:streinu:2005], which runs in $O(n^2)$ time. As the $(k,\ell)$-sparsity has a wide range of applications in rigidity theory and scene analysis (see e.g. [@whiteley:hand; @Whiteley:1997]), it is recognized as an important open problem to improve the $O(n^2)$ upper-bound for computing the rank of the $(k,\ell)$-sparsity matroid (see e.g., [@demaine2008 Open Problem 4.1]). To the best of our knowledge, our result is the first sub-quadratic algorithm for approximating the rank of the $(k,\ell)$-sparsity matroid. #### Applications It is elementary to see that a graph is a forest if and only if it is $(1,1)$-sparse, and the concept of $(1,1)$-fullness coincides with the connectivity of graphs. As a variant of the commonly studied trees or forests, a graph is called a [pseudoforest]{} if each connected component contains at most one cycle [@gabow:1992]. It is known that a graph is a pseudoforest if and only if it is $(1,0)$-sparse [@gabow:1988]. As we mentioned above, Nash-Williams [@nash1964] proved that a graph contains $k$ edge-disjoint spanning trees if and only if it is $(k,k)$-full. Motivated by an application to rigidity theory, Whiteley [@Whiteley:1997] or Haas [@haas:2002] proved a generalization of Nash-Williams’ theorem to $(k,\ell)$-sparse graphs by mixing trees and pseudoforests. Our result leads to constant-time testers for these properties. Another important application of $(k,\ell)$-sparse graphs is the rigidity of graphs. A classical theorem by Laman [@laman:1970] implies that a $(2,3)$-full graph has a special property of being a generically rigid bar-joint framework on the plane, by regarding each vertex as a joint and each edge as a bar. More precisely, the deficiency between $2n-3$ and the rank of the $(2,3)$-sparsity matroid is equal to the degree of freedoms of the graph in the plane. It is further proved by Whiteley [@whiteley:88] that the $(2,2)$-sparsity matroid characterizes the generic rigidity of graphs embedded on a torus or a cylinder while the $(2,1)$-sparsity does the generic rigidity of graphs on the surface of a cone. For a general $d$-dimensional case, the $({d+1 \choose 2}, {d+1\choose 2})$-sparsity matroid characterizes the generic rigidity of special types of structural models, called body-bar frameworks [@tay:84] and body-hinge frameworks [@whiteley:88]. Although a combinatorial characterization of 3-dimensional generic rigidity of graphs has not been found yet (see e.g. [@whiteley:hand; @Whiteley:1997]), a characterization of an important special class, called [molecular graphs]{}, has been proved recently. In terms of graph theory, a molecular graph means the square $G^2$ of a graph $G$ as the rigidity of a molecule can be modeled by the rigidity of the square of a graph by identifying each atom as a vertex and each covalent bond as an edge (see e.g., [@thorpe:2005; @Whiteley:2005]). Tay and Whiteley [@tay:whiteley:84], or more formally, Jackson and Jord[á]{}n [@jackson:08], conjectured that $G^2$ is generically rigid in 3-dimensional space if and only if $5G$ is $(6,6)$-full. Here $5G$ denotes the graph obtained from $G$ by duplicating each edge by five parallel copies. Recently, Katoh and Tanigawa [@molecular] solved this conjecture affirmatively. In fact, based on this theory, the pebble game algorithm for checking $(6,6)$-fullness (runs in $O(n^2)$ time) is implemented in several softwares (e.g., [@amato; @flexweb; @kinari]) to compute the degree of freedoms of proteins. In this sense our super-efficient approximation algorithm for computing the degree of freedoms of molecules could bring a totally new approach in the protein flexibility analysis and the similarity search in the protein data base. #### Organization and proof overview In Section \[sec:pre\], we review properties of ${\cal M}_{k,\ell}(G)$. Then, in Sections \[sec:matching\] and \[sec:sparsity\], we first describe how to test $(k,\ell)$-fullness. To test whether $G$ is $(k,\ell)$-full, we develop a $(1,\epsilon n)$-approximation algorithm for $\rho_{k,\ell}(E)$ running in constant time (Theorem \[thr:approximation-to-mkl\]). A natural way to estimate the rank of ${\mathcal{M}}_{k,\ell}(G)$ is locally simulating the greedy algorithm, i.e., we add edges one by one, and if a newly added edge forms a circuit, we discard it. The main obstacle to simulate this algorithm is that, in general, we cannot detect any circuit in constant time. For example, a circuit in ${\mathcal{M}}_{1,1}(G)$ corresponds to a cycle in $G$. However, there is a $d$-regular graph in which any cycle is of length $\Omega(\log_d n)$. Thus, we need to estimate the rank without seeing any circuit. We mention that, for ${\mathcal{M}}_{1,1}(G)$, it is known that $\rho_{1,1}(E) = n-c$ holds where $c$ is the number of connected components. Using this fact, [@CRT01] gave an algorithm to estimate $\rho_{1,1}(E)$. However, for general $k$ and $\ell$, there is no such formula. Our strategy to overcome this issue is as follows: First, we remove constant-size circuits w.r.t. ${\mathcal{M}}_{k,\ell}(G)$, and let $G' = (V, E')$ be the resulting graph. We can show that $\rho_{k,\ell}(E) = \rho_{k,\ell}(E')$. A crucial fact is that $\rho_{k,\ell}(E')$ is close to $\rho_{k,0}(E')$. Thus, it amount to estimate $\rho_{k,0}(E')$ efficiently. It is known that $\rho_{k,0}(E')$ equals the size of the maximum matching of an auxiliary graph, and we can compute the maximum matching with a constant-time approximation algorithm for the maximum matching [@YYI09]. In Section \[sec:orientability\], we provide a constant-time tester for $(k,\ell)$-ec-orientability. Our algorithm is based on a characterization of the number of edges we need to add to make a graph $(k,\ell)$-ec-orientable by Frank and Kir[á]{}ly [@FK03]. Although this characterization is not so simple as the case of the edge-connectivity augmentation problem, we are able to show that, if $G$ is $\epsilon$-far, either there are many small evidences or $G$ is globally sparse which can be measured by $(k,k)$-fullness (Theorem \[thm:key2\]). As mentioned in introduction, the $(k,\ell)$-ec-orientability has strong relations to the sparsity as well as to the conventional edge-connectivity. Indeed, our algorithm can be considered as a combination of the idea of Yoshida and Ito for testing connectivity and the algorithm for testing $(k,k)$-sparsity given in Section \[sec:sparsity\]. In Section \[sec:lower-bound\], we prove linear lower bounds of one-sided error testers. In [@Ore10], Orenstein proved linear lower bounds of one-sided error tester for $(k,0)$-ec-orientability (or equivalently, $(k,k)$-fullness). Orenstein’s proof made use of Tutte-and-Nash-Williams’ tree packing theorem (see Theorem \[thm:tutte\]), which is a special property of $(k,k)$-fullness. We can however show that Orenstein’s approach can be applied to the general case of $\ell$ by the use of graph operations that preserve $(k,\ell)$-fullness. Preliminaries {#sec:pre} ============= For an integer $n$, we denote by $[n]$ the set $\{1,\ldots,n\}$. Let $G = (V,E)$ be a graph. For a vertex set $S \subseteq V$, $G[S]$ denote the subgraph of $G$ induced by $S$. For an edge set $F \subseteq E$, we define $V_G(F)$ as the set of vertices incident to $F$. For a vertex set $S,T \subseteq V$, we define $E_G(S,T) = \{uv \in E\mid u \in S, v \in T\}$ and $d_G(S,T)=\|E_G(S,T)\|$. If $T=V\setminus S$, we abbreviate them as $E_G(S)$ and $d_G(S)$, respectively. For a vertex $v \in V$, we use $E_G(v)$ and $E_G(v,T)$ instead of $E_G(\{v\})$ and $E_G(\{v\},T)$, respectively. Also, we define $\Gamma_G(S)$ as the set of vertices in $V \setminus S$ adjacent to some vertex in $S$. When the context is clear, we omit the subscripts. Let $f:2^E\rightarrow \mathbb{R}$ be a set function on a finite set $E$. $f$ is called [submodular]{} if $f(X)+f(Y)\geq f(X\cap Y)+f(X\cup Y)$ holds for any $X,Y\subseteq V$, and $f$ is called [non-decreasing]{} if $f(X)\leq f(Y)$ for any $X\subseteq Y\subseteq V$. Edmonds and Rota [@edmonds:1966] observed (and Pym and Perfect [@pym] formally proved) that an integer-valued non-decreasing submodular function $f:2^E\rightarrow \mathbb{Z}$ [induces]{} a matroid on $E$, where $F\subseteq E$ is independent if and only if $\|F'\|\leq f(F')$ for every non-empty $F'\subseteq F$. For a graph $G=(V,E)$ and integers $k \geq 1,\ell \geq 0$, we define a function $f_{k,\ell}:2^E\rightarrow \mathbb{Z}$ by $f_{k,\ell}(F)=k\|V(F)\|-\ell$ for $F\subseteq E$. It is known (and easy to show anyway) that $f_{k,\ell}$ is non-decreasing and submodular. Thus, $f_{k,\ell}$ induces a matroid on $E$, that is, the $(k,\ell)$-sparsity matroid ${\cal M}_{k,\ell}(G)$ defined in introduction. The rank function and the closure operator are denoted by $\rho_{k,\ell}$ and ${\rm cl}_{k,\ell}$, respectively. We note that $\rho_{k,\ell}(F)$ equals the size of the largest $(k,\ell)$-sparse edge set contained in $F$. This implies that $G$ is $(k,\ell)$-tight iff the rank of ${\cal M}_{k,\ell}(G)$ is $kn-\ell$. A set $F\subseteq E$ is called a $(k,\ell)$-connected set if, for any pair $e,e'\in F$, $F$ has a circuit of ${\cal M}_{k,\ell}(G)$ that contains $e$ and $e'$. For simplicity of the description, a singleton $\{e\}$ is also considered as a $(k,\ell)$-connected set. A maximal $(k,\ell)$-connected set w.r.t. edge inclusion is called a $(k,\ell)$-connected component. The following property of $(k,\ell)$-connected sets is just a restatement of a general fact on matroid-connectivity for our purpose. \[prop:property1\] Let $G=(V,E)$ be a graph and $k \geq 1, \ell \geq 0$ be integers with $2k-\ell\geq 1$. Then, ${\cal M}_{k,\ell}(G)$ has the following properties: (i) : For two $(k,\ell)$-connected sets $F_1$ and $F_2$ with $F_1\cap F_2\neq \emptyset$, $F_1\cup F_2$ is $(k,\ell)$-connected. (ii) : We can uniquely partition $E$ into $(k,\ell)$-connected components $\{C_1,\dots, C_t\}$, and the following relation holds: $$\label{eq:connected_component} \rho_{k,\ell}(E)=\sum_{i=1}^t \rho_{k,\ell}(C_i).$$ Proofs can be found in e.g., [@Oxl92 Chapter 4]. A $(k,\ell)$-connected set (or component) is called trivial if it is singleton, otherwise non-trivial. We remark that $\{e\}$ is a trivial $(k,\ell)$-connected component if and only if $e$ is a coloop in ${\cal M}_{k,\ell}(G)$ (i.e., every base contains $e$) since ${\cal M}_{k,\ell}(G)$ has no loop (in the matroid sense) if $2k-\ell\geq 1$. Hence, if we denote the family of non-trivial $(k,\ell)$-connected components in ${\cal M}_{k,\ell}(G)$ by $\{C_1,\dots, C_s\}$, then implies $$\label{eq:connected_component2} \rho_{k,\ell}(E)=\left\|E\setminus \bigcup_{i=1}^s C_i\right\|+\sum_{i=1}^s \rho_{k,\ell}(C_i).$$ We also need the following known properties of ${\cal M}_{k,\ell}(G)$. (Since they are so fundamental, we present proofs for completeness.) \[lmm:property2\] Let $G=(V,E)$ be a graph and $k \geq 1, \ell \geq 0$ be integers with $2k-\ell\geq 1$. Then, ${\cal M}_{k,\ell}(G)$ has the following properties: (i) : For any circuit $C$ of ${\cal M}_{k,\ell}(G)$, $\rho_{k,\ell}(C)=f_{k,\ell}(C)$. (ii) : For any non-trivial $(k,\ell)$-connected set $F\subseteq E$, $\rho_{k,\ell}(F)=f_{k,\ell}(F)$. Namely, $F$ is $(k,\ell)$-full. A proof for (i): Since $C$ is a minimal dependent set, $\|C\|>f_{k,\ell}(C)$ and $\|C\|-1=\|C-e\|\leq f_{k,\ell}(C-e)\leq f_{k,\ell}(C)$ for any $e\in C$. This implies $\|C\|=f_{k,\ell}(C)+1$. Thus, $\rho_{k,\ell}(C)=\|C\|-1=f_{k,\ell}(C)$. A proof for (ii): Suppose $\rho_{k,\ell}(F)<f_{k,\ell}(F)$. Then, there is an edge $uv\notin F$ with $u,v\in V(F)$ such that $\rho_{k,\ell}(F+uv)=\rho_{k,\ell}(F)+1$. Let us take two distinct edges $e$ and $e'$ of $F$ incident to $u$ and $v$, respectively. (It is easy to see that such two edges exist since $F$ is $(k,\ell)$-connected.) Since $F$ is $f$-connected, there is a circuit $C\subseteq F$ that contains $e$ and $e'$. Then, by (i) and by $f_{k,\ell}(C+uv)=f_{k,\ell}(C)$, we obtain $\rho_{k,\ell}(C+uv)\leq f_{k,\ell}(C+uv)=f_{k,\ell}(C)=\rho_{k,\ell}(C)$, implying $\rho_{k,\ell}(C+uv)=\rho_{k,\ell}(C)$. In other words, $uv$ is contained in the closure of $C$. This contradicts $\rho_{k,\ell}(F+uv)=\rho_{k,\ell}(F)+1$. We also need the following relation between ${\cal M}_{k,\ell}(G)$ and ${\cal M}_{k,\ell'}(G)$ with distinct $\ell$ and $\ell'$. \[lmm:property3\] Let $G=(V,E)$ be a graph and $k \geq 1, \ell \geq 0$ be integers with $2k-\ell\geq 1$. Then, any $(k,\ell)$-sparse set $F\subseteq E$ is $(k,\ell')$-sparse for any $\ell'\leq \ell$. For any nonempty $F'\subseteq F$, we have $\|F'\|\leq k\|V(F')\|-\ell\leq k\|V(F)\|-\ell'$. Finally, we give the formal definition of the bounded-degree model. In the bounded-degree model with a degree bound $d$, we consider graphs with maximum degree at most $d$. A graph $G=(V,E)$ of $n$ vertices is represented by an oracle ${\mathcal{O}}_G$ satisfying the followings: - For each vertex $v \in V$, there exists an injection $\pi_v: E_G(v) \to [d]$ such that $\pi_v$ is an injection. - The oracle ${\mathcal{O}}_G$, on two numbers $u \in V, i\in \mathbb{N}$, returns $v$ such that $(u,v)\in E$ and $\pi_u((u,v)) = i$. If no such vertex $v$ exists, it returns a special character $\bot$. An edge $e = uv$ is called the $i$-th edge of $u$ if $\pi_u(e) = i$. Algorithms are given $V$, $n$, $d$, and the access to ${\mathcal{O}}_G$ beforehand. For an error parameter $\epsilon > 0$, a graph is called $\epsilon$-far from a property $P$, if we must add or remove at least $\frac{\epsilon dn}{2}$ edges to make $G$ satisfy $P$. Approximating the rank of ${\cal M}_{k,0}(G)$ {#sec:matching} ============================================= \[sec:k0\] Let $k \geq 1$ be an integer. In this section, we present a constant-time approximation algorithm for the rank $\rho_{k,0}(E)$ of ${\mathcal{M}}_{k,0}(G)$ for a graph $G=(V,E)$. A crucial fact is that computing $\rho_{k,0}(E)$ can be reduced to computing the size of a maximum matching in an auxiliary bipartite graph $G_k$ obtained from $G$. The vertex set of $G_k$ is $E \cup (V \times [k])$ where $E$ and $V \times [k]$ form a partition, and $G_k$ has an edge between $e \in E$ and $(v, i) \in V \times [k]$ iff $e$ is incident to $v$ in the original graph $G$ (see Figure \[fig:auxiliary\]). From the celebrated Hall’s marriage theorem, the following result easily follows (see e.g., [@Ima83] for more details): ![(a) $G$ and (b) $G_2$.[]{data-label="fig:auxiliary"}](auxiliary_graph.eps) (a) ![(a) $G$ and (b) $G_2$.[]{data-label="fig:auxiliary"}](auxiliary_graph2.eps) (b) \[prop:pseudo\_forest\] Let $G=(V,E)$ be a graph and $k$ be an integer. Then, $G_k$ contains a matching covering $F \subseteq E$ if and only if $F$ is $(k,0)$-sparse. Proposition \[prop:pseudo\_forest\] implies that the rank of ${\cal M}_{k,0}(G)$ is equal to the size of a maximum matching in $G_k$. We use the following algorithm. \[lmm:approximation-to-matching\] In the bounded-degree model with a degree bound $d$, there exists a $(1,\epsilon n)$-approximation algorithm for the size of the maximum matching of a graph with query complexity $d^{O(1/\epsilon^2)}(\frac{1}{\epsilon})^{O(1/\epsilon)}$. To run the algorithm given in Lemma \[lmm:approximation-to-matching\] on $G_k$, we want to make an oracle access ${\mathcal{O}}_{G_k}$ to $G_k$ using the oracle access ${\mathcal{O}}_G$ to $G$. However, since we do not have a method to access $E$ directly, the vertex set $E \cup (V \times [k])$ is inconvenient to design ${\mathcal{O}}_{G_k}$. To deal with this issue, we use a slightly different auxiliary graph, which is essentially equivalent to the previous auxiliary graph. First, we introduce arbitrary ordering among vertices. We call $(v, i) \in V \times [d]$ valid if the $i$-th edge incident to $v$ exists and the vertex $v$ is the larger one in the endpoints of the edge, and invalid if otherwise. Then, we define a graph $G_k = (U_k \cup V_k, E_k)$ where $$\begin{aligned} U_k &=& \{(0, v, i) \mid v \in V, i \in [d]\},\\ V_k &=& \{(1, v, i) \mid v \in V, i \in [k]\}, \\ E_k &=& \{ ((0, u, i), (1, v, j)) \in U_k \times V_k \mid \text{if $(u, i)$ is valid and the corresponding edge is incident to $v$ in $G$} \}.\end{aligned}$$ For a vertex $(b, v, i)$ in $G_k$, the first bit $b$ is used to distinguish whether the vertex is in $U_k$ or $V_k$. We can see $G_k$ constructed here is isomorphic to the graph obtained from the previous auxiliary graph by adding singleton vertices. \[lmm:approximation-to-mk0\] Let $k \geq 1$ be an integer. In the bounded-degree model with a degree bound $d$, for a graph $G$ of $n$ vertices, there exists a $(1,\epsilon n)$-approximation algorithm for the rank of ${\mathcal{M}}_{k,0}(G)$ with query complexity $(k+d)^{O(1/\epsilon'^2)}(\frac{1}{\epsilon'})^{O(1/\epsilon')}$ where $\epsilon' = \frac{\epsilon}{k+d}$. Let $n$ and $m$ be the number of vertices and edges in $G = (V, E)$, respectively. The number of vertices in $G_k$ is $n' := kn + dn = (k+d)n$. Also, the maximum degree of $G_k$ is $d' := \max(2k, d) = O(k + d)$. Using the oracle access ${\mathcal{O}}_G$ to $G$, we make an oracle access ${\mathcal{O}}_{G_k}$ to $G_k$ on which we will run the algorithm given in Lemma \[lmm:approximation-to-matching\]. For a query ${\mathcal{O}}_{G_k}((b, v, i), j)$, we do the following. Suppose that $b = 0$, which means that $(b, v, i)$ is a vertex in $U_k$. We can check whether $(v, i)$ is valid by asking ${\mathcal{O}}_G$ once. If $(v, i)$ is invalid, we return $\bot$. Suppose that $(v, i)$ is valid and it corresponds to an edge $e = uv$ where $v > u$. If $j \leq k$, we return $(1, u, j)$. If $j > k$, we return $(1, v, j - k + 1)$. Suppose that $b = 1$, which means that $(b, v, i)$ is a vertex in $V_k$. If there is no $j$-th edge incident to $v$, we return $\bot$. Let $e = uv$ be the $j$-th edge incident to $v$. If $v > u$, we return $(0, v, j)$. If $v < u$ and $e$ is the $j'$-th edge of $u$, we return $(0, u, j')$. Here, we can find $j'$ by asking ${\mathcal{O}}_G$ at most $d$ times. To summarize, we can simulate the oracle access ${\mathcal{O}}_{G_k}$ by asking ${\mathcal{O}}_G$ at most $d+1$ times. To approximate the rank of ${\mathcal{M}}_{k,0}(G)$ with an additive error $\epsilon n$, we run the algorithm given in Lemma \[lmm:approximation-to-matching\] on ${\mathcal{O}}_{G_k}$ after replacing $\epsilon$ by $\epsilon' = \frac{\epsilon}{k+d}$. The query complexity becomes $d\cdot d'^{O(1/\epsilon'^2)}(\frac{1}{\epsilon'})^{O(1/\epsilon')}$, and the lemma holds. Approximating the rank of ${\cal M}_{k,\ell}(G)$ {#sec:sparsity} ================================================ In this section, we describe a constant-time approximation algorithm for the rank of ${\mathcal{M}}_{k,\ell}(G)$ for a graph $G = (V, E)$. For a given error value $\epsilon$, let $t$ be a constant determined later. We say that a subset $S\subseteq E$ is [large]{} if $\|S\|\geq t$; otherwise called [small]{}. For an edge $e = uv$ and an integer $r > 0$, let $G_r(e)$ be the graph induced by the set of vertices whose distance to $u$ or $v$ is at most $r$. Also, let $E_r(e)$ be the set of edges in $G_r(e)$. The core of our approximation algorithm is an efficient implementation of an algorithm $(e)$ that (approximately) decides whether a given edge $e \in E$ is in a large $(k,\ell)$-connected set or not. As a subroutine, we first prepare an algorithm called $(e)$ in Algorithm 1 and then show $(e)$ in Algorithm 2. $S = \{e\}$. $S=S\cup C$. return Large (a special symbol). return $S$. $S = \{e\}$. return Large. check $f$. $S=S\cup \textsf{SmallCircuits}(f)$ \[line\] return Large. return $S$. The following sequence of lemmas shows structural properties of outputs of $(e)$ and $(e)$. \[lmm:circuits\] For any $e\in E$, $\textsf{SmallCircuits}(e)$ and $\textsf{Component}(e)$ are small $(k,\ell)$-connected sets unless they return Large. Let $S=\textsf{SmallCircuits}(e)$. If $S=\{e\}$, then $S$ is a trivial $(k,\ell)$-connected set. If $\|S\|>1$, then $S$ is the union of circuits containing $e$. By Proposition \[prop:property1\](i), $S$ is $(k,\ell)$-connected. The latter claim similarly follows from Proposition \[prop:property1\](i) since $\textsf{Component}(e)$ is the union of $\textsf{SmallCircuit}(e')$ for all $e'\in \textsf{Component}(e)$. We define a relation $\sim$ on $E$ such that $e\sim f$ for $e,f\in E$ if and only if ${\cal M}_{k,\ell}(G)$ has a small circuit that contains $e$ and $f$. \[lmm:large\] Suppose that $\textsf{Component}(e)=\mathbf{Large}$. Then, there is a large $(k,\ell)$-connected set $S$ containing $e$ such that, for each $f\in S$, - $e\sim f$, or - $e\sim f'\sim f$ for some $f'\in S$. If $\textsf{SmallCircuits}(e)$ returns Large, then the union of small circuits containing $e$ forms a large $(k,\ell)$-connected set. This set satisfies the property of the statement. Thus, assume $\textsf{SmallCircuits}(e)\neq \mathbf{Large}$. Since $\textsf{Component}(e)$ returns Large, we encounter either one of the following two situations at the end of Algorithm 2: a small $(k,\ell)$-connected set $S$ with $e\in S$ contains an edge $f$ such that (i) $\textsf{SmallCircuits}(f)$ returns Large or (ii) $\textsf{SmallCircuits}(f)$ is small but $S\cup \textsf{SmallCircuits}(f)$ is large. In both cases, let $S_{f}$ be the union of all small circuits containing $f$. Then, $S\cup S_{f}$ is a desired large $(k,\ell)$-connected set. \[lmm:maximal\] Let $e\in E$. Suppose that $\textsf{Component}(e)$ does not return Large. Then, every small $(k,\ell)$-connected set intersecting $\textsf{Component}(e)$ is contained in $\textsf{Component}(e)$. Let $S=\textsf{Component}(e)$. Suppose that ${\cal M}_{k,\ell}(G)$ has a small $(k,\ell)$-connected set $S'$ such that $S\cap S'\neq\emptyset$ and $S'\setminus S\neq \emptyset$. Take $f\in S\cap S'$ and $f'\in S'\setminus S$. Since $f\sim f'$, we have $f'\in \textsf{SmallCircuts}(f)$. By Line \[line\] of Algorithm 2, we obtain $f'\in \textsf{Component}(e)=S$, a contradiction. \[lmm:relation\] For any $e\in E$ with $\textsf{Component}(e)\neq \mathbf{Large}$ and for any $f\in \textsf{Component}(e)$, $\textsf{Component}(e)=\textsf{Component}(f)$. Let $S=\textsf{Component}(e)$. Suppose that $\textsf{Component}(f)= \mathbf{Large}$. Then, by Lemma \[lmm:large\], there exists a large $(k,\ell)$-connected set $S_f$ containing $f$ such that, for each $f'\in S_f$, $f\sim f'$ or $f\sim f''\sim f'$ holds for some $f''\in S_f$. In particular, $S$ contains every element of $S_f$ since $e\sim f$ and $\textsf{Component}(e)$ never return Large during Algorithm 2. This contradicts that $S$ is small. Thus, $\textsf{Component}(f)$ is a small $(k,\ell)$-connected set by Lemma \[lmm:circuits\]. Lemma \[lmm:maximal\] now implies $\textsf{Component}(f)\subseteq S$ and $S\subseteq \textsf{Component}(f)$. Let $L=\{e\in E\mid \textsf{Component}(e)=\mathbf{Large}\}$, and let $\{S_1,S_2,\dots, S_m\}$ be the set of subsets of $E$ such that $S_i=\textsf{Component}(e)$ for some $e\in E$. Then, by Lemma \[lmm:relation\], $\{L,S_1,\dots,S_m\}$ forms a partition of $E$. Our testing algorithm directly follows from the next theorem. \[thm:key\] Let $\{L,S_1,\dots, S_m\}$ be the partition of $E$ defined as above. For each $i$ with $1\leq i \leq m$, let $B_i$ be a base of $S_i$ in ${\mathcal{M}}_{k,\ell}(G)$, and let $E' = L \cup \bigcup_{i=1}^m B_i$. Then, $\rho_{k,0}(E')-\frac{\ell dn}{t}\leq \rho_{k,\ell}(E)\leq \rho_{k,0}(E')$. Since $B_i$ is a base of $S_i$ in ${\cal M}_{k,\ell}(G)$, we have $S_i\subseteq {\rm cl}_{k,\ell}(B_i)\subseteq {\rm cl}_{k,\ell}(E')$ for each $i$. This implies $\rho_{k,\ell}(E')=\rho_{k,\ell}(E)$. Also, by Lemma \[lmm:property3\], we have $\rho_{k,\ell}(E')\leq \rho_{k,0}(E')$. To see $\rho_{k,0}(E')-\frac{\ell dn}{t}\leq \rho_{k,\ell}(E')$, recall that $(k,\ell)$-connected components of ${\cal M}_{k,\ell}(G)\|E'$ partitions $E'$ by Proposition \[prop:property1\](ii) (where ${\cal M}_{k,\ell}(G)\|E'$ denotes the restriction of ${\cal M}_{k,\ell}(G)$ to $E'$). We have the following properties of these connected sets. \[claim1\] Any $e\in L$ is contained in a large $(k,\ell)$-connected component in ${\cal M}_{k,\ell}(G)\|E'$. Let us take a large $(k,\ell)$-connected set $S_e$ of ${\cal M}_{k,\ell}(G)$ satisfying the property of Lemma \[lmm:large\] for $e$. Suppose, for a contradiction, that $\textsf{Component}(f)$ returns a small $(k,\ell)$-connected set $S_f$ for some $f\in S_e$. By a property of $S_e$, for every $f'\in S_e$ we have $f\sim f_1\sim e\sim f_2 \sim f'$ for some $f_1, f_2\in S_e$. As $\textsf{Component}(f)$ never return Large, we have $S_e\subseteq S_f$ according to Algorithm 2, contradicting that $S_f$ is small. Thus, each element of $S_e$ is included in $L$. This implies that $S_e$ remains in $E'$. Namely, $S_e$ exists as a large $(k,\ell)$-connected set even in ${\cal M}_{k,\ell}(G)\|E'$, and $e$ is contained in a large $(k,\ell)$-connected component in ${\cal M}_{k,\ell}(G)\|E'$. \[claim2\] Every non-trivial $(k,\ell)$-connected component in ${\cal M}_{k,\ell}(G)\|E'$ is large. To see this, suppose that there is a non-trivial small $(k,\ell)$-connected component $C$ in ${\cal M}_{k,\ell}(G)\|E'$. By Claim \[claim1\], each element of $L$ belongs to a large $(k,\ell)$-connected component in ${\cal M}_{k,\ell}(G)\|E'$. This implies $C\subseteq \bigcup_{i=1}^m B_i$. Also, since $B_i$ is independent in ${\cal M}_{k,\ell}(G)$, $C$ must intersect at least two sets among $\{B_1,\dots, B_m\}$. In particular, $C$ intersects at least two sets among $\{S_1,\dots, S_m\}$. Since $C$ is a small $(k,\ell)$-connected set in ${\cal M}_{k,\ell}(G)$, this contradicts Lemma \[lmm:maximal\]. Let $\{C_1,C_2,\dots,C_s\}$ be the family of non-trivial $(k,\ell)$-connected components in ${\cal M}_{k,\ell}(G)\|E'$. Note that $s\leq \frac{dn}{t}$ holds by Claim \[claim2\]. Therefore, $$\begin{aligned} \rho_{k,\ell}(E') &=& \|E'\setminus \bigcup_{i=1}^s C_i\| + \sum_{i=1}^s \rho_{k,\ell}(C_i) \quad \text{(by \eqref{eq:connected_component2})} \\ &=& \|E'\setminus \bigcup_{i=1}^s C_i\| + \sum_{i=1}^s (k\|V(C_i)\|-\ell) \quad \text{(by Lemma~\ref{lmm:property2}(ii))} \\ &\geq & \|E'\setminus \bigcup_{i=1}^s C_i\| + \sum_{i=1}^s k\|V(C_i)\| - \frac{\ell dn}{t} \quad \text{(by $s\leq \frac{dn}{t}$)}. \end{aligned}$$ On the other hand, $$\begin{aligned} \rho_{k,0}(E') &\leq& \|E'\setminus \bigcup_{i=1}^s C_i\| + \sum_{i=1}^s \rho_{k,0}(C_i) \quad \text{(by the submodularity of $\rho_{k,0}$)} \\ &\leq& \|E'\setminus \bigcup_{i=1}^s C_i\| + \sum_{i=1}^s k\|V(C_i)\|. \end{aligned}$$ This completes the proof. \[thr:approximation-to-mkl\] Let $G=(V,E)$ be a graph with $n$ vertices, and $k \geq 1, \ell \geq 0$ be integers with $2k - \ell \geq 1$. In the bounded-degree model with a degree bound $d$, there exists a $(1,\epsilon n)$-approximation algorithm for the rank of ${\mathcal{M}}_{k,\ell}(G)$ with query complexity $(k+d)^{O(1/\epsilon'^2)}(\frac{1}{\epsilon'})^{O(1/\epsilon')}$ where $\epsilon' = \frac{\epsilon}{k+d\ell}$. Let $G' = (V, E')$ where $E'$ is as in Theorem \[thm:key\]. Set $t=\frac{\ell d}{\epsilon}$. Our algorithm computes $\rho_{k,0}(E')$ based on the algorithm given in Lemma \[lmm:approximation-to-mk0\] for the error threshold $\epsilon$ and just returns this value. By Lemma \[lmm:approximation-to-mk0\] and Theorem \[thm:key\], this value approximates $\rho_{k,\ell}(E)$ with additive error $\epsilon n$. Therefore, if we can make an oracle access ${\mathcal{O}}_{G'}$ to the graph $G'$, we are done. For a query ${\mathcal{O}}_{G'}(v, i)$, we return a value as follows. If ${\mathcal{O}}_{G}(v, i) = \bot$, we return $\bot$. Suppose that ${\mathcal{O}}_{G}(v, i) = e$. Then, we invoke $(e)$. If $(e)$ returns Large, we return $e$. Otherwise, we take any base $B$ of the returned set of $(e)$ by an existing algorithm. We return $e$ if $e \in B$ and return $\bot$ if otherwise. Note that for another edge $f \in S$, we use the same base $B$. To analyze the query complexity, note that, during $(e)$, we perform queries ${\mathcal{O}}_G(v,i)$ only for vertices $v$ in $G_{3t}(e)$. So, to perform $(e)$, we need $d^{3t} = d^{3\ell d/\epsilon}$ queries to ${\mathcal{O}}_G$. In total, we need $d^{3\ell d/\epsilon}(k+d)^{O(1/\epsilon'^2)}(\frac{1}{\epsilon'})^{O(1/\epsilon')}=(k+d)^{O(1/\epsilon'^2)}(\frac{1}{\epsilon'})^{O(1/\epsilon')}$, where $\epsilon' = \frac{\epsilon}{k+d\ell}$. Theorem \[thr:test-k-l-fullness\] directly follows from Theorem \[thr:approximation-to-mkl\]. Testing $(k, \ell)$-edge-connected-orientability {#sec:orientability} ================================================ In this section, we present a tester for the $(k,\ell)$-edge-connected-orientability of a graph $G=(V,E)$. A multiset ${\cal F}=\{V_1,\dots, V_s\}$ of subsets of $V$ is said to be [regular]{} if each element of $V$ belongs to the same number of subsets in ${\cal F}$. For a regular multiset ${\mathcal{F}}=\{V_1,\dots, V_s\}$ of subsets of $V$, let $d_G({\mathcal{F}})=\sum_{i=1}^s \frac{d_G(V_i)}{2}$. If ${\mathcal{F}}$ is a partition of $V$, $d_G({\mathcal{F}})$ amounts to the number of edges connecting distinct subsets of ${\mathcal{F}}$. In [@F80], Frank proved a characterization of the orientability of graphs satisfying a so-called supermodular covering condition. This theorem includes the following characterization of the $(k,\ell)$-ec-orientability as a special case (see e.g., [@FK02; @FK03] for more detail). \[thm:orientability\] Let $G=(V,E)$ be a graph. Then, $G$ admits a $(k,\ell)$-edge-connected-orientation if and only if $d_G({\mathcal{F}})\geq k(\|{\mathcal{F}}\|-1)+\ell$ for any partition $\cal F$ of $V$ into non-empty subsets with $\|{\mathcal{F}}\|\geq 2$. This theorem motivates us to look at the following deficiency function: $$\begin{aligned} \label{eq:eta1} \eta_{k,\ell}(G) = \max\{0,\max\{k(\|{\mathcal{F}}\| - 1) + \ell - d_G({\mathcal{F}})\mid \text{ a partition ${\mathcal{F}}$ of $V$ with $\|{\mathcal{F}}\|\geq 2$}\}\},\end{aligned}$$ Then, $G$ admits a $(k,\ell)$-ec-orientation if and only if $\eta_{k,\ell}(G)=0$. Notice that, if $\ell=0$, we have $k(\|{{\mathcal{F}}}\|-1)-d_G({{\mathcal{F}}})=0$ for ${\mathcal{F}}=\{V\}$. We thus redefine $\eta_{k,0}(G)$, for convenience, by $$\label{eq:eta2} \eta_{k,0}(G)=\max\{k(\|{\mathcal{F}}\|-1)-d_G({\mathcal{F}}) \mid \text{ a partition ${\mathcal{F}}$ of $V$}\}.$$ ($\eta_{k,\ell}(G)$ remains if $\ell>0$.) Tutte [@tutte61] and Nash-Williams [@nash61] proved a special relation between $\eta_{k,0}(G)$ and the arbolicity of $G$. Specifically, Tutte-and-Nash-Williams tree packing theorem can be described in terms of ${\cal M}_{k,k}(G)$ as follows. \[thm:tutte\] Let $G=(V,E)$ be a graph and $k \geq 1$ be an integer. Then, $$\label{eq:tutte} \rho_{k,k}(E)=k(n-1)-\eta_{k,0}(G).$$ Notice that $$\begin{aligned} \eta_{k,\ell}(G)&=\max\{k(\|{\mathcal{F}}\| - 1) + \ell - d_G({\mathcal{F}})\mid \text{ a partition ${\cal F}$ of $V$ with $\|{\mathcal{F}}\|\geq 2$}\} \\ &= \max\{k(\|{\mathcal{F}}\| - 1) - d_G({\mathcal{F}}) \mid \text{ a partition ${\cal F}$ of $V$ with $\|{\mathcal{F}}\|\geq 2$}\} + \ell \\ &\leq \eta_{k,0}(G)+\ell, \end{aligned}$$ where the equality holds if $\eta_{k,0}(G)>0$. Hence, we also have $\eta_{k,0}(G)\leq \eta_{k,\ell}(G)$. Namely, $$\label{eq:eta_relation} \eta_{k,0}(G)\leq \eta_{k,\ell}(G)\leq \eta_{k,0}(G)+\ell.$$ Since $\eta_{k,0}(G)$ can be computed from $\rho_{k,k}(G)$ by , the approximation algorithm for $\rho_{k,k}(G)$ proposed in Theorem \[thr:approximation-to-mkl\] can be modified to compute $\eta_{k,\ell}(G)$. \[crl:approximation-to-eta\] Let $G$ be a graph with $n$ vertices, and $k \geq 1,\ell \geq 0$ be integers with $2k-\ell \geq 1$. In the bounded-degree model with a degree bound $d$, there exists a $(1,\ell+\epsilon n)$-approximation algorithm for $\eta_{k,\ell}(G)$ with query complexity $(k+d)^{O(1/\epsilon'^2)}(\frac{1}{\epsilon'})^{O(1/\epsilon')}$ where $\epsilon' = \frac{\epsilon}{dk}$. For testing $(k,\ell)$-ec-orientability, we need a certificate for deciding whether $G$ is $\epsilon$-far from $(k,\ell)$-ec-orientable. This part relies on a structural property of the connectivity argumentation problem proved by Frank and Kir[á]{}ly [@FK03]. A family $\{X_1,\dots, X_s\}$ of subsets of $X\subseteq V$ is called a [co-partition]{} of $X$ if $\{V\setminus X_1, \dots, V\setminus X_s\}$ forms a partition of $V\setminus X$. Also, for two multisets ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$, ${\mathcal{F}}_1+{\mathcal{F}}_2$ denotes their union as a multiset. \[thm:fk03-augment\] Let $G=(V,E)$ be a graph. $G$ can be made $(k,\ell)$-ec-orientable by adding $\gamma$ edges iff the following two conditions hold: (A) : $ \gamma \geq k(\|{\mathcal{F}}\|-1)+\ell - d_G({\mathcal{F}}) $ for every partition ${\mathcal{F}}$ of $V$ with $\|{\cal F}\|\geq 2$. (B) : $ 2\gamma \geq \|{\mathcal{F}}_1\|k + \|{\mathcal{F}}_2\|\ell - d_G({\mathcal{F}})$ for every multiset ${\mathcal{F}}= {\mathcal{F}}_1+ {\mathcal{F}}_2$ satisfying the following three conditions: $$\begin{aligned} &\text{${\mathcal{F}}_1$ is a partition of some $X \subset V$,} \nonumber \\ &\text{${\mathcal{F}}_2$ is a co-partition of $V\setminus X$,} \label{eq:partition} \\ &\text{every member of ${\mathcal{F}}_2$ is the complement of the union of some members of ${\mathcal{F}}_1$.} \nonumber\end{aligned}$$ By Corollary \[crl:approximation-to-eta\], the condition (A) is efficiently checkable. The non-trivial part is an algorithm for checking the second condition. Let $$\xi_{k,\ell}(G)=\max_{{\mathcal{F}}={\mathcal{F}}_1+{\mathcal{F}}_2}\{\|{\mathcal{F}}_1\|k + \|{\mathcal{F}}_2\|\ell - d_G({\mathcal{F}})\}$$ where the maximum is taken over all multisets ${\mathcal{F}}= {\mathcal{F}}_1+{\mathcal{F}}_2$ satisfying . Our goal is to approximate $\xi_{k,\ell}(G)$ efficiently. To simplify $\xi_{k,\ell}$, we need some terminology. For two partitions ${\mathcal{P}}_1$ and ${\mathcal{P}}_2$ of $X\subseteq V$, ${\mathcal{P}}_1$ is said to be a [refinement]{} of ${\mathcal{P}}_1$ if each member of ${\mathcal{P}}_2$ is the union of some members of ${\mathcal{P}}_1$. A regular multiset ${\mathcal{P}}$ of subsets of $X$ is called a [double-partition]{} of $X$ if ${\mathcal{P}}$ is written as ${\mathcal{P}}={\mathcal{P}}_1+{\mathcal{P}}_2$ for some partitions ${\mathcal{P}}_1$ and ${\mathcal{P}}_2$ of $X$ such that ${\mathcal{P}}_1$ is a refinement of ${\mathcal{P}}_2$. We should note the following relation between a double-partition and a multiset satisfying . Let ${\mathcal{F}}={\mathcal{F}}_1+{\mathcal{F}}_2$ be a family of subsets satisfying with a partition ${\mathcal{F}}_1$ of $X$ and a co-partition ${\mathcal{F}}_2$ of $V\setminus X$. Let ${\mathcal{P}}_1={\mathcal{F}}_1$ and ${\mathcal{P}}_2=\{V\setminus X'\mid X'\in {\mathcal{F}}_2\}$. Then, ${\mathcal{P}}_1$ and ${\mathcal{P}}_2$ are partitions of $X$ and ${\mathcal{P}}_1$ is a refinement of ${\mathcal{P}}_2$. Also, carefully counting the number of edges contributed to $d_G({\cal F})$, we have $d_G({\mathcal{F}})=d_G({\mathcal{P}}_1+{\mathcal{P}}_2)$. Thus, using $$\begin{aligned} \eta_{k,0}(G)&=\max\{k(\|{\mathcal{P}}\|-1)-d_G({{\mathcal{P}}})\mid \text{ a partition ${\mathcal{P}}$ of $V$} \} \\ &= \max\left\{k(\|{\mathcal{P}}\|-1)-\sum_{i=1}^s \frac{d_G(X_i)}{2}\mid \text{ a partition ${\mathcal{P}}=\{X_1,\dots, X_s\}$ of $V$} \right\} \\ &= \max\left\{\sum_{i=1}^s \left(k-\frac{d_G(X_i)}{2}\right)-k\mid \text{ a partition ${\mathcal{P}}=\{X_1,\dots, X_s\}$ of $V$} \right\}\end{aligned}$$ we obtain $$\begin{aligned} \xi_{k,\ell}(G)&=\max\{\|{\mathcal{F}}_1\|k + \|{\mathcal{F}}_2\|\ell - d_G({\mathcal{F}})\mid \text{ a family }{\mathcal{F}}={\mathcal{F}}_1+{\mathcal{F}}_2 \text{ satisfying \eqref{eq:partition}}\} \\ &=\max\{\|{\mathcal{P}}_1\|k + \|{\mathcal{P}}_2\|\ell - d_G({\mathcal{P}})\mid \text{ a double-partition } {\mathcal{P}}={\mathcal{P}}_1+{\mathcal{P}}_2 \text{ of some } X\subset V \} \\ \begin{split}&=\max \bigg\{\sum_{i=1}^s \bigg(\ell-d_G(X_i)+\max\bigg\{ \sum_{j=1}^{s_i} (k-d_{G[X_i]}(X_{i,j})/2) \mid \text{ a partition $\{X_{i,1},\dots, X_{i,s_i}\}$ of $X_i$} \bigg\}\bigg) \\ & \hspace{25em} \mid \text{ a sub-partition $\{X_1,\dots,X_s\}$ of $V$} \bigg\} \end{split} \\ &=\max\bigg\{\sum_{i=1}^s (k+\ell-d_G(X_i)+\eta_{k,0}(G[X_i])) \mid \text{ a sub-partition $\{X_1,\dots, X_s\}$ of $V$} \bigg\}.\end{aligned}$$ Let $g_{k,\ell}(X)=k+\ell-d(X)+\eta_{k,0}(G[X])$ for $X\subseteq V$. Then, we have $$\label{eq:xi} \xi_{k,\ell}(G)=\max\bigg\{\sum_{i=1}^s g_{k,\ell}(X_i)\mid \text{ a sub-partition ${\cal P}=\{X_1,\dots, X_s\}$ of $V$} \bigg\}.$$ We say that $X\subseteq V$ is [deficient]{} if $g_{k,\ell}(X)>0$. By Theorem \[thm:fk03-augment\] and , $g_{k,\ell}(X)\leq 0$ holds for every $X$ with $\emptyset\neq X\subsetneq V$ if $G$ is $(k,\ell)$-ec-orientable. The following theorem is a key result for developing a constant-time tester. \[thm:key2\] For a given $\epsilon$, let $c=\frac{\epsilon^2d^2}{16k\ell}$ and $t=\frac{4\ell}{\epsilon d}$. Suppose that $\xi_{k,\ell}(G)\geq \epsilon dn$. Then, at least one of the followings holds: (i) : There are at least $cn$ disjoint small deficient sets, where a set is called small if the cardinality is less than $t$; (ii) : $\eta_{k,0}(G)\geq \frac{1}{4}\epsilon dn$. Namely, $G$ is $\frac{\epsilon}{2}$-far from $(k,k)$-fullness. Let ${\cal P}=\{X_1,\dots, X_s\}$ be a sub-partition of $V$ that maximizes the right hand side of . Since the maximum is taken over all sub-partitions of $V$, we may assume $g_{k,\ell}(X_i)>0$ for all $1\leq i\leq t$. Let us divide ${\cal P}$ into two subsets ${\cal P}_{\rm small}$ and ${\cal P}_{\rm large}$ depending on whether it is small or not. Notice that for each $X\in {\cal P}_{\rm small}$ we have $g_{k,\ell}(X)=k+\ell-d_G(X)+\eta_{k,0}(G[X])\leq kt$ since $\eta_{k,0}(G[X])\leq k(\|X\|-1)\leq kt-2k$. Suppose that (i) does not happen. Then, by $\xi_{k,\ell}(G)\geq \epsilon dn$ and $\sum_{X\in {\cal P}_{\rm small}} g_{k,\ell}(X)\leq ktcn$, we have $$\label{eq:large_deficient1} \sum_{X\in {\cal P}_{\rm large}}g_{k,\ell}(X)\geq (\epsilon d-ktc)n.$$ We now prove the following relation between $\eta_{k,0}$ and $g_{k,\ell}$, which gives us a lower bound on $\eta_{k,0}(G)$. $$\label{eq:large_deficient2} \eta_{k,0}(G)\geq \frac{1}{2}\sum_{X\in {\cal P}_{\rm large}} (g_{k,\ell}(X)-\ell).$$ Recall that $\eta_{k,0}(G)$ is the number of edges we need to add to make $G$ $(k,k)$-full. Hence, we can take a new graph $H=(V,E_H)$ on $V$ such that $\|E_H\|=\eta_{k,0}(G)$ and $G'=(V,E\cup E_H)$ is $(k,k)$-full. We need the following formulae. \[claim:5\_1\] For any $X\subseteq V$, (a) : $\eta_{k,0}(G'[X])+k\leq d_{G'}(X)$, and (b) : $\eta_{k,0}(G[X])\leq \eta_{k,0}(G'[X])+i_{H}(X)$, where $i_H(X)=\|\{uv\in E_H\mid u,v\in X\}\|$. Let ${\cal P}_X'$ be a partition of $X$ such that $\eta_{k,0}(G'[X])=k(\|{\cal P}_X'\|-1)-d_{G'[X]}({\cal P}_X')$. Let ${\cal F}={\cal P}_X'\cup\{V\setminus X\}$. Then ${\mathcal{F}}$ is a partition of $V$. Since $G'$ is $(k,k)$-full, we have $$\begin{aligned} 0=\eta_{k,0}(G')&\geq k(\|{\mathcal{F}}\|-1)-d_{G'}({\mathcal{F}}) \\ &=k(\|{\mathcal{P}}_X'\|-1)+k-d_{G'}(X)-d_{G'[X]}({\mathcal{P}}_X') \\ &=\eta_{k,0}(G'[X])+k-d_{G'}(X),\end{aligned}$$ implying (a). On the other hand, let ${\cal P}_X$ be a partition of $X$ such that $\eta_{k,0}(G[X])=k(\|{\cal P}_X\|-1)-d_{G[X]}({\cal P}_X)$. Since $d_{G[X]}({\cal P}_X)+i_H(X)\geq d_{G'[X]}({\cal P}_X)$, we have $$\begin{aligned} \eta_{k,0}(G[X])&=k(\|{\mathcal{P}}_X\|-1)-d_{G[X]}({\mathcal{P}}_X) \\ &\leq k(\|{\mathcal{P}}_X\|-1)-d_{G'[X]}({\mathcal{P}}_X)+i_H(X) \\ &\leq \eta_{k,0}(G'[X])+i_H(X).\end{aligned}$$ In total, we have $$\begin{aligned} \sum_{X\in {\mathcal{P}}_{\rm large}}(g_{k,\ell}(X)-\ell)&=\sum_{X\in {\cal P}_{\rm large}}(\eta_{k,0}(G[X])+k-d_G(X)) \\ &\leq \sum_{X\in {\cal P}_{\rm large}}(\eta_{k,0}(G'[X])+i_H(X)+k-d_G(X)) \\ &\leq \sum_{X\in {\cal P}_{\rm large}}(d_{G'}(X)+i_H(X)-d_G(X)) \\ &= \sum_{X\in {\cal P}_{\rm large}}(d_{H}(X)+i_H(X)) \\ &\leq 2\|E_H\|=2\eta_{k,0}(G).\end{aligned}$$ Thus, we obtain . Moreover, since there are at most $\frac{n}{t}$ large sets among ${\cal P}$, implies $$\label{eq:large_deficient3} \eta_{k,0}(G)\geq \sum_{X\in {\cal P}_{\rm large}} \frac{g_{k,\ell}(X)}{2} -\frac{\ell n}{2t}.$$ Combining and , we finally have $\eta_{k,0}(G)\geq \frac{1}{2}(\epsilon d-ktc-\frac{\ell}{t})n=\frac{\epsilon dn}{4}$. This completes the proof. A testing algorithm for the $(k,\ell)$-ec-orientability of a graph $G=(V,E)$ is given in Algorithm 3. In Line 7, $V_t(v)$ denotes the set of vertices whose distances to $v\in V$ are at most $t$. Take any $\epsilon''$ such that $\epsilon''<\epsilon$. Run a $(1,\frac{\epsilon'' dn}{4})$-approximation algorithm for $\eta_{k,0}(G)$. reject $G$. Choose a set $S$ of $\frac{8k\ell}{\epsilon^2d^2}$ vertices uniformly at random from $G$. compute $X_v={\rm argmax}\{g_{k,\ell}(X):X\subseteq V_{t}(v), X\neq \emptyset\}$ with $t=\frac{4\ell}{\epsilon d}$. reject $G$. accept $G$. We prove that Algorithm \[alg:k-l-orientability\] can be implemented with query complexity $(k+d)^{O(1/\epsilon'^2)}(\frac{1}{\epsilon'})^{O(1/\epsilon')}$ where $\epsilon'=\max(\frac{\epsilon }{dk},\frac{d \epsilon }{\ell})$, and correctly tests the $(k,\ell)$-ec-orientability of $G$. For Line 2, we use an approximation algorithm mentioned in Corollary \[crl:approximation-to-eta\] that runs in $(k+d)^{O(1/\epsilon'^2)}(\frac{1}{\epsilon'})^{O(1/\epsilon')}$ queries. For Line 7, we use an algorithm for minimizing $g_{k,\ell}$ on $V_t(v)$. Let $\hat{g}_{k,\ell}:2^{V_t(v)}\rightarrow \mathbb{Z}$ be the function defined by, for each $X\subseteq V_t(v)$, $$\hat{g}_{k,\ell}(X)=\begin{cases} -\infty & \text{if } X=\emptyset \\ g_{k,\ell}(X) & \text{otherwise}. \end{cases}$$ Since $g_{k,\ell}$ is a supermodular function (see Appendix \[sec:supermodularity\]), it is easy to observe that $\hat{g}_{k,\ell}$ is an intersecting supermodular function (i.e., $f(X)+f(Y)\leq f(X\cap Y)+f(X\cup Y)$ holds for any $X,Y\subseteq V_t(v)$ with $X\cap Y\neq \emptyset$). Thus, to perform Line 7, we call a polynomial time algorithm for minimizing an intersecting submodular function (see e.g., [@Schrijver; @fujishige]). By $\|V_t(v)\|\leq d^t$, the query complexity taken in the for-loop is upper bounded by $O(\frac{d^2}{\epsilon'^2})\cdot d^{O(1/\epsilon')}$. Thus, Algorithm \[alg:k-l-orientability\] can be implemented with query complexity $(k+d)^{O(1/\epsilon'^2)}(\frac{1}{\epsilon'})^{O(1/\epsilon')}$. To see the correctness, assume first that $G$ is $(k,\ell)$-ec-orientable. Then, we have $0=\eta_{k,\ell}(G)\geq \eta_{k,0}(G)$ by (\[eq:eta\_relation\]). Since $x^\leq \eta_{k,0}(G)$ with probability at least $\frac{2}{3}$, $x^\leq 0$ holds. Namely, the algorithm does not reject $G$ at Line 4 with probability $\frac{2}{3}$. Also, since $g_{k,\ell}(G)\leq 0$ for every $X$ with $\emptyset\neq X\subsetneq V$ by Theorem \[thm:fk03-augment\], the algorithm never rejects $G$ at Line 9. Thus, Algorithm 3 accepts $G$ with probability at least $\frac{2}{3}$. Conversely, suppose that $G$ is $\epsilon$-far. Then, by Theorem \[thm:fk03-augment\] and (\[eq:eta\_relation\]), $\eta_{k,0}(G)+\ell\geq \eta_{k,\ell}(G)\geq \frac{\epsilon dn}{2}$ or $\xi_{k,\ell}(G)\geq \epsilon dn$ holds. Since $\ell <\frac{\epsilon dn}{4}$ (otherwise $n$ becomes constant and we can apply any existing polynomial time algorithm), $\eta_{k,0}(G)\geq \frac{\epsilon dn}{4}$ or $\xi_{k,\ell}(G)\geq \epsilon dn$ holds. In total, combining this with Theorem \[thm:key2\], $\eta_{k,0}(G)\geq \frac{\epsilon dn}{4}$ holds or $G$ has at least $cn$ disjoint deficient sets of size at most $t$, where $c=\frac{\epsilon^2d^2}{16k\ell}$. If $\eta_{k,0}(G)\geq \frac{\epsilon dn}{4}$, then $x^\geq \eta_{k,0}(G)-\frac{\epsilon''dn}{4}>0$ holds and $G$ is rejected in Line 4; Otherwise, the probability that we choose some vertex in a deficient set of size at most $t$ in Line 5 is at least $$1-\left(\frac{n-cn}{n}\right)^{\frac{2}{c}}\geq 1-\frac{1}{e^2}\geq \frac{2}{3},$$ and Algorithm 3 rejects $G$ with probability at least $\frac{2}{3}$ in Line 9. Linear Lower Bounds for One-Sided Error Testers {#sec:lower-bound} =============================================== In this section, we prove Theorems \[thr:lower-k-l-fullness\] and \[thr:lower-k-l-ec-orientability\]. As for $(k,\ell)$-ec-orientability, in the bounded-degree model, Orenstein [@Ore10] showed a liner lower bound of one-sided error tester for $(k,0)$-ec-orientability where $k \geq 2$. We can easily modify his proof to achieve Theorem \[thr:lower-k-l-ec-orientability\] by using Theorem \[thm:orientability\]. Thus, we omit the detail in this paper. He also showed that there is a one-sided error tester for $(1,0)$-ec-orientability. We cannot extend the lower bound to the case $\ell \geq k$ since, in such a case, $(k,\ell)$-ec-orientability coincides with the $(k+\ell)$-edge-connectivity, and we have one-sided error testers for it [@GR02]. In what follows, we consider lower bounds for testing $(k,\ell)$-fullness. Let $k \geq 2,\ell \geq 0$ be integers with $2k - \ell \geq 1$. We mention that, when $k = 1$, it is easy to make one-sided error testers (see Appendix \[sec:trivial-testers\]). Note that a one-sided error tester cannot reject a graph until it has found an evidence that the graph is not $(k,\ell)$-full, i.e., an $\epsilon$-far graph cannot be $(k,\ell)$-full no matter how we add edges in the unseen part of the graph. With this observation, Orenstein [@Ore10] constructed a graph which is $\epsilon$-far from $(k,k)$-full while if one has seen only $\beta n$ vertices for some constant $\beta$, one can add edges so that the resulting graph is $(k,k)$-full. His construction relies on the Tutte-Nash-Williams tree-packing theorem (Theorem \[thm:tutte\]), which is a special property of $(k,k)$-fullness. We complete the work by showing the existence of such a graph for general $(k,\ell)$-fullness. First, we define a $(\beta,\gamma)$-expander* as a graph $G=(V,E)$ such that for any $S \subseteq V, \|S\| \leq \beta \|V\|$, we have that $\|\Gamma(S)\| \geq \gamma \|S\|$. The following lemma states that such graphs indeed exist. \[lmm:expander\] Let $d \geq 3$ be an integer. Then, there exists a $d$-regular $(\beta,d-2)$-expander graph for some universal constant $\beta > 0$. \[lmm:far\] Let $G$ be the $(2k-1)$-regular expander graph of $n$ vertices given in Lemma \[lmm:expander\]. When $n$ is sufficiently large, $G$ is $\epsilon$-far from $(k,\ell)$-fullness for $\epsilon = O(\frac{1}{k})$. Note that any $(k,\ell)$-full graph must have at least $kn - \ell$ edges. However, $G$ has $\frac{(2k-1)n}{2}$ edges. Thus, to make $G$ $(k,\ell)$-full, we need to add at least $kn - \ell - \frac{(2k-1)n}{2} = \frac{n}{2} - \ell$ edges. Thus, the lemma holds. The following is a well-known graph operation that preserves $(k,\ell)$-fullness. \[lmm:combine\] Let $G = (V,E)$ be a $(k,\ell)$-full graph. We introduce a new vertex $v$ and connect $v$ and distinct $k$ vertices of $V$ by new edges. Then, the resulting graph is also $(k,\ell)$-full. We also need the following graph operation for the case $k=2$. \[lmm:combine2\] Let $G=(V,E)$ be a $(2,\ell)$-full graph with $\|V\|\geq 2$. We introduce a cycle graph $G'=(U,C)$ consisting of new vertices $U=\{u_1,u_2,\dots, u_s\}$ and then connect each new vertex $u_i$ to a vertex $v_i\in V$ so that $v_i\neq v_j$ for some $1\leq i<j\leq s$. Then, the resulting graph is also $(2,\ell)$-full. It is sufficient to consider the case when $G$ is $(2,\ell)$-tight. Let $E_{U,V}=\{u_iv_i\mid 1\leq i\leq s\}$. Note that the total number of edges amounts to $\|E\|+\|C\|+\|E_{U,V}\|=2\|V\|-\ell+\|U\|+\|U\|=2\|V\cup U\|-\ell$. Suppose that the resulting graph is not $(2,\ell)$-tight. Then, there is an edge subset $F$ that violates the counting condition, i.e., $\|F\|>f_{2,\ell}(F)$. We split $F$ into three parts; $F_{U}=F\cap C$, $F_{U,V}=F\cap E_{U,V}$ and $F_V=F\cap E$. Since $G'$ is a cycle, we have $\|F_U\|\leq \|V_{G'}(F)\|$. Also, each vertex $u_i\in U$ is incident to only one vertex in $V$, $\|F_{U,V}\|\leq \|V_{G'}(F)\|$. Thus, if $F_V\neq \emptyset$, we have $\|F\| = \|F_U\| + \|F_{U,V}\| + \|F_V\| \leq \|V_{G'}(F)\| + \|V_{G'}(F)\| + 2\|V_G(F)\| - \ell = 2\|V_{G'}(F)\cup V_G(F)\| - \ell=f_{2,\ell}(F)$. Therefore, $F_V=\emptyset$ must hold, but a simple counting argument shows that any subset of $C\cup E_{U,V}$ cannot violate the counting condition, which is a contradiction. Let $G = (V,E_G)$ be the $(2k-1)$-regular $(\beta,2k-3)$-expander graph of $n$ vertices given in Lemma \[lmm:expander\]. From Lemma \[lmm:far\], $G$ is $O(\frac{1}{k})$-far from $(k,\ell)$-fullness. Suppose that an algorithm ${\mathcal{A}}$ has queried $\beta n$ times. and let $V_{{\mathcal{A}}} \subseteq V$ be the set of vertices involved with those queries. That is, for every $v \in V_{{\mathcal{A}}}$, there was a query of the form ${\mathcal{O}}_G(v,i)$ for some $i \in [d]$, or ${\mathcal{O}}_G$ returns an edge incident to $v$. Clearly, $\|V_{{\mathcal{A}}}\| \leq \beta n$ holds. Let $S = V \setminus V_{{\mathcal{A}}}$. We take any $(k,\ell)$-full graph $H=(S,E_H)$ on $S$ using new edges. Then, we consider the graph $G' = (V, E_G \cup E_H)$. We show that $G'$ is $(k,\ell)$-full. This means that any algorithm cannot reject $G$ just by seeing $\beta n$ edges. We know that $G'[S]$ is $(k,\ell)$-full since $H$ is $(k,\ell)$-full. To show that entire $G'$ is $(k,\ell)$-full, we iteratively enlarge $S$ keeping that $G'[S]$ is $(k,\ell)$-full. Let $\overline{S}=V\setminus S$. We have the following two cases. - If there exists a vertex $v \in \overline{S}$ such that $\|\Gamma(v) \cap S\| \geq k$, then $G'[S+v]$ is also $(k,\ell)$-full by Lemma \[lmm:combine\]. Thus, we replace $S$ by $S + v$. - If every vertex $v \in \overline{S}$ satisfies $\|\Gamma(v) \cap S\| < k$, then we have $d_{G'}(v,S) < k$ for any $v \in \overline{S}$. Since $G'$ contains a $(\beta,2k-3)$-expander, we have that $d_{G'}(\overline{S}) \geq (2k-3)\|\overline{S}\|$. However, the assumption implies that $d_{G'}(\overline{S}) = \sum_{v \in \overline{S}}d_{G'}(v,S) < k\|\overline{S}\|$. Combining those inequalities, we have $k = 2$. Furthermore, by $d_{G'}(v,S)<k$, we have $d_{G'}(v,S) = 1$ for every $v \in \overline{S}$. Note that the degree of any vertex $v \in \overline{S}$ is $2k-1 = 3$. This means that $G'[\overline{S}]$ consists of disjoint cycles and we have an edge from each vertex $v \in \overline{S}$ to $S$. Let us take such a cycle and let $U$ be the vertex set of this cycle. Then, by $\|\Gamma(U)\| \geq \|U\|$, we can apply Lemma \[lmm:combine\] to claim that $G'[S\cup U]$ is $(2,\ell)$-full. Thus, we replace $S$ by $S\cup U$. For any of those two cases, we can enlarge $S$ until $S$ becomes $V$. Thus, the theorem holds. Concluding Remarks ================== The concept of $(k,\ell)$-sparsity can be generalized as follows. For a hypergraph $H=(V,{\cal E})$, let $\mathbf{k}:V\rightarrow \mathbb{Z}_+$ and $\ell\in \mathbb{Z}_+$. We define a function $f_{\mathbf{k},\ell}:2^{\cal E}\rightarrow \mathbb{Z}_+$ by $f_{\mathbf{k},\ell}({\cal E}')=\sum_{v\in V({\cal E}')}\mathbf{k}(v)-\ell$ for ${\cal E}'\subseteq {\cal E}$, where $V({\cal E}')=\bigcup_{X\in {\cal E}'}X$. It is easy to see that $f_{\mathbf{k},\ell}$ is non-decreasing and submodular, and thus $f_{\mathbf{k},\ell}$ induces a matroid, ${\cal M}_{\mathbf{k},\ell}(H)$, on ${\cal E}$. It is easy to generalize our approximation algorithm to that for the rank of ${\cal M}_{\mathbf{k},\ell}(H)$ by just modifying the auxiliary graph $G_k$ defined in Section \[sec:matching\]. Our tester for the $(k,\ell)$-fullness of a graph $G$ approximates the rank of ${\mathcal{M}}_{k,\ell}(G)$. It might be interesting to know for which matroid we can approximate the rank of it in constant time. In particular, can we approximate the rank of a matrix with entries in $\mathbb{F}_2$? We note that the $(k,k)$-fullness of a graph can be decided by checking the rank of the union of $k$ graphic matroids. This problem is usually solved via a matroid intersection problem. This leaves us questions: for which matroids can we approximate the rank of their union, and for which matroids ${\mathcal{M}}_1,{\mathcal{M}}_2$ can we approximate the size of the largest common independent set in ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ in constant queries? One-Sided Error Testers for $(1,0)$-Fullness and $(1,1)$-Fullness {#sec:trivial-testers} ================================================================= In this section, we give one-sided error testers for $(1,0)$-fullness and $(1,1)$-fullness. \[lmm:far-from-(1,0)-full\] Let $G$ be a graph $\epsilon$-far from $(1,0)$-fullness. Then, there are at least $\frac{\epsilon dn}{4}$ connected components of size at most $\frac{4}{\epsilon d}$ containing no cycle. Note that a graph is $(1,0)$-full iff each connected component in the graph contains a cycle. Thus, there are at least $\frac{\epsilon dn}{2}$ connected components containing no cycle in $G$. Then, it is easy to observe that the lemma holds. \[lmm:far-from-(1,1)-full\] Let $G$ be a graph $\epsilon$-far from $(1,1)$-fullness. Then, there are at least $\frac{\epsilon dn}{4}$ connected components of size at most $\frac{4}{\epsilon d}$. Note that a graph is $(1,1)$-full iff the graph is connected. Thus, there are at least $\frac{\epsilon dn}{2}$ connected components in $G$. Then, it is easy to observe that the lemma holds. In the bounded-degree model with a degree bound $d$, There are one-sided error testers for $(1,0)$-fullness and $(1,1)$-fullness with query complexity $O(\frac{1}{\epsilon^2 d})$. We describe the algorithm for $(1,1)$-fullness. Let $S$ be a set of $\frac{8}{\epsilon d} $ vertices chosen uniformly at random from an input graph. For each chosen vertex, we perform BFS from the vertex until we reach $\frac{4}{\epsilon d} + 1$ vertices. If the BFS cannot reach $\frac{4}{\epsilon d} + 1$ vertices for some vertex in $S$, we reject the graph. Otherwise, we accept the graph. Clearly, the query complexity of the algorithm is at most $\frac{8}{\epsilon d} \cdot (\frac{4}{\epsilon d}+1) \cdot d = O(\frac{1}{\epsilon^2 d})$. Since $(1,1)$-full graph is connected, we can reach $n$ vertices from any vertex. Thus, the algorithm always accepts $(1,1)$-full graph. Suppose that a graph is $\epsilon$-far from $(1,1)$-fullness. From Lemma \[lmm:far-from-(1,1)-full\], the probability that we choose some vertex in a connected component of size at most $\frac{4}{\epsilon d}$ is at least $$\begin{aligned} 1 - \left(1 - \frac{\epsilon dn}{4n}\right)^{\frac{8}{\epsilon d}} \geq 1 - \frac{1}{e^{2}} \geq \frac{2}{3}. \end{aligned}$$ For such vertex, the BFS cannot reach $\frac{4}{\epsilon d}$ vertices. Thus, the algorithm rejects a graph $\epsilon$-far from $(1,1)$-fullness with probability at least $\frac{2}{3}$. We can construct a tester for $(1,0)$-fullness in a similar way using Lemma \[lmm:far-from-(1,0)-full\]. Supermodularity of $g_{k,\ell}$ {#sec:supermodularity} =============================== For each $X\subseteq V$, we have $$\begin{aligned} g_{k,\ell}(X)&=k+\ell-d_G(X)+\eta_{k,0}(G[X_i]) \\ &=k+\ell-\frac{d_G(X)}{2}+\max\left\{\sum_{j=1}^s\left(k-\frac{d_G(X_j)}{2}\right)\mid \text{ a partition } \{X_1,\dots, X_s\} \text{ of } X \right\} \\ &=k+\ell-\frac{d_G(X)}{2}+\max\left\{\sum_{j=1}^s h(X_j)\mid \text{ a partition } \{X_1,\dots, X_s\} \text{ of } X \right\},\end{aligned}$$ where $h:2^{V}\rightarrow \mathbb{Z}$ is defined as $h(X):=k-\frac{d_G(X)}{2}$ for $X\subseteq V$. Note that $h$ is a supermodular function and $\hat{h}(X):=\max\left\{\sum_{j=1}^s h(X_j)\mid \text{ a partition } \{X_1,\dots, X_s\} \text{ of } X \right\}$ is the so-called Dilworth truncation of $h$, which is known to be supermodular again (see e.g., [@Schrijver Chapter 48]). Since $d_G$ is submodular, $g_{k,\ell}$ is a supermodular. [^1]: School of Informatics, Kyoto University. `itohiro@kuis.kyoto-u.ac.jp` [^2]: Research Institute for Mathematical Sciences, Kyoto University. Supported by Grant-in-Aid for JSPS Research Fellowships for Young Scientists. `tanigawa@kurims.kyoto-u.ac.jp` [^3]: School of Informatics, Kyoto University, and Preferred Infrastructure, Inc. Supported by MSRA Fellowship 2010. `yyoshida@kuis.kyoto-u.ac.jp`
--- abstract: 'We present modified versions of existing criteria for the density property and the volume density property of complex manifolds. We apply this methods to show the (volume) density property for a family of manifolds given by $x^2y=a(\bar z) + xb(\bar z)$ with $\bar z =(z_0,\ldots,z_n)\in\C^{n+1}$ and volume form $\d x/x^2\wedge \d z_0\wedge\ldots\wedge\d z_n$. The key step is showing that in certain cases transitivity of the action of (volume preserving) holomorphic automorphisms implies the (volume) density property, and then giving sufficient conditions for the transitivity of this action. In particular, we show that the Koras-Russell Cubic Threefold $\lbrace x^2y + x + z_0^2 + z_1^3=0\rbrace$ has the density property and the volume density property.' address: 'Mathematisches Institut, Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland.' author: - Matthias Leuenberger title: '(Volume) Density Property of a family of complex manifolds including the Koras-Russell Cubic' --- Introduction ============ The density property and the volume density property is a property of Stein manifolds with a huge amount of applications in complex geometry in several variables. It was introduced by Varolin in [@varo-dens]. The fact that $\C^n$ has the density property for $n\geq2$ was already used by Andersén and Lempert in [@AL] where they showed that holomorphic automorphisms can be approximated by some special family of automorphisms (called shear automorphisms). Varolin realized that the main observation of Andersén and Lempert may be formalized and can be applied to more general complex manifolds and different problems. Rosay and Forstnerič contributed also a lot to this progress in [@FR]. This area of complex analysis in several variables is nowadays called Andersén-Lempert theory. The numerous applications of the density property are due to the Main Theorem of Andersén-Lempert theory which states that on manifolds with density property any local phase flow on a Runge domain can be approximated uniformly on compacts by global automorphisms. The analogous statement holds in the volume preserving case. For a deeper view into this topic we refer to the comprehensive texts [@kaku-state; @kut; @fo]. Let $X$ be a Stein manifold. If the Lie algebra $\lie(\chvf(X))$ generated by complete (= globally integrable) holomorphic vector fields $\chvf(X)$ on $X$ is dense (in compact-open topology) in the Lie algebra of all holomorphic vector fields $\hvf(X)$ on $X$ then $X$ has the density property. Let $X$ be a Stein manifold equipped with a holomorphic volume form $\omega$. If the Lie algebra $\lie(\chvfw(X))$ generated by complete volume preserving (= vanishing $\omega$-divergence) holomorphic vector fields $\chvfw(X)$ on $X$ is dense in the Lie algebra of all volume preserving holomorphic vector fields $\hvfw(X)$ on $X$ then $X$ has the volume density property. Recall that a vector field $\nu$ is called volume preserving if the Lie derivative $L_\nu \omega$ vanishes where the Lie derivative is given by the formula $L_\nu = \d i_\nu + i_\nu \d$ and $i_\nu$ is the interior product of a form with $\nu$. In Section \[sec-crit\] we present a criterion that is showing (volume) density property. For the definition of (semi-)compatible pairs and ($\omega$-)generating sets see Definitions \[def-scomp\], \[def-comp\] and \[def-generate\]. For a vector field $\nu$ and a point $p\in X$ we denote by $\nu [p] \in\T_pX$ the tangent vector of $\nu$ at $p$. \[criterion\] (1) Let $X$ be a Stein manifold such that the holomorphic automorphisms $\aut(X)$ act transitively on $X$. If there are compatible pairs $(\nu_i,\mu_i)$ such that there is a point $p\in X$ where the vectors $\mu_i[p]$ form a generating set of $\mathrm{T}_pX$ then $X$ has the density property. \(2) Let $X$ be a Stein manifold with a holomorphic volume form $\omega$ such that the volume preserving holomorphic automorphisms $\autw(X)$ act transitively on $X$ and $\mathrm{H}^{n-1}(X,\C)=0$ (where $n=\dim X$). If there are semi-compatible pairs $(\nu_i,\mu_i)$ of volume preserving vector fields such that there is a point $p\in X$ where the vectors $\nu_i[p]\wedge\mu_i[p]$ form a generating set of $\mathrm{T}_pX\wedge\mathrm{T}_pX$ then $X$ has the volume density property. Note that this criterion and its proof is very much inspired by the criteria in [@kaku-density; @kaku-volume2] for the algebraic (volume) density property (see Definition \[def-adp\]). Actually, e.g. for (1), the only difference is that instead of requiring the algebraic automorphisms to act transitively on $X$ we require the holomorphic automorphisms to act transitively. In Section \[sec-trans\] we investigate the transitivity of the action by (volume preserving) holomorphic automorphisms $\aut(X)$ (resp. $\autw(X)$) where $X$ is given by $x^2y=a(\bar z) + xb(\bar z)$ with $\bar z =(z_0,\ldots,z_n)\in\C^{n+1}$ for some $n\geq 0$, $\deg_{z_0}(a)\leq 2$ and $\deg_{z_0}(b)\leq 1$. We show that (after possibly reordering the $z_i$) the condition 1. There is some $k\geq0$ such that $\deg_{z_i}(a)\leq 2$ and $\deg_{z_i}(b)\leq 1$ for all $i\leq k$ and for all common zeroes $\bar q = (q_0,\ldots,q_n)$ of $a,\frac{\partial a}{\partial z_0},\ldots,\frac{\partial a}{\partial z_k}$, we have $b(\bar q)\neq 0$, and there is some $j\leq k$ such that $\frac{\partial a}{\partial z_j}$ does not vanish along the curve $\lbrace z_i=q_i \text{ for all } i\neq j \rbrace \subset \C^{n+1}$. is a sufficient condition for $\aut(X)$ to act transitively on $X$. If additionally 1. There is some $k\geq0$ such that $\deg_{z_i}(a)\leq 2$ and $\deg_{z_i}(b)\leq 1$ for all $i\leq k$ and there is no $c\in\C^$ for which the polynomials $\frac{\partial a}{\partial{z_i}} + c\frac{\partial b}{\partial{z_i}}$ for $i\leq k$ are all constant to zero. holds, then also $\autw(X)$ acts transitively (Proposition \[prop-trans\]). In Section \[sec-pos\] we apply Theorem \[criterion\] to these kind of surfaces for $n>0$. This leads to our Main Theorem. Let $n\geq 0$ and $a,b\in\C[z_0,\ldots,z_n]$ such that $\deg_{z_0}(a)\leq 2$, $\deg_{z_0}(b)\leq 1$ and not both of $\deg_{z_0}(a)$ and $\deg_{z_0}(b)$ are equal to zero. Let $\bar z = (z_0,\ldots z_n)$. Then the hypersurface $X=\lbrace x^2y=a(\bar z) + xb(\bar z)\rbrace$ has the density property provided that the holomorphic automorphisms $\aut(X)$ act transitively on $X$. In particular $X$ has the density property if (A) holds or if $n=0$. Moreover, if $\H^{n+1}(X,\C)=0$ and the volume preserving holomorphic automorphisms $\autw(X)$ act transitively on $X$ then $X$ has the volume density property for the volume form $\omega=\d x/x^2\wedge \d z_0\wedge\ldots\wedge \d z_n$. In particular the transitivity condition holds if (A) and (B) hold or if $n=0$. The proof of the Main Theorem is finished in Section \[sec-zero\] where the case $n=0$ is done by explicit calculations, not using the methods described in Section 2. It is worth pointing out that the Main Theorem together with Corollary \[cor-koras\] implies that the Koras-Russell Cubic Threefold $C = \lbrace x^2y + x + z_0^2 + z_1^3=0\rbrace$ has the (volume) density property. The Threefold $C$ is a famous example of a affine variety which is diffeomorphic to $\R^6$ but not algebraically isomorphic to $\C^3$, e.g. see [@ma]. As an affine algebraic variety $C$ (in particular the algebraic automorphism group of $C$) is well understood, e.g. see [@dumopo]. For example it is know that the algebraic automorphisms does not act transitively on $C$. The density property implies that the situation in the holomorphic context is completely different. However, it is still unclear if $C$ is biholomorphic to $\C^3$. Related to this question there is a conjecture of Tóth and Varolin. The conjecture [@va] states that a manifold which has the density property and which is diffeomorphic to $\C^n$ is automatically biholomorphic to $\C^n$. If the conjecture holds then the Main theorem would imply that $C$ is isomorphic to $\C^3$. Proof of Theorem \[criterion\] {#sec-crit} ============================== Let $X$ be a Stein manifold of dimension $n$, and let $\O_X$ be the sheaf of holomorphic functions on $X$. Preliminaries ------------- Let $\F$ be a coherent sheaf of $\O_X$-modules, and let $s_1,\ldots,s_N \in \F(X)$ be global sections. The following lemmas are standard applications of sheaf theory. \[lem-pre1\] Let $p\in X$, and let $\mu_p\subset\O_X(X)$ be corresponding ideal. If the elements $s_i + \mu_p\F(X)$ span the vector space $\F(X)/\mu_p\F(X)$ then the localizations $(s_i)_p$ generate the stalk $\F_p$. Let $\G_p$ be the $(\O_X)_p$-submodule of $\F_p$ generated by $(s_1)_p,\ldots, (s_N)_p$. The assumption implies that we have $(\G_p + \mu_p\F_p)/\mu_p\F_p= \F_p/\mu_p\F_p$. Let $\L_p = \F_p/\G_p$ then a short calculation using the isomorphism theorems for modules shows that we have $\L_p / \mu_p\L_p = 0$. The ring $(\O_X)_p$ is local, so by the Nakayama Lemma we may lift a basis of $\L_p/\mu_p\L_p$ to a generating set of $\L_p$, and thus $\L_p = 0$. This yields $\G_p=\F_p$ which shows the claim. \[lem-pre2\] If the elements $(s_i)_p$ generate the stalks $\F_p$ for all points $p\in X$ then every global section $\nu\in\F(X)$ is of the form $\sum f_i s_i$ for some global holomorphic functions $f_i\in \O_X(X)$. Consider the morphisms of sheaves $\varphi: \O_X^N \rightarrow \F$ given by $(f_i)\mapsto \sum f_i s_i$. By assumption $\varphi$ is surjective on the level of stalk. Therefore we get the following short exact sequence of coherent sheaves: $$0 \rightarrow \ker \varphi \rightarrow \O_X^N \rightarrow \F \rightarrow 0.$$ This yields the following long exact sequence: $$\cdots\rightarrow \H^0(X,\O_X^N) \rightarrow \H^0(X,\F)\rightarrow \H^1(X,\ker \varphi)\rightarrow \cdots.$$ By Theorem B of Cartan we have $\H^1(X,\ker\varphi)=0$. Thus the left map is surjective, and therefore every global section $\nu \in \H^0(X,\F)=\F(X)$ is of the desired form. Recall that a domain $Y\subset X$ is called Runge if all holomorphic function on $Y$ can be approximated uniformly on compacts $K\subset Y$ by global holomorphic functions on $X$. \[lem-pre3\] Let $Y\subset X$ be a domain of $X$ which is Runge and Stein. If the elements $(s_i)_p$ generate the stalks $\F_p$ for all points $p\in Y$ then every global section $\nu\in\F(X)$ can be uniformly approximated on compacts $K\subset Y$ by global sections of the form $\sum f_i s_i$ for some global holomorphic functions $f_i\in\O_X(X)$. Let $\nu\vert_Y\in\F(Y)$ be the restriction of $\nu$ to $Y$. By Lemma \[lem-pre2\] we have $\nu\vert_Y = \sum g_i s_i\vert_Y$ for some holomorphic functions $g_i\in\O_X(Y)$ on $Y$. Since $Y$ is a Runge domain we may approximate the functions $g_i$ by global functions $f_i\in\O_X(X)$ uniformly on compacts $K\subset Y$. Thus the global section $\nu$ can be approximated by sections $\sum f_is_i$ uniformly on compacts $K\subset Y$. Criterion for (volume) density property --------------------------------------- The following definition is due to [@kaku-density], but adapted to the holomorphic case. \[def-generate\] Let $p\in X$. A set $U\subset \T_pX$ is called generating set for $\T_pX$ if the orbit of $U$ of the induced action of the stablilizer $\aut(X)_p$ contains a basis of $\T_pX$. If $X$ has a volume form $\omega$ then a set $U\subset \T_pX\wedge\T_pX$ is called $\omega$-generating set for $\T_pX\wedge \T_pX$ if the orbit of $U$ of the induced action of the stabilizer $\autw(X)_p$ contains a basis of $\T_pX\wedge\T_pX$. The next proposition a powerful criterion for the density property. It is a generalization of Theorem 2 in [@kaku-density] and the proof is similar. \[prop-crit\] Let $X$ be a Stein manifold such that $\aut(X)$ acts transitively on $X$. Assume that there are complete vector fields $\nu_1,\ldots,\nu_N \in \chvf(X)$ which generate a submodule that is contained in the closure of $\lie(\chvf(X))$ and assume that there is a point $p\in X$ such that the tangent vectors $\nu_i[p]\in \mathrm{T}_pX$ are a generating set for the tangent space $\mathrm{T}_pX$. Then $X$ has the density property. We may assume that the vectors $\nu_i[p]$ contain a basis of $\T_pX$. Indeed, the vectors $\nu_i[p]$ are a generating set of $\T_pX$. Thus after adding some pull backs of some vector fields $\nu_i$ by automorphisms in $\aut(X)_p$ we get the desired basis of $\T_pX$. Let $A\varsubsetneq X$ be the analytic subset of points where the vectors $\nu_i[a]$ do not span the whole tangent space $\T_aX$. Let $K\subset X$ be a compact set. After replacing $K$ by its $\O_X$-convex hull we may assume that $K$ is $\O_X$-convex. Let $Y$ be a neighborhood of $K$ which is Stein and Runge, and moreover such that the closure of $Y$ is compact. See e.g. the beginning of the proof of Theorem 7 in [@ty] for the existence of such a $Y$. After adding finitely many complete vector fields to $\nu_1,\ldots,\nu_N$ we get that $Y\cap A = \emptyset$. Indeed, since the closure of $Y$ is compact, $Y \cap A$ is a finite union of irreducible analytic subsets. Let $A_0\subset A$ be an irreducible component of maximal dimension. Pick any $a\in A_0$ and $\phi\in\aut(X)$ such that $\phi(a)\in Y\setminus A$. Since the vectors $\nu_i[\phi(a)]$ span the tangent space $\T_{\phi(a)}X$ the vectors $(\phi^\nu_i)[a]$ span the tangent space $\T_aX$. Thus after adding some of the pull backs to $\nu_1,\ldots,\nu_N$ the component $A_0 \cap Y$ is replaced by finitely many components of lower dimension. Repeating the same procedure, inductively we get after finitely steps a list of complete vector fields $\nu_1,\ldots,\nu_N$ such that $A\cap Y =\emptyset$. Let $\F$ be the coherent sheaf corresponding to the tangent bundle. The fact that the vectors $\nu_i[a]$ span $\T_aX$ for all $a\in Y$ translates to the fact the elements $\nu_i + \mu_a\F$ span the vector space $\F/\mu_a\F$ for all $a\in Y$, where $\mu_a$ is the maximal ideal of $a$. Thus by Lemma \[lem-pre1\] the assumption of Lemma \[lem-pre3\] holds. Therefore ever vector field on $X$ can be approximated uniformly on $K$ by elements of the form $\sum f_i \nu_i$ for some regular functions $f_i \in \O_X(X)$. By assumption the submodule generated by $\nu_1,\ldots,\nu_N$ is contained in the closure of $\lie(\chvf(X))$ (note that this property still holds after enlarging the list $\nu_1,\ldots,\nu_N$ in the procedure above). Therefore every holomorphic vector field is in the closure of $\lie(\chvf(X))$. For the volume case the proof can be found in [@kakuconference]. For completeness indicate a proof here. We introduce the following isomorphisms. The article [@kaku-volume2] is a good reference for this methods. Note the reference is written only for the algebraic case, however all isomorphisms exist identically in the holomorphic case. Let $\omega$ be a holomorphic volume form. For $0\leq i\leq n$ let $\sC_i(X)$ be the vector space of holomorphic $i$-forms on $X$. Moreover let $\Z_i(X)\subset\sC_i(X)$ be the vector space of closed $i$-forms, and let $\B_i(X)\subset\Z_i(X)$ be the vector space of exact $i$-forms. Then we have the isomorphism: $$\Phi: \hvf(X) \ \tilde\rightarrow \ \sC_{n-1}(X), \quad \nu \mapsto i_v\omega,$$ and in the same spirit we also have the isomorphism $\Psi$ induced by: $$\Psi: \hvf(X)\wedge\hvf(X) \ \tilde\rightarrow \ \sC_{n-2}(X), \quad \nu\wedge\mu \mapsto i_\nu i_\mu \omega.$$ By definition $\hvfw(X)$ consists of those vector fields $\nu$ such that $L_\nu\omega=\d i_\nu\omega = 0$ holds and thus the isomorphism $\Phi$ restricts to an isomorphisms $$\Theta = \Phi\vert_{\hvfw(X)}: \hvfw(X) \ \tilde\rightarrow \ \Z_{n-1}(X).$$ Moreover, we consider the outer differential $$D: \sC_{n-2}(X)\twoheadrightarrow \B_{n-1}(X)$$ of $(n-2)$-forms. \[lem-3.1\] Let $\alpha,\beta\in\hvfw(X)$. Then $i_{[\alpha,\beta]}\omega = \d i_\alpha i_\beta \omega$. Proposition 3.1 in [@kaku-volume2]. The next proposition is in some sense the version Proposition \[prop-crit\] in the volume preserving case. It is an adaption of Theorem 1 in [@kaku-volume2]. \[prop-critw\] Let $X$ be a Stein manifold such that $\autw(X)$ acts transitively on $X$. Assume that every class of $\H^{n-1}(X,\C)=\Z_{n-1}(X)/\B_{n-1}(X)$ contains an element of $\Theta(\lie(\chvfw(X)))$. Moreover, assume that there are complete vector fields $\nu_1,\ldots,\nu_N \in \chvfw(X)$ and $\mu_1,\ldots,\mu_N\in\chvfw(X)$ such that the submodule of $\hvf(X)\wedge\hvf(X)$ generated by the elements $\nu_j\wedge\mu_j$ is contained in the closure of $\lie(\chvfw(X))\wedge\lie(\chvfw(X))$, and assume that there is a point $p\in X$ such that the vectors $\nu_j[p]\wedge\mu_j[p]$ are a $\omega$-generating set for the vector space $\mathrm{T}_pX\wedge\mathrm{T}_pX$. Then $X$ has the volume density property. Let $K\subset X$ be a compact set. By the identical arguments as in the proof of Proposition \[prop-crit\] we see that every element of $\hvf(X)\wedge\hvf(X)$ may be uniformly approximated on $K$ by elements contained in $\lie(\chvfw(X))\wedge\lie(\chvfw(X))$. Let $\zeta\in\hvfw(X)$. By the first assumption we may, after subtracting an element of $\Theta(\lie(\chvfw(X)))$, assume that $\Theta(\zeta)\in\B_{n-1}(X)$. Thus $\Theta(\zeta) = D(\Psi(\gamma))$ for some $\gamma \in \hvf(X)\wedge\hvf(X)$. Let us approximate $\gamma$ uniformly on $K$ by elements of the form $\sum \alpha_i\wedge \beta_i \in \lie(\chvfw(X))\wedge\lie(\chvfw(X))$ with $\alpha_i,\beta_i \in \lie(\chvfw(X))$. We use Lemma \[lem-3.1\] to see that $$D (\Psi(\sum\alpha_i\wedge \beta_i)) = \sum \d i_{\alpha_i} i_{\beta_i} \omega = \sum i_{[\alpha_i,\beta_i]} \omega\in \Theta(\lie(\chvfw(X))).$$ Since $\Theta(\zeta)=i_\zeta\omega$ can be approximated uniformly on $K$ by elements of the form $\sum i_{[\alpha_i,\beta_i]} \omega = \Theta(\sum [\alpha_i,\beta_i])$ the vector field $\zeta$ can be approximated uniformly on $K$ by elements of the form $\sum [\alpha_i,\beta_i] \in \lie(\chvfw(X))$ which proves the proposition. Semi-compatible and compatible pairs ------------------------------------ This section provides a way to find the submodules which are required in Propositions \[prop-crit\] and \[prop-critw\]. The parts \[def-scomp\] - \[lem-comp\] are e.g. from [@kaku-density] and [@kaku-volume2] and are here adapted to the holomorphic case. For a vector field $\nu$ and a regular function $f$ we denote by $\nu(f)$ the regular function which is obtained by applying $\nu$ as a derivation and $\ker \nu$ is the kernel of this linear map. \[def-scomp\] A semi-compatible pair is a pair $(\nu,\mu)$ of complete vector fields such that the closure of the linear span of the product of the kernels $\ker\nu \cdot\ker\mu$ contains a non-trivial ideal $I \subset\O_X(X)$. We call $I$ the ideal of $(\nu,\mu)$. \[lem-scomp\] Let $(\nu,\mu)$ be a semi-compatible pair of volume preserving vector fields and $I$ be its ideal. Then the submodule of $\hvf(X)\wedge\hvf(X)$ given by $I\cdot(\nu\wedge\mu)$ is contained in the closure of $\lie(\chvfw(X))\wedge\lie(\chvfw(X))$ Let $\tau=(\sum f_ig_i)\cdot(\nu\wedge\mu)$ with $f_i\in\ker\nu$ and $g_i\in\ker\mu$ be an arbitrary element of $\mathrm{span}\lbrace\ker\nu \cdot\ker\mu\rbrace\cdot(\nu\wedge\mu)$. Since $f_i\in\ker\nu$ we have $f_i\nu\in\chvfw(X)$ for all $i$ and similarly $g_i\nu\in\chvfw(X)$ for all $i$. Thus $\tau=\sum f_i\nu\wedge g_i\mu\in\lie(\chvfw(X))\wedge\lie(\chvfw(X))$. Therefore the closure of $\mathrm{span}\lbrace\ker\nu \cdot\ker\mu\rbrace\cdot(\nu\wedge\mu)$ is contained in the closure of $\lie(\chvfw(X))\wedge\lie(\chvfw(X))$, and the claim follows. \[def-comp\] A semi-compatible pair $(\nu,\mu)$ is called a compatible pair if there is an holomorphic function $h\in\O_X(X)$ with $\nu(h)\in\ker\nu\setminus 0$ and $h\in\ker\mu$. Note that this condition in particular implies that $h\nu$ is a complete vector field. \[lem-comp\] Let $(\nu,\mu)$ be a compatible pair, and let $I$ be its ideal and $h$ its function. Then the submodule of $\hvf(X)$ given by $I\cdot \nu(h)\cdot \mu$ is contained in the closure of $\lie(\chvf(X))$. Let $f\in\ker\nu$ and $g\in\ker\mu$, then $f\nu,fh\nu,g\mu,gh\nu\in\chvf(X)$. A standard calculation shows $$[f\nu,gh\mu]-[fh\nu,g\mu]=fg\nu(h)\mu\in\lie(\chvf(X)).$$ Thus an arbitrary element $\sum (f_i g_i)\nu(h)\mu\in \mathrm{span}\lbrace\ker\nu \cdot\ker\mu\rbrace\cdot\nu(h)\cdot\mu$ with $f_i\in\ker\nu$ and $g_i\in\ker\mu$ is contained in $\lie(\chvf(X))$ which concludes the proof. (1) Let $I_i$ be the ideals and $h_i$ the functions of the pairs $(\nu_i,\mu_i)$ and pick any non-trivial $f_i\in I_i\cdot\nu_i(h_i)$ for every $i$. Since the set of points $p\in X$ where the vector fields $\mu_i[p]$ are a generating set is open and non-empty there is some $q\in X$ where the vector fields $f_i(q)\mu_i[q]$ are a generating set for $\T_qX$. By Lemma \[lem-comp\] the module generated by the vector fields $f_i\mu_i$ is contained in the closure of $\lie(\chvf(X))$ and thus by Proposition \[prop-crit\] the manifold $X$ has the density property. (2) Process as in (1): Let $I_i$ be the ideals of the pairs $(\nu_i,\mu_i)$ and pick any non-trivial $f_i\in I_i$ for every $i$. Since the set of points $p\in X$ where the elements $\nu_i[p]\wedge\mu_i[p]$ are an $\omega$-generating set is open and non-empty there is a $q\in X$ where the vector fields $f_i(q)\cdot(\nu_i[q]\wedge\mu_i[q])$ are an $\omega$-generating set for $\T_qX\wedge\T_qX$. By Lemma \[lem-scomp\] the module generated by the elements $f_i\cdot(\nu_i\wedge\mu_i)$ is contained in the closure of $\lie(\chvfw(X))\wedge\lie(\chvfw(X))$. Thus by Proposition \[prop-critw\] the manifold $X$ has the volume density property (the first condition of Proposition \[prop-critw\] on the cohomology group is trivially fulfilled since $\H^{n-1}(X,\C)=0$ by assumption). We conclude this section by two remarks. These two remarks are just for general information and are not used later in this article. Clearly Theorem \[criterion\](2) still holds if we have that every class of $\H^{n-1}(X,\C)$ contains an element of $\Theta(\lie(\chvfw(X)))$ as in Proposition \[prop-critw\] instead of $\H^{n-1}(X,\C)=0$. Also, note that this condition is equivalent to the condition that every class of $\H^{n-1}(X,\C)$ contains an element of $\Theta(\chvfw(X))$. Indeed, by Lemma \[lem-3.1\] all Lie brackets represent the trivial class of $\H^{n-1}(X,\C)$. There is another class of compatible pairs. Sometimes a semi-compatible pair $(\nu,\mu)$ is also called compatible if there exists a function $h\in\O_X(X)$ with $\nu(h)\in\ker\nu\setminus 0$ and $\mu(h)\in\ker\mu\setminus 0$. For this version the identity $[f\nu,gh\mu]-[fh\nu,g\mu]=fg(\nu(h)\mu-\mu(h)\nu)$ implies that there would be a version of Theorem \[criterion\] where we allow compatible pairs of this kind such that the vectors $\nu(h)\mu - \mu(h)\nu$ take part in constructing the generating sets. Transitivity of the $\aut(X)$- and $\autw(X)$-action {#sec-trans} ==================================================== Let $n\geq k\geq 0$ and $a,b\in\C[z_0,\ldots,z_n]$ such that $\deg_{z_i}(a)\leq2$ and $\deg_{z_i}(b)\leq 1$ for all $i\leq k$. Let $\bar z = (z_0,\ldots z_n)$ and $X=\lbrace x^2y=a(\bar z) + xb(\bar z)\rbrace$ with the holomorphic volume form $\omega = \d x/x^2\wedge\d z_0\wedge\ldots\wedge \d z_n$[^1]. Consider the following vector fields on $X$: $$v_x^i = \left(\frac{\partial a}{\partial z_i}+x\frac{\partial b}{\partial z_i}\right)\frac{\partial}{\partial y} + x^2\frac{\partial}{\partial z_i} \quad \text{and} \quad v_y^j = \left(\frac{\partial a}{\partial z_j}+x\frac{\partial b}{\partial z_j}\right)\frac{\partial}{\partial x} + \left( 2xy - b(\bar z)\right) \frac{\partial}{\partial z_j}$$ for $0\leq i \leq n$ and $0\leq j \leq k$ and moreover let $$v_z = a(\bar z)x\frac{\partial}{\partial x}-\left(2a(\bar z)y -xyb(\bar z) + b(\bar z)^2\right)\frac{\partial}{\partial y}.$$ \[lem-complete\] $f(\bar z)v_z\in\chvf(X)$, $f(x,z_0,..\hat{z_i}..,z_n)v_x^i \in \chvfw(X)$ and $f(y,z_0,..\hat{z_j}..,z_n)v_y^j \in \chvfw(X)$ for $f:\C^{n+1}\rightarrow \C$, $0\leq i \leq n$ and $0\leq j \leq k$. It is easy to see that the vector fields $v_x^i$ are locally nilpotent, so they are in particular complete. The coefficients of $v_y^j$ are linear in $x$ and $z_j$ for all $0\leq j\leq k$ so the flow equation is a linear differential equation, and thus has a global solution. The vector field $v_z$ is complete since we may first solve the linear and uncoupled differential equation for $x$. Then the differential equation for $y$ becomes linear and uncoupled as well, and thus we have a global solution. It is left to show that the vector fields $v_x^i$ and $v_y^j$ are volume preserving. A standard calculation shows that $$\begin{aligned} i_{v_x^i}\omega & = &(-1)^{i+1}\d x\wedge \d z_0 \wedge ..\widehat{\d z_i}..\wedge \d z_n, \\ i_{v_y^j}\omega &= &\frac{1}{x^2}\left(\frac{\partial a}{\partial z_j}+x\frac{\partial b}{\partial z_j}\right)\d z_0 \wedge \ldots \wedge \d z_n + \\ & & \frac{(-1)^{j+1}}{x^2} ( 2xy -b(\bar z) ) \d x\wedge \d z_0 \wedge..\widehat{\d z_j}..\wedge \d z_n \\ & = & (-1)^j \ \d \left( \frac{a(\bar z) + xb(\bar z)}{x^2} \right) \wedge \d z_0 \wedge ..\widehat{\d z_j}..\wedge \d z_n \\ & = & (-1)^j \d y \wedge \d z_0 \wedge ..\widehat{\d z_j}..\wedge \d z_n.\end{aligned}$$ Thus $L_{v_x^i}\omega = \d i_{v_x^i}\omega = 0$ and $L_{v_y^j}\omega = \d i_{v_y^j}\omega = 0$ which shows that they are volume preserving. Multiplying with a kernel element doesn’t affect these properties. The group $\aut(X)$ acts transitively on $X\setminus \lbrace x=0\rbrace$. Moreover $\autw(X)$ acts transitively on $X\setminus \lbrace x=0\rbrace$ provided that condition (B) from the introduction holds. Use the flow of the vector fields $v_x^i$ to get a transitive action on the fibers $\lbrace x=c\neq0\rbrace$ and the fields $v_z$ to connect the fibers. Thus we see that $\aut(X)$ acts transitively on $X\setminus \lbrace x=0\rbrace$. If condition (B) holds then we may also use the vector fields $v_y^j$ to connect the fibers. Indeed, for every $c\neq 0$ there is a point $p\in \lbrace x=c\rbrace$ and a vector field $v_y^j$ such that $v_y^j[p]$ points outwards of the fiber. Thus in this case $\autw(X)$ acts transitively on $X\setminus \lbrace x=0\rbrace$. If (A) from the introduction holds then for every point $p\in X\cap \lbrace x=0\rbrace$ there is some $\phi\in\autw(X)$ such that $\phi(p)\notin\lbrace x=0\rbrace$. By (A) we have that for any point $p=(0,y_0,\bar q)\in \lbrace x=0\rbrace\cap X$ at least one of the polynomials $b,\frac{\partial a}{\partial z_0},\ldots,\frac{\partial a}{\partial z_k}$ does not vanish at $\bar q$. For a non-vanishing $\frac{\partial a}{\partial z_j}$ the vector field $v_y^j$ points outwards from $\lbrace x=0 \rbrace$ at the point $p$. Thus the flow of $v_y^j$ moves $p$ away from $\lbrace x = 0\rbrace$. If all polynomials $\frac{\partial a}{\partial z_0},\ldots,\frac{\partial a}{\partial z_k}$ vanish at $\bar q=(q_0,\ldots,q_n)$ then $b$ is non-vanishing at $\bar q$ and moreover there is a $j\leq k$ such that $\frac{\partial a}{\partial z_j}$ does not vanish along the curve $\lbrace z_i=q_i \text{ for all } i\neq j \rbrace \subset \C^{n+1}$. This means that $v_y^j$ is non-vanishing at $p$. Assume the orbit of $p$ by the flow of $v_y^j$ is contained in $\lbrace x=0\rbrace$ then the set $\lbrace x=0,y=y_0,z_i=q_i \text{ for all } i\neq j \rbrace \subset \C^{n+3}$ would contained in $X$ and tangent to $v_y^j$, which is not the case. These two lemmas combined give the following proposition using the conditions from the introduction. \[prop-trans\] Assume that (A) holds. Then $\aut(X)$ acts transitively on $X$. Assume that additionally (B) holds. Then $\autw(X)$ acts transitively on $X$. \[cor-koras\] The volume preserving automorphisms act transitively on the Koras-Russell cubic $C=\lbrace x^2y + x + z_0^2 + z_1^3=0\rbrace$. We have $\frac{\partial a}{\partial z_0} = -2z_0$ and $b(z_0,z_1)=-1$. Thus it is easy to see that (A) and (B) hold, and thus the vector fields $v_x^0,v_x^1$ and $v_y^0$ induce a transitive action on $C$. The transitivity of the action by automorphisms a priori need not to be achieved by the vector fields $v_x^i,v_y^i,v_z$ only (which is equivalent to condition (A) from the introduction). There could be further automorphisms. For example depending on $a$ and $b$ there could be automorphisms of the form $(x,y,\bar z)\mapsto(x,\gamma y,\lambda(\bar z))$ where $\lambda$ is an automorphisms of $\C^{n+1}$ with the property that $a(\lambda(\bar z))+xb(\lambda(\bar z))=\gamma(a(\bar z) + xb(\bar z))$ for some $\gamma\in\C^$. A similar statement holds for transitivity by volume preserving automorphisms. The Main Theorem for $n>0$ {#sec-pos} ========================== Let $n>0$, $n\geq k\geq 0$ and $a,b\in\C[z_0,\ldots,z_n]$ such that $\deg_{z_i}(a)\leq2$ and $\deg_{z_i}(b)\leq 1$ for all $i\leq k$. Moreover, assume that not both of $\deg_{z_0}(a)$ and $\deg_{z_0}(b)$ are equal to zero. Let $\bar z=(z_0,\ldots z_n)$ and $X=\lbrace x^2y=a(\bar z) + xb(\bar z)\rbrace$ with the holomorphic volume form $\omega = \d x/x^2\wedge\d z_0\wedge\ldots\wedge \d z_n$. Consider, again, the following vector fields on $X$: $$v_x^i = \left(\frac{\partial a}{\partial z_i}+x\frac{\partial b}{\partial z_i}\right)\frac{\partial}{\partial y} + x^2\frac{\partial}{\partial z_i} \quad \text{and} \quad v_y^j = \left(\frac{\partial a}{\partial z_j}+x\frac{\partial b}{\partial z_j}\right)\frac{\partial}{\partial x} + \left( 2xy - b(\bar z)\right) \frac{\partial}{\partial z_j}$$ for $0\leq i \leq n$ and $0\leq j \leq k$. \[lem-pairs\] Let $0\leq i\leq n$ and $0\leq j \leq k$. Then $(v_x^i,v_y^j)$ are compatible pairs for $i\neq j$. First we show that $(v_x^i,v_y^j)$ are semi-compatible pairs. Indeed the kernel of $v_x^i$ is contained in the functions depending on $x,z_0,..\hat z_i ..,z_n$ and the kernel of $v_y^j$ is contained in the functions depending on $y,z_0,..\hat z_j ..,z_n$ thus the closure of $\mathrm{span}\lbrace\ker v_x^i\cdot \ker v_y^j\rbrace$ is equal to $\O_X(X)$ and in particular contains an ideal. For $(v_x^i,v_y^j)$ being a compatible pair we need a function $h\in\ker v_y^j$ such that $v_x^i(h)\in\ker v_x^i\setminus 0$. The function $h=z_i$ does the job. \[lem-genset\] For a generic point $p\in X$ the vector $v_y^0[p]$ is a generating set for $\mathrm{T}_pX$ and the set $v_x^n[p]\wedge v_y^0[p]$ is a $\omega$-generating set for $\mathrm{T}_pX\wedge \mathrm{T}_pX$. Note that the first statement also true for $n=0$. Let $p=(x_0, y_0 ,\bar q)$ where $\bar q=(q_0,\ldots,q_n)$ be such that $x_0\neq 0$ and $\frac{\partial a(\bar q)}{\partial z_0}+x_0\frac{\partial b(\bar q)}{\partial z_0}\neq0$. For a complete vector field $\nu\in\chvf(X)$ and kernel element $f\in\ker\nu$ with $f(p)=0$ we get an induced action (by the time one flow map of $f\nu$) on $\T_pX$ given by $v\mapsto v+v(f)\nu[p]$. Let $\nu_i = v_x^i$ and $f_i = x - x_0$ for $0\leq i\leq n$. Thus the orbit of $v_y^0[p]$ under the $\autw(X)_p$-action contains the vectors $v_y^0[p] + \left(\frac{\partial a(\bar q)}{\partial z_0}+x_0\frac{\partial b(\bar q)}{\partial z_0}\right)v_x^i[p]$. Therefore the orbit contains a basis for $\T_pX$. Since $f\in\ker v_x^n$ we have that $v_x^n[p]\mapsto v_x^n[p]$ under the actions given by $(x-x_0)v_x^i$. So in particular, similarly we have that the orbit of $v_x^n[p]\wedge v_y^0[p]$ contains a basis for $v_x^n[p]\wedge \T_pX$. Considering now the actions given by the vector fields $(z_n - q_n)v_x^i$ for $i\leq n-1$. We get the actions $v_x^n[p]\mapsto v_x^n[p] + x_0^2v_x^i[p]$, and thus we see that the orbit of the $\autw(X)_p$-action contains a basis for $v_x^i[p]\wedge \T_pX$ for all $i\leq n$. Together they build then together a basis for $\T_pX\wedge \T_pX$. By Lemma \[lem-pairs\] there exists a point $p\in X$ and compatible pairs $(\nu_i,\mu_i)$ such that the vectors $\mu_i[p]$ are a generating set for $\T_pX$ and the elements $\nu_i[p]\wedge\mu_i[p]$ are a $\omega$-generating set for $\T_pX\wedge\T_pX$ thus by Theorem \[criterion\] the claim is proven. Proposition \[prop-trans\] proves the “in particular” part. We never used that $a$ and $b$ are polynomials. In fact, the Main Theorem also holds if $a$ and $b$ are polynomial in $z_0$ and analytic in $z_1,\ldots,z_n$. The condition that $\H^{n+1}(X,\C)=0$ in the Main Theorem could be omitted in the case when every class of $\H^{n+1}(X,\C)$ contains an element of $\Theta(\lie(\chvfw(X))$ as in Proposition \[prop-critw\]. However this cannot be achieved by the complete vector fields $f\cdot v_x^i$ and $g\cdot v_y^j$ since they are all mapped to the zero class by $\Theta$ (see the calculation in the proof of Lemma \[lem-complete\]). So the existence of other complete volume preserving vector fields would be required. If $a(\bar z)$ is reduced (i.e. has connected fibers) and $\lbrace a(\bar z) = 0\rbrace\subset\C^{n+1}$ is smooth then the condition $\H^{n+1}(X,\C)$ is actually equivalent to the condition that $\tilde\H^{n-1}(\lbrace a(\bar z) = 0\rbrace,\C) = 0$. Indeed, if $Y=\lbrace uv = a(\bar w)\rbrace$ then the map $X\rightarrow Y$ given by $(x,y,\bar z)\mapsto (x, xy -b(\bar z),\bar z)$ is an affine modification in the sense of [@za]. Theorem 3.1 of [@za] shows that $\H^(X,\C)=\H^(Y,\C)$. Moreover, Proposition 4.1 of [@za] states that we have $$\tilde\H^(Y,\C) = \tilde\H^{-2}\Big(\lbrace a(\bar z)=0\rbrace,\C\Big)$$ which proofs the statement of the remark. The Main Theorem for $n=0$ {#sec-zero} ========================== Let $a,b\in\C$, and let $X_{a,b}=\lbrace x^2y = z^2 - b + ax \rbrace$. Note that $X_{a,b}$ is smooth if and only if $a$ and $b$ are not both equal to zero. Also, the conditions (A) and (B) from the introduction hold automatically if $X_{a,b}$ is smooth. Therefore the volume preserving automorphisms act transitively on $X_{a,b}$. The following proposition is mostly taken from [@po], and among others it shows that $X_{a,b}$ is algebraically isomorphic to $X_{1,0}$, $X_{0,1}$ or $X_{1,1}$. Moreover it shows that $X_{0,1}$ and $X_{1,1}$ are biholomorphic. \[standard\] (a) Let $a\in\C$ and $b\in\C^$ then (i) $X_{a,1}\algiso X_{a,b}$ and $X_{b,a}\algiso X_{1,a}$, (ii) $X_{0,1}\holiso X_{a,1}$, where $\algiso$ means isomorphic as algebraic surfaces and $\holiso$ isomorphic as complex manifolds. (b) Let $X = \lbrace x^2y=p(x,z)\rbrace$ be a smooth hypersurface with $p\in\C[x,z]$ and $\deg_zp(0,z)\leq 2$ then (i) if $\deg_zp(0,z)= 0$ then $X\algiso\C^\times \C$, (ii) if $\deg_zp(0,z)= 1$ then $X\algiso\C^2$, (iii) if $p(0,z)$ has a zero with multiplicity 2 then $X\algiso X_{1,0}$, (iv) if $p(0,z)$ has two different zeroes then there is a unique $a\in\lbrace 0,1\rbrace$ such that $X\algiso X_{a,1}$. Theorem 9 from [@po] gives the isomorphisms in (a)(i). The biholomorphic map in (a)(ii) is given by $$(x,y,z)\mapsto \left(x,e^{-ax}y +\frac{e^{-ax}+ax -1}{x^2},e^{-\frac{a}{2}x}z\right).$$ Theorem 5 from [@po] states that $X$ is algebraically isomorphic to $x^2y = s(z) + xt(z)$ for some $s,t\in\C[z]$ with $\deg t \leq \deg s -2$ Following the given algorithm we see that $s(z)=p(0,z)$ which is then, after a linear change in the $z$-coordinate, given by (i) $1$, (ii) $z$, (iii) $z^2$ or (iv) $z^2-1$. The isomorphisms in (b) are then easily found and the uniqueness in (b)(iv) follows from Theorem 9 from [@po]. Despite of Proposition \[standard\](a) we will start working on $X_{a,b}$ for general $a,b\in\C$. It turns out that this is more convenient for most arguments. \[rem-standard\] Every function $f\in\C[X_{a,b}]$ can be written uniquely as $$f(x,y,z)=\sum_{i=1}^\infty x^ia_i(z) + \sum_{i=1}^\infty xy^ib_i(z) + \sum_{i=1}^\infty y^ic_i(z) + d(z),$$ indeed replace every $x^2y$ by $z^2 - b + ax$. Alternatively $f$ can be written uniquely as $$f(x,y,z)= f_1(x,y) + zf_2(x,y),$$ indeed replace every $z^2$ by $x^2y + b -ax$. We will use a tool in order to proof the (volume) density property, namely the algebraic (volume) density property. Note that the algebraic (volume) density property implies the (volume) density property, see [@kaku-volume]. \[def-adp\] Let $X$ be an affine algebraic manifold. If the Lie algebra $\lie(\cavf(X))$ generated by complete algebraic vector fields $\cavf(X)$ on $X$ is equal to the Lie algebra of all algebraic vector fields $\avf(X)$ on $X$ then $X$ has the algebraic density property. Let $X$ be an affine algebraic manifold equipped with an algebraic volume form $\omega$. If the Lie algebra $\lie(\cavfw(X))$ generated by complete volume preserving algebraic vector fields $\cavfw(X)$ on $X$ is equal to the Lie algebra of all volume preserving algebraic vector fields $\hvfw(X)$ on $X$ then $X$ has the algebraic volume density property. The volume density property --------------------------- Let $a,b\in\C$ such that not both equal to zero. For proving the volume density property for $X_{a,b}= \lbrace x^2y=z^2 - b + ax\rbrace$ with respect to $\omega = \d x/x^2\wedge \d z$ we will need the following two vector fields: $$v_x=2z\frac{\partial}{\partial y} + x^2\frac{\partial}{\partial z}, \quad v_y=2z\frac{\partial}{\partial x} + (2xy-a)\frac{\partial}{\partial z}.$$ Lemma \[lem-complete\] translates into: \[complete\] $x^kv_x\in\cavfw(X_{a,b})$ and $y^kv_y\in\cavfw(X_{a,b})$ for $k\geq0$. \[correspondance\] Let $v\in\avfw(X_{a,b})$. Then the 1-form $i_v \omega$ is exact and $i_v \omega = \d f$ defines a bijection between algebraic volume preserving vector fields and algebraic functions modulo constants. The functions corresponding to $x^kv_x$ and $y^kv_y$ are given by the equations $$(k+1)i_{(x^kv_x)}\omega=-\d x^{k+1} \text{ and } (k+1)i_{(y^kv_k)}\omega =\d y^{k+1}.$$ The correspondence is given by the isomorphism $\Theta$ the map $D$ in Section \[sec-crit\] using the fact that $\H^1(X_{a,b},\C) = 0$. The same correspondence was also used in [@le]. The triviality of $\H^1(X_{a,b},\C)$ follows from the fact that $X_{a,b}$ is an affine modification of $\C^2$ along the divisor $2\cdot\lbrace x=0\rbrace$ with center at the ideal $(x^2,z^2 - b + ax)$ using the notation from [@za]. If $b\neq 0$ then Proposition 3.1 from [@za] shows that $X_{a,b}$ is simply connected (since $\C^2$ is simply connected). If $b = 0$ then Theorem 3.1 from [@za] shows in a similar way that $\H^1(X_{a,0},\C) = \H^1(\C^2,\C)$, and thus is trivial. For the two identities we make the calculations: $$(k+1)i_{(x^kv_x)}\omega= (k+1)x^k i_{v_x}\omega=-(k+1)x^k\d x = -\d x^{k+1},$$ $$(k+1)i_{(y^kv_y)}\omega= (k+1)y^k i_{v_y}\omega= (k+1)y^k \left( \frac{2z}{x^2}\d z - \frac{2xy -a}{x^2}\d x\right) =$$ $$= (k+1)y^k\d \frac{z^2 - b + ax}{x^2}=(k+1)y^k\d y = \d y^{k+1}.$$ Recall that for a vector field $\nu$ and a regular function $f$ we denote by $\nu(f)$ the regular function which is obtained by applying $\nu$ as a derivation. \[bracket\] Let $v_1,v_2\in\avfw(X_{a,b})$ and $i_{v_1} \omega =\d f$ then $i_{[v_2,v_1]}\omega=\d v_2(f)$. In particular, if $f$ corresponds to a vector field in $\lie(\cavfw(X_{a,b}))$ then $x^kv_x(f)$ and $y^kv_y(f)$ correspond to a vector field in $\lie(\cavfw(X_{a,b}))$. The identity $i_{[v_2,v_1]}\omega=\d v_2(f)$ is shown in Lemma 3.2 in [@le]. Let $i,j,k\geq 0$. Then $$\begin{aligned} \label{4} v_x(y^{j+1}) &=& 2(j+1)y^j z,\\ \label{5} v_x(y^{j+1}z^{k+1})&=&y^jz^k(2(j+1)z^2+(k+1)(z^2 - b + ax)),\\ \label{1} y^jv_y(z^{k+1})&=&(k+1)y^jz^k(2xy-a),\\ \label{2} v_y(x^{i+1}) &=& 2(i+1)x^i z,\\ \label{3} v_y(x^{i+1}z^{k+1})&=& x^iz^k\left( 2(i+1)z^2 + (k+1)(2z^2 - 2b +ax)\right). \end{aligned}$$ The lemma is proven by straight forward calculations. \[svdp\] Smooth surfaces $X_{a,b}=\lbrace x^2y=z^2 - b + ax\rbrace$ have the algebraic volume density property with respect to $\omega=\mathrm{d}x/x^2\wedge\mathrm{d}z$. Let $L$ be the set of function that corresponds to the Lie algebra of complete vector fields. By Lemma \[correspondance\] we already have $x^i\in L$ and $y^i\in L$ for $i\geq0$. We need to show that all functions on $X$ (modulo constants) are contained in $L$. It is enough to show that (a) $x^iz^{k+1} \in L$, (b) $xy^{j+1}z^k\in L$ and (c) $y^{j+1}z^{k+1}\in L$ for all $i,j,k\geq 0$. First we show (a) $x^iz^{k+1}\in L$:* The statement (a) is also true for $k=-1$ by Lemma \[correspondance\]. Lemma \[bracket\] shows that $v_y(x^{i+1})\in L$. Therefore by (\[2\]) we get $2(i+1)x^iz\in L$ and thus $x^iz\in L$ for $i\geq 0$ which is the statement for $k=0$. Let us assume that the statement is true for $k-1$ and for $k$. Then, by Lemma \[bracket\] we have $v_y(x^{i+1}z^k)\in L$. By the induction assumption and (\[3\]) we have also $x^{i}z^{k+1}\in L$ which concludes the proof of (a) $x^iz^{k+1}\in L$ inductively for all $i,k\geq 0$. The next step is to show (b) $xy^{j+1}z^k\in L$ and (c) $y^{j+1} z^{k+1}$ for $k=0$: Note that (c) holds also for $k=-1$ by Lemma \[correspondance\]. By Lemma \[bracket\] we have $v_x(y^{j+2})\in L$, and thus by (\[4\]) $y^{j+1}z\in L$ which proofs statement (c) for $j\geq 0$ and $k=0$. By the same lemma we have $y^jv_y(z)\in L$. Thus by (\[1\]) and (c) we have $xy^{j+1}\in L$ proving statement (b) for $j\geq0$ and $k=0$. The last step is to show (b) $xy^{j+1}z^k\in L$ and (c) $y^{j+1} z^{k+1}$ for arbitrary $k$: Let us assume that (b) and (c) hold for $k$ and moreover (c) holds for $k-1$. By Lemma \[bracket\] and the induction assumption we have $v_x(y^{j+1}z^{k+1})\in L$ for all $j\geq0$. Thus by the induction assumption and (\[5\]) we also have $y^jz^{k+2}\in L$ for all $j\geq0$ which is statement (c) for $k+1$. Similarly, we have $y^jv_y(z^{k+2})\in L$, and thus by (\[1\]) and the induction assumption we get $xy^{j+1}z^{k+1}\in L$ for all $j\geq0$. This is statement (b) for $k+1$. Thus by induction over $k$ the statements (b) and (c) are shown. Let $X=\lbrace x^2y=p(x,z)\rbrace$ with $p\in\C[x,z]$ and $\deg(p(0,z))\leq 0$ be a smooth surface, then $X$ has the algebraic volume density property for the volume form $\d x / x^2 \wedge dz$. If $\deg(p(0,z))\in\lbrace 1,2 \rbrace$ then by Proposition \[standard\] the surface $X$ is algebraically isomorphic to some surface $X_{a,b}$ or to $\C^2$. Since on these surfaces there are no non-constant invertible regular functions two different algebraic volume forms differ only by multiplication with a constant. So the isomorphism induces a natural bijection between algebraic volume preserving vector fields on $X$ and $X_{a,b}$ (resp. $\C^2$). Thus algebraic volume density property is preserved under algebraic isomorphisms. Therefore Proposition \[svdp\] and the well known fact that $\C^2$ has the algebraic volume density property concludes this case. If $\deg(p(0,z))=0$ then by Proposition \[standard\] the surface $X$ is algebraically isomorphic to $\C^\times \C$. For any algebraic volume form $\eta$ on $\C^\times \C$ there is an algebraic automorphism such that the pull back of $\eta$ is equal to $\eta_0 = \d u/u\wedge \d v$. Indeed for an arbitrary $\eta = au^k \d u \wedge \d v$ with $a\in\C^$ and $k\in\mathbb{Z}$ the pull back of $\eta$ by $(u,v)\mapsto (u,a^{-1}u^{-k-1}v)$ is $\eta_0$. Apply Theorem 1 of [@kaku-volume2] to the semi-compatible pair $(u\cdot\partial/\partial u,\partial/\partial v)$ to see that $\C^\times\C$ with $\eta_0$ has the algebraic volume density property and thus $\C^\times \C$ has the algebraic volume density property for all algebraic volume forms. The density property -------------------- We will show the density property for the surfaces $X_{1,b}=\lbrace x^2y = z^2-b+x\rbrace$ with $b\in\C$. We will use the following vector fields on $X_{1,b}$: $$v_x=2z\frac{\partial}{\partial y} + x^2\frac{\partial}{\partial z}, \quad v_y=2z\frac{\partial}{\partial x} + (2xy-1)\frac{\partial}{\partial z},$$$$v_z=z^2x\frac{\partial}{\partial x} - \left(2(z^2-b)y -axy + a^2\right)\frac{\partial}{\partial y}.$$ For an algebraic vector field $\nu$ the $\omega$-divergence $\div \nu$ is the regular function given by $\d i_\nu\omega = (\div\nu)\cdot \omega$. \[div’\] We have $x^kv_x,y^kv_y\in\cavfw(X_{1,b})$ and $zx^kv_x,z^kv_z\in\cavf(X_{1,b})$ for $k\geq0$. Moreover $\div zx^kv_x = x^{k+2}$ and $\div z^kv_z = - z^{k+2}$. Lemma \[complete\] says $x^kv_x,y^kv_y\in\cavfw(X_0)$ and $zx^kv_x\in\cavf(X_{1,b})$. For the statement about the divergence we make the calculations $$\d i_{zx^kv_x}\omega = -\d zx^k\d x = x^k\d x\wedge\d z = x^{k+2}\omega,$$ $$\d i_{z^kv_z}\omega = \d \frac{z^{k+2}}{x}\d z = \frac{-z^{k+2}}{x^2}\d x \wedge \d z = -z^{k+2}\omega,$$ and thus prove the statement. The following lemma is well known. \[div\] For $v_1,v_2\in\avf(X_{1,b})$ we have $$\div [v_1,v_2] =v_1(\div v_2) - v_2(\div v_1),$$ in particular we have $$\div [x^kv_x,v_2] = x^kv_x(\div v_2) \quad \mathrm{and} \quad \div [y^kv_y,v_2] = y^kv_y(\div v_2)$$ for any $k\geq0$. Moreover for $f\in\C[X_{1,b}]$: $$\div fv = f\div v + v(f).$$ \[lem-v\_ysubmod\] Let $f\in\C[X_{1,b}]$. Then $fv_y\in\lie(\cavf(X_{1,b}))$. Let $E\subset\C[X_{1,b}]$ be given by $E=\lbrace \div\nu: \nu\in\lie(\cavf(X_{1,b}))\rbrace$. It is enough to show that for every $f\in\C[X_{1,b}]$ we have $\div(fv_y)\in E$. Indeed, then $fv_y - \nu \in\avfw(X_{1,b})$ for some $\nu\in\lie(\cavf(X_{1,b}))$. Thus by Proposition \[svdp\] we have $fv_y\in\lie(\cavf(X_{1,b}))$. By Lemma \[div’\] we have $x^{i+2}\in E$ and $z^{i+2}\in E$ for all $i\geq 0$. Thus by Lemma \[div\] we get $v_y(x^{i+2})=2(i+2)x^{i+1}z\in E$, and therefore $v_y(xz)=2z^2 + 2x^2y -x = 4z^2 - 2b + x \in E$. Since $z^2\in E$ we get $x-2b\in E$, and thus $v_y(x-2b)=2z\in E$. Altogether we have $x^iz\in E$, $x^{i+2}\in E$ and $x-2b\in E$ for all $i\geq 0$. Let $f=x^iy^jz^k$ for $i,j\geq 0$ and $k\in\lbrace 0,1\rbrace$. If $f\neq xy^j$ then we have $$\div (fv_y) = y^jv_y(x^iz^k) \in E$$ by Lemma \[div\] because $x^iz^k\in E$ by the above. If $f=xy^j$ then $$\div (fv_y) = y^jv_y(x) = y^jv_y(x-2b) \in E$$ by the same arguments. Thus the lemma is proven since any regular function is a sum of such monomials by Remark \[rem-standard\]. \[prop-surfdp\] The surface $X_{1,b}$ has the density property. By Lemma \[lem-genset\] the tangent vectors of $v_y$ are a generating set for the tangent space $\T_qX_{1,b}$ at a generic point $q\in X_{1,b}$. By Lemma \[lem-v\_ysubmod\] the $\C[X_{1,b}]$- submodule generated by $v_y$ is contained in $\lie(\cavf(X_{1,b}))$. Thus the $\O_{X_{1,b}}(X_{1,b})$- submodule generated by $v_y$ is contained in the closure of $\lie(\cavf(X_{1,b}))$. Therefore by Lemma \[lem-genset\] and Proposition \[prop-crit\] the surface $X_{1,b}$ has the density property. Let $X=\lbrace x^2y=p(x,z)\rbrace$ with $p\in\C[x,z]$ and $\deg(p(0,z))\leq 2$ be a smooth surface. Then $X$ has the density property. By Proposition \[standard\] the surface $X$ is biholomorphic to $\C^\times \C$, $\C^2$, $X_{1,0}$ or $X_{1,1}$. It is well known that the first two have the density property. The two other surfaces have the density property by Proposition \[prop-surfdp\]. [10]{} E. Anders[é]{}n and L. Lempert. . , 110(2):371–388, 1992. A. Dubouloz, L. Moser-Jauslin, and P.-M. Poloni. . , 59(3):679–694, 2010. F. Forstnerič. , volume 56 of [ [Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics \[Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics\]]{}]{}. Springer, Heidelberg, 2011. F. Forstnerič and J.-P. Rosay. . , 112(2):323–349, 1993. S. Kaliman and F. Kutzschebauch. . , 172(1):71–87, 2008. S. Kaliman and F. Kutzschebauch. . , 181(3):605–647, 2010. S. Kaliman and F. Kutzschebauch. . In [[Affine algebraic geometry]{}]{}, volume 54 of [[CRM Proc. Lecture Notes]{}]{}, pages 85–122. Amer. Math. Soc., Providence, RI, 2011. S. [Kaliman]{} and F. [Kutzschebauch]{}. . , Jan. 2012. arXiv1201.4769, to appear in Transfom. Groups. S. Kaliman and M. Zaidenberg. . , 4(1):53–95, 1999. F. [Kutzschebauch]{}. . , Sept. 2013. arXiv1309.3070. F. Kutzschebauch and S. Kaliman. . In [[Complex Analysis and Geometry]{}]{}, Aug. 2014. Springer Proceedings in Mathematics & Statistics, KSCV10, Gyeongju, Korea. F. [Kutzschebauch]{} and M. [Leuenberger]{}. . , Nov. 2013. arXiv1311.1075, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5). L. Makar-Limanov. . , 96(part B):419–429, 1996. P.-M. Poloni. . , 16(2):579–597, 2011. T. Ritter. . , 23(2):571–597, 2013. . T[ó]{}th and D. Varolin. . , 142(5):1308–1326, 2006. D. Varolin. . , 11(1):135–160, 2001. [^1]: It is a general fact that on every hypersurface $\lbrace P = 0\rbrace\subset\C^N$ there is a natural volume form given by $\omega = \left(\frac{\partial P}{\partial x_i}\right)^{-1} \d x_1\wedge\ldots\wedge\widehat{\d x_i}\wedge\ldots\wedge \d x_N$.
--- author: - Wen Hu - Zhi Wang - Lifeng Sun bibliography: - 'mylib\_short.bib' title: A Measurement Study of TCP Performance for Chunk Delivery in DASH ---
--- abstract: 'This is an informal elementary introduction to buildings – what they are and where they come from.' address: 'Department of Mathematics, University of York, York YO10 5DD, United Kingdom.' author: - Brent Everitt bibliography: - 'version3.bib' title: 'A (very short) introduction to buildings' --- This is an informal elementary introduction to buildings – written for, and by, a non-expert. The aim is to get to the definition of a building and feel that it is an entirely natural thing. To maintain the lecture style examples have replaced proofs. The notes at the end indicate where these proofs can be found. Most of what we say has its origins in the work of Jacques Tits, and our account borrows heavily from the books of Abramenko and Brown [@Abramenko_Brown08] and Ronan [@Ronan09]. Section \[lecture1\] illustrates all the essential features of a building in the context of an example, but without mentioning any building terminology. In principle anyone could read this. Sections \[lecture2\]-\[lecture4\] firm-up and generalize these specifics: Coxeter groups appear in §\[lecture2\], chambers systems in §\[lecture3\] and the definition of a building in §\[lecture4\]. Section \[lecture5\] addresses where buildings come from by describing the first important example: the spherical building of an algebraic group. The flag complex of a vector space {#lecture1} ================================== Let $V$ be a three dimensional vector space over a field $k$. Let $\Delta$ be the graph with vertices the non-trivial proper subspaces of $V$, and an edge connecting the vertices $V_i$ and $V_j$ whenever $V_i$ is a subspace of $V_j$: $$\xymatrix{ \begin{pspicture}(0,0)(2.65,0.1) %\showgrid \rput(0,-0.4){ \psline[](0.5,.5)(2,.5) \pscircle[fillstyle=solid,fillcolor=black](2,.5){0.15} \pscircle[fillstyle=solid,fillcolor=white](0.5,.5){0.15} \rput(0.05,.5){$V_i$}\rput(2.45,.5){$V_j$}} \end{pspicture} \ar@{<=>}[r] & V_i\subset V_j. }$$ Figure \[fig:flagcomplex\] shows the graph $\Delta$ when $k$ is the field of orders $q=2$ and $3$. There are $1+q+q^2$ one dimensional subspaces – illustrated by the white vertices – and $1+q+q^2$ two dimensional subspaces, illustrated by the black vertices. Each one dimensional space is contained in $1+q$ two dimensional spaces and each two dimensional space contains $1+q$ one dimensional spaces. The duality here might remind the reader of projective geometry. Call the edges $V_i\subset V_j$ of $\Delta$ chambers. Some more structure can be wrung out of this picture: there is an “$\gS_3$-valued metric”, with $\gS_3$ the symmetric group, that gives the shortest route(s) through $\Delta$ between any two chambers. To see how, suppose $c,c'$ are chambers and we want a shortest route of edges connecting them: $$\xymatrix{ c=V_1\subset V_2 \ar@{~>}[rr]^-{\rm{shortest\,\,route}} && c'=V_1'\subset V_2'. }$$ Make $c$ and $c'$ as different as possible by assuming that $V_1\not=V_1'$, $V_2\not=V_2'$ and $V_2\cap V_2'$ is a line different from $V_1,V_1'$. Changing notation, let $L_1,L_2,L_3$ be lines with $L_1=V_1$, $L_3=V_1'$ and $L_2=V_2\cap V_2'$. One then gets $V_2=L_1+L_2$ and $V_2'=L_2+L_3$. (0,0)(12,6) (3,3)[ (0,0) ]{} (9,3)[ (0,0) ]{} We get a small piece of $\Delta$, a local picture containing $c,c'$, as in Figure \[fig:localpicture\]. The field $k$ wasn’t mentioned at all in the previous paragraph, so this is the local picture for $\Delta$ over any field. The global picture gets more complicated however as the field $k$ gets bigger as Figure \[fig:flagcomplex\] illustrates. Say that chambers are $i$-adjacent if any difference between them occurs only in the $i$-th position, so $V_1\subset V_2\supset V_1',\, (V_1\not=V_1')$ are a pair of $1$-adjacent chambers and $V_2\supset V_1\subset V_2',\,(V_2\not= V_2')$ a pair of $2$-adjacent chambers (a chamber is also $i$-adjacent to itself for any $i$). Place the label $i$ on a vertex of the local picture in Figure \[fig:localpicture\] if the two chambers meeting at the vertex are $i$-adjacent. The shortest routes from $c$ to $c'$ in the local picture are given by $$\xymatrix{ c\ar[r]^-{s_2s_1s_2}_-{s_1s_2s_1} & c' }$$ where the route $s_1s_2s_1$ means cross a $1$-labeled vertex, then a $2$-labeled vertex and then a $1$-labeled vertex. Routes are read from left to right, although it obviously doesn’t matter with the two above. These routes then take values in the symmetric group $\gS_3$ by letting $s_1=(1,2)$ and $s_2=(2,3)$, so that both $s_1s_2s_1$ and $s_2s_1s_2$ give the permutation $(1,3)\in\gS_3$. Our actions will always be on the left, so in particular permutations in $\gS_3$ are composed from right to left. Define the $\gS_3$-distance between $c,c'$ to be $\delta(c,c')=(1,3)$. For an arbitrary pair of chambers define $\delta(c,c')$ to be the element of $\gS_3$ obtained by situating the chambers $c,c'$ in some local picture and taking the shortest route(s) as in Figure \[fig:localpicture\]. The resulting map $\delta:\Delta\times\Delta\rightarrow \gS_3$ can be thought of as a metric on $\Delta$ taking values in $\gS_3$. We will see in §\[lecture4\] why this map is well defined and doesn’t depend on which local picture we choose containing $c,c'$, although an ad-hoc argument shows that an element of $\gS_3$ can be associated in a canonical fashion to any pair of chambers. Take the $c,c'$ above and write $$c=0\subset L_1\subset L_1+L_2\subset V=V_0\subset V_1\subset V_2\subset V_3$$ and $c'=V'_0\subset\cdots\subset V'_3$ similarly. For each $i$ the filtration $V_0\subset V_1\subset V_2\subset V_3$ of $V$ induces a filtration of the one dimensional quotient $V'_i/V'_{i-1}$: $$\label{eq:13} (V'_i\cap V_0)/V'_{i-1}\subset\cdots\subset (V'_i\cap V_3)/V'_{i-1}$$ (by $(V'_i\cap V_0)/V'_{i-1}$, etc, we mean the image of $V'_i\cap V_0$ under the quotient map $V\rightarrow V/V'_{i-1}$). Any filtration of a one dimensional space must start with a sequence of trivial subspaces and end with a sequence of $V'_i/V'_{i-1}$’s. At some point in the middle the filtration jumps from being zero dimensional to one dimensional; for the $c,c'$ above: $$ ----- ----------------- ------------------------ ------------------------------------------------------------------------------------- $i$ $V'_i/V'_{i-1}$ filtration (\[eq:13\]) “jump index" $j$\ $1$&$L_3$&$0\subset 0\subset 0\subset L_3$&$3$\ $2$&$(L_2+L_3)/L_3$&$0\subset 0\subset (L_2+L_3)/L_3\subset (L_2+L_3)/L_3$&$2$\ $3$&$V/(L_2+L_3)$&$0\subset V/(L_2+L_3)\subset V/(L_2+L_3)\subset V/(L_2+L_3)$&$1$\ ----- ----------------- ------------------------ ------------------------------------------------------------------------------------- $$ Defining $\pi(i)=j$ gives $\pi=(1,3)\in\gS_3$. Summarizing: #### First rough definition of a building A building is a set of chambers with $i$-adjacency between them, the $i$ coming from some set $S$, together with a “$W$-valued metric” for $W$ some group. (0,0)(12,5.8) (1.25,1.5)[ (1.5,1.5) (1.5,3)[$\scriptstyle{L_1+L_3}$]{} (3.15,0.9)[$\scriptstyle{L_2+L_3}$]{} (-0.15,0.9)[$\scriptstyle{L_1+L_2}$]{} (0.1,2.1)[$\scriptstyle{L_1}$]{} (1.5,-0.05)[$\scriptstyle{L_2}$]{} (2.9,2.1)[$\scriptstyle{L_3}$]{} (0.2,1.5)[${\red c}$]{}(2.8,1.5)[${\red c'}$]{} (1.5,2.45)[$\scriptstyle{1}$]{} (2.3,2.05)[$\scriptstyle{2}$]{} (2.3,0.95)[$\scriptstyle{1}$]{} (1.5,0.55)[$\scriptstyle{2}$]{} (0.7,0.95)[$\scriptstyle{1}$]{} (0.7,2.05)[$\scriptstyle{2}$]{} (1.5,3.7)[$\scriptstyle{s_2s_1s_2}$]{} (1.5,-0.75)[$\scriptstyle{s_1s_2s_1}$]{} ]{} (-0.75,0)[ (9.5,3)[ (0,0) ]{} (-0.5,0)[(7.3,3.8)[${\red c}$]{}(10.1,0.4)[${\red c'}$]{}]{} ]{} #### Returning to the running example, the symmetric group $\gS_3$ is a reflection group, with Figure \[fig:localpicture\] and the resulting metric $\delta$ coming from the geometry of these reflections. To see why suppose we have a three dimensional Euclidean space – a real vector space with an inner product. Let $v_1,v_2,v_3$ be an orthonormal basis and let $\gS_3$ act on the space by permuting coordinates: $\pi\cdot v_i:=v_{\pi(i)}$ for $\pi\in\gS_3$ (and extend linearly). This action is not essential as the vector $v=v_1+v_2+v_3$ is fixed by all $\pi\in\gS_3$. This can be gotten around by passing to the perp space $${\textstyle v^\perp=\{\sum\lambda_i v_i\,\|\,\sum\lambda_i=0\}}.$$ The picture to keep in mind is the following, where $v^\perp$ is translated off the origin to make it easier to see: $$\begin{pspicture}(0,0)(14.3,3.5) %\showgrid \rput(0,0){ \rput(4.6,1.75){ \rput(0,0){\BoxedEPSF{symmetric.group1.eps scaled 225}} } \rput(4.1,0.4){$v_1$}\rput(5.75,1.7){$v_2$}\rput(3.1,2.5){$v_3$} \rput(5.7,0.6){${\red s_1}$}\rput(4.1,3.3){${\red s_1}$} \rput(5,2.5){$v^\perp+\frac{1}{3}v$} \psbezier[showpoints=false,linecolor=red]{->}(3.3,3.2)(3,3.75)(4,3.75)(3.7,3.2) \psbezier[showpoints=false,linecolor=red]{<->}(5,0.4)(5.5,0.5)(5.6,1)(5.5,1.2) } \rput(0,0){ \rput(9.2,1.75){ \rput(0,0){\BoxedEPSF{symmetric.group2.eps scaled 225}} } \rput(10.2,0.4){${\red s_1}$}\rput(9.1,2.9){${\red s_2}$} } \end{pspicture}$$ The element $s_1=(1,2)$ acts as on the left – as the reflection in the plane with equation $x_1-x_2=0$. Similarly $s_2=(2,3)$ and $(1,3)$ are reflections in the planes $x_2-x_3=0$ and $x_1-x_3=0$. These three planes chop the intersection of $v^\perp+\frac{1}{3}v$ with the positive quadrant into a triangle with its boundary barycentrically subdivided (or hexagon). So we start to see the local picture of Figure \[fig:localpicture\] coming from the geometry of these reflecting hyperplanes. Putting $v^\perp$ into the plane of the page decomposes the plane into six infinite wedge-shaped regions: $$\begin{pspicture}(0,0)(\textwidth,3) %\showgrid \rput(2.4,1.5){ \rput(-2,-0.8){{\red chambers}} % \psline{->}(-1.2,-1)(-0.5,-0.75) % \psline{->}(-2,-0.8)(-1,0) \rput(0.6,-1.4){${\red s_1}$}\rput(1.95,0.9){${\red s_2}$} \rput(0,0){\BoxedEPSF{fig6a.eps scaled 500}} } \rput(-0.5,0){ \psline{->}(4.9,1.5)(6.1,1.5) \rput(5.5,1.75){intersect} \rput(5.5,1.25){with $S^1$} } \rput(7.25,1.5){ \rput(0,0){\BoxedEPSF{fig6b.eps scaled 500}} } \rput(9.25,1.5){$\approx$} \rput(11.25,1.5){ \rput(-1.6,-0.8){{\red chambers}} \rput(0,0){\BoxedEPSF{fig6c.eps scaled 500}} } \end{pspicture}$$ In the theory of reflection groups (§\[lecture2\]) these regions are also called chambers. The chambers of our local picture are gotten back by intersecting these regions with the sphere $S^1$. (In the next dimension up we can still draw pictures of some of these objects. Let $V$ be four dimensional over $k$ and $\Delta$ the two dimensional simplicial complex with vertices the non-trivial subspaces of $V$, edges (or $1$-simplicies) the pairs $V_i\subset V_j$ and $2$-simplicies the triples $V_i\subset V_j\subset V_k$. We can get the local picture by working backwards from a symmetric group action like we did above. If we have a four dimensional Euclidean space with orthonormal basis $v_1,v_2,v_3,v_4$, then the convex hull of the $v_i$ is a tetrahedron lying in the hyperplane $v^\perp+\frac{1}{4}v$ where $v=v_1+v_2+v_3+v_4$. The six reflecting hyperplanes of the $\gS_4$-action have equations $x_i-x_j=0$ and slice the boundary of the tetrahedron barycentrically. Identifying the hyperplane with three dimensions and intersecting the whole picture with the sphere $S^2$, we end up with Figure \[fig:localpicture2\] (left). Flattening it out, we can retrospectively label the simplicies by lines $L_1,L_2,L_3,L_4\in V$ and the spaces they generate.) Returning to the hexagon, the $\gS_3$-action turns out to be regular on the chambers, i.e. given chambers $c,c'$ there is a unique $\pi\in\gS_3$ with $\pi c=c'$. This is most easily seen by brute force: fix a “fundamental” chamber $c_0$ and show that the six elements of $\gS_3$ send it to the six chambers in the decomposition above. In particular there is a one-one correspondence between the chambers and the elements of $\gS_3$ given by $\pi\in\gS_3\leftrightarrow\mbox{chamber }\pi c_0$. This correspondence gives the adjacency labelings of the hexagonal local picture of Figure \[fig:localpicture\]: choose the fixed chamber $c_0$ to be one of the two regions bounded by the reflecting lines for $s_1$ and $s_2$. Starting with the edge of the hexagon contained in $c_0$, label its vertices by the corresponding reflections as below left: $$\begin{pspicture}(0,0)(\textwidth,3.5) %\showgrid \rput(1.75,1.75){ \rput(0,0){\BoxedEPSF{fig7a.eps scaled 500}} \rput(0.75,1.25){$c_0$} \rput(-0.35,1.25){$s_1$}\rput(1.4,0.5){$s_2$} } \rput(-0.25,0){ \rput(6,1.75){ \rput(0,0){\BoxedEPSF{fig7b.eps scaled 500}} \rput(0.4,1.25){$c_0$} \rput(-0.75,1.25){$s_1$}\rput(1,0.5){$s_2$} \rput(-1.2,0){${\red \pi}$} } \psline{->}(7.5,1.75)(8.8,1.75) \rput(8.15,2){gives} \rput(11,1.75){ \rput(0,0){\BoxedEPSF{fig7c.eps scaled 500}} \rput(0.3,1.25){$s_1$}\rput(1.45,0.45){$s_2$} \rput(-1.4,-0.5){$s_1$}\rput(-1.4,0.45){$s_2$} \rput(-.8,0){$c_1$}\rput(.825,0){$c_2$} \rput(1.45,-0.5){$s_1$}\rput(0.3,-1.45){$s_2$} \rput(-1.4,1.5){${\red\pi_1}$}\rput(1.9,0.8){${\red\pi_2}$} }} \end{pspicture}$$ Now transfer this labeled edge to the other chambers using the $\gS_3$-action as in the picture above middle; the result is shown above right, where the $i$’s have become $s_i$’s. Vertices on opposite ends of the same line have different labels because the antipodal map $x\mapsto -x$ is not in the action of $\gS_3$ on the plane $v^\perp$. Finally, to get the metric $\delta$ observe that if $c$ is some chamber of the local picture in Figure \[fig:localpicture\] and $\pi\in\gS_3$ sends $c_0$ to $c$, then $\pi=s_{i_1}\ldots s_{i_k}$ where $s_{i_k},\ldots,s_{i_1}$ are the labels (read from left to right) on the vertices crossed in a path in the hexagon from $c_0$ to $c$. So for chambers $c_1,c_2$ we have $\delta(c_1,c_2)=\pi_1^{-1}\pi_2$ where $c_i=\pi_i c_0$. For our original $c_1,c_2$ we have $\pi_1=s_1s_2$, $\pi_2=s_2$, hence $\delta(c_1,c_2)=s_2s_1s_2$ as shown in the picture above. (0,0)(12,5) (6,4) (2,2.5) (4.55,4)[$\approx$]{} (1.3,0)[ (8,2.5) (8.1,5)[$L_1$]{}(10.65,1.25)[$L_2$]{}(8.25,0)[$L_3$]{}(5.35,1.25)[$L_4$]{} (9.95,3)[$L_1+L_2$]{}(9.8,0.5)[$L_2+L_3$]{}(7.5,2.5)[$L_1+L_3$]{} (10.1,2.1)[$L_1+L_2+L_3$]{} ]{} #### Second rough definition of building A building is a set of chambers with $i$-adjacency, the $i$ coming from some set $S$, together with a $W$-valued metric $\delta$, for $W$ a reflection group generated by $S$ and $\delta$ arising from the geometry of $W$. #### In the next sections we will make precise and general the ideas in this rough definition, but working in the reverse order: we start with reflection groups (§\[lecture2\]), then an abstract version of chambers and adjacency (§\[lecture3\]) and finally $W$-valued metrics (§\[lecture4\]). Reflection Groups and Coxeter Groups {#lecture2} ==================================== Reflection groups arise as the symmetries of familiar geometric objects; Coxeter groups are an abstraction of them. This section covers the basics. All vector spaces and linear maps here are over the reals $\R$. A reflection of a finite dimensional vector space $V$ is a linear map $s:V\rightarrow V$ for which there is a decomposition $$\label{eq:1} V=H_s\oplus L_s$$ where $H_s$ is a hyperplane (a codimension one subspace); $L_s$ is one dimensional; the restriction of $s$ to $H_s$ is the identity; and the restriction to $L_s$ is the map $x\mapsto -x$. Thus a reflection fixes pointwise a mirror $H_s$, the reflecting hyperplane of $s$, and acts as multiplication by $-1$ in some direction (the reflecting line) not lying in the mirror. In particular $s$ is invertible and an involution[^1]. A reflection group $W$ is a subgroup of $GL(V)$ generated by finitely many reflections. \[example:orthogonal\] The most familiar kind of reflections are the orthogonal ones for which we further assume that $V$ is a Euclidean space, i.e. is equipped with an inner product. Then $s$ is orthogonal if in the decomposition (\[eq:1\]) the line $L_s=H_s^\perp$, the orthogonal complement. In particular $L_s$, and hence the reflection, is determined by the reflecting hyperplane, unlike a general reflection where both the hyperplane and the line are needed. $$\begin{pspicture}(0,0)(4,3) %\showgrid \rput(0.5,0){ \rput(1.5,1.5){\BoxedEPSF{orthogonal.reflection.eps scaled 600}} \rput(1.5,2.5){$v$}\rput(1.3,0.5){$-v$} \rput(0.9,2.7){$L_v$}\rput(2.8,1.6){$H_v$} \rput(0.6,1.6){$s_v$} } \end{pspicture}$$ If $s$ is orthogonal then for any vector $v$ in $L_s$ we have $s:v\mapsto -v$ with $v^\perp$ fixed pointwise. Thus an orthogonal reflection $s$ can be specified by just a non-zero vector $v$, as the reflection with $H_s=v^\perp$ and $L_s$ spanned by $v$. We write $s=s_v$, $H_s=H_v$, $L_s=L_v$, and by choosing a sensible basis one gets that an orthogonal reflection is an orthogonal map of the Euclidean space. Let $\HH=\{H_{v_1},\ldots,H_{v_m}\}$ be hyperplanes in Euclidean $V$ and $W$ the reflection group generated by the orthogonal reflections $s_{v_1},\ldots,s_{v_m}$. As an exercise the reader can show that if $W\HH=\HH$, i.e. $g H_{v_i}=H_{v_j}$ for all $g\in W$ and all $v_i$, then $W$ is finite (hint: $\|W\|\leq (2m)!$). It turns out (although this is harder) that $\HH$ then consists of all the reflecting hyperplanes of $W$. \[example:finite\] Let $V$ be Euclidean with an orthonormal basis $v_1,\ldots,v_{n+1}$ and $\HH$ the hyperplanes $H_{ij}:=(v_i-v_j)^\perp$ for $1\leq i\not=j\leq n+1$ (in other words, $H_{ij}$ is the hyperplane with equation $x_i-x_j=0$). The reflection $s_{v_i-v_j}$ sends $v_i-v_j$ to $v_j-v_i$, thus swapping the vectors $v_i$ and $v_j$. Any other basis vector is orthogonal to $v_i-v_j$, so lies in $H_{ij}$, and is fixed. Thus if $\pi=(i,j)\in\gS_{n+1}$ then $s_{v_i-v_j}H_{k\ell}=H_{\pi(k),\pi(\ell)}$. Now let $W$ be the group generated by the reflections $s_{v_i-v_j}$. We have just shown that $W\HH=\HH$, so $W$ is a finite reflection group by the exercise above. Indeed, $W$ is the symmetric group $\gS_{n+1}$ acting by permuting coordinates as in §\[lecture1\]. To make this identification we have already seen that each $s_{v_i-v_j}$, and so every element of $W$, permutes the basis vectors $v_1,\ldots,v_{n+1}$. This gives a homomorphism $W\rightarrow\gS_{n+1}$. Injectivity of this homomorphism follows as the $v_i$ span $V$ and surjectivity as the transpositions $(i,j)$ generate $\gS_{n+1}$. The convex hull of the $v_i$ is the standard $n$-simplex, barycentrically subdivided by its $n(n-1)$ hyperplanes of reflectional symmetry (the $\HH$), each of which is a reflecting hyperplane of $W$. This is the picture we had for $n=2$ and $n=3$ in §\[lecture1\]. Finite reflection groups are often called spherical as the geometrical realisation of their Coxeter complexes (the boundary of the barycentrically divided $n$-simplex in this case; see Example \[example:coxeter.complexes\] for the general definition) are spheres. \[example:affine\] Let $V$ be $2$-dimensional and consider reflections $s_0,s_1$ where the reflecting hyperplanes and lines are shown below left (there is no inner product this time). The reflecting hyperplanes are different but both have the same reflecting line: $L_{s_0}=L=L_{s_1}$. If $W$ is the group generated by $s_0,s_1$ then $W$ leaves invariant any affine line parallel to $L$ as the $s_i$ do. But if $\HH=\{H_{s_0},H_{s_1}\}$ then $W\HH\not=\HH$ as $s_0H_{s_1}\not\in\HH$. Indeed, we must expand $\HH$ to the infinite set shown below right before it becomes $W$-invariant: $$\begin{pspicture}(0,0)(\textwidth,3.5) %\showgrid \rput(-1,0){ \rput(3,1.5){\BoxedEPSF{affine.group1.eps scaled 500}} \rput(2.6,2.8){$H_{s_0}$}\rput(4.3,2.8){$H_{s_1}$} \rput(4.5,1.35){$L_{s_0}=L_{s_1}$} \rput(4.5,0.25){$W=\langle s_0,s_1\rangle$} } \rput(5.5,0){ \rput(3,1.5){\BoxedEPSF{affine.group2.eps scaled 500}} } \rput(-1.5,0){ \rput(10,3.2){$s_0$}\rput(11,3.2){$s_1$} \rput(9,3.2){$s_0s_1s_0$}\rput(12,3.2){$s_1s_0s_1$} \rput(8,3.2){$\ldots$}\rput(13,3.2){$\ldots$} \rput(13,1.5){invariant affine line} \psline{->}(13,1.65)(13,2.3) \rput(7.5,1.4){$L$} } \end{pspicture}$$ In fact, by identifying the invariant affine line with the reals, $W$ is isomorphic to the group of “affine reflections” of $\R$ in the integers $\Z$, i.e. to the group of transformations of $\R$ generated by the maps $s_n:x\mapsto 2n-x$ for $n\in\Z$. The element $s_1s_0$ acts on the affine line as the translation $x\mapsto x+2$ so has infinite order. In particular $W$ is infinite. This also follows from $\HH$ being infinite as the reflections in the hyperplanes in $\HH$ are the $W$-conjugates of $s_0,s_1$. \[hyperbolic.reflection\] Let $V$ be $3$-dimensional and again there is no inner product. Let $a,b,c$ be real numbers such that $a^2+b^2>c^2$, and consider the reflection $s$ with reflecting hyperplane $H_s$ having the equation $ax+by-cz=0$ and reflecting line $L_s$ spanned by the vector $v=(a,b,c)$. Then $v$ lies on the outside of the pair of cones with equation $z^2=x^2+y^2$ and $H_s$ passes through the interior of this cone: $$ (0,0)(,4) (2.5,1.8)[ (0,0) (1.5,-1.8)[$z^2=x^2+y^2$]{} (-2,0.5)[$H_s$]{}(2.5,1)[$v$]{} ]{} (8.5,1.8)[ (0,0) (0.8,2.3)[$s$]{} (2.8,1)[$x^2+y^2-z^2=-1$]{} ]{} $$ One can check that $s$ leaves invariant each sheet of the two sheeted hyperboloid with equation $x^2+y^2-z^2=-1$. Either sheet is a model for the hyperbolic plane. Intersecting $H_s$ with the top sheet gives a hyperbola – a straight line of hyperbolic geometry – and $s$ is the “hyperbolic reflection" of the plane in this line[^2]. Returning to the finite orthogonal case, let $V$ be Euclidean, $\HH=\{H_i\}_{i\in T}$ a finite set of hyperplanes and $W=\langle s_i\rangle_{i\in T}$ the group generated by the orthogonal reflections in the $H_i$. Suppose also that $W\HH=\HH$, so $W$ is finite and $\HH$ is the set of all reflecting hyperplanes of $W$ as above. For each $i\in T$ choose a linear functional $\aa_i\in V^$ with $H_i=\ker\aa_i$. The choice of $\aa_i$ is unique upto scalar multiple and $H_i$ consists of those $v\in V$ with $\aa_i(v)=0$. The two sides (or half-spaces) of the hyperplane consist of the $v$ with $\aa_i(v)>0$ or the $v$ with $\aa_i(v)<0$. Fix an $T$-tuple $\ve=(\ve_i)_{i\in T}$, with $\ve_i\in\{\pm 1\}$, and consider the set $$\label{eq:2} c=c(\ve)=\{v\in V\,\|\, %\aa_i(v)=\lambda_v\ve_i (\lambda_v>0) \ve_i\aa_i(v)>0\text{ for all }i \}.$$ So each $\aa_i(v)$ is non-zero and $\aa_i(v),\ve_i$ have the same sign for all $i$. If this set is non-empty then call it a chamber* of $W$. A non-empty set of the form $$\label{eq:3} a=a(\ve)=\{v\in V\,\|\,\aa_{i_0}(v)=0\text{ for some }i_0, %\text{ and }\aa_i(v)=\lambda_v\ve_i (\lambda_v>0)\text{ for %}i\not=i_0 \text{ and $\ve_i\aa_i(v)>0$ for all $i\not=i_0$} \}$$ is called a panel. Here is the example from §\[lecture1\]: $$\begin{pspicture}(0,0)(\textwidth,4) %\showgrid \rput(3,2){ \rput(0,0){\BoxedEPSF{fig11a.eps scaled 600}} } \rput(-3.3,0){ \rput(4.25,0.8){{\red chambers}} \rput(7.4,3){$\scriptstyle{+++}$} \rput(8,2){$\scriptstyle{+-+}$} \rput(7.4,1){$\scriptstyle{+--}$} \rput(6.2,3){$\scriptstyle{-++}$} \rput(5.8,2){$\scriptstyle{-+-}$} \rput(6.2,1){$\scriptstyle{---}$} \rput(7.5,3.5){$\aa_1$}\rput(8.35,3.2){$\aa_2$}\rput(8.7,1.3){$\aa_3$} } \rput(-1,0){ \rput(10.5,2){ \rput(0,0){\BoxedEPSF{fig11b.eps scaled 600}} } \rput(3,0){ \rput(6.8,0.75){$0--$} \rput(6.8,3.25){$0++$} \rput(5.6,1.4){$-0-$} \rput(8,1.4){$+-0$} \rput(5.6,2.6){$-+0$} \rput(8,2.6){$+0+$} \rput(10,2){{\red panels}} }} \end{pspicture}$$ where there are three hyperplanes in $\HH$ and the $\aa_i$ are chosen so that $\aa_i(v)>0$ for those $v$ on the side indicated by the arrow. The chambers are marked by their $T$-tuples. There are $2^3$ $T$-tuples but only $6$ chambers because the tuples $++-$ and $--+$ give empty sets in (\[eq:2\]). Extend the notation to include panels (\[eq:3\]) by placing a $0$ in the $i_0$-th position. There are then $3.2^2$ such tuples but only $6$ give non-empty panels, with two lying on each reflecting line. There is an obvious notion of adjacency between chambers suggested by these pictures. Say that $a$ is a panel of the chamber $c$ if the corresponding $T$-tuples are identical except in one position where the tuple for $a$ has a $0$. It turns out that this can also be defined topologically: $a$ is a panel of $c$ exactly when $\bar{a}\subset\bar{c}$, the closures of these sets with respect to the usual topology on $V$. Chambers $c_1$ and $c_2$ are then adjacent if they share a common panel. In the Example from §\[lecture1\], chambers are adjacent when they share a common edge. The adjacency relation can be refined by bringing the reflection group $W$ into the picture. In §\[lecture1\] we saw that $\gS_3$ acts regularly as a reflection group on the chambers. This turns out to be true in general for the $W$-action on the chambers: given chambers $c,c'$ there is a unique $g\in W$ with $g c=c'$. Fix one of the chambers $c_0$. Then the regular action gives the chambers are in one-one correspondence with the elements of $W$ via $g\in W\leftrightarrow \text{chamber }gc_0$. Now let $S=\{s_1,\ldots,s_n\}$ be those reflections in $W$ whose hyperplanes $H_1,\ldots,H_n$ are spanned by a panel of the fixed chamber $c_0$. Thus $S=\{s_1,s_2\}$ for the $c_0$ in the example from §\[lecture1\]: $$\begin{pspicture}(0,0)(\textwidth,4) %\showgrid \rput(3,2){ \rput(0,0){\BoxedEPSF{fig12a.eps scaled 600}} \rput(0.5,1){$c_0$} \rput(-0.5,2){$s_1$}\rput(2,1){$s_2$} } \rput(9.5,2){ \rput(0,0){\BoxedEPSF{fig12b.eps scaled 600}} \rput(0.7,1){$c_0$}\rput(-0.7,1){$sc_0$} \rput(-0.7,-1){$c_1$}\rput(0.7,-1){$c_2$} \rput(-0.4,2){$s$} \rput(-1.1,0){${\red g}$} } \end{pspicture}$$ Suppose $c_1,c_2$ are a pair of adjacent chambers as above right. Then there is a $g\in W$ with $c_1=gc_0$. Translating the picture back to $c_0$ we have $g^{-1}c_1=c_0$ and $g^{-1}c_2$ are adjacent chambers, and the common panel of $c_1,c_2$ is sent by $g^{-1}$ to a common panel of $c_0$ and $g^{-1}c_2$ (these are most easily seen using the topological version of adjacency). If $s\in S$ is the reflection in the hyperplane spanned by the common panel of $c_0$ and $g^{-1}c_2$, then the chamber $g^{-1}c_2$ is the same as the chamber $sc_0$. Thus $c_1=gc_0$, $c_2=(gs)c_0$, and we have the following more refined description of adjancey: $$\label{eq:4} \text{the chambers adjacent to the chamber $gc_0$ are the $(gs)c_0$ for $s\in S$.}$$ When $S=\{s_1,\ldots,s_n\}$ we say that chambers $gc_0$ and $gs_ic_0$ are $i$-adjacent. In our running example, the chambers adjacent to $gc_0$ are $gs_1c_0$ and $gs_2c_0$, and these are the two that were $1$- and $2$-adjacent to $gc_0$ in §\[lecture1\]. #### Coxeter groups We motivate the definition of Coxeter group by quoting two facts, staying with the assumptions above where $W$ is generated by orthogonal reflections in finitely many hyperplanes $\HH$ with $W\HH=\HH$: #### Fact 1. The group $W$ is generated by the reflections $s\in S$ in those hyperplanes spanned by a panel of the fixed chamber $c_0$. #### In our running example we can see a how a proof might work using induction on the “distance” of a chamber from $c_0$. If $g$ is an element of $W$ then there is a chamber adjacent to the chamber $gc_0$ that is closer to $c_0$ than $gc_0$ is. If this closer chamber is $g'c_0$ say, then by (\[eq:4\]) we have $g=g's$ for some $s\in S$. Repeat the process until $g$ completely decomposes as a word in the $s\in S$. #### Fact 2. With respect to the generators $S$ the group $W$ admits a presentation $$\label{eq:5} \langle s\in S\,\|\, (s_i s_j)^{m_{ij}}=1\rangle$$ where the $m_{ij}\in\Z^{\geq 1}$ and are such that $m_{ij}=m_{ji}$, and $m_{ij}=1$ if and only if $i=j$ (so in particular, $s_i^2=1$). #### If $s_i$ and $s_j$ are reflections in $W$ finite, then the element $s_i s_j$ has finite order $m_{ij}\geq 2$. So the relations in the presentation (\[eq:5\]) certainly hold. The content of Fact 2 is that these relations suffice. Geometrically, $s_i s_j$ is a rotation “about” the intersection $H_i\cap H_j$ of the corresponding hyperplanes. In Example \[example:finite\] we have $S=\{s_1,\ldots,s_n\}$ where $s_i=s_{v_i-v_{i+1}}$. The $s_is_{i+1}$ have order $3$ and all other $s_is_j$ have order $2$. Moreover $W$ is isomorphic to $\gS_{n+1}$ via the map induced by $(i,i+1)\mapsto s_i$. Our running example of the action of $\gS_3$ on $3$-dimensional $V$ is the $n=2$ case of this. Here is the promised abstraction of reflection group: a group $W$ is called a Coxeter group if it admits a presentation (\[eq:5\]) with respect to some finite $S$, where the $m_{ij}\in\Z^{\geq 1}\cup\{\infty\}$ satisfy the rules following (\[eq:5\]). Sometimes the dependency on the relations $S$ is emphasized and $(W,S)$ is called a Coxeter system. We want the new concept to cover all the examples we have seen so far in this section, including the affine group in Example \[example:affine\] where the element $s_1s_0$ had infinite order. This is why in the definition of Coxeter group the conditions on the $m_{ij}$ are relaxed to allow them to be infinite. A relation $(s_i s_j)^{m_{ij}}=1$ is omitted from the presentation when $m_{ij}=\infty$. There is a standard shorthand for a Coxeter presentation (\[eq:5\]) called the Coxeter symbol. This is a graph with nodes the $s_i\in S$, and where nodes $s_i$ and $s_j$ are joined by an edge labeled $m_{ij}$ if $m_{ij}\geq 4$, an unlabeled edge if $m_{ij}=3$ and no edge when $m_{ij}=2$: $$\begin{pspicture}(0,0)(1,1) \pscircle[fillstyle=solid,fillcolor=white](0,0.8){0.125} \pscircle[fillstyle=solid,fillcolor=white](1,0.8){0.125} \rput(0.5,0.2){$m_{ij}=2$} \end{pspicture} \quad\quad\quad \begin{pspicture}(0,0)(1,1) \psline(0,0.8)(1,0.8) \pscircle[fillstyle=solid,fillcolor=white](0,0.8){0.125} \pscircle[fillstyle=solid,fillcolor=white](1,0.8){0.125} \rput(0.5,0.2){$m_{ij}=3$} \end{pspicture} \quad\quad\quad \begin{pspicture}(0,0)(1,1) \psline(0,0.8)(1,0.8) \pscircle[fillstyle=solid,fillcolor=white](0,0.8){0.125} \pscircle[fillstyle=solid,fillcolor=white](1,0.8){0.125} \rput(0.5,1){$m_{ij}$} \rput(0.5,0.2){$m_{ij}\geq 4$} \end{pspicture}$$ The examples from §\[lecture1\] and Example \[example:affine\] are then: $$\begin{pspicture}(0,0)(\textwidth,4) %\showgrid \rput(3.5,2){ \rput(0,0){\BoxedEPSF{fig13.eps scaled 600}} \rput(-0.3,1.85){$s_1$}\rput(2,1){$s_2$} } \rput(0.5,2){ \psline(0,0)(1,0) \pscircle[fillstyle=solid,fillcolor=white](0,0){0.125} \pscircle[fillstyle=solid,fillcolor=white](1,0){0.125} \rput(0,-0.3){$s_1$}\rput(1,-0.3){$s_2$} %\rput(0.5,0.2){$\scriptstyle{m_{\aaj}}$} } \rput(11.5,1){ \psline(0,0)(1,0) \pscircle[fillstyle=solid,fillcolor=white](0,0){0.125} \pscircle[fillstyle=solid,fillcolor=white](1,0){0.125} \rput(0,-0.3){$s_0$}\rput(1,-0.3){$s_1$} \rput(0.5,0.2){$\infty$} } \rput(9.5,2){ \rput(0,0){\BoxedEPSF{affine.group3.eps scaled 500}} \rput(0,1.2){$s_0$}\rput(1,1.2){$s_1$} } \end{pspicture}$$ \[remark:titsrep\] What is the relationship between the concrete reflection groups defined at the beginning of this section and the abstract Coxeter groups defined at the end? The answer is that the Coxeter groups are discrete reflection groups: for a Coxeter system $(W,S)$ one can construct a faithful representation $(W,S)\rightarrow GL(V)$ for some vector space $V$, where the $s\in S$ act on $V$ as reflections, and the image of $(W,S)$ is a discrete subgroup of $GL(V)$. Chamber Systems and Coxeter Complexes {#lecture3} ===================================== We have seen several examples of sets of chambers with different kinds of adjacency between them. This section introduces the formalization of this idea: chamber systems. A chamber system over a finite set $I$ is a set $\Delta$ equipped with equivalence relations $\sim_i$, one for each $i\in I$. The $c\in\Delta$ are the chambers and two chambers are $i$-adjacent when $c\sim_i c'$. The generic picture to keep in mind is below where chambers are $i$-adjacent if they share a common $i$-labeled edge. Thus, $c_0\sim_1 c_1,c_0\sim_2 c_2$, etc. $$\begin{pspicture}(0,0)(\textwidth,3) %\showgrid \rput(6.25,1.5){ \rput(0,0){\BoxedEPSF{chamber.complex.eps scaled 500}} \rput(0.425,0.675){${\scriptstyle 1}$} \rput(-0.375,0.675){${\scriptstyle 2}$} \rput(0,0){${\scriptstyle 3}$} \rput(0,0.4){${\scriptstyle c_0}$} \rput(0.75,0.8){${\scriptstyle c_1}$} \rput(-0.75,0.8){${\scriptstyle c_2}$} \rput(0,-0.5){${\scriptstyle c_3}$} } %\rput(2,0.3){$c_0\sim_1 c_1,c_0\sim_2 c_2$, etc} \end{pspicture}$$ A gallery in a chamber system $\Delta$ is a sequence of chambers $$\label{eq:6} c_0\sim_{i_1} c_1\sim_{i_2}\cdots\sim_{i_k} c_k$$ with $c_{j-1}$ and $c_j$ $i_j$-adjacent and $c_{j-1}\not=c_{j}$. The last condition is a technicality to help with the accounting. We say that the gallery (\[eq:6\]) has type $i_1i_2\ldots i_k$, and write $c_0\rightarrow_f c_k$ where $f=i_1i_2\ldots i_k$. If $J\subseteq I$ then a $J$-gallery is a gallery of type $i_1i_2\ldots i_k$ with the $i_j\in J$. A subset $\Delta'\subseteq\Delta$ of chambers is $J$-connected when any two $c,c'\in\Delta'$ can be joined by a $J$-gallery that is contained in $\Delta'$. The $J$-residues of $\Delta$ are the $J$-connected components and they have rank $\|J\|$. Thus the chambers themselves are the rank $0$ residues. The rank $1$ residues are the equivalence classes of the equivalence relations $\sim_i$ as $i$ runs through $I$. Call these rank $1$ residues the panels of $\Delta$. The chamber system itself has rank $\|I\|$. A morphism $\aa:\Delta\rightarrow\Delta'$ of chamber systems (both over the same set $I$) is a map of the chambers of $\Delta$ to the chambers of $\Delta'$ that preserves $i$-adjacence for all $i$: if $c\sim_i c'$ in $\Delta$ then $\aa(c)\sim_i \aa(c')$ in $\Delta'$. An isomorphism is a bijective morphism whose inverse is also a morphism. \[example:chambers:generic\] The local picture from §\[lecture1\] (below left) is a chamber system over $I=\{1,2\}$, with chambers the edges, and two chambers $i$-adjacent when they share a common $i$-labeled vertex. The $\{i\}$-residues, or panels, are the pairs of edges having a $i$-labeled vertex in common; in particular each panel contains exactly two chambers and there is a one-one correspondence between the panels and the vertices: $$\begin{pspicture}(0,0)(\textwidth,3) %\showgrid \rput(2,1.5){ \rput(0.6,0.1){\BoxedEPSF{fig3e.eps scaled 500}} %\rput(-2.25,-0.8){{\red $1$-incident}} %\rput(-1.75,-1.15){{\red chambers}} \rput(2.75,1.05){{\red $\{2\}$-residue or panel}} \rput(-1.425,-1.525){ \rput(1.5,2.45){$\scriptstyle{1}$} \rput(2.3,2.05){$\scriptstyle{2}$} \rput(2.3,0.95){$\scriptstyle{1}$} \rput(1.5,0.55){$\scriptstyle{2}$} \rput(0.7,0.95){$\scriptstyle{1}$} \rput(0.7,2.05){$\scriptstyle{2}$} } } \rput(9,1.5){ \rput(0,0){\BoxedEPSF{affine.coxeter.eps scaled 500}} \rput(0.75,-1.35){${\scriptstyle 2}$} \rput(0.75,1.35){${\scriptstyle 2}$} \rput(-1.55,0){${\scriptstyle 2}$} \rput(-0.75,-1.35){${\scriptstyle 1}$} \rput(0,0){${\scriptstyle 3}$} \rput(-0.75,1.35){${\scriptstyle 1}$} \rput(1.55,0){${\scriptstyle 1}$} \rput(0.425,0.675){${\scriptstyle 1}$} \rput(-0.375,0.675){${\scriptstyle 2}$} \rput(-1.2,0.675){${\scriptstyle 3}$} \rput(-0.375,-0.675){${\scriptstyle 2}$} \rput(-1.2,-0.675){${\scriptstyle 3}$} \rput(-1.95,0.675){${\scriptstyle 1}$} \rput(0.425,-0.675){${\scriptstyle 1}$} \rput(-1.95,-0.675){${\scriptstyle 1}$} \rput(1.95,0.675){${\scriptstyle 2}$} \rput(1.2,0.675){${\scriptstyle 3}$} \rput(1.95,-0.675){${\scriptstyle 2}$} \rput(1.2,-0.675){${\scriptstyle 3}$} } \end{pspicture}$$ The example above right has chambers the $2$-simplicies, $I=\{1,2,3\}$, and two chambers $i$-adjacent when they share a common $i$-labeled edge. The six highlighted $2$-simplicies are a $\{2,3\}$-residue and the pair of $2$-simplicies a $\{1\}$-residue or panel (so again, each panel contains two chambers). The six chambers in the rank $2$ residue have a single common vertex at their center, and there is a one-one correspondence between the rank $2$ residues and the vertices; similarly there is a one-one correspondence between the panels and the edges. So the chambers are the maximal dimensional simplicies and the residues correspond to the lower dimensional ones. We will return to this point below. \[example:flagcomplexes\] Generalizing the example of §\[lecture1\], let $V$ be an $n$-dimensional vector space over a field $k$. A flag is a sequence of subspaces $V_{i_0}\subset\cdots\subset V_{i_k}$ with $V_{i_j}$ a proper subspace of $V_{i_{j+1}}$. Let $\Delta$ be the chamber system over $I=\{1,\ldots,n-1\}$ whose chambers are the maximal flags $V_1\subset\cdots\subset V_{n-1}$ with $\dim V_i=i$, and where $$(V_1\subset\cdots\subset V_{n-1}) \sim_i (V'_1\subset\cdots\subset V'_{n-1})$$ when $V_j=V'_j$ for $j\not= i$, i.e. any difference between the maximal flags occurs only in the $i$-th position. The chambers in the panel (or $\{i\}$-residue) containing $V_1\subset\cdots\subset V_{n-1}$ correspond to the $1$-dimensional subspaces of the $2$-dimensional space $V_{i+1}/V_{i-1}$. If $k$ is finite of order $q$ then each panel thus contains $q+1$ chambers; if $k$ is infinite then each panel contains infinitely many chambers. \[example:coxeter.complexes\] In §\[lecture2\] we defined chambers, panels and $i$-adjacence for a finite reflection group $W$ acting on a Euclidean space: the chambers were in one-one correspondence with the elements of $W$ via $g\leftrightarrow gc_0$ ($c_0$ a fixed fundamental chamber), and $gc_0$ and $g'c_0$ were $i$-adjacent when $g'=gs_i$. Now let $(W,S)$ be a Coxeter system with $S=\{s_i\}_{i\in I}$. The Coxeter complex $\Delta_W$ is the chamber system over $I$ with chambers the elements of $W$ and $$\label{eq:8} g\sim_i g'\text{ if and only if }g'=gs_i\text{ in $W$}.$$ Thus $g\sim_i gs_i$ and also $gs_i\sim_i gs_is_i=g$. The $\{i\}$-panel containing $g$ is thus $\{g,gs_i\}$, so each panel contains exactly two chambers (which can be thought of as lying on either side of the panel). This is the picture the geometry was giving us in §\[lecture2\]. A gallery in $\Delta_W$ has the form $$g\sim_{i_1} gs_{i_1} \sim_{i_2}gs_{i_1}s_{i_2}\sim\cdots\sim_{i_k}gs_{i_1}s_{i_2}\ldots s_{i_k}.$$ If $f=i_1i_2\ldots i_k$ and $s_f=s_{i_1}s_{i_2}\ldots s_{i_k}$, then there is a gallery $g\rightarrow_f g'$ in $\Delta_W$ exactly when $g'=gs_f$ in $W$. If $s_i,s_j\in S$ then starting at the chamber $g$ we can set off in the two directions given by the galleries: $$g\sim_{i} gs_{i}\sim_{j}gs_{i}s_{j}\sim_{i}gs_{i}s_{j}s_i \cdots \hspace{0.5cm} \text{ and } \hspace{0.5cm} g\sim_{j} gs_{j}\sim_{i}gs_{j}s_{i}\sim_{j}gs_{j}s_{i}s_j \cdots$$ If the order of $s_i s_j$ is finite, then $(s_i s_j)^{m_{ij}}=1$ is equivalent to the relation $$s_i s_j s_i\cdots=s_j s_i s_j\cdots,$$ where there are $m_{ij}$ symbols on both sides, so the two galleries above, despite starting out in opposite directions, nevertheless end up at the same place: the chamber $g s_i s_j s_i\cdots=gs_j s_i s_j\cdots$. Thus the $\{i,j\}$-residues in $\Delta_W$ are circuits containing $2m_{ij}$ chambers when $s_is_j$ has finite order. If the order is not finite then the residue is an infinitely long line of chambers stretching in “both directions” from $g$. The two Coxeter groups from the end of §\[lecture2\] have Coxeter complexes illustrating both these phenomena: $$ (0,0)(,3) (2.5,1.5)[ (0,0) (-1.475,-1.5)[ (1.5,2.45)[$\scriptstyle{1}$]{} (2.3,2.05)[$\scriptstyle{2}$]{} (2.3,0.95)[$\scriptstyle{1}$]{} (1.5,0.55)[$\scriptstyle{2}$]{} (0.7,0.95)[$\scriptstyle{1}$]{} (0.7,2.05)[$\scriptstyle{2}$]{} ]{} (0.6,1.175)[$1$]{} (-0.6,1.15)[$s_1$]{} (1.3,0)[$s_2$]{} (-1.4,0)[$s_1s_2$]{} (0.7,-1.15)[$s_2s_1$]{} (-1.5,-1.15)[$s_1s_2s_1=s_2s_1s_2$]{} ]{} (4,0.5)[ (0,0)(1,0) (0,0)[0.125]{} (1,0)[0.125]{} (0,-0.3)[$s_1$]{}(1,-0.3)[$s_2$]{} ]{} (8.5,2)[ (0,0) (-3.3,-.3)[$\scriptstyle{1}$]{} (-2.2,-.3)[$\scriptstyle{0}$]{} (-1.1,-.3)[$\scriptstyle{1}$]{} (0,-.3)[$\scriptstyle{0}$]{} (1.1,-.3)[$\scriptstyle{1}$]{} (2.2,-.3)[$\scriptstyle{0}$]{} (3.3,-.3)[$\scriptstyle{1}$]{} (-2.75,0.2)[$s_0s_1s_0$]{} (-1.65,0.2)[$s_0s_1$]{} (-0.555,0.2)[$s_0$]{} (0.55,0.2)[$1$]{} (1.65,0.2)[$s_1$]{} (2.75,0.2)[$s_1s_0$]{} ]{} (8,1)[ (0,0)(1,0) (0,0)[0.125]{} (1,0)[0.125]{} (0,-0.3)[$s_0$]{}(1,-0.3)[$s_1$]{} (0.5,0.2)[$\infty$]{} ]{} $$ #### Aside. In all our pictures of chamber systems, the chambers, panels and lower dimensional cells have been simplicies. It turns out that chamber systems are particularly nice examples of simplicial complexes where the chambers are the maximal dimensional simplicies. Moreover in all the chamber systems arising in these lectures there is a correspondence between the lower dimensional simplicies and the residues. To see why, recall that an abstract simplicial complex $X$ with vertex set $V$ is a collection of subsets of $V$ such that $$\text{(a). }\ss\in X\text{ and }\tau\subset\ss\Rightarrow\tau\in X \text{ and } \text{(b). }\{v\}\in X\text{ for all }v\in V.$$ A $\ss=\{v_0,\ldots,v_k\}$ is a $k$-simplex of the simplicial complex $X$. The empty set $\varnothing$ is by convention the unique simplex of dimension $-1$. Now let $\Delta$ be a chamber system over $I$ and let $V$ be the set of residues of rank $\|I\|-1$ (recall that there is only one residue of rank $\|I\|$, namely $\Delta$ itself). Then let $X_\Delta$ be the simplicial complex with vertex set $V$ and such that if $R_0,\ldots,R_k$ are rank $\|I\|-1$ residues then $$\ss=\{R_0,\ldots,R_k\}\text{ is a $k$-simplex of }X_\Delta \Leftrightarrow \bigcap R_i\not=\varnothing.$$ In other words, $X_\Delta$ is the nerve of the covering of $\Delta$ by rank $\|I\|-1$ residues. Take the empty intersection to be the union $\bigcup_V R_i$, and observe that the maximum dimension a simplex can have is $\|I\|-1$. If $\Delta$ is the flag complex chamber system of Example \[example:flagcomplexes\] with chambers the maximal flags, then the $k$-simplicies of $X_\Delta$ correspond to the flags $V_{i_0}\subset\cdots\subset V_{i_k}$ containing $k+1$ subspaces. We illustrate with the Coxeter complex $\Delta_W$ of the Coxeter system $(W,S)$ with the symbol shown below left. Some elements of $W$ have been written down in a suggestive pattern, grouped into three rank $2$ residues. The simplicial complex $X_\Delta$ acquires a $2$-simplex from these residues as any two intersect in a residue of rank $1$ and all three intersect in a residue of rank $0$. In fact $X_W$ is the infinite tiling of the plane from Example \[example:chambers:generic\]: $$\begin{pspicture}(0,0)(\textwidth,5) %\showgrid \rput(1,1){ \rput(0,0){\BoxedEPSF{fig22.eps scaled 450}} \rput(-0.7,0.95){$s_1$}\rput(0,-0.95){$s_2$}\rput(0.7,0.95){$s_3$} \rput(0,1.3){$(W,S)$} } \rput(5,2.5){ \rput(0,0){\BoxedEPSF{nerve.eps scaled 650}} \rput(0,1.25){$s_1s_3$} \rput(-1,-0.55){$s_1s_2$} \rput(1,-0.55){$1$} \rput(0,0){$s_1$} \rput(2,0){$s_3$} \rput(2,1.25){$s_3s_1$} \rput(1,1.85){$s_1s_3s_1$} \rput(-1,1.85){$s_1s_3s_2$} \rput(-2,0){$s_1s_2s_3$} \rput(-2,1.25){$s_1s_2s_3s_2$} \rput(1,-1.7){$s_2$} \rput(-1,-1.7){$s_1s_2s_1$} \rput(0,-2.3){$s_2s_1$} \rput(4.2,1.5){{\red $\{1,3\}$-residue}} \rput(-4,1.5){{\red $\{2,3\}$-residue}} \rput(3.5,-0.7){{\red $\{1,2\}$-residue}} \rput(-2.25,-1.75){$\Delta_W$} } \rput(11,2){ \rput(0,0){\BoxedEPSF{fig21a.eps scaled 500}} \rput(1,1.5){$X_\Delta$} } \end{pspicture}$$ It would seem from this example that if $R_0,\ldots,R_k$ are rank $\|I\|-1$ residues over $J_0,\ldots,J_k$ with $\bigcap R_i\not=\varnothing$, then $\bigcap R_i$ is a residue over $\bigcap J_i$. In fact this is always true for a Coxeter complex and indeed any building, although not for an arbitrary chamber system. As $\bigcap J_i$ has $\|I\|-(k+1)$ elements, there is a one-one correspondence between the simplicies of $X_\Delta$ and the residues of $\Delta$: $$\text{codimension }\ell\text{ simplicies }\ss=\{R_0,\ldots,R_{m}\} \leftrightarrow \text{ residues }\bigcap_{i=0}^{m} R_i\text{ of rank }\ell,$$ where $m=\|I\|-(\ell+1)$. So for buildings the chambers of a chamber system $\Delta$ are the top dimensional simplicies of $X_\Delta$, with the lower dimensional simplicies given by the residues. #### Returning to the general discussion, we now have all the properties of chamber systems that we need. We finish the section by defining a $W$-valued metric on a Coxeter complex $\Delta_W$. If $(W,S)$ is a Coxeter system and $f=i_1i_2\ldots i_k$ with $s_f=s_{i_1}s_{i_2}\ldots s_{i_k}$, then we have seen that there is a gallery $g\rightarrow_f g'$ in $\Delta_W$ exactly when $g'=gs_f$ in $W$. Call such a gallery minimal* if there is no gallery in $\Delta_W$ from $g$ to $g'$ that passes through fewer chambers. Call an expression $s_f=s_{i_1}s_{i_2}\ldots s_{i_k}$ reduced if there is no expression in $W$ for $s_f$ involving fewer $s$’s (counted with multiplicity). Thus a gallery $g\rightarrow_f g'$ is minimal if and only if the expression $s_f$ is reduced. Define $\delta_W:\Delta_W\times\Delta_W\rightarrow W$ by $\delta_W(g,g')=g^{-1}g'$. Then $$\label{eq:7} \delta_W(g,g')=s_f \Leftrightarrow g'=gs_f \Leftrightarrow \text{ there is a gallery }g\rightarrow_f g'.$$ Moreover, $\delta_W(g,g')$ is reduced if and only if the gallery $g\rightarrow_f g'$ is minimal. A slight relaxation will define the metric on an arbitrary building. Here are two examples, one of which is our running one: $$\begin{pspicture}(0,0)(\textwidth,4) %\showgrid \rput(3.8,2.25){ \rput(0,0){\BoxedEPSF{fig3d.eps scaled 500}} \rput(-1.475,-1.5){ \rput(1.5,2.45){$\scriptstyle{1}$} \rput(2.3,2.05){$\scriptstyle{2}$} \rput(2.3,0.95){$\scriptstyle{1}$} \rput(1.5,0.55){$\scriptstyle{2}$} \rput(0.7,0.95){$\scriptstyle{1}$} \rput(0.7,2.05){$\scriptstyle{2}$} } %\rput(0.6,1.175){$\id$} %\rput(-0.6,1.15){$s_1$} \rput(1.75,0){$g'=s_2$} \rput(-1.9,0){$g=s_1s_2$} %\rput(0.7,-1.15){$s_2s_1$} %\rput(-0.95,-1.15){$s_1s_2s_1$} } \rput(0.5,3.5){ \psline(0,0)(1,0) \pscircle[fillstyle=solid,fillcolor=white](0,0){0.125} \pscircle[fillstyle=solid,fillcolor=white](1,0){0.125} \rput(0,-0.3){$s_1$}\rput(1,-0.3){$s_2$} %\rput(0.5,0.2){$\scriptstyle{m_{\aa\bb}}$} } \rput(2.2,0.5){$\delta_W(g,g')=s_2s_1s_2\,\,(=s_1s_2s_1 )$} \rput(10.5,2.25){ \rput(0,0){\BoxedEPSF{affine.coxeter3a.eps scaled 500}} \rput(-1.55,0){${\scriptstyle 2}$} \rput(0,0){${\scriptstyle 3}$} \rput(1.55,0){${\scriptstyle 1}$} \rput(0.425,0.675){${\scriptstyle 1}$} \rput(-0.375,0.675){${\scriptstyle 2}$} \rput(-1.2,0.675){${\scriptstyle 3}$} \rput(-0.375,-0.675){${\scriptstyle 2}$} \rput(-1.2,-0.675){${\scriptstyle 3}$} \rput(0.425,-0.675){${\scriptstyle 1}$} \rput(1.2,0.675){${\scriptstyle 3}$} \rput(1.2,-0.675){${\scriptstyle 3}$} \rput(-1.6,-0.5){$g$}\rput(1.45,0.4){$g'$} } \rput(9.8,0.5){$\delta_W(g,g')=s_2s_3s_2s_1s_3\,\,(=s_3s_2s_1s_3s_1=\text{etc} )$} \rput(7.25,3){ \rput(0,0){\BoxedEPSF{fig22.eps scaled 450}} \rput(-0.7,0.95){$s_1$}\rput(0,-0.95){$s_2$}\rput(0.7,0.95){$s_3$} %\rput(0,1.3){$(W,S)$} } \end{pspicture}$$ #### Another way to draw chamber systems. A chamber system over $I$ can be drawn as a graph whose edges are “coloured” by $I$. The vertices of the graph are the chambers, and two vertices are joined by an edge labeled $i\in I$ iff the corresponding chambers are $i$-adjacent. These graphs are essentially the $1$-skeletons of the duals of our simplicial complexes. If $\Delta_W$ is the Coxeter complex of the Coxeter system $(W,S)$ then this graph is the Cayley graph of $W$ with respect to the generating set $S$. Figure \[fig:permutohedron\] (left) shows the graph for the local picture of the flag complex of a four dimensional space of Figure \[fig:localpicture2\] (or the Coxeter complex of (0,0)(2.5,0.3) (0.25,0.1)[ (0,0)(1,0) (1,0)(2,0) (0,0)[0.125]{} (1,0)[0.125]{} (2,0)[0.125]{} ]{} ) and (right) the graph for the Coxeter complex of the group of symmetries of the dodecahedron (with Coxeter symbol (0,0)(2.5,0.3) (0.25,0.1)[ (0,0)(1,0) (1,0)(2,0) (0,0)[0.125]{} (1,0)[0.125]{} (2,0)[0.125]{} (1.5,0.15)[${\scriptstyle 5}$]{} ]{} ). (0,0)(12,6.5) (2.5,3.25) (9,3.25) Buildings and Apartments {#lecture4} ======================== Let $(W,S)$ be a Coxeter system with $S=\{s_i\}_{i\in I}$. A building of type $(W,S)$ is a chamber system $\Delta$ over $I$ such that: (B1). : every panel of $\Delta$ contains at least two chambers; (B2). : $\Delta$ has a $W$-valued metric $\delta:\Delta\times\Delta\rightarrow W$ such that if $s_f=s_{i_1}\ldots s_{i_k}$ is a reduced expression in $W$ then $$\delta(c,c')=s_f\Leftrightarrow\mbox{ there is a gallery }c\rightarrow_f c'\mbox{ in }\Delta.$$ There is at least one building for every Coxeter system $(W,S)$, namely the Coxeter complex $\Delta_W$ with $\delta=\delta_W$ in (\[eq:7\]), hence (B2). For (B1) we observed in Example \[example:coxeter.complexes\] that the panels in $\Delta_W$ have the form $\{g,gs\}$ for $g\in W$ and $s\in S$. Such a building, where each panel has the minimum possible number of chambers, is said to be thin. It turns out that the thin buildings are precisely the Coxeter complexes. \[example:Anbuilding\] For $(W,S)$ having this symbol ($n-1$ vertices) we put a $W$-valued metric on the flag complex of Example \[example:flagcomplexes\]. First identify $(W,S)$ with $\gS_n$ as in §\[lecture2\], with $s_i\mapsto (i,i+1)$ for $1\leq i\leq n-1$. Let $$c=(V_1\subset\cdots\subset V_{n-1}) \text{ and } c'=(V'_1\subset\cdots\subset V'_{n-1})$$ be chambers and write $V_0=V'_0=0$, $V_n=V'_n=V$. We can define $\delta(c,c')\in\gS_n$ using the filtration of $V'_i/V'_{i-1}$ of §\[lecture1\] in the obvious way. Alternatively, for $1\leq i\leq n$, let $$\pi(i)=\min\{j\,\|\,V'_i\subset V'_{i-1}+V_j\}$$ and define $\delta(c,c')=\pi$. We show that we have a building (when $\dim V=3$) at the end of this section. \[example:affinebuilding\] An affine building has type $(W,S)$ an affine reflection group as in Example \[example:affine\]. Taking this example, with $S=\{s_0,s_1\}$ and Coxeter symbol (0,0)(1.5,0.4) (0.25,0.1)[ (0,0)(1,0) (0,0)[0.125]{} (1,0)[0.125]{} (0.5,0.15)[$\infty$]{} ]{} , let $\Delta$ be the chamber system over $I=\{0,1\}$ shown below – an infinite 3-valent tree. The edges are the chambers, and two chambers are $0$-adjacent when they share a common black vertex and $1$-adjacent when they share a common white vertex. Each panel thus contains three chambers, hence (B1). The Coxeter complex $\Delta_W$ is in Example \[example:coxeter.complexes\] (also a tree). $$\begin{pspicture}(0,0)(\textwidth,5) %\showgrid \rput(3.25,0){ \rput(3,2.5){ \rput(0,0){\BoxedEPSF{affine.building3.eps scaled 450}} } \rput(6.5,2.75){ \rput(0,0){\BoxedEPSF{affine.building1a.eps scaled 450}} } \rput(6.5,2.25){ \rput(0,0){\BoxedEPSF{affine.building1b.eps scaled 450}} } \rput(7.7,2.8){$=0$-adjacent}\rput(7.7,2.3){$=1$-adjacent} \rput(3,4){$\Delta$} } \end{pspicture}$$ To define the $W$-metric on $\Delta$ recall that in a tree there is a unique path between chambers without “backtracking”: a backtrack is a path that crosses an edge and then immediately comes back across the edge again. For chambers $c,c'\in\Delta$, match this unique path between $c$ and $c'$ with the same path starting at $1$ in the Coxeter complex $\Delta_W$: $$\begin{pspicture}(0,0)(\textwidth,3) %\showgrid \rput(6.25,1.5){ \rput(0,0){\BoxedEPSF{affine.building4.eps scaled 500}} } \rput(6.25,3){{\red unique path}} \rput(11.25,2){$\Delta$}\rput(11.25,0.3){$\Delta_W$} \rput(3.6,0.6){$1$}\rput(9,0.6){$g$} \rput(3.6,1.7){$c$}\rput(8.9,1.7){$c'$} \end{pspicture}$$ and define $\delta(c,c')$ to be the resulting $g$. To see (B2), let $\delta(c,c')=g\in W$ and suppose that $g=s_{j_1}\ldots s_{j_\ell}$. Then by (\[eq:7\]) there is a gallery in $\Delta_W$ from $1$ to $g$ of type $j_1\ldots j_\ell$. As $\Delta_W$ is also a tree this gallery differs from the unique minimal one only by backtracks. First transfer this minimal gallery to $\Delta$ to get the minimal gallery from $c$ to $c'$, and then transfer the backtracks to obtain a gallery of type $j_1\ldots j_\ell$ from $c$ to $c'$. Conversely if there is a gallery from $c$ to $c'$ of type $j_i\ldots j_\ell$ with $s_{j_1}\ldots s_{j_\ell}$ reduced, then in particular no two consecutive $s$’s are the same and so the gallery has no backtracks. Thus it is the unique minimal gallery from $c$ to $c'$ giving $\delta(c,c')=s_{j_1}\ldots s_{j_\ell}$ by definition. In a Coxeter complex we have $\delta_W(c,c')=s_{i_1}\ldots s_{i_k}$ if and only if there is a gallery of type $i_1\ldots i_k$ from $c$ to $c'$, but in an arbitrary building there is the extra condition that the word $s_{i_1}\ldots s_{i_k}$ be reduced. We can see why in the example above: if there is a gallery of type $i_1\ldots i_k$ from $c$ to $c'$ with $s_{i_1}\ldots s_{i_k}$ not reduced, then $\delta(c,c')$ need not necessarily be $s_{i_1}\ldots s_{i_k}$. For example, if we have three adjacent chambers: $$\begin{pspicture}(0,0)(3,2.75) %\showgrid \rput(0,-.25){ \rput(1.5,1.5){ \rput(0,0){\BoxedEPSF{affine.building2.eps scaled 500}} } \rput(1.3,1){$c$}\rput(1,2.2){$c'$} \rput(0.4,1){${\scriptstyle{\red f=1}}$} \rput(2.5,1){${\scriptstyle{\red f'=11}}$}} \end{pspicture}$$ then there is a gallery of type $1$ from $c$ to $c'$ with $s_1$ reduced, hence $\delta(c,c')=s_1$. The non-reduced gallery $c\rightarrow_{11} c'$ does not give $\delta(c,c')=s_1s_1$, as $s_1s_1=1\not=s_1$. Examples \[example:Anbuilding\]-\[example:affinebuilding\] are our first of thick buildings: one where every panel contains at least three chambers. “Thick” is generally taken to be synonymous with interesting. It turns out that there are quite naturally arising Coxeter groups for which there are no thick buildings. One such example is the group of reflectional symmetries of a regular dodecahedron having symbol (0,0)(2.5,0.3) (0.25,0.1)[ (0,0)(1,0) (1,0)(2,0) (0,0)[0.125]{} (1,0)[0.125]{} (2,0)[0.125]{} (1.5,0.15)[${\scriptstyle 5}$]{} ]{} . In §\[lecture1\] (as well as Example \[example:affinebuilding\]) we defined the $W$-metric $\delta$ by situating a pair of chambers $c,c'$ inside a copy of the Coxeter complex $\Delta_W$ and transferring the metric $\delta_W$ defined in (\[eq:7\]). We need to see that this process is well defined – although this is obvious in Example \[example:affinebuilding\] – and that the resulting $\delta$ satisfies (B2). This leads to an alternative definition of building (Theorem \[theorem:second\_building\_definition\] below) based on this idea of defining $\delta$ locally. Let $(\Delta,\delta)$ and $(\Delta',\delta')$ be buildings of type $(W,S)$ and $X\subset (\Delta,\delta),Y\subset (\Delta',\delta')$ be subsets. A morphism $\aa:X\rightarrow Y$ is an isometry when it preserves the $W$-metrics: for all chambers $c,c'$ in $X$ we have $\delta'(\aa(c),\aa(c'))=\delta(c,c')$. A simple example is if $g_0\in W$, then $g\mapsto g_0g$ is an isometry $\Delta_W\rightarrow\Delta_W$. The following result guarantees the existence of copies of the Coxeter complex in a building: \[theorem:isometries\] Let $\Delta$ be a building of type $(W,S)$ and $X$ a subset of the Coxeter complex $\Delta_W$. Then any isometry $X\rightarrow\Delta$ extends to an isometry $\Delta_W\rightarrow\Delta$. An apartment in a building $\Delta$ of type $(W,S)$ is an isometric image of the Coxeter complex $\Delta_W$, i.e. a subset of the form $\aa(\Delta_W)$ for $\aa:\Delta_W\rightarrow\Delta$ some isometry. Apartments are precisely the local pictures we saw in §\[lecture1\]. We are particularly interested in the following two consequences of Theorem \[theorem:isometries\]: $$\label{eq:9} \text{Any two chambers $c,c'$ lie in some apartment $A$.}$$ (If $\delta(c,c')=g\in W$, then $X=(1,g)\subset\Delta_W\mapsto(c,c')\subset\Delta$ is an isometry. It extends by Theorem \[theorem:isometries\] to an isometry $\Delta_W\rightarrow\Delta$ and hence an apartment containing $c,c'$.) So the $W$-metric on $\Delta$ can be recovered from the metric on the Coxeter complex; moreover, the metrics on overlapping Coxeter complexes agree on the overlaps: $$\label{eq:10} \text{If chambers $c,c'\in A$ and $c,c'\in A'$ then there is an isometry $A\rightarrow A'$ fixing $A\cap A'$.}$$ (We leave this to the reader with the following hints: use the apartments to get an isometry $A\rightarrow A'$ fixing a chamber $c_0\in A\cap A'$; then show that every chamber in the intersection is fixed by showing that in an apartment there is a unique chamber a given $W$-distance from $c_0$.) It turns out that any chamber system covered by sufficiently many Coxeter complexes in a sufficiently nice way so that (\[eq:9\]) and (\[eq:10\]) hold can be made into a building by patching together the local metrics on the Coxeter complexes ala §\[lecture1\]. To formulate this properly we need to replace isometries by maps not involving metrics. Let $\Delta,\Delta'$ be chamber systems over the same set $I$. We leave it as an exercise to show that (i). $\aa:(\Delta,\delta)\rightarrow (\Delta',\delta')$ is an isometry of buildings if and only if $\aa:\Delta\rightarrow\Delta'$ is an injective morphism of chamber systems, and (ii). $\aa$ is a surjective isometry of buildings if and only if $\aa$ an isomorphism of chamber systems. \[theorem:second\_building\_definition\] Let $(W,S)$ be a Coxeter system with $S=\{s_i\}_{i\in I}$ and $\Delta$ a chamber system over $I$. Suppose $\Delta$ contains a collection $\{A_\aa\}$ of sub-chamber systems over $I$, called apartments, with each subsystem isomorphic (as a chamber system) to the Coxeter complex $\Delta_W$. Suppose also that (B1$^\prime$). : any two chambers $c,c'$ of $\Delta$ are contained in some apartment $A$, and (B2$^\prime$). : if chambers $c,c'\in A_\aa$ and $\in A_\bb$, then there is an isomorphism $A_\aa\rightarrow A_\bb$ fixing $A_\aa\cap A_\bb$. Define $\delta:\Delta\times\Delta\rightarrow W$ by $\delta(c,c')=\delta_W(\aa(c),\aa(c'))$ where $\aa:\Delta_W\rightarrow A$ is an isomorphism with $c,c'\in A$. Then $(\Delta,\delta)$ is a building of type $(W,S)$. The chamber system structure on the flag complex $\Delta$ of §\[lecture1\] was given there (and in Example \[example:flagcomplexes\], where we saw that $\Delta$ is thick). If $L_1,L_2,L_3$ are lines in $V$ spanned by independent vectors, then we get a hexagonal configuration as in §\[lecture1\]. Let the apartments be all the hexagons obtained in this way. If $c=V_1\subset V_2$ and $c'=V_1'\subset V_2'$ are chambers, then they can be situated in an apartment by extending $V_1,V_1'$ to a set $L_1,L_2,L_3$ of independent lines. If $V_1\not=V_1'$, $V_2\not=V_2'$ and $V_2\cap V_2'$ is a line different from $V_1,V_1'$ as for the $c,c'$ of §\[lecture1\], then this extension is unique, so $c,c'$ lie in a unique apartment. Otherwise (e.g. if $V_2\cap V_2'$ is one of $V_2$ or $V_2'$) there is some choice. In any case, if $L_1,L_2,L_3$ and $L'_1,L'_2,L'_3$ are two such extensions corresponding to apartments $A_\aa,A_\beta$ containing $c,c'$, then any $g\in GL(V)$ with $g(L_i)=L'_i$ induces an isomorphism $A_\aa\rightarrow A_\beta$ that fixes $A_\aa\cap A_\beta$. Spherical Buildings {#lecture5} =================== So far our supply of thick buildings is a little disappointing: only the flag complex of §\[lecture1\] and the affine building of Example \[example:affinebuilding\]. In this section we considerably increase the library by extracting a building from the structure of a reductive algebraic group. These guys really are the motivating examples of buildings. Call a building of type $(W,S)$ spherical when the Coxeter system $(W,S)$ is spherical (i.e. finite). It turns out that there is a uniform construction of a large class of thick spherical buildings. To motivate this we reconstruct the flag complex building $\Delta$ of §\[lecture1\] inside the general linear group $G=GL(V)\cong GL_3(k)$. First, let $P\subset G$ be the subgroup of permutation matrices – those matrices with exactly one $1$ in each row and column and all other entries $0$; alternatively, the $a_\pi=\sum_j e_{\pi j,j}$, where $\pi\in\gS_3$ and $e_{ij}$ is the $3\times 3$ matrix with a $1$ in the $ij$-th position and $0$’s elsewhere. The map $\pi\mapsto a_\pi$ is an isomorphism $\gS_3\rightarrow P$ with $$\label{eq:12} s_1=(1,2)\mapsto \left(\begin{array}{ccc} 0&1&0\\ 1&0&0\\ 0&0&1 \end{array}\right) \text{ and } s_2=(2,3)\mapsto \left(\begin{array}{ccc} 1&0&0\\ 0&0&1\\ 0&1&0 \end{array}\right).$$ For the rest of this section we will blur the distinction between the symmetric group $\gS_3$, the group of permutation matrices $P$, and the Coxeter system $(W,S)$ with the symbol (0,0)(1.5,0.3) (0.25,0.1)[ (0,0)(1,0) (0,0)[0.125]{} (1,0)[0.125]{} ]{} . Assume for the moment that: (G1). : The action of $G$ on the flag complex $\Delta$ given by $a:V_1\subset V_2\mapsto aV_1\subset aV_2$ for $a\in G$, is by chamber system isomorphisms (hence via isometries by the comments immediately prior to Theorem \[theorem:second\_building\_definition\]). (G2). : Fix $g\in (W,S)$ and let $X(g)=\{(c,c')\in\Delta\times\Delta\,\|\,\delta(c,c')=g\}$. Then for any $g$ the diagonal action $a:(c,c')\mapsto (ac,ac')$ of $G$ on $X(g)$ is transitive (thus $G$ acts transitively on the ordered pairs of chambers a fixed $W$-distance apart). (G3). : Let $A_0\subset\Delta$ be the apartment given by the lines $L_i=\langle e_i\rangle$ with $\{e_1,e_2,e_3\}$ the usual basis for $V$, and $c_0$ the chamber $\langle e_1\rangle\subset\langle e_1,e_2\rangle$ – see Figure \[fig:apartment\]. Then $P$ acts on $A_0$. Moreover, the isometry $\Delta_W\rightarrow A_0$, $g\mapsto gc_0$ is equivariant with respect to the $(W,S)$-action $g\stackrel{g_0}{\mapsto}g_0g$ on the Coxeter complex $\Delta_W$ and the $P$-action on the apartment $A_0$ (thus, the $(W,S)$-action on $\Delta_W$ is the same as the $P$-action on $A_0$). These three allow us to reconstruct the chambers, adjacency and $\gS_3$-metric of $\Delta$ inside $G$: (0,0)(3,3.5) (0,0.25)[ (1.5,1.5) (1.5,3.05)[${\langle e_1, e_3\rangle}$]{} (3.25,0.9)[${\langle e_2, e_3\rangle}$]{} (-0.25,0.9)[${\langle e_1, e_2\rangle}$]{} (0,2.1)[${\langle e_1\rangle}$]{} (1.5,-0.05)[${\langle e_2\rangle}$]{} (3,2.1)[${\langle e_2\rangle}$]{} (0.2,1.5)[$\red{c_0}$]{}]{} #### Reconstructing the chambers of $\Delta$ in $G$. For $a\in G$ we have $ac_0=c_0$ with $c_0=\langle e_1\rangle\subset\langle e_1,e_2\rangle$, exactly when $$a\in B:= \left\{\left( \begin{array}{ccc} \blob&\blob&\blob\\ 0&\blob&\blob\\ 0&0&\blob \end{array} \right) \in G \right\},$$ the subgroup of upper triangular matrices. It is easy to show that (G2) is equivalent to (G2a): the $G$-action on $\Delta$ is transitive on the chambers, and (G2b): for any $g\in (W,S)$ the action of the subgroup $B$ is transitive on the chambers $c$ such that $\delta(c_0,c)=g$. Combining (G2a) with the fact that the chamber $c_0$ has stabilizer $B$, we get a 1-1 correspondence between the chambers of $\Delta$ and the left cosets $G/B$: $$\xymatrix{ \text{chambers }ac_0\in\Delta\ar@{<->}[r]^-{\text{1-1}} & \text{cosets }aB\in G/B. }$$ #### Reconstructing the $i$-adjacency. Let $c_1,c_2\in\Delta$ be $1$-adjacent chambers: $c_1=V_1\subset V_2$ and $c_2=V_1'\subset V_2$, and let $c_i=a_ic_0$ with the $a_i\in G$. Then $a_1^{-1}a_2$ stabilizes the subspace $\langle e_1,e_2\rangle$, hence $$\label{eq:11} a_1^{-1}a_2\in \left\{\left( \begin{array}{ccc} \blob&\blob&\blob\\ \blob&\blob&\blob\\ 0&0&\blob \end{array} \right) \in G \right\}.$$ The reader can show that for $s_1$ the permutation matrix in (\[eq:12\]), the subgroup of matrices in (\[eq:11\]) is the disjoint union $B\langle s_1\rangle B:=B \cup Bs_1B$, where $BaB=\{bab'\,\|\,b,b'\in B\}$ is a double coset. Thus, if we are to replace the chambers $c_1,c_2$ by the cosets $a_1 B,a_2 B$, then we need to replace $c_1\sim_1 c_2$ by $a_1^{-1}a_2\in B\langle s_1\rangle B$. Similarly $$c_1\sim_2 c_2 \text{ exactly when the } c_i=a_ic_0\text{ with } a_1^{-1}a_2\in \left\{\left( \begin{array}{ccc} \blob&\blob&\blob\\ 0&\blob&\blob\\ 0&\blob&\blob \end{array} \right) \in G \right\} =B\langle s_2\rangle B.$$ #### Reconstructing the $\gS_3$-metric $\delta$. Let $c_1,c_2\in\Delta$ be chambers with $c_i=a_ic_0$. Suppose that $\delta(c_1,c_2)=g\in (W,S)$. As $G$ is acting by isometries (G1), we have $\delta(c_0,a_1^{-1}a_2c_0)=g$. In the Coxeter complex $\Delta_W$ we have by (\[eq:7\]) that $\delta_W(1,g)=g$, so that by (G3), $\delta(c_0,gc_0)=g$ also. Thus by (G2b) there is a $b\in B$ with $(bc_0,bgc_0)=(c_0,a_1^{-1}a_2c_0)$, so in particular, $bgc_0=a_1^{-1}a_2c_0$. As the elements of $G$ sending $c_0$ to $bgc_0$ are precisely the coset $bgB$, we get $a_1^{-1}a_2\in bgB\subset BgB$. Conversely, if $a_1^{-1}a_2\in BgB$ then $$\delta(c_1,c_2) =\delta(a_1c_0,a_2c_0) =\delta(c_0,a_1^{-1}a_2c_0) =\delta(c_0,bgb'c_0) =\delta(c_0,bgc_0),$$ for some $b\in B$, and so $$\delta(c_0,bgc_0) =\delta(bc_0,bgc_0) =\delta(c_0,gc_0) =\delta_W(1,g) =g,$$ (the first as $B$ stabilizes $c_0$, the second by (G1) and the third by (G3)). We conclude that $$\delta(c_1,c_2)=g\in (W,S)\text{ if and only if }a_1^{-1}a_2\in BgB.$$ Summarizing, let the left cosets $G/B$ be a chamber system over $I=\{1,2\}$ with adjacency defined by $a_1B\sim_ia_2B$ iff $a_1^{-1}a_2\in B\langle s_i\rangle B$ and $\gS_3$-metric $\delta(a_1B,a_2B)=g$ iff $a_1^{-1}a_2\in BgB$. Then $G/B$ is a building of type (0,0)(1.4,0.3) (0.25,0.1)[ (0,0)(1,0) (0,0)[0.125]{} (1,0)[0.125]{} ]{} , isomorphic to the flag complex of §\[lecture1\]. We leave it to the reader to show that the assumptions (G1)-(G3) hold (hint: for (G2) with $\delta(c_1,c_2)=\delta(c_1',c_2')$, situate $c_1,c_2$ in a hexagon as in §\[lecture1\] and $c_1',c_2'$ similarly. Then use the fact that $GL(V)$ acts transitively on ordered bases of $V$). We are feeling our way towards a class of groups in which we can mimic this reconstruction of the flag complex. It turns out to be convenient to formulate the class abstractly first, and then bring in the natural examples later. A Tits system or $BN$-pair for a group $G$ is a pair of subgroups $B$ and $N$ of $G$ satisfying the following axioms: (BN0). : $B$ and $N$ generate $G$; (BN1). : the subgroup $T=B\cap N$ is normal in $N$, and the quotient $N/T$ is a Coxeter system $(W,S)$ for some $S=\{s_i\}_{i\in I}$; (BN2). : for every $g\in W$ and $s\in S$ the product of double cosets[^3] $BsB\cdot BgB\subset BgB\,\bigcup\, BsgB$; (BN3). : for every $s\in S$ we have $sBs\not= B$. The group $W$ is called the Weyl group of $G$, and is in general not finite. \[example:generallinear\] $G=GL_n(k)$; $B=$ the upper triangular matrices in $G$; $N=$ the monomial matrices in $G$ (those having exactly one non-zero entry in each row and column), $$T=\{\text{diag}(t_1,\ldots,t_n)\,\|\,t_1\ldots t_n\not=0\},$$ and $W=$ the permutation matrices with $$\begin{pspicture}(0,0)(\textwidth,3.75) %\showgrid \rput(-0.75,-0.2){ \rput(7,2){$s_i= \left( \begin{array}{cccccccc} 1&&&&&&&\\ &\vrule width 4mm height 0 mm depth 0mm&&&&&\\ &&1&&&&&\\ &&&0&1&&&\\ &&&1&0&&&\\ &&&&&1&&\\ &&&&&&\vrule width 4mm height 0 mm depth 0mm&\\ &&&&&&&1 \end{array} \right)$} \rput[c](6.9,1.6){\psframe(0,0)(0.85,0.85)} \rput(-0.6,-0.1){\psline[linewidth=1pt,linestyle=dotted](6.1,3.4)(6.9,2.9)} \rput(0.1,0.1){\rput(2.2,-2.25){\psline[linewidth=1pt,linestyle=dotted](6.1,3.4)(6.9,2.9)}} } \end{pspicture}$$ for $i\in\{1,\ldots,n-1\}$, where the number of $1$’s on the diagonal before the $2\times 2$ block is $i-1$. Let $e_i$ be the $n$-column vector $(0,\ldots,1,\ldots,0)^T$ with the $1$ in the $i$-th position and $L_i=\{te_i\,\|\,t\in k\}$. Then $N$ permutes the set of lines $\{L_1,\ldots,L_n\}$ and $W$ is isomorphic to the symmetric group on this set (hence $\cong\gS_n$). This example is misleadingly special in that the extension $1\rightarrow T\rightarrow N\rightarrow W\rightarrow 1$ splits, so that the Weyl group $W$ can be realised, via the permutation matrices, as a subgroup of $G$. In general this doesn’t happen. \[theorem:generalized.flag.varieties\] Let $G$ be a group with a $BN$-pair and let $\Delta$ be a chamber system over $I$ with chambers the cosets $G/B$ and adjacency defined by $a_1B\sim_i a_2B$ iff $a_1^{-1}a_2\in B\langle s_i\rangle B$. Define a $W$-metric by $\delta(a_1B,a_2B)=g\in W$ iff $a_1^{-1}a_2\in BgB$. Then $(\Delta,\delta)$ is a thick building of type $(W,S)$. \[example:symplectic\] $G=$ the symplectic group $\sp_{2n}(k)=\{g\in GL_{2n}(k)\,\|\,g^TJg=J\}$ where $$J= \left( \begin{array}{cc} 0&I_n\\ -I_n&0 \end{array} \right),$$ with $I_n$ the $n\times n$ identity matrix; $B=$ the upper triangular matrices in $\sp_{2n}(k)$; $N=$ the monomial matrices in $\sp_{2n}(k)$, and $$T=\{\text{diag}(t_1,\ldots,t_n,t_1^{-1},\ldots,t_n^{-1})\,\|\,t_i\not=0\}.$$ Let $\{e_1,\ldots,e_n,\ov{e}_1,\ldots,\ov{e}_n\}$ be $2n$-column vectors $(0,\ldots,1,\ldots,0)^T$ with the $1$ in the $i$-th position for $e_i$ and the $(i+n)$-th position for $\ov{e}_i$. Let $L_i=\{te_i\,\|\,t\in k\}$ and $\ov{L}_i=\{t\ov{e}_i\,\|\,t\in k\}$, writing $\ov{\ov{L}}=L$. Then $N$ permutes the set $\{L_1,\ldots,L_n,\ov{L}_1,\ldots,\ov{L}_n\}$ and $W$ is isomorphic to the “signed” permutations $\gS_{\pm n}=\{\pi\in\gS_{2n}\,\|\,\pi(\ov{L}_i)=\ov{\pi(L_i)}\}$. This can be reformulated geometrically as follows. Let $V$ be a $2n$-dimensional space over $k$ and $(u,v)$ a symplectic form on $V$ – a non-degenerate alternating bilinear form[^4]. Let $O(V)$ be those linear maps preserving the form, i.e. $O(V)=\{g\in GL(V)\,\|\,(g(u),g(v))=(u,v)\text{ for all }u,v\in V\}$. The form can be defined on a basis $\{e_1,\ldots,e_n,\ov{e}_1,\ldots,\ov{e}_n\}$ by $$(e_i,e_j)=0=(\ov{e}_i,\ov{e}_j)\text{ and }(e_i,\ov{e}_j)=\delta_{ij}=-(\ov{e}_j,e_i),$$ so that $O(V)\cong \sp_{2n}(k)$. Call a subspace $U\subset V$ totally isotropic if $(u,v)=0$ for all $u,v\in U$. It turns out that the maximal totally isotropic subspaces are $n$-dimensional. A (maximal) flag in $V$ is a sequence of totally isotropic subspaces $V_1\subset\cdots\subset V_n$ with $\dim V_i=i$. Let $\Delta$ be the chamber system with chambers these flags and adjacencies over $I=\{1,\ldots,n\}$ as in the flag complex of Example \[example:flagcomplexes\]: $(V_1\subset\cdots\subset V_{n}) \sim_i (V'_1\subset\cdots\subset V'_{n})$ when $V_j=V'_j$ for $j\not= i$. Let $c_0$ be the chamber $$\langle e_1\rangle \subset \langle e_1,e_2\rangle \subset\cdots\subset \langle e_1,e_2,\ldots,e_n\rangle$$ and $A_0$ the set of images of $c_0$ under the signed permutations $\gS_{\pm n}=\{\pi\in\gS_{2n}\,\|\,\pi(\ov{e}_i)=\ov{\pi(e_i)}\}$ (writing $\ov{\ov{e}}=e$). Finally, let $\{A_\aa\}$ be the set of images of $A_0$ under $\sp_{2n}(k)$. Then this set of apartments $\Delta$ gives a building isomorphic to the spherical building of $\sp_{2n}(k)$ arising from Theorem \[theorem:generalized.flag.varieties\] and Example \[example:symplectic\]. (0,0)(,10) (4.65,5.25)[ (0,0) ]{} (10.8,1.5)[ (0,0) (0,0)[$A_0$]{} (1,1)[$L_1$]{}(-1,1)[$L_2$]{} (-1,-1)[$\ov{L}_1$]{}(1,-1)[$\ov{L}_2$]{} (0,1.35)[$L_1+L_2$]{}(0,-1.35)[$\ov{L}_1+\ov{L}_2$]{} (1.75,0)[$L_1+\ov{L}_2$]{}(-1.75,0)[$\ov{L}_1+L_2$]{} ]{} We finish where we started by drawing a picture. Let $V$ be four dimensional over the field of order $2$ and equipped with symplectic form $(u,v)$. Let $\Delta$ be the graph with vertices the proper non-trivial totally isotropic subspaces of $V$, with an edge connecting the (white) one dimensional vertex $V_i$ to the (black) two dimensional vertex $V_j$ whenever $V_i$ is a subspace of $V_j$. Any one dimensional subspace (of which there are 15) is totally isotropic, and is contained in 3 two dimensional totally isotropic subspaces, each of which in turn contains 3 one dimensional subspaces. There are thus 15 two dimensional vertices. The local pictures/apartments are octagons (or barycentrically subdivided diamonds). The apartment $A_0$ above has white vertices $L_1,L_2,\ov{L}_1,\ov{L}_2$, using the notation of Example \[example:symplectic\], and black vertices $L_1+L_2,L_1+\ov{L}_2, \ov{L}_1 +L_2$ and $\ov{L}_1+\ov{L}_2$. See Figure \[fig:flagcomplex2\]. \[remark:BNpair\] Examples \[example:generallinear\] and \[example:symplectic\] are of classical groups of matrices. This can be generalized. Let $k=\ov{k}$ be algebraically closed and $G$ a connected algebraic group defined over $k$. Suppose also that $G$ is reductive, i.e. that its unipotent radical is trivial. Let $B$ be a Borel subgroup (a maximal closed connected soluble subgroup) and $T\subset B$ a maximal torus – a subgroup isomorphic to $(k^\times)^m$ for some $m$. Finally, let $W=N/T$ be the Weyl group of $G$, where $N$ is the normalizer in $G$ of $T$. This is isomorphic to a finite Coxeter group $(W,S)$ with $S=\{s_i\}_{i\in I}$. The result is a $BN$-pair for $G$. For a general non-algebraically closed $k$ a $BN$-pair can still be extracted from $G$, but one has to tread more carefully. Notes and References {#notes-and-references .unnumbered} ==================== As mentioned in the Introduction, most of what we have said has its origins in the work of Tits, and we start by listing his (many) original contributions. Coxeter groups as a notion first appeared in his 1961 mimeographed notes, Groupes et géométries de Coxeter. These were reproduced in [@Wolf pages 740–754]. The name is a homage to [@Coxeter35]. The Bourbaki volume [@Bourbaki02] dealing with Coxeter groups was produced after “numerous conversations” with Tits. Buildings as simplicial complexes go back to the very beginnings of the subject, but the first complete account can be found in [@Tits74]. Buildings as chamber systems with a $W$-metric have their origins in [@Tits81]. The earliest reference to $BN$-pairs that we could find in Tits’s work is in [@Tits62]; they start to prove an essential tool in [@Tits64]. #### Section \[lecture1\]. This is mostly folklore. The reader is to be minded of projective geometry as $\Delta$ is the incidence graph of the standard projective plane over $k$. The ad-hoc argument (essentially the Jordan-Hölder Theorem) for associating the permutation $(1,3)$ to the pair of chambers is from [@Abramenko_Brown08 §4.3]. #### Section \[lecture2\]. Standard references on reflection groups and Coxeter groups are [@Bourbaki02] (still the only place you can find some things), [@Humphreys90] and [@Kane01]. The definition of reflection in (\[eq:1\]) is from [@Bourbaki02 V.2.2]. That $\HH$ consists of all the reflecting hyperplanes of $W$ is [@Humphreys90 Proposition 1.14]. The general theory of finite reflection groups, including their classification, can be found in Chapters 1 and 2 of [@Humphreys90]. Example \[example:affine\], although fairly standard, is taken from [@Abramenko_Brown08 §2.2.2]. The general theory of affine groups is in [@Humphreys90 Chapter 4]. For the hyperboloid or Minkowski model of hyperbolic space, hyperbolic lines, etc, see [@Ratcliffe06 Chapter 3]. The standard reference on hyperbolic reflection groups is [@Vinberg85]. The treatment of chambers, panels and adjacency is taken from [@Abramenko_Brown08 §1.1.4]. That $W$ acts regularly on the chambers is [@Humphreys90 Theorem 1.12]. Fact 1 is [@Humphreys90 Theorem 1.5] and Fact 2 is [@Humphreys90 Theorem 1.9]. For the general theory of Coxeter groups see [@Humphreys90 Chapter 5]. The representation $(W,S)\rightarrow GL(V)$ described in Remark \[remark:titsrep\] is called the geometric or reflectional or Tits representation, and is one of the crucial results of [@Wolf]. See [@Humphreys90 §5.3] for its definition; faithfulness is [@Humphreys90 Corollary 5.4] or [@Abramenko_Brown08 Theorem 2.59] (where it is also shown that the image in $GL(V)$ of $(W,S)$ is discrete). #### Section \[lecture3\]. Apart from the aside, this section is based mainly on Chapters 1-2 of [@Ronan09]; the initial chamber system notions and Example \[example:flagcomplexes\] are directly from [@Ronan09 §1.1]. Chapter 2 of this book is entirely devoted to Coxeter complexes. A thorough exploration of the general connections between chambers systems and simplicial complexes is given in [@Abramenko_Brown08 Appendix A]. The building specific set-up is in [@Abramenko_Brown08 §5.6]. The construction of the simplicial complex $X_\Delta$ as the nerve of the covering by rank $\|I\|-1$ residues is [@Abramenko_Brown08 Exercise 5.98]. The statement about the intersection of residues being a residue is [@Abramenko_Brown08 Exercise 5.32]. The edge coloured graph way of viewing chamber systems is a point of view adopted in [@Weiss03]. #### Section \[lecture4\]. This section is based on Chapter 3 of [@Ronan09] from which the definition of building is taken. That the Coxeter complexes comprise the thin buildings is from [@Ronan09 §3.2]. The alternative definition of the permutation associated to a pair of chambers of a flag complex in Example \[example:Anbuilding\] is taken from [@Weiss03 Example 7.4]. The infinite 3-valent tree of Example \[example:affinebuilding\] is an example of a building that does not have much structure as a combinatorial object. Nevertheless it can be constructed in an interesting way from a vector space over a field with a discrete valuation (and as such is an important special case of the Bruhat-Tits theory [@Bruhat_Tits72]) in the following way. Let $K$ be a non-archimedean local field with residue field $k$ and valuation ring $A$ (for example $K$ is the $p$-adics $\Q_p$ with $k=\Z/p\Z$ and $A$ the $p$-adic integers). If $V$ is a $2$-dimensional vector space over $K$, then a lattice $L\subset V$ is a free $A$-module of rank $2$. Consider the equivalence classes $\Lambda$ of lattices under the relation $L\sim Lx$ for $x\in K^\times$, and let $\Delta$ be the graph with vertices these classes and an edge joining $\Lambda,\Lambda'$ iff there are $L\in\Lambda,L'\in\Lambda'$ with $L'\subset L$ and $L/L'\cong k$. Then $\Delta$ is a tree, and Example \[example:affinebuilding\] is the case where $k$ has two elements ($K=\Q_2$ for example). See [@Serre03 II.1.1] for details. In general there is a construction that extracts a $BN$-pair, and an affine building, from an algebraic group defined over such a $K$, and Example \[example:affinebuilding\] is such an affine building for $\sl_2\Q_2$. For affine buildings in general see [@Weiss09]. The fact that the affine building for $\sl_2\Q_p$ is a tree was used by Serre to reprove a theorem of Ihara that a torsion free lattice in $\sl_2\Q_p$ is a free group. A theorem of Walter Feit and Graham Higman [@Feit_Higman64] has consequence that a finite thick building has type $(W,S)$ a finite reflection group where each irreducible component of $W$ is of type $A_n,B_n/C_n, D_n,E_6,E_7,E_8,F_4,G_2$ or $I_2(8)$ (see [@Abramenko_Brown08 Theorem 6.94]; see [@Humphreys90 Chapter 2] for a description of these types of finite reflection group). Hence there can be no finite thick buildings of type the symmetry group of the dodecahedron, for which $(W,S)$ has type $H_3$. That there are no infinite thick buildings of type $H_3$ is shown in [@Tits77]. Theorem \[theorem:isometries\] is [@Ronan09 Theorem 3.6] and Theorem \[theorem:second\_building\_definition\] is [@Ronan09 Theorem 3.11]. Prior to [@Tits81] axioms (B1$^\prime$) and (B2$^\prime$) of Theorem \[theorem:second\_building\_definition\] provided the standard definition of building. #### Section \[lecture5\]. This section is based on [@Ronan09 Chapter 5]. Properties (G1)-(G3) are the specialization to $GL_3$ of a strongly transitive group action [@Ronan09 §5.1]. The argument that reconstructs the $W$-metric is taken from the proof of [@Ronan09 Theorem 5.2]. The axioms for a $BN$-pair are from [@Ronan09 §5.1]. A proof that Example \[example:generallinear\] is a $BN$-pair using nothing but row and column operations can be found in [@Abramenko_Brown08 §6.5]. Theorem \[theorem:generalized.flag.varieties\] is [@Ronan09 Theorem 5.3]. The flag complex of a symplectic space is from [@Ronan09 Chapter 1]. Figure \[fig:flagcomplex2\] has several names: in graph theory circles it is called Tutte’s eight-cage, and is the unique smallest cubic graph with girth $8$ (where these minimal $8$-circuits are, of course, the apartments). It is a pleasantly mindless exercise to label the vertices of the Figure with the totally isotropic subspaces (hint: start with the $8$-circuit at the top as the apartment $A_0$). There is also a very simple construction that goes back to Sylvester (1844) – this (and much else) is engagingly described in [@Coxeter58]. There are $30$ odd permutations of order $2$ in $\gS_6$: $15$ transpositions – like $(1,2)$ – and $15$ products of three disjoint transpositions, like $(1,2)(3,4)(5,6)$. Let these be the vertices of the eight-cage, and join a vertex $\ss$ in one of these two groups to the three $\tau_1,\tau_2,\tau_3$ in the other group for which $\ss=\tau_1\tau_2\tau_3$. That the $B$ (Borel subgroup) and $N$ (normalizer of a maximal torus) extracted from a reductive group $G$ in Remark \[remark:BNpair\] are a $BN$-pair for $G$ is shown in [@Humphreys75 §29.1]. #### Further reading. Surely the shortest introduction to buildings is [@Brown02]; [@Brown91], [@Rousseau09] and [@Tits75] are slightly longer. The book [@Abramenko_Brown08] is a greatly expanded version of [@Brown89], while [@Ronan09] is an updated version of the 1988 original. A nice introduction to spherical buildings, including an account of Tits’s classification [@Tits74] of the thick spherical buildings of type $(W,S)$ for $\|S\|\geq 3$, is [@Weiss03]; the sequel [@Weiss09] treats affine buildings. [^1]: We will have no need for them in these notes, but one can reflect a vector space over an arbitrary field $k$: the definition is identical except that the restriction of $s$ to the reflecting line $L_s$ is the map $x\mapsto\zeta x$, where $\zeta$ is a primititve root of unity in $k$. The only such $\zeta$ in $\R$ is $-1$, hence the definition we have given of real reflections. By contrast a complex reflection can have any finite order. [^2]: Although there is no inner product in Examples \[example:affine\] and \[hyperbolic.reflection\], it is possible to endow $V$ with a bilinear form so that the reflections are “orthogonal" with respect to this form. [^3]: A $g\in W$ is not an element of $G$ but a coset $\ov{g}T$ for some representative in $\ov{g}\in N$ for $g$. As $T\subset B$, if $\ov{g}_1T=\ov{g}_2T$ then $B\ov{g}_1B=B\ov{g}_2B$, so we can unambiguously write $BgB$ to mean $B\ov{g}B$. [^4]: Alternating means $(u,u)=0$ for all $u$, and non-degenerate that $V^\perp=\{0\}$.
--- abstract: 'We show that the linking number of two homologically trivial disjoint $p$ and $(D-p-1)$-dimensional submanifolds of a $D$-dimensional manifold can be derived from the topologically massive $BC$ theory in low energy regime.' author: - \| M' arcio A. M. Gomes[^1] and R. R. Landim[^2]\ Universidade Federal do Ceará - Departamento de Física\ C.P. 6030, 60470-455 Fortaleza-Ce, Brazil title: '[Linking number from a topologically massive p-form theory]{}' --- PACS: 11.15.-q, 11.10.Ef, 11.10.Kk Keywords: topological mass generation; gauge theories; antisymmetric tensor gauge fields; arbitrary space-time dimensions. Antisymmetric tensor fields arise in string theory [@green] and supergravity [@nieu] and play an important role in dualization [@spa; @spa1; @wots]. They can be viewed as the components of a p-form field $B$ given by B=B\_[\_[1]{}...\_[p]{}]{}dx\^[\_[1]{}]{}...dx\^[\_[p]{}]{}. The theory involving a $p$-form field $B$ and a $(D-p-1)$-form field $C$ was first introduced by Horowitz [@hor1] and Blau and Thompson [@blau1]. Horowitz’s theory does not involve any local dynamics. He was in fact interested in generalizing Witten’s idea [@wit] – who proved the equivalence between the three dimensional Einstein action and the non-abelian Chern-Simons term – to an arbitrary dimension. Horowitz treated a class of models that are invariant under diffeomorphism, and that naturally bring “three dimensional gravity included as a special case”. In [@hor2], Horowitz and Srednicki used the same model to provide a definition of generalized linking number of $p$-dimensional and $\left( D-p-1\right)$-dimensional surfaces in a $D$-dimensional manifold. Later, making use of variational method, Oda and Yahikozawa [@oda0] obtained the same result and generalized it to the nonabelian case. The introduction of dynamical terms for a $p$-form field $B$ and a $(D-p-1)$-form field $C$ leads to topologically massive theories for abelian [@oda1] and non-abelian [@oda2] gauge theories. These theories are a generalization of the topological mass generation mechanism in three dimensions proposed by Deser, Jackiw and Templeton with the Chern-Simons term [@jackiw]. This also generalizes the abelian topological mass mechanism in $D=4$ constructed with a $2$-form and a vector field with a $BF$ term [@lah]. We emphasize here that the non-abelian construction proposed in [@oda2] does not describes a topolocally massive $BF$ model in $D$ dimensions. The authors did not include the Yang-Mills term, since they consider a flat connection. The non-abelian topological massive Yang-Mills theory with no flat connection was constructed in [@hwang] and [@landim1], in four and $D$ dimensions, respectively. In this paper we analyze the local effects in the correlation function $\left<B(x)C(y)\right>$ of the topologically massive abelian $BC$ model integrated over two homologically trivial disjoint submanifolds. We show that the linking number can be derived from the topologically massive $BC$ theory, in the low energy regime generalizing in part the results in [@hor2] and extending to $D$ dimensions the $3$-dimensional case [@sorella4]. We follow closely the notation and conventions adopted in [@landim2]. We use the form representation for fields with the usual Hodge $$ operator, which maps a $p$-form into a $(D-p)$-form and $\ast\ast=(-1)^{p(D-p)+1}$. The adjoint operator acting in a $p-$form is defined as $d^{\dagger }=(-1)^{Dp+D}d$ [@nakahara], where $d=dx^\mu (\partial /\partial x^\mu )$ is the exterior derivative and $D$ is the dimension of a flat manifold $\cal{M}_D$ without boundary with metric $g_{\mu \nu }=\mbox{diag}(-++\cdots +++)$. The inner product of two $p$-forms fields $A$ and $B$ are defined by =A(x)B(x)=\_MA(x)\_[\_1..\_p]{}B(x)\^[\_1..\_p]{}d\^Dx. The $\ast d$ operator maps a $p$-form into a $(D-p-1)$-form and has the properties $$\begin{aligned} &&\left(\Omega_p,\ast d\Omega_{D-p-1}\right)=(-1)^{Dp+1}(\Omega_{D-p-1},\ast d\Omega_p),\label{ast1}\\ &&\left(\Omega_p,\ast d \ast d\omega_p\right)=\left(\omega_p,\ast d \ast d\Omega_p\right)\label{ast2},\end{aligned}$$ for any $p$ and $(D-p-1)$-form. We use from now on the rules to forms functional calculus developed in [@marcio]: =\^D\_p(x-y), with $\delta^D_p(x-y)$ is defined in terms of usual Dirac delta function: $$\begin{aligned} \delta^D_p(x-y)=\frac{1}{p!}\delta^D(x-y)g_{\mu_1\nu_1}..g_{\mu_p\nu_p}dx^{\mu_1} \wedge..\wedge dx^{\mu_p}\otimes dy^{\nu_1}\wedge..\wedge dy^{\nu_p}.\end{aligned}$$ The linking number between two disjoint submanifolds of $\cal{M}_D$ can be defined as L( U,V)=\_[U]{}\_[W]{}\_[p]{}\^[D]{}( x-y), where $U$ and $V$ are boundaries of submanifolds $Z$ and $W$, namely, $U=\partial Z$ and $V=\partial W$. In this expression, $x$ and $y$ are points of $U$ and $W$ respectively, and the $\ast$ operator acts on the part of $\delta _{p}^{D}\left(x-y\right)$ defined on $W$. We start with the following classical abelian action [@oda2], S=\_[\_[D]{}]{}( (-1)\^r H\_[B]{}H\_[B]{}+(-1)\^s H\_[C]{}H\_[C]{}+mBdC), where $r=Dp+p+D$, $s=Dp+p+1$, $B$ is a $p$-form field, $C$ is a $(D-p-1)$-form field both with canonical dimension $(D-2)/2$ and $H_{B}$, $H_{C}$ are their respective field strengths $$\begin{aligned} H_{B} &=&dB, \label{HB} \\ H_{C} &=&dC, \label{HC}\end{aligned}$$ all them real-valued and $m$ is a mass parameter. The factor $(-1)$ in front of the kinetic terms is required in order to have a positive kinetic energy in the Hamiltonian. As claimed in [@oda2], the model just describe a topologically massive $BC$ model denoted by $TMBC$. Note that for $D=4$ and $p=1$ we recover the topologically massive $BF$ model [@lah]. The action is clearly invariant under the gauge transformations $$\begin{aligned} \delta B &=&d\Omega, \label{deltaB} \\ \delta C &=&d\Theta, \label{deltaC}\end{aligned}$$ where $\Omega $ and $\Theta $ are $(p-1)$-form and $(D-p-2)$-form gauge parameters. These gauge transformations are reducible, i.e.,$\Omega ^{\prime }$ and $\Theta ^{\prime }$ given by $$\begin{aligned} \Omega ^{\prime } &=&\Omega +d\omega, \label{omega} \\ \Theta ^{\prime } &=&\Theta +d\theta, \label{theta}\end{aligned}$$ are also honest gauge parameters satisfying and respectively, since $d^{2}=0$. Naturally, the same holds to $\omega $, $\theta $, etc. So, in order to construct the action to be quantized, one has to introduce ghosts and ghosts for ghosts and so on. Let us write the action in a more compact form. We introduce a doublet $\Phi(x)$, with $B(x)$ and $C(x)$ being the components fields: (x)= ( [c]{} \_1(x)\ \_2(x)\ ) = ( [c]{} B(x)\ C(x)\ ). The inner product between two doublets is defined by =+=. Then, making the use of Eqs. and , we have S\_[TMBC]{}=, where = ( [cc]{} (-1)\^[Dp+D+1]{}d/m & 1\ (-1)\^[Dp+1]{}&(-1)\^[Dp+D+1]{}d/m\ ). We are interested in the computation of \_[TMBC]{}. In order to obtain this correlation function, we must deal with the gauge-fixed action. The gauge fixed action becomes, $$\label{SGF} S_{gf} =\frac{m}{2}\inner{\Phi}{\ast ^{-1}d\cal{O}\Phi} +\inner{L}{d\ast\Phi} +\ldots,$$ where the dublet $$\label{dubNL} L(x) = \left( \begin{array}{c} L_{1}(x) \\ L_{2}(x) \\ \end{array} \right),$$ is the Nakanishi-Lautrup field introduced to implement the evaluation of path integral. Note that $L_{1}$ and $L_{2}$ are a $(D-p+1)$-form and a $(p-2)$-form respectively. The functional is written as Z=Xe\^[iS\_[gf]{}]{}, where $\cal{D}X=\cal{D}B\cal{D}C\cal{D}L_1\cal{D}L_2\cdots$ is the functional measure. From the functional identities X=0, and $$\frac{1}{Z}\int \cal{D}X\frac{\delta }{\delta \Phi \left( y\right) }\left[ L\left( x\right) e^{iS_{gf}}\right] =0,$$ we have im i +\_[p,D-p-1]{}\^[D]{}( x-y) =0, = 0, where $$\label{dubdelta} \delta _{p,D-p-1}^{D}\left( x-y\right)= \frac{\delta \Phi \left( x\right) }{\delta \Phi \left( y\right) }= \left( \begin{array}{c} \delta _{p}^{D}(x-y) \\ \delta _{D-p-1}^{D}(x-y) \\ \end{array} \right),$$ and the correlation function of two dublets is taken as being =( [c]{}\ \ ). To compute these correlation functions, we must invert the operator $\cal{O}$. But $\cal{O}^{-1}$ has local and non-local terms. To get rid of non-local terms, one has to be concerned with low energy regime. So, to get the local terms of $\cal{O}^{-1}$, we expand in powers of $\ast d/m$: \^[-1]{}= ( [cc]{} (-1)\^[D+1]{}d/m & (-1)\^[Dp+1]{}\ 1&(-1)\^[D+1]{}d/m\ ), where =\_[n=0]{}\^(-1)\^[n(Dp+1)]{}(d/m)\^[2n]{}. Since $L$ is Nakanishi-Lautrup field, $\cor{L(x)}{L(y)}=0$. Then, from Eq. , we have =0, and consequently, $$\begin{aligned} \left<\int_U B(x)\int_W \ast d\ast L_1(y)\right>=0.\end{aligned}$$ Using this identity and Eq. we arrive at m+(-1)\^[Dp+D+1]{}=iL(U,V), where we have used the Eq. . To evaluate the second term of the equation above, we take $x\ne y$ in the equation and apply $\cal{O}^{-1}$ on it: =. In low energy regime $\cal{O}^{-1}d\ast=\beta d\ast$, where = ( [cc]{} 0& (-1)\^[Dp+1]{}\ 1&0\ ). Writing Eq. in components and integrating over $U$ and $V$, it is clear that ==0. So, we finally have that iL(U,V)=m. We must enforce that this remarkable result was deduced restricting ourselves to low energy regime. Otherwise, non-local terms would appear and could jeopardize our analysis. [Acknowledgments]{} 0.5cm We wish to thank J. R. Goncalves for reading the manuscript. 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\#1\#2\#3[Ap. J. \#1, \#2 (\#3)]{} \#1\#2\#3[Ap. J. Lett. \#1, \#2 (\#3)]{} \#1\#2\#3[Phys. Lett. B\#1, \#2 (\#3)]{} \#1\#2\#3[Phys. Rev. D\#1, \#2 (\#3)]{} \#1\#2\#3[Phys. Rev. Lett. \#1, \#2 (\#3)]{} \#1\#2\#3[Nucl. Phys. B\#1, \#2 (\#3)]{} \#1\#2\#3[JHEP \#1, \#2 (\#3)]{} \#1\#2\#3[Nature, \#1, \#2 (\#3)]{} \#1\#2\#3[ Astropart. Phys. \#1, \#2 (\#3)]{} Introduction {#introduction .unnumbered} ============ Despite many experimental and theoretical efforts, the origin of almost 95$\%$ of the universe remains unknown. About one third of the content of the Universe seems to be made of matter but only a few percent is to attribute to ordinary matter. The rest, called Dark Matter (DM), is expected to be made of neutral weakly-interacting massive particles. Their mass is generally believed to be above the proton mass. However, recent studies have pointed out that lighter particles are possible, and perhaps even more promising from the astrophysical point of view. Light Dark Matter particles are supposed to annihilate into pairs of leptons-anti leptons. When they lose their energy and decay (or annihilate, as in the case of electrons and positrons), they produce energetic photons that can be detected by modern gamma-ray telescopes. The total DM annihilation cross-section can be written as $ \sigma v_r \equiv a + b v^2.$ In this expression, $v_r$ and $v$ are the DM relative and individual velocity, respectively. In the case of the LDM scenario, this cross-section must satisfy two constraints in order to be compatible with current observations: Firstly, it must have the correct value to explain why Dark Matter constitutes precisely 23$\%$ of the universe [@wmap]. This value is about $(\sigma v_r)_{\vert prim} \sim 10^{-26} \ \mbox{cm}^3\ \mbox{s}^{-1}$, and it is valid in the early universe (when $v\sim c/3$). Secondly, it must decrease with time: $(\sigma v_r)_{MW}$ in the Milky Way must be $\sim 10^{-5}$ times smaller than $(\sigma v_r)_{\vert prim}$ to avoid an overproduction of low-energy gamma rays within our galaxy [@bens]. The combination of these two constraints yields $a \lesssim 10^{-31} (m_{dm}/\rm{MeV})^2$ cm$^3$ s$^{-1}$ and $b \sim 10^{-25} (m_{dm}/\rm{MeV})^2$ cm$^3$ s$^{-1}$, valid at any time. From the particle physics point of view, the problem is to find a candidate with the appropriate annihilation cross section. A simple solution consists in coupling DM particles to a new neutral gauge boson ($\zp$). In this case, there is no a-term ($a=0$) and the b-term can be set to the correct value by imposing specific values of the $\zp$ couplings [@bf]. Strictly speaking, DM could also be coupled to new particles $F$ having a mass $m_F > 100$ GeV (to satisfy accelerator limits). The net effect is to introduce an a-term in $\sigma v_r$, which must be smaller than $10^{-31} (m_{dm}/\rm{MeV})^2$ cm$^3$ s$^{-1}$ to satisfy the gamma-ray constraint [@bf]. The LDM scenario has been shown to provide an elegant explanation of the 511 keV emission line detected at the center of our galaxy. Many experiments had measured the intensity of this emission over the years, but they lacked the resolution to determine its morphology. Recently, however, INTEGRAL/SPI experiment provided a surface brightness map which unambiguously indicates the presence of an extended source located at the galactic center [@integral]. The observed flux is in good agreement with previous data [@previous], and the morphology is well reproduced by a 2D-Gaussian with a Full Width Half Maximum (FWHM) of $\sim 10$ deg. This gamma-ray emission line can be interpreted in terms of $e^+ e^-$ annihilations, although the origin of low-energy positrons remains a matter of heated debate. Standard explanations (e.g. astrophysical sources) have been proposed, but most of them seem excluded by the value of the bulge-to-disc ratio [@pjean] or rely on rather controversial hypotheses (jet emission [@Prantzos], positron propagation [@Bertone], etc.). In contrast, the LDM scenario seems to be a quite appealing and simple explanation despite it certainly involves more exotic physics [@511]. In this framework, the DM particles (with a mass $m_{dm} < m_{\mu}$) would annihilate into electron-positron pairs. Assuming a microgauss-scale magnetic field, the latter remain confined in the galactic center, losing all their kinetic energy by collisions with the baryonic material and eventually annihilating into monoenergetic photons of 511 keV. Here, we shall see that the LDM scenario must have two important ingredients. Both should be of experimental/theoretical interest. Can the morphology of the emission be well reproduced by LDM annihilations? =========================================================================== The answer is given in Figure \[figSPI\], where we plotted the predictions of the LDM model, compared to the INTEGRAL/SPI results. We used a NFW profile to describe the dark matter halo of the Milky way, and a Gaussian source with FWHM between 6 and $18\deg$ (reported 2-$\sigma$ confidence limits) to fit the observational data. Assuming any reasonable mass distribution for our galaxy (i.e. $\rho\sim r^{-3}$ at large radii) and using a realistic description of its velocity dispersion profile, we obtain that LDM annihilations produce a significant amount of positrons, even far away from the galactic center. However, the 511 keV emission is expected to be mostly from the bulge, since the baryon density in the outer parts of the halo is simply too small for the positrons to lose their kinetic energy. For a fixed total flux, one obtains less photons from the outer regions when a cuspy density profile is assumed, in better agreement with INTEGRAL/SPI. On the other hand, the dark matter profile cannot be too cuspy, because the possibility of a point-like source is excluded by the data with a high confidence level. A thorough analysis of which halo models might be compatible with the observed morphology of the 511 keV line emission will be presented elsewhere. For a NFW profile [@nfw] with $\rho_s=0.183$ GeVcm$^{-3}$ and $r_s=25$kpc, the total flux at the Earth ($r_\odot=8.5$kpc) is $$\Phi_{tot}= m_{MeV}^{-2} ( 130\,a_{26}+ 1.37\!\times\!10^{-4}\,b_{26} )\ {\rm cm}^{-2}\,{\rm s}^{-1}$$ that we expressed in terms of the dimensionless parameters $m_{MeV}\equiv m_{dm}/(1\,{\rm MeV})$, $a_{26}\equiv a/(10^{-26}\,{\rm cm}^{-3}\,{\rm s}^{-1})$ and $b_{26}\equiv b/(10^{-26}\,{\rm cm}^{-3}\,{\rm s}^{-1})$. Due to our naive prescription for positron propagation (each $e^+$ is assumed to instantly yield two 511 keV photons), most of the emission comes from large angular distances from the galactic center. A fairer comparison with SPI data can be obtained by restricting ourselves to the inner $16\deg$. This roughly corresponds to the instrument’s field of view, and encloses most of the detected emission. Using the NFW density profile, we now obtain $$\Phi_{16}= m_{MeV}^{-2} (46.3\,a_{26}+2.98\!\times\!10^{-5}\,b_{26})\ {\rm cm}^{-2}\,{\rm s}^{-1}. \label{eqF}$$ The surface brightness beyond $16 \deg$ is expected to be below $m_{MeV}^{-2}(70a_{26}+7\!\times\!10^{-5}b_{26})\ {\rm cm}^{-2}\,{\rm s}^{-1}\,{\rm sr}^{-1}$, or even smaller because of the low baryon density outside the galactic center. This is likely to be too faint to be detected by SPI, so a NFW profile is able to fit the observed 511 keV emission if the total flux is equated to expression (\[eqF\]). A more precise assessment of the best-fit values of the coefficients $a_{26}$, $b_{26}$ and $m_{MeV}$ requires a realistic treatment of the SPI instrumental response matrix. A preliminary analysis hinted that a “pure” velocity-dependent cross-section (i.e. $\zp$ exchange only) could be entirely responsible for the 511 keV signal. Here, we show that the emission profile arising from this term is shallower than the one due to the $a$-term. This is a generic conclusion, regardless of the particular model assumed for the Milky Way mass. Therefore, our present conclusion is that, unlike stated in previous studies, the $a$-term should be included indeed. Moreover, accounting for the observed flux (with a $b$-term only) would require $m_{dm}<1$ MeV, which would have a tremendous impact on primordial nucleosynthesis. Setting $\Phi_{16}=10^{-3}\ {\rm cm}^{-2}\ {\rm s}^{-1}$ (the INTEGRAL’s result), we conclude that $a_{26}$ must be of the order of $$\begin{aligned} a_{26} &\sim& 2.16 \ 10^{-5} \ m_{MeV}^{2} \label{eqMdm} \\ \mbox{or} \ \ (\sigma v_r)_{MW} &\sim& 2.16 \ 10^{-31} \ m_{MeV}^{2} \ \mbox{cm}^3 \ \mbox{s}^{-1} \nonumber\end{aligned}$$ in order to explain the 511 keV line, assuming that LDM annihilations constitute the main source of low-energy positrons. Finally, we note that: i) fermionic LDM candidates cannot account for the observed flux, because $a_{26} \ll 10^{-7} \ m_{MeV}^2$ for these particles (except perhaps in the marginal case where they would exchange a light gauge boson with axial couplings to ordinary matter [@llbfs]; in this case $m_{dm}$ should be $\sim$ 10 MeV), ii) LDM annihilations should proceed through both F and $\zp$ exchanges. The extra gauge boson is necessary to obtain the correct relic density, while the F exchange is needed to explain the the 511 keV signal. We confirm that decaying DM cannot fit the emission in a NFW profile [@Hooper]. The need for (fermionic) $F$ particles is of interest for both atomic and particle physics experiments. Their contribution to the muon and electron anomalous magnetic moment ($g-2)_{\mu,e}$ is given by $\delta a_{\mu, e} \sim \frac{f_l f_r}{16 \pi^2} \frac{m_{\mu, e}}{m_{F_{\mu, e}}}$. Noting that the quantity $\frac{f_l f_r}{m_{F_{e}}}$ also enters the expression of $\sigma v_r$, we obtain $(\sigma v_r)_{MW} \simeq (0.864, 3.456) \ 10^{-31} \left(\frac{\delta a_{e}}{10^{-12}}\right)^2 $ cm$^3$ s$^{-1}$ (depending on whether DM is made of self-conjugate particles or not). $(\sigma v_r)_{MW}$ is dominated by its a-term. The above expression matches eq.\[eqMdm\] when $\delta a_e = (a_e^{exp} - a_e^{th}) \simeq (1.58,0.79) \ 10^{-12} \ m_{MeV}$ (respectively). Yet, we expect the experimental value $a_e^{exp}$ to be somewhat larger than the theoretical estimate $a_e^{th}$. It turns out that there exists a small discrepancy between the theoretical prediction and the experimental measurement. The latter is about $ \delta a_e \sim (3.44-3.49) \ 10^{-11}$ (the first number is obtained from the positron g-2, while the second one is from the electrons) [@Marciano]. This discrepancy is generally “disregarded” because the new physics processes that are generally considered are supposed to yield a much smaller contribution. Here, we see that DM particles with a mass of $m_{dm} \sim (21.8,43.6)$ MeV (depending on whether DM is made of self-conjugate particles or not) could surprisingly explain the discrepancy. On the other hand, greater masses would yield a too large value of $\delta a_e$. The discrepancy between $a_e^{exp}$ and $a_e^{th}$ thus appears for the first time related to a new physics process. To determine $a_e^{th}$, we used the fine structure constant obtained from the Quantum Hall effect $\alpha_{QH}$. The latter is seen to be the most accurate experimental determination of $\alpha$. However, it is never used by particle physicists, since it leads to an unexplained discrepancy. To obtain a perfect agreement between theory and observations, one instead “forces” $a_e^{th}$ to match $a_e^{exp}$. The value of $\alpha$ thus obtained (denoted $\alpha_{st}$) is quoted in the international reference CODATA [@Mohr], and it is used to get theoretical estimates of other processes. But this procedure yields wrong results if the discrepancy between $a_e^{exp}$ and $a_e^{th}$ is due to new physics. If it is so, then one should use the experimental value of $\alpha$ (for example $\alpha_{QH}$) instead of $\alpha_{st}$ to make theoretical predictions. The difference between $\alpha_{st}^{-1}$ and $\alpha_{QH}^{-1}$ is about $ 4 \ 10^{-6}$. This changes $a_e^{th}$ and presumably also $a_{\mu}^{th}$ (at least slightly). The latter was found to be smaller than the experimental value by a few $10^{-9}$ units [@E821]. This discrepancy is widely considered as a possible case for new physics and its precise value is of crucial interest. So, it would be useful to determine $a_{\mu}^{th}$ by taking $\alpha = \alpha_{QH}$ and estimate the discrepancy again. We based our previous estimate of $\delta a_e = a_e^{exp}-a_e^{th}$ on $\alpha_{QH}$. Other experimental values of $\alpha$ can be found in the literature (obtained notably from the measurement of the Rydberg constant, Josephson effect and muonium). The most precise values are thought to come from the Quantum Hall effect and the Rydberg constant, and they both give $a_e^{exp}-a_e^{th}>0$. Nevertheless, if we use the other two values we obtain a negative discrepancy, which cannot be explained by the presence of F particles, at least not in the simple form considered here (although they are not excluded neither). Assuming universality and $F$ exchange, we obtain a relationship between the electron and muon g-2: $$\left(\frac{\delta a_{\mu}}{10^{-9}}\right) = 2.1 \ (m_{Fe}/m_{F \mu}) \ \left(\frac{\delta a_{e}}{10^{-11}}\right). \label{eq1}$$ Plugging the measured discrepancy for $\delta a_e$ into the above expression and assuming $m_{Fe} = m_{F \mu}$, we find $\delta a_{\mu} \sim 7 \ 10^{-9}$. The E821 experiment measured $\delta a_{\mu} \sim (2.7 \pm 1.04) \ 10^{-9}$ (using $e^+ e^-$ data). One can therefore explain both the anomalous values of the muon and electron g-2 by introducing a set of $F$ particles satisfying $m_{Fe} \sim m_{F \mu}/x$, with presumably $x \gtrsim 2$. The correct approach to determine $x$ more precisely would be to estimate $a_{\mu}^{th}$ by using $\alpha_{QH}$. But as mentioned earlier, this has not been done yet. Note that, to our knowledge, it is the first time that a possible connexion between $\delta a_e$ and $\delta a_{\mu}$ is established. One could make a prediction for the tau $g-2$. However present experiments still lack the sensitivity to challenge the Standard Model predictions. Unambiguous signatures of $F$ particles could be detected in the Large Hadron Collider (LHC), unless they turn out to be heavier than a few TeV. They could be produced through $e^+ e^-$ collisions and detected through their two-body decay (i.e. $F$ going into electrons and DM particles). It is worthwhile to precise that $F$ particles are required to explain the 511 keV line. If the LDM scenario is not the explanation to this emission, then the spin-1 boson becomes the only ingredient that is really needed for LDM to be viable. Hints of the presence of a light gauge boson may also have been detected [@moi]. Assuming universal couplings, it was found that, albeit small, the $\zp$ couplings could be “large” enough to modify the neutrino-quark elastic scattering cross sections when the Z’ mass is about a few GeV. NuTeV collaboration measured these cross sections and found small deviations [@nutev]. QCD corrections (isospin violations, strange sea asymmetry) could well be the explanation of this anomaly [@Kretzer]. However, QCD uncertainties are still very large and, in some cases, even increase the anomaly. Therefore, it is worthwhile to consider the existence of a light gauge boson seriously. Figure \[nacio\] illustrates how a light $\zp$ can solve the NuTeV anomaly. (Note that a QCD explanation does not exclude the existence of a $\zp$, but it certainly sets an upper limit on its mass.) The case without universality seems even more interesting as one could perhaps find evidence for a light $\zp$ in high energy colliders! The measurement of the electron $g-2$ restricts the coupling $f_e$ to be less than $f_e \lesssim 5.7 \ 10^{-1} \ (\frac{\delta a_{e}}{10^{-11}})^{1/2} \ (\frac{m_{\zp}}{\rm{GeV}})$. Also, the product of the $\zp$ couplings to neutrinos and electrons must not exceed $[f_e f_{\nu}]^{max} \sim 5.388 \, 10^{-7} \ (\frac{m_{\zp}}{\rm{GeV}})^2,$ according to the very precise measurement of the elastic scattering of muon neutrinos on electrons by the CHARM II experiment [@charmii]. CHARM II’s results are in good agreement with the Standard Model (SM) predictions. However, the mean experimental value seems to be slightly different from the SM expectations. It is quite surprising but very interesting to see that the introduction of a new gauge boson with couplings $f_e f_{\nu} \simeq [f_e f_{\nu}]^{max}$ allows one to reduce this discrepancy! Let us assume $f_e \sim f_q$ (although it may be quite challenging to build a theory that predicts $f_{\nu} < f_e \sim f_q$). On one hand, the NuTeV anomaly can be explained by taking $m_{\zp}<$ GeV. On the other hand, predictions for $e^+ e^- \rightarrow e^+ e^-$ and $e^+ e^- \rightarrow q \bar{q}$ cross sections will now differ from the SM expectations by a few percent. These two cross sections have been well measured at LEP II. The intriguing point is that small deviations have been found in a preliminary analysis [@lepii]. It is probably too soon to conclude that these deviations are i) significant, ii) indeed due to new physics. However, if they turn out to be confirmed, the effect of a light gauge boson on these cross-sections would become of crucial interest for the astro/particle/astroparticle physics communities. [![The left panel represents how NuTeV’s observable $R_{num}$ is affected by a $\zp$. $R_{num}$ values in presence of a $\zp$ are symbolised by the colours ranging from black to yellow. The Standard Model predicts $R_{num}= \sum_{u,d} \left[ G_F c_v^q c_a^q \right] = 3.2072 \ 10^{-6}$ (black region) while NuTeV finds $3.1507 \ 10^{-6}$. The parameters of the gauge boson that impressively fit the NuTeV anomaly (without error bars) are represented by the blue curve with $f_q f_{\nu}= f^2 \ 10^{-6} \ (m_{\zp}/\rm{GeV})^2$. On the right panel, we show our predictions for $R_{lept}$, using $f_e f_{\nu} = f^2 \ 10^{-6} \ (m_{\zp}/\rm{GeV})^2$.[]{data-label="nacio"}](dplot.eps "fig:"){width="4.cm"} ![The left panel represents how NuTeV’s observable $R_{num}$ is affected by a $\zp$. $R_{num}$ values in presence of a $\zp$ are symbolised by the colours ranging from black to yellow. The Standard Model predicts $R_{num}= \sum_{u,d} \left[ G_F c_v^q c_a^q \right] = 3.2072 \ 10^{-6}$ (black region) while NuTeV finds $3.1507 \ 10^{-6}$. The parameters of the gauge boson that impressively fit the NuTeV anomaly (without error bars) are represented by the blue curve with $f_q f_{\nu}= f^2 \ 10^{-6} \ (m_{\zp}/\rm{GeV})^2$. On the right panel, we show our predictions for $R_{lept}$, using $f_e f_{\nu} = f^2 \ 10^{-6} \ (m_{\zp}/\rm{GeV})^2$.[]{data-label="nacio"}](rlep.eps "fig:"){width="4cm"}]{} However, since a QCD explanation is possible, one can do a test that does not involve quarks but only leptons (so as to get rid of the QCD corrections). It is based on an already proposed experiment which aims to measure the ratio $R_{lept}=\sigma_{\nu_{\mu} e}/(\sigma_{\nu_{e} e}+\sigma_{\bar{\nu_{\mu}} e})$, where $\sigma_{\nu_{\mu} e}$, $\sigma_{\nu_{e} e}$ and $\sigma_{\bar{\nu_{\mu}} e}$ are the muon neutrino-, electron neutrino- and muon anti neutrino-electron elastic scattering cross-sections, respectively [@Imlay]. If a deviation is measured (as shown in Figure 3 on the right) then it is likely to be due to the presence of a light $\zp$. If not, this may exclude it or, alternatively, impose even stricter limits on the $\zp$ couplings. This test is independent on $m_{\zp}$ or universality assumptions; it probes the maximal value for $[f_e f_{\nu}]^{max}$. If a deviation is found, one has to refer to the NuTeV findings (and evaluate the size of the QCD corrections) to get an estimate of $m_{\zp}$. Conclusion {#conclusion .unnumbered} ========== Our study indicates that the LDM scenario could have already shown up in astrophysical and particle physics processes. Direct evidences are needed, but the presence of heavy fermionic particles $F$ may have been already detected though the anomalous value of the electron $g-2$. As a major consequence, the value of the fine structure constant quoted in the CODATA and used for theoretical estimates of various particle physics processes could be wrong. Instead $\alpha$ could be close to its “direct” experimental value! Quantities like the theoretical estimate of the muon $g-2$ (subject of great debate since a few years), may have to be computed again. The existence of both $F$ and $\zp$ (if one relaxes the universality assumptions) could be challenged in high energy colliders. Their discovery would hint in the direction of $N=2$ supersymmetry [@mirror]. Acknowledgement The authors would like to thank I. de la Calle for his PAW expertise and M. Langer for interesting discussions. [99]{} C. L. Bennett [et al.]{}, “First Year Wilkinson Microwave Anisotropy Probe (WMAP)”, arXiv:astro-ph/0302207. C. Boehm, T. A. Ensslin and J. Silk, J. Phys. G [30]{} (2004) 279 \[arXiv:astro-ph/0208458\]. C. Boehm and P. Fayet, Nucl. Phys. B [683]{}, 219 (2004) \[arXiv:hep-ph/0305261\]. J. Knodlseder, [et.al.]{}, Accepted for publication in A&A, arXiv:astro-ph/0309442; P. Jean [et al.]{}, arXiv:astro-ph/0309484; D. D. Dixon [et al.]{}, arXiv:astro-ph/9703042; P. A. Milne, J. D. Kurfess, R. L. Kinzer and M. D. Leising, arXiv:astro-ph/0106157. G. Weidenspointner [et al.]{}, arXiv:astro-ph/0406178, and references therein. N. Prantzos, arXiv:astro-ph/0404501. G. Bertone, A. Kusenko, S. Palomares-Ruiz, S. Pascoli and D. Semikoz, arXiv:astro-ph/0405005. C. Boehm, D. Hooper, J. Silk and M. Casse, J.Paul Phys. Rev. Lett. [92]{}, 101301 (2004) \[arXiv:astro-ph/0309686\]. J. F. Navarro, C. S. Frenk and S. D. White, Astrophys. J. 490, 493 (1997). D. Hooper and L. T. Wang, arXiv:hep-ph/0402220; C. Picciotto and M. Pospelov, arXiv:hep-ph/0402178. P. Fayet, arXiv:hep-ph/0403226. W. J. Marciano, Int. J. Mod. Phys. A [19S1]{} (2004) 77, and references therein. P. J. Mohr and B. N. Taylor, Rev. Mod. Phys. [72]{}, 351 (2000). G. W. Bennett [et al.]{} \[Muon g-2 Collaboration\], Phys. Rev. Lett. [92]{}, 161802 (2004) \[arXiv:hep-ex/0401008\]. P. Vilain [et al.]{} \[CHARM-II Collaboration\], Phys. Lett. B [335]{}, 246 (1994). C. Boehm, arXiv:hep-ph/0405240; S. Davidson, S. Forte, P. Gambino, N. Rius and A. Strumia, JHEP [0202]{} (2002) 037 \[arXiv:hep-ph/0112302\]. K. S. McFarland and S. O. Moch, arXiv:hep-ph/0306052. S. Kretzer, F. Olness, J. Pumplin, D. Stump, W. K. Tung and M. H. Reno, arXiv:hep-ph/0312322. \[LEP Collaborations\], arXiv:hep-ex/0212036. R. Imlay and G. J. VanDalen, J. Phys. G [29]{}, 2647 (2003). P. Fayet, ; Proc. 1980 Karpacz School (Harwood, 1981) p.115; ; ; Nucl. Phys. B [187]{} (1981) 184; Phys. Lett. B [95]{} (1980) 285.
--- abstract: \| A repetition is a sequence of symbols in which the first half is the same as the second half. An edge-coloring of a graph is repetition-free or nonrepetitive if there is no path with a color pattern that is a repetition. The minimum number of colors so that a graph has a nonrepetitive edge-coloring is called its Thue edge-chromatic number. We improve on the best known general upper bound of $4\Delta-4$ for the Thue edge-chromatic number of trees of maximum degree $\Delta$ due to Alon, Grytczuk, Haluszczak and Riordan (2002) by providing a simple nonrepetitive edge-coloring with $3\Delta-2$ colors. author: - 'André Kündgen[^1]' - Tonya Talbot bibliography: - 'repfree-tree-coloring-final.bib' nocite: '[@]' title: 'Nonrepetitive edge-colorings of trees' --- Introduction {#sec:intro} ============ A [repetition]{} is a sequence of even length (for example $abacabac$), such that the first half of the sequence is identical to the second half. In 1906 Thue [@T1] proved that there are infinite sequences of 3 symbols that do not contain a repetition consisting of consecutive elements in the sequence. Such sequences are called [Thue sequences]{}. Thue studied these sequences as words that do not contain any square words $ww$ and the interested reader can consult Berstel [@B2; @B1] for some background and a translation of Thue’s work using more current terminology. Thue sequences have been studied and generalized in many views (see the survey of Grytczuk [@G]), but in this paper we focus on the natural generalization of the Thue problem to Graph Theory. In 2002 Alon, Grytczuk, Ha[ł]{}uszczak and Riordan [@AGHR] proposed calling a coloring of the edges of a graph [nonrepetitive]{} if the sequence of colors on any open path in $G$ is nonrepetitive. We will use $\pi'(G)$ to denote the [Thue chromatic index]{} of a graph $G$, which is the minimum number of colors in a nonrepetitive edge-coloring of $G$. In [@AGHR] the notation $\pi(G)$ was used for the Thue chromatic index, but by common practice we will instead use this notation for the [Thue chromatic number]{}, which is the minimum number of colors in a nonrepetitive coloring of the [vertices]{} of $G$. Their paper contains many interesting ideas and questions, the most intriguing of which is if $\pi(G)$ is bounded by a constant when $G$ is planar. The best result in this direction is due to Dujmovi[ć]{}, Frati, Joret, and Wood [@DFJW] who show that for planar graphs on $n$ vertices $\pi(G)$ is $O(\log n)$. Conjecture 2 from [@AGHR] was settled by Currie [@C] who showed that for the $n$-cycle $C_n$, $\pi(C_n)=3$ when $n\ge 18$. One of the conjectures from [@AGHR] that remains open is whether $\pi'(G)=O(\Delta)$ when $G$ is a graph of maximum degree $\Delta$. At least $\Delta$ colors are always needed, since nonrepetitive edge-colorings must give adjacent edges different colors. In this paper we study the seemingly easy question of nonrepetitive edge-colorings of trees. Thue’s sequence shows that if $P_n$ is the path on $n$ vertices, then $\pi'(P_n)=\pi(P_{n-1})\le 3$. (Keszegh, Patk[ó]{}s, and Zhu [@KPZ] extend this to more general path-like graphs.) Using Thue sequences Alon, Grytczuk, Ha[ł]{}uszczak and Riordan [@AGHR] proved that every tree of maximum degree $\Delta\geq 2$ has a nonrepetitive edge-coloring with $4(\Delta-1)$ colors and stated that the same method can be used to obtain a nonrepetitive vertex-coloring with 4 colors. However, while the star $K_{1,t}$ is the only tree whose vertices can be colored nonrepetitively with fewer than 3 colors, it is still unknown which trees need 3 colors, and which need 4 (see Bre[š]{}ar, Grytczuk, Klav[ž]{}ar, Niwczyk, Peterin [@BGKNP].) Interestingly Fiorenzi, Ochem, Ossona de Mendez, and Zhu [@FOOZ] showed that for every integer $k$ there are trees that have no nonrepetitive vertex-coloring from lists of size $k$. Up to this point the only paper we are aware of that narrows the large gap between the trivial lower bound of $\Delta$ colors in a nonrepetitive edge-coloring of a tree of maximum degree $\Delta$ and the $4\Delta-4$ upper bound from [@AGHR] is by Sudeep and Vishwanathan [@SV]. We will describe their results in the next section. The main result of this paper is to give the first nontrivial improvement of the upper bound from [@AGHR]. \[thm:main\] If $G$ is a tree of maximum degree $\Delta$, then $\pi'(G)\le 3\Delta-2$. We will give a proof of this theorem in Section \[sec:main\] using a coloring method we describe in Section \[sec:derived\] . We discuss some possible ways for further improvements in Section \[sec:improve\]. Trees of small height {#sec:small} ===================== A $k$-ary tree is a tree with a designated root and the property that every vertex that is not a leaf has exactly $k$ children. The $k$-ary tree in which the distance from the root to every leaf is $h$ is denoted by $T_{k,h}$. For convenience we will assume that the vertices in $T_{k,h}$ are labeled as suggested in Figures \[fig:2,2\] and \[fig:2,3\] with the root labeled 1, its children labeled $2,\dots,k+1$, their children $k+2,\dots k^2+k+1$ and so on. This allows us to write $u<v$ if $u$ is to the left or above $v$, and also gives the vertices at each level (distance from the root) a natural left to right order. To obtain bounds on the Thue chromatic index of general trees $G$ of maximum degree $\Delta\ge 2$ it suffices to study $k$-ary trees for $k=\Delta-1$, since $G$ is a subgraph of $T_{k,h}$ for sufficiently large $h$. Of course the Thue sequence shows that for $h>4$ we have $\pi'(T_{1,h})=\pi'(P_h)=3$, and it is similarly obvious that $\pi'(T_{k,1})=\pi'(K_{1,k})=k$. It is easy to see that the next smallest tree $T_{2,2}$ already requires 4 colors, and Figure \[fig:2,2\] shows the only two such 4-colorings up to isomorphism. The Masters thesis of the second author [@thesis] contains a proof of the fact that the type II coloring of $T_{2,2}$ extends to a unique 4-coloring of $T_{2,3}$ whereas the type I coloring extends to exactly 5 non-isomorphic 4-colorings of $T_{2,3}$, one of which we show in Figure \[fig:2,3\]. It is furthermore shown that none of these 6 colorings can be extended to $T_{2,4}$. In fact $\pi'(T_{2,4})=5$ as we can easily extend the coloring from Figure \[fig:2,3\] by using color 5 on one of the two new edges at every vertex from $8$ through $15$, and (for example) using colors 1,1,3,4,2,3,2,3 on the other edges in this order. On a more general level, Sudeep and Vishwanathan [@SV] proved that $\pi'(T_{k,2})=\lfloor \frac32 k\rfloor+1$ (compare also Theorem 4 of [@BLMSS]) and $\pi'(T_{k,3})>\frac{\sqrt5 +1}2k>1.618k$. Their lower bounds follow from counting arguments, whereas the construction for $h=2$ consists of giving the edges at the first level colors $0,1,\dots, k-1$ and using all the $\lfloor k/2\rfloor+1$ remaining colors below each vertex at level 1. The remaining $m=\lceil k/2\rceil -1$ edges below the edge of color $i$ are colored with $i+1 { {~\rm mod~}}k, i+2 { {~\rm mod~}}k,\dots,i+m { {~\rm mod~}}k$, in other words cyclically. To explain the general upper bound of Alon, Grytczuk, Ha[ł]{}uszczak and Riordan [@AGHR] we let $T_k$ denote the infinite $k$-ary tree. It is not difficult to see that $\pi'(T_k)$ is the minimum number of colors needed to color $T_{k,h}$ for every $h\ge 1$. They prove that $\pi'(T_k)\le 4k$ by giving a nonrepetitive edge-coloring of $T_k$ on $4k$ colors as follows: Starting with a Thue-sequence $123231\dots$ insert 4 as every third symbol to obtain a nonrepetitive sequence $S=124324314\dots$ that also does not contain a [palindrome]{}, that is a sequence of length at least 2 that reads forwards the same as backwards, such as 121. Now color the edges with a common parent at distance $h-1$ from the root with $k$ different copies $s^{(1)},\dots ,s^{(k)}$ of the symbol $s$ in position $h$ of $S$. For example, the type II coloring in Figure \[fig:2,2\] is isomorphic to the first two levels of this coloring of $T_2$ if we replace $1^{(1)},1^{(2)},2^{(1)},2^{(2)}$ by $1,2,3,4$ respectively. It is now easy to verify that this coloring has no repetitively colored paths that are monotone ([i.e.]{} have all vertices at different levels) since $S$ is nonrepetitive, and none with a turning point ([i.e.]{} a vertex whose two neighbors on the path are its children) since $S$ is palindrome-free. Sudeep and Vishwanathan noted the gap between the bounds $1.618k<\pi'(T_k)\le 4k$, and stated their belief that both can be improved. Even for $k=2$ the gap $3.2<\pi'(T_2)\le 8$ is large. Whereas obviously $\pi'(T_2)\ge \pi'(T_{2,4})=5$ is not hard to obtain, the specific question of showing that $\pi'(T_2)<8$ is already raised in [@AGHR] at the end of Section 4.2. Theorem \[thm:main\] implies that indeed $\pi'(T_2)\le 7$. On the other hand, improving on the lower bound of 5 (if that is possible) would require different ideas from those in [@SV] because [@thesis] presents a nonrepetitive 5-coloring of $T_{2,10}$ as Example 3.2.6. Derived colorings {#sec:derived} ================= In this section, which can also be found in [@thesis], we present a way to color the edges of $T_k$ that is different from that used by Alon, Grytczuk, Ha[ł]{}uszczak and Riordan [@AGHR]. While their idea is in some sense the natural generalization of the type II coloring in the sense that the coloring precedes by level, our coloring generalizes the type I coloring by moving diagonally. The fact that the type I colorings could be extended in 5 nonisomorphic ways, whereas the extension of the type II coloring was unique encourages this notion. Let $S=s_1,s_2, \dots$ be a sequence. The edge-coloring of a $k$-ary tree $T$ derived* from $S$ is obtained as follows: The edges incident with the root receive colors $s_1,s_2,\dots ,s_k$ going from left to right in this order. If $v$ is any vertex other than the root and if the edge between $v$ and its parent has color $s_i$, then the edges between $v$ and its children receive colors $s_{i+1},s_{i+2},\dots ,s_{i+k}$ again going from left to right in this order. To color the edges of the infinite $k$-ary tree $T_k$ in this fashion we need $S$ to be infinite. To color the edges of $T_{k,h}$ it suffices for the length of $S$ to be at least $kh$ (which is rather small considering that there about $k^h$ edges) as each level will use $k$ entries of $S$ more than the previous level (on the edges incident with the right-most vertex). For example the type I coloring of $T_{2,2}$ is the coloring derived from $S=1,2,3,4$, whereas the coloring of $T_{2,3}$ in Figure \[fig:2,3\] is derived from $S=1,2,3,4,1,2$. The next definition will enable us to characterize infinite sequences whose derived coloring is nonrepetitive. \[def:kspecial\] Let $S=s_1,s_2, \dots $ be a (finite or infinite) sequence. A sequence of indices $i_1,i_2, \dots, i_{2r}$ is called [$k$-bad]{} for $S$ if there is an $m$ with $1 < m \leq 2r$ such that the following four conditions hold: a) $s_{i_1},s_{i_2}, \dots ,s_{i_{2r}}$ is a repetition b) $i_1 > i_2> \dots > i_m < i_{m+1} < i_{m+2} < \dots < i_{2r}$ c) $\|i_j-i_{j+1}\| \leq k$ for all $j$ with $1 \leq j <2r$ d) $i_{m+1}<i_m+k$ if $m<2r$. $S$ is called [[$k$-special]{}]{} if it has no $k$-bad sequence of indices. The following proposition says something about the structure of a [[$k$-special]{}]{} sequence, namely that identical entries must be at least $2k$ apart. \[prop:distance\] A sequence $S$ has a $k$-bad sequence of length at most four with $m\le 3$ if and only if $s_i=s_j$ for some $i<j< i+2k$. For the back direction observe that if $j\le i+k$, then the sequence of indices $j,i$ is $k$-bad with $m=2$. If $i+k\le j< i+2k$, then the sequence $i+k-1,i,i+k-1,j$ is $k$-bad with $m=2$. For the forward direction, observe that if $i_1,i_2$ is $k$-bad (necessarily with $m=2$), then we can let $j=i_1$ and $i=i_2$. If $i_1,i_2,i_3,i_4$ is $k$-bad with $m=2$ then we let $i=i_2$ and $j=i_4$ and observe that $i<i_3<j\le i_3+k\le i+2k-1$. So we may assume that $i_1,i_2,i_3,i_4$ is $k$-bad with $m=3$. If $i_2=i_4$, then we let $i=i_3$ and $j=i_1$ and obtain $i<i_2<j\le i_4+k-1=i_2+k-1\le i+2k-1$ as desired. Otherwise $i_2, i_4$ are distinct numbers $x$ with $i_3<x\le i_3+k$ and we can let $\{i,j\}=\{i_2,i_4\}$. We are now ready to prove the following. \[thm:kspecial\] An infinite sequence $S$ is [[$k$-special]{}]{} if and only if the edge-coloring of $T_k$ derived from $S$ is nonrepetitive. $(\Rightarrow)$ Suppose that a [[$k$-special]{}]{} sequence $S$ creates a repetition on a path $P=v_0,v_1,\dots, v_{2r}$ in $T_k$, that is $R=c(v_0v_1),c(v_1v_2),\dots,c(v_{2r-1}v_{2r})$ satisfies $c(v_iv_{i+1})=c(v_{i+r}v_{i+r+1})$ for $0 \leq i \leq r-1$. Observe that $c(v_jv_{j+1})=s_{i_{j+1}}$ where $0 \leq j \leq 2r-1$, for some $s_{i_{j+1}} \in S$. There are two possibilities; $v_0,v_1,\dots, v_{2r}$ is monotone or it has a single turning point. Case 1: Suppose $v_0,v_1,\dots, v_{2r}$ is monotone.\ If $v_0,v_1,v_2 \dots, v_{2r}$ is monotone then we may assume $v_0>v_1>v_2>\dots>v_{2r}$. Since $v_j>v_{j+1}$ we know that $v_j$ is the child of $v_{j+1}$ so we have that $i_j>i_{j+1}$ and $\|i_j-i_{j+1}\|\leq k$. The subsequence $s_{i_1},s_{i_2},\dots, s_{i_{2r}}$ is a repetition, so that $i_1,\dots,i_{2r}$ is $k$-bad with $m=2r$, a contradiction. Case 2: Suppose $v_0,v_1,\dots, v_{2r}$ has a turning point $v_m$ for some $m$ with $0<m<2r$. By the definition of a turning point $v_{m-1}$ and $v_{m+1}$ are the children of $v_m$, and thus $v_0>v_1>\dots>v_{m-1} >v_m <v_{m+1}< \dots <v_{2r}$. We may also assume without loss of generality that $v_{m-1}<v_{m+1}$. Observe that $v_0,v_1,\dots, v_{m}$ is moving towards the root and $v_m,v_{m+1},\dots, v_{2r}$ is moving away from the root. Let $c(v_jv_{j+1})=s_{i_{j+1}}$. We will show that $i_1>i_2>\dots >i_{m-1}>i_m<i_{m+1}<\dots< i_{2r}$ and that this sequence is $k$-bad for $S$. Since $v_{j-1} > v_j > v_{j+1}$ for $1 \leq j < m$ we know that $v_j$ is the child of $v_{j+1}$ and the parent of $v_{j-1}$ so we have $i_j>i_{j+1}$ and $\|i_j-i_{j+1}\|\leq k$. Similarly, since $v_{j-1}<v_j<v_{j+1}$ for $m < j < 2r$ we know that $v_{j}$ is the child of $v_{j-1}$ and the parent of $v_{j+1}$ so $i_j<i_{j+1}$ and $\|i_j-i_{j+1}\| \leq k$. Finally, since $v_m$ is the parent of $v_{m-1}$ and $v_{m+1}$ so $\|i_m-i_{m+1}\|<k$ and $i_m < i_{m+1}$ since we assumed $v_{m-1} < v_{m+1}$. The subsequence $s_{i_1},s_{i_2},\dots, s_{i_{2r}}$ is a repetition, leading to the contradiction that $i_1,\dots,i_{2r}$ is $k$-bad. $(\Leftarrow)$ We proceed by contrapositive. So suppose $S$ has a $k$-bad sequence ${i_1},{i_2}, \dots ,{i_{2r}}$. We will show that there is a path on vertices $v_0,v_1,v_2,\dots, v_{2r}$ with $c(v_jv_{j+1})=s_{i_{j+1}}$ where the color pattern $c(v_0v_1),c(v_1v_2)\dots,c(v_{2r-1}v_{2r})$ is a repetition in the derived edge-coloring of $T_k$. The left child of a vertex $v$ is the child with the smallest label, and we will denote this child as $v'$. Observe that if $c(vp(v))=s_\alpha$, then $c(vv')=s_{\alpha+1}$. If $m=2r$ then we start at the root and successively go to the left child of the current vertex until we find a vertex $v_{2r}$ such that $c(v_{2r}v_{2r}')=s_{i_{2r}}$ and let $v_{2r-1}=v_{2r}'$. Let $v_{2r-2}$ be the child of $v_{2r-1}$ with $c(v_{2r-1}v_{2r-2})=s_{i_{2r-1}}$ (this exists since $\|i_j-i_{j+1}\| \leq k$). We continue in this way until we have found $v_{0}$. Now observe that the color pattern of $v_0,v_1,\dots, v_{2r}$ is $s_{i_1},s_{i_2}, \dots ,s_{i_{2r}}$ as desired. If $m < 2r$ then we start at the root and successively go to the left child of the current vertex until we find a vertex $v_m$ such that $c(v_mv'_m)=s_{i_m}$ and let $v_{m-1}=v_{m}'$. Let $v_{m+1}$ be the child of $v_m$ with $c(v_mv_{m+1})=s_{i_{m+1}}$ (this exists since $i_m<i_{m+1}<i_m+k$). Now, for $0 \leq p \leq (m-1)$ we successively find a child $v_{p-1}$ of $v_p$ such that $c(v_pv_{p-1})=s_{i_p}$. The existence of $v_{p-1}$ is guaranteed by the fact $\|i_p-i_{p-q}\|\leq k$ as in the case $m=2r$. For $m+1\leq q \leq 2r$ we successively find a child $v_{q+1}$ of $v_q$ such that $c(v_qv_{q+1})=s_{i_{q-1}}$ which we can do since $\|i_q-i_{q+1}\|\leq k$. Now observe that the color pattern of $v_0,v_1,\dots, v_{2r}$ is $s_{i_1},s_{i_2}, \dots ,s_{i_{2r}}$ as desired. \[rem:finite\] Observe that the proof of the forward direction also works for the finite case $T_{k,h}$, a fact we will use in Section \[sec:improve\]. However, the back direction need not hold in this case: We already mentioned that the coloring derived from $S=1,2,3,4,1,2$ in Figure \[fig:2,3\] is nonrepetitive (see also $k=2$ in Proposition \[2k\]), but this sequence $S$ is not 2-special, because the index-sequence $3,1,2,3,5,6$ is $2$-bad. Thus to get a good upper bound on $\pi'(T_k)$ we just need an infinite [[$k$-special]{}]{} sequence with few symbols. As every $2k$ consecutive elements must be distinct, the following simple idea turns out to be useful: from a sequence $S$ on $q$ symbols we can form a sequence $S^{(w)}$ on $qw$ symbols by replacing each symbol $t$ in $S$ by a block $T=t^{(0)},t^{(1)},\dots t^{(w-1)}$ of $w$ symbols. In [@thesis] it is shown that if $S$ is nonrepetitive and palindrome-free then $S^{(k)}$ is [$k$-special]{}. This gives a new proof of the result from [@AGHR] that $\pi'(T_k)\le 4k$. In the next section we will improve on that. Main result {#sec:main} =========== We begin with the simple observation, that if $S$ is a sequence then $S^{(k+1)}=S^+$ has the property that if $i,j$ are indices with $s^+_i=x^{(u)}$ and $s^+_j=y^{(v)}$ then $i<j\le i+k$ implies that either $x=y$ and $u<v$, or $s^+_i$ and $s^+_j$ are in consecutive blocks $XY$ of $S^+$ and $u>v$. In other words we can tell whether we are moving left or right through the sequence just by looking at the superscripts (as long as consecutive symbols in $S$ are distinct.) As a starting point we immediately get the following result. \[cor:3k+3\] For all $k\ge 1$, $\pi'(T_k)\le 3k+3$. It is enough to show that $S^+$ on $3(k+1)$ is [[$k$-special]{}]{} whenever $S$ is an infinite Thue sequence on 3 symbols. Suppose there is a $k$-bad sequence of indices $i_1,\dots,i_{2r}$. Since every sequence of $2(k+1)$ consecutive symbols in $S^+$ is distinct we get that $r>1$ by Proposition \[prop:distance\]. If $m<2r$, then we can find an index $j$ such that $i_j>i_{j+1}$ and $i_{r+j}<i_{r+j+1}$ with $s_{i_j}=s_{i_{r+j}}=x^{(u)}$ and $s_{i_{j+1}}=s_{i_{r+j+1}}=y^{(v)}$. Indeed, if $2<m\le r$ we let $j=1$, and otherwise we let $j=m-r$. In this case $x=y$ and $u\le v$ would violate $i_j>i_{j+1}\ge i_j-k$, whereas $u\ge v$ would violate $i_{r+j}<i_{r+j+1}\le i_{r+j}+k$. Similarly if $x\neq y$, then $u\ge v$ would violate $i_j>i_{j+1}\ge i_j-k$, whereas $u\le v$ would violate $i_{r+j}<i_{r+j+1}\le i_{r+j}+k$. It remains to observe that in the case when $m=2r$ the sequence $s_{i_1},s_{i_2}, \dots ,s_{i_{2r}}$ in $S^+$ yields a repetition in $S$ by erasing the superscripts and merging identical consecutive terms where necessary. This bound can be improved to $3k+2$ by removing all symbols of the form $a^{(0)}$ from $S^+$ for one of the symbols $a$ from $S$ and showing that the resulting sequence is still [$k$-special]{}. However, we can do a bit better. In fact, Theorem \[thm:main\] follows directly from our main result in this section. \[thm:3k+1\] There are arbitrarily long [[$k$-special]{}]{} sequences on $3k+1$ symbols. One difficulty is that removing two symbols from $S^+$ can easily result in the sequence not being [[$k$-special]{}]{} anymore. To make the proof work we need to start with a Thue sequence with additional properties. The following result was proved by Thue [@T2] and reformulated by Berstel [@B2; @B1] using modern conventions. \[thm:Thue+\] There are arbitrarily long nonrepetitive sequences with symbols $a, b, c$ that do not contain $aba$ or $bab$. To give an idea of how such a sequence can be found, observe that it must be built out of blocks of the form $ca, cb, cab,$ and $cba$ which we denote by $x, y, z, u$, respectively. (In fact, Thue primarily studied two-way infinite sequences, but for our purposes we may simply assume our sequence starts with $c$.) We first build a sufficiently long sequence on the 5 symbols $A, B, C, D, E$ by starting with the sequence “B” and then in each step simultaneously replacing each letter as follows: Replace A B C D E --------- -------- ----- ------ ------ ------ by BDAEAC BDC BDAE BEAC BEAE In the resulting sequence we then let $A=zuyxu$, $B=zu$, $C=zuy$, $D=zxu$, $E=zxy$. Lastly we replace $x$, $y$, $z$ and $u$ as aforementioned. For example, from $B$ we obtain $BDC$, and then after a second step $BDCBEACBDAE$. This translates to the intermediate sequence $zuzxuzuyzuzxyzuyxuzuyzuzxuzuyxuzxy$, which gives us the desired sequence $cabcbacabcacbacabcbacbcabcbacabcacbcabcbacbcacbacabcbacbcabcbacabcacbacabcbacbcacbacabcacb$. It is worth pointing out that Thue’s work goes deeper in that he essentially characterizes all two-way infinite sequences that meet the conditions from Theorem \[thm:Thue+\] as well as several other related sequences. We also want to mention that the $A,B,C$ in the following proof have nothing to do with the $A,B,C$ in the previous paragraph, but we wanted to maintain the notation used in [@B2; @B1]. Start with an infinite sequence $S$ in the form of Theorem \[thm:Thue+\] and replace each occurrence of $c$ by a block $C$ of $k+1$ consecutive symbols $c^{(0)},c^{(1)},\dots,c^{(k)}$, whereas we replace each occurrence of $a$ or $b$ by shorter blocks $A=a^{(1)},\dots,a^{(k)}$ and $B=b^{(1)},\dots,b^{(k)}$ respectively. We claim that the resulting sequence $S'$ on $3k+1$ symbols is [$k$-special]{}. So suppose there is a $k$-bad sequence of indices $i_1,\dots,i_{2r}$. As before when $m=2r$ the sequence $s_{i_1},s_{i_2}, \dots ,s_{i_{2r}}$ in $S'$ yields a repetition in $S$ by erasing the superscripts and merging identical consecutive terms where necessary, as we can not “jump” over any of the blocks $A, B$ or $C$ in $S'$. So we may assume that $1<m<2r$, and since every $2k$ consecutive elements are distinct Proposition \[prop:distance\] implies that $r>2$. [Claim:]{} If there is an index $j$ with $0<j< r$ such that $i_j>i_{j+1}$ and $i_{r+j}<i_{r+j+1}$, then $s_{i_j}=s_{i_{r+j}}=x^{(u)}$ and $s_{i_{j+1}}=s_{i_{r+j+1}}=y^{(u)}$ for $1\le u\le k$ and $\{x,y\}=\{a,b\}$. Consequently, $i_j-i_{j+1}=k=i_{r+j+1}-i_{r+j}$. Indeed, $s_{i_j}=s_{i_{r+j}}=x^{(u)}$ and $s_{i_{j+1}}=s_{i_{r+j+1}}=y^{(v)}$ for some $u,v,x,y$. If $x=y$, then $u\le v$ would violate $i_j>i_{j+1}\ge i_j-k$, whereas $u\ge v$ would violate $i_{r+j}<i_{r+j+1}\le i_{r+j}+k$. Thus $x\neq y$. Now $u> v$ would violate $i_j>i_{j+1}\ge i_j-k$, whereas $u< v$ would violate $i_{r+j}<i_{r+j+1}+k$. So we may assume that $u=v$. If $x=c$, then this would violate $i_{j}>i_{j+1}\ge i_j+k$ (as the presence of $c^{(0)}$ means that the distance is $k+1$). Similarly if $y=c$, then this violates $i_{r+j}<i_{r+j+1}\le i_{r+j}+k$. Hence we must have $\{x,y\}=\{a,b\}$ finishing the proof of the claim. If $r<m<2r$, then we can apply the claim with $j=m-r$ and obtain consequently that $i_{m+1}-i_m=k$, in direct contradiction to condition d) from Definition \[def:kspecial\]. So we suppose that $2 \le m\le r$. In this case we will let $j=m-1$ in our claim and we may assume due to the symmetry of $S$ in $a,b$ that $x=a$ and $y=b$. Thus for some $u$ with $1\le u\le k$ we get $s_{i_{m-1}}=a^{(u)}=s_{i_{m+r-1}}$ and $s_{i_m}=b^{(u)}=s_{i_{m+r}}$. If $m>2$, then we may apply the claim again with $j=m-2$ to obtain that $s_{i_{m-2}}=b^{(u)}=s_{i_{m+r-2}}$. However, the fact that $i_{m-2}>i_{m-1}>i_m$ correspond to symbols $b^{(u)},a^{(u)},b^{(u)}$ means that $S'$ must have consecutive blocks $BAB$, yielding a contradiction to the fact that in $S$ we had no consecutive symbols $bab$. So we may assume that $m=2$. Since $r>2$ and $s_{i_2}=b^{(u)}$ and $i_2<\dots<i_r$ we have that for $3\le j\le r$ either all $s_{i_j}$ are of the form $b^{(u_j)}$ or there is a smallest index $j$ such that $s_{i_j}=x^{(u_j)}$ for some $x\neq b$. In the first case it follows that there must be consecutive blocks $BAB$ (yielding a contradiction) such that $i_1$ and $i_{r+1}$ are in the $A$ block, $i_2,\dots i_r$ are in the first $B$-block and $i_{r+2},\dots,i_{2r}$ are in the second. In the second case it follows that since there must be blocks $BA$ with $i_1$ in $A$ and $i_2$ in $B$, that $i_j$ must be in the $A$ block again, that is $s_{i_j}=a^{(u_j)}$. However, since $i_{r+1}<\dots <i_{r+j}$ it follows that there must be consecutive blocks $ABA$ in $S'$ (our final contradiction), such that $i_{r+1}$ is in the first $A$ block, $i_{r+j}$ in the second and $i_{r+2}, \dots, i_{r+j-1}$ are in the $B$ block. [[$k$-special]{}]{} sequences on at most $3k$ symbols {#sec:improve} ===================================================== One possible way to improve on Theorem \[thm:main\] is to study [[$k$-special]{}]{} sequences on at most $3k$ symbols. The sequence $S_{n,c}=1,2,\dots,n,1,2,\dots c$ for $n>c\ge 0$ turns out to be a key example in this situation. Recall that by Proposition \[prop:distance\] the entries in a block of length $2k$ of a [[$k$-special]{}]{} sequence must all be distinct. Thus, if we let $f_k(n)$ denote the maximum length of a [[$k$-special]{}]{} sequence $S$ on $n$ symbols, then this observation immediately implies that $f_k(n)=n$ when $n<2k$ and up to isomorphism the only sequence achieving this value is $S_{n,0}$. When $n\ge 2k$ we can furthermore assume without loss of generality that if $S$ is nonrepetitive on $n$ symbols, then $S_i=i$ for $1\le i\le 2k$ (just like $S_{n,c}$.) If $n=2k$ then it follows from Proposition \[prop:distance\] that a sequence achieving $f_k(2k)$ must be of the form $S_{2k,c}$. It is easy to check $S_{2k,1}$ is in fact [$k$-special]{}, whereas $S_{2k,2}$ contains the $k$-bad index sequence $k+1,1,2,k+1,2k+1,2k+2$, which yields the repetition $k+1,1,2,k+1,1,2$. Thus $f_k(2k)=2k+1$ with $S_{2k,1}$ being the unique sequence achieving this value. This $k$-bad index sequence also explains why we could not have consecutive blocks $ABA$ or $BAB$ in our construction for Theorem \[thm:3k+1\] . For the remaining range we get \[prop:2k+\] a) If $n\ge 2k$, then $S_{n,n-k}$ has a $k$-bad sequence only when $n=2k$ and such a sequence must have $2=m<r$. b) If $n\ge 2k+1$, then $f_k(n)\ge 2n-k$. It suffices to prove the first statement, as it immediately implies the second. So suppose $n\ge 2k$ and $I=i_1,\dots,i_{2r}$ is a $k$-bad sequence of indices for some $m$. If $m=2r$, then $I$ is decreasing and so the fact that $s_{i_j}=s_{i_{j+r}}$ for all $1\le j\le r$ implies that $i_1>\dots >i_r\ge n+1$ and $n-k\ge i_{r+1}>\dots>i_{2r}$, yielding the contradiction $i_r-i_{r+1}>k$. So we may assume that $m<2r$. If $m>r$, then let $m'=m-r$. Since $s_{i_m}=s_{i_{m'}}$ and $i_{m'}>i_m$, it follows that $i_{m}=i_{m'}-n\in\{1,\dots,n-k\}$. Since $i_{m'}\ge n,$ $i_m\le n-k$ and for all $j$ we have $\|i_j-i_{j+1}\|\le k$ it follows that there must be some $j$ with $m'<j<m$ such that $i_j\in\{n-k+1,\dots, n\}$. Since $I$ yields a repetition with $i_1>\dots >i_m$, but the symbol $s_{i_j}=i_j$ is unique in $S_{n,n-k}$ we conclude that $i_j=i_{j+r}$. It follows that $j=m'+1$, since otherwise $i_{m'}>i_{j-1}>i_{j}$ and $i_m<i_{j+r-1}<i_{j+r}$ would contradict $s_{i_{j-1}}=s_{i_{j+r-1}}$ as the sets $\{s_{i_j+1},s_{i_j+2},\dots, s_{i_{m'}-1}\}$ and $\{s_{i_m+1}, s_{i_m+2},\dots s_{i_j-1}\}$ are disjoint. Now $j=m'+1$ implies that $i_{m'}-k=i_{j-1}-k\le i_j=i_{j+r}=i_{m+1}\le i_m+k-1$, and since $i_{m'}=i_m+n$ we get $n\le 2k-1$, a contradiction. If $m\le r$, then let $m'=m+r$. It follows again that $i_{m'}=i_m+n$, and that there must be some $j$ such that $i_j=i_{j+r}\in\{n-k+1,\dots, n\}$ and $j<m<j+r$. Thus $m'>j+r$ this time. It follows that $j=m-1$, since otherwise $i_{j}>i_{j+1}>i_{m}$ and $i_{j+r}<i_{j+r+1}<i_{m'}$ would contradict $s_{i_{j+1}}=s_{i_{j+r+1}}$ as the sets $\{s_{i_m+1}, s_{i_m+2},\dots s_{i_j-1}\}$ and $\{s_{i_j+1},s_{i_j+2},\dots, s_{i_{m'}-1}\}$ are still disjoint. Now $j=m-1$ implies that $i_{m}+k=i_{j+1}+k\ge i_j=i_{j+r}=i_{m'-1}\ge i_{m'}-k$, and since $i_{m'}=i_m+n$ we get $n\le 2k$, a contradiction unless $n=2k$. In this case also $i_m+k=i_j=i_{j+r}=i_{m'}-k=x$ for some $k+1\le x\le n=2k$. If we have $m>2$ then $j-1=m-2\ge 1$ and we consider $i_{j-1}$. Since $i_{j+r-1}<i_{j+r}$ and $k+1=n-k+1\le s_{i_j}\le n=2k$ implies that $s_{i_{j+r-1}}\in\{x-k,x-k+1,\dots,x-1\}$. Similarly $i_{j-1}>i_j$ implies that $s_{i_{j-1}}\in\{x+1,x+2,\dots,n\}\cup\{1,2,\dots,k-(n-x)=x-k\}$. Since $s_{i_{j+r-1}}=s_{i_{j-1}}$ it now follows that this value must be $x-k=i_m$. Hence $i_{j+r-1}=i_m$ and thus $m=j+r-1=(m-1)+r-1$. This implies the contradiction $2=r\ge m>2$. Hence $m=2$ and the fact that $r>2$ follows from Proposition \[prop:distance\] and the fact that the distance between identical labels is $2k$. We believe that for in Proposition \[prop:2k+\] b) equality holds when $2k<n<3k$. An exhaustive search by computer shows that this is the case when $2k<n<3k$ with $n\le 16$. Moreover $S_{2k+1,k+1}$ turns out to be the unique sequence achieving $f_k(2k+1)=3k+2$, whereas for $2k+2\le n<3k$ a typical sequence achieving $f_k(n)$ is obtained by permuting the last $n-k$ entries of $S_{n,n-k}$. \[2k\] The coloring of $T_{k,3}$ derived from $S_{2k,k}$ is nonrepetitive. If the coloring of $T_{k,3}$ derived from $S_{2k,k}$ contains a repetition of length $2r$, then as in the proof of Theorem \[thm:kspecial\] it follows that there must be a $k$-bad sequence of $2r$ indices. From Proposition \[prop:2k+\] a) it now follows that $r> m=2$. Since a longest path in $T_{k,3}$ has 6 edges we must have $r=3$. However, any repetition of length 6 would have to connect two leaves and turn around at the root, and as such would have $m=3$, a contradiction. Combining everything we know so far we get \[cor:small\] If $h\ge 3$, then $\pi'(T_{k,h})\le \lceil\frac{h+1}2k\rceil$. If $h=3$, then the result follows from Proposition \[2k\]. For $h>3$ we can apply Proposition \[prop:2k+\] b) with $n=\lceil\frac{h+1}2k\rceil$. Since $2n-k\ge hk$ it now follows from Remark \[rem:finite\] that the coloring of $T_{k,h}$ derived from $S_{n,n-k}$ is nonrepetitive. The bound in Corollary \[cor:small\] is better than that derived from Theorem \[thm:3k+1\] when $h\le 5$ and we obtain the following table of values for $\pi'(T_{h,k})$, where the presence of two values denotes a lower and an upper bound. The values marked by an asterisk were confirmed by computer search. The programs used are based on those found in [@thesis] and the Python code is available at http://public.csusm.edu/akundgen/Python/Nonrepetitive.py $k\backslash h$ 1 2 3 4 5 6-10 $h\geq 11$ ----------------- ---------- -------------------------- ------------- -------------------------- ------------ -------------- -------------- -- 1 1 2 2 3 3 3 3 2 2 4 4 5 $5^$ $5^$ 5,7 3 3 5 $6^$ $6^$ 6,9 6,10 6,10 4 4 7 $7^*$ 7,10 7,12 7,13 7,13 5 5 8 9,10 9,13 9,15 9,16 9,16 $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $k$ $k$ $\lfloor 1.5k \rfloor$+1 $1.61k, 2k$ $1.61k,\lceil2.5k\rceil$ $1.61k,3k$ $1.61k,3k+1$ $1.61k,3k+1$ It is worth noting that even though it may be possible to use derived colorings to improve individual columns of this table by a more careful argument (as we did in Proposition \[2k\]), this seems unlikely to work for $\pi'(T_k)$ in general. Theorem \[thm:kspecial\] implies that the infinite sequence from which we derive the coloring must be [[$k$-special]{}]{}, and while we were able to provide such a sequence on $3k+1$ symbols, it seems unlikely that there are such sequences on $3k$ symbols. An exhaustive search shows that for $k\le 5$ the maximum length of a $k$-special sequence on $n=3k$ symbols is $5k+3$, which is only 3 more than the length of $S_{n,n-k}$. The $k!$ examples achieving this value are all of the strange form $[1,2k],1,[2k+1,3k],x_1,[k+2,2k],1,x_2,x_3,\dots,x_k,x_1,2k+1$ where $\{x_1,\dots,x_k\}=\{2,\dots,k+1\}$ and $[a,b]$ denotes $a,a+1,a+2\dots,b$. In other words they are $S_{3k,2k+1}$ with the last $2k+1$ entries permuted and with 1 and $x_1$ inserted after positions $2k$ and $3k$. A more promising next step would be to try to improve the lower bounds for $\pi'(T_{k,h})$ for $h=3,4,5$. \[sec:biblio\] [^1]: Supported by ERC Advanced Grant GRACOL, project no. 320812.
--- abstract: 'We present the theory of the Josephson effect in nanotube dots where an $SU(4)$ symmetry can be realized. We find a remarkably rich phase diagram that significantly differs from the $SU(2)$ case. In particular, $\pi$-junction behavior is largely suppressed. We analytically obtain the Josephson current in various parameter regions: (i) in the Kondo regime, covering the full crossover from $SU(4)$ to $SU(2)$, (ii) for weak tunnel couplings, and (iii) for large BCS gap. The transition between these regions is studied numerically.' author: - 'A. Zazunov,$^{1}$ A. Levy Yeyati,$^2$ and R. Egger$^1$' title: 'Josephson effect for ${\bf SU(4)}$ carbon nanotube quantum dots' --- Several experimental groups have recently started to study the Josephson effect in ultra-small nanostructures,[@nazarov] where the supercurrent can be tuned via the gate voltage dependence of the electronic levels of the nanostructure. An important system class where supercurrents have been successfully observed [@tubeexp] is provided by carbon nanotube (CNT) quantum dots. In many cases, the experimental results compare quite well to predictions based on modeling the CNT dot as a spin-degenerate electronic level with $SU(2)$ spin symmetry, where the presence of a repulsive on-dot charging energy $U$ may allow for a (normal-state) Kondo effect. Depending on the ratio $T_K/\Delta$, where $\Delta$ is the energy gap in the superconducting electrodes and $T_K$ the Kondo temperature, theory[@glazman; @rozhkov; @slaveboson; @su2numeric; @largedelta; @vecino] predicts a transition between a unitary (maximum) Josephson current for $\Delta\ll T_K$, possible thanks to the survival of the Kondo resonance in that limit, and a $\pi$-junction regime for $\Delta\gg T_K$, where the critical current is small and negative, i.e., the junction free energy $F(\varphi)$ has a minimum at phase difference $\varphi=\pi$ as opposed to the more common $0$-junction behavior. Recent progress has paved the way for the fabrication of very clean CNTs, resulting in a new generation of quantum transport experiments and thereby revealing interesting physics, e.g., spin-orbit coupling effects[@mceuen] or incipient Wigner crystal behavior.[@bockrath] In ultra-clean CNTs, the orbital degree of freedom ($\alpha=\pm$) reflecting clockwise and anti-clockwise motion around the CNT circumference (i.e., the two $K$ points) is approximately conserved when electrons enter or leave the dot.[@aguado] Due to the combined presence of this orbital “pseudo-spin” (denoted in the following by $T$) and the true electronic spin ($S$), an enlarged $SU(4)$ symmetry group can be realized. In addition, a purely orbital $SU(2)$ symmetry arises when a Zeeman field is applied. Experimental support for this scenario has already been published [@su4exp] (for the case of semiconductor dots, see Ref. ), and several aspects have been addressed theoretically.[@aguado; @su4theory] In particular, the $SU(4)$ Kondo regime is characterized by an enhanced Kondo temperature and exotic local Fermi liquid behavior, where the Kondo resonance is asymmetric with respect to the Fermi level. However, so far both experiment and theory have only studied the case of normal-conducting leads, where conventional linear response transport measurements cannot reliably distinguish the $SU(4)$ from the $SU(2)$ scenario.[@su4theory] Here we provide the first theoretical study of the Josephson effect for interacting quantum dots with (approximate) $SU(4)$ symmetry, and find drastic differences compared to the standard $SU(2)$ picture. In the Kondo limit, a qualitatively different current-phase relation (CPR) is found, with the critical current smaller by a factor $\approx 0.59$. The usual $\pi$-junction behavior is largely suppressed, but new phases do appear and time-reversal symmetry can be spontaneously broken. Our predictions can be tested using state-of-the-art experimental setups, and offer clear signatures of the $SU(4)$ symmetry in very clean CNT quantum dots. Model and formal solution.— We study a quantum dot ($H_d$) contacted via a standard tunneling Hamiltonian ($H_t$) to two identical superconducting electrodes ($H_{L/R}$), $H=H_d+H_t+H_L+H_R$. We assume that the dot has a spin- and orbital-degenerate electronic level $\epsilon_{\alpha\sigma}= \epsilon$ with identical intra- and inter-orbital charging energy $U$,[@foot1] $H_d = \epsilon \hat n + U \hat n(\hat n-1) /2$ with $\hat n=\sum_{\alpha\sigma} d^\dagger_{\alpha \sigma} d^{}_{\alpha\sigma}$, where $d^\dagger_{\alpha\sigma}$ creates a dot electron with spin $\sigma=\uparrow,\downarrow=\pm$ and orbital pseudo-spin projection $\alpha$. Since the $\alpha=\pm$ states are related by time-reversal symmetry (clockwise and anti-clockwise states are exchanged), we take the lead Hamiltonian as $$H_{j} = \sum_{{\bm k}\alpha\sigma} \xi_{\bm k} \ c_{j{\bm k}\alpha\sigma}^\dagger c_{j{\bm k}\alpha\sigma}^{} + \sum_{{\bm k} \alpha} \left( \Delta e^{\mp i \frac{\varphi}{2}} \ c^\dagger_{j{\bm k}\alpha\uparrow} c^\dagger_{j,-{\bm k},-\alpha, \downarrow} + {\rm h.c.} \right),$$ where $c^\dagger_{j{\bm k}\alpha\sigma}$ creates an electron with wavevector ${\bm k}$ in lead $j=L/R$, and $\xi_{\bm k}$ is the single-particle energy. The tunneling Hamiltonian is $H_t = \sum_{j{\bm k}\sigma,\alpha\alpha'} \left(t\delta_{\alpha\alpha'} + \tilde t \delta_{\alpha,-\alpha'}\right) c^\dagger_{j{\bm k}\alpha\sigma} d^{}_{\alpha'\sigma} + {\rm h.c.},$ where $t$ ($\tilde t$) describes orbital (non-)conserving tunneling processes. Following standard steps,[@rozhkov] the noninteracting lead fermions can now be integrated out. The partition function $Z(\varphi)=e^{-\beta F(\varphi)}$ at inverse temperature $\beta$ then reads (we often set $e=\hbar=1$) $$\label{partition} Z(\varphi) = {\rm Tr}_{d} \left( e^{-\beta H_d} {\cal T} e^{-\int_0^\beta d\tau d\tau'\ D^\dagger(\tau) \Sigma(\tau-\tau') D(\tau') } \right) ,$$ where the trace extends over the dot Hilbert space, ${\cal T}$ denotes time ordering, and we use the Nambu bispinor $D=(d^{}_{e\uparrow}, d^\dagger_{e\downarrow}, d^{}_{o\uparrow}, d^\dagger_{o\downarrow})$ with even/odd linear combinations of the orbital states, $d_{e\sigma} = (d_{+,\sigma}+ d_{-,\sigma})/\sqrt{2}$ and $d_{o\sigma}=\sigma (d_{+,\sigma}-d_{-,\sigma})/\sqrt{2}$. In this basis, the self-energy $\Sigma(\tau)$ representing the BCS leads is diagonal in orbital space. With the orbital mixing angle $\theta=2\tan^{-1} (\tilde t/t)$ and the normal-state density of states $\nu_0 = 2\sum_{\bm k} \delta(\xi_{\bm k})$, the even/odd channels are characterized by the hybridization widths $\Gamma_{\nu=e,o} = (1\pm \sin\theta)\ \Gamma$ with $\Gamma = \pi \nu_0 (t^2+{\tilde t}^2)$. In what follows, we study the zero-temperature limit and assume the wide-band limit[@nazarov] for the leads. The Fourier transformed self-energy is then expressed in terms of the $2\times 2$ Nambu matrices $\Sigma_{\nu=e,o}(\omega) = \frac{\Gamma_\nu}{\sqrt{\omega^2+\Delta^2}} \left( \begin{array}{cc} -i\omega & \Delta \cos\frac{\varphi}{2} \\ \Delta \cos\frac{\varphi}{2} & -i\omega\end{array} \right).$ The result (\[partition\]) will now be examined in several limits. We start with the strong-correlation limit $U\to \infty$, and later address the case of finite $U$. Note that Eq. (\[partition\]) for $\theta=0$ corresponds to the $SU(4)$ symmetric case while for $\theta=\pi/2$ there is only one conducting channel with non-zero transmission which, under certain conditions, corresponds to the usual $SU(2)$ model. Deep Kondo limit.— Let us first discuss the Kondo limit $T_K\gg \Delta$ in the quarter-filled case, $\epsilon<0$ and $\langle \hat n\rangle \approx 1$. The Kondo temperature is given by $T_K=D \exp(\pi\epsilon/4\Gamma)$[@aguado] with bandwidth $D$. As in the $SU(2)$ case,[@glazman] the Josephson current at $T=0$ can be computed from local Fermi liquid theory, either using phase shift arguments or an equivalent mean-field slave-boson treatment.[@slaveboson] The latter approach yields the self-consistent dot level $\tilde\epsilon$ and thereby the transmission probability for channel $\nu=e,o$,[@aguado] $${\cal T}_\nu = \frac{(1\pm \sin\theta)^2 T_K^2}{\tilde \epsilon^2 + (1\pm \sin\theta)^2 T_K^2}, \quad \frac{\tilde \epsilon}{T_K} = \frac{(1-\sin\theta)^{(\sin\theta+1)/4}}{(1+\sin\theta)^{(\sin\theta-1)/4}}.$$ In the $SU(4)$ case ($\theta=0$), we have ${\cal T}_e={\cal T}_o=1/2$, while the $SU(2)$ limit ($\theta=\pi/2$) has a decoupled odd channel, ${\cal T}_e=1$ and ${\cal T}_o=0$. The CPR covering the crossover from the $SU(4)$ to the $SU(2)$ Kondo regime then follows as $$I(\varphi) = \frac{e\Delta}{2\hbar} \sum_{\nu=e,o} \frac{{\cal T}_\nu \sin\varphi}{\sqrt{1-{\cal T}_\nu \sin^2\frac{\varphi}{2}}}.$$ The known $SU(2)$ result[@glazman] is recovered for $\theta=\pi/2$. The $SU(4)$ CPR has a completely different shape, as shown in Fig. \[fig1\]. We note that the critical current $I_c={\rm max} [I(\varphi)]$ is suppressed by the factor $2-\sqrt{2}\approx 0.59$ relative to the unitary limit $e\Delta/\hbar$ reached for the $SU(2)$ dot. The Josephson current in the deep Kondo regime is thus very sensitive to the $SU(4)$ vs $SU(2)$ symmetry. Perturbation theory in $\Gamma$.— Next we address the opposite limit of very small $\Gamma\ll \Delta$, where lowest-order perturbation theory in $\Gamma$ applies. After some algebra, Eq. (\[partition\]) for $\theta=0$ yields the CPR of a tunnel junction, $I(\varphi)=I_c\sin(\varphi)$, where the critical current is $$\label{perturb} I_c = \left( 4\Theta(\epsilon)-\Theta(-\epsilon) \right) F(\|\epsilon\|/\Delta) \ I_0,$$ with the Heaviside function $\Theta$, the current scale $I_0 = \Delta (\Gamma/\pi \Delta)^2$, and (see also Ref. ) $$F(x) = \frac{(\pi/2)^2 (1-x) - {\rm arccos}^2 x}{2x (1-x^2)}.$$ In this $U\to \infty$ limit, the dot contains one electron for (finite) $\epsilon<0$, and thus we have spin $S=1/2$. Equation (\[perturb\]) shows that such a magnetic junction displays a $\pi$-phase. For the $SU(4)$ case, the ratio $I_c(-\|\epsilon\|)/I_c(\|\epsilon\|)=-1/4$ is twice smaller than in the $SU(2)$ case, i.e., $\pi$-junction behavior tends to be suppressed. This tendency is also confirmed for $U\ll \Delta$ (see below), where the $\pi$-phase is in fact essentially absent. The factor $1/4$ can be understood in simple terms by counting the number of possible processes leading to a Cooper-pair transfer through the dot.[@spivak; @shimizu]. When $\epsilon>0$, there are four possibilities corresponding to the quantum numbers $(\alpha,\sigma)$ of the first electron entering the dot. However, for $\epsilon<0$ there is only one possibility since an electron already occupies the dot, and then only one specific choice of $(\alpha,\sigma)$ allows for Cooper pair tunneling. This argument is readily generalized to the $SU(2N)$ case, where the above ratio of critical currents is obtained as $-1/2N$. Effective Hamiltonian for $\Delta\to \infty$.— The partition function (\[partition\]) simplifies considerably when $\Delta$ exceeds all other energy scales of interest. Then the dynamics is always confined to the subgap region (Andreev states), and quasiparticle tunneling processes from the leads (continuum states) are negligible. In particular, this allows to study the case $U\ll \Delta$. In fact, for $\Delta\to \infty$, with the Cooper pair operators $b^\dagger_1 = d^\dagger_{e\uparrow} d^\dagger_{o\downarrow}$ and $b^\dagger_2 = d^\dagger_{o\uparrow} d^\dagger_{e\downarrow}$, Eq. (\[partition\]) is equivalently described by the effective dot Hamiltonian $$\label{delinf} H_\infty = H_d + \cos(\varphi/2) \left [ \Gamma_e b_1 + \Gamma_o b_2 + {\rm h.c.} \right ].$$ The resulting Hilbert space can be decomposed into three decoupled sectors[@footnot2] according to spin $S$ and orbital pseudo-spin $T$ (notice that these quantities are localized on the dot for $\Delta\to \infty$). The ground-state energy $E_g(\varphi)= {\rm min}(E_{(S,T)})$ then determines the Josephson current $I(\varphi)=2\partial_\varphi E_g(\varphi)$. (i) The $(S,T)=0$ sector is spanned by the four states $\{ \|0\rangle, b_1^\dagger\|0\rangle, b_2^\dagger\|0\rangle, b_1^\dagger b_2^\dagger\|0\rangle\}$, where $\|0\rangle$ is the empty dot state. The matrix representation reads $$H_{(S,T)=0} = \left( \begin{array}{cccc} 0 & \Gamma_e \cos\frac{\varphi}{2} & \Gamma_o \cos\frac{\varphi}{2} & 0 \\ \Gamma_e \cos\frac{\varphi}{2} & E_2 & 0 & \Gamma_o\cos\frac{\varphi}{2}\\ \Gamma_o \cos\frac{\varphi}{2} & 0 & E_2 & \Gamma_e\cos\frac{\varphi}{2}\\ 0 & \Gamma_o \cos\frac{\varphi}{2} & \Gamma_e\cos\frac{\varphi}{2} & E_4 \end{array} \right),$$ with the eigenenergies $E_n= \epsilon n + U n(n-1)/2$ of the decoupled dot. The lowest energy $E_{(S,T)=0}=E_2+z$ then follows from the smallest root of the quartic equation $\prod_\pm \left( z^2 - 2 z U - (\Gamma_e\pm \Gamma_o)^2 \cos^2\frac{\varphi}{2} \right) = (E_4 z/2)^2.$ (ii) The $(S,T)=1/2$ sector can be decomposed into four subspaces with one or three electrons according to $S_z=\pm 1/2$ and Cooper pair channel $\nu=e,o$. The Hamiltonian is $H^{(\nu)}_{(S,T)=1/2}=\left( \begin{array}{cc} E_1 & \Gamma_\nu\cos\frac{\varphi}{2} \\ \Gamma_\nu \cos\frac{\varphi}{2} & E_3 \end{array}\right)$, where $H^{(e)}_{(S,T)=1/2}$ operates in the subspace spanned by $\{ d^\dagger_{o\uparrow}\|0\rangle, b_1^\dagger d^\dagger_{o\uparrow}\| 0\rangle \}$ for $S_z=+1/2$, and $\{ d^\dagger_{e\downarrow}\|0\rangle, b_1^\dagger d^\dagger_{e\downarrow}\| 0\rangle \}$ for $S_z=-1/2$. (Similarly, the subspaces corresponding to $H^{(o)}_{(S,T)=1/2}$ are obtained by letting $d^\dagger_{\nu\sigma}\to d^\dagger_{\nu,-\sigma}$ and $b_1^\dagger\to b_2^\dagger$.) With $\Gamma_e\ge \Gamma_o$, the lowest energy is $E_{(S,T)=1/2}= \left( E_1+E_3 - \left[(E_3-E_1)^2 + 4\Gamma_e^2 \cos^2 (\varphi/2)\right]^{1/2} \right)/2.$ (iii) Finally, the $(S,T)=(1,0)$ sector is spanned by the two uncoupled two-particle states $d^{\dagger}_{e,\sigma} d^\dagger_{o,\sigma}\|0\rangle$, with $\varphi$-independent energy $E_{S=1,T=0}=E_2$. In addition, there are two decoupled $(S,T)=(0,1)$ states $d_{\nu\uparrow}^\dagger d_{\nu\downarrow}^\dagger\|0\rangle$ with the same energy $E_2$. In the limit $\Delta\to \infty$, this $(S,T)=1$ sector is energetically unfavorable except possibly at $\varphi=\pi$. Phase diagram for $\Delta\gg \Gamma$.— Next we discuss the resulting phase diagram in the $SU(4)$ limit ($\theta=0$). The result for $\Delta\to \infty$ is shown in Fig. \[fig2\] in the $U-\epsilon$ plane. The phases are classified according to the three sectors defined above.[@footn] The reported phases are specific for the $SU(4)$ symmetry and are qualitatively different from the standard $SU(2)$ case. We observe that the $\varphi$-dependence of the $\Delta\to \infty$ ground-state energy implies $0$-junction behavior for both $S=0$ and $S=1/2$. While the magnetic $S=1/2$ sector often represents a $\pi$-junction,[@glazman; @rozhkov; @su2numeric] in multi-level dots there is no direct connection between the spin and the sign of the Josephson coupling.[@shimizu] The $\pi$-phase found under perturbation theory \[Eq. (\[perturb\]) for $\epsilon<0$\] is in fact restricted to the regime $U\gg \Delta$, while for $U\ll \Delta$, the $S=1/2$ state displays a $0$-phase. In the intermediate regime one should therefore observe a crossover between those two behaviors. Interestingly, there are parameter regions with a spin/pseudo-spin transition as $\varphi$ varies. For instance, the “black” regions in Fig. \[fig2\] correspond to a mixed state with $(S,T)=0$ at $\varphi=0$ and $(S,T)=1/2$ at $\varphi=\pi$, while for the “blue” region, the ground state is in the $(S,T)=0$ sector except at $\varphi=\pi$ where it crosses to the $(S,T)=1$ sector. We find that these phases are also observable at finite $\Delta \agt \Gamma$, where we have employed two complementary approaches. First, a full numerical solution is possible when approximating each electrode by a single site (zero-bandwidth limit), which can provide a satisfactory, albeit not quantitative, understanding of the phase diagram.[@vecino] Second, one can go beyond the above $\Delta\to \infty$ limit by including cotunneling processes in a systematic way. Both approaches give essentially the same results, and here we only show results from the single-site model. As can be observed in Fig. \[fig3\](a), the overall features of the $\Delta\to\infty$ phase diagram are reproduced for finite $\Delta$, with somewhat shifted boundaries between the different regions. In particular, in the “green” ($S=1/2$) regime, this calculation captures the mentioned transition from a $0$-junction at $\Delta\gg U$ to a $\pi$-junction at $\Delta\ll U$, as illustrated in Fig. \[fig3\](b). Consequently, for finite $\Delta$, the “black” phase may now have lowest energy at $\varphi=\pi$, implying the $\pi'$-phase [@rozhkov; @su2numeric; @vecino] indicated in “red” in Fig. \[fig3\](a). Finally, for the junctions with $U/\Gamma=10.5$ and $11$ in Fig. \[fig3\](b), the ground state is realized at phase difference $0<\varphi<\pi$, which implies that time-reversal symmetry is spontaneously broken here. To conclude, we have studied the Josephson current in $SU(4)$ symmetric quantum dots, including the crossover to the standard $SU(2)$ symmetric case. Contrary to normal-state transport, the supercurrent is very sensitive to the symmetry group, and should allow to observe clear signatures of the $SU(4)$ state in ultra-clean CNT dots. In particular, the $\pi$-phase is largely suppressed, the CPR in the Kondo limit has a distinctly different shape and a smaller critical current, and the phase diagram turns out to be quite rich. In addition, following Ref. , we expect a strongly reduced thermal noise in the deep $SU(4)$ Kondo regime since (in contrast to the $SU(2)$ case) there are two channels with imperfect transmission. Future theoretical work is needed to give a quantitative understanding of the crossover between the various regimes discussed above. This work was supported by the SFB TR/12 of the DFG, the EU network HYSWITCH, the ESF network INSTANS and by the Spanish MICINN under contracts FIS2005-06255 and FIS2008-04209. [99]{} Yu.V. Nazarov and Ya.M. 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Le Hur, P. Simon, and D. Loss, Phys. Rev. B [75]{}, 035332 (2007); C.A. Büsser and G.B. Martins, ibid. [75]{}, 045406 (2007); C. Mora, X. Leyronas, and N. Regnault, Phys. Rev. Lett. [100]{}, 036604 (2008); P. Vitushinsky, A.A. Clerk, and K. Le Hur, ibid. [100]{}, 036603 (2008); F.B. Anders et al., ibid. [100]{}, 086809 (2008). It is straightforward to allow for orbital or Zeeman fields, or for more general interactions. We also consider identical tunnel couplings between the dot and both electrodes. Asymmetries produce similar effects as the orbital mixing $\tilde t$ in $H_t$.[@aguado] T. Novotn[' y]{}, A. Rossini, and K. Flensberg, Phys. Rev. B 72, 224502 (2005). B.I. Spivak and S.A. Kivelson, Phys. Rev. B [43]{}, 3740 (1991). Y. Shimizu, H. Horii, Y. Takane, and Y. Isawa, J. Phys. Soc. Jpn. [5]{}, 1525 (1998); A.V. Rozhkov, D.P. Arovas, and F. Guinea, Phys. Rev. B [64]{}, 233301 (2001). Different phases can be labeled by $S^2 + T^2$, and we use the notation $(S,T)=0$ for $S=T=0$, $(S,T)=1/2$ for $S=T=1/2$, and $(S,T)=1$ for both $(S,T)=(1,0)$ and $(0,1)$. Strictly speaking, precisely at $\Delta=\infty$ there are degeneracies that make the classification ambiguous. However, at finite $\Delta$, Fig. \[fig3\] indicates that the reported phases are stable. We have also confirmed their stability for $\theta\ne 0$. A. Mart[í]{}n-Rodero, A. Levy Yeyati, and F.J. Garc[í]{}a-Vidal, Phys. Rev. B [53]{}, R8891 (1996).
--- abstract: 'There is currently much interest in the two-axis countertwisting spin squeezing Hamiltonian suggested originally by Kitagawa and Ueda, since it is useful for interferometry and metrology. No analytical solution valid for arbitrary spin values seems to be available. In this article we systematically consider the issue of the analytical solvability of this Hamiltonian for various specific spin values. We show that the spin squeezing dynamics can be considered to be analytically solved for angular momentum values upto $21/2$, i.e. for $21$ spin half particles. We also identify the properties of the system responsible for yielding analytic solutions for much higher spin values than based on naive expectations. Our work is relevant for analytic characterization of squeezing experiments with low spin values, and semi-analytic modeling of higher values of spins.' author: - 'M. Bhattacharya' title: 'Analytical solvability of the two-axis countertwisting spin squeezing Hamiltonian' --- Introduction ============ The two-axis countertwisting spin-squeezing Hamiltonian $(\hbar=1)$ $$\label{eq:HTA} H_{TA}=\frac{\chi}{2i}\left(J_{+}^{2}-J_{-}^{2}\right),$$ was originally proposed by Kitagawa and Ueda [@Kitagawa1991; @Kitagawa1993]. In Eq. (\[eq:HTA\]) $J_{\pm}$ are the angular momentum raising and lowering operators [@SakuraiBook], $i=\sqrt{-1}$ and $\chi$ is a constant. The quantity $J$ may refer to a real angular momentum or a pseudospin $J=N/2$ representing the collective squeezing of $N$ spin half systems [@Ma2011]. The Hamiltonian $H_{TA}$ yields maximal squeezing, with a squeezing angle independent of system size or evolution time. Experimental implementation has not yet been achieved, while a number of theoretical proposals have been put forward [@Berry2002; @Liu2011; @Nemoto2014; @Opatrny2014; @Huang2015; @Das2015; @Li2015]. A general analytic solution to the Hamiltonian for arbitrary angular momentum does not seem to be available [@Jafarpour2008; @Pathak2008]. Solutions to the dynamics for low values of spin are of interest to experiments with trapped ions [@Meyer2001] and quantum magnets [@Toth2009], for example. In Ref. [@Ma2011] a bound of spin $3/2$ was stated as the maximum angular momentum value for which analytic solutions can be found. We presume this was based on the fact that the Hamiltonian matrix in that case is of dimension $2\times 3/2+1=4,$ leading to a characteristic polynomial of quartic order, which is the highest degree for which algebraic solutions can generally be found [@StewartBook]. In Ref. [@Jafarpour2008] the existence of analytical solutions (with the additional presence of an external field) up to spin $10$ was reported and explicit expressions were provided for spin $1$. No explanation was given of the unexpected fact that solutions for spins much larger than $3/2$ were found, nor was the maximum value of spin for which analytical solutions could be found supplied. In the present article we systematically analyze the question of analytical solvability of the two-axis countertwisting spin squeezing Hamiltonian of Eq. (\[eq:HTA\]). We extend the bound for solvability to spin $21/2$, i.e. $21$ spin half particles (we note that in Ref. [@Jafarpour2008] spin $10$ corresponds to $10$ spin half particles). We point out that critical roles in the solvability of the model for large spin values are played by the chiral symmetry and sparsity of the Hamiltonian matrix. Our approach requires only the use of matrix representations of the angular momentum operators and the evaluation the time evolution operator [@Zela2014]. Our results may be useful for experiments with small number of spins. Some properties of $H_{TA}$ =========================== In this section we go over some properties of the Hamiltonian $H_{TA}$ of Eq. (\[eq:HTA\]) which are relevant to the discussion of analytical solvability. First, it can be seen readily by using $$J_{+}=J_{-}^{\dagger},$$ that $H_{TA}$ is Hermitian, implying that its eigenvalues are real. Further, by using $$J_{\pm}=J_{x}\pm i J_{y},$$ we can rewrite $$H_{TA}=\chi\left(J_{x}J_{y}+J_{y}J_{x}\right).$$ Now let us consider a rotation by the angle $\pi$ about the $J_{y}$ axis. This rotation leaves $J_{y}$ unaffected, but reverses the sign of $J_{x}$, i.e. $$e^{-i\pi J_{y}}H_{TA}e^{i\pi J_{y}}=-H_{TA},$$ which can be written as the anticommutation relation $$\label{eq:AntiC} \{H_{TA},e^{i\pi J_{y}}\}=0.$$ This relation implies that $e^{i\pi J_{y}}$ is a chiral symmetry of $H_{TA}$ [@Mishkat2014]. In practical terms, the implication of the anticommutation is that eigenvalues of $H_{TA}$ occur as signed pairs $\pm \lambda_{1},\pm \lambda_{2},\ldots$ (A brief proof is provided in the Appendix for the reader’s convenience). Therefore if $H_{TA}$ can be represented by an even dimensional matrix, then the characteristic polynomial of $H_{TA}$ $$P(\lambda)=\left\|H_{TA}-\lambda I\right\|=0,$$ is even in $\lambda$, where $I$ is the unit matrix of the same dimension as $H_{TA}$. If instead $H_{TA}$ is of odd dimension, then its characteristic polynomial is $\lambda$ times a polynomial even in $\lambda$. In this case there is necessarily a zero eigenvalue, and all other eigenvalues are signed pairs. Specific examples will be given below. As will be verified with these examples, the property of chiral symmetry contributes to giving a simple form to the characteristic polynomials for even large spins, and making them analytically solvable. For completeness we note that the Hamiltonian considered by the authors of Ref. [@Jafarpour2008] $$H_{f}=H_{TA}+\Omega J_{z},$$ where $\Omega$ represents an external field along the $z$ axis, also possesses the chiral symmetry indicated above. This partly explains the solvability of the squeezing model $H_{f}$ analytically for up to spin $10$. Finally, by using the matrix elements in the basis $\|j,m\rangle$ where $J_{z}$ is diagonal $$\langle j,m'\|J_{\pm}\|j,m\rangle = \sqrt{j(j+1)-m(m'\pm 1)},$$ it can be readily verified that $H_{TA}$ is a rather sparse matrix, i.e. most of its elements are zero. This feature also contributes to simplifying the form of the characteristic polynomial. The time evolution operator =========================== We now consider the time evolution operator $$\label{eq:UTA} U_{TA}=e^{-iH_{TA}t}.$$ Since $U_{TA}$ determines the spin dynamics completely, the model is analytically solvable if a matrix representation for $U_{TA}$ can be found, with all entries determined analytically. A straightforward way to implement this is to diagonalize $H_{TA}$. If the eigenvalues of $H_{TA}$ can be found analytically, the diagonal and analytic form of $U_{TA}$ follows. However, the calculation of expectation values then requires the relevant initial state to be rotated by the unitary matrix that diagonalizes $H_{TA}.$ As these matrices can be quite cumbersome and require the determination and careful handling of the eigenvectors of $H_{TA},$ we follow instead an equivalent procedure that only deals with the eigenvalues of $H_{TA},$ but avoids referring to its eigenvectors. Our approach is to expand the right hand side of Eq. (\[eq:UTA\]) in a Taylor series. The termination of that series is actually guaranteed since $H_{TA}$ is a finite dimensional matrix. This guarantee comes from the Cayley-Hamilton theorem, which states that every square matrix satisfies its own characteristic equation, which in turn implies that any power of the $H_{TA}$ can be expressed in terms of the matrix powers $(H_{TA})^{0},(H_{TA})^{1},\ldots, (H_{TA})^{2J},$ where $J$ is the associated spin. The entries of the terminating matrix representation of $U_{TA}$, it turns out, can be written as functions of the eigenvalues of $H_{TA}$ [@Zela2014], see below. Therefore, if the eigenvalues can be calculated analytically, then the spin dynamics can be found exactly. For reference, we quote the explicit expression for $U_{TA}$ in the case where $H_{TA}$ possesses nondegenerate eigenvalues $\lambda_{k}, k=1,2,\ldots$ (the degenerate case can be handled via some straightforward modifications) [@Zela2014]) $$\label{eq:UTAZela} U_{TA}=\displaystyle \sum_{k = 1}^{2j+1}e^{-i\lambda_{k}t} \displaystyle\prod_{{\substack{n=1 \\ n\neq k}}}^{2j+1} \left(\frac{H_{TA}-\lambda_{n}}{\lambda_{k}-\lambda_{n}}\right).$$ Results ======= Solvability ----------- In Table 1. we show for various spins the characteristic polynomial $P(\lambda)$ of $H_{AT}$. We make some comments on the entries in this table. For $J=1/2$, the eigenvalues are both zero, and it can be verified that $$H_{TA}(J=1/2)= \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \\ \end{array} \right),$$ consistent with the observation, first made by Kitagawa and Ueda, that a single spin half particle cannot be squeezed [@Kitagawa1993]. Generally for half-integer spin, $P(\lambda)$ is even in $\lambda$ and some of the eigenvalues are degenerate. In Table 1, we have indicated which spin values lead to degeneracy in the eigenvalues of $H_{TA}$, so that the appropriate procedure may be used for obtaining $U_{TA}$. Generally, if any factor in the characteristic polynomial repeats, then there is degeneracy in the energy spectrum. To make this identification rigorous, we have calculated the discriminant of $P(\lambda)$, which returns a zero value if degeneracy is present. The combination of chiral symmetry and matrix sparseness leads to rather compact expressions for the spin half-integer $P(\lambda)$’s. Indeed it is remarkable that $P(\lambda)$ for $J=21/2$ fits on a single line. For integer spin, $P(\lambda)$ is a polynomial even in $\lambda$, times a single factor of $\lambda$. There seems to be no degeneracy in general and the polynomials are less compact in form than in the half-integer spin case. Upto $J=9/2$ it is evident that the roots of the polynomials can be found analytically, since the factors are of degree $4$ or less, and solvability by radicals is guaranteed by the Abel-Ruffini theorem [@StewartBook]. While spins $5$ to $13/2$ have factors sextic in $\lambda$, since these polynomials are even in $\lambda$, they can be thought of as being cubics in $\lambda^{2},$ which can be solved analytically. The same reasoning applies to the octic factors in the polynomials for spins $7$ to $17/2.$ Finally, spin $9$ to $21/2$ contain factors of degree $10$, which can be considered as being polynomials of degree $5$ in $\lambda^{2}.$ While the roots of such factors cannot be found algebraically, they can be stated in terms of hypergeometric functions [@King1991]. However, for $J=11$, there is a factor of degree $12$ in $\lambda$ (i.e. of degree $6$ in $\lambda^{2}$). While the roots of polynomials of degree $6$ and higher can be found in terms of modular functions (for example), they involve transcendental functions, and we will consider them not to be of closed form and therefore analytically unsolvable [@BoydBook]. We note that for $J>21/2$ the polynomial roots can be found numerically and inserted in Eq. (\[eq:UTAZela\]), thus yielding a semianalytic solution for any spin value. Spin squeezing -------------- To compactly illustrate our results, we present the details for $J=2$ ($4$ spin half particles), a case for which there seem to be no explicit results in the literature. In this instance, the matrix representations are given by $$J_{+}= \left( \begin{array}{ccccc} 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & \sqrt{6} & 0 & 0 \\ 0 & 0 & 0 & \sqrt{6} & 0 \\ 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)$$ $J_{-}=J_{+}^{\dagger}$, and $$H_{TA}=i\chi \left( \begin{array}{ccccc} 0 & 0 & -\sqrt{6} & 0 & 0 \\ 0 & 0 & 0 & -3 & 0 \\ \sqrt{6} & 0 & 0 & 0 & -\sqrt{6} \\ 0 & 3 & 0 & 0 & 0 \\ 0 & 0 & \sqrt{6} & 0 & 0 \\ \end{array} \right),$$ The eigenvalues of $H_{TA}$ can be read off from Table 1. as $0,\pm 3,\pm 2\sqrt{3}$. The eigenvectors are $(-1, 0, i\sqrt{2}, 0,1)/2, (-1, 0, -i\sqrt{2}, 0, 1)/2, (0, i, 0, 1, 0)/\sqrt{2}$, $(0, -i, 0, 1, 0)/\sqrt{2}$, and $(1, 0, 0, 0, 1)/\sqrt{2}$ in no specific order. The time evolution operator is $$U_{TA}= \left( \begin{array}{ccccc} \cos^{2}(\sqrt{3}\chi t)& 0 & -\sin(2\sqrt{3}\chi t)/2 & 0 & \sin^{2}(\sqrt{3}\chi t) \\ 0 & \cos(3\chi t) & 0 & -\sin(3\chi t) & 0 \\ \sin(2\sqrt{3}\chi t)/2 & 0 & \cos(2\sqrt{3}\chi t) & 0 & -\sin(2\sqrt{3}\chi t)/2 \\ 0 & \sin(3\chi t) & 0 & \cos(3\chi t) & 0 \\ \sin^{2}(\sqrt{3}\chi t) & 0 & \sin(2\sqrt{3}\chi t)/2 & 0 & \cos^{2}(\sqrt{3}\chi t) \\ \end{array} \right).$$ The time evolution of any operator $\mathcal{O}$ is given by $\mathcal{O}(t) = U_{TA}^{-1}\mathcal{O}U_{TA}.$ Using this relation we can find the time-evolved quantities $J_{y}(t), J_{y}^{2}(t),$ etc., and variances such as $$\left(\Delta J_{y,}(t)\right)^{2}=\langle J_{y,}^{2}(t)\rangle-\langle J_{y}(t)\rangle^{2},$$ etc. Starting from the initial state $$\ket{i}=e^{i\pi J_{y}/2}\ket{j=2,m=2}=\frac{1}{2}\ket{\frac{1}{2},1,\sqrt{\frac{3}{2}},1,\frac{1}{2}},$$ i.e. the stretched state along $z$ rotated by $90^\circ$ about the $y$ axis [@Kitagawa1993], we find the squeezing parameters following Wineland et al. [@Wineland1992; @Wineland1994] $$\begin{aligned} \label{eq:WineSqueeze1} \xi_{y}&=&\sqrt{4}\frac{\langle \left(\Delta J_{y}(t)\right)\rangle}{\left\|\langle J_{x}(t)\rangle \right\|},\nonumber\\ &=&\frac{\sqrt{2}(17-6\cos6\chi t-6\cos2\sqrt{3}\chi t+3\cos4\sqrt{3}\chi t)^{1/2}} {\left\|\cos3\chi t(1+3\cos2\sqrt{3}\chi t)+\sqrt{3}\sin3\chi t\sin2\sqrt{3}\chi t\right\|},\nonumber\\\end{aligned}$$ and ----------------- -------------------------------------------------------------------------------------------------------------------------------------------- ----------------------- $J$ $P(\lambda)$ $\mathrm{Degenerate}$ \[0.5ex\] $1/2$ $\lambda^{2}$ Yes $1$ $\lambda (1-\lambda^{2})$ No $3/2$ $(\lambda^{2}-3)^{2}$ Yes $2$ $-\lambda (\lambda^{2}-3)(\lambda^{2}-12)$ No $5/2$ $\lambda^{2}(\lambda^{2}-28)^{2}$ Yes $3$ $-\lambda (\lambda^2-60) (\lambda^2-6 \lambda-15) (\lambda^2+6 \lambda-15)$ No $7/2$ $(\lambda^{4} - 126 \lambda^{2}+945)^{2}$ Yes $4$ $-\lambda (\lambda^{2}-28)(\lambda^{2}-208)$ $\times (\lambda^{2}+10\lambda-63)(\lambda^{2}-10\lambda-63)$ No $9/2$ $\lambda^{2}(\lambda^{4}-396\lambda^{2}+19008)^{2}$ Yes $5$ $-\lambda (\lambda^{2}-108)(\lambda^{2}-528)$ $\times (\lambda^{6}-651\lambda^{4}+65619\lambda^{2}-455625)$ No $11/2$ $(\lambda^{6} - 1001 \lambda^{4}+172315\lambda^{2}-2338875)^{2}$ Yes $6$ $-\lambda (\lambda^{2}-336)(\lambda^{4}-1176\lambda^{2}+55440)$ $\times (\lambda^{6}-1491\lambda^{4}+421155\lambda^{2}-12006225)$ No $13/2$ $\lambda^{2}(\lambda^{6}-2184\lambda^{4}+1012752\lambda^{2}-74794752)^{2}$ Yes $7$ $-\lambda (\lambda^{2}-784)(\lambda^{4}-2296\lambda^{2}+353808)$ $\times (\lambda^{8}-3108\lambda^{6}+2236710\lambda^{4}-328692196\lambda^{2}+3773030625)$ No $15/2$ $(\lambda^{8}-4284\lambda^{6}+4488102\lambda^{4}-1062230652 \lambda^{2}+22347950625)^{2}$ Yes $8$ $-\lambda (\lambda^{4}+6624\lambda^{2}+1900800)(\lambda^{4}+16704\lambda^{2}+28753920)$ $\times (\lambda^{8}+23184\lambda^{6}+138054240\lambda^{4}+204233529600\lambda^{2}+33886369440000)^{2}$ No $17/2$ $\lambda^{2}(\lambda^{8}-7752\lambda^{6}+16263696\lambda^{4}-9531032320\lambda^{2}+995361177600)^{2}$ Yes $9$ $-\lambda (\lambda^{4}-7056\lambda^{2}+6441984)(\lambda^{4}-3096\lambda^{2}+668304)$ $\times (\lambda^{10}-10197\lambda^{8}+29403594\lambda^{6}-25878927978\lambda^{4}+5213177173701\lambda^{2}-88322873900625)$ No $19/2$ $(\lambda^{10}-13167\lambda^{8}+50640282\lambda^{6}-62764022286\lambda^{4}+19627235976789\lambda^{2}-584689432201875)^{2}$ Yes $10$ $-\lambda (\lambda^{4}-5456\lambda^{2}+3165184)(\lambda^{6}-11396 \lambda^{4}+20438704\lambda^{2}-2031480000)$ $\times (\lambda^{10}-16797\lambda^{8}+84869994 \lambda^{6}-145160193178\lambda^{4}+68747106284901\lambda^{2}-3870591128105625)$ No $21/2$ $\lambda^{2}(\lambda^{10}-21252\lambda^{8}+140008176\lambda^{6}-329460868800\lambda^{4}+241815611520000\lambda^{2}-33685691719680000)^{2}$ Yes $11$ $-\lambda (\lambda^{4}-8976\lambda^{2}+10644480)(\lambda^{6}-17556\lambda^{4}+55226160\lambda^{2}-15437822400)$ $\times (\lambda^{12}-26598\lambda^{10}+225185103\lambda^{8}-712278892116\lambda^{6}$ $+768687668037135\lambda^{4} No -202420859545362150\lambda^{2}4712996874211250625)$ ----------------- -------------------------------------------------------------------------------------------------------------------------------------------- ----------------------- : Characteristic Polynomials of $H_{TA}/\chi$ \[table:table1\] $$\begin{aligned} \label{eq:WineSqueeze2} \xi_{z}&=&\sqrt{4}\frac{\langle \left(\Delta J_{z}(t)\right)\rangle}{\left\|\langle J_{x}(t)\rangle \right\|},\nonumber\\ &=&\frac{2[7-3\cos4\sqrt{3}\chi t-(\sin6\chi t+\sqrt{3}\sin2\sqrt{3}\chi t)^{2}]^{1/2}} {\left\|\cos3\chi t(1+3\cos2\sqrt{3}\chi t)+\sqrt{3}\sin3\chi t\sin2\sqrt{3}\chi t\right\|}.\nonumber\\\end{aligned}$$ Squeezing occurs when the squeezing parameter is less than $1$. The parameter $\xi_{y}$ is plotted versus time in Fig. \[fig:P1\]. There is no squeezing in the $y$ quadrature for the duration shown. ![(Color online) The squeezing parameter $\xi_{y}$ \[Eq. (\[eq:WineSqueeze1\])\] as a function of the dimensionless time $\chi t$ for $J=2$.[]{data-label="fig:P1"}](P1.eps){width="40.00000%"} The parameter $\xi_{z}$ is plotted in Fig. \[fig:P2\]. Squeezing in the $z$ quadrature can be seen for two short intervals in the diagram. ![(Color online) The squeezing parameter $\xi_{z}$ \[Eq. (\[eq:WineSqueeze2\])\] as a function of the dimensionless time $\chi t$ for $J=2$.[]{data-label="fig:P2"}](P2.eps){width="40.00000%"} ![(Color online) The correlation $\langle J_{x}J_{z}+J_{z}J_{x}\rangle$ \[Eq. (\[eq:corr\])\] as a function of the dimensionless time $\chi t$ for $J=2$.[]{data-label="fig:P3"}](P3.eps){width="40.00000%"} For completeness we mention that correlations between the various spin components, which are relevant to squeezing [@Ma2011], can also be found analytically for this solvable model. For example, plotted in Fig. \[fig:P3\] is the quantity $$\begin{aligned} \label{eq:corr} \langle J_{x}J_{z}+J_{z}J_{x}\rangle&= &\frac{3}{2}\cos\sqrt{3}\chi t\nonumber\\ &&\times\left[\left(1-\sqrt{3}\right)\sin \left(3-\sqrt{3}\right)\chi t\right.\nonumber\\ &&+\left.\left(1+\sqrt{3}\right)\sin\left(3+\sqrt{3}\right)\chi t\right].\\ \nonumber\end{aligned}$$ Conclusion ========== We have shown that the two-axis counter-twisting spin squeezing Hamiltonian can be solved analytically for up to angular momentum $21/2.$ We have discussed the properties of the Hamiltonian that lead to such a high degree of solvability. From our results the axis of optimum squeezing can be found readily. Our methods can also be used to find useful quantities such as entanglement measures, in closed form. Future work will investigate the effects of decoherence on the solutions. We would like to thank K. Hazzard for stimulating discussions. Appendix ======== In this Appendix we show that the anticommutation of Eq.(\[eq:AntiC\]) implies the pairing of eigenvalues of $H_{TA}$. Consider an eigenvector $\psi_{+}$ of $H_{TA}$ with an eigenvalue $\lambda$, i.e. $$\label{eq:PlusH} H_{TA}\psi_{+}=\lambda\psi_{+},$$ Multiplying from the left by $e^{i\pi J_{y}}$ and using the anticommutation of Eq. (\[eq:AntiC\]), we find the left-hand side of Eq. (\[eq:PlusH\]) reads $$\label{eq:LHS} e^{i\pi J_{y}}H_{TA}\psi_{+} = - H_{TA}e^{i\pi J_{y}}\psi_{+},$$ while the right-hand side reads $$\label{eq:RHS} e^{i\pi J_{y}}(\lambda\psi_{+}) = \lambda(e^{i\pi J_{y}}\psi_{+}).$$ Equating the right hand sides of Eqs. (\[eq:LHS\]) and (\[eq:RHS\]), we arrive at $$H_{TA}(e^{i\pi J_{y}}\psi_{+}) = -\lambda(e^{i\pi J_{y}}\psi_{+}),$$ which implies that $$\psi_{-} = e^{i\pi J_{y}}\psi_{+},$$ is an eigenfunction of $H_{TA}$ with an eigenvalue of $-\lambda$. Thus the anticommutation of the operator $e^{i\pi J_{y}}$ with $H_{TA}$ leads to the $\pm \lambda$ pairing of eigenvalues in the spectrum of $H_{TA}$. [27]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , (, , ). , (, , ). , ** (, , ).
--- abstract: 'The equation of motion of an extended object in spacetime reduces to an ordinary differential equation in the presence of symmetry. By properly defining of the symmetry with notion of cohomogeneity, we discuss the method for classifying all these extended objects. We carry out the classification for the strings in the five-dimensional anti-de Sitter space by the effective use of the local isomorphism between $\operatorname{{\it SO}}(4,2)$ and $\operatorname{{\it SU}}(2,2)$. We present a general method for solving the trajectory of the Nambu-Goto string and apply to a case obtained by the classification, thereby find a new solution which has properties unique to odd-dimensional anti-de Sitter spaces. The geometry of the solution is analized and found to be a timelike helicoid-like surface.' author: - Tatsuhiko Koike - Hiroshi Kozaki - Hideki Ishihara date: 'February, 2008' title: 'Strings in five-dimensional anti-de Sitter space with a symmetry' --- Introduction ============ Existence and dynamics of extended objects play important roles in various stages in cosmology. Examples of extended objects include topological defects, such as strings and membranes, and the Universe as a whole embedded in a higher-dimensional spacetime in the context of the brane-world universe model [@RanSun99PRL]. The trajectory of an extended object forms a hypersurface in the spacetime which is determined by a partial differential equation (PDE). For example, a test string is described by the Nambu-Goto equation which is a PDE in two dimensions. Because the dynamics is more complicated than that of a particle, one usually cannot obtain general solutions. One way to find exact solutions is to assume symmetry. The simplest solutions to such a PDE are homogeneous ones, in which case the problem reduces to a set of algebraic equations. However, the solutions do not have much variety and the dynamics is trivial. One may expect that if we assume “less” homogeneity, the equation still remains tractable and the solutions have enough variety to include nontrivial configurations and dynamics of physical interest. The [[cohomogeneity-one]{}]{} objects give such a class, which helps us to understand the basic properties of the extended objects and serves as a base camp to explore their general dynamics. For a string, stationarity is a special case of the [[cohomogeneity one]{}]{} condition. Some stationary configurations of the Nambu-Goto strings are obtained in the Schwarzchild spacetime [@FSZH]. Even in the Minkowski space, many nontrivial [[cohomogeneity-one]{}]{} solutions of the string were recently found [@ogawa; @IshKoz05PRD]. A [[cohomogeneity-one]{}]{} object is defined, roughly speaking, as the one whose world sheet is homogeneous except in one direction. Then any covariant PDE governing such an object reduces to an ordinary differential equation (ODE), which can easily be solved analytically, or at least, numerically. A solution represents the dynamics of a spatially homogeneous object, or the nontrivial configuration of a stationary object, depending on the homogeneous “direction” is spacelike or timelike. The case of null homogeneous “direction” should also give new intriguing models. In this paper, we treat strings in the five-dimensional anti-de Sitter space ${{AdS}}^5$. The choice of the spacetime is to meet the recent interest in higher-dimensional cosmology, including the brane-world universe model, and in string theory, though the method developed here is applicable to any background spacetime. A particular example which has recently been attracting much attention is the string in a spacetime with large extra dimensions, which are suggested e.g. by the brane-world model. A detailed investigation [@Jackson:2004zg] suggests that the reconnection probability for this type of strings is significantly suppressed. Then, contrary to what had usually been believed, the strings in the Universe can stay long enough to be considered stationary. Therefore classifying [[cohomogeneity-one]{}]{} strings and solving dynamics thereof are important for examining the roles of the string in cosmology. We first give the classification of all [[cohomogeneity-one]{}]{} strings which is valid for any covariant equation of motion. Then, in the case of Nambu-Goto strings, we give a general method for solving the trajectory. The method can be easily applied to the cases of other equations of motion. We demonstrate the procedure and give explicit solutions in some particular cases. In the classification, we make use of the local isomorphism between $\operatorname{{\it SO}}(4,2)$ and $\operatorname{{\it SU}}(2,2)$ in an essential way. The latter group is easier to treat because the dimensionality of the matrix is lower and because the Jordan decomposition of complex matrices is simpler than that of real ones. Therefore, though a similar classification of Killing fields is found in literature in the context of constructing quotient spaces of the anti-de Sitter space [@HolPel97CQG], we present an alternative proof based on the classification of $H$-[[anti-selfadjoint]{}]{} matrices in the Appendix. In Sec. \[general\], we give a method for the classification of all [[cohomogeneity-one]{}]{} strings in general, and a method for solving the equations of motions for Nambu-Goto strings. The latter can be easily applied to other equations of motion. In Sec. \[loc iso\], The useful relation of the isometry group ${{\operatorname{{\it SO}}(4,2)_0}}$ and ${{\operatorname{{\it SU}}(2,2)}}$ is briefly explained. We give the classification of the [[cohomogeneity-one]{}]{} strings in the anti-de Sitter space in Sec. \[classification\]. In Sec. \[hopf\], we demonstrate the method presented in Sec. \[general\] by an example. There we solve the Nambu-Goto equation and examine the geometry of its world sheet. Sec. \[conc\] is devoted for conclusion. In this paper, a spacetime $({{\cal M}},g)$ is a manifold ${{\cal M}}$ endowed with a Lorentzian metric $g$. We denote by $G$ the identity component of the isometry group of $({{\cal M}},g)$, and by ${{\mathfrak g}}$ its Lie algebra. We use the unit such that the speed of light and Newton’s constant are one. ![To solve a trajectory of the [[cohomogeneity-one]{}]{} string is to find a curve $C$ in ${{\cal M}}$ which projects to a geodesic $c$ on ${{\cal O}}$.[]{data-label="fig-bundle"}](c1string){width=".6\linewidth"} General treatment of [[cohomogeneity-one]{}]{} strings {#general} ====================================================== In this section, we develop a general method for classifying [[cohomogeneity-one]{}]{} objects and solving their dynamics in an arbitrary spacetime $({{\cal M}},g)$. Let us start with the definition of the [[cohomogeneity-one]{}]{} objects. We say that a $m$-dimensional hypersurface ${{\cal S}}$ in ${{\cal M}}$ is of [[cohomogeneity one]{}]{} if it is foliated by $(m-1)$-dimensional submanifolds ${{\cal S}}_\sigma$ labeled by a real number $\sigma$ and there is a subgroup $K$ of $G$ which preserves the foliation and acts transitively on ${{\cal S}}_\sigma$. In particular, the hypersurfaces ${{\cal S}}_\sigma$’s are embedded homogeneously in ${{\cal M}}$. A [[cohomogeneity-one]{}]{} object has a world sheet which is a [[cohomogeneity-one]{}]{} hypersurface. In this paper, we focus on the case that the extended objects are strings, so that $m=2$, and $K$ is a one-parameter group $(\phi_\tau)_{\tau\in{{\mathbb R}}}$ of isometries. First, let us consider how to classify the [[cohomogeneity-one]{}]{} strings. Given a one-dimensional subgroup $K\subset G$ and a point $p\in M$, the equations of motion determines a unique world sheet of a [[cohomogeneity-one]{}]{} object. The dynamics of the two strings can be considered the same if there is an isometry sending one of their trajectories, ${{\cal S}}$, to the other, ${{\cal S}}'$. In this paper, we identify the two dynamics if we can do so gradually, namely, if there is a one-parameter group of isometries $(\phi'_\lambda)_{\lambda\in[0,1]}$ such that $\phi'_0$ is the identity and $\phi'_1({{\cal S}})={{\cal S}}'$. We therefore classify the [[cohomogeneity-one]{}]{} strings up to isometry connected to the identity. In terms of Killing vector fields, it is to classify the Killing vector field $\xi$ generating $K$ up to scalar multiplication and up to isometry. Namely, $\xi$ and $a\phi_\xi$ are equivalent if there exists $\phi\in G$ and $a\ne0$. To put it more algebraically, the task is to find ${{\mathfrak g}}/\operatorname{Ad}_G$ up to scalar multiplication. Second, let us give a formalism to solve the dynamics and the configuration of the [[cohomogeneity-one]{}]{} strings. We assume that the string is described by the Nambu-Goto action $$\begin{aligned} S=\int_S\sqrt{-g_{ab}dx^adx^b}. \end{aligned}$$ The orbit space of the string with the symmetry group $K$ is defined by ${{\cal O}}:={{\cal M}}/K$, i.e., by identifying all the points on each Killing orbit in ${{\cal M}}$. The submanifolds ${{\cal S}}_\sigma$ mentioned above are the preimages $\pi^{-1}(x)$ of a point $x\in{{\cal O}}$. One can endow ${{\cal O}}$ with a metric $h$ so that the projection $\pi: ({{\cal M}},g)\to ({{\cal O}},h)$ is an orthogonal projection, or more precisely, a Riemannian submersion. The metric $h$ is given by $$\begin{aligned} h_{ab}:=g_{ab}-\xi_a\xi_b/f, \label{hab}\end{aligned}$$ where $f:=\xi^a\xi_a$. This metric has the Euclidean signature if the Killing vector $\xi$ is timelike, i.e., if $f<0$, and the Lorentzian signature if $\xi$ is spacelike, i.e., if $f>0$. Carrying out the integration along $\xi$ in the Nambu-Goto action, one obtains $$\begin{aligned} S=\int_c \sqrt{-f h_{ab}dx^adx^b}, \label{line_action}\end{aligned}$$ where $c$ is a curve on ${{{\cal O}}}$. Thus the problem of the string reduces to finding geodesics on the orbit space ${{\cal O}}$ with the metric $-fh$. For convenience, we adopt a modified action $$\begin{aligned} S=\int_c d\sigma{{\left( -\frac1{\alpha}f h_{ab}\dot x^a\dot x^b+{\alpha} \right)}}, \label{line_action_2}\end{aligned}$$ where an overdot denotes the differentiation by $\sigma$. The action [(\[line\_action\_2\])]{} derives the same geodesic equations as [(\[line\_action\])]{} and retains the invariance under reparametrization of $\sigma$. The function ${\alpha}$ is the norm of the tangent vector. The two-dimensional world sheet of the string is the preimage $\pi^{-1}(c)$ of the geodesic $c$ on $({{\cal O}},-fh)$. However, it is sometimes more convenient to find a [lift]{} curve ${C}$ on ${{\cal M}}$ whose projection $\pi({C})$ is a geodesic on $({{\cal O}},-fh)$ than to find a geodesic on $({{\cal O}},-fh)$ (Fig. \[fig-bundle\]). The Hopf string in Sec. \[hopf\] is such an example. In the case, the trajectory of the string is given by $$\begin{aligned} {{\cal S}}&=\pi^{-1}(\pi({C})) {\nonumber \\ }&= \{\phi_\tau({C}(\sigma));\, (\tau,\sigma)\in {{\mathbb R}}^2\}. \label{eq-general-trajectory}\end{aligned}$$ Note that the last expression in [(\[eq-general-trajectory\])]{} depends on the objects in ${{\cal M}}$ only. Thus the trajectory ${{\cal S}}$ can be viewed as a foliation by mutually isometric curves $\phi_\tau\circ{C}$ labeled by $\tau$. After one obtains the solutions of the equation of motion, one may want to classify their trajectories up to isometry. This can be done by identifying $C$ (or ${{\cal S}}$) which are related by [ homogeneity-preserving isometries]{}. We say that an isometry $\Phi$ is homogeneity-preserving if it preserves the action of $K$, i.e., if it satisfies $$\begin{aligned} \Phi\circ K\circ\Phi^{-1}=K. \end{aligned}$$ The homogeneity-preserving isometries form a group. In algebraic terms, the group is the normalizer of $K$ in the group $G$ of isometries on ${{\cal M}}$, which is denoted by $N_G(K)$. Its Lie algebra is the idealizer of ${{{\mathfrak k}}}$ in ${{{\mathfrak g}}}$ which is denoted by $N_{{{\mathfrak g}}}({{\mathfrak k}})$. We note that in the special case that $\Phi$ commutes with the action of $K$, [[[i.e.]{}]{}]{} when $\Phi$ is in the [centralizer]{} $Z_G(K)$ of $K$ in $G$, the squared norm of $\xi$ must be invariant under $\Phi$. This can be seen from $\Phi_f =\Phi_* (g_{ab}\xi^a\xi^b) =(\Phi_g_{ab})\xi^a\xi^b +g_{ab}(\Phi_\xi^a)\xi^b +g_{ab}\xi^a(\Phi_\xi^b) =f $, where we have used $\Phi_g_{ab}=g_{ab}$ and $\Phi_\xi^a=\xi^a$. The whole procedure of solving the dynamics is explicitly carried out for an example in Sec. \[hopf\]. ${{AdS}}^5$ and its isometry group {#loc iso} ================================== Hereafter in this paper, we assume that the spacetime $({{\cal M}},g)$ is the five-dimensional anti-de Sitter space ${{AdS}}^5$, or its universal cover ${\widetilde}{{{AdS}}^5}$. The former space has closed timelike curves which in the latter space are “opened up” to infinite nonclosed curves. The latter is usually more suitable when we discuss cosmology, but we will not distinguish them strictly in the following. The space ${{AdS}}^5$ is the most easily expressed as a pseudo-sphere $$\begin{aligned} {\overline}{\psi} {\psi}=-1 \label{eq-AdS}\end{aligned}$$ in the pseudo-Euclidean space ${E}^{4,2}$ whose metric is $dS^2=l^2d{\overline}{\psi}d{\psi}$, where we have used complex coordinates ${\psi}:=({\psi}^0,{\psi}^1,{\psi}^2)^T\in{{\mathbb C}}^3$, and have defined ${\overline}{\psi}:={\psi}^\dagger\zeta$ and $\zeta:={{\mbox{\rm diag}}}[-1,1,1]$. The isometry group of ${{AdS}}^5$ is ${{\operatorname{{\it SO}}(4,2)}}$ acting on $(s,t,x,y,z,w)^T\in{{\mathbb R}}^6$, where ${\psi}^0:=s+it$, ${\psi}^1:=x+iy$, and ${\psi}^2:=z+iw$. In the classification of the strings, however, we take advantage of the isomorphism ${{\operatorname{{\it SO}}(4,2)_0}}\simeq{{\operatorname{{\it SU}}(2,2)}}/\{\pm1\}$ and work with ${{\operatorname{{\it SU}}(2,2)}}$. Let $V$ be the vector space whose elements are complex symmetric matrices of the form $$\begin{aligned} p &= \begin{bmatrix} 0&{({\psi}^0)^}& -{\psi}^2 & {({\psi}^1)^}\\ {({\psi}^0)^}&0&{\psi}^2 & {({\psi}^2)^} \\ {\psi}^2 & -{\psi}^1&0&{\psi}^0\\ -{({\psi}^1)^} & -{({\psi}^2)^}&{\psi}^0&0 \end{bmatrix} {\nonumber \\ }&=s i{\sigma_{z}}{\otimes}{\sigma_{y}}+t 1{\otimes}{\sigma_{y}} +x i{\sigma_{y}}{\otimes}{\sigma_{z}} {\nonumber \\ }&\qquad +y{\sigma_{x}}{\otimes}1 -zi{\sigma_{y}}{\otimes}{\sigma_{x}}+w {\sigma_{x}}{\otimes}{\sigma_{y}}, \label{eq-SU22rep-2}\end{aligned}$$ where ${\sigma_{x}}$, ${\sigma_{y}}$ and ${\sigma_{z}}$ are the Pauli matrices and $1$ is the $2\times2$ identity matrix. The action of an element of ${{\operatorname{{\it SO}}(4,2)_0}}$ on $E^{4,2}$ corresponds to the action of $U\in{{\operatorname{{\it SU}}(2,2)}}$ on $V$ in the following way [@Yok90 p106]: $$\begin{aligned} &p\mapsto UpU^T. \label{eq-SU22rep-1} \end{aligned}$$ The Lie algebra ${{{{\mathfrak s}}{{\mathfrak u}}(2,2)}}$ of ${{\operatorname{{\it SU}}(2,2)}}$ consists of the matrices $X=$ satisfying $X\eta+\eta {X^\dag}=0$, where ${\eta}:={{\mbox{\rm diag}}}[1,1,-1,-1]$. The explicit form is $$\begin{aligned} X={\left[ \begin{array}{cc} {\beta} & {\gamma} \\ {{\gamma^\dag}} & {\delta} \end{array}\right] }, \end{aligned}$$ where $\gamma$ is a $2\times2$ complex matrix, and $\beta$ and $\delta$ are $2\times2$ anti-Hermitian matrices. The infinitesimal transformation for [(\[eq-SU22rep-1\])]{} is given by the action of $X\in{{{{\mathfrak s}}{{\mathfrak u}}(2,2)}}$ as $$\begin{aligned} p\mapsto Xp+pX^T=\{X_S,p\}+[X_A,p], \end{aligned}$$ where $X_S:=(X+X^T)/2$ and $X_A:=(X-X^T)/2$ are the symmetric and antisimmetric parts, respectively, of $X$. The correspondence between the ${{{{\mathfrak s}}{{\mathfrak u}}(2,2)}}$ and ${{{{\mathfrak s}}{{\mathfrak o}}(4,2)}}$ infinitesimal transformations are given in Table \[tab-cor\], where $X=(e_1{\otimes}e_2)/2$. In the table, $J_{xy}$ denotes the rotation in the $xy$ plane, $L$ denotes the rotation in the $st$ plane, $K_x$ denotes the $t$-boost in the $x$ direction, ${\widetilde}K_w$ denotes the $s$-boost in the $w$ direction, etc. 1 ${\sigma_{x}}$ ${\sigma_{y}}$ ${\sigma_{z}}$ ------------------ -------------------- ------------------- ------------------- ------------------- $1/i$ $J_{yz}$ $J_{zx}$ $J_{xy}$ ${\sigma_{x}}$ $K_w$ ${\widetilde}K_x$ ${\widetilde}K_y$ ${\widetilde}K_z$ ${\sigma_{y}}$ $-{\widetilde}K_w$ $K_x$ $K_y$ $K_z$ ${\sigma_{z}}/i$ $L$ $J_{wx}$ $J_{wy}$ $J_{wz}$ : Correspondence between the ${{{{\mathfrak s}}{{\mathfrak u}}(2,2)}}$ and ${{{{\mathfrak s}}{{\mathfrak o}}(4,2)}}$ transformations.[]{data-label="tab-cor"} The classification {#classification} ================== In this section, we obtain the classification of the [[cohomogeneity-one]{}]{} strings in ${{AdS}}^5$. As discussed in Sec. \[general\], the classification is to find ${{\mathfrak g}}/\operatorname{Ad}_G$ up to scalar multiplication, where $G={{\operatorname{{\it SO}}(4,2)_0}}$. Because ${{\operatorname{{\it SO}}(4,2)_0}}$ is isomorphic to ${{\operatorname{{\it SU}}(2,2)}}/\{\pm1\}$ as is seen in Sec. \[loc iso\], ${{{{\mathfrak s}}{{\mathfrak o}}(4,2)}}/\operatorname{Ad}_{{\operatorname{{\it SO}}(4,2)_0}}$ is isomorphic to ${{{{\mathfrak s}}{{\mathfrak u}}(2,2)}}/\operatorname{Ad}_{{\operatorname{{\it SU}}(2,2)}}$. Thus the classification is to find ${{{{\mathfrak s}}{{\mathfrak u}}(2,2)}}/\operatorname{Ad}_{{\operatorname{{\it SU}}(2,2)}}$ up to scalar multiplication. However, the equivalence classes ${{{{\mathfrak s}}{{\mathfrak u}}(2,2)}}/\operatorname{Ad}_{{\operatorname{{\it SU}}(2,2)}}$ is known as in the Lemma below, so that we can easily classify the [[cohomogeneity-one]{}]{} strings by further identifying the equivalence classes by scalar multiplications. We begin with introducing some terms which is necessary to state the Lemma. Let $H$ be an invertible Hermitian matrix. The [$H$-adjoint]{} of a square matrix $A$ is defined by ${{A}^{\star}}:=H^{-1}{A^\dag}H$. A matrix $A$ is called [$H$-[[selfadjoint]{}]{}]{} when ${{A}^{\star}}=A$, [$H$-[[anti-selfadjoint]{}]{}]{} when ${{A}^{\star}}=-A$, and [$H$-unitary]{} when $A{{A}^{\star}}={{A}^{\star}} A=1$. We say that matrices $A$ and $B$ are [$H$-unitarily similar]{} and write $A{\stackrel{H}\sim}B$ if there exists an $H$-unitary matrix $W$ satisfying $B=WAW^{-1}$. In these terms, ${{\operatorname{{\it SU}}(2,2)}}$ is the group of unimodular ${\eta}$-unitary matrices and ${{{{\mathfrak s}}{{\mathfrak u}}(2,2)}}$ is the Lie algebra of traceless ${\eta}$-[[anti-selfadjoint]{}]{} matrices. Thus, from the discussion in Sec. \[general\], our task of classifying [[cohomogeneity-one]{}]{} strings is to classify the elements of ${{{{\mathfrak s}}{{\mathfrak u}}(2,2)}}$ up to equivalence relation ${\stackrel{\rm \eta}\sim}$ and up to scalar multiplication. Let us introduce another equivalence relation closely related to the one above. Let $(A,H)$ be a pair of a complex matrix and an invertible Hermitian matrix $H$. The pairs $(A,H)$ and $(A',H')$ are said [unitarily similar]{} if there is a complex matrix $W$ such that $ A'=WAW^{-1}, H'=WHW^\dagger $ [GLR83]{}. This is an equivalence relation and will be denoted by $(A,H){\sim}(A',H')$. Note that $A{\stackrel{\rm \eta}\sim}A'$ is equivalent to $(A,{\eta}){\sim}(A',{\eta})$. Let $A$ be an $H$-[[selfadjoint]{}]{} matrix. Then if $\lambda$ is an eigenvalue of $A$, so is its complex conjugate $\lambda^$. Let $J_0(\lambda)$ be the Jordan block with eigenvalue $\lambda$ and let $$\begin{aligned} J(\lambda):= \begin{cases} J_0(\lambda), & \text{$\lambda$ is real}, \\ {{\mbox{\rm diag}}}[J_0(\lambda), J_0(\lambda^)], & \text{$\lambda$ is non-real}. \end{cases}\end{aligned}$$ Now we can state the Lemma [GLR83]{}. If $A$ is $H$-[[selfadjoint]{}]{}, then $(A,H){\sim}(J,P)$ with $$\begin{aligned} J&={{\mbox{\rm diag}}}{{\left[ J(\lambda_1),\ldots, J(\lambda_\beta) \right]}}, \\ P&= {{\mbox{\rm diag}}}{{\left[ {\epsilon}_1 P_1,\ldots, {\epsilon}_\alpha P_\alpha, P_{\alpha+1},\ldots, P_\beta \right]}}, {\nonumber \\ }{\epsilon}_j&=\pm1, \quad P_j={\tiny \begin{bmatrix} {0}&{}&1\\{}&{\lddots}&{}\\1&{}&{\large 0} \end{bmatrix} },\end{aligned}$$ where $\lambda_1,\ldots,\lambda_\alpha$ are the real eigenvalues of $A$, $\lambda_{\alpha+1},\lambda_{\alpha+1}^,\ldots,\lambda_\beta,\lambda_\beta^$ are the non-real eigenvalues of $A$, and the size of $P_j$ is the same as that of $J(\lambda_j)$. For any $X\in{{{{\mathfrak s}}{{\mathfrak u}}(2,2)}}$, there is a pair $(J,P)$ in the Lemma such that $(X/i,\eta)\sim(J,P)$, because $X/i$ is ${\eta}$-[[selfadjoint]{}]{}. We will denote the [type]{} of $X$ by $$\begin{aligned} \text{\rm Type}(X):= ({\epsilon}_1d_1,\ldots {\epsilon}_\alpha d_\alpha\|d_{\alpha+1}/2,\ldots,d_{\beta}/2), \end{aligned}$$ where $d_j:=\dim J(\lambda_j)$. \[If there is either no real ($\alpha=0$) or no non-real ($\alpha=4$) eigenvalues, we put a 0 in the corresnponding slot.\] We combine all the types with the same $d_j$ and call it the (major) type $[d_1,\ldots d_\alpha\|d_{\alpha+1}/2,\ldots,d_{\beta}/2]$, and we call $({\epsilon}_1,\cdots,{\epsilon}_\alpha)$ the minor type. In the Theorem below, $J_{xy}$ denotes spatial rotations in the $xy$ plane, $K_z$ denotes the boost with respect to the time $t$ in the $z$ direction, ${\widetilde}K_w$ denotes the boost with respect to the time $s$ in the $w$ direction, $L$ denotes the rotation in the $st$ plane, etc. ------------------------------------------------------------------------------------------- Type Killing vector field $\xi$ --------------- ------------------------------------------------------------------------ -- $(4\|0)$ ${\displaystyle}{K_x+{\widetilde}K_y+J_{xy}+L} +2(J_{yz}+ K_z)$ $\pm(3,1\|0)$ ${\displaystyle}K_x+{\widetilde}K_y+J_{yz} \pm J_{xw} + a(J_{xy}-L\mp J_{zw})$ $(2,2\|0)$ $K_x+ L+a J_{yz}$ $(2,-2\|0)$ $K_x+J_{xy}+a J_{zw}$ $(2,1,1\|0)$ $K_x+{\widetilde}K_y+J_{xy}+L+a\, J_{zw}+b\,(J_{xy}- L)$ $(1,1,1,1\|0)$ $a\, L+b\, J_{xy}+c\,J_{zw}$ ($a^2+b^2+c^2=1$) $(2\|1)$ $K_x+{\widetilde}K_y+L+J_{xy} +a J_{zw} + b(K_y+{\widetilde}K_x)$ $(1,1\|1)$ $K_x+{\widetilde}K_y+b\,(L-J_{xy})+c\, J_{zw}$ $(0\|2)$ $K_x+J_{xy}+a\,{\widetilde}K_z$ ($a\ne0$) $(0\|1,1)$ $aK_x+b\,{\widetilde}K_y+c\,J_{zw} \quad (b\ne\pm a,\ a^2+b^2+c^2=1)$ ------------------------------------------------------------------------------------------- : The classification of [[cohomogeneity-one]{}]{} strings. The types of the generator of ${{\operatorname{{\it SU}}(2,2)}}$ and the corresponding Killing vector fields $\xi$ on ${{AdS}}^5$.[]{data-label="tab-res"} \[th-main\] Any one-dimensional connected Lie group of isometries of ${{AdS}}^5$ is generated by one of the nine types of $\xi$ in Table \[tab-res\] up to isometry of ${{AdS}}^5$ connected to the identity, where $a,b,c$ are real numbers, and the double-signs must be taken in the same order in each expression. The proof is given in the Appendix. Note in Table \[tab-res\] that Type $(0\|1,1)$ would become Type $(1,1,\|1)$ (with $b=0$) if one set $b=\pm a$ and that Type $(0\|2)$ would become Type $(2,-2\|0)$ (with $a=0$) if one set $a=0$. The Hopf string {#hopf} =============== In this section, we choose a type from the classified strings in the Theorem and find its trajectory. We assume that the string obeys the Nambu-Goto equation and apply the general procedure presented in Sec. \[general\]. The example also shows that working with the lift curves as explained in Sec. \[general\] can make the calculations and geometric interpretation of the trajectory simple and transparent. We shall say that a [Hopf string]{} is a [[cohomogeneity-one]{}]{} string which is homogeneous under the change of the overall phase in the complex coordinates $\psi$ defined in Sec. \[loc iso\]: $$\begin{aligned} {\psi}\mapsto e^{i\tau}{\psi}, \quad \tau\in{{\mathbb R}}. \label{eq-hopf-symmetry}\end{aligned}$$ This isometry is the simultaneous rotations in the $st$, $xy$, and $zw$ planes. The Killing vector field $\xi$ is proportional to $L+J_{xy}+J_{zw}$ and falls into Type $(1,1,1,1\|0)$ with the condition $a=b=c$. The Killing orbits are closed timelike curves in ${{AdS}}^5$. In the universal cover ${\widetilde}{{{AdS}}^5}$, they are not closed and the string solution represents a stationary string. Let us find the configurations of the Hopf string by solving the action principle [(\[line\_action\_2\])]{} and finding the geodesics on $({{\cal O}},-fh)$. We first see that the orbit space $({{\cal O}},h)$ is a Riemannian manifold, since $\xi$ is timelike. Then, from the fact that $f=\xi^a\xi_a$ is a constant (which we set $-1$), we find that solving the geodesics on $({{\cal O}},-fh)$ is nothing but solving geodesics on $({{\cal O}},h)$. One could either introduce some coordinate system on ${{\cal O}}$ to solve [(\[line\_action\_2\])]{} directly or make an ansatz with some coordinate system on ${{AdS}}^5$ to solve [(\[line\_action\])]{}. Both methods work well but would lead to somewhat complicated equations. In what follows, we would take the advantage of the symmetry, especially the complex structure, of $E^{4,2}$ and find the lift curves on the spacetime ${{AdS}}^5$ which project to the geodesics on $({{\cal O}},-fh)$, as was explained in Sec. \[general\]. The metric $h$ in [(\[hab\])]{} for the Hopf string is the usual flat metric $d{\overline}\psi d\psi$ with the contribution from the phase change being subtracted. With the constraint [(\[eq-AdS\])]{}, $h$ can be written as $$\begin{aligned} h=l^2d{\overline}{\psi}(1-P)d{\psi}, \end{aligned}$$ where $P:=-{\psi}{\overline}{\psi}$ is the normal projection along ${\psi}$. This is the same as the Fubini-Study metric on a projective space ${{\mathbb C}}P^2$ except that we started with an indefinite scalar product $\zeta={{\mbox{\rm diag}}}[-1,1,1]$ in [(\[eq-AdS\])]{} and in $dS^2=l^2d{\overline}\psi d\psi$, while the usual Fubini-Study metric is defined by means of a positive definite scalar product. We shall also call $h$ as the Fubini-Study metric here and shall denote the Riemannian manifold $({{\cal O}},h)$ by ${{\mathbb C}}P_-^2$. The fibration ${{\mathbb C}}P_-^2\simeq {{AdS}}^5/U(1)$ is the generalization of the Hopf fibration to the case of indefinite scalar product [@fn-a]. Thus the problem of finding Nambu-Goto strings has reduced to solving geodesics on ${{\mathbb C}}P_-^2$. Our action [(\[line\_action\_2\])]{} for the Hopf string becomes $$\begin{aligned} S=\int_C d\sigma{{\left( \frac1{\alpha}\dot{{\overline}{\psi}}(1+{\psi}{\overline}{\psi})\dot{{\psi}}+{\alpha} +\mu(1+{\overline}{\psi}{\psi}) \right)}}, \label{line_action_hopf}\end{aligned}$$ where $\mu$ is a Lagrange multiplier. This is the action for geodesics on ${{\cal O}}$ written in terms of the coordinates $\psi$ in $E^{4,2}$. The action [(\[line\_action\_hopf\])]{} has a $U(1)$ gauge invariance ${\psi}(\sigma)\mapsto e^{i\theta(\sigma)}{\psi}(\sigma)$ [@fn-c] which corresponds to the freedom in the choice of a lift. This gauge degree of freedom is used to simplify the calculation. In particular, we shall show that each geodesic on ${{\cal O}}$ for the Hopf string can always be written in a proper gauge as the projection of a [geodesic]{} on ${{AdS}}^5$. The Euler-Lagrange equations are the constraint [(\[eq-AdS\])]{} and $$\begin{aligned} \dot{{\overline}{\psi}}(1+{\psi}{\overline}{\psi})\dot{{\psi}}={\alpha}^2, \qquad & \label{eq-hopf-norm-Z} \\ -{{\left( \frac1{\alpha}(1+{\psi}{\overline}{\psi})\dot {\psi} \right)}}\parbox[c][4ex]{1em}{}^\bullet +\frac1\alpha\dot{{\overline}\psi}\psi\dot\psi +\mu {\psi}&=0. \label{eq-hopf-ddZ}\end{aligned}$$ Multiplying ${\overline}{\psi}$ on [(\[eq-hopf-ddZ\])]{} from the left and using the constraint [(\[eq-AdS\])]{}, one obtains an equation which merely determines $\mu$. On the other hand, the time derivative of [(\[eq-AdS\])]{} implies that ${\overline}{\psi}\dot {\psi}$ is pure imaginary. This value can be changed by the gauge transformation ${\psi}\mapsto e^{i\theta(\sigma)}{\psi}(\sigma)$. We can always choose the gauge ${{\rm Re}}{\overline}{\psi}\dot {\psi}=0$ which under the constraint [(\[eq-AdS\])]{} implies $$\begin{aligned} {\overline}{\psi}\dot {\psi}=0. \label{eq-parallelity}\end{aligned}$$ Geometrically, [(\[eq-parallelity\])]{} means that the curve ${C}$ on ${{\cal M}}$ is [horizontal]{}, namely, it is orthogonal, with respect to $g$, to the fiber $\pi^{-1}(\pi\circ{C}(\sigma))$ at each point on ${C}$. Multiplying $1+{\psi}{\overline}{\psi}$ on [(\[eq-hopf-ddZ\])]{} from the left, and using [(\[eq-AdS\])]{} and [(\[eq-parallelity\])]{}, one obtains the geodesic equation for the Fubini-Study metric, $$\begin{aligned} (1+{\psi}{\overline}{\psi}){{\left( \frac{\dot {\psi}}{\alpha} \right)}}\parbox[c][4ex]{1em}{}^\bullet=0. \label{eq-hopf-geodeq}\end{aligned}$$ Choosing the parameter of the curve to be the proper length so that ${\alpha}\equiv1$, one can write [(\[eq-hopf-geodeq\])]{} in a particularly simple form. Since [(\[eq-hopf-norm-Z\])]{} and [(\[eq-parallelity\])]{} imply ${\overline}{\psi}\ddot {\psi}=-\dot{{\overline}{\psi}}\dot {\psi}=-1$, [(\[eq-hopf-geodeq\])]{} yields $$\begin{aligned} \ddot {\psi}={\psi}. \end{aligned}$$ One can immediately solve the equation to obtain $$\begin{aligned} &{\psi}(\sigma)=A\cosh\sigma+B\sinh\sigma, \label{eq-hopf-geod-lift} \\ &{\overline}AA=-1, \quad {\overline}AB=0, \quad {\overline}BB=1, \label{eq-hopf-geod-lift-cond}\end{aligned}$$ where $A,B\in{{\mathbb C}}^3$. The projection $\pi\circ{C}$ of the curves ${C}:\sigma\mapsto\psi(\sigma)$ expressed by [(\[eq-hopf-geod-lift\])]{} are geodesics on ${{\cal O}}$. Some remarks are in order. First, the geodesics on the four-dimensional manifold ${{\cal O}}$ should contain seven independent real constants: the initial position and the direction of the initial velocity. One sees that $\pi\circ{C}$ actually contains seven independent real constants since we have twelve real constants, four constraints [(\[eq-hopf-geod-lift-cond\])]{} and one redundancy, i.e., the phase of $\psi(0)$. Second, the lift curve [(\[eq-hopf-geod-lift\])]{} is a [horizontal geodesic]{} on ${{AdS}}^5$. A special feature of the Hopf string is that one can always choose a lift curve $C$—the horizontal lift in this case— of a geodesic $c$ on the orbit space $({{\cal O}},-fh)$ so that $C$ [is also a geodesic on $({{\cal M}},g)$]{}. Third, a horizontal geodesic ${C}$ on ${{AdS}}^5$ is the intersection of ${{AdS}}^5$ and a two-dimensional plane through the origin in $E^{4,2}$, which corresponds to the great circle in the case of positive definite metric. Thus the hyperbolic curve [(\[eq-hopf-geod-lift\])]{} is unique up to isometry, for any choice of $A$ and $B$. Furthermore, ${C}$ is a Killing orbit on ${{AdS}}^5$. Now the world sheet ${{\cal S}}$ of the Hopf string can be written down easily. From [(\[eq-general-trajectory\])]{}, [(\[eq-hopf-symmetry\])]{} and [(\[eq-hopf-geod-lift\])]{}, we have $$\begin{aligned} \psi(\tau,\sigma)=e^{i\tau} {{\left( A\cosh\sigma+B\sinh\sigma \right)}}, \label{eq-hopf-trajectory}\end{aligned}$$ where $A$ and $B$ satisfy the condition [(\[eq-hopf-geod-lift-cond\])]{}. To describe geometry of the world sheet ${{\cal S}}$ in more detail, let us introduce a new time coordinate $T$ on ${{AdS}}^5$ defined by $$\begin{aligned} T=\arg\psi^0=\arg(s+it). \end{aligned}$$ In ${\widetilde}{{{AdS}}^5}$, $T$ runs from $-\infty$ to $\infty$. The $T={\rm constant}$ hypersurfaces embedded in ${\widetilde}{{{AdS}}^5}$ are Cauchy surfaces. The Killing field $\xi=d/d\tau$ drives the simultaneous rotations in the $xy$ and $zw$ planes while going up along the $T$ axis. Thus the world sheet of the Hopf string can be viewed pictorially as the surface swept by a boomerang [(\[eq-hopf-geod-lift\])]{} flying up while rotating (Fig. \[fig-hopf\]). Let us reduce the degrees of freedom of $A$ and $B$ in [(\[eq-hopf-geod-lift\])]{} by the homogeneity-preserving isometries and canonicalize them, as was explained in Sec. \[general\]. The Lie algebra $N_{{{\mathfrak g}}}({{\mathfrak k}})$ of the [[homogeneity-preserving]{}]{} isometries is the vector space spanned by $$\begin{aligned} &\xi,\; L-J_{xy},\; L+J_{wz},\; J_{yz}+J_{wx},\;J_{zx}+J_{wy},\;{\nonumber \\ }&{\widetilde}K_z+K_w,\;K_z-{\widetilde}K_w,\; {\widetilde}K_x+K_y,\;K_x-{\widetilde}K_y. \label{eq-gen}\end{aligned}$$ In fact, all generators [(\[eq-gen\])]{} commutes with $\xi$. The isometries generated by [(\[eq-gen\])]{} map the solution [(\[eq-hopf-trajectory\])]{} to another isometric one. First, using the isometries generated by $L$, $J_{xy}$ and $J_{wz}$, one can make a general $A\in{{\mathbb C}}^3$ in [(\[eq-hopf-geod-lift-cond\])]{} to be real, i.e., to have no $t$, $y$, $w$ components. Then, by using ${\widetilde}K_z+K_w$ and ${\widetilde}K_x+K_y$, one has $A=(1,0,0)^T$. Next, we canonicalize $B$ by the isometries which leaves this $A$ unchanged. By ${\overline}AB=0$, $B$ must have the form $B=(0,B^1,B^2)$. By using $J_{xy}$ and $J_{wz}$, one can make $B^1$ and $B^2$ real. Finally, by using $J_{zx}+J_{wy}$, one has $B=(0,1,0)^T$, where $\alpha\in{{\mathbb R}}$. As a result, the trajectory [(\[eq-hopf-trajectory\])]{} can be written up to isometry as $$\begin{aligned} \begin{pmatrix} T\\x\\y\\z\\w \end{pmatrix} = \begin{pmatrix} {\tau}\\ {\sinh\sigma\cos\tau}\\ {\sinh\sigma\sin\tau}\\ 0\\0 \end{pmatrix}, \label{eq-ws}\end{aligned}$$ where we have used $T=\arg(s+it)$. In particular, the world sheet has no parameter and is unique. We can therefore say that the Hopf string has [rigidity]{}. ![The world sheet of the Hopf string in the case in the coordinates $(T,x,y)$. The other coordinates $z$ and $w$ vanish. []{data-label="fig-hopf"}](fig-HopfString1-v2){width=".9\linewidth"} Fig. \[fig-hopf\] shows the worls sheet of the Hopf string. This is a helicoid swept by a rotating rod passing through the $T$ axis. This surface is periodic in $T$ direction with period $\pi$. The similar helical motion of an infinite curve in the Minkowski space has a cylinder outside of which the trajectory becomes tachyonic (spacelike). In the anti-de Sitter case, however, the trajectory is always timelike because the physical time passing with the unit difference in $T$ becomes large when the curve is far from the $T$ axis in Fig. \[fig-hopf\]. Let us summarize some special features of the Hopf string. (i) The Killing vector $\xi$ has a constant squared norm. (ii) The orbit space $({{\cal O}},-fh)$ for Nambu-Goto Hopf string inherits the complex structure of ${E}^{4,2}$, over which ${{AdS}}^5$ admits a Hopf fibration. (iii) The orbit space $({{\cal O}},-fh)$ is homogeneous and is highly symmetric. (iv) The world sheet of the string is homogeneously embedded and is flat intrinsically. (v) The world sheet of the string is rigid, [[i.e.]{}]{}, it is unique up to isometry. Among anti-de Sitter spaces, a Killing field satisfying (i) or (ii) exists only in the [odd-dimensional]{} ones. In the case of ${{AdS}}^5$, the only Killing vector satisfying (i) is $L+J_{xy}\pm J_{zw}$ up to scaling and rotation of the spatial axes [@fn-b]. The condition (i) is partially a reason for (ii) and (iii). In the case of the Hopf string, the homogeneity-preserving isometry group $N_G(K)$ equals the centralizer $Z_G(K)$. On the other hand, $Z_G(K)$ must preserve $f$ (Sec. \[general\]). Thus (i) in general suggests high symmetry of $({{\cal O}},-fh)$. In the case of Hopf string, the isometry group of the orbit space is an eight-dimensional group. In fact, The vector fields [(\[eq-gen\])]{} except the first one $\xi$ form a closed Lie algebra and act on $({{\cal O}},-fh)$ as Killing fields. As for (iv), one finds that the resulting world sheet [(\[eq-ws\])]{} for the Hopf string is invariant under the infinitesimal isometry ${\widetilde}K_x+K_{y}$ of ${{AdS}}^5$. Since $\xi$ and ${\widetilde}K_x+K_{y}$ commute, the world sheet ${{\cal S}}$ is acted by ${{\mathbb R}}^2$ and is homogeneous. This implies that ${{\cal S}}$ is flat intrinsically, namely, ${{\cal S}}$ is the two-dimensional Minkowski space embedded in ${{AdS}}^5$. This can also be verified by a direct computation of the intrinsic metric. The high symmetry (iii) implies (v) for the Hopf string. Incidentally, stationary strings in ${{AdS}}^4$ [@ads4] does not have rigidity. They would most naturally correspond in ${{AdS}}^5$ to the cases $\xi\propto L+b J_{xy}$, which are in the same Type $(1,1,1,1\|0)$ as the Hopf string but with different parameters. These suggest that the Hopf string is similar to the string with simple time translation invariance in the Minkowski space. The Hopf string is the only solution in ${{AdS}}^5$ which shares all of the properties (i), (iii), (iv) and (v) with the flat string in the Minkowski space. Conclusion {#conc} ========== The [[cohomogeneity-one]{}]{} symmetry reduces the partial differential equation governing the dynamics of an extended object in the spacetime ${{\cal M}}$ to an ordinary differential equation. With applications in higher-dimensional cosmology in mind, we have presented the procedure to classify all [[cohomogeneity-one]{}]{} strings and solve their trajectories with a given equation of motion. The former is to classify the Killing vector fields up to isometry, and the latter is to solve geodesics on the orbit space $({{\cal O}},-fh)$ which is the quotient space of ${{\cal M}}$ by the symmetry group $K$. We have carried out the classification in the case that the spacetime is the five-dimensional anti-de Sitter space, by an effective use of the local isomorphism of $\operatorname{{\it SO}}(4,2)$ and $\operatorname{{\it SU}}(2,2)$ and of the notion of $H$-similarity. Assuming that the string obeys the Nambu-Goto equation, we have solved the world sheet of one of the strings, which we call the Hopf string, in the classification. The problem has reduced to find geodesics on the orbit space $({{\cal O}},h)$. By using a technique similar to the one used in quntum information theory and working on the lift curves in ${{\cal M}}$, we have obtained a new solution which describes the trajectories of the Hopf string. They are timelike helicoid-like surfaces around the time axis which is unique up to isometry of ${{AdS}}^5$. We can say that the Hopf string is the simplest example of string in the anti-de Sitter space which corresponds to a straight static string in the Minkowski space. The Killing vector field defining the symmetry of the string is homogeneous in the spacetime and has a constant norm. This greatly simplifies solving the geodesics on the orbit space and the world sheet becomes homogeneous and rigid, as we have seen in Sec. \[hopf\]. The simplicity of the Hopf strings suggests that they were common in the Universe and played significant roles, if the Universe is higher-dimensional or is a brane-world. We would like to remark that although we now have all types where the equations of motion reduce to ordinary differential equations this does not in general imply solvability. The solvability problem is nontrivial and strongly related to the structure of the orbit spaces. A systematic analysis will be presented in a future work. Finally, we would like to remark that the classification presented here will be the basis for that of higher-dimensional [[cohomogeneity-one]{}]{} objects. The procedure is the following: (i) for each of the Killing vector field $\xi$ classified in Table \[tab-res\], enumerate how one can add new independent Killing vector fields $\xi^{(1)}$, ..., $\xi^{(n)}$ such that $\xi$, $\xi^{(1)}$ ..., $\xi^{(n)}$ form a closed Lie algebra ${{\mathfrak k}}'$; (ii) reduce the degrees of freedom of ${{\mathfrak k}}'$ by using the isometries which preserve $\xi$, thus classifying the Lie algebras ${{\mathfrak k}}'$; (iii) examine the orbits in the spacetime generated by ${{\mathfrak k}}'$. Acknowledgment {#acknowledgment .unnumbered} ============== The work is partially supported by Keio Gijuku Academic Development Funds (T. K.). Appendix: Proof of the Theorem {#proof .unnumbered} ============================== Let $X$ be an ${\eta}$-[[anti-selfadjoint]{}]{} matrix $X$. The Lemma implies that $(X/i,{\eta}){\sim}(J,P)$ with some $(J,P)$. On the other hand, if ${\eta}=W PW^\dagger$, the definition of unitary similarity implies $(J,P){\sim}(WJW^{-1},{\eta})$. Thus $(X/i,{\eta}){\sim}(WJW^{-1},{\eta})$ so that $X{\stackrel{\rm \eta}\sim}iWJW^{-1}$. We therefore can carry out the classification by the following procedure: (i) enumerate $(J,P)$ in the Lemma such that there exists $W$ satisfying ${\eta}=WP{W^\dag}$, (ii) construct $X_0=iWJW^{-1}$, (iii) translate $X_0$ back to the Killing vector field $\xi$ in ${{\operatorname{{\it SO}}(4,2)_0}}$ by Table \[tab-cor\]. In some cases, however, the canonical pairs $(J,P)$ and $(J',P')$ correspond to $X_0$’s which generate an identical Lie group. This happens when $(J',P'){\sim}(\alpha J,P)$ with a nonzero real number $\alpha$. Thus it is important to know how a pair $(\alpha J_j({\lambda_{j}}),P_j)$ can be canonicalized. For $\alpha>0$, we simply have $(\alpha J_j({\lambda_{j}}),P_j){\sim}(J_j(\alpha {\lambda_{j}}),P_j)$, so that they generate an identical group. Thus we focus on $(- J_j({\lambda_{j}}),P_j)$ in the following. When $d_j$ is odd, we have $$\begin{aligned} \label{eq-dimP-odd} (-J({\lambda_{j}}),P_j){\sim}(J(-{\lambda_{j}}),P_j). \end{aligned}$$ This can be seen by applying a similarity transformation by ${{\mbox{\rm diag}}}[1,-1,1,\cdots]$. When $d_j$ is even, we have $$\begin{aligned} \label{eq-dimP-even} (-J({\lambda_{j}}),P_j){\sim}(J(-{\lambda_{j}}),-P_j), \end{aligned}$$ which can be shown by applying a similarity transformation by ${{\mbox{\rm diag}}}[1,-1,1,\cdots]$, etc. In the special case of $d_j=2$ and ${\lambda_{j}}\in{{\mathbb C}}$, not only [(\[eq-dimP-even\])]{} but also [(\[eq-dimP-odd\])]{} holds because $-J({\lambda_{j}})=J(-{\lambda_{j}})$. The relation between $(J,P)$ and $(J,-P)$ is also important. Let us show that their corresponding Killing vector fields are related by a reflection $r:(t,x)\mapsto(-t,-x)$, which is a transformation in $SO(4,2)$ which is not connected to the identity (hence is not used in the equivalence relation ${\stackrel{\rm \eta}\sim}$). When $(J,P){\sim}(X_0,{\eta})$, we have $(J,-P){\sim}(-X_0',{\eta})$ with $X_0':=-{U}X{U}^{-1}$ and ${U}:={\sigma_{y}}{\otimes}{\sigma_{x}}$, because ${U}{\eta}{{U}^\dag}=-{\eta}$. On the other hand, one can read off from [(\[eq-SU22rep-2\])]{} that the transformation $p\mapsto-{U}p({U}^{-1})^T=-{U}p{U}^T$ is a reflection along the $t$ and $x$ axes. Thus the Killing vector field $\xi$ corresponding to $X_0$ and the one $\xi'$ corresponding to $'X_0$ are related by $\xi'=r_\xi$. Let us find the relation of the minor types within each major type by using the results above. We denote by an equal sign if two minor types are related by $\eta$-unitary similarity which should be considered identical, and by ${\stackrel{\rm sc}\sim}$ if two minor types are related by a scalar multiplication. For Type $[4\|0]$, it follows from [(\[eq-dimP-even\])]{} that $(+){\stackrel{\rm sc}\sim}(-)$, which is invariant under $r$ (though the parameters change). For Type $[3,1\|0]$, there are two minor types $(++)$ and $(--)$ which are not related by scalar multiplication but by the reflection $r$ (hence not equivalent in the classification). For Type $[2,2\|0]$, it follows from [(\[eq-dimP-even\])]{} that $(++){\stackrel{\rm sc}\sim}(--)$, which is invariant under $r$. by a simple reordering, we have $(+-)=(-+)$, which is invariant under $r$. For Type $[2,1,1\|0]$, by reordering, there are at most two minor types $(++-)$ and $(--+)$. Furthermore, we have $(++-){\stackrel{\rm sc}\sim}(--+)$, by applying [(\[eq-dimP-even\])]{} to all blocks. It is invariant under $r$. Type $[1,1,1,1\|0]$ has only one minor type (by reordering). For Type $[2\|1]$, we have $(+){\stackrel{\rm sc}\sim}(-)$ by applying [(\[eq-dimP-even\])]{} to the first block and [(\[eq-dimP-odd\])]{} to the second block, yielding $({{\mbox{\rm diag}}}[J_1,J_2],{{\mbox{\rm diag}}}[-P_1,P_2]){\sim}(-{{\mbox{\rm diag}}}[J_1',J_2'],{{\mbox{\rm diag}}}[P_1,P_2])$. Type $[1,1\|1]$ has a unique minor type (by reordering). Type $[0\|2]$ and Type $[0\|1,1]$ have a unique minor type. Let us demonstrate the concrete calculation for Type $[2\|1]$ (the other types can be found in a similar manner). We have, because $J$ is traceless, $J={{\mbox{\rm diag}}}{{\left[ {\left[ \begin{array}{cc} {a} & {1} \\ {0} & {a} \end{array}\right] }, -a+bi,-a-bi \right]}}$, where $a$ and $b$ are real numbers, and $P={{\mbox{\rm diag}}}{{\left[ \pm{\left[ \begin{array}{cc} {0} & {1} \\ {1} & {0} \end{array}\right] }, {\left[ \begin{array}{cc} {0} & {1} \\ {1} & {0} \end{array}\right] } \right]}}$. As discussed above, however, it suffices to consider the plus sign. Let us choose $W=S_{23}\cdot{{\mbox{\rm diag}}}[R(\pi/2),R(-\pi/2)]$ where $R(\theta)= \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} $ and $S_{23}= {\tiny \begin{bmatrix} 1&0&0&0\\ 0&0&1&0\\ 0&1&0&0\\ 0&0&0&1 \end{bmatrix} }. $ Then $X_0=iWJW^{-1}= {\tiny \begin{bmatrix} i(a+ 1/2) & 0 & i/2 & 0\\ 0 & -ia & 0 & b\\ -i/2 & 0 & i(a- 1/2) & 0\\ 0 & b & 0 & -ia \end{bmatrix} } = -a\,(1/i){\otimes}{\sigma_{z}}+b\,{\sigma_{x}}{\otimes}\frac{1-{\sigma_{z}}}2 - \frac{({\sigma_{y}}+({\sigma_{z}}/i)){\otimes}(1+{\sigma_{z}})}4 $. By Table \[tab-cor\], we find that $X_0$ corresponds to the ${{{{\mathfrak s}}{{\mathfrak o}}(4,2)}}$ transformation $ \xi= {-{\widetilde}K_w+K_z+L+J_{wz}} +a J_{xy} + b(K_w-{\widetilde}K_z), $ where we have rescaled $\xi$ (by $-4$) and redefined $a$ and $b$. [99]{} L. Randall and R. Sundrum, [Phys. Rev. Lett.]{} [83]{}, (1999) 3370. V. P. Frolov, V. Skarzhinsky, A. Zelnikov and O. Heinrich, Phys. Lett. B [224]{}, 255 (1989). V. P. Frolov, S. Hendy and J. P. De Villiers, Class. Quant. Grav. [14]{}, 1099 (1997). K. Ogawa, H.Ishihara, H. Kozaki, H. Nakano, and I. Tanaka, “Gravitational radiation from stationary rotating cosmic strings”, [Proceeding of 15th JGRG workshop]{}, ed. by T.Shiromizu et al. (Tokyo, 2005) 159. H. Ishihara and H. Kozaki, [Phys. Rev.]{} [ D72]{}, 061701(R) (2005). M. G. Jackson, N. T. Jones and J. Polchinski, J. High Energy Phys. [10]{}, 013 (2005). S. Holst and P. Peldan, [Class. Quantum Grav.]{}, [14]{} (1997) 3442. I. Yokota, [Classical simple Lie groups]{} (Gendai-Sugakusha, 1990) (in Japanese). L. Gohberg, P. Lancaster and L. Rodman, [Matrices and indefinite scalar products]{}, (Birkhäuser Verlag, [1983]{}). Also the variable $\mu$ changes by the gauge transformation: $\mu\mapsto \mu-(1/\alpha)(2\dot\theta{{\rm Im}}{\overline}\psi\dot\psi+\dot\theta^2{\overline}\psi\psi)$. If one also admits the spatial reflection, an isometry which is not connected to the identity, one sees that $L+J_{xy}+J_{zw}$ is the only Killing vector of constant norm. Alternatively, one can treat the case with $L+J_{xy}-J_{zw}$ in the same manner as in the present section by considering $(\psi^0,\psi^1,(\psi^2)^)$ instead of $(\psi^0,\psi^1,\psi^2)$. A. L. Larsen and N. Sánchez, [Phys. Rev.]{} [D51]{}, 6929; H. J. de Vega and I. L. Egusquiza, [Phys. Rev.]{} [D54]{}, 7513. In quantum mechanics, the global phase of the state vector is irrelevant so that one works on the projective space. One obtains the Fubini-Study metric on ${{\mathbb C}}P^n$ by subtracting the contribution of the phase change from the usual inner product on the Hilbert space of state vectors. See J. Anandan and Y. Aharonov, [Phys. Rev. Lett.]{} [65]{}, 1697 (1990). For a physical application, see, e.g., A. Carlini, A. Hosoya, T. Koike and Y. Okudaira, [Phys. Rev. Lett.]{} [96*]{}, 060503 (2006). The geodesic equation is derived and solved in a similar manner.
--- abstract: 'Non-convex regularizers usually improve the performance of sparse estimation in practice. To prove this fact, we study the conditions of sparse estimations for the sharp concave regularizers which are a general family of non-convex regularizers including many existing regularizers. For the global solutions of the regularized regression, our sparse eigenvalue based conditions are weaker than that of L1-regularization for parameter estimation and sparseness estimation. For the approximate global and approximate stationary (AGAS) solutions, almost the same conditions are also enough. We show that the desired AGAS solutions can be obtained by coordinate descent (CD) based methods. Finally, we perform some experiments to show the performance of CD methods on giving AGAS solutions and the degree of weakness of the estimation conditions required by the sharp concave regularizers.' address: \| Department of Automation, Tsinghua University, Beijing 100084, P.R.China\ State Key Lab of Intelligent Technologies and Systems\ Tsinghua National Laboratory for Information Science and Technology(TNList)\ author: - Zheng Pan - Changshui Zhang bibliography: - 'papers.bib' title: 'Relaxed Sparse Eigenvalue Conditions for Sparse Estimation via Non-convex Regularized Regression' --- Sparse estimation ,non-convex regularization ,sparse eigenvalue ,coordinate descent Introduction ============ High-dimensional estimation concerns the parameter estimation problems in which the dimensions of parameters are comparable to or larger than the sampling size. In general, high-dimensional estimation is ill-posed. Additional prior knowledge about the structure of the parameters is usually needed to obtain consistent estimations. In recent years, tremendous research works have demonstrated that the prior on sparsity of the true parameters can lead to good estimators, e.g., the well-known work of compressed sensing [@candes2011probabilistic] and its extensions to general high-dimensional inference [@negahban2012unified]. For high-dimensional sparse estimation, sparsity is usually imposed as sparsity-encouraging [@candes2008enhancing] regularizers for linear regression methods. Many regularizers have been proposed to describe the prior of sparsity, e.g., $\ell_0$-norm, $\ell_1$-norm, $\ell_q$-norm with $0<q<1$, smoothly clipped absolute deviation (SCAD) penalty [@fan2001variable], log-sum penalty (LSP) [@candes2008enhancing], minimax concave penalty (MCP) [@zhang2010nearly] and Geman penalty (GP) [@geman1995nonlinear; @trzasko2009relaxed]. Except $\ell_1$-norm, all of these sparsity-encouraging regularizers are non-convex. Non-convex regularizers were proposed to improve the performance of sparse estimation in many applications, e.g., image inpainting and denoising [@shi2011nonconvex], biological feature selection [@breheny2011coordinate; @shen2012likelihood], MRI [@candes2008enhancing; @chartrand2009fast; @trzasko2007sparse; @trzasko2009highly; @trzasko2008fixed] and CT [@ramirez2011nonconvex; @sidky2007image]. However, it still lacks theoretical explanation for the improvement on sparse estimation for non-convex regularizers. This paper aims to establish such a theoretical analysis. In the field of sparse estimation, the following three problems are typically studied. In this paper, we mainly study the first two problems. 1. Sparseness estimation: whether the estimation is as sparse as the true parameters; 2. Parameter estimation: whether the estimation is accurate in the sense that the error between the estimation and the true parameter is small under some metric; 3. Feature selection: whether the estimation correctly identities the non-zero components of the true parameters. For the sparseness estimation, the non-convex regularizers give better approximations to $\ell_0$-norm than the convex ones. They are more probable to encourage the regularized regression to yield sparser estimations than the convex regularizers. For example, $\ell_q$-regularization can give the sparsest consistent estimations even when $\ell_1$-regularization fails [@foucart2009sparsest]. However, $\ell_q$-norm has infinite derivatives at zero and zero vector is always a trivial local minimizer of the regularized regression. The non-convex regularizers with finite derivatives can remedy the numerical problem of $\ell_q$-norm, e.g., LSP, SCAD and MCP. These regularizers can also give sparser solutions for more general situations than $\ell_1$-regularization in experiments [@candes2008enhancing] and in theory [@fan2001variable; @trzasko2009relaxed; @zhang2010nearly; @zhang2011general]. For the parameter estimation, a lot of applications and experiments have demonstrated that many non-convex regularizers give good estimations with far less sampling sizes than $\ell_1$-norm as the regularizers [@candes2008enhancing; @chartrand2009fast; @ramirez2011nonconvex; @sidky2007image; @trzasko2007sparse; @trzasko2009highly; @trzasko2008fixed]. In theory, the requirements for the sampling sizes are essentially the requirements for design matrix or, rather, estimation conditions. A weaker estimation condition means less sampling size needed or weaker requirements on design matrix. Weaker estimation conditions are important for the application in which the data dimension is very high while the sampling is expensive or restrictive. Theoretically, all of the non-convex regularizers mentioned above admit accurate parameter estimations under appropriate conditions, e.g., $\ell_q$-norm [@foucart2009sparsest], MCP [@zhang2010nearly], SCAD [@zhang2010nearly] and general non-convex regularizers [@zhang2011general]. There are mainly two types of estimation conditions. The first is sparse eigenvalue (SE) conditions, e.g., the restricted isometry property (RIP) [@candes2011probabilistic; @candes2005decoding] and the SE used by @foucart2009sparsest and @zhang2011sparse. The second is restricted eigenvalue (RE) conditions, e.g., the $\ell_2$-restricted eigenvalue ($\ell_2$-RE) [@bickel2009simultaneous; @koltchinskii2009dantzig] and restricted invertibility factor (RIF) [@ye2010rate; @zhang2011general]. Based on SE, @foucart2009sparsest gave a weaker estimation condition for $\ell_q$-norm than $\ell_1$-norm. @trzasko2009relaxed established a universal RIP condition for general non-convex regularizers including $\ell_1$-norm. Since the conditions proposed by @trzasko2009relaxed are regularizer-independent, it can not be weakened for non-convex regularizers unfortunately. The definition of SE is regularizer-independent while the RE is dependent on the regularizers. RE can give a regularizer-dependent estimation condition for general regularizers, e.g., the $\ell_2$-RE based work by @negahban2012unified and the RIF based work by @zhang2011general. However, the optimization for non-convex regularizers is difficult. It usually cannot be guaranteed to achieve a global optimum for general non-convex regularizers. Nevertheless, some optimization methods can lead to local optimums, e.g., coordinate descent [@breheny2011coordinate; @mazumder2011sparsenet] and iterative reweighted (or majorization-minimization) methods [@candes2008enhancing; @hunter2005variable; @zhang2013multi; @zhang2010analysis], homotopy [@zhang2010nearly], difference convex (DC) methods [@shen2012likelihood; @shen2013constrained] and proximal methods [@gong2013general; @pan2013high]. Hence, it is meaningful to analyze the performance of sparse estimation for these non-optimal optimization methods. For example, the multi-stage relaxation methods [@zhang2013multi; @zhang2010analysis] and its one-stage version the adaptive LASSO [@huang2008adaptive; @zou2006adaptive] replace the regularizers with their convex relaxations using majorization-minimization. Compared with LASSO, the multi-stage relaxation methods improve the performance on parameter estimation [@zhang2010analysis]. @zhang2011general use the solutions of LASSO as the initialization and continue to optimize by gradient descent. It is stated that LASSO followed by gradient descent can output an approximate global and stationary solution which is identical to the unique sparse local solution and the global solution. The multi-stage relaxation methods, the “LASSO + gradient descent” methods and the homotopy methods need the same SE or RE conditions as LASSO. The DC methods [@shen2013constrained] and the proximal methods [@pan2013high] need to know the sparseness of its solutions in advance to ensure the performance of parameter estimation, but these two methods cannot control the sparseness of its solutions explicitly. Based on the related work, we make the following contributions: - For a general family of non-convex regularizers, we propose new SE based estimation conditions which are weaker than that of $\ell_1$-norm. As far as we know, our estimation conditions are the weakest ones for general non-convex regularizers. The proposed conditions approach the SE conditions of $\ell_0$-regularized regression as the regularizers become closer and closer to $\ell_0$-norm. We also compare our SE conditions with RE conditions. For $\ell_1$-regularized regression, RE based estimation conditions are less severe than that based on SE [@bickel2009simultaneous]. However, for the case of non-convex regularizers, their relationship changes. For proper non-convex regularizers, SE conditions become weaker than RE conditions, because SE conditions can be greatly weakened from $\ell_1$-norm to non-convex regularizers while RE conditions remain the same. - Under the proposed SE conditions, we establish upper bounds for the estimation error in $\ell_2$-norm. The error bounds are on the same order as that of $\ell_1$-regularized regression. It means that although the proposed SE conditions are weakened, the parameter estimation performance is not weakened. With appropriate additional conditions, we further give the results of sparseness estimations, which show the non-convex regularized regression give estimations with the sparseness on the same order as the true parameters. - Like the global solutions of non-convex regularized regression, we show that the approximate global and approximate stationary (AGAS) solutions [@zhang2011general] also theoretically guarantee accurate parameter estimation and sparseness estimation. The error bounds of parameter estimation are on the order of noise level and the degrees of approximating the stationary solutions and the global optimums. If the degrees of these two approximations are comparable to the noise level, the theoretical performance on parameter estimation and sparseness estimation is also comparable to that of global solutions. Furthermore, the required estimation conditions are almost the same as that of global solutions, which means the estimation conditions for AGAS solutions are also weaker than that required by $\ell_1$-norm. The estimation result on AGAS solutions is useful for application since it shows the robustness of the non-convex regularized regression to the inaccuracy of the solutions and gives a theoretical guarantee for the numerical solutions. - Under a mild SE condition, the approximate global (AG) solutions are obtainable and the approximation error is bounded by the prediction error. If the prediction error is small, the solution will be a good approximate global solution. The algorithms which control the sparseness of the solutions explicitly are suitable to give good AG solutions, e.g., OMP [@tropp2007signal] and GraDeS [@garg2009gradient]. For an AG solution, the coordinate descent (CD) methods update it to be approximate stationary (AS) without destroying its AG property. CD have been applied to regularized regression with non-convex regularizers [@breheny2011coordinate; @mazumder2011sparsenet]. However, the previous works did not allow the non-convex regularizers to approximate $\ell_0$-norm arbitrarily. Our analysis does not have such restriction on the non-convex regularizers . Denotation. We use $\bar{\mathcal T}$ to denote the complement of the set $\mathcal T$ and $\|\mathcal T\|$ to denote the number of elements in $\mathcal T$. For an index set $\mathcal T \in \{1, 2, \cdots, p\}$, $\theta_{\mathcal T}$ denotes the restriction of $\theta=(\theta_1, \theta_2, \cdots, \theta_p)$ on $\mathcal T$, i.e., $\theta_{\mathcal T} = (\theta_i: i\in \mathcal T)$. The support $\text{supp}(\theta)$ of a vector $\theta$ is defined as the index set composed of the non-zero components’ indices of $\theta$, i.e., $\text{supp}(\theta)=\{i : \theta_i \not = 0\}$. The $\ell_0$-norm of the vector $\theta$ is the number of non-zero components of $\theta$, i.e., $\\|\theta\\|_0 = \|\text{supp}(\theta)\|$. Preliminaries {#sec:5} ============= We first formulate the sparse estimation problems. Suppose we have $n$ samples $(y_1,z_1), (y_2,z_2), \cdots, (y_n,z_n) $, where $y_i \in \mathbb R$ and $z_i\in \mathbb R^p$ for $i=1,\cdots,n$. Let $X=(z_1, \cdots, z_n)^T \in \mathbb R^{n \times p}$ and $y=(y_1, \cdots, y_n)^T \in \mathbb R^{n}$. We assume there exists an $s$-sparse true parameter $\theta^$ which is supported on $\mathcal S$ and satisfies $y = X \theta^ + e$ with a small noise $e \in \mathbb R^n$. In this paper, we assume that the energy of the noise is limited by a known level $\epsilon$, i.e., $\\|e\\|_2 \le \epsilon$. For Gaussian noise $e \sim \mathcal N(0,\sigma I_n)$, this assumption is satisfied for $\epsilon = \sigma \sqrt{n+2\sqrt{n\log n}}$ with the probability at least $1-1/n$ [@cai2009recovery]. We focus on using the following regularized regression to recover $\theta^$ from $y$. This method uses the solutions of the following regularized regression as the estimations to the true parameters. $$\label{eqn:1} \hat \theta = \arg\min_{\theta \in \mathbb R^p} \mathcal F(\theta),$$ where $\mathcal F(\theta) = \mathcal L(\theta) + \mathcal R(\theta)$. $\mathcal L(\theta) = \\|y - X \theta\\|_2^2/(2n)$ is the prediction error. $\mathcal R(\theta)$ is a non-convex regularizer. In this paper, we only study the component-decomposable* regularizer, i.e., $\mathcal R(\theta) = \sum_{i=1}^p r(\|\theta_i\|)$. We call $r(u)$ the basis function of $\mathcal R(\theta)$. Table \[table:1\] lists the basis functions of some popular regularizers. For the basis functions in Table \[table:1\], $r(u)$ has the formulation $r(u)=\lambda^2 r_0(u/\lambda;\gamma)$ where $r_0(u;\gamma)$ is a non-decreasing concave function over $[0,+\infty)$ and $\gamma$ is a parameter to describe the “degree of concavity”, i.e., $r(u)$ changes from linear function of $u$ to the indicator function $I_{\{u \not = 0\}}$ as $\gamma$ varies from $+\infty$ to $0$ (except $\ell_1$-norm). -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Name Basis Functions Zero gap $\lambda^$ --------------- --------------------------------------------------------------------------------------------------- -------------------------------------------------------- --------------------------------------------------------------------------------- $\ell_1$-norm $r(u)=\lambda u$ 0 $\lambda^ = \lambda$ $\ell_q$-norm $r(u)= \lambda^2 (u /\lambda)^q, \gamma = \log(1/(1-q))$ $\lambda (q(1-q)/\xi)^{1/(2-q)}$ $\lambda^* = \lambda (2-q) \left( \frac{2(1-q)}{\xi} \right)^{\frac{q-1}{2-q}}$ SCAD $r(u)= \lambda \int_0^u \min \left\{1, \left(1- \frac{x/\lambda-1}{\gamma} \right)_+ \right\} dx$ 0 $\lambda^* = \lambda$ for $\xi=1$ LSP $r(u)=\lambda^2 \log\left(1+ \frac{u}{\lambda \gamma}\right)$ $\max\{ \lambda (1/\sqrt{\xi} - \gamma),0\}$ $\lambda^* \le \lambda \sqrt{2 \xi \log (1+2/(\xi\gamma^2))}$ MCP $r(u)=\lambda \int_0^u \left(1- \frac{x}{\lambda \gamma}\right)_+ dx$ $\lambda \sqrt{\gamma/\xi} $ $\lambda^* = \lambda \min\{\sqrt{\gamma \xi },1\}$ GP $r(u) = \lambda^2 u/(\lambda \gamma + u)$ $\max\{ \lambda (\sqrt[3]{2\gamma /\xi} - \gamma),0\}$ $\lambda^* = \left\{ \begin{array}{ll} \lambda(\sqrt{2\xi} - \xi \gamma/2), & \xi\gamma^2 \le 2\\ \lambda/\gamma, & \xi\gamma^2 >2\\ \end{array} \right. $ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- : Examples of popular regularizers. The second column is the basis functions of the regularizers. The third column is the zero gaps of the global solutions when the regularizers are $\xi$-sharp concave. The fourth column is the values of $\lambda^$. Section \[sec:13\] gives the proof for the result on $\lambda^$ of LSP.[]{data-label="table:1"} Throughout this paper, we assume the basis function $r(u)$ satisfy the following properties. All of them hold for the basis functions in Table \[table:1\]. 1. $r(0)=0$; 2. $r(u)$ is non-decreasing; 3. $r(u)$ is concave over $[0,+\infty)$; 4. $r(u)$ is continuous and piecewise differentiable. We use $\dot r(u+)$ and $\dot r(u-)$ to denote the right and left derivatives. 5. $r(u)$ has the formulation $r(u) = \lambda^2 r_0(u/\lambda; \gamma)$, where $r_0(u;\gamma)$ is parameterized by $\gamma$ and is independent of $\lambda$. In this paper, the weaker SE based estimation conditions need two important properties: zero gap and null consistency [@zhang2011general]. Zero gap means the true parameters and the estimations are strong in the sense that the minimal magnitude of the non-zero components cannot be too close to zero. Null consistency requires that the regularized regression in Eqn. (\[eqn:1\]) is able to identify the true parameter $\theta^$ exactly when $\theta^ = 0$ and the error $e$ is inflated by a factor of $1/\eta>1$. \[def:5\] We say $\theta\in \mathbb R^p$ has a zero gap $u_0$ for some $u_0 \ge 0$ if $\min\{ \|\theta_i\|: i\in \text{supp}(\theta)\} \ge u_0$. \[def:6\] Let $\eta\in(0,1)$. We say the regularized regression in Eqn. (\[eqn:1\]) is $\eta$-null consistent if $ \min_{\theta} \\|X \theta - e/\eta\\|_2^2/(2n) + \mathcal R(\theta) = \\|e/\eta\\|_2^2/(2n)$. In order to guarantee the above two properties, we propose the following assumption, named sharp concavity. Sharp concavity is important for our analysis because zero gap and null consistency can be derived from it. \[def:4\] We say a basis function $r(u)$ satisfies $C$-sharp concavity condition over an interval $\mathcal I$ if $r(u) > u \dot r(u-) + C u^2/2 $ holds for any $u \in \mathcal I$, where $C$ is a positive constant. We also say $r(u)$ is $C$-sharp concave over $\mathcal I$ and a regularizer $\mathcal R(\theta)$ is $C$-sharp concave if its basis function is $C$-sharp concave. Strictly concave functions can only satisfy $r(u) > u \dot r(u-)$. However, if the left-derivative $\dot r(u-)$ decreases so fast that it admits a margin proportional to $u^2$ in some interval $\mathcal I$, the concave functions guarantee the sharp concavity. $C$-sharp concavity is satisfied over $(0,u_0)$ if $r(u)$ is strongly concave (or $-r(u)$ is strongly convex) over $(0,u_0)$ , i.e., for any $t_1,t_2 \in (0,u_0)$ and $\alpha\in [0,1]$, $$\label{eqn:57} r(\alpha t_1 + (1-\alpha) t_2) \ge \alpha r(t_1) + (1-\alpha) r(t_2) + \frac{1}{2} C \alpha (1-\alpha) (t_1-t_2)^2.$$ Section \[sec:14\] shows that sharp concavity only needs Eqn. (\[eqn:57\]) holds for $t_1=0$ and any $t_2\in(0,u_0)$, which means that the sharp concavity is weaker than the strong concavity. For example, MCP is $((1+a)\gamma)^{-1}$-sharp concave over $(0,\sqrt{1+a}\lambda \gamma)$ for any $a>0$. Whereas, the strong concavity does not hold over $(\lambda \gamma, \sqrt{1+a}\lambda \gamma)$. Besides, $\ell_q$-norm holds $q(1-q)(u_0/\lambda)^{q-2}$-sharp concavity over $(0,u_0)$; LSP satisfies $\lambda^2/(\lambda \gamma + u_0)^2$-sharp concavity over $(0, u_0)$; GP is $2 \lambda^3 \gamma / (\lambda \gamma + u_0)^3$-sharp concave over $(0, u_0)$. Let $x_i$ be the i-th column of $X$ and $$\xi = \max_{1\le i \le p} \\|x_i\\|_2^2/n.$$ We observe that $\xi$-sharp concavity derives non-trivial zero gaps and null consistency. \[thm:4\] If $r(u)$ is $\xi$-sharp concave over $(0,u_0)$, any global solution of Problem (\[eqn:1\]) has a zero gap no less than $u_0$, i.e., $\|\hat\theta_i\| \ge u_0$ for any $i\in \mathrm{supp}(\hat\theta)$. Table \[table:1\] lists the zero gaps of $\hat\theta$ when the basis functions are $\xi$-sharp concave. \[thm:10\] Let $r(u)$ be $\xi$-sharp concave over $(0,u_0)$. The $\eta$-null consistency condition is satisfied if $r(u_0) \ge \frac{1}{2n\eta^2} \\|e\\|_2^2 $. @zhang2011general give a probabilistic condition for null consistency when $X$ is drawn from Gaussian distributions. However, our condition is deterministic from the view of $X$. It is easy to check whether our condition holds. For the case of $r(u)=\lambda^2 r_0(u/\lambda;\gamma)$, the condition of Theorem \[thm:10\] is $\lambda \ge \eta^{-1} b_0 \\|e\\|_2 / \sqrt{n}$, where $b_0 =1/\sqrt{2 r_0(u_0/\lambda;\gamma)}$ is a constant if $u_0 =O(\lambda)$ (all the regularizers in Table \[table:1\] satisfy $u_0 = O(\lambda)$). Hence, we assume $$\label{eqn:20} \lambda = \eta^{-1} b_0 \epsilon/\sqrt{n}$$ in this paper, so that the $\eta$-null consistency holds. In addition, we define $$\label{eqn:15} \lambda^* = \inf_{u>0}\{\xi u/2 + r(u)/u\}.$$ $\lambda^$ provides a natural normalization of $\lambda$ [@zhang2011general]. Table \[table:1\] lists the values of $\lambda^$ of the regularizers. We observe $\lambda^* = O(\lambda)$ from Table \[table:1\]. In general, for $r(u)=\lambda^2 r_0(u/\lambda;\gamma)$, we can define a constant $a_\gamma$ (independent to $\lambda$), $$\label{eqn:21} a_\gamma = \inf_{u>0} \{\xi u/2 + r_0(u;\gamma)/u\},$$ so that $\lambda^* = a_\gamma \lambda $. Thus, we have $$\label{eqn:58} \lambda^* = \eta^{-1}a_\gamma b_0 \epsilon/\sqrt{n}.$$ If the basis function $r(u)$ is linear over $(0, u)$ for some $u>0$, it is not sharp concave, e.g., SCAD and truncated $\ell_1$-norm [@zhang2010analysis]. We name such regularizers that are linear near the origin as weak non-convex regularizers. The zero gaps of the global solutions with such regularizers cannot be guaranteed to be strictly positive. Sparse Estimation of Global Solutions {#sec:1} ===================================== In this section, we show our results on the SE based sparse estimation. For an integer $t\ge 1$, we say that $\kappa_-(t)$ and $\kappa_+(t)$ are the minimum and maximum sparse eigenvalues(SE) of a matrix $X$ if $$\label{eqn:51} \kappa_-(t) \le \frac{\\|X\Delta\\|_2^2}{ n\\|\Delta\\|_2^2} \le \kappa_+(t) \text{ for any } \Delta \text{ with } \\|\Delta\\|_0 \le t.$$ The SE is related to the restricted isometry constant (RIC) $\delta_{t}$ [@candes2011probabilistic; @candes2005decoding], which satisfies $1-\delta_t \le \\|X\Delta\\|_2^2/(n\\|\Delta\\|_2^2) \le 1+\delta_t$ for all $\Delta$ with $\\|\Delta\\|_0 \le t$. Thus, it follows that $\delta_t = (\kappa_+(t) - \kappa_-(t)) / (\kappa_+(t) + \kappa_-(t))$, where $\delta_t$ is actually the RIC of the scaled matrix $2 X / (\kappa_+(t) + \kappa_-(t))$. We employ SE since it allows $\kappa_+(t) \ge 2$ and avoids the scaling problem of RIC [@foucart2009sparsest]. In order to show the typical values of $\kappa_+(t)$ and $\kappa_-(t)$, we compute them and their ratio $\kappa_+(t)/\kappa_-(t)$ for the standard Gaussian $n \times p$ matrix[^1], where we fix $p=10~000$, $n=500$, $1000$, $1500$, $2000$ and $t$ varies from $1$ to $n$. It should be noted that $\kappa_+(t)$ and $\kappa_-(t)$ cannot be obtained efficiently. We use the following approximation method: For a matrix $X \in \mathbb R^{n\times p}$, we randomly sample its 100 submatrices $X_1, X_2, \cdots, X_{100} \in \mathbb R^{n\times t}$ composed by $t$ columns of $X$ and regard $\tilde\kappa_+(t) = \max_i \lambda_{\max}(X_i^T X_i)$ and $\tilde\kappa_-(t) = \min_i \lambda_{\min} (X_i^T X_i)$ as the approximations for $\kappa_+(t)$ and $\kappa_-(t)$, where $\lambda_{\max}(A)$ and $\lambda_{\min}(A)$ mean the maximal and minimal eigenvalues of $A$. Actually, $\tilde\kappa_+(t) \le \kappa_+(t)$ and $\tilde\kappa_-(t) \ge \kappa_-(t)$. For each $n$ and $t$, we generate 100 standard Gaussian matrices and compute the maximums, minimums and the means of the values of $\tilde\kappa_+(t)$, $\tilde\kappa_-(t)$ and $\tilde\kappa_+(t)/\tilde\kappa_-(t)$ for the 100 trials. Figure \[fig:4\] illustrates the results. The variances of $\tilde\kappa_+(t)$, $\tilde\kappa_-(t)$ and $\tilde\kappa_+(t)/\tilde\kappa_-(t)$ with the same $n$ and $t$ are small since the corresponding lines for the maximum, minimum and mean values are close to each other. However, $\tilde\kappa_+(t)/\tilde\kappa_-(t)$ grows very fast as $t$ grows or $n$ decreases. Based on SE, we establish the following parameter estimation result for global solutions of non-convex regularized regression. Let $\hat \rho_0$ and $\rho_0^$ be the zero gaps of the global solution $\hat\theta$ and the true parameter $\theta^$ respectively. Denote $$\label{eqn:60} \rho_0 =\min\{ \hat \rho_0, \rho^* \}.$$ \[thm:1\] Suppose the following conditions hold. 1. $r(u)$ is invertible for $u\ge 0$ and $ r^{-1}(u/s_1) / r^{-1}(u/s_2)$ is a non-decreasing function of $u$ for any $s_2 \ge s_1 \ge 1$; 2. The regularized regression satisfies $\eta$-null consistency; 3. The following SE condition holds for some integer $t \ge \alpha s$, $$\label{eqn:5} \kappa_+(2t)/ \kappa_-(2t) < 4(\sqrt{2}-1) H_r(\rho_0, \alpha, s, t) +1,$$ where $s=\\|\theta^\\|_0$, $\alpha = \frac{1+\eta}{ 1- \eta}$, $H_r(\rho_0, \alpha, s, t) = \sqrt{ \frac{s}{t} } \frac{r^{-1}( r(\rho_0)/s)}{ r^{-1}(\alpha r(\rho_0)/t)}$ for $\rho_0>0$ and $H_r(0, \alpha, s, t) = \lim_{\rho\to 0+} H_r(\rho, \alpha, s, t)$. Then, $$\label{eqn:9} \\|\hat\theta - \theta^\\|_2 \le C_1 \lambda^,$$ where $C_1 = \frac{(1+\sqrt{2})(1+\eta)\sqrt{t}}{\kappa_-(2t)} \frac{H_r(\rho_0, \alpha, s, t) + 1/2}{H_r(\rho_0, \alpha, s, t) - (1+\sqrt{2})(\kappa_+(2t) / \kappa_-(2t) - 1)/4}$. Since $\lambda^$ is on the order of noise level $\epsilon$ (Eqn. (\[eqn:58\])), the estimation error $\\|\hat\theta - \theta^\\|_2$ is at most on the order of noise level. We give a detailed discussion on Theorem \[thm:1\] in Section \[sec:12\]. Before the discussion, we first show a corollary given in Section \[sec:12\], which shows that our SE condition only needs $\kappa_-(t) >0$ with $t=O(s)$. This SE condition is much weaker than that of $\ell_1$-norm. In fact, it is almost optimal since it is the same as the estimation condition of $\ell_0$-regularization [@foucart2009sparsest; @zhang2011general]. \[coro:1\] Let the condition 1 and 2 of Theorem \[thm:1\] hold and $H_r(\rho_0,\alpha,s,\alpha s+1) \to \infty$ as $\gamma\to 0$. If $\kappa_-(2\alpha s +2) >0$, there exists $\gamma>0$ such that $\\|\hat\theta - \theta^\\|_2 \le O(\lambda^)$. In addition to the error bound in Theorem \[thm:1\], we hope that the regularized regressions yield enough sparse solutions. We extend the results from @zhang2011general and show that the global solutions are sparse under appropriate conditions. \[thm:8\] Suppose the conditions of Theorem \[thm:1\] hold. Consider $l_0>0$ and integer $m_0>0$ such that $$\sqrt{ 2 t \kappa_+(m_0) r(C_2 (1+\eta) \lambda^)/m_0 } + \\|X^T e/n\\|_\infty < \dot r(l_0-),$$ where $C_2$ is defined in Eqn. (\[eqn:13\]). Then, $\|\text{supp}(\hat\theta) \backslash \mathcal S\| \le m_0 + t r( C_2 (1+\eta) \lambda^* )/ r(l_0)$. \[coro:2\] Suppose the basis function $r(u) = \lambda^2 r_0(u/\lambda)$ and the conditions of Theorem \[thm:8\] hold with $t=(\alpha+1)s$, $m_0=\beta_0 s$ and $l_0 = \beta_1 \lambda$ for some $\beta_0,~\beta_1>0$. Let $C_3 = C_2 (1+\eta)a_\gamma $ where $C_2$ is the same as Theorem \[thm:8\] and $a_\gamma$ is defined in Eqn. (\[eqn:21\]). If $$\label{eqn:32} \frac{2(\alpha+1) \kappa_+(\beta_0 s)}{\beta_0} < \frac{(\dot r_0(\beta_1 -) - \eta a_\gamma)^2}{r_0(C_3)},$$ then $$\label{eqn:33} \|\text{supp}(\hat\theta) \backslash \mathcal S\| \le (\beta_0 + (\alpha+1) r_0(C_3)/r_0(\beta_1)) s.$$ Example for Corollary \[coro:2\]. Consider the example of LSP with $r_0(u) = \log(1+u/\gamma)$ and $\beta_1=\sqrt{\gamma}$. Suppose the columns of $X$ are normalized so that $\xi=1$. Section \[sec:13\] shows that $a_\gamma \le \sqrt{2 \log (1+2/\gamma^2)}$. Thus, the right hand of Eqn (\[eqn:32\]) is larger than $$\frac{\left( 1/(1+\sqrt{\gamma}) - \eta \sqrt{2 \gamma \log(1+2/\gamma^2)} \right)^2}{ \gamma \log ( 1+ \gamma^{-1} C_2(1+\eta) \sqrt{2 \log(1 + 2/\gamma^2) } )}$$ Thus, as $\gamma$ goes to 0, the right side of Eqn. (\[eqn:32\]) is arbitrarily large. Eqn. (\[eqn:32\]) holds for enough small $\gamma$. The right side of Eqn. (\[eqn:33\]) is $\beta_0 s + O(s)$ as $\gamma \to 0$. Hence, we can freely select $\beta_0$ satisfying Eqn. (\[eqn:32\]) with enough small $\gamma$. For example, if $\beta_0=1/s$, Eqn. (\[eqn:32\]) holds for enough small $\gamma$ and Eqn. (\[eqn:33\]) becomes $$\label{eqn:46} \|\text{supp}(\hat\theta)/\mathcal S\| \le 1 + s(\alpha+1) \frac{ \log ( 1+ \gamma^{-1} C_2(1+\eta) \sqrt{2 \log(1 + 2/\gamma^2) } )}{ \log (1 + 1/\sqrt{\gamma})}.$$ The right side of Eqn (\[eqn:46\]) is at most on the order of $s$ when $\gamma$ is close to zero. Discussion on Theorem \[thm:1\] {#sec:12} =============================== This section gives some detailed discussion on Theorem \[thm:1\]. Invertible approximate regularizers ----------------------------------- If $r(u)$ is not invertible, e.g., MCP, we can design invertible basis function to approximate it. For example, we can use the following invertible function, named Approximate MCP, to approximate MCP. $$\label{eqn:26} r(u) = \left\{\begin{array}{ll} \lambda u - u^2/(2\gamma), & 0 \le u \le \lambda \gamma (1-\phi),\\ \frac{1}{2} \lambda^2 \gamma (1-\phi^2) \left( \frac{u}{\lambda\gamma (1-\phi)} \right)^{2\phi/(1+\phi)}, & u > \lambda \gamma(1-\phi), \end{array}\right.$$ where $\phi\in (0,1)$. Approximate MCP is concatenated by the part of MCP over $[0,\lambda\gamma(1-\phi)]$ and the part of $\ell_{q}$-norm over $(\lambda\gamma(1-\phi), \infty)$ with $q = 2\phi/(1+\phi)$. When $\phi \to 0$, $r(u)$ will become the basis function of MCP. We will address the method to obtain Eqn. (\[eqn:26\]) in Section \[sec:8\]. Any other non-invertible regularizers in Table \[table:1\] can be approximated in the same way. Non-decreasing property of $ r^{-1}(u/s_1) / r^{-1}(u/s_2)$ ----------------------------------------------------------- It can be verified that all the regularizers in Table \[table:1\] or their invertible approximate ones (in the way of Eqn. (\[eqn:26\])) satisfy the non-decreasing property of $\frac{ r^{-1}(u/s_1) }{ r^{-1}(u/s_2)}$ for any $s_2\ge s_1 >0$. In fact, for derivative basis functions, this non-decreasing property is equal to that $u\dot r(u)/r(u)$ is a non-increasing function of $u$. Non-sharp concave regularizers ------------------------------ If $r(u)$ is not $\xi$-sharp concave, e.g., SCAD or LSP with $\gamma^2 > 1/\xi$, we cannot guarantee $\hat\theta$ has a positive zero gap. In this case, the condition 2 (null consistency) of Theorem \[thm:1\] can be guaranteed by the $\ell_2$-regularity conditions [@zhang2011general] and the condition 3 becomes $ \kappa_+(2\alpha s)/ \kappa_-(2 \alpha s) < 1.65/\sqrt{\alpha} +1$ with $t=\alpha s$, which also belongs to the $\ell_2$-regularity conditions. Hence, without $\xi$-sharp concavity, Theorem \[thm:1\] still holds. Intuitively, non-sharp concave regularizers need the same estimation conditions as $\ell_1$-regularization since they cannot approximate $\ell_0$-norm arbitrarily. Relaxed SE based estimation conditions -------------------------------------- Much more relaxed estimation conditions are sufficient for $\xi$-sharp concave regularizers. Suppose $r(u)$ is $\xi$-sharp concave over $(0,\rho_0)$ with $0< \rho_0 \le \min_{i \in \mathcal S} \|\theta^_i\|$. In this case, $H_r(\rho_0,\alpha, s,t )$ can become arbitrarily large for proper regularizers so that the SE condition in Eqn. (\[eqn:5\]) is much weaker than the SE conditions of $\ell_1$-regularized regression. We have shown in Figure \[fig:4\] that $\tilde\kappa_+(t)/\tilde\kappa_-(t)$ ($\le \kappa_+(t)/\kappa_-(t)$) increases very fast as $t$ increases or $n$ decreases. Thus, a weaker constraint on $\kappa_+(2t) / \kappa_-(2t)$ in Eqn. (\[eqn:5\]) is very important for sparse estimation problems. Here, we give the examples of approximate MCP, $\ell_q$-norm and LSP. For approximate MCP, Eqn. (\[eqn:16\]) gives its $H_r(\rho_0,\alpha,s,t)$ (see Section \[sec:8\]). $$\label{eqn:16} H_r(\rho_0, \alpha, s, t) = \alpha^{-1/2} (t/(\alpha s))^{\frac{1}{2\phi}},$$ where we set $\gamma \xi = \frac{\phi}{1+\phi} (\alpha/t)^{1/\phi}$. For $\ell_q$-norm, the SE conditions can be written as $$\label{eqn:45} \frac{\kappa_+(2t)}{\kappa_-(2t)} < 1 + \frac{4(\sqrt{2}-1)}{\sqrt{\alpha}} \left(\frac{t}{\alpha s}\right)^{1/q-1/2}.$$ When $\alpha =1$, Eqn. (\[eqn:45\]) is identical to the estimation condition of @foucart2009sparsest. Hence, @foucart2009sparsest can be regarded as a special case of our theory. For LSP, we have $$\label{eqn:18} H_r(\rho_0, \alpha , s,t) = \sqrt{\frac{s}{t}} \frac{(1+\rho_0/(\lambda \gamma))^{\frac{1}{s}}-1}{(1+\rho_0/(\lambda \gamma))^{\frac{\alpha}{t}}-1} = \sqrt{\frac{s}{t}} \frac{(\gamma\sqrt{\xi})^{-1/s} -1 }{(\gamma\sqrt{\xi})^{-\alpha/t} -1}.$$ It should be noted that $H_r(\rho_0, \alpha , s,t) \to \infty$ as $\gamma \to 0$ for approximate MCP, $\ell_q$-norm and LSP. Figure \[fig:5\] shows some special cases of $H_r(\rho_0,\alpha,s,t)$ for these three regularizers and $\ell_1$-norm. In Figure \[fig:5\], the SE conditions in Eqn. (\[eqn:5\]) are much weaker than that of $\ell_1$-norm. Theorem \[thm:1\] reveals that the upper bound constraint for $\kappa_+(2t)/\kappa_-(2t)$ tends to infinity as $\gamma \to 0$ for proper non-convex regularizers. It implies that if $$\label{eqn:28} \kappa_-(2t) = \inf_\theta \left\{ \frac{\\|X\Delta\\|^2_2}{n\\|\Delta\\|^2_2} : \\|\Delta\\|_0 \le 2t \right\} >0,$$ there exists $\gamma>0$ so that the SE condition (Eqn. (\[eqn:5\])) is satisfied. Based on this observation, we have Corollary \[coro:1\]. In Corollary \[coro:1\], $\kappa_-(2\alpha s +2) >0$ holds if the columns of $X$ are in general position[^2] and $2\alpha s +2 \le n$, which is almost optimal in the sense that it is the same as the SE condition of $\ell_0$-regularized regression [@zhang2011general]. Comparison between SE and RE {#sec:7} ---------------------------- Like SE, RE is also popular to construct estimation conditions. There are some variants of RE, e.g., $\ell_2$-RE [@bickel2009simultaneous; @koltchinskii2009dantzig] and RIF [@ye2010rate; @zhang2011general]. It can derive a simple expression to the parameter estimation and the corresponding estimation condition. \[def:7\] For $\alpha \ge 1$, a regularizer $\mathcal R$, an index set $\mathcal S \subset \{1, \cdots p\}$ and its complement set $\bar{\mathcal S}$, the $\ell_2$-RE is defined as $$\label{eqn:29} \text{RE}^{\mathcal R} (\alpha,\mathcal S) = \inf_{\Delta} \left\{ \frac{\\|X\Delta\\|_2^2}{n\\|\Delta\\|_2^2}: \mathcal R(\Delta_{\bar{\mathcal S}}) \le \alpha \mathcal R(\Delta_{\mathcal S}) \right\}.$$ \[def:1\] For $\tau\ge 1$, $\alpha \ge 1$, a regularizer $\mathcal R$, an index set $\mathcal S \subset \{1, \cdots p\}$, RIF is defined as $$\text{RIF}_{\tau}^{\mathcal R}(\alpha,\mathcal S) = \inf_{\Delta} \left\{ \frac{ \|\mathcal S\|^{1/\tau} \\|X^T X \Delta \\|_{\infty}}{ n \\|\Delta\\|_\tau } : \mathcal R(\Delta_{\bar{\mathcal S}}) \le \alpha \mathcal R(\Delta_{\mathcal S}) \right\}.$$ \[thm:9\] Suppose $\eta$-null consistency condition holds and $\alpha = (1+\eta)/(1-\eta)$. Then, $\\|\hat \theta - \theta^\\|_2 \le \frac{ 2\alpha\sqrt{s}}{ \text{RE}^{\mathcal R} (\alpha, \mathcal S) } \dot r(0+)$. For any $\tau \ge 1$, $\\|\hat \theta - \theta^\\|_\tau \le \frac{(1+\eta)\lambda^ s^{1/\tau} }{\text{RIF}^{\mathcal R}_\tau (\alpha, \mathcal S)}$. The estimation conditions based on RE require that $\text{RE}^{\mathcal R} (\alpha, \mathcal S) > 0$ or $ \text{RIF}^{\mathcal R}_\tau (\alpha, \mathcal S) > 0$. The same conclusion also can be obtained for $\ell_1$-regularized regression [@negahban2012unified; @zhang2011general]. What we are interested in is whether non-convex regularizers allow a larger value of $\text{RE}^{\mathcal R}(\alpha, \mathcal S)$ than $\ell_1$-norm, i.e., whether $\text{RE}^{\mathcal R}(\alpha, \mathcal S)>0$ becomes weaker by employing non-convex regularizers. Define $\Omega(\beta)=\{\Delta \in \mathbb R^n: \mathcal R(\beta \Delta_{\bar{\mathcal S}}) \le \alpha \mathcal R(\beta \Delta_{\mathcal S}), \\|\Delta\\|_2=1\} $ for $\beta>0 $. The concavity of $r(u)$ gives that $\dot r(0+)u \ge r(u) \ge u \dot r(u-)$, which derives that $$\Omega(\beta) \supset \{\Delta \in \mathbb R^n: \dot r(0+) \beta \\|\Delta_{\bar{\mathcal S}}\\|_1 \le \alpha \beta \langle \|\Delta_{\mathcal S}\|, \dot r(\|\beta \Delta_{\mathcal S}\|-) \rangle , \\|\Delta\\|_2=1 \} ,$$ where $\|\Delta_{\mathcal S}\|$ is the vector composed of the absolute values of the components of $\Delta_{\mathcal S}$, i.e., $\|\Delta_{\mathcal S}\| = (\|\Delta_i\|: i\in \mathcal S)$. In the same way, $\dot r(\|\beta \Delta_{\mathcal S}\|-) = (\dot r(\|\beta \Delta_i\|-): i\in \mathcal S)$. Thus, we give an upper bound to $\text{RE}^{\mathcal R} (\alpha,\mathcal S)$: $$\begin{aligned} \text{RE}^{\mathcal R} (\alpha,\mathcal S) & = \inf_{\beta > 0, \Delta \in \mathbb R^ p}\{ \frac{\\|X \Delta\\|_2^2}{n \\|\Delta\\|_2^2}: \Delta \in \Omega(\beta)\} \nonumber\\ & \le \inf_{\beta > 0, \Delta \in \mathbb R^ p} \{ \frac{\\|X \Delta\\|_2^2}{n \\|\Delta\\|_2^2} : \dot r(0+) \\|\Delta_{\bar{\mathcal S}}\\|_1 \le \alpha \langle \|\Delta_{\mathcal S}\|, \dot r(\|\beta \Delta_{\mathcal S}\|-) \rangle, \\|\Delta\\|_2=1 \} \nonumber\\ & \stackrel{(\beta\to 0+)}{\le} \inf_{\Delta \in \mathbb R^ p} \{ \frac{\\|X \Delta\\|_2^2}{n \\|\Delta\\|_2^2}: \\|\Delta_{\bar{\mathcal S}}\\|_1 \le \alpha \\|\Delta_{\mathcal S}\\|_1, \\|\Delta\\|_2=1 \} \nonumber\\ & = \text{RE}^{\ell_1} (\alpha,\mathcal S) \nonumber\end{aligned}$$ $\text{RE}^{\mathcal R} (\alpha,\mathcal S) \le \text{RE}^{\ell_1} (\alpha,\mathcal S)$ means that the RE based condition of non-convex regularized regression $\text{RE}^{\mathcal R} (\alpha,\mathcal S)>0$ is not relaxed. @negahban2012unified put an additional constraint $\mathcal U(\epsilon) = \{\Delta: \\|\Delta\\| \ge \epsilon\}$ to the definition of RE. This constraint avoids the bad case $\Delta \to 0$. However, it still cannot guarantee to provide larger RE for non-convex regularizers than $\ell_1$-norm. For example, let $t_1, t_2$ and $t_3$ satisfy that $\|t_1\| + \|t_2\| \le 2\|t_3\|$ and $\alpha=2$, $\mathcal S=\{3\}$ and $\bar{\mathcal S}=\{1,2\}$. Thus, the concavity of $r(u)$ implies that $r(\|t_1\|) + r(\|t_2\|) \le 2r((\|t_1\| + \|t_2\|)/2) \le 2r(\|t_3\|)$. For this case, $\{\Delta: \\|\Delta_{\bar{\mathcal S}}\\|_1 \le \alpha \\|\Delta_{\mathcal S}\\|_1\} \subset \{\Delta: \mathcal R (\Delta_{\bar{\mathcal S}}) \le \alpha \mathcal R (\Delta_{\mathcal S}) \}$. Thus, $\text{RE}^{\mathcal R} (\alpha, \mathcal S) \le \text{RE}^{\ell_1} (\alpha, \mathcal S)$. For RIF, we have the same result. Although non-convex regularizers give better approximations to $\ell_0$-norm, the RE of non-convex regularizers cannot be guaranteed to be lager than that of $\ell_1$-norm. The framework of RE does not leave space to relax the estimation condition for non-convex regularizers. The only difference between the definitions of SE and RE lies in the constraints for $\Delta$. The two constraints $\{\Delta: \\|\Delta\\|_0 \le 2t\}$ and $\{\Delta: \mathcal R(\Delta_{\bar{\mathcal S}}) \le \alpha \mathcal R(\Delta_{\mathcal S})$ do not contain each other. However, we observe that $\kappa_-(2t) \ge \min_{\|\mathcal T\|\le s} \text{RE}^{\mathcal R} ((2t-s)/s ,\mathcal T) \ge \min_{\|\mathcal T\|\le s} \text{RE}^{\mathcal R} (2\alpha-1+2/s,\mathcal T)$ for $t \ge \alpha s+1$. When $\eta$ is small and $s\gg 2$, $2\alpha -1 +2/s$ is close to $\alpha$ and $\min_{\|\mathcal T\|\le s} \text{RE}^{\mathcal R} (2\alpha-1+2/s,\mathcal T) \approx \min_{\|\mathcal T\|\le s} \text{RE}^{\mathcal R} (\alpha,\mathcal T)$. Hence, with proper regularizers, the SE condition in Eqn. (\[eqn:28\]) is a weaker condition than $\min_{\|\mathcal T\| \le s} \text{RE}^{\mathcal R} (\alpha ,\mathcal T)>0$. We can also compare RE and our SE conditions with the help of the failure bound of RIC $\delta_{2s} = 1/\sqrt{2}$ for $\ell_1$-minimization recovery [@Davies2009Restricted], where $\ell_1$-minimization recovery includes the basis pursuit [@chen1999atomic] and Dantzig selector [@candes2007dantzig]. The failure bound means that for any $\varepsilon>0$ there exists $X \in \mathbb R^{(p-1)\times p}$ with $\delta_{2s}<1/\sqrt{2} + \varepsilon$ where $\ell_1$-minimization recovery fails. On the other hand, $\ell_1$-minimization recovery succeeds when $\text{RE}^{\ell_1}(\alpha,\mathcal S) >0$ [@bickel2009simultaneous], like $\ell_1$-regularized regression (Theorem \[thm:9\]). Thus, $\min_{\|\mathcal T\| \le s} \text{RE}^{\ell_1}(\alpha,\mathcal T) = 0$ if $\delta_{2s} \ge 1/\sqrt{2}$, i.e., $\kappa_+(2s)/\kappa_-(2s) \ge 3+ 2\sqrt{2}$. Since non-convex regularizers cannot weaken RE conditions, $\kappa_+(2s)/\kappa_-(2s) \ge 3+ 2\sqrt{2}$ also causes $\min_{\|\mathcal T\| \le s} \text{RE}^{\mathcal R}(\alpha,\mathcal T) = 0$ for non-convex regularizers. On the contrary, our SE conditions, e.g., $\kappa_-(2\alpha s+2)>0$, still hold with proper non-convex regularizers even when $\kappa_+(2s)/\kappa_-(2s) \ge 3+ 2\sqrt{2}$. Comparison with the conditions for feature selection ---------------------------------------------------- @shen2013constrained gave a necessary condition for consistent feature selection, which can be relaxed further to $\kappa_-(s) > C\log p/n$ with a constant $C>0$ independent of $p,~ s,~ n$. This necessary condition needs $\kappa_+(s) / \kappa_-(s)$ to be upper bounded by a constant which is independent of the regularizers. For their DC algorithm based methods, they tightened the conditions to that $\kappa_+(2\tilde s)/\kappa_-(2\tilde s)$ is upper bounded, where $\tilde s$ is the number of non-zero components of the solutions given by their methods. This condition cannot be verified until the solutions are given. However, our SE conditions do not depend on the sparseness of the practical solutions (see Section \[sec:6\]). Sparse Estimation of AGAS Solutions {#sec:6} =================================== For Problem (\[eqn:1\]), it is practical to obtain a solution which is approximate global (AG) (Definition \[def:2\]) and approximate stationary (AS) (Definition \[def:3\]). We show in this section that this kind of solutions also give good estimation to the true parameters. \[def:2\] Given $\mu\ge 0$, we say $\tilde \theta$ is a $(\theta^, \mu)$-approximate global solution of $\min_\theta \mathcal F (\theta)$ if $\mathcal F (\tilde \theta) \le \mathcal F (\theta^) + \mu$. \[def:3\] Given $\nu\ge 0$, we say $\tilde \theta$ is a $\nu$-approximate stationary solution of $\min_\theta \mathcal F (\theta)$ if the directional derivative of $\mathcal F$ at $\tilde \theta$ in any direction $d \in \mathbb R^{p}$ with $\\|d\\|_2=1$ is no less than $-\nu$, i.e., $\mathcal F'(\theta; d) \ge -\nu$. The directional derivative is defined as $\mathcal F'(\theta;d) = \liminf_{\lambda \downarrow 0} (\mathcal F(\theta+\lambda d) - \mathcal F(\theta) )/\lambda$ for any $\theta \in \mathbb R^p$ and $d\in \mathbb R^p$. For Problem (\[eqn:1\]), $\mathcal F'(\theta;d) =d^T \nabla \mathcal L(\theta) + \sum_{i=1}^p \mathcal R'(\theta_i;d_i)$. The following theorem gives the parameter estimation result with AGAS solutions. Let $\tilde u_0 \ge 0$ be the zero gap of $\tilde \theta$ and $\tilde \rho_0 =\min\{ \tilde u_0, \min_{i \in \text{supp}(\theta^)} \|\theta^_i\|\} $. \[thm:3\] Suppose the following conditions hold for the regularized regression. 1. $\tilde \theta$ is a $(\theta^, \mu)$-AG solution and $\nu$-AS solution. 2. $r(u)$ is invertible for $u \ge 0$ and $r^{-1}(u/s_1) / r^{-1}(u/s_2)$ is a non-decreasing function w.r.t. $u$ for any $s_2 \ge s_1 >0$; 3. The regularized regression satisfies $\eta$-null consistency; 4. The following SE condition holds for some integer $t \ge \alpha s+1$, $$\label{eqn:6} \kappa_+(2t) / \kappa_-(2t) < 4(\sqrt{2}-1) G_r(\tilde\rho_0,\alpha, s,t)+ 1,$$ where $\alpha = \frac{1+\eta}{ 1- \eta}$, $G_r(\tilde\rho_0,\alpha, s,t) = \frac{\sqrt{st}}{t-1} \frac{r^{-1}(r(\tilde\rho_0)/ s)}{r^{-1}( \alpha r(\tilde\rho_0)/(t-1))}$ for $\tilde\rho_0 >0$ and $G_r(0,$ $\alpha, s,t) = \lim_{\rho \to 0+} G_r(\rho, \alpha ,s,t)$. Then, $\\|\tilde \theta - \theta^\\|_2 \le C_4 \tilde\epsilon + C_5 r^{-1}(\frac{\mu}{1-\eta})$, where $\tilde\epsilon=\dot r(0+) + \eta \lambda^* + \nu$ and $C_4,~C_5$ are positive constants. $C_4$ and $C_5$ are defined in Eqn. (\[eqn:41\]) and (\[eqn:42\]). The condition 2, 3 and 4 are almost the same as the three conditions of Theorem \[thm:1\] except the slightly different requirements for $t$ and the definition of $G_r(\tilde\rho_0, \alpha, s, t)$. Consequently, the discussion in Section \[sec:12\] is also suitable for this theorem: 1. The non-invertible basis functions can be approximated by approximate invertible basis functions; 2. Without $\xi$-sharp concavity, the condition 4 of Theorem \[thm:3\] is almost the same as RIP conditions in @foucart2009sparsest; 3. With $\xi$-sharp concavity and a positive zero gap (we show in Theorem \[thm:5\] that our CD methods guarantee the positive zero gaps), SE based estimation conditions can be much relaxed. Theorem \[thm:3\] shows that the error bounds of parameter estimation are mainly determined by four parts: the slope of $r(u)$ at zero $\dot r(0+)$, the parameter $\lambda^* = O(\epsilon/\sqrt{n})$, the degree of approximating the stationary solutions $\nu$ and the degree of approximating the global optimums $r^{-1}(\mu/(1-\eta))$. If $r(u)=\lambda^2 r_0(u/\lambda;\gamma)$ and $r_0(u; \gamma)$ has a finite derivative at zero, we know that $\dot r(0+) = \lambda \dot r_0(0+;\gamma)$, e.g., $\dot r(0+) = \lambda$ for MCP. Since $\lambda = O(\epsilon/\sqrt{n})$ by Eqn. (\[eqn:20\]) in this paper, the estimation error bound is actually $$\\|\tilde \theta - \theta^* \\|_2 \le O(\epsilon/\sqrt{n}) + O(\nu) + O(r^{-1}(\mu/(1-\eta))).$$ According to Theorem \[thm:3\], we do not need to solve Problem (\[eqn:1\]) exactly. A good suboptimal solution is enough to give good parameter estimation. Even, we do not need a strictly stationary solution since Theorem \[thm:3\] allows a margin $\nu$. So, the non-convex regularized regression is robust to the inaccuracy of the solutions, which is important for numerical computation. It should be noted that $\dot r(0+)$ is required to be finite in Theorem \[thm:3\], which forbids the regularizers with infinite $\dot r(0+)$, e.g., $\ell_0$-norm and $\ell_q$-norm ($0<q<1$). It may be due to the strongly NP-hard property brought by $\ell_0$-norm and $\ell_q$-norm regularized regression [@chen2011complexity]. Similar to Theorem \[thm:8\], we give the following sparseness estimation result for AGAS solutions. The proof is the same as that of Theorem \[thm:8\]. \[thm:2\] Suppose the conditions of Theorem \[thm:3\] hold. Let $b=(t-1) r \left( c_4 \tilde\epsilon + c_5 r^{-1}\left( \frac{\mu}{1-\eta}\right)\right)$, where $c_4$ and $c_5$ are defined in Eqn. (\[eqn:43\]) and Eqn. (\[eqn:44\]). Consider $l_0>0$ and integer $m_0>0$ such that $$\sqrt{ \frac{2\kappa_+(m_0)}{m_0} (\frac{\mu}{1-\eta} + b) } + \\|X^T e/n\\|_\infty \le \dot r(l_0-).$$ Then, $\|\text{supp}(\tilde \theta) \backslash \mathcal S\| < m_0 + b/r(l_0)$. The sparseness of AGAS solutions is also affected by $\tilde\epsilon=\dot r(0+) + \eta \lambda^* + \nu$ and $\mu$. Theorem \[thm:2\] can also derive a similar conclusion as Corollary \[coro:2\]. For an AGAS solution with small $\nu$ and $\mu$, the sparseness of the solution is on the order of $s$, just like the global solutions. Approximate Global Solutions {#sec:3} ---------------------------- We need AG solutions in Theorem \[thm:3\] and Theorem \[thm:2\]. The methods to obtain such solutions are crucial consequently. Instead of restricting to the solutions given by a specific algorithm, we use the prediction error $\\|X\theta^0 - y\\|_2^2/(2n)$ to give a quality guarantee for any solution $\theta^0$ that is regarded as an AG solution. \[thm:11\] Suppose $\theta^0$ is an $s_0$-sparse vector with the prediction error $\mu_0^2 = \\|X \theta^0 - y\\|_2^2/(2n)$. If $\kappa_-(s+s_0)>0$, then $\theta^0$ is a ($\theta^$, $\mu$)-AG solution where $$\mu =\mu_0^2 + (s+s_0) r \left( \frac{\sqrt{2} \mu_0 + \epsilon/\sqrt{n}}{\sqrt{(s+s_0) \kappa_{-}(s+s_0)}} \right)$$ \[coro:3\] Suppose $\theta^0$ is an $s_0$-sparse vector with the prediction error $\mu_0 = \zeta \epsilon/\sqrt{n}$ for some $\zeta \ge 0$ and the basis function has the formulation $r(u) = \lambda^2 r_0(u/\lambda)$ with $\lambda=\eta^{-1} b_0 \epsilon / \sqrt{n}$. Then, $\theta^0$ is a $(\theta^, C_6 \epsilon^2/n)$-AG solution where $$C_6 = \zeta^2 + \frac{(s+s_0) b_0^2}{\eta^2} r_0 \left( \frac{(1+ \sqrt{2} \zeta)\eta }{ b_0 \sqrt{(s+s_0) \kappa_{-}(s+s_0)}} \right) .$$ The methods that explicitly control the sparseness of its solutions are suitable for giving the AG solutions, e.g., OMP [@tropp2007signal] and GraDeS [@garg2009gradient]. However, we do not need the strong conditions for consistent parameter estimation for these methods, e.g., $\delta_{2s}<1/3$ for GraDeS [@garg2009gradient] or $ (\kappa_+(1)/\kappa_-(t)) \log (\kappa_+(s)/\kappa_-(t))$ grows sub-linearly as $t$ for OMP [@zhang2011sparse]. In fact, Theorem \[thm:11\] only requires $\kappa_{-}(s+s_0)>0$. Hence, $s_0$ can be large enough to make $\mu_0$ to be small. The relationship between $\mu_0$ and $s_0$ depends on the employed method and the design matrix $X$. Even with a bad value of $\mu$ in the initialization, we can decrease it further by CD methods as stated in Section \[sec:4\]. Approximate Stationary Solutions with Zero Gap {#sec:4} ---------------------------------------------- Theorem \[thm:3\] also requires the solution to be $\nu$-AS and has a positive zero gap. General gradient descent algorithms can provide stationary solutions but they cannot ensure a positive zero gap. However, we observe that the coordinate descent (CD) methods can yield AS solutions and all of these solutions have positive zero gaps under proper sharp concavity conditions. In every step, CD only optimizes for one dimension, i.e., $$\label{eqn:22} \theta^{(k)}_i = \arg\min_{u \in \mathbb R} \mathcal F((\theta^{(k)}_1, \cdots, \theta^{(k)}_{i-1}, u, \theta^{(k-1)}_{i+1}, \cdots, \theta^{(k-1)}_{p} )^T ) + \frac{\psi}{2}(u - \theta_i^{(k-1)})^2,$$ where $k$ is the number of iterations, $i=1, \cdots,p$ and $\psi>0$ is a positive constant. The constant $\psi$ plays a role of balance between decreasing $\mathcal F(\theta)$ and not going far from the previous step. The above CD method is also called proximal coordinate descent. For Problem (\[eqn:1\]), the CD methods iterate as follows. $$\label{eqn:23} \theta^{(k)}_i = \arg\min_{u \in \mathbb R} \frac{1}{2} \left( \frac{\\|x_i\\|_2^2}{n} + \psi \right) \left( u - \frac{\psi \theta_i^{(k-1)} + x_i^T \omega_i^{(k)}/n}{\psi + \\|x_i\\|_2^2/n } \right)^2 + r(\|u\|),$$ where $x_i$ is the i-th column of the design matrix $X$ and $\omega_i^{(k)} = y - \sum_{j<i} x_j \theta^{(k)}_j - \sum_{j>i} x_j \theta^{(k-1)}_j$. Problem (\[eqn:23\]) is a non-convex but only one-dimensional problem. All of its solutions are between $0$ and $\frac{\psi \theta_i^{(k-1)} + x_i^T \omega_i^{(k)}/n}{\psi + \\|x_i\\|_2^2/n }$. We assume that Problem (\[eqn:23\]) can be exactly solved. If Problem (\[eqn:23\]) has more than one minimizer, any one of them can be selected as $\theta_i^{(k)}$. In this paper, CD methods stop iterating if $$\label{eqn:55} \\|\theta^{(k)} - \theta^{(k-1)}\\|_2 \le \tau,$$ where $\tau>0$ is a small tolerance proportional to the value $\nu$ (see Theorem \[thm:6\]). \[thm:5\] If $r(u)$ is $(\xi+\psi)$-sharp concave over $(0,u_0)$, then $\theta^{(k)}_i \ge u_0$ or $\theta^{(k)}_i = 0$ for any $k=1,2,\cdots$ and any $i=1,\cdots, p$. The above zero gap property of CD is a corollary of Theorem \[thm:4\]. The sharp concavity condition of Theorem \[thm:5\] is a little stronger than the requirements of Theorem \[thm:1\]. Nonetheless, we can set $\psi$ to be small to narrow the difference between the sharp concavity conditions of Theorem \[thm:1\] and Theorem \[thm:5\]. Besides the zero gap, we show in the following theorem that CD methods simultaneously give AS solutions and keep them to be still AG solutions. \[thm:6\] $\{\mathcal F(\theta^{(k)})\}$ is a non-increasing sequence and converges; For any $\nu>0$ and with $\tau = \nu/(\sqrt{p}(\psi + p \xi))$, CD stops within $k=1 + \frac{2p (\psi + p \xi)^2 \mathcal F(\theta^{(0)})}{\psi \nu^2}$ iterations and outputs a $\nu$-AS solution, where $p$ is the number of columns of the design matrix $X$. Theorem \[thm:6\] shows CD methods give a further decrease to the value $\mu$ of AG property and guarantees the $\nu$-AS property, which is necessary for sparse estimation in Theorem \[thm:3\] and Theorem \[thm:2\]. This theorem also gives an upper bound for $\nu$, i.e., $$\label{eqn:56} \nu \le \sqrt{p}(\psi + p \xi)\\|\theta^{(k)} - \theta^{(k-1)}\\|_2,$$ where $k$ is the number of iterations. Usually, we hope $\nu$ is on the order of $\lambda^$ so that $\tilde\epsilon = \dot r(0+) + \eta \lambda^ + \nu = O(\lambda^) = O(\epsilon/\sqrt{n})$ in Theorem \[thm:3\]. CD has been applied to the non-convex regularized regression by @breheny2011coordinate and @mazumder2011sparsenet . However, their non-convex regularizers are restrictive because they need Eqn. (\[eqn:23\]) to be strictly convex for $\psi=0$. They could not deal with the MCP with $\gamma \le 1$, the SCAD with $\gamma \le 2$ or the LSP with $\gamma \le 1$. Compared with them, the conclusions of Theorem \[thm:6\] are weaker but they are enough to obtain $\nu$-AS solutions and the regularizers can approximate $\ell_0$-norm arbitrarily. Experiment {#sec:11} ========== In this section, we experimentally show the performance of CD methods on giving AGAS solutions and the degree of weakness of the estimation conditions required by the sharp concave regularizers. AGAS solutions {#sec:10} -------------- In Section \[sec:6\], we prove that $\mu$ is monotonously decreasing, $\nu$ tends to 0 and the zero gap $\tilde u_0$ is maintained in each iteration of CD algorithm. We experimentally show these in this part. We set the dimension of the parameter as $p=1000$, the number of non-zero components of $\theta^$ (the true parameter) $s=\log p$. We randomly choose $s$ indices as the non-zero components. The non-zero components are i.i.d. drawn from $\mathcal N(0,1)$ and those belonging to $(-0.1,0.1)$ are promoted to $\pm 0.1$ according to their signs. The elements of the design matrix $X \in \mathbb R^{n\times p}$ are i.i.d. drawn from $\mathcal N(0,1)$, where $n=10s\log p$. The noise $e$ is drawn from $\mathcal N(0,I_n)$ and is normalized such that $\epsilon=\\|e\\|_2=0.01$. We fix $\gamma=0.1$ and $\eta=0.01$ for all the non-convex regularizers (LSP, MCP and GP) and use Eqn. (\[eqn:20\]) to choose $\lambda$. For CD algorithm, we set $\psi=0.1$. The CD algorithm is initialized with zero vectors and terminated when $\nu$ is below $10^{-3}$ (we set $\tau = 10^{-3}/(\sqrt{p} (\psi + p \xi))$ by Theorem \[thm:6\]) or the number of iterations is over 500. For each regularizer, we run CD for 100 trials with independent true parameters and design matrices. We illustrate the boxplots for $\tilde u_0$, $\mu$ and $\nu$ of each iteration in Figure \[fig:1\]. The left column shows that CD methods maintain the zero gaps in each iteration as stated in Theorem \[thm:5\]. The middle column shows $\mathcal F(\theta^{(k)}) - \mathcal F(\theta^)$ decrease to zero for most of trials in 100 iterations. The right column shows that most of the solutions are very close to stationary solutions within 100 iterations. Weaker Conditions for Sparse Estimation --------------------------------------- We show the performance of non-convex regularizers for sparse estimation in this part. For an estimation $\tilde\theta$, three criterions are used to describe the performance of sparse estimation: 1. sparseness $\\|\tilde\theta\\|_0$; 2. Relative recovery error (RRE) $\\|\tilde\theta - \theta^\\|_2/\\|\theta^\\|_2$; 3. Support recovery rate (SRR) $\| \text{supp}(\tilde\theta) \cap \text{supp}(\theta^)\| / \|\text{supp}(\tilde\theta) \cup \text{supp}(\theta^)\|$. A weaker estimation condition than convex regularizers can be verified by achieving a more accurate sparseness, lower RRE or higher SRR with less sampling size. We fix the dimension of the parameters and the sparseness of the true parameters and we vary the sampling size $n$ to compare the three criterions between convex regularizers ($\ell_1$-norm, implemented by FISTA [@beck2009fast]) and non-convex regularizers (LSP, MCP and GP).As Figure \[fig:8\] shows, non-convex regularizers give much more accurate sparseness estimation, lower RREs and higher SRRs than $\ell_1$-regularization. Among the three non-convex regularizers, the performance of sparse estimation is similar to each other. ![The sparseness (left), RRE (middle) and SRR (right) corresponding to the regularizers(LSP, MCP, GP and $\ell_1$-norm). The true parameters, the design matrices and the noises are generated in the same way as Section \[sec:10\] except that $p=10~000$, $s=100$ and $n$ varies from $s$ to $15s$. The parameter of the regularizers $\gamma$ is set as $10^{-7}$. We use the OMP [@tropp2007signal] to generate an initial solution for CD with at most $(n-s)$ non-zero components. The parameters of CD $\psi = 0.1$ and the stopping criterion of CD is the same as Section \[sec:10\]. Every data point is the average of 100 trials of CD methods. For each regularizer and each $n$, we select $\lambda$ from $10^{-6},~ 10^{-5}, \cdots,~ 10$ such that it gets the smallest average RRE of the 100 trials.[]{data-label="fig:8"}](paramest_sparsity.pdf "fig:"){width="32.00000%"} ![The sparseness (left), RRE (middle) and SRR (right) corresponding to the regularizers(LSP, MCP, GP and $\ell_1$-norm). The true parameters, the design matrices and the noises are generated in the same way as Section \[sec:10\] except that $p=10~000$, $s=100$ and $n$ varies from $s$ to $15s$. The parameter of the regularizers $\gamma$ is set as $10^{-7}$. We use the OMP [@tropp2007signal] to generate an initial solution for CD with at most $(n-s)$ non-zero components. The parameters of CD $\psi = 0.1$ and the stopping criterion of CD is the same as Section \[sec:10\]. Every data point is the average of 100 trials of CD methods. For each regularizer and each $n$, we select $\lambda$ from $10^{-6},~ 10^{-5}, \cdots,~ 10$ such that it gets the smallest average RRE of the 100 trials.[]{data-label="fig:8"}](paramest_rre.pdf "fig:"){width="32.00000%"} ![The sparseness (left), RRE (middle) and SRR (right) corresponding to the regularizers(LSP, MCP, GP and $\ell_1$-norm). The true parameters, the design matrices and the noises are generated in the same way as Section \[sec:10\] except that $p=10~000$, $s=100$ and $n$ varies from $s$ to $15s$. The parameter of the regularizers $\gamma$ is set as $10^{-7}$. We use the OMP [@tropp2007signal] to generate an initial solution for CD with at most $(n-s)$ non-zero components. The parameters of CD $\psi = 0.1$ and the stopping criterion of CD is the same as Section \[sec:10\]. Every data point is the average of 100 trials of CD methods. For each regularizer and each $n$, we select $\lambda$ from $10^{-6},~ 10^{-5}, \cdots,~ 10$ such that it gets the smallest average RRE of the 100 trials.[]{data-label="fig:8"}](paramest_srr.pdf "fig:"){width="32.00000%"}\ Single-Pixel Camera ------------------- We compare non-convex regularizers and $\ell_1$-norm in the application of single-pixel camera [@duarte2008single]. In this application, we need to recover an image from a small fraction of pixels of an image, which is a similar task to image inpainting [@mairal2008sparse]. Since most of natural images have sparse Discrete Cosine Transformations (DCT), we can recover the image by solving the problem $\min_\theta \\|y-M \text{vec}(\theta)\\|_2^2/(2n) + \mathcal R(\text{vec}(D[\theta]))$, where $y$’s components are the known pixels, $\theta$ is the estimated image, $M$ is a mask matrix indicating the positions of the known pixels, $D[\theta]$ is the 2D-DCT of $\theta$ and $\text{vec}(\theta)$ is the vectorization of $\theta$. Denote $\Theta = D[\theta]$ and we rewrite the problem in the form of Problem (\[eqn:1\]) $\min_\Theta \\|y-M \text{vec}(D^{-1} [\Theta] ) \\|_2^2/(2n) + \mathcal R(\text{vec}(\Theta))$, where $D^{-1}[\Theta]$ is the inverse 2D-DCT of $\Theta$. Figure \[fig:2\](a) shows the test image (size 256$\times$256). We randomly choose 25% pixels of it as $y$. The PSNRs of LSP ($\gamma=10^{-7}$) and $\ell_1$-norm are compared in Figure \[fig:2\](d), where LSP has higher PSNRs than $\ell_1$-norm for all $\lambda$s in the figure. The PSNRs of LSP are more robust to $\lambda$ than $\ell_1$-norm. Figure \[fig:2\](b) and (c) illustrate the recovered images by LSP and $\ell_1$-norm with the best PSNRs. The image produced by LSP is of better quality than the one created by $\ell_1$-norm. \ Conclusion ========== This paper establishes a theory for sparse estimation with non-convex regularized regression. The framework of non-convex regularizers in this paper is general and especially suitable for sharp concave regularizers. For proper sharp concave regularizers, both global solutions and AGAS solutions can give good parameter estimation and sparseness estimation. The proposed SE based estimation conditions are weaker than that of $\ell_1$-norm. To obtain AGAS solutions, we give a prediction error based guarantee for AG property and prove that CD methods yield the desired AGAS solutions. Our theory explains the improvements on sparse estimation from $\ell_1$-regularization to non-convex regularization. Our work can serve as a guideline for the further study on designing regularizers and developing algorithms for non-convex regularization. Technical Proofs ================ We first provide two lemmas. The first is Lemma 1 of @zhang2011general. \[lem:9\] Let $\hat\theta$ be a global optima of Problem (\[eqn:1\]). We have $$\label{eqn:11} \\|X^T (X\hat \theta - y)/n\\|_\infty \le \lambda^.$$ Under the $\eta$-null consistency condition, we further have $$\label{eqn:19} \\|X^T e /n\\|_\infty \le \eta \lambda^.$$ \[lem:8\] 1. $r(u)$ is subadditive, i.e., $r(u_1+u_2) \le r(u_1) + r(u_2), ~ \forall u_1, u_2 \ge 0.$ 2. For any $\forall u > 0$ and any $d \in \partial r(u)$, $\dot r(0+) \ge \dot r(u-) \ge d \ge \dot r(u+) \ge 0.$ Proof.* 1. Since $r(u)$ is concave, it follows that $\forall u_1,u_2 \ge 0$, $\frac{ u_1}{ u_1 + u_2} r( u_1+ u_2) + \frac{ u_2}{ u_1 + u_2} r(0) \le r( u_1)$ and $ \frac{u_2}{ u_1 + u_2} r(u_1 + u_2) + \frac{u_1}{u_2 + u_2} r(0) \le r(u_2)$. Summing up the two inequalities gives $r(u_1 + u_2) \le r(u_1) + r(u_2)$. 2\. Invoking the subadditivity, we have $[r(u-\Delta u) - r(u)]/\Delta u \le r(\Delta u)/\Delta u$ for $\Delta u>0 $ and $u \ge \Delta u$. Let $\Delta u \to 0$. Then $\dot r(0+) \ge \dot r(u-)$. The concavity of $r(u)$ yields that $\frac{r(u) - r(u-\Delta u)}{\Delta u} \ge \frac{r(u + \Delta u) - r(u)}{ \Delta u}$ for $\Delta u>0$. From the definition of subgradient of concave function, we have $\Delta u \cdot d \ge r(u+\Delta u) - r(u)$ and $-\Delta u \cdot d \ge r(u - \Delta u) - r(u)$ for any $\Delta u>0$. Hence, $\frac{r(u) - r(u-\Delta u)}{\Delta u} \ge d\ge \frac{r(u + \Delta u) - r(u)}{ \Delta u}$. Let $\Delta u \to 0$ and then the lemma follows. $\blacksquare$ Sharp concavity and strong concavity {#sec:14} ------------------------------------ Invoking Eqn. (\[eqn:57\]) with $\alpha>0$, $t_1=0$ and $t_2= t>0$, we have $r((1-\alpha)t) \ge (1-\alpha)r(t) + C\alpha (1-\alpha)t^2/2$, which implies $$r(t) \ge t \cdot \frac{r(t) - r(t-\alpha t)}{\alpha t} + C(1-\alpha)t^2/2.$$ Let $\alpha \to 0$. Sharp concavity follows. The upper bound of $\lambda^$ for LSP {#sec:13} -------------------------------------- Define $U>0$ such that $\frac{U^2}{\log(1+U)} = \frac{2}{\xi \gamma^2}$. Let $u=U\lambda \gamma$ and we have $\lambda^ \le \lambda (\frac{\xi \gamma U}{2} + \frac{\log(1+U)}{\gamma U}) = \lambda \sqrt{2 \xi \log(1+U)}$. Note that $U \le \frac{U^2}{\log(1+U)} = \frac{2}{\xi \gamma^2}$. Hence, $\lambda^* \le \lambda \sqrt{2 \xi \log(1+\frac{2}{\xi \gamma^2})}$. Also, $a_\gamma \le \sqrt{2 \xi \log(1+\frac{2}{\xi \gamma^2})}$. Proof of Theorem \[thm:4\] -------------------------- $\hat\theta$ minimizes $\frac{1}{2n}\\|y-X\theta\\|_2^2 + \mathcal R(\theta)$, therefore the subgradient at $\hat \theta$ contains zero, i.e., $ \|x_i^T(X\hat \theta - y)/n\| \le \dot r(\|\hat\theta_i\|-)$ for any $i\in \mathrm{supp}(\hat\theta)$. Define $\bar \theta = (\hat \theta_1, \cdots,\hat \theta_{i-1}, 0 ,\hat \theta_{i+1} ,\cdots,\hat \theta_n) $. We have $\frac{1}{2n} \\|y-X\hat \theta\\|_2^2 + \mathcal R(\hat \theta) \le \frac{1}{2n} \\|y-X \bar \theta\\|_2^2 + \mathcal R(\bar \theta)$, which implies $2n r(\|\hat \theta_i\|) \le \hat \theta_i^2 \\|x_i\\|_2^2 + 2 \hat \theta_i x_i^T (y-X\hat \theta) \le \hat \theta_i^2 \\|x_i\\|_2^2 + 2 \|\hat \theta_i\| \| x_i^T (y-X\hat \theta)\| \le n \xi \hat \theta_i^2 + 2n \|\hat \theta_i\| \dot r(\|\hat\theta_i\|-)$. If $\hat \theta_i \in (0,u_0)$, this inequality contradicts with $\xi$-sharp concavity condition. $\blacksquare$ Proof of Theorem \[thm:10\] --------------------------- We assume that $\theta=0$ is not a minimizer of $\min_{\theta} \frac{1}{2n} \\|X \theta - e/\eta\\|_2^2 + \mathcal R(\theta)$ while $\hat\theta_\eta\not=0$ is a minimizer. Therefore, $\frac{1}{2n\eta^2} \\|e\\|_2^2 > \frac{1}{2n}\\|X \hat \theta_\eta - e/\eta\\|_2^2 + \mathcal R(\hat\theta_\eta)$. Since $r(u)$ is $\xi$-sharp concave over $(0,u_0)$, the non-zero components of $\hat\theta_\eta$ has magnitudes larger than $u_0$. Thus, $\frac{1}{2n}\\|X \hat \theta_\eta - e/\eta\\|_2^2 + \mathcal R(\hat\theta_\eta) \ge r(u_0) \ge \frac{1}{2n\eta^2 }\\|e\\|_2^2$. It contradicts with the assumption. $\blacksquare$ Proof of Theorem \[thm:1\] {#sec:9} -------------------------- Let $\Delta=\hat \theta - \theta^* $, $\mathcal S = \mathrm{supp}(\theta^)$, $s=\|\mathcal S\|$ and $\mathcal T$ be any index set with $\|\mathcal T\| \le s$. Let $i_1, i_2, \cdots$ be a sequence of indices such that $i_k \in \bar{\mathcal T}$ for $k \ge 1$ and $\|\Delta_{i_1}\| \ge \|\Delta_{i_2}\| \ge \|\Delta_{i_3}\| \ge \cdots$. Given an integer $t \ge s$, we partition $\bar{\mathcal T}$ as $\bar{\mathcal T} = \cup_{i\ge 1}\mathcal T_i$ such that $\mathcal T_1 = \{i_1, \cdots, i_t\}$, $\mathcal T_2 = \{ i_{t+1}, \cdots, i_{2t}\}$, $\cdots$. Define $\Sigma = \sum_{i\ge 2} \\|\Delta_{\mathcal T_i}\\|_2$, $\alpha = (1+\eta) / (1-\eta)$. Before the proof, we introduce the following three lemmas. Lemma \[lem:3\] is a special case of Lemma \[lem:7\] with $\mu = 0$. \[lem:3\] Under $\eta$-null consistency, $\frac{1}{2n} \\|X \Delta\\|_2^2 + \mathcal R(\Delta_{\bar{\mathcal S}}) \le \alpha \mathcal R(\Delta_{\mathcal S})$. \[lem:2\] $r(\Sigma/\sqrt{t}) \le \mathcal R(\Delta_{\bar{\mathcal T}}) / t.$ Proof.* For any $i\in \mathcal T_k$ and $j\in \mathcal T_{k-1}$ ($k \ge 2$), we have $\|\Delta_i\| \le \|\Delta_j\|$. Thus, $r(\|\Delta_i\|) \le \mathcal R(\Delta_{\mathcal T_{k-1}})/t$, i.e., $\|\Delta_i\|^2 \le ( r^{-1}(\mathcal R(\Delta_{\mathcal T_{k-1}})/t))^2$. It follows that $r( \\|\Delta_{\mathcal T_k}\\|_2 / \sqrt{t} ) \le \mathcal R(\Delta_{\mathcal T_{k-1}})/t$. Thus, $\mathcal R(\Delta_{\bar{\mathcal T}})/ t \ge \sum_{k \ge 2} \mathcal R(\Delta_{\mathcal T_{k-1}})/t \ge \sum_{k \ge 2} r( \\|\Delta_{\mathcal T_k}\\|_2 / \sqrt{t} ) \ge r(\Sigma/\sqrt{t}). ~\blacksquare $ \[lem:1\] Under $\eta$-null consistency, $$\label{eqn:3} \max\{ \\|\Delta_{\mathcal T}\\|_2, \\|\Delta_{\mathcal T_1}\\|_2 \} \le \frac{1+\sqrt{2}}{2 \kappa_-(2t)} \left[ \frac{\kappa_+(2t) - \kappa_-(2t)}{2} \Sigma + \sqrt{t} (1+\eta)\lambda^* \right].$$ Proof. By Lemma \[lem:9\], we have $\\| X^T X \Delta/n\\|_\infty \le \\|X^T(X \hat \theta - y)/n\\|_\infty + \\| X^T e/n\\|_\infty \le \lambda^* + \eta \lambda^$. We modify the Eqn. (12) in @foucart2009sparsest to the following inequality. $$\frac{1}{n} \left< X \Delta,X( \Delta_{\mathcal T} + \Delta_{\mathcal T_1}) \right> \le ( \\|\Delta_{\mathcal T}\\|_1 + \\|\Delta_{\mathcal T_1}\\|_1 ) \\|\frac{1}{n} X^T X \Delta\\|_\infty \le \sqrt{t} (1+\eta) \lambda^ (\\|\Delta_{\mathcal T}\\|_2 + \\|\Delta_{\mathcal T_1}\\|_2).$$ Then, following the proof of Theorem 3.1 in @foucart2009sparsest, Eqn. (\[eqn:3\]) follows.$\blacksquare$ Next, we turn to the proof Theorem \[thm:1\]. Let $\kappa_- = \kappa_-(2t)$, $\kappa_+ = \kappa_+(2t)$, $H_r = H_r(\rho_0,\alpha,s,t)$ and $\varrho = (1+\sqrt{2}) (\kappa_+ / \kappa_- -1) /4 $. There are two cases according to the difference of supports of $\hat \theta$ and $\theta^$. Case 1: $\text{supp}(\hat \theta) = \text{supp}(\theta^)$. For this case, we have $\Delta_i=0$ for $i\in \bar{\mathcal S}$ and $\Sigma = 0$, with which and Lemma \[lem:1\], we obtain that $\\|\Delta\\|_2 = \\|\Delta_{\mathcal S}\\|_2 \le c_1 \lambda^,$ where $c_1 = (1+\sqrt{2}) (1+\eta) \sqrt{t} / (2\kappa_-) $. Case 2: $\text{supp}(\hat \theta) \not= \text{supp}(\theta^)$. Let $\mathcal T$ be the indices of the first $s$ largest components of $\Delta$ in the sense of magnitudes. From the concavity of $r(u)$, $\mathcal R(\Delta_{\mathcal T}) \le s r(\\|\Delta_{\mathcal T}\\|_1/s) \le s r(\\|\Delta_{\mathcal T}\\|_2/\sqrt{s})$. By Lemma \[lem:1\], we have $$\label{eqn:17} \mathcal R(\Delta_{\mathcal T}) \le s r \left( \frac{\\|\Delta_{\mathcal T}\\|_2}{\sqrt{s}} \right) \le s r \left( \frac{1+\sqrt{2}}{2\sqrt{s} \kappa_- } \left( \frac{\kappa_+ - \kappa_-}{2} \Sigma + \sqrt{t}(1+\eta) \lambda^\right) \right).$$ Combining with Lemma \[lem:3\] and \[lem:2\], it follows that $$\label{eqn:35} r^{-1}\left( \frac{ \mathcal R(\Delta_{\mathcal T})}{s}\right) - \varrho \sqrt{\frac{t}{s}} r^{-1}\left( \frac{ \alpha \mathcal R(\Delta_{\mathcal T})}{t} \right) \le \frac{(1+\sqrt{2}) (1+\eta)}{2 \kappa_-} \sqrt{\frac{t}{s}} \lambda^.$$ By the definition of $\rho_0$ in Eqn. (\[eqn:60\]) and $\text{supp}(\hat \theta) \not= \text{supp}(\theta^)$, there exists $j$ satisfying $\|\Delta_j\| \ge \rho_0$, which implies $\mathcal R(\Delta_{\mathcal T}) \ge r(\rho_0)$. Since $\frac{r^{-1}(u/s)}{r^{-1}(\alpha u/t)}$ is a non-decreasing function of $u$, we have that $$\frac{r^{-1}(\mathcal R(\Delta_{\mathcal T})/s)}{r^{-1}( \alpha \mathcal R( \Delta_{\mathcal T})/t)} \ge \frac{r^{-1}(r(\rho_0)/s)}{r^{-1}(\alpha r(\rho_0)/t)} = \sqrt{\frac{t}{s}} H_r(\rho_0,\alpha,s,t).$$ for $\rho_0>0$. If $\rho_0=0$, the left hand of the above inequality still holds since $H_r(0,\alpha,s,t) = \lim_{\rho\to 0+} H_r(\rho,\alpha,s,t)$. Under the condition $H_r - \varrho > 0$, we have $$\label{eqn:40} r^{-1}\left( \alpha \mathcal R(\Delta_{\mathcal S}) / t \right) \le r^{-1}\left( \alpha \mathcal R(\Delta_{\mathcal T}) / t \right) \le C_2 (1+\eta) \lambda^,$$ where $$\label{eqn:13} C_2 = \frac{1+\sqrt{2}}{2 (H_r - \varrho)\kappa_- } .$$ Hence, we have $\Sigma \le \sqrt{t} C_2 (1+\eta) \lambda^$ by Lemma \[lem:3\] and Lemma \[lem:2\]. Invoking Lemma \[lem:1\] and $\\|\Delta\\|_2 \le \\|\Delta_{\mathcal T}\\|_2 + \\|\Delta_{\mathcal T_1}\\|_2 + \Sigma$, the conclusion follows with some algebra. $\blacksquare$ Proof of Theorem \[thm:8\] -------------------------- The proof is similar to Theorem 2 in @zhang2011general except that we bound $\mathcal R(\Delta_{\mathcal S})$ and $\\|X \Delta\\|_2^2/(2n)$ as follows. By Eqn. (\[eqn:40\]), we have $\mathcal R(\Delta_{\mathcal S}) \le \frac{t}{\alpha} r(C_2 (1+\eta)\lambda^ )$ and $\frac{1}{2n} \\|X \Delta\\|_2^2 \le \alpha \mathcal R(\Delta_{\mathcal S}) \le t r(C_2 (1+\eta) \lambda^)$. The method to obtain Eqn. (\[eqn:26\]) and (\[eqn:16\]) {#sec:8} ------------------------------------------------------- Suppose $r(u) = C u^q~(0<q\le 1)$ for $u\ge \lambda \gamma (1-\phi)$. The continuity and the concavity of $r(u)$ require that $C (\lambda \gamma(1-\phi))^q = 0.5\lambda^2 \gamma(1-\phi^2)$ and $Cq (\lambda \gamma (1-\phi))^{q-1} \le \lambda \phi$. Thus, it is feasible that $q=2\phi/(1+\phi)$ and $C = 0.5 \lambda^2 \gamma (1-\phi^2)/ (\lambda \gamma (1-\phi))^q$. Eqn. (\[eqn:26\]) follows. For this setting for $C$ and $q$, $r(u)$ is $\xi$-sharp concave over $(0,\rho_0)$ with $\rho_0 = \lambda \gamma (1-\phi) (\frac{\phi}{\xi\gamma (1+\phi)})^{(1+\phi)/2}$. We observe that $r(\rho_0)/s \ge \lambda^2 \gamma (1-\phi^2)/2 = \alpha r(\rho_0)/t$ holds under the condition that $\frac{\alpha}{t} (\frac{\phi}{\gamma \xi (1+\phi)})^\phi =1$, i.e., $\gamma \xi= \frac{\phi}{1+\phi} (\alpha/t)^{1/\phi}$. Thus, $r^{-1} (\alpha r(\rho_0) / t ) = \lambda \gamma (1-\phi)$ and $r^{-1} ( r(\rho_0)/s) = \lambda \gamma (1-\phi) (t/(\alpha s))^{1/q}$ with $q=2\phi/(1+\phi)$. Then, Eqn. (\[eqn:16\]) follows. Proof of Theorem \[thm:9\] -------------------------- Let $\Delta = \hat \theta - \theta^$. By Lemma \[lem:3\] in Section \[sec:9\], we have $\text{RE}^{\mathcal R}(\alpha, \mathcal S) \\|\Delta\\|_2^2 \le \\|X \Delta\\|_2^2/n$ and $\text{RIF}^{\mathcal R}_\tau (\alpha, \mathcal S) \\|\Delta\\|_\tau \le s^{1/\tau} \\|X^T X \Delta\\|_\infty/n$.Invoking null consistency, we have $ e^T X\Delta/n \le \eta\\|X \Delta\\|_2^2/(2n) + \eta \mathcal R(\Delta)$. Then, $$\begin{array}{ll} 0 & \ge \mathcal L(\theta^* + \Delta) - \mathcal L(\theta^) + \mathcal R(\theta^ + \Delta) - \mathcal R(\theta^) \\ & \ge \\|X \Delta\\|_2^2/(2n) - e^T X \Delta/n + \mathcal R(\Delta_{\bar{\mathcal S}}) - \mathcal R(\Delta_{\mathcal S}) \\ & \ge (1-\eta) \\|X \Delta\\|_2^2/(2n) - (1+\eta) \mathcal R(\Delta) \\ & \ge (1-\eta) \\|\Delta\\|_2^2 \text{RE}^{\mathcal R}(\alpha, \mathcal S)/2 - (1+\eta) \sqrt{s} \dot r(0+) \\|\Delta\\|_2. \end{array}$$ Hence, we obtain $\\|\Delta\\|_2 \le \frac{ 2\alpha\sqrt{s}}{ \text{RE}^{\mathcal R} (\alpha, \mathcal S) } \dot r(0+)$. By Lemma \[lem:9\], $\\|X^T X \Delta/n\\|_\infty \le \\|X^T (X \hat \theta -y)/n\\|_\infty + \\|X^T e/n\\|_\infty \le (1+\eta)\lambda^$. By the definition of RIF, we have $\\|\Delta\\|_\tau \le \frac{(1+\eta)\lambda^* s^{1/\tau} }{\text{RIF}^{\mathcal R}_\tau (\alpha, \mathcal S)}$. $\blacksquare$ Proof of Theorem \[thm:3\] -------------------------- The proof needs the following two lemmas, which are extensions of Lemma \[lem:3\] and Lemma \[lem:1\] The notations are the same as Section \[sec:9\] except that $\Delta = \tilde\theta - \theta^$. \[lem:7\] Suppose $\tilde \theta$ is a $(\theta^, \mu)$-approximate global solution and the regularized regression satisfies the $\eta$-null consistency condition. Then, $\\|X \Delta \\|_2^2/(2n) + \mathcal R(\Delta_{\bar{\mathcal S}}) \le \alpha \mathcal R(\Delta_{\mathcal S}) + \mu/(1-\eta)$. Proof. Invoking $\eta$-null consistency condition, we have $e^T X\Delta/n \le \eta \\|X \Delta\\|_2^2/(2n) + \eta \mathcal R(\Delta)$. Since $\tilde\theta = \theta^+ \Delta $ is a $(\theta^, \mu)$-approximate global solution, we have $$\begin{array}{ll} \mu & \ge \mathcal L(\theta^+\Delta) - \mathcal L(\theta^) + \mathcal R(\theta^+\Delta) - \mathcal R(\theta^)\\ & \ge \\|X\Delta\\|_2^2/(2n) - e^T X\Delta/n + \mathcal R(\Delta_{\bar{\mathcal S}}) - \mathcal R(\Delta_{\mathcal S}) \\ & \ge (1-\eta) \\|X \Delta\\|_2^2/(2n) -\eta \mathcal R(\Delta) + \mathcal R(\Delta_{\bar{\mathcal S}}) - \mathcal R(\Delta_{\mathcal S}) \end{array}$$ Hence, the conclusion follows. $\blacksquare$ \[lem:6\] Under $\eta$-null consistency, $$\label{eqn:7} \max\{ \\|\Delta_{\mathcal T}\\|_2, \\|\Delta_{\mathcal T_1}\\|_2 \} \le \frac{1+\sqrt{2}}{2 \kappa_-(2t)} \left[ \frac{\kappa_+(2t) - \kappa_-(2t)}{2} \Sigma + \sqrt{t} \tilde\epsilon \right].$$ Proof. Since $\tilde \theta$ is a $\nu$-AS solution, we have $\\| X^T(X \tilde \theta - y)/n \\|_\infty \le \dot r(0+) + \nu$. From the triangle inequality and Eqn. (\[eqn:19\]), we have $\\|X^T X \Delta/n\\|_\infty \le \\|X^T(X \tilde \theta - y)/n\\|_\infty + \\|X^T e/n\\|_\infty \le \dot r(0+) + \eta \lambda^* + \nu = \tilde\epsilon$. Eqn. (\[eqn:7\]) follows with the same analysis as the proof of Lemma \[lem:1\]. $\blacksquare$ Next, we turn to the proof of Theorem \[thm:3\]. The proof is similar to that of Theorem \[thm:1\]. Here we only provide some important steps. Let $\kappa_- = \kappa_-(2t)$, $\kappa_+ = \kappa_+(2t)$, $G_r = G_r(\tilde\rho_0,\alpha,s,t)$ and $\varrho = (1+\sqrt{2}) (\kappa_+ / \kappa_- -1) /4 $. Case 1: $\text{supp}(\tilde \theta) = \text{supp}(\theta^)$. Similar to Case 1 of Theorem \[thm:1\], we have $\\|\Delta\\|_2 = \\|\Delta_{\mathcal S}\\|_2 \le c_3 \tilde\epsilon$ where $c_3 = (1+\sqrt{2}) \sqrt{t} / (2\kappa_-)$. Case 2: $\text{supp}(\tilde \theta) \not= \text{supp} (\theta^)$. Similar to Eqn. (\[eqn:35\]), we have $$\label{eqn:36} r^{-1} \left( \frac{\mathcal R(\Delta_{\mathcal T})}{s} \right) - \varrho \sqrt{\frac{t}{s}} r^{-1} \left( \frac{\alpha \mathcal R(\Delta_{\mathcal T})}{t} + \frac{\mu}{(1-\eta)t} \right) \le \frac{1+\sqrt{2}}{2\kappa_-} \sqrt{\frac{t}{s}} \tilde\epsilon$$ Since $r(u)$ is non-decreasing and concave, $r^{-1}(u)$ is convex. Therefore, $$\label{eqn:37} r^{-1} \left( \frac{\alpha \mathcal R(\Delta_{\mathcal T})}{t} + \frac{\mu}{(1-\eta)t} \right) \le \frac{t-1}{t} r^{-1} \left( \frac{\alpha \mathcal R(\Delta_{\mathcal T})}{t-1} \right) + \frac{1}{t} r^{-1} \left( \frac{\mu}{1-\eta}\right).$$ We observe that $$\label{eqn:38} \frac{r^{-1}(\mathcal R(\Delta_{\mathcal T})/s)}{r^{-1}( \alpha \mathcal R(\Delta_{\mathcal T})/(t-1))} \ge G_r \frac{t-1}{\sqrt{st}}$$ Combining Eqn. (\[eqn:36\])-(\[eqn:38\]), we know that under the condition of Eqn. (\[eqn:6\]), $$\label{eqn:50} r^{-1} ( \alpha \mathcal R(\Delta_{\mathcal S})/(t-1)) \le c_4 \tilde\epsilon + c_5 r^{-1}( \mu/(1-\eta))/(t-1),$$ where $$\label{eqn:43} c_4 =\frac{t}{t-1} \frac{1+\sqrt{2}}{2 (G_r - \varrho) \kappa_- }$$ and $$\label{eqn:44} c_5 = \varrho/(G_r-\varrho).$$ Hence, we have $\Sigma \le \sqrt{t}c_4 \tilde\epsilon + \frac{c_5+1}{\sqrt{t}} r^{-1} \left( \frac{\mu}{1-\eta}\right)$. With this and Lemma \[lem:6\], it follows that $\\|\Delta\\|_2 \le C_4 \tilde\epsilon + C_5 r^{-1} \left( \frac{\mu}{1-\eta}\right)$, where $$\label{eqn:41} C_4 = \frac{\sqrt{t}(1+\sqrt{2})}{\kappa_-} \frac{G_r + \varrho/(t-1) + 0.5t/(t-1)}{ G_r - \varrho} \ge c_3,$$ $$\label{eqn:42} C_5 = \frac{ (2\varrho + 1) G_r}{\sqrt{t} ( G_r - \varrho) }. ~\blacksquare$$ Proof of Theorem \[thm:11\] --------------------------- Let $\Delta^0= \theta^0 - \theta^$. We have $\\|X \Delta^0\\|_2 \le \\|X\Delta^0 - e\\|_2 + \epsilon \le \mu_0 \sqrt{2n} +\epsilon$. So, $$\begin{array}{ll} \mu & = \mathcal L(\theta^0) -\mathcal L(\theta^) + \mathcal R(\theta^+ \Delta^0) -\mathcal R(\theta^) \\ & \le \mathcal L(\theta^0) + \mathcal R(\Delta^0) \\ & \le \mu_0^2 + (s+s_0) r(\\|\Delta^0\\|_2/\sqrt{s+s_0}) \\ & \le \mu_0^2 + (s+s_0) r(\\|X \Delta^0\\|_2/\sqrt{n \kappa_-(s+s_0) (s+s_0}) \\ & \le \mu_0^2 + (s+s_0) r((\epsilon/\sqrt{n}+ \sqrt{2} \mu_0)/\sqrt{(s+s_0) \kappa_-(s+s_0)}) \nonumber. ~\blacksquare \end{array}$$ Proof of Theorem \[thm:6\] -------------------------- For any $i=1, \cdots, p-1$, let $z_{k,i} = (\theta_1^{(k)}, \cdots, \theta_i^{(k)}, \theta_{i+1}^{(k-1)}, \cdots, \theta_p^{(k-1)})^T$ and $z_{k,0} = \theta^{(k-1)}$, $z_{k,p} = \theta^{(k)}$. By the definition of $\theta_i^{(k)}$ in Eqn (\[eqn:22\]), we have $$\label{eqn:53} \mathcal F( z_{k, i} ) \le \mathcal F( z_{k, i} ) + \psi ( \theta^{(k)}_{i} - \theta^{(k-1)}_{i})^2/2 \le \mathcal F( z_{k, i-1} ).$$ Thus, $\mathcal F( \theta^{(k)}) = \mathcal F(z_{k,p}) \le \mathcal F( z_{k, i} ) \le \mathcal F( z_{k, i} )+ \psi ( \theta^{(k)}_i - \theta^{(k-1)}_i)^2/2 \le \mathcal F( z_{k, 0}) = \mathcal F( \theta^{(k-1)})$. Note that $\mathcal F(\theta^{(k)}) \ge 0$ for any $k$. Thus, $\{\mathcal F(\theta^{(k)})\}$, as well as $\{\mathcal F( z_{k, i} )\}$ and $\{\mathcal F( z_{k, i} )+ \psi ( \theta^{(k)}_i - \theta^{(k-1)}_i)^2/2\} $ are non-increasing sequences and converge to the same non-negative value. Summing up the right inequality of Eqn. (\[eqn:53\]) from $i=1$ to $p$, we have $\\|\theta^{(k)} - \theta^{(k-1)}\\|_2^2 \le 2(\mathcal F(\theta^{(k-1)}) - \mathcal F(\theta^{(k)}) )/\psi$. Summing up from $k=1$ to $K$, we have $$\label{eqn:59} \min_{1\le k \le K} \\|\theta^{(k)} - \theta^{(k-1)}\\|_2^2 \le \frac{\sum_{k=1}^K \\|\theta^{(k)} - \theta^{(k-1)}\\|_2^2}{K} \le \frac{2\mathcal F(\theta^{(0)})}{\psi K}$$ The directional derivative of Eqn. (\[eqn:22\]) at $\theta_i^{(k)}$ is non-negative, i.e., $$\label{eqn:54} d_i x_i^T (Xz_{k,i} - y)/n + \mathcal R'(\theta_i^{(k)};d_i) + \psi(\theta^{(k)}_i - \theta^{(k-1)}_i)d_i \ge 0$$ for any $d_i \in \mathbb R$. Summing up Eqn. (\[eqn:54\]) from $i=1$ to $p$, we have for any $d\in \mathbb R^p$ $$\begin{array}{ll} 0 & \le \sum_{i=1}^p \psi(\theta^{(k)}_i - \theta^{(k-1)}_i)d_i + \mathcal R'(\theta^{(k)}; d) + \sum_{i=1}^p d_i x_i^T (Xz_{k,i} - y)/n \\ & \le \psi \\|d\\|_\infty \\|\theta^{(k)} - \theta^{(k-1)}\\|_1 + \mathcal R'(\theta^{(k)}; d) \\ & ~~~~~~ + d^T \nabla \mathcal L(\theta^{(k)}) + \sum_{i=1}^p \sum_{j=i+1}^p d_i (\theta_j^{(k-1)} - \theta_j^{(k)}) x_i^T x_j/n \\ & \le \mathcal F'(\theta^{(k)}; d) + \psi \\|d\\|_\infty \\|\theta^{(k)} - \theta^{(k-1)}\\|_1 + \xi \\|d\\|_\infty \sum_{i=1}^p \sum_{j=i+1}^p \|\theta_j^{(k-1)} - \theta_j^{(k)}\| \\ & \le \mathcal F'(\theta^{(k)};d) + (\psi + p \xi)\\|d\\|_\infty \\|\theta^{(k)} - \theta^{(k-1)}\\|_1 \\ & \le \mathcal F'(\theta^{(k)};d) + (\psi + p \xi)\sqrt{p} \\|d\\|_\infty \\|\theta^{(k)} - \theta^{(k-1)}\\|_2 \end{array}$$ Hence, $\mathcal F'(\theta^{(k)};d) \ge - (\psi + p \xi)\sqrt{p} \\|d\\|_\infty \\|\theta^{(k)} - \theta^{(k-1)}\\|_2$. When CD stops iteration, $\\|\theta^{(k)} - \theta^{(k-1)}\\|_2 \le \tau = \nu/((\psi + p \xi)\sqrt{p})$ and $\\|\theta^{(j)} - \theta^{(j-1)}\\|_2 \ge \tau$ for $j\le k-1$, which implies $\mathcal F'(\theta^{(k)};d) \ge -\nu$ for any $\\|d\\|_2=1$. Invoking Eqn. (\[eqn:59\]), we have $\tau^2 \le 2\mathcal F(\theta^{(0)})/(\psi(k-1))$. Thus, $k\le 2p (\psi + p\xi)^2 \mathcal F(\theta^{(0)})/(\psi \nu^2) +1 $. $\blacksquare$ [^1]: The elements are i.i.d. drawn from the standard Gaussian distribution $\mathcal N(0,1)$. [^2]: General position means any $n$ columns of $X$ are linear independent. The columns of $X$ are in general position with probability 1 if the elements of $X$ are i.i.d. drawn from some distribution, e.g., Gaussian.
--- abstract: 'In complex systems, the interplay between nonlinear and stochastic dynamics, e.g., J. Monod’s necessity and chance, gives rise to an evolutionary process in Darwinian sense, in terms of discrete jumps among attractors, with punctuated equilibrium, spontaneous random “mutations” and “adaptations”. On an evlutionary time scale it produces sustainable diversity among individuals in a homogeneous population rather than convergence as usually predicted by a deterministic dynamics. The emergent discrete states in such a system, i.e., attractors, have natural robustness against both internal and external perturbations. Phenotypic states of a biological cell, a mesoscopic nonlinear stochastic open biochemical system, could be understood through such a perspective.' author: - \| Hong Qian\ Department of Applied Mathematics\ University of Washington\ Seattle, WA 98195, U.S.A. bibliography: - 'spb.bib' title: 'Stochastic Physics, Complex Systems and Biology[^1]' --- = 6.1 in = 8.5 in = -0.5 in = -0.5 in = 0.29 in Biological systems and processes are complex. One of the hallmarks of complex behavior is uncertainties, either in the causes of an occurred event, or in predicting its future [@mackey_rmp; @ge_dill]. This “feel” of complexity is intimately related to the following issue [@jjh_94]: When a system consists of only a few degrees of freedom, say $x_1$, $x_2$ and $x_3$, a complete description of the “trajectory” of $(x_1,x_2,x_3)(t)$ for all $t$ consitutes a full understanding of the system. However, when a system has a million of degrees of freedom, $\vx(t)=\{x_i(t)\|1\le i\le 10^6\}$, a complete description of the $\vx(t)$ is not informative at all! One needs to find an particular “angle” to synthesize the large amount of data, or a “pattern” to obtain a summary. In classical physics of inanimated matters with relatively homogeneous individuals, this is accomplished by introducing the concept of [distribution]{} together with macroscopic thermodynamic quantities, giving rise to the discipline of statistical thermodynamics. In modern cellular biology, this is known as “data interpretation with respect to biological functions”: Usually a narrative in addition to the data is required [@knight]. The foregoing brief discussion points to a key departure from the classical physics of Newton and Laplace [@prigogine]: A rational choice of mathematical descriptions of biological systems and processes requires a probabilistic view of the dynamics, which provides both individual-based and distribution-based perspectives. Studying system dynamics in terms of stochastics, either due to intrinsic uncertainties, lack of full knowledge, or due to a need for organizing large amount of data, is the basis of what we call [stochastic physics]{}. What is Stochastic Physics {#what-is-stochastic-physics .unnumbered} ========================== Modern sciences emphasize quantitative representation of experimental observations, widely known as [mathematical modeling]{}. Along this line, there are two types of modeling: the [data-driven]{} and the [mechanism based]{} models. In the history of physics, Kepler’s model (laws) was the most celebrated example of the former, while Newton’s theory of universal gravity, which “explains” Kepler’s results, is the canonical example of a mechanism. In fact, the very term [mechanism]{} was derived from the word [mechanics]{}. In biology, Mendel’s model (law) was the former, and Hardy-Weinberg’s theory was the latter. The difference between the example in physics and the example in biology is that the latter has to take into account of uncertainties. Data-driven modeling incorporating uncertainties gives rise to the entire field of statistics - and bioinformatics and financial engineering are two most active branches of studies in recent years. This leads straight to the question “where is the mechanism based modeling with uncertainty”? Stochastic physics is precisely the answer to this calling. In sociology and economics, this type of modeling is called [agent-based]{}, and in finance it is called [behavior finance]{}. In applied mathematics, statistics is associted with [data-driven modeling]{} and stochastic process is associted with [population distribution based mechanistic modeling]{}. In physics, statistical physics has traditionally dealt with more on state of matters in equilibrium rather than dynamics of open, driven systems. Nevertheless, it is a shining example of successful stochastic modeling. Nonlinear Physics and Stochastic Physics {#nonlinear-physics-and-stochastic-physics .unnumbered} ======================================== Stochastic physics shares many of the concepts and concerns of the nonlinear physics that has gone before it: They both are focused on dynamics of a system [@haken_book]. Technically, for nonlinear systems exhibiting chaotic dynamics, a characterization based on distribution turns out to be more appropriate [@lasota]. Data analyses of chaotic signals also constantly employ methods from statistics [@abarbanel_rmp; @tong_book]. Stochastic dynamics in linear systems and nonlinear systems are fundamentally different [@qian_pccp; @qian_nonl]. The former can be essentially represented by a Gaussian process, which was extensively studied by eminent physicsts like Uhlenbeck, Chandrasekhar, and Onsager [@wax_book; @onsager_53; @fox]. But stochastic dynamics [per se]{} is not the reason for complex behavior. A Gaussian process has certain unpredictability, nevertheless the ultimate fate of the dynamics is all the same: It fluctuates around its mean value. However, when one faces a strongly nonlinear system with stochasticity, one has to talk about [evolution]{}, evolution process in Darwin’s sense with punctuated equilibrium and spontaneous random “mutations” and “adaptations”. This is one of the profound insights derived from the studies of nonlinear stochastic systems: The fluctuations in a nonlinear system with multiple attractors make rare events, something with infinitesimal probability from a determinsitic stand point, an sure occurance with probability one in an “evolutionary” time scale [@ge_qian_11; @qian_ge_mcb]. This picture fits J. Monod’s notion of chance and necessity [@monod_book; @haken_book]. Furthermore, when encountering external environmental changes, nonlinear multi-stable systems exhibit adaptation by enhanced rate of transition into the “favored attractors”; and ultimately exhibit “rupture” - the nonlinear catastrophe scenario in the presence of stochasticity [@shapiro_qian_bpc_97]. Newton-Laplace’s dynamics gives us a sense of convergence. For strongly nonlinear stochastic dynamics, the validity of the converging dynamics is only on a rather limited time scale. In an evolutionary time scale, divergent dynamics emerges. This, we believe, is a philosophical implication derived from stochastic physics [@prigogine]. Stochastic Physics and Quantitative Biology {#stochastic-physics-and-quantitative-biology .unnumbered} =========================================== Physics and computer science (CS) are two cornerstones of modern, engineering world. Therefore, it is not surprising that they support the most quantitative aspects of biology. Yet, upon a more careful reflection, one realizes that thinkings in both physics and CS are in odd with that of biologists: Physics considers systems that can be described with a few variables, known as “information poor” according to J.J. Hopfield [@jjh_94], and CS, while deals with much more complex problems, nevertheless in terms of perfect logics with almost infinite precision. Biological systems are information rich, and biological processes are not about percision or optimal, but rather about functional and survival. The studies of biological cells, the universal building block of living organisms, also have two foundations that echoed physics and CS: biochemistry and genomics. Biochemistry is founded on the tradition of physics, via the investigations of macromolecular structures and dynamics and biochemical reactions, while genomics heavily utilizes concepts and methods from CS, i.e. coding, information, discrete mathematics, leading to the emergence of bioinformatics in recent years. The heavy influences of physics and CS in biological thinking, in fact in all 21-century modern thinkings, is unmistakable. Nowadays, even the studies of biochemical reaction systems are usually about their information logic flow. Known as signal transduction, it provides a clear link between biochemistry within a cell, to perceived function. However, one often forgets that information is only an abstract term; its physical bases have to be either energy or material. In cell biology, they are represented by the structure and states of macromolecules. The information logic flow aspects of biochemical reaction is our “models” and “interpretations” of a biological organism based on our understanding of its engineering functions! It is a “narrative” cell biologists provide to understand a complex reality [@petermoore]. This reveals an important gap in the current dominant thinking of cell biology: the link between the physics of molecules, the chemistry of reactions, and the information logic flow they represents. It is widely recognized that investigations into this link require statistical physics and molecular thermodynamics in [small systems]{} with [dynamics]{} [@phillips_06; @bustamante_05; @qian_jbp_12]. Filling this gap has been called for as [the systems biology of cells]{} [@palsson_04]. Though yet to be proven, it is not difficult to see that the stochastic physics approach as described above has the potential to be a powerful, quantitative language of cellular dynamics and other biological systems [@qian_arbp_12]. The stochastic physics approach to biology relies more on mechanistic understanding of biological systems and processes than on high-throughput large data sets. It is a powerful tool to generate working hypotheses in a rigorous way. In current biological research, one often states that “we like to know how it works”. However, a scientifically more sound statement should be “we like to know whether it works in [this]{} way?”. This goes back to the hypothesis-driven research with strong inference [@beard_09]. Taking uncertainties into account, stochastic modeling is based on one’s mechanistic understanding, and relies on mathematical deduction to generate precise hypothesis. It will be an indispensible tool in biological research on par with data-driven bioinformatics. Cellular Biology and Theory of Evolution {#cellular-biology-and-theory-of-evolution .unnumbered} ======================================== Based on the Modern Synthesis of Darwin’s theory of evolution, the current population genetics and genomics [@koonin] attribute the molecular basis of biological variations to different DNA sequences, which is inheritable through Mendelian genetics and Watson-Crick base-pairing mechanism. Biochemistry, however, has been always considered as merely a deterministic mechanics that executes the instructions coded in the DNA [@alberts_98]. Recent laboratory measurements on [stochastic gene expression]{} in single cells with single-molecule sensitivity, however, has broken the genomic monoplay of biological variations [@siggia_science_02; @xie_nature_06]. Stochasticity has been increasingly recognized as a key aspect of cellular molecular biology. In terms of Darwin’s evolution, Kirschner and Gerhart have maintained that the essential role of cellular and organismal biology is to provide phenotypic variations with plausible molecular mechanisms that bridge genomes and lives [@K_and_G_book]. The tenants of stochasitc physics fit this perspective. In particular, the mathematical theory of stochastic processes has revealed a rich thermodynamic structure in any stochastic dynamics based on Markov formalism [@ge_qian_10]. The thermodynamic theory clearly distinguishes a closed stochastic system which reaches an equilibrium distribution with detailed balance, and an open, driven stochastic system which reaches a nonequilibrium steady state [@zqq; @gqq; @jqq_book; @bertalanffy]. It has been firmly established that the latter corresponds to precisely cellular biochemical systems upon which continuous chemical driving forces are applied. The conversion of chemical energy into heat in isothermal cellular systems can be characterized by entropy production rate [@qian_arpc_07; @wangjin_08]. The external energy supply, as the “environment condition” for an open system, is the thermodynamic necessity for self-organization and complex behavior [@qian_arpc_07]. Thermodynamics, however, can only tell what is possible and what is not; but it does not tell what is feasible and what is the mechanism. For the latter, detailed “molecular mechanisms” have to be developed. There is clearly a dichotomy between the nature vs. nurture for the function of a biochemical system. A stochastic description of dynamics provides a unique tool to understand the occurrence of sequential events, i.e., kinetics, in terms of the “most probable path” [@wang_mpp; @wang_pnas_11; @ge_qian_ijmpb]. There is a growing interest in understanding cell differentiation including stem cell differentiation and reprograming, isogenetic variations, and even cancer carcinogenesis from an evolutionary perspective at the cellular level [@weinberg; @aoping_07; @wang_pnas_11]. The mathematical theory of evolution and population genetics has long been based on stochastic processes [@ewens_book; @aoping_05; @aoping_08]. Therefore, the stochastic physics approach to cellular biochemical dynamics provides a natural unifying framework to further this exciting new frontier of biological science. A stochastic physics based quantitative understanding of cellular biology, in return, will provide a paradigm for studying other complex systems [@qian_nonl; @zqq; @qian_decomp]. [^1]: The 1st Gordon Research Conference on “Stochastic Physics in Biology”, chaired by K.A. Dill, was held on January 23-28, 2011, in Ventura, CA.
--- abstract: 'A Rabi dimer is used to model a recently reported circuit quantum electrodynamics system composed of two coupled transmission-line resonators with each coupled to one qubit. In this study, a phonon bath is adopted to mimic the multimode micromechanical resonators and is coupled to the qubits in the Rabi dimer. The dynamical behavior of the composite system is studied by the Dirac-Frenkel time-dependent variational principle combined with the multiple Davydov D$_{2}$ ansätze. Initially all the photons are pumped into the left resonator, and the two qubits are in the down state coupled with the phonon vacuum. In the strong qubit-photon coupling regime, the photon dynamics can be engineered by tuning the qubit-bath coupling strength $\alpha$ and photon delocalization is achieved by increasing $\alpha$. In the absence of dissipation, photons are localized in the initial resonator. Nevertheless, with moderate qubit-bath coupling, photons are delocalized with quasiequilibration of the photon population in two resonators at long times. In this case, high frequency bath modes are activated by interacting with depolarized qubits. For strong dissipation, photon delocalization is achieved via frequent photon-hopping within two resonators and the qubits are suppressed in their initial down state.' address: - '$^{1}$ School of Physics and Energy, Shenzhen University, Shenzhen 518060, China' - '$^{2}$ School of Materials Science and Engineering, Nanyang Technological University, Singapore 639798' - '$^3$ Department of Physics, Chongqing University, Chongqing 404100, China' - '$^4$ School of Science, Inner Mongolia University of Science and Technology, Baotou 041010, China' author: - 'Fulu Zheng$^{1, 2}$, Yuyu Zhang$^{3}$, Lu Wang$^{2,4}$, Yadong Wei$^{1}$ and Yang Zhao$^{2}$[^1]' title: Engineering Photon Delocalization in a Rabi Dimer with a Dissipative Bath --- Introduction ============ In circuit quantum electrodynamics (QED) systems, superconducting qubits are strongly coupled with microwave photons in resonators or transmission lines [@Wallraff2004; @Blais2004; @Schmidt2010; @Hwang2016; @Gu2017]. Since conceived in 2004 [@Wallraff2004; @Blais2004], circuit QED architectures have been designed and fabricated as research platforms in quantum computation [@Blais2004; @Niskanen2007; @Helmer2009; @Buluta2009; @Buluta2011; @Houck2012; @Georgescu2014; @Noh2016] and quantum information [@You2005; @Nielsen2010; @You2011; @Devoret2013]. Due to high flexibility and tunability, circuit QED devices offer the possibility to simulate light-matter interactions in quantum systems with an integrated circuit [@Wallraff2004; @Girvin; @Houck2012; @Schmidt2013; @Noh2016]. Experiments focus on engineering the coupling between a single resonator and a qubit for controllable single-resonator systems [@schuster; @hofheinz; @bishop]. A key challenge is to carry out quantum simulations of strongly correlated photons of coupled-resonator systems by controlling the inter-resonator photon coupling and device-environment interactions [@majer; @gerace]. It gives rise to an interesting phenomenon of photon self-trapping due to the competition between the qubit-photon coupling and the inter-resonator photon hopping, which has been realized in experiment using transmon qubits and two coupled transmission-line resonators [@Raftery2014]. Another QED system composed of two coupled nonlinear resonators has been fabricated for quantum amplification [@Eichler2014]. Described as a Bose-Hubbard dimer, this device can also be used for photon generation [@Leonski2004; @Miranowicz2006; @Liew2010; @Bamba2011; @Bamba2011_2]. Recent theoretical studies model the tunnel-coupled resonators each containing a qubit as a Jaynes-Cumming (JC) dimer, which is the smallest possible coupled-resonator system [@Hartmann2006; @ciuti; @Raftery2014; @Schmidt2010; @Hwang2016]. The JC Hamiltonian describes a QED system with weak qubit-photon coupling, which omits the counter-rotating-wave (CRW) interactions between the qubit and the photon mode [@Jaynes1963]. Experimental progress has made it possible to achieve ultra-strong coupling [@Wallraff2004; @Niemczyk2010; @Forn-Diaz2010; @Fedorov2010; @Yoshihara2017], where the qubit-photon coupling strength is comparable to the resonator frequency. In this regime, the JC model is invalid and the CRW terms play a crucial role in systems with strongly correlated photons [@Braak2011; @LeBoite2016; @LeBoite2017; @Wangxin2017; @Garziano2015; @Garziano2016; @Kockum2017]. The quantum Rabi model with the CRW interactions considered is expected to provided different physics [@Rabi1936; @Rabi1937]. Beyond the JC dimer, Hwang [et al.]{} studied the phase transition of photons in a Rabi dimer [@Hwang2016]. In experimental realizations, fabricated QED systems suffer from ineluctable dissipation stemming from device-environment interactions. The Markovian Lindblad master equation has been applied to describe the photon and qubit dynamics of the JC dimer with the photon decay and the qubit decoherence taken into consideration [@Schmidt2010; @Raftery2014]. It is found that dissipation can favor the photon localization in the initial resonator. Although the phase diagram for the photons in a Rabi dimer has been constructed, dissipation induced effects on the dynamics of the photons and the qubits in a Rabi dimer are still not well-understood [@Hwang2016]. Since the early application of the Markovian Lindblad master equation to describing the dissipative dynamics in the JC dimer [@Schmidt2010], increasing attention has been attracted to studying influences of dissipation on various QED systems [@Beaudoin2011; @Guo2011; @Nissen2012; @Coto2015; @LeBoite2016; @Casteels2017; @Nazir2016; @LeBoite2017]. Most of these studies are conducted by adopting master equations to capture the photon and the qubit dynamics of the QED systems, where environmental effects are considered in a phenomenological manner. However, the interplay between the QED devices and their surroundings is too complex to be modeled by a few dissipative parameters in the methods based on the Markovian Lindblad master equation [@Beaudoin2011; @Kaer2013; @Nazir2016]. In addition to being affected by bath induced dissipation, the operation of QED devices can benefit from interactions with their surroundings [@Hohenester2009; @Calic2011; @Roy2011; @Carmele2013; @Kaer2013; @Kaer2013_PRB; @Pirkkalainen2013; @Muller2015; @Nazir2016; @Liu2016; @Roy-Choudhury2017; @Gustin2017; @Hornecker2017]. For instance, Hohenester [et al.]{} observed phonon-assisted transition from quantum dot (QD) excitons to photons in nanocavity [@Hohenester2009]. Recently it has been found that exciton-phonon coupling favors single-photon generation in QD-nanocavity systems [@Muller2015]. Therefore, proper treatments of the system-bath interactions are needed. ![Schematic of the dissipative circuit QED system studied in this work. Photons can hop between two transmission line resonators with a tunneling rate $J$. In each resonator, a qubit interacts with the photon mode with a coupling strength $g$. A phonon bath is used to model the multimode micromechanical resonators and two qubits are coupled to the phonon bath with a strength $\alpha$. []{data-label="Fig1_sketch"}](sketch_1.jpg) Compared with the master equation approach which traces out the bath degrees of freedom (DOFs) in evaluating the reduced density matrices of the target system, the time-dependent variational principle (TDVP) with the Davydov ansätze can capture simultaneously the system and the bath dynamics [@Dirac1930; @Frenkel1934; @Zhao2000_1; @Zhao2000_2; @Zhao2012; @Sun2010; @Luo2010]. This approach has been recently extended to include multiple Davydov trial states [@Zhou2015; @Zhou2016; @Huang2016; @Chen2017; @Huang2017_2; @Wang2017], producing numerically exact results in a broad parameter regime. Applications have been made to simulate various dynamical processes, such as the dynamics of the Holstein polaron and the spin-boson model [@Sun2010; @Luo2010; @Zhou2015; @Zhou2016; @Huang2016; @Chen2017; @Huang2017_2; @Wang2017], energy transfer in photosynthetic systems [@Ye2012; @Huynh2013; @Sun2014; @Sun2015; @Chen2015; @Somoza2016; @Somoza2017], singlet fission dynamics [@Fujihashi2017; @Huang2017; @Fujihashi2017_2], the Landau-Zener transition [@Huang2018_LZ], and the qubit and photon dynamics in circuit QED system [@Fujihashi2017_2]. In addition to the dynamical behavior, the ground state of the Rabi model has been explored with the variational method [@Shore1973_ad2; @Chen1989_ad1; @Stolze1990_ad3; @Hwang2010_ad4]. The trial wave functions were constructed with the displaced oscillator state, the displaced squeezed state, and superpositions of those states. As a superposition of two displaced-squeezed states, a trial state was proposed by Hwang and Choi to capture the squeezing effect of the Rabi model in the ultrastrong coupling regime [@Hwang2010_ad4]. In this work, combining the TDVP with the multiple Davydov D$_{2}$ ansätze, we present a comprehensive study of bath induced effects on the dynamical behavior of a Rabi dimer. Instead of adding dissipation terms into the Lindblad master equation, we model the environmental influences by coupling the qubits to a phonon bath, producing indirect photon-bath interactions. Tuning the qubit-bath coupling strength, modulations on the photon dynamics by the phonon bath can be studied. In addition to the dynamics of the photons and the qubits, the temporal evolution of individual bath modes is depicted explicitly. The reminder of the paper is structured as follows. The Hamiltonian and the methodology applied in this work are described in Section \[sec:Section-II\], including an introduction to the multiple Davydov D$_{2}$ ansätze and the TDVP. Observables of interest and parameter configurations are also discussed in this section. Section \[sec:Section-III\] documents all the numerical results, covering photon dynamics, qubit polarization and phonon mode populations in various parameter regimes. Concluding remarks are drawn in Section \[sec:Section-IV\]. Model and Methodology \[sec:Section-II\] ======================================== Hamiltonian of the hybrid system -------------------------------- As illustrated in Fig. \[Fig1\_sketch\], we consider a dissipative circuit QED device composed of two coupled transmission line resonators with each interacting with a qubit. The device is modeled as a Rabi dimer and the environmental effects on the device are simulated by coupling the qubits to multimode micromechanical resonators. A phonon bath is adopted to describe the multimode micromechanical resonators. Therefore, the total Hamiltonian for the hybrid system contains three terms $$\label{eq:Htot} H=H_{\textrm{RD}}+H_\textrm{B}+H_{\textrm{BQ}}.$$ The Rabi dimer can be described by the following Hamiltonian ($\hbar=1$) $$\label{eq:HRD} H_{\textrm{RD}}=H_{\textrm{Rabi},\textrm{L}}+H_{\textrm{Rabi},\textrm{R}}-J(a_{\textrm{L}}^{\dagger}a_{\textrm{R}}+a_{\textrm{R}}^{\dagger}a_{\textrm{L}}),$$ where $J$ is the photon tunneling amplitude, and $H_{{\rm Rabi},i}$ ($i=\textrm{L},\textrm{R}$) are the left (L) and right (R) Rabi Hamiltonian, given by [@Rabi1936; @Rabi1937; @Braak2011; @Zhong2017] $$\label{Hrabi} H_{\textrm{Rabi},i=\textrm{L}/\textrm{R}} = \frac{\Delta_{i}}{2} \sigma_{z}^{i} + \omega_{i} a_{i}^{\dagger} a_{i} - g_{i} ( a_{i}^{\dagger} + a_{i} ) \sigma_{x}^{i}.$$ Here, $\Delta_{i}$ and $\omega_{i}$ are the energy spacing of the qubits and the frequency of the photon mode in the $i$th Rabi system, respectively. $\sigma_{x}^{i}$ and $\sigma_{z}^{i}$ are the usual Pauli matrices, and $a_{i}$ ($a_{i}^{\dagger}$) is the annihilation (creation) operator of the $i$th photon mode. $g_{i}$ characterizes the strength of coupling between the qubits and the photons. In this study, two Rabi sites are assumed to be identical, i.e., $\Delta_{\textrm{L}}=\Delta_{\textrm{R}}=\Delta$, $\omega_{\textrm{L}}=\omega_{\textrm{R}}=\omega_{0}$, and $g_{\textrm{L}}=g_{\textrm{R}}=g$. The environmental effects on the Rabi dimer are modeled by coupling two qubits to a common phonon bath $$\label{Hb} H_\textrm{B}=\sum_{k} \omega_{k} b_{k}^{\dagger} b_{k}$$ with an interaction Hamiltonian $$\label{Hbq} H_{\textrm{BQ}}=\sum_{k} \phi_{k} (b_{k}^{\dagger}+b_{k})(\sigma_{z}^{\textrm{L}}+\sigma_{z}^{\textrm{R}})$$ where $b_{k}$ ($b_{k}^{\dagger}$) is the annihilation (creation) operator of the $k$th bath mode with frequency $\omega_{k}$, and $\phi_{k}$ is the strength of coupling between the $k$th mode and the qubits. The qubit-bath coupling is characterized by the spectral function, $$J(\omega )=\sum_{k} \phi _{k}^{2} \delta( \omega - \omega _{k} )=2 \alpha \omega _{c}^{1-s} \omega ^{s} e^{-\omega/\omega_{c}},$$ with $\omega _{c}$ being the cut-off frequency and the dimensionless parameter $\alpha$ quantifying the qubit-bath coupling strength. In our calculations, the photon tunneling $J$ is much smaller than the frequencies of the qubits and the photon modes. As a result of the small $J$, the energy spectrum of the Rabi dimer contains energy levels with small energy gaps. Low frequency bath modes have to be taken into account in our calculations, as these modes may be at resonance with some transitions in the Rabi dimer. It has been demonstrated that the logarithmic discretization procedure is suitable to parameterize the low frequency bath modes with balanced numerical accuracy and efficiency [@WangLu2016]. Therefore, a Sub-Ohmic bath ($s=0.5$) is adopted in this work, and a logarithmic discretization method is used to obtain $\omega_{k}$ and $\phi_{k}$. The cut-off frequency for the bath modes is set to $\omega_{c}=\omega_{0}$, and the maximum frequency used in the discretization is $\omega_{\textrm{max}}=20~\omega_{c}$. To verify our choice of the Sub-Ohmic bath, we have performed test calculations with Ohmic and Super-Ohmic spectral densities. As shown in Fig. S1 in Supporting Information, it is found that the bath-induced effects on the photon dynamics are independent of the bath types. The multiple Davydov D$_2$ ansätze ---------------------------------- The multiple Davydov D$_2$ ansätze have been applied to study static and dynamic properties of various systems, producing excellent numerical efficiency and accuracy in a broad parameter regime [@Zhou2015; @Zhou2016; @Huang2016; @Chen2017; @Huang2017_2; @Wang2017; @Fujihashi2017; @Huang2017; @Fujihashi2017_2; @Huang2018_LZ]. In principle, the multiple Davydov D$_2$ ansätze can give numerically exact results with a sufficiently high multiplicity. In this study, both the off-diagonal qubit-photon coupling and the diagonal qubit-phonon coupling are included in the system Hamiltonian (\[eq:Htot\]). Therefore, the multiple Davydov D$_2$ ansätze are employed to probe the time evolution of the composite system $$\begin{aligned} \label{eq:MD2} \|{\rm D}_{2}^{M}(t)\rangle &=& \sum_{n=1}^{M} \Big[ A_{n} (t)\|\uparrow\uparrow\rangle + B_{n} (t) \|\uparrow\downarrow\rangle + C_{n} (t) \|\downarrow\uparrow\rangle \nonumber\\ &&~~~~+ D_{n} (t) \|\downarrow\downarrow\rangle \Big] \bigotimes \|\mu_{n}\rangle_{\textrm{L}}\|\nu_{n}\rangle_{\textrm{R}} \|\eta_{n}\rangle_{\textrm{B}},\end{aligned}$$ where $\|\uparrow \downarrow \rangle=\| \uparrow \rangle_{\textrm{L}} \otimes \| \downarrow \rangle_{\textrm{R}}$ with $\uparrow$ $(\downarrow)$ indicating the up (down) state of the qubits. $\|\mu_{n}\rangle_{\textrm{L}}$ and $\|\nu_{n}\rangle_{\textrm{R}}$ are coherent states of the photon modes $$\begin{aligned} \|\mu_{n}\rangle_{\textrm{L}} & = & \exp\left[\mu_{n} (t) a_{\textrm{L}}^{\dagger}-\mu_{n}^{\ast} (t) a_{\textrm{L}}\right]\|0\rangle_{\textrm{L}},\\ \|\nu_{n}\rangle_{\textrm{R}} & = & \exp\left[\nu_{n} (t) a_{\textrm{R}}^{\dagger}-\nu_{n}^{\ast} (t) a_{\textrm{R}}\right]\|0\rangle_{\textrm{R}},\end{aligned}$$ where $\|0\rangle_{\textrm{L}(\textrm{R})}$ is the vacuum state of the left (right) resonator. $\|\eta_{n}\rangle_{\textrm{B}}$ is the coherent state of the phonon bath $$\label{eta} \|\eta_{n}\rangle_{\textrm{B}} = \exp \left[ \sum_{k}\eta_{nk} (t) b_{k}^{\dagger}-\eta_{nk}^{\ast} (t) b_{k} \right] \|0\rangle_{\textrm{B}}$$ with $\|0\rangle_{\textrm{B}}$ being the vacuum state of the bath. In Eq. (\[eq:MD2\]), $A_{n}(t)$, $B_{n}(t)$, $C_{n}(t)$, $D_{n}(t)$, $\mu_{n}(t)$, $\nu_{n}(t)$, and $\eta_{nk}(t)$ are time-dependent variational parameters to be determined via the TDVP. The physical significance of these variational parameters is clear. For instance, $A_{n}$ is the probability amplitude in the state $\|\uparrow\uparrow\rangle\|\mu_{n}\rangle_{\textrm{L}}\|\nu_{n}\rangle_{\textrm{R}}\|\eta_{n}\rangle_{\textrm{B}}$, $\mu_{n}$ ($\nu_{n}$) is the displacement of the left (right) photon mode, and $\eta_{nk}$ is the displacement of the $k$th bath mode. The time-dependent variational principle ---------------------------------------- The dynamics of Hamiltonian (\[eq:Htot\]) is obtained from the Dirac-Frenkel time-dependent variational principle. The equations of motion for all the variational parameters can be derived from $$\label{DiracFrenkel} \frac{d}{dt} \bigg( \frac{\partial L}{ \partial \dot{\alpha}^{}_{n}} \bigg) - \frac{\partial L}{ \partial \alpha^{}_{n}} =0.$$ Here, $\alpha_{n}$ are the variational parameters, i.e., $A_{n}(t)$, $B_{n}(t)$, $C_{n}(t)$, $D_{n}(t)$, $\mu_{n}(t)$, $\nu_{n}(t)$, and $\eta_{nk}(t)$ in this work. The Lagrangian $L$ is written as $$\label{Lagrangian} L = \frac{i}{2} \langle {\rm D}_{2}^{M}(t) \| \frac{\overrightarrow{\partial}}{\partial t} - \frac{\overleftarrow{\partial}}{\partial t} \| {\rm D}_{2}^{M}(t) \rangle - \langle {\rm D}_{2}^{M}(t) \| H \| {\rm D}_{2}^{M}(t) \rangle.$$ Observables ----------- Combining the TDVP with the multiple Davydov D$_2$ ansätze, we are capable of investigating the bath induced dynamics of a Rabi dimer with specific contribution from individual bath modes presented explicitly. In order to study the localization/delocalization of the photons, we calculate the photon numbers in two resonators from expressions $$\begin{aligned} N_{\textrm{L}}(t) & = & \langle{\rm D}_{2}^{M}(t)\| a_{\textrm{L}}^{\dagger} a_{\textrm{L}} \|{\rm D}_{2}^{M}(t) \rangle \nonumber\\ & = & \sum_{l,n}^{M} \Big[ A_{l}^{\ast}(t) A_{n}(t) + B_{l}^{\ast}(t) B_{n}(t) + C_{l}^{\ast}(t) C_{n}(t) \nonumber\\ &&~~~~~~~ + D_{l}^{\ast}(t) D_{n}(t) \Big] \mu_{l}^{\ast}(t) \mu_{n}(t) S_{ln}(t) , \\ N_{\textrm{R}}(t) & = & \langle{\rm D}_{2}^{M}(t)\|a_{\textrm{R}}^{\dagger}a_{\textrm{R}}\|{\rm D}_{2}^{M}(t) \rangle \nonumber \\ & = & \sum_{l,n}^{M} \Big[ A_{l}^{\ast}(t) A_{n}(t) + B_{l}^{\ast}(t) B_{n}(t) + C_{l}^{\ast}(t) C_{n}(t) \nonumber\\ &&~~~~~~~ + D_{l}^{\ast}(t) D_{n}(t) \Big] \nu_{l}^{\ast}(t) \nu_{n}(t) S_{ln}(t) ,\end{aligned}$$ where $S_{ln} (t)$ is the Debye-Weller factor $$\begin{aligned} S_{ln} & = & \exp\left[\mu_{l}^{\ast}(t) \mu_{n}(t)-\frac{1}{2} \|\mu_{l}(t)\|^2-\frac{1}{2}\|\mu_{n}(t)\|^2\right] \cdot \nonumber\\ && \exp\left[ \nu_{l}^{\ast}(t) \nu_{n}(t) - \frac{1}{2} \|\nu_{l}(t)\|^2 - \frac{1}{2} \|\nu_{n}(t)\|^2 \right] \cdot \nonumber\\ &&\exp \left[ \sum_{k} \Big( \eta_{lk}^{\ast}(t) \eta_{nk}(t) - \frac{1}{2} \|\eta_{lk}(t)\|^2 - \frac{1}{2} \|\eta_{nk}(t)\|^2 \Big) \right]. \nonumber\\\end{aligned}$$ The photon imbalance is then calculated as $Z(t)=N_{\textrm{L}}(t)-N_{\textrm{R}}(t)$ and the normalized photon imbalance is $\tilde{Z}(t)=Z(t)/\big(N_{\textrm{L}}(t)+N_{\textrm{R}}(t)\big)$. These quantities are used to characterize the localization and delocalization of the photons. In addition to photon dynamics, the time evolution of the qubit states is recorded during the simulations by measuring the qubit polarization via $$\begin{aligned} \langle\sigma_{z}^{\textrm{L}}(t)\rangle & = & \langle{\rm D}_{2}^{M}(t)\|\sigma_{z}^{\textrm{L}}\|{\rm D}_{2}^{M}(t) \rangle \nonumber \\ & = & \sum_{l,n}^{M} \Big[ A_{l}^{\ast}(t)A_{n}(t) + B_{l}^{\ast}(t)B_{n}(t) \nonumber\\ &&- C_{l}^{\ast}(t)C_{n}(t) - D_{l}^{\ast}(t)D_{n}(t) \Big] S_{ln}(t),\\ \langle\sigma_{z}^{\textrm{R}}(t)\rangle & = & \langle{\rm D}_{2}^{M}(t)\|\sigma_{z}^{\textrm{R}}\|{\rm D}_{2}^{M}(t) \rangle \nonumber \\ & = & \sum_{l,n}^{M} \Big[ A_{l}^{\ast}(t)A_{n}(t) - B_{l}^{\ast}(t)B_{n}(t) \nonumber\\ &&+ C_{l}^{\ast}(t)C_{n}(t) - D_{l}^{\ast}(t)D_{n}(t) \Big] S_{ln}(t).\end{aligned}$$ As given in Hamiltonian (\[eq:Htot\]) and illustrated in Fig. \[Fig1\_sketch\], the qubits serve as the bridge to connect the photon and the phonon modes, transmitting the bath induced impacts to the photons. Combining influences from the photons and the bath, our calculated qubit dynamics reflects the complex interactions between the photon modes and the phonon bath. Thanks to the methodology adopted here, the temporal evolution of the phonon bath can also be obtained explicitly. To reveal the participation of individual phonon modes in the Rabi dimer dynamics, we calculate the population on the $k$th mode as follows $$\begin{aligned} N_{k}^{\textrm{B}}(t) &=& \langle{\rm D}_{2}^{M}(t)\| b_{k}^{\dagger} b_{k}\|{\rm D}_{2}^{M}(t) \rangle \nonumber \\ &=& \sum_{l,n}^{M} \Big[ A_{l}^{\ast}(t) A_{n}(t) + B_{l}^{\ast}(t) B_{n}(t) + C_{l}^{\ast}(t) C_{n}(t) \nonumber\\ &&~~~~~~~~+ D_{l}^{\ast}(t) D_{n}(t) \Big] \eta_{lk}^{\ast}(t) \eta_{nk}(t) S_{ln}(t).\end{aligned}$$ Through interacting with the qubits, the phonon bath gradually gains sufficient energy from the Rabi dimer to affect the dynamics of the photons and the qubits. In return, the influences of the QED system on the bath modes can be investigated by calculating the populations dynamics $N_{k}^{\textrm{B}}(t)$. ![Time evolution of the photon imbalance $Z (t)$ with the qubits coupled to the phonon bath with three coupling strengths $\alpha=0, 0.01 \omega_{0}, 0.1 \omega_{0}$. The qubit-photon coupling strength is $g=0.01~\omega_{0}$ and the inter-resonator photon hopping rate is $J=0.01~\omega_{0}$. Sixty phonon modes are considered in the calculations.[]{data-label="Fig_2_ZandNLR_g001"}](Fig2_PI_J001g001_M8N60.jpg) Parameter configurations and initial conditions ----------------------------------------------- In this study, we discuss the case in which the qubits and the photons are at resonance ($\Delta /\omega _{0}=1$). In contrast to the JC dimer, the photon phase diagram in a Rabi dimer is more complicated, and is determined by the competitive effects of $J$ and $g$. For a bare Rabi dimer, a critical photon tunneling rate $J_{c}\approx 0.03~\omega_{0}$ has been proposed [@Hwang2016], above which the photons are always delocalized regardless of the qubit-photon coupling strength $g$. If $J<J_{c}$, the photon dynamics in a Rabi dimer is found to undergo double phase transitions as the qubit-photon coupling strength $g$ increases [@Hwang2016]. The first transition is from a delocalized phase to a localized phase, then the second one takes place from the localized phase to a new delocalized phase. Photons hop between two resonators in the first delocalized phase, while photons are quasi-equilibrated over two resonators in the second delocalized phase. In order to elucidate the phonon-bath induced effects on the photon dynamics in the Rabi dimer, we choose three combinations of $J$ and $g$, and perform simulations for each parameter configuration with varying qubit-bath coupling strength $\alpha$. In Case , both $J$ and $g$ are assigned with a value of $0.01~\omega_{0}$, yielding photon delocalization over two resonators in a bare Rabi dimer [@Hwang2016]. In Case , the photon tunneling amplitude $J$ is set to $0.02~\omega_{0}$ and the qubit-photon coupling strength $g=0.3~\omega_{0}$. The photons in a Rabi dimer are found to be trapped in the initial resonator in this parameter scheme [@Hwang2016]. In Case , we parameterize the system with $J=0.05~\omega_{0}$ and $g=0.3~\omega_{0}$. As $J > J_{c}$ in this situation, a photon delocalization phase is found for a Rabi dimer [@Hwang2016]. As adopted in common experiments [@Raftery2014], the photon tunneling rate $J$ used in our work is relatively weak. Therefore, the quadrature-quadrature coupling between the two photon modes [@Rossatto2016; @WangYM2016] is neglected in our model, as shown in Eq. (\[eq:HRD\]). Initially, a fully localized photon state is prepared by pumping $N(0)=20$ photons into the left resonator while keeping the right one in a photon vacuum. Experimently it is nontrivial to prepare such an initial photon state. Raftery and coworkers have accomplished such a state through three steps [@Raftery2014]. First, the qubits are detuned by fast flux pulses, turning off the photon-photon interactions. Then an initialization pulse is used to populate the linear resonator modes. Finally, after a variable time delay, the qubits are biased into resonance and the photon-photon interaction is turned on. Along these procedures, one can prepare an initial state with any desired photon imbalance [@Raftery2014]. In contrast to the photons, the qubits in the two resonators start to evolve from their down states and the phonon bath is initially in a vacuum state. Our calculations are numerically robust regarding the model parameters and it has been verified by preliminary calculations where small deviations were added to the parameters \[see Fig. S2 in Supporting Information\]. Although the photon behavior is dependent on the initial photon number, the bath-induced effects on the photon dynamics can be captured by our approach for the case with smaller initial photon number \[see Fig. S3 in Supporting Information for the calculations with $N(0)=5$\]. The numerical convergence also has been carefully checked in preliminary calculations with the sample results shown in Fig. S3 in Supporting Information. Results and discussion {#sec:Section-III} ====================== With the qubits coupled to a phonon bath, the dynamics of a Rabi dimer will be modulated by the phonon bath in various parameter regimes. Dynamics results calculated by our TDVP approach are presented in this section. In addition to the photon propagation and the qubit polarization, dynamics of individual phonon modes is also explicitly examined. Case : Weak qubit-photon coupling with weak photon tunneling {#SecIIIA} ------------------------------------------------------------ ![Effects of the qubit-bath coupling on the qubit polarization with $g=0.01~\omega_{0}$ and $J=0.01~\omega_{0}$. Sixty phonon modes are considered in the calculations.[]{data-label="Fig5_sigmaz_g001"}](Fig3_sz_J001g001_M8N60.jpg) If the qubit-photon coupling is weak, the photon dynamics is mainly determined by the photon tunneling rate. With a $J$ comparable to $g$, the photons hop between two resonators, yielding photon delocalization [@Hwang2016]. Aiming to study the bath effects on the delocalized photon phase in a Rabi dimer, we first investigate the system dynamics with a weak inter-resonator photon tunneling rate $J=0.01~\omega_{0}$ and a weak qubit-photon coupling strength $g=0.01~\omega_{0}$. The photon imbalance $Z (t)$ is shown in Fig. \[Fig\_2\_ZandNLR\_g001\] for three qubit-bath coupling strengths ($\alpha=0, 0.01, 0.1$). The photons of a bare Rabi dimer ($\alpha=0$) are delocalized over two resonators, producing Josephson oscillations of $Z (t)$ [@Hwang2016; @Guo2011; @Larson2011]. If the qubits are weakly coupled to the bath ($\alpha\neq0$), there are no remarkable bath-induced influences on the photon dynamics, as the photons cannot “feel" the indirect impact from the phonon bath with weak qubit-photon coupling. The photon dynamics is mainly determined by the photon tunneling rate $J$, yielding a Josephson oscillation period of $T_{J}=2\pi/2J$. ![Population dynamics of the bath modes with the qubit-bath coupling $\alpha$ being 0.01 (a), and 0.1 (b). The qubit-photon coupling strength is $g=0.01~\omega_{0}$ and the inter-resonator photon hopping rate is $J=0.01~\omega_{0}$. Sixty phonon modes are considered in the calculations.[]{data-label="Fig8_Popbath_g001"}](Fig4_Popbath_J001g001_M8N60.jpg) In addition to the photon number, the qubit state is another often measured quantity [@Schmidt2010]. Compared with the photon dynamics, which is almost unaffected by indirect coupling to the phonon bath, the populations of the qubit states are dramatically modulated by the phonon bath, as the qubits are directly coupled to the bath. The time evolution of the qubit polarization is depicted in Fig. \[Fig5\_sigmaz\_g001\] for qubit-bath coupling strengths $\alpha=0$, $0.01$ and $0.1$. In a bare Rabi dimer, two qubits can be excited from the down to the up state via the interaction with the delocalized photons. If all photons are transferred to one resonator, qubit flipping is accelerated in this resonator and is slowed down in the other. Similar phenomenon has been observed in a JC dimer in a semiclassical approximation [@Schmidt2010]. As can be seen from the Rabi Hamiltonian (\[Hrabi\]), a considerable amount of photons are required to flip the qubits if the qubit-photon coupling is weak. Once the qubit-bath coupling is switched on, oscillations of qubit polarization are greatly suppressed and the two qubits are constrained in the down state, which is similar to the frozen spin in a two-level dissipative system [@Zhang2010]. The stronger the qubit-bath coupling is, the more severe the confinement is. This phenomenon can be explained by the quantum Zeno effect. Conventionally the quantum Zeno effect refers to the suppression of quantum evolution by frequent measurements [@Koshino2005; @Harrington2017]. A recent scheme treats the system-environment interactions as “quasi-measurements" [@Ai2013] which also can lead to quantum Zeno effect [@Harrington2017]. In our model, the qubit-bath coupling works as the quasi-measurements of the state of the qubits, hindering qubit flipping. The frozen qubits weaken the effective qubit-photon interaction, contributing to decoupling of the photons from the qubits. ![(a) Specific coupling strength $\phi_{k}$ between the $k$th phonon mode and the qubits. (b) Population of individual bath modes at $t=0.1$. As illustrated by the blue vertical dashed line, stronger coupling between the bath mode and the qubits leads to higher population of the corresponding mode in the short time region. The qubit-photon coupling strength is $g=0.01~\omega_{0}$ and the inter-resonator photon hopping rate is $J=0.01~\omega_{0}$. Sixty phonon modes are considered in the calculations.[]{data-label="Fig_5_PhikPopBath_g001"}](Fig5_Phik_Popbath_t01_J001g001_M8N60.jpg) In contrast to the density matrix methods that trace out the DOFs of the bath, the TDVP with the Davydov ansätze can treat the DOFs of the qubits, the photons and the bath modes explicitly, making it possible to study the time evolution of each bath mode. In order to investigate the participation of individual bath modes in the Rabi dimer dynamics, we calculate the population of the bath modes, and the results with $g=0.01~\omega_{0}$ and $J=0.01~\omega_{0}$ are shown in Fig. \[Fig8\_Popbath\_g001\]. The shape of the population distribution over all modes is independent of the qubit-bath coupling, while the population magnitude increases with $\alpha$. For short times, the bath mode population is associated with the strength of coupling between specific bath mode and the qubits. As shown in Fig. \[Fig\_5\_PhikPopBath\_g001\], the phonon modes with frequencies near $\omega_{k} = 1.5~\omega_{0}$ have relatively strong coupling to the qubits, giving rise to high populations of these modes at short times. Within this short time interval, there is energy flowing into the phonon bath. As the time evolves, it is difficult to transfer more energy to the bath as the qubits cannot flip. It follows that the energy within the bath redistributes by gradually populating the low-frequency modes. ![ (a) Effects of the qubit-bath coupling $\alpha$ on the time evolution of the photon imbalance $Z$. (b)-(d) Time evolution of the photon numbers in the left ($N_{\textrm{L}}$) and right ($N_{\textrm{R}}$) resonators with different qubit-bath coupling $\alpha$. The qubit-photon coupling strength is $g=0.3~\omega_{0}$ and the inter-resonator photon hopping rate is $J=0.02~\omega_{0}$. Sixty phonon modes are considered in the calculations.[]{data-label="Fig_3_ZandNLR_g02"}](Fig6_PN_J002g03_M4N60_2.jpg) Case : Strong qubit-photon coupling with weak photon tunneling -------------------------------------------------------------- By increasing the qubit-photon coupling, Hwang and coworkers have shown that the photons undergo a transition from a delocalized phase to a localized one if the photon tunneling amplitude is weak [@Hwang2016]. Configuring the Rabi dimer with $J=0.02 ~\omega_{0}$ and $g=0.3~\omega_{0}$, we study the bath induced effects on the photon dynamics and the results are presented in Fig. \[Fig\_3\_ZandNLR\_g02\]. In this parameter regime, the photons are localized in the left resonator of a bare Rabi dimer ($\alpha=0$), a result of the qubit-photon interactions. With non-negligible $g$, the photon dynamics is dominated by the nonlinearity of the Rabi dimer spectrum, which prevents the photons from hopping to the right resonator. This phenomenon is known as the photon-blockade effect [@Imamoglu1997; @Birnbaum2006; @Lang2011; @Ridolfo2012; @Hwang2016; @LeBoite2016]. Once the dissipation is taken into account by switching on the qubit-bath coupling $\alpha$, the photons are no longer localized in the initial resonator. For a qubit-bath coupling strength of $\alpha=0.07$, the photons are found to escape from the left resonator through two channels. On one hand, the photons tunnel to the right resonator at short times. On the other hand, some photons are dissipated due to the coupling to the phonon bath via the qubits, producing a gradually decreasing total photon number. At long times, only a small portion of photons are conserved and delocalized in two resonators. The photon delocalization is characterized by quasiequilibration of the photon population with two resonators having almost the same number of photons. Therefore, the photon imbalance oscillates around zero with decreasing amplitudes as time evolves. With stronger coupling, e.g., $\alpha=0.16$, there is almost no loss of the total photon number, since the indirect photon-bath coupling is weakened by the qubits frozen in the down state. Surprisingly, the photons are delocalized via frequent hopping within two resonators, which is similar to that for lower values of $J$ and $g$, e.g., $J=0.01~\omega_{0}$ and $g=0.01~\omega_{0}$ \[see Fig. \[Fig\_2\_ZandNLR\_g001\]\]. Comparing the photon localization for $\alpha=0$ and the photon delocalization for $\alpha=0.16$, we find that the photon confinement due to the qubit-photon coupling can be eliminated by strong qubit-bath coupling. From this set of calculations, we can clearly see that the dissipative bath induces two different forms of photon delocalization in the strong coupling regime, which is controlled by tuning the qubit-bath coupling strength $\alpha$. For instance, a moderate $\alpha$ can be applied to achieve the distinguished photon delocalization with quasiequilibration of the photons, which is usually observed in the deep-strong coupling regime without dissipation [@Hwang2016]. If the qubit-bath coupling is strong, the total photon number is conserved and the photons are delocalized over two resonators via hopping. ![Effects of the qubit-bath coupling on the qubit polarizations with $g=0.3~\omega_{0}$ and $J=0.02~\omega_{0}$. Sixty phonon modes are considered in the calculations.[]{data-label="Fig6_sigmaz_g02"}](Fig7_sz_J002g03_M4N60.jpg) In order to understand how the phonon bath impacts the photon dynamics, we study the qubit dynamics by calculating the qubit polarization. Results are collected in Fig. \[Fig6\_sigmaz\_g02\]. For a Rabi dimer without the phonon bath, the photons are trapped in the left resonator and help flip the left qubit frequently, as shown in Fig. \[Fig6\_sigmaz\_g02\] (a). In contrast to the left qubit, the polarization of the right qubit oscillates around the value of $-\frac{1}{2}$, indicating that the right qubit tends to reside in its down state throughout. If the qubit-bath coupling is moderate ($\alpha=0.07$), the right qubit stays in the down state at short times ($tJ<0.6$). In this regime, the flipping of the left qubit is quenched, promoting the photon delocalization \[see Fig. \[Fig\_3\_ZandNLR\_g02\] (c)\]. Stimulated by the photons flowing to the right resonator, the right qubit gradually acquires sufficient energy to flip between its down and up states. In the time interval of $0.6< tJ < 2.0$, two qubits frequently flip, and a bridge between the photons and the phonon bath is established, producing a rapid decay of the total photon number and a sharp increase of the bath mode population \[see Fig. \[Fig9\_Popbath\_g02\] (a)\]. At long times ($tJ>2.0$), two qubits are depolarized with $\sigma_z^{\textrm{L}(\textrm{R})}\sim0$, which is useful to characterize the photon delocalization with quasiequilibration of photons in two resonators [@Hwang2016]. ![Population dynamics of the bath modes with the qubit-bath coupling $\alpha$ being 0.07 (a), and 0.16 (b). The qubit-photon coupling strength is $g=0.3~\omega_{0}$ and the inter-resonator photon hopping rate is $J=0.02~\omega_{0}$. Sixty phonon modes are considered in the calculations.[]{data-label="Fig9_Popbath_g02"}](Fig8_Popbath_J002g03_M4N60.jpg) If the qubit-bath coupling is stronger, e.g., $\alpha=0.16$, two qubits stay in the down states most of the time. This phenomenon originates from the fact that the diagonal qubit-bath coupling considerably increases the effective bias of the qubits and removes the qubit-photon resonance, preventing qubit flipping. Through the off-diagonal qubit-photon coupling \[see Eq. (\[Hrabi\])\], localized photons stimulate qubit flipping, which can help trap the photons in return. As the two qubits are confined in the down state, the effects of the qubit-photon interactions on the photon dynamics can be eliminated. It seems that the photon modes are decoupled from the subsystem composed of the qubits and the phonon bath. Therefore, the photon hopping between the two resonators is mainly due to the photon tunneling. Bridging the photons and the bath modes, the qubit can tune the indirect coupling between the photons and the bath via qubit flipping. If the qubit is localized in one state, bath induced effects cannot be experienced by the photons. Hence, there is no detectable decay of the total photon number in this scenario, as is shown in Fig. \[Fig\_3\_ZandNLR\_g02\] (d). At short times, the left qubit flips, while the right qubit stays in the down state \[see Fig. \[Fig6\_sigmaz\_g02\] (c)\]. In this time interval, the photons are completely localized in the left resonator while helping flip the left qubit. In contrast, the phonon bath is initiated from a vacuum state and cannot confine the left qubit in its down state within a short time. In order to elucidate contributions of individual bath modes to the manipulation of the photon and qubit dynamics in the Rabi dimer, we calculate the bath mode populations, and the results are illustrated in Fig. \[Fig9\_Popbath\_g02\]. In the moderate qubit-bath coupling case ($\alpha=0.07$), several low frequency bath modes get excited with small populations in short times ($tJ<0.6$). In the interval of $0.6<tJ<2.0$, the modes with frequency around $\omega_{k}=2.0~\omega_{0}$ are promptly populated. Within this period, two qubits can flip, effectively turning on the indirect photon-bath coupling and leading to a rapid reduction of the total photon number. Increasing population in the phonon modes with $\omega_{k}\sim2.0~\omega_{0}$ is contributed from the energy transferred by the two qubits from the photons to the phonon modes. Similar phenomenon has been reported recently by Garziano and coworkers in a model where a photon mode is coupled to two separate atoms [@Garziano2015; @Garziano2016]. When the photon frequency is twice the atomic transition frequency, they found that the two atoms can be simultaneously excited by the photon, and this process is reversible [@Garziano2015; @Garziano2016]. In our study, the phonon modes with $\omega_{k}\sim2.0~\omega_{0}$ are populated through coupling to two qubits, analogous to the reversible process demonstrated in Ref. [@Garziano2016], i.e., two atoms can jointly emit a single photon during downward transitions from their excited states. With the passing of time, more bath modes are excited with frequencies ranging from $0.05~\omega_{0}$ to $2.0~\omega_{0}$. If the qubit-bath coupling is strong, i.e., $\alpha=0.16$, the bath-induced effects are too weak to impact the photons as the qubits are confined in the down state. Therefore, the phonon dynamics is similar to the weak qubit-photon coupling case (see Fig. \[Fig8\_Popbath\_g001\]), where the population gradually flows to the low frequency modes. From aforementioned results and discussions, it is found that the photon dynamics in a Rabi dimer can be controlled by manipulating the qubit state. Schmidt and coworkers demonstrated that the photon phase transition can be detected by measuring the qubit state [@Schmidt2010]. Recently, Baust [et al.]{} have achieved tunable coupling between two resonators by controlling the state of the qubit that connects the resonators [@Baust2015]. In their experiment, the qubit population is controlled by a microwave drive [@Baust2015]. Nori and coworkers have proposed a multioutput single-photon device by coupling two resonators to a qubit [@Wang2016]. They have also studied the phonon blockade in nanomechanical resonators coupled to a qubit [@Liu2010; @Wang2016_2]. Some optomechanical systems have been conceived with the photon mode, the mechanical resonator, and the two-level system coupled together [@Restrepo2017; @Restrepo2014; @Holz2015; @Akram2015; @ZhouBY2016]. Pirkkalainen and coworkers have fabricated a QED device coupled with phonons, in which a superconducting transmon qubit is coupled to a microwave cavity and a micromechanical resonator [@Pirkkalainen2013; @Pirkkalainen2015]. The qubit-phonon coupling strength also can be measured and regulated [@Pirkkalainen2013; @Xiang2013; @Xiong2015; @Pirkkalainen2015; @Hartke2017; @Rouxinol2016]. In our study, the qubits in a Rabi dimer are coupled to a multimode phonon bath. By tuning the qubit-bath coupling, the qubit state can impact the photon dynamics via the qubit-photon interaction. Thanks to the advances in nanofabrication, multimode micromechanical resonators can be manufactured to serve as the phonon bath [@Massel2012; @Mian2012]. The hybrid QED system proposed in this work is experimentally feasible. Such devices can be fabricated not only to control the photon dynamics in the QED systems as demonstrated here, but also to serve as a platform for fundamental studies of quantum physics, such as the photon-qubit-phonon interactions in QED systems. ![ (a) Effects of the qubit-bath coupling $\alpha$ on the time evolution of the photon imbalance $Z$. (b)-(d) Time evolution of the photon numbers in the left ($N_{\textrm{L}}$) and right ($N_{\textrm{R}}$) resonators with different qubit-bath coupling $\alpha$. The qubit-photon coupling strength is $g=0.3~\omega_{0}$ and the inter-resonator photon hopping rate is $J=0.05~\omega_{0}$. Sixty phonon modes are considered in the calculations.[]{data-label="Fig_9_ZandNLR_J005"}](Fig9_PI_J005g03_longtime.jpg) ![Effects of the qubit-bath coupling on the qubit polarizations with $g=0.3~\omega_{0}$ and $J=0.05~\omega_{0}$. Sixty phonon modes are considered in the calculations.[]{data-label="Fig10_sigmaz_J005"}](Fig10_sz_J005g03_longtime.jpg) Case : Strong qubit-photon coupling with moderate photon tunneling ------------------------------------------------------------------ ![Population dynamics of the bath modes with the qubit-bath coupling $\alpha$ being 0.1 (a), and 0.3 (b). The qubit-photon coupling strength is $g=0.3~\omega_{0}$ and the inter-resonator photon hopping rate is $J=0.05~\omega_{0}$. Sixty phonon modes are considered in the calculations.[]{data-label="Fig11_Popbath_J005"}](Fig11_Popbath_J005g03_longtime.jpg) It has been reported that the photons are delocalized in the parameter configuration of $J=0.05 ~\omega_{0}$ and $g=0.3~\omega_{0}$ [@Hwang2016]. The bath effects on the photon and qubit dynamics in this configuration are presented here. As shown in Fig. \[Fig\_9\_ZandNLR\_J005\] (a), the photon imbalance for a bare Rabi dimer oscillates around zero with decreasing amplitudes, indicating photon delocalization with quasiequilibration of photons in two resonators at long times. Coupling the qubits to the phonon bath with a strength of $\alpha=0.1$ accelerates the decay of the total photon number, although the photon imbalance is similar to that of the bare Rabi dimer. In this parameter configuration, the two qubits can freely flip \[see Fig. \[Fig10\_sigmaz\_J005\] (b)\] and the bath induced dissipation is effective on the photons, leading to continuous decrease of the total photon number. Many high frequency modes are activated by interacting with the Rabi dimer \[see Fig. \[Fig11\_Popbath\_J005\] (a)\]. Increasing the qubit-bath coupling strength $\alpha$ to 0.3, the photons are still delocalized over the two resonators and the decay of the total photon number is greatly decelerated, as can be seen in Fig. \[Fig\_9\_ZandNLR\_J005\] (d). In this case, two qubits remain in the down states \[see Fig. \[Fig10\_sigmaz\_J005\] (c)\] and it is difficult to build indirect photon-phonon coupling. In contrast to the case of $\alpha=0.1$ where many high-frequency phonon modes are excited, the phonon population gradually flows to the low frequency modes with $\alpha=0.3$, as shown in Fig. \[Fig11\_Popbath\_J005\] (b). Concluding remarks {#sec:Section-IV} ================== Equipped with the multiple Davydov D$_2$ ansätze, the Dirac-Frenkel time-dependent variational principle is applied to investigate the bath induced effects on the dynamics of a Rabi dimer by coupling the qubits to a common phonon bath. The dynamics of the photons, the qubits and the bath modes are studied in various parameter regimes. Through extensive calculations, it is found that the photon dynamics in a Rabi dimer can be controlled by manipulating the qubit states via tuning the qubit-bath coupling. For weak qubit-photon coupling, the photon dynamics is almost unaffected by the phonon bath, and the photons are delocalized by hopping between two resonators. However, the qubits are frozen in the initial down state due to their strong coupling to the bath. In the strong coupling regime with weak photon tunneling, the photons are localized in the initial resonator in the absence of the dissipation. With inclusion of the environmental influences, the total photon number is reduced, and photons are delocalized in two resonators. At long times, the two resonators have almost the same number of photons, accompanied by the depolarization of the qubits, $\sigma_z^{\textrm{L}(\textrm{R})}\sim0$. For a large dissipation strength $\alpha$, the indirect photon-bath coupling is suppressed by the qubits frozen in their down state. Consequently, the photons can freely hop between the two resonators via the inter-resonator tunneling rate $J$, producing a photon delocalization with oscillating photon imbalance. The two kinds of photon delocalization can be achieved by tuning the qubit-bath interaction amplitudes. These intriguing features are attributed to the environmental effects in the strong qubit-photon coupling regime. It is expected that the hybrid QED device proposed in here can be fabricated in future experiments, which will provide a platform for the studies of quantum physics. Supporting Information {#supporting-information .unnumbered} ====================== Supporting Information is available from the Wiley Online Library or from the author. 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--- abstract: 'The progenitor systems of Type-Ia supernovae (SNe Ia) are yet unknown. The collisional-triple SN Ia progenitor model posits that SNe Ia result from head-on collisions of binary white dwarfs (WDs), driven by dynamical perturbations by the tertiary stars in mild-hierarchical triple systems. To reproduce the Galactic SN Ia rate, at least $\sim 30-55$ per cent of all WDs would need to be in triple systems of a specific architecture. We test this scenario by searching the Gaia DR2 database for the postulated progenitor triples. Within a volume out to 120pc, we search around Gaia-resolved double WDs with projected separations up to 300au, for physical tertiary companions at projected separations out to 9000au. At 120pc, Gaia can detect faint low-mass tertiaries down to the bottom of the main sequence and to the coolest WDs. Around 27 double WDs, we identify zero tertiaries at such separations, setting a 95 per cent confidence upper limit of 11 per cent on the fraction of binary WDs that are part of mild hierarchical triples of the kind required by the model. As only a fraction (likely $\sim 10$ per cent) of all WDs are in $<300$au WD binaries, the potential collisional-triple progenitor population appears to be at least an order of magnitude (and likely several) smaller than required by the model.' author: - \| Na’ama Hallakoun[^1] and Dan Maoz\ School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 6997801, Israel\ bibliography: - 'gaiatriples.bib' date: 'Accepted XXX. Received YYY; in original form ZZZ' title: 'Limits on a population of collisional-triples as progenitors of Type-Ia supernovae' --- \[firstpage\] binaries: visual – white dwarfs – supernovae: general Introduction {#sec:Intro} ============ A major unsolved puzzle in astrophysics is the identity of the progenitors of Type-Ia supernovae (SNe Ia; see @Maoz_2014 [@Livio_2018; @Wang_2018], for a review). A number of progenitor scenarios have been considered over the years. Among them, several have involved the collision of two white dwarfs (WDs) in a variety of configurations and environments [e.g. @Benz_1989; @Thompson_2011]. In particular, @Katz_2012 and @Kushnir_2013 have proposed that SNe Ia result from head-on collisions of WDs in “mild-hierarchical” triple systems. The triple system consists of an inner double-WD binary with separation $a\lesssim 300$au, orbited by a roughly solar-mass tertiary star in an orbit with pericentre separation $\sim 3-10$ times the inner-binary’s separation. Using numerical integration of the evolution of such 3-body systems and assuming a uniform distribution of inclinations, @Katz_2012 found that, in about 5 per cent of all such systems (the 5 per cent within a small range around a high inclination between the inner and outer orbits), the outer tertiary stochastically drives a Kozai-Lidov perturbation of the inner pair’s orbital eccentricity. Within a few Gyr, the eccentricity lands on a high-enough value to send the inner pair on a head-collision. Given that the actual distribution of inclinations of the systems that survive as a wide double WD with a tertiary companion is unlikely to be uniform, this 5 per cent is most likely an upper limit of the fraction of systems that undergo a collision. @Kushnir_2013 showed that the compression undergone by the WDs upon collision could be effective both at igniting a thermonuclear carbon detonation, and in producing about 0.5[M$_\odot$]{} of radioactive $^{56}$Ni in the explosion, as observed in typical SN Ia events. The range of masses of the WDs that collide could also reproduce the observed range of SN Ia luminosities and their correlation with light-curve evolution time. The asymmetry of the system at the time of explosion predicts double-peaked emission line profiles in SN Ia spectra during the nebular phase in a fraction of events, for which there may be some observational evidence [@Dong_2015; @Dong_2018; @Kollmeier_2019; @Vallely_2019]. However, a major challenge for the collisional-triple model is to produce a collision rate that can match the SN Ia rate in our Galaxy. The Milky Way, if it is a typical Sbc galaxy, has a SN Ia rate per unit stellar mass of $(1.12\pm0.35)\times 10^{-13}$ yr$^{-1}$[M$_\odot$]{}$^{-1}$ [@Li_2011; @Graur_2017].[^2] The stellar mass density in the Solar neighborhood is $0.085\pm 0.010$[M$_\odot$]{}pc$^{-3}$ [@Mcmillan_2011], and the WD number density is $0.0045\pm 0.0004$ pc$^{-3}$ [@Hollands_2018], giving a stellar-mass to WD number ratio of $18.9\pm 2.8$[M$_\odot$]{}WD$^{-1}$. The SN Ia rate per WD is therefore $(2.1\pm0.7)\times 10^{-12}$yr$^{-1}$, and the fraction of all WDs that explode as SNe Ia during the 10 Gyr-lifetime of the Galaxy is $(2.1\pm 0.7)$ per cent. Considering then, that in the collisional-triple model at most 5 per cent of suitable triple systems undergo a SN Ia inducing collision, implies that at least $\sim 30-55$ per cent of all WDs need to be in suitable triple systems, if this mechanism is to explain all SNe Ia [see also @Papish_2016]. While the observational picture on stellar multiplicity is far from clear yet, and even more so the situation regarding the multiplicity of stellar remnants such as WDs, there are already question marks as to the plausibility of such a high frequency of triples with a specific architecture \[$a<300$au, $r_{\rm peri, out}=(3-10)a$\]. The fraction of triple stars (of all configurations) among main-sequence A-through-K type stars has been estimated at $5-15$ per cent [@Tokovinin_2008; @Duchene_2013; @Tokovinin_2014; @Leigh_2013 and references therein], or $15-30$ per cent [@Moe_2017], and likely only a fraction of those have the required architectures. @Klein_2017 estimated the occurrence frequency of the inner binary WD component of the collisional-triple model, by analysing two published studies of $>1 M_\odot$ stars (that will eventually evolve into WDs): an adaptive optics survey of A stars by @DeRosa_2014, in which $>1$[M$_\odot$]{} companions within 400au were searched for; and a radial-velocity survey by @Mermilliod_2007 of $>1$[M$_\odot$]{} giants, which was searched for $>1$[M$_\odot$]{} companions within 3au. @Klein_2017 conclude that $15-20$ per cent of the intermediate mass stars that become WDs are in binaries that can constitute the inner component of collisional-triple systems, and therefore even if all such binaries had a mild hierarchical tertiary, there would still be a factor 2 shortage of collisional-triple progenitors for SNe Ia. Even if triple main-sequence systems are abundant, a challenge of the model that was already acknowledged by @Katz_2012 is that triple systems with the suitable architecture and relative orbital inclination to induce to a collision of the inner binary will undergo such a collision already when the stars are on the main sequence (without an ensuing SN Ia), effectively eliminating all the SN Ia progenitor systems. This conclusion has emerged also from binary population synthesis calculations [@Hamers_2013; @Toonen_2018]. @Katz_2012 raised the possibility that angular momentum loss by the system due to asymmetric mass loss during the evolution to the WD stage, or perturbations by passing stars, could “reset” the relative orbital inclinations of the system, and thus solve the problem. In this paper, we address more directly the subject of the putative triple-progenitor population of SNe Ia, by searching the Gaia DR2 database [@Gaia_2016; @GaiaDR2_2018] specifically for double WDs orbited by a tertiary star, as envisaged in the collisional-triple model. Gaia search {#sec:Gaia} ============= We search for triple systems akin to those required by the collisional-triple model using two approaches: first, by identifying resolved WD binaries with projected separations $<300$au, and searching their surroundings for tertiaries with projected separations $<9000$au; and, second, by searching for tertiaries projected within $3000$au of unresolved, double-WD, separation $a\sim 0.1-1$au, candidates identified via radial-velocity variations. Our search for triples is conservatively inclusive in several respects. First, by pre-selecting WD binaries, and then asking what fraction of those binaries have a tertiary, we obtain an upper limit on the fraction of all WDs in triples, since not all WDs are in binaries. Second, because of projection effects, some of the selected inner binaries will have physical separations $a>300$au, and some of the counted tertiaries will be at physical separations beyond the tertiary separation range required by the model. By including all of these systems, we will obtain a conservative upper limit on the true fraction of of WDs that are in triples with the architecture required by the model. Resolved Gaia triples ----------------------- Following @GentileFusillo_2019, we start by identifying all WD candidates using an initial colour-magnitude cut in Gaia DR2 data: $$\begin{aligned} \texttt{parallax\_over\_error} &> 1\\ M_G &> 5\\ M_G &> 5.93 + 5.047\times \left(G_\textrm{BP}-G_\textrm{RP}\right)\\ M_G &> 6 \times \left(G_\textrm{BP}-G_\textrm{RP}\right)^3+\nonumber\\ &- 21.77\times \left(G_\textrm{BP}-G_\textrm{RP}\right)^2 +\nonumber\\ &+27.91\times \left(G_\textrm{BP}-G_\textrm{RP}\right)+0.897\\ \left(G_\textrm{BP}-G_\textrm{RP}\right) &< 1.7\end{aligned}$$ where $M_G = \texttt{phot\_g\_mean\_mag}+5\times\log_{10}\left(\texttt{parallax}\right)-10$ is the absolute magnitude in the $G$ band. This results in $8,144,732$ sources. We further select only those within a distance of 120pc: $$\texttt{parallax} > 8.33$$ leaving $313,871$ sources. We follow @GaiaHRD_2018 and remove astrometric artefacts by requiring: $$\label{eq:astrometric} \sqrt{\frac{\chi^2}{\nu'-5}}<1.2 \textrm{max}\left(1,~ e^{-0.2\left(G-19.5\right)}\right)$$ where $\chi^2$ is `astrometric_chi2_al` and $\nu'$ is `astrometric_n_good_obs_al` in the Gaia database, leaving $86,182$ sources. Since the $G_\textrm{BP}$ and $G_\textrm{RP}$ fluxes are calculated by integrating over low-resolution spectra, they are more prone to contamination from nearby sources, compared to the $G$-band flux, which is measured by photometric profile fitting. Following @ElBadry_2018 and @Evans_2018, we filter out sources that might be contaminated by nearby sources by limiting the total $G_\textrm{BP}$ and $G_\textrm{RP}$ excess compared to the $G$ band: $$\begin{aligned} \texttt{phot\_bp\_rp\_excess\_factor} &> 1.0 + 0.015 \left(G_\textrm{BP}-G_\textrm{RP}\right)^2\\ \texttt{phot\_bp\_rp\_excess\_factor} &< 1.3 + 0.06 \left(G_\textrm{BP}-G_\textrm{RP}\right)^2\end{aligned}$$ leaving $26,591$ sources. We further follow @ElBadry_2018 by selecting only sources with high signal-to-noise ratio photometry, i.e., $<2$ per cent flux uncertainty in the $G$ band, and $<5$ per cent in both the $G_\textrm{BP}$ and $G_\textrm{RP}$ bands: $$\begin{aligned} \texttt{phot\_g\_mean\_flux\_over\_error} &> 50 \\ \texttt{phot\_rp\_mean\_flux\_over\_error} &> 20 \\ \texttt{phot\_bp\_mean\_flux\_over\_error} &> 20\end{aligned}$$ leaving $17,410$ sources. These criteria assure that we only select sources that fall with high certainty within in the WD region of the colour-magnitude diagram. Finally, we select only sources where the relative error on the parallax is smaller than $10$ per cent: $$\texttt{parallax\_over\_error}>10$$ resulting in $17,395$ sources. We then choose all physical-double resolved WDs within 120pc with projected separations $<300$au, by requiring $$\frac{\theta}{\textrm{arcsec}} \leq 10^{-3}\frac{s}{\textrm{au}}\frac{\varpi}{\textrm{mas}}$$ where $\theta$ is the angular separation, $s=300$au is the projected separation, and $\varpi$ is the parallax. Following @ElBadry_2018 [eq. 2] we further select only pairs with consistent parallaxes: $$\label{eq:distance} \Delta d - 2s \leq 3 \sigma_{\Delta d},$$ where $\Delta d = \left\| 1/\varpi_1 - 1/\varpi_2 \right\|$ pc, $s = \theta /\varpi_1$ pc, and $\sigma_{\Delta d}= \sqrt{\sigma_{\varpi,1}^2/\varpi_1^4 + \sigma_{\varpi,2}^2/\varpi_2^4}$ pc. Here $\varpi_i$ and $\sigma_{\varpi,i}$ are the parallax of the $i$th target and its standard error in arcsec, respectively, and the angular separation, $\theta$, is in radians. This means that we inclusively count a system as a WD binary, even if its component distances differ at the $3 \sigma_{\Delta d}$ level. The search results in $27$ WD pairs, listed in Table \[tab:DWD\], out of the $17,395$ WDs that are nearer than 120pc. The relative projected velocity differences between the members of each pair, also listed in Table \[tab:DWD\], are well below the $\sim 6$kms$^{-1}$ maximum that is possible between bound solar-mass objects, indicating these are real, bound, pairs. Fig. \[fig:Resolved\] shows the WD-pair locations on the Gaia colour-magnitude diagram. We note, following @ElBadry_2018 and @Arenou_2018, that the above 27 resolved pairs are a small fraction of the actual number of WD pairs with separations $<300$au, which is likely to be at least 10 per cent [@Maoz_2018]. First, Gaia is incomplete to binaries at angular separations $\theta\lesssim 0.7$, corresponding to 84au at 120pc. More importantly, the Gaia $G_\textrm{BP}-G_\textrm{RP}$ colour is not available, in most cases, for both components of doubles with projected separations $\theta<2$ (240au at 120pc), selecting against the identification of one or both stars as WDs. We further note that extending the WD sample to 200 or 300pc did not result in any additional $<300$au WD pairs, likely because of the limitations imposed by the resolution, which become more severe with distance. However, our strategy for testing the collisional-triple model does not require a complete census of WD binaries; to the contrary, we select only a small minority of double-WDs that Gaia can detect, but then perform a thorough search for tertiaries around these binaries. By limiting our sample of WD pairs to 120pc, we expect that, at the limiting Gaia magnitude $G \sim 20$, we are sensitive and largely complete to tertiary companions corresponding to the faintest, lowest-mass, main-sequence stars, as well as the coolest WDs. As shown by @GentileFusillo_2019 [fig. 19], the 100pc Gaia DR2 WD sample is complete up to $M_G \sim 16$, indeed corresponding to all but the coolest few percent of WDs with $M_G > 16$ [@Isern_2019; @Tremblay_2019]. This completeness likely applies also at the relatively brighter luminosities near the bottom of the main sequence ($M_G \sim 14$, $T_\textrm{eff} \lesssim 3000$K), corresponding to $\lesssim 0.2$[M$_\odot$]{} M-dwarfs [@Hillenbrand_2004]. However, a more thorough future assessment of the main-sequence completeness at 120pc will place our conclusions, below, on a surer footing. ![Gaia colour-magnitude diagram for resolved double WDs within 120pc, with projected separations $s<300$au. Pairs are connected by black solid lines, where the photometric primary (secondary) is marked by a blue triangle (red circle). None of the pairs have a tertiary companion with a projected separation $s<9000$au. The number density in this parameter space of the full 120pc WD sample from Gaia ($17,395$ sources) is shown in grayscale for reference.[]{data-label="fig:Resolved"}](Resolved_pairs.pdf){width="\columnwidth"} [c c c c c c c c]{} GaiaID & $\varpi$ (mas) & $d$ (pc) & $\mu_\textrm{RA}$ (masyr$^{-1}$) & $\mu_\textrm{Dec}$ (masyr$^{-1}$) & $v_\textrm{RA}$ (kms$^{-1}$) & $v_\textrm{Dec}$ (kms$^{-1}$)\ [c c c c c c c c]{} GaiaID & $\varpi$ (mas) & $d$ (pc) & $\mu_\textrm{RA}$ (masyr$^{-1}$) & $\mu_\textrm{Dec}$ (masyr$^{-1}$) & $v_\textrm{RA}$ (kms$^{-1}$) & $v_\textrm{Dec}$ (kms$^{-1}$)\ Next, we search for tertiary companions around the inner double WDs, by querying the entire Gaia DR2 database. Assuming a uniform distribution of orbital eccentricities, as supported by observations [@Duchene_2013; @Tokovinin_2016], the mean eccentricity is $\left<e\right>=1/2$, and the apocentre separation is therefore, on average, $\left(1+e\right)/\left(1-e\right)=3$ times the pericentre separation. The apocentre separation of the tertiary star in the collisional model can thus be as high as $\sim 30a$, and we therefore search for tertiaries projected within 9000au of the inner, $a<300$au, double WDs. For the tertiaries we include sources only if they satisfy $\texttt{parallax\_over\_error} > 5$, Eq. \[eq:astrometric\] (astrometric artefacts removal), and Eq. \[eq:distance\] (distance from the first WD in the inner binary), but this time we allow only a $2 \sigma_{\Delta d}$ difference in the distances. None of the 27 resolved inner WD binaries has a candidate tertiary companion projected within 9000au. With Poisson statistics, this null detection implies, at 95 per cent confidence, the existence of $<3$ tertiaries, or an upper limit of $<11$ per cent on the fraction of $a<300$au-separation double WDs with a tertiary companion with $r_\textrm{peri,out}=\left(3-10\right)a$. Gaia tertiaries to SPY double WDs ----------------------------------- To further explore the potential triple SN Ia progenitor landscape, we have searched for tertiaries around a second sample of close double-WD binaries, this time unresolved WD binary candidates from the Supernova Progenitor surveY [SPY; @Napiwotzki_2001]. The SPY survey was a few-epoch spectroscopic survey of $\sim$800 bright ($V\sim 16$mag) WDs conducted with the European Southern Observatory 8m Very Large Telescope, with the objective of using radial-velocity differences between epochs to identify close double-WD systems that will merge within a Hubble time. In @Maoz_2017 we measured the maximal changes in radial velocity ([$\Delta{\rm RV}_{\rm max}$]{}) between epochs, and modelled the observed [$\Delta{\rm RV}_{\rm max}$]{} statistics via Monte Carlo simulations [@Maoz_2012; @Badenes_2012], to constrain the population characteristics of double WDs. We identified 43 high-[$\Delta{\rm RV}_{\rm max}$]{} systems as likely double-WD candidates [see table 1 of @Maoz_2017], and an additional double-lined system that was not included in the candidate list because of its low [$\Delta{\rm RV}_{\rm max}$]{} (HE0315$-$0118). Since the Gaia DR2 astrometric model does not take binarity and potential astrometric wobble into account, it might fail, or not fail, but report spurious results in case of unresolved binaries [e.g. @Hollands_2018]. We have verified that full five-parameter astrometric solutions are reported for all 44 SPY systems. To further check for spurious distances, we compared the astrometric distances derived from Gaia’s parallaxes with the photometric distances for these WDs. We used the spectroscopic effective temperatures and surface gravities derived by @Koester_2009 to generate synthetic WD model spectra using the spectral synthesis program <span style="font-variant:small-caps;">Synspec</span> [version 50; @Hubeny_2011], based on model atmospheres created by the <span style="font-variant:small-caps;">Tlusty</span> program [version 205; @Hubeny_1988; @Hubeny_1995; @Hubeny_2017a]. The luminosities were then estimated by assuming a mass-radius relation [‘thick’ hydrogen-dominated carbon-oxygen WD cooling tracks[^3] from @Fontaine_2001]. Finally, the photometric distances were calculated using the $V$/$B$-band magnitudes reported in @Koester_2009. We find that the photometric distances of all of the SPY WDs are consistent with their Gaia distances, with a scatter $\sim 10$ per cent, and with few or no obvious outliers that could be indicative of spurious Gaia distances. Nonetheless, future Gaia data releases that will include binarity in the astrometric model are required in order to verify that none of the WD distances are erroneous due to unresolved binarity. As with the previous sample, we have searched with Gaia for tertiaries within 3000au of these 44 candidate close-double-WDs from SPY. Contrary to the previous sample, the triple systems turned up here are not actual candidate progenitors for the collisional-triple scenario, since their inner separations are of order $0.1-4$au, whereas the tertiary separation is of order $10^{3-4}$ larger, rather than just $3-30$ times larger, as required by the model. The tertiaries within only 120au, which would be have been relevant to the model, are well below the Gaia angular resolution limit at the typical distances of the SPY WDs ($\sim 10-250$pc). Nevertheless, it is instructive to estimate the frequency of these “extreme” (rather than “mild”) hierarchical triples, to learn about the triple population in this different hierarchy range. Among the 44 SPY double-WD candidates, we find four with tertiaries projected within $<3000$au separation, satisfying $\texttt{parallax\_over\_error} > 5$, Eq. \[eq:astrometric\] (astrometric artefacts removal), and Eq. \[eq:distance\] (distance from the first WD in the inner binary, within $2 \sigma_{\Delta d}$). We list them in Table \[tab:SPY\]. Their location on the Gaia colour-magnitude diagram is shown in Fig. \[fig:SPY\]. At 3000au separation, the maximum velocity difference due to orbital motion is $\sim 1.5$kms$^{-1}$, and again all four candidate tertiaries are consistent with a velocity difference that is lower than this, arguing that these are true bound systems. The M-dwarf tertiary we find for HE0516$-$1804 could be the source of the infrared excess we found in its photometry in @Maoz_2017. We note that the known M-dwarf tertiary candidate of WD0326$-$273 [@Nelemans_2005 and references therein] failed to pass our consistent parallax criterion (Eq. \[eq:distance\]), with $\sigma_{\Delta d} \sim 5$. This result suggests that $\lesssim 9$ per cent of separation-$a\sim 0.1-4$au double WDs have tertiaries within separations of 3000au. Again, the population of tertiaries considered around this sample is much more extensive than those with a role in the collisional-triple SN Ia model. ![Candidate unresolved double WDs from the SPY sample (blue triangles) with a tertiary stellar companion in Gaia with projected separations $s<3000$au (red circles). Pairs are connected by dashed lines. The number density of stars from a random Gaia sample and the full 120pc WD sample is shown in grayscale.[]{data-label="fig:SPY"}](SPY_triples.pdf){width="\columnwidth"} [c c c c c c c c c]{} Name & GaiaID & $\varpi$ (mas) & $d$ (pc) & $\mu_\textrm{RA}$ (masyr$^{-1}$) & $\mu_\textrm{Dec}$ (masyr$^{-1}$) & $v_\textrm{RA}$ (kms$^{-1}$) & $v_\textrm{Dec}$ (kms$^{-1}$)\ Discussion ========== We have used Gaia to search for candidate triple stellar systems similar to those envisaged in the collisional triple SN Ia progenitor model, and to assess their abundance. Gaia, due to its angular resolution limits, is strongly biased against detection of the separation $a<300$au inner WD binaries of the proposed triple systems. We have therefore focused on two samples of “known” double WDs, and used Gaia to search for a putative wide tertiary member out to 9000au separations, where Gaia completeness is high. Our first inner-double-WD sample consists of 27 WD pairs with projected separations $s<300$au, found in the Gaia DR2 database among $\sim 17,400$ WDs within a 120pc distance. These 27 systems are the small fraction of the actual number of double WDs having such separations that are detected as binaries, because of Gaia’s low sensitivity in this regime. Around this sample of 27 Gaia-selected inner-WD-binaries, we find no tertiary stars of any kind (WD or main-sequence, down to the lowest stellar masses) out to 9000au projected separations. Such systems, had they been found, could correspond closely in separation and hierarchy to the SN Ia progenitors in the model. Their non-detection sets a 95 per-cent-confidence upper limit of 11 per cent on the fraction of $a<300$au WD binaries that are orbited by a tertiary with pericentre separation $3-10$ times the inner-binary separation. @Maoz_2018 have combined the [$\Delta{\rm RV}_{\rm max}$]{} results of @Badenes_2012 and @Maoz_2017, to deduce that the fraction of WDs in binaries in the range 0.01 to 4au is $10\pm2.5$ per cent, with a separation distribution in this range consistent with a power law, $a^\alpha$, with $\alpha=-1.3\pm0.2$. @ElBadry_2018, analysing the resolved binary WD population that they find with Gaia, and after accounting for selection and incompleteness effects, find a power-law separation distribution with a slope, again, close to $\alpha=-1$ (i.e. equal numbers per logarithmic interval in separation) in the range $a=50-1500$au. Considering then that there is a similar number of decades in separation between $0.01-4$au [@Maoz_2018] and between $1-300$au (the range of inner-binary separations in the @Katz_2012 model), implies that $\sim 10$ per cent of all WDs are in binaries with $a=1-300$au. Combined with our present result that, at most, 11 per cent of such binaries host a tertiary companion out to 9000au, we conclude that the fraction among WDs of triple systems with the architecture required by the collisional-triple model is $<1$ per cent, at the 95 per cent confidence level. This triple fraction is at least a factor $\sim 30-55$ times lower than the fraction required by the model ($\sim 30-55$ per cent, see Section \[sec:Intro\]). A second double-WD sample that we have studied consists of 44 unresolved, separation $a\sim 0.1-4$au candidate WD binaries that have emerged, based on radial-velocity variations, from SPY. We searched the SPY candidate WD binaries for tertiary companions, this time out to 3000au separations, i.e., a much larger range than relevant for the collisional model. Here, we found four candidate tertiary companions (two main-sequence stars, a WD, and an uncertain case, perhaps a subdwarf) with actual projected separations from 190 to 1650au. Considering the WD binarity fraction of roughly 4 per cent for every decade of separation (see above), the 9 per cent fraction of tertiaries (of all kinds) in this separation range around short-orbit WD binaries is not surprising. Combined with the abundance of inner-WD binaries, we again get a 1 per cent upper limit on the fraction of WDs that are in triples, consisting of an inner double WD plus a tertiary (although now triples of a different hierarchy than those required by the collisional-triple model). We conclude that there is at least a 1.5 order of magnitude deficit, and likely more, in the fraction of WDs that are in the types of triple systems required by the collisional-triple model. The double WDs in our first, $s<300$au, sample have actual separations in the $100-300$au range. A remaining “hiding place” for the progenitor systems of the model could be in the unexplored parameter range of $a\sim 1-100$au WD binaries. A large fraction of all WDs would need to be in binaries with this separation range, and a large fraction of those same binaries would need to have tertiaries at separations $(3-10)a$. The first condition is unlikely to be true, given the similar separation distributions (about flat in log separation, see above) followed by binary WDs within the $a=0.01-4$au range and the $a=50-1500$au range. The second condition is also unlikely, given the rarity of tertiary stars around WD binaries with those separation ranges, as shown here. A different progenitor and explosion scenario is most probably at the root of SN Ia explosions. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Boaz Katz, Julio Chanamé, J. J. Hermes, and the anonymous referee for valuable comments. This work was supported by a grant from the Israel Science Foundation (ISF). This work has made use of data from the European Space Agency (ESA) mission [Gaia]{} (<https://www.cosmos.esa.int/gaia>), processed by the Gaia Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the [Gaia]{} Multilateral Agreement. This research made use of <span style="font-variant:small-caps;">astropy</span>[^4], a community-developed core <span style="font-variant:small-caps;">python</span> package for Astronomy [@Astropy_2013; @Astropy_2018], <span style="font-variant:small-caps;">astroquery</span> [@Astroquery_2013], <span style="font-variant:small-caps;">matplotlib</span> [@Hunter_2007], <span style="font-variant:small-caps;">numpy</span> [@Numpy_2006; @Numpy_2011], <span style="font-variant:small-caps;">uncertainties</span>[^5], a <span style="font-variant:small-caps;">python</span> package for calculations with uncertainties by Eric O. Lebigot, and <span style="font-variant:small-caps;">topcat</span> [@Taylor_2005], a tool for operations on catalogues and tables. \[lastpage\] [^1]: E-mail: <naama@wise.tau.ac.il> (NH) [^2]: For a total stellar mass of $(6.4\pm0.6)\times 10^{10}$[M$_\odot$]{} [@Mcmillan_2011], the Galactic rate is $(7.2\pm2.3)\times 10^{-3}$yr$^{-1}$, or a SN Ia every $100-200$yr. This is broadly consistent with the four known Galactic events over the last millennium that were likely or certain SNe Ia (SN1006, SN1572-Tycho, SN1604-Kepler, and G1.9+03). [^3]: <http://www.astro.umontreal.ca/bergeron/CoolingModels/> [^4]: <http://www.astropy.org> [^5]: <http://pythonhosted.org/uncertainties/>
--- abstract: \| A variety of powerful results have been shown for the chromatic number of triangle-free graphs. Three noteworthy bounds are in terms of the number of vertices, edges, and maximum degree given by Nilli (2000), Gimbel & Thomassen (2000), and Johansson. There have been comparatively fewer works extending these types of bounds to graphs with a small number of triangles. One noteworthy exception is a result of Alon et. al (1999) bounding the chromatic number for graphs with low degree and few triangles per vertex; this bound is nearly the same as for triangle-free graphs. This type of parametrization is much less rigid, and has appeared in dozens of combinatorial constructions. In this paper, we show a similar type of result for $\chi(G)$ as a function of the number of vertices $n$, the number of edges $m$, as well as the triangle count (both local and global measures). Our results smoothly interpolate between the generic bounds true for all graphs and bounds for triangle-free graphs. Our results are tight for most of these cases; we show how an open problem of Johnson (2009) can resolve the small remaining gap in our bounds. author: - 'David G. Harris[^1]' title: Some results on chromatic number as a function of triangle count --- \[section\] \[theorem\][Proposition]{} \[theorem\][Conjecture]{} \[theorem\][Observation]{} \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Definition]{} Introduction ============ In this paper, we will show bounds on the chromatic number of undirected graphs $G = (V,E)$ which have relatively few triangles (a triangle is a triple of vertices $x,y,z \in V$ where all three edges $(x,y), (x,z), (y,z)$ are present in $G$). A variety of powerful results have been shown for the chromatic number of triangle-free graphs. Two noteworthy bounds, which we explore further in this paper, are in terms of the number of vertices and edges. \[triangle-free-thm1\] Suppose $G$ is triangle-free and has $n$ vertices. Then $\chi(G) \leq O \Bigl( \sqrt{ \frac{n}{\log n}} \Bigr)$. \[triangle-free-thm2\] Suppose $G$ is triangle-free and has $m$ edges. Then $\chi(G) \leq O \Bigl( \frac{m^{1/3}}{\log^{2/3} m} \Bigr)$. Another powerful bound for triangle-free graphs can be given in terms of the maximum degree:[^2] \[triangle-free-d\] Suppose $G$ is triangle-free and has maximum degree $d$. Then $\chi(G) \leq O \Bigl( \frac{d}{\log d} \Bigr)$ These bounds are much smaller than would be possible for a generic graph (with no restriction on the number of triangles); in those cases one can only show the bounds $$\chi(G) \leq \min(n, \sqrt{m}, d)$$ The requirement that a graph has no triangles is quite rigid. There have been comparatively fewer works which relax the triangle-free condition to allow a small number of triangles. One noteworthy exception to this is the result of [@aks]: \[aks-thm\] Suppose $G$ is a graph with maximum degree $d$ and each vertex incident on at most $y$ triangles, where $1 \leq y \leq d^2/2$. Then $$\chi(G) \leq O( \frac{d}{\ln(d^2/y)} )$$ Thus, for instance, if $y < d^{2 - \Omega(1)}$, then $\chi(G)$ is roughly the same as if $G$ had no triangles at all. The result of [@aks] has been used in dozens of combinatorial constructions. Roughly speaking, if $G$ is produced in a somewhat “random” or “generic” way, then $G$ will have relatively few triangles; only a few extremal cases (such as a graph containing a $d$-clique) give the full triangle count. In this paper, we show upper bounds for $\chi(G)$ as a function of the number of vertices $n$, the number of edges $m$, as well as the triangle count. Our results smoothly interpolate between the generic bounds true for all graphs, and the triangle-free bounds of Theorems \[triangle-free-thm1\] and \[triangle-free-thm2\]. We will see here as well that graphs with few triangles have nearly the same behavior as graphs with no triangles. This is convenient because triangle counts (local and global) can be computed easily in polynomial time; this is quite different from computing $\chi(G)$ directly (which is NP-hard). We say that a vertex $v \in V$ has local triangle count $y$, if there are $y$ pairs of vertices $u, w \in V$ such that $(v,u), (v,w), (u,w) \in E$. We say that $G$ has local triangle bound $y$ if every vertex has local triangle count at most $y$. We shall show the following bounds on the chromatic number: Suppose a graph $G$ has $m$ edges, $n$ vertices, $t$ triangles, and local triangle bound $y$. Then $$\chi(G) \leq \min(a_1, a_2, a_3, a_4, a_5, a_6)$$ where $$\begin{aligned} &a_1 = O \bigl( \sqrt{ \frac{n}{\log n}} + \frac{n^{1/3} y^{1/3}}{\log^{2/3} (n^2/y)} \bigr) && a_2 = O \bigl( \frac{m^{1/3}}{\log^{2/3} m} + \frac{m^{1/4} y^{1/4}}{\log^{3/4} (m/y)} \bigr) \\ &a_3 = O \bigl( \sqrt{\frac{n}{\log n}} + \frac{t^{1/3} \log \log(t^2/y^3)}{\log^{2/3}(t^2/y^3)} \bigr) && a_4 = O \bigl( \frac{m^{1/3}}{\log^{2/3} m} + \frac{t^{1/3} \log \log(t^2/y^3)}{\log^{2/3}(t^2/y^3)} \bigr) \\ &a_5 = \sqrt{\frac{n}{\log n}} + (6^{1/3} + o(1)) t^{1/3} && a_6= \frac{m^{1/3}}{\log^{2/3} m} + (6^{1/3} + o(1)) t^{1/3} \end{aligned}$$ Furthermore, colorings with the indicated bounds can be constructed by a randomized polynomial-time algorithm. We show in Section \[lb-sec\] that the bounds $a_1, a_2$ are tight up to constant factors, and the bounds $a_5, a_6$ are tight up to second-order terms. The bounds $a_3, a_4$ are tight up to factors of $\log \log (t^2/y^3)$; we will show in Section \[conj-sec\] that this small gap can be resolved via a conjecture of Johnson [@johnson]. Note on constructive algorithms ------------------------------- All our bounds on chromatic number can be realized constructively; that is, there are randomized algorithms which construct such colorings in expected polynomial time. Our proofs will use a few previous results on chromatic number, which did not have constructive algorithms stated directly. For completeness, we state these here with a proof sketch. \[aks-thm2\] Suppose $G$ has maximum degree $d$ and local triangle bound $y$, where $1 \leq y \leq d^2/2$. Then there is a randomized polynomial-time algorithm producing a coloring with $O( \frac{d}{\ln(d^2/y)} )$ colors. The original proof of Theorem \[aks-thm\] uses a result on constructing coloring with $O(d/\log d)$ colors for triangle-free graphs. This has been made into a constructive algorithm in [@pettie]. The other main probabilistic tool used in Theorem \[aks-thm\] is the Lovász Local Lemma; the algorithm of Moser & Tardos [@mt] makes this constructive in a fairly straightforward way. Vu [@vu] extended Theorem \[aks-thm\] to list-chromatic number: \[aks-thm3\] Suppose $G$ has maximum degree $d$ and local triangle bound $y$, where $1 \leq y \leq d^2/2$. If each vertex has a palette of size $\frac{c d}{\log(d^2/y)}$, where $c$ is a universal constant, then there is a randomized polynomial-time algorithm to producing a list-coloring. \[turan-thm\] Suppose $G$ is a graph with $n$ vertices and $m$ edges. Then $G$ has an independent set of size at least $\frac{n}{2m/n + 1}$; furthermore, such an independent set can be constructed in polynomial time. Notation -------- We use $\ln$ to refer to the natural logarithm, while the expression $\log$ always refers to the truncated logarithm, i.e. $$\log(x) = \begin{cases} \ln(x) & x > e \\ 1 & x \leq e \end{cases}$$ We write $f {\lesssim}g$ to mean that $f = O(g)$. We write $f \approx g$ if $f = \Theta(g)$. Given a graph $G$ on a vertex set $V$ and a subset $B \subseteq V$, we use $\chi(B)$ as a shorthand for $\chi(G[B])$, where $G[B]$ denotes the subgraph induced on $B$. We let $N(v)$ denote the neighborhood of $v$, i.e. the set of vertices $w$ such that $(w,v) \in E$. The degree of $v$ is $\text{deg}(v) = \|N(v)\|$. Bounds in terms of $n$ ====================== We begin by showing bounds of $\chi(G)$ as a function of triangle information (local and global) as well the vertex count $n$. These generalize a result of [@ajtai], which show a similar bound for chromatic number as a function of the vertex count for triangle-free graphs. All of these bounds will have the following form: we show $$\chi(G) {\lesssim}\sqrt{ \frac{n}{\log n} } + f(n, y, t)$$ where $f$ is a function which depends on $n$ as well as $y$ and/or $t$; when $y,t$ are sufficiently small, $f(n,y,t)$ has a negligible contribution. This shows that, as long as the number of triangles is sufficiently small, then $\chi(G)$ will have approximately the same behavior as if $G$ had no triangles at all. \[prop0\] Suppose $G$ has $n$ vertices and local triangle bound $y$. Then $$\chi(G) {\lesssim}\sqrt{ \frac{n}{\log n}} + \frac{n^{1/3} y^{1/3}}{\log^{2/3} (n^2/y)}$$ Further, such a coloring can be produced in expected polynomial time.[^3] Let $f = \log(n^2/y)$. Simple analysis shows that it suffices to show that $\chi(G) {\lesssim}\sqrt{n / \log n}$ for $y \leq \sqrt{n \log n}$ and $\chi(G) {\lesssim}\frac{ (n y)^{1/3} }{f^{2/3}}$ for $y \geq \sqrt{n \log n}$. Our plan is to repeatedly remove independent sets from $G$, as long as its maximum degree exceeds $d$, where $d$ is some parameter to be chosen. Initially let $G_0 = G$. Now, repeat the following for $i = 0, 1, 2, \dots, k$: if all the vertices of $G_i$ have degree $\leq d$, then abort the process and color the residual graph $G_i$ using Theorem \[aks-thm2\]; this requires $O( \frac{d}{\log(d^2/y)})$ colors. Otherwise, select some vertex $v_i$ which has degree $> d$ in $G_i$, and let $I_i$ be a maximum independent set of the neighborhood of $v$ in $G_i$. Color all the vertices of $I_i$ with a new color, and let $G_{i+1} = G_i - I_i$. The total number of colors used during this process is at most $O( \frac{d}{\log (d^2/y)} + k)$. To bound $k$, observe the following. Each vertex $v_i$ has degree $> d$ in $G_i$, so the neighborhood of $v_i$ contains $> d$ vertices. By hypothesis, the neighborhood of $v_i$ contains $\leq y$ edges. Hence, by Theorem \[turan-thm\], one can construct an independent set of size $\geq \frac{d}{y/d + 1}$. So we have $\|I_i\| \geq \frac{d}{y/d + 1}$ for $i = 1, \dots, k$. As $\sum \|I_i\| \leq n$, it follows that $k \leq \frac{n (y/d + 1)}{d}$. So, overall we have $\chi {\lesssim}\frac{d}{\log(d^2/y)} + \frac{n (y/d + 1)}{d}$. It remains now to select the parameter $d$. First let us suppose that $y \leq \sqrt{n \log n}$. In this case, take $d = \sqrt{n \log n}$. Then $$\begin{aligned} \chi(G) &{\lesssim}\frac{\sqrt{n} (1 + \frac{y}{\sqrt{n \log n}})}{\sqrt{\log n}} + \frac{\sqrt{n \log n}}{\log \frac{n \log n}{y}} {\lesssim}\sqrt{\frac{n}{\log n}}\end{aligned}$$ Next, suppose that $y > \sqrt{n \log n}$. In this case, take $d = (n y f)^{1/3}$. As $y > \sqrt{n f}$ we have $$\begin{aligned} \chi(G) &{\lesssim}\frac{n (1 + \frac{y}{(n y f)^{1/3}})}{(n y f)^{1/3}} + \frac{(n y f)^{1/3}}{\log \frac{(n y f)^{2/3}}{y}} \leq \frac{2 n^{1/3} y^{1/3}}{f^{2/3}} + \frac{(n y f)^{1/3}}{\log( f^{1/3} e^f )} {\lesssim}\frac{n^{1/3} y^{1/3}}{f^{2/3}}\end{aligned}$$ Next, we will show bounds on $\chi(G)$ given information both about the global triangle count and the local triangle bound. We begin with a useful lemma. \[ff-lemma\] Suppose that a graph $G$ has local triangle bound $y$, and there is a partition of the vertices $V = A_1 \sqcup A_2 \sqcup \dots \sqcup A_k$ such that, for all $1 \leq i \leq j \leq k$, every $v \in A_i$ satisfies $\|N(v) \cap A_j\| \leq d x^{i - j}$ for some parameters $d, x \geq 1$. Then $$\chi(G) {\lesssim}\frac{d \lceil \log_x \log(d^2/y) \rceil}{\log (d^2/y)};$$ furthermore, such a coloring can be found in polynomial time. In particular, if $x \geq 1 + \Omega(1)$, then $$\chi(G) {\lesssim}\frac{d \log \log (d^2/y)}{\log(d^2/y)}$$ Define $f = \log(d^2/y)$. We will first show that this result hold for $x \geq f$. In this case, we want to show that $\chi(G) {\lesssim}d/f$. We will allocate a palette of size $c d/f$ to each vertex $v$, where $c$ is a constant to be determined. We proceed for $j = k, k-1, \dots, 1$, attempting to list-color $G[A_j]$. Let us consider the state at a given value $j$ and for a given $v \in A_j$. At this stage, only the neighbors of $v$ in $A_{j+1}, A_{j+2}, \dots, A_k$ have been colored so far; the total number of such neighbors is at most $$d x^{-1} + d x^{-2} + \dots \leq d (\frac{1}{f} + \frac{1}{f^2} + \frac{1}{f^3}) \leq 2 d/f$$ Thus, the vertex $v$ remains with $c d / f - 2 d/f = (c - 2) d/f$ colors in its palette. So every vertex in $A_j$ has a residual palette of size $(c-2) d /f$. If $c$ is a sufficiently large constant, then Theorem \[aks-thm3\] allows us to extend the coloring to $A_j$. Now that we have shown this result holds for $x = f$, let us show that it holds for an arbitrary value $x < f$. Set $s = \lceil \log_x f \rceil$ and for each $i = 1, \dots, s$ define $B_i = A_i \cup A_{i+s} \cup A_{i+2 s} \cup \dots$. Observe that each $G[B_i]$ satisfies the hypotheses of this Lemma with the vertex-partition $A_i, A_{i+s}, A_{i+2 s}, \dots$ and with parameters $d' = d, x' = x^s \geq f$. Thus $\chi(B_i) {\lesssim}d/f$ and so $$\chi(G) \leq \chi(B_1) + \dots + \chi(B_s) {\lesssim}s d/f = \lceil \log_x \log(d^2/y) \rceil \times \frac{d}{\log (d^2/y)}$$ \[ttprop2\] Suppose a graph $G$ has $n$ vertices, $t$ triangles, and local triangle bound $y$. Then $$\chi(G) {\lesssim}\sqrt{ \frac{n}{\log n}} + \frac{t^{1/3} \log \log (t^2/y^3)}{\log^{2/3}(t^2/y^3)}$$ Let $f = \log(t^2/y^3)$. We begin by showing the following slightly weaker bound: $$\label{trt1} \chi(G) {\lesssim}\sqrt{n} + \frac{t^{1/3} \log f}{f^{2/3} }$$ Let $A_i$ denote the set of vertices in $G$ with between $2^i$ and $2^{i+1}$ triangles and let $d$ be a parameter which we will specify shortly. Let $k = \log_2 d$. Suppose that for $i \geq k$ there is a vertex $v \in A_i$ with more than $2^{(i-j)/2} d$ neighbors in $A_j$ where $j \geq i$. Then apply Theorem \[turan-thm\] to $N(v) \cap A_j$. This yields an independent set $I$ of size $\|I\| \geq \frac{2^{(i-j)/2} d}{1 + \frac{2^{i+2}}{2^{(i-j)/2} d}} {\gtrsim}2^{-j} d^2$. Assign each vertex in $I$ to one new color and remove $I$ from the graph. Since each $w \in I$ is incident on at least $2^j$ triangles, this removes $\Omega(d^2)$ triangles. Note that this process of removing vertices may cause the membership of the sets $A_l$ to change. Overall the total number of new colors that can be produced this way is $O(t/d^2)$. Next, suppose that this process has finished, and there no more vertices $v \in A_i$ which have more than $2^{(i-j)/2} d$ neighbors in $A_j$. Apply Lemma \[ff-lemma\] to color $G[A_{k+1} \cup A_{k+2} \cup \dots ]$; this satisfies the requirements of that theorem with the parameter $x = \sqrt{2}$. So it can be colored using $O( \frac{d \log \log (d^2/y)}{\log (d^2/y)})$ colors. Finally, apply Theorem \[prop0\] to color $G[A_1 \cup \dots \cup A_k]$. This graph has maximum of $2^{k+1} \leq O(d)$ triangles per vertex, hence it can be colored using $O( \sqrt{\frac{n}{\log n}} + \frac{(n d)^{1/3}}{\log^{2/3}(n^2/d)})$ colors. Putting all these terms together, $$\chi(G) {\lesssim}\frac{d \log \log (d^2/y)}{\log (d^2/y)} + \sqrt{\frac{n}{\log n}} + \frac{(n d)^{1/3}}{\log^{2/3}(n^2/d)} + \frac{t}{d^2}$$ Now set $d = (f t)^{1/3} + \sqrt{n}$. Observe that $d^2/y \geq \frac{(t \log(t^2/y^3))^{2/3}}{y} \approx (t^{2/3}/y) \log( t^{2/3}/y)$. Thus $\log(d^2/y) {\gtrsim}f$. Similarly, $\log(n^2/d) {\gtrsim}\log n$. Then we compute these terms in turn. $$\begin{aligned} \frac{t}{d^2} &= \frac{t^{1/3}}{ (\sqrt{n} + (t \log(t^2/y^3))^{1/3} )^2 } {\lesssim}\frac{t^{1/3}}{\log^{2/3} (t^2/y^3)} \\ \frac{(n d)^{1/3}}{\log^{2/3}(n^2/d)} &{\lesssim}\frac{\sqrt{n}}{\log^{2/3} n} + \frac{n^{1/3} t^{1/9} f^{1/9}}{\log^{2/3} n} \\ \frac{d \log \log (d^2/y)}{\log (d^2/y)} &{\lesssim}\frac{d \log f}{f} {\lesssim}\frac{t^{1/3} \log f}{f^{2/3}} + \frac{ \sqrt{n} \log f}{f}\end{aligned}$$ We see that $$\chi(G) {\lesssim}\frac{t^{1/3} \log f}{f^{2/3}} + \sqrt{n} + \frac{n^{1/3} t^{1/9} f^{1/9}}{\log^{2/3} n}$$ Observe that when $t \leq n^{3/2} \sqrt{\log n}$, we have $\frac{n^{1/3} t^{1/9} f^{1/9}}{\log^{2/3} n} {\lesssim}\sqrt{\frac{n}{\log n}}$. When $t > n^{3/2} \sqrt{\log n}$, then $$\begin{aligned} \frac{n^{1/3} t^{1/9} f^{1/9}}{\log^{2/3} n} &\leq \frac{t^{1/3}}{f^{2/3}} \times \frac{f^{7/9} n^{1/3}}{t^{2/9} \log^{2/3} n} \leq \frac{t^{1/3}}{f^{2/3}} \times (f/\log n)^{7/9} \leq \frac{t^{1/3}}{f^{2/3}}\end{aligned}$$ Thus, either way, $\frac{n^{1/3} t^{1/9} f^{1/9}}{\log^{2/3} n}$ is dominated by the other two terms. So we have shown that (\[trt1\]) holds. We now show the result claimed by the theorem. Let $z = t/n$, let $A$ denote the vertices of $G$ which are incident on at least $z$ triangles, and let $B$ denote the vertices incident on at most $z$ triangles. As $\|A\| {\lesssim}t/z$, Proposition \[ttprop1a\] gives $$\chi(A) {\lesssim}\sqrt{ \frac{t/z}{\log(t/z)}} + \frac{t^{1/3}}{\log^{2/3} (t^2/y^3)} = \sqrt{ \frac{n}{\log n} } + \frac{t^{1/3} \log f}{f^{2/3}}$$ By the bound (\[trt1\]) we have $$\chi(B) {\lesssim}\sqrt{ \frac{n}{\log n}} + \frac{n^{1/3} z^{1/3}}{\log^{2/3} (n^2/z)} = \sqrt{ \frac{n}{\log n}} + \frac{t^{1/3}}{\log^{2/3} (n^3/t)}$$ Now observe that $n^3/t \geq n^3/(n y) = n^2/y \geq t^{2/3}/y$, and so $\log^{2/3} (n^3/t) \approx f$. So $\chi(B) {\lesssim}\sqrt{ \frac{n}{\log n} } + \frac{t^{1/3}}{f^{2/3}}$. Putting these terms together, $$\chi(G) \leq \chi(A) + \chi(B) {\lesssim}\sqrt{ \frac{n}{\log n} } + \frac{t^{1/3} \log f}{f^{2/3}}$$ A result on independence number ------------------------------- Recently, Bohman & Mubayi [@bohman] have investigated the relation between independence number and the number of copies of $K_s$. In particular, for $s = 3$, they discuss extremal bounds relating $\alpha(G)$ and triangle count. In this section, we show that the results may be obtained as immediate corollaries of Theorem \[prop0\]. Note however that Theorem \[prop0\] requires the use of heavy-duty results of Johansson and [@aks], whereas [@bohman] uses more elementary methods. If $G$ has $n$ vertices and at most $t$ triangles, then $$\alpha(G) {\gtrsim}\begin{cases} \sqrt{n \log n} & \text{if $t \leq n^{3/2} \sqrt{\log n}$} \\ \frac{n}{t^{1/3}} \log^{2/3} (n/t^{1/3}) & \text{if $t \geq n^{3/2} \sqrt{\log n}$} \end{cases}$$ Furthermore, this bound is tight up to constant factors for any value of $n,t$. Set $f = \log(n/t^{1/3})$. To show the upper bound, let $S$ denote the set of vertices which are incident upon at most $y = 10 t/n$ triangles. Clearly, $\|S\| {\gtrsim}n$. Applying Theorem \[prop0\] to $G[S]$, we have $$\chi(G[S]) {\lesssim}\sqrt{ \frac{n}{\log n} } + \frac{t^{1/3}}{f^{2/3}}$$ Therefore, $G[S]$ has an independent set of size at least $\|S\|/\chi(G[S])$. As $\|S\| \approx n$, this achieves lower bound on $\alpha(G)$. To show tightness for $t \geq n^{3/2} \sqrt{\log n}$, use Proposition \[kim-thm2\] (which we defer to Section \[lb-sec\]) with parameters $k = n^2 f^{1/3}/t^{2/3}, i = t^{2/3}/(n f^{1/3})$. It is clear that $k {\gtrsim}1$, and the fact that $i {\gtrsim}1$ follows from $t \geq n^{3/2} \sqrt{\log n}$. Property (B4) ensures that the resulting graph has independence number $O( \sqrt{k \log k} ) {\lesssim}(n/t^{1/3}) f^{1/6} \log^{1/2} ( f^{1/3} n^2/t^{2/3})$. Note that $\log(f^{1/3} n^2/t^{2/3}) = \log (f^{1/3} e^{2 f}) {\lesssim}f$. Therefore, the resulting graph has independence number $O( \frac{n f^{2/3}}{t^{1/3}})$ as desired. To show the lower bound for $t \leq n^{3/2} \sqrt{\log n}$, we use Proposition \[kim-thm2\] with $k = n, i = 1$. Bounds in terms of $m$ ====================== In this section, we show some bounds in terms of the edge count $m$, as well as triangle count (local and global). These generalizes results of [@nilli] and [@gimbel], which show similar bounds for the chromatic number as a function of $m$ for triangle-free graphs. As for the vertex-based bounds, all our bounds will have the form $$\chi(G) {\lesssim}\frac{m^{1/3}}{\log^{2/3} m} + f(m,t,y)$$ for some function $f$. When $t, y$ are small, then $f(m,t,y)$ is negligible and so this estimate is essentially the same as if $G$ had no triangles at all. \[prop0a\] Suppose $G$ has $m$ edges and local triangle bound $y$. Then $$\chi(G) {\lesssim}\frac{m^{1/3}}{\log^{2/3} m} + \frac{m^{1/4} y^{1/4}}{\log^{3/4} (m/y)}$$ Let $f = \log (m/y)$. Let $A$ denote the vertices of degree greater than $d$, and let $B$ denote the vertices of degree $\leq d$, where $d$ is some parameter to be chosen. As $\|A\| \leq m/d$, Theorem \[prop0\] gives $$\chi(A) {\lesssim}\sqrt{ \frac{m/d}{\log(m/d)}} + \frac{(m/d)^{1/3} y^{1/3}}{\log^{2/3} ( (m/d)^2/y ) }$$ By Theorem \[aks-thm2\] we have $$\chi(B) {\lesssim}\frac{d}{\log (d^2/y)}$$ Suppose first that $y > (m \log m)^{1/3}$. Then let $d = (m y f)^{1/4}$. We have $\chi(B) {\lesssim}(m y f)^{1/4} / \log(\sqrt{f e^f}) {\lesssim}m^{1/4} y^{1/4} f^{-3/4}$. To compute $\chi(A)$: $$\begin{aligned} \chi(A) &{\lesssim}\sqrt{ \frac{m/d}{\log(m/d)}} + \frac{(m/d)^{1/3} y^{1/3}}{\log^{2/3} ( (m/d)^2/y ) } \\ &{\lesssim}\frac{m^{3/8} f^{-1/8} y^{-1/8}}{\log^{1/2} ( m^{1/2} g^{1/4} f^{-1/4})} + \frac{m^{3/8} m^{1/4} y^{1/4} f^{-1/2}}{ \log^{2/3} (e^{3/2 f} f^{-1/12})} \\ &{\lesssim}\frac{m^{3/8} f^{-1/8} y^{-1/8}}{f^{1/2}} + \frac{m^{3/8} m^{1/4} y^{1/4} f^{-1/2}}{ f^{2/3} } \\ &{\lesssim}m^{1/3} f^{-2/3} + m^{1/4} y^{1/4} f^{-3/4} \qquad \text{as $y > (m f)^{1/3}$}\end{aligned}$$ Next, suppose that $y \leq (m \log m)^{1/3}$. Then $y' = (m \log m)^{1/3}$ is also a valid upper bound on the local triangle count, and so (as we have already shown) this gives $$\begin{aligned} \chi(G) {\lesssim}\frac{m^{1/3}}{\log^{2/3} m} + \frac{m^{1/4} (m \log m)^{1/12}}{\log^{3/4} (m/(m \log m)^{1/3})} {\lesssim}\frac{m^{1/3}}{\log^{2/3} m} + \frac{m^{1/3}}{\log^{2/3} m}\end{aligned}$$ as desired. \[ttprop3\] Suppose a graph $G$ has $m$ edges, $t$ triangles, and local triangle bound $y$. Then $$\chi(G) {\lesssim}\frac{m^{1/3}}{\log^{2/3} m} + \frac{t^{1/3} \log \log(t^2/y^3)}{\log^{2/3}(t^2/y^3)}$$ Let $f = \log(t^2/y^3)$. Let $A$ denote the set of vertices incident on at least $z = \frac{f^{1/3} t^{2/3}}{m^{1/3}}$ triangles, and let $B$ denote the remaining vertices; here $z$ is a parameter to be determined. Then $\|A\| {\lesssim}t/z$. By Theorem \[ttprop2\], we have $\chi(A) {\lesssim}\frac{t^{1/3} \log f}{f} + \sqrt{ \frac{t/z}{\log(t/z)} }$. Here $t/z = (m t/\log m)^{1/3}$ and hence $\log(t/z) {\gtrsim}\log m$ so $\sqrt{ \frac{t/z}{\log(t/z)} } {\lesssim}m^{1/6} t^{1/6} / (\log m)^{2/3}$. By Theorem \[prop0a\], we have $\chi(B) {\lesssim}\frac{m^{1/3}}{\log^{2/3} m} + \frac{(m z)^{1/4}}{\log^{3/4}(m/z)}$. As $m/z = \frac{m^{4/3}}{f^{1/3} t^{2/3}} \geq \frac{m^{1/3}}{f^{1/3}}$ we have $\log(m/z) {\gtrsim}\log m$ and hence $\frac{(m z)^{1/4}}{\log^{3/4}(m/z)} {\lesssim}m^{1/6} t^{1/6} / (\log m)^{2/3}$. We now claim that $m^{1/6} t^{1/6} / (\log m)^{2/3} {\lesssim}\frac{t^{1/3} \log f}{f} + \frac{m^{1/3}}{\log^{2/3} m}$, which will give the result of this theorem. When $t \leq m$, we have $m^{1/6} t^{1/6} / (\log m)^{2/3} \leq \frac{m^{1/3}}{\log^{2/3} m}$. When $t > m$, we have that $m^{1/6} t^{1/6} / (\log m)^{2/3} \leq \frac{t^{1/3}}{\log^{2/3} m}$. As $f {\lesssim}\log m$, this shows that $m^{1/6} t^{1/6} / (\log m)^{2/3} {\lesssim}\frac{t^{1/3}}{f^{2/3}}$. Lower bounds {#lb-sec} ============ We next show matching lower bounds. We will show that Theorems \[prop0\] and \[prop0a\] are asymptotically tight for all admissible values of $m,n,y$, while Theorem \[ttprop2\] is tight up to factors of $\log \log(t^2/y^3)$ for all admissible values of $m,n,y,t$. The situation for Theorem \[ttprop3\] is slightly more complicated; in general, the bounds given by Theorem \[ttprop3\] and Theorem \[prop0a\] are incomparable. For a given value of $m,n,y,t$, we show that either Theorem \[ttprop3\] or Theorem \[prop0a\] is asymptotically tight (the latter up to a factor of $\log \log(t^2/y^3)$). The lower bounds apply even for the fractional chromatic number (this is always less than or equal to the ordinary chromatic number; see Section \[conj-sec\] for more details). We begin by recalling a result of [@kim]: \[kim-thm\] For any integer $n \geq 1$, there exists a graph $H_n$ on $n$ vertices with the following properties: 1. $H_n$ is triangle-free 2. Each vertex has degree at most $O(\sqrt{n \log n})$ 3. $\alpha(H_n) \leq O(\sqrt{n \log n})$ (where $\alpha(G)$ denotes the size of the maximum independent set of $G$) 4. $H_n$ has chromatic number $\chi(H_n) \geq \Omega(\sqrt{\frac{n}{\log n}})$. (We note that item (A2) is not stated explicitly in the paper [@kim], but can be deduced in terms of “Property 1” in that paper.) Following a strategy of [@aks], we may construct a blow-up graph, which is an version of $H_n$ inflated to have a small number of triangles, using the following construction: \[kim-thm2\] For any real numbers $k,i {\gtrsim}1$, there is a graph $H_{k,i}$ with the following properties: 1. $H_{k,i}$ contains $O(k i)$ vertices 2. Each vertex has triangle count at most $O(i^2 \sqrt{k \log k})$. 3. Each vertex has degree at most $O(i \sqrt{k \log k})$ 4. $\alpha(H_{k,i}) \leq O(\sqrt{k \log k})$. 5. The fractional chromatic number of $H_{n,i}$ satisfies $\chi_{\text{f}} (H_{k,i}) {\gtrsim}i \sqrt{ \frac{k}{\log k}}$. We first show this assuming that $k, i$ are integers. We replace each vertex of $H_k$ with a clique on $i$ vertices. For every edge $(x,y) \in H_{n}$, we place an edge between all the corresponding copies of $x, y$ in $H_{k,i}$, a total of $i^2$ edges. Now (B1) follows immediately from the fact that $H_k$ contains $k$ vertices and (B3) follows immediately from (A2). To show (B2). Suppose that $x',y',z'$ form a triangle of $H_{k,i}$, where $x'$ is some fixed vertex of $H_{k,i}$. These vertices correspond to vertices $x,y,z$ of $H_k$. If $x,y,z$ are distinct, then this would imply that there are edges $(x,y), (y,z), (x,z)$ in $H_k$ which is impossible as $H_k$ is triangle-free. In the next case, suppose that $y = z = x$. The total number of such triangles is at most $i^2$ (since $x$ is fixed from $x'$ and $y', z'$ must lie in the same clique as $x'$). In the next case, suppose $y = z \neq x$. Then there must be an edge in $H_k$ from $x$ to $y$. There are at most $O(\sqrt{k \log k})$ choices of $y$ and once $y$ is fixed, at most $i^2$ choices for $y', z'$. In the final case, suppose $y = x \neq z$. Again there must be an edge in $H_k$ from $x$ to $z$. There are at most $O(\sqrt{k \log k})$ choices for $z$ and at most $i^2$ choices for $y', z'$. Adding all these cases, we see that there are $O(i^2 \sqrt{k \log k})$ triangles. To show (B4). Observe that if $I$ is an independent set of $H_{k,i}$, then all of its vertices must correspond to distinct vertices of $H_k$, and it must correspond to an independent set of $H_k$. So $\|I\| \leq O(\sqrt{k \log k})$. The bound (B5) follows from (B1), (B4) and the general bound on fractional chromatic number $\chi_{\text{f}} (G) \geq \frac{ \|V(G)\|}{\alpha(G)}$. When $k, i$ are not integers, then apply the above construction using $\lceil i \rceil, \lceil k \rceil$ in place of $i, k$. The condition that $i, k {\gtrsim}1$ ensures that $\lceil i \rceil \approx i$ and $\lceil k \rceil \approx k$. \[lb0\] For any integers $n, y \leq n^2, t \leq n y$, there is a graph $G$ with at most $n$ vertices, at most $t$ triangles, local triangle bound $y$, and such that $$\chi(G) \geq \chi_{\text{f}}(G) {\gtrsim}\sqrt{ \frac{n}{\log n}} + \frac{t^{1/3}}{\log^{2/3} (t^2/y^3)}$$ By rescaling, it suffices to show that a graph has this value of $\chi_{\text{f}} (G)$ and $O(n)$ vertices, $O(t)$ triangles, and $O(y)$ local triangle bound. We will break this into a number of cases. Define $f = \log(t^2/y^3)$. If $f = 1$, then we need to construct a graph $G$ with $\chi_{\text{f}}(G) {\gtrsim}\sqrt{ \frac{n}{\log n} } + t^{1/3}$ and $O(n)$ vertices. If $\sqrt{n/\log n} > t^{1/3}$, then simply take $G = H_n$. If $ \sqrt{n/\log n} < t^{1/3}$, take $G$ to be the complete graph on $t^{1/3}$ vertices. This clearly has $O(t)$ triangles and $O(n)$ vertices. Also, it has a local triangle bound of $t^{2/3} < y$. So we suppose that $f \geq 1$. If $\sqrt{n/\log n} > \frac{t^{1/3}}{f^{2/3}}$, this is easily achieved by taking $G = H_n$. Thus we assume throughout that $\sqrt{n / \log n} \leq \frac{t^{1/3}}{f^{2/3}}$. Note that as $t \leq n y$ we have $\frac{t^{1/3}}{f^{2/3}} \leq \frac{ (n y)^{1/3}}{\log^{2/3}(n^2/y)}$. We thus claim that $y \geq \sqrt{n \log n}$. For, $\frac{ (n y)^{1/3}}{\log^{2/3}(n^2/y)}$ is an increasing function of $y$, and so if $y < \sqrt{n \log n}$ then we would have $\frac{ (n y)^{1/3}}{\log^{2/3}(n^2/y)} < \sqrt{ \frac{n}{\log n}} \leq \frac{t^{1/3}}{f^{2/3}}$, which we have already ruled out. Then apply Proposition \[kim-thm2\] with $$i = \frac{y}{f^{1/3} t^{1/3}}, \qquad k = \frac{f^{1/3} t^{4/3}}{y^2}$$ We must first show that $i,k {\gtrsim}1$. Observe that $\frac{y}{f^{1/3} t^{1/3}}$ is a decreasing function of $t$, and as $t \leq n y$ this implies that $\frac{y}{f^{1/3} t^{1/3}} \geq \frac{y^{2/3}}{n^{1/3} \log(n^2/y)^{1/3}}$. Our assumption that $y \geq \sqrt{n \log n}$ implies that this is at least $1$. Similarly, the assumption that $t^2 \geq y^3$ implies that $ \frac{f^{1/3} t^{4/3}}{y^2} \geq 1$. With these observations, one can immediately observe that $H_{k,i}$ has $O(t)$ triangles, $O(y)$ triangles per vertex, and $O(t y)$ vertices, and $\chi_{\text{f}}(H_{k,i}) \geq \Omega( i \sqrt{ \frac{k}{\log k}} )$. Observe that $k = t^2/y^3 \times \frac{f^{1/3} y}{t^{2/3}} \leq f \log^{1/3} f$. Thus $\log k = \Theta(f)$. So $i \sqrt{ \frac{k}{\log k}} = \Omega(t^{1/3}/f^{2/3})$ as desired. Finally, observe that $H_{k,i}$ has $O(i k) {\lesssim}t y {\lesssim}n$ vertices. \[lb0a\] For any integers $n, y \leq n^2$, there is a graph $G$ with at most $n$ vertices and local triangle bound $y$, such that $$\chi(G) \geq \chi_{\text{f}}(G) {\gtrsim}\sqrt{ \frac{n}{\log n}} + \frac{(n y)^{1/3}}{\log^{2/3} (n^2/y)}$$ Apply Proposition \[lb0a\] with $t = n y$. We next show a lower bound on $\chi(G)$ as a function of $t, m, y$. We first observe a simple relationship that must hold between these quantities: Suppose $G$ has $m$ edges, $t$ triangles, and local triangle bound $y$. Then $t \leq m \sqrt{y}$. Let $a_v$ denote the number of triangles incident on each vertex $v \in V$. Then $t = \frac{\sum_v a_v}{3}$. Also, we have $a_v \leq y$ and $a_v \leq \binom{d_v}{2}$ for each vertex $v$, where $d_v$ denotes the degree of $v$. Finally, $\sum_v d_v = 2 m$. Thus, we have that $$t \leq \frac{\sum_v \min(d^2/2, y)}{3}$$ Note that $\min(\binom{x}{2}, y) \leq \frac{x y}{\sqrt{2 y}}$, so $t \leq \sum_v \frac{d_v y}{3 \sqrt{2 y}} \leq \frac{2 m y}{3 \sqrt{2 y}} \leq m \sqrt{y}$. Given any integers $m, y \leq m, t \leq m \sqrt{y}$, there is a graph $G$ with at most $m$ edges, at most $t$ triangles, local triangle bound $y$, and $$\chi(G) \geq \chi_{\text{f}}(G) {\gtrsim}\min \Bigl( \frac{(m y)^{1/4}}{\log^{3/4}(m/y)}, \frac{t^{1/3}}{\log^{2/3}(t^2/y^3)} \Bigr) + \frac{m^{1/3}}{\log^{2/3} m}$$ Let $f = \log(m/y)$ and let $g = \log(t^2/y^3)$. Case I: $\bm{t \geq \frac{g^2 (m y)^{3/4}}{f^{9/4}}}$. Then we need to construct $G$ such that $\chi_{\text{f}} (G) {\gtrsim}\frac{(m y)^{1/4}}{f^{3/4}} + \frac{m^{1/3}}{\log^{2/3} m}$. We can construct a triangle-free graph with $\Omega(m^{1/3}/\log^{2/3} m)$ edges, so we may assume that $ y \geq \frac{f^3 m^{1/3}}{\log^{8/3} m}$. Apply Proposition \[kim-thm2\] with $$i = \frac{y^{3/4}}{(m f)^{1/4}} \qquad k = m/y$$ We must show that $i, k {\gtrsim}1$. Clearly $k \geq 1$. If $y \geq \sqrt{m}$, then $i \geq m^{1/8}/f^{1/4} {\gtrsim}1$; otherwise, then $f \approx \log m$. Using the bound $y \geq \frac{f^3 m^{1/3}}{\log^{8/3} m}$, we see that $i \geq \frac{f^2}{\log^2 m} {\gtrsim}1$. One may easily verify that this graph has $m$ edges and local triangle bound $y$. It has at most $i k y$ triangles. As $t \geq \frac{g^2 (m y)^{3/4}}{f^{9/4}}$, it suffices to show that $i k y {\lesssim}\frac{g^2 (m y)^{3/4}}{f^{9/4}}$. This reduces to showing that $f {\lesssim}g$. To see this, observe that $t^2/y^3 \geq \frac{g^4 (m/y)^{3/2}}{f^{9/2}} \geq \frac{g^4 \exp(f)}{f^{9/2}}$. This implies that $\log(t^2/y^3) {\gtrsim}\log( e^{f/2}) {\gtrsim}f$ as desired. Case II: $\bm{t \leq \frac{g^2 (m y)^{3/4}}{f^{9/4}}}$. Then we need to construct $G$ such that $\chi_{\text{f}} (G) {\gtrsim}\frac{t^{1/3}}{g^{2/3}} + \frac{m^{1/3}}{\log^{2/3} m} $. We can construct a triangle-free graph with $\Omega(m^{1/3}/\log^{2/3} m)$ edges, so we may assume that $ t \geq \frac{m g^2}{\log^2 m}$. If $y \geq t^{2/3}$, then $g = 1$ and our goal is achieved by taking $G$ to be a clique on $\sqrt{y}$ vertices. So we assume $y \leq t^{2/3}$. Apply Proposition \[kim-thm2\] with $$i = \frac{y}{(g t)^{1/3}}, k = \frac{g^{1/3} t^{4/3}}{y^2}$$ The bound $k {\gtrsim}1$ follows immediately from the bound $y \leq t^{2/3}$. If $y > m^{0.51}$, then $i \geq \frac{m^{0.51}}{ (g m^{3/2})^{1/3}} {\gtrsim}1$. Let us thus assume that $y \leq m^{0.51}$, in which case $f = \log(m/y) \approx \log m$. Note that we have the two bounds $$\frac{m g^2}{\log^2 m} \leq t \leq \frac{g^2 (m y)^{3/4}}{f^{9/4}}$$ These together imply that $y \geq \frac{f^3 m^{1/3}}{\log^{8/3} m} {\gtrsim}\frac{(\log m)^3 m^{1/3}}{\log^{8/3} m} = (m \log m)^{1/3}$. So $$i {\gtrsim}\frac{(m \log m)^{1/3}}{(g t)^{1/3}} \geq \frac{(m \log m)^{1/3}}{((\log m) m^{3/2})^{1/3}} = 1$$ We easily verify that this graph has $O(t)$ triangles and local triangle bound $O(y)$. Finally, we need to verify that $G$ has $O(m)$ edges. Property (B3) implies that $G$ has at most $i k \times O(i \sqrt{k \log k})$ edges. Note that here $k = g^{1/3} t^{4/3}/y^2 = t^2/y^3 \times g \frac{y}{t^{2/3}} = \sqrt{\exp(g)}/g$. So $\log k \approx g$. So $G$ has $O(\frac{g^{1/3} t^{4/3} / y} {\lesssim}(g/f)^3 m$ edges. So, we need to show that $g {\lesssim}f$; we assume without loss of generality that $g$ is large than any fixed constant. Now observe that $t^2/y^3 \leq \frac{g^4 (m/y)^{3/2}}{f^{9/2}} \leq 2 g^4 \exp(f)$. Taking logarithms (and noting that $\log$ agrees with $\ln$ for $g$ sufficiently large), we see that $g \leq \log 2 + 4 \log g + f$. As $g - 4 \log g - \log 2 \approx g$ for $g$ sufficiently large, this implies that $g {\lesssim}f$ as desired. Getting the correct coefficient of $t^{1/3}$ ============================================ Suppose we have no information on the local triangle counts; in this case, Theorem \[ttprop2\] would give us a bound (solely in terms of $n$ and $t$) of the form $$\chi(G) \leq O( t^{1/3} + \sqrt{\frac{n}{\log n}} )$$ for a graph with $n$ vertices and $t$ triangles. This bound is clearly tight up to constant factors. In this section, we compute a more exact formula, which gives us the correct coefficient of the term $t^{1/3}$. The correct coefficient of the term $\sqrt{\frac{n}{\log n}}$ is not currently known even for triangle-free graphs. We begin by showing a weaker bound. \[tw-prop1\] If $G$ has $n$ vertices and $t$ triangles, then $\chi(G) \leq 100 \sqrt{n} + (6 t)^{1/3}$. Let us define $f(x,y) = 100 \sqrt{n} + (6 t)^{1/3}$, and let $d = \lfloor f(n, t) \rfloor$. Suppose that some vertex $v$ of $G$ has degree strictly less than $d$. In that case, apply the induction hypothesis to $G - v$, obtaining a coloring using at most $\lfloor f(n-1, t) \rfloor \leq d$ colors. At least one of those colors does not appear in a neighbor of $v$, so we can extend the coloring to $v$ as desired. So we suppose that $G$ has minimum degree $d$. Among all vertices, select one vertex $v$ which participates in the minimum number $y$ of triangles. Select a set $Y$ consisting of exactly $d$ neighbors of $v$; let $r$ denote the number of edges in $G[Y]$. It must be the case that $r \leq y$ and every vertex in $G[Y]$ is incident upon at least $y$ triangles. By Theorem \[turan-thm\], $G[Y]$ contains an independent set $I$ of size at least $\frac{d}{2 r/d + 1} \geq s$, where we define $s = \frac{d}{2 y/d + 1}$. Assign all vertices in $I$ one new color, and recurse on $G - I$. As $v$ is chosen to be incident on the minimum number of triangles, every vertex of $I$ is itself incident on at least $y$ triangles; furthermore as $I$ is independent these triangles are all distinct. So $I$ is incident on at least $y s$ triangles, which are all removed in $G - I$. Thus $G - I$ has at most $n - s$ vertices and at most $t - y s$ triangles. By induction hypothesis $\chi(G - I) \leq f(n - s, t - y s)$ and so $$\begin{aligned} \chi(G) &\leq 1 + f(n - s, t - y s) = 1 + 100 \sqrt{n - s} + 6^{1/3} (t - y s)^{1/3} \\ &\leq 1 + 1 + 100 \sqrt{n} - \frac{50 s}{\sqrt{n}} + 6^{1/3} (t^{1/3} - \frac{y s}{3 t^{2/3}}) = f(n,t) + 1 - \frac{50 s}{\sqrt{n}} - \frac{6^{1/3} y s}{3 t^{2/3}}\end{aligned}$$ We want to show that $\chi(G) \leq f(n,t)$; thus, it suffices to show that $$\label{yy1} \frac{50 s}{\sqrt{n}} + \frac{6^{1/3} y s}{3 t^{2/3}} \geq 1$$ Substituting in the value for $s$, simple algebraic manipulations show that this is equivalent to showing: $$\label{yy2} d^2 ( \frac{150}{\sqrt{n}} + \frac{6^{1/3} y}{t^{2/3}}) \geq 3 (d + 2 y)$$ Now, observe that the LHS and RHS of (\[yy2\]) are both linear functions of $y$. As $y$ is a non-negative real number, it suffices to show that (\[yy2\]) holds at the extreme values $y = 0, y \rightarrow \infty$ respectively. When $y = 0$, then (\[yy2\]) reduces to: $$\label{yy3} d \frac{150}{\sqrt{n}} \geq 3$$ We have $d \geq \lfloor 100 \sqrt{n} \rfloor \geq 50 \sqrt{n}$. Thus (\[yy3\]) easily holds. When $y \rightarrow \infty$, then (\[yy2\]) reduces to: $$\label{yy4} d^2 (\frac{6^{1/3}}{t^{2/3}}) \geq 6$$ Note that $d \geq (6 t)^{1/3}$, and so again this is easily seen to hold. This completes the induction. \[t-prop2\] Suppose that $G$ contains $t$ triangles and $n$ vertices. Then $$\chi(G) \leq O \Bigl( \sqrt{\frac{n}{\log n}} + \frac{t^{1/3} (\log \log n)^{3/2}}{\log n} \Bigr) + (6 t)^{1/3} = O( \sqrt{\frac{n}{\log n}} ) + (6^{1/3} + o(1)) t^{1/3}$$ Let $y = t^{1/3} \log^2 n$. Let $A$ denote the vertices of $G$ which are incident on at least $y$ triangles, and let $a$ be the number of triangles contained in $G[A]$. As $\|A\| \leq 3 t/y$, Proposition \[tw-prop1\] shows that $\chi(A) \leq O( \sqrt{t/y} )+ (6 a)^{1/3}$. Let $B = V - A$, and suppose $G[B]$ contains $b$ triangles. By Theorem \[ttprop2\] we have $$\chi(B) {\lesssim}\sqrt{\frac{n}{\log n}} + \frac{(t-a)^{1/3} \log \log((t-a)^2/y^3)}{\log^{2/3}((t-a)^2/y^3)}$$ Putting these terms together, we have $$\chi(G) \leq (6 a)^{1/3} + O \Bigl( \sqrt{\frac{n}{\log n}} + \frac{b^{1/3} \log \log(b^2/y^3)}{\log^{2/3}(b^2/y^3)} + \frac{t^{1/3}}{\log n} \Bigr)$$ As $a + b \leq t$, we have $$\label{b-e0} \chi(G) \leq (6 t)^{1/3} + O \Bigl( \sqrt{\frac{n}{\log n}} - \frac{2^{1/3} b}{3^{2/3} t^{2/3}} + \frac{b^{1/3} \log \log(b^2/y^3)}{\log^{2/3}(b^2/y^3)} + \frac{t^{1/3}}{\log n} \Bigr)$$ Our next task is to show an upper bound on the quantity $$\label{b-e1} -\frac{2^{1/3} b}{3^{2/3} t^{2/3}} + \frac{b^{1/3} \log \log(b^2/y^3)}{\log^{2/3}(b^2/y^3)}$$ If $b \leq t/\log^3 n$, this this is clearly at most $\frac{t^{1/3}}{\log n}$. Otherwise, we have $b^2/y^3 \geq t / \log^{12} n$. Again, if $t \leq n$ then Theorem \[t-prop2\] easily holds, so $\log (b^2/y^3) {\gtrsim}\log n$, and $$-\frac{2^{1/3} b}{3^{2/3} t^{2/3}} + \frac{b^{1/3} \log \log(b^2/y^3)}{\log^{2/3}(b^2/y^3)} \leq O( \frac{b}{t^{2/3}} + \frac{b^{1/3} \log \log n}{\log^{2/3} n})$$ and simple calculus shows that is bounded by $O( \frac{t^{1/3} (\log \log n)^{3/2}}{\log n} )$. Substituting this bound into (\[b-e0\]) gives the claimed result. \[t-prop3\] Suppose that $G$ contains $t$ triangles and $m$ edges. Then $$\chi(G) \leq O \Bigl( \frac{m^{1/3}}{\log^{2/3} m} + \frac{t^{1/3} (\log \log m)^{3/2}}{\log m} \Bigr) + (6 t)^{1/3} = O \Bigl( \frac{m^{1/3}}{\log^{2/3} m} \Bigr) + (6^{1/3} + o(1)) t^{1/3}$$ The proof is nearly identical to Theorem \[t-prop2\], except that we apply Theorem \[ttprop3\] instead of Theorem \[ttprop2\]. By considering a clique of $(6 t)^{1/3}$ vertices or a triangle-free graph, we can easily see that Theorems \[t-prop2\] and \[t-prop3\] are tight up to lower-order terms. Conjectured tight bounds {#conj-sec} ======================== The following conjecture seems natural: \[conj1\] Suppose $G$ has $n$ vertices, $t$ triangles, and local triangle bound $y$. Then $$\chi(G) {\lesssim}\frac{t^{1/3}}{\log^{2/3} (t^2/y^3) } + \sqrt{ \frac{n}{\log n}}$$ This conjecture would strengthen Theorem \[ttprop2\], would give Proposition \[prop0\] as a special case (as $t \leq n y$), and would match the lower bound Proposition \[lb0\]. As further evidence for Conjecture \[conj1\], we show that it relates to an open problem of Johnson [@johnson] regarding fractional chromatic number and Hall ratio, two graph parameters we will define next. For any graph $G$ and any weight function $w: V \rightarrow \mathbf R_{+}$ on its vertices, one may define the weighted independence number $\alpha(G,w)$ as the largest possible value of $\sum_{v \in I} w(v)$ over all independent sets $I$; note that when $w$ is the constant function this is the ordinary independence number. We may then define the fractional chromatic number of $G$ as $$\chi_{\text{f}}(G) = \sup_{w: V \rightarrow \mathbf R_{+}} \frac{ w(V) }{ \alpha(G,w) }$$ where the supremum is taken over all such weighting functions and we define $w(V) := \sum_{v \in V} w(v)$. A similar graph parameter is the Hall ratio, which is defined as $$\rho(G) = \sup_{U \subseteq V} \frac{ \|U\| }{ \alpha(G[U]) }$$ where the supremum is taken over all vertex subsets $U$. Note that every subset $U \subseteq V$ corresponds to a weighting function $w$ which is the indicator function of $U$; thus $\rho(G) \leq \chi_{\text{f}}(G)$. Also, given a weighting function $w$ and an $r$-coloring of $G$, the lowest-weight color class is an independent set of weight at most $w(V)/r$; thus $\chi_{\text{f}}(G) \leq \chi(G)$. In [@johnson], Johnson investigated the relation between $\chi_{\text{f}}(G)$ and $\rho(G)$; further details appear in [@barnett]. One intriguing open problem of that paper is whether $\chi_{\text{f}}(G)/\rho(G)$ is unbounded. Johnson conjectured that it was not, but did not have any counterexamples. Motivated by this conjecture (and the failure to find a counter-example), we make the opposite conjecture to Johnson. \[rho-conjecture\] For any graph $G$, we have $\chi_{\text{f}}(G) \leq O( \rho(G) )$. In this section, we will show how Conjecture \[rho-conjecture\] implies Conjecture \[conj1\]. \[rhobound0\] Suppose that $G$ is triangle-free and is $d$-degenerate. Then $\rho(G) {\lesssim}\frac{d}{\log d}$. Let $U \subseteq V$. Since $G$ is $d$-degenerate, $U$ has average degree $d$. Let $W \subseteq U$ be chosen so that $\|W\| \geq \|U\|/2$ and $\|W\|$ has maximum degree $2 d$. Since $G[W]$ is triangle-free we have $$\chi(G[W]) {\lesssim}\frac{d}{\log d}$$ Thus, $$\alpha(G) \geq \alpha(G[U]) \geq \alpha(G[W]) {\gtrsim}\frac{ \|W\| }{ \frac{d}{\log d} } {\gtrsim}\frac{ \|U\| }{ \frac{d}{\log d} }$$ \[fconj\] Suppose that $G$ is triangle-free and is $d$-degenerate. Then $\chi_{\text{f}}(G) {\lesssim}\frac{d}{\log d}$. Conjecture \[fconj\] follows immediately from Proposition \[rhobound0\] and Conjecture \[rho-conjecture\]. As another piece of evidence for Conjecture \[fconj\], we note that [@aks] showed that a triangle-free, $d$-degenerate graph may have $\chi(G)$ as large as $d$; the graph $G$ which achieves this indeed has $\chi_{\text{f}}(G) {\lesssim}\frac{d}{\log d}$. Regardless of the truth of the Conjecture \[rho-conjecture\] (and heuristically, the viewpoint of Johnson seems more reasonable), we believe that Conjecture \[fconj\] is natural and intriguing. \[fconj1\] Suppose that Conjecture \[fconj\] holds. Suppose that $G$ is $d$-degenerate and has local triangle bound $y$. Then $\chi_{\text{f}}(G) {\lesssim}\frac{d}{\log(d^2/y)}$. We need to show that there is a probability distribution $\Omega$ over independent sets, such that when $I \sim \Omega$ every vertex $v \in V$ has $P(v \in I) {\gtrsim}\frac{\log(d^2/y)}{d}$. Since $G$ is $d$-degenerate, we may fix an orientation of the edges such that each vertex has out-degree at most $d$. Consider the following two-part process. First, let us draw a vertex subset $U \subseteq V$, in which each vertex goes into $U$ with probability $0.1/\sqrt{y}$. Next, if there any vertex $u$ in $G[U]$ participates in a triangle, or has more than $d/\sqrt{y}$ out-neighbors also in $U$, then we discard $u$; let $W \subseteq U$ be the resulting vertex set after discarding. By construction, $G[W]$ is triangle-free and has an orientation with maximum out-degree $d/\sqrt{y}$. Therefore, $G[W]$ is $r$-degenerate for $r = O(d/\sqrt{y})$. Using Conjecture \[fconj\], select an independent set $I \subseteq W$ wherein every vertex $w \in W$ has $P(w \in I) {\gtrsim}\frac{\log r}{r}$. Consider now some vertex $v \in V$. This will survive to $W$ if $v \in U$, and $v$ has no triangles in $G[U]$, and $v$ has at most $d/\sqrt{y}$ out-neighbors also in $U$. The probability that $v \in U$ is $0.1/\sqrt{y}$. Conditional on this event, the expected number of out-neighbors of $v$ going into $U$ is at most $0.1 d/\sqrt{y}$ and the expected number of triangles of $v$ surviving to $G[U]$ is at most $y \times (0.1/\sqrt{y})^2$. Therefore, by Markov’s inequality, the probability that $v$ is discarded from $U$ is at most $1/10 + 1/100$. In total, we see that $P(v \in W) {\gtrsim}1/\sqrt{y}$. Conditional on $v \in W$, we have $P( v \in I) {\gtrsim}\frac{\log r}{r} \approx \frac{ \log(d^2/y) }{d/\sqrt{y}}$. Overall, $$P(v \in I) {\gtrsim}\frac{1}{\sqrt{y}} \times \frac{ \log(d^2/y) }{d/\sqrt{y}} = \frac{\log(d^2/y)}{d}$$ \[ttprop1a\] Suppose a graph $G$ has $n$ vertices, $t$ triangles, and local triangle bound $y$. If Conjecture \[fconj\] holds, then $$\chi(G) {\lesssim}\sqrt{n} + \frac{t^{1/3}}{\log^{2/3}(t^2/y^3)}$$ We will show that $$\label{trt2} \chi(G) \leq C \Bigl( \sqrt{n} + \frac{t^{1/3}}{\log^{2/3}(t^2/y^3)} \Bigr)$$ by induction on $n$, where $C$ is some sufficiently large universal constant. Let us set $f = \log(t^2/y^3)$ and $d = (t f)^{1/3} + \sqrt{n}$. We may assume that $n, t$, and $f$ are larger than any needed constant; otherwise, this follows from Theorem \[ttprop2\] if the constant $C$ is sufficiently large. We will first show that there is an independent set $I$ of $G$ which is incident upon either $\Omega(\sqrt{n})$ vertices or $\Omega((t f)^{2/3})$ triangles. In order to show this, we consider two cases: whether $G$ is or is not $d$-degenerate. CASE I: $\bm{G}$ is not $\bm{d}$-degenerate. Then there is some set of vertices $U$ such that $G[U]$ has minimum degree at $d$. Let $w \in U$ be the vertex of $U$ which is incident upon the fewest triangles of $G$; suppose that $w$ is incident upon $k$ triangles and has degree at least $d$. If $k \leq d$, then Turán’s Theorem gives an independent set $I \subseteq N(w) \cap U$ with $\|I\| \geq \frac{d}{1 + \frac{k}{d}} {\gtrsim}d \geq \sqrt{n}$, as desired. If $k \leq d$, then Turán’s Theorem gives an independent set $I \subseteq N(w) \cap U$ with $\|I\| \geq \frac{d}{1 + \frac{k}{d}} {\gtrsim}d^2/k$. Since $w$ is incident on the smallest number of triangles of $U$, every vertex in $I$ must be incident upon at least $k$ triangles. So $I$ is incident upon at least $\|I\| k {\gtrsim}d^2 \geq (t f)^{2/3}$ triangles, as desired. CASE II: $\bm{G}$ is $\bm{d}$-degenerate and $\bm{\sqrt{n} < (t f)^{1/3}}$. Define $w(v)$ to be the number of triangles incident upon $v$. By Proposition \[fconj1\], there is an independent set $I$ with $\sum_{v \in I} w(v) {\gtrsim}\frac{\sum_{v \in V} w(v)}{d/\log(d^2/y)} {\gtrsim}\frac{t \log(d^2/y)}{d}$. As $(t f)^{1/3} > \sqrt{n \log n}$, observe that $$\frac{\log (d^2/y)}{d} {\gtrsim}\frac{\log((t f)^{2/3}/y)}{(t f)^{1/3}} {\gtrsim}\frac{\log(e^f f^2)}{(t f)^{1/3}} \approx \frac{f^{2/3}}{t^{1/3}}$$ Thus, the number of triangles incident upon the vertices of $I$ is at least $\sum_{v \in I} w(v) {\gtrsim}(t f)^{2/3}$. CASE III: $\bm{G}$ is $\bm{d}$-degenerate and $\bm{\sqrt{n} \geq (t f)^{1/3}}$. Since $G$ is $d$-degenerate, we have $\chi(G) \leq d$. Hence $G$ has an independent set of size at least $n/d {\gtrsim}\sqrt{n}$. To finish the proof, let $I$ be an independent set incident on either $\Omega(\sqrt{n})$ vertices or $\Omega((t f)^{2/3})$ triangles. We assign all the vertices in $I$ one new color, remove $I$ from the graph, and apply our induction hypothesis to color $G - I$. If $\|I\| \geq c \sqrt{n}$, this gives $$\chi(G) \leq 1 + C ( \sqrt{n - c \sqrt{n}} + \frac{t^{1/3}}{\log^{2/3}(t^2/y^3)} ) \leq 1 + C ( \sqrt{n} + \frac{t^{1/3}}{\log^{2/3}(t^2/y^3)} ) - \frac{c \sqrt{n}}{2 \sqrt{n}}$$ This is at most $C (\sqrt{n} + \frac{t^{1/3}}{\log^{2/3}(t^2/y^3)} )$ for $C$ a sufficiently large constant, and the induction proceeds in this case. Next, suppose that $I$ is incident upon at least $r$ triangles, where $r = c (t f)^{2/3}$ for some constant $c$. These triangles are all removed in $G - I$, and so by induction hypothesis we have $$\chi(G) \leq 1 + C ( \sqrt{n} + \frac{(t - r)^{1/3}}{\log^{2/3}((t - r)^2/y^3)} )$$ Now consider the function $F(x) = x^{1/3}/\log^{2/3} (x^2/y^3)$. For $x \geq e^2 y^{3/2}$, this function is increasing concave-down. We claim that $t - r \geq e^2 y^{3/2}$. For, observe that $r {\lesssim}t^{0.9}$ and so $t - r \geq t/2$ for $t$ above a sufficiently large constant. Thus, it suffices to show that $t \geq 2 e^2 y^{3/2}$, which follows from our assumption that $f$ exceeds any sufficiently large constant. Thus, we may upper-bound the expression $F(t-r)$ by its tangent line $F(t) - r F'(t)$, yielding $$\chi(G) \leq 1 + C ( \sqrt{n} + F(t) ) \leq 1 + C( \sqrt{n} + \frac{t^{1/3}}{f^{2/3}} - r \frac{ f - 4 }{t^{2/3} f^2} )$$ To show that $\chi(G) \leq C ( \sqrt{n} + \frac{t^{1/3}}{f^{2/3}})$ it thus suffices to show that $$1 - C c \frac{(t f)^{2/3} (f - 4)}{3 t^{2/3} f^{5/3}} ) \leq 0$$ By taking $C$ to be sufficiently large, this follows immediately from our assumption that $f$ is larger than any needed constant. Thus the induction again holds in this case. From Proposition \[ttprop1a\], we can tighten many of the bounds in our paper. We omit the proofs since they are essentially identical to proofs we have already encountered. Suppose a graph $G$ has $n$ vertices, $m$ edges, $t$ triangles, and local triangle bound $y$. If Conjecture \[fconj\] holds then we have the following bounds: 1. $\chi(G) {\lesssim}\sqrt{ \frac{n}{\log n}} + \frac{t^{1/3}}{\log^{2/3}(t^2/y^3)}$. 2. $\chi(G) {\lesssim}\frac{t^{1/3}}{\log^{2/3}(t^2/y^3)} + \frac{m^{1/3}}{\log^{2/3} m}$ 3. $\chi(G) \leq O \Bigl( \sqrt{\frac{n}{\log n}} + \frac{t^{1/3}}{\log n} \Bigr) + (6 t)^{1/3} = O( \sqrt{\frac{n}{\log n}} ) + (6^{1/3} + o(1)) t^{1/3}$ Acknowledgments =============== Thanks to Vance Faber, Paul Burkhardt, Louis Ibarra, and Aravind Srinivasan for helpful suggestions, discussions, and proofreading. Thanks to Seth Pettie for some clarifications about Theorem \[triangle-free-d\]. Thanks to Hsin-Hao Su for some discussions about fractional chromatic number. [1]{} Ajtai, M., Komlós, J., Szemerédi, E.: A note on Ramsey numbers. Journal of Combinatorial Theory Series A 29, pp. 354-360 (1980) Alon, N., Krivelevich, M., Sudakov, B.: Coloring graphs with sparse neighborhoods. Journal of Combinatorial Theory, Series B 77.1, pp. 73-82 (1999) Barnett, J: The fractional chromatic number and the Hall ratio. PhD Thesis (2016) Bohman, T., Mubayi, D.: Independence number of graphs with a prescribed number of cliques. arxiv:1801.01091 (2018) Gimbel, J., Thomassen, C.: Coloring triangle-free graphs with fixed size. Discrete Mathematics 219-1, pp. 275-277 (2000) Johnson, P. D.: The fractional chromatic number, the Hall ratio, and the lexicographic product. Discrete Mathematics 309-14, pp. 4746-4749 (2009) Kim, J. H.: The Ramsey number $R(3,t)$ has order of magnitude $t^2/\log t$. Random Structures & Algorithms 7-3, pp. 173-207 (1995). Molloy, M., Reed, B.: Graph colouring and the probabilistic method. Algorithms and Combinatorics, Springer (2001) Molloy, M.: The list chromatic number of graphs with small clique number. arxiv:1701.09133 (2017) Moser, R., Tardos, G.: A constructive proof of the general Lovász Local Lemma. Journal of the ACM 57.2, p. 11 (2010) Nilli, A.: Triangle-free graphs with large chromatic number. Discrete Mathematics 211-1, pp. 261-262 (2000) Pettie, S., Su, H.: Distributed coloring algorithms for triangle-free graphs. Information and Computation 243, pp, 263-280 (2015) Vu, V.: A general upper bound on the list chromatic number of locally sparse graphs. Combinatorics, Probability, and Computing 11, pp. 103-111 (2002) [^1]: Department of Computer Science, University of Maryland, College Park, MD 20742. Email: `davidgharris29@gmail.com` [^2]: Theorem \[triangle-free-d\] is attributed to Johansson, as attributed by [@molloy]. See [@molloy2] for a more recent proof, which also gives bounds on the constant term. [^3]: For the remainder of this paper, we will not explicitly state that the colorings can be produced in expected polynomial time, if it is clear from context.
--- abstract: \| We show that the logarithmic version of the syntomic cohomology of Fontaine and Messing for semistable varieties over $p$-adic rings extends uniquely to a cohomology theory for varieties over $p$-adic fields that satisfies $h$-descent. This new cohomology - syntomic cohomology - is a Bloch-Ogus cohomology theory, admits period map to étale cohomology, and has a syntomic descent spectral sequence (from an algebraic closure of the given field to the field itself) that is compatible with the Hochschild-Serre spectral sequence on the étale side. In relative dimension zero we recover the potentially semistable Selmer groups and, as an application, we prove that Soulé’s étale regulators land in the potentially semistable Selmer groups. Our construction of syntomic cohomology is based on new ideas and techniques developed by Beilinson and Bhatt in their recent work on $p$-adic comparison theorems. author: - 'Jan Nekovář, Wies[ł]{}awa Nizio[ł]{}' title: 'Syntomic cohomology and $p$-adic regulators for varieties over $p$-adic fields' --- [^1] Introduction ============ In this article we define syntomic cohomology for varieties over $p$-adic fields and use it to study the images of Soulé’s étale regulators. Contrary to all the previous constructions of syntomic cohomology (see below for a brief review) we do not restrict ourselves to varieties coming with a nice model over the integers. Statement of the main result ---------------------------- Recall that, for varieties proper and smooth over a $p$-adic ring of mixed characteristic, syntomic cohomology (or its non-proper variant: syntomic-étale cohomology) was introduced by Fontaine and Messing [@FM] in their proof of the Crystalline Comparison Theorem as a natural bridge between crystalline cohomology and étale cohomology. It was generalized to log-syntomic cohomology for semistable varieties by Kato [@Kas]. For a log-smooth scheme $X$ over a complete discrete valuation ring $V$ of mixed characteristic $(0,p)$ and a perfect residue field, and for any $r\geq 0$, rational log-syntomic cohomology of $X$ can be defined as the “filtered Frobenius eigenspace” in log-crystalline cohomology, i.e., as the following mapping fiber $$\label{first} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X,r):={\operatorname{Cone} }\big({\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X,{{\mathcal J}}^{[r]})\lomapr{1-{\varphi}_r}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X)\big)[-1],$$ where ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(\cdot,{{\mathcal J}}^{[r]})$ denotes the rational log-crystalline cohomology (i.e., over ${\mathbf Z}_p)$ of the $r$’th Hodge filtration sheaf ${{\mathcal J}}^{[r]}$ and ${\varphi}_r$ is the crystalline Frobenius divided by $p^r$. This definition suggested that the log-syntomic cohomology could be the sought for $p$-adic analog of Deligne-Beilinson cohomology. Recall that, for a complex manifold $X$, the latter can be defined as the cohomology ${\mathrm {R} }\Gamma(X,{\mathbf Z}(r)_{{{\mathcal{D}}}})$ of the Deligne complex ${\mathbf Z}(r)_{{{\mathcal{D}}}}$: $$0\to {\mathbf Z}(r)\to \Omega^1_X\to\Omega^2_X\to\ldots \to \Omega^{r-1}_X\to 0$$ And, indeed, since its introduction, log-syntomic cohomology has been used with some success in the study of special values of $p$-adic $L$-functions and in formulating $p$-adic Beilinson conjectures (cf. [@BE] for a review). The syntomic cohomology theory with ${\mathbf{Q}_p}$-coefficients $R\Gamma_{{ \operatorname{syn} }}(X_h,r)$ ($r\geq 0$) for arbitrary varieties – more generally, for arbitrary essentially finite diagrams of varieties – over the $p$-adic field $K$ (the fraction field of $V$) that we construct in this article is a generalization of Fontaine-Messing(-Kato) log-syntomic cohomology. That is, for a semistable scheme [^2] ${{\mathcal{X}}}$ over $V$ we have ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}({{\mathcal{X}}},r)\simeq{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)$, where $X$ is the largest subvariety of ${{\mathcal{X}}}_K$ with trivial log-structure. An analogous theory $R\Gamma_{{ \operatorname{syn} }}(X_{{\overline{K} },h},r)$ ($r\geq 0$) exists for (diagrams of) varieties over ${\overline{K} }$, where ${\overline{K} }$ is an algebraic closure of $K$. Our main result can be stated as follows. \[main1\] For any variety $X$ over $K$, there is a canonical graded commutative dg ${\mathbf{Q}_p}$-algebra ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,)$ such that 1. it is the unique extension of log-syntomic cohomology to varieties over $K$ that satisfies $h$-descent, i.e., for any hypercovering $\pi: Y_{\jcdot}\to X$ in $h$-topology, we have a quasi-isomorphism $$\pi^:{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,)\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(Y_{\jcdot,h},).$$ 2. it is a Bloch-Ogus cohomology theory [@BO]. 3. for $X={\operatorname{Spec} }(V)$, $H^_{{ \operatorname{syn} }}(X_h,r)\simeq H^_{{\operatorname{st} }}(G_K,{{\mathbf Q}}_p(r))$, where $H^i_{{\operatorname{st} }}(G_K, -)$ denotes the ${\operatorname{Ext} }$-group ${\operatorname{Ext} }^i({\mathbf{Q}_p}, -)$ in the category of (potentially) semistable representations of $G_K={\operatorname{Gal} }({\overline{K} }/K)$. 4. There are functorial syntomic period morphisms $$\rho_{{ \operatorname{syn} }}: R\Gamma_{{ \operatorname{syn} }}(X_h,r)\to R\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)),\qquad \rho_{{ \operatorname{syn} }}: R\Gamma_{{ \operatorname{syn} }}(X_{{\overline{K} },h},r) \to R\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r))$$ compatible with products which induce quasi-isomorphisms $$\tau_{\leq r} R\Gamma_{{ \operatorname{syn} }}(X_h,r) \stackrel{\sim}{\to} \tau_{\leq r} R\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)),\qquad \tau_{\leq r} R\Gamma_{{ \operatorname{syn} }}(X_{{\overline{K} },h},r) \stackrel{\sim}{\to} \tau_{\leq r} R\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)).$$ 5. The Hochschild-Serre spectral sequence for étale cohomology $$^{{\operatorname{\acute{e}t} }}E^{i,j}_2 = H^i(G_K,H^j(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r))) \Longrightarrow H^{i+j}(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r))$$ has a syntomic analog $$^{{ \operatorname{syn} }}E^{i,j}_2 = H^i_{{\operatorname{st} }}(G_K,H^j(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r))) \Longrightarrow H^{i+j}_{{ \operatorname{syn} }}(X_{h},r).$$ 6. There is a canonical morphism of spectral sequences ${}^{{ \operatorname{syn} }}E_t \to {}^{{\operatorname{\acute{e}t} }}E_t$ compatible with the syntomic period map. 7. There are syntomic Chern classes $${c}_{i,j}^{{ \operatorname{syn} }} \colon K_j(X) \to H^{2i-j}_{{ \operatorname{syn} }}(X_{h},i)$$ compatible with étale Chern classes via the syntomic period map. Construction of syntomic cohomology ----------------------------------- We will now sketch the proof of Theorem \[main1\]. Recall first that a little bit after log-syntomic cohomology had appeared on the scene, Selmer groups of Galois representations – describing extensions in certain categories of Galois representations – were introduced by Bloch and Kato [@BK] and linked to special values of $L$-functions. And a syntomic cohomology (in the good reduction case), a priori different than that of Fontaine-Messing, was defined in [@NS] and by Besser in [@BS] as a higher dimensional analog of the complexes computing these groups. The guiding idea here was that just as Selmer groups classify extensions in certain categories of “geometric” Galois representations their higher dimensional analogs – syntomic cohomology groups – should classify extensions in a category of “$p$-adic motivic sheaves”. This was shown to be the case for $H^1$ by Bannai [@Ban] who has also shown that Besser’s (rigid) syntomic cohomology is a $p$-adic analog of Beilinson’s absolute Hodge cohomology [@BE0]. Complexes computing the semistable and potentially semistable Selmer groups were introduced in [@JH] and [@FPR]. Their higher dimensional analog can be written as the following homotopy limit $$\label{first2} {\mathrm {R} }\Gamma^{\prime}_{{ \operatorname{syn} }}(X,r):=\left[ \begin{aligned}\xymatrix@=40pt{ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_0)\ar[d]^{N}\ar[r]^-{(1-{\varphi}_r,\iota_{{\mathrm{dR}}})} & {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_0)\oplus {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_K)/F^r\ar[d]^{(N,0)}\\ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_0)\ar[r]^-{1-{\varphi}_{r-1}} & {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_0) }\end{aligned}\right]$$ where $X_0$ is the special fiber of $X$, ${\mathrm {R} }\Gamma_{{\mathrm{HK}}}(\cdot)$ is the Hyodo-Kato cohomology, $N$ denotes the Hyodo-Kato monodromy, and ${\mathrm {R} }\Gamma_{{\mathrm{dR}}}(\cdot)$ is the logarithmic de Rham cohomology. The map $\iota_{{\mathrm{dR}}}$ is the Hyodo-Kato morphism that induces a quasi-isomorphism $\iota_{{\mathrm{dR}}}:{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_0)\otimes_{K_0}K\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_K)$, for $K_0$ - the fraction field of Witt vectors of the residue field of $V$. Using Dwork’s trick, we prove (cf. Proposition \[reduction1\]) that the two definitions of log-syntomic cohomology are the same, i.e., that there is a quasi-isomorphism $$\alpha_{{ \operatorname{syn} }}: {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X,r)\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}^{\prime}(X,r).$$ It follows that log-syntomic cohomology groups vanish in degrees strictly higher than $2\dim X_K +2$ and that, if $X={\operatorname{Spec} }(V)$, then $H^i{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X,r)\simeq H^i_{{\operatorname{st} }}(G_K,{{\mathbf Q}}_p(r))$. The syntomic cohomology for varieties over $p$-adic fields that we introduce in this article is a generalization of the log-syntomic cohomology of Fontaine and Messing. Observe that it is clear how one can try to use log-syntomic cohomology to define syntomic cohomology for varieties over fields that satisfies $h$-descent. Namely, for a variety $X$ over $K$, consider the $h$-topology of $X$ and recall that (using alterations) one can show that it has a basis consisting of semistable models over finite extensions of $V$ [@BE1]. By $h$-sheafifying the complexes $Y\mapsto {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(Y,r)$ (for a semistable model $Y$) we get syntomic complexes ${{\mathcal{S}}}(r)$. We define the ([arithmetic]{}) syntomic cohomology as $${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r):={\mathrm {R} }\Gamma(X_h,{{\mathcal{S}}}(r)).$$ A priori it is not clear that the so defined syntomic cohomology behaves well: the finite ramified field extensions introduced by alterations are in general a problem for log-crystalline cohomology. For example, the related complexes ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_h,{{\mathcal J}}^{[r]})$ are huge. However, taking Frobenius eigenspaces cuts off the “noise” and the resulting syntomic complexes do indeed behave well. To get an idea why this is the case, $h$-sheafify the complexes $Y\mapsto {\mathrm {R} }\Gamma^{\prime}_{{ \operatorname{syn} }}(Y,r)$ and imagine that you can sheafify the maps $\alpha_{{ \operatorname{syn} }}$ as well. We get sheaves ${{\mathcal{S}}}^{\prime}(r)$ and quasi-isomorphisms $\alpha_{{ \operatorname{syn} }}:{{\mathcal{S}}}(r)\stackrel{\sim}{\to}{{\mathcal{S}}}^{\prime}(r)$. Setting ${\mathrm {R} }\Gamma^{\prime}_{{ \operatorname{syn} }}(X_h,r):={\mathrm {R} }\Gamma(X_h,{{\mathcal{S}}}^{\prime}(r))$ we obtain the following quasi-isomorphisms $$\label{first3} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)\simeq {\mathrm {R} }\Gamma^{\prime}_{{ \operatorname{syn} }}(X_h,r)\simeq \left[ \begin{aligned}\xymatrix@=40pt{ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_h)\ar[d]^{N}\ar[r]^-{(1-{\varphi}_r,\iota_{{\mathrm{dR}}})} & {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_h)\oplus {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_K)/F^r\ar[d]^{(N,0)}\\ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_h)\ar[r]^-{1-{\varphi}_{r-1}} & {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_h) }\end{aligned}\right]$$ where ${\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_h)$ denotes the Hyodo-Kato cohomology (defined as $h$-cohomology of the presheaf: $Y\mapsto {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(Y_0)$) and ${\mathrm {R} }\Gamma_{{\mathrm{dR}}}(\cdot)$ is the Deligne’s de Rham cohomology [@De]. The Hyodo-Kato map $\iota_{{\mathrm{dR}}}$ is the $h$-sheafification of the logarithmic Hyodo-Kato map. It is well-known that Deligne’s de Rham cohomology groups are finite rank $K$-vector spaces; it turns out that the Hyodo-Kato cohomology groups are finite rank $K_0$-vector spaces: we have a quasi-isomorphism ${\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_h)\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_{{\overline{K} },h})^{G_K}$ and the geometric Hyodo-Kato groups $H^{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_{{\overline{K} },h})$ are finite rank $K_0^{{\operatorname{nr} }}$-vector spaces, where $K_0^{{\operatorname{nr} }}$ is the maximal unramified extension of $K_0$ (see (\[trivialization\]) below). It follows that syntomic cohomology groups vanish in degrees higher than $2\dim X_K+2$ and that syntomic cohomology is, in fact, a generalization of the classical log-syntomic cohomology, i.e., for a semistable scheme ${{\mathcal{X}}}$ over $V$ we have ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}({{\mathcal{X}}},r)\simeq{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)$, where $X$ is the largest subvariety of ${{\mathcal{X}}}_K$ with trivial log-structure. This follows from the quasi-isomorphism $\alpha_{{ \operatorname{syn} }}$: logarithmic Hyodo-Kato and de Rham cohomologies (over a fixed base) satisfy proper descent and the finite fields extensions that appear as the “noise” in alterations do not destroy anything since logarithmic Hyodo-Kato and de Rham cohomologies satisfy finite Galois descent. Alas, we were not able to sheafify the map $\alpha_{{ \operatorname{syn} }}$. The reason for that is that the construction of $\alpha_{{ \operatorname{syn} }}$ uses a twist by a high power of Frobenius – a power depending on the field $K$. And alterations are going to introduce a finite extension of $K$ – hence a need for higher and higher powers of Frobenius. So instead we construct directly the map $\alpha_{{ \operatorname{syn} }}:{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)\to{\mathrm {R} }\Gamma^{\prime}_{{ \operatorname{syn} }}(X_h,r) $. To do that we show first that the syntomic cohomological dimension of $X$ is finite. Then we take a semistable $h$-hypercovering of $X$, truncate it at an appropriate level, extend the base field $K$ to $K^{\prime}$, and base-change everything to $K^{\prime}$. There we can work with one field and use the map $\alpha_{{ \operatorname{syn} }}$ defined earlier. Finally, we show that we can descend. Syntomic period maps -------------------- We pass now to the construction of the period maps from syntomic to étale cohomology that appear in Theorem \[main1\]. They are easier to define over ${\overline{K} }$, i.e., from the [geometric]{} syntomic cohomology. In this setting, things go smoother with $h$-sheafification since going all the way up to ${\overline{K} }$ before completing kills a lot of “noise” in log-crystalline cohomology. More precisely, for a semistable scheme ${{\mathcal{X}}}$ over $V$, we have the following canonical quasi-isomorphisms [@BE2] $$\label{trivialization} \iota_{{\operatorname{cr} }}: {\mathrm {R} }\Gamma_{{\mathrm{HK}}}({{\mathcal{X}}}_{{\overline{V} }})^{\tau}_{B^+_{{\operatorname{cr} }}}\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}({{\mathcal{X}}}_{{\overline{V} }}),\quad \iota_{{\mathrm{dR}}}: {\mathrm {R} }\Gamma_{{\mathrm{HK}}}({{\mathcal{X}}}_{{\overline{V} }})^{\tau}_{{\overline{K} }}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\mathrm{dR}}}({{\mathcal{X}}}_{{\overline{K} }}),$$ where ${\overline{V} }$ is the integral closure of $V$ in ${\overline{K} }$, $B^+_{{\operatorname{cr} }}$ is the crystalline period ring, and $\tau$ denotes certain twist. These quasi-isomorphisms $h$-sheafify well: for a variety $X$over $K$, they induce the following quasi-isomorphisms [@BE2] $$\label{trivialization1} \iota_{{\operatorname{cr} }}: {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_{{\overline{K} },h})^{\tau}_{B^+_{{\operatorname{cr} }}}\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_{{\overline{K} },h}),\quad \iota_{{\mathrm{dR}}}: {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_{{\overline{K} },h})^{\tau}_{{\overline{K} }}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} }}),$$ where the terms have obvious meaning. Since Deligne’s de Rham cohomology has proper descent (by definition), it follows that $h$-crystalline cohomology behaves well. That is, if we define crystalline sheaves ${{\mathcal J}}^{[r]}_{{\operatorname{cr} }}$ and ${{\mathcal{A}}}_{{\operatorname{cr} }}$ on $X_{{\overline{K} },h}$ by $h$-sheafifying the complexes $Y\mapsto {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Y,{{\mathcal J}}^{[r]})$ and $Y\mapsto {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Y)$, respectively, for $Y$ which are a base change to ${\overline{V} }$ of a semistable scheme over a finite extension of $V$ (such schemes $Y$ form a basis of $X_{{\overline{K} },h}$) then the complexes ${\mathrm {R} }\Gamma(X_{{\overline{K} },h},{{\mathcal J}}^{[r]})$ and ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_{{\overline{V} },h}):={\mathrm {R} }\Gamma(X_{{\overline{K} },h},{{\mathcal{A}}}_{{\operatorname{cr} }})$ generalize log-crystalline cohomology (in the sense described above) and the latter one is a perfect complex of $B^+_{{\operatorname{cr} }}$-modules. We obtain syntomic complexes ${{\mathcal{S}}}(r)$ on $X_{{\overline{K} },h}$ by $h$-sheafifying the complexes $Y\mapsto {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(Y,r)$ and (geometric) syntomic cohomology by setting ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{{\overline{K} },h},r):={\mathrm {R} }\Gamma(X_{{\overline{K} },h},{{\mathcal{S}}}(r))$. They fit into the analog of the exact sequence (\[first\]) and, by the above, generalize log-syntomic cohomology. To construct the syntomic period maps $$\label{periodss} \rho_{{ \operatorname{syn} }}: {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{{\overline{K} },h},r)\to {\mathrm {R} }\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p(r)),\quad \rho_{{ \operatorname{syn} }}: {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{h},r)\to {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p(r))$$ consider the syntomic complexes ${{\mathcal{S}}}_n(r)$: the mod-$p^n$ version of the syntomic complexes ${{\mathcal{S}}}(r)$ on $X_{{\overline{K} },h}$. We have the distinguished triangle $${{\mathcal{S}}}_n(r)\to {{\mathcal J}}_{{\operatorname{cr} },n}^{[r]}\lomapr{p^r-{\varphi}} {{\mathcal{A}}}_{{\operatorname{cr} },n}$$ Recall that the filtered Poincaré Lemma of Beilinson and Bhatt [@BE2], [@BH] yields a quasi-isomorphism $\rho_{{\operatorname{cr} }}: J_{{\operatorname{cr} },n}^{[r]}\stackrel{\sim}{\to} {{\mathcal J}}_{{\operatorname{cr} },n}^{[r]}$, where $J^{[r]}_{{\operatorname{cr} }}\subset A_{{\operatorname{cr} }}$ is the $r$’th filtration level of the period ring $A_{{\operatorname{cr} }}$. Using the fundamental sequence of $p$-adic Hodge Theory $$0\to {\mathbf Z}/p^n(r)^{\prime}\to J^{<r>}_{{\operatorname{cr} },n}\lomapr{1-{\varphi}_r} A_{{\operatorname{cr} },n}\to 0,$$ where ${\mathbf Z}/p^n(r)^{\prime}:=(1/(p^aa!){\mathbf Z}_p(r))\otimes {\mathbf Z}/p^n$ and $a$ denotes the largest integer $\leq r/(p-1)$, we obtain the syntomic period map $\rho_{{ \operatorname{syn} }}:{{\mathcal{S}}}_n(r)\to {\mathbf Z}/p^n(r)^{\prime}$. It is a quasi-isomorphism modulo a universal constant. It induces the geometric syntomic period map in (\[periodss\]), and, by Galois descent, its arithmetic analog. To study the descent spectral sequences from Theorem \[main1\], we need to consider the other version of syntomic cohomology, i.e., the complexes $$\label{first31} {\mathrm {R} }\Gamma^{\prime}_{{ \operatorname{syn} }}(X_{{\overline{K} },h},r):=\left[\begin{aligned}\xymatrix@C=40pt{ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_{{\overline{K} },h})\otimes_{K_0^{{\operatorname{nr} }}} B^+_{{\operatorname{st} }}\ar[r]^-{(1-{\varphi}_r,\iota_{{\mathrm{dR}}})}\ar[d]^N & {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_{{\overline{K} },h})\otimes_{K_0^{{\operatorname{nr} }}} B^+_{{\operatorname{st} }}\oplus ({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} }})\otimes _{{\overline{K} }}B^+_{{\mathrm{dR}}})/F^r\ar[d]^{(N,0)}\\ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_{{\overline{K} },h})\otimes_{K_0^{{\operatorname{nr} }}} B^+_{{\operatorname{st} }}\ar[r]^{1-{\varphi}_{r-1}} & {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_{{\overline{K} },h})\otimes _{K_0^{{\operatorname{nr} }}}B^+_{{\operatorname{st} }} }\end{aligned}\right]$$ where $B^+_{{\operatorname{st} }}$ and $B^+_{{\mathrm{dR}}}$ are the semistable and de Rham $p$-adic period rings, respectively. We also have a syntomic period map $$\label{first4} \rho_{{ \operatorname{syn} }}^{\prime}:{\mathrm {R} }\Gamma^{\prime}_{{ \operatorname{syn} }}(X_{{\overline{K} },h},r)\to {\mathrm {R} }\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p(r))$$ that is compatible with the map $\rho_{{ \operatorname{syn} }}$ via $\alpha_{{ \operatorname{syn} }}$. To describe how it is constructed, recall that the crystalline period map of Beilinson induces compatible Hyodo-Kato and de Rham period maps [@BE2] $$\label{first5} \rho_{{\mathrm{HK}}}:{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_{{\overline{K} },h})\otimes _{K_0^{{\operatorname{nr} }}}B^+_{{\operatorname{st} }}{\to} {\mathrm {R} }\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p)\otimes B_{{\operatorname{st} }}^+,\quad \rho_{{\mathrm{dR}}}: {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_K)\otimes_K B_{{\mathrm{dR}}}^+{\to}{\mathrm {R} }\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p)\otimes B_{{\mathrm{dR}}}^+$$ Applying them to the above homotopy limit, removing all the pluses from the period rings, reduces the homotopy limit to the complex $$\label{first32} \left[\begin{aligned}\xymatrix@C=40pt{ {\mathrm {R} }\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf{Q}_p}(r))\otimes B_{{\operatorname{st} }}\ar[r]^-{(1-{\varphi}_r,\iota_{{\mathrm{dR}}})}\ar[d]^N & {\mathrm {R} }\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf{Q}_p}(r))\otimes B_{{\operatorname{st} }}\oplus ({\mathrm {R} }\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf{Q}_p}(r))\otimes B_{{\mathrm{dR}}})/F^r\ar[d]^{(N,0)}\\ {\mathrm {R} }\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf{Q}_p}(r))\otimes B_{{\operatorname{st} }}\ar[r]^{1-{\varphi}_{r-1}} & {\mathrm {R} }\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf{Q}_p}(r))\otimes B_{{\operatorname{st} }} }\end{aligned}\right]$$ By the familiar fundamental exact sequence $$0\to {{\mathbf Q}}_p(r)\to B_{{\operatorname{st} }}\verylomapr{(N,1-{\varphi}_r,\iota)} B_{{\operatorname{st} }}\oplus B_{{\operatorname{st} }}\oplus B_{{\mathrm{dR}}}/F^r\veryverylomapr{(1-{\varphi}_{r-1})-N}B_{{\operatorname{st} }}\to 0$$ the above complex is quasi-isomorphic to ${\mathrm {R} }\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf{Q}_p}(r))$. This yields the syntomic period morphism from (\[first4\]). We like to think of geometric syntomic cohomology as represented by the complex from (\[first31\]) and of geometric étale cohomology as represented by the complex (\[first32\]). From the above constructions we derive several of the properties mentioned in Theorem \[main1\]. The quasi-isomorphisms (\[first5\]) give that $$H^i_{{\mathrm{HK}}}(X_{{\overline{K} },h})\simeq D_{{\mathrm{pst}}}(H^i(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf{Q}_p}(r))),\quad H^i_{{\mathrm{HK}}}(X_h)\simeq D_{{\operatorname{st} }}(H^i(X_{{\operatorname{\acute{e}t} }},{\mathbf{Q}_p}(r))),$$ where $D_{{\mathrm{pst}}}$ and $D_{{\operatorname{st} }}$ are the functors from [@FPR]. This combined with the diagram (\[first3\]) immediately yields the spectral sequence ${}^{{ \operatorname{syn} }}E_t$ since the cohomology groups of the total complex of $$\left[ \begin{aligned}\xymatrix@=40pt{ H^j_{{\mathrm{HK}}}(X_h)\ar[d]^{N}\ar[r]^-{(1-{\varphi}_r,\iota_{{\mathrm{dR}}})} & H^j_{{\mathrm{HK}}}(X_h)\oplus H^j_{{\mathrm{dR}}}(X_K)/F^r\ar[d]^{(N,0)}\\ H^j_{{\mathrm{HK}}}(X_h)\ar[r]^-{1-{\varphi}_{r-1}} & H^j_{{\mathrm{HK}}}(X_h) }\end{aligned}\right]$$ are equal to $H^_{{\operatorname{st} }}(G_K,H^j(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf{Q}_p}(r)))$. Moreover, the sequence of natural maps of diagrams $(\ref{first3})\to (\ref{first31})\stackrel{\rho_{{ \operatorname{syn} }}}{\to} (\ref{first32})$ yields a compatibility of the syntomic descent spectral sequence with the Hochschild-Serre spectral sequence in étale cohomology (via the period maps). We remark that, in the case of proper varieties with semistable reduction, this fact was announced in [@JB]. Looking again at the period map $\rho_{{ \operatorname{syn} }}: (\ref{first31}){\to} (\ref{first32})$ we see that truncating all the complexes at level $r$ will allow us to drop $+$ from the first diagram. Hence we have $$\rho_{{ \operatorname{syn} }}:\tau_{\leq r}{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{{\overline{K} },h},r)\stackrel{\sim}{\to} \tau_{\leq r}{\mathrm {R} }\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p(r))$$ To conclude that we have $$\rho_{{ \operatorname{syn} }}:\tau_{\leq r}{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)\stackrel{\sim}{\to} \tau_{\leq r}{\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p(r))$$ as well, we look at the map of spectral sequences ${}^{{ \operatorname{syn} }}E\to {}^{{\operatorname{\acute{e}t} }}E$ and observe that, in the stated ranges of the Hodge-Tate filtration we have $H^_{{\operatorname{st} }}(G_K,\cdot)=H^(G_K,\cdot)$ (a fact that follows, for example, from the work of Berger [@BER]). $p$-adic regulators ------------------- As an application of Theorem \[main1\], we look at the question of the image of Soulé’s étale regulators $$r^{{\operatorname{\acute{e}t} }}_{r,i}: K_{2r-i-1}(X)_0\to H^1(G_K,H^i(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p(r))),$$ where $K_{2r-i-1}(X)_0:=\ker(c^{{\operatorname{\acute{e}t} }}_{r,i+1}:K_{2r-i-1}(X)\to H^{i+1}(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p(r)))$, inside the Galois cohomology group. We prove that The regulators $r^{{\operatorname{\acute{e}t} }}_{r,i}$ factor through the group $H^1_{{\operatorname{st} }}(G_K,H^i(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p(r)))$. As we explain in the article, this fact is known to follow from the work of Scholl [@Sc] on “geometric” extensions associated to $K$-theory classes. In our approach, this is a simple consequence of good properties of syntomic cohomology and the existence of the syntomic descent spectral sequence. Namely, as can be easily derived from the presentation (\[first3\]), syntomic cohomology has projective space theorem and homotopy property [^3] hence admits Chern classes from higher $K$-theory. It can be easily shown that they are compatible with the étale Chern classes via the syntomic period maps. The factorization we want in the above theorem follows then from the compatibility of the two descent spectral sequences. Notation and Conventions ------------------------ Let $V$ be a complete discrete valuation ring with fraction field $K$ of characteristic 0, with perfect residue field $k$ of characteristic $p$, and with maximal ideal ${\mathfrak m}_K$. Let $v$ be the valuation on $K$ normalized so that $v(p)=1$. Let ${\overline{K} }$ be an algebraic closure of $K$ and let $\overline{V}$ denote the integral closure of $V$ in ${\overline{K} }$. Let $W(k)$ be the ring of Witt vectors of $k$ with fraction field $K_0$ and denote by $K_0^{{\operatorname{nr} }}$ the maximal unramified extension of $K_0$. Denote by $e_K$ the absolute ramification index of $K$, i.e., the degree of $K$ over $K_0$. Set $G_K={\operatorname{Gal} }(\overline {K}/K)$ and let $I_K$ denote its inertia subgroup. Let ${\varphi}$ be the absolute Frobenius on $W(\overline {k})$. We will denote by $V$, $V^{\times}$, and $V^0$ the scheme ${\operatorname{Spec} }(V)$ with the trivial, canonical (i.e., associated to the closed point), and $({\mathbf N}\to V, 1\mapsto 0)$ log-structure respectively. For a log-scheme $X$ over $W(k)$, $X_n$ will denote its reduction mod $p^n$, $X_0$ will denote its special fiber. Unless otherwise stated, we work in the category of integral quasi-coherent log-schemes. In general, we will not distinguish between simplicial abelian groups and complexes of abelian groups. Let $A$ be an abelian category with enough projective objects. In this paper $A$ will be the category of abelian groups or ${\mathbf Z}_p$-, ${\mathbf Z}/p^n$-, or ${{\mathbf Q}}_p$-modules. Unless otherwise stated, we work in the (stable) $\infty$-category ${{\mathcal{D}}}(A)$, i.e., stable $\infty$-category whose objects are (left-bounded) chain complexes of projective objects of $A$. For a readable introduction to such categories the reader may consult [@MG], [@Lu2 1]. The $\infty$-derived category is essential to us for two reasons: first, it allows us to work simply with the Beilinson-Hyodo-Kato complexes; second, it supplies functorial homotopy limits. Many of our constructions will involve sheaves of objects from ${{\mathcal{D}}}(A)$. The reader may consult the notes of Illusie [@IL] and Zheng [@Zhe] for a brief introduction to the subject and [@Lu1], [@Lu2] for a thorough treatment. Parts of this article were written during our visits to the Fields Institute in Toronto. The second author worked on this article also at BICMR, Beijing, and at the University of Padova. We would like to thank these institutions for their support and hospitality. This article was inspired by the work of Beilinson on $p$-adic comparison theorems. We would like to thank him for discussions related to his work. Luc Illusie and Weizhe Zheng helped us understand the $\infty$-category theory involved in Beilinson’s work and made their notes [@IL], [@Zhe] available to us – we would like to thank them for that. We have also profited from conversations with Laurent Berger, Amnon Besser, Bharghav Bhatt, Bruno Chiarellotto, Pierre Colmez, Frédéric Déglise, Luc Illusie, and Weizhe Zheng - we are grateful for these exchanges. Moreover, we would like to thank Pierre Colmez for reading and correcting parts of this article. Special thanks go to Laurent Berger and Frédéric Déglise for writing the Appendices. Preliminaries ============= In this section we will do some preparation. In the first part, we will collect some relevant facts from the literature concerning period rings, derived log de Rham complex, and $h$-topology. In the second part, we will prove vanishing results in Galois cohomology and a criterium comparing two spectral sequences that we will need to compare the syntomic descent spectral sequence with the étale Hochschild-Serre spectral sequence. The rings of periods -------------------- Let us recall briefly the definitions of the rings of periods $B_{{\operatorname{cr} }}$, $B_{{\mathrm{dR}}}$, $B_{{\operatorname{st} }}$ of Fontaine [@F1]. Let $A_{{\operatorname{cr} }}$ denote the Fontaine’s ring of crystalline periods [@F1 2.2,2.3]. This is a $p$-adically complete ring such that $A_{{\operatorname{cr} },n}:=A_{{\operatorname{cr} }}/p^n$ is a universal PD-thickening of $\overline{V}_n$ over $W_n(k)$. Let $J_{{\operatorname{cr} },n}$ denote its PD-ideal, $A_{{\operatorname{cr} },n}/J_{{\operatorname{cr} },n}=\overline{V}_n$. We have $$A_{{\operatorname{cr} },n}=H^0_{{\operatorname{cr} }}({\operatorname{Spec} }(\overline{V}_n)/W_n(k)), \quad B^+_{{\operatorname{cr} }}:=A_{{\operatorname{cr} }}[1/p],\quad B_{{\operatorname{cr} }}:=B^+_{{\operatorname{cr} }}[t^{-1}],$$ where $t$ is a certain element of $B^+_{{\operatorname{cr} }}$ (see [@F1] for a precise definition of $t$). The ring $B^+_{{\operatorname{cr} }}$ is a topological $K_0$-module equipped with a Frobenius ${\varphi}$ coming from the crystalline cohomology and a natural $G_K$-action. We have that ${\varphi}(t)=pt$ and that $G_K$ acts on $t$ via the cyclotomic character. Let $$B^+_{{\mathrm{dR}}:}=\invlim_r({\bold Q}\otimes \invlim_n A_{{\operatorname{cr} },n}/ J_{{\operatorname{cr} },n}^{[r]}),\quad B_{{\mathrm{dR}}}:=B^+_{{\mathrm{dR}}}[t^{-1}].$$ The ring $B^+_{{\mathrm{dR}}}$ has a discrete valuation given by the powers of $t$. Its quotient field is $B_{{\mathrm{dR}}}$. We set $F^nB_{{\mathrm{dR}}}=t^nB^+_{{\mathrm{dR}}}$. This defines a descending filtration on $B_{{\mathrm{dR}}}$. The period ring $B_{{\operatorname{st} }}$ lies between $B_{{\operatorname{cr} }}$ and $B_{{\mathrm{dR}}}$ [@F1 3.1]. To define it, choose a sequence of elements $s=(s_n)_{n\geq 0}$ of $\overline{V}$ such that $s_0=p$ and $s_{n+1}^p=s_n$. Fontaine associates to it an element $u_{s }$ of $B^+_{{\mathrm{dR}}}$ that is transcendental over $B^+_{{\operatorname{cr} }}$. Let $B^+_{{\operatorname{st} }}$ denote the subring of $B_{{\mathrm{dR}}}$ generated by $B^+_{{\operatorname{cr} }}$ and $u_{s}$. It is a polynomial algebra in one variable over $B^+_{{\operatorname{cr} }}$. The ring $B^+_{{\operatorname{st} }}$ does not depend on the choice of $s$ (because for another sequence $s^{\prime}=(s^{\prime}_n)_{n\geq 0}$ we have $u_s-u_{s^\prime}\in {\mathbf Z}_pt\subset B_{{\operatorname{cr} }}^+$). The action of $G_K$ on $B^+_{{\mathrm{dR}}}$ restricts well to $B^+_{{\operatorname{st} }}$. The Frobenius ${\varphi}$ extends to $B^+_{{\operatorname{st} }}$ by ${\varphi}(u_{s})=pu_{s}$ and one defines the monodromy operator $N:B^+_{{\operatorname{st} }}\to B^+_{{\operatorname{st} }} $ as the unique $B^+_{{\operatorname{cr} }}$-derivation such that $Nu_{s}=-1$. We have $N{\varphi}=p{\varphi}N$ and the short exact sequence $$0\to B^+_{{\operatorname{cr} }}\to B^+_{{\operatorname{st} }}\stackrel{N}{\to}B^+_{{\operatorname{st} }}\to 0$$ Let $B_{{\operatorname{st} }}=B_{{\operatorname{cr} }}[u_{s }]$. We denote by $\iota$ the injection $\iota:B^+_{{\operatorname{st} }}\hookrightarrow B^+_{{\mathrm{dR}}}$. The topology on $B_{{\operatorname{st} }}$ is the one induced by $B_{{\operatorname{cr} }}$ and the inductive topology; the map $\iota$ is continuous (though the topology on $B_{{\operatorname{st} }}$ is not the one induced from $B_{{\mathrm{dR}}}$). Derived log de Rham complex --------------------------- In this subsection we collect a few facts about the relationship between crystalline cohomology and de Rham cohomology. Let $S$ be a log-PD-scheme on which $p$ is nilpotent. For a log-scheme $Z$ over $S$, let ${\mathrm {L} }\Omega^{\scriptscriptstyle\bullet}_{Z/S}$ denote the derived log de Rham complex (see [@BE1 3.1] for a review). This is a commutative dg ${{\mathcal O}}_S$-algebra on $Z_{{\operatorname{\acute{e}t} }}$ equipped with a Hodge filtration $F^m$. There is a natural morphism of filtered commutative dg ${{\mathcal O}}_S$-algebras [@BE2 1.9.1] $$\label{kappamap} \kappa:\quad {\mathrm {L} }\Omega^{\scriptscriptstyle\bullet}_{Z/S}\to {\mathrm {R} }u_{Z/S}({{\mathcal O}}_{Z/S}),$$ where $u_{Z/S}: Z_{{\operatorname{cr} }}\to Z_{{\operatorname{\acute{e}t} }}$ is the projection from the log-crystalline to the étale topos. The following theorem was proved by Beilinson [@BE2 1.9.2] by direct computations of both sides. \[beilinson\] Suppose that $Z,S$ are fine and $f:Z\to S$ is an integral, locally complete intersection morphism. Then (\[kappamap\]) yields quasi-isomorphisms $$\kappa_m:\quad {\mathrm {L} }\Omega^{\scriptscriptstyle\bullet}_{Z/S}/F^m\stackrel{\sim}{\to} {\mathrm {R} }u_{Z/S}({{\mathcal O}}_{Z/S}/{{\mathcal J}}^{[m]}_{Z/S}).$$ Recall [@BH Def. 7.20] that a log-scheme is called G-log-syntomic if it is log-syntomic and the local log-smooth models can be chosen to be of Cartier type. The next theorem, finer than Theorem \[beilinson\], was proved by Bhatt [@BH Theorem 7.22] by looking at the conjugate filtration of the l.h.s. \[bhatt\] Suppose that $f:Z\to S$ is G-log-syntomic. Then we have a quasi-isomorphism $$\kappa:\quad {\mathrm {L} }\Omega^{\scriptscriptstyle\bullet}_{Z/S}\stackrel{\sim}{\to} {\mathrm {R} }u_{Z/S}({{\mathcal O}}_{Z/S}).$$ Combining the two theorems above, we get a filtered version: Suppose that $f:Z\to S$ is G-log-syntomic. Then we have a quasi-isomorphism $$F^m{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet}_{Z/S}\stackrel{\sim}{\to} {\mathrm {R} }u_{Z/S}({{\mathcal J}}^{[m]}_{Z/S}).$$ Consider the following commutative diagram with exact rows $$\begin{CD} F^m{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet}_{Z/S} @>>> {\mathrm {L} }\Omega^{\scriptscriptstyle\bullet}_{Z/S} @>>>{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet}_{Z/S}/F^m \\ @VVV @VV\wr V @VV\wr V\\ {\mathrm {R} }u_{Z/S}({{\mathcal J}}^{[m]}_{Z/S})@>>> {\mathrm {R} }u_{Z/S}({{\mathcal O}}_{Z/S}) @>>> {\mathrm {R} }u_{Z/S}({{\mathcal O}}_{Z/S}/{{\mathcal J}}^{[m]}_{Z/S}). \end{CD}$$ and use the above theorems of Bhatt and Beilinson. Let $X$ be a fine, proper, log-smooth scheme over $V^{\times}$. Set $$\begin{aligned} {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet,\wedge}_{X/W(k)}){\widehat}{\otimes}{\mathbf Q}_p :=({\operatorname{holim} }_n{\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet,\wedge}_{X_n/W_n(k)}))\otimes {\mathbf Q}\end{aligned}$$ and similarly for complexes over $V^{\times}$. Here the hat over derived log de Rham complex refers to the completion with respect to the Hodge filtration (in the sense of prosystems). For $r\geq 0$, consider the following sequence of maps $$\label{composition1} \begin{aligned} {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_K)/F^r & \stackrel{\sim}{\leftarrow}{\mathrm {R} }\Gamma(X,{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet}_{X/V^{\times}}/F^r)_{{\mathbf Q}}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet}_{X/V^{\times}}/F^r){\widehat}{\otimes}{\mathbf{Q}_p}\\ & \stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X,{{\mathcal O}}_{X/V^{\times}}/{{\mathcal J}}^{[r]}_{X/V^{\times}})_{\mathbf Q} \leftarrow {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X,{{\mathcal O}}_{X/W(k)}/{{\mathcal J}}^{[r]}_{X/W(k)})_{\mathbf Q} \end{aligned}$$ The first quasi-isomorphism follows from the fact that the natural map ${\mathrm {L} }\Omega_{X_K/K_0}^{\scriptscriptstyle\bullet}/F^r\stackrel{\sim}{\to} \Omega_{X_K/K_0}^{\scriptscriptstyle\bullet}/F^r$ is a quasi-isomorphism since $X_K$ is log-smooth over $K_0$. The second quasi-isomorphism follows from $X$ being proper and log-smooth over $V^{\times}$, the third one from Theorem \[beilinson\]. Define the map $$\gamma_r^{-1}:\quad {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X,{{\mathcal O}}_{X/W(k)}/{{\mathcal J}}^{[r]}_{X/W(k)})_{\mathbf Q}\to {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_K)/F^r$$ as the composition (\[composition1\]). \[Langer\] Let $X$ be a fine, proper, log-smooth scheme over $V^{\times}$. Let $r\geq 0$. There exists a canonical quasi-isomorphism $$\gamma_r:\quad {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_K)/F^r \stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_{{\operatorname{\acute{e}t} }},{{\mathcal O}}_{X/W(k)}/{{\mathcal J}}^{[r]}_{X/W(k)})_{\mathbf Q}$$ It suffices to show that the last map in the composition (\[composition1\]) is also a quasi-isomorphism. By Theorem \[beilinson\], this map is quasi-isomorphic to the map $$({\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet,\wedge}_{X/W(k)}){\widehat}{\otimes}{\mathbf{Q}_p})/F^r \to ({\mathrm {R} }\Gamma (X_{{\operatorname{\acute{e}t} }},{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet,\wedge}_{X/V^{\times}}){\widehat}{\otimes}{\mathbf{Q}_p})/F^r$$ Hence it suffices to show that the natural map $${\operatorname{gr} }^i_{F}{\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet,\wedge}_{X/W(k)}){\widehat}{\otimes}{\mathbf{Q}_p}\to {\operatorname{gr} }^i_F{\mathrm {R} }\Gamma (X_{{\operatorname{\acute{e}t} }},{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet,\wedge}_{X/V^{\times}}){\widehat}{\otimes}{\mathbf{Q}_p}$$ is a quasi-isomorphism for all $i\geq 0$. Fix $n\geq 1$ and $i\geq 0$ and recall [@BE1 1.2] that we have a natural identification $${\operatorname{gr} }^i_F{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet}_{X_n/W_n(k)}\stackrel{\sim}{\to}L\Lambda^i_X(L_{X_n/W_n(k)})[-i],\quad {\operatorname{gr} }^i_F{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet}_{X_n/V^{\times}_n}\stackrel{\sim}{\to}L\Lambda^i_X(L_{X_n/V^{\times}_n})[-i],$$ where $L_{Y/S}$ denotes the relative log cotangent complex [@BE1 3.1] and $L\Lambda_X(\scriptscriptstyle\bullet)$ is the nonabelian left derived functor of the exterior power functor. The distinguished triangle $${{\mathcal O}}_X\otimes_VL_{V^{\times}_n/W_n(k)}\to L_{X_n/X_n(k)}\to L_{X_n/V^{\times}_n}$$ yields a distinguished triangle $$L\Lambda_X^i({{\mathcal O}}_X\otimes_VL_{V^{\times}_n/W_n(k)})[-i]\to {\operatorname{gr} }^i_F{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet}_{X_n/W_n(k)}\to {\operatorname{gr} }^i_F{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet}_{X_n/V^{\times}_n}$$ Hence we have a distinguished triangle $${\operatorname{holim} }_n {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},L\Lambda_X^i({{\mathcal O}}_X\otimes_VL_{V^{\times}_n/W_n(k)}))\otimes{\mathbf Q}[-i] \to {\operatorname{gr} }^i_{F}{\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet,\wedge}_{X/W(k)}){\widehat}{\otimes}{\mathbf{Q}_p}\to {\operatorname{gr} }^i_F{\mathrm {R} }\Gamma (X_{{\operatorname{\acute{e}t} }},{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet,\wedge}_{X/V^{\times}}){\widehat}{\otimes}{\mathbf{Q}_p}$$ It suffices to show that the term on the left is zero. But this will follow as soon as we show that $L_{V^{\times}_n/W_n(k)}$ is annihilated by $p^c$, where $c$ is a constant independent of $n$. To show this recall that $V$ is a log complete intersection over $W(k)$. If $\pi$ is a generator of $V/W(k)$ , $f(t)$ its minimal polynomial then (c.f. [@Ol 6.9]) $L_{V^{\times}/W(k)}$ is quasi-isomorphic to the cone of the multiplication by $f^{\prime}(\pi)$ map on $V$. Hence $L_{V^{\times}/W(k)}$ is acyclic in non-zero degrees, $H^0L_{V^{\times}/W(k)}=\Omega_{V^{\times}/W(k)}$ is a cyclic $V$-module and we have a short exact sequence $$0\to \Omega_{V/W(k)}\to \Omega_{V^{\times}/W(k)}\to V/{\mathfrak m}_K\to 0$$ Since $\Omega_{V/W(k)}\simeq V/{{\mathcal{D}}}_{K/K_0}$, where ${{\mathcal{D}}}_{K/K_0}$ is the different, we are done. Versions of the above corollary appear in various degrees of generality in the proofs of the $p$-adic comparison theorems (c.f. [@KM Lemma 4.5], [@Ln Lemma 2.7]). They are proved using computations in crystalline cohomology. We find the above argument based on Beilinson comparison Theorem \[beilinson\] particularly conceptual and pleasing. $h$-topology ------------ In this subsection we review terminology connected with $h$-topology from Beilinson papers [@BE1], [@BE2], [@BH]; we will use it freely. Let $\mathcal{V}ar_K$ be the category of varieties (i.e., reduced and separated schemes of finite type) over a field $K$. An [arithmetic pair]{} over $K$ is an open embedding $j:U\hookrightarrow \overline{U}$ with dense image of a $K$-variety $U$ into a reduced proper flat $V$-scheme $\overline{U}$. A morphism $(U,\overline{U})\to (T,\overline{T})$ of pairs is a map $\overline{U}\to \overline{T}$ which sends $U$ to $T$. In the case that the pairs represent log-regular schemes this is the same as a map of log-schemes. For a pair $(U,\overline{U})$, we set $V_U:=\Gamma(\overline{U},{{\mathcal O}}_{\overline{U}})$ and $K_U:=\Gamma(\overline{U}_K,{{\mathcal O}}_{\overline{U}})$. $K_U$ is a product of several finite extensions of $K$ (labeled by the connected components of $\overline{U}$) and $V_U$ is the product of the corresponding rings of integers. We will denote by ${{\mathcal{P}}}_K^{ar}$ the category of arithmetic pairs over $K$. A [semistable pair]{} ([ss-pair]{}) over $K$ [@BE1 2.2] is a pair of schemes $(U,\overline{U})$ over $(K,V)$ such that (i) $\overline{U}$ is regular and proper over $V$, (ii) $\overline{U}\setminus U$ is a divisor with normal crossings on $\overline{U}$, and (iii) the closed fiber $\overline{U}_0$ of $\overline{U}$ is reduced. Closed fiber is taken over the closed points of $V_U$. We will think of ss-pairs as log-schemes equipped with log-structure given by the divisor $\overline{U}\setminus U$. The closed fiber $\overline{U}_0$ has the induced log-structure. We will say that the log-scheme $(U,\overline{U})$ is [split]{} over $V_U$. We will denote by ${{\mathcal{P}}}_K^{ss}$ the category of ss-pairs over $K$. A semistable pair is called [strict]{} if the irreducible components of the closed fiber are regular. We will often work with the larger category ${{\mathcal{P}}}_K^{\log}$ of log-schemes $(U,\overline{U})\in {{\mathcal{P}}}^{ar}_K$ log-smooth over $V_U^{\times}$. A [semistable pair]{} ([ss-pair]{}) over ${\overline{K} }$ [@BE1 2.2] is a pair of connected schemes $(T,\overline{T})$ over $({\overline{K} },\overline{V})$ such that there exists an ss-pair $(U,\overline{U})$ over $K$ and a ${\overline{K} }$-point $\alpha: K_U\to {\overline{K} }$ such that $(T,\overline{T})$ is isomorphic to the base change $(U_{{\overline{K} }},\overline{U}_{\overline{V}})$. We will denote by ${{\mathcal{P}}}_{{\overline{K} }}^{ss}$ the category of ss-pairs over ${\overline{K} }$. A [geometric pair]{} over $K$ is a pair $(U,\overline{U})$ of varieties over $K$ such that $\overline{U}$ is proper and $U\subset \overline{U}$ is open and dense. We say that the pair $(U,{\overline{U}})$ is a [nc-pair]{} if $\overline{U}$ is regular and ${\overline{U}}\setminus U$ is a divisor with normal crossings in ${\overline{U}}$; it is [strict nc-pair]{} if the irreducible components of $U\setminus {\overline{U}}$ are regular. A morphism of pairs $f:(U_1,{\overline{U}}_1)\to (U,{\overline{U}})$ is a map ${\overline{U}}_1\to {\overline{U}}$ that sends $U_1$ to $U$. We denote the category of nc-pairs over $K$ by ${{\mathcal{P}}}_K^{{nc}}$. For a field $K$, the $h$-topology (c.f. [@SV],[@BE1 2.3]) on $\mathcal{V}ar_K$ is the coarsest topology finer than the Zariski and proper topologies. [^4] It is stronger than the étale and proper topologies . It is generated by the pretopology whose coverings are finite families of maps $\{Y_i\to X\}$ such that $Y:=\coprod Y_i\to X$ is a universal topological epimorphism (i.e., a subset of $X$ is Zariski open if and only if its preimage in $Y$ is open). We denote by $\mathcal{V}ar_{K,h},X_h$ the corresponding $h$-sites. For any of the categories ${{\mathcal{P}}}$ mentioned above let $\gamma: (U,{\overline{U}})\to U$ denote the forgetful functor. Beilinson proved [@BE1 2.5] that the categories ${{\mathcal{P}}}^{{nc}}$, $({{\mathcal{P}}}_K^{ar},\gamma)$ and $({{\mathcal{P}}}_K^{ss},\gamma)$ form a base for $\mathcal{V}ar_{K,h}$. One can easily modify his argument to conclude the same about the categories $({{\mathcal{P}}}_K^{\log},\gamma)$. Galois cohomology ----------------- In this subsection we review the definition of (higher) semistable Selmer groups and prove that in stable ranges they are the same as Galois cohomology groups. Our main references are [@F2], [@F3], [@CF], [@BK], [@FPR], [@JH]. Recall [@F2], [@F3] that a $p$-adic representation $V$ of $G_K$ (i.e., a finite dimensional continuous ${{{\mathbf Q}}}_p$-vector space representation) is called [semistable]{} (over $K$) if $\dim_{K_0} (B_{{\operatorname{st} }}\otimes_{{\mathbf Q}_p} V )^{G_K} = \dim_{{\mathbf Q}_p} (V )$. It is called [potentially semistable]{} if there exists a finite extension $K^{\prime}$ of $K$ such that $V\|G_{K^{\prime}}$ is semistable over $K^{\prime}$. We denote by ${\operatorname{Rep} }_{{\operatorname{st} }}(G_K)$ and ${\operatorname{Rep} }_{{\mathrm{pst}}}(G_K)$ the categories of semistable and potentially semistable representations of $G_K$, respectively. As in [@F2 4.2] a ${\varphi}$-module over $K_0$ is a pair $(D,{\varphi})$, where $D$ is a finite dimensional $K_0$-vector space, ${\varphi}={\varphi}_D$ is a ${\varphi}$-semilinear automorphism of $D$; a $({\varphi},N)$-module is a triple $(D,{\varphi},N)$, where $(D,{\varphi})$ is a ${\varphi}$-module and $N=N_V$ is a $K_0$-linear endomorphism of $D$ such that $N{\varphi}=p{\varphi}N$ (hence $N$ is nilpotent). A filtered $({\varphi},N)$-module is a tuple $(D,{\varphi},N,F^{\scriptscriptstyle\bullet})$, where $(D,{\varphi},N)$ is a $({\varphi}, N)$-module and $F^{\scriptscriptstyle\bullet} $ is a decreasing finite filtration of $D_K$ by $K$-vector spaces. There is a notion of a [(weakly) admissible]{} filtered $({\varphi}, N)$-module [@CF]. Denote by $MF^{{\operatorname{ad} }}_K({\varphi},N)\subset MF_K({\varphi},N)$ the categories of admissible filtered $({\varphi}, N)$-modules and filtered $({\varphi}, N)$-modules, respectively. We know [@CF] that the pair of functors $D_{{\operatorname{st} }}(V)=(B_{{\operatorname{st} }}\otimes_{{\mathbf Q}_p}V)^{G_K}$, $V_{{\operatorname{st} }}(D)=(B_{{\operatorname{st} }}\otimes_{K_0}D)^{{\varphi}={ \operatorname{Id} },N=0}\cap F^0(B_{{\mathrm{dR}}}\otimes_{K}D_K)$ defines an equivalence of categories $MF_K^{{\operatorname{ad} }}({\varphi},N)\simeq {\operatorname{Rep} }_{{\operatorname{st} }}(G_K)$. For $D\in MF_K({\varphi},N)$, set $$C_{{\operatorname{st} }}(D):= \left[\begin{aligned}\xymatrix@C=36pt{D\ar[d]^N \ar[r]^-{(1-{\varphi},{ \operatorname{can} })} & D\oplus D_K/F^0\ar[d]^{(N,0)}\\ D\ar[r]^{1-p{\varphi}} & D }\end{aligned}\right]$$ Here the brackets denote the total complex of the double complex inside the brackets. Consider also the following complex $$C^+(D):= \left[\begin{aligned}\xymatrix@C=40pt{D\otimes_{K_0}B^+_{{\operatorname{st} }}\ar[d]^N \ar[r]^-{(1-{\varphi},{ \operatorname{can} }\otimes \iota)} & D\otimes_{K_0}B^+_{{\operatorname{st} }}\oplus (D_K\otimes_{K}B^+_{{\mathrm{dR}}})/F^0\ar[d]^{(N,0)}\\ D\otimes_{K_0}B^+_{{\operatorname{st} }}\ar[r]^{1-p{\varphi}} & D\otimes_{K_0}B^+_{{\operatorname{st} }}}\end{aligned}\right]$$ Define $C(D)$ by omitting the superscript $+$ in the above diagram. We have $C_{{\operatorname{st} }}(D)=C(D)^{G_K}$. Recall [@JH 1.19], [@FPR 3.3] that to every $p$-adic representation $V$ of $G_K$ we can associate a complex $$C_{{\operatorname{st} }}(V):\quad D_{{\operatorname{st} }}(V)\veryverylomapr{(N,1-{\varphi},\iota)}D_{{\operatorname{st} }}(V)\oplus D_{{\operatorname{st} }}(V)\oplus t_V\veryverylomapr{(1-p{\varphi})-N} D_{{\operatorname{st} }}(V)\to 0\cdots$$ where $t_V:=(V\otimes_{{\mathbf Q}_p} (B_{{\mathrm{dR}}}/B^+_{{\mathrm{dR}}}))^{G_K}$ [@FPR I.2.2.1]. The cohomology of this complex is called $H^_{{\operatorname{st} }}(G_K,V)$. If $V$ is semistable then $C_{{\operatorname{st} }}(V)=C_{{\operatorname{st} }}(D_{{\operatorname{st} }}(V))$ hence $H^(C_{{\operatorname{st} }}(D_{{\operatorname{st} }}(V)))=H^_{{\operatorname{st} }}(G_K,V)$. If $V$ is potentially semistable the groups $H^_{{\operatorname{st} }}(G_K,V)$ compute Yoneda extensions of ${\mathbf Q}_p$ by $V$ in the category of potentially semistable representations [@FPR I.3.3.8]. In general [@FPR I.3.3.7], $H^0_{{\operatorname{st} }}(G_K,V)\stackrel{\sim}{\to} H^0(G_K,V)$ and $H^1_{{\operatorname{st} }}(G_K,V)\hookrightarrow H^1(G_K,V)$ computes ${\operatorname{st} }$-extensions[^5] of ${\mathbf Q}_p$ by $V$. \[basics\] Let $D\in MF_K({\varphi}, N)$. Note that 1. $H^0(C(D))=V_{{\operatorname{st} }}(D)$; 2. for $i\geq 2$, $H^i(C^+(D))=H^i(C(D))=0$ (because $N$ is surjective on $B^+_{{\operatorname{st} }}$ and $B_{{\operatorname{st} }}$); 3. if $F^1D_K=0$ then $F^0(D_K\otimes _KB^+_{{\mathrm{dR}}})=F^0(D_K\otimes _KB_{{\mathrm{dR}}})$ (hence the map of complexes $C^+(D)\to C(D)$ is an injection); 4. if $D=D_{{\operatorname{st} }}(V)$ is admissible then we have quasi-isomorphisms $$C(D)\stackrel{\sim}{\leftarrow}V\otimes_{{\mathbf Q}_p}[B_{{\operatorname{cr} }}\verylomapr{(1-{\varphi},{ \operatorname{can} })}B_{{\operatorname{cr} }}\oplus B_{{\mathrm{dR}}}/F^0]\stackrel{\sim}{\leftarrow}V\otimes_{{\mathbf Q}_p}(B_{{\operatorname{cr} }}^{{\varphi}=1}\cap F^0)=V$$ and the map of complexes $C_{{\operatorname{st} }}(D)\to C(D)$ represents the canonical map $H^i_{{\operatorname{st} }}(G_K,V)\to H^i(G_K,V)$. ([@F1 Theorem II.5.3]) If $X\subset B_{{\operatorname{cr} }}\cap B_{{\mathrm{dR}}}^+$ and ${\varphi}(X)\subset X$ then ${\varphi}^2(X)\subset B_{{\operatorname{cr} }}^+$. If $D\in MF_K({\varphi}, N)$ and $F^1D_K=0$ then $H^0(C(D)/C^+(D))=0$. We will argue by induction on $m$ such that $N^m=0$. Assume first that $m=1$ (hence $N=0$). We have $$\begin{aligned} C(D)/C^+(D) & =\left[\begin{aligned}\xymatrix@C=40pt{D\otimes_{K_0}(B_{{\operatorname{st} }}/B^+_{{\operatorname{st} }})\ar[r]^-{(1-{\varphi},{ \operatorname{can} }\otimes\iota)}\ar[d]^{1\otimes N} & D\otimes_{K_0}(B_{{\operatorname{st} }}/B^+_{{\operatorname{st} }})\oplus D_K\otimes_{K}(B_{{\mathrm{dR}}}/B^+_{{\mathrm{dR}}})\ar[d]^{(1\otimes N,0)}\\ D\otimes_{K_0}(B_{{\operatorname{st} }}/B^+_{{\operatorname{st} }})\ar[r]^{1-p{\varphi}} & D\otimes_{K_0}(B_{{\operatorname{st} }}/B^+_{{\operatorname{st} }}) }\end{aligned}\right]\\ & \stackrel{\sim}{\leftarrow} [D\otimes_{K_0}(B_{{\operatorname{cr} }}/B^+_{{\operatorname{cr} }})\verylomapr{(1-{\varphi},{ \operatorname{can} })} D\otimes_{K_0}(B_{{\operatorname{cr} }}/B^+_{{\operatorname{cr} }})\oplus D_K\otimes_{K}(B_{{\mathrm{dR}}}/B^+_{{\mathrm{dR}}})]\end{aligned}$$ Write $D=\oplus_{i=1}^{r}K_0d_i$ and, for $1\leq i\leq r$, consider the following maps $$p_i:H^0(C(D)/C^+(D))=(D\otimes_{K_0}((B_{{\operatorname{cr} }}\cap B_{{\mathrm{dR}}}^+)/B^+_{{\operatorname{cr} }}))^{{\varphi}=1}\subset \oplus_{i=1}^rd_i\otimes ((B_{{\operatorname{cr} }}\cap B_{{\mathrm{dR}}}^+)/B^+_{{\operatorname{cr} }})\stackrel{{\operatorname{pr} }_i}{\to}(B_{{\operatorname{cr} }}\cap B_{{\mathrm{dR}}}^+)/B^+_{{\operatorname{cr} }}$$ Let $Y_a$, $a\in H^0(C(D)/C^+(D))$, denote the $K_0$-subspace of $(B_{{\operatorname{cr} }}\cap B_{{\mathrm{dR}}}^+)/B^+_{{\operatorname{cr} }}$ spanned by $p_1(a),\ldots,p_r(a)$. We have $(p_1(a),\ldots ,p_r(a))^T=M{\varphi}(p_1(a),\ldots ,p_r(a))^T$, for $M\in GL_r(K_0)$. Hence ${\varphi}(Y_a)\subset Y_a$. Let $X_a\subset B_{{\operatorname{cr} }}\cap B^+_{{\mathrm{dR}}}$ be the inverse image of $Y_a$ under the projection $B_{{\operatorname{cr} }}\cap B^+_{{\mathrm{dR}}}\to (B_{{\operatorname{cr} }}\cap B^+_{{\mathrm{dR}}})/B_{{\operatorname{cr} }}^+$ (naturally $B^+_{{\operatorname{cr} }}\subset X_a$). Then ${\varphi}(X_a)\subset X_a + B^+_{{\operatorname{cr} }}=X_a.$ By the above lemma ${\varphi}^2(X_a)\subset B^+_{{\operatorname{cr} }}$. Hence ${\varphi}^2(Y_a)=0$ and (applying $M^{-2}$) $Y_a=0$. This implies that $a=0$ and $H^0(C(D)/C^+(D))=0$, as wanted. For general $m>0$, consider the filtration $D_1\subset D$, where $D_1:=\ker(N)$ with induced structures. Set $D_2:= D/D_1$ with induced structures. Then $D_1,D_2\in MF_{K}({\varphi},N)$; $N^i$ is trivial on $D_1$ for $i=1$ and on $D_2$ for $i=m-1$. Clearly $F^1D_{1,K}=F^1D_{2,K}=0$. Hence, by Remark \[basics\].3, we have a short exact sequence $$0\to C(D_1)/C^+(D_1)\to C(D)/C^+(D)\to C(D_2)/C^+(D_2)\to 0$$ By the inductive assumption $H^0(C(D_1)/C^+(D_1))=H^0(C(D_2)/C^+(D_2))=0$. Hence $H^0(C(D)/C^+(D))=0$, as wanted. If $D\in MF_K({\varphi}, N)$ and $F^1D_K=0$ then $H^0(C^+(D))=H^0(C(D))=V_{{\operatorname{st} }}(D)$ ($\subset D\otimes_{K_0}B^+_{{\operatorname{st} }}$) and $H^1(C^+(D))\hookrightarrow H^1(C(D))$. \[MF1\] If $D\in MF^{{\operatorname{ad} }}_K({\varphi}, N)$ and $F^1D_K=0$ then $$H^i(C^+(D))=H^i(C(D))= \begin{cases} V_{{\operatorname{st} }}(D) &i=0\\ 0 & i\neq 0 \end{cases}$$ (i.e., $C^+(D)\stackrel{\sim}{\to} C(D)$). A filtered $({\varphi},N,G_K)$-module is a tuple $(D,{\varphi},N,\rho, F^{\scriptscriptstyle\bullet})$, where 1. $D$ is a finite dimensional $K_0^{{\operatorname{nr} }}$-vector space; 2. ${\varphi}: D \to D$ is a Frobenius map; 3. $N : D \to D$ is a $K_0^{{\operatorname{nr} }}$-linear monodromy map such that $N{\varphi}= p{\varphi}N$; 4. $\rho$ is a $K_0^{{\operatorname{nr} }}$-semilinear $G_K$-action on $D$ (hence $\rho\|I_K$ is linear) that factors through a finite quotient of the inertia $I_K$ and that commutes with ${\varphi}$ and $N$; 5. $F^{\scriptscriptstyle\bullet}$ is a decreasing finite filtration of $D_K:=(D\otimes _{K_0^{{\operatorname{nr} }}}{\overline{K} })^{G_K}$ by $K$-vector spaces. Morphisms between filtered $({\varphi},N, G_K)$-modules are $K_0^{{\operatorname{nr} }}$-linear maps preserving all structures. There is a notion of a [(weakly) admissible]{} filtered $({\varphi}, N,G_K)$-module [@CF], [@F3]. Denote by $MF_K^{{\operatorname{ad} }}({\varphi},N,G_K)\subset MF_K^{}({\varphi},N,G_K)$ the categories of admissible filtered $({\varphi}, N,G_K)$-modules and filtered $({\varphi}, N,G_K)$-modules, respectively. We know [@CF] that the pair of functors $D_{{\mathrm{pst}}}(V)=\injlim_H(B_{{\operatorname{st} }}\otimes_{{\mathbf Q}_p}V)^{H}$, $H\subset G_K$ - an open subgroup, $V_{{\mathrm{pst}}}(D)=(B_{{\operatorname{st} }}\otimes_{K_0^{{\operatorname{nr} }}}D)^{{\varphi}={ \operatorname{Id} },N=0}\cap F^0(B_{{\mathrm{dR}}}\otimes_{K}D_K)$ define an equivalence of categories $MF_K^{{\operatorname{ad} }}({\varphi},N,G_K)\simeq {\operatorname{Rep} }_{{\mathrm{pst}}}(G_K)$. For $D\in MF_K({\varphi},N,G_K)$, set [^6] $$C_{{\mathrm{pst}}}(D):= \left[\begin{aligned}\xymatrix@C=36pt{D_{{\operatorname{st} }}\ar[d]^N \ar[r]^-{(1-{\varphi},{ \operatorname{can} })} & D_{{\operatorname{st} }}\oplus D_{K}/F^0\ar[d]^{(N,0)}\\ D_{{\operatorname{st} }}\ar[r]^{1-p{\varphi}} & D_{{\operatorname{st} }} }\end{aligned}\right]$$ Here $D_{{\operatorname{st} }}:=D^{G_K}$. Consider also the following complex (we set $D_{{\overline{K} }}:=D\otimes_{K_0^{{\operatorname{nr} }}}{\overline{K} }$) $$C^+(D):= \left[\begin{aligned}\xymatrix@C=40pt{D\otimes_{K_0^{{\operatorname{nr} }}}B^+_{{\operatorname{st} }}\ar[d]^N \ar[r]^-{(1-{\varphi},{ \operatorname{can} }\otimes \iota)} & (D\otimes_{K_0^{{\operatorname{nr} }}}B^+_{{\operatorname{st} }})\oplus (D_{{\overline{K} }}\otimes_{{\overline{K} }}B^+_{{\mathrm{dR}}})/F^0\ar[d]^{(N,0)}\\ D\otimes_{K_0^{{\operatorname{nr} }}}B^+_{{\operatorname{st} }}\ar[r]^{1-p{\varphi}} & D\otimes_{K_0^{{\operatorname{nr} }}}B^+_{{\operatorname{st} }}}\end{aligned}\right]$$ Define $C(D)$ by omitting the superscript $+$ in the above diagram. We have $C_{{\mathrm{pst}}}(D)=C(D)^{G_K}$. \[pst=st\] If $V$ is potentially semistable then $C_{{\operatorname{st} }}(V)=C_{{\mathrm{pst}}}(D_{{\mathrm{pst}}}(V))$ hence $H^(C_{{\mathrm{pst}}}(D_{{\mathrm{pst}}}(V)))=H^_{{\operatorname{st} }}(G_K,V)$. \[resolution2\] If $D=D_{{\mathrm{pst}}}(V)$ is admissible then we have quasi-isomorphisms $$C(D)\stackrel{\sim}{\leftarrow}V\otimes_{{\mathbf Q}_p}[B_{{\operatorname{cr} }}\verylomapr{(1-{\varphi},{ \operatorname{can} })}B_{{\operatorname{cr} }}\oplus B_{{\mathrm{dR}}}/F^0]\stackrel{\sim}{\leftarrow}V\otimes_{{\mathbf Q}_p}(B_{{\operatorname{cr} }}^{{\varphi}=1}\cap F^0)=V$$ and the map of complexes $C_{{\mathrm{pst}}}(D)\to C(D)$ represents the canonical map $H^i_{{\operatorname{st} }}(G_K,V)\to H^i(G_K,V)$. \[resolution3\] If $D\in MF^{{\operatorname{ad} }}_K({\varphi}, N, G_K)$ and $F^1D_K=0$ then $$H^i(C^+(D))\stackrel{\sim}{\to}H^i(C(D))= \begin{cases} V_{{\mathrm{pst}}}(D) &i=0\\ 0 & i\neq 0 \end{cases}$$ (i.e., $C^+(D)\stackrel{\sim}{\to} C(D)$). By Remark \[resolution2\] we have $C(D)\simeq V_{{\mathrm{pst}}}(D)[0]$. To prove the isomorphism $H^i(C^+(D))\stackrel{\sim}{\to}H^i(C(D))$, $i\geq 0$, take a finite Galois extension $K^{\prime}/K$ such that $D$ becomes semistable over $K^{\prime}$, i.e., $I_{K^{\prime}}$ acts trivially on $D$. We have $(D^{\prime},{\varphi},N)\in MF_{K^{\prime}}^{{\operatorname{ad} }}({\varphi},N)$, where $D^{\prime}:=D^{G_{K^{\prime}}}$ and (compatibly) $D\simeq D^{\prime}\otimes_{K_0^{\prime}}K_0^{{\operatorname{nr} }}$, $F^{\scriptscriptstyle\bullet}D^{\prime}_{K^{\prime}}\simeq F^{\scriptscriptstyle\bullet}D_K\otimes_KK^{\prime}$. It easily follows that $C^+(D)=C^+(K^{\prime},D^{\prime})$ and $C(D) = C(K^{\prime}, D^{\prime})$. Since $F^1D^{\prime}_{K^{\prime}}=0$, our corollary is now a consequence of Corollary \[MF1\] \[resolution33\] If $D\in MF^{{\operatorname{ad} }}_K({\varphi}, N, G_K)$ and $F^1D_K=0$ then, for $i\geq 0$, the natural map $$H^i_{{\operatorname{st} }}(G_K,V_{{\mathrm{pst}}}(D))\stackrel{\sim}{\to} H^i(G_K,V_{{\mathrm{pst}}}(D))$$ is an isomorphism. Both sides satisfy Galois descent for finite Galois extensions. We can assume, therefore, that $D = D_{{\operatorname{st} }}(V)$ for a semistable representation $V$ of $G_K$. For $i=0$ we have (even without assuming $F^1D_K=0$) $H^0(C_{{\operatorname{st} }}(D)) = H^0(C(D)^{G_K}) = H^0(C(D))^{G_K} = V^{G_K}$. For $i=1$ the statement is proved in [@BER Thm. 6.2, Lemme 6.5]. For $i=2$ it follows from the assumption $F^1D_K=0$ (by weak admissibility of $D$) that there is a $W(k)$-lattice $M \subset D$ such that ${\varphi}^{-1}(M) \subset p^2 M$, which implies that $1 - p{\varphi}= -p{\varphi}(1 - p^{-1}{\varphi}^{-1}) : D \to D$ is surjective, hence $H^2(C_{{\operatorname{st} }}(D)) = 0$ (cf. the proof of [@BER Lemme 6.7]). The proof of the fact that $H^2(G_K, V) = 0$ if $F^1D_K=0$ was kindly communicated to us by L. Berger; it is reproduced in Appendix A (c.f. Theorem \[main\]). For $i > 2$ both terms vanish. Comparison of spectral sequences {#Jan} -------------------------------- The purpose of this subsection is to prove a derived category theorem (Theorem \[speccomp\]) that we will use later to relate the syntomic descent spectral sequence with the étale Hochschild-Serre spectral sequence (cf. Theorem \[stHS\]). Let $D$ be a triangulated category and $H : D \to A$ a cohomological functor to an abelian category $A$. A finite collection of adjacent exact triangles (a “Postnikov system" in the language of [@GM IV.2, Ex. 2]) $$\label{postnikov} \xymatrix@C=10pt@R=15pt{ & Y^0 \ar[dr] && Y^1 \ar[dr] &&&& Y^n \ar[dr] &\\ X = X^0 \ar[ur] && X^1 \ar [ur] \ar[ll]^(.45){[1]} && X^2 \ar[ll]^{[1]} && X^n \ar@{.}[ll] \ar[ru] && X^{n+1} = 0 \ar[ll]^(.55){[1]}\\}$$ gives rise to an exact couple $$D_1^{p,q} = H^q(X^p) = H(X^p[q]),\qquad E_1^{p,q} = H^q(Y^p) \Longrightarrow H^{p+q}(X).$$ The induced filtration on the abutment is given by $$F^p H^{p+q}(X) = {\operatorname{Im} }\left(D_1^{p,q} = H^q(X^p) \to H^{p+q}(X)\right).$$ In the special case when $A$ is the heart of a non-degenerate $t$-structure $(D^{\leq n}, D^{\geq n})$ on $D$ and $H = \tau_{\leq 0} \tau_{\geq 0}$, the following conditions are equivalent: 1. $E_2^{p,q} = 0$ for $p\not= 0$; 2. $D_2^{p,q} = 0$ for all $p, q$; 3. $D_r^{p,q} = 0$ for all $p, q$ and $r > 1$; 4. the sequence $0 \to H^q(X^p) \to H^q(Y^p) \to H^q(X^{p+1}) \to 0$ is exact for all $p, q$; 5. the sequence $0 \to H^q(X) \to H^q(Y^0) \to H^q(Y^1) \to \cdots$ is exact for all $q$; 6. the canonical map $H^q(X) \to E_1^{{\scriptscriptstyle\bullet},q}$ is a quasi-isomorphism, for all $q$; 7. the triangle $\tau_{\leq q} X^p \to \tau_{\leq q} Y^p \to \tau_{\leq q} X^{p+1}$ is exact for all $p, q$. From now on until the end of \[Jan\] assume that $D = D(A)$ is the derived category of $A$ with the standard $t$-structure and that $X^i, Y^i \in D^+(A)$, for all $i$. Furthermore, assume that $f : A \to A^\prime$ is a left exact functor to an abelian category $A^\prime$ and that $A$ admits a class of $f$-adapted objects (hence the derived functor ${\mathrm {R} }f : D^+(A) \to D^+(A^\prime)$ exists). Applying ${\mathrm {R} }f$ to (\[postnikov\]) we obtain another Postnikov system, this time in $D^+(A^\prime)$. The corresponding exact couple $$\label{firstcouple} {}^I D_1^{p,q} = ({\mathrm {R} }^q f)(X^p),\qquad {}^I E_1^{p,q} = ({\mathrm {R} }^q f)(Y^p) \Longrightarrow ({\mathrm {R} }^{p+q}f)(X)$$ induces filtration $${}^I F^p ({\mathrm {R} }^{p+q}f)(X) = {\operatorname{Im} }\left({}^I D_1^{p,q} = ({\mathrm {R} }^q f)(X^p) \to ({\mathrm {R} }^{p+q}f)(X) \right).$$ Our goal is to compare (\[firstcouple\]), under the equivalent assumptions (\[ass\]), to the hypercohomology exact couple $$\label{secondcouple} {}^{II} D_2^{p,q} = ({\mathrm {R} }^{p+q}f)(\tau_{\leq q-1} X),\qquad {}^{II} E_2^{p,q} = ({\mathrm {R} }^p f)(H^q(X)) \Longrightarrow ({\mathrm {R} }^{p+q}f)(X)$$ for which $${}^{II} F^p ({\mathrm {R} }^{p+q}f)(X) = {\operatorname{Im} }\left({}^{II} D_2^{p-1,q+1} = ({\mathrm {R} }^{p+q} f)(\tau_{\leq q} X) \to ({\mathrm {R} }^{p+q}f)(X) \right).$$ \[speccomp\] Under the assumptions (\[ass\]) there is a natural morphism of exact couples $(u, v) : ({}^I D_2, {}^I E_2) \to ({}^{II} D_2, {}^{II} E_2)$. Consequently, we have ${}^I F^p \subseteq {}^{II} F^p$ for all $p$ and there is a natural morphism of spectral sequences ${}^I E_r^{,} \to {}^{II} E_r^{,}$ ($r > 1$) compatible with the identity map on the common abutment. [Step 1:]{} we begin by constructing a natural map $u : {}^I D_2 \to {}^{II} D_2$. For each $p > 0$ there is a commutative diagram in $D^+(A^\prime)$ $$\xymatrix{ ({\mathrm {R} }^{p+q}f)((\tau_{\leq q} Y^{p-1})[-p]) \ar [r] \ar[d]^\wr & ({\mathrm {R} }^{p+q}f)((\tau_{\leq q} X^p)[-p]) \ar[r] \ar[d]^\wr & ({\mathrm {R} }^{p+q}f)(\tau_{\leq q} X) \ar[d]^{\alpha_{II}} \\ {}^I E_1^{p-1,q} = ({\mathrm {R} }^{p+q}f)(Y^{p-1}[-p]) \ar[r]^(.54){k_1} & {}^I D_1^{p,q} = ({\mathrm {R} }^{p+q}f)(X^p[-p]) \ar [ru]^{u^\prime} \ar[r]^(.62){\alpha_I} & ({\mathrm {R} }^{p+q}f)(X)\\}$$ whose both rows are complexes. This defines a map $u^\prime : {}^I D_1^{p,q} \to {}^{II} D_2^{p-1,q+1}$ such that $u^\prime k_1 = 0$ and $\alpha_{II} u ^\prime = \alpha_I$ (hence ${}^I F^p = {\operatorname{Im} }(\alpha_I) \subseteq {\operatorname{Im} }(\alpha_{II}) = {}^{II} F^p$). By construction, the diagram (with exact top row) $$\xymatrix{ {}^I E_1^{p,q-1} \ar[r]^{k_1} \ar[rd]^0 & {}^I D_1^{p+1,q-1} \ar[r]^(.55){i_1} \ar[d]^{u^\prime} & {}^I D_1^{p,q} \ar[d]^{u^\prime}\\ & {}^{II} D_2^{p,q} \ar[r]^(.45){i_2} & {}^{II} D_2^{p-1,q+1}\\}$$ is commutative for each $p\geq 0$, which implies that the map $$u = u^\prime i_1^{-1} : {}^I D_2^{p,q} = i_1({}^I D_1^{p+1,q-1}) \to {}^{II} D_2^{p,q}$$ is well-defined and satisfies $u i_2 = i_2 u$. [Step 2:]{} for all $q$, the canonical quasi-isomorphism $H^q(X) \to E_1^{{\scriptscriptstyle\bullet},q}$ induces natural morphisms $$\begin{aligned} v^\prime : {}^I E_2^{p,q} &= H^p(i \mapsto ({\mathrm {R} }^q f)(Y^i)) \to H^p(i \mapsto f(H^q(Y^i))) \to ({\mathrm {R} }^p f)(i \mapsto H^q(Y^i))\\ &= ({\mathrm {R} }^p f)(E_1^{{\scriptscriptstyle\bullet},q}) \stackrel\sim\longleftarrow ({\mathrm {R} }^p f)(H^q(X)) = {}^{II} E_2^{p,q};\\\end{aligned}$$ set $v = (-1)^p v^\prime : {}^I E_2^{p,q} \to {}^{II} E_2^{p,q}$. It remains to show that $u$ and $v$ are compatible with the maps $${}^? D_2^{p-1,q+1} \stackrel {j_2} \longrightarrow {}^? E_2^{p,q} \stackrel {k_2} \longrightarrow {}^? D_2^{p+1,q} \qquad\qquad (? = I, II).$$ [Step 3:]{} for any complex $M^{\scriptscriptstyle\bullet}$ over $A$ denote by $Z^i(M^{\scriptscriptstyle\bullet}) = {\operatorname{Ker} }(\delta^i : M^i \to M^{i+1})$ the subobject of cycles in degree $i$. If $M^{\scriptscriptstyle\bullet}$ is a resolution of an object $M$ of $A$, then each exact sequence $$\label{cycle} 0 \longrightarrow Z^p(M^{\scriptscriptstyle\bullet}) \longrightarrow M^p \stackrel{\delta^p}\longrightarrow Z^{p+1}(M^{\scriptscriptstyle\bullet}) \longrightarrow 0 \qquad\qquad (p\geq 0)$$ can be completed to an exact sequence of resolutions $$\xymatrix@C=15pt@R=15pt{ 0 \ar[r] & Z^p(M^{\scriptscriptstyle\bullet}) \ar[r] \ar[d]^{{ \operatorname{can} }} & M^p \ar[r] \ar[d]^{{ \operatorname{can} }} & Z^{p+1}(M^{\scriptscriptstyle\bullet}) \ar[r] \ar[d]^{-{ \operatorname{can} }} & 0\\ 0 \ar[r] & (\sigma_{\geq p}(M^{\scriptscriptstyle\bullet}))[p] \ar[r] & (\sigma_{\geq p} {\rm Cone}(M^{\scriptscriptstyle\bullet} \stackrel {\rm id} \to M^{\scriptscriptstyle\bullet}))[p] \ar[r] & (\sigma_{\geq p+1}(M^{\scriptscriptstyle\bullet}))[p+1] \ar[r] & 0.\\}$$ By induction, we obtain that the following diagram, whose top arrow is the composition of the natural maps $Z^i \to Z^{i-1}[1]$ induced by (\[cycle\]), commutes in $D^+(A)$. $$\label{sign} \xymatrix@C=10pt@R=15pt{ Z^p(M^{\scriptscriptstyle\bullet}) \ar[r] \ar[d]^{{ \operatorname{can} }} & Z^0(M^{\scriptscriptstyle\bullet})[p] = M[p] \ar[d]^{(-1)^p{ \operatorname{can} }}\\ (\sigma_{\geq p}(M^{\scriptscriptstyle\bullet}))[p] \ar[r]^(.58){{ \operatorname{can} }} & M^{\scriptscriptstyle\bullet}[p]\\}$$ We are going to apply this statement to $M = H^q(X)$ and $M^{\scriptscriptstyle\bullet} = E_1^{\scriptscriptstyle\bullet,q}$, when $Z^p(M^{\scriptscriptstyle\bullet}) = D_1^{p,q} = H^q(X^p)$ and $Z^0(M^{\scriptscriptstyle\bullet}) = H^q(X)$. [Step 4:]{} we are going to investigate ${}^I E_2^{p,q}$. Complete the morphism $Y^p \to Y^{p+1}$ to an exact triangle $U^p \to Y^p \to Y^{p+1}$ in $D^+(A)$ and fix a lift $X^p \to U^p$ of the morphism $X^p \to Y^p$. There are canonical epimorphisms $$\label{epi} ({\mathrm {R} }^q f)(U^p) \twoheadrightarrow {\operatorname{Ker} }(({\mathrm {R} }^q f)(Y^p) \stackrel {j_1 k_1} \longrightarrow ({\mathrm {R} }^q f)(Y^{p+1})) = Z^p({}^I E_1^{\scriptscriptstyle\bullet,q}) \twoheadrightarrow {}^I E_2^{p,q}$$ and the map $$k_2 : {}^I E_2^{p,q} \to {}^I D_2^{p+1,q} = {\operatorname{Ker} }({}^I D_1^{p+1,q} \stackrel {j_1} \longrightarrow {}^I E_1^{p+1,q})$$ is induced by the restriction of $k_1 : {}^I E_1^{p,q} \to {}^I D_1^{p+1,q}$ to $Z^p({}^I E_1^{\scriptscriptstyle\bullet,q})$. The following octahedron (in which we have drawn only the four exact faces) $$\xymatrix@C=15pt@R=15pt{ X^{p+2} \ar[rd]_{[1]} && Y^{p+1} \ar[ll]\\ & X^{p+1} \ar[ru] \ar[dl] &\\ X^p[1] \ar[rr]^{[1]} && Y^p \ar[lu]\\} \qquad\qquad \xymatrix@C=15pt@R=15pt{ X^{p+2} \ar[dd]_{[1]} && Y^{p+1} \ar[dl]\\ & U^p[1] \ar[lu] \ar[dr]^{[1]} &\\ X^p[1] \ar[ru] && Y^p \ar[uu]\\}$$ shows that the triangle $X^p \to U^p \to X^{p+2}[-1]$ is exact and the diagrams $$\xymatrix@C=15pt@R=15pt{ U^p[1] \ar[r] \ar[d] & Y^p[1] \ar[d]\\ X^{p+2} \ar[r] & X^{p+1}[1]\\} \qquad\qquad \xymatrix@C=15pt@R=15pt{ ({\mathrm {R} }^q f)(U^p) \ar[r] \ar[d] & Z^p({}^I E_1^{\scriptscriptstyle\bullet,q}) \ar[d]_{k_1}\\ ({\mathrm {R} }^q f)(X^{p+2}[-1]) = {}^I D_2^{p+2,q-1} \ar[r]^(.72){i_1} & {}^I D_2^{p+1,q}\\}$$ commute. The previous discussion implies that the composite map $$({\mathrm {R} }^q f)(U^p) \twoheadrightarrow Z^p({}^I E_1^{\scriptscriptstyle\bullet,q}) \twoheadrightarrow {}^I E_2^{p,q} \stackrel {k_2} \longrightarrow {}^I D_2^{p+1,q} \stackrel u \to {}^{II} D_2^{p+1,q} = ({\mathrm {R} }^q f)((\tau_{\leq q-1} X)[p+1])$$ is obtained by applying ${\mathrm {R} }^q f$ to $$\label{comp1} \tau_{\leq q}\, U^p \to \tau_{\leq q}(X^{p+2}[-1]) = (\tau_{\leq q-1} X^{p+2})[-1] \to (\tau_{\leq q-1}\, X)[p+1].$$ [Step 5:]{} all boundary maps $H^q(X^{p+2}[-1]) \to H^q(X^p)$ vanish by (\[ass\]), which means that the following triangles are exact. $$\tau_{\leq q}\, X^p \to \tau_{\leq q}\, U^p \to \tau_{\leq q}(X^{p+2}[-1]) = (\tau_{\leq q-1}\, X^{p+2})[-1]$$ The commutative diagram $$\xymatrix@C=15pt@R=15pt{ \tau_{\leq q}\, U^p \ar[r] & H^q(U^p)[-q] \ar[r] & {\operatorname{Ker} }\left(H^q(Y^p) \to H^q(Y^{p+1})\right)[-q]\\ \tau_{\leq q}\, X^p \ar[r] \ar[u] & H^q(X^p)[-q] \ar[u] \ar@2{-}[ru] &\\}$$ gives rise to an octahedron $$\xymatrix@C=10pt@R=15pt{ V^p \ar[rd]_{[1]} && H^q(X^p)[-q] \ar[ll]\\ & \tau_{\leq q}\, U^p \ar[ru] \ar[dl] &\\ \tau_{\leq q}(X^{p+2}[-1]) \ar[rr]^{[1]} && \tau_{\leq q}\,X^p \ar[lu]\\} \qquad \xymatrix@C=10pt@R=15pt{ V^p \ar[dd]_{[1]} && H^q(X^p)[-q] \ar[dl]\\ & (\tau_{\leq q-1}\, X^p)[1] \ar[lu] \ar[dr]^{[1]} &\\ X^p[1] \ar[ru] && Y^p \ar[uu]\\}$$ In particular, the following diagram commutes. $$\label{comp2} \xymatrix@C=15pt@R=15pt{ \tau_{\leq q}\, U^p \ar[r] \ar[d] & H^q(X^p)[-q] \ar[d]\\ \tau_{\leq q}(X^{p+2}[-1]) \ar[r] & (\tau_{\leq q-1}\, X^p)[1]\\}$$ [Step 6:]{} the diagram (\[sign\]) implies that the composition of $v : {}^I E_2^{p,q} \to {}^{II} E_2^{p,q}$ with the second epimorphism in (\[epi\]) is equal to the composite map $$\begin{aligned} &Z^p({}^I E_1^{\scriptscriptstyle\bullet,q}) = {\operatorname{Ker} }\left(({\mathrm {R} }^q f)(\tau_{\leq q}\,Y^p) \to ({\mathrm {R} }^q f)(\tau_{\leq q}\,Y^{p+1})\right)\to\\ &\to {\operatorname{Ker} }\left(({\mathrm {R} }^q f)(H^q(Y^p)[-q]) \to ({\mathrm {R} }^q f)(H^q(Y^{p+1})[-q])\right) =\\ &= ({\mathrm {R} }^q f)(Z^p(E_1^{\scriptscriptstyle\bullet,q})[-q]) \to ({\mathrm {R} }^q f)(Z^0(E_1^{\scriptscriptstyle\bullet,q})[-q+p]) = ({\mathrm {R} }^p f)(H^q(X)) = {}^{II} E_2^{p,q}.\\\end{aligned}$$ As a result, the composition of $v$ with (\[epi\]) is obtained by applying ${\mathrm {R} }^q f$ to $$\label{comp3} \tau_{\leq q}\, U^p \to H^q(X^p)[q] \to H^q(X)[-q+p].$$ Consequently, the composite map $${}^I D_1^{p,q} = ({\mathrm {R} }^q f)(\tau_{\leq q}\, X^p) \stackrel {j_1} \longrightarrow Z^p({}^I E_1^{\scriptscriptstyle\bullet,q}) \twoheadrightarrow {}^I E_2^{p,q} \stackrel v \to {}^{II} E_2^{p,q}$$ is given by applying ${\mathrm {R} }^q f$ to $$\tau_{\leq q}\, X^p \to H^q(X^p)[q] \to H^q(X)[-q+p],$$ hence is equal to $j_2 u^\prime$. It follows that $v j_2 = v j_1 i_1^{-1} = j_2 u^\prime i_1^{-1} = j_2 u$. [Step 7:]{} the diagram (\[comp2\]) implies that the map (\[comp1\]) coincides with the composition of (\[comp3\]) with the canonical map $H^q(X)[-q+p] \to (\tau_{\leq q-1}\, X)[p+1]$, hence $u k_2 = k_2 v$. Theorem is proved. If $K^{\scriptscriptstyle\bullet}$ is a bounded below filtered complex over $A$ (with a finite filtration) $$K^{\scriptscriptstyle\bullet} = F^0 K^{\scriptscriptstyle\bullet} \supset F^1 K^{\scriptscriptstyle\bullet} \supset \cdots \supset F^n K^{\scriptscriptstyle\bullet} \supset F^{n+1} K^{\scriptscriptstyle\bullet} = 0,$$ then the objects $$X^p = F^p K^{\scriptscriptstyle\bullet}[p],\qquad Y^p = (F^p K^{\scriptscriptstyle\bullet}/F^{p+1} K^{\scriptscriptstyle\bullet})[p] = gr^p_F(K^{\scriptscriptstyle\bullet})[p] \in D^+(A)$$ form a Postnikov system of the kind considered in (\[postnikov\]). The corresponding spectral sequences are equal to $$E_1^{p,q} = H^{p+q}(gr^p_F(K^{\scriptscriptstyle\bullet})) \Longrightarrow H^{p+q}(K^{\scriptscriptstyle\bullet}),\qquad {}^I E_1^{p,q} = ({\mathrm {R} }^{p+q}f)(gr^p_F(K^{\scriptscriptstyle\bullet})) \Longrightarrow ({\mathrm {R} }^{p+q}f)(K^{\scriptscriptstyle\bullet}).$$ In the special case when $K^{\scriptscriptstyle\bullet}$ is the total complex associated to a first quadrant bicomplex $C^{\scriptscriptstyle\bullet, \scriptscriptstyle\bullet}$ and the filtration $F^p$ is induced by the column filtration on $C^{\scriptscriptstyle\bullet, \scriptscriptstyle\bullet}$, then the complex $f(K^{\scriptscriptstyle\bullet})$ over $A^\prime$ is equipped with a canonical filtration $(f F^p)(f(K^{\scriptscriptstyle\bullet})) = f(F^p K^{\scriptscriptstyle\bullet})$ satisfying $$gr^p_{f(F)}(f(K^{\scriptscriptstyle\bullet})) = f(gr^p_F(K^{\scriptscriptstyle\bullet})).$$ Under the assumptions (\[ass\]), the corresponding exact couple $${}^f D_1^{p,q} = H^{p+q}(f(F^p K^{\scriptscriptstyle\bullet})),\qquad {}^f E_1^{p,q} = H^{p+q}(gr^p_{f(F)}(f(K^{\scriptscriptstyle\bullet}))) = H^{p+q}(f(gr^p_F(K^{\scriptscriptstyle\bullet}))) \Longrightarrow H^{p+q}(f(K^{\scriptscriptstyle\bullet}))$$ then naturally maps to the exact couple (\[firstcouple\]), hence (beginning from $(D_2, E_2)$) to the exact couple (\[secondcouple\]), by Theorem \[speccomp\]. Syntomic cohomology =================== In this section we will define the arithmetic and geometric syntomic cohomologies of varieties over $K$ and ${\overline{K} }$, respectively, and study their basic properties. Hyodo-Kato morphism revisited ----------------------------- We will need to use the Hyodo-Kato morphism on the level of derived categories and vary it in $h$-topology. Recall that the original morphism depends on the choice of a uniformizer and a change of such is encoded in a transition function involving exponential of the monodromy. Since the fields of definition of semistable models in the bases for $h$-topology change we will need to use these transitions functions. The problem though is that in the most obvious (i.e., crystalline) definition of the Hyodo-Kato complexes the monodromy is (at best) homotopically nilpotent - making the exponential in the transition functions impossible to define. Beilinson [@BE2] solves this problem by representing Hyodo-Kato complexes using modules with nilpotent monodromy. In this subsection we will summarize what we need from his approach. At first a quick reminder. Let $(U,\overline{U})$ be a log-scheme, log-smooth over $V^{\times}$. For any $r\geq 0$, consider its absolute (meaning over $W(k)$) log-crystalline cohomology complexes $$\begin{aligned} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]})_n: & ={\mathrm {R} }\Gamma(\overline{U}_{{\operatorname{\acute{e}t} }},{\mathrm {R} }u_{U^{\times}_n/W_n(k)}{{\mathcal J}}^{[r]}_{U^{\times}_n/W_n(k)}),\quad {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]}):={\operatorname{holim} }_n{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]})_n,\\ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]})_{{\mathbf Q}}: & ={\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]})\otimes{{\mathbf Q}}_p,\end{aligned}$$ where $U^{\times}$ denotes the log-scheme $(U,\overline{U})$ and $u_{U^{\times}_n/W_n(k)}: (U^{\times}_n/W_n(k))_{{\operatorname{cr} }}\to \overline{U}_{{\operatorname{\acute{e}t} }}$ is the projection from the log-crystalline to the étale topos. For $r\geq 0$, we write ${{\mathcal J}}^{[r]}_{U^{\times}_n/W_n(k)}$ for the r’th divided power of the canonical PD-ideal ${{\mathcal J}}_{U^{\times}_n/W_n(k)}$; for $r\leq 0$, we set ${{\mathcal J}}^{[r]}_{U^{\times}_n/W_n(k)}:={{\mathcal O}}_{U^{\times}_n/W_n(k)}$ and we will often omit it from the notation. The absolute log-crystalline cohomology complexes are filtered commutative dg algebras (over $W_n(k)$, $W(k)$, or $K_0$). The canonical pullback map $${\mathrm {R} }\Gamma(\overline{U}_{{\operatorname{\acute{e}t} }},{\mathrm {R} }u_{U^{\times}_n/W_n(k)}{{\mathcal J}}^{[r]}_{U^{\times}_n/W_n(k)})\stackrel{\sim}{\to} {\mathrm {R} }u_{U^{\times}_n/{\mathbf Z}/p^n}{{\mathcal J}}^{[r]}_{U^{\times}_n/{\mathbf Z}/p^n})$$ is a quasi-isomorphism. In what follows we will often call both the “absolute crystalline cohomology”. Let $W(k)<t_l> $ be the divided powers polynomial algebra generated by elements $t_l$, $l\in {\mathfrak m}_K/ {\mathfrak m}^2_K\setminus \{0\}$, subject to the relations $t_{al}=[\overline{a}]t_l,$ for $a\in V^$, where $[\overline{a}]\in W(k)$ is the Teichmüller lift of $\overline{a}$ - the reduction mod $\mathfrak m_K$ of $a$. Let $R_{V}$ (or simply $R$) be the $p$-adic completion of the subalgebra of $W(k)<t_l>$ generated by $t_l$ and $t_l^{ie_K}/i!$, $i\geq 1$. For a fixed $l$, the ring $R$ is the following $W(k)$-subalgebra of $K_0[[t_l]]$: $$\begin{aligned} R =\{\sum_{i=0}^{\infty} a_i\frac{t_l^i}{\lfloor i/e_K\rfloor !}\mid a_i\in W(k), \lim_{i\rightarrow \infty}a_i=0\}.\end{aligned}$$ One extends the Frobenius ${\varphi}_R$ (semi-linearly) to $R$ by setting ${\varphi}_R(t_l)=t_l^p$ and defines a monodromy operator $N_R$ as a $W(k)$-derivation by setting $ N_R(t_l)=-t_l. $ Let $E:={\operatorname{Spec} }(R)$ equipped with the log-structure generated by the $t_l$’s. We have two exact closed embeddings $$i_0:W(k)^0\hookrightarrow E,\quad i_{\pi}: V^{\times}\hookrightarrow E.$$ The first one is canonical and induced by $ t_l\mapsto 0$. The second one depends on the choice of the class of the uniformizing parameter $\pi\in {\mathfrak m}_K/p{\mathfrak m}_{K}$ up to multiplication by Teichmüller elements. It is induced by $t_l\mapsto [\overline{l/\pi}]\pi$. Assume that $(U,\overline{U})$ is of Cartier type (i.e., the special fiber $\overline{U}_0$ is of Cartier type). Consider the log-crystalline and the Hyodo-Kato complexes (cf. [@BE2 1.16]) $${\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R,{{\mathcal J}}^{[r]})_n: ={\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})_{n}/R_n,{{\mathcal J}}^{[r]}_{\overline{U}_n/R_n}),\quad {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_n:={\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})_0/W_n(k)^0).$$ Let $ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R,{{\mathcal J}}^{[r]})$ and ${\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})$ be their homotopy inverse limits. The last complex is called the [Hyodo-Kato]{} complex. The complex ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)$ is $R$-perfect and $${\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)_n\simeq {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)\otimes^{L}_RR_n\simeq {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)\otimes^L{\mathbf Z}/p^n.$$ In general, we have ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R,{{\mathcal J}}^{[r]})_n\simeq {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R,{{\mathcal J}}^{[r]})\otimes^L{\mathbf Z}/p^n$. The complex ${\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})$ is $W(k)$-perfect and $${\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_n\simeq {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})\otimes^L_{W(k)}W_n(k)\simeq {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})\otimes^L{\mathbf Z}/p^n.$$ We normalize the monodromy operators $N$ on the rational complexes ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)_{{{\mathbf Q}}}$ and ${\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}}$ by replacing the standard $N$ [@HK 3.6] by $N_R:=e_{K}^{-1}N$. This makes them compatible with base change. The embedding $i_0: (U,\overline{U})_0\hookrightarrow (U,\overline{U}) $ over $i_0: W_n(k)^0\hookrightarrow E_n$ yields compatible morphisms $i^_{0,n}: {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)_n\to {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_n$. Completing, we get a morphism $$i^_0: {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)\to {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U}),$$ which induces a quasi-isomorphism $i^_0:{\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)\otimes ^L_RW(k)\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})$. All the above objects have an action of Frobenius and these morphisms are compatible with Frobenius. The Frobenius action is invertible on ${\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{\mathbf Q}$. The map $i^_{0}: {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)_{{\mathbf Q}}\to {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}}$ admits a unique (in the classical derived category) $W(k)$-linear section $\iota_{\pi}$ [@BE2 1.16], [@Ts 4.4.6] that commutes with ${\varphi}$ and $N$. The map $\iota_{\pi}$ is functorial and its $R$-linear extension is a quasi-isomorphism $$\iota: R\otimes_{W(k)}{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R) _{{\mathbf Q}}.$$ The composition (the [Hyodo-Kato map]{}) $$\iota_{{\mathrm{dR}},\pi}:=\gamma_r^{-1}i^_{\pi}\cdot\iota_{\pi}:\quad {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}}\to {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K}),$$ where $$\gamma_r^{-1}:\quad {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal O}}/{{\mathcal J}}^{[r]})_{{\mathbf Q}}\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K})/F^r$$ is the quasi-isomorphism from Corollary \[Langer\], induces a $K$-linear functorial quasi-isomorphism (the [Hyodo-Kato quasi-isomorphism]{}) [@Ts 4.4.8, 4.4.13] $$\label{HKqis} \iota_{{\mathrm{dR}},\pi}: {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})\otimes_{W(k)}K\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K})$$ We are going now to describe the Beilinson-Hyodo-Kato morphism and to study it on a few examples. Let $S_n={\operatorname{Spec} }({\mathbf Z}/p^n)$ equipped with the trivial log-structure and let $S={\operatorname{Spf} }({\mathbf Z}_p)$ be the induced formal log-scheme. For any log-scheme $Y\to S_1$ let $D_{{\varphi}}((Y/S)_{{\operatorname{cr} }},{{\mathcal O}}_{Y/S})$ denote the derived category of Frobenius ${{\mathcal O}}_{Y/S}$-modules and $D_{{\varphi}}^{{pcr}}(Y/S)$ its thick subcategory of perfect F-crystals, i.e., those Frobenius modules that are perfect crystals [@BE2 1.11]. We call a perfect F-crystal $({{\mathcal{F}}},{\varphi})$ [non-degenerate]{} if the map $L{\varphi}^({{\mathcal{F}}})\to {{\mathcal{F}}}$ is an isogeny. The corresponding derived category is denoted by $D_{{\varphi}}^{{pcr}}(Y/S)^{{nd}}$. It has a dg category structure [@BE2 1.14] that we denote by ${{\mathcal{D}}}_{{\varphi}}^{{pcr}}(Y/S)^{{nd}}$. We will omit $S$ if understood. Suppose now that $Y$ is a fine log-scheme that is affine. Assume also that there is a PD-thickening $P={\operatorname{Spf} }R$ of $Y$ that is formally smooth over $S$ and such that $R$ is a $p$-adically complete ring with no $p$-torsion. Let $f:Z\to Y$ be a log-smooth map of Cartier type with $Z$ fine and proper over $Y$. Beilinson [@BE2 1.11,1.14] proves the following theorem. \[kk1\] The complex ${{\mathcal{F}}}:=Rf_{{\operatorname{cr} }}({{\mathcal O}}_{Z/S})$ is a non-degenerate perfect F-crystal. Let $D_{{\varphi},N}(K_0)$ denote the bounded derived category of $({\varphi}, N)$-modules. By [@BE2 1.15], it has a dg category structure that we will denote by ${{\mathcal{D}}}_{{\varphi},N}(K_0)$. We call $({\varphi},N)$-module [effective]{} if it contains a $W(k)$-lattice preserved by ${\varphi}$ and $N$. Denote by ${{\mathcal{D}}}_{{\varphi},N}(K_0)^{{eff}}\subset {{\mathcal{D}}}_{{\varphi},N}(K_0)$ the bounded derived category of the abelian category of effective modules. Let $f:Y\to k^0$ be a log-scheme. We think of $k^0$ as $W(k)^{\times}_1$. Then the map $f$ is given by a $k$-structure on $Y$ plus a section $l=f^(\overline{p})\in\Gamma(Y,M_Y)$ such that its image in $\Gamma(Y,{{\mathcal O}}_Y)$ equals $0$. We will often write $f=f_l,l=l_f$. Beilinson proves the following theorem [@BE2 1.15]. \[kk2\] 1. There is a natural functor $$\label{kwaku-kwik} \epsilon_{f}=\epsilon_l: {{\mathcal{D}}}_{{\varphi},N}(K_0)^{{eff}}\to {{\mathcal{D}}}^{{pcr}}_{{\varphi}}(Y)^{{nd}}\otimes {{\mathbf Q}}.$$ 2. $\epsilon_{f}$ is compatible with base change, i.e., for any $\theta: Y^{\prime}\to Y$ one has a canonical identification $\epsilon_{f\theta}\stackrel{\sim}{\to}L\theta^_{{\operatorname{cr} }}\epsilon_{f}$. For any $a\in k^, m\in {\mathbf Z}_{>0}$, there is a canonical identification $\epsilon_{al^m}(V,{\varphi},N)\stackrel{\sim}{\to}\epsilon_l(V,{\varphi},mN)$. 3. Suppose that $Y$ is a local scheme with residue field $k$ and nilpotent maximal ideal, $M_Y/{{\mathcal O}}^_Y={\mathbf Z}_{>0}$, and the map $f^:M_{k^0}/k^\to M_Y/{{\mathcal O}}^_Y$ is injective. Then (\[kwaku-kwik\]) is an equivalence of dg categories. In particular, we have an equivalence of dg categories $$\epsilon:=\epsilon_{\overline{p}}: {{\mathcal{D}}}_{{\varphi},N}(K_0)^{{eff}}\stackrel{\sim}{\to}{{\mathcal{D}}}^{{pcr}}_{{\varphi}}(k^0)^{{nd}}\otimes{{\mathbf Q}}$$ and a canonical identification $\epsilon_f=Lf^_{{\operatorname{cr} }}\epsilon$. On the level of sections the functor (\[kk2\]) has a simple description [@BE2 1.15.3]. Assume that $Y={\operatorname{Spec} }(A/J)$, where $A$ is a $p$-adic algebra and $J$ is a PD-ideal in $A$, and that we have a PD-thickening $i:Y\hookrightarrow T={\operatorname{Spf} }(A)$. Let $\lambda_{l,n}$ be the preimage of $l$ under the map $\Gamma(T_n,M_{T_n})\to i_\Gamma(Y,M_Y)$. It is a trivial $(1+J_n)^{\times}$-torsor. Set $\lambda_A:=\invlim{_n}\Gamma(T_n,\lambda_{l,n})$. It is a $(1+J)^{\times}$-torsor. Let $\tau_{A_{{\mathbf Q}}}$ be the [Fontaine-Hyodo-Kato]{} torsor: $A_{{\mathbf Q}}$-torsor obtained from $\lambda_A$ by the pushout by $(1+J)^{\times}\stackrel{\log}{\to}J\to A_{{\mathbf Q}}$. We call the ${\mathbb G}_a$-torsor ${\operatorname{Spec} }A_{{\mathbf Q}}^{\tau}$ over ${\operatorname{Spec} }A_{{\mathbf Q}}$ with sections $\tau_{A_{{\mathbf Q}}}$ the same name. Denote by $N_{\tau}$ the $A_{{\mathbf Q}}$-derivation of $A_{{\mathbf Q}}^{\tau}$ given by the action of the generator of ${\operatorname{Lie}}_{{\mathbb G_a}}$. Let $M$ be an $({\varphi}, N)$-module. Integrating the action of the monodromy $N_M$ we get an action of the group ${\mathbb G}_a$ on $M$. Denote by $M^{\tau}_{A_{{\mathbf Q}}}$ the $\tau_{A_{{\mathbf Q}}}$-twist of $M_{A_{{\mathbf Q}}}:=M\otimes_{K_0}A_{{\mathbf Q}}$. It can be represented as the module of maps $v:\tau_{A_{{\mathbf Q}}}\to M_{A_{{\mathbf Q}}}$ that are $A_{{{\mathbf Q}}}$-equivariant, i.e., such that $v(\tau+a)=\exp(aN)(v(\tau))$, $\tau\in \tau_{A_{{\mathbf Q}}}$, $a\in A_{{\mathbf Q}}$. We can also write $$M^{\tau}_{A_{{\mathbf Q}}}=(M\otimes_{K_0}A^{\tau}_{{\mathbf Q}})^{{\mathbb G}_a}=(M\otimes_{K_0}A^{\tau}_{{\mathbf Q}})^{N=0},$$ where $N:=N_M\otimes 1+ 1\otimes N_{\tau}$. Now, by definition, $$\label{isom} \epsilon_f(M)(Y,T)=M^{\tau}_{A_{{\mathbf Q}}}$$ The algebra $A^{\tau}_{{\mathbf Q}}$ has a concrete description. Take the natural map $a:\tau_{A_{{\mathbf Q}}}\to A^{\tau}_{{\mathbf Q}}$ of $A_{{\mathbf Q}}$-torsors which maps $\tau\in \tau_{A_{{\mathbf Q}}}$ to a function $a({\tau})\in A^{\tau}_{{\mathbf Q}}$ whose value on any $\tau^{\prime}\in \tau_{A_{{\mathbf Q}}}$ is $\tau-\tau^{\prime}\in A_{{\mathbf Q}}$. This map is compatible with the logarithm $\log: (1+J)^{\times}\to A$. The algebra $A^{\tau}_{A_{{\mathbf Q}}}$ is freely generated over $A_{{\mathbf Q}}$ by the elements $a({\tau})$ for all $\tau\in\tau_{A_{{\mathbf Q}}}$; the $A_{{\mathbf Q}}$-derivation $N_{\tau}$ is defined by $N_{\tau}(a({\tau}))=-1$. That is, we can write $$\begin{aligned} A^{\tau}_{A_{{\mathbf Q}}} =A_{{\mathbf Q}}<a({\tau})>,\quad \tau\in \tau_{A_{{\mathbf Q}}};\quad N_{\tau}(a({\tau}))=-1\end{aligned}$$ For every lifting ${\varphi}_T$ of Frobenius to $T$ we have ${\varphi}^_T\lambda_A=\lambda_A^p$. Hence Frobenius ${\varphi}_T$ extends canonically to a Frobenius ${\varphi}_{\tau}$ on $A_{{\mathbf Q}}^{\tau}$ in such a way that $N_\tau{\varphi}_{\tau}=p{\varphi}_{\tau} N_{\tau}$. The isomorphism (\[isom\]) is compatible with Frobenius. \[standard\] As an example, consider the case when the pullback map $f^:{{\mathbf Q}}=(M_{k^0}/k^)^{{\operatorname{gp} }}\otimes{{\mathbf Q}}\stackrel{\sim}{\to} (\Gamma(Y,M_Y)/k^)^{{\operatorname{gp} }}\otimes{{\mathbf Q}}$ is an isomorphism. We have a surjection $v: (\Gamma(T,M_T)/k^)^{{\operatorname{gp} }}\otimes {{\mathbf Q}}\to {{\mathbf Q}}$ with the kernel $\log: (1+J)^{\times}_{{\mathbf Q}}\stackrel{\sim}{\to} J_{{\mathbf Q}}=A_{{{\mathbf Q}}}$. We obtain an identification of $A_{{\mathbf Q}}$-torsors $\tau_{A_{{\mathbf Q}}}\simeq v^{-1}(1)$. Hence every non-invertible $t\in \Gamma(T,M_T)$ yields an element $t^{1/v(t)}\in v^{-1}(1)$ and a trivialization of $\tau_{A_{{\mathbf Q}}}$. For a fixed element $t^{1/v(t)}\in v^{-1}(1)$, we can write $$\begin{aligned} A^{\tau}_{A_{{\mathbf Q}}} =A_{{\mathbf Q}}[a (t^{1/v(t))}],\quad \tau\in \tau_{A_{{\mathbf Q}}}, \quad N_{\tau}(a( t^{1/v(t))})=-1\end{aligned}$$ For an $({\varphi}, N)$-module $M$, the twist $M^{\tau}_{A_{{\mathbf Q}}}$ can be trivialized $$\begin{aligned} \beta_t: M\otimes_{K_0 }A_{{\mathbf Q}}& \stackrel{\sim}{\to} M^{\tau}_{A_{{\mathbf Q}}}=(M\otimes_{K_0}A_{{\mathbf Q}}[a( t^{1/v(t)})])^{N=0} \\ m & \mapsto \exp(N_M(m) a(t^{1/v(t))})\end{aligned}$$ For a different choice $t_1^{1/v(t_1)}\in v^{-1}(1)$, the two trivializations $\beta_t,\beta_{t_1}$ are related by the formula $$\beta_{t_1}=\beta_t\exp(N_M(m)a(t_1,t)),\quad a(t_1,t)=a(t_1)/v(t_1)-a(t)/v(t).$$ Consider the map $f:V^{\times}_1\to k^0$. By Theorem \[kk2\], we have the equivalences of dg categories $$\begin{aligned} \epsilon: & \quad {{\mathcal{D}}}_{{\varphi},N}(K_0)^{{eff}}\stackrel{\sim}{\to}{{\mathcal{D}}}^{{pcr}}_{{\varphi}}(k^0)^{{nd}}\otimes{{\mathbf Q}},\\ \epsilon_{f}=Lf^_{{\operatorname{cr} }}\epsilon: & \quad {{\mathcal{D}}}_{{\varphi},N}(K_0)^{{eff}}\stackrel{\sim}{\to}{{\mathcal{D}}}^{{pcr}}_{{\varphi}}(V_1^{\times})^{{nd}}\otimes{{\mathbf Q}}\end{aligned}$$ Let $Z_1\to V_1^{\times}$ be a log-smooth map of Cartier type with $Z_1$ fine and proper over $V_1$. By Theorem \[kk1\] $Rf_{{\operatorname{cr} }}({{\mathcal O}}_{Z_1/{\mathbf Z}_p})$ is a non-degenerate perfect F-crystal on $V_{1,{\operatorname{cr} }}$. Set $${\mathrm {R} }\Gamma^{B}_{{\mathrm{HK}}}(Z_1):=\epsilon^{-1}_{f}Rf_{{\operatorname{cr} }}({{\mathcal O}}_{Z_1/{\mathbf Z}_p})_{{\mathbf Q}}\in {{\mathcal{D}}}_{{\varphi},N}(K_0).$$ We will call it the [Beilinson-Hyodo-Kato]{} complex [@BE2 1.16.1]. \[crucial\] To get familiar with the Beilinson-Hyodo-Kato complexes we will work out some examples. 1. Let $g:X\to V^{\times}$ be a log-smooth log-scheme, proper, and of Cartier type. Adjunction yields a quasi-isomorphism $$\begin{aligned} \label{adjunction} \epsilon_f{\mathrm {R} }\Gamma^{B}_{{\mathrm{HK}}}(X_1)=\epsilon_f\epsilon^{-1}_{f}Rg_{{\operatorname{cr} }}({{\mathcal O}}_{X_1/{\mathbf Z}_p})_{{\mathbf Q}}\stackrel{\sim}{\to} Rg_{{\operatorname{cr} }}({{\mathcal O}}_{X_1/{\mathbf Z}_p})_{{\mathbf Q}}\end{aligned}$$ Evaluating it on the PD-thickening $V^{\times}_1\hookrightarrow V^{\times}$ (here $A=V$, $J=pV$, $l=\overline{p}$, $\lambda_V=p(1+J)^{\times}$, $\tau_K=p(1+J)^{\times}\otimes_{(1+J)^{\times}}K$), we get a map $$\begin{aligned} {\mathrm {R} }\Gamma^{B}_{{\mathrm{HK}}}(X_1)^{\tau}_K & =\epsilon_f{\mathrm {R} }\Gamma^{B}_{{\mathrm{HK}}}(X_1)(V^{\times}_1\hookrightarrow V^{\times}) \stackrel{\sim}{\to} Rg_{{\operatorname{cr} }}({{\mathcal O}}_{X_1/{\mathbf Z}_p})(V^{\times}_1\hookrightarrow V^{\times})_{{\mathbf Q}}={\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_1/V^{\times})_{{\mathbf Q}}\\ & \simeq {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X/V^{\times})_{{\mathbf Q}}\simeq {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_K)\end{aligned}$$ We will call it the [Beilinson-Hyodo-Kato]{} map [@BE2 1.16.3] $$\label{HK1} \iota^B_{{\mathrm{dR}}}:\quad {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)^{\tau}_K\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_K)$$ Recall that $${\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)^{\tau}_K=({\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)\otimes_{K_0}K[a(\tau)])^{N=0},\quad \tau\in\tau_K$$ This makes it clear that the Beilinson-Hyodo-Kato map is not only functorial for log-schemes over $V^{\times}$ but, by Theorem \[kk2\], it is also compatible with base change of $V^{\times}$. Moreover, if we use the canonical trivialization by $p$ $$\begin{aligned} \beta=\beta_p:\quad {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)_K & \stackrel{\sim}{\to}{\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X)_K^{\tau} =({\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)\otimes_{K_0}K[a(p)])^{N=0}\\ & x\mapsto \exp(N(x)a(p))\end{aligned}$$ we get that the composition (which we we also call the Beilinson-Hyodo-Kato map and denote by $\iota^B_{{\mathrm{dR}}}$) $$\iota^B_{{\mathrm{dR}}}=\iota^B_{{\mathrm{dR}}}\beta:\quad {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)\to {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_K)$$ is functorial and compatible with base change. 2. Evaluating the map (\[adjunction\]) on the PD-thickening $V^{\times}_1\hookrightarrow E$ associated to a uniformizer $\pi$ (here $A=R$, $l=\overline{p}$), we get a map $$\label{kappar} \kappa_R:\quad {\mathrm {R} }\Gamma^{B}_{{\mathrm{HK}}}(X_1)^{\tau}_{R_{{\mathbf Q}}} \stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X/R)_{{\mathbf Q}}$$ as the composition $$\begin{aligned} {\mathrm {R} }\Gamma^{B}_{{\mathrm{HK}}}(X_1)^{\tau}_{R_{{\mathbf Q}}} & =\epsilon_f{\mathrm {R} }\Gamma^{B}_{{\mathrm{HK}}}(X_1)(V^{\times}_1\hookrightarrow E) \stackrel{\sim}{\to} Rg_{{\operatorname{cr} }}({{\mathcal O}}_{X_1/{\mathbf Z}_p})(V^{\times}_1\hookrightarrow E)_{{\mathbf Q}}={\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_1/R)_{{\mathbf Q}}\\ & \simeq {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X/R)_{{\mathbf Q}}\end{aligned}$$ We have $${\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)^{\tau}_{R_{{\mathbf Q}}}=({\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)\otimes_{K_0}R_{{{\mathbf Q}}}[a(\tau)])^{N=0},\quad \tau\in\tau_{R_{{\mathbf Q}}}$$ Since the map $\kappa_R$ is compatible with the log-connection on $R$ it is also compatible with the normalized monodromy operators. Specifically, if we define the monodromy on the left hand side of (\[kappar\]) as $$\begin{aligned} N:\quad {\mathrm {R} }\Gamma^{B}_{{\mathrm{HK}}}(X_1)^{\tau}_{R_{{\mathbf Q}}} & \to {\mathrm {R} }\Gamma^{B}_{{\mathrm{HK}}}(X_1)^{\tau}_{R_{{\mathbf Q}}},\\ \sum_I m_{\tau_I}\otimes r_{\tau_I}a^{k_I}(\tau_I) & \mapsto \sum_I (N_M(m_{\tau_I})\otimes r_{\tau_I}a^{k_I}(\tau_I) + m_{\tau_I}\otimes N_R(r_{\tau_I})a^{k_I}(\tau_I))\end{aligned}$$ the two operators will correspond under the map $\kappa_R$. The exact immersion $i_{\pi}: V^{\times}\hookrightarrow E$, yields a commutative diagram $$\xymatrix{ {\mathrm {R} }\Gamma^{B}_{{\mathrm{HK}}}(X_1)^{\tau}_{R_{{\mathbf Q}}}\ar[r]^{\sim}\ar[d]^{i^_{\pi}} & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X/R)_{{\mathbf Q}}\ar[d]^{i^_{\pi}}\\ {\mathrm {R} }\Gamma^{B}_{{\mathrm{HK}}}(X_1)^{\tau}_{K}\ar[r]^{\sim} & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X/V^{\times})_{{\mathbf Q}}}$$ Note that here we do not have a canonical trivialization: we have $\lambda_R=ut_{\pi}^{e_K}(1+J)^{\times}$, where $u\in R$ is such that $u\pi^{e_K}$ lifts $\overline{p}$, but $u$ is not unique. 3. Consider the log-scheme $k^0_1$: the scheme ${\operatorname{Spec} }(k)$ with the log-structure induced by the exact closed immersion $i:k^0_1\hookrightarrow V^{\times}_1$. We have the commutative diagram $$\xymatrix{ X_0\ar@{^{(}->}[r]^i\ar[d]^{g_0} & X_1\ar[d]^g\\ k^0_1\ar[rd]_{f_0}\ar@{^{(}->}[r]^i & V^{\times}_1\ar[d]^f\\ & k^0 }$$ The morphisms $f, f_0$ map $\overline{p}$ to $\overline{p}$. By log-smooth base change we have a canonical quasi-isomorphism $Li^Rg_{{\operatorname{cr} }}({{\mathcal O}}_{X_1/{{\mathbf Z}_p} })\simeq Rg_{0{\operatorname{cr} }}({{\mathcal O}}_{X_1/{{\mathbf Z}_p}})$. By Theorem \[kwaku-kwik\] we have the equivalence of dg categories $$\epsilon_{f_0}: \quad {{\mathcal{D}}}_{{\varphi},N}(K_0)^{{eff}}\stackrel{\sim}{\to}{{\mathcal{D}}}^{{pcr}}_{{\varphi}}(k^0_1)^{{nd}}\otimes{{\mathbf Q}},\quad \epsilon_{f_0}=Li^\epsilon_f.$$ This implies the natural quasi-isomorphisms $$\begin{aligned} {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1) & =\epsilon^{-1}_fRg_{{\operatorname{cr} }}({{\mathcal O}}_{X_1/{{\mathbf Z}_p}})_{{\mathbf Q}}\simeq \epsilon^{-1}_{f_0}Li^Rg_{{\operatorname{cr} }}({{\mathcal O}}_{X_1/{{\mathbf Z}_p}})_{{\mathbf Q}}\\ & \simeq \epsilon_{f_0}^{-1}Rg_{0{\operatorname{cr} }}({{\mathcal O}}_{X_0/{{\mathbf Z}_p}})_{{\mathbf Q}}\end{aligned}$$ Hence, by adjunction, $$\epsilon_{f_0}{\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1) =\epsilon_{f_0}\epsilon_{f_0}^{-1}Rg_{0{\operatorname{cr} }}({{\mathcal O}}_{X_0/{{\mathbf Z}_p}})_{{\mathbf Q}}\simeq Rg_{0{\operatorname{cr} }}({{\mathcal O}}_{X_0/{{\mathbf Z}_p}})_{{\mathbf Q}}$$ We will evaluate both sides on the PD-thickening $ k^0_1\hookrightarrow W(k)^0$. Here we write the log-structure on $W(k)^0$ as associated to the map $\Gamma(V^{\times},M_{V^{\times}})\to k\to W(k)$. We take $A=W(k),$ $l=\overline{p}$, $J=pW(k)$, $\lambda_{W(k)}=\overline{p}(1+pW(k))^{\times}$, $\tau_{K_0}=\overline{p}(1+pW(k))^{\times}\otimes_{(1+pW(k))^{\times}}K_0$. We get a quasi-isomorphism $$\kappa:\quad {\mathrm {R} }\Gamma^{B}_{{\mathrm{HK}}}(X_1)^{\tau} \stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X)_{{\mathbf Q}}$$ as the composition $$\begin{aligned} {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)^{\tau} & =\epsilon_{f_0}{\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1) (k^0_1\hookrightarrow W(k)^0)\simeq Rg_{0{\operatorname{cr} }}({{\mathcal O}}_{X_0/{{\mathbf Z}_p}})(k^0_1\hookrightarrow W(k)^0)_{{\mathbf Q}}\\ & ={\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_0/W(k)^0)_{{\mathbf Q}}={\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X)_{{\mathbf Q}}\end{aligned}$$ To compare the monodromy operators on both sides of the map $\kappa$, note that by Theorem \[kk2\], we have the canonical identification $$Rg_{0{\operatorname{cr} }}({{\mathcal O}}_{X_0/{{\mathbf Z}_p}})_{{\mathbf Q}}\simeq \epsilon_{f_0}({\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1),N) \simeq \epsilon_{\overline{p}}({\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1) ,e_KN)$$ Hence, from the description of the Hyodo-Kato monodromy in [@HK 3.6], it follows easily that the map $\kappa$ pairs the operator $N$ on ${\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)^{\tau}$ defined by $$N(\sum_I m_{\tau_I}\otimes r_{\tau_I}a^{k_I}(\tau_I))=\sum_I (N_M(m_{\tau_I})\otimes r_{\tau_I}a^{k_I}(\tau_I) + m_{\tau_I}\otimes N_R(r_{\tau_I})a^{k_I}(\tau_I)),$$ with the normalized Hyodo-Kato monodromy on ${\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X)_{{\mathbf Q}}$. Composing the map $\kappa$ with the trivialization $$\begin{aligned} \beta=\beta_{p}:\quad {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1) & \stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)^{\tau}_{K_0} =({\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)[a(\overline{p})])^{N=0}\\ & x\mapsto \exp(N(x)a(\overline{p}))\end{aligned}$$ we get a quasi-isomorphism between Beilinson-Hyodo-Kato complexes and the (classical) Hyodo-Kato complexes. $$\begin{aligned} \label{B=K} \kappa=\beta\kappa:\quad {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X)_{{\mathbf Q}}\end{aligned}$$ The trivialization above is compatible with Frobenius and the normalized monodromy hence so is the quasi-isomorphism (\[B=K\]). It is clearly functorial and, by Theorem \[kk2\], compatible with base change. By functoriality (Theorem \[kk2\]), the morphism of PD-thickenings (exact closed immersion) $i_0: (k^0_1\hookrightarrow W(k)^0)\hookrightarrow (V^{\times}_1\hookrightarrow E)$ yields the left square in the following diagram $$\xymatrix{ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_1/R)_{{\mathbf Q}}\ar[r]^{i^_0} & {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X)_{{\mathbf Q}}\ar[r]^{\iota_{\pi}} & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_1/R)_{{\mathbf Q}}\\ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)^{\tau}_{R_{{\mathbf Q}}}\ar[u]^{\wr}_{\kappa_R}\ar[r]^{i^_0} & {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)^{\tau}_{K_0}\ar[u]^{\wr}_{\kappa}\ar[r]^{\iota_{\pi}} & {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)^{\tau}_{R_{{\mathbf Q}}}\ar[u]^{\wr}_{\kappa_R} }$$ In the right square the bottom map $\iota_{\pi}$ is induced by the natural map $ K_0\to R$ and the pullback $i^_0:\tau_{R_{{\mathbf Q}}}\to \tau_{W(k)^0_{{\mathbf Q}}}$. It is a (right) section to $i_0^$ and it (together with the vertical maps) commutes with Frobenius. By uniqueness of the top map $\iota_{\pi}$ this makes the right square commute in the $\infty$-derived category (of abelian groups). It is easy to check that we have the following commutative diagram $$\xymatrix{ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)^{\tau}\ar[r]^{\iota_{\pi}} & {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)^{\tau}_{R_{{\mathbf Q}}}\ar[r]^{i_{\pi}^} & {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)^{\tau}_K\\ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)\ar[u]^{\beta_p}_{\wr}\ar[rr]^{{ \operatorname{can} }} & & {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_1)_K\ar[u]^{\beta_p}_{\wr} }$$ and that the composition of maps on the top of it is equal to the map induced by the canonical map $K_0\to K$ and the diagram $\tau_{W(k)^0_{{\mathbf Q}}}\stackrel{i^_0}{\leftarrow}\tau_{R_{{\mathbf Q}}}\stackrel{i^_{\pi}}{\to}\tau_{K}.$ Combining the commutative diagrams in parts (2) and (3) of this example we get the following commutative diagram. $$\xymatrix{ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X)\ar[r]^{\iota_{\pi}} & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_1/R)_{{\mathbf Q}}\ar[r]^{i_{\pi}^} & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_1/V^{\times})_{{\mathbf Q}}\\ {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_1)^{\tau} \ar[u]^{\wr}_{\kappa}\ar[r]^{\iota_{\pi}} & {\mathrm {R} }\Gamma(X_1)_{R_{{\mathbf Q}}}^{\tau} \ar[r]^{i^_{\pi}}\ar[u]^{\wr}_{\kappa_R} & {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_1)_K^{\tau}\ar[u]^{\wr}_{\iota^B_{{\mathrm{dR}}}}\\ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_1)\ar[u]^{\beta_p}_{\wr}\ar[rr]^{{ \operatorname{can} }} & & {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_1)_K\ar[u]^{\beta_p}_{\wr} }$$ Since the composition of the top maps is equal to the Hyodo-Kato map $\iota_{{\mathrm{dR}}}$ and the bottom maps is just the canonical map ${\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_1)\to {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_1)_K$ we obtain that the Hyodo-Kato and the Beilinson-Hyodo-Kato maps are related by a natural quasi-isomorphism, i.e., that the following diagram commutes. $$\label{Beilinson=HK} \xymatrix{ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X)\ar[r]^{\iota_{{\mathrm{dR}},\pi}} & {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{K})\\ {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_1)\ar[ru]_{\iota_{{\mathrm{dR}}}^B}\ar[u]^{\wr}_{\kappa} }$$ The above examples can be generalized [@BE2 1.16]. It turns out that the relative crystalline cohomology of all the base changes of the map $f$ can be described using the Beilinson-Hyodo-Kato complexes [@BE2 1.16.2]. Namely, let $\theta:Y\to V^{\times}_1$ be an affine log-scheme and let $T$ be a $p$-adic PD-thickening of $Y$, $T={\operatorname{Spf} }(A),$ $Y={\operatorname{Spec} }(A/J)$. Denote by $f_Y:Z_{1Y}\to Y$ the $\theta$-pullback of $f$. Beilinson proves the following theorem [@BE2 1.16.2]. \[Bthm\] 1. The $A$-complex ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Z_{1Y}/T,{{\mathcal O}}_{Z_{1Y/T}})$ is perfect, and one has $${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Z_{1Y}/T_n,{{\mathcal O}}_{Z_{1Y/T_n}})={\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Z_{1Y}/T,{{\mathcal O}}_{Z_{1Y/T}})\otimes^{L}{\mathbf Z}/p^n.$$ 2. There is a canonical Beilinson-Hyodo-Kato quasi-isomorphism of $A_{{\mathbf Q}}$-complexes $$\kappa_{A_{{\mathbf Q}}}^B: {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(Z_1)^{\tau}_{A_{{\mathbf Q}}}\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Z_{1Y}/T,{{\mathcal O}}_{Z_{1Y/T}})_{{\mathbf Q}}$$ If there is a Frobenius lifting ${\varphi}_T$, then $\kappa^B_{A_{{\mathbf Q}}}$ commutes with its action. Log-syntomic cohomology ----------------------- We will study now (rational) log-syntomic cohomology. Let $(U,\overline{U})$ be log-smooth over $V^{\times}$. For $r\geq 0$, define the mod $p^n$, completed, and rational log-syntomic complexes $$\begin{aligned} \label{log-syntomic} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_n & := {\operatorname{Cone} }({\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]})_n\verylomapr{p^r-{\varphi}} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_n)[-1],\\ {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r) & :={\operatorname{holim} }_n{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_n,\notag\\ {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{\mathbf Q} & := {\operatorname{Cone} }({\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]})_{{\mathbf Q}}\verylomapr{1-{\varphi}_r} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_{{\mathbf Q}})[-1].\notag\end{aligned}$$ Here the Frobenius ${\varphi}$ is defined by the composition $$\begin{aligned} {\varphi}: {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]})_n & \to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_n \stackrel{\sim}{\to }{\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})_1/W(k))_n \stackrel{{\varphi}}{\to} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})_1/W(k))_n\\ & \stackrel{\sim}{\leftarrow}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_n\end{aligned}$$ and ${\varphi}_r:={\varphi}/p^r$. The mapping fibers are taken in the $\infty$-derived category of abelian groups. The direct sums $$\bigoplus _{r\geq 0}{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_n,\quad \bigoplus_{r\geq 0} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r),\quad \bigoplus_{r \geq 0} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{{\mathbf Q}}$$ are graded $E_{\infty}$ algebras over ${\mathbf Z}/p^n$, ${\mathbf Z}_p$, and ${{\mathbf Q}}_p$, respectively [@HS 1.6]. The rational log-syntomic complexes are moreover graded commutative dg algebras over ${{\mathbf Q}}_p$ [@HS 4.1], [@MG 3.22], [@Lu2]. Explicit definition of syntomic product structure can be found in [@Ts 2.2]. We have $ {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_n\simeq {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)\otimes^L{\mathbf Z}/p^n. $ There is a canonical quasi-isomorphism of graded $E_{\infty}$ algebras $$\begin{aligned} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_n \stackrel{\sim}{\to}{\operatorname{Cone} }({\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_n\verylomapr{(p^r-{\varphi},{ \operatorname{can} })} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_n\oplus {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal O}}/{{\mathcal J}}^{[r]})_n)[-1].\end{aligned}$$ Similarly in the completed and rational cases. Since, by Corollary \[Langer\], there is a quasi-isomorphism $$\gamma_r^{-1}:\quad {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal O}}/{{\mathcal J}}^{[r]})_{{\mathbf Q}}\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K})/F^r,$$ we have a particularly nice canonical description of rational log-syntomic cohomology $$\begin{aligned} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{{\mathbf Q}} \stackrel{\sim}{\to} [{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_{{\mathbf Q}}\verylomapr{(1-{\varphi}_r,\gamma_r^{-1})} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_{{\mathbf Q}}\oplus {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K})/F^r)],\end{aligned}$$ where square brackets stand for mapping fiber. For arithmetic pairs $(U,\overline{U})$ that are log-smooth over $V^{\times}$ and of Cartier type this can be simplified further by using Hyodo-Kato complexes (cf. Proposition \[reduction1\] below). To do that, consider the following sequence of maps of homotopy limits. Homotopy limits are taken in the $\infty$-derived category. We will describe the coherence data only if they are nonobvious. $$\begin{aligned} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{\mathbf Q} & \stackrel{\sim}{\to} \xymatrix@C=36pt{[{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_{\mathbf Q}\ar[r]^-{(1-{\varphi}_r,\gamma_r^{-1})} & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_{\mathbf Q}\oplus {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K})/F^r ]}\\ & \stackrel{\sim}{\to} \left[\begin{aligned}{\xymatrix{{\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)_{\mathbf Q}\ar[rr]^-{(1-{\varphi}_r,i^_{\pi}\gamma_r^{-1})}\ar[d]^{N} && {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)_{\mathbf Q}\oplus {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K})/F^r \ar[d]^{(N,0)}\\ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)_{\mathbf Q}\ar[rr]^-{1-{\varphi}_{r-1}} && {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)_{\mathbf Q}}}\end{aligned}\right]\\ & \stackrel{\iota_{\pi}}{\leftarrow} \left[\begin{aligned}\xymatrix@C=40pt{{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{\mathbf Q}\ar[r]^-{(1-{\varphi}_r,\iota_{{\mathrm{dR}},\pi})}\ar[d]^{N} & {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{\mathbf Q} \oplus {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K}) /F^r\ar[d]^{(N,0)}\\ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{\mathbf Q}\ar[r]^{1-{\varphi}_{r-1}} & {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{\mathbf Q}}\end{aligned}\right]\end{aligned}$$ The first map was described above. The second one is induced by the distinguished triangle $${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})\to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)\stackrel{N}{\to}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)$$ The third one - by the section $\iota_{\pi}: {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{\mathbf Q}\to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)_{\mathbf Q}$ (notice that $\iota_{{\mathrm{dR}},\pi}=\gamma_{r}^{-1}i^_{\pi}\iota_{\pi}$). We will show below that the third map is a quasi-isomorphism. Set $C_{{\operatorname{st} }}({\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})\{r\})$ equal to the last homotopy limit in the above diagram. \[reduction1\] Let $(U,\overline{U})$ be an arithmetic pair that is log-smooth over $V^{\times}$ and of Cartier type. Let $r\geq 0$. Then the above diagram defines a canonical quasi-isomorphism. $$\alpha_{{ \operatorname{syn} },\pi}:\quad {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{\mathbf Q}\stackrel{\sim}{\to} C_{{\operatorname{st} }}({\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})\{r\}).$$ We need to show that the map $\iota_{\pi}$ in the above diagram is a quasi-isomorphism. Define complexes ($r\geq -1$) $$\begin{aligned} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R,r):= & {\operatorname{Cone} }({\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)_{{\mathbf Q}}\stackrel{1-{\varphi}_r}{\longrightarrow}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)_{{\mathbf Q}})[-1],\\ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U},r):= & {\operatorname{Cone} }({\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}}\stackrel{1-{\varphi}_r}{\longrightarrow}{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}})[-1]\end{aligned}$$ It suffices to prove that the following maps $$\label{reduction} i^_0: {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R,r) \stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U},r),\quad \iota_{\pi}: {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U},r) \stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R,r)$$ are quasi-isomorphisms. Since $i^_0\iota_{\pi}={ \operatorname{Id} }$, it suffices to show that the map $i^_0$ is a quasi-isomorphism. Base-changing to $W(\overline{k})$, we may assume that the residue field of $V$ is algebraically closed. It suffices to show that, for $i\geq 0$, $t\geq -1$, in the commutative diagram $$\begin{CD} H^i_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}}@>p^t-{\varphi}>> H^i_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}}\\ @AA i^_0 A @AA i^_0 A\\ H^i_{{\operatorname{cr} }}((U,\overline{U})/R)_{{\mathbf Q}}@>p^t-{\varphi}>>H^i_{{\operatorname{cr} }}((U,\overline{U})/R)_{{\mathbf Q}} \end{CD}$$ the vertical maps induce isomorphisms between the kernels and cokernels of the horizontal maps. Since the $W(k)$-linear map $\iota_{\pi}$ commutes with ${\varphi}$ and its $R$-linear extension is a quasi-isomorphism $$\iota_{\pi}: R\otimes_{W(k)}{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R) _{{\mathbf Q}}$$ it suffices to show that in the following commutative diagram $$\begin{CD} H^i_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}}@>p^t-{\varphi}>> H^i_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}}\\ @AA i_0\otimes{ \operatorname{Id} }A @AA i_0 \otimes{ \operatorname{Id} }A \\ R\otimes_{W(k)}H^i_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}}@>p^t-{\varphi}>> R\otimes_{W(k)}H^i_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}}\end{CD}$$ the vertical maps induce isomorphisms between the kernels and cokernels of the horizontal maps. This will follow if we show that the following map $$I\otimes_{W(k)}H^i_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}}\stackrel{p^t-{\varphi}}{\longrightarrow} I\otimes_{W(k)}H^i_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}},$$ for $I\subset R$ - the kernel of the projection $i_0:R_{\mathbf Q}\to K_0$, $t_l\mapsto 0$, is an isomorphism. We argue as Langer in [@Ln p. 210]. Let $M:=H^i_{{\mathrm{HK}}}(U,\overline{U})/tor$. It is a lattice in $H^i_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}}$ that is stable under Frobenius. Consider the formal inverse $\psi:=\sum_{n\geq 0}(p^{-t}{\varphi})^n$ of $1-p^{-t}{\varphi}$. It suffices to show that, for $y\in I\otimes_{W(k)}M$, $\psi(y)\in I\otimes_{W(k)}M$. Fix $l$ and let $T^{\{k\}}:=t_l^k/\lfloor k/e_K\rfloor !$. We will show that, for any $m\in M$, $\psi(T^{\{k\}}\otimes m)\in I\otimes_{W(k)}M$ and the infinite series converges uniformly in $k$. We have $$(p^{-t}{\varphi})^n(T^{\{k\}}\otimes m )=\frac{\lfloor kp^n/e_K\rfloor !}{\lfloor k/e_K\rfloor !p^{tn}}T^{\{kp^n\}}\otimes m{^{\prime}}$$ and ${\operatorname{ord} }_p(\lfloor kp^n/e_K\rfloor !/\lfloor k/e_K\rfloor !)\geq p^{n-1}$. Hence $\frac{\lfloor kp^n/e_K\rfloor !}{\lfloor k/e_K\rfloor !p^{tn}}$ converges $p$-adically to zero, uniformly in $k$, as wanted. It was Langer [@Ln p.193] (cf. [@JS Lemma 2.13] in the good reduction case) who observed the fact that while, in general, the crystalline cohomology ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,{\overline}{U})$ behaves badly (it is “huge”), after taking “filtered Frobenius eigenspaces” we obtain syntomic cohomology ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,{\overline}{U},r)_{{{\mathbf Q}}}$ that behaves well (it is “small”). In [@JB 3.5] this phenomena is explained by relating syntomic cohomology to the complex $C_{{\operatorname{st} }}({\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,{\overline}{U})\{r\})$. \[reduction21\] The construction of the map $\alpha_{{ \operatorname{syn} },\pi}$ depends on the choice of the uniformizer $\pi$ what makes $h$-sheafification impossible. We will show now that there is a functorial and compatible with base change quasi-isomorphism $\alpha^{\prime}_{{ \operatorname{syn} }}$ between rational syntomic cohomology and certain complexes built from Hyodo-Kato cohomology and de Rham cohomology that $h$-sheafify well. Set $$\begin{aligned} \alpha^{\prime}_{{ \operatorname{syn} }}:\quad {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{\mathbf Q} & \stackrel{\sim}{\to} [{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},r)\lomapr{\gamma_r^{-1}} {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K})/F^r ]\\ & \stackrel{\beta}{\to} [{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U},r)^{N=0}\lomapr{\iota^{\prime}_{{\mathrm{dR}}}} {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K}) /F^r]\end{aligned}$$ Here the two morphisms $\beta$ and $\iota^{\prime}_{{\mathrm{dR}}}$ are defined as the following compositions $$\begin{aligned} \beta:\quad & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},r)\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U_0,\overline{U}_0,r)\stackrel{\sim}{\to} [{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U},r)]^{N=0}\\ \iota^{\prime}_{{\mathrm{dR}}}:\quad & [{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U},r)]^{N=0}\stackrel{\beta}{\leftarrow} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},r)\stackrel{\gamma_r^{-1}}{\to}{\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K}),\end{aligned}$$ where $[\cdots ]^{N=0}$ denotes the mapping fiber of the monodromy. The map $\beta$ is a quasi-isomorphism because so is each of the intermediate maps. To see this for the map $i^_0: {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},r)\to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U_0,\overline{U}_0,r)$, consider the following factorization $$F^m: {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},r)\stackrel{i^_0}{\to} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U_0,\overline{U}_0,r) \stackrel{\psi_m}{\to}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},r)$$ of the $m$’th power of the Frobenius, where $m$ is large enough. We also have $ i^_0\psi_m=F^m$. Since Frobenius is a quasi-isomorphism on ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},r)$ and ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U_0,\overline{U}_0,r)$ both $i^_0$ and $\psi_m$ are quasi-isomorphisms as well. The second morphism in the sequence defining $\beta$ is a quasi-isomorphism by an argument similar to the one we used in the proof of Proposition \[reduction1\]. Define the complex $$C^{\prime}_{{\operatorname{st} }}({\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})\{r\}):= [{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U},r)^{N=0}\lomapr{\iota^{\prime}_{{\mathrm{dR}}}} {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K}) /F^r].$$ We have obtained a quasi-isomorphism $$\alpha^{\prime}_{{ \operatorname{syn} }}:\quad {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{\mathbf Q}\stackrel{\sim}{\to} C^{\prime}_{{\operatorname{st} }}({\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})\{r\})$$ It is clearly functorial but it is also easy to check that it is compatible with base change (of the base $V$). Define the complex $$C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(U,\overline{U})\{r\}):= [{\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(U_1,\overline{U}_1,r)^{N=0}\lomapr{\iota^{B}_{{\mathrm{dR}}}} {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K}) /F^r].$$ From the commutative diagram (\[Beilinson=HK\]) we obtain the natural quasi-isomorphisms $$\begin{aligned} \gamma: & \quad C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(U,\overline{U})\{r\}) \stackrel{\sim}{\to}C_{{\operatorname{st} }}({\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})\{r\})\\ \alpha_{{ \operatorname{syn} },\pi}^B:=\gamma^{-1}\alpha_{{ \operatorname{syn} },\pi}: & \quad {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{\mathbf Q} \stackrel{\sim}{\to}C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(U,\overline{U})\{r\})\end{aligned}$$ We will show now that log-syntomic cohomology satisfies finite Galois descent. Let $(U,\overline{U})$ be a fine log-scheme, log-smooth over $V^{\times}$, and of Cartier type. Let $r\geq 0$. Let $K^{\prime}$ be a finite Galois extension of $K$ and let $G={\operatorname{Gal} }(K{^{\prime}}/K)$. Let $(T,\overline{T})=(U\times_{V}{V^{\prime}},\overline{U}\times _{V}{V^{\prime}})$, $V^{\prime}$ - the ring of integers in $K^{\prime}$, be the base change of $(U,\overline{U})$ to $(K{^{\prime}},V{^{\prime}})$, and let $f: (T,\overline{T})\to (U,\overline{U})$ be the canonical projection. Take $R=R_{V}$, $N$, $e$, $\pi$ associated to $V$. Similarly, we define $R{^{\prime}}:=R_{V{^{\prime}}}$, $N{^{\prime}}$, $e{^{\prime}}$, $\pi{^{\prime}}$. Write the map $\alpha^B_{{ \operatorname{syn} },\pi}$ as $$\xymatrix{ {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{\mathbf Q}\ar[d]^{\alpha^B_{{ \operatorname{syn} },\pi}}_{\wr}\ar[r]^-{\sim}_-h & [{\mathrm {R} }\Gamma^{B,\tau}_{{\mathrm{HK}}}((U,\overline{U})_{R},r)^{N=0}\ar[r]^{i^_{\pi}} & {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K}) /F^r]\\ C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(U,\overline{U}) \{r\})\ar[r]^{\sim} & [{\mathrm {R} }\Gamma^{B}_{{\mathrm{HK}}}(U,\overline{U},r)^{N=0}\ar[r]^{\iota^B_{{\mathrm{dR}}}}\ar[u]_{\wr}^{\iota_{\pi}\beta} & {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K}) /F^r]\ar@{=}[u] }$$ Here we defined the map $h$ as the composition $$\label{h} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{\mathbf Q}\to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)_{{\mathbf Q}}\stackrel{\sim}{\leftarrow} {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(U_1,\overline{U}_1)_{R_{{\mathbf Q}}}^{\tau}$$ From the construction of the Beilinson-Hyodo-Kato map $\iota_{{\mathrm{dR}}}^B: {\mathrm {R} }\Gamma^{B}_{{\mathrm{HK}}}(T_1,\overline{T}_1)\to {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(T,\overline{T}_{K^{\prime}}) $ it follows that it is $G$-equivariant; hence the complex $C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(T,\overline{T})\{r\})$ is equipped with a natural $G$-action. We claim that the map $\alpha^B_{{ \operatorname{syn} },\pi^\prime}$ induces a natural map $$\begin{aligned} \tilde{\alpha}^B_{{ \operatorname{syn} },\pi^\prime}: & \quad {\mathrm {R} }\Gamma(G,{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(T,\overline{T},r)_{{\mathbf Q}})\to {\mathrm {R} }\Gamma(G,C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(T,\overline{T})\{r\})),\\ \tilde{\alpha}^B_{{ \operatorname{syn} },\pi^\prime} & := (1/\|G\|)\sum_{g\in G}\alpha^B_{{ \operatorname{syn} },g(\pi^\prime)} \end{aligned}$$ To see this it suffices to show that, for every $g\in G$, we have a commutative diagram $$\begin{CD} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(T,\overline{T},r)_{{\mathbf Q}}@>\alpha^B_{{ \operatorname{syn} },\pi^\prime}>> C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(T,\overline{T})\{r\})\\ @VV g^* V @VV g^* V\\ {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(T,\overline{T},r)_{{\mathbf Q}}@>\alpha^B_{{ \operatorname{syn} },g(\pi^\prime)}>> C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(T,\overline{T})\{r\}) \end{CD}$$ We accomplish this by constructing natural morphisms $$\begin{aligned} g^: \quad & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((T,\overline{T})/R_{\pi{^{\prime}}}{^{\prime}})\to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((T,\overline{T})/R_{g(\pi{^{\prime}})}{^{\prime}}),\\ g^: \quad & {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(T_1,\overline{T}_1)_{R^\prime_{\pi^{\prime}}}^{\tau}\to {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(T_1,\overline{T}_1)_{R^\prime_{g(\pi^{\prime})}}^{\tau} \end{aligned}$$ that are compatible with the maps in (\[h\]) that define $h$, the maps $\iota_?$ and $i^_?$, and the trivialization $\beta$. We define the pullbacks $g^$ from a map $g:R{^{\prime}}_{\pi{^{\prime}}}\to R{^{\prime}}_{g(\pi{^{\prime}})}$ constructed by lifting the action of $g$ from $V_1^\prime$ to $R{^{\prime}}$ by setting $g(T{^{\prime}})=[x_g]T{^{\prime}}$, $x_g=\overline{\pi{^{\prime}}/g(\pi{^{\prime}})}$. This map is compatible with Frobenius and monodromy. The induced pullbacks $g^$ are clearly compatible with the map $i^_0$ (hence the maps $\iota_?$), the maps $i^_{\pi{^{\prime}}}$, $i^_{g(\pi{^{\prime}})}$, and the trivialization $\beta$. From the construction of the Beilinson-Hyodo-Kato map, the pullbacks $g^$ are also compatible with the maps $\kappa_{R^\prime_?}$; hence with the map $h$, as wanted. \[hypercov11\] 1. The following diagram commutes in the (classical) derived category. $$\begin{CD} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{{\mathbf Q}} @> f^>> {\mathrm {R} }\Gamma(G,{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(T,\overline{T},r)_{{\mathbf Q}})\\ @VV\alpha^B_{{ \operatorname{syn} },\pi}V @VV\tilde{\alpha}^B_{{ \operatorname{syn} },\pi^{\prime}}V\\ C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(U,\overline{U}) \{r\}) @> f^* >> {\mathrm {R} }\Gamma(G,C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(T,\overline{T})\{r\})) \end{CD}$$ 2. The natural map $$f^: {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{{\mathbf Q}} \stackrel{\sim}{\to} {\mathrm {R} }\Gamma(G,{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(T,\overline{T},r)_{{\mathbf Q}})$$ is a quasi-isomorphism. The second claim of the proposition follows from the first one and the fact that the Hyodo-Kato and de Rham cohomologies satisfy finite Galois decent. Since everything in sight is functorial and satisfies finite unramified Galois descent we may assume that the extension $K{^{\prime}}/K$ is totally ramified. First, we will construct a $G$-equivariant (for the trivial action of $G$ on $R$) map $$f^: {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R,r)^{N=0} \to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((T,\overline{T})/R{^{\prime}},r)^{N{^{\prime}}=0}$$ such that the following diagram commutes $$\label{diag1} \begin{CD} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},r)@> f^>> {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(T,\overline{T},r)\\ @VV\wr V @VV\wr V\\ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R,r)^{N=0} @> f^>> {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((T,\overline{T})/R{^{\prime}},r)^{N{^{\prime}}=0}\\ @A\wr A\iota_{\pi} A @A\wr A\iota_{\pi^{\prime}} A\\ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U},r)^{N=0}@> f^* > \sim > {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(T,\overline{T},r)^{N{^{\prime}}=0} \end{CD}$$ Note that the bottom map is an isomorphism because $f^$ acts trivially on the Hyodo-Kato complexes. The commutativity of the above diagram and the quasi-isomorphisms (\[reduction\]) will imply that a totally ramified Galois extension does not change the log-crystalline complexes ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},r)$ and ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R,r)^{N=0}$. Let $e_1$ be the ramification index of $V{^{\prime}}/V$. Set $v=(\pi{^{\prime}})^{e_1}\pi^{-1}$, and choose an integer $s$ such that $(\pi{^{\prime}})^{p^s}\in pV{^{\prime}}$. Set $T:=t_{\pi}, T{^{\prime}}:=t_{\pi{^{\prime}}}$ and define the morphism $a: R\to R{^{\prime}}$ by $T\mapsto (T{^{\prime}})^{e_1}[\overline{v}]^{-1}$. Since $V{^{\prime}}_1$ and $V_1$ are defined by $pR+T^{e}R$ and by $pR{^{\prime}}+(T{^{\prime}})^{e{^{\prime}}}R{^{\prime}}$, respectively, $a$ induces a morphism $a_1: V_{1}\to V{^{\prime}}_1$. We have $F^sa_1=F^sf_1$, where $F$ is the absolute Frobenius on ${\operatorname{Spec} }(V_{1})$. Notice that in general $f_1\neq a_1$ if $v[\overline{v}]^{-1}\ncong 1 \mod pV{^{\prime}}$. The morphism ${\varphi}_R^sa: {\operatorname{Spec} }(R{^{\prime}})\to {\operatorname{Spec} }(R)$ is compatible with $F^sf_1:{\operatorname{Spec} }(V{^{\prime}}_1)\to{\operatorname{Spec} }(V_{1})$ and it commutes with the operators $N$ and $p^sN{^{\prime}}$. We have the following commutative diagram $$\xymatrix{ (T,\overline{T})_1\ar[rr]^-{F^sf_1} \ar[d] && (U,\overline{U})_1\ar[d]\\ {\operatorname{Spec} }(V^{\prime}_1)\ar[rr] ^-{F^sa_1=F^sf_1}\ar[d]&& {\operatorname{Spec} }(V_{1})\ar[d]\\ {\operatorname{Spec} }(R^{\prime}) \ar[rr] ^-{{\varphi}_R^sa}&& {\operatorname{Spec} }(R) }$$ Hence the commutative diagram of distinguished triangles $$\label{brrrr} \begin{CD} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_{{\mathbf Q}}@>>> {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)_{{\mathbf Q}}@> eN >> {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)_{{\mathbf Q}}\\ @VV f^F^s V @VV f^F^sV @VV p^se_1f^F^sV \\ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(T,\overline{T})_{{\mathbf Q}}@>>> {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((T,\overline{T})/R{^{\prime}})_{{\mathbf Q}}@> e^{\prime}N{^{\prime}}>> {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((T,\overline{T})/R{^{\prime}})_{{\mathbf Q}}. \end{CD}$$ To see how this diagram arises we may assume (by the usual Čech argument) that we have a fine affine log-scheme $X_n/V^{\times}_n$ that is log-smooth over $V^{\times}_n$. We can also assume that we have a lifting of $X_n\hookrightarrow Z_n$ over ${\operatorname{Spec} }(W_n(k)[T])$ (with the log-structure coming from $T$) and a lifting of Frobenius ${\varphi}_Z$ on $Z_n$ that is compatible with the Frobenius ${\varphi}_R$. Recall [@Kas Lemma 4.2] that the horizontal distinguished triangles in the above diagram arise from an exact sequence of complexes of sheaves on $X_{n,{\operatorname{\acute{e}t} }}$ $$\label{monodromyC} 0\to C_V{^{\prime}}[-1]\verylomapr{\wedge {\operatorname{dlog} }T} C_V\to C_V{^{\prime}}\to 0$$ where $C_V:=R_n\otimes _{W_n(k)[T]}\Omega{^{\cdot }}_{Z_n/W_n(k)}$ and $C_V{^{\prime}}:=R_n\otimes _{W_n(k)[T]}\Omega{^{\cdot }}_{Z_n/W_n(k)[T]}$. Now consider the base change of $Z_n/W_n(k)[T]$ by the map $F^sa:{\operatorname{Spec} }(W_n(k)[T{^{\prime}}])\to{\operatorname{Spec} }(W_n(k)[T])$ and the related complexes (\[monodromyC\]). We get a commutative diagram of complexes of sheaves on $X_{n,{\operatorname{\acute{e}t} }}$ (note that $X_{V{^{\prime}},n,{\operatorname{\acute{e}t} }}=X_{n,{\operatorname{\acute{e}t} }}$) $$\begin{CD} 0@>>> C_{V{^{\prime}}}{^{\prime}}[-1] @>\wedge{\operatorname{dlog} }T{^{\prime}}>>C_{V{^{\prime}}} @>>> C_{V{^{\prime}}}{^{\prime}}@>>> 0\\ @. @AA p^se_1a^{\varphi}_Z^sA @AA a^{\varphi}_Z^s A @AA a^{\varphi}_Z^s A @.\\ 0 @>>> C_V{^{\prime}}[-1] @> \wedge{\operatorname{dlog} }T >>C_V @>>> C_V{^{\prime}}@>>> 0 \end{CD}$$ Hence diagram (\[brrrr\]). Combining diagram (\[brrrr\]) with Frobenius we obtain the following commutative diagram $$\begin{CD} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},r) @< F^s << {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},r)@> f^F^s >> {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(T,\overline{T},r)\\ @VV\wr V @VV\wr V @VV\wr V\\ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R,r)^{N=0} @<(F^s, p^sF^s) <<{\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R,r)^{N=0} @> (a^F^s, p^sa^F^s) >> {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((T,\overline{T})/R{^{\prime}},r)^{N{^{\prime}}=0}\\ @V\wr V i_0^* V @V\wr V i_0^* V @V\wr V i_0^* V\\ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U},r)^{N=0} @< (F^s,p^sF^s) <\sim <{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U},r)^{N=0} @> (F^s,p^sF^s) >\sim > {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(T,\overline{T},r)^{N{^{\prime}}=0} \end{CD}$$ It follows that all the maps in the above diagram are quasi-isomorphisms. We define the map $$f^: {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R,r)^{N=0}\to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((T,\overline{T})/R{^{\prime}},r)^{N{^{\prime}}=0}$$ by the middle row. Since, for any $g\in G$, we have $x_g^{e_1}(\overline{v})^{-1}=g(\overline{v})^{-1}$, the map $f^$ is $G$-equivariant. In the (classical) derived category, this definition is independent of the constant $s$ we have chosen. Since $i^_0$ is a quasi-isomorphism and $i^_0\iota_{}={ \operatorname{Id} }$, the diagram (\[diag1\]) commutes as well, as wanted. We define the map $$\label{BHK} f^: {\mathrm {R} }\Gamma^{B,\tau}_{{\mathrm{HK}}}((U,\overline{U})/R,r)^{N=0}\to {\mathrm {R} }\Gamma^{B,\tau}_{{\mathrm{HK}}}((T,\overline{T})/R{^{\prime}},r)^{N{^{\prime}}=0}$$ in an analogous way. By the above diagram and by compatibility of the Beilinson-Hyodo-Kato constructions with base change and with Frobenius, the two pullback maps $f^$ are compatible via the morphism $h$, i.e., the following diagram commutes $$\xymatrix{ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},r) \ar[r]\ar[d]^{f^} & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R,r)^{N=0}\ar[d]^{f^} & {\mathrm {R} }\Gamma^{B,\tau}_{{\mathrm{HK}}}((U,\overline{U})/R,r)^{N=0} \ar[l]_{\kappa_R}\ar[d]^{f^} \\ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(T,\overline{T},r)\ar[r] & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((T,\overline{T})/R{^{\prime}},r)^{N{^{\prime}}=0} & {\mathrm {R} }\Gamma^{B,\tau}_{{\mathrm{HK}}}((T,\overline{T})/R{^{\prime}},r)^{N{^{\prime}}=0}\ar[l]_{\kappa_{R^\prime} } }$$ From the analog of diagram (\[diag1\]) for the Beilinson-Hyodo-Kato complexes and by the universal nature of the trivialization at $\overline{p}$ we obtain that the pullback map $f^$ is compatible with the maps $\beta\iota_?$. It remains to show that we have a commutative diagram $$\xymatrix{ {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(U,\overline{U},r)^{N=0}\ar[r]^{f^}_{\sim}\ar[d]^{\iota^B_{{\mathrm{dR}}}} & {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(T,\overline{T},r)^{N^\prime=0}\ar[d]^{\iota^B_{{\mathrm{dR}}}}\\ {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K})/F^r \ar[r]^{f^} & {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(T,\overline{T}_{K^{\prime}})/F^r }$$ But this follows since the Beilinson-Hyodo-Kato map is compatible with base change. Arithmetic syntomic cohomology ------------------------------ We are now ready to introduce and study arithmetic syntomic cohomology, i.e., syntomic cohomology over $K$. Let ${{\mathcal J}}^{[r]}_{{\operatorname{cr} }}$, ${{\mathcal{A}}}_{{\operatorname{cr} }}$, and ${{\mathcal{S}}}(r)$ for $r\geq 0$ be the $h$-sheafifications on $\mathcal{V}ar_{K}$ of the presheaves sending $(U,\overline{U})\in {{\mathcal{P}}}^{ss}_{K}$ to $ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},J^{[r]})$, $ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})$, and ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)$, respectively. Let ${{\mathcal J}}^{[r]}_{{\operatorname{cr} },n}$, ${{\mathcal{A}}}_{{\operatorname{cr} },n}$, and ${{\mathcal{S}}}_n(r)$ denote the $h$-sheafifications of the mod-$p^n$ versions of the respective presheaves. We have $${{\mathcal{S}}}_n(r)\simeq {\operatorname{Cone} }({{\mathcal J}}^{[r]}_{{\operatorname{cr} },n}\stackrel{p^r-{\varphi}}{\longrightarrow}{{\mathcal{A}}}_{{\operatorname{cr} },n})[-1],\quad {{\mathcal{S}}}(r)\simeq {\operatorname{Cone} }({{\mathcal J}}^{[r]}_{{\operatorname{cr} }}\stackrel{p^r-{\varphi}}{\longrightarrow}{{\mathcal{A}}}_{{\operatorname{cr} }})[-1] .$$ For $r\geq 0$, define ${{\mathcal{S}}}(r)_{\mathbf Q}$ as the $h$-sheafification of the preasheaf sending ss-pairs $(U,\overline{U})$ to ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{\mathbf Q}$. We have $${{\mathcal{S}}}(r)_{\mathbf Q}\simeq {\operatorname{Cone} }({{\mathcal J}}^{[r]}_{{\operatorname{cr} },{\mathbf Q}}\lomapr{1-{\varphi}_r}{{\mathcal{A}}}_{{\operatorname{cr} },{\mathbf Q}})[-1]$$ For $X\in \mathcal{V}ar_{K}$, set ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)_n={\mathrm {R} }\Gamma(X_h,{{\mathcal{S}}}_n(r))$, ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r):={\mathrm {R} }\Gamma(X_h,{{\mathcal{S}}}(r)_{\mathbf Q})$. We have $$\begin{aligned} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)_n & \simeq {\operatorname{Cone} }({\mathrm {R} }\Gamma(X_h,{{\mathcal J}}^{[r]}_{{\operatorname{cr} },n})\stackrel{p^r-{\varphi}}{\longrightarrow}{\mathrm {R} }\Gamma(X_h,{{\mathcal{A}}}_{{\operatorname{cr} },n}))[-1],\\ {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r) & \simeq {\operatorname{Cone} }({\mathrm {R} }\Gamma(X_h,{{\mathcal J}}^{[r]}_{{\operatorname{cr} },{\mathbf Q}})\stackrel{1-{\varphi}_r}{\longrightarrow}{\mathrm {R} }\Gamma(X_h,{{\mathcal{A}}}_{{\operatorname{cr} },{\mathbf Q}}))[-1] .\end{aligned}$$ We will often write ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_h)$ for ${\mathrm {R} }\Gamma(X_h,{{\mathcal{A}}}_{{\operatorname{cr} }})$ if this does not cause confusion. Let ${{\mathcal{A}}}_{{\mathrm{HK}}}$ be the $h$-sheafification of the presheaf $(U,\overline{U})\mapsto {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{\mathbf Q}$ on ${{\mathcal{P}}}_{K}^{ss}$; this is an $h$-sheaf of $E_{\infty}$ $ K_0$-algebras on ${\mathcal V}ar_{K}$ equipped with a ${\varphi}$-action and a derivation $N$ such that $N{\varphi}=p{\varphi}N$. For $X\in {\mathcal V}ar_{K}$, set ${\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_h):={\mathrm {R} }\Gamma(X_h,{{\mathcal{A}}}_{{\mathrm{HK}}})$. Similarly, we define $h$-sheaves ${{\mathcal{A}}}^B_{{\mathrm{HK}}}$ and the complexes ${\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h):={\mathrm {R} }\Gamma(X_h,{{\mathcal{A}}}^B_{{\mathrm{HK}}})$. The maps $\kappa: {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(U_1,\overline{U}_1)\to {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{\mathbf Q}$ $h$-sheafify and we obtain functorial quasi-isomorphisms $$\begin{aligned} \kappa: \quad {{\mathcal{A}}}^B_{{\mathrm{HK}}} \stackrel{\sim}{\to} {{\mathcal{A}}}_{{\mathrm{HK}}},\quad \kappa:\quad {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h) \stackrel{\sim}{\to } {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_h). \end{aligned}$$ The complexes ${{\mathcal J}}^{[r]}_{{\operatorname{cr} },n}$ and ${{\mathcal{S}}}_n(r)$ (and their completions) have a concrete description. For the complexes $ {{\mathcal J}}^{[r]}_{{\operatorname{cr} },n}$: we can represent the presheaves $(U,{\overline}{U})\mapsto {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,{\overline}{U},{{\mathcal J}}^{[r]}_n)$ by Godement resolutions (on the crystalline site), sheafify them for the $h$-topology on ${{\mathcal{P}}}^{ss}_K$, and then move them to ${\mathcal V}ar_K$. For the complexes ${{\mathcal{S}}}_n(r)$: the maps $p^r-{\varphi}$ can be lifted to the Godement resolutions and their mapping fiber (defining ${{\mathcal{S}}}_n(r)(U,{\overline}{U})$) can be computed in the abelian category of complexes of abelian groups. To get ${{\mathcal{S}}}_n(r)$ we $h$-sheafify on ${{\mathcal{P}}}^{ss}_K$ and pass to ${\mathcal V}ar$. Let, for a moment, $K$ be any field of characteristic zero. Consider the presheaf $(U,\overline{U})\mapsto {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}):={\mathrm {R} }\Gamma(\overline{U},\Omega^{\scriptscriptstyle\bullet}_{(U,\overline{U})})$ of filtered dg $K$-algebras on ${{\mathcal{P}}}^{nc}_K$. Let ${{\mathcal{A}}}_{{\mathrm{dR}}}$ be its $h$-sheafification. It is a sheaf of filtered $K$-algebras on $\mathcal{V}ar_K$. For $X\in \mathcal{V}ar_K$, we have the Deligne’s de Rham complex of $X$ equipped with Deligne’s Hodge filtration: ${\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h):={\mathrm {R} }\Gamma(X_h,{{\mathcal{A}}}_{{\mathrm{dR}}})$. Beilinson proves the following comparison statement. ([@BE1 2.4]) \[deRham1\] 1. For $(U,\overline{U})\in {{\mathcal{P}}}^{nc}_K$, the canonical map ${\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U})\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U_h)$ is a filtered quasi-isomorphism. 2. The cohomology groups $H^i_{{\mathrm{dR}}}(X_h):=H^i{\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h)$ are $K$-vector spaces of dimension equal to the rank of $H^i(X_{\overline{K},{\operatorname{\acute{e}t} }},{\mathbf Q}_p)$. \[blowup\] For a geometric pair $(U,\overline{U})$ over $K$ that is saturated and log-smooth, the canonical map $${\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U})\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U_h)$$ is a filtered quasi-isomorphism. Recall [@NR Theorem 5.10] that there is a log-blow-up $(U,\overline{T})\to (U,\overline{U})$ that resolves singularities of $(U,\overline{U})$, i.e., such that $(U,{\overline}{T}) \in {{\mathcal{P}}}^{{nc}}_K$. We have a commutative diagram $$\xymatrix{ {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{T})\ar[r]^{\sim} &{\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U_h)\\ {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U})\ar[u]^{\wr}\ar[ur]& }$$ The vertical map is a filtered quasi-isomorphism; the horizontal map is a filtered quasi-isomorphism by the above proposition. Our corollary follows. Return now to our $p$-adic field $K$. \[descent\] By construction the complexes ${\mathrm {R} }\Gamma(X_h,{{\mathcal J}}^{[r]}_{{\operatorname{cr} },{\mathbf Q}})$, ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)$, ${\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_h)$, ${\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)$, and ${\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h)$ satisfy $h$-descent. In particular, since $h$-topology is finer than the étale topology, they satisfy Galois descent for finite extensions. Hence, for any finite Galois extension $K_1/K$, the natural maps $${\mathrm {R} }\Gamma^_{?}(X_h)\stackrel{\sim}{\to}{\mathrm {R} }\Gamma (G,{\mathrm {R} }\Gamma^_{?}(X_{K_1,h})),\quad ?={{\operatorname{cr} }}, {{ \operatorname{syn} }},{\mathrm{HK}}, {\mathrm{dR}}; \quad =B,\emptyset$$ where $G={\operatorname{Gal} }(K_1/K)$, are (filtered) quasi-isomorphisms. Since $G$ is finite, it follows that the natural maps $${\mathrm {R} }\Gamma^_{{\mathrm{HK}}}(X_h)\otimes_{K_0}K_{1,0}\stackrel{\sim}{\to}{\mathrm {R} }\Gamma^_{{\mathrm{HK}}}(X_{K_1,h}),\quad {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h)\otimes_{K}K_1\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{K_1,h})$$ are (filtered) quasi-isomorphisms as well. Recall [@BE2 2.5], Proposition \[isomorphism\], that for a fine, log-scheme $X$, log-smooth over $V^{\times}$, and of Cartier type we have a quasi-isomorphism ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_{\overline{V}},{{\mathcal J}}^{[r]}_{X_{\overline{V}}/W(k)})_{\mathbf Q}\simeq {\mathrm {R} }\Gamma(X_{{\overline{K} },h},{{\mathcal J}}^{[r]}_{{\operatorname{cr} }})_{\mathbf Q}$. We can descend this result to $K$ but on the level of rational log-syntomic cohomology; the key observation being that the field extensions introduced by the alterations are harmless since, by Proposition \[hypercov11\], log-syntomic cohomology satisfies finite Galois descent. Along the way we will get an analogous comparison quasi-isomorphism for the Hyodo-Kato cohomology. \[hypercov\] For any arithmetic pair $(U,\overline{U})$ that is fine, log-smooth over $V^{\times}$, and of Cartier type, and $r\geq 0$, the canonical maps $${\mathrm {R} }\Gamma_{{\mathrm{HK}}}^(U,\overline{U})_{\mathbf Q} \stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^(U_h),\quad {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{\mathbf Q} \stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U_h,r)$$ are quasi-isomorphisms. It suffices to show that for any $h$-hypercovering $(U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet})\to (U,\overline{U})$ by pairs from ${{\mathcal{P}}}^{\log}_{K}$ the natural maps $${\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}} \to {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet})_{{\mathbf Q}},\quad {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{{\mathbf Q}} \to {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet},r)_{{\mathbf Q}}$$ are (modulo taking a refinement of $(U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet})$) quasi-isomorphisms. For the second map, since we have a canonical quasi-isomorphism $${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{\mathbf Q}\stackrel{\sim}{\to} {\operatorname{Cone} }({\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},r)_{\mathbf Q}\to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal O}}/{{\mathcal J}}^{[r]})_{\mathbf Q})[-1]$$ it suffices to show that, up to a refinement of the hypercovering, we have quasi-isomorphisms $${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal O}}/{{\mathcal J}}^{[r]})_{\mathbf Q}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet},{{\mathcal O}}/{{\mathcal J}}^{[r]})_{\mathbf Q},\quad {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},r)_{\mathbf Q}\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet},r)_{\mathbf Q} .$$ For the first of these maps, by Corollary \[Langer\] this amounts to showing that the following map is a quasi-isomorphism $${\mathrm {R} }\Gamma(\overline{U}_K,\Omega^{\scriptscriptstyle\bullet}_{(U,\overline{U}_K)})/F^r \stackrel{\sim}{\to} {\mathrm {R} }\Gamma(\overline{U}_{\scriptscriptstyle\bullet,K},\Omega^{\scriptscriptstyle\bullet}_{(U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet,K})})/F^r.$$ But, by Corollary \[blowup\] this map is quasi-isomorphic to the map $${\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U_h)/F^r\to {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U_{\scriptscriptstyle\bullet,h})/F^r,$$ which is clearly a quasi-isomorphism. Hence it suffices to show that, up to a refinement of the hypercovering, we have quasi-isomorphisms $${\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{\mathbf Q}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet})_{\mathbf Q},\quad {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},r)_{\mathbf Q}\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet},r)_{\mathbf Q} .$$ Fix $t\geq 0$. To show that $H^t{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},r)_{\mathbf Q}\stackrel{\sim}{\to}H^t{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet},r)_{\mathbf Q} $ is a quasi-isomorphism we will often work with $(t+1)$-truncated $h$-hypercovers. This is because $\tau_{\leq t}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet},r)\simeq \tau_{\leq t}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet})_{\leq t+1},r)$, where $(U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet})_{\leq t+1}$ denotes the $t+1$-truncation. Assume first that we have an $h$-hypercovering $(U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet})\to (U,\overline{U})$ of arithmetic pairs over $K$, where each pair $(U_i,\overline{U}_i)$, $i\leq t+1$, is log-smooth over $V^{\times}$ and of Cartier type. We claim that then already the maps $$\label{firsteq} \tau_{\leq t}{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{\mathbf Q}\stackrel{\sim}{\to} \tau_{\leq t}{\mathrm {R} }\Gamma_{{\mathrm{HK}}}((U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet})_{\leq t+1})_{\mathbf Q};\quad \tau_{\leq t}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_{\mathbf Q}\stackrel{\sim}{\to} \tau_{\leq t}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet})_{\leq t+1})_{\mathbf Q}$$ are quasi-isomorphisms. To see the second quasi-isomorphism consider the following commutative diagram of distinguished triangles ($R=R_V$) $$\begin{CD} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U}) @>>> {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)@> N >> {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)\\ @VVV @VVV @VVV \\ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet})_{\leq t+1})@>>> {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet})_{\leq t+1}/R)@> N >> {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet})_{\leq t+1}/R) \end{CD}$$ It suffices to show that the two right vertical arrows are rational quasi-isomorphisms in degrees less or equal to $t$. But we have the $R$-linear quasi-isomorphisms $$\iota: R\otimes_{W(k)}{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}}\stackrel{\sim}{\to}{\mathrm {R} }\Gamma((U,\overline{U})/R)_{{\mathbf Q}}, \quad \iota: R\otimes_{W(k)}{\mathrm {R} }\Gamma_{{\mathrm{HK}}}((U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet})_{\leq t+1})_{{\mathbf Q}}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma((U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet})_{\leq t+1}/R)_{ {\mathbf Q}}.$$ Hence to show both quasi-isomorphisms (\[firsteq\]), it suffices to show that the map $$\tau_{\leq t}{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}}\to \tau_{\leq t}{\mathrm {R} }\Gamma_{{\mathrm{HK}}}((U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet})_{\leq t+1})_{{\mathbf Q}}$$ is a quasi-isomorphism. Tensoring over $K_0$ with $K$ and using the Hyodo-Kato quasi-isomorphism (\[HKqis\]) we reduce to showing that the map $$\tau_{\leq t}{\mathrm {R} }\Gamma(\overline{U}_K,\Omega{^{\cdot }}_{(U,\overline{U}_K)})\to \tau_{\leq t} {\mathrm {R} }\Gamma(\overline{U}_{\scriptscriptstyle\bullet K, \leq t+1},\Omega{^{\cdot }}_{(U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet,K})_{\leq t+1}})$$ is a quasi-isomorphism. And this we have done above. To treat the general case, set $X=(U,\overline{U}), Y_{\scriptscriptstyle\bullet}=(U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet})$. We will do a base change to reduce to the case discussed above. We may assume that all the fields $K_{n,i}$, $K_{U_n}\simeq \prod K_{n,i}$ are Galois over $K$. Choose a finite Galois extension $(V{^{\prime}},K{^{\prime}})/(V,K)$ for $K{^{\prime}}$ Galois over all the fields $K_{n,i}$, $n\leq t+1$. Write $N_X(X_{V{^{\prime}}})$ for the “Čech nerve” of $X_{V{^{\prime}}}/X$. The term $N_X(X_{V{^{\prime}}}) _n$ is defined as the $(n+1)$-fold fiber product of $X_{V{^{\prime}}}$ over $X$: $N_X(X_{V{^{\prime}}}) _n=(U\times_{K}K^{\prime,n+1},(\overline{U}\times_{V}V^{\prime,n+1})^{{\operatorname{norm} }})$, where $V^{\prime,n+1},K^{\prime,n+1}$ are defined as the $(n+1)$-fold product of $V{^{\prime}}$ over $V$ and of $K{^{\prime}}$ over $K$, respectively. Normalization is taken with respect to the open regular subscheme $U\times_{K}K^{\prime,n+1}$. Note that $N_X(X_{V{^{\prime}}}) _n\simeq (U\times_{K}K{^{\prime}}\times G^{n},\overline{U}\times_{V}V{^{\prime}}\times G^{n})$, $G={\operatorname{Gal} }(K{^{\prime}}/K)$. Hence it is a log-smooth scheme over $V^{\prime,\times}$, of Cartier type. The augmentation $N_X(X_{V{^{\prime}}})\to X$ is an $h$-hypercovering. Consider the bi-simplicial scheme $Y_{\scriptscriptstyle\bullet}\times_{X}N_X(X_{V{^{\prime}}})_{\scriptscriptstyle\bullet} $, $$\begin{aligned} (Y_{\scriptscriptstyle\bullet}\times_{X}N_X(X_{V{^{\prime}}})_{\scriptscriptstyle\bullet}) _{n,m}:=Y_n\times_{X}N_X(X_{V{^{\prime}}}) _m& \simeq (U_n\times_UU\times_{K}K^{\prime,m+1},(\overline{U}_n\times_{\overline{U}}(\overline{U}\times_{V}V^{\prime,m+1})^{{\operatorname{norm} }})^{{\operatorname{norm} }})\\ & \simeq \coprod_i(U_n\times_{K_{n,i}}K_{n,i}\times_{K}K^{\prime,m+1},\overline{U}_n\times_{V_{n,i}}(V_{n,i}\times_{V}V^{\prime,m+1})^{{\operatorname{norm} }}). \end{aligned}$$ Hence $(Y_{\scriptscriptstyle\bullet}\times_{X}N_X(X_{V{^{\prime}}})_{\scriptscriptstyle\bullet}) _{n,m}\in {{\mathcal{P}}}^{\log}_{K}$. For $n,m\leq t+1$, we have $$(Y_{\scriptscriptstyle\bullet}\times_{X}N_X(X_{V{^{\prime}}})_{\scriptscriptstyle\bullet})_{n,m} \simeq \coprod_i(U_n\times_{K_{n,i}}K{^{\prime}}\times G_{n,i}\times G^{m}, \overline{U}_n\times_{V_{n,i}}V{^{\prime}}\times G_{n,i}\times G^{m}),$$ where $G_{n,i}={\operatorname{Gal} }(K_{n,i}/K)$. It is a log-scheme log-smooth over $V^{\prime, \times}$, of Cartier type. Consider now its diagonal $Y_{\scriptscriptstyle\bullet}\times_{X}N_X(X_{V{^{\prime}}}):=\Delta (Y_{\scriptscriptstyle\bullet}\times_{X}N_X(X_{V{^{\prime}}})_{\scriptscriptstyle\bullet})$. It is an $h$-hypercovering of $X$ refining $Y_{\scriptscriptstyle\bullet}$ such that, for $n\leq t+1$, $(Y_{\scriptscriptstyle\bullet}\times_{X}N_X(X_{V{^{\prime}}}))_n $ is log-smooth over $V^{\prime, \times}$, of Cartier type. It suffices to show that the compositions $$\begin{aligned} \label{secondeq} {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X)_{\mathbf Q} & \stackrel{}{\to} {\mathrm {R} }\Gamma_{{\mathrm{HK}}}(Y_{\scriptscriptstyle\bullet})_{\mathbf Q} \stackrel{{\operatorname{pr} }_1^}{\to}{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(Y_{\scriptscriptstyle\bullet}\times_{X}N_X(X_{V{^{\prime}}}))_{\mathbf Q};\\ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X,r)_{\mathbf Q} & \stackrel{}{\to} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Y_{\scriptscriptstyle\bullet},r)_{\mathbf Q} \stackrel{{\operatorname{pr} }_1^}{\to}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Y_{\scriptscriptstyle\bullet}\times_{X}N_X(X_{V{^{\prime}}}),r)_{\mathbf Q} \notag \end{aligned}$$ are quasi-isomorphisms in degrees less or equal to $t$. Using the commutative diagram of bi-simplicial schemes $$\begin{CD} Y_{\scriptscriptstyle\bullet}\times_{X}N_X(X_{V{^{\prime}}}) @>\Delta >> Y_{\scriptscriptstyle\bullet}\times_{X}N_X(X_{V{^{\prime}}})_{\scriptscriptstyle\bullet} @>{\operatorname{pr} }_1 >> Y_{\scriptscriptstyle\bullet}\\ @. @VV{\operatorname{pr} }_2 V @VVV\\ @. N_X(X_{V{^{\prime}}}) @> f >> X \end{CD}$$ we can write the second composition as $${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X,r)_{\mathbf Q} \stackrel{f^}{\to} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(N_X(X_{V{^{\prime}}}),r)_{\mathbf Q} \stackrel{{\operatorname{pr} }_2^}{\to}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Y_{\scriptscriptstyle\bullet}\times_{X}N_X(X_{V{^{\prime}}})_{\scriptscriptstyle\bullet},r)_{\mathbf Q} \stackrel{\Delta^}{\to}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Y_{\scriptscriptstyle\bullet}\times_{X}N_X(X_{V{^{\prime}}}),r)_{\mathbf Q}$$ We claim that all of these maps are quasi-isomorphisms in degrees less or equal to $t$. The map $\Delta^$ is a quasi-isomorphism (in all degrees) by [@Fr Prop. 2.5]. For the second map, fix $n\leq t+1$ and consider the induced map ${\operatorname{pr} }_2: (Y_{\scriptscriptstyle\bullet}\times_{X}N_X(X_{V{^{\prime}}})_{\scriptscriptstyle\bullet}) _{\scriptscriptstyle\bullet,n}\to N_X(X_{V{^{\prime}}}) _n$. It is an $h$-hypercovering whose $(t+1)$-truncation is built from log-schemes, log-smooth over $(V{^{\prime}},K{^{\prime}})$, of Cartier type. It suffices to show that the induced map $\tau_{\leq t}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(N_X(X_{V{^{\prime}}})_n,r)_{\mathbf Q} \stackrel{{\operatorname{pr} }_2^}{\to}\tau_{\leq t}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}((Y_{\scriptscriptstyle\bullet}\times_{X}N_X(X_{V{^{\prime}}}))_{\scriptscriptstyle\bullet,n},r)_{\mathbf Q}$ is a quasi-isomorphism. Since all maps are defined over $K{^{\prime}}$, this follows from the case considered at the beginning of the proof. To prove that the map $f^: {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X,r)_{\mathbf Q} \to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(N_X(X_{V{^{\prime}}}),r)_{\mathbf Q} $ is a quasi-isomorphism consider first the case when the extension $V^{\prime}/V$ is unramified. Then $ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_{V^{\prime}}) \simeq {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X)\otimes_{W(k)}W(k^{\prime})$ and the map $f^$ is a quasi-isomorphism by finite étale descent for crystalline cohomology. Assume now that the extension $V^{\prime}/V$ is totally ramified and let $\pi$ and $\pi^{\prime}$ be uniformizers of $V$ and $V^{\prime}$, respectively. Consider the target of $f^$ as a double complex. To show that $f^$ is a quasi-isomorphism it suffices to show that, for each $s\geq 0$, the sequence $$0\to H^s{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X,r)_{\mathbf Q} \stackrel{f^}{\to} H^s{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(N_X(X_{V{^{\prime}}})_0,r)_{\mathbf Q} \stackrel{d_0^}{\to}H^s{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(N_X(X_{V{^{\prime}}})_1,r)_{\mathbf Q} \stackrel{d_1^}{\to}H^s{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(N_X(X_{V{^{\prime}}})_2,r)_{\mathbf Q}\ldots$$ is exact. Embed it into the following diagram $$\xymatrix{ 0 \ar[r] & H^s{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X,r)_{\mathbf Q} \ar[d]^{\wr}_{\alpha^B_{{ \operatorname{syn} },\pi}}\ar[r]^-{f^} & H^s{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(N_X(X_{V{^{\prime}}})_0,r)_{\mathbf Q} \ar[d]^{\wr}_{\tilde{\alpha}^B_{{ \operatorname{syn} },\pi^{\prime}}}\ar[r]^{d_0^} & H^s{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(N_X(X_{V{^{\prime}}})_1,r)_{\mathbf Q}\ar[d]^{\wr}_{\tilde{\alpha}^B_{{ \operatorname{syn} },\pi^{\prime}}} \ar[r] & \\ 0 \ar[r] & H^s{\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X,r)_{\mathbf Q}^{N=0}\ar[r]^-{ f^} & H^s{\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(N_X(X_{V{^{\prime}}})_0,r)_{\mathbf Q}^{N^{\prime}=0} \ar[r]^{d_0^} & H^s{\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(N_X(X_{V{^{\prime}}})_1,r)_{\mathbf Q}^{N^{\prime}=0} \ar[r] & }$$ Note that, since all the maps $d_i^$ are induced from automorphisms of $V{^{\prime}}/V$, by the proof of Proposition \[hypercov11\] (take the map $f$ used there to be a given automorphism $g\in G={\operatorname{Gal} }(K^{\prime}/K)$ and $\pi^{\prime}$, $g(\pi^{\prime})$ for the uniformizers of $V^{\prime}$) and the proof of Proposition \[reduction1\], we get the vertical maps above that make all the squares commute. Hence it suffices to show that the following sequence of Hyodo-Kato cohomology groups is exact: $$0 \to H^s{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X)_{\mathbf Q} \stackrel{f^}{\to} H^s{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(N_X(X_{V{^{\prime}}})_0)_{\mathbf Q} \stackrel{ d_0^}{\to}H^s{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(N_X(X_{V{^{\prime}}})_1)_{\mathbf Q} \stackrel{ d_1^}{\to}H^s{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(N_X(X_{V{^{\prime}}})_2)_{\mathbf Q} \to$$ But this sequence is isomorphic to the following sequence $$0 \to H^s{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X)_{\mathbf Q} \stackrel{f^}{\to} H^s{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_{V{^{\prime}}})_{\mathbf Q} \stackrel{ d_0^}{\to}H^s{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_{V{^{\prime}}})_{\mathbf Q} \times G \stackrel{ d_1^}{\to}H^s{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_{V{^{\prime}}})_{\mathbf Q} \times G^2\to$$ representing the (augmented) $G$-cohomology of $H^s{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X)_{\mathbf Q}$. Since $G$ is finite, this complex is exact in degrees at least $1$. It remains to show that $H^0(G,H^s{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_{V{^{\prime}}})_{{\mathbf Q}})\simeq H^s{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X)_{{\mathbf Q}}$. Since $K{^{\prime}}/K$ is totally ramified, we have $H^s{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_{V{^{\prime}}})\simeq H^s{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X)$. Hence the action of $G$ on $H^s{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_{V{^{\prime}}})$ is trivial and we get the right $H^0$ as well. We have proved the second quasi-isomorphism from (\[secondeq\]). Notice that along the way we have actually proved the first quasi-isomorphism. For $X\in {\mathcal V}ar_K$, we define a canonical $K_0$-linear map ([the Beilinson-Hyodo-Kato morphism]{}) $$\iota^B_{{\mathrm{dR}}}: \quad {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h){\to}{\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h)$$ as sheafification of the map $\iota^B_{{\mathrm{dR}}}: {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(U_1,\overline{U}_1){\to}{\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_K) $. It follows from Proposition \[HKdR\] that we prove in the next section that the cohomology groups $H^i_{{\mathrm{HK}}}(X_h):=H^i{\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)$ are finite rank $K_0$-vector spaces and that they vanish for $i>2\dim X$. This implies the following lemma. \[dim\] The syntomic cohomology groups $H^i_{{ \operatorname{syn} }}(X_h,r):=H^i{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)$ vanish for $i > 2\dim X+2$. The map $\iota_{{\mathrm{dR}}}^{\prime}:{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U},r)^{N=0}\to {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K})/F^r$ from Remark \[reduction21\] sheafifies and so does the quasi-isomorphism $\alpha^{\prime}_{{ \operatorname{syn} }}: {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{{\mathbf Q}}\stackrel{\sim}{\to}C^{\prime}_{{\operatorname{st} }}({\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})\{r\})$. Hence ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)$ is quasi-isomorpic via $\alpha^{\prime}_{{ \operatorname{syn} }}$ to the mapping fiber $$C^{\prime}_{{\operatorname{st} }}({\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_h)\{r\}):= [{\mathrm {R} }\Gamma_{{\mathrm{HK}}}(X_h,r)^{N=0}\lomapr{\iota_{{\mathrm{dR}}}^{\prime}} {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h)/F^r]$$ The statement of the lemma follows. For $X\in {\mathcal V}ar_{K}$ and $r\geq 0$, define the complex $$C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)\{r\}):= \left[ \begin{aligned}\xymatrix{{\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)\ar[rr]^-{(1-{\varphi}_r,\iota^B_{{\mathrm{dR}}})}\ar[d]^{N} & & {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h) \oplus {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h) /F^r\ar[d]^{(N,0)}\\ {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)\ar[rr]^-{1-{\varphi}_{r-1}} & & {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)}\end{aligned}\right]$$ \[reduction2\] For $X\in {\mathcal V}ar_{K}$ and $r\geq 0$, there exists a canonical (in the classical derived category) quasi-isomorphism $$\alpha_{{ \operatorname{syn} }}: {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)\stackrel{\sim}{\to}C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)\{r\}).$$ Moreover, this morphism is compatible with finite base change (of the field $K$). To construct the map $\alpha_{{ \operatorname{syn} }}$, take a number $t\geq 2\dim X+2$ and let $Y_{\scriptscriptstyle\bullet}\to X$, $Y_{\scriptscriptstyle\bullet}=(U_{\scriptscriptstyle\bullet},\overline{U}_{\scriptscriptstyle\bullet})$, be an $h$-hypercovering of $X$ by ss-pairs over $K$. Choose a finite Galois extension $(V{^{\prime}},K{^{\prime}})/(V,K)$ and a uniformizer $\pi^\prime$ of $V^\prime$ as in the proof of Proposition \[hypercov\]. Keeping the notation from that proof, refine our hypercovering to the $h$-hypercovering $Y_{\scriptscriptstyle\bullet}\times_V {V^{\prime}}\to X_{K^{\prime}}$. Then the truncation $(Y_{\scriptscriptstyle\bullet}\times_V {V^{\prime}})_{\leq t+1}$ is built from log-schemes log-smooth over $V^{\prime,\times}$ and of Cartier type. We have the following sequence of quasi-isomorphisms $$\begin{aligned} \gamma_{\pi^\prime}: {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{K{^{\prime}},h}) &\stackrel{\sim}{\leftarrow} \tau_{\leq t}{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{K{^{\prime}},h}) \stackrel{\sim}{\to}\tau_{\leq t}{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}((U_{\scriptscriptstyle\bullet}\times_K {K^{\prime}})_{\leq t+1,h})\stackrel{\sim}{\leftarrow} \tau_{\leq t}{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}((Y_{\scriptscriptstyle\bullet}\times_V {V^{\prime}})_{\leq t+1})_{\mathbf Q}\\ & \stackrel{\sim}{\to}C_{{\operatorname{st} }}(\tau_{\leq t}{\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}((Y_{\scriptscriptstyle\bullet}\times_V {V^{\prime}})_{\leq t+1})\{r\})\stackrel{\sim}{\to}C_{{\operatorname{st} }}(\tau_{\leq t}{\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}((U_{\scriptscriptstyle\bullet}\times_K {K^{\prime}})_{\leq t+1,h})\{r\})\\ & \stackrel{\sim}{\leftarrow} C_{{\operatorname{st} }}(\tau_{\leq t}{\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{K^{\prime},h})\{r\})\stackrel{\sim}{\to}C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{K^{\prime},h})\{r\}) \end{aligned}$$ The first quasi-isomorphism follows from Lemma \[dim\]. The third and fifth quasi-isomorphisms follow from Proposition \[hypercov\]. The fourth quasi-isomorphism (the map $\tilde{\alpha}_{{ \operatorname{syn} },\pi^\prime}^B$), since all the log-schemes involved are log-smooth over $V^{\prime,\times}$ and of Cartier type, follows from Proposition \[reduction1\]. Now, set $G:={\operatorname{Gal} }(K{^{\prime}}/K)$. Passing from $\gamma_{\pi^\prime}$ to its $G$-fixed points we obtain the map $$\alpha_{{ \operatorname{syn} }}:=\alpha_{{ \operatorname{syn} },\pi^\prime}:\quad {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{h})\to C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{h})\{r\})$$ as the composition $${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{h})\to {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{K{^{\prime}},h})^G\stackrel{\gamma_{\pi^\prime}}{\to}C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{K^{\prime},h})\{r\})^G\stackrel{\sim}{\leftarrow} C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{K,h})\{r\})$$ It remains to check that so defined map is independent of all choices. For that, it suffices to check that, in the above construction, for a finite Galois extension $(V_1,K_1)$ of $(V^{\prime},K^{\prime})$, $H={\operatorname{Gal} }(K_1/K^\prime)$, the corresponding maps $\alpha_{{ \operatorname{syn} },?}: {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{h})\to C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)\{r\}) $ are the same in the classical derived category (note that this includes trivial extensions). Easy diagram chase shows that this amounts to checking that the following diagram commutes $$\xymatrix{ {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}((Y_{\scriptscriptstyle\bullet}\times_V {V^{\prime}})_{\leq t+1})_{\mathbf Q}\ar[r]^-{\sim}_-{\alpha_{{ \operatorname{syn} },\pi^{\prime}}}\ar[d] & C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}((Y_{\scriptscriptstyle\bullet}\times_V {V^{\prime}})_{\leq t+1})\{r\}) \ar[d]\\ {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}((Y_{\scriptscriptstyle\bullet}\times_V {V_1})_{\leq t+1})_{\mathbf Q}^H\ar[r]^-{\sim}_-{\alpha_{{ \operatorname{syn} },\pi_1}} & C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}((Y_{\scriptscriptstyle\bullet}\times_V {V_1})_{\leq t+1})\{r\})^H }$$ But this we have shown in Proposition \[hypercov11\]. For the compatibility with finite base change, consider a finite field extension $L/K$. We can choose in the above a Galois extension $K^{\prime}/K$ that works for both fields. We get the same maps $\gamma_{\pi^{\prime}}$ for both $L$ and $K$. Consider now the following commutative diagram. The top and bottom rows define the maps $\alpha^L_{{ \operatorname{syn} },\pi^{\prime}}$ and $\alpha^K_{{ \operatorname{syn} },\pi^{\prime}}$, respectively. $$\xymatrix{ {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{L,h})\ar[r] & {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{K{^{\prime}},h})^{G_L}\ar[r]^-{\gamma_{\pi^\prime}} & C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{K^{\prime},h})\{r\})^{G_L} & C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{L,h})\{r\})\ar[l]_-{\sim}\\ {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{h})\ar[u]\ar[r] & {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{K{^{\prime}},h})^G\ar[u] \ar[r]^-{\gamma_{\pi^\prime}} & C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{K^{\prime},h})\{r\})^G\ar[u] & C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{K,h})\{r\})\ar[l]_-{\sim}\ar[u] }$$ This proves the last claim of our proposition. Geometric syntomic cohomology ----------------------------- We will now study geometric syntomic cohomology, i.e., syntomic cohomology over ${\overline{K} }$. Most of the constructions related to syntomic cohomology over $K$ have their analogs over ${{\overline{K} }}$. We will summarize them briefly. For details the reader should consult [@Ts], [@BE2]. For $(U,\overline{U})\in {{\mathcal{P}}}^{ss}_{{\overline{K} }}$, $r\geq 0$, we have the absolute crystalline cohomology complexes and their completions $$\begin{aligned} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]})_n: & ={\mathrm {R} }\Gamma_{{\operatorname{cr} }}(\overline{U}_{{\operatorname{\acute{e}t} }},{\mathrm {R} }u_{\overline{U}_n/W_n(k)}{{\mathcal J}}^{[r]}_{\overline{U}_n/W_n(k)}),\quad {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]}): = {\operatorname{holim} }_n{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]})_n,\\ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]})_{{\mathbf Q}}: & ={\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]})\otimes{{\mathbf Q}}_p\end{aligned}$$ By [@BE2 Theorem 1.18], the complex ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})$ is a perfect $A_{{\operatorname{cr} }}$-complex and ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_n\simeq {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})\otimes^{L}_{A_{{\operatorname{cr} }}}{A_{{\operatorname{cr} }}}/p^n\simeq {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})\otimes^{L}{\mathbf Z}/p^n$. In general, we have ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]})_n\simeq {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]})\otimes^{L}{\mathbf Z}/p^n$. Moreover, $J^{[r]}_{{\operatorname{cr} }}={\mathrm {R} }\Gamma_{{\operatorname{cr} }}({\operatorname{Spec} }({\overline{K} }),{\operatorname{Spec} }(\overline{V}),{{\mathcal J}}^{[r]})$ [@Ts 1.6.3,1.6.4]. The absolute log-crystalline cohomology complexes are filtered commutative dg algebras (over $A_{{\operatorname{cr} },n}$, $A_{{\operatorname{cr} }}$, or $A_{{\operatorname{cr} },{{\mathbf Q}}}$). For $r\geq 0$, the mod-$p^n$, completed, and rational log-syntomic complexes ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_n$, ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)$, and ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{{\mathbf Q}}$ are defined by analogs of formulas (\[log-syntomic\]). We have ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_n\simeq {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)\otimes^{L}{\mathbf Z}/p^n$. Let ${{\mathcal J}}^{[r]}_{{\operatorname{cr} }}$, ${{\mathcal{A}}}_{{\operatorname{cr} }}$, and ${{\mathcal{S}}}(r)$ be the $h$-sheafifications on $\mathcal{V}ar_{{\overline{K} }}$ of the presheaves sending $(U,\overline{U})\in {{\mathcal{P}}}^{ss}_{{\overline{K} }}$ to $ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]})$, $ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})$, and ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)$, respectively. Let ${{\mathcal J}}^{[r]}_{{\operatorname{cr} },n}$, ${{\mathcal{A}}}_{{\operatorname{cr} },n}$, and ${{\mathcal{S}}}_n(r)$ denote the $h$-sheafifications of the mod-$p^n$ versions of the respective presheaves; and let ${{\mathcal J}}^{[r]}_{{\operatorname{cr} },{{\mathbf Q}}}$, ${{\mathcal{A}}}_{{\operatorname{cr} },{{\mathbf Q}}}$, ${{\mathcal{S}}}(r)_{{\mathbf Q}}$ be the $h$-sheafification of the rational versions of the same presheaves. For $X\in \mathcal{V}ar_{{\overline{K} }}$, set ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_h):={\mathrm {R} }\Gamma(X_h,{{\mathcal{A}}}_{{\operatorname{cr} }})$. It is a filtered (by ${\mathrm {R} }\Gamma(X_h,{{\mathcal J}}^{[r]}_{{\operatorname{cr} }})$, $r\geq 0$, ) $E_{\infty}$ $A_{{\operatorname{cr} }}$-algebra equipped with the Frobenius action ${\varphi}$. The Galois group $G_K$ acts on ${\mathcal V}ar_{{\overline{K} }}$ and it acts on $X\mapsto {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_h)$ by transport of structure. If $X$ is defined over $K$ then $G_K$ acts naturally on ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_h)$. For $r\geq 0$, set ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)_n={\mathrm {R} }\Gamma(X_h,{{\mathcal{S}}}_n(r))$, ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r):={\mathrm {R} }\Gamma(X_h,{{\mathcal{S}}}(r)_{{\mathbf Q}})$. We have $$\begin{aligned} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)_n & \simeq {\operatorname{Cone} }({\mathrm {R} }\Gamma(X_h,{{\mathcal J}}^{[r]}_{{\operatorname{cr} },n})\stackrel{p^r-{\varphi}}{\longrightarrow}{\mathrm {R} }\Gamma(X_h,{{\mathcal{A}}}_{{\operatorname{cr} },n}))[-1],\\ {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r) & \simeq {\operatorname{Cone} }({\mathrm {R} }\Gamma(X_h,{{\mathcal J}}^{[r]}_{{\operatorname{cr} },{\mathbf Q}})\stackrel{1-{\varphi}_r}{\longrightarrow}{\mathrm {R} }\Gamma(X_h,{{\mathcal{A}}}_{{\operatorname{cr} },{\mathbf Q}}))[-1] .\end{aligned}$$ The direct sum $\bigoplus_{r\geq 0}{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)$ is a graded $E_{\infty}$ algebra over ${\mathbf Z}_p$. Let $\overline{f}: Z_1\to {\operatorname{Spec} }(\overline{V}_1)^{\times}$ be an integral, quasi-coherent log-scheme. Suppose that $\overline{f}$ is the base change of $\overline{f}_L:Z_{L,1}\to {\operatorname{Spec} }({{\mathcal O}}_{L,1})^{\times}$ by $\theta_1: {\operatorname{Spec} }(\overline{{{\mathcal O}}_{L,1}})^{\times}\to{\operatorname{Spec} }({{\mathcal O}}_{L,1})^{\times}$, for a finite extension $L/K$. That is, we have a map $\theta_{L,1}: Z_1\to Z_{L,1}$ such that the square $(\overline{f},\overline{f}_L,\theta_1,\theta_{L,1})$ is Cartesian. Assume that $\overline{f}_L$ is log-smooth of Cartier type and that the underlying map of schemes is proper. Such data $(L,Z_1,\theta_{L,1})$ form a directed set $\Sigma_1$ and, for a morphism $(L{^{\prime}},Z{^{\prime}}_1,\theta{^{\prime}}_{L^\prime,1})\to (L,Z_1,\theta_{L,1})$, we have a canonical base change identification compatible with ${\varphi}$-action [@BE2 1.18] $${\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(Z_{L,1})\otimes_{L_0}L{^{\prime}}_0\stackrel{\sim}{\to} {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(Z{^{\prime}}_{L^\prime,1}).$$ These identifications can be made compatible with respect to $L$, so we can set $${\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(Z_1):= \dirlim_{\Sigma_1}{\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(Z_{L,1})$$ It is a complex of $({\varphi},N)$-modules over $K^{{\operatorname{nr} }}_0$, functorial with respect to morphisms of $Z_1$. Consider the scheme $E_{{\operatorname{cr} }}:={\operatorname{Spec} }(A_{{\operatorname{cr} }})$. We have $E_{{\operatorname{cr} },1}={\operatorname{Spec} }(\overline{V}_1)$ and we equip $E_{{\operatorname{cr} },1}$ with the induced log-structure. This log-structure extends uniquely to a log-structure on $E_{{\operatorname{cr} },n}$ and the PD-thickening ${\operatorname{Spec} }(\overline{V})^{\times}_1\hookrightarrow E_{{\operatorname{cr} },n}$ is universal over ${\mathbf Z}/p^n$. Set $E_{{\operatorname{cr} }}:={\operatorname{Spec} }(A_{{\operatorname{cr} }})$ with the limit log-structure. Since we have [@BE2 1.18.1] $${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Z_1)\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Z_1/E_{{\operatorname{cr} }}),$$ Theorem \[Bthm\] yields a canonical quasi-isomorphism of $B^+_{{\operatorname{cr} }}$-complexes (called [the crystalline Beilinson-Hyodo-Kato quasi-isomorphism]{}) $$\iota^B_{{\operatorname{cr} }}:\quad {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Z_1)^{\tau}_{B^+_{{\operatorname{cr} }}}\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Z_1)_{{{\mathbf Q}}}$$ compatible with the action of Frobenius. But we have $${\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Z_1)^{\tau}_{B^+_{{\operatorname{cr} }}}= ({\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Z_1)\otimes_{K_0^{{\operatorname{nr} }}}A_{{\operatorname{cr} },{{\mathbf Q}}}^{\tau})^{N=0}$$ and there is a canonical isomorphism $A_{{\operatorname{cr} },{{\mathbf Q}}}^{\tau}\stackrel{\sim}{\to} B_{{\operatorname{st} }}^+$ that is compatible with Frobenius and monodromy. This implies that the above quasi-isomorphism amounts to a quasi-isomorphism of $B^+_{{\operatorname{cr} }}$-complexes $$\iota^B_{{\operatorname{cr} }}:\quad {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Z_1)_{B^+_{{\operatorname{st} }}}\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Z_1)\otimes_{A_{{\operatorname{cr} }}}^{L}B^+_{{\operatorname{st} }}$$ compatible with the action of ${\varphi}$ and $N$. The crystalline Beilinson-Hyodo-Kato map can be canonically trivialized at $[\tilde{p}]$, where $\tilde{p}$ is a sequence of $p^n$’th roots of $p$,: $$\begin{aligned} \beta=\beta_{[\tilde{p}]}: {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Z_1)\otimes_{K_0^{{\operatorname{nr} }}}B^+_{{\operatorname{cr} }} & =({\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Z_1)\otimes_{K_0^{{\operatorname{nr} }}}B^+_{{\operatorname{cr} }}[a([\tilde{p}])])^{N=0}\\ x & \mapsto \exp(N(x)a([\tilde{p}]))\end{aligned}$$ This trivialization is compatible with Frobenius and monodromy. Suppose now that $\overline{f}_1:Z_1\to {\operatorname{Spec} }(\overline{V}_1)^{\times}$ is a reduction mod $p$ of a log-scheme $\overline{f}:Z\to {\operatorname{Spec} }(\overline{V})^{\times}$. Suppose that $\overline{f}$ is the base change of $\overline{f}_L:Z_{L}\to {\operatorname{Spec} }({{\mathcal O}}_{L})^{\times}$ by $\theta: {\operatorname{Spec} }(\overline{{{\mathcal O}}_{L}})^{\times}\to{\operatorname{Spec} }({{\mathcal O}}_{L})^{\times}$, for a finite extension $L/K$. That is, we have a map $\theta_{L}: Z\to Z_{L}$ such that the square $(\overline{f},\overline{f}_L,\theta,\theta_{L})$ is Cartesian. Assume that $\overline{f}_L$ is log-smooth of Cartier type and that the underlying map of schemes is proper. Such data $(L,Z,\theta_{L})$ form a directed set $\Sigma$ and the reduction mod $p$ map $\Sigma\to \Sigma_1$ is cofinal. The Beilinson-Hyodo-Kato quasi-isomorphisms (\[HK1\]) are compatible with morphisms in $\Sigma$ and their colimit yields a natural quasi-isomorphism (called again the [Beilinson-Hyodo-Kato quasi-isomorphism]{}) $$\iota^B_{{\mathrm{dR}}}:\quad {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Z_1)_{{\overline{K} }}^{\tau}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma(Z_{{\overline{K} }},\Omega^{\scriptscriptstyle\bullet}_{Z/{\overline{K} }}).$$ The trivializations by $p$ are also compatible with the maps in $\Sigma$ hence we obtain the Beilinson-Hyodo-Kato maps $$\iota^B_{{\mathrm{dR}}}:=\iota^B_{{\mathrm{dR}}}\beta_{p}:\quad {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Z_1)\to {\mathrm {R} }\Gamma(Z_{{\overline{K} }},\Omega^{\scriptscriptstyle\bullet}_{Z/{\overline{K} }}).$$ For an ss-pair $(U,\overline{U})$ over ${\overline{K} }$, set ${\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(U,\overline{U}):={\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}((U,\overline{U})_1)$. Let ${{\mathcal{A}}}^B_{{\mathrm{HK}}}$ be $h$-sheafification of the presheaf $(U,\overline{U})\mapsto {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(U,\overline{U})$ on ${{\mathcal{P}}}^{ss}_{{\overline{K} }}$. This is an $h$-sheaf of $E_{\infty}$ $K_0^{{\operatorname{nr} }}$-algebras equipped with a ${\varphi}$-action and locally nilpotent derivation $N$ such that $N{\varphi}=p{\varphi}N$. For $X\in\mathcal{V}ar_{{\overline{K} }}$, set ${\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h):={\mathrm {R} }\Gamma(X_h,{{\mathcal{A}}}^B_{{\mathrm{HK}}}). $ 1. For any $(U,\overline{U})\in {{\mathcal{P}}}^{ss}_{{\overline{K} }}$, the canonical maps $$\label{isomorphism} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]})_{\mathbf Q}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma(U_h,{{\mathcal J}}^{[r]}_{{\operatorname{cr} }})_{\mathbf Q},\quad {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(U,\overline{U})\stackrel{\sim}{\to}{\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(U_h)$$ are quasi-isomorphisms. 2. For every $X\in \mathcal{V}ar_{{\overline{K} }}$, the cohomology groups $H^n_{{\operatorname{cr} }}(X_h):=H^n{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_h)_{\mathbf Q}$, resp. $H^n_{{\mathrm{HK}}}(X_h):= H^n{\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)$, are free $B^{+}_{{\operatorname{cr} }}$-modules, resp. $K_0^{{\operatorname{nr} }}$-modules, of rank equal to the rank of $ H^n(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p)$. Only the filtered statement in part (1) for $r > 0$ requires argument since the rest has been proven by Beilinson in [@BE2 2.4]. Take $r >0$. To prove that we have a quasi-isomorphism ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[r]})_{\mathbf Q}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma(U_h,{{\mathcal J}}^{[r]}_{{\operatorname{cr} }})_{\mathbf Q}$ it suffices to show that the map ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal O}}/{{\mathcal J}}^{[r]})_{\mathbf Q}\stackrel{}{\to} {\mathrm {R} }\Gamma(U_h,{{\mathcal{A}}}_{{\operatorname{cr} }}/{{\mathcal J}}^{[r]}_{{\operatorname{cr} }})_{\mathbf Q}$ is a quasi-isomorphism. Since, for an ss-pair $(T,{\overline}{T})$ over $K$, by Corollary \[Langer\], ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(T,\overline{T},{{\mathcal O}}/{{\mathcal J}}^{[r]})_{\mathbf Q}\simeq {\mathrm {R} }\Gamma(\overline{T}_K,\Omega^{\scriptscriptstyle\bullet}_{(T,\overline{T}_K)}/F^r)$ this is equivalent to showing that the map ${\mathrm {R} }\Gamma(\overline{U}_K,\Omega^{\scriptscriptstyle\bullet}_{(U,\overline{U}_K)}/F^r)\to {\mathrm {R} }\Gamma(U_h,{{\mathcal{A}}}_{{\mathrm{dR}}}/F^r)$ is a quasi-isomophism. And this follows from Proposition \[deRham1\]. \[HKdR\] Let $X\in {\mathcal V}ar_{K}$. The natural projection $\varepsilon: X_{{\overline{K} },h}\to X_h$ defines pullback maps $$\label{qis11} \varepsilon^: {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)\to {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})^{G_K},\quad \varepsilon^: {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h)\to {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} },h})^{G_K}.$$ These are (filtered) quasi-isomosphisms. Notice that the action of $G_K$ on ${\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{\overline{K},h})\{r\}$ and $ {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} },h})$ is smooth, i.e., the stabilizer of every element is an open subgroup of $G_K$. We will prove only the first quasi-isomorphism - the proof of the second one being analogous. By Proposition \[hypercov\], it suffices to show that for any ss-pair over $K$ the natural map $${\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(U_1,\overline{U}_1) \to {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}((U,\overline{U})\otimes _K{\overline{K} })^{G_K}$$ is a quasi-isomorphism. Passing to a finite extension of $K_U$, if necessary, we may assume that $(U,\overline{U})$ is log-smooth of Cartier type over a finite Galois extension $K_U$ of $K$. Then $${\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}((U,\overline{U})\otimes _K{\overline{K} }) \simeq {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(U_1,\overline{U}_1)\otimes_{K_{U,0}}K_0^{{\operatorname{nr} }}\times H,\quad H={\operatorname{Gal} }(K_U/K).$$ Taking $G_K$-fixed points of this quasi-isomorphism we obtain the first quasi-isomorphism of (\[qis11\]), as wanted. Let $(U,\overline{U})$ be an ss-pair over ${\overline{K} }$. Set $$\begin{aligned} {\mathrm {R} }\Gamma_{{\mathrm{dR}}}^{\natural}(U,\overline{U}):= & {\mathrm {R} }\Gamma(\overline{U}_{{\operatorname{\acute{e}t} }},{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet,\wedge}_{(U,\overline{U})/W(k)}),\quad {\mathrm {R} }\Gamma_{{\mathrm{dR}}}^{\natural}(U,\overline{U})_n: = {\mathrm {R} }\Gamma_{{\mathrm{dR}}}^{\natural}(U,\overline{U})\otimes^{{\mathbb L}}{\mathbf Z}/p^n\simeq {\mathrm {R} }\Gamma(\overline{U}_{{\operatorname{\acute{e}t} }},{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet,\wedge}_{(U,\overline{U})_n/W_n(k)}),\\ {\mathrm {R} }\Gamma_{{\mathrm{dR}}}^{\natural}(U,\overline{U}){\widehat}{\otimes}{\mathbf Z}_p:= & {\operatorname{holim} }_n {\mathrm {R} }\Gamma_{{\mathrm{dR}}}^{\natural}(U,\overline{U})_n,\quad {\mathrm {R} }\Gamma_{{\mathrm{dR}}}^{\natural}(U,\overline{U}){\widehat}{\otimes}{\mathbf Q}_p:= ({\mathrm {R} }\Gamma_{{\mathrm{dR}}}^{\natural}(U,\overline{U}){\widehat}{\otimes}{\mathbf Z}_p)\otimes {\mathbf Q}.\end{aligned}$$ These are $F$-filtered $E_{\infty}$ algebras. Take the associated presheaves on ${{\mathcal{P}}}^{ss}_{\overline{K}}$. Denote by ${{\mathcal{A}}}^{\natural}_{{\mathrm{dR}}}$, ${{\mathcal{A}}}^{\natural}_{{\mathrm{dR}},n},{{\mathcal{A}}}^{\natural}_{{\mathrm{dR}}}{\widehat}{\otimes}{\mathbf Z}_p,{{\mathcal{A}}}^{\natural}_{{\mathrm{dR}}}{\widehat}{\otimes}{\mathbf Q}_p$ their sheafifications in the $h$-topology of $\mathcal{V}ar_{{\overline{K} }}$. These are sheaves of $F$-filtered $E_{\infty}$ algebras (viewed as the projective system of quotients modulo $F^i$). Set $A_{{\mathrm{dR}}}:={\mathrm {L} }\Omega^{\scriptscriptstyle\bullet,\wedge}_{\overline{V}/V}$. By [@BE1 Lemma 3.2] we have $A_{{\mathrm{dR}}}={{\mathcal{A}}}^{\natural}_{{\mathrm{dR}}}({\operatorname{Spec} }({\overline{K} }))={\mathrm {R} }\Gamma^{\natural}_{{\mathrm{dR}}}({\overline{K} },\overline{V})$. The corresponding $F$-filtered algebras $A_{{\mathrm{dR}},n}$, $A_{{\mathrm{dR}}}{\widehat}{\otimes}{\mathbf Z}_p$, $A_{{\mathrm{dR}}}{\widehat}{\otimes}{\mathbf Q}_p$ are acyclic in nonzero degrees and the projections $\cdot/F^{m+1}\to \cdot/F^m$ are surjective. Thus (we set $\lim_F:={\operatorname{holim} }_F$) $$\begin{aligned} A^{\diamond}_{{\mathrm{dR}},n} & :=\lim_FA_{{\mathrm{dR}},n}=\invlim_{m}H^0(A_{{\mathrm{dR}},n}/F^m), \quad A^{\diamond}_{{\mathrm{dR}}}:=\lim_F(A_{{\mathrm{dR}}}{\widehat}{\otimes}{{\mathbf Z}}_p)=\invlim_{m}H^0(A_{{\mathrm{dR}}}{\widehat}{\otimes}{{\mathbf Z}}_p/F^m)\\ \lim_FA_{{\mathrm{dR}}}{\widehat}{\otimes}{\mathbf Q}_p & =\invlim_{m}H^0(A_{{\mathrm{dR}}}{\widehat}{\otimes}{\mathbf Q}_p/F^m)=B_{{\mathrm{dR}}}^+, \quad A_{{\mathrm{dR}}}{\widehat}{\otimes}{\mathbf Q}_p/F^m=B^+_{{\mathrm{dR}}}/F^m \end{aligned}$$ For any $(U,\overline{U})$ over ${\overline{K} }$, the complex ${\mathrm {R} }\Gamma_{{\mathrm{dR}}}^{\natural}(U,\overline{U})$ is an $F$-filtered $E_{\infty}$ filtered $A_{{\mathrm{dR}}}$-algebra hence $\lim_F{\mathrm {R} }\Gamma_{{\mathrm{dR}}}^{\natural}(U,\overline{U})_n$ is an $A_{{\mathrm{dR}},n}^{\diamond}$-algebra, $\lim_F({\mathrm {R} }\Gamma_{{\mathrm{dR}}}^{\natural}(U,\overline{U}){\widehat}{\otimes}{{\mathbf Q}}_p)$ is a $B^+_{{\mathrm{dR}}}$-algebra, etc. We have canonical morphisms $$\kappa^{\prime}_{r,n}:\quad {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_n\to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_n/F^r\stackrel{\sim}{\to }{\mathrm {R} }\Gamma_{{\mathrm{dR}}}^{\natural}(U,\overline{U})_n/F^r$$ In the case of $({\overline{K} },\overline{V})$, from Theorem \[beilinson\], we get isomorphisms $\kappa^{\prime}_{r,n}=\kappa^{-1}_r:A_{{\operatorname{cr} },n}/J^{[r]}\stackrel{\sim}{\to} A_{{\mathrm{dR}},n}/F^r$. Hence $A_{{\mathrm{dR}}}^{\diamond}$ is the completion of $A_{{\operatorname{cr} }}$ with respect to the $J^{[r]}$-topology. For $X\in\mathcal{V}ar_{{\overline{K} }}$, set ${\mathrm {R} }\Gamma_{{\mathrm{dR}}}^{\natural}(X_h):={\mathrm {R} }\Gamma(X_h,{{\mathcal{A}}}_{{\mathrm{dR}}}^{\natural}). $ Since $A_{{\mathrm{dR}},{\mathbf Q}}={\overline{K} }$, for any variety $X$ over ${\overline{K} }$, we have a filtered quasi-isomorphism of ${\overline{K} }$-algebras [@BE1 3.2] $ {\mathrm {R} }\Gamma^{\natural}_{{\mathrm{dR}}}(X_h)_{\mathbf Q}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h) $ obtained by $h$-sheafification of the quasi-isomorphism $$\label{deRham11} {\mathrm {R} }\Gamma^{\natural}_{{\mathrm{dR}}}(U,\overline{U})_{\mathbf Q}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,{\overline}{U}_{{\mathbf Q}}).$$ Concerning the $p$-adic coefficients, we have a quasi-isomorphism $$\label{gamma} \gamma_r: ({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{h})\otimes_{{\overline{K} }}B^+_{{\mathrm{dR}}}) /F^r\stackrel{\sim}{\to} {\mathrm {R} }\Gamma(X_{h},{{\mathcal{A}}}^{\natural}_{{\mathrm{dR}}}{\widehat}{\otimes}{\mathbf Q}_p) /F^r$$ To define it, consider, for any ss-pair $(U,\overline{U})$ over ${\overline{K} }$, the natural map ${\mathrm {R} }\Gamma^{\natural}_{{\mathrm{dR}}}(U,\overline{U})\to {\mathrm {R} }\Gamma^{\natural}_{{\mathrm{dR}}}(U,\overline{U}){\widehat}{\otimes}{\mathbf Z}_p$. It yields, by extension to $A_{{\mathrm{dR}}}{\widehat}{\otimes}{\mathbf Q}_p$ and by the quasi-isomorphism (\[deRham11\]), a quasi-isomorphism of $F$-filtered ${\overline{K} }$-algebras [@BE2 3.5] $$\gamma: {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U})_{\mathbf Q}\otimes_{{\overline{K} }}(A_{{\mathrm{dR}}}{\widehat}{\otimes}{\mathbf Q}_p)\stackrel{\sim}{\to}{\mathrm {R} }\Gamma^{\natural}_{{\mathrm{dR}}}(U,\overline{U}){\widehat}{\otimes}{\mathbf Q}_p$$ Its mod $F^r$-version $\gamma_r$ after $h$-sheafification yields the quasi-isomorphism $$\gamma_r: ({{\mathcal{A}}}_{{\mathrm{dR}}}\otimes_{{\overline{K} }}B_{{\mathrm{dR}}}^+)/F^r\stackrel{\sim}{\to} {{\mathcal{A}}}_{{\mathrm{dR}}}^{\natural}{\widehat}{\otimes}{\mathbf Q}_p/F^r$$ Passing to ${\mathrm {R} }\Gamma(X_h,\scriptscriptstyle\bullet)$ we get the quasi-isomorphism (\[gamma\]). For $X\in {\mathcal V}ar_{{\overline{K} }}$, we have canonical quasi-isomorphisms $$\iota^B_{{\operatorname{cr} }}: {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)^{\tau}_{B^+_{{\operatorname{cr} }}}\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_{h})_{\mathbf Q},\quad \iota^B_{{\mathrm{dR}}}: {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)^{\tau}_{{\overline{K} }}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h),$$ compatible with the ${\operatorname{Gal} }({\overline{K} }/K)$-action. Here ${}^{\tau}_{B^+_{{\operatorname{cr} }}}$ and ${}^{\tau}_{{\overline{K} }}$ denote the $h$-sheafification of the crystalline and the de Rham Beilinson-Hyodo-Kato twists [@BE2 2.5.1]. Trivializing the first map at $[\tilde{p}]$ and the second map at $p$ we get the Beilinson-Hyodo-Kato maps $$\begin{aligned} \iota^B_{{\operatorname{cr} }}:=\iota^B_{{\operatorname{cr} }}\beta_{[\tilde{p}]}: {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)\otimes_{K_0^{{\operatorname{nr} }}}{B^+_{{\operatorname{cr} }}}\to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_{h})_{\mathbf Q},\quad \iota_{{\mathrm{dR}}} :=\iota_{{\mathrm{dR}}}\beta_p: {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)\to {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h).\end{aligned}$$ Using the quasi-isomorphism $\kappa_r^{-1}: {{\mathcal{A}}}_{{\operatorname{cr} },{\mathbf Q}} /{{\mathcal J}}^{[r]}_{{\operatorname{cr} },{\mathbf Q}}\stackrel{\sim}{\to} ({{\mathcal{A}}}^{\natural}_{{\mathrm{dR}}}{\widehat}{\otimes}{\mathbf Q}_p)/F^r$ from Theorem \[beilinson\], we obtain the following quasi-isomorphisms of complexes of sheaves on $X_{{\overline{K} },h}$ (brackets denote homotopy limits) $$\begin{aligned} {{\mathcal{S}}}(r)_{\mathbf Q} & \stackrel{\sim}{\to} \xymatrix{[{{\mathcal J}}^{[r]}_{{\operatorname{cr} },{\mathbf Q}}\ar[r]^-{1-{\varphi}_r} & {{\mathcal{A}}}_{{\operatorname{cr} },{\mathbf Q}}]} \stackrel{\sim}{\to} \xymatrix{[{{\mathcal{A}}}_{{\operatorname{cr} },{\mathbf Q}}\ar[rr]^-{(1-{\varphi}_r,{ \operatorname{can} })} & & {{\mathcal{A}}}_{{\operatorname{cr} },{\mathbf Q}}\oplus {{\mathcal{A}}}_{{\operatorname{cr} },{\mathbf Q}} /{{\mathcal J}}^{[r]}_{{\operatorname{cr} },{\mathbf Q}}]}\\ & \stackrel{\sim}{\leftarrow}\xymatrix{[{{\mathcal{A}}}_{{\operatorname{cr} },{\mathbf Q}}\ar[rr]^-{(1-{\varphi}_r,\kappa_r^{-1})} & & {{\mathcal{A}}}_{{\operatorname{cr} },{\mathbf Q}}\oplus ({{\mathcal{A}}}^{\natural}_{{\mathrm{dR}}}{\widehat}{\otimes}{\mathbf Q}_p)/F^r ]} \end{aligned}$$ Applying ${\mathrm {R} }\Gamma(X_{h},\scriptscriptstyle\bullet)$ and the quasi-isomorphism $\gamma_r^{-1}:{\mathrm {R} }\Gamma(X_{h},{{\mathcal{A}}}^{\natural}_{{\mathrm{dR}}}{\widehat}{\otimes}{\mathbf Q}_p) /F^r\stackrel{\sim}{\to} ({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{h})\otimes_{{\overline{K} }}B^+_{{\mathrm{dR}}}) /F^r$ from (\[gamma\]) we obtain the following quasi-isomorphisms $$\begin{aligned} \label{kwak1} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{h},r) & \stackrel{\sim}{\to} \xymatrix{[{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_{h})_{\mathbf Q}\ar[rr]^-{(1-{\varphi}_r,\kappa_r^{-1})} & & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_{h})_{\mathbf Q}\oplus {\mathrm {R} }\Gamma(X_{h},{{\mathcal{A}}}^{\natural}_{{\mathrm{dR}}}{\widehat}{\otimes}{\mathbf Q}_p) /F^r]}\\ & \stackrel{\sim}{\to} \xymatrix{[{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_{h})_{\mathbf Q}\ar[rr]^-{(1-{\varphi}_r,\gamma_{r}^{-1}\kappa_r^{-1} )} & & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_{h})_{\mathbf Q}\oplus ({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{h})\otimes_{{\overline{K} }}B_{{\mathrm{dR}}}^+) /F^r]}\notag\end{aligned}$$ For any $(U,\overline{U})\in{{\mathcal{P}}}^{ss}_{{\overline{K} }}$, the canonical map $${\mathrm {R} }\Gamma _{{ \operatorname{syn} }}(U,\overline{U},r)_{{\mathbf Q}}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }} (U_h,r)$$ is a quasi-isomorphism. Arguing as above we find quasi-isomorphisms $$\begin{aligned} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{{\mathbf Q}}& \stackrel{\sim}{\to} \xymatrix{[{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_{\mathbf Q}\ar[rr]^-{(1-{\varphi}_r,\kappa_r^{-1})} & & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_{\mathbf Q}\oplus ({\mathrm {R} }\Gamma^{\natural}(U,\overline{U}){\widehat}{\otimes}{\mathbf Q}_p) /F^r]}\\ & \stackrel{\sim}{\to} \xymatrix{[{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_{\mathbf Q}\ar[rr]^-{(1-{\varphi}_r,\gamma_{r}^{-1}\kappa_r^{-1})} & & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_{\mathbf Q}\oplus ({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U})\otimes_{{\overline{K} }}B_{{\mathrm{dR}}}^+) /F^r]}\end{aligned}$$ Comparing them with quasi-isomorphisms (\[kwak1\]) we see that it suffices to check that the natural maps $${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})_{\mathbf Q} \stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U_h)_{\mathbf Q},\quad {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}) \stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U_h),$$ are (filtered) quasi-isomorphism. But this is known by Proposition \[isomorphism\] and Proposition \[deRham1\]. Consider the following composition of morphisms $$\begin{aligned} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{h},r) & \stackrel{\sim}{\to} \left[\xymatrix@C=36pt{{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_{h})_{\mathbf Q}\ar[rr]^-{(1-{\varphi}_r,\gamma^{-1}_r\kappa_r^{-1})} & & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_{h})_{\mathbf Q}\oplus ({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{h})\otimes_{{\overline{K} }}B^+_{{\mathrm{dR}}}) /F^r}\right]\notag\\ \label{Cccc} & \stackrel{\sim}{\leftarrow} \left[\begin{aligned}\xymatrix@C=50pt{{\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{h})\otimes_{K_0^{{\operatorname{nr} }}}B_{{\operatorname{st} }}^+\ar[r]^-{(1-{\varphi}_r,\iota^B_{{\mathrm{dR}}}\otimes\iota)}\ar[d]^{N} & {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{h})\otimes_{K_0^{{\operatorname{nr} }}}B_{{\operatorname{st} }}^+\oplus ({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{h})\otimes_{{\overline{K} }}B^+_{{\mathrm{dR}}}) /F^r\ar[d]^{(N,0)}\\ {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{h})\otimes_{K_0^{{\operatorname{nr} }}}B_{{\operatorname{st} }}^+\ar[r]^{1-{\varphi}_{r-1}} & {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{h})\otimes_{K_0^{{\operatorname{nr} }}}B_{{\operatorname{st} }}^+}\end{aligned}\right] \end{aligned}$$ The second quasi-isomorphism uses the map $$({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{h})\otimes_{K_0^{{\operatorname{nr} }}}B_{{\operatorname{st} }}^+)^{N=0}={\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{h})^{\tau}_{B^+_{{\operatorname{cr} }}}\stackrel{\iota^B_{{\operatorname{cr} }}}{\to}{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_{h})_{\mathbf Q}$$ (that is compatible with the action of $N$ and ${\varphi}$) and the following lemma. \[HK-compatibility\] The following diagrams commute $$\xymatrix{ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_{h})_{\mathbf Q}\otimes_{B^+_{{\operatorname{cr} }}}B_{{\operatorname{st} }}^+\ar[rr]^-{\gamma^{-1}_r\kappa_r^{-1}\otimes \iota} & & ({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{h})\otimes_{{\overline{K} }}B^+_{{\mathrm{dR}}}) /F^r & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_{h})_{\mathbf Q}\otimes_{A_{{\operatorname{cr} }}}B_{{\mathrm{dR}}} \ar[r]^{\gamma_{{\mathrm{dR}}}}_{\sim} & {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{h})\otimes_{{\overline{K} }}B_{{\mathrm{dR}}} \\ {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{h})\otimes_{K_0^{{\operatorname{nr} }}}B_{{\operatorname{st} }}^+\ar[rru]_-{\iota^B_{{\mathrm{dR}}}\otimes\iota}\ar[u]^{\iota^B_{cr}}_{\wr} & & &{\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{h})\otimes_{K_0^{{\operatorname{nr} }}}B_{{\operatorname{st} }}\ar[u]^{\iota^B_{{\operatorname{cr} }}\otimes\iota}\ar[ur]_{\iota^B_{{\mathrm{dR}}}\otimes\iota}& }$$ Here $\gamma_{{\mathrm{dR}}}$ is the map defined by Beilinson in [@BE2 3.4.1]. We will start with the left diagram. It suffices to show that it canonically commutes with $X_h$ replaced by any ss-pair $\overline {Y}=(U,{\overline}{U})$ over ${\overline{K} }$ – a base change of an ss-pair $Y$ split over $(V,K)$. Proceeding as in Example \[crucial\], we obtain the following diagram in which all squares but the one in the left bottom clearly commute. $$\xymatrix{ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Y_1)^{\tau}_K\ar[d]^{{ \operatorname{Id} }\otimes 1} \ar[r]^{\iota^B_{K}} & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Y_1/V^{\times})_{{\mathbf Q}}/F^r\ar[d] & {\mathrm {R} }\Gamma(Y_{{\operatorname{\acute{e}t} }},{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet,\wedge}_{Y/V^{\times}}){\widehat}{\otimes}{{\mathbf Q}}_p/F^r \ar[l]_{\kappa_r}^{\sim}\ar[d] & {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(Y_K)/F^r\ar[l]_-{\gamma_r}^-{\sim}\ar[d]\\ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B({\overline}{Y}_1)^{\tau}_{{\overline{K} }}\otimes_{{\overline{K} }} B_{{\mathrm{dR}}}^+\ar[r]^{\iota_{{\overline{K} }}^B\otimes \kappa_r} & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(\overline{Y}_1/V^{\times})_{{\mathbf Q}}/F^r & {\mathrm {R} }\Gamma(\overline{Y}_{{\operatorname{\acute{e}t} }},{\mathrm {L} }\Omega^{\scriptscriptstyle\bullet,\wedge}_{\overline{Y}/V^{\times}}){\widehat}{\otimes}{{\mathbf Q}}_p/F^r \ar[l]_{\kappa_r}^{\sim} & ({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(\overline{Y}_K)\otimes _{{\overline{K} }}B_{{\mathrm{dR}}}^+)/F^r\ar[l]_{\gamma_r}^-{\sim}\ar[ld]_-{\gamma_r}^{\sim}\\ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B({\overline}{Y}_1)^{\tau}_{B^+_{{\operatorname{cr} }}}\otimes_{B^+_{{\operatorname{cr} }}}B^+_{{\operatorname{st} }}\ar[u]^{\delta}\ar[r]^{\iota^B_{{\operatorname{cr} }}\otimes \kappa_r\iota} & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(\overline{Y}_1/A_{{\operatorname{cr} }})_{{\mathbf Q}}/F^r\ar[u]^{\wr} & ({\mathrm {R} }\Gamma^{\natural}_{{\mathrm{dR}}}(\overline{Y}){\widehat}{\otimes}{{\mathbf Q}}_p)/F^r \ar[l]_{\kappa_r}^{\sim}\ar[u]^{\wr} }$$ Here we think $B^+_{{\mathrm{dR}}}/F^m={\mathrm {R} }\Gamma^{\natural}_{{\mathrm{dR}}}({\overline}{K},{\overline{V} }){\widehat}{\otimes}{{\mathbf Q}}_p$ and the map $\delta$ is defined as the composition $$\delta:\quad {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B({\overline}{Y}_1)^{\tau}_{B^+_{{\operatorname{cr} }}}\otimes_{B^+_{{\operatorname{cr} }}}B^+_{{\operatorname{st} }}= ({\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B({\overline}{Y}_1)\otimes_{K_0^{{\operatorname{nr} }}} B^+_{{\operatorname{st} }})^{N=0}\otimes_{B^+_{{\operatorname{cr} }}}B^+_{{\operatorname{st} }}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B({\overline}{Y}_1)\otimes_{K_0^{{\operatorname{nr} }}}B^+_{{\operatorname{st} }}\lomapr{\beta_p\otimes \iota} {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B({\overline}{Y}_1)^{\tau}_{{\overline{K} }}\otimes_{{\overline{K} }} B_{{\mathrm{dR}}}^+$$ Recall that for the map $\iota_{{\mathrm{dR}}}^B: {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Y_1)^{\tau}_K\to {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(Y_K)/F^r$ we have $\iota^B_{{\mathrm{dR}}}=\gamma_r^{-1} \kappa_r^{-1}\iota^B_K$. Everything in sight being compatible with change of the ss-pairs $Y$ - more specifically with maps in the directed system $\Sigma$ - if this diagram commutes so does its $\Sigma$ colimit and the left diagram in the lemma for the pair $(U,{\overline}{U})$. It remains to show that the left bottom square in the above diagram commutes. To do that consider the ring ${\widehat}{A}_n$ defined as the PD-envelope of the closed immersion $${\overline{V} }_1^{\times}\hookrightarrow A_{{\operatorname{cr} },n}\times _{W_n(k)}V^{\times}_n$$ That is, ${\widehat}{A}_n$ is the product of the PD-thickenings $({\overline{V} }_1^{\times}\hookrightarrow A_{{\operatorname{cr} },n})$ and $(V^{\times}_1 \hookrightarrow V^{\times}_n)$ over $(W_1(k)\hookrightarrow W_n(k))$. By [@BE2 Lemma 1.17], this makes $\overline{V}_1^{\times}\hookrightarrow {\widehat}{A}_{{\operatorname{cr} },n}$ into the universal PD-thickening in the log-crystalline site of $\overline{V}_1^{\times}$ over $V_n^{\times}$. Let ${\widehat}{A}:=\injlim_n{\widehat}{A}_{{\operatorname{cr} },n}$ with the limit log-structure. Set ${\widehat}{B}^+_{{\operatorname{cr} }}:={\widehat}{A}_{{\operatorname{cr} }}[1/p]$. Using Theorem \[Bthm\] , we obtain a canonical quasi-isomorphism $$\iota^B_{{\widehat}{B}^+_{{\operatorname{cr} }}}: {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(\overline{Y}_1)^{\tau}_{{\widehat}{B}^+_{{\operatorname{cr} }}}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}({\overline}{Y}_1/{\widehat}{A}_{{\operatorname{cr} }})_{{\mathbf Q}}$$ By construction, we have the maps of PD-thickenings $$\xymatrix{ (V^{\times}_1\hookrightarrow V^{\times}) & ( \overline{V}^{\times}_1\hookrightarrow {\widehat}{A}_{{\operatorname{cr} }})\ar[l]_{{\operatorname{pr} }_1}\ar[r]^{{\operatorname{pr} }_2} & (\overline{V}^{\times}_1\hookrightarrow A_{{\operatorname{cr} }}) }$$ Consider the following diagram $$\xymatrix{ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(\overline{Y}_1)^{\tau}_{{\widehat}{B}^+_{{\operatorname{cr} }}}\ar[ddd]_{\iota^B_{{\widehat}{B}^+_{{\operatorname{cr} }}}} & & {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(\overline{Y}_1)_{{\overline{K} }}^{\tau}\otimes_{{\overline{K} }} B_{{\mathrm{dR}}}^+/F^r \ar[ll]_{{\operatorname{pr} }^_1\otimes{\operatorname{pr} }^_1\kappa_r}\ar[ddd]^{\iota_{{\overline{K} }}^B\otimes \kappa_r}\\ & {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(\overline{Y}_1)^{\tau}_{B^+_{{\operatorname{cr} }}}\ar[d]^{\iota^B_{{\operatorname{cr} }}} \ar[lu]^{{\operatorname{pr} }_2^}\ar[ru]^{\delta} & \\ & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(\overline{Y}_1/A_{{\operatorname{cr} }})_{{\mathbf Q}}/F^r\ar[ld]_{{\operatorname{pr} }^_2} ^{\sim}\ar[rd]^{\sim} &\\ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(\overline{Y}_1/{\widehat}{A}_{{\operatorname{cr} }})_{{\mathbf Q}}/F^r & & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(\overline{Y}_1/V^{\times})_{{\mathbf Q}}/F^r \ar[ll]^{{\operatorname{pr} }_1^}_{\sim} }$$ The bottom triangle commutes since $ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(\overline{Y}_1/A_{{\operatorname{cr} }})= {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(\overline{Y}_1/W(k))$. The pullback maps $$\begin{aligned} {\operatorname{pr} }_1^:\quad & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(\overline{Y}_1/V^{\times}) \stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(\overline{Y}/{\widehat}{A}_{{\operatorname{cr} }}),\\ {\operatorname{pr} }_2^: \quad & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(\overline{Y}/A_{{\operatorname{cr} }})_{{\mathbf Q}}/F^r\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(\overline{Y}/{\widehat}{A}_{{\operatorname{cr} }})_{{\mathbf Q}}/F^r\end{aligned}$$ are quasi-isomorphisms. Indeed, in the case of the first pullback this follows from the universal property of ${\widehat}{A}_{{\operatorname{cr} }}$; in the case of the second one - it follows from the commutativity of the bottom triangle since the right slanted map is a quasi-isomorphism as shown by the first diagram in our proof. The left trapezoid and the big square commute by the definition of the Beilinson-Bloch-Kato maps. To see that the top triangle commutes it suffices to show that for an element $$x\in {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(\overline{Y}_1)^{\tau}_{B^+_{{\operatorname{cr} }}}= ({\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(\overline{Y}_1)\otimes_{K_0^{{\operatorname{nr} }}}B^+_{{\operatorname{st} }})^{N=0},\quad x=b\sum_{i\geq 0} N^i(m) a([\tilde{p}])^{[i]}, m\in {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}({\overline}{Y}_1),b\in B^+_{{\operatorname{cr} }},$$ we have ${\operatorname{pr} }^_2(x)={\operatorname{pr} }^_1\delta(x)$. Since $\iota(a([\tilde{p}]))=\log([\tilde{p}]/p)$ [@F1 4.2.2], we calculate $$\begin{aligned} \delta(x)=\delta(b\sum_{i\geq 0} N^i(m) a([\tilde{p}])^{[i]}) & = b\sum_{i\geq 0}(\sum_{j\geq 0} N^{i+j}(m)a(p)^{[j]})\log([\tilde{p}]/p)^{[i]}\\ & =b\sum_{k\geq 0} N^{k}(m)(a(p)+\log([\tilde{p}]/p))^{[k]}\end{aligned}$$ Since in ${\widehat}{B}^+_{{\operatorname{cr} }}$ we have $[\tilde{p}]=([\tilde{p}]/p)p$ and $[\tilde{p}]/p\in 1+J_{{\widehat}{B}^+_{{\operatorname{cr} }}}$, it follows that $a([\tilde{p}])=\log([\tilde{p}]/p)+a(p)$ and $${\operatorname{pr} }^_1\delta(x)={\operatorname{pr} }^_1(b\sum_{k\geq 0} N^{k}(m)(a(p)+\log([\tilde{p}]/p))^{[k]})= b\sum_{k\geq 0} N^{k}(m)a([\tilde{p}])^{[k]}={\operatorname{pr} }_2^ (b\sum_{k\geq 0} N^{k}(m)a([\tilde{p}])^{[k]})={\operatorname{pr} }_2^(x),$$ as wanted. It follows now that the right trapezoid in the above diagram commutes as well and that so does the left diagram in our lemma. To check the commutativity of the right diagram, consider the following map obtained from the maps $\kappa^{\prime}_{r,n}$ by passing to $F$-limit $$\kappa^{\prime}_n: \quad {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(\overline{Y})_n\otimes^L_{A_{{\operatorname{cr} },n}}A_{{\mathrm{dR}},n}\stackrel{\sim}{\to}\invlim_F {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(\overline{Y})_n/F^r$$ By [@BE2 3.6.2], this is a quasi-isomorphism. Beilinson [@BE2 3.4.1] defines the map $$\gamma_{{\mathrm{dR}}}:\quad {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(\overline{Y})_{{\mathbf Q}}\otimes_{A_{{\operatorname{cr} }}}B_{{\mathrm{dR}}}^+\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(\overline{Y}_K)\otimes_{{\overline{K} }}B^+_{{\mathrm{dR}}}$$ by $B^+_{{\mathrm{dR}}}$-linearization of the composition $\invlim_r(\gamma_r^{-1}\kappa_r^{-1}){\operatorname{holim} }_n\kappa^{\prime}_n$. We have $$\gamma_{{\mathrm{dR}}}=\gamma_r^{-1}\kappa^{-1}_r:{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(\overline{Y})_{{\mathbf Q}}\to ({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(\overline{Y}_K)\otimes_{{\overline{K} }}B^+_{{\mathrm{dR}}})/F^r$$ Hence the commutativity of the right diagram follows from that of the left one. Let $C^+({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{h})\{r\})$ denote the second homotopy limit in the diagram (\[Cccc\]); denote by $C({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{h})\{r\})$ the complex $C^+({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{h})\{r\})$ with all the pluses removed. We have defined a map $\alpha_{{ \operatorname{syn} }}: {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{h},r)\to C^+({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{h})\{r\})$ and proved the following proposition. There is a functorial $G_K$-equivariant quasi-isomorphism $$\alpha_{{ \operatorname{syn} }}:\quad {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{h},r)={\mathrm {R} }\Gamma(X_{h},{{\mathcal{S}}}(r)_{\mathbf Q})\simeq C^+({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{h})\{r\}).$$ Relation between syntomic cohomology and étale cohomology ========================================================= In this section we will study the relationship between syntomic and étale cohomology in both the geometric and the arithmetic situation. Geometric case -------------- We start with the geometric case. In this subsection, we will construct the geometric syntomic period map from syntomic to étale cohomology. We will prove that in the torsion case, on the level of $h$-sheaves it is a quasi-isomorphism modulo a universal constant; in the rational case – it induces an isomorphism on cohomology groups in a stable range. Finally, we will construct the syntomic descent spectral sequence. We will first recall the de Rham and Crystalline Poincaré Lemmas of Beilinson and Bhatt [@BE1], [@BE2], [@BH]. (de Rham Poincaré Lemma [@BE1 3.2]) \[derham\] The maps $A_{{\mathrm{dR}}}\otimes ^{{L}}{\mathbf Z}/p^n \to {{\mathcal{A}}}_{{\mathrm{dR}}}^{\natural}\otimes ^{{L}}{\mathbf Z}/p^n $ are filtered quasi-isomorphisms of $h$-sheaves on ${\mathcal V}ar_{\overline{K}}$. (Filtered Crystalline Poincaré Lemma [@BE2 2.3], [@BH Theorem 10.14]) The map $J^{[r]}_{{\operatorname{cr} },n}\to {{\mathcal J}}^{[r]}_{{\operatorname{cr} },n}$ is a quasi-isomorphism of $h$-sheaves on ${\mathcal V}ar_{{\overline{K} }}$. We have the following map of distinguished triangles $$\begin{CD} J^{[r]} _{{\operatorname{cr} },n}@>>> A_{{\operatorname{cr} },n} @>>> A_{{\operatorname{cr} },n}/J^{[r]}_{{\operatorname{cr} },n} \\ @VVV @VV\wr V @VV\wr V\\ {{\mathcal J}}^{[r]}_{{\operatorname{cr} },n}@>>> {{\mathcal{A}}}_{{\operatorname{cr} },n} @>>> {{\mathcal{A}}}_{{\operatorname{cr} },n}/{{\mathcal J}}^{[r]}_{{\operatorname{cr} },n} \end{CD}$$ The middle map is a quasi-isomorphism by the Crystalline Poincaré Lemma proved in [@BE2 2.3]. Hence it suffices to show that so is the rightmost map. But, by [@BE2 1.9.2], this map is quasi-isomorphic to the map $A_{{\mathrm{dR}},n}/F^r\to {{\mathcal{A}}}^{\natural}_{{\mathrm{dR}},n}/F^r$. Since the last map is a quasi-isomorphism by the de Rham Poincaré Lemma (\[derham\]) we are done. We will now recall the definitions of the crystalline, Beilinson-Hyodo-Kato, and de Rham period maps [@BE2 3.1], [@BE1 3.5]. Let $X\in {\mathcal V}ar_{{\overline{K} }}$. To define the crystalline period map $$\rho_{{\operatorname{cr} }}: {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_h)\to {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Z}_p){\widehat}{\otimes} A_{{\operatorname{cr} }},$$ consider the natural map $\alpha_n: {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_h)\to {\mathrm {R} }\Gamma(X_h,{{\mathcal{A}}}_{{\operatorname{cr} },n})$ and the composition $$\beta_n: \quad {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Z}_p(r))\otimes^{L}_{{\mathbf Z}_p}A_{{\operatorname{cr} },n}\stackrel{\sim}{\to}{\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},A_{{\operatorname{cr} },n}) \stackrel{\sim}{\to}{\mathrm {R} }\Gamma(X_{h},A_{{\operatorname{cr} },n}) \stackrel{\sim}{\to} {\mathrm {R} }\Gamma(X_{h},{{\mathcal{A}}}_{{\operatorname{cr} },n}).$$ Set $\rho_{{\operatorname{cr} },n}:=\beta_n^{-1}\alpha_n$ and $\rho_{{\operatorname{cr} }}:={\operatorname{holim} }_n\rho_{{\operatorname{cr} },n}$. The Hyodo-Kato period map $$\rho_{{\mathrm{HK}}}:{\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)^{\tau}_{B^+_{{\operatorname{cr} }}}\to {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p)\otimes B^+_{{\operatorname{cr} }},\quad \rho_{{\mathrm{HK}}}=\rho_{{\operatorname{cr} },{\mathbf Q}}\iota^B_{{\operatorname{cr} }},$$ is obtained by composing the map $\rho_{{\operatorname{cr} }, {\mathbf Q}}$ with the quasi-isomorphism $\iota^B_{{\operatorname{cr} }}: {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)^{\tau}_{B^+_{{\operatorname{cr} }}}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_h)_{{\mathbf Q}}$. The maps $\rho_{{\operatorname{cr} }}, \rho_{{\mathrm{HK}}}$ are morphisms of $E_{\infty}$ $A_{{\operatorname{cr} }}$- and $B^+_{{\operatorname{cr} }}$-algebras equipped with a Frobenius action; they are compatible with the action of the Galois group $G_K$. To define the de Rham period map $\rho_{{\mathrm{dR}}}:{\mathrm {R} }\Gamma(X_h)\otimes_{{\overline{K} }}B_{{\mathrm{dR}}}^+\to {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p)\otimes B_{{\mathrm{dR}}}^+$ consider the compositions $$\begin{aligned} \alpha:\, & {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h)\stackrel{\sim}{\to}{\mathrm {R} }\Gamma^{\natural}_{{\mathrm{dR}}}(X_h)\otimes {\mathbf Q}\to {\mathrm {R} }\Gamma^{\natural}_{{\mathrm{dR}}}(X_h){\widehat}{\otimes} {\mathbf Q}_p,\\ \beta:\, & {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Z})\otimes^{\mathbf L}A_{{\mathrm{dR}}}\stackrel{\sim}{\to}{\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},A_{{\mathrm{dR}}})\to {\mathrm {R} }\Gamma(X_h,A_{{\mathrm{dR}}})\to {\mathrm {R} }\Gamma(X_h,{{\mathcal{A}}}_{{\mathrm{dR}}}^{\natural})={\mathrm {R} }\Gamma^{\natural}_{{\mathrm{dR}}}(X_h). \end{aligned}$$ After tensoring the map $\beta$ with ${\mathbf Z}/p^n$ and using the de Rham Poincaré Lemma we get a quasi-isomorphism $$\beta_n: {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Z}/p^n)\otimes^{\mathbf L}A_{{\mathrm{dR}}}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma^{\natural}_{{\mathrm{dR}}}(X_h)\otimes^{\mathbf L}{\mathbf Z}/p^n.$$ Set $\beta_{\mathbf Q}:={\operatorname{holim} }_n\beta_n\otimes {\mathbf Q}$ and $\rho_{{\mathrm{dR}}}:=\beta^{-1}\alpha$. This is a morphism of filtered $E_{\infty}$ $B^+_{{\mathrm{dR}}}$-algebras, compatible with $G_K$-action. ([@BE2 3.2], [@BE1 3.6])For $X\in {\mathcal V}ar_{{\overline{K} }}$, we have canonical quasi-isomorphisms $$\begin{aligned} \rho_{{\operatorname{cr} }}: \,& {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_h)\otimes_{A_{{\operatorname{cr} }}}B_{{\operatorname{cr} }}\stackrel{\sim}{\to} {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p)\otimes B_{{\operatorname{cr} }},\quad \rho_{{\mathrm{HK}}}:\, {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)^{\tau}_{B_{{\operatorname{cr} }}}\stackrel{\sim}{\to}{\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p)\otimes B_{{\operatorname{cr} }},\\ \rho_{{\mathrm{dR}}}: \,& {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h)\otimes_{{\overline{K} }}{B_{{\mathrm{dR}}}}\stackrel{\sim}{\to}{\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p)\otimes B_{{\mathrm{dR}}}.\end{aligned}$$ Pulling back $\rho_{{\mathrm{HK}}}$ to the Fontaine-Hyodo-Kato ${\mathbb G}_a$-torsor ${\operatorname{Spec} }(B_{{\operatorname{st} }})/{\operatorname{Spec} }(B_{{\operatorname{cr} }})$ we get a canonical quasi-isomorphism of $B_{{\operatorname{st} }}$-complexes $$\rho_{{\mathrm{HK}}}: {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)\otimes_{K_0^{{\operatorname{nr} }}}B_{{\operatorname{st} }}\stackrel{\sim}{\to}{\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p)\otimes B_{{\operatorname{st} }}$$ compatible with the $({\varphi},N)$-action and with the $G_K$-action on ${\mathcal V}ar_{{\overline{K} }}$. \[period-compatibility\] The period morphisms are compatible, i.e., the following diagrams commute. $$\xymatrix{ {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)\otimes_{K_0^{{\operatorname{nr} }}}B_{{\operatorname{st} }}\ar[r]^-{\iota^B_{{\mathrm{dR}}}\otimes\iota}\ar[d]^{\rho_{{\mathrm{HK}}}} & {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h)\otimes_{{\overline{K} }}B_{{\mathrm{dR}}}\ar[d]^{\rho_{{\mathrm{dR}}}} & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_h)\otimes_{A_{{\operatorname{cr} }}}B_{{\mathrm{dR}}}\ar[d]_{\rho_{{\operatorname{cr} }}\otimes{ \operatorname{Id} }_{B_{{\mathrm{dR}}}}} & {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h)\otimes_{{\overline{K} }}B_{{\mathrm{dR}}} \ar[dl]^{\rho_{{\mathrm{dR}}}}\ar[l]_{\gamma_{{\mathrm{dR}}}}^{\sim} \\ {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p)\otimes B_{{\operatorname{st} }}\ar[r]^-{1\otimes \iota} & {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p)\otimes B_{{\mathrm{dR}}} & {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p)\otimes B_{{\mathrm{dR}}}\\ }$$ The second diagram commutes by [@BE2 3.4]. The commutativity of the first one can be reduced, by the equality $\rho_{{\mathrm{HK}}}=\rho_{{\operatorname{cr} }}\iota^B_{{\operatorname{cr} }}$ and the second diagram above, to the commutativity of the right diagram in Lemma \[HK-compatibility\]. We will now define the syntomic period map $$\rho_{{ \operatorname{syn} }}:\, {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)_{\mathbf Q}\to {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)),\quad r\geq 0.$$ Set ${\mathbf Z}/p^n(r)^{\prime}:=(1/(p^aa!){\mathbf Z}_p(r))\otimes{\mathbf Z}/p^n$, where $a$ is the largest integer $\leq r/(p-1)$. Recall that we have the fundamental exact sequence [@Ts Theorem 1.2.4] $$0\to {\mathbf Z}/p^n(r)^{\prime}\to J_{{\operatorname{cr} },n}^{<r>}\lomapr{1-{\varphi}_r}A_{{\operatorname{cr} },n}\to 0,$$ where $$J_n^{<r>}:= \{x\in J_{n+s}^{[r]}\mid {\varphi}(x)\in p^rA_{{\operatorname{cr} },n+s}\}/p^n ,$$ for some $s\geq r$. Set $S_n(r):={\operatorname{Cone} }(J^{[r]}_{{\operatorname{cr} },n}\lomapr{p^r-{\varphi}} A_{{\operatorname{cr} },n})[-1]$. There is a natural morphism of complexes $S_n(r)\to{\mathbf Z}/p^n(r)^{\prime}$ (induced by $p^r $ on $J_{{\operatorname{cr} },n}^{[r]}$ and ${ \operatorname{Id} }$ on $A_{{\operatorname{cr} },n}$) , whose kernel and cokernel are annihilated by $p^r$. The Filtered Crystalline Poincaré Lemma implies easily the following Syntomic Poincaré Lemma. 1. For $0\leq r\leq p-2$, there is a unique quasi-isomorphism ${\mathbf Z}/p^n(r)\stackrel{\sim}{\longrightarrow}{{\mathcal{S}}}_n(r)$ of complexes of sheaves on ${\mathcal V}ar_{{\overline{K} },h}$ that is compatible with the Crystalline Poincaré Lemma. 2. There is a unique quasi-isomorphism $S_n(r)\stackrel{\sim}{\to}{{\mathcal{S}}}_n(r)$ of complexes of sheaves on ${\mathcal V}ar_{{\overline{K} },h}$ that is compatible with the Crystalline Poincaré Lemma. We will prove the second claim - the first one is proved in an analogous way. Consider the following map of distinguished triangles $$\xymatrix{ {{\mathcal{S}}}_n(r)\ar[r] & {{\mathcal J}}^{[r]}_{{\operatorname{cr} },n}\ar[r]^{p^r-{\varphi}} & {{\mathcal{A}}}_{{\operatorname{cr} },n}\\ S_n(r)\ar[r]\ar@{-->}[u]& J^{[r]}_{{\operatorname{cr} },n}\ar[u]^{\wr}\ar[r]^{p^r-{\varphi}} & A_{{\operatorname{cr} },n}\ar[u]^{\wr} }$$ The triangles are distinguished by definition. The vertical continuous arrows are quasi-isomorphisms by the Crystalline Poincaré Lemma. They induce the dash arrow that is clearly a quasi-isomorphism. Consider the natural map $\alpha_n: {\mathrm {R} }\Gamma(X_h,{{\mathcal{S}}}(r))\to {\mathrm {R} }\Gamma(X_h,{{\mathcal{S}}}_n(r))$ and the zig-zag $$\beta_n:\, {\mathrm {R} }\Gamma(X_h,{{\mathcal{S}}}_n(r))\leftarrow {\mathrm {R} }\Gamma(X_{h},S_n(r))\to {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Z}/p^n(r)^{\prime})\stackrel{\sim}{\leftarrow} {\mathrm {R} }\Gamma(X_{h},{\mathbf Z}/p^n(r)^{\prime}).$$ Set $\beta:=({\operatorname{holim} }_n\beta_{n})\otimes {\mathbf Q}$; note that this is a quasi-isomorphism. Set $$\rho_{{ \operatorname{syn} }}:=p^{-r}\beta\alpha:\quad {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)\to {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)),$$ where $\alpha:=({\operatorname{holim} }_n\alpha_{n})\otimes {\mathbf Q}$. The period map $\rho_{{ \operatorname{syn} }}$ is a map of $E_{\infty}$ algebras over ${{\mathbf Q}}_p$ compatible with the action of the Galois group $G_K$. The syntomic period map has a different, more global definition that we find very useful. Define the map $\rho_{{ \operatorname{syn} }}^{\prime}$ by the following diagram. $$\xymatrix@C=40pt{ {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)\ar[r]^{\sim} \ar[d]^{\rho_{{ \operatorname{syn} }}^{\prime}}& [{\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_h)_{{\mathbf Q}}\ar[r]^-{(1-{\varphi}_r,\gamma^{-1}_r\kappa^{-1}_r)}\ar[d]^{\rho_{{\operatorname{cr} }}} & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_h)_{{\mathbf Q}}\oplus {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h)/F^r]\ar[d]^{\rho_{{\operatorname{cr} }}+\rho_{{\mathrm{dR}}}}\\ {\mathrm {R} }\Gamma_{{\operatorname{\acute{e}t} }}(X,{{\mathbf Q}}_p(r))\ar[r]^-{\sim} & [{\mathrm {R} }\Gamma_{{\operatorname{\acute{e}t} }}(X,{{\mathbf Q}}_p(r))\otimes B_{{\operatorname{cr} }}\ar[r]^-{(1-{\varphi}_r, { \operatorname{can} }) } & {\mathrm {R} }\Gamma_{{\operatorname{\acute{e}t} }}(X,{{\mathbf Q}}_p(r))\otimes B_{{\operatorname{cr} }}\oplus {\mathrm {R} }\Gamma_{{\operatorname{\acute{e}t} }}(X,{{\mathbf Q}}_p(r))\otimes B_{{\mathrm{dR}}}/F^r] }$$ This definition makes sense since the following diagram commutes. $$\xymatrix{ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(X_h)_{{\mathbf Q}}\ar[r]^{\gamma^{-1}_r\kappa^{-1}_r}\ar[d]^{\rho_{{\operatorname{cr} }}} & {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h)/F^r\ar[d]^{\rho_{{\mathrm{dR}}}}\\ {\mathrm {R} }\Gamma_{{\operatorname{\acute{e}t} }}(X,{{\mathbf Q}}_p(r))\otimes B_{{\operatorname{cr} }}\ar[r]^-{{ \operatorname{can} }} & {\mathrm {R} }\Gamma_{{\operatorname{\acute{e}t} }}(X,{{\mathbf Q}}_p(r))\otimes B_{{\mathrm{dR}}}/F^r }$$ The syntomic period morphisms $\rho_{{ \operatorname{syn} }}$ and $\rho_{{ \operatorname{syn} }}^{\prime}$ are homotopic by a homotopy compatible with the $G_K$-action (and, unless necessary, we will not distinguish them in what follows). These two facts follow easily from the definitions. For $X\in {\mathcal V}ar_K$, we have a quasi-isomorphism $$\label{definition1} \alpha_{{\operatorname{\acute{e}t} }}: {\mathrm {R} }\Gamma_{}(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r))\stackrel{\sim}{\to} C({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\})$$ that we define as the inverse of the following composition of quasi-isomorphisms (square brackets denote complex) $$\begin{aligned} C({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}) & \twomapr{\rho}{\sim}{\mathrm {R} }\Gamma_{}(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Q}_p)\otimes_{{\mathbf Q}_p} [B_{{\operatorname{st} }}\verylomapr{(N,1-{\varphi}_r,\iota)}B_{{\operatorname{st} }}\oplus B_{{\operatorname{st} }}\oplus B_{{\mathrm{dR}}}/F^r\veryverylomapr{(1-{\varphi}_{r-1})-N} B_{{\operatorname{st} }}]\\ & \stackrel{\sim}{\leftarrow}{\mathrm {R} }\Gamma_{}(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Q}_p)\otimes_{{\mathbf Q}_p}C(D_{{\operatorname{st} }}({\mathbf Q}_p(r))) \stackrel{\sim}{\leftarrow}{\mathrm {R} }\Gamma_{}(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)).\end{aligned}$$ The last quasi-isomorphism is by Remark \[basics\]. The map $\rho$ is defined using the period morphisms $\rho_{{\mathrm{HK}}}$ and $\rho_{{\mathrm{dR}}}$ and their compatibility (Corollary \[period-compatibility\]). The map $\alpha_{{\operatorname{\acute{e}t} }}$ is compatible with the action of $G_K$. For a variety $X\in {\mathcal V}ar_{K}$, we have a canonical, compatible with the action of $G_K$, quasi-isomorphism $$\rho_{{ \operatorname{syn} }}: \tau_{\leq r}{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{{\overline{K} },h},r)\stackrel{\sim}{\to} \tau_{\leq r}{\mathrm {R} }\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)).$$ The Bousfield-Kan spectral sequences associated to the homotopy limits defining the complexes $C^+(H^j_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\})$ and $C(H^j_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\})$ form the following commutative diagram $$\xymatrix{ ^+E^{i,j}_2=H^i(C^+(H^j_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}))\ar[d]^{{ \operatorname{can} }}\ar@{=>}[r] & H^{i+j}(C^+({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}))\ar[d]^{{ \operatorname{can} }}\\ E^{i,j}_2=H^i(C(H^j_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}))\ar@{=>}[r] & H^{i+j}(C({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\})) }$$ We have $D_j=H^j_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}\in MF_K^{{\operatorname{ad} }}({\varphi}, N,G_K)$. For $j\leq r$, $F^{1}D_{j,K}=F^{1-(r-j)}H^j_{{\mathrm{dR}}}(X_{h})=0$. Hence, by Corollary \[resolution3\], we have $^+E^{i,j}_2\stackrel{\sim}{\to} E^{i,j}_2$. This implies that $\tau_{\leq r}C^+({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\})\stackrel{\sim}{\to} \tau_{\leq r}C({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\})$. Since $\rho_{{\mathrm{HK}}}=\rho_{{\operatorname{cr} }}\iota^B_{{\operatorname{cr} }}$, we check easily that we have the following commutative diagram $$\label{compatibility0} \begin{CD} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{{\overline{K} },h},r)@>\sim >\alpha_{{ \operatorname{syn} }}> C^+({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\})\\ @VV\rho_{{ \operatorname{syn} }}V @VV{ \operatorname{can} }V\\ {\mathrm {R} }\Gamma_{}(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r))@>\sim >\alpha_{{\operatorname{\acute{e}t} }}> C({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}) \end{CD}$$ It follows that $\rho_{{ \operatorname{syn} }}: \tau_{\leq r}{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{{\overline{K} },h},r)\stackrel{\sim}{\to} \tau_{\leq r}{\mathrm {R} }\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)) $, as wanted. Let $X\in {\mathcal V}ar_K$. The natural projection $\varepsilon: X_{{\overline{K} },h}\to X_h$ defines pullback maps $$\varepsilon^: {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)\to {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h}),\quad \varepsilon^: {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h)\to {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} },h}).$$ By construction they are compatible with the monodromy operator, Frobenius, the action of the Galois group $G_K$, and filtration. It is also clear that they are compatible with the Beilinson-Hyodo-Kato morphisms, i.e., that the following diagram commutes $$\xymatrix{ {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)\ar[r]^{\iota^B_{{\mathrm{dR}}}}\ar[d]^{\varepsilon^} & {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_h)\ar[d]^{\varepsilon^}\\ {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\ar[r]^{\iota^B_{{\mathrm{dR}}}} & {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} },h}). }$$ It follows that we can define a canonical pullback map $$\varepsilon^:\quad C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)\{r\})\to C^+({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{\overline{K},h})\{r\}).$$ \[compatibilit01\] Let $r\geq 0$. The following diagram commutes in the derived category. $$\xymatrix{ {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)\ar[r]^-{\alpha_{{ \operatorname{syn} }}}\ar[d]^{\varepsilon^} & C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)\{r\})\ar[d]^{\varepsilon^}\\ {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{{\overline{K} },h},r)\ar[r]^-{\alpha_{{ \operatorname{syn} }}} & C^+({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{\overline{K},h})\{r\}). }$$ Take a number $t\geq 2\dim X +2$ and choose a finite Galois extension $(V^{\prime},K^{\prime})/(V,K)$ (see the proof of Proposition \[hypercov\]) such that we have an $h$-hypercovering $Z_{\scriptscriptstyle\bullet}\to X_{K^{\prime}}$ with $(Z_{\scriptscriptstyle\bullet})_{\leq t+1}$ built from log-schemes log-smooth over $V^{\prime, \times}$ and of Cartier type. Since the top map $\alpha_{{ \operatorname{syn} }}$ is compatible with base change (c.f. Proposition \[reduction2\]) it suffices to show that the diagram in the lemma commutes with $X$ replaced by $(Z_{\scriptscriptstyle\bullet})_{\leq t+1}$. By Propositions \[isomorphism\], \[hypercov\], and \[deRham1\], this reduces to showing that, for an ss-pair $(U,{\overline}{U})$ split over $V$, the following diagram commutes canonically in the $\infty$-derived category (we set $Y:=(U,{\overline}{U}), {\overline}{Y}:=Y_{{\overline{V} }}$, $\pi$ - a fixed uniformizer of $V$). $$\xymatrix{ {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(Y,r)_{{\mathbf Q}}\ar[r]^-{\alpha^B_{{ \operatorname{syn} },\pi}}\ar[d]^{\varepsilon^} & C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(Y)\{r\})\ar[d]^{\varepsilon^}\\ {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(Y_{{\overline{K} }},r)_{{\mathbf Q}}\ar[r]^-{\alpha_{{ \operatorname{syn} }}} & C^+({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(Y_{\overline{K}})\{r\}). }$$ From the uniqueness property of the homotopy fiber functor, it suffices to show that the following diagram commutes canonically in the $\infty$-derived category. $$\xymatrix{ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Y)_{{\mathbf Q}}\ar[r] \ar[d] & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Y/R)^{N=0} _{{\mathbf Q}}& {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Y_1)^{\tau,N=0}_{R_{{\mathbf Q}}}\ar[l]_{\iota_{\pi}}^{\sim} & {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Y_1)^{N=0}\ar[l]_{\beta}^{\sim}\ar[ld]\\ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}({\overline}{Y})_{{\mathbf Q}}& {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B({\overline}{Y}_1)^{\tau,N=0}_{B^+_{{\operatorname{cr} }}}\ar[l]_-{\iota^B_{{\operatorname{cr} }}} ^-{\sim}& ({\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B({\overline}{Y}_1)\otimes _{K_0^{{\operatorname{nr} }}}B^+_{{\operatorname{st} }})^{N=0}\ar@{=}[l] }$$ To do that we will need the ring of periods ${\widehat}{A}_{{\operatorname{st} }}$ [@Ts p.253]. Set $${\widehat}{A}_{{\operatorname{st} },n}=H^0_{{\operatorname{cr} }}(\overline{V}_{n}^{\times}/R_{n}), \quad {\widehat}{A}_{{\operatorname{st} }}=\invlim_nH^0_{{\operatorname{cr} }}(\overline{V}_{n}^{\times}/R_{n}).$$ The ring ${\widehat}{A}_{{\operatorname{st} },n}$ has a natural action of $G_K$, Frobenius ${\varphi}$, and a monodromy operator $N$. It is also equipped with a PD-filtration $F^i{\widehat}{A}_{{\operatorname{st} },n}=H^0_{{\operatorname{cr} }}(\overline{V}_{n}^{\times}/R_{n},{{\mathcal J}}_{{\operatorname{cr} },n}^{[i]})$. We have a morphism $A_{{\operatorname{cr} },n}\to {\widehat}{A}_{{\operatorname{st} },n}$ induced by the map $H^0_{{\operatorname{cr} }}(\overline{V}_{n}/W_n(k))\to H^0_{{\operatorname{cr} }}(\overline{V}_{n}^{\times}/R_{n})$. It is compatible with the Galois action, the Frobenius, and the filtration. The natural map $R_{n}\to {\widehat}{A}_{{\operatorname{st} },n}$ is compatible with all the structures. We can view ${\widehat}{A}_{{\operatorname{st} },n}$ as the PD-envelope of the closed immersion $$\overline{V}_n^{\times}\hookrightarrow A_{{\operatorname{cr} },n}\times_{W_n(k)}W_n(k)[X]^{\times}$$ defined by the map $\theta: A_{{\operatorname{cr} },n}\to \overline{V}_n$ and the projection $W_n(k)[X] \to \overline{V}_n$, $X\mapsto \pi$. This makes $\overline{V}_1^{\times}\hookrightarrow {\widehat}{A}_{{\operatorname{st} },n}$ into a PD-thickening in the crystalline site of $\overline{V}_1$. Set ${\widehat}{B}^+_{{\operatorname{st} }}:={\widehat}{A}_{{\operatorname{st} }}[1/p]$. Commutativity of the last diagram will follow from the following commutative diagram $$\xymatrix{ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Y)_{{\mathbf Q}}\ar[d] \ar[rr] & &{\mathrm {R} }\Gamma_{{\operatorname{cr} }}({\overline}{Y})_{{\mathbf Q}}\ar[dl]^{\sim}\\ {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(Y/R)^{N=0} _{{\mathbf Q}}\ar[r] & {\mathrm {R} }\Gamma_{{\operatorname{cr} }}({\overline}{Y}/{\widehat}{A}_{{\operatorname{st} }})^{N=0}_{{\mathbf Q}}\\ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Y_1)^{\tau,N=0}_{R_{{\mathbf Q}}}\ar[u]^{\iota_{\pi}}_{\wr}\ar[r]& {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Y_1)^{\tau,N=0}_{{\widehat}{B}^+_{{\operatorname{st} }}} \ar[u]^{\iota^B_{{\widehat}{B}^+_{{\operatorname{st} }}}}_{\wr} &{\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Y_1)^{\tau,N=0}_{B^+_{{\operatorname{cr} }}}\ar[l]\ar[uu]^{\iota_{{\operatorname{cr} }}^B}_{\wr}\\ & {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Y_1)^{N=0} \ar[ur]\ar[ul]^{\beta}_{\sim} }$$ as soon as we show that the map ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}({\overline}{Y})_{{\mathbf Q}}\to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}({\overline}{Y}/{\widehat}{A}_{{\operatorname{st} }})^{N=0}_{{\mathbf Q}}$ is a quasi-isomorphism. Notice that the map $\iota^B_{{\widehat}{B}^+_{{\operatorname{st} }}}$ is a quasi-isomorphism by Theorem \[Bthm\]. Hence using the Beilinson-Hyodo-Kato maps $\iota^B_{{\widehat}{B}^+_{{\operatorname{st} }}}$ and $ \iota_{{\operatorname{cr} }}^B$ this reduces to proving that the canonical map $ {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Y_1)^{\tau,N=0}_{B^+_{{\operatorname{cr} }}}\to {\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(Y_1)^{\tau,N=0}_{{\widehat}{B}^+_{{\operatorname{st} }}} $ is a quasi-isomorphism. In fact, we claim that for any $({\varphi}, N)$-module $M$ we have an isomorphism $M_{B_{{\operatorname{cr} }}^{+}}^{\tau,N=0}\stackrel{\sim}{\to} M_{{\widehat}{B}_{{\operatorname{st} }}^{+}}^{\tau,N=0}.$ Indeed, assume first that the monodromy $N_M$ is trivial. We calculate $$\begin{aligned} M^{\tau}_{B^+_{{\operatorname{cr} }}} & =(M\otimes _{K_0}B^{+,\tau}_{{\operatorname{cr} }})^{N^{\prime}=0}=M\otimes _{K_0}(B^{+,\tau}_{{\operatorname{cr} }})^{N_{\tau}=0}=M\otimes_{K_0} B^+_{{\operatorname{cr} }},\quad N^{\prime}= N_M\otimes 1 +1\otimes N_{\tau}=1\otimes N_{\tau},\\ M^{\tau}_{{\widehat}{B}^+_{{\operatorname{st} }}} & =(M\otimes _{K_0}{\widehat}{B}^{+,\tau}_{{\operatorname{st} }})^{N^{\prime}=0}=M\otimes_{K_0} ({\widehat}{B}^{+,\tau}_{{\operatorname{cr} }})^{N_{\tau}=0}=M\otimes _{K_0}{\widehat}{B}^+_{{\operatorname{st} }}\end{aligned}$$ Hence $M_{B_{{\operatorname{cr} }}^{+}}^{\tau,N=0}=M\otimes _{K_0}B^+_{{\operatorname{cr} }}$ and $M_{{\widehat}{B}_{{\operatorname{st} }}^{+}}^{\tau,N=0}=M\otimes _{K_0}({\widehat}{B}^+_{{\operatorname{st} }})^{N=0}=M\otimes _{K_0}B^+_{{\operatorname{cr} }}$, where the last equality is proved in [@Ts Lemma 1.6.5]. We are done in this case. In general, we can write $M\otimes _{K_0}B^+_{{\operatorname{st} }}\stackrel{\sim}{\leftarrow} M^{\prime}\otimes _{K_0}B^+_{{\operatorname{st} }}$ for a $({\varphi},N)$-module $M^{\prime}$ such that $N_{M^{\prime}}=0$ (take for $M^{\prime}$ the image of the map $M\to M\otimes _{K_0}B^+_{{\operatorname{st} }}$, $m\mapsto \exp(N_M(m)u)$, for $u\in B^+_{{\operatorname{st} }}$ such that $B^+_{{\operatorname{st} }}=B^+_{{\operatorname{cr} }}[u]$, $N_{\tau}(u)=-1$). Similarly, using the fact that the ring $B^+_{{\operatorname{st} }}$ is canonically (and compatibly with all the structures) isomorphic to the elements of ${\widehat}{B}^+_{{\operatorname{st} }}$ annihilated by a power of the monodromy operator [@Kas 3.7], we can write in a compatible way $M\otimes _{K_0}{B}^+_{{\operatorname{st} }}\stackrel{\sim}{\leftarrow} M^{\prime}\otimes _{K_0}{\widehat}{B}^+_{{\operatorname{st} }}$ for the same module $M^{\prime}$. We obtained a commutative diagram $$\begin{CD} M_{B_{{\operatorname{cr} }}^{+}}^{\tau,N=0}@>>> M_{{\widehat}{B}_{{\operatorname{st} }}^{+}}^{\tau,N=0} \\ @VV\wr V @VV\wr V\\ M_{B_{{\operatorname{cr} }}^{+}}^{\prime \tau,N=0}@>\sim>> M_{{\widehat}{B}_{{\operatorname{st} }}^{+}}^{\prime \tau,N=0} \end{CD}$$ that reduces the general case to the case of trivial monodromy on $M$ that we treated above. Let $X\in {\mathcal V}ar_K$, $r\geq 0$. Set $$\begin{aligned} C_{{\mathrm{pst}}}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{\overline{K},h})\{r\}):= \left[\begin{aligned} \xymatrix@C=50pt{{\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})^{G_K}\ar[r]^-{(1-{\varphi}_r,\iota^B_{{\mathrm{dR}}})}\ar[d]^{N} & {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})^{G_K}\oplus ({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} },h})/F^r)^{G_K}\ar[d]^{(N,0)}\\ {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})^{G_K}\ar[r]^{1-{\varphi}_{r-1}}& {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})^{G_K}}\end{aligned}\right] \end{aligned}$$ The above makes sense since the action of $G_K$ on ${\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{\overline{K},h})\{r\}$ and $ {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} },h})$ is smooth. In particular, we have $$\begin{aligned} H^j({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{\overline{K},h})\{r\}^{G_K}) \simeq H^j({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{\overline{K},h})\{r\})^{G_K},\quad H^j({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} },h})^{G_K}) & \simeq H^j({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} },h}))^{G_K}. \end{aligned}$$ Consider the canonical pullback map $$\varepsilon^: C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)\{r\})\stackrel{\sim}{\to} C_{{\mathrm{pst}}}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{\overline{K},h})\{r\}).$$ By Proposition \[HKdR\], this is a quasi-isomorphism. This allows us to construct a canonical spectral sequence (the [syntomic descent spectral sequence]{}) $$\label{kwak2} \xymatrix{ ^{{ \operatorname{syn} }}E^{i,j}_2=H^i_{{\operatorname{st} }}(G_K,H^j(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)))\ar@{=>}[r] & H^{i+j}_{{ \operatorname{syn} }}(X_{h},r) }$$Indeed, the Bousfield-Kan spectral sequences associated to the homotopy limits defining complexes $C_{{\mathrm{pst}}}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}))$ and $C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{h})\{r\}))$ give us the following commutative diagram $$\xymatrix{ ^{{\mathrm{pst}}} E^{i,j}_2=H^i(C_{{\mathrm{pst}}}(H^j_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}))\ar@{=>}[r] & H^{i+j}(C_{{\mathrm{pst}}}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}))\\ ^{{ \operatorname{syn} }} E^{i,j}_2=H^i(C_{{\operatorname{st} }}(H^j_{{\mathrm{HK}}}(X_{h})\{r\}))\ar[u]^{\wr}_{\varepsilon^}\ar@{=>}[r] & H^{i+j}(C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{h})\{r\}))\ar[u]^{\wr}_{\varepsilon^} }$$ Since, by Proposition \[reduction2\], we have $\alpha_{{ \operatorname{syn} }}: H^{i+j}_{{ \operatorname{syn} }}(X_h,r) \stackrel{\sim}{\to}H^{i+j}(C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_h)\{r\}))$, we have obtained a spectral sequence $$\xymatrix{ E^{i,j}_2=H^i(C_{{\mathrm{pst}}}(H^j_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}))\ar@{=>}[r] & H^{i+j}_{{ \operatorname{syn} }}(X_{h},r) }$$ It remains to show that there is a canonical isomorphism $$\label{eq1} H^i(C_{{\mathrm{pst}}}(H^j_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}))\simeq H^i_{{\operatorname{st} }}(G_K,H^j(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r))).$$ But, we have $D_j=H^j_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}\in MF_K^{{\operatorname{ad} }}({\varphi}, N,G_K)$, $V_{{\mathrm{pst}}}(D_j)\simeq H^j(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{{\mathbf Q}}(r))$, and $D_{{\mathrm{pst}}}(H^j(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{{\mathbf Q}}(r)))\simeq D_j$. Hence isomorphism (\[eq1\]) follows from Remark \[pst=st\] and we have obtained the spectral sequence (\[kwak2\]). Arithmetic case --------------- In this subsection, we define the arithmetic syntomic period map by Galois descent from the geometric case. Then we show that, via this period map, the syntomic descent spectral sequence and the étale Hochschild-Serre spectral sequence are compatible. Finally, we show that this implies that the arithmetic syntomic cohomology and étale cohomology are isomorphic in a stable range. Let $X\in {\mathcal V}ar_K$. For $r\geq 0$, we define the canonical syntomic period map $$\rho_{{ \operatorname{syn} }}: \quad {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)\to {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)),$$ as the following composition $$\begin{aligned} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r) & ={\mathrm {R} }\Gamma(X_h,{{\mathcal{S}}}(r))_{{\mathbf Q}}\to {\operatorname{holim} }_n {\mathrm {R} }\Gamma(X_{h},{{\mathcal{S}}}_n(r))_{{\mathbf Q}}\stackrel{\varepsilon^}{\to} {\operatorname{holim} }_n {\mathrm {R} }\Gamma (G_K, {\mathrm {R} }\Gamma(X_{{\overline{K} },h},{{\mathcal{S}}}_n(r)))_{{\mathbf Q}}\\ & \lomapr{p^{-r}\beta} {\operatorname{holim} }_n {\mathrm {R} }\Gamma (G_K, {\mathrm {R} }\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Z}/p^n(r)^{\prime}))_{{\mathbf Q}}\stackrel{\sim}{\leftarrow}{\operatorname{holim} }_n {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Z}/p^n(r)^{\prime})_{{\mathbf Q}}={\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)).\end{aligned}$$ It is a morphism of $E_{\infty}$ algebras over ${{\mathbf Q}}_p$. The syntomic period map $\rho_{{ \operatorname{syn} }}$ is compatible with the syntomic descent and the Hochschild-Serre spectral sequences. \[stHS\] For $X\in {{\mathcal{V}}}ar_K$, $r\geq 0$, there is a canonical map of spectral sequences $$\xymatrix{ ^{{ \operatorname{syn} }}E^{i,j}_2=H^i_{{\operatorname{st} }}(G_K,H^j(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)))\ar[d]^{{ \operatorname{can} }}\ar@{=>}[r] & H^{i+j}_{{ \operatorname{syn} }}(X_{h},r)\ar[d]^{\rho_{{ \operatorname{syn} }}}\\ ^{{\operatorname{\acute{e}t} }}E^{i,j}_2=H^i(G_K,H^j(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)))\ar@{=>}[r] & H^{i+j}(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)) }$$ We work in the (classical) derived category. The Bousfield-Kan spectral sequences associated to the homotopy limits defining complexes $C({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\})$ and $C_{{\mathrm{pst}}}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\})$, and Theorem \[speccomp\] give us the following commutative diagram of spectral sequences $$\xymatrix{ ^{II}E^{i,j}_2=H^i(G_K,C(H^j_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}))\ar@{=>}[r] & H^{i+j}(G_K,C({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}))\\ ^{{\mathrm{pst}}} E^{i,j}_2=H^i(C_{{\mathrm{pst}}}(H^j_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}))\ar[u]^{\delta}\ar@{=>}[r] & H^{i+j}(C_{{\mathrm{pst}}}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}))\ar[u]^{\delta} }$$ More specifically, in the language of Section \[Jan\], set $X=C({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\})$ (hopefully, the notation will not be too confusing). Filtering complex $X$ in the direction of the homotopy limit we obtain a Postnikov system (\[postnikov\]) with $Y^i=0$, $i\geq 3$, and $$\begin{aligned} Y^0 & ={\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}\otimes_{K_0^{{\operatorname{nr} }}} B_{{\operatorname{st} }},\\ Y^1 & ={\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r-1\}\otimes_{K_0^{{\operatorname{nr} }}} B_{{\operatorname{st} }}\oplus ({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}\otimes_{K^{{\operatorname{nr} }}_0} B_{{\operatorname{st} }}\oplus ({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} }})\otimes_{{\overline{K} }} B_{{\mathrm{dR}}})/F^r),\\ Y^2 & ={\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r-1\}\otimes _{K_0^{{\operatorname{nr} }}}B_{{\operatorname{st} }}.\end{aligned}$$ Still in the setting of Section \[Jan\], take for $A$ the abelian category of sheaves of abelian groups on the pro-étale site ${\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }}$ of Scholze [@Sch 3]. We work with the pro-étale site to make sense of the continuous cohomology ${\mathrm {R} }\Gamma(G_K,\cdot)$. If the reader is willing to accept that this is possible then he can skip the tedious parts of the proof involving passage to the pro-étale site (and existence of continuous sections). Recall that there is a projection map $\nu: {\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }}\to {\operatorname{Spec} }(K)_{{\operatorname{\acute{e}t} }}$ such that, for an étale sheaf ${{\mathcal{F}}}$, we have the quasi-isomorphism $\nu^: {{\mathcal{F}}}\stackrel{\sim}{\to}{\mathrm {R} }\nu_\nu^{{\mathcal{F}}}$ [@Sch 3.17]. More generally, for a topological $G_K$-module $M$, we get a sheaf $\nu M$ on ${\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }}$ by setting, for a profinite $G_K$-set $S$, $\nu M(S)={\operatorname{Hom} }_{{\operatorname{cont} },G_K}(S,M)$, and Scholze shows that there is a canonical quasi-isomorphism $H^({\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }},\nu M)\simeq H^_{{\operatorname{cont} }}(G_K,M)$ [@Sch 3.7]. In this proof we will need this kind of quasi-isomorphisms for complexes $M$ as well and this will require extra arguments. For that observe that the functor $\nu$ is left exact. To study right exactness, we can look at the stalks at points $x_S$ corresponding to profinite sets $S$ with a free $G_K$-action [@Sch 3.8] and write $S=S/G_K\times G_K$. Then, for any $G_K$-module $T$, we have $(\nu T)_{x_S}={\operatorname{Hom} }_{{\operatorname{cont} }}(S/G_K,T)$. It follows that, for a surjective map $T_1\to T_2$ of $G_K$-modules, the pullback map $\nu T_1\to \nu T_2$ is also surjective if the original map had a continuous set-theoretical section. This is a criterium familiar from continuous cohomology and we will use it often. We will see the complex $X$ as a complex of sheaves on the site ${\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }}$ in the following way: represent ${\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})$ and ${\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} }})$ by (filtered) perfect complexes of $K_0^{{\operatorname{nr} }}$- and ${\overline{K} }$-modules, respectively, think of $X$ as $\nu X$, and work on the pro-étale site. This makes sense, i.e., functor $\nu$ transfers (filtered) quasi-isomorphisms of representatives of ${\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})$ and ${\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} }})$ to quasi-isomorphisms of the corresponding sheaves $\nu X$. To see this look at the Postnikov system of sheaves on ${\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }}$ obtained by pulling back by $\nu$ the above Postnikov system. Now, look at the stalks at points $x_S$ as above and note that we have $(\nu Y^0)_{x_S}={\operatorname{Hom} }_{{\operatorname{cont} }}(S/G_K,Y^0)$. Conclude that, by perfectness of the Beilinson-Hyodo-Kato complexes, quasi-isomorphisms of representatives of ${\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})$ yield quasi-isomorphisms of the sheaves $\nu Y^0$. By a similar argument, we get the analogous statement for $Y^2$. For $Y^1$, we just have to show that filtered quasi-isomorphisms of representatives of ${\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} }})$ yield quasi-isomorphisms of the sheaves $\nu (({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} }})\otimes_{{\overline{K} }}B_{{\mathrm{dR}}})/F^r)$. Again, we look at stalks at the points $x_S$. By compactness of $S/G_K$ we may replace $({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} }})\otimes_{{\overline{K} }}B_{{\mathrm{dR}}})/F^r$ by $(t^{-i}{\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} }})\otimes_{{\overline{K} }}B^+_{{\mathrm{dR}}})/F^r$, for some $i\geq 0$, where, using devissage, we can again argue by (filtered) perfection of ${\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{{\overline{K} }})$. Observe that the same argument shows that ${{\mathcal{H}}}^j(\nu Y^i)\simeq \nu H^j(Y^i)$, for $i=0,1,2$. The above Postnikov system gives rise to an exact couple $$D_1^{i,j}={{\mathcal{H}}}^j(X^i),\quad E_1^{i,j}={{\mathcal{H}}}^j(Y^i) \Longrightarrow {{\mathcal{H}}}^{i+j}(X)$$ This is the Bousfield-Kan spectral sequence associated to $X$. Consider now the complex $X_{{\mathrm{pst}}}:=C_{{\mathrm{pst}}}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\})$. We claim that the canonical map $$\begin{aligned} C_{{\mathrm{pst}}}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{\overline{K},h})\{r\}) \stackrel{\sim}{\to} C({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{\overline{K},h})\{r\})^{G_K} \end{aligned}$$ is a quasi-isomorphism (recall that taking $G_K$-fixed points corresponds to taking global sections on the pro-étale site). In particular, that the term on the right hand side makes sense. To see this, it suffices to show that the canonical maps $$\begin{aligned} ({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{\overline{K},h})/F^r)^{G_K} & \stackrel{\sim}{\to} (({\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{\overline{K},h})\otimes_{{\overline{K} }}B_{{\mathrm{dR}}})/F^r)^{G_K},\\ {\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{\overline{K},h})^{G_K} & \stackrel{\sim}{\to} ({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{\overline{K},h})\otimes_{K_0^{{\operatorname{nr} }}}B_{{\operatorname{st} }})^{G_K}\end{aligned}$$ are quasi-isomorphisms and to use the fact that the action of $G_K$ on ${\mathrm {R} }\Gamma_{{\mathrm{HK}}}^B(X_{{\overline{K} },h})$ is smooth. The fact that the first map is a quasi-isomorphism follows from the filtered quasi-isomorphism ${\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X)\otimes_K{\overline{K} }\stackrel{\sim}{\to}{\mathrm {R} }\Gamma_{{\mathrm{dR}}}(X_{\overline{K},h})$ and the fact that $B_{{\mathrm{dR}}}^{G_K}=K$. Similarly, the second map is a quasi-isomorphism because, by [@F1 4.2.4], ${\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{\overline{K},h})$ is the subcomplex of those elements of ${\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{\overline{K},h})\otimes_{K^{{\operatorname{nr} }}_0}B_{{\operatorname{st} }}$ whose stabilizers in $G_K$ are open. Taking the $G_K$-fixed points of the above Postnikov system we get an exact couple $${}^{{\mathrm{pst}}} D_1^{i,j}=H^j(X^i_{{\mathrm{pst}}}),\quad {}^{{\mathrm{pst}}}E_1^{i,j}=H^j(Y^i_{{\mathrm{pst}}}) \Longrightarrow H^{i+j}(X_{{\mathrm{pst}}})$$ corresponding to the Bousfield-Kan filtration of the complex $X_{{\mathrm{pst}}}$. On the other hand, applying ${\mathrm {R} }\Gamma({\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }},\cdot)$ to the same Postnikov system we obtain an exact couple $${}^{I}D_1^{i,j}=H^j({\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }},X^i),\quad {}^{I}E_1^{i,j}=H^j({\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }},Y^i) \Longrightarrow H^{i+j}({\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }},X)$$ together with a natural map of exact couples $({}^{{\mathrm{pst}}} D_1^{i,j}, {}^{{\mathrm{pst}}}E_1^{i,j})\to ({}^{I}D_1^{i,j}, {}^{I}E_1^{i,j})$. We also have the hypercohomology exact couple $${}^{II}D_2^{i,j}=H^{i+j}({\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }},\tau_{\leq j-1}X),\quad {}^{II}E_2^{i,j}=H^i({\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }},{{\mathcal{H}}}^j(X)) \Longrightarrow H^{i+j}({\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }},X)$$ Theorem \[speccomp\] gives us a natural morphism of exact couples $({}^{I}D^{i,j}_2,{}^{I}E^{i,j}_2)\to ({}^{II}D^{i,j}_2,{}^{II}E^{i,j}_2)$ – hence a natural morphism of spectral sequences ${}^{I}E_2^{i,j}\to {}^{II}E_2^{i,j}$ compatible with the identity map on the common abutment – if our original Postnikov system satisfies the equivalent conditions (\[ass\]). We will check the condition (4), i.e., that the following long sequence is exact for all $j$ $$0\to {{\mathcal{H}}}^j(X)\to {{\mathcal{H}}}^j(Y^0)\to {{\mathcal{H}}}^j(Y^1)\to {{\mathcal{H}}}^j(Y^2)\to 0$$ For that it is enough to show that 1. ${{\mathcal{H}}}^j(\nu Y^i)\simeq \nu H^j(Y^i)$, for $i=0,1,2$; 2. ${{\mathcal{H}}}^j(\nu X)\simeq \nu H^j(X)$; 3. the following long sequence of $G_K$-modules $$0\to H^j(X)\to H^j(Y^0)\to H^j(Y^1)\to H^j(Y^2)\to 0$$ is exact; 4. the pullback $\nu$ preserves its exactness. The assertion in (1) was shown above. The sequence in (3) is equal to the top sequence in the following commutative diagram (where we set $M=H^j_{{\mathrm{HK}}}(X_{{\overline{K} },h})$, $M_{{\mathrm{dR}}}=H^j_{{\mathrm{dR}}}(X_{{\overline{K} },h})$, $E= H^j(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p)$). $$\xymatrix@C=40pt{ H^j(X)\ar[d]^{\alpha_{{\operatorname{\acute{e}t} }}^{-1}}_{\wr}\ar[r] & M\otimes_{K_0^{{\operatorname{nr} }}} B_{{\operatorname{st} }}\ar[r]^-{(N,1-{\varphi}_r,\iota)}\ar[d]^{\rho_{{\mathrm{HK}}}}_{\wr} & M\otimes_{K_0^{{\operatorname{nr} }}}( B_{{\operatorname{st} }} \oplus B_{{\operatorname{st} }})\oplus (M_{{\mathrm{dR}}}\otimes_{{\overline{K} }} B_{{\mathrm{dR}}})/F^r\ar[r]^-{(1-{\varphi}_{r-1})-N}\ar[d]^{\rho_{{\mathrm{HK}}}+\rho_{{\mathrm{HK}}}+\rho_{{\mathrm{dR}}}}_{\wr}& M\otimes_{K_0^{{\operatorname{nr} }}} B_{{\operatorname{st} }}\ar[d]^{\rho_{{\mathrm{HK}}}}_{\wr} \\ E(r)\ar@{^{(}->}[r] & E\otimes B_{{\operatorname{st} }}\ar[r]^-{(N,1-{\varphi}_r,\iota)} & E\otimes (B_{{\operatorname{st} }}\oplus B_{{\operatorname{st} }})\oplus E\otimes B_{{\mathrm{dR}}}/F^r\ar@{->>}[r]^-{(1-{\varphi}_{r-1})-N} & E\otimes B_{{\operatorname{st} }} }$$ Since the bottom sequence is just a fundamental exact sequence of $p$-adic Hodge Theory, the top sequence is exact, as wanted. To prove assertion (4), we pass to the bottom exact sequence above and apply $\nu$ to it. It is easy to see that it enough now to show that the following surjections have continuous ${{\mathbf Q}}_p$-linear sections $$\begin{aligned} B_{{\operatorname{st} }}\stackrel{N}{\to}B_{{\operatorname{st} }},\quad B_{{\operatorname{cr} }}\verylomapr{(1-{\varphi}_r,{ \operatorname{can} })}B_{{\operatorname{cr} }}\oplus B_{{\mathrm{dR}}}/F^r.\end{aligned}$$ For the monodromy, write $B_{{\operatorname{st} }}=B_{{\operatorname{cr} }}[u_s]$ and take for a continuous section the map induced by $bu_s^i\mapsto -(b/(i+1))u_s^{i+1}$, $b\in B_{{\operatorname{cr} }}$. For the second map, the existence of continuous section was proved in [@BK 1.18]. For a different argument: observe that an analogous statement was proved in [@Col Prop. II.3.1] with $B_{\max}$ in place of $B_{{\operatorname{cr} }}$ as a consequence of the general theory of $p$-adic Banach spaces. We will just modify it here. Write $A_i=t^{-i}B^+_{{\operatorname{cr} }}$ and $B_i=t^{-i}B^+_{{\operatorname{cr} }}\oplus t^{-i}B^+_{{\mathrm{dR}}}/t^r$ for $i\geq 1$. These are $p$-adic Banach spaces. Observe that $B_i\subset B_{i+1}$ is closed. Indeed, it is enough to show that $tB^+_{{\operatorname{cr} }}\subset B^+_{{\operatorname{cr} }}$ is closed. But we have $tB^+_{{\operatorname{cr} }}=\bigcap_{n\geq 0}\ker(\theta\circ {\varphi}^n)$. It follows [@Col Prop. I.1.5] that we can find a closed complement $C_{i+1}$ of $B_i$ in $B_{i+1}$. Set $f=(1-{\varphi}_r,{ \operatorname{can} }):B_{{\operatorname{cr} }}\to B_{{\operatorname{cr} }}\oplus B_{{\mathrm{dR}}}/F^r$. We know that $f$ maps $A_i$ onto $B_i$. Write $t^{-i}B^+_{{\operatorname{cr} }}\oplus t^{-i}B^+_{{\mathrm{dR}}}/t^r=B_1\oplus (\oplus_{j=2}^{i-1}C_j)$. By [@Col Prop. I.1.5], we can find a continuous section $s_1: B_1\to A_1$ of $f$ and, if $i\geq 2$, a continuous section $s_i: C_i\to A_i$ of $f$. Define the map $s: t^{-i}B^+_{{\operatorname{cr} }}\oplus t^{-i}B^+_{{\mathrm{dR}}}/t^r \to B_{{\operatorname{cr} }}$ by $s_1$ on $B_1$ and by $s_i$ on $C_i$ for $i\geq 2$. Taking inductive limit over $i$ we get our section of $f$. To prove assertion (2), take a perfect representative of the complex ${\mathrm {R} }\Gamma(X_{{\overline{K} }, {\operatorname{\acute{e}t} }},{\mathbf Z}_p(r))$. Consider the complex $Z={\mathrm {R} }\Gamma(X_{{\overline{K} }, {\operatorname{\acute{e}t} }},{{\mathbf Q}}_p(r))$ as a complex of sheaves on ${\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }}$. As before, we see that this makes sense and we easily find that (canonically) ${{\mathcal{H}}}^j(Z)\simeq \nu H^j(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p(r))$. To prove (2), it is enough to show that we can also pass with the map $\alpha_{{\operatorname{\acute{e}t} }}: {\mathrm {R} }\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p(r))\stackrel{\sim}{\to} C({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\})$ to the site ${\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }}$. Looking at its definition (cf. (\[definition1\])) we see that we need to show that the period quasi-isomorphisms $\rho_{{\operatorname{cr} }}, \rho_{{\mathrm{HK}}}, \rho_{{\mathrm{dR}}}$ as well as the quasi-isomorphism $${{\mathbf Q}}_p(r)\stackrel{\sim}{\to}[ B_{{\operatorname{st} }}\verylomapr{(N,1-{\varphi}_r,\iota)} B_{{\operatorname{st} }}\oplus B_{{\operatorname{st} }}\oplus B_{{\mathrm{dR}}}/F^r\veryverylomapr{(1-{\varphi}_{r-1})-N}B_{{\operatorname{st} }}]$$ can be lifted to the pro-étale site. The last fact we have just shown. For the crystalline period map $\rho_{{\operatorname{cr} }}$ this follows from the fact that it is defined integrally and all the relevant complexes are perfect. For the Hyodo-Kato period map $\rho_{{\mathrm{HK}}}$ - it follows from the case of $\rho_{{\operatorname{cr} }}$ and from perfection of complexes involved in the definition of the Beilinson-Hyodo-Kato map. For the de Rham period map $\rho_{{\mathrm{dR}}}$ this follows from perfection of the involved complexes as well as from the exactness of ${\operatorname{holim} }_n$ (in the definition of $\rho_{{\mathrm{dR}}}$) on the pro-étale site of $K$ (cf. [@Sch 3.18]). We define the map of spectral sequences $\delta:=(\delta_D,\delta):=({}^{{\mathrm{pst}}} D_2^{i,j}, {}^{{\mathrm{pst}}}E_2^{i,j})\to ({}^{II}D^{i,j}_2,{}^{II}E^{i,j}_2)$ – that we stated at the beginning of the proof – as the composition of the two maps constructed above $$\delta:\quad ({}^{{\mathrm{pst}}} D_2^{i,j}, {}^{{\mathrm{pst}}}E_2^{i,j})\to({}^{I}D^{i,j}_2,{}^{I}E^{i,j}_2)\to ({}^{II}D^{i,j}_2,{}^{II}E^{i,j}_2).$$ To get the spectral sequence from the theorem we need to pass from $^{II}E_2$ to the Hochschild-Serre spectral sequence. To do that consider the hypercohomology exact couple $${}^{{\operatorname{\acute{e}t} }}D_2^{i,j}=H^{i+j}({\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }},\tau_{\leq j-1}Z),\quad {}^{{\operatorname{\acute{e}t} }}E_2^{i,j}=H^i({\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }},{{\mathcal{H}}}^j(Z)) \Longrightarrow H^{i+j}({\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }},Z)$$ and, via $\alpha_{{\operatorname{\acute{e}t} }}^{-1}$, a natural morphism of exact couples $({}^{II}D^{i,j}_2,{}^{II}E^{i,j}_2)\to ({}^{{\operatorname{\acute{e}t} }}D^{i,j}_2,{}^{{\operatorname{\acute{e}t} }}E^{i,j}_2)$ – hence a natural morphism of spectral sequences ${}^{II}E_2^{i,j}\to {}^{{\operatorname{\acute{e}t} }}E_2^{i,j}$ compatible with the map $\alpha_{{\operatorname{\acute{e}t} }}^{-1}$ on the abutment. We have a quasi-isomorphism $\psi: {\mathrm {R} }\Gamma({\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }},Z)\stackrel{\sim}{\to} {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p(r))$ defined as the composition $$\begin{aligned} \psi:\quad {\mathrm {R} }\Gamma({\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }},{\mathrm {R} }\Gamma(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p(r))) & \stackrel{\sim}{\to}{{\mathbf Q}}\otimes{\operatorname{holim} }_n {\mathrm {R} }\Gamma(G_K,{\mathrm {R} }\Gamma (X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Z}/p^n(r)))\\ & ={{\mathbf Q}}\otimes {\operatorname{holim} }_n {\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Z}/p^n(r))={\mathrm {R} }\Gamma (X_{{\operatorname{\acute{e}t} }},{{\mathbf Q}}(r))\end{aligned}$$ We have obtained the following natural maps of spectral sequences $$\xymatrix{^{{ \operatorname{syn} }}E^{i,j}_2=H^i_{{\operatorname{st} }}(G_K,H^j(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)))\ar[d]_{\wr}\ar@{=>}[r] & H^{i+j}_{{ \operatorname{syn} }}(X_{h},r)\ar[d]^{\alpha_{{ \operatorname{syn} }}}_{\wr}\\ E^{i,j}_2=H^i(C_{{\operatorname{st} }}(H^j_{{\mathrm{HK}}}(X_{h})\{r\}))\ar[d]^{\alpha_{{\operatorname{\acute{e}t} }}^{-1}\delta\varepsilon^}\ar@{=>}[r] & H^{i+j}(C_{{\operatorname{st} }}({\mathrm {R} }\Gamma^B_{{\mathrm{HK}}}(X_{h})\{r\}))\ar[d]^{\psi \alpha_{{\operatorname{\acute{e}t} }}^{-1}\delta \varepsilon^}\\ ^{{\operatorname{\acute{e}t} }}E^{i,j}_2=H^i(G_K,H^j(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)))\ar@{=>}[r] & H^{i+j}(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)) }$$ It remains to show that the right vertical composition $\gamma: H^{i+j}_{{ \operatorname{syn} }}(X_{h},r)\to H^{i+j}(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r))$ is equal to the map $\rho_{{ \operatorname{syn} }}$. Since we have the equality $\alpha_{{ \operatorname{syn} }}=\rho_{{ \operatorname{syn} }}\alpha_{{\operatorname{\acute{e}t} }}$ (in the derived category) from (\[compatibility0\]) and, by Lemma \[compatibilit01\], $\varepsilon^\alpha_{{ \operatorname{syn} }}=\alpha_{{ \operatorname{syn} }}\varepsilon^$, the map $\gamma$ can be written as the composition $$\begin{aligned} \tilde{\rho}_{{ \operatorname{syn} }}:\quad H^{i+j}_{{ \operatorname{syn} }}(X_{h},r)\stackrel{\varepsilon^}{\to}H^{i+j}({\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }},\nu {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{{\overline{K} },h},r)) & \stackrel{\rho_{{ \operatorname{syn} }}}{\to} H^{i+j}({\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }},\nu{\mathrm {R} }\Gamma (X_{{\overline{K} },{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p(r)))\\ & \stackrel{\psi}{\to} H^{i+j}(X_{{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p(r)),\end{aligned}$$ where the period map $\rho_{{ \operatorname{syn} }}$ is understood to be on sheaves on ${\operatorname{Spec} }(K)_{{\operatorname{pro\acute{e}t} }}$. There is no problem with that since we care only about the induced map on cohomology groups. It is easy now to see that $\tilde{\rho}_{{ \operatorname{syn} }}=\rho_{{ \operatorname{syn} }}$, as wanted. If $X$ is proper and smooth, it is known that the étale Hochschild-Serre spectral sequence degenerates, i.e., that ${}^{{\operatorname{\acute{e}t} }}E_2={}^{{\operatorname{\acute{e}t} }}E_{\infty}$. It is very likely that so does the syntomic descent spectral sequence in this case, i.e., that ${}^{{ \operatorname{syn} }}E_2={}^{{ \operatorname{syn} }}E_{\infty}$. For $X\in {\mathcal V}ar_{K}$, we have a canonical quasi-isomorphism $$\rho_{{ \operatorname{syn} }}: \quad\tau_{\leq r}{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_{h},r)_{\mathbf Q}\stackrel{\sim}{\to} \tau_{\leq r}{\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r)).$$ By Theorem \[stHS\], the syntomic descent and the Hochschild-Serre spectral sequence are compatible. We have $D_j=H^j_{{\mathrm{HK}}}(X_{{\overline{K} },h})\{r\}\in MF_K^{{\operatorname{ad} }}({\varphi}, N,G_K)$. For $j\leq r$, $F^{1}D_{j,K}=F^{1-(r-j)}H^j_{{\mathrm{dR}}}(X_{h})=0$. Hence, by Proposition \[resolution33\], we have $^{{ \operatorname{syn} }}E^{i,j}_2\stackrel{\sim}{\to} {}^{{\operatorname{\acute{e}t} }}E^{i,j}_2$. This implies that $\rho_{{ \operatorname{syn} }}: \tau_{\leq r}{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)\stackrel{\sim}{\to} \tau_{\leq r}{\mathrm {R} }\Gamma(X_{{\operatorname{\acute{e}t} }},{\mathbf Q}_p(r))$, as wanted. All of the above automatically extends to finite diagrams of $K$-varieties, hence to essentially finite diagrams of $K$-varieties (i.e., the diagrams for which every truncation of their cohomology $\tau_{\leq n}$ is computed by truncating the cohomology of some finite diagram). This includes, in particular, simplicial and cubical varieties. Syntomic regulators =================== In this section we prove that Soulé’s étale regulators land in the semistable Selmer groups. This will be done by constucting syntomic regulators that are compatible with the étale ones via the period map and by exploting the syntomic descent spectral sequence. Construction of syntomic Chern classes -------------------------------------- We start with the construction of syntomic Chern classes. This will be standard once we prove that syntomic cohomology satisfies projective space theorem and homotopy property. In this subsection we will work in the (classical) derived category. For a fine log-scheme $(X,M)$, log-smooth over $V^{\times}$, we have the log-crystalline and log-syntomic first Chern class maps of complexes of sheaves on $X_{{\operatorname{\acute{e}t} }}$ [@Ts 2.2.3] $$\begin{aligned} c_1^{{\operatorname{cr} }}: j_{{\mathcal O}}^_{X_{{ \operatorname{tr} }}}\stackrel{\sim}{\to} M^{{\operatorname{gp} }}\to M^{{\operatorname{gp} }}_n\to R\varepsilon_{{\mathcal J}}^{[1]}_{X_n/W_n(k)} [1], & \quad c_1^{{\operatorname{st} }}: j_{{\mathcal O}}^_{X_{{ \operatorname{tr} }}}\stackrel{\sim}{\to} M^{{\operatorname{gp} }}\to M^{{\operatorname{gp} }}_n\to R\varepsilon_{{\mathcal J}}^{[1]}_{X_n/R_n}[1],\\ c_1^{{\mathrm{HK}}}: j_{{\mathcal O}}^_{X_{{ \operatorname{tr} }}}\stackrel{\sim}{\to} M^{{\operatorname{gp} }}\to M^{{\operatorname{gp} }}_0\to R\varepsilon_{{\mathcal J}}^{[1]}_{X_0/W_n(k)^0}[1], & \quad c_1^{{ \operatorname{syn} }}: j_{{\mathcal O}}^_{X_{{ \operatorname{tr} }}}\stackrel{\sim}{\to} M^{{\operatorname{gp} }}\to {{\mathcal{S}}}(1)_{X,{{\mathbf Q}}} [1].\end{aligned}$$ Here $\varepsilon$ is the projection from the corresponding crystalline site to the étale site. The maps $c_1^{{\operatorname{cr} }}, c_1^{{\operatorname{st} }},$ and $c_1^{{ \operatorname{syn} }}$ are clearly compatible. So are the maps $c_1^{{\operatorname{st} }}$ and $ c_1^{{\mathrm{HK}}}$. For ss-pairs $(U,\overline{U})$ over $K$, we get the induced functorial maps $$\begin{aligned} c_1^{{\operatorname{cr} }}:\Gamma(U,{{\mathcal O}}_U^) \stackrel{\sim}{\leftarrow}\Gamma(\overline{U},j_{{\mathcal O}}_U^)\to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U}, {{\mathcal J}}^{[1]})[1], & \quad c_1^{{\operatorname{st} }}:\Gamma(U,{{\mathcal O}}_U^) \to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R, {{\mathcal J}}^{[1]})[1],\\ c_1^{{\mathrm{HK}}}:\Gamma(U,{{\mathcal O}}_U^) \to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})_0/W_n(k)^0,{{\mathcal J}}^{[1]})[1], & \quad c_1^{{ \operatorname{syn} }}: \Gamma(U,{{\mathcal O}}_U^) \to {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},1)_{{{\mathbf Q}}}[1].\end{aligned}$$ For $X\in {\mathcal V}ar_K$ we can glue the absolute log-crystalline and log-syntomic classes to obtain the absolute crystalline and syntomic first Chern class maps $$c_1^{{\operatorname{cr} }}: {{\mathcal O}}^_{X_h}\to {{\mathcal J}}_{{\operatorname{cr} },X}[1],\quad c_1^{{ \operatorname{syn} }}: {{\mathcal O}}^_{X_h}\to {{\mathcal{S}}}(1)_{X,{{\mathbf Q}}}[1].$$ They induce (compatible) maps $$\begin{aligned} c_1^{{\operatorname{cr} }}: {\operatorname{Pic} }(X) & =H^1(X_{{\operatorname{\acute{e}t} }},{{\mathcal O}}^_X)\to H^1(X_{h},{{\mathcal O}}^_X)\stackrel{c_1^{{\operatorname{cr} }}}{\to }H^2(X_h,{{\mathcal J}}_{{\operatorname{cr} }}),\\ c_1^{{ \operatorname{syn} }}: {\operatorname{Pic} }(X) & =H^1(X_{{\operatorname{\acute{e}t} }},{{\mathcal O}}^_X)\to H^1(X_{h},{{\mathcal O}}^_X)\stackrel{c_1^{{ \operatorname{syn} }}}{\to }H^2_{{ \operatorname{syn} }}(X_h,1).\end{aligned}$$ Recall that, for a log-scheme $(X,M)$ as above, we also have the log de Rham first Chern class map $$c_1^{{\mathrm{dR}}}: j_{{\mathcal O}}^_{X_{{ \operatorname{tr} }}}\stackrel{\sim}{\to} M^{{\operatorname{gp} }}\to M^{{\operatorname{gp} }}_n\stackrel{{\operatorname{dlog} }}{\to} \Omega^{\scriptscriptstyle\bullet}_{(X,M)_n/V^{\times}_n} [1].$$ For ss-pairs $(U,\overline{U})$ over $K$, it induces maps $$c_1^{{\mathrm{dR}}}:\Gamma(U,{{\mathcal O}}_U^) \stackrel{\sim}{\leftarrow}\Gamma(\overline{U},j_{{\mathcal O}}_U^)\to {\mathrm {R} }\Gamma(\overline{U},\Omega^{\scriptscriptstyle\bullet}_{(U,\overline{U})/V^{\times}} )[1].\\$$ By the map ${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},{{\mathcal J}}^{[1]})\to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})\to {\mathrm {R} }\Gamma(\overline{U},\Omega^{\scriptscriptstyle\bullet}_{(U,\overline{U})/V^{\times}} )$ they are compatible with the absolute log-crystalline and log-syntomic classes [@Ts 2.2.3]. \[compatibility\] For strict ss-pairs $(U,\overline{U})$ over $K$, the Hyodo-Kato map and the Hyodo-Kato isomorphism $$\iota: H^2_{{\mathrm{HK}}}(U,\overline{U})_{\mathbf Q}\to H^2_{{\operatorname{cr} }}((U,\overline{U})/R)_{\mathbf Q},\quad \iota_{{\mathrm{dR}},\pi}: H^2_{{\mathrm{HK}}}(U,\overline{U})_{\mathbf Q}\otimes_{K_0}K\stackrel{\sim}{\to} H^2(\overline{U}_{K},\Omega^{\scriptscriptstyle\bullet}_{(U,\overline{U}_{K})/K})$$ are compatible with first Chern class maps. Since $\iota_{{\mathrm{dR}},\pi}=i^_{\pi}\iota\otimes{ \operatorname{Id} }$ and the map $i^_{\pi}$ is compatible with first Chern classes, it suffices to show the compatibility for the Hyodo-Kato map $\iota$. Let ${{\mathcal{L}}}$ be a line bundle on $U$. Since the map $\iota$ is a section of the map $i^_0:H^2_{{\operatorname{cr} }}((U,\overline{U})/R)_{\mathbf Q}\to H^2_{{\mathrm{HK}}}(U,\overline{U})_{\mathbf Q}$ and the map $i^_0$ is compatible with first Chern classes, we have that the element $\zeta\in H^2_{{\operatorname{cr} }}((U,\overline{U})/R)_{\mathbf Q}$ defined as $\zeta=\iota(c_1^{{\mathrm{HK}}}({{\mathcal{L}}}))-c_1^{{\operatorname{st} }}({{\mathcal{L}}})$ lies in $ TH^2_{{\operatorname{cr} }}((U,\overline{U})/R)_{\mathbf Q}$. Hence $\zeta=T\gamma$. Since the map $\iota$ is compatible with Frobenius and ${\varphi}(c_1^{{\mathrm{HK}}}({{\mathcal{L}}}))=pc_1^{{\mathrm{HK}}}({{\mathcal{L}}})$, ${\varphi}(c_1^{{\operatorname{st} }}({{\mathcal{L}}}))=pc_1^{{\operatorname{st} }}({{\mathcal{L}}})$, we have ${\varphi}(\zeta)=p\zeta$. Since ${\varphi}(T\gamma)=T^p{\varphi}(\gamma)$ this implies that $\gamma\in\bigcap_{n=1}^{\infty}T^nH^2_{{\operatorname{cr} }}((U,\overline{U})/R)_{\mathbf Q}$, which is not possible unless $\gamma$ (and hence $\zeta$) are zero. But this is what we wanted to show. We have the following projective space theorem for syntomic cohomology. \[projective\] Let ${{\mathcal{E}}}$ be a locally free sheaf of rank $d+1$, $d\geq 0$, on a scheme $X\in {\mathcal V}ar_{K}$. Consider the associated projective bundle $\pi:{\mathbb P}({{\mathcal{E}}})\to X$. Then we have the following quasi-isomorphism of complexes of sheaves on $X_h$ $$\begin{aligned} \bigoplus_{i=0}^d{c}_1^{{ \operatorname{syn} }}({{\mathcal O}}(1))^i\cup\pi^: \quad \bigoplus_{i=0}^d {{\mathcal{S}}}(r-i)_{X,{\mathbf Q}}[-2i] \stackrel{\sim}{\to} R\pi_ {{\mathcal{S}}}(r)_{{\mathbb P}({{\mathcal{E}}}),{\mathbf Q}}, \quad 0\leq d \leq r.\end{aligned}$$ Here, the class ${c}_1^{{ \operatorname{syn} }}({{\mathcal O}}(1))\in H^2_{{ \operatorname{syn} }}({\mathbb P}({{\mathcal{E}}})_h, 1)$ refers to the class of the tautological bundle on ${\mathbb P}({{\mathcal{E}}})$. By (tedious) checking of many compatibilities we will reduce the above projective space theorem to the projective space theorems for the Hyodo-Kato and the filtered de Rham cohomologies. To prove our proposition it suffices to show that for any ss-pair $(U,\overline{U})$ over $K$ and the projective space $\pi: {\mathbb P}^d_{\overline{U}}\to\overline{U} $ of dimension $d$ over $\overline{U}$ we have a projective space theorem for syntomic cohomology ($a\geq 0$) $$\bigoplus_{i=0}^d{c}_1^{{ \operatorname{syn} }}({{\mathcal O}}(1))^i\cup\pi^: \quad \bigoplus_{i=0}^d H^{a-2i}_{{ \operatorname{syn} }}(U_h,r-i)\stackrel{\sim}{\to} H^a_{{ \operatorname{syn} }}({\mathbb P}^d_{U,h},r), \quad 0\leq d \leq r.$$ By Proposition \[hypercov\] and the compatibility of the maps $ H^{}_{{ \operatorname{syn} }}(U,\overline{U},j)_{{\mathbf Q}}\stackrel{\sim}{\to} H^{}_{{ \operatorname{syn} }}(U_h,j)_{{\mathbf Q}}$ with products and first Chern classes, this reduces to proving a projective space theorem for log-syntomic cohomology, i.e., a quasi-isomorphism of complexes $$\begin{aligned} \bigoplus_{i=0}^d{c}_1^{{ \operatorname{syn} }}({{\mathcal O}}(1))^i\cup\pi^: \quad \bigoplus_{i=0}^d H^{a-2i}_{{ \operatorname{syn} }}(U,\overline{U},r-i)_{{\mathbf Q}}\stackrel{\sim}{\to} H^a_{{ \operatorname{syn} }}({\mathbb P}^d_{U},{\mathbb P}^d_{\overline{U}},r)_{\mathbf Q}, \quad 0\leq d \leq r,\end{aligned}$$ where the class ${c}_1^{{ \operatorname{syn} }}({{\mathcal O}}(1))\in H^2_{{ \operatorname{syn} }}({\mathbb P}^d_{U},{\mathbb P}^d_{\overline{U}}, 1)$ refers to the class of the tautological bundle on ${\mathbb P}^d_{\overline{U}}$. By the distinguished triangle $${\mathrm R}\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{{\mathbf Q}} \to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U},r)_{{\mathbf Q}}\stackrel{}{\to} {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_{K})/F^r$$ and its compatibility with the action of $c_1^{{ \operatorname{syn} }}$, it suffices to prove the following two quasi-isomorphisms for the twisted absolute log-crystalline complexes and for the filtered log de Rham complexes ($0\leq d \leq r$) $$\begin{aligned} \bigoplus_{i=0}^d{c}_1^{{\operatorname{cr} }}({{\mathcal O}}(1))^i\cup\pi^: &\quad \bigoplus_{i=0}^d H^{a-2i}_{{\operatorname{cr} }}(U,\overline{U},r-i)_{{\mathbf Q}} \stackrel{\sim}{\to} H^a_{{\operatorname{cr} }}({\mathbb P}^d_{U},{\mathbb P}^d_{\overline{U}},r)_{\mathbf Q}, \\ \bigoplus_{i=0}^d{c}_1^{{\mathrm{dR}}}({{\mathcal O}}(1))^i\cup\pi^: & \quad \bigoplus_{i=0}^d F^{r-i}H^{a-2i}_{{\mathrm{dR}}}(U,\overline{U}_K) \stackrel{\sim}{\to} F^{r}H^a_{{\mathrm{dR}}}({\mathbb P}^d_{U},{\mathbb P}^d_{\overline{U}_K}).\end{aligned}$$ For the log de Rham cohomology, notice that the above map is quasi-isomorphic to the map [@BE1 3.2] $$\bigoplus_{i=0}^d{c}_1^{{\mathrm{dR}}}({{\mathcal O}}(1))^i\cup\pi^: \quad \bigoplus_{i=0}^d F^{r-i}H^{a-2i}_{{\mathrm{dR}}}(U) \stackrel{\sim}{\to} F^{r}H^a_{{\mathrm{dR}}}({\mathbb P}^d_{U}).$$ Hence well-known to be a quasi-isomorphism. For the twisted log-crystalline cohomology, notice that since Frobenius behaves well with respect to ${c}_1^{{\operatorname{cr} }}$, it suffices to prove a projective space theorem for the absolute log-crystalline cohomology $H^{}_{{\operatorname{cr} }}(U,\overline{U})_{{\mathbf Q}} $. $$\bigoplus_{i=0}^d{c}_1^{{\operatorname{cr} }}({{\mathcal O}}(1))^i\cup\pi^: \quad \bigoplus_{i=0}^d H^{a-2i}_{{\operatorname{cr} }}(U,\overline{U})_{{\mathbf Q}} \stackrel{\sim}{\to} H^a_{{\operatorname{cr} }}({\mathbb P}^d_{U},{\mathbb P}^d_{\overline{U}})_{\mathbf Q}$$ Without loss of generality we may assume that the pair $(U,\overline{U})$ is split over $K$. By the distinguished triangle $${\mathrm {R} }\Gamma_{{\operatorname{cr} }}(U,\overline{U})\to {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R)\stackrel{N}{\to} {\mathrm {R} }\Gamma_{{\operatorname{cr} }}((U,\overline{U})/R))$$ and its compatibility with the action of $c^{{\operatorname{cr} }}_1({{\mathcal O}}(1))$ (cf. [@Ts Lemma 4.3.7]), it suffices to prove a projective space theorem for the log-crystalline cohomology $H^{}_{{\operatorname{cr} }}((U,\overline{U})/R)_{{\mathbf Q}} $. Since the $R$-linear isomorphism $\iota: H^{}_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}} \otimes R_{\mathbf Q}\stackrel{\sim}{\to}H^{}_{{\operatorname{cr} }}((U,\overline{U})/R)_{{\mathbf Q}} $ is compatible with products [@Ts Prop. 4.4.9] and first Chern classes (cf. Lemma \[compatibility\]) we reduce the problem to showing the projective space theorem for the Hyodo-Kato cohomology. $$\bigoplus_{i=0}^d{c}_1^{{\mathrm{HK}}}({{\mathcal O}}(1))^i\cup\pi^: \quad \bigoplus_{i=0}^d H^{a-2i}_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}} \stackrel{\sim}{\to} H^a_{{\mathrm{HK}}}({\mathbb P}^d_{U},{\mathbb P}^d_{\overline{U}})_{\mathbf Q}$$ Tensoring by $K$ and using the isomorphism $ \iota_{{\mathrm{dR}},\pi}: H^{}_{{\mathrm{HK}}}(U,\overline{U})_{{\mathbf Q}} \otimes_{K_0} K\stackrel{\sim}{\to}H^{}_{{\mathrm{dR}}}(U,\overline{U}_K)$ that is compatible with products [@Ts Cor. 4.4.13] and first Chern classes (cf. Lemma \[compatibility\]) we reduce to checking the projective space theorem for the log de Rham cohomology $ H^{}_{{\mathrm{dR}}}(U,\overline{U}_K) $. And we have done this above. The above proof proves also the projective space theorem for the absolute crystalline cohomology. \[projective1\] Let ${{\mathcal{E}}}$ be a locally free sheaf of rank $d+1$, $d\geq 0$, on a scheme $X\in {\mathcal V}ar_{K}$. Consider the associated projective bundle $\pi:{\mathbb P}({{\mathcal{E}}})\to X$. Then we have the following quasi-isomorphism of complexes of sheaves on $X_h$ $$\begin{aligned} \bigoplus_{i=0}^d{c}_1^{{\operatorname{cr} }}({{\mathcal O}}(1))^i\cup\pi^: \quad \bigoplus_{i=0}^d {{\mathcal J}}^{[r-i]}_{X,{\mathbf Q}}[-2i] \stackrel{\sim}{\to} R\pi_ {{\mathcal J}}^{[r]}_{{\mathbb P}({{\mathcal{E}}}),{\mathbf Q}}, \quad 0\leq d \leq r.\end{aligned}$$ Here, the class ${c}_1^{{\operatorname{cr} }}({{\mathcal O}}(1))\in H^2({\mathbb P}({{\mathcal{E}}})_h, {{\mathcal J}}_{{\operatorname{cr} }})$ refers to the class of the tautological bundle on ${\mathbb P}({{\mathcal{E}}})$. For $X\in {\mathcal V}ar_K$, using the projective space theorem (cf. Theorem \[projective\]) and the Chern classes $$c_0^{{ \operatorname{syn} }}: {{\mathbf Q}}_p\stackrel{{ \operatorname{can} }}{\to} {{\mathcal{S}}}(0)_{X_{{{\mathbf Q}}}},\quad c_1^{{ \operatorname{syn} }}: {{\mathcal O}}_{X_h}^\to {{\mathcal{S}}}(1)_{X_{{{\mathbf Q}}}}[1],$$ we obtain syntomic Chern classes $c_i^{{ \operatorname{syn} }}({{\mathcal{E}}})$, for any locally free sheaf ${{\mathcal{E}}}$ on $X$. Syntomic cohomology has homotopy invariance property. \[homotopy\] Let $X\in {\mathcal V}ar_K$ and $f: {\mathbb A}^1_X \to X$ be the natural projection from the affine line over $X$ to $X$. Then, for all $r\geq 0$, the pullback map $$f^:\,{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X_h,r)\lomapr{\sim}{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}({\mathbb A}^1_{X,h},r)$$ is a quasi-isomorphism. Localizing in the $h$-topology of $X$ we may assume that $X=U$ - the open set of an ss-pair $(U,\overline{U})$ over $K$. Consider the following commutative diagram. $$\begin{CD} {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U,\overline{U},r)_{\mathbf Q}@> f^>>{\mathrm {R} }\Gamma_{{ \operatorname{syn} }}({\mathbb A}^1_{U},{\mathbb P}^1_{\overline{U}},r)_{\mathbf Q}\\ @VV \wr V @VV\wr V\\ {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(U_h,r) @> f^>> {\mathrm {R} }\Gamma_{{ \operatorname{syn} }}({\mathbb A}^1_{U,h},r) \end{CD}$$ The vertical maps are quasi-isomorphisms by Proposition \[hypercov\]. It suffices thus to show that the top horizontal map is a quasi-isomorphism. By Proposition \[reduction1\], this reduces to showing that the map $$C_{{\operatorname{st} }}({\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{\mathbf Q}\{r\})\stackrel{ f^}{\to}C_{{\operatorname{st} }}({\mathrm {R} }\Gamma_{{\mathrm{HK}}}({\mathbb A}^1_{U},{\mathbb P}^1_{\overline{U}})_{\mathbf Q}\{r\})$$ is a quasi-isomorphism. Or, that the map $f: ({\mathbb A}^1_{U},{\mathbb P}^1_{\overline{U}})\to (U,\overline{U})$ induces a quasi-isomorphism on the Hyodo-Kato cohomology and a filtered quasi-isomorphism on the log de Rham cohomology: $${\mathrm {R} }\Gamma_{{\mathrm{HK}}}(U,\overline{U})_{\mathbf Q}\stackrel{ f^}{\to}{\mathrm {R} }\Gamma_{{\mathrm{HK}}}({\mathbb A}^1_{U},{\mathbb P}^1_{\overline{U}})_{\mathbf Q},\quad {\mathrm {R} }\Gamma_{{\mathrm{dR}}}(U,\overline{U}_K)\stackrel{ f^}{\to}{\mathrm {R} }\Gamma_{{\mathrm{dR}}}({\mathbb A}^1_{U},{\mathbb P}^1_{\overline{U}_K})$$ Without loss of generality we may assume that the pair $(U,\overline{U})$ is split over $K$. Tensoring with $K$ and using the Hyodo-Kato quasi-isomorphism we reduce the Hyodo-Kato case to the log de Rham one. The latter follows easily from the projective space theorem and the existence of the Gysin sequence in log de Rham cohomology. The above implies that syntomic cohomology is a Bloch-Ogus theory. A proof of this fact was kindly communicated to us by Frédéric Déglise and is contained in Appendix B, Proposition \[Bloch-Ogus\]. For a scheme $X$, let $K_(X)$ denote Quillen’s higher $K$-theory groups of $X$. For $X\in {\mathcal V}ar_K$, $i,j\geq 0$, there are functorial syntomic Chern class maps $$c_{i,j}^{{ \operatorname{syn} }}: K_j(X) \rightarrow H^{2i-j}_{{ \operatorname{syn} }}(X_h,i).$$ Recall the construction of the classes $c_{i,j}^{{ \operatorname{syn} }}$. First, one constructs universal classes $C^{{ \operatorname{syn} }}_{i,l}\in H^{2i}_{{ \operatorname{syn} }}(B_{\scriptscriptstyle\bullet}GL_{l,h},i)$. By a standard argument, projective space theorem and homotopy property show that $$H^_{{ \operatorname{syn} }}(B_{\scriptscriptstyle\bullet}GL_{l,h},)\simeq H^_{{ \operatorname{syn} }}(K,)[x^{{ \operatorname{syn} }}_1,\ldots,x^{{ \operatorname{syn} }}_l],$$ where the classes $x^{{ \operatorname{syn} }}_i\in H^{2i}_{{ \operatorname{syn} }}(B_{\scriptscriptstyle\bullet}GL_{l,h},i)$ are the syntomic Chern classes of the universal locally free sheaf on $B_{\scriptscriptstyle\bullet}GL_{l}$ (defined via a projective space theorem). For $l\geq i$, we define $$C^{{ \operatorname{syn} }}_{i,l}=x^{{ \operatorname{syn} }}_i\in H^{2i}_{{ \operatorname{syn} }}(B_{\scriptscriptstyle\bullet}GL_{l,h},i).$$ The classes $C^{{ \operatorname{syn} }}_{i,l}\in H^{2i}_{{ \operatorname{syn} }}(B_{\scriptscriptstyle\bullet}GL_{l,h},i)$ yield compatible universal classes (see [@Gi p. 221]) $C^{{ \operatorname{syn} }}_{i,l}\in H^{2i}_{{ \operatorname{syn} }}(X,GL_l({{\mathcal O}}_X),i)$, hence a natural map of pointed simplicial sheaves on $X_{ZAR}$, $C^{{ \operatorname{syn} }}_i:B_{\scriptscriptstyle\bullet}GL({{\mathcal O}}_X)\to {{\mathcal{K}}}(2i,{{\mathcal{S}}}^{\prime}(i){}_{X})$, where ${{\mathcal{K}}}$ is the Dold–Puppe functor of $\tau_{\geq 0}{{\mathcal{S}}}^{\prime}(i){}_{X}[2i]$ and ${{\mathcal{S}}}^{\prime}(i){}_{X}$ is an injective resolution of ${{\mathcal{S}}}(i){}_{X}:=R\varepsilon_{{\mathcal{S}}}(i)_{{{\mathbf Q}}}$, $\varepsilon: X_h\to X_{{\operatorname{Zar} }}$. The characteristic classes $c^{{ \operatorname{syn} }}_{i,j}$ are now defined [@Gi 2.22] as the composition $$\begin{aligned} K_j(X) & \to H^{-j}(X,{\mathbf Z}\times B_{\scriptscriptstyle\bullet}GL({{\mathcal O}}_X)^+) \to H^{-j}(X,B_{\scriptscriptstyle\bullet}GL({{\mathcal O}}_X)^+)\\ & \stackrel{C^{{ \operatorname{syn} }}_i}{\longrightarrow} H^{-j}(X, {{\mathcal{K}}}(2i,{{\mathcal{S}}}^{\prime}(i){}_{X})) \stackrel{h_j}{\rightarrow} H^{2i-j}_{{ \operatorname{syn} }}(X_h,i),\end{aligned}$$ where $B_{\scriptscriptstyle\bullet}GL({{\mathcal O}}_X)^+$ is the (pointed) simplicial sheaf on $X$ associated to the $+\,$- construction [@S 4.2]. Here, for a (pointed) simplicial sheaf ${{\mathcal{E}}}_{\scriptscriptstyle\bullet} $ on $X_{ZAR}$, $H^{-j}(X,{{\mathcal{E}}}_{\scriptscriptstyle\bullet} ) =\pi_j({\mathrm {R} }\Gamma (X_{ZAR},{{\mathcal{E}}}_{\scriptscriptstyle\bullet} ))$ is the generalized sheaf cohomology of ${{\mathcal{E}}}_{\scriptscriptstyle\bullet} $ [@Gi 1.7]. The map $h_j$ is the Hurewicz map: $$\begin{aligned} H^{-j}(X, {{\mathcal{K}}}(2i,{{\mathcal{S}}}^{\prime}(i){}_{X} )) & = \pi_j({{\mathcal{K}}}(2i,{{\mathcal{S}}}^{\prime}(i)(X))) \!\stackrel{h_j}{\rightarrow}\! H_j({{\mathcal{K}}}(2i,{{\mathcal{S}}}^{\prime}(i)(X)))\\ & = H_j({{\mathcal{S}}}^{\prime}(i)(X)[2i])= H^{2i-j}_{{ \operatorname{syn} }}(X_h,i).\end{aligned}$$ \[syn-et\] The syntomic and the étale Chern classes are compatible, i.e., for $X\in {\mathcal V}ar_K$, $j\geq 0, 2i-j\geq 0$, the following diagram commutes $$\xymatrix{ & K_j(X)\ar[ld]_{c^{{ \operatorname{syn} }}_{i,j}}\ar[rd]^{c^{{\operatorname{\acute{e}t} }}_{i,j}} &\\ H^{2i-j}_{{ \operatorname{syn} }}(X_h,i)\ar[rr]^{\rho_{{ \operatorname{syn} }}} & & H^{2i-j}_{{\operatorname{\acute{e}t} }}(X,{{\mathbf Q}}_p(i))}$$ We can pass to the universal case ($X=B_{\scriptscriptstyle\bullet}GL_l:=B_{\scriptscriptstyle\bullet}GL_l/K$, $l\geq 1$). We have $$\begin{aligned} H^_{{ \operatorname{syn} }}(B_{\scriptscriptstyle\bullet}GL_{l,h},) & \simeq H^_{{ \operatorname{syn} }}(K,)[x^{{ \operatorname{syn} }}_1,\ldots,x^{{ \operatorname{syn} }}_l],\\ H^_{{\operatorname{\acute{e}t} }}(B_{\scriptscriptstyle\bullet}GL_l,) & \simeq H^_{{\operatorname{\acute{e}t} }}(K,)[x^{{\operatorname{\acute{e}t} }}_1,\ldots,x^{{\operatorname{\acute{e}t} }}_l]\end{aligned}$$ By the projective space theorem and the fact that the syntomic period map commutes with products it suffices to check that $\rho_{{ \operatorname{syn} }}(x_1^{{ \operatorname{syn} }})=x_1^{{\operatorname{\acute{e}t} }}$ and that the syntomic period map $\rho_{{ \operatorname{syn} }}$ commutes with the classes $c_0^{{ \operatorname{syn} }}: {{\mathbf Q}}_p\to {{\mathcal{S}}}(0)_{{\mathbf Q}}$ and $c_0^{{\operatorname{\acute{e}t} }}: {{\mathbf Q}}_p\to {{\mathbf Q}}_p(0)$. The statement about $c_0$ is clear from the definition of $\rho_{{\operatorname{cr} }}$; for $c_1$ consider the canonical map $f: B_{\scriptscriptstyle\bullet}GL_l\to B_{\scriptscriptstyle\bullet}GL_{l,{\overline{K} }}$ and the induced pullback map $$f^_{{\operatorname{\acute{e}t} }}:\quad H^_{{\operatorname{\acute{e}t} }}(B_{\scriptscriptstyle\bullet}GL_l,)=H^_{{\operatorname{\acute{e}t} }}(K,)[x_1,\ldots,x_l]\to H^_{{\operatorname{\acute{e}t} }}(B_{\scriptscriptstyle\bullet}GL_{l,{\overline{K} }},)={{\mathbf Q}}_p[\overline{x}_1,\ldots,{\overline}{x}_l]$$ that sends the Chern classes $x^{{\operatorname{\acute{e}t} }}_i$ of the universal vector bundle to the classes $\overline{x}^{{\operatorname{\acute{e}t} }}_i$ of its pullback. It suffices to show that $f^_{{\operatorname{\acute{e}t} }}\rho_{{ \operatorname{syn} }}(C_{1,1}^{{ \operatorname{syn} }})=C_{1,1}^{{\operatorname{\acute{e}t} }}$. But, by definition, $f^_{{\operatorname{\acute{e}t} }}\rho_{{ \operatorname{syn} }}=\rho_{{ \operatorname{syn} }}f^_{{ \operatorname{syn} }}$ and, by construction, we have the following commutative diagram $$\xymatrix{ H^2_{{ \operatorname{syn} }}(B_{\scriptscriptstyle\bullet}{\mathbb G}_{m,h},1)\ar[r]^{{ \operatorname{can} }}\ar[d]^{\rho_{{ \operatorname{syn} }}} & H^2_{{\operatorname{cr} }}(B_{\scriptscriptstyle\bullet}{\mathbb G}_{m,{\overline{K} },h})\ar[d]^{\rho_{cr}}\\ H^2_{{\operatorname{\acute{e}t} }}(B_{\scriptscriptstyle\bullet}{\mathbb G}_{m,{\overline{K} }},{{\mathbf Q}}_p(1))\ar[r] & H^2_{{\operatorname{\acute{e}t} }}(B_{\scriptscriptstyle\bullet}{\mathbb G}_{m,{\overline{K} }}, B^+_{{\operatorname{cr} }})= H^2_{{\operatorname{\acute{e}t} }}(B_{\scriptscriptstyle\bullet}{\mathbb G}_{m,{\overline{K} }},{{\mathbf Q}}_p(1))\otimes B^+_{{\operatorname{cr} }} }$$ where the bottom map sends the generator of ${{\mathbf Q}}_p(1)$ to the element $t\in B^+_{{\operatorname{cr} }}$ associated to it. Since the syntomic and the crystalline Chern classes are compatible, it suffices to show that, for a line bundle ${{\mathcal{L}}}$, $\rho_{{\operatorname{cr} }}(c_1^{{\operatorname{cr} }}({{\mathcal{L}}}))=c_1^{{\operatorname{\acute{e}t} }}({{\mathcal{L}}})\otimes t$. But this is [@BE2 3.2]. Image of étale regulators ------------------------- In this subsection we show that Soule’s étale regulators factor through the semistable Selmer groups. Let $X\in {\mathcal V}ar_K$. For $2r-i-1\geq 0$, set $$K_{2r-i-1}(X)_0:=\ker(K_{2r-i-1}(X)\lomapr{c^{{\operatorname{\acute{e}t} }}_{r,i+1}} H^0(G_K,H^{i+1}_{{\operatorname{\acute{e}t} }}(X_{{\overline{K} }},{{\mathbf Q}}_p(r))))$$ Notice that, for $2r-i-1 >0$, we have $K_{2r-i-1}(X)_0=K_{2r-i-1}(X)$. Write $r^{{\operatorname{\acute{e}t} }}_{r,i}$ for the map $$r^{{\operatorname{\acute{e}t} }}_{r,i}:K_{2r-i-1}(X)_0\to H^1(G_K,H^{i}_{{\operatorname{\acute{e}t} }}(X_{{\overline{K} }},{{\mathbf Q}}_p(r)))$$ induced by the Chern class map $c^{{\operatorname{\acute{e}t} }}_{r,i+1}$ and the Hochschild-Serre spectral sequence map $\delta: H^{i+1}_{{\operatorname{\acute{e}t} }}(X,{{\mathbf Q}}_p(r))_0\to H^1(G_K,H^i_{{\operatorname{\acute{e}t} }}(X_{{\overline{K} }},{{\mathbf Q}}_p(r)))$, where we set $H^{i+1}_{{\operatorname{\acute{e}t} }}(X,{{\mathbf Q}}_p(r))_0:=\ker( H^{i+1}_{{\operatorname{\acute{e}t} }}(X,{{\mathbf Q}}_p(r))\to H^{i+1}_{{\operatorname{\acute{e}t} }}(X_{{\overline{K} }},{{\mathbf Q}}_p(r))) $. \[Tony\] The map $r^{{\operatorname{\acute{e}t} }}_{r,i}$ factors through the subgroup $$H^1_{{\operatorname{st} }}(G_K,H^{i+1}_{{\operatorname{\acute{e}t} }}(X_{{\overline{K} }},{{\mathbf Q}}_p(r)))\subset H^1(G_K,H^{i+1}_{{\operatorname{\acute{e}t} }}(X_{{\overline{K} }},{{\mathbf Q}}_p(r))).$$ By Proposition \[syn-et\], we have the following commutative diagram $$\xymatrix{K_{2r-i-1}(X)\ar[d]^{c^{{ \operatorname{syn} }}_{r,i+1}}\ar[dr]^{c^{{\operatorname{\acute{e}t} }}_{r,i+1}}\\ H^{i+1}_{{ \operatorname{syn} }}(X_h,r)\ar[r]^-{\rho_{{ \operatorname{syn} }}} & H^{i+1}_{{\operatorname{\acute{e}t} }}(X,{{\mathbf Q}}_p(r))\ar[r] & H^{i+1}_{{\operatorname{\acute{e}t} }}(X_{{\overline{K} }},{{\mathbf Q}}_p(r)) }$$ Hence the Chern class map $c^{{ \operatorname{syn} }}_{r,i+1}:K_{2r-i-1}(X)\to H^{i+1}_{{ \operatorname{syn} }}(X_h,r)$ factors through $H^{i+1}_{{ \operatorname{syn} }}(X_h,r)_0:=\ker(H^{i+1}_{{ \operatorname{syn} }}(X_h,r)\stackrel{\rho_{{ \operatorname{syn} }}}{\to} H^{i+1}(X_{{\overline{K} },{\operatorname{\acute{e}t} }},{{\mathbf Q}}_p(r)))$. Compatibility of the syntomic descent and the Hochschild-Serre spectral sequences (cf. Theorem \[stHS\]) yields the following commutative diagram $$\xymatrix{ K_{2r-i-1}(X)_0\ar[d]^{c^{{ \operatorname{syn} }}_{r,i+1}}\ar[dr]^{c^{{\operatorname{\acute{e}t} }}_{r,i+1}}\\ H^{i+1}_{{ \operatorname{syn} }}(X_h,r)_0\ar[d]^{\delta}\ar[r]^{\rho_{{ \operatorname{syn} }}} & H^{i+1}_{{\operatorname{\acute{e}t} }}(X,{{\mathbf Q}}_p(r))_0\ar[d]^{\delta}\\ H^1_{{\operatorname{st} }}(G_K,H^{i}(X_{{\overline{K} },h},{{\mathbf Q}}_p(r)))\ar[r]^{{ \operatorname{can} }} & H^1(G_K,H^{i}_{{\operatorname{\acute{e}t} }}(X_{{\overline{K} }},{{\mathbf Q}}_p(r))) }$$ Our theorem follows. The question of the image of Soulé’s regulators $r_{r,i}^{{\operatorname{\acute{e}t} }}$ was raised by Bloch-Kato in [@BK] in connection with their Tamagawa Number Conjecture. Theorem \[Tony\] is known to follow from the constructions of Scholl [@Sc]. The argument goes as follows. Recall that for a class $y\in K_{2r-i-1}(X)_0$ he constructs an explicit extension $E_y\in {\operatorname{Ext} }^1_{{{\mathcal{M}}}{{\mathcal{M}}}_K}({\mathbf Q}(-r),h^{i}(X))$ in the category of mixed motives over $K$. The association $y\mapsto E_y$ is compatible with the étale cycle class and realization maps. By the de Rham Comparison Theorem, the étale realization $r^{{\operatorname{\acute{e}t} }}_{r,i}(y)$ of the extension class $E_y$ in $${\operatorname{Ext} }^1_{G_K}({\mathbf Q}_p(-r),H^{i}(X_{{\overline{K} }},{{\mathbf Q}}_p))=H^1(G_K,H^i_{{\operatorname{\acute{e}t} }}(X_{{\overline{K} }},{{\mathbf Q}}_p(r)))$$ is de Rham, hence potentially semistable by [@BER], as wanted. Vanishing of $H^2(G_K,V)$ by Laurent Berger =========================================== Let $V$ be a ${\mathbf{Q}_p}$-linear representation of $G_K$. In this appendix we prove the following theorem. \[main\] If $V$ is semistable and all its Hodge-Tate weights are $\geq 2$, then $H^2(G_K,V)=0$. Let ${\mathrm{D}}(V)$ be Fontaine’s $({\varphi},\Gamma)$-module attached to $V$ [@Fon]. It comes with a Frobenius map ${\varphi}$ and an action of $\Gamma_K$. Let $H_K = {\operatorname{Gal} }({\overline{K} }/K(\mu_{p^\infty}))$ and let $I_K = {\operatorname{Gal} }({\overline{K} }/K^{{\operatorname{nr} }})$. The injectivity of the restriction map $H^2(G_K,V) \to H^2(G_L,V)$ for $L/K$ finite allows us to replace $K$ by a finite extension, so that we can assume that $H_K I_K = G_K$ and that $\Gamma_K \simeq {\mathbf{Z}_p}$. Let $\gamma$ be a topological generator of $\Gamma_K$. Recall (§I.5 of [@CC99]) that we have a map $\psi : {\mathrm{D}}(V) \to {\mathrm{D}}(V)$. Ideally, our proof of this theorem would go as follows. We use the Hochschild-Serre spectral sequence $$H^i(G_K/I_K,H^j(I_K,V\|_{I_K})) \Rightarrow H^{i+j}(G_K,V)$$ and, interpreting Galois cohomology in terms of $({\varphi},\Gamma)$-modules, we compute that $H^2(I_K,V\|_{I_K})=0$ and $H^1(I_K,V\|_{I_K})=\hat{K}^{{\operatorname{nr} }}\otimes_KD_{{\mathrm{dR}}}(V)$. We conclude since, by Hilbert 90, $H^1(G_K/I_K,H^1(I_K,V\|_{I_K})) =0$. However, we do not, in general, have Hochschild-Serre spectral sequences for continuous cohomology. We mimic thus the above argument with direct computations on continuous cocycles (again using $({\varphi}, \Gamma)$-modules). Laurent Berger is grateful to Kevin Buzzard for discussions related to the above spectral sequence. \[cc\] 1. If $V$ is a representation of $G_K$, then there is an exact sequence $$0 \to {\mathrm{D}}(V)^{\psi=1} / (\gamma-1) \to H^1(G_K,V) \to ({\mathrm{D}}(V)/(\psi-1))^{\Gamma_K} \to 0;$$ 2. We have $H^2(G_K,V) = {\mathrm{D}}(V)/(\psi-1,\gamma-1)$. See I.5.5 and II.3.2 of [@CC99]. \[psimsur\] We have ${\mathrm{D}}(V\|_{I_K})/(\psi-1)=0$ Since $V\|_{I_K}$ corresponds to the case when $k$ is algebraically closed, see the proof of Lemma VI.7 of [@L01]. Let $\gamma_I$ denote a generator of $\Gamma_{\widehat{K}^{{\operatorname{nr} }}}$. \[psigam\] The natural map ${\mathrm{D}}(V\|_{I_K})^{\psi=1} / (\gamma_I-1) \to ({\mathrm{D}}(V\|_{I_K}) / (\gamma_I-1))^{\psi=1}$ is an isomorphism if $V^{I_K}=0$. This map is part of the six term exact sequence that comes from the map $\gamma_I-1$ applied to $0 \to {\mathrm{D}}(V\|_{I_K})^{\psi=1} \to {\mathrm{D}}(V\|_{I_K}) \xrightarrow{\psi-1} {\mathrm{D}}(V\|_{I_K}) \to 0$. Its kernel is included in ${\mathrm{D}}(V\|_{I_K})^{\gamma_I=1}$ which is $0$, since $V^{I_K}=0$ (note that the inclusion $(\widehat{K}^{{\operatorname{nr} }} \otimes V)^{G_K} \subseteq (\widehat{\mathcal E}^{{\operatorname{nr} }} \otimes V)^{G_K} = {\mathrm{D}}(V)^{G_K}$ is an isomorphism). Suppose that $x \in {\mathrm{D}}(V)/(\psi-1,\gamma-1)$. If $\tilde{x} \in {\mathrm{D}}(V)$ lifts $x$, then Lemma \[psimsur\] gives us an element $y\in{\mathrm{D}}(V \|_{I_K})$ such that $(\psi-1)y = \tilde{x}$. Define a cocycle $\delta(x) \in Z^1(G_K/I_K, {\mathrm{D}}(V \|_{I_K})^{\psi=1} / (\gamma_I-1))$ by $\delta(x) : \overline{g} \mapsto (g-1)(y)$ if $g \in G_K$ lifts $\overline{g} \in G_K/I_K$. \[hsss\] If $V^{I_K}=0$, then the map $$\delta : {\mathrm{D}}(V)/(\psi-1,\gamma-1) \to H^1(G_K/I_K, ({\mathrm{D}}(V\|_{I_K}) / (\gamma_I-1))^{\psi=1})$$ is well-defined and injective. We first check that $\delta(x)(g) \in ({\mathrm{D}}(V\|_{I_K}) / (\gamma_I-1))^{\psi=1}$. We have $(\psi-1)(g-1)(y) = (g-1)(x)$. If we write $g = ih \in I_KH_K$, then $(g-1)x = (ih-1)x=(i-1)x \in (\gamma_I-1) {\mathrm{D}}(V\|_{I_K})$ since $\gamma_I-1$ divides the image of $i-1$ in ${\mathbf{Z}_p}{[\![ \Gamma_{\widehat{K}^{{\operatorname{nr} }}} ]\!]}$. This implies that $\delta(x)(g) \in ({\mathrm{D}}(V\|_{I_K}) / (\gamma_I-1))^{\psi=1}$. We now check that $\delta(x)$ does not depend on the choices. If we choose another lift $g' \in G_K$ of $\overline{g} \in G_K/I_K$, then $g'=ig$ for some $i \in I_K$ and $(g'-1)y-(g-1)y = (i-1)gy \in (\gamma_I-1) {\mathrm{D}}(V\|_{I_K})$ since $\gamma_I-1$ divides the image of $i-1$ in ${\mathbf{Z}_p}{[\![ \Gamma_{\widehat{K}^{{\operatorname{nr} }}} ]\!]}$. If we choose another $y'$ such that $(\psi-1)y'=\tilde{x}$, then $y-y' \in {\mathrm{D}}(V\|_{I_K})^{\psi=1}$ so that $\delta$ and $\delta'$ are cohomologous. Finally, if $\tilde{x}'$ is another lift of $x$, then $\tilde{x}' - \tilde{x} = (\gamma-1)a + (\psi-1)b$ with $a,b \in {\mathrm{D}}(V)$. We can then take $y' = y + b + (\gamma_G-1)c$ where $(\psi-1)c=a$. We then have $(g-1)y'=(g-1)y + (g-1)b + (\gamma_G-1)(g-1)c$. Since $G_K=I_K H_K$, we can write $g=ih$ and $(g-1)b=(i-1)b$. Using $G_K=I_K H_K$ once again, we see that $I_K \to G_K/H_K$ is surjective, so that we can identify $\gamma_I$ and $\gamma_G$. The resulting cocycle is then cohomologous to $\delta(x)$. This proves that $\delta$ is well-defined. We now prove that $\delta$ is injective. If $\delta(x)=0$, then using Lemma \[psigam\] there exists $z \in {\mathrm{D}}(V\|_{I_K})^{\psi=1}$ such that $\delta(x)(\overline{g})$ is the image of $(g-1)(z)$ in ${\mathrm{D}}(V\|_{I_K})^{\psi=1} / (\gamma_I-1)$. This implies that $(g-1)(y-z) \in (\gamma_I-1) {\mathrm{D}}(V\|_{I_K})^{\psi=1}$. Applying $\psi-1$ gives $(g-1)\tilde{x} = 0$ so that $\tilde{x} \in {\mathrm{D}}(V)^{G_K} \subset V^{I_K}=0$. The map $\delta$ is therefore injective. \[expbij\] If $V$ is semistable and the weights of $V$ are all $\geq 2$, then $\exp_V : {\mathrm{D}_{\mathrm{dR}}}(V\|_{I_K}) \to H^1(I_K,V)$ is an isomorphism. Apply Thm. 6.8 of [@BER] to $V\|_{I_K}$. We can replace $K$ by $K_n$ for $n \gg 0$ and use the fact that if $H^2(G_{K_n},V) = 0$, then $H^2(G_K,V) = 0$ since the restriction map is injective. In particular, we can assume that $H_K I_K = G_K$ and that $\Gamma_K$ is isomorphic to ${\mathbf{Z}_p}$. By item (2) of Lemma \[cc\], we have $H^2(G_K,V) = {\mathrm{D}}(V)/(\psi-1,\gamma-1)$, and so by Proposition \[hsss\] above, it is enough to prove that $$H^1(G_K/I_K, ({\mathrm{D}}(V\|_{I_K}) / (\gamma_I-1))^{\psi=1}) = 0.$$ Lemma \[psigam\] tells us that $({\mathrm{D}}(V\|_{I_K}) / (\gamma_I-1))^{\psi=1} = {\mathrm{D}}(V\|_{I_K})^{\psi=1} / (\gamma_I-1)$. Since ${\mathrm{D}}(V\|_{I_K})/(\psi-1)=0$ by Lemma \[psimsur\], item (1) of Lemma \[cc\] tells us that ${\mathrm{D}}(V\|_{I_K})^{\psi=1} / (\gamma-1) = H^1(I_K,V)$. The map $\exp_V : {\mathrm{D}_{\mathrm{dR}}}(V\|_{I_K}) \to H^1(I_K,V)$ is an isomorphism by Lemma \[expbij\], and this isomorphism commutes with the action of $G_K$ since it is a natural map. We therefore have $H^1(I_K,V) = \widehat{K}^{{\operatorname{nr} }} \otimes_K {\mathrm{D}_{\mathrm{dR}}}(V)$ as $G_K$-modules. It remains to observe that the cocycle $\delta(x) \in Z^1(G_K/I_K, \widehat{K}^{{\operatorname{nr} }} \otimes_K {\mathrm{D}_{\mathrm{dR}}}(V))$ is continuous and that $H^1(G_K/I_K, \widehat{K}^{{\operatorname{nr} }})=0$ by taking a lattice, reducing modulo a uniformizer of $K$, and applying Hilbert 90. The Syntomic ring spectrum by Frédéric Déglise ============================================== In this appendix, we explain why syntomic cohomology as defined in this paper is representable by a motivic ring spectrum in the sense of Morel and Voevodsky’s homotopy theory. More precisely, we will exhibit a monoid object ${{\mathcal{S}}}$ of the triangulated category of motives with ${\mathbf{Q}_p}$-coefficients (see below), $DM$, such that for any variety $X$ and any pair of integers $(i,r)$, $$H^i_{{ \operatorname{syn} }}(X_h,r)={\operatorname{Hom} }_{DM}(M(X),{{\mathcal{S}}}(r)[i]).$$ In fact, it is possible to apply directly [@DM1 Th. 1.4.10] to the graded commutative dg-algebra ${\mathrm {R} }\Gamma_{{ \operatorname{syn} }}(X,)$ of Theorem \[main1\] in view of the existence of Chern classes established in Section 5.1. However, the use of $h$-topology in this paper makes the construction of ${\mathbb{E}}_{synt}$ much more straight-forward and that is what we explain in this appendix. Reformulating slightly the original definition of Voevodsky (see [@V1]), we introduce: Let $\operatorname{PSh}(K,{\mathbf{Q}_p})$ be the category of presheaves of ${\mathbf{Q}_p}$-modules over the category of varieties. Let $C$ be a complex in $\operatorname{PSh}(K,{\mathbf{Q}_p})$. We say: 1. $C$ is $h$-local if for any h-hypercovering $\pi:Y_\bullet \rightarrow X$, the induced map: $$C(X) \rightarrow{\pi^} \mathrm{Tot}^\oplus(C(Y_\bullet))$$ is a quasi-isomorphism; 2. $C$ is ${\mathbb A^1}$-local if for any variety $X$, the map induced by the projection: $$H^i(X_h,C) \rightarrow H^i({\mathbb A^1}_{X,h},C)$$ is an isomorphism. We define the triangulated category ${DM^{eff}}_h(K,{\mathbf{Q}_p})$ of effective $h$-motives as the full subcategory of the derived category $D(\operatorname{PSh}(K,{\mathbf{Q}_p}))$ made by the complexes which are $h$-local and ${\mathbb A^1}$-local. Equivalently, we can define this category as the ${\mathbb A^1}$-localization of the derived category of $h$-sheaves on $K$-varieties (see [@CD3], Sec. 5.2 and more precisely Prop. 5.2.10, Ex. 5.2.17(2)). Recall also from loc. cit., that there are derived tensor products and internal ${\operatorname{Hom} }$ on ${DM^{eff}}_h(K,{\mathbf{Q}_p})$. For any integer $r \geq 0$, the syntomic sheaf* ${{\mathcal{S}}}(r)$ is both $h$-local (by definition) and ${\mathbb A^1}$-local (Prop. \[homotopy\]). Thus it defines an object of ${DM^{eff}}_h(K,{\mathbf{Q}_p})$ and for any variety $X$, one has an isomorphism: $${\operatorname{Hom} }_{{DM^{eff}}_h(K,{\mathbf{Q}_p})}({\mathbf{Q}_p}(X),{{\mathcal{S}}}(r)[i])= {\operatorname{Hom} }_{D(\operatorname{PSh}(K,{\mathbf{Q}_p}))}({\mathbf{Q}_p}(X),{{\mathcal{S}}}(r)[i]) =H^{i,r}_{{ \operatorname{syn} }}(X)$$ where ${\mathbf{Q}_p}(X)$ is the presheaf of $K$-vector spaces represented by $X$. Thus, the representability assertion for syntomic cohomology is obvious in the effective setting. Recall that one defines the Tate motive in ${DM^{eff}}_h(K,{\mathbf{Q}_p})$ as the object ${\mathbf{Q}_p}(1):={\mathbf{Q}_p}(\mathbb P^1)/{\mathbf{Q}_p}(\{\infty\})[-2]$. Given any complex object $C$ of ${DM^{eff}}_h(K,{\mathbf{Q}_p})$, we put: $C(n):=C \otimes {\mathbf{Q}_p}(1)^{\otimes,n}$. One should be careful that this notation is in conflict with that of ${{\mathcal{S}}}(r)$ considered as an effective $h$-motive, as the natural twist on syntomic cohomology is unrelated to the twist of $h$-motives. To solve this matter, we are led to consider the following notion of Tate spectrum, borrowed from algebraic topology according to Morel and Voevodsky. A Tate $h$-spectrum (over $K$ with coefficients in ${\mathbf{Q}_p}$), is a sequence ${\mathbb{E}}=(E_i,\sigma_i)_{i \in \mathbb N}$ such that: - for each $i \in \mathbb N$, $E_i$ is a complex of $\operatorname{PSh}(K,{\mathbf{Q}_p})$ equipped with an action of the symmetric group $\Sigma_i$ of the set with $i$-element, - for each $i \in \mathbb N$, $\sigma_i:E_i(1) \rightarrow E_{i+1}$ is a morphism of complexes – called the suspension map in degree $i$, - For any integers $i \geq 0$, $r>0$, the map induced by the morphisms $\sigma_i,\cdots,\sigma_{i+r}$: $$E_i(r) \rightarrow E_{i+r}$$ is compatible with the action of $\Sigma_i \times \Sigma_r$, given on the left by the structural $\Sigma_i$-action on $E_i$ and the action of $\Sigma_r$ via the permutation isomorphism of the tensor structure on $C(\operatorname{PSh}(K,{\mathbf{Q}_p}))$, and on the right via the embedding $\Sigma_i \times \Sigma_r \rightarrow \Sigma_{i+r}$. A morphism of Tate $h$-spectra $f:{\mathbb{E}}\rightarrow \mathbb F$ is a sequence of $\Sigma_i$-equivariant maps $(f_i:E_i \rightarrow F_i)_{i \in \mathbb N}$ compatible with the suspension maps. The corresponding category will be denoted by $\operatorname{Sp_h}(K,{\mathbf{Q}_p})$. There is an adjunction of categories: $$\label{eq:suspension} \Sigma^\infty:C\big(\operatorname{PSh}(K,{\mathbf{Q}_p})\big) \leftrightarrows \operatorname{Sp_h}(K,{\mathbf{Q}_p}):\Omega^\infty$$ such that for any complex $K$ of $h$-sheaves, $\Sigma^\infty C$ is the Tate spectrum equal in degree $n$ to $C(n)$, equiped with the obvious action of $\Sigma_n$ induced by the symmetric structure on tensor product and with the obvious suspension maps. A morphism of Tate spectra $(f_i:E_i \rightarrow F_i)_{i \in \mathbb N}$ is a level quasi-isomorphism if for any $i$, $f_i$ is a quasi-isomorphism. A Tate spectrum ${\mathbb{E}}$ is called a $\Omega$-spectrum if for any $i$, $E_i$ is $h$-local and ${\mathbb A^1}$-local and the map of complexes $$E_i \rightarrow \underline{{\operatorname{Hom} }}({\mathbf{Q}_p}(1),E_{i+1})$$ is a quasi-isomorphism. We define the triangulated category ${DM}_h(K,{\mathbf{Q}_p})$ of $h$-motives over $K$ with coefficients in ${\mathbf{Q}_p}$ as the category of Tate $\Omega$-spectra localized by the level quasi-isomorphisms. The category of $h$-motives notably enjoys the following properties: 1. The adjunction of categories induces an adjunction of triangulated categories: $$\Sigma^\infty:{DM^{eff}}_h(K,{\mathbf{Q}_p}) \leftrightarrows {DM}_h(K,{\mathbf{Q}_p}):\Omega^\infty$$ such that for a Tate $\Omega$-spectrum ${\mathbb{E}}$, and any integer $r \geq 0$, $\Omega^\infty({\mathbb{E}}(r))=E_r$ (see [@CD3 Sec. 5.3.d, and Ex. 5.3.31(2)]). Given any variety $X$, we define the (stable) $h$-motive of $X$ as $M(X):=\Sigma^\infty {\mathbf{Q}_p}(X)$. 2. There exists a symmetric closed monoidal structure on ${DM}(K,{\mathbf{Q}_p})$ such that $\Sigma^\infty$ is monoidal and such that $\Sigma^\infty {\mathbf{Q}_p}(1)$ admits a tensor inverse (see [@CD3 Sec. 5.3, and Ex. 5.3.31(2)]). By abuse of notations, we put: ${\mathbf{Q}_p}=\Sigma^\infty {\mathbf{Q}_p}$. 3. The triangulated monoidal category ${DM}_h(K,{\mathbf{Q}_p})$ is equivalent to all known version of triangulated categories of mixed motives over ${\operatorname{Spec} }(K)$ with coefficients in ${\mathbf{Q}_p}$ (see [@CD3 Sec. 16, and Th. 16.1.2]). In particular, it contains as a full subcategory the category ${DM}_{gm}(K) \otimes {\mathbf{Q}_p}$ obtained from the category of Voevodsky geometric motives ([@FSV chap.5]) by tensoring ${\operatorname{Hom} }$-groups with ${\mathbf{Q}_p}$ (see [@CD3 Cor. 16.1.6, Par. 15.2.5]). With that definition, the construction of a Tate spectrum representing syntomic cohomology is almost obvious. In fact, we consider the sequence of presheaves $${{\mathcal{S}}}:=({{\mathcal{S}}}(r), r \in \mathbb N),$$ where each ${{\mathcal{S}}}(r)$ with the trivial action of $\Sigma_r$. According to the first paragraph of Section 5.1, we can consider the first Chern class of the canonical invertible sheaf $\mathbb P^1$: $\bar c \in H^2_{{ \operatorname{syn} }}(\mathbb P^1_K,1)=H^2(R\Gamma((\mathbb P^1_K)_h,{{\mathcal{S}}}(1))$. Take any lift $c:{\mathbf{Q}_p}(\mathbb P^1_K) \rightarrow {{\mathcal{S}}}(1)[2]$ of this class. By definition of the Tate twist, it defines an element ${\mathbf{Q}_p}(1) \rightarrow {{\mathcal{S}}}(1)$ still denoted by $c$. We define the suspension map: $${{\mathcal{S}}}(r) \otimes {\mathbf{Q}_p}(1) \lomapr{{ \operatorname{Id} }\otimes c} {{\mathcal{S}}}(r) \otimes {{\mathcal{S}}}(1) \xrightarrow{\mu} {{\mathcal{S}}}(r+1)$$ where $\mu$ is the multiplication comming from the graded dg-structure on ${{\mathcal{S}}}()$. Because this dg-structure is commutative, we obtain that these suspension maps induce a structures of a Tate spectrum on ${{\mathcal{S}}}$. Moreover, ${{\mathcal{S}}}$ is a Tate $\Omega$-spectrum because each ${{\mathcal{S}}}(r)$ is $h$-local and ${\mathbb A^1}$-local, and the map obtained by adjunction from $\sigma_r$ is a quasi-isomorphism because of the projective bundle theorem for $\mathbb P^1$ (an easy case of Proposition \[projective\]). Now, by definition of ${DM}_h(K,{\mathbf{Q}_p})$ and because of property (DM1) above, for any variety $X$, and any integers $(i,r)$, we get: $${\operatorname{Hom} }_{{DM}_h(K,{\mathbf{Q}_p})}(M(X),{{\mathcal{S}}}(r)[i]) ={\operatorname{Hom} }_{{DM^{eff}}_h(K,{\mathbf{Q}_p})}({\mathbf{Q}_p}(X),\Omega^\infty({{\mathcal{S}}}(r))[i]) =H^i_{{ \operatorname{syn} }}(X_h,r).$$ Moreover, the commutative dg-structure on the complex ${{\mathcal{S}}}()$ induces a monoid structure on the associated Tate spectrum. In other words, ${{\mathcal{S}}}$ is a ring spectrum (strict and commutative). This construction is completely analogous to the proof of [@DM1 Prop. 1.4.10]. In particular, we can apply all the constructions of [@DM1 Sec. 3] to the ring spectrum ${{\mathcal{S}}}$. Let us summarize this briefly: \[Bloch-Ogus\] 1. Syntomic cohomology is covariant with respect to projective morphisms of smooth varieties (Gysin morphisms in the terminology of [@DM1]). More precisely, to a projective morphism of smooth $K$-varieties $f: Y\to X$ one can associate a Gysin morphism in syntomic cohomology $$f_: H^n_{{ \operatorname{syn} }}(Y_h,i)\to H^{n-2d}_{{ \operatorname{syn} }}(X_h,i-d),$$ where $d$ is the dimension of $f$. 2. The syntomic regulator over ${\mathbf{Q}_p}$ is induced by the unit $\eta:{\mathbf{Q}_p}\rightarrow {{\mathcal{S}}}$ of the ring spectrum ${{\mathcal{S}}}$: $$\begin{aligned} r_{{ \operatorname{syn} }}:H^{r,i}_M(X) \otimes {\mathbf{Q}_p}=&{\operatorname{Hom} }_{{DM}_h(K,{\mathbf{Q}_p})}(M(X),{\mathbf{Q}_p}(r)[i]) \\ & \longrightarrow {\operatorname{Hom} }_{{DM}_h(K,{\mathbf{Q}_p})}(M(X),{{\mathcal{S}}}(r)[i])=H^i_{{ \operatorname{syn} }}(X_h,r).\end{aligned}$$ It is compatible with product, pullbacks and pushforwards. 3. The syntomic cohomology has a natural extension to $h$-motives[^7]: $${DM}_h(K,{\mathbf{Q}_p})^{op} \rightarrow D({\mathbf{Q}_p}), \quad M \mapsto {\operatorname{Hom} }_{{DM}_h(K,{\mathbf{Q}_p})}(M,{{\mathcal{S}}})$$ and the syntomic regulator $r_{{ \operatorname{syn} }}$ can be extended to motives. 4. There exists a canonical syntomic Borel-Moore homology $H^{{ \operatorname{syn} }}_(-,)$ such that the pair of functor $(H_{{ \operatorname{syn} }}^(-,),H^{{ \operatorname{syn} }}_(-,))$ defines a Bloch-Ogus theory. 5. To the ring spectrum ${{\mathcal{S}}}$ there is associated a cohomology with compact support satisfying the usual properties. For points (1) and (2), we refer the reader to [@DM1 Sec. 3.1] and for the remaining ones to [@DM1 Sec. 3.2]. Note that the construction of the syntomic ring spectrum ${{\mathcal{S}}}$ in ${DM}_h(K,{\mathbf{Q}_p})$ automatically yields the general projective bundle theorem (already obtained in Prop. \[projective\]). More generally, the ring spectrum ${{\mathcal{S}}}$ is oriented* in the terminology of motivic homotopy theory. Thus, besides the theory of Gysin morphisms, this gives various constructions – symbols, residue morphisms – and yields various formulas – excess intersection formula, blow-up formulas (see [@Deg8] for more details). 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[^4]: The latter is generated by a pretopology whose coverings are proper surjective maps [^5]: Extension $0\to V_1\to V_2\to V_3\to 0$ is called ${\operatorname{st} }$ if the sequence $0\to D_{{\operatorname{st} }}(V_1)\to D_{{\operatorname{st} }}(V_2)\to D_{{\operatorname{st} }}(V_3)\to 0$ is exact. [^6]: We hope that the notation below will not lead to confusion with the semistable case in general but if in doubt we will add the data of the field $K$ in the latter case. [^7]: and in particular to the usual Voevodsky geometrical motives by (DM3) above.
--- abstract: \| We describe the INTEGRAL Burst Alert System (IBAS): the automatic software developed at the INTEGRAL Science Data Center to allow the rapid distribution of the coordinates of the Gamma-Ray Bursts detected by INTEGRAL. IBAS is implemented as a ground based system, working on the near-real time telemetry stream. It is expected that the system will detect more than one GRB per month in the field of view of the main instruments. Positions with an accuracy of a few arcminutes will be distributed to the community for follow-up observations within a few tens of seconds of the event. The system will also upload commands to optimize the possible detection of bursts in the visible band with the INTEGRAL Optical Monitor Camera. author: - 'S. Mereghetti' - 'D.I. Cremonesi' - 'J. Borkowski' title: The INTEGRAL Burst Alert System --- \[1999/12/02 1.01 (PWD)\] Introduction ============ The origin of gamma-ray bursts (GRB’s) remained one of the great mysteries of high energy astrophysics for more than 25 years after the publication of their discovery (Klebesadel et al. 1973). The puzzle became even more complicated after the first results obtained with the BATSE instrument. This detector, especially designed to significantly increase the GRB sample, led to the unexpected result of a non-euclidean LogN-LogS coupled with a uniform angular distribution in the sky (see, e.g., Fishman 1995, and references therein). It soon became clear that the only way to make a significant advance in the field was the possibility of identifying the counterparts of GRB at other wavelengths. The instruments on board INTEGRAL, though not specifically optimized to observe GRB’s, offer the possibility of rapidly obtaining accurate positions of the GRB’s observed by chance in their large field of view. It was therefore proposed to implement a ”burst alert system” in order to allow rapid multi-wavelength follow-ups (Pedersen et al. 1997). Such a proposal was boosted by the exciting results obtained with the BeppoSAX satellite (Costa et al. 1997, van Paradijs et al. 1997), that clearly demonstrated the capabilities of a coded mask instrument, similar to the INTEGRAL ones, in quickly localizing GRB’s. Overview of the INTEGRAL Burst Alert System (IBAS) =================================================== Since no on-board GRB detection system is foreseen on INTEGRAL, the search for GRB’s will be performed on ground by means of a near real time analysis running at the INTEGRAL Science Data Center (ISDC). The telemetry data, received at the MOC, will be transmitted on a 128 kbs dedicated line to the ISDC. Here the relevant data packets will be extracted and immediately fed into a dedicated software system called IBAS (Integral Burst Alert System), independent from the main data processing pipeline of the ISDC (Mereghetti et al. 1999). It is expected that ISGRI, the first layer of the IBIS instrument, operating in the 15-300 keV energy range, will yield the best chances to detect a large number of GRB’s and to accurately determine their positions. The SPI instrument, with a similar sensitivity and large field of view, can also detect GRB’s, but its angular resolution is not as good as the IBIS one. Relatively strong bursts in the central part of the field of view will be detectable by SPI and IBIS, therefore a simultaneous trigger in both instruments could be used to increase the confidence in the reality of the event. However, due to the different sensitivity curves as a function of energy and source direction, it is likely that several GRB’s will trigger only in a single detector. For this reason the IBAS validation process will not require a multiple detection to confirm the occurrence of a GRB. The first step of the GRB search will be based on a simple monitoring of the incoming count rates, without resorting to more complex image analysis. In practice this will be done by looking for significant excesses with respect to a running average, in a way similar to traditional on-board triggering algorithms. In fact any transient source strong enough to appear as a significant new peak in the deconvolved images will also produce a detectable excess in the overall count rate. The search will be simultaneously performed in many different time scales and energy ranges, to optimize the sensitivity to GRB’s with different characteristics. When a candidate event is detected, a process of image analysis shall start to verify the origin of the count rate variation and to ensure that the event was not caused by an instrumental malfunctioning (e.g. a bad detector pixel) or by a background variation (see Figures \[lcurve\], \[grb\], \[hotpix\]). Images shall be accumulated for different time intervals, deconvolved with very fast algorithms, and compared to the pre-burst reference images in order to detect the appearance of the GRB as a new source. This last step is of fundamental importance, since in general different sources will be present in the field of view. The image analysis will be based on the time intervals, derived from the GRB light curve, that optimize the signal to noise ratio. If the event is genuine, the satellite attitude information will be applied to derive a sky position that is then automatically transmitted, by e-mail and/or direct TCP/IP socket, to all the subscribed users. In addition, if the GRB is located in the sky region covered by the OMC, an appropriate telecommand will be generated and sent to the satellite to reconfigure its observing parameters (see Section 6). Because full event validation and localization might require a longer time, we foresee different levels of alert messages providing increasingly accurate and reliable information. These messages will be configured in such a way to allow an easy filtering by the users in order to react only to the situations that best fit their needs. Number of expected GRB’s ======================== A very simple estimate of the number of GRB’s expected within the field of view of the INTEGRAL instruments can be done by scaling the total rate of events measured by BATSE ($\sim$666 GRB year$^{-1}$, Paciesas et al. 1999). For the field of view of IBIS ($\sim$30$^{\circ} \times$ 30$^{\circ}$), this yields 666 $\times$ (0.23 sterad / 4$\pi$) $\sim$ 12 GRB year$^{-1}$. A more accurate estimate must take into account the varying sensitivity within the IBIS field of view and the different energy range of the BATSE detectors. This can be done by convolving the IBIS sensitivity and solid angle as a function of off-axis angle with the BATSE LogN-LogP relations, converted to the appropriate energy range assuming an average GRB spectral shape. It turns out that such computations do not significantly change the above rough result. In fact we obtain expected rates of $\sim$13, 10 and 8 GRB year$^{-1}$, within the ISGRI field of view, adopting respectively the LogN-LogP corresponding to BATSE trigger times $\Delta$T of 1 s, 256 ms and 64 ms. The major uncertainty on the derived GRB rates is related to the extrapolation of the BATSE LogN-LogP curves down to the ISGRI sensitivity ($\sim$0.1 ph cm$^{-2}$ s$^{-1}$, 50-300 keV peak flux for $\Delta$T=1 s). Such an extrapolation depends on the poorly known spectral shape of very weak GRB’s. It must be noted that these estimates do not take into account the likely existence of GRB’s with different characteristics than those observed by BATSE. For instance it is clear that BATSE had a limited sensitivity for events shorter than 64 ms, as well as a strong bias against the detection of long, slowly rising GRB’s. It it therefore very likely that the figures reported above will be an underestimate, and that a few events per month will be available through IBAS for rapid multi-wavelength follow-up observations. Location Accuracy ================= The source location accuracy (SLA) of coded mask imaging systems depends on the signal to noise ratio of the source. For sources detected with a high statistical significance the SLA can be a small fraction of the angular resolution. Theoretical evaluations, confirmed by several independent simulations, have shown that for ISGRI a SLA smaller than 30$''$ (90% confidence level) can be obtained for a signal to noise ratio of 30 (see Figure \[sla\]). In such cases, the final accuracy on the GRB location is also affected by the uncertainties on the satellite attitude (see below). Of course, most of the detected GRB’s will have relatively small signal to noise ratios, resulting in typical uncertainties, dominated by the photon statistics, of the order of $\sim$2-3$'$. The expected INTEGRAL attitude accuracy depends on three cases: (1) attitude during slews, (2) attitude at the start of a stable pointing period, and (3) attitude from about ten minutes after the start to the end of a stable pointing period. For case (1) a slew path simulator generates predictions of the pointing direction every 10 seconds. In most circumstances these predictions should be accurate to within 10$'$ (3 $\sigma$). The average slew rate during the Galactic Plane scans and during the normal dithering observations will be of the order of $\sim$1$^{\circ}$/min. Therefore, it will be possible to detect GRB’s even during slews, although their location accuracy will be worse. For case (2) the attitude is based upon a prediction of the expected position with respect to the previously commanded slew. In most cases (i.e. slews shorter than 2$^{\circ}$) these predicted values will have an accuracy of the order of the star-tracker/instrument alignment uncertainty (30$''$ or less). For larger slews the predicted attitude will have an error smaller than 5$\pm$1$'$ (3$\sigma$). Approximately 5 minutes after the start of a pointing (case 3) a “snapshot” attitude reconstruction will be completed based upon the first down-linked star-tracker map. The result shall be made available to the IBAS within a maximum delay of 10 minutes since the start of the stable pointing, yielding, as above, an attitude with accuracy $\lsim$30$''$. IBAS time performance ===================== Using simulated telemetry data, we have obtained the following performance figures for the current prototype version of IBAS running on a SUN ULTRA 10 workstation (440 MHz clock, 256 Mbytes RAM). Of course the final version will run on a faster workstation exclusively dedicated to the IBAS system. The first IBAS steps (telemetry receipt, extraction and sorting of the relevant data packets, photon binning for the trigger search) require less than 0.1 s. The speed of the triggering algorithm depends on the duration of the smallest time interval considered and on the number of timescales. For example, sampling on bins of 1, 4, 16, 64 and 256 ms, with a resolution of 1 ms it is still possible to process the telemetry at a speed twice faster than that of the real-time incoming data. The most time consuming tasks are those related to the image analysis. For this reason, the first part of imaging after a trigger is based on a detection algorithm able to discover new sources by only considering (ghost) peaks in the fully coded field of view. This can be done within $\sim$200 ms for ISGRI and it allows to discriminate between false triggers and true events. Only when a likely point source is detected, a more thorough analysis will be done to locate the GRB. The deconvolution of the whole (totally plus partially coded) field of view and localization of the true source peak currently requires $\sim$5 s (we expect to improve this performance with a new optimized version of the code). Thus it should be possible to send the first GRB alert within a few seconds after the trigger time. The delay between the trigger time and the GRB onset is of course dependent on the intensity and time profile of the event, but the IBAS simultaneous sampling in different timescales should ensure a minimum delay in most cases. To this time budget one has to add the time required for the telemetry transmission from the satellite to the ISDC, that under normal circumstances should be smaller than $\sim$30 s. Thus, in many cases, we foresee to be able to generate [first level]{} alerts while the GRB is still ongoing. Optical Monitor Recomanding =========================== The Optical Monitor Camera (OMC) on board INTEGRAL has a field of view of 5$^{\circ} \times$5$^{\circ}$, but owing to the limited allocated telemetry ($\sim$1.7 kbit s$^{-1}$) only the data from a number of preselected windows around sources of interest will be downloaded during normal operations. IBAS will check whether the derived GRB position falls within the OMC field of view. In such a case, the appropriate telecommand with the definition of a new window centered on the interesting region will be generated and sent to the satellite. This will allow to quickly observe the GRB/afterglow emission in the optical band. The OMC observation should consist of many frames with short integration times to permit variability studies and to increase the sensitivity for very intense but short outbursts. The expected limiting magnitude is of the order of V$\sim$14-15 for an integration time of 10 s, and V$\sim$12-13 for 1 s. Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful to several people that have contributed to the IBAS project, including G.Bazzaro, S.Brandt, D.Jennings, R.Hudec, H.Pedersen, A.Pellizzoni, M.Pohl, T.Courvoisier and R.Walter. The IBAS development is supported by the Italian Space Agency. JB was supported by the Polish Committee for Scientific Research (KBN) under grant No. 2P03C00619p02. Costa E. et al., 1997, [Nature]{} [387]{}, 783. Fishman G.J., 1995, [PASP]{} [107]{}, 1145. Klebesadel R., Strong I., Olsen R., 1993, [ApJ]{} [182]{}, L85. Mereghetti S., et al. 1999, [Proc. 3nd INTEGRAL Workshop - Astro. Lett. and Communications,]{} [39]{}, 301. Paciesas W.S., et al., 1999, [ApJSS]{} [122]{}, 465. Pedersen H., Jennings D., Mereghetti S., Teegarden B., 1997, [Proc. 2nd INTEGRAL Workshop]{} [ESA SP-382]{}, 433. van Paradijs J. et al., 1997, [Nature]{} [386]{}, 686.
--- author: - 'N. Oppermann[^1]' - 'H. Junklewitz' - 'G. Robbers' - 'M.R. Bell' - 'T.A. Enßlin' - 'A. Bonafede' - 'R. Braun' - 'J.C. Brown' - 'T.E. Clarke' - 'I.J. Feain' - 'B.M. Gaensler' - 'A. Hammond' - 'L. Harvey-Smith' - 'G. Heald' - 'M. Johnston-Hollitt' - 'U. Klein' - 'P.P. Kronberg' - 'S.A. Mao' - 'N.M. McClure-Griffiths' - 'S.P. O’Sullivan' - 'L. Pratley' - 'T. Robishaw' - 'S. Roy' - 'D.H.F.M. Schnitzeler' - 'C. Sotomayor-Beltran' - 'J. Stevens' - 'J.M. Stil' - 'C. Sunstrum' - 'A. Tanna' - 'A.R. Taylor' - 'C.L. Van Eck' bibliography: - 'FD\_map.bib' date: 'Received DD MMM. YYYY / Accepted DD MMM. YYYY' title: An improved map of the Galactic Faraday sky --- Introduction ============ Magnetic fields are ubiquitous in the interstellar medium. They are likely to play a major dynamical role in the evolution of galaxies. It is by comparing theoretical predictions and simulations to observations of galactic magnetic fields that their generation and dynamical role can be understood [see e.g. @beck-2011 and references therein]. It is natural to look first and foremost at our own galaxy, the Milky Way, and try to study its magnetic field. However, its observation is complicated by a number of effects. The magnetic field is a three-dimensional vector field that varies on multiple scales throughout the Galaxy. Thus, a large number of measurements of the field would be needed to determine even its large-scale properties. Furthermore, virtually any observation suffers from a projection effect as local effects add up along the line of sight. And finally the magnetic field cannot be measured directly, so that related observables have to be used. These observables, however, are not only sensitive to the magnetic field itself but also to other quantities which are not necessarily better understood, introducing ambiguities when inferring properties of the magnetic field. The intensity of synchrotron radiation is sensitive to the strength of the magnetic field component orthogonal to the line of sight, however it is modulated by the density of cosmic ray electrons [e.g. @ginzburg-1965]. The direction of this magnetic field component can be studied via the polarization direction of synchrotron radiation and thermal dust emission [e.g. @gardner-1966; @lazarian-2003]. A magnetic field component along the line of sight, on the other hand, gives rise to the effect of Faraday rotation [e.g. @nicholson-1983; @gardner-1966; @burn-1966]. The strength of this effect is influenced not only by the magnetic field but also by the density of thermal electrons. Furthermore, when observing this effect for extragalactic sources, it contains contributions non only from the Galaxy, but rather from every position along the line of sight to the source with a non-vanishing magnetic field and thermal electron density. In order to find an unambiguous terminology capturing these subtleties, we introduce the concept of Faraday depth, which depends on position and is independent of any astrophysical source. The Faraday depth corresponding to a position at a distance $r_0$ from an observer is given by a line of sight integral, $$\phi(r_0)=\frac{e^3}{2\pi m_e^2c^4}\int_{r_0}^0\mathrm{d}r~n_\mathrm{e}(r)B_r(r),$$ over the thermal electron density $n_\mathrm{e}$ and the line of sight component of the magnetic field $B_r$. Here, $e$ and $m_e$ are the electron charge and mass and $c$ is the speed of light. The Galactic Faraday depth is therefore exactly this integral, where the lower boundary is the outer edge of the Milky Way. It is this integral that contains the information on the Galactic magnetic field. The observational consequence of Faraday rotation on a single linearly polarized source is a rotation of its plane of polarization about an angle that is proportional to the square of the wavelength. The proportionality constant is equal to the source’s Faraday depth, i.e the above integral expression, where the lower boundary is now the source’s position. Often, the assumption that the observed polarized radiation stems from a single source is made implicitly and a linear fit to the position angle of the plane of polarization as a function of the squared wavelength is made. We refer to the slope of such a $\lambda^2$-fit as rotation measure (RM). In the case of a single source this is the same as the source’s Faraday depth. However, the polarized radiation will in general be emitted over a range of physical distances and also over a range of Faraday depths, and the position angle will no longer vary linearly with $\lambda^2$. This emission spectrum in Faraday space can be recovered using the technique of RM synthesis [@burn-1966; @brentjens-2005]. In this work we create a map of the Galactic Faraday depth using both data that are based on RM synthesis and data that are based on linear $\lambda^2$-fits. Neither measures the Galactic Faraday depth exclusively and we use the term Faraday rotation data when referring to data values without specifying whether they are rotation measures or the result of a synthesis study. A review of early work on the inference of features of the regular component of the Galactic magnetic field from RM measurements is included in the work of @frick-2001. Some of the studies are done by @morris-1964 [@gardner-1969; @vallee-1973; @ruzmaikin-1977; @ruzmaikin-1978; @simard-normandin-1979; @andreasian-1980; @andreasian-1982; @inoue-1981; @sofue-1983; @vallee-1983a; @agafonov-1988; @clegg-1992; @han-1994; @han-1997], as well as @rand-1989 [@rand-1994] who use RM data of pulsars, and @seymour-1966 [@seymour-1984] who uses spherical harmonics to obtain an all-sky RM map. Some of the more recent studies aiming to constrain the Galactic magnetic field using rotation measures of extragalactic radio sources include the ones by @brown-2001 [@mao-2010; @kronberg-2011; @pshirkov-2011], as well as @brown-2003b [@brown-2007; @nota-2010; @vaneck-2011], who supplement extragalactic RMs with pulsar rotation measures. @weisberg-2004 [@vallee-2005; @vallee-2008; @han-2006; @men-2008] rely entirely on pulsar rotation measures for estimating the Galactic magnetic field, while @sun-2008 [@jansson-2009; @jaffe-2010] use rotation measures of extragalactic sources in combination with synchrotron polarization and intensity data. Recent attempts to create an all-sky map of Faraday rotation measure were made by @frick-2001 [@johnston-hollitt-2004a; @dineen-2005; @xu-2006]. However, due to the limited number of data points available at the time, their reconstructions are limited to the largest-scale features. A rather sophisticated attempt is made by @short-2007, who use Monte Carlo Markov Chain methods and account for uncertainty in the noise covariance while avoiding the direct involvement of covariance matrices. Realistic attempts to create all-sky maps including smaller-scale features have been possible only since @taylor-2009 published the NRAO VLA Sky Survey (NVSS) [@condon-1998] rotation measure catalog that contains data on sources distributed roughly equally over the sky at declinations larger than $-40^\circ$. One such attempt is made in the same publication where the data are simply smoothed to cover the celestial sphere in regions where data are taken. Another attempt has been made by @oppermann-2011a, using a more sophisticated signal reconstruction algorithm which takes into account spatial correlations without oversmoothing any maxima or minima. The NVSS rotation measure catalog is, however, suboptimal in two respects. It lacks data in a large region in the southern sky below the declination of $-40^\circ$ due to the position of the observing telescope (VLA) and its rotation measure values were deduced using only two nearby frequency channels (see Table \[tab:datasets\]). This increases the risk of introducing offsets of integer multiples of $\pi$ in the rotation angle, as discussed by @sunstrum-2010, and makes it impossible to detect any deviations from a proportionality to $\lambda^2$ in the polarization angle. Thus, sources with a non-trivial Faraday spectrum could not be identified and were assigned a possibly misleading RM value. In this work we aim to create a map of the Galactic Faraday depth that summarizes the current state of knowledge. To this end we combine the NVSS rotation measure catalog of @taylor-2009 with several other catalogs of Faraday rotation data of polarized extragalactic radio sources, increasing the spatial coverage and further constraining the signal also in regions where several data sets overlap. We improve on the map of @oppermann-2011a by using this more extensive data set and by using an extended version of the reconstruction algorithm which takes into account uncertainties in the noise covariance, presented by @oppermann-2011b. The resulting all-sky map of the Galactic Faraday depth will be useful in many respects. On the one hand, all-sky information can help in bringing forth global features of the underlying physics, such as the Galactic magnetic field or the electron distribution. On the other hand, an all-sky map can also be useful when studying local or extragalactic features. It could, for example, serve as a look-up table for Galactic contributions to the Faraday depth when studying extragalactic objects. The remainder of this paper is organized as follows: In Sect. \[sec:algorithm\] we briefly review the main features of the extended critical filter algorithm that we use in our map making procedure and discuss how it is applied to the situation at hand. The data sets entering the reconstruction are listed in Sect. \[sec:datasets\] and the results are presented in Sect. \[sec:results\]. In the results section, we also include a brief discussion of the reconstructed angular power spectrum. We summarize our findings in Sect. \[sec:conclusions\]. Reconstruction algorithm {#sec:algorithm} ======================== In order to reconstruct the Galactic Faraday depth from the point source measurements, we use the extended critical filter formalism that was presented by @oppermann-2011b. This filter is based on the critical filter that was used for the reconstruction by @oppermann-2011a and derived by @ensslin_frommert-2011 and @ensslin_weig-2010 within the framework of information field theory developed by @ensslin-2009. Signal model ------------ The signal model we use is the same as the one used by @oppermann-2011a. We review the essentials briefly. In the inference formalism we employ, it is assumed that a linear relationship, subject to additive noise, exists between the observed data $d$ and the signal field $s$ that we try to reconstruct, i.e. $$d=Rs+n.$$ Here, the response operator $R$ describes the linear dependence of the data onto the signal. Formally, the signal could be a continuous field, e.g. some field like the Galactic Faraday depth on the celestial sphere. In practice, however, the best we can hope for is to reconstruct a discretized version of such a field, i.e. a pixelized sky-map. In this case, one can think of the signal field $s$ on the sphere as a vector of dimension $N_\mathrm{pixels}$, each component of which corresponds to one pixel, and the whole set of data points $d$ as another vector of dimension $N_\mathrm{data}$. The response operator then becomes a matrix of dimension $N_\mathrm{data}\times N_\mathrm{pixels}$ and $n$ is another vector of dimension $N_\mathrm{data}$ that contains the noise contributions to each data point. Next, we specify the definitions of the signal field and the response matrix for our specific application. The critical filter algorithm, as well as the extended critical filter, is intended to reconstruct statistically isotropic and homogeneous random signal fields. We briefly recapture the meaning of this. It is assumed in the derivation of the filter formulas [see @oppermann-2011b for details], that the signal field that describes nature is one realization of infinitely many possible ones. Further, it is assumed that some of these possibilities are a priori more likely to be realized in nature than others, i.e. a prior probability distribution function on the space of all possible signal realizations is defined. We assume this probability distribution to ba a multivariate Gaussian with an autocorrelation function $S(\hat{n},\hat{n}')$. Here, $\hat{n}$ and $\hat{n}'$ denote two positions on the celestial sphere. Now assuming statistical homogeneity and isotropy means assuming that $S(\hat{n},\hat{n}')$ depends only on the angle between the two positions $\hat{n}$ and $\hat{n}'$. This means that the correlation of the value of the signal field at one position with another one at a certain distance depends only on this distance, not on the position on the sphere (homogeneity) and not on the direction of their separation (isotropy). Note, however, that we are making this assumption only for the prior probability distribution, i.e. the inherent probability for signal realizations. The data can (and do) break this symmetry, making the posterior probability distribution, i.e. the probability for a signal realization given the measured data, anisotropic. Furthermore, any single realization of a signal with isotropic statistics can appear arbitrarily anisotropic. Extremely anisotropic realizations will, however, be a priori more unlikely than others. For this reason we divide out the most obvious largest scale anisotropy introduced by the presence of the Galactic disk. We do this by defining our signal as $$s(l,b)=\frac{\phi(l,b)}{p(b)},$$ i.e. the dimensionless ratio of the Galactic Faraday depth $\phi$ and a variance profile $p$ that is a function of Galactic latitude only. We use this simplistic ansatz for the Galactic variance profile in order to account for the largest scale anisotropies without using any specific Galactic model in the analysis. The profile function is calculated in a multi-step procedure. In the first step, we sort the data points into bins of Galactic latitude and calculate the root mean square value for the Faraday rotation data of each bin, disregarding any information on Galactic longitude of the data points. We then smooth these values with a kernel with $10^\circ$ FWHM[^2] to form an initial profile function $\tilde{p}$. In the second step, we reconstruct the signal field, resulting in a map $\tilde{m}$ and the corresponding $1\sigma$ uncertainty map $\hat{\tilde{D}}^{1/2}$. We use these to calculate the corresponding posterior mean of the squared Faraday depth according to $$\left<\phi^2\right>_{\mathcal{P}(s\|d)}=\tilde{p}^2\tilde{m}^2+\tilde{p}^2\hat{\tilde{D}}.$$ The posterior mean is the ensemble average over all possible signal configurations weighted with their posterior probability distribution $\mathcal{P}(s\|d)$, i.e. their probability given the measured data, and is denoted by $\left<\cdot\right>_{\mathcal{P}(s\|d)}$. From this expected map of the squared Faraday depth, we then calculate a new variance profile $p$, now using the pixel values of the map instead of the data points. A few data points were added before repeating this final step yet another time. The final reconstruction is then conducted with the resulting profile function. Both the initial variance profile and the one used in the final reconstruction are shown in Fig. \[fig:profile\]. The drop-off toward the Galactic poles of the first-guess profile function is less pronounced since the relatively high noise component of the Faraday rotation data in these regions enters in the root mean square that is calculated from the data points. The variance profile as calculated from the final results is also shown in Fig. \[fig:profile\]. Having introduced the Galactic variance profile, we can now specify the response operator. In our application, the response matrix $R$ needs to contain both the multiplication of the signal field with this profile function and the probing of the resulting Faraday depth in the directions of the point sources. It is a matrix of dimension $N_\mathrm{data}\times N_\mathrm{pixels}$. Each row corresponds to one data point and each column to one pixel of the sky map. Here, the row corresponding to the $i$-th data point contains a non-zero entry only in the column corresponding to the pixel in which the $i$-th observed extragalactic source lies, modeling the probing of the Faraday depth in the observed directions. This entry is the value of the Galactic variance profile $p$ at the latitude of the pixel, effectively rescaling the local signal field value into a Faraday depth. Furthermore, we assume Gaussian priors both for the signal and for the noise with covariance matrices $S$ and $N$, respectively. Since our signal field is assumed to be statistically homogeneous and isotropic, its covariance matrix $S$ is completely determined by its angular power spectrum[^3] $(C_\ell)_\ell$, $\ell=0,1,\dots,\ell_\mathrm{max}$. The minimum length scale $\ell_\mathrm{max}$ is determined by the finite resolution of the discretization. Assuming uncorrelated noise for all data points, the noise covariance $N$ becomes diagonal. The diagonal entries are given by the variance calculated from the error bars given in the data catalogs, modified to account for the expected average extragalactic contribution, $$\sigma^2=\sigma_\mathrm{(measurement)}^2+\sigma_\mathrm{(extragalactic)}^2. \label{eq:sigmacorr}$$ We include a multiplicative correction factor $\eta$ that will be determined during the reconstruction, making the diagonal entry of $N$ corresponding to the $i$-th data point $$N_{ii}=\eta_i\sigma_i^2.$$ As the extragalactic contribution, we use the value $\sigma_\mathrm{(extragalactic)}=6.6~\mathrm{rad}/\mathrm{m}^{2}$, motivated by the study of @schnitzeler-2010. Reasons for a deviation of $\eta$ from unity could be a general under-estimation of the measurement error, as was discussed for the NVSS catalog by @stil-2011, a misestimation of the extragalactic contribution, a multi-component Faraday depth spectrum, but also the presence of an offset of an integer multiple of $\pi$ in the rotation angle. The extended critical filter ---------------------------- The extended critical filter [see @oppermann-2011b] is a method to simultaneously reconstruct the signal, its covariance, given here by its angular power spectrum $(C_\ell)_\ell$, and the noise covariance, given here by the correction factors $(\eta_i)_i$. To this end, inverse Gamma distributions are assumed as priors for the parameters of the covariances, i.e. $$\mathcal{P}(C_\ell)=\frac{1}{q_\ell\Gamma(\alpha_\ell-1)}\left(\frac{C_\ell}{q_\ell}\right)^{-\alpha_\ell}\exp\left(-\frac{q_\ell}{C_\ell}\right)$$ and $$\mathcal{P}(\eta_i)=\frac{1}{r_i\Gamma(\beta_i-1)}\left(\frac{\eta_i}{r_i}\right)^{-\beta_i}\exp\left(-\frac{r_i}{\eta_i}\right), \label{eq:etaprior}$$ and all these parameters are assumed to be independent. We choose $\alpha_\ell=1$ for the parameter describing the slope of the power law and $q_\ell=0$ for the parameter giving the location of the exponential low-amplitude cutoff, turning the prior for each $C_\ell$ into Jeffreys prior which is flat on a logarithmic scale, enforcing the fact that we have no a priori information on the power spectrum. For the prior of the correction factors we choose the parameter $\beta_i=2$, since we already have information on the expected noise covariance from the data catalogs. We adapt the value of $r_i$ such that the a priori expectation value of $\log\eta$ becomes $0$, thereby conforming with the catalogs. With these values, the actual filtering process consists of iterating the three equations[^4] $$\label{eq:WF} m=DR^\dagger N^{-1}d,$$ $$\label{eq:Cl} C_\ell=\frac{1}{2\ell+1}\mathrm{tr}\left(\left(mm^\dagger+D\right)S_\ell^{-1}\right),$$ and $$\label{eq:eta} \eta_i=\frac{1}{2\beta_i-1}\left[2r_i+\frac{1}{\sigma_i^2}\left(\left(d-Rm\right)_i^2+\left(RDR^\dagger\right)_{ii}\right)\right]$$ until convergence is reached. Here, $m$ is the reconstructed signal map, the $\dagger$-symbol denotes a transposed quantity, and $D=\left(S^{-1}+R^\dagger N^{-1}R\right)^{-1}$ is the so-called information propagator [@ensslin-2009]. The matrix $S_\ell^{-1}$ projects a signal vector onto the $\ell$-th length-scale by keeping only the degrees of freedom represented by spherical harmonics components with the appropriate azimuthal quantum number. Although we have chosen $\beta_i=2$ for our reconstruction, we leave the parameter unspecified in these equations, since we later compare our results to those obtained with $\beta\neq2$ (see Sect. \[sec:noise-rec\]). The three equations can be qualitatively explained. Eq. links the reconstructed map to the data. It consists of a response over noise weighting of the data and an application of the information propagator to the result. The information propagator combines knowledge about the observational procedure encoded in the response matrix $R$ and the noise covariance matrix $N$ with information on the signal’s correlation structure contained in the signal covariance matrix $S$. It is used in Eq. to reconstruct the map at a given location by weighting the contributions of all data points using this information. The information propagator is also (approximatively) the covariance matrix of the posterior probability distribution. Therefore, it can be used to obtain a measure for the uncertainty of the map estimate. The 1$\sigma$ uncertainty of the map estimate in the $j$-th pixel is given by $\hat{D}^{1/2}_j=D_{jj}^{1/2}$. Eq. estimates the angular power spectrum from two contributions. The first term in the trace gives the power contained within a reconstructed map, while the second term compensates for the power lost in the filtering procedure generating this map. This second contribution is not contained in the map calculated via Eq. since the data are not informative enough to determine the locations of all features. In a very similar fashion, Eq. estimates the correction factors for the error bars also from two main contributions. The first contribution uses simply the difference between the observed data and the data expected from the reconstructed map and the second contribution compensates partly for the attraction the data exhibit onto the map in the reconstruction step which lets some fraction of the noise imprint itself onto the map. Both contributions are rescaled by the inverse noise variance to turn this estimate of the noise variance into a correction factor. There is a third term in Eq. that is solely due to the prior we chose for $\eta$. It prevents the error bars from vanishing in case a data point is by chance in perfect agreement with the map. For a detailed derivation of these formulas, the reader is referred to @oppermann-2011b. We include a smoothing step for the angular power spectrum in each step of the iteration, where we smooth with a kernel with $\Delta_\ell=8$ FWHM, lowering $\Delta_\ell$ for the lowest $\ell$-modes. This is done to avoid a possible perception threshold on scales with little power in the data [see @ensslin_frommert-2011]. The smoothing step is also justified by the fact that none of the underlying physical fields, i.e. the thermal electron density and the line of sight component of the magnetic field, are expected to have vastly different power on neighboring scales. Data sets {#sec:datasets} ========= Table \[tab:datasets\] summarizes the data catalogs that we use for the reconstruction. Altogether, the catalogs contain 41330 measurements of the Faraday rotation of extragalactic point sources. Fig. \[fig:datadist\] shows their distribution on the sky. The coverage is clearly far from complete, especially at declinations below $-40^\circ$ where the Taylor-catalog does not provide any data. However, 24% of the data points from the other catalogs lie within this region, so some toeholds are present even there. The densely sampled region that stands out at the top of the empty patch in Fig. \[fig:datadist\] is Centaurus A, studied in the Feain-catalog. The relative scarcity of data points near the Galactic plane is due to numerous depolarization effects caused by nearby structures in the magneto-ionic medium, as explained by @stil-2007. We use only extragalactic sources, and not pulsar rotation measures, since this ensures that each measurement contains the full Galactic Faraday depth. ![\[fig:datadist\]Distribution of the data points on the sky. Shown is a <span style="font-variant:small-caps;">HEALPix</span> map at a resolution of $N_\mathrm{side}=128$, using Galactic coordinates. The map is centered on the Galactic center, latitudes increase upward, and longitudes increase to the left. Each black pixel contains at least one data point.](datadist.pdf){width="\columnwidth"} Since the regions of coverage of the different catalogs overlap some of the data points have the same underlying radio source. While this does not constitute a problem for the reconstruction algorithm, it does in principle lead to noise correlations since the intrinsic Faraday rotation of this source, which is part of the noise in our formalism, enters each of these data points in the same way. We ignore this effect in favor of a greatly simplified analysis. The combination of the response matrix and the inverse noise covariance matrix in Eq. corresponds to an inverse noise weighted averaging of all data points that fall within one sky pixel. If the error bars were only due to the intrinsic Faraday rotation of the sources, this would amount to an underestimation of the error bar by a factor $1/\sqrt{k}$ for a source that appears in $k$ different catalogs. In reality, the intrinsic Faraday rotation constitutes only a fraction of the total error budget. The effect is therefore smaller. Some of the catalogs listed in Table \[tab:datasets\] are themselves compilations of earlier measurements. As a consequence, some individual observations are contained in several of the catalogs. We have removed data points where we suspect such duplications so that each observation is used only once. Note that this does not apply to different observations of the same source, as discussed above. The number of data points given in Table \[tab:datasets\] is the effective number of data points that we use in our analysis from the respective catalog. Any variation of the Galactic Faraday depth within one pixel of our map can naturally not be reconstructed. Such variations on very small scales have been detected by @braun-2010 for a region around $(l,b)\approx(94^\circ,-21^\circ)$. Should several sources fall within a pixel in such a region, our algorithm will yield an appropriate average value for the pixel and increase the error bars of the data points until they are consistent with this average value. The sources studied in the Bonafede-catalog and some of the sources in the Clarke-catalog lie within or behind galaxy clusters. They are therefore expected to have an increased extragalactic contribution to their measured Faraday rotation. In order to take the cluster contribution into account, we have corrected the error bars of these points accoring to $$\sigma_{\mathrm{(corrected)}}^2=\sigma^2+\sigma_{\mathrm{(cluster)}}^2.$$ To estimate the cluster contribution $\sigma_{\mathrm{(cluster)}}$, @bonafede-2010 studied resolved background sources for which several independent RM measurements are possible. $\sigma_{\mathrm{(cluster)}}$ was then identified with the empirical value of the standard deviation of these measurements. @clarke-2001 estimated the cluster contribution by comparing the RM values of sources within the cluster to those of sources behind the cluster. The Johnston-Hollit-B-catalog also contains sources associated with galaxy clusters. However, due to the low density of sources, an estimation of the cluster contribution is not possible in this case. We expect a fraction of the other sources to be affected by clusters as well. However, since information on which sources exactly are affected is missing in general, we leave it to our algorithm to increase the error bars of the appropriate data points. The same problem exists in principle for satellite galaxies of the Milky Way, such as the Large and Small Magellanic Clouds. We do not attempt to separate their contribution to the Faraday depth from the one of the Milky Way, so that the map we reconstruct is strictly speaking not a pure map of the Galactic Faraday depth, but rather a map of the Faraday depth of the Milky Way and its surroundings. Due to our use of spatial correlations in the reconstruction algorithm, the Faraday depth contribution intrinsic to the sources will, however, be largely removed. ![image](map.pdf){width="17cm"} ![image](phi.pdf){width="17cm"} Furthermore, some of the sources will have a non-trivial Faraday spectrum, i.e. they exhibit polarized emission at more than one Faraday depth. While the technique of RM synthesis [@burn-1966; @brentjens-2005] is able to make out these sources, such features are not described by a $\lambda^2$-fit, which may thus lead to an erroneous rotation measure value. This problem becomes more severe if the number of frequencies used in the fit is low. In the limit of two frequencies, multi-component Faraday spectra necessarily go unnoticed. We use the data points obtained by $\lambda^2$-fits of only a few frequencies nevertheless, and leave it to the reconstruction algorithm to increase the error bars of those with an underlying multi-component spectrum accordingly. Results {#sec:results} ======= All results shown here are calculated at a <span style="font-variant:small-caps;">HEALPix</span>[^5] resolution of $N_\mathrm{side}=128$, i.e. the maps contain 196608 pixels. The minimum angular scale that we consider is $\ell_\mathrm{max}=383$, corresponding roughly to half a degree. These results are publicly available and can be downloaded from <http://www.mpa-garching.mpg.de/ift/faraday/>. The maps that we show are all centered on the Galactic center with positive Galactic latitudes at the top and positive Galactic longitudes plotted to the left. Map --- Figure \[fig:map\] shows the reconstructed dimensionless signal map $m$ and an estimate for its uncertainty, given by $\hat{D}^{1/2}$. The same for the physical Galactic Faraday depth $pm$, i.e. the signal multiplied by the Galactic variance profile, is shown in Fig. \[fig:phi\]. As expected, the signal reconstruction is more uncertain in regions that lack data. Furthermore, the uncertainty in Fig. \[fig:map\] tends to be smaller in the Galactic plane. This is due to the higher signal response brought along by the Galactic variance profile in this area. When considering the uncertainty of the final map of the Faraday depth, i.e. the bottom panel of Fig. \[fig:phi\], this feature gets turned around. The values within the Galactic plane now tend to be more uncertain than the ones near the poles. Note, however, that this is the absolute uncertainty. Since the Galactic Faraday depths are greater for lines of sight through the Galactic disk as well, the relative uncertainty is smaller there. This corresponds roughly to the uncertainty shown in the bottom panel of Fig. \[fig:map\] which can be interpreted as the uncertainty of the Galactic Faraday depth relative to the value of the Galactic variance profile at the specific latitude. Also, the uncertainty is only high in the Galactic plane in pixels that do not contain any data. In the pixels that contain measurements, the uncertainty is comparable to the error bars of the data. It should be noted, however, that due to the approximations made in the derivation of the filter formulas [for details, see @oppermann-2011b], the presented 1$\sigma$ intervals cannot be interpreted as containing $68\%$ of the correct pixel values of the signal. @oppermann-2011b found in their mock tests, that about $50\%$ of the correct pixel values lie within this range. In general, Fig. \[fig:map\] is better suited to make out localized features away from the Galactic plane. The most striking of these features is the quadrupole-like structure on large scales that favors positive Faraday depths in the upper left and lower right quadrant and negative Faraday depths in the upper right and lower left quadrant. This has been observed in measurements of Faraday rotation in the past, first by @simard-normandin-1980, and has often been claimed to be due to a toroidal component of the large scale Galactic magnetic field that changes sign over the Galactic plane [see e.g @han-1997]. Recent studies by @wolleben-2010 and @mao-2010 have shown, however, that this pattern is probably at least partly due to local features of the interstellar medium in the solar neighborhood. At Galactic longitudes beyond roughly $\pm100^\circ$, this pattern turns into a dipolar structure, favoring negative values at the left edge of the map and positive ones on the very right, as noted previously by @kronberg-2011. This might be a signature of a toroidal magnetic field component that does not change sign over the Galactic plane. But of course this could also be a local effect, independent of the large scale magnetic field. ![\[fig:annotated\]Same as the top panel of Fig. \[fig:map\], with markings around the regions discussed in the text. The letters labeling the regions are used for reference in the main text. Dashed lines denote lines of constant Galactic longitude or latitude. Their angular separation is $30^\circ$.](annotated.pdf){width="\columnwidth"} Many other features are visible in the top panel of Fig. \[fig:map\]. We have marked some of the features that have already been discussed in the literature in Fig. \[fig:annotated\] for easier reference. @simard-normandin-1980 identified three large regions (A, B, and C in Fig. \[fig:annotated\]) with large angular size that stand out in Galactic Faraday depth amplitude. @stil-2011 narrowed the definitions of the regions A and C down to their more striking parts using the NVSS RM catalog. Region A is a large area of negative Galactic Faraday depth localized roughly at $80\degr < l < 150\degr$, $-40\degr < b < -20\degr$. This region is seen in the direction of radio Loop II, but there is little evidence that the two are associated. The high-longitude boundary of region A coincides with part of the edge of Loop II. However, pulsar rotation measures suggest that Region A extends more than 3 kpc along the line of sight [@simard-normandin-1980], which suggests that region A is a much larger structure. Region B of @simard-normandin-1980 is associated with the Gum nebula. @vallee-1983b and @stil-2007 identified a large magnetic shell in the area. The arc of positive Galactic Faraday depth around $250~\mathrm{rad}/\mathrm{m}^2$ at $-120\degr < l < -90\degr$, $b \approx 13^\circ$ (region b1 in Fig. \[fig:annotated\]) coincides with the northern H$\alpha$ arc of the Gum nebula. A small excess in Galactic Faraday depth (region b2 in Fig. \[fig:annotated\]) is associated with the nearby HII region RCW 15 ($l = -125^\circ$, $b = -7^\circ$). Region C is an area of positive Galactic Faraday depth in the range $33\degr < l < 68^\circ$, $10\degr < b < 35^\circ$ near the boundary of Radio Loop I. @wolleben-2010 found diffuse polarized emission at a Faraday depth of $60~\mathrm{rad}/\mathrm{m}^2$ at $l \approx 40^\circ$, $b \approx 30^\circ$ with associated HI structure, and interpreted this structure as part of a separate super shell around a subgroup of the Sco-Cen (Sco OB2\_2). Besides the Gum nebula, some extended HII regions at intermediate Galactic latitude can be identified in the form of a localized excess in Galactic Faraday depth [@stil-2007; @harvey-smith-2011]. The HII regions Sh 2-27 around $\zeta$ Oph at $l= 8^\circ, b = 23.5^\circ$ (region d in Fig. \[fig:annotated\]) and Sivan 3 around $\alpha$ Cam at $l = 144.5^\circ$, $b = 14^\circ$ (region e in Fig. \[fig:annotated\]) stand out as isolated regions of negative Galactic Faraday depth, while Sh 2-264 around $\lambda$ Ori (region f in Fig. \[fig:annotated\]) is visible as a positive excess at $l = 195$, $b = -12$. @stil-2011 presented an image of H$\alpha$ intensity with rotation measure data overplotted. Some large shells are also visible in the image of the Galactic Faraday depth. The Galactic anti-center direction is the most favourable direction to see these large structures, because it is less crowded than the inner Galaxy and the line of sight makes a large angle with the large-scale magnetic field. The North Polar Spur (region g in Fig. \[fig:annotated\]) is the notable exception toward the inner Galaxy. The filament of positive Galactic Faraday depth at $180^\circ < l < 200^\circ$, $b \approx -50^\circ$ (region h in Fig. \[fig:annotated\]) is associated with the wall of the Orion-Eridanus superbubble [@heiles-1976; @brown-1995]. A large arc of positive Galactic Faraday depth (region i in Fig. \[fig:annotated\]) rises north of the Galactic plane at around $l \approx 95^\circ$ up to $b \approx 65^\circ$ around $l = 180^\circ$ and curves back to the Galactic plane at around $l = 210^\circ$ [@stil-2011]. This arc of positive Galactic Faraday depth traces the intermediate-velocity arch of atomic hydrogen gas identified by [@kuntz-1996]. @xu-2006 reported RM excesses in the direction of the nearby Perseus-Pisces and Hercules super clusters. The higher sampling provided by the new Faraday rotation data catalogs has revealed high-latitude structures in the Galactic Faraday depth that warrant further investigation in the effect of the Galactic foreground. Many more small- and intermediate-scale features are visible in the top panel of Fig. \[fig:map\]. A detailed analysis of these features is left for future work. Reconstruction of the noise covariance {#sec:noise-rec} -------------------------------------- The extended critical filter adapts the correction factors $(\eta_i)_i$, introduced in Sect. \[sec:algorithm\], so as to make the error bars of the data conform with the local map reconstruction. This is influenced by the surrounding data points and the angular power spectrum, which is in turn reconstructed using the entirety of the data. @oppermann-2011b showed that allowing for this adaptation of the error bars leads to a slight oversmoothing of the reconstructed map since small-scale features that are only supported by individual data points get easily misinterpreted as noise. In our reconstruction, we find that the median correction factor is $\eta^\mathrm{(med)}=0.56$. This indicates that the bulk of the data points are rather consistent with one another and therefore with the reconstruction as well. As a consequence, their error bars are not enlarged but rather slightly decreased by the algorithm. Oversmoothing can therefore not be a serious issue for the map as a whole. This is supported by the geometric mean of the correction factors, for which we find $\eta^{\mathrm{(geom)}}=0.75$. This corresponds to the arithmetic mean on a logarithmic scale and its prior expectation value was tuned to be one. Looking at the arithmetic mean on a linear scale, we find $\eta^\mathrm{(mean)}=6.40$, indicating that there are at least a few data points whose error bars get corrected upward significantly. In fact, there are $134$ data points with $\eta_i>400$, meaning that the error bar has been increased by a factor of more than $20$. These are isolated outliers in the data that are not consistent with their surroundings. Figure \[fig:hist\] shows the final distribution of $\eta$-values. The bulk of these values lie around $\eta=1$ or even slightly below. Only relatively few data points have highly increased error bars (note the logarithmic scale of the vertical axis in Fig. \[fig:hist\]). Also plotted in Fig. \[fig:hist\] is the distribution of $\eta$-values that resulted from a reconstruction in which the slope parameter in the prior for the correction factors was chosen to be $\beta=3$, as well as the prior probability distributions corresponding to $\beta=2$ and $\beta=3$. This shows two things. The resulting distribution does not change much when the value of $\beta$ is changed and both distributions are better represented by a prior with $\beta=2$. Our choice for $\beta$ is thus justified. The data points with $\eta\gg1$ do not appear to be spatially clumped, making it improbable that any extended physical features that are present in the data are lost due to the increase in the assumed noise covariance. Any real features that might mistakenly be filtered out in this procedure can be expected to be smaller or comparable in size to the distance to the next data point, i.e. one or two pixels or about one degree in most parts of the sky. The data points with strongly corrected error bars are predominantly located near the Galactic plane. This can be clearly seen in Fig. \[fig:hist\_latitude\], where we plot the distribution of the correction factors for three latitude bins separately. While the difference in the distributions for the polar regions and the intermediate latitude bin is not very big, the data points around the Galactic disk clearly are more likely to have correction factors at the high end. At least in some cases these high $\eta$-values can be interpreted as correcting an offset in the rotation angle of $\pi$ that has escaped the observational analysis. Others might be due to a high level of polarized emissivity within the Galactic disk that can lead to misleading RM fits. Another reason for high $\eta$-values is a higher extragalactic contribution to the measured Faraday rotation, caused e.g. by magnetic fields in galaxy clusters. This last reason, however, would not be expected to show any statistical latitude dependence. As mentioned earlier, a non-trivial emission spectrum in Faraday space is hard to identify when using linear $\lambda^2$-fits to obtain RM values. We therefore compare the distributions of the correction factors for data points from $\lambda^2$-fits and the ones for data points that stem from RM synthesis studies in Fig. \[fig:hist\_method\]. From the histograms it can indeed be seen that the data from $\lambda^2$-fits are more likely to have a high $\eta$-value, as expected. ![\[fig:oldcomp\]Comparison of the reconstruction of the dimensionless signal to earlier results. The top panel shows the reconstructed signal field of @oppermann-2011a, the middle panel shows the same as the top panel of Fig. \[fig:map\], only coarsened to a resolution of $N_\mathrm{side}=64$ to match the resolution of the old reconstruction. The bottom panel shows the difference between the upper panel and the middle panel.](oldcomp.pdf){width="\columnwidth"} Figure \[fig:oldcomp\] shows a comparison of our reconstructed signal map with the reconstruction of @oppermann-2011a, where the critical filter formalism was used without accounting for uncertainties in the noise covariance and only data from the Taylor-catalog were used. The differences that can be seen are twofold. On the one hand, our map shows structure due to the additional data points that we use, most prominently at declinations below $-40^\circ$. On the other hand, some of the features present in the older map have vanished since they were supported only by a single data point which has been interpreted as being noise-dominated by our algorithm. These features appear prominently both in the old map and in the difference map, where our newly reconstructed map has been subtracted from the old one. They have the same sign in both these maps. Also, our new reconstruction is less grainy. This is a combined effect of the higher resolution that we use and the adaptation of error bars during our reconstruction. Power spectrum -------------- The reconstructed angular power spectrum of the dimensionless signal field is shown in Fig. \[fig:angspec\]. It is well described by a power law. A logarithmic least square fit, which is also shown in Fig. \[fig:angspec\], yields a spectral index of $2.17$, i.e. $$C_\ell\propto\ell^{-2.17}, \label{eq:Clfit}$$ where we have taken scales down to $\ell=300$ into account. Note that due to the typical distance of neighboring data points of roughly one degree, structures smaller than this angular size, corresponding to $\ell\gtrsim180$, will in general not be reconstructed and we might therefore be missing some power on the smallest scales. However, some data points have smaller angular separations and we therefore have some information on the angular power spectrum up to $\ell_\mathrm{max}=383$. Also shown in Fig. \[fig:angspec\] is a comparison with the angular power spectra of the maps that @dineen-2005 reconstructed. They created three separate maps from three different RM catalogs. We used the spherical harmonics components of their maps[^6], transformed them to position space, and then divided them by our Galactic variance profile. We plot the angular power spectra of the three resulting dimensionless maps. Evidently, both the slope and the normalization of the spectra are in agreement with our result. @haverkorn-2003 study the angular power spectrum of rotation measures of diffuse polarized radio emission from the local interstellar medium in two regions of the sky on scales $400<\ell<1500$. They fit power laws with exponents close to $-1$, i.e. $C_\ell\propto\ell^{-1}$, significantly larger than our result. This is not necessarily a contradiction, however, since a flattening of the angular power spectrum on scales that are too small for our analysis could explain both results. Furthermore, we take into account the full line of sight through the galaxy by using only extragalactic sources, so the volume that we probe is significantly larger than the one probed by @haverkorn-2003. In order to compare our result to other earlier papers, we consider the second order structure function for the dimensionless signal field, $$\begin{aligned} \label{eq:structurefunction} D_s(\vartheta)&=\left<\left(s(\hat{n})-s(\hat{n}')\right)^2\right>_{\mathcal{P}(s)}\nonumber\\ &=2\left(S_{\hat{n}\hat{n}}-S_{\hat{n}\hat{n}'}\right),\end{aligned}$$ where $\vartheta=\arccos(\hat{n}\cdot\hat{n}')$ and $\hat{n}$ and $\hat{n}'$ are two directions in the sky. Here, $S$ denotes the signal covariance matrix and the angle brackets denote a prior ensemble average. Since we assume statistical homogeneity and isotropy for the signal field, $S_{\hat{n}\hat{n}}$ does not depend on $\hat{n}$, $S_{\hat{n}\hat{n}'}$ depends only on $\vartheta$, and both terms are completely determined by the angular power spectrum. This also allows us to exchange the usual spatial average with an ensemble average in Eq. . The resulting structure function is plotted in Fig. \[fig:structurefunction\]. Using the final angular power spectrum of our reconstruction (the solid line in Fig. \[fig:angspec\]), we find a broken power law with exponents $0.65$ for small angles and $0.26$ for large angles with the transition occuring around $\vartheta=5^\circ$ (the solid line in Fig. \[fig:structurefunction\]). The power law fit to the angular power spectrum (the dashed line in Fig. \[fig:angspec\]) leads to a structure function that can be approximated by a single power law with exponent $0.39$ (the dashed line in Fig. \[fig:structurefunction\]). @minter-1996 found that the structure function derived from their observations is well described by a power law with exponent $0.64$ for angular scales of $\vartheta>1^\circ$. @sun-2004 study the structure function in three different regions within the Galactic plane and in the vicinity of the North Galactic pole. An inverse noise weighted average of their power law indices yields a value of $0.11$. @haverkorn-2006a and @haverkorn-2008 study observations through interarm regions in the Galactic plane separately from observations along Galactic arms. They find flat structure functions for the observations along Galactic arms. @haverkorn-2006a find a weighted mean power law index of $0.55$ for the structure functions derived from observations through interarm regions, while @haverkorn-2008 find an inverse-noise weighted mean power law index of $0.40$. @haverkorn-2003 find flat structure functions for the two regions that they study. @roy-2008 find a structure function for the region around the Galactic center that is constant on scales above $\vartheta=0.7^\circ$ and exhibits a power law behavior with an exponent of $0.7$ on smaller scales. @stil-2011 fit broken power laws with the breaking point at $\vartheta=1^\circ$ to the structure functions they extract from the NVSS rotation measure catalog [@taylor-2009]. They find power law indices that vary spatially. Taking an inverse-noise weighted average of their power law indices for the regions that they study in detail yields $0.37$ for $\vartheta>1^\circ$ and $0.59$ for $\vartheta<1^\circ$. These observational results indicate that the slope of the structure function varies from region to region. Our result is insensitive to these variations since our structure function is just a description of the prior for the dimensionless signal, for which we have assumed statistical isotropy. It can therefore be interpreted as a mean structure function across the whole sky. The observations that yield non-flat structure functions are in rough agreement with the slopes that we fit in Fig. \[fig:structurefunction\]. The dependence of the structure function slope on Galactic latitude [e.g. @simonetti-1984; @sun-2004] is partly removed in our analysis by the division through the Galactic variance profile. Note that @simonetti-1984 [@simonetti-1986] already suspected a break in the structure function at roughly five degrees. However, existing studies have not shown convincing evidence for this. ### Consequences for the 3D fields As an illustrative thought experiment, assume that an observer is sitting in the middle of a spherical distribution of magnetoionic medium. Let $\tilde{\varphi}(\vec{x})\propto n_\mathrm{e}(\vec{x})B_r(\vec{x})$ be the product of the local thermal electron density and the line of sight component of the magnetic field as a function of 3D position $\vec{x}$, i.e. the differential contribution to the Faraday depth that this observer is measuring. We model this field as factorizing into two parts, $$\tilde{\varphi}(\vec{x})=\bar{\varphi}(r)\varphi(\vec{x}).$$ The first part is a spherically symmetric contribution, whose functional dependence on the radial distance from the observer is known, and the second part is assumed to be a realization of a statistically homogeneous and isotropic random field, i.e. $$\label{eq:phistatistics} \left<\varphi(\vec{k})\varphi^(\vec{k}')\right>=\left(2\pi\right)^3\delta^{(3)}(\vec{k}-\vec{k}')P_\varphi(k),$$ where the angle-brackets denote an average over all possible field realizations, $P_\varphi(k)$ is the Fourier power spectrum[^7] that describes the statistics of $\varphi$, and $k=\left\|\vec{k}\right\|$. Using the simplest form of $\bar{\varphi}(r)$, namely a constant within some finite radius $r_0$, i.e. $$\bar{\varphi}(r)=\left\{ \begin{array}{cc} \varphi_0 & \textrm{if}~r<r_0\\ 0 & \textrm{else} \end{array} \right.,$$ and a power law for the Fourier power spectrum, $$P_\varphi(k)\propto k^{-\alpha},$$ we calculated the angular power spectrum of the Faraday depth that the observer would measure and compared the result numerically with Eq. . We find that the two agree well if one chooses $\alpha$ roughly equal to the power law index that was found for the angular power spectrum, i.e. $2.17$ in this case. A similar thought experiment has been conducted by @simonetti-1984. They assume a Fourier power spectrum $P_\varphi(k)\propto\exp\left(-k^2/k_1^2\right)\left(1+k^2/k_0^2\right)^{\alpha/2}$, i.e. a power law with a low-wavenumber cutoff at $k_0$ and a high-wavenumber cutoff at $k_1$, and calculate the expected structure function. In the power law regime, i.e. $1/k_1\ll r_0\sin\vartheta\ll 1/k_0$, they find $D_s(\vartheta)\propto\vartheta^{\alpha-2}$ to lowest order in $\vartheta$. Extending this study to independent variations in the thermal electron density and the magnetic field component along the line of sight, each described by a power law power spectrum with the same index $\alpha$, @minter-1996 found the same dependence on $\vartheta$.[^8] Our intermediate fit of $D_s(\vartheta)\propto\vartheta^{0.39}$ (see Fig. \[fig:structurefunction\]) therefore corresponds to $\alpha=2.39$, in rough agreement with our numerical finding from the power spectrum analysis. @armstrong-1995 have used observations of effects of interstellar radio scintillation [see also @rickett-1977; @rickett-1990], as well as pulsar dispersion measures, to constrain the power spectrum describing the fluctuations of the thermal electron density in the local interstellar medium. They found a Kolmogorov-type power spectrum, i.e. a power law index of $\alpha=11/3$ in the present notation. This result was combined by @minter-1996 with their own observations of rotation measures of extragalactic sources. Since they do not find the slope expected from the Kolmogorov power law in the structure function of the rotation measure they observe, they conclude that the outer scale of the Kolmogorov-type turbulence is smaller than the smallest scale probed by their RM observations. They fit model structure functions for the variations of the thermal electron density and the magnetic field to their own observations of RM, as well as observations of $\mathrm{H}\alpha$ intensity and $\mathrm{H}\alpha$ velocity performed by @reynolds-1980, while also taking into account the results of @armstrong-1995 on smaller scales. This procedure leads to an estimate of the angular scale corresponding to the outer scale of the turbulence in the region of their observations of $\vartheta^{\mathrm{(out)}}\lesssim0.1^\circ$. Although the outer scale of the turbulence may well vary across the Galaxy, it is probably safe to assume that the scales larger than one degree that are mainly probed by the observations used in this work, are not dominated by three-dimensional turbulence. Whether or not the simple power law behavior of the angular power spectrum in Eq. points to some sort of interaction between the fluctuations on different scales is at the moment an open question. In any case it is clear that the simplifying assumptions made in the thought experiments presented above are far from the truth in the Galactic setting. A more realistic study will likely have to involve numerical magneto-hydrodynamical simulations of the interstellar medium, which have become more and more sophisticated over the last years [see e.g. @avillez-2007; @kissmann-2008; @burkhart-2009; @tofflemire-2011]. Cross-checking the angular power spectrum of the Faraday depth that is predicted by such a simulation against Eq. might be a good indicator of how realistic the simulation actually is. For this, an empiric variance profile would have to be calculated from the simulated observations to create a dimensionless signal field comparable to our reconstruction. Numerical studies will also be able to show whether the simple power law that we find for the angular power spectrum is a functional form that arises generically or an outcome that needs certain ingredients. This may then enable a physical interpretation of the angular power spectrum that we find. On the other hand, if simulations show that different physical processes are needed to create the fluctuation power on different angular scales, our result will directly constrain the relative strength of these processes. Conclusions {#sec:conclusions} =========== We have presented a map of the Galactic Faraday depth that summarizes the current state of knowledge, along with its uncertainty. For the map reconstruction we have used the extended critical filter, a state-of-the-art algorithm, yielding a result that is robust against individual faulty measurements. It is this robustness, along with the usage of the most complete data set on the Faraday rotation of extragalactic sources to date, and the high resolution that we are therefore able to reach, that make our map an improvement over existing studies. Along with the map, the reconstruction algorithm yields the angular power spectrum of the underlying signal field, $C_\ell\propto\ell^{-2.17}$, which is in agreement with earlier work. We have discussed the implications of this power spectrum for the statistics of the 3D quantities involved in a greatly simplified scenario and suggested future work on simulations with the possibility of checking predicted angular power spectra against our observational result. All products of this work, i.e. the maps and their uncertainties, as well as the angular power spectrum, are made available to the community[^9] for further analysis, interpretation, and for use in other work where the Galactic Faraday depth plays a role. The authors would like to thank Steven R. Spangler for the valuable contributions he made to this paper as a referee. N.O. thanks Marco Selig and Maximilian Ullherr for fruitful discussions during the genesis of this work. Some of the results in this paper have been derived using the <span style="font-variant:small-caps;">HEALPix</span> [@gorski-2005] package. The calculations were performed using the <span style="font-variant:small-caps;">SAGE</span> [@sage] mathematics software. This research has made use of NASA’s Astrophysics Data System. This research was performed in the framework of the DFG Forschergruppe 1254 “Magnetisation of Interstellar and Intergalactic Media: The Prospects of Low-Frequency Radio Observations”. Basic research in radio astronomy at the Naval Research Laboratory is funded by 6.1 Base funding. B.M.G. and T.R. acknowledge the support of the Australian Research Council through grants FF0561298, FL100100114 and FS100100033. M.J.-H. and L.P. acknowledge support via Victoria University of Wellington Faculty of Science and Marsden Development Fund research grants awarded to M.J.-H. The Australia Telescope Compact Array is part of the Australia Telescope National Facility which is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO. This paper includes archived data obtained through the Australia Telescope Online Archive[^10]. identifyer telescope survey \# observed wavelengths frequency range / MHz method \# data points catalog reference survey reference -------------------- ------------------ ------------ ------------------------- ----------------------- ----------------- ---------------- ------------------- ------------------ Bonafede VLA 3-5 various $\lambda^2$-fit 7 (1) Broten various various various $\lambda^2$-fit 121+3/2 (2) Brown CGPS DRAO ST CGPS 4 1403-1438 $\lambda^2$-fit 380 (3) (4) Brown SGPS ATCA SGPS 12 1332-1436 $\lambda^2$-fit 148 (5) (6),(7) Clarke VLA 4,6 1365-4885 $\lambda^2$-fit 125 (8),(9) Clegg VLA 6 1379-1671 $\lambda^2$-fit 56 (10) Feain ATCA Cent. A 24 1280-1496 RM synthesis 281 (11) (12) Gaensler ATCA SGPS test 9 1334-1430 $\lambda^2$-fit 18 (13) Hammond ATCA 23 1332-1524 RM synthesis 88 (14) Heald WSRT WSRT-SINGS 1024 1300-1763 RM synthesis 57 (15) (16) Hennessy VLA 4 1362-1708 $\lambda^2$-fit 17 (17) Johnston-Hollitt A ATCA 23 1292-1484 RM synthesis 68 (18) Johnston-Hollitt B ATCA 4 1384-6176 $\lambda^2$-fit 12 (19),(20) Kato Nobeyama 4 8800-10800 $\lambda^2$-fit 1 (21) Kim various various various $\lambda^2$-fit 20+1/2 (22) Klein VLA & Effelsberg B3/VLA 4 1400-10600 $\lambda^2$-fit 143 (23) (24),(25) Lawler various various various $\lambda^2$-fit 3 (26) (27) Mao SouthCap ATCA 32 1320-2432 RM synthesis 329 (28) Mao NorthCap WSRT 16 1301-1793 RM synthesis 400 (28) Mao LMC ATCA 14 1324-1436 RM synthesis 188 (29),(30) Mao SMC ATCA 14 1324-1436 $\lambda^2$-fit 62 (31) Minter VLA 4 1348-1651 $\lambda^2$-fit 98 (32) Oren VLA 4,6 various $\lambda^2$-fit 51+4/2 (33) O’Sullivan ATCA 100 1100-2000 RM synthesis 46 (34) Roy ATCA & VLA 4 and more various $\lambda^2$-fit 67 (35) Rudnick VLA 2 1440-1690 $\lambda^2$-fit 17+2/2 (36) Schnitzeler ATCA 12 1320-11448 RM synthesis 178 (37) Simard-Normandin various various various $\lambda^2$-fit 535+6/2 (38) Tabara various various various $\lambda^2$-fit 62+3/2 (39) Taylor VLA NVSS 2 1344-1456 $\lambda^2$-fit 37543 (40) (41) Van Eck VLA 14 1353-1498 RM synthesis 194 (42) Wrobel VLA 6 1373-1677 $\lambda^2$-fit 5+1/2 (43) : \[tab:datasets\]Details of the data sets used for the map reconstruction. [^1]: [^2]: @oppermann-2011a experiment with different smoothing lengths and find that a factor two difference does not matter for the end result. We chose $10^\circ$ by visual inspection of the smoothness of the resulting profile. [^3]: The angular power spectrum is defined by $C_\ell=\left<s_{\ell m}s_{\ell m}^\right>_{\mathcal{P}(s)}$, where $s_{\ell m}$ denotes the signal’s spherical harmonic component of a certain azimuthal quantum number $\ell$ and an arbitrary magnetic quantum number $m$, the asterisk denotes complex conjugation, and the angular brackets denote an ensemble average weighted with the prior probability distribution. [^4]: This is the first order version of the extended critical filter. See @oppermann-2011b for details. [^5]: The <span style="font-variant:small-caps;">HEALPix</span> package is available from <http://healpix.jpl.nasa.gov>. [^6]: @dineen-2005 provide their results at <http://astro.ic.ac.uk/~pdineen/rm_maps/>. [^7]: Note that the definition of the Fourier power spectrum made in Eq. corresponds to what is sometimes referred to as the 3D power spectrum, i.e. the variance of the field $\varphi$ at each position $\vec{x}$ in real space can be calculated as $\left<\varphi^2(\vec{x})\right>\propto\int_0^\infty\mathrm{d}k~k^2P_\varphi(k)$. [^8]: @minter-1996 assume a rectangular shape for $\varphi_0$ instead of a spherical one. [^9]: See <http://www.mpa-garching.mpg.de/ift/faraday/> for a fits-file containing all the results and an interactive map to explore the Galactic Faraday sky. [^10]: <http://atoa.atnf.csiro.au>
The physical properties of atomic gases change dramatically when quantum degeneracy is reached, i.e. when the ground state population approaches unity [@grif:95]. Recent successes in reaching quantum degeneracy with Bose gases [@ande:95; @davi:95; @brad:97] have relied on non-adiabatic, irreversible methods such as laser and evaporative cooling. The possibility of changing the ground state population by an adiabatic change in the trapping potential had been overlooked for quite some time [@kett:92]. Indeed, in the case of an ideal gas, adiabatic changes in the [strength]{} of the trapping potential do not change the ground state population [@reif:65; @footnote:1]. However, Pinkse and collaborators [@pink:97] recently showed, both theoretically and experimentally, that by changing the [form]{} of the trapping potential, the population in the ground state can be changed adiabatically. For a non-degenerate gas the ground state population is identical to the phase-space density $\Gamma = n \lambda_T^3$, where $n$ is the density of the gas and $\lambda_{T}$ is the thermal de Broglie wavelength. Within the type of trap deformations considered in Ref. [@pink:97] the maximum increase of phase-space density is limited to a factor of 20. In this Letter, we show that a more general deformation of the trapping potential can increase the phase-space density by an arbitrary factor, and we describe an implementation of this scheme using a combination of magnetic and optical forces. Furthermore, we demonstrate the ability to cross the Bose-Einstein condensation (BEC) phase transition and to change the condensate fraction of a Bose gas in a reversible way. [Adiabatic increase in phase-space density]{}. The type of trap deformations which we study can be understood with the following “two-box” model. Consider a classical gas of $N$ atoms confined in a box of volume $V_{0}= V_{1} + V_{2}$ with an initial phase-space density $\Gamma_0$. Suppose that the potential within a sub-volume $V_{2}$ of the box is lowered to a final well-depth $U$. In this final potential, the gas equilibrates at a temperature $T$, and the density in $V_2$ will be higher than that in $V_1$ by the Boltzmann factor $e^{U/k_B T}$. Using the condition of adiabaticity and constant particle number, one obtains the relative increase of phase-space density in $V_{2}$ compared to that in $V_{0}$ before compression: $$\ln \left(\Gamma_2/\Gamma_0\right) = \frac{U/k_B T}{1 + (V_2 / V_1) e^{U/k_B T}}~. \label{classical}$$ For deep potential wells, where $U/k_B T \gg \ln(V_1/V_2)$, there is no increase in phase-space density since all of the gas becomes confined in $V_2$, and the adiabatic deformation corresponds simply to a uniform compression of the gas. For shallow potential wells ($U/k_B T \ll \ln(V_1/V_2)$), the phase-space density in $V_2$ increases as $e^{U/k_B T}$. As $U$ is varied between these limits, the phase-space density increase reaches a maximum which is greater than $( V_1/V_2 )^{1/2}$. Thus, by choosing an extreme ratio of volumes $V_1/V_{2}$, an arbitrarily large increase in phase-space density $\Gamma_2/\Gamma_0 $ is possible. To demonstrate this phase-space density increase in a gas of trapped atoms, a narrow potential well (analogous to $V_{2}$) was added to a broad harmonic potential (corresponding to $V_{1}$) by focusing a single infrared laserbeam at the center of a magnetic trap. First a gas of atomic sodium was evaporatively cooled to a temperature higher than the BEC phase transition temperture in the cloverleaf magnetic trap [@mewe:96]. The number of atoms and their temperature were adjusted by varying the final radio frequency (rf) used in the rf-evaporative cooling stage [@kett:96]. Afterwards, the magnetically trapped cloud was decompressed by slowly reducing the currents in the magnetic trapping coils. Time-of-flight absorption imaging was used to characterize the cloud. The total number of atoms $N$ was determined by integrating the column density across the cloud, and $T$ was determined by one-dimensional Gaussian fits to the wings of the density distribution. From these, we determined the fugacity $z$ of the gas by the relation $g_3(z) = N (\hbar \bar\omega / k_B T)^3$, and then its phase-space density by $\Gamma_0 = g_{3/2}(z)$, where $g_{n}(z) =\sum \limits_{i=1}^{\infty} z^{i}/i^{n}$ [@footnote:2]. Here, $\bar\omega$ is the geometric-mean trapping frequency of the magnetic trap, as determined by [in situ]{} measurements [@stam:98b]. These phase-space density measurements were calibrated with images from magnetically trapped clouds at the phase transition. The optical setup was similar to that used in Ref. [@stam:98a]. The infrared laser power was gradually ramped-up from zero to a power $P_c$ at which the onset of BEC was seen in time-of-flight images of clouds released from the deformed trap; this implied $\Gamma_f = g_{3/2}(1)=2.612$ for the final phase-space density. The depth of the optical potential well was given by $U_c/k_{B} = 37 \mu\text{K}~ P_c / w_{0}^{2} ~(\mu\text{m}^{2}/\text{mW})$, where $w_{0}$ is the $1/e^{2}$ beam-waist radius at the focus. The ramp-up time was made long enough to ensure that the trap deformation was adiabatic, but also short enough to minimize heating and trap loss. Ramp-up times of up to 10 s were used. Fig. \[fig1\]a shows the increases in maximum phase-space density which were measured at three different settings of the trap parameters. A maximum increase by a factor of 50 was obtained. Condensates were observed in clouds with temperatures as high as 5 $\mu$K. Further increases were hindered by limitations in laser power and by limits to the ramp-up time set by the various heating and loss processes in the deformed trap. The scatter in the data is primarily due to statistical errors in our measurements of $U_c/k_B T$, at the level of 30%, which arise from measurements of $P_c$ and $w_0$, and in the determination of the transition point. The well-depth $ U_{c} $ required to reach BEC can be understood by a simple model depicted in Fig. \[fig1\]. We begin with a gas in a harmonic trap above the BEC transition temperature, i.e. its chemical potential $\mu < 0$ (Fig. \[fig1\]b). Its initial phase-space density is given by $\Gamma_0 = g_{3/2}(z) $, where $z = e^{\mu/k_B T}$ is the fugacity. In Fig. \[fig1\]c, a narrow potential with depth $U_{c}$ is added until $ U_c = - \mu$, at which point the gas begins to Bose condense. The phase-space density thus reaches the critical value $g_{3/2}(1)$, so that the increase in maximum phase-space density is given by $$\frac{\Gamma_f}{\Gamma_0} = \frac{g_{3/2}(1)}{g_{3/2}(\exp({-U_c/k_B T}))} \quad . \label{simple_theory_increase}$$ This prediction, shown in Fig. \[fig1\]a, describes our data well, and accounts for the universal behaviour of our measurements over a wide range of temperatures and well-depths. Note that in this simple model we made the implicit assumption that the initial and final temperatures of the cloud were equal. We can remove this assumption by considering instead the condition of constant entropy. The entropy per non-condensed particle in a harmonically confined Bose gas is determined uniquely by its fugacity $z$ [@pink:97]: $${\frac{S}{N}}(z) = 4 \frac{g_4(z)}{g_3(z)} - \ln z \quad . \label{harmonic_spern}$$ This equation describes the entropy of the gas before compression, with the fugacity given by $z_0$. After compression, because of the small volume of the potential well, the entropy per particle is approximately that of a harmonically trapped gas (Eq. (\[harmonic\_spern\])) with fugacity $z_f = e^{-U_c / k_B T}$. Here $T$ is the [final]{} temperature of the gas, which is generally higher than the initial temperature [@footnote:3]. Constant entropy then implies $z_0 = z_f$, and thus one obtains Eq. (\[simple\_theory\_increase\]) where $T$ is now the final temperature, as we have plotted in Fig. \[fig1\]a. One may also consider the process of adiabatically increasing the phase-space density as a change in the density of states $D(\epsilon)$ of the system. By increasing the well-depth in a small region of the trap, we lowered only the energy of the ground state and a few excited states. Thus $\Gamma$, a local quantity, is maximally increased as the ground state energy is brought ever closer to the chemical potential, while the entropy, a global property of the gas, is unchanged by the minimal modification of $D(\epsilon)$. The fact that global properties of the gas are not affected by the trap deformation can also be seen in the momentum distributions probed by time-of-flight imaging [@mewe:96; @ensh:96]. The onset of BEC in the combined optical and magnetic trap is signaled only by the formation of a condensate peak. The remaining thermal cloud is well fit by a Maxwell-Boltzmann distribution, which describes a magnetically trapped cloud far from condensation (Fig. \[fig2\]a). In contrast, at the BEC transition in the harmonic magnetic trap, the momentum distribution of the thermal cloud is clearly Bose-enhanced at low momenta (Fig. \[fig2\]b). [Adiabatic condensation]{}. We now turn to the studies of adiabatic, i.e., reversible, condensate formation. A cloud of about $50 \times 10^6$ atoms was evaporatively cooled down to the the transition temperature in the magnetic trap, at trap frequencies of $\omega_r = 2 \pi \times 20\, \text{Hz}$ and $\omega_z = 2 \pi \times 13\, \text{Hz}$ in the radial and axial direction, respectively. The power of the infrared laser beam (of radius $w_0 = 20\, \mu\text{m}$), was ramped up over 1 s and held at a constant power for a dwell time of 1.5 s. Condensate fractions as small as 1% could be distinguished from the normal fraction by their anisotropic expansion in time-of-flight images [@ande:95; @davi:95]. The condensate number $N_0$ was determined by subtracting out the thermal cloud background using Gaussian fits to the thermal cloud in regions where the condensate was clearly absent. As shown in Fig. 3, the adiabatic trap deformation yielded condensate fractions of up to 15%; we observed condensate fractions of 30% with different trap settings. By varying the dwell time, we confirmed that clouds for which $U/ k_B T < 1.5$ suffered no significant losses of condensate number due to three-body decay or heating, whereas those points with higher values of $U / k_B T$ were affected by such losses. For an ideal Bose gas, the result of this adiabatic change can be understood as follows. Before compression, the cloud of $N$ particles at the BEC transition has an entropy $S_i$ given by Eq. (\[harmonic\_spern\]) as $S_i = N \times 4 g_4(1) / g_3(1)$. After compression, the situation is similar to that indicated in Fig. 1c, i.e. because of the small volume of the attractive well, the cloud is well-described as a harmonically trapped gas with $\mu = -U$. Thus, accounting for the fact that condensate particles carry no entropy and again using Eq. (\[harmonic\_spern\]), which gives the entropy per [non-condensed]{} particle, we equate the entropy before and after compression and obtain $$\frac{N_0}{N} = 1 - \frac{4 g_4(1) / g_3(1)}{4 g_4(e^{-U/k_B T}) / g_3(e^{-U/k_B T}) + U/k_B T}. \label{ideal_cf}$$ However, this simple prediction does not describe our findings well. The theory described above has two shortcomings. First, the approximation of using Eq. (\[harmonic\_spern\]) for the deformed trap is not strictly valid. However, calculations which accounted for the true shape of the deformed potential changed the prediction of Eq. (\[ideal\_cf\]) only for $U/k_B T > 1$, and only slightly improved the fit to our data. A second shortcoming is the neglect of interactions. It has been shown that in harmonic potentials, the condensate fraction in a Bose gas with repulsive interactions is reduced in comparison to that of an ideal gas [@finite_t; @nara:98]. To estimate this effect in our system, we use the “semi-ideal” model of Ref. [@nara:98]. The thermal cloud is described as an ideal gas for which the chemical potential is raised by $ g n_{0} = 4\pi \hbar^{2} a n_{0} /m $, where $n_{0}$ is the maximum condensate density, $m$ the mass of sodium, and $a = 2.75$ nm its scattering length [@ties:96]. This simply corresponds to using Eq. (\[ideal\_cf\]) with the substitution $U \rightarrow U - g n_0$. We determined $n_{0}$ using the Gross-Pitaevskii equation in the Thomas-Fermi limit [@baym:96], and a harmonic approximation for the deformed trap potential at its center. This approach predicts a significant reduction of the condensate fraction (Fig. 3, dashed line), and the improved agreement with our data is strong evidence for this effect. In contrast to related studies in purely harmonic traps [@mewe:96; @ensh:96], which did not show evidence for interaction effects, this depletion is strongly enhanced by the shape of the potential we are using. The mean-field energy of the condensate $g n_0$ is large because the condensate forms in the tight optical potential, while the transition temperature $T_c$ is small since it is determined by the weak magnetic potential. The reversibility of crossing the BEC phase transition was demonstrated by preparing a magnetically trapped cloud just above $T_{c}$. We then sinusoidally modulated the power of the infrared light at 1 Hz, between zero and 7 mW. This modulation frequency was significantly smaller than the magnetic trap frequencies ($\omega_{r} = 2 \pi\times 48~\text{Hz}$ and $\omega_{z} = 2 \pi \times 16~\text{Hz}$). These low frequencies and a large optical focus ($w_{0}=18~\mu $m) were used to minimize trap loss due to inelastic collisions. During the first seven condensation cycles the condensate fraction oscillated between zero and 6% (Fig. 4); later probing showed repeated condensation for at least 15 cycles. The peak of these oscillations decreased slowly in time. The temperature also oscillated, with an amplitude of about 100 nK, while gradually rising by about 10 nK/s. This heating and the decrease of the peak condensate fraction are consistent with similar behaviour in clouds held at a constant infrared power, which result from beam jitter, spontaneous scattering, and three-body decay[@stam:98a]. Thus, within the stability limitations of our optical setup, the repeated crossing of the BEC phase transition appears fully adiabatic. This method of creating condensates provides insight into their formation, which was recently studied experimentally [@mies:98] and theoretically [@gard:97]. For example, in the experiment described above, the condensate fraction was found to lag about 70 ms behind the modulation of the laser power, which is a measure for the formation time. In other experiments, by switching on the infrared light instantly, we observed condensation on timescales much faster than the oscillation periods in the magnetic trap and along the weakly confining axis of the optical trap. The resulting condensates showed striations in time-of-flight images, indicating that the condensates formed into excited states of the deformed potential. Such studies of shock-condensation might give new insight into the formation of quasi-condensates and condensation into excited states [@kaga:92; @gard:98]. In conclusion we have demonstrated the adiabatic Bose-Einstein condensation of an ultracold gas of atomic sodium. Changes in the trapping potential resulted in large phase-space density increases and allowed for repeated crossings of the BEC phase transition. This method allows for detailed studies of condensate formation and the phase transition. The combined trapping potential widens the range of trap parameters over which BEC can be studied. This was used to strikingly enhance the role of interactions, and led to higher transition temperatures (up to 5 $\mu$K) than achieved in purely magnetic traps. We thank Michael R. Andrews for discussions. This work was supported by the Office of Naval Research, NSF, Joint Services Electronics Program (ARO), NASA, and the David and Lucile Packard Foundation. A. P. C. acknowledges additional support from the NSF, D. M. S. -K. from JSEP, and J. S. from the Alexander von Humboldt-Foundation. [99]{} A. Griffin, D.W. Snoke, and S. Stringari, [ Bose-Einstein Condensation]{} (Cambridge University Press, Cambridge, 1995); Georgia Southern University BEC home page, http://amo.phy.gasou.edu/bec.html M.H. Anderson [et al.]{}, Science [269]{}, 198 (1995). K.B. Davis [et al.]{}, Phys. Rev. Lett. [75]{}, 3969 (1995). C.C. Bradley, C.A. Sackett, and R.G. Hulet, Phys. Rev. Lett. [78]{}, 985 (1997), see also: C.C.Bradley [et al.]{}, Phys. Rev. Lett. [75]{}, 1687 (1995). W. Ketterle and D.E. Pritchard, Phys. Rev. A [46]{}, 4051 (1992). F. Reif, [Fundamentals of Statistical and Thermal Physics ]{} (McGraw-Hill Book Company, New York, 1965) For weakly interacting Bose gases this argument has to be modified; see: M. Houbiers, H.T.C Stoof, and E.A. Cornell, Phys. Rev. [A 56]{}, 2041 (1997). P. W. H. Pinkse [et al.]{}, Phys. Rev. Lett. [78]{}, 990 (1997). M.-O. Mewes [et al.]{}, Phys. Rev. Lett. [77]{}, 416 (1996). W. Ketterle and N.J. van Druten, in [Advances in Atomic, Molecular, and Optical Physics]{}, edited by B. Bederson and H. Walther (Academic Press, San Diego, 1996), Vol. 37, p 181. This assumes an ideal Bose gas and harmonic confinement; see: V. Bagnato, D. E. Pritchard, and D. Kleppner, Phys. Rev. [A 35]{}, 4354 (1987). D. M. Stamper-Kurn [et al.]{}, preprint cond-mat/9801262 (unpublished). D. M. Stamper-Kurn [et al.]{}, Phys. Rev. Lett. [80]{}, 2027 (1998). K. Huang, [Statistical Mechanics]{}, second edition (Wiley, New York, 1987). This approximation is valid for shallow potential wells ($U/k_B T \ll \ln(V_1/V_2)$), as discussed in the two-box model. Numerical calculations confirmed its validity for all data in Fig. \[fig1\]a. Note that the effective volume of the magnetic trap ($V_1$) increases with temperature. J. R. Ensher [et al.]{}, Phys. Rev. Lett. [77]{}, 4984 (1996). V.V. Goldman, I.F. Silvera, and A.J. Leggett, Phys. Rev. B [24]{},2870 (1981); R.J. Dodd, K. Burnett, M. Edwards, and C.W. Clark, Acta Physica Polonica [93]{},45 (1998); S. Giorgini, L.P. Pitaevskii and S. Stringari, Phys. Rev. A [54]{}, R4633 (1996). M. Naraschewski and D.M. Stamper-Kurn, preprint cond-mat/9803216 (unpublished). E. Tiesinga [et al.]{}, J. Res. Natl. Inst. Stand. Technol. [ 101]{}, 505 (1996). G. Baym and C.J. Pethick, Phys. Rev. Lett [76]{}, 6 (1996). H.-J. Miesner [et al.]{}, Science [279]{}, 1005 (1998). C.W. Gardiner, P. Zoller, R.J. Ballagh, and M.J. Davis, Phys. Rev. Lett. [79]{}, 1793 (1997). Yu. Kagan, B.V. 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--- abstract: 'We prove that the $2$-category ${\ensuremath{\mathsf{Grt}}}$ of Grothendieck abelian categories with colimit preserving functors and natural transformations is a bicategory of fractions in the sense of Pronk of the $2$-category ${\ensuremath{\mathsf{Site}}}$ of linear sites with continuous morphisms of sites and natural transformations. This result can potentially be used to make the tensor product of Grothendieck categories from earlier work by Lowen, Shoikhet and the author into a bi-monoidal structure on ${\ensuremath{\mathsf{Grt}}}$.' address: 'Universiteit Antwerpen, Departement Wiskunde-Informatica, Middelheimcampus, Middelheimlaan 1, 2020 Antwerp, Belgium' author: - Julia Ramos González title: Grothendieck categories as a bilocalization of linear sites --- Introduction ============ Grothendieck categories are arguably the best-behaved and most studied large abelian categories, second only to module categories which are their first examples. They play a fundamental role in algebraic geometry since the Grothendieck school, and are center stage in non-commutative algebraic geometry since the work of Artin, Stafford, Van den Bergh and others (see, for example, [@artintatevandenbergh], [@artinzhang2], [@staffordvandenbergh]). By the Gabriel-Popescu Theorem, Grothendieck categories can be viewed as “linear topoi”, that is, as categories of sheaves on linear sites. The aim of this paper is to study the relation between Grothendieck categories and linear sites on a bicategorical level. Throughout the paper $k$ will be a commutative ring. Let ${\ensuremath{\mathsf{Grt}}}$ denote the 2-category of $k$-linear Grothendieck categories with colimit preserving $k$-linear functors and $k$-linear natural transformations. Let ${\ensuremath{\mathsf{Site}}}$ denote the 2-category of $k$-linear sites with $k$-linear continuous morphisms of sites and $k$-linear natural transformations[^1]. Given a linear site $({\mathfrak{a}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}})$ we can naturally associate to it a Grothendieck category, namely its category of sheaves ${\ensuremath{\mathsf{Sh}} }({\mathfrak{a}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}})$. In addition, given a continuous morphism $f: ({\mathfrak{a}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) {\longrightarrow}({\mathfrak{b}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$ between sites, it naturally induces a colimit preserving functor $f^s: {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) {\longrightarrow}{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$ between the corresponding categories of sheaves (see §\[seclinearsites\]). In particular, among continuous morphisms there is a distinguished class of the so-called LC morphisms (see below), which induce equivalences between the corresponding categories of sheaves. Observe that, from the Gabriel-Popescu theorem, it follows that every Grothendieck category can be realised as a category of sheaves on a linear site . Moreover, every colimit preserving functor between two categories of sheaves can be obtained as a “roof” of functors coming from continuous morphisms between sites, where the “reversed arrows” are equivalences induced by LC morphisms (see below). These observations make it natural to view ${\ensuremath{\mathsf{Grt}}}$ as a kind of “localization” of ${\ensuremath{\mathsf{Site}}}$ at the class of LC morphisms. In this paper we make this idea precise by using the localization of bicategories with respect to a class of $1$-morphisms developed by Pronk in [@pronk] and further analysed by Tommasini in the series of papers [@tommasini1; @tommasini2; @tommasini3]. Our main result is the following: \[mainresult\] There exists a pseudofunctor $$\Phi: {\ensuremath{\mathsf{Site}}}{\longrightarrow}{\ensuremath{\mathsf{Grt}}}$$ which sends LC morphisms to equivalences in ${\ensuremath{\mathsf{Grt}}}$, such that the pseudofunctor $$\tilde{\Phi}: {\ensuremath{\mathsf{Site}}}[{\ensuremath{\mathsf{LC}}}^{-1}] {\longrightarrow}{\ensuremath{\mathsf{Grt}}}$$ induced by $\Phi$ via the universal property of the bicategory of fractions is an equivalence of bicategories. The paper is structured as follows. In sections §\[seclinearsites\] and §\[sectionbicategoriesfractions\] we provide the introductory material required for the rest of the paper. Linear sites and Grothendieck categories are the linear counterpart of Grothendieck sites and Grothendieck topoi. In §\[seclinearsites\] we recall the basic notions of the corresponding linearized topos theory based on [@lowenlin] and [@lowen-ramos-shoikhet]. More concretely, we define continuous and cocontinuous morphism of linear sites and analyse the corresponding induced functors between the sheaf categories. In particular, we focus on the class of LC morphisms, as they play a fundamental role for the rest of the paper. Next, in §\[sectionbicategoriesfractions\], we revisit some basic notions and results from [@pronk] and [@tommasini1] on bicategories and their localizations with respect to classes of 1-morphisms, which we will refer to as bilocalizations from now on. The tecnical core of the paper that allows the proof of is developed in §\[section2catssitesgroth\], §\[sectionlocalizingLC\] and §\[sectionlocalizationallsites\]. In §\[section2catssitesgroth\] we show that the natural map that assigns to each site its category of sheaves and to each continuous morphism between sites the induced colimit preserving functor between the categories of sheaves extends to a pseudofunctor $$\Phi: {\ensuremath{\mathsf{Site}}}{\longrightarrow}{\ensuremath{\mathsf{Grt}}}.$$ In addition, the class ${\ensuremath{\mathsf{LC}}}$ of LC morphisms admits a calculus of fractions in ${\ensuremath{\mathsf{Site}}}$ and hence we have a bilocalization ${\ensuremath{\mathsf{Site}}}[{\ensuremath{\mathsf{LC}}}^{-1}]$. This is done in §\[sectionlocalizingLC\]. In particular, $\Phi$ sends LC morphisms to equivalences in ${\ensuremath{\mathsf{Grt}}}$, and hence, by the universal property of the bilocalization we obtain a pseudofunctor from the bilocalization to ${\ensuremath{\mathsf{Grt}}}$: $$\tilde{\Phi}: {\ensuremath{\mathsf{Site}}}[{\ensuremath{\mathsf{LC}}}^{-1}] {\longrightarrow}{\ensuremath{\mathsf{Grt}}}.$$ In §\[sectionlocalizationallsites\], based on [@tommasini3], we show that the pseudofunctor $\Phi: {\ensuremath{\mathsf{Site}}}{\longrightarrow}{\ensuremath{\mathsf{Grt}}}$ fulfills the necessary and sufficient conditions for the induced pseudofunctor $\tilde{\Phi}: {\ensuremath{\mathsf{Site}}}[{\ensuremath{\mathsf{LC}}}^{-1}] {\longrightarrow}{\ensuremath{\mathsf{Grt}}}$ to be an equivalence of bicategories, which finishes the proof of . Our original interest in representing ${\ensuremath{\mathsf{Grt}}}$ as a localization of ${\ensuremath{\mathsf{Site}}}$ is of geometrical nature and comes from [@lowen-ramos-shoikhet], where a tensor product of Grothendieck categories is defined in terms of a tensor product of linear sites. In particular, the fact that LC morphisms are closed under the tensor product of sites [@lowen-ramos-shoikhet Prop 3.14] is key to the proof of the independence of the tensor product of Grothendieck categories from the sheaf representations chosen. Combining with this fact makes it natural to think that the monoidal structure on ${\ensuremath{\mathsf{Site}}}$ could be transferred to ${\ensuremath{\mathsf{Grt}}}$ via the bilocalization, following the same principle as the monoidal localization of ordinary categories from [@day]. This is briefly addressed in §\[tensorproductvialocalization\]. The reader can easily check that the methods and arguments used along the paper are also available in the classical set-theoretical setup of topos theory. Hence, if we denote by ${\ensuremath{\mathsf{Topoi}}}$ the $2$-category of Grothendieck topoi with colimit preserving functors and natural transformations and by ${\ensuremath{\mathsf{GrSite}}}$ the $2$-category of Grothendieck sites with continuous morphisms and natural transformations, we can state the corresponding analogue of above: There exists a pseudofunctor $$\Phi: {\ensuremath{\mathsf{GrSite}}}{\longrightarrow}{\ensuremath{\mathsf{Topoi}}}$$ which sends LC morphisms to equivalences, such that the pseudofunctor $$\tilde{\Phi}: {\ensuremath{\mathsf{GrSite}}}[{\ensuremath{\mathsf{LC}}}^{-1}] {\longrightarrow}{\ensuremath{\mathsf{Topoi}}}$$ induced by $\Phi$ via the universal property of the bicategory of fractions is an equivalence of bicategories. Acknowledgement. I am very grateful to Wendy Lowen for many interesting discussions, the careful reading of this manuscript and her valuable suggestions. I would also like to thank Ivo Dell’Ambrogio, Boris Shoikhet and Enrico Vitale for useful comments on bicategories of fractions and Matteo Tommasini, whose explanations on the behaviour of $2$-morphisms after localization have been essential in order to obtain the main result of this paper. Linear sites and Grothendieck categories {#seclinearsites} ======================================== Linear sites and Grothendieck categories are the linear counterpart of the classical notions of Grothendieck sites and Grothendieck topoi from [@artingrothendieckverdier1]. We proceed now to give a brief account on this linearized version of topos theory. We refer the reader to [@lowenlin §2] for further details. Let ${\mathfrak{a}}$ be a small $k$-linear category and $A \in {\mathfrak{a}}$ an object. A sieve on $A$ is a subobject $R$ of the representable module ${\mathfrak{a}}(-,A)$ on $A$ in the category ${\ensuremath{\mathsf{Mod}}}({\mathfrak{a}}) \coloneqq {\ensuremath{\mathsf{Fun}}}_k({\mathfrak{a}}^{{\ensuremath{\mathsf{op}}}}, {\ensuremath{\mathsf{Mod}}}(k))$. Given $F = (f_i: A_i \rightarrow A)_{i \in I}$ a family of morphisms in ${\mathfrak{a}}$, the sieve generated by $F$ is the smallest sieve $R$ on $A$ such that $f_i \in R(A_i)$ for all $i \in I$. We denote it by $\langle F \rangle = \langle f_i \rangle_{i \in I}$. A cover system $\mathscr{R}$ on ${\mathfrak{a}}$ consists of providing for each $A \in {\mathfrak{a}}$ a family of sieves ${\ensuremath{\mathscr{R}}}(A)$ on $A$. The sieves in a cover system ${\ensuremath{\mathscr{R}}}$ are called covering sieves or simply covers (for ${\ensuremath{\mathscr{R}}}$). We will say that a family $(f_i: A_i {\longrightarrow}A)_{i\in I}$ is a cover if the sieve $\langle f_i \rangle_{i \in I}$ it generates is a cover. A cover system ${\ensuremath{\mathscr{T}}}$ on ${\mathfrak{a}}$ is called a $k$-linear topology if it fulfills the linearized version of the well-known identity, pullback and glueing axioms [@lowenlin §2.2]. In particular, a $k$-linear site is a pair $({\mathfrak{a}},{\ensuremath{\mathscr{T}}})$ where ${\mathfrak{a}}$ is a small $k$-linear category and ${\ensuremath{\mathscr{T}}}$ is a $k$-linear topology on ${\mathfrak{a}}$. As in the classical setting, one defines presheaves and sheaves as follows. \[defsheaf\] A presheaf $F$ on $({\mathfrak{a}}, {\ensuremath{\mathscr{T}}})$ is simply an ${\mathfrak{a}}$-module, i.e. $F$ is an object in ${\ensuremath{\mathsf{Mod}}}({\mathfrak{a}})$. A sheaf $F$ on $({\mathfrak{a}}, {\ensuremath{\mathscr{T}}})$ is a presheaf such that the restriction functor $$F(A) \cong {\ensuremath{\mathsf{Mod}}}({\mathfrak{a}})({\mathfrak{a}}(-,A), F) {\longrightarrow}{\ensuremath{\mathsf{Mod}}}({\mathfrak{a}})(R, F)$$ is an isomorphism for all $A \in {\mathfrak{a}}$ and all covering sieves $R \in {\ensuremath{\mathscr{T}}}(A)$. We denote by $${\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}) \subseteq {\ensuremath{\mathsf{Mod}}}({\mathfrak{a}})$$ the full subcategory of $k$-linear sheaves. In analogy with the classical setting, given a $k$-linear category ${\mathfrak{a}}$, a $k$-linear topology ${\ensuremath{\mathscr{T}}}$ on ${\mathfrak{a}}$ is said to be subcanonical if all the representable presheaves are sheaves for ${\ensuremath{\mathscr{T}}}$. The finest $k$-linear topology on ${\mathfrak{a}}$ for which all representable presheaves are sheaves is called the canonical topology on ${\mathfrak{a}}$. We now proceed to give in more detail the corresponding linear notions of morphisms of sites and the morphisms of linear topoi induced by them. This is a linearized version of [@artingrothendieckverdier1 Exposé iii]. Given a $k$-linear functor $f: {\mathfrak{a}}{\longrightarrow}{\mathfrak{b}}$ between two $k$-linear categories ${\mathfrak{a}}$ and ${\mathfrak{b}}$, we have the following induced functors between their module categories: - $f^: {\ensuremath{\mathsf{Mod}}}({\mathfrak{b}}) {\longrightarrow}{\ensuremath{\mathsf{Mod}}}({\mathfrak{a}}): F \longmapsto F \circ f$; - Its left adjoint, denoted by $f_{!}: {\ensuremath{\mathsf{Mod}}}({\mathfrak{a}}) {\longrightarrow}{\ensuremath{\mathsf{Mod}}}({\mathfrak{b}})$; - Its right adjoint, denoted by $f_: {\ensuremath{\mathsf{Mod}}}({\mathfrak{a}}) {\longrightarrow}{\ensuremath{\mathsf{Mod}}}({\mathfrak{b}})$. \[defcontinuous\] Consider $k$-linear sites $({\mathfrak{a}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}})$ and $({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$. A $k$-linear functor $f: {\mathfrak{a}}{\longrightarrow}{\mathfrak{b}}$ is continuous, if any of the following equivalent properties hold: 1. The functor $f^:{\ensuremath{\mathsf{Mod}}}({\mathfrak{b}}) {\longrightarrow}{\ensuremath{\mathsf{Mod}}}({\mathfrak{a}}) : F \longmapsto F\circ f$ preserves sheaves; 2. There exists a functor $f_s: {\ensuremath{\mathsf{Sh}} }({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}}) {\longrightarrow}{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}})$ such that the diagram $$\label{comsquare1} \begin{tikzcd} {\ensuremath{\mathsf{Mod}}}({\mathfrak{a}}) &{\ensuremath{\mathsf{Mod}}}({\mathfrak{b}}) \arrow[l,"f^"']\\ {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) \arrow[u,hook,"i_{{\mathfrak{a}}}"] & {\ensuremath{\mathsf{Sh}} }({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}}) \arrow[l,"f_s"] \arrow[u,hook,"i_{{\mathfrak{b}}}"'] \end{tikzcd}$$ commutes; 3. There exists a colimit preserving functor $f^s: {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) {\longrightarrow}{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$ such that the diagram $$\label{comsquare2} \begin{tikzcd} {\mathfrak{a}}\arrow{r}{f} \arrow[d,hook,"Y_{{\mathfrak{a}}}"'] &{\mathfrak{b}}\arrow[d,hook,"Y_{{\mathfrak{b}}}"]\\ {\ensuremath{\mathsf{Mod}}}({\mathfrak{a}}) \arrow[r,"f_!"] \arrow[d,"\#_{{\mathfrak{a}}}"'] &{\ensuremath{\mathsf{Mod}}}({\mathfrak{b}}) \arrow[d,"\#_{{\mathfrak{b}}}"]\\ {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) \arrow{r}{f^s} & {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) \end{tikzcd}$$ commutes, where $Y_{{\mathfrak{a}}}: {\mathfrak{a}}{\lhook\joinrel\longrightarrow}{\ensuremath{\mathsf{Mod}}}({\mathfrak{a}})$, $Y_{{\mathfrak{b}}}: {\mathfrak{b}}{\lhook\joinrel\longrightarrow}{\ensuremath{\mathsf{Mod}}}({\mathfrak{b}})$ are the corresponding Yoneda embeddings and $\#_{{\mathfrak{a}}}:{\ensuremath{\mathsf{Mod}}}({\mathfrak{a}}) {\longrightarrow}{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}})$, $\#_{{\mathfrak{b}}}:{\ensuremath{\mathsf{Mod}}}({\mathfrak{b}}) {\longrightarrow}{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$ are the corresponding sheafification functors. In addition, if any of the previous properties holds, we necessarily have that $f^s \dashv f_s$ and $$f^s \cong \#_{{\mathfrak{b}}} \circ f_! \circ i_{{\mathfrak{a}}}.$$ Consider $k$-linear sites $({\mathfrak{a}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}})$ and $({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$. A $k$-linear functor $f: {\mathfrak{a}}{\longrightarrow}{\mathfrak{b}}$ is cocontinuous, if any of the following equivalent properties hold: 1. For each object $A \in {\mathfrak{a}}$ and each covering sieve $R \in {\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}} (f(A))$, there exists a covering sieve $S \in {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}(A)$ with $f S \subseteq R$. 2. The functor $f_: {\ensuremath{\mathsf{Mod}}}({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) {\longrightarrow}{\ensuremath{\mathsf{Mod}}}({\mathfrak{b}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$ preserves sheaves. In addition, if any of the previous properties holds we have: 1. The functor $\widetilde{f}^ = \#_{{\mathfrak{a}}} \circ f^* \circ i_{{\mathfrak{b}}}$ is colimit preserving and exact and the diagram $$\label{comsquare3} \begin{tikzcd} {\ensuremath{\mathsf{Mod}}}({\mathfrak{a}}) \arrow[d,"\#_{{\mathfrak{a}}}"'] &{\ensuremath{\mathsf{Mod}}}({\mathfrak{b}}) \arrow[d,"\#_{{\mathfrak{b}}}"] \arrow[l,"f^"']\\ {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) & {\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}}) \arrow{l}{\widetilde{f}^} \end{tikzcd}$$ is commutative up to canonical isomorphism. 2. There exists a functor $\widetilde{f}_:{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) {\longrightarrow}{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$ such that the diagram $$\label{comsquare4} \begin{tikzcd} {\ensuremath{\mathsf{Mod}}}({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) \arrow{r}{f_} &{\ensuremath{\mathsf{Mod}}}({\mathfrak{b}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}}) \\ {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) \arrow[r,"\widetilde{f}_"'] \arrow[u,hook,"i_{{\mathfrak{a}}}"] & {\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}}) \arrow[u,hook,"i_{{\mathfrak{b}}}"'] \end{tikzcd}$$ commutes up to canonical isomorphism and $\widetilde{f}^ \dashv \widetilde{f}_$ is an adjoint pair. In [@lowenlin] the term cover continuous* is used for what we call here cocontinuous. Recall that a $k$-linear Grothendieck abelian category ${\ensuremath{\mathcal{C}}}$ is a cocomplete abelian $k$-linear category with a generator and exact filtered colimits. The fact that the categories of sheaves over linear sites (or linear Grothendieck topoi) are precisely the Grothendieck categories can be deduced from Gabriel-Popescu theorem [@gabrielpopescu] together with the main result in [@borceuxquinteiro]. Indeed, Gabriel-Popescu theorem characterizes Grothendieck categories as the localizations of presheaf categories of linear sites, that is subcategories of presheaf categories whose embedding functor has a left exact left adjoint. On the other hand, from [@borceuxquinteiro Thm 1.5] one deduces that categories of sheaves are precisely the localizations of presheaf categories of linear categories. Thus, the combination of the two results provides a linear counterpart of the classical Giraud Theorem that characterizes Grothendieck topoi in the classical setting. Observe, nevertheless, that the classical Gabriel-Popescu theorem does not provide us with all the possible realizations of Grothendieck categories as categories of linear sheaves. Such result is provided by the generalization of Gabriel-Popescu theorem in [@lowenGP]: given a Grothendieck category ${\ensuremath{\mathcal{C}}}$, it characterizes the linear functors $u:{\mathfrak{a}}{\longrightarrow}{\ensuremath{\mathcal{C}}}$ such that the functor $${\mathfrak{c}}{\longrightarrow}{\ensuremath{\mathsf{Mod}}}({\mathfrak{a}}): C \longmapsto {\ensuremath{\mathcal{C}}}(u(-),C)$$ is a localization. We now introduce a distinguished class of continuous morphisms, called LC morphisms (see [@lowen-ramos-shoikhet Def 3.4]), where LC stands for “Lemme de comparison” (see [@lowenGP §4]). In particular, the functoriality of these morphisms with respect to the tensor product of linear sites constructed in [@lowen-ramos-shoikhet §2.4] is the key point in order to provide a well-defined tensor product of Grothendieck categories expressed in terms of realizations as sheaf categories [@lowen-ramos-shoikhet §4.1]. \[defLC\] Consider a $k$-linear functor $f: {\mathfrak{a}}{\longrightarrow}{\mathfrak{c}}$. 1. Suppose ${\mathfrak{c}}$ is endowed with a cover system ${\ensuremath{\mathscr{T}}}_{{\mathfrak{c}}}$. We say that $f: {\mathfrak{a}}{\longrightarrow}({\mathfrak{c}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{c}}})$ satisfies - if for every $C \in {\mathfrak{c}}$ there is a covering family $(f(A_i) {\longrightarrow}C)_i$ for ${\ensuremath{\mathscr{T}}}_{{\mathfrak{c}}}$. 2. Suppose ${\mathfrak{a}}$ is endowed with a cover system ${\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}$. We say that $f: ({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) {\longrightarrow}{\mathfrak{c}}$ satisfies - if for every $c: f(A) {\longrightarrow}f(A')$ in ${\mathfrak{c}}$ there exists a covering family $a_i: A_i {\longrightarrow}A$ for ${\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}$ and $f_i: A_i {\longrightarrow}A'$ with $cf(a_i) = f({f}_i)$; - if for every $a: A {\longrightarrow}A'$ in ${\mathfrak{a}}$ with $f(a) = 0$ there exists a covering family $a_i: A_i {\longrightarrow}A$ for ${\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}$ with $aa_i = 0$. 3. Suppose ${\mathfrak{a}}$ and ${\mathfrak{c}}$ are endowed with cover systems ${\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}$ and ${\ensuremath{\mathscr{T}}}_{{\mathfrak{c}}}$ respectively. We say that $f: ({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) {\longrightarrow}({\mathfrak{c}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{c}}})$ satisfies - if $f$ satisfies (G) with respect to ${\ensuremath{\mathscr{T}}}_{{\mathfrak{c}}}$, (F) and (FF) with respect to ${\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}$, and we further have ${\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}} = f^{-1} {\ensuremath{\mathscr{T}}}_{{\mathfrak{c}}}$. The following gives a characterization of LC morphisms between linear sites in terms of continuity and cocontinuity, and it will be used in §\[sectionlocalizingLC\]. \[LCcandcc\] Consider a morphism of sites $f: ({\mathfrak{a}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) {\longrightarrow}({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$ satisfying *(G)* with respect to ${\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}}$ and *(F), (FF)* with respect to ${\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}$. Then, the following are equivalent: 1. The morphism $f$ satisfies *(LC)* (with respect to the topologies ${\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}$ and ${\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}}$); 2. The morphism $f$ is continuous and cocontinuous. Assume (1) holds. The fact that $f$ is continuous is given by Lemme de comparison [@lowenGP Cor 4.5] and the fact that $f$ is cocontinuous follows from [@lowenlin Lem 2.15]. We prove now the converse. As by hypothesis $f$ is cocontinuous and satisfies (F) and (FF), we have that $f^{-1}{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}} \subseteq {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}$ by [@lowenlin Prop 2.16]. Now, as $f$ is continuous, by applying the linear counterpart of [@artingrothendieckverdier1 Exposé iii, Prop 1.6], we have that $f({\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) \subseteq {\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}}$. Consequently, by applying $f^{-1}$ we have that ${\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}} \subseteq f^{-1}f {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}} \subseteq f^{-1}({\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$, which concludes the argument. \[cocontandcontLC\] Let $f: ({\mathfrak{a}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) {\longrightarrow}({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$ be an LC morphism. As it is continuous and cocontinuous, one can consider the induced functors between the corresponding sheaf categories both as a continuous and as a cocontinuous morphism. An easy check shows that those are related as follows: $$\widetilde{f}_* \cong f^s,$$ and hence $$\widetilde{f}^* \cong f_s.$$ Our interest in LC morphisms is twofold. Firstly, they induce equivalences between the corresponding sheaf categories [@lowenGP Cor 4.5]. Secondly, making essential use of LC morphisms we are able recover any colimit preserving functor between Grothendieck categories as being induced by a roof of continuous morphisms of linear sites. More precisely: \[roofthm\] Let $({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}})$ and $({\mathfrak{b}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$ be linear sites and consider a colimit preserving functor $F: {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) {\longrightarrow}{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$. Then, there exist a subcanonical site $({\mathfrak{c}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{c}}})$ and a diagram $$\begin{tikzcd} &{\mathfrak{c}}\\ {\mathfrak{a}}\arrow[ur,"f"] &&{\mathfrak{b}}, \arrow[ul,"w",swap] \end{tikzcd}$$ where $f$ is a continuous morphism and $w$ is an LC morphism, such that $$\begin{tikzcd} {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) \arrow[rr,"F"] \arrow[dr,"f^s",swap] &&{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})\\ &{\ensuremath{\mathsf{Sh}} }({\mathfrak{c}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{c}}}) \arrow[ur,"\tilde{w}^",swap] \end{tikzcd}$$ is a commutative diagram up to isomorphism. This theorem is a slight generalization of [@stacksproject Tag 03A2] in the linear setting, where the result is provided for geometric morphisms between Grothendieck topoi (i.e. adjunctions of functors between Grothendieck topoi where the left adjoint is left exact). Our version focuses on the left adjoints, or equivalently on the colimit preserving functors, without requiring them to be left exact (i.e. without requiring the adjunction to be a geometric morphism). Observe that we call LC morphism what in [@stacksproject] is called special cocontinuous functor* (see ). The proof can be obtained along the lines of [@stacksproject Tag 032A]. We will just provide, for convenience of the reader, the construction of the site $({\mathfrak{c}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{c}}})$ and the morphisms $f$ and $w$, as these constructions will be frequently used throughout the paper. Take ${\mathfrak{c}}$ to be the full $k$-linear subcategory of ${\ensuremath{\mathsf{Sh}} }({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$ with the following set of objects $${\mathrm{Obj}}({\mathfrak{c}})=\{\#_{{\mathfrak{b}}}({\mathfrak{b}}(-,B))\}_{B\in {\mathfrak{b}}} \cup \{F(\#_{{\mathfrak{a}}}({\mathfrak{a}}(-,A)))\}_{A \in {\mathfrak{a}}}.$$ We endow it with the topology ${\ensuremath{\mathscr{T}}}_{{\mathfrak{c}}}$ induced from the canonical topology in ${\ensuremath{\mathsf{Sh}} }({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$. Then we define $f: {\mathfrak{a}}{\longrightarrow}{\mathfrak{c}}$ as the composition $F \circ \#_{{\mathfrak{a}}} \circ Y_{{\mathfrak{a}}}$ and $w: {\mathfrak{b}}{\longrightarrow}{\mathfrak{c}}$ as the composition $\#_{{\mathfrak{b}}} \circ Y_{{\mathfrak{b}}}$. Bicategories of fractions {#sectionbicategoriesfractions} ========================= In this section we recall the main notions and results on localizations of bicategories from [@pronk] and [@tommasini1]. In general we will follow the notations and terminology from [@pronk] with the exception that, following a more standard terminology, we will call pseudofunctor what in [@pronk] is called homomorphism of bicategories. We first fix some notations for the rest of the paper. Given a bicategory, we denote the vertical composition of $2$-morphisms by $\bullet$ and the horizontal composition of $2$-morphisms by $\circ$. In particular, given a diagram $$\begin{tikzcd}[column sep= 50pt ] A \arrow[r,bend left=40, "f"] \arrow[r,bend right=40,swap,"f"] \arrow[r, phantom,"\Downarrow {\text{Id}}_{f}", pos=0.6] &B \arrow[r,bend left=40, "g"] \arrow[r,bend right=40, swap, "h"] \arrow[r, phantom,"\Downarrow \alpha", pos=0.55] &C \end{tikzcd}$$ in a bicategory ${\ensuremath{\mathcal{C}}}$, we denote by $\alpha \circ f$ to the horizontal composition $\alpha \circ {\text{Id}}_f$. We will recall some important definitions for the rest of the paper. \[defequiv\] Given a 1-morphism $f: A {\longrightarrow}B$ in a bicategory ${\ensuremath{\mathcal{C}}}$, we say it is an equivalence (or internal equivalence in the terminology of [@tommasini1]) if there exists another 1-morphism $g: B {\longrightarrow}A$ and two invertible $2$-morphisms $\alpha: {\text{Id}}_A \Rightarrow g \circ f$ and $\beta: f \circ g \Rightarrow {\text{Id}}_B$ satisfying the triangle identities, i.e. the compositions $$\begin{tikzcd}[row sep=small] f \arrow[r, Rightarrow,"f \circ \alpha"] & f \circ g \circ f \arrow[r,Rightarrow, "\beta \circ f"] & f;\\ g \arrow[r,Rightarrow,"\alpha \circ g"] & g \circ f \circ g \arrow[r,Rightarrow,"g \circ \beta"] & g \end{tikzcd}$$ are the identity on $f$ and on $g$ respectively. \[defpropertiespseudofunctor\] A pseudofunctor $\Phi: {\ensuremath{\mathcal{A}}}{\longrightarrow}{\ensuremath{\mathcal{B}}}$ between two bicategories is - essentially surjective on objects if and only if for all $B \in {\mathrm{Obj}}({\ensuremath{\mathcal{B}}})$ there exists an $A \in{\mathrm{Obj}}({\ensuremath{\mathcal{A}}})$ such that there is an equivalence $\Phi(A) \cong B$ in ${\ensuremath{\mathcal{B}}}$; - essentially full if for all $A, A' \in {\mathrm{Obj}}({\ensuremath{\mathcal{A}}})$ the functor $$\Phi_{A,A'}: {\ensuremath{\mathcal{A}}}(A, A') {\longrightarrow}{\ensuremath{\mathcal{B}}}(\Phi(A), \Phi(A'))$$ is essentially surjective ; - fully faithful on 2-morphisms if for all $A, A' \in {\mathrm{Obj}}({\ensuremath{\mathcal{A}}})$ the functor $$\Phi_{A,A'}: {\ensuremath{\mathcal{A}}}(A, A') {\longrightarrow}{\ensuremath{\mathcal{B}}}(\Phi(A), \Phi(A'))$$ is fully faithful; - an equivalence of bicategories if it is essentially surjective on objects, essentially full and fully faithful on 2-morphisms. In [@pronk §2] a localization theory for bicategories along a class of 1-morphisms is developed generalizing the well-known localization of ($1$-)categories due to Gabriel-Zisman [@gabriel-zisman]. In particular, the bicategory of fractions in loc.cit. is defined and constructed by means of a right calculus of fractions. Observe that one could analogously develop the theory for a left calculus of fractions, as it is done in the $1$-categorical case. More precisely, a class of ($1$-)morphisms admits a left calculus of fractions if and only if the same class of ($1$-)morphisms in the opposite (bi)category admits a right calculus of fractions. Recall that the opposite bicategory (or transpose bicategory in the terminology of [@benabou]) is given by reversing the 1-morphisms and keeping the direction of the $2$-morphisms. In our case, we will use a left calculus of fractions, hence we introduce the analogous results from [@pronk] for a left calculus of fractions. [[@pronk §2.1]]{} Let ${\ensuremath{\mathcal{C}}}$ be a bicategory. We say a class $\mathsf{W}$ of 1-morphisms on ${\ensuremath{\mathcal{C}}}$ admits a left calculus of fractions if it satisfies: - All equivalences belong to $\mathsf{W}$; - $\mathsf{W}$ is closed under composition of 1-morphisms; - Every solid diagram $$\begin{tikzcd} {\mathfrak{a}}\\ {\mathfrak{c}}\arrow[u,"f"] \arrow[r,"w"'] & {\mathfrak{b}}\end{tikzcd}$$ in ${\ensuremath{\mathcal{C}}}$ with $w \in \mathsf{W}$ can be completed to a square $$\begin{tikzcd} {\mathfrak{a}}\arrow[r,dotted,"v"] & {\mathfrak{d}}\\ {\mathfrak{c}}\arrow[u,"f"] \arrow[r,"w"'] & {\mathfrak{b}}\arrow[u,dotted,"g"'] \arrow[ul, start anchor={[xshift=-1.3ex, yshift=+1.3ex]}, end anchor={[xshift=+1.3ex, yshift=-1.3ex]}, Rightarrow,"\alpha"'] \end{tikzcd}$$ where $\alpha$ is an invertible $2$-morphism and $v \in \mathsf{W}$; - - Given two morphisms $f,g : B {\longrightarrow}A$, a morphism $w: B' {\longrightarrow}B$ in $\mathsf{W}$ and a 2-morphism $\alpha: f \circ w \Rightarrow g \circ w$, there exists a morphism $v: A {\longrightarrow}A'$ in $\mathsf{W}$ and a 2-morphism $\beta: v \circ f \Rightarrow v \circ g$ such that $v \circ \alpha = \beta \circ w$; - if $\alpha$ is an isomorphism, we require $\beta$ to be an isomorphism too; and - given another pair $v': A {\longrightarrow}A'$ in $\mathsf{W}$ and $\beta': v' \circ f \Rightarrow v' \circ g$ satisfying condition (1), there exist 1-morphisms $u,u': A' {\longrightarrow}A''$ with $u \circ v$, $u' \circ v'$ in $\mathsf{W}$ and an invertible $2$-morphism $\epsilon: u \circ v \Rightarrow u' \circ v'$ such that the diagram $$\begin{tikzcd} u \circ v \circ f \arrow[r,"u \circ \beta"] \arrow[d,"\epsilon \circ f"'] & u \circ v \circ g \arrow[d,"\epsilon \circ g"]\\ u' \circ v' \circ f \arrow[r,"u' \circ \beta'"'] &u' \circ v' \circ g \end{tikzcd}$$ is commutative; - $\mathsf{W}$ is closed under invertible $2$-morphisms. The first axiom can be weakened as is done in [@tommasini1]. [[@pronk §2]]{} Given a category ${\ensuremath{\mathcal{C}}}$ and a class of 1-morphisms $\mathsf{W}$ in ${\ensuremath{\mathcal{C}}}$ admitting a left calculus of fractions, a bilocalization of ${\ensuremath{\mathcal{C}}}$ along $\mathsf{W}$ is a pair $({\ensuremath{\mathcal{C}}}[\mathsf{W}^{-1}],\Psi)$, where ${\ensuremath{\mathcal{C}}}[\mathsf{W}^{-1}]$ is a bicategory and $\Psi: {\ensuremath{\mathcal{C}}}{\longrightarrow}{\ensuremath{\mathcal{C}}}[\mathsf{W}^{-1}]$ is a pseudofunctor such that: 1. $\Psi$ sends elements in $\mathsf{W}$ to equivalences; 2. Composition with $\Psi$ gives an equivalence of bicategories $${\mathrm{Hom}}({\ensuremath{\mathcal{C}}}[\mathsf{W}^{-1}], {\ensuremath{\mathcal{D}}}) {\longrightarrow}{\mathrm{Hom}}_{\mathsf{W}}({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{D}}})$$ for each bicategory ${\ensuremath{\mathcal{D}}}$, where ${\mathrm{Hom}}$ denotes the bicategory of pseudofunctors (see [@benabou §8]) and ${\mathrm{Hom}}_{\mathsf{W}}$ its full sub-bicategory of elements sending $\mathsf{W}$ to equivalences. Observe that, in particular, ${\ensuremath{\mathcal{C}}}[\mathsf{W}^{-1}]$ is unique up to equivalence of bicategories [@pronk §3.3]. In [@pronk §2] a detailed construction for $({\ensuremath{\mathcal{C}}}[\mathsf{W}^{-1}],\Psi)$ is provided for a right calculus of fractions and in [@tommasini1] a simplified version of this construction is provided, less dependent of the axiom of choice. By inverting the direction of $1$-morphisms one gets the analogous construction of the bilocalization for a left calculus of fractions. The 2-category of Grothendieck categories and the 2-category of sites {#section2catssitesgroth} ===================================================================== Fix a universe ${\ensuremath{\mathcal{U}}}$. For a ${\ensuremath{\mathcal{U}}}$-small $k$-linear site $({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}})$, the category ${\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}})$ is defined with respect to the category ${\ensuremath{\mathcal{U}}}$-${\ensuremath{\mathsf{Mod}}}(k)$ of ${\ensuremath{\mathcal{U}}}$-small $k$-modules. Let ${\ensuremath{\mathsf{Site}}}$ denote the 2-category of ${\ensuremath{\mathcal{U}}}$-small $k$-linear sites with $k$-linear continuous morphisms of sites and $k$-linear natural transformations. By definition, a ${\ensuremath{\mathcal{U}}}$-Grothendieck category is a $k$-linear abelian category with a ${\ensuremath{\mathcal{U}}}$-small set of generators, ${\ensuremath{\mathcal{U}}}$-small colimits and exact ${\ensuremath{\mathcal{U}}}$-small filtered colimits. Let ${\ensuremath{\mathcal{V}}}$ be a larger universe such that all the categories ${\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}})$ are ${\ensuremath{\mathcal{V}}}$-small and let ${\ensuremath{\mathsf{Grt}}}$ denote the 2-category of $k$-linear ${\ensuremath{\mathcal{V}}}$-small ${\ensuremath{\mathcal{U}}}$-Grothendieck categories. Up to equivalence, ${\ensuremath{\mathsf{Grt}}}$ is easily seen to be independent of the choice of ${\ensuremath{\mathcal{V}}}$. In the rest of the paper, we will omit the universes ${\ensuremath{\mathcal{U}}}$ and ${\ensuremath{\mathcal{V}}}$ from our notations and terminology. \[enriched\] Observe that ${\ensuremath{\mathsf{Site}}}$ and ${\ensuremath{\mathsf{Grt}}}$ are actually enriched 2-categories, more precisely $k$-linear 2-categories in the sense of [@ganter-kapranov Def 2.4 & 2.5]. Observe that equivalences (see above) in ${\ensuremath{\mathsf{Grt}}}$ are just the colimit preserving $k$-linear functors which are equivalences of categories in the usual sense, while equivalences in ${\ensuremath{\mathsf{Site}}}$ are just the $k$-linear continuous morphisms which are equivalences of categories in the usual sense. We denote by ${\ensuremath{\mathsf{LC}}}$ the family of LC morphisms in ${\ensuremath{\mathsf{Site}}}$ (see ). The 2-category ${\ensuremath{\mathsf{Grt}}}$ is related to ${\ensuremath{\mathsf{Site}}}$ in a natural way. Indeed, we define a pseudofunctor $$\label{twofunctorsitegroth} \Phi: {\ensuremath{\mathsf{Site}}}{\longrightarrow}{\ensuremath{\mathsf{Grt}}}$$ as follows: - Given a site $({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}})$, we define $$\Phi({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) = {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}),$$ which is a Grothendieck category; - Given a continuous map between two sites $f: ({\mathfrak{a}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) {\longrightarrow}({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$, we define $$\Phi(f) = \begin{tikzcd} {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) \arrow[r,"f^s"] &{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}}), \end{tikzcd}$$ which is colimit preserving; - Given two continuous morphisms $f,g : ({\mathfrak{a}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}) {\longrightarrow}({\mathfrak{b}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$ and a natural transformation $\alpha: f \Rightarrow g$, we define a natural transformation $$\label{thenaturaltransform} \Phi(\alpha)= \alpha^s: f^s \Rightarrow g^s$$ as follows. For any $F \in {\ensuremath{\mathsf{Sh}} }({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$ and any $A \in {\mathfrak{a}}$, we have the following morphism: $$(\alpha_s)_F (A) \coloneqq F(\alpha_A): g_s(F)(A) = F(g(A)) {\longrightarrow}F(f(A)) = f_s(F)(A)$$ which is $k$-linear and natural in $A$ and $F$ and hence it defines a natural transformation $\alpha_s: g_s \Rightarrow f_s$. We define $\alpha^s$ as the natural transformation corresponding to $\alpha_s$ via the natural adjunctions. More precisely, for all $F \in {\ensuremath{\mathsf{Sh}} }({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$ and all $G \in {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}})$ we have a composition: $$\begin{tikzcd} {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}})(G,g_s(F)) \arrow[r,"(\alpha_s)_F \circ \: -"] &{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}})(G, f_s(F)) \arrow[d,"\cong"]\\ {\ensuremath{\mathsf{Sh}} }({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})(g^s(G),F) \arrow[u,"\cong"] &{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})(f^s(G), F) \end{tikzcd}$$ where the vertical functors are the adjunctions. Observe this composition is natural in $F$ and $G$. Consequently, there is an induced 2-morphism $f^s \Rightarrow g^s$, and this is the 2-morphism we denote by $\alpha^s$. One can easily check these data indeed define a pseudofunctor: - Given any two sites $({\mathfrak{a}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}), ({\mathfrak{b}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}})$ the map $$\Phi_{{\mathfrak{a}},{\mathfrak{b}}}: {\ensuremath{\mathsf{Site}}}({\mathfrak{a}}, {\mathfrak{b}}) {\longrightarrow}{\ensuremath{\mathsf{Grt}}}({\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}), {\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}))$$ induced by $\Phi$ is a functor. Indeed, consider a continuous morphism $f: {\mathfrak{a}}{\longrightarrow}{\mathfrak{b}}$, trivially the 2-morphism ${\text{Id}}_{f}: f \Rightarrow f$ is mapped to ${\text{Id}}_{f^s}: f^s \Rightarrow f^s$. Now consider $\alpha: f \Rightarrow g$ and $\beta: g \Rightarrow h$ and their vertical composition $\beta \bullet \alpha:f \Rightarrow h$. One has that $$\left[ \alpha_s \bullet \beta_s \right]_G(A) = G(\alpha_A) \circ G(\beta_A) = G((\alpha \bullet \beta)_A )= \left[ (\beta \bullet \alpha)_s\right]_G(A)$$ for all $G \in {\ensuremath{\mathsf{Sh}} }({\mathfrak{b}})$ and all $A \in {\mathfrak{a}}$. Hence, by adjunction, $\Phi_{{\mathfrak{a}},{\mathfrak{b}}}$ preserves compositions. - Let ${\mathfrak{a}}= ({\mathfrak{a}},{\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}})$ be a site and consider its identity morphism ${\text{Id}}_{{\mathfrak{a}}}$. One has that $({\text{Id}}_{{\mathfrak{a}}})_s = {\text{Id}}_{{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}})}$ is the identity functor of ${\ensuremath{\mathsf{Sh}} }({\mathfrak{a}})$. Hence, by adjunction, $${\text{Id}}_{{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}})} \cong ({\text{Id}}_{{\mathfrak{a}}})^s,$$ which gives us the unitor of $\Phi$. Consider now two continous morphisms $f:{\mathfrak{a}}{\longrightarrow}{\mathfrak{b}}$ and $g: {\mathfrak{b}}{\longrightarrow}{\mathfrak{c}}$ in ${\ensuremath{\mathsf{Site}}}$. By definition, we have that $$(g\circ f)^s \cong g^s \circ f^s,$$ which provides the associator of $\Phi$. - It can be readily seen, using the fact that adjoints are unique up to unique isomorphism, that the unitor and associator of $\Phi$ fulfill the corresponding coherence axioms. Observe that $\Phi$ sends LC morphisms to equivalences. This is a direct consequence of the Lemme de comparaison. Hence, if ${\ensuremath{\mathsf{LC}}}$ admits a left calculus of fractions in ${\ensuremath{\mathsf{Site}}}$, we will get, by the universal property of bilocalizations, a pseudofunctor $$\tilde{\Phi}: {\ensuremath{\mathsf{Site}}}[{\ensuremath{\mathsf{LC}}}^{-1}] {\longrightarrow}{\ensuremath{\mathsf{Grt}}}.$$ Recall from that ${\ensuremath{\mathsf{Site}}}$ and ${\ensuremath{\mathsf{Grt}}}$ are $k$-linear $2$-categories, and observe that $\Phi$ is also a $k$-linear pseudofunctor. While in this paper we only need the bilocalization to exist as an ordinary bicategory, it is possible to show that in this case the bilocalization automatically satisfies the universal property of an “enriched bilocalization” (and in particular, the induced functor $\tilde{\Phi}$ is automatically $k$-linear), where we use the term in analogy with the enriched localizations from [@wolff]. Bilocalization of the 2-category of sites with respect to LC morphisms {#sectionlocalizingLC} ====================================================================== In this section we prove that ${\ensuremath{\mathsf{LC}}}$ admits a left calculus of fractions in ${\ensuremath{\mathsf{Site}}}$. First, we fix the following notations. Given a site $({\mathfrak{a}}, {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}})$, it will usually be denoted simply by ${\mathfrak{a}}$ for the sake of brevity. We will denote by $i_{{\mathfrak{a}}}: {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}) {\lhook\joinrel\longrightarrow}{\ensuremath{\mathsf{Mod}}}({\mathfrak{a}})$ the natural inclusion and by $\#_{{\mathfrak{a}}}:{\ensuremath{\mathsf{Mod}}}({\mathfrak{a}}) {\longrightarrow}{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}})$ the corresponding sheafification functor. The indexes will be omitted if the site we are working with is clear from the context. Furthermore, given an object $A \in {\mathfrak{a}}$ we will denote $h_A = {\mathfrak{a}}(-,A)$ the corresponding representable presheaf and by $h^{\#}_A = \#({\mathfrak{a}}(-,A)) $ its sheafification. We adopt these notations in order to simplify the formulas that will appear further in the paper. \[CondLF1\] Condition *LF1* holds for ${\ensuremath{\mathsf{LC}}}$ in ${\ensuremath{\mathsf{Site}}}$. Take an equivalence $f \in {\ensuremath{\mathsf{Site}}}({\mathfrak{a}}, {\mathfrak{b}})$, and denote by $g: {\mathfrak{b}}{\longrightarrow}{\mathfrak{a}}$ its quasi-inverse. Then, it is easy to see that the induced functor $$f_s: {\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}) {\longrightarrow}{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}})$$ is an equivalence of Grothendieck categories, with quasi-inverse given by $g_s$. We prove that $f$ belongs to ${\ensuremath{\mathsf{LC}}}$. Property (G) follows from the fact that $f$ is essentially surjective, and properties (F) and (FF) follow immediately from the fact that $f$ is fully-faithful. By , it only remains to prove that $f$ is cocontinuous. One can easily see that $$f^: {\ensuremath{\mathsf{Mod}}}({\mathfrak{b}}) {\longrightarrow}{\ensuremath{\mathsf{Mod}}}({\mathfrak{a}})$$ is also an equivalence with quasi-inverse given by $g^: {\ensuremath{\mathsf{Mod}}}({\mathfrak{a}}) {\longrightarrow}{\ensuremath{\mathsf{Mod}}}({\mathfrak{b}})$. Hence we have that $f_* \cong f_! \cong g^$ by unicity of adjoint functors. Observe then that $$f_ \circ i_{{\mathfrak{a}}} \cong g^* \circ i_{{\mathfrak{a}}} = i_{{\mathfrak{b}}} \circ g_s,$$ which implies that $f_\|_{{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}})}$ takes values in ${\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}) \subseteq {\ensuremath{\mathsf{Mod}}}({\mathfrak{b}})$. Consequently $f$ is cocontinuous. \[CondLF2\] Condition LF2* holds for ${\ensuremath{\mathsf{LC}}}$ in ${\ensuremath{\mathsf{Site}}}$. The composition of continuous morphisms is again continuous, and the analogous statement is true for cocontinuous morphisms. So we only have to see that the composition of two LC morphisms again fulfills properties (G), (F) and (FF) and we conclude by . Consider $v: {\mathfrak{a}}{\longrightarrow}{\mathfrak{b}}$ and $w: {\mathfrak{b}}{\longrightarrow}{\mathfrak{c}}$ two LC morphisms between sites. Property (G) for $w \circ v$ follows immediately from the fact that both $v$ and $w$ have property (G) and that the composition of coverings is again a covering (this is a direct consequence of the glueing axiom [@lowenlin §2.2 ]). We now prove property (F). Consider a morphism $$c: (w \circ v) (A) {\longrightarrow}(w \circ v) (A')$$ in ${\mathfrak{c}}$. As $w$ has property (F), we know there exist a covering $\{r_i: B_i {\longrightarrow}v(A)\}_i$ in ${\mathfrak{b}}$ and morphisms $b_i:B_i {\longrightarrow}v(A')$ such that $$\label{propF} c \circ w(r_i) = w(b_i).$$ Now, as $v$ has property (G), we can find covering families $\{s_{ij}:v(A_{ij}) {\longrightarrow}B_i\}_j$ in ${\mathfrak{b}}$ for all $i$. Now consider the morphisms $r_i \circ s_{ij}: v(A_{ij}) {\longrightarrow}v(A)$ and $b_i \circ s_{ij}:v(A_{ij}) {\longrightarrow}v(A')$ for each $i,j$. As $v$ has property (F), we know there exist coverings $\{t_{ijk} : A_{ijk} {\longrightarrow}A_{ij}\}_{k}$ and $\{t'_{ijk}: A'_{ijk} {\longrightarrow}A_{ij}\}_{k}$ and morphisms $a_{ijk}: A_{ijk} {\longrightarrow}A$ and $a'_{ijk}: A'_{ijk} {\longrightarrow}A'$ in ${\mathfrak{a}}$, such that $$\begin{aligned} r_i \circ s_{ij} \circ v(t_{ijk}) &= v(a_{ijk})\\ b_i \circ s_{ij} \circ v(t'_{ijk}) &= v(a'_{ijk}) \end{aligned}$$ for all $i,j,k$. Consider now the intersection of the two covering sieves generated by $\{t_{ijk} : A_{ijk} {\longrightarrow}A_{ij}\}_{k}$ and $\{t'_{ijk}: A'_{ijk} {\longrightarrow}A_{ij}\}_{k}$ in ${\mathfrak{a}}$ for each $i,j$ and take a covering family $\{u_{ijk}:A_{ijk} {\longrightarrow}A_{ij}\}_k$ generating this sieve. Then we have that $$\begin{aligned} r_i \circ s_{ij} \circ v(u_{ijk}) &= v(\bar{a}_{ijk})\\ b_i \circ s_{ij} \circ v(u_{ijk}) &= v(\bar{\bar{a}}_{ijk}) \end{aligned}$$ for morphisms $\bar{a}_{ijk}: A_{ijk} {\longrightarrow}A$ and $\bar{\bar{a}}_{ijk}: A_{ijk} {\longrightarrow}A'$. Observe that the family formed by the compositions $\{r_i \circ s_{ij} \circ v(u_{ijk})\}_{i,j,k}$ is a covering because it is a composition of coverings (note that $\{v(u_{ijk})\}_k$ is a covering of $v(A_{ij})$ in ${\mathfrak{b}}$ because $\{u_{ijk}\}_k$ is a covering of $A_{ij}$ in ${\mathfrak{a}}$ and $v$ is an LC morphism). Consequently $\{\bar{a}_{ijk}\}_{i,j,k}$ is a covering in ${\mathfrak{a}}$ of $A$, because $v$ is an LC morphism and $\{v(\bar{a}_{ijk})\}_{i,j,k}$ is a covering on ${\mathfrak{b}}$. Hence precomposing with $w(s_{ij} \circ v(u_{ijk}))$ in both terms of (\[propF\]) we have that: $$c \circ w(r_i) \circ w(s_{ij} \circ v(u_{ijk})) = w(b_i) \circ w(s_{ij} \circ v(u_{ijk}))$$ for all $i,j,k$. Observe that the first term is equal to $c \circ (w\circ v)(\bar{a}_{ijk})$ and the second term is equal to $(w \circ v)(\bar{\bar{a}}_{ijk})$. This proves (F) for $w \circ v$. To conclude, we prove property (FF). Consider a morphism $a:A {\longrightarrow}A'$ in ${\mathfrak{a}}$ such that $(w \circ v)(a) = 0$. As $w$ has property (FF), there is a covering $\{r_i:B_i {\longrightarrow}v(A)\}_i$ in ${\mathfrak{b}}$ such that $$\label{propFF} v(a) \circ r_i = 0$$ for all $i$. As $v$ has property (G), for each $i$ there exists a covering $\{s_{ij}: v(A_{ij}) {\longrightarrow}B_i\}_j$ in ${\mathfrak{b}}$. Consider the covering $\{r_i \circ s_{ij}: v(A_{ij}) {\longrightarrow}v(A)\}_{i,j}$ of $v(A)$ in ${\mathfrak{b}}$ given by the composition. As $v$ has property (F), for each $i,j$ there exist a covering $\{t_{ijk}:A_{ijk} {\longrightarrow}A_{ij}\}_k$ in ${\mathfrak{a}}$ and a family of morphisms $\bar{a}_{ijk}: A_{ijk} {\longrightarrow}A$ such that: $$\label{confirmcovering} r_i \circ s_{ij} \circ v(t_{ijk}) = v(\bar{a}_{ijk}).$$ Then, precomposing in both terms of (\[propFF\]) with $s_{ij} \circ v(t_{ijk})$, one has that $$v(a) \circ r_i \circ s_{ij} \circ v(t_{ijk}) = v(a \circ \bar{a}_{ijk})= 0.$$ Eventually, as $v$ has property (FF), we know that for every $i,j,k$ there exists a covering $\{u_{ijkl}:A_{ijkl} {\longrightarrow}A_{ijk}\}_{l}$ in ${\mathfrak{a}}$ such that: $$a \circ \bar{a}_{ijk} \circ u_{ijkl} = 0$$ But the family $\{\bar{a}_{ijk} \circ u_{ijkl}:A_{ijkl} {\longrightarrow}A\}_{ijkl}$ is a covering because it is a composition of coverings (observe in (\[confirmcovering\]) that, as $v$ is LC, $\{v(t_{ijk})\}_{k}$ is a covering in ${\mathfrak{b}}$ and hence so is $\{v(\bar{a}_{ijk})\}_{i,j,k}$ and thus $\{\bar{a}_{ijk}\}_{i,j,k}$ is a covering in ${\mathfrak{a}}$). Hence we conclude the argument. \[CondLF3\] Condition *LF3* holds for ${\ensuremath{\mathsf{LC}}}$ in ${\ensuremath{\mathsf{Site}}}$. Assume we have a solid diagram $$\begin{tikzcd} {\mathfrak{b}}\\ {\mathfrak{c}}\arrow[u,"w"] \arrow[r,"f"'] & {\mathfrak{a}}\end{tikzcd}$$ in ${\ensuremath{\mathsf{Site}}}$ with $w \in {\ensuremath{\mathsf{LC}}}$. We have to prove that it can be completed to a square $$\begin{tikzcd} {\mathfrak{b}}\arrow[r,dotted,"g"] \arrow[dr, start anchor={[xshift=+1.3ex, yshift=-1.3ex]}, end anchor={[xshift=-1.3ex, yshift=+1.3ex]}, Rightarrow,"\alpha"]& {\mathfrak{d}}\\ {\mathfrak{c}}\arrow[u,"w"] \arrow[r,"f"'] & {\mathfrak{a}}\arrow[u,dotted,"v"'] \end{tikzcd}$$ where $\alpha$ is an invertible $2$-morphism and $v \in {\ensuremath{\mathsf{LC}}}$. Consider the following morphism of Grothendieck categories $$\begin{tikzcd} {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}) \arrow[r, "f_s"] &{\ensuremath{\mathsf{Sh}} }({\mathfrak{c}}) \arrow[r,"\tilde{w}_"] &{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}) \end{tikzcd}$$ induced by $f$ and $w$, whose left adjoint is given by $f^s \circ \tilde{w}^: {\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}) {\longrightarrow}{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}})$ (see §\[seclinearsites\]). We apply the roof theorem () to this latter morphism. Take ${\mathfrak{d}}$ the site defined as the full subcategory of ${\ensuremath{\mathsf{Sh}} }({\mathfrak{a}})$ with objects $\{h^{\#}_A\}_{A \in {\mathfrak{a}}} \cup \{(f^s \circ \tilde{w}^)(h^{\#}_{B})\}_{B \in {\mathfrak{b}}}$ and we endow it with the topology induced by the canonical topology in ${\ensuremath{\mathsf{Sh}} }({\mathfrak{a}})$. Then consider the following morphisms: $$\begin{tikzcd} {\mathfrak{b}}\arrow[r,"g"] & {\mathfrak{d}}\\ & {\mathfrak{a}}\arrow[u,"v"'] \end{tikzcd}$$ defined $v = \#_{{\mathfrak{a}}} \circ Y_{{\mathfrak{a}}}$ and $g = (f^s \circ \tilde{w}^) \circ \#_{{\mathfrak{b}}} \circ Y_{{\mathfrak{b}}}$. Now, as $w$ is an LC morphism by hypothesis, and thus continuous and cocontinuous, we have the following chain of invertible $2$-morphisms (see §\[seclinearsites\] for the properties of continuous and cocontinuous functors): $$\begin{aligned} g \circ w &= f^s \circ \tilde{w}^* \circ \#_{{\mathfrak{b}}} \circ Y_{{\mathfrak{b}}} \circ w \\ &= f^s \circ \#_{{\mathfrak{c}}} \circ w^\circ i_{{\mathfrak{b}}} \circ \#_{{\mathfrak{b}}} \circ Y_{{\mathfrak{b}}} \circ w\\ &= f^s \circ \#_{{\mathfrak{c}}} \circ w^\circ i_{{\mathfrak{b}}} \circ w^s \circ \#_{{\mathfrak{c}}} \circ Y_{{\mathfrak{c}}}\\ &= f^s \circ \#_{{\mathfrak{c}}} \circ i_{{\mathfrak{c}}} \circ w_s \circ w^s \circ \#_{{\mathfrak{c}}} \circ Y_{{\mathfrak{c}}}\\ &\cong f^s \circ \#_{{\mathfrak{c}}} \circ Y_{{\mathfrak{c}}}\\ &= \#_{{\mathfrak{a}}} \circ Y_{{\mathfrak{a}}} \circ f\\ &= v \circ f \end{aligned}$$ Hence, if we take $\alpha$ to be this invertible $2$-morphism, we conclude the argument. \[CondLF5\] Condition *LF5* holds for ${\ensuremath{\mathsf{LC}}}$ in ${\ensuremath{\mathsf{Site}}}$. Consider two morphisms $v,w: {\mathfrak{a}}{\longrightarrow}{\mathfrak{b}}$ in ${\ensuremath{\mathsf{Site}}}$ such that $w \in {\ensuremath{\mathsf{LC}}}$. Assume we also have an invertible $2$-morphism $\alpha: v \Rightarrow w$. We want to prove that $v$ also belongs to ${\ensuremath{\mathsf{LC}}}$. Consider an object $B \in {\mathfrak{b}}$. By hypothesis, there exists a covering $\{r_i: w(A_i) {\longrightarrow}B \}_i$ and we can consider the associated covering $$\{v(A_i) \xrightarrow{\alpha_{A_i}} w(A_i) \xrightarrow{r_i} B \}$$ by composing with the isomorphism $\alpha_{A_i}$ given by the invertible $2$-morphism (recall that any isomorphism generates a covering sieve in a topology, the corresponding representable one). This proves property (G) for $v$. Consider now a morphism $b: v(A) {\longrightarrow}v(A')$ in ${\mathfrak{b}}$. Then we can take the morphism $\alpha_{A'} \circ b \circ \alpha_{A}^{-1}: w(A) {\longrightarrow}w(A')$, and as property (F) holds for $w$, we have that there exists a covering $\{s_i:A_i {\longrightarrow}A\}_i$ and morphisms $a_i: A_i {\longrightarrow}A'$ in ${\mathfrak{a}}$, such that $$\alpha_{A'} \circ b \circ \alpha_{A}^{-1} \circ w(s_i) = w(a_i)$$ for all $i$. Consequently: $$v(a_i) = \alpha_{A'}^{-1} \circ w(a_i) \circ \alpha_{A_i} = \alpha_{A'}^{-1} \circ \alpha_{A'} \circ b \circ \alpha_{A}^{-1} \circ w(s_i) \circ \alpha_{A_i} = b \circ v(s_i),$$ which proves that property (F) holds for $v$. Take now $a:A {\longrightarrow}A'$ in ${\mathfrak{a}}$ such that $v(a)= 0$. This implies that $w(a) = 0$ and hence there exists a covering $\{t_i:A_i {\longrightarrow}A \}_i$ such that $a \circ t_i = 0$ for all $i$, which proves that property (FF) holds for $v$. It remains to prove that ${\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}} = v^{-1}{\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}}$. We have that ${\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}} = w^{-1} {\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}}$ by hypothesis, hence given a covering sieve $\langle r_i: A_i {\longrightarrow}A \rangle$ of $A$ in ${\mathfrak{a}}$, we have that $\langle r_i \rangle \in {\ensuremath{\mathscr{T}}}_{{\mathfrak{a}}}(A)$ if and only if $\langle w(r_i): w(A_i) {\longrightarrow}w(A) \rangle \in {\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}}(w(A))$. On the other hand, it is easy to see that $\langle w(r_i) \rangle \in {\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}}(w(A))$ if and only if $$\langle \alpha_A^{-1} \circ w(r_i) \circ \alpha_{A_i} = v(r_i): v(A_i) {\longrightarrow}v(A) \rangle \in {\ensuremath{\mathscr{T}}}_{{\mathfrak{b}}}(v(A)),$$ because $\alpha_{A_i}, \alpha_A$ are isomorphisms. This concludes the argument. \[CondLF4\] Condition *LF4* holds for ${\ensuremath{\mathsf{LC}}}$ in ${\ensuremath{\mathsf{Site}}}$. First we prove (1) holds. Given two continuous morphisms $f,g : {\mathfrak{b}}{\longrightarrow}{\mathfrak{a}}$ in ${\ensuremath{\mathsf{Site}}}$, an LC morphism $w: {\mathfrak{b}}' {\longrightarrow}{\mathfrak{b}}$ and a 2-morphism $\alpha: f \circ w \Rightarrow g \circ w$, we have to prove that there exists an LC morphism $v: {\mathfrak{a}}{\longrightarrow}{\mathfrak{a}}'$ and a 2-morphism $\beta: v \circ f \Rightarrow v \circ g$ such that $v \circ \alpha = \beta \circ w$. Consider the morphisms $$\begin{tikzcd} {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}) \arrow[r,"(f \circ w)_s"] &{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}') &\text{and} &{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}) \arrow[r,"(g \circ w)_s"] &{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}') \end{tikzcd}$$ between the corresponding sheaf categories, with left adjoints $(f \circ w)^s: {\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}') {\longrightarrow}{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}})$ and $(g \circ w)^s:{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}') {\longrightarrow}{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}})$ respectively. We now perform a similar construction to that on the roof theorem. Take the full subcategory ${\mathfrak{a}}'$ of ${\ensuremath{\mathsf{Sh}} }({\mathfrak{a}})$ with $${\mathrm{Obj}}({\mathfrak{a}}') = \{h^{\#}_{A}\}_{A \in {\mathfrak{a}}} \cup \{(f \circ w)^s (h^{\#}_{B'})\}_{B' \in {\mathfrak{b}}'} \cup \{(g \circ w)^s (h^{\#}_{B'})\}_{B' \in {\mathfrak{b}}'},$$ and endow it with the topology given by the restriction of the canonical topology in ${\ensuremath{\mathsf{Sh}} }({\mathfrak{a}})$. In particular, ${\mathfrak{a}}'$ is subcanonical. We construct the following roofs $$\begin{tikzcd} &{\mathfrak{a}}' &&&{\mathfrak{a}}'\\ {\mathfrak{b}}' \arrow[ur,"r_f"] &&{\mathfrak{a}}\arrow[ul,"v"'] &{\mathfrak{b}}' \arrow[ur,"r_g"] &&{\mathfrak{a}}\arrow[ul,"v"'] \end{tikzcd}$$ where $v = \#_{{\mathfrak{a}}} \circ Y_{{\mathfrak{a}}}$, $r_f =(f \circ w)^s \circ \#_{{\mathfrak{b}}'} \circ Y_{{\mathfrak{b}}'}$ and $r_g= (g \circ w)^s \circ \#_{{\mathfrak{b}}'} \circ Y_{{\mathfrak{b}}'}$. One can easily see, following the same arguments as in the roof theorem, that $v$ is an LC morphism and that $(f \circ w)^s \cong \tilde{v}^* \circ (r_f)^s$ and $(g \circ w)^s \cong \tilde{v}^* \circ (r_g)^s$. We have chosen this special ${\mathfrak{a}}'$ in order to have the same site on the top of both roofs, but the reasoning to prove that these roofs behave as the usual roof construction is not affected by this enlargement of the top category, as it remains to be small. Now that $v$ is constructed, we proceed to build the 2-morphism $\beta: v \circ f \Rightarrow v \circ g$. First observe that given $\alpha: f \circ w \Rightarrow g \circ w$ we have $\alpha^s: (f \circ w)^s \Rightarrow (g \circ w)^s$ the induced 2-morphism described in . In particular, one has that: $$\label{alphasrepres} \begin{tikzcd} {\mathrm{Hom}}_{{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}')}(h^{\#}_{(g \circ w)(A)},F) \arrow[r,"- \: \circ h^{\#}_{\alpha_A}"] \arrow[d,"\cong"] &{\mathrm{Hom}}_{{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}')}(h^{\#}_{(f\circ w)(A)}, F) \arrow[d,"\cong"]\\ F((g \circ w)(A)) \arrow[r,"F(\alpha_A)"] \arrow[d,"\cong"] &F((f \circ w)(A)) \arrow[d,"\cong"]\\ {\mathrm{Hom}}_{{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}})}(h^{\#}_A,(g \circ w)_s(F)) \arrow[r,"(\alpha_s)_F \circ \: -"] \arrow[d,"\cong"] &{\mathrm{Hom}}_{{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}})}(h^{\#}_A, (f\circ w)_s(F)) \arrow[d,"\cong"]\\ {\mathrm{Hom}}_{{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}')}((g \circ w)^s(h^{\#}_A),F) \arrow[r,"- \: \circ (\alpha^s)_{h^{\#}_A}"] &{\mathrm{Hom}}_{{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}')}((f\circ w)^s(h^{\#}_A), F) \end{tikzcd}$$ is a commutative diagram. Now observe that: $$\begin{aligned} v \circ f &= \#_{{\mathfrak{a}}} \circ Y_{{\mathfrak{a}}} \circ f = f^s \circ \#_{{\mathfrak{b}}} \circ Y_{{\mathfrak{b}}},\\ v \circ g &= \#_{{\mathfrak{a}}} \circ Y_{{\mathfrak{a}}} \circ g = g^s \circ \#_{{\mathfrak{b}}} \circ Y_{{\mathfrak{b}}}. \end{aligned}$$ But notice that, as $w$ is an LC morphism, $w^s: {\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}') {\longrightarrow}{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}})$ is an equivalence with quasi-inverse given by $w_s:{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}) {\longrightarrow}{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}')$. Hence we have that $$\label{definingbeta} \begin{aligned} v \circ f &\cong (f \circ w)^s (w_s \circ \#_{{\mathfrak{b}}} \circ Y_{{\mathfrak{b}}})\\ v \circ g &\cong (g \circ w)^s (w_s \circ \#_{{\mathfrak{b}}} \circ Y_{{\mathfrak{b}}}). \end{aligned}$$ We define $\beta$ as the composition $$\begin{tikzcd} v \circ f \arrow[r,Rightarrow,"\cong"] &(f \circ w)^s \circ w_s \circ \#_{{\mathfrak{b}}} \circ Y_{{\mathfrak{b}}} \arrow[r,Rightarrow] &(g \circ w)^s \circ w_s \circ \#_{{\mathfrak{b}}} \circ Y_{{\mathfrak{b}}} \arrow[r,Rightarrow,"\cong"] &v \circ g \end{tikzcd}$$ where the second 2-morphism is given by $\alpha^s \circ (w_s \circ \#_{{\mathfrak{b}}} \circ Y_{{\mathfrak{b}}})$. Let’s now check that $v \circ \alpha = \beta \circ w$. First observe that have the following commutative diagram $$\begin{tikzcd}[column sep=50pt] h^{\#}_{(g \circ w)(B')} \arrow[r,"\beta_{w(B')}"] \arrow[d,"\cong"] &h^{\#}_{(f \circ w)(B')} \arrow[d,"\cong"]\\ (g\circ w)^s \circ w_s \circ h^{\#}_{w(B')} \arrow[r,"(\alpha^s)_{w_s (h^{\#}_{w(B')})}"] \arrow[d,"\cong"] &(f\circ w)^s \circ w_s \circ h^{\#}_{w(B')} \arrow[d,"\cong"]\\ (g \circ w)^s(h^{\#}_{B'}) \arrow[r,"(\alpha^s)_{h^{\#}_{B'}}"] \arrow[d,equal] &(f\circ w)^s(h^{\#}_{B'}) \arrow[d,equal]\\ h^{\#}_{(g \circ w)(B')} \arrow[r,"h^{\#}_{\alpha_{B'}}"] &h^{\#}_{(f \circ w)(B')}, \end{tikzcd}$$ where the last commutative square comes from the commutative diagram (\[alphasrepres\]) above. It is easily seen that the vertical compositions are the identity. Hence we have that $\beta_{w(B')} = h^{\#}_{\alpha_{B'}} = v(\alpha_{B'})$ for all $B' \in {\mathfrak{b}}'$, which concludes the argument. We prove now (2). Assume $\alpha$ is an invertible $2$-morphism. Then so are $\alpha_s$ and $\alpha^s$, and hence $\alpha^s \circ (t_s \circ w)$ is also an invertible $2$-morphism. As $\beta$ is obtained from $\alpha^s \circ (w_s \circ v)$ via pre- and postcomposing (vertically) with invertible $2$-morphisms, we conclude the argument. Finally, we prove (3). Assume there exists another $v': {\mathfrak{a}}{\longrightarrow}{\mathfrak{a}}'$ in ${\ensuremath{\mathsf{LC}}}$ and another 2-morphism $\beta': v' \circ f \Rightarrow v'\circ g$ with $v' \circ \alpha = \beta' \circ w$. We have to prove that there exist morphisms $u,u': {\mathfrak{a}}' {\longrightarrow}{\mathfrak{a}}''$ such that $u \circ v \in {\ensuremath{\mathsf{LC}}}$ and $u' \circ v' \in {\ensuremath{\mathsf{LC}}}$, and an invertible $2$-morphism $\epsilon: u \circ v \Rightarrow u'\circ v'$ such that the following diagram $$\label{LF4part3} \begin{tikzcd} u \circ v \circ f \arrow[r,"u \circ \beta"] \arrow[d,"\epsilon \circ f"] &u \circ v \circ g \arrow[d,"\epsilon \circ g"]\\ u' \circ v' \circ f \arrow[r,"u' \circ \beta'"] &u' \circ v' \circ g \end{tikzcd}$$ commutes. Consider the equivalence $v'^s \circ v_s: {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}') {\longrightarrow}{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}')$, whose quasi-inverse is given by $v^s \circ v'_s$. We consider the associated roof construction for $v^s \circ v'_s$: $$\begin{tikzcd} &{\mathfrak{a}}'' \\ {\mathfrak{a}}' \arrow[ur,"u'"] &&{\mathfrak{a}}', \arrow[ul,"u"'] \end{tikzcd}$$ with ${\mathfrak{a}}'' = \{ h^{\#}_{A'}\}_{A' \in {\mathfrak{a}}'} \cup \{ v^s \circ v'_s (h^{\#}_{A'}) \}_{A' \in {\mathfrak{a}}'}$, $u = \#_{{\mathfrak{a}}'} \circ Y_{{\mathfrak{a}}'}$ and $u' = (v^s \circ v'_s) \circ \#_{{\mathfrak{a}}'} \circ Y_{{\mathfrak{a}}'}$. In particular, $u$ is in ${\ensuremath{\mathsf{LC}}}$ and as so was $v$ by assumption, so it follows from that $u \circ v$ belongs to ${\ensuremath{\mathsf{LC}}}$. Now let’s construct $\epsilon$. For each $A \in {\mathfrak{a}}$ we have: $$u \circ v = \#_{{\mathfrak{a}}'} \circ Y_{{\mathfrak{a}}'} \circ v = v^s \circ \#_{{\mathfrak{a}}} \circ Y_{{\mathfrak{a}}} \cong (v^s \circ v'_s \circ v'^s) \circ \#_{{\mathfrak{a}}} \circ Y_{{\mathfrak{a}}} \cong (v^s \circ v'_s)(\#_{{\mathfrak{a}}'} \circ Y_{{\mathfrak{a}}'} \circ v') = u' \circ v'.$$ Let’s denote this composition of invertible $2$-morphisms by $\epsilon: u \circ v \Rightarrow u' \circ v'$. Observe first that from above, it follows that $u' \circ v'$ also belongs to ${\ensuremath{\mathsf{LC}}}$. To finish the argument, it remains to check that the diagram (\[LF4part3\]) above is commutative. Evaluating in any object $B \in {\mathfrak{b}}$ we obtain the following commutative diagram. $$\label{bigcomdiag} \begin{tikzcd}[row sep=small, column sep=60pt] u \circ v \circ f (B) \arrow[r,"u (\beta_B)"] \arrow[d,equals] &u \circ v \circ g(B) \arrow[d,equals]\\ h^{\#}_{v \circ f(B)} \arrow[r,"h^{\#}_{\beta_B}"] \arrow[d,"\cong"] &h^{\#}_{v \circ g(B)} \arrow[d,"\cong"]\\ (v \circ f \circ w)^s (w_s(h^{\#}_B)) \arrow[r,"(\beta \circ w)^s_{w_s (h^{\#}_B)}"] \arrow[d,equals] &(v \circ g \circ w)^s (w_s(h^{\#}_B)) \arrow[d,equals]\\ (v \circ f \circ w)^s (w_s (h^{\#}_B)) \arrow[r,"(v \circ \alpha)^s_{w_s h^{\#}_B}"] \arrow[d,"\cong"] &(v \circ f \circ w)^s (w_s (h^{\#}_B)) \arrow[d,"\cong"]\\ (v^s \circ v'_s) \circ (v'^s \circ f \circ w)^s (w_s (h^{\#}_B)) \arrow[r,"(v^s \circ v'_s) \circ (v' \circ \alpha)^s_{w_s h^{\#}_B}"] \arrow[d,equals] &(v \circ v'_s) \circ (v'^s \circ f \circ w)^s (w_s (h^{\#}_B)) \arrow[d,equals]\\ (v^s \circ v'_s) \circ (v'^s \circ f \circ w)^s (w_s (h^{\#}_B)) \arrow[r,"(v^s \circ v'_s) \circ (\beta' \circ w)^s_{w_s h^{\#}_B}"] \arrow[d,"\cong"] &(v^s \circ v'_s) \circ (v'^s \circ g \circ w)^s (w_s (h^{\#}_B)) \arrow[d,"\cong"]\\ (v^s \circ v'_s) \circ (v'^s \circ f)^s (h^{\#}_B)\arrow[r,"(v^s \circ v'_s) \circ \beta'^s_{h^{\#}_B}"] \arrow[d,equal] &(v^s \circ v'_s) \circ (v'^s \circ g)^s (h^{\#}_B) \arrow[d,equal]\\ (v^s \circ v'_s) (h^{\#}_{(v' \circ f)(B)})\arrow[r,"(v^s \circ v'_s) \circ h^{\#}_{\beta'_B}"] \arrow[d,equal] &(v^s \circ v'_s) (h^{\#}_{(v' \circ g)(B)}) \arrow[d,equal]\\ u' \circ v' \circ f (B) \arrow[r,"u' (\beta'_B)"] &u' \circ v' \circ g(B) \end{tikzcd}$$ Observe that the left vertical composition in the diagram equals $\epsilon_{f(B)}$ and the right vertical composition equals $\epsilon_{g(B)}$, which concludes our argument. Finally, we are in the position to prove the following. ${\ensuremath{\mathsf{LC}}}$ admits a left calculus of fractions in ${\ensuremath{\mathsf{Site}}}$. The statement follows from above. Hence we can localize ${\ensuremath{\mathsf{Site}}}$ with respect to ${\ensuremath{\mathsf{LC}}}$ and obtain the bilocalization ${\ensuremath{\mathsf{Site}}}[{\ensuremath{\mathsf{LC}}}^{-1}]$. The 2-category of Grothendieck categories as a bilocalization of the 2-category of sites {#sectionlocalizationallsites} ======================================================================================== In this section we prove the main result of the paper. \[mainresultbis\] There exists a pseudofunctor $$\Phi: {\ensuremath{\mathsf{Site}}}{\longrightarrow}{\ensuremath{\mathsf{Grt}}}$$ which sends LC morphisms to equivalences in ${\ensuremath{\mathsf{Grt}}}$, such that the pseudofunctor $$\tilde{\Phi}: {\ensuremath{\mathsf{Site}}}[{\ensuremath{\mathsf{LC}}}^{-1}] {\longrightarrow}{\ensuremath{\mathsf{Grt}}}$$ induced by $\Phi$ via the universal property of the bicategory of fractions is an equivalence of bicategories. Let ${\ensuremath{\mathcal{C}}}$ be a bicategory and $\mathsf{W}$ a class of $1$-morphisms in ${\ensuremath{\mathcal{C}}}$ that admits a calculus of left fractions. Given a bicategory ${\ensuremath{\mathcal{D}}}$ and a pseudofunctor $\Phi: {\ensuremath{\mathcal{C}}}{\longrightarrow}{\ensuremath{\mathcal{D}}}$ sending $1$-morphisms that belong to $\mathsf{W}$ to equivalences in ${\ensuremath{\mathcal{D}}}$, we have that $\Phi$ induces a pseudofunctor $\tilde{\Phi}: {\ensuremath{\mathcal{C}}}[\mathsf{W}^{-1}] {\longrightarrow}{\ensuremath{\mathcal{D}}}$ by the universal property of bilocalizations. A characterization of the pseudofunctors $\Phi$ such that $\tilde{\Phi}$ is an equivalence of bicategories (in the case of a right bicategory of fractions) is provided in [@tommasini3] . The characterization makes use of the right saturation of a class of morphisms introduced in [@tommasini2]. We formulate below an analogue for a left calculus of fractions. Let $\mathsf{W}$ be a class of $1$-morphisms in the bicategory ${\ensuremath{\mathcal{C}}}$. The (left) saturation $\mathsf{W}_{\mathrm{sat}}$ of $\mathsf{W}$ is the class of all $1$-morphisms $f : A {\longrightarrow}B$ in ${\ensuremath{\mathcal{C}}}$, such that there exists a pair of objects $C,D \in {\ensuremath{\mathcal{C}}}$ and a pair of morphisms $g : B {\longrightarrow}C$ and $h : C {\longrightarrow}D$, such that both $g \circ f$ and $h \circ g$ belong to $\mathsf{W}$. We say that $\mathsf{W}$ is (left) saturated if $\mathsf{W} = \mathsf{W}_{\mathrm{sat}}$. In analogy to [@tommasini2 Rem 2.3] in the case of right saturation, we have the following statement for the left saturation. \[leftsaturation\] If the class of morphisms $\mathsf{W}$ admits a left calculus of fractions, then $\mathsf{W} \subseteq \mathsf{W}_{\mathrm{sat}}$. \[mainresultbilocalization\] Let ${\ensuremath{\mathcal{C}}}$ be a bicategory and $\mathsf{W}$ a class of $1$-morphisms in ${\ensuremath{\mathcal{C}}}$ that admits a calculus of left fractions. Given a bicategory ${\ensuremath{\mathcal{D}}}$ and a pseudofunctor $\Phi: {\ensuremath{\mathcal{C}}}{\longrightarrow}{\ensuremath{\mathcal{D}}}$ sending $1$-morphisms that belong to $\mathsf{W}$ to equivalences in ${\ensuremath{\mathcal{D}}}$, we have that $\Phi$ induces an equivalence of bicategories $\tilde{\Phi}: {\ensuremath{\mathcal{C}}}[\mathsf{W}^{-1}] \overset{\cong}{{\longrightarrow}} {\ensuremath{\mathcal{D}}}$ if and only if: - $\Phi$ is essentially surjective on objects; - Given objects $C_1,C_2 \in {\ensuremath{\mathcal{C}}}$ and an equivalence $e: \Phi(C_2) \overset{\cong}{{\longrightarrow}} \Phi(C_1)$, there exits an object $C_3 \in {\ensuremath{\mathcal{C}}}$, a pair of morphisms $w_1: C_1 {\longrightarrow}C_3$ in $\mathsf{W}$ and $w_2: C_2 {\longrightarrow}C_3$ in $\mathsf{W}_{\mathrm{sat}}$, an equivalence $e':\Phi(C_3) {\longrightarrow}\Phi(C_1)$ in ${\ensuremath{\mathcal{D}}}$ and a pair of invertible $2$-morphisms $\delta_1,\delta_2$ as follows $$\begin{tikzcd} \Phi(C_2) \arrow[dr,"\Phi(w_2)"'] \arrow[drr, bend left, "e"] \arrow[drr, phantom, description, "\Downarrow \delta_2", pos=0.6,yshift=1ex]\\ &\Phi(C_3) \arrow[r, "e'"] &\Phi(C_1).\\ \Phi(C_1)\arrow[ur,"\Phi(w_1)"] \arrow[urr, bend right,"{\text{Id}}_{\Phi(C_1)}"'] \arrow[urr, phantom, description, "\Downarrow \delta_1", pos=0.6, yshift=-1ex] \end{tikzcd}$$ - Given objects $C \in {\ensuremath{\mathcal{C}}}$ and $D \in {\ensuremath{\mathcal{D}}}$ and a morphism $f: \Phi(C) {\longrightarrow}D$, there exists an object $C' \in {\ensuremath{\mathcal{C}}}$, a morphism $g: C {\longrightarrow}C'$ in ${\ensuremath{\mathcal{C}}}$, an equivalence $e:\Phi(C') \overset{\cong}{{\longrightarrow}} D$ in ${\ensuremath{\mathcal{D}}}$ and an invertible $2$-morphism $\alpha: f \Rightarrow e \circ \Phi(g)$. - Given objects $C, C' \in {\ensuremath{\mathcal{C}}}$, two $1$-morphisms $f_1, f_2: C {\longrightarrow}C'$ in ${\ensuremath{\mathcal{C}}}$ and two $2$-morphisms $\gamma_1,\gamma_2: f_1 \Rightarrow f_2$ such that $\Phi(\gamma_1) = \Phi(\gamma_2)$, there exits an object $C'' \in {\ensuremath{\mathcal{C}}}$ and a $1$-morphism $w: C' {\longrightarrow}C''$ in $\mathsf{W}$ such that $w \circ \gamma_1 = w \circ \gamma_2$. - Given objects $C,C' \in {\ensuremath{\mathcal{C}}}$, a pair of morphisms $f_1,f_2: C {\longrightarrow}C'$ and a $2$-morphism $\alpha: \Phi(f_1) \Rightarrow \Phi(f_2)$, then, there is an object $C'' \in {\ensuremath{\mathcal{C}}}$, a morphism $w: C' {\longrightarrow}C''$ in $\mathsf{W}$ and a $2$-morphism $\beta: w \circ f_1 \Rightarrow w \circ f_2$ such that $\Phi(w) \circ \alpha = \psi^{\Phi}_{w,f_2} \bullet \Phi(\beta) \bullet (\psi^{\Phi}_{w,f_1})^{-1} $, where $\psi^{\Phi}$ denotes the associator of the pseudofunctor $\Phi$. We need to use this set of necessary and sufficient conditions from [@tommasini3] as the set of sufficient conditions provided by [@pronk Prop 24] is not satisfied in our case. We are now in the position to prove . Recall from §\[section2catssitesgroth\] that we have a pseudofunctor $\Phi: {\ensuremath{\mathsf{Site}}}{\longrightarrow}{\ensuremath{\mathsf{Grt}}}$ that sends LC morphisms to equivalences. Hence by the universal property of the bilocalization, we have an induced pseudofunctor $$\tilde{\Phi}: {\ensuremath{\mathsf{Site}}}[{\ensuremath{\mathsf{LC}}}^{-1}] {\longrightarrow}{\ensuremath{\mathsf{Grt}}}.$$ Consequently, if the pseudofunctor $\Phi: {\ensuremath{\mathsf{Site}}}{\longrightarrow}{\ensuremath{\mathsf{Grt}}}$ satisfies properties *B1* to *B5* from , we conclude the argument. This is done in below. \[propertiesB\] The pseudofunctor $\Phi: {\ensuremath{\mathsf{Site}}}{\longrightarrow}{\ensuremath{\mathsf{Grt}}}$ satisfies properties *B1* to *B5* in above. We know every Grothendieck category can be realised as a category of sheaves on a site (see §\[seclinearsites\]), hence $\Phi$ is essentially surjective on objects, which proves B1. We now prove B2. Consider two sites ${\mathfrak{a}}_1,{\mathfrak{a}}_2$ and an equivalence $e: {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}_2) \overset{\cong}{{\longrightarrow}} {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}_1)$ in ${\ensuremath{\mathsf{Grt}}}$. We apply the roof theorem to the functor $e$. Let ${\mathfrak{a}}_3$ be the site with objects $${\mathrm{Obj}}({\mathfrak{a}}_3) =\{h^{\#}_{A_1}\}_{A_1 \in {\mathfrak{a}}_1} \cup \{e(h^{\#}_{A_2})\}_{A_2 \in {\mathfrak{a}}_2} \subseteq {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}_1),$$ and the topology induced by the canonical topology in ${\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}_1)$. We have the roof construction $$\begin{tikzcd} &{\mathfrak{a}}_3 \\ {\mathfrak{a}}_2 \arrow[ur,"w_2"] &&{\mathfrak{a}}_1, \arrow[ul,"w_1"'] \end{tikzcd}$$ where $w_1 = \#_{{\mathfrak{a}}_1} \circ Y_{{\mathfrak{a}}_1}$ and $w_2= e \circ \#_{{\mathfrak{a}}_2} \circ Y_{{\mathfrak{a}}_2}$. By the roof theorem, $w_1$ is an LC morphism. On the other hand, we have that $$e \cong \widetilde{(w_1)}^* \circ (w_2)^s \cong (w_1)_s \circ (w_2)^s,$$ where the first step is given by the roof theorem and the second by . Observe that $(w_1)_s$ is an equivalence because $w_1$ is an LC morphism. Then, as $e$ is an equivalence by hypothesis, $(w_2)^s$ is also and equivalence and hence so is $(w_2)_s$. In summary, we have a morphism $w_2: {\mathfrak{a}}_2 {\longrightarrow}{\mathfrak{a}}_3$ with ${\mathfrak{a}}_3$ subcanonical and such that $(w_2)_s$ is an equivalence. Then, it follows from [@lowenGP Cor 4.5] that $w_2$ is an LC morphism. Hence, in particular, as ${\ensuremath{\mathsf{LC}}}\subseteq {\ensuremath{\mathsf{LC}}}_{\mathrm{sat}}$ by , we have that $w_2 \in {\ensuremath{\mathsf{LC}}}_{\mathrm{sat}}$. Consider now the equivalence $$e' : {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}_3) \overset{\cong}{{\longrightarrow}} {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}_1),$$ given by $e'= \widetilde{(w_1)}^$. Then we can then choose $\delta_2$ to be the isomorphism: $$e \cong e' \circ (w_2)^s = e' \circ \Phi(w_2)$$ given by the roof decomposition of $e$. On the other hand, we have that $$e' \circ \Phi(w_1) = \widetilde{(w_1)}^ \circ (w_1)^s \cong (w_1)_s \circ (w_1)^s \cong {\text{Id}}_{{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}_1)},$$ where the second step follows from , and the last from the fact that $w_1$ is LC, and hence $(w_1)^s$ is an equivalence with quasi-inverse given by $(w_1)_s$. Then, we can denote by $\delta_1$ the horizontal composition of this chain of invertible $2$-morphisms, which concludes the argument. We now proceed to prove B3. Fix a site ${\mathfrak{b}}$, a Grothendieck category ${\ensuremath{\mathcal{A}}}$ and a $1$-morphism $f:{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}) {\longrightarrow}{\ensuremath{\mathcal{A}}}$ in ${\ensuremath{\mathsf{Grt}}}$. We choose a site ${\mathfrak{a}}$ such that we have an equivalence $e': {\ensuremath{\mathcal{A}}}\overset{\cong}{{\longrightarrow}} {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}})$. Consider the morphism $e' \circ f:{\ensuremath{\mathsf{Sh}} }({\mathfrak{b}}) {\longrightarrow}{\ensuremath{\mathsf{Sh}} }({\mathfrak{a}})$ in ${\ensuremath{\mathsf{Grt}}}$ and its associated roof decomposition $$\begin{tikzcd} &{\mathfrak{c}}\\ {\mathfrak{b}}\arrow[ur,"g"] &&{\mathfrak{a}}. \arrow[ul,"w",swap] \end{tikzcd}$$ We then have an equivalence $\tilde{w}_* \circ e':{\ensuremath{\mathcal{A}}}\overset{\cong}{{\longrightarrow}} {\ensuremath{\mathsf{Sh}} }({\mathfrak{a}}) \overset{\cong}{{\longrightarrow}} {\ensuremath{\mathsf{Sh}} }({\mathfrak{c}})$. Choose $e''$ a quasi-inverse of $e'$ and consider $e = e'' \circ \tilde{w}^: {\ensuremath{\mathsf{Sh}} }({\mathfrak{c}}) \overset{\cong}{{\longrightarrow}} {\ensuremath{\mathcal{A}}}$ which is a quasi-inverse of $\tilde{w}^ \circ e'$. Then, by the roof theorem, we have that $e' \circ f \cong \tilde{w}^* \circ g^s$ and hence, by postcomposing with $e''$ on both the left and the right hand side, we have an invertible $2$-morphism $f \cong e \circ g^s = e \circ \Phi(g)$, which finishes the argument. We now prove B4. Fix two sites ${\mathfrak{a}}, {\mathfrak{b}}$, two continuous morphisms $f_1,f_2: {\mathfrak{a}}{\longrightarrow}{\mathfrak{b}}$ and a pair of 2-morphisms $\gamma_1,\gamma_2:f_1 \Rightarrow f_2$ such that $$(\gamma_1)^s = (\gamma_2)^s: (f_1)^s \Rightarrow (f_2)^s.$$ Consider ${\mathfrak{c}}$ the site with objects $\{h^{\#}_{B}\}_{B \in {\mathfrak{b}}}$ with the topology induced by the canonical topology in ${\ensuremath{\mathsf{Sh}} }({\mathfrak{b}})$ and $w = \#_{{\mathfrak{b}}} \circ Y_{{\mathfrak{b}}}: {\mathfrak{b}}{\longrightarrow}{\mathfrak{c}}$ the corresponding LC morphism. Observe that $((\gamma_1)^s)_{h^{\#}_A} = ((\gamma_2)^s)_{h^{\#}_A}$ for all $A \in {\mathfrak{a}}$. This implies, applying the commutative diagram , that $h^{\#}_{(\gamma_1)_A} = h^{\#}_{(\gamma_2)_A}$ for all $A \in {\mathfrak{a}}$. Hence, we have that $(w \circ \gamma_1)_A = (w \circ \gamma_2)_A$ for all $A \in {\mathfrak{a}}$ and natural in $A$, which concludes the argument. Finally, we prove property B5. Consider two sites ${\mathfrak{a}}, {\mathfrak{b}}$, two continuous morphisms $f_1,f_2: {\mathfrak{a}}{\longrightarrow}{\mathfrak{b}}$ and a $2$-morphism $\alpha: f_1^s \Rightarrow f_2^s$. Let ${\mathfrak{c}}$ the site with objects $\{h^{\#}_{B}\}_{B \in {\mathfrak{b}}} \subseteq {\ensuremath{\mathsf{Sh}} }({\mathfrak{b}})$ and the topology induced by the canonical topology in ${\ensuremath{\mathsf{Sh}} }({\mathfrak{b}})$ and the LC morphism $w = \#_{{\mathfrak{b}}} \circ Y_{{\mathfrak{b}}}: {\mathfrak{b}}{\longrightarrow}{\mathfrak{c}}$. Take $\beta: w\circ f_1 \Rightarrow w \circ f_2$ the $2$-morphism given by $$w \circ f_1 = \#_{{\mathfrak{b}}} \circ Y_{{\mathfrak{b}}} \circ f_1 = f_1^s \circ \#_{{\mathfrak{a}}} \circ Y_{{\mathfrak{a}}} \overset{\alpha \circ \#_{{\mathfrak{a}}} \circ Y_{{\mathfrak{a}}}}{\Longrightarrow} f_2^s \circ \#_{{\mathfrak{a}}} \circ Y_{{\mathfrak{a}}} = \#_{{\mathfrak{b}}} \circ Y_{{\mathfrak{b}}} \circ f_2 = w \circ f_2.$$ Then, we have that $$\begin{tikzcd}[column sep= 80pt] w^s \circ f_1^s \arrow[r,Rightarrow,"w^s \circ \alpha"] &w^s \circ f_2^s \arrow[d, Rightarrow, "(\psi^{S}_{w,f_2})^{-1}", "\cong"'] \\ (w \circ f_1)^s \arrow[r,Rightarrow,"(\psi^{S}_{w,f_2})^{-1} \bullet (w^s \circ \alpha)\bullet \psi^{S}_{w,f_1}"] \arrow[u, Rightarrow, "\psi^{S}_{w,f_1}", "\cong"'] \arrow[d,equal] & (w \circ f_2)^s \arrow[d,equal] \\ (f_1^s \circ \#_{{\mathfrak{a}}} \circ Y_{{\mathfrak{a}}})^s \arrow[r,Rightarrow,"(\alpha \circ\#_{{\mathfrak{a}}} \circ Y_{{\mathfrak{a}}})^s "]&(f_2^s \circ \#_{{\mathfrak{a}}} \circ Y_{{\mathfrak{a}}})^s \end{tikzcd}$$ is a commutative diagram of $2$-morphisms. But observe that the composition $$(w \circ f_1)^s = (f_1^s \circ \#_{{\mathfrak{a}}} \circ Y_{{\mathfrak{a}}})^s \overset{(\alpha \circ\#_{{\mathfrak{a}}} \circ Y_{{\mathfrak{a}}})^s}{\Longrightarrow} (f_2 \circ \#_{{\mathfrak{a}}} \circ Y_{{\mathfrak{a}}})^s = (w \circ f_2)^s$$ is just $\beta^s$ by definition, hence $$w^s \circ \alpha = \psi^{S}_{w,f_2} \bullet \beta^s \bullet (\psi^{S}_{w,f_1})^{-1},$$ which concludes the argument. Monoidal bilocalization {#tensorproductvialocalization} ======================= Let ${\ensuremath{\mathcal{C}}}$ be a category and $\mathsf{W}$ a class of morphisms which admits a calculus of fractions in the sense of Gabriel-Zisman [@gabriel-zisman]. Then, as it is proven in [@day], if ${\ensuremath{\mathcal{C}}}$ has a symmetrical monoidal structure such that $\mathsf{W}$ is closed under tensoring, then the localization ${\ensuremath{\mathcal{C}}}[\mathsf{W}^{-1}]$ has a monoidal structure such that the localization functor ${\ensuremath{\mathcal{C}}}{\longrightarrow}{\ensuremath{\mathcal{C}}}[\mathsf{W}^{-1}]$ is a monoidal functor. This is what in [@day] is referred to as monoidal localization. It is reasonable to believe that an analogous result for monoidal bicategories and bilocalizations holds true, and we plan to return to this topic in the future. In this section, we briefly sketch a possible application of the main result of this paper given that we have a satisfactory theory of monoidal bilocalization available. In [@lowen-ramos-shoikhet], we define a tensor product of linear sites, which is seen to define a symmetric monoidal structure on the bicategory ${\ensuremath{\mathsf{Site}}}$ $$\boxtimes: {\ensuremath{\mathsf{Site}}}\times {\ensuremath{\mathsf{Site}}}{\longrightarrow}{\ensuremath{\mathsf{Site}}}: (({\mathfrak{a}}, {\ensuremath{\mathcal{T}}}_{{\mathfrak{a}}}), ({\mathfrak{b}}, {\ensuremath{\mathcal{T}}}_{{\mathfrak{b}}})) \longmapsto ({\mathfrak{a}}\otimes {\mathfrak{b}}, {\ensuremath{\mathcal{T}}}_{{\mathfrak{a}}} \boxtimes {\ensuremath{\mathcal{T}}}_{{\mathfrak{b}}}).$$ We further showed in loc. cit. that the class of LC morphisms is closed under $\boxtimes$, hence we obtain an induced bi-pseudofunctor $$\tilde{\boxtimes}: {\ensuremath{\mathsf{Site}}}[{\ensuremath{\mathsf{LC}}}^{-1}] \times {\ensuremath{\mathsf{Site}}}[{\ensuremath{\mathsf{LC}}}^{-1}] = ({\ensuremath{\mathsf{Site}}}\times {\ensuremath{\mathsf{Site}}})[({\ensuremath{\mathsf{LC}}}\times {\ensuremath{\mathsf{LC}}})^{-1}] {\longrightarrow}{\ensuremath{\mathsf{Site}}}[{\ensuremath{\mathsf{LC}}}^{-1}]$$ which a general theory of monoidal bilocalization would yield to define a monoidal structure in the bicategorical sense. This structure could then be transferred to the equivalent bicategory ${\ensuremath{\mathsf{Grt}}}$ (in a non-canonical way). Note that in [@lowen-ramos-shoikhet], we use the roof construction in order to obtain a well-defined (up to equivalence) tensor product of Grothendieck categories, at least on the level of the categories, and we show that this tensor product is a special case of the tensor product of locally presentable categories which is known to be bi-functorial and monoidal. Further, we provide an alternative approach to these issues in [@ramosthesis], making use of canonical bicolimit presentations of Grothendieck categories. \[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{} [10]{} Théorie des topos et cohomologie étale des schémas. [T]{}ome 1: [T]{}héorie des topos, Springer-Verlag, Berlin, 1972, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. 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Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35 Springer-Verlag New York, Inc., New York, 1967, x+168 pp. N. Ganter and M. Kapranov, Representation and character theory in 2-categories, Adv. Math. 217 (2008), no. 5, 2268–2300. W. Lowen, Linearized topologies and deformation theory, Topology Appl. 200 (2016), 176–211. , A generalization of the [G]{}abriel-[P]{}opescu theorem, J. Pure Appl. Algebra 190 (2004), no. 1-3, 197–211. W. Lowen, J. Ramos González and B. Shoikhet, On the tensor product of linear sites and Grothendieck categories, Int. Math. Res. Not. IMRN, 2017, [https://doi.org/10.1093/imrn/rnx072]{}. D. Pronk, Etendues and stacks as bicategories of fractions, Compositio Math. 102 (1996), no. 3, 24–303. J. Ramos González, On the tensor product of large categories, PhD thesis in preparation, University of Antwerp. The Stacks Project Authors, Stacks Project, 2017, [http://stacks.math.columbia.edu]{}. J. T. Stafford and M. van den Bergh, Noncommutative curves and noncommutative surfaces, Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 2, 171–216 (electronic). M. Tommasini,Some insights on bicategories of fractions: representations and compositions of 2-morphisms Theory Appl. Categ. 31 (2016), Paper No. 10, 257–329. , Some insights on bicategories of fractions II - Right saturations and induced pseudofunctors between bicategories of fractions, [arXiv:1410.5075 \[math.CT\]]{}. , Some insights on bicategories of fractions III - Equivalences of bicategories of fractions, [arXiv:1410.6395 \[math.CT\]]{}. H. Wolff, ${\ensuremath{\mathcal{V}}}$-fractional categories, Cahiers Topologie Géom. Différentielle 16 (1975), no. 2, 149–168. [^1]: We address size issues in §\[section2catssitesgroth\].
--- author: - 'N. Schartel[^1]' title: \| Space Astronomy for the mid-21st Century:\ Robotically Maintained Space Telescopes --- Introduction ============ Today, about 20% of the articles in major astronomical journals (Astronomy and Astrophysics, The Astrophysical Journal and Monthly Notices of the Royal Astronomical Society), refer to X-rays in their title or abstract. This illustrates the fact that X-ray observations have become one of our main tools of astrophysical research. This development is based on a fleet of satellites with three observatory-type facilities currently operating: $\;$ Chandra (Weisskopf et al. 2002), XMM-Newton ([@Jansen2001]) and Suzaku ([@Mitsuda2007]). During the last thirty years the astronomical community had almost permanent access to X-ray observations through a multitude of national and international missions, e.g. HEAO-1, Einstein, EXOSAT, Ginga, ROSAT, ASCA and BeppoSax. Active astronomers expect to have permanent access to the X-ray sky with observatory-class facilities $\;$ to satisfy their scientific needs. Although the funding agencies recognize this demand as demonstrated by the high priority that the next generation X-ray telescopes received in US Decadal Survey and in ESA’s Cosmic Vision, the future is far from certain due to budget constraints and the requirement to explore other observational areas $\;$ through space born facilities, e.g. the gravitational waves. Even if there would be an approved next-generation X-ray telescope, the basic expectation, permanent access to the X-ray sky would only be temporarily be satisfied due to the restricted lifetime of space missions. In this paper we try to explore if ground based astronomy and its recent history can be a helpful guide. In section \[p2\] we outline basic development in ground based astronomy and we present in section \[p3\] a concept for space mission. In section \[p4\] we discuss the technical and organizational feasibility followed by some concluding remarks in section \[p5\]. Ground observatories {#p2} ==================== Over the last century ground based astronomical observing facilities were the subject to a major developmental process. Today, first class observatories are generally funded either nationally or even multi-nationally, e.g. ESO, ALMA, etc. The primary characteristic of a telescope is its collecting area, and substantial scientific impact can be expected if it is possible to increase the collecting area by factor of ten (greater than five as minimum). The development cycle for such mirror advances is some thirty years. Instrumentations develop at a much faster rate. Within some ten years, instrumentation shows impressive progress. Construction of a telescope mounting, a dome and the mirror itself are the major costs. In comparison to these costs, the instrumentation requires only minor resources. Operations are an important aspect because their costs have to be paid annually. Generally, an observatory supports several telescopes allowing the minimization of the operational costs due to a high amount of synergy between operations of the individual telescopes. A leading observatory has to provide a new primary telescope every thirty years in order to continue to be scientifically competitive, whereas the instrumentation is replaced on shorter time scales. In general, older telescopes continue to be operated reaching operational lifetimes of some sixty years maximizing the scientific return on the primary investment. Outline of a concept {#p3} ==================== If we want to transfer concepts of ground based observatories into space then we must consider space borne telescopes with operational lifetimes of forty to sixty years. It is obvious that such telescopes will need maintenance in space. Only robotic maintenance missions can be expected to provide these at reasonable costs. Beside repair and replacement of failing components, and substitution of consumables, they will transport and install new instruments such that the scientific value of the observatory $\,$ remains competitive. The robotically maintained space telescope $\,$ should include the following components: 1. Selection of roboticaly maintained space telescope: It is impossible to predict the astronomical requirements or the development of scientific instruments on timescales of thirty to sixty years. Therefore, the main criteria for the decision to build a robotic maintained space telescope must be the progress achieved in the engineering, construction and manufacturing of the satellite and the mirror and specifically the possibility to increase of collecting area of the mirror. We can assume that an increase in the effective area by a factor of 10 ($>$5) will bring substantial scientific progress and justify the selection of a mission. A further important criterion is the achievable sensitivity in comparison with other wavelengths $\,$ as the scientific merit $\,$ increases with wider wavelength coverage. Depending on the wavelength, an increase of the spatial resolution or other performance parameter may be an important addition. 2. The space telescope: The primary goal is a long lifetime, i.e. a designed lifetime of some thirty years with a potential to operate for sixty years. Because it can not be expected that a mission will operate for such a long time without any failure, the mission must be constructed for robotic maintenance in space. Most important will be the possibility to replace consumables and the possibility to replace key spacecraft components, e.g. gyroscopes or reaction wheels. From the scientific side it is most important that the spacecraft allows a simple exchange of the scientific instruments. 3. Robotic maintenance missions: Assuming a typical instrumental development cycle $\,$ of ten years, $\,$ robotic maintenance missions $\,$ may visit the space telescope once each decade. The maintenance spacecraft must be able to perform small repairs, substitute key spacecraft elements, refill consumables and specifically to replace scientific instruments. The replacement of the scientific instrument is of utmost importance to ensure the scientific competitiveness of the mission. 4. Operations: Given the long lifetime of the space telescope it is of importance that the ground operations run with extremely low costs. These can be ensured by a highly automated $\,$ and autonomous telescope and onboard instruments. Another aspect is the software used for ground operations. The challenge here is that the software has to be useable over many decades. A further important issue is the amount of funds the community can contribute to the scientific aspects of the operation of the mission, e.g. instrument calibration etc.. \[figure1\] Feasibility of a robotically maintained space telescope {#p4} ======================================================= Almost no challenge for robotically maintained space telescopes is new, unsolved or not already a hot topic of development and engineering and therefore is expected to be available within a decade. In the following section we give some comments on the arguments brought forward above and provide examples to illustrate the claims: 1. Selection of $\,$ roboticaly maintained space telescope: $\,$ [@Lang2010] illustrated impressively that astronomical research was and is fundamentally driven from unexpected detections which were possible only due to technological advance. The thirty years long research $\,$ to identify the nature of gamma ray bursts ([@Klebesadel1973] and [@vanParadijs1997]), illustrates the importance to have comparable sensitivity at different wavelengths combined with adequate spatial resolution. A good example is the current debate on the electro-magnetic $\,$ counterparts of gravitational wave sources. Specifically $\,$ the X-ray energies are considered most promising [@Komossa2010], implying that a gravitat–ional wave telescope should be operated simultaneously with a large X-ray telescope. Only when we achieve simultaneous coverage at different wavelengths will the astronomical progress be optimal and efficient. 2. The space telescope: $\,$ With respect to the space telescope, the foremost question is the scientific competitiveness of the collecting area on timescales of thirty to sixty years. Fig. 1 demonstrates that the effective area of hypothetical microcalorimeter spectrometers on board of XMM-Newton would exceed the effective area of the currently operating Reflection Grating Spectrometers (RGS, [@denHerder2001]) by more than a factor ten. Both instruments can reasonably be assumed to have comparable spectral resolution. Currently, there is no plan for a mission that has spatial resolution comparable to Chandra. Therefore, there will be a continual scientific demand that can only be satisfied by Chandra observations and there is not successor likely be to happen for some decades. A further example is ROSAT [@Truemper1982], which operated from 1990 to 1999. Its grasp in the energy range from 0.1 to 2.0 keV was significantly higher that the grasp of XMM-Newton. In combination with the low background due to ROSATs orbit, large surveys aiming for cosmological studies could be performed with an operating ROSAT in about half the time required by XMM-Newton. These three examples illustrate that a conservation of telescop collecting areas for some thirty years is scientifically compelling. Chandra and XMM-Newton are now operating for twelve years [@Santos-Leo2009]. IUE reached an operational lifetime of eighteen years [@Wamsteker2000] and HST is now twenty-two years old [@Dalcanton2009]. Therefore, lifetimes of sixty years are within the range of technological possibilities. HST is the only astronomical satellite which was constructed for maintenance in space and several components and instruments were replaced by astronauts. Most probably robotically maintained space telescopes will require a modular construction of the spacecraft $\,$ allowing easy replacements of complete units and therefore reducing the complexity of the robotic maintenance missions. Such modular elements may offer the possibility of general cost reduction because the modules might be used by several robotically maintained scientific and commercial satellites. The planning for the replacement of scientific instruments with maintenance missions significantly reduces the risk in the development of the primary generation of instruments that can constraint to established technology. 3. Robotic maintenance missions: The engineering of the spacecraft for robotic maintenance is the most challenging task for this concept. To development the required engineering skills for a single specific mission or task would most probably be inefficient due to the high costs. Robotic spacecrafts must maintain a wide range of commercial and scientific missions and may either be reused or built in large numbers. However, international, intergovernmental, national as well as private companies are currently developing such spacecrafts and the main issue is to utilize these developments also for scientific purposes. Examples for developments in this direction are the Special Purpose Dexterous Manipulator of MacDonald, Dettwiler and Associates (MDA), the Robotic Refueling Mission of NASA, the \$280 million contract between Intelsat and MDA to develop robots for simple repairs and refueling of geostationary communication satellites and the Deutsche Orbital Servicing Mission of DLR for repair and non-destructive capturing of satellites. It is important to see that such robotic maintenance mission will have very low costs, e.g. a refueling mission is estimated to cost some thirty to fifty million dollars. With respect to the exchange of scientific instruments it is important that scientific instruments generally have very low mass in comparison to the weight of the spacecraft and its mirror. ROSAT had a mass of some 2,500 kg, XMM-Newton has a mass of 3,800 kg and Chandra has a mass of 4,800 kg, respectively. For comparison a X-ray microcalorimeter spectrometer with cryogenic cooling has a mass of some 400 kg and a polarimeter has a mass of some 10 kg. The overall risk in association with the development and employment of new instruments is significantly reduced because the robotic $\,$ maintenance mission $\,$ would be easier $\,$ and cheaper than the primary mission and can be delayed. 4. Operations: Operational costs are important due to the long lifetime of the mission. Ideally astronomers should be able to operate $\,$ the satellite $\,$ within their research work. There are strong development lines in this direction. $\,$ Now, many astronomical satellites $\,$ are contin–uously operating for days without interaction with the ground station. The Swift satellite already allows astronomers to change the pointing direction with a cell phone call. Instruments can be developed such that no pointing direction will put them on risk and the instruments can be able to cope autonomously with unexpected source fluxes $\,$ or radiation environment. $\,$ The maintenance of operational and scientific software over decades is challenging. $\,$ The solution can only be to avoid mission specific software as far as possible and be in a modular system structure. Many examples of task specific software are already applied in astrophysical and operational contexts. Ideally the individual missions would define their specific requirements simply in the form of configuration files whereas the software itself is unchanged. Ideally, astronomers could easily install the software by themselves. The HEASARC remote proposal submission system is an example for such a task specific software component. Scientific analysis software packages, like MIDAS or IRAF, are further examples of widely used mission independent software. Although the investments to establish a full set of operations and scientific software along the engineering concepts is high, the pay off would be threefold: (a) astronomers would not have to spend time to familiarize themselves with new software, (b) the costs of further science operation center development would be drastically reduced, and (c) the development risk would basically be eliminated. Conclusions {#p5} =========== Most of the examples provided come from X-ray astronomy and reflect experiences of the author. However, the concept of robotically maintained space observatories itself are naturally wavelength independent. The concept may be applied at any wavelength where the mirror capacities are sufficiently advanced such that the number of potential targets does not naturally constrain the lifetime of the mission. The application of the concept further depends on the ratio of instrument mass to space telescope mass and the development cycle of the instruments. The concept depends on the progress in robotic engineering, but currently enormous efforts and progress are being made. The scientific community is asked to consider and utilize the progress made in robotics for their future space missions. I thank Brian McBreen, who convinced me to formulate my ideas of robotically maintained space telescopes and to share them with the community. I would like to thank R. Gonz[á]{}lez-Riestra for calculating the RGS effective area and M. Smith for his help with the XMM-Newton filter transmission. Dalcanton, J.J.: 2009, Natur 457, 41 den Herder, et al.: 2001, A&A 365, L7 Jansen, F., et al.: 2001, A&A 365, L1 Klebesadel, R. W., Strong, I. B. & Olson, R. A.: 1973, ApJ 182, L85 Komossa, S.: 2010, IAU Symposium 267, 451-457 Lang, K.R.: 2010, Science 327, 39 Mitsuda, et al.: 2007, PASJ 59S, 1 Santos-Lleo, M., et al.: Nature 462, 997 Str[ü]{}der, L., et al.: 2001, A&A 365, L18 Tr[ü]{}mper, J.: 1982, AdSpR 2, 241 van Paradijs, J., et al.: 1997, Nature 386, 686 Wamsteker, W., et al.: 2000, Ap&SS 273, 155 Weisskopf, M. C., et al.: 2002, PASP 114, 1 Effective area of hypothetical X-ray microcalorimeter spectrometer onboard of XMM-Newton {#p6} ======================================================================================== XMM-Newton has three X-ray telescopes. To estimate the total effective area of hypothetical X-ray microcalorimeter spectrometers onboard XMM-Newton we considered three components: the effective area of the telescopes, the filter transmission and the quantum efficiency of the microcalorimeter. The pn-CCD camera [@Strueder2001] is operated in the focal plane of X-ray telescope 3 which is the telescope without reflection grating array unit. We use the on-axis effective area of telescope 3 as provided in the current calibration file XRT3\_XAREAE F\_0011.CCF[^2]. We approximate the transmission of the optical filters of the microcalorimeter through the transmission of the thick filter of pn (EPN\_FILTERTRANSX\_0014.CCF$^1$). We assume a quantum efficiency (including filling factor and ’dead’ detector elements) of 0.93 for energies below 4.5 keV which decreases with $\sim$0.11 keV$^{-1}$ for higher energies. The obtained total effective area is added as on line material (xmm\_cal.fits). [^1]: Corresponding author: [^2]: http://xmm2.esac.esa.int/external/xmm\_sw\_cal/calib/rel\_notes/index. shtml
--- abstract: 'Given a bounded domain $\Omega$ in $\mathbb{R}^N$, $N\geq 1$ we study the asymptotic behavior as $\varepsilon \to 0$ of the eigencurves of $$-\Delta_p u_\varepsilon=\alpha_\varepsilon m(\tfrac{x}{\varepsilon})(u_\varepsilon^+ )^{p-1} - \beta_\varepsilon n(\tfrac{x}{\varepsilon})(u_\varepsilon^- )^{p-1} \quad \textrm{ in } \Omega$$ with Dirichlet boundary conditions, where $m$ and $n$ are bounded periodic weights. In this work we obtain accurate bounds of the convergence rates of these curves to some limit curves as ${\varepsilon}\to 0$.' address: 'Departamento de Matemática FCEN - Universidad de Buenos Aires and IMAS - CONICET. Ciudad Universitaria, Pabellón I (1428) Av. Cantilo s/n. Buenos Aires, Argentina.' author: - 'Ariel M. Salort' bibliography: - 'Biblio.bib' title: 'Homogenization of [F]{}u[č]{}[í]{}k eigenvalues by optimal partition methods' --- Introduction ============ Given a bounded domain $\Omega$ in ${\mathbb R}^N$, $N\geq 1$ we study the asymptotic behavior as $\varepsilon \to 0$ of the spectrum of the following asymmetric elliptic problem $$\begin{aligned} \label{P1} \bigg\{ \begin{array}{ll} -\Delta_p u_\varepsilon=\alpha_\varepsilon m_\varepsilon(u_\varepsilon^+ )^{p-1} - \beta_\varepsilon n_\varepsilon(u_\varepsilon^- )^{p-1} &\quad \textrm{ in } \Omega\\ u_{\varepsilon}=0 &\quad \mbox{on }\partial \Omega. \end{array}\end{aligned}$$ Here, $\Delta_p u:=div(\|\nabla u\|^{p-2}\nabla u)$ denotes the $p-$Laplace operator with $1<p<\infty$ and, as usual, $u^\pm:=\max\{\pm u, 0\}$. The parameters $\alpha_\varepsilon$ and $\beta_\varepsilon$ are real numbers depending on $\varepsilon>0$. Here the family of functions $m_\varepsilon$ and $n_\varepsilon$ are given in terms of $Q-$periodic functions, $Q$ being the unit cube in $R^N$, in the form $m_{\varepsilon}(x)=m(x/{\varepsilon})$ and $n_{\varepsilon}(x)=n(x/{\varepsilon})$. The functions $m$ and $n$ are assumed to be positive and uniformly bounded away from zero and infinity, that is, there are constants $\theta_-$, $\theta_+$ such that $$\label{cotas} 0<\theta_- \leq m(x), n(x) \leq \theta_+ < +\infty.$$ It is well-known that as $\varepsilon \to 0$, $$\label{limi} m_\varepsilon(x)\rightharpoonup \bar m=\fint_Q m(x)\, dx, \quad n_\varepsilon(x)\rightharpoonup \bar n=\fint_Q n(x)\, dx \quad\textrm{ weakly* in }L^\infty(\Omega).$$ Problem was widely studied for a fixed value of ${\varepsilon}>0$: see for instance Arias and Campos [@AR1], Drabek [@DRA], Reichel and Walter [@RE1], Rynne and Walter [@RYN], for positive weights; Alif and Gossez [@ALIF1], Leadi and Marcos [@LIAM] for indefinite weights. For a fixed ${\varepsilon}>0$, the Fučík spectrum of is defined as the set $$\Sigma_{\varepsilon}=\Sigma_{\varepsilon}(m_{\varepsilon},n_{\varepsilon}):=\{(\alpha_{\varepsilon},\beta_{\varepsilon}) \in {\mathbb R}^2\colon (\ref{P1}) \textrm{ has a nontrivial solution} \}.$$ Moreover, we say that a nontrivial function $u_{\varepsilon}\in W^{1,p}_0(\Omega)$ is an eigenfunction of associated to $(\alpha_{\varepsilon},\beta_{\varepsilon})\in {\mathbb R}^+\times {\mathbb R}^+$ if it satisfies the weak formulation $$\label{debil} \int_\Omega \|\nabla u\|^{p-2} \nabla u \cdot \nabla v \, dx = \int_\Omega (\alpha m_{\varepsilon}(x) (u_{\varepsilon}^+)^{p-1} v - \beta_{\varepsilon}n(x) (u_{\varepsilon}^-)^{p-1} v)\,dx$$ for all $v\in W^{1,p}_0(\Omega)$. Observe that when both weights are the same, let us say, $r_{\varepsilon}$, and both parameters are equal, let us say, ${\lambda}_{\varepsilon}$, equation (\[P1\]) becomes the weighted $p-$laplacian eigenvalue problem with Dirichlet boundary conditions, i.e., $$\begin{aligned} \label{Plap} \begin{cases} -\Delta_p u_{\varepsilon}= {\lambda}_{\varepsilon}r_{\varepsilon}\|u_{\varepsilon}\|^{p-2}u_{\varepsilon}&\quad \textrm{in } \Omega\\ u_{\varepsilon}=0 &\quad \mbox{on }\partial \Omega. \end{cases}\end{aligned}$$ One immediately observe that $\Sigma_{\varepsilon}$ contains the trivial lines ${\lambda}_{1}(m_{\varepsilon})\times {\mathbb R}$ and ${\mathbb R}\times {\lambda}_{1}(n_{\varepsilon})$, being ${\lambda}_1(r_{\varepsilon})$ the first eigenvalue of . In contrast with the one-dimensional case, where a full description of the spectrum is obtained, when $N>1$ it is only known the existence of a curve $\mathcal{C}_{\varepsilon}$ beyond the trivial lines, see [@AR1; @AR3]. Such curve can be written by considering its intersection with the line of slope $s\in {\mathbb R}^+$ passing through the origin in ${\mathbb R}^2$ as $$\begin{aligned} \label{def_ce} {\mathcal{C}}_{\varepsilon}=\mathcal{C}_{{\varepsilon}}(m_\varepsilon,n_\varepsilon):=\{(\alpha_\varepsilon(s),\beta_\varepsilon(s)), s\in{\mathbb R}^+\}.\end{aligned}$$ The authors in [@AR3] deal with a variational characterization for $\alpha(s)$ and $\beta(s)$. When $\varepsilon \to 0$ the following natural limit problem for (\[P1\]) is obtained $$\label{pron1limite.gral} \bigg\{ \begin{array}{ll} -\Delta_p u=\alpha_0 \bar m (u^+ )^{p-1} - \beta_0 \bar m (u^- )^{p-1} &\quad \textrm{ in } \Omega \\[0.05 cm] u=0 &\quad \textrm{ on } \partial \Omega \end{array}$$ where $\bar m$ and $\bar n$ are given in (\[limi\]), and whose corresponding first nontrivial curve is denoted by $$\mathcal{C}_0=\mathcal{C}_0(m_0,n_0):=\{(\alpha_0(s),\beta_0(s)), s\in{\mathbb R}^+\}.$$ In this context, in the previous work [@SA-fucik] it was stated the convergence of ${\mathcal{C}}_{\varepsilon}$ to the limit curve ${\mathcal{C}}_0$ (even for non-periodic weights) in the sense that $$\alpha_\varepsilon(s)\to \alpha_0(s) \quad \mbox{and} \quad \beta_\varepsilon(s)\to \beta_0(s)$$ as ${\varepsilon}\to 0$, for each fixed $s\in{\mathbb R}^+$. Moreover, by using the variational characterization of ${\mathcal{C}}_{\varepsilon}$ and ${\mathcal{C}}_0$ provided by [@AR3], it was established the convergence rates of the curves: \[teo\_1.viejo\] Given ${\varepsilon}>0$ and $s\in {\mathbb R}^+$, let $(\alpha_{\varepsilon}(s),\beta_{\varepsilon}(s))\in {\mathcal{C}}_{\varepsilon}$ and $(\alpha_0(s),\beta_0(s))\in {\mathcal{C}}_0$. Then the following estimates hold $$\begin{aligned} \label{estim1} \|\alpha_\varepsilon(s) - \alpha_0(s)\|\leq \begin{cases} c{\varepsilon}s &\quad s\geq 1\\ c{\varepsilon}s^{-2} &\quad s< 1, \end{cases} \quad \|\beta_\varepsilon(s) - \beta_0(s)\| \leq \begin{cases} c{\varepsilon}s^2 &\quad s\geq 1\\ c{\varepsilon}s^{-1} &\quad s< 1 \end{cases}\end{aligned}$$ where $c$ is a computable constant independent on ${\varepsilon}$ and $s$. Nevertheless, since estimates do not depend on $p$, we suspect that Theorem \[teo\_1.viejo\] does not turn to be enough accurate. Our first aim in this paper is to refine by using an alternative characterization of the curves. By following the arguments of [@terra] it is possible to define $\mathcal{C}_{\varepsilon}$ by minimizing the first eigenvalue of weighted $p-$Laplacian problems over all possible partition of the kind $$\mathcal{P}=\{\{\omega_+,\omega_-\}\subset \Omega \, : \, \omega_\pm \mbox{ is open and connected, }\omega_+\cap \omega_-=\emptyset\},$$ see Theorem \[teocurva\] in Section \[sec.fucik.rn\] for the precise statement. Such optimal partition characterization reduces our analysis to studying the homogenization rates of the first eigenvalue of the weighted $p-$laplacian. Our first result reads as follows. \[main1\] Given ${\varepsilon}>0$ and $s\in {\mathbb R}^+$, let $(\alpha_{\varepsilon}(s),\beta_{\varepsilon}(s))\in {\mathcal{C}}_{\varepsilon}$ and $(\alpha_0(s),\beta_0(s))\in {\mathcal{C}}_0$. Then the following estimates hold $$\begin{aligned} \|\alpha_\varepsilon(s) - \alpha_0(s)\|\leq \begin{cases} C{\varepsilon}s^\frac{1}{p} &\quad s\geq 1\\ C{\varepsilon}s^{-1-\tfrac{1}{p}} &\quad s< 1 \end{cases} \qquad \|\beta_\varepsilon(s) - \beta_0(s)\| \leq \begin{cases} C{\varepsilon}s^{1+\frac{1}{p}} &\quad s\geq 1\\ C{\varepsilon}s^{-\frac{1}{p}} &\quad s< 1 \end{cases}\end{aligned}$$ where $C$ is a constant independent on ${\varepsilon}$ and $s$. A careful computation allow us to compute explicitly the constant in Theorem \[main1\] as $$\label{cte} C=\left(\frac{\theta_+}{\theta_-}\right)^{1+\frac{1}{p}}\mu_2(\Omega)^{1+\frac{1}{p}} \max\{C_m,C_n\}$$ where $\mu_2$ is the second eigenvalue of the Dirichlet $p-$laplacian in $\Omega$ and $$\label{ctee} C_r= p\frac{\sqrt{N}}{2} \\|r-\bar r\\|_{L^\infty({\mathbb R}^N)} \theta_+ (\theta_-)^{-\frac{1}{p}-2}.$$ In the second part of the work we deal with the homogenization of the one-dimensional version of , i.e., $$\begin{aligned} \label{P1.1D} \bigg\{ \begin{array}{ll} -\Delta_p u_\varepsilon=\alpha_\varepsilon m_\varepsilon(u_\varepsilon^+ )^{p-1} - \beta_\varepsilon n_\varepsilon(u_\varepsilon^- )^{p-1} &\quad \textrm{ in } (a,b)\subset {\mathbb R}\\ u_{\varepsilon}(a)=u_{\varepsilon}(b)=0. \end{array}\end{aligned}$$ Problem was introduced in the ’70s by Dancer and [F]{}učík (see [@DAN; @FUCiK-libro]) for constant weights and a fixed value of ${\varepsilon}>0$. These authors were interested in problems with jumping nonlinearities, and obtained that the nontrivial solutions consist in a family of hyperbolic-like curves. The existence of similar curves in the spectrum was proved later for non-constant weights by Rynne in [@RYN], together with several properties about simplicity of zeros. The asymptotic behavior of the curves was studied in [@PiS]. For sign-changing weights, similar results were obtained by Alif and Gossez, see [@ALIF1]. The main advantage with regard to the higher dimensional case is the fact of knowing the structure of the whole spectrum and precise information about the curves. By means of shooting arguments Rynne proved that the spectrum of can be described as an union of curves $$\Sigma_{\varepsilon}(m_{\varepsilon},n_{\varepsilon}):= \bigcup_{k\in {\mathbb N}_0} \mathcal{C}_{k,{\varepsilon}}.$$ Here, ${\mathcal{C}}_{0,{\varepsilon}}={\mathcal{C}}_{0,{\varepsilon}}^+ \cup {\mathcal{C}}_{0,{\varepsilon}}^-$ are the trivial lines, which are given by ${\lambda}_{1}(m_{\varepsilon})\times {\mathbb R}$, and ${\mathbb R}\times {\lambda}_{1}(n_{\varepsilon})$, respectively, being ${\lambda}_1(r_{\varepsilon})$ the first eigenvalue of the Dirichlet $p-$laplacian with weight $r_{\varepsilon}$. These curve are characterized for having eigenfunctions which do not change signs. The remaining curves are made as the union $\mathcal{C}_{k,{\varepsilon}}=\mathcal{C}_{k,{\varepsilon}}^+ \cup \mathcal{C}_{k,{\varepsilon}}^-$, where $\mathcal{C}_{k,{\varepsilon}}^+$ (resp. $\mathcal{C}_{k,{\varepsilon}}^-$) it is composed of pairs whose corresponding eigenfunctions have $k$ internal zeros, and positive (resp. negative) slope at $x=a$. As ${\varepsilon}\to 0$, the following natural limit problem for is obtained $$\begin{aligned} \label{P1.1D.lim} \bigg\{ \begin{array}{ll} -\Delta_p u=\alpha_0 \bar m(u^+ )^{p-1} - \beta_0 \bar n(u^- )^{p-1} &\quad \textrm{ in } (a,b)\\ u(a)=u(b)=0, \end{array}\end{aligned}$$ where $\bar m$ and $\bar n$ are given in (\[limi\]). Similarly, its corresponding spectrum is composed as the union $$\Sigma_0(n_0,n_0):= \bigcup_{k\in {\mathbb N}_0} \mathcal{C}_{k,0} ,$$ with curves $\mathcal{C}_{k,0} $ satisfying analogous properties to $\mathcal{C}_{k,{\varepsilon}} $. In order to describe the curves in the spectrum of $\Sigma_{\varepsilon}$ and $\Sigma_0$ we denote $(\alpha_{k,{\varepsilon}}(s),\beta_{k,{\varepsilon}}(s))$ and $(\alpha_{k,0}(s),\beta_{k,0}(s))$ the intersection of the curves $\mathcal{C}_{k,{\varepsilon}}$ and $\mathcal{C}_{k,0}$ with the line of slope $s$ passing through the origin, respectively. Under these considerations, in [@FBPS-fucik] it was studied the behavior of the eigencurves of as ${\varepsilon}$ approaches zero. It was proved that for each $k\in {\mathbb N}_0$, the curve ${\mathcal{C}}_{k,{\varepsilon}}$ converges to ${\mathcal{C}}_{k,0}$ in the sense that $$\label{convv} \alpha_{k,{\varepsilon}}(s)\to \alpha_{k,0}(s) \quad \mbox{and} \quad \beta_{k,\varepsilon}(s)\to \beta_{k,0}(s)$$ as ${\varepsilon}\to 0$, for each fixed $s\in{\mathbb R}^+$. Again, following [@terra] it is possible to obtain a representation of the curves $\mathcal{C}_{k,{\varepsilon}}$ and $\mathcal{C}_{k,0}$ by minimizing the first eigenvalue of weighted $p-$laplacian problems over all possible partition of the kind $$\mathcal{P}_{k+1}\, : \, \{a=t_0<t_1<\ldots<t_{k+1}=b\},$$ see Theorem \[teo\_part\_1d\] in Section \[sec.1d\] for the precise statement. Such characterization allow us to prove the following result concerning to the convergence rates of : \[teo-1d\] Given ${\varepsilon}>0$ and $s\in {\mathbb R}^+$, let $(\alpha_{\varepsilon}(s),\beta_{\varepsilon}(s))\in {\mathcal{C}}_{k,{\varepsilon}}$ and $(\alpha_0(s),\beta_0(s))\in {\mathcal{C}}_{k,0}$. Then the following estimates hold $$\begin{aligned} \|\alpha_\varepsilon(s) - \alpha_0(s)\|\leq \begin{cases} C{\varepsilon}k^{p+1}s^\frac{1}{p} &\, s\geq 1\\ C{\varepsilon}k^{p+1} s^{-1-\tfrac{1}{p}} &\, s< 1, \end{cases} \,\, \|\beta_\varepsilon(s) - \beta_0(s)\| \leq \begin{cases} C{\varepsilon}k^{p+1} s^{1+\frac{1}{p}} &\, s\geq 1\\ C{\varepsilon}k^{p+1} s^{-\frac{1}{p}} &\, s< 1 \end{cases}\end{aligned}$$ where $(\alpha_0(s),\beta_0(s))\in \mathcal{C}_{k,0}$ and $$C=\left(\frac{\theta_+}{\theta_-}\right)^{1+\frac{1}{p}} \left(\frac{\pi_p}{b-a} \right)^{1+p} \max\{C_m,C_n\},$$ being $C_m$ and $C_n$ given in . Observe that when we specialize Theorem \[teo-1d\] with both weight functions being the same $1-$periodic function $r(\tfrac{x}{{\varepsilon}})$, and both parameters being the same, that is, ${\lambda}_{\varepsilon}=\alpha_{\varepsilon}=\beta_{\varepsilon}$, it follows that $s=1$, and we recover the homogenization rates for the eigenvalue convergence of $$\label{er1} -\Delta_p u_{\varepsilon}= {\lambda}_{\varepsilon}r_{\varepsilon}\|u_{\varepsilon}\|^{p-2}u_{\varepsilon}\quad \mbox{in }(a,b), \qquad u_{\varepsilon}(a)=u_{\varepsilon}(b)=0,$$ to the limit problem $$\label{er2} -\Delta_p u = {\lambda}_0 \bar r \|u\|^{p-2}u \quad \mbox{in }(a,b), \qquad u(a)=u(b)=0$$ as ${\varepsilon}\to 0$, which has been widely studied, see for instance [@FBPS1; @FBPS2; @FBPS3]. More precisely, in particular Theorem \[teo-1d\] states that $$\|{\lambda}_{k}(r_{\varepsilon})-{\lambda}_{k}(\bar r)\|\leq ck^{p+1}{\varepsilon}$$ where ${\lambda}_{k}(r_{\varepsilon})$ is $k-$th eigenvalue of , ${\lambda}_{k}(\bar r)$ is the $k-$th eigenvalue of , and $c$ is a constant independent on $k$ and ${\varepsilon}$. The paper is organized as follows: in Section \[sec.fucik.rn\] we state some properties concerning to the first nontrivial curve in the Fu[č]{}[í]{}k spectrum for $N\geq 1$; in Section \[seccion.part\] we deal with the proof of Theorem \[main1\]; in Section \[sec.1d\] we study the one-dimensional Fu[č]{}[í]{}k eigencurves; finally in Section \[sect1d.p\] we provide a proof for Theorem \[teo-1d\]. The Fu[č]{}[í]{}k spectrum in ${\mathbb R}^N$ {#sec.fucik.rn} ============================================= As we pointed in the introduction, given $\Omega\subset{\mathbb R}^N$, $N\geq 1$, and functions $m$ and $n$ satisfying , the structure of the spectrum $\Sigma(m,n)$ of the following asymmetric equation $$\begin{aligned} \label{P10} \bigg\{ \begin{array}{ll} -\Delta_p u=\alpha m(u^+ )^{p-1} - \beta n(u^- )^{p-1} &\quad \textrm{ in } \Omega\\ u=0 &\quad \mbox{on }\partial \Omega. \end{array}\end{aligned}$$ is not completely understood, even in the constant weight case. Immediately one can check that $\Sigma(m,n)$ contains the lines ${\lambda}_1(m)\times {\mathbb R}$ and ${\mathbb R}\times {\lambda}_1(n)$. Here, given a function $r$ satisfying , ${\lambda}_1(r)$ denotes the first eigenvalue of $$\begin{aligned} \label{p.lap} \begin{cases} -\Delta_p u = {\lambda}r \|u\|^{p-2}u &\quad \textrm{in } \Omega\\ u=0 &\quad \mbox{on }\partial \Omega. \end{cases}\end{aligned}$$ The first eigenvalue of can be written variationally by minimizing the following quotient over all the non-zero functions belonging to $W^{1,p}_0(\Omega)$ $$\label{variac} {\lambda}_1(r)=\inf \frac{\int_\Omega \|\nabla u\|^p \, dx}{\int_\Omega r(x)\|u\|^p\, dx}.$$ When $r\equiv 1$ we just write $\mu_1$ to denote . When it is precise to empathize the dependence on the domain we will write ${\lambda}_k(r,\Omega)$ and $\mu_k(\Omega)$ to denote the $k-$th variational eigenvalue of . In [@AR1; @AR3] it was shown the existence of a first variational nontrivial curve ${\mathcal{C}}_1(m,n)$ given by minimizing the Rayleigh quotient associated to along a family of sign-changing paths. More precisely, the authors in [@AR3] proved that $$C_1(m,n)=\{(\alpha(s),\beta(s)), s\in {\mathbb R}^+\}$$ where $(\alpha(s),\beta(s))$ is the intersection between $\Sigma(m,n)$ and the line of slope $s$ passing through the origin in ${\mathbb R}^2$. Each component is given by $$\alpha(s)=c(m,sn), \qquad \beta(s)=s\alpha(s)$$ where $$c(m,n)=\inf_{\gamma \in \Gamma} \max_{u\in \gamma[-1,1]} \frac{\int_\Omega \|\nabla u\|^p\, dx}{\int_\Omega (m (u^+)^p+n (u^-)^p)\, dx}$$ and $\Gamma:=\{\gamma \in C([-1,1])\colon \gamma(-1)\geq 0 \mbox{ and } \gamma(1)\leq 0 \}$. Later on, problem was considered for the case $p=2$ and constant weights in [@terra]. In that paper the authors, among other things, obtain an alternative representation of ${\mathcal{C}}_1$ in the framework of optimal partitions, see Theorem 1.2 in [@terra]. However, as the they notice (in Remark 2.2, [@terra]) the procedure leading to the proof of such result can be trivially adapted for any $p\geq 2$ and by considering weights. Therefore, the result corresponding to can be stated as follows. \[teocurva\] The first nontrivial curve ${\mathcal{C}}_1(m,n)$ in the spectrum of can be written as $${\mathcal{C}}(m,n)=\{(\alpha(s),\beta(s)), s\in {\mathbb R}^+\},\qquad \mbox{ with } \quad \alpha(s)=s^{-1}c(s), \quad \beta(s)=c(s)$$ where $$\label{el_c} c(s):=\inf_{(\omega_i)\in \mathcal{P}_2} \max\{s{\lambda}_{1}(m,\omega_+),{\lambda}_{1}(n,\omega_-)\}$$ and $$\mathcal{P}_2=\{(\omega_+,\omega_-\}\subset \Omega \, : \, \omega_i \mbox{ is open and connected, }\omega_+\cap \omega_-=\emptyset\}.$$ Moreover, for every $s>0$ there exists $u\in W^{1,p}_0(\Omega)$ such that $(\{u^+>0\},\{u^->0\})$ achieves $c(s)$. In order to prove our main result we state some properties concerning to the curve ${\mathcal{C}}_1$. First, we establish bounds for points belonging to ${\mathcal{C}}_1$ in terms of the parameter $s$ and the auxiliary function $\gamma:{\mathbb R}^+ \to {\mathbb R}^+$ defined as $$\label{gama} \gamma(s)= \bigg\{ \begin{array}{ll} 1 &\mbox{ if }\ s \geq 1 \\ s^{-1} &\mbox{ if }\ s < 1. \end{array}$$ \[lema.2\] Given $s\in {\mathbb R}^+$, let $(\alpha(s),\beta(s))\in \mathcal C_1(m,n)$. Then $$\alpha(s) \leq \theta_-^{-1} \mu_2(\Omega) \gamma(s), \qquad \beta(s) \leq \theta_-^{-1} \mu_2(\Omega) s \gamma(s)$$ where $\gamma$ is defined in and $\mu_2$ denotes the second eigenvalue of the $p-$laplacian in $\Omega$ with Dirichlet boundary conditions. In the following lemma we consider the first eigenvalue of the $p-$laplacian on nodal domains of eigenfunctions corresponding to points belonging to ${\mathcal{C}}_1$. \[lema.3\] Given $s\in {\mathbb R}^+$, let $(\alpha(s),\beta(s))\in {\mathcal{C}}_1(m,n)$ and let $u$ be a corresponding eigenfunction. If we denote $\omega_\pm=supp(u^\pm)$, then $$\mu_1(\omega_+) \leq C\gamma(s), \qquad \mu_1(\omega_-)\leq Cs\gamma(s)$$ where $C=\frac{\theta_+}{\theta_-}\mu_2(\Omega)$ and $\gamma(s)$ is given in . By taking $v=u^+$ in the weak formulation of we obtain that $$\begin{aligned} \label{ef1} \int_{\omega_+} \|\nabla u^+\|^p \, dx &= \int_{\Omega} \|\nabla u^+\|^p \, dx =\alpha \int_{\Omega} m \|u^+\|^p\, dx = \alpha\int_{\omega_+} m \|u^+\|^p\, dx, \end{aligned}$$ from where it follows that $\alpha={\lambda}_1(m,\omega_+)$ and $u\|_{\omega_+}\in W^{1,p}_0(\omega_+)$ is an eigenfunction associated to ${\lambda}_1(m,\omega_+)$. Since $$\frac{1}{\theta_+}\frac{\int_{\omega_+} \|\nabla v\|^p}{\int_{\omega^+} \|v\|^p}\leq \frac{\int_{\omega_+} \|\nabla v\|^p}{\int_{\omega_+}m \|v\|^p} \leq \frac{1}{\theta_-}\frac{\int_{\omega_+} \|\nabla v\|^p}{\int_{\omega_+} \|v\|^p}$$ for all $v \in W^{1,p}_0(\omega_+)$, from it follows that $$\tfrac{1}{\theta_+}\mu_1(\omega_+ ) \leq {\lambda}_1(m,\omega_+) \leq \tfrac{1}{\theta_-}\mu_1(\omega_+ ),$$ and the desired inequality follows by using Lemma \[lema.2\]. Analogously, by using $u^-$ as a test function in the weak formulation of the another inequality is obtained. proof of the results for $N\geq 1$ {#seccion.part} ================================== Before proving our main result, we state an auxiliary results concerning to the homogenization of eigenvalues of the weighted $p$-laplacian. Given a bounded domain $\Omega\subset {\mathbb R}^N$ and a function $r$ satisfying , we denote ${\lambda}_{1}(r_{\varepsilon})$ the first eigenvalue of $$\begin{aligned} \label{Plap1.ve} \begin{cases} -\Delta_p u_{\varepsilon}= {\lambda}_{\varepsilon}r(\tfrac{x}{{\varepsilon}}) \|u_{\varepsilon}\|^{p-2}u_{\varepsilon}&\quad \textrm{in } \Omega\\ u_{\varepsilon}=0 &\quad \mbox{on }\partial \Omega. \end{cases} \end{aligned}$$ When an explicit emphasis on the domain is required, we denote ${\lambda}_{1}(r_{\varepsilon},\Omega)$ the first eigenvalue of ; additionally, when $r_{\varepsilon}\equiv 1$ we write $\mu_{1}(\Omega)$ instead of ${\lambda}_{1}(1,\Omega)$. As we pointed in the introduction, convergence rates in the homogenization of the [F]{}učík spectrum are closely related with the convergence rates in the homogenization of eigenvalues of the $p-$laplacian. Given a $Q-$periodic function $r$ satisfying , $Q$ being the unit cube in ${\mathbb R}^N$, as ${\varepsilon}\to 0$ the following limit problem for is obtained $$\begin{aligned} \label{Plap1.lim} \begin{cases} -\Delta_p u_0= {\lambda}_{0} \bar r \|u_0\|^{p-2}u_0 &\quad \textrm{in } \Omega\\ u_0=0 &\quad \mbox{on }\partial \Omega, \end{cases}\end{aligned}$$ where $\bar r$ is the average of $r$ over $Q$. The eigenvalue ${\lambda}_1(r_{\varepsilon})$ converges to the first eigenvalue of . Furthermore, the rate of the convergence of ${\lambda}_{1}(r_{\varepsilon})$ is stated in the following result. \[teo\_1\] Given a $Q-$periodic function $r$ satisfying , let us denote ${\lambda}_{1}(r_{\varepsilon})$ and ${\lambda}_{1}(\bar r)$ the first eigenvalue of equations and , respectively. Then $$\|{\lambda}_{1}(r_{\varepsilon},\Omega)-{\lambda}_{1}(\bar r,\Omega)\|\leq C_r \mu_1(\Omega)^{\frac{1}{p}+1} \varepsilon$$ with $C_r$ given by $$C_r= p\frac{\sqrt{N}}{2} \\|r-\bar r\\|_{L^\infty({\mathbb R}^N)} \theta_+ (\theta_-)^{-\frac{1}{p}-2}.$$ We are ready to prove our main result in this section. According to Theorem \[teocurva\] the curve $\mathcal{C}_{\varepsilon}$ associated to is given by $$\mathcal{C}_\varepsilon:=\{(\alpha_{\varepsilon}(s),\beta_\varepsilon(s)),\, s\in{\mathbb R}^+\} = \{(s^{-1}c_{\varepsilon}(s) , c_{\varepsilon}(s)),\, s\in {\mathbb R}^+\}$$ where $$\label{el_c_eps} c_{\varepsilon}(s):=\inf_{(\xi_+,\xi_-)\in \mathcal{P}_2} \max\{s{\lambda}_1(m_{\varepsilon},\xi_+),{\lambda}_1(n_{\varepsilon},\xi_-)\}.$$ In a similar way the limit curve $\mathcal{C}_0$ associated to is given by $$\mathcal{C}_0:=\{(\alpha_0(s),\beta_0(s)),\, s\in{\mathbb R}^+\} = \{(s^{-1}c_0(s) , c_0(s)),\, s\in {\mathbb R}^+\}$$ where $$\label{el_c_0} c_0(s):=\inf_{(\xi_+,\xi_-) \in \mathcal{P}_2} \max\{s{\lambda}_1(\bar m,\xi_+),{\lambda}_1(\bar n,\xi_-)\}.$$ Let $(\omega_+,\omega_-)\in \mathcal{P}_2$ be a partition such that $$c_0(s)= \max\{s{\lambda}_1(\bar m,\omega_+),{\lambda}_1(\bar n,\omega_-)\}.$$ By putting $(\omega_+,\omega_-)$ in it follows that $$\begin{aligned} \label{eq.c.0} \begin{split} c_{\varepsilon}(s)&\leq \max\{s{\lambda}_1(m_{\varepsilon},\omega_+),{\lambda}_1(n_{\varepsilon},\omega_-)\}. \end{split}\end{aligned}$$ Now, Theorem \[teo\_1\] allows as to bound ${\lambda}_1(m_{\varepsilon},\omega_+)$ and ${\lambda}_1(n_{\varepsilon},\omega_-)$ in terms of ${\lambda}_1(\bar m,\omega_+)$ and ${\lambda}_1(\bar n,\omega_-)$, from where we bound as $$\begin{aligned} \label{eq.c.1} \begin{split} \max\{&s\big({\lambda}_1(\bar m ,\omega_+)+ C_m \mu_1(\omega_+)^{\frac{1}{p}+1} \varepsilon \big),{\lambda}_1(\bar n,\omega_-) + C_n \mu_1(\omega_-)^{\frac{1}{p}+1} \varepsilon \}\leq \\ &\leq \max\{s{\lambda}_1(\bar m ,\omega_+) ,{\lambda}_1(\bar n,\omega_-)\}+C_1{\varepsilon}\max\{ s\mu_1(\omega_+)^{\frac{1}{p}+1},\mu_1(\omega_-)^{\frac{1}{p}+1} \} \\ &= c_0(s) +C_1{\varepsilon}\max\{ s\mu_1(\omega_+)^{\frac{1}{p}+1},\mu_1(\omega_-)^{\frac{1}{p}+1} \} \end{split} \end{aligned}$$ where $C_1=\max\{C_m,C_n\}$. In the another hand, by using Lemma \[lema.3\] we obtain that $$\begin{aligned} \label{eq.c.11} \begin{split} \max\{ &s\mu_1(\omega_+)^{ \frac{1}{p}+1},\mu_1(\omega_-)^{\frac{1}{p}+1} \}\leq \\ & \leq \left(\frac{\theta_+}{\theta_-}\right)^{1+\frac{1}{p}} \max\{s (\mu_2(\omega_+)\gamma(s))^{\frac{1}{p}+1},(s\mu_2(\omega_-)\gamma(s))^{\frac{1}{p}+1} \}\\ &\leq C_2 \max\{s \gamma(s)^{\frac{1}{p}+1},(s\gamma(s))^{\frac{1}{p}+1} \}\\ &\leq C_2 \gamma(s)^{1+\frac{1}{p}} s \max\{1,s^\frac{1}{p}\}, \end{split} \end{aligned}$$ where $C_2=\left(\frac{\theta_+}{\theta_-}\mu_2(\Omega)\right)^{1+\frac{1}{p}}$. Collecting – we obtain that $$\begin{aligned} \label{eq.c.2} \begin{split} c_{\varepsilon}(s)\leq c_0(s)+ C {\varepsilon}\gamma(s)^{1+\frac{1}{p}}s\max\{1,s^\frac{1}{p}\}, \end{split}\end{aligned}$$ where $C=C_1C_2$. Interchanging the roles of $c_{\varepsilon}(s)$ and $c_0(s)$ we similarly obtain that $$\begin{aligned} \label{eq.c.21} c_0(s)\leq c_{\varepsilon}(s)+ C {\varepsilon}\gamma(s)^{1+\frac{1}{p}}s\max\{1,s^\frac{1}{p}\},\end{aligned}$$ where $C$ is the same constant that in . Mixing up and it follows that $$\|c_{\varepsilon}(s)-c_0(s)\|\leq C {\varepsilon}\gamma(s)^{1+\frac{1}{p}}s\max\{1,s^\frac{1}{p}\}.$$ Now, from Theorem \[teocurva\] we get $$\begin{aligned} \|\beta_{\varepsilon}(s)-\beta_0(s)\|&=\|c_{\varepsilon}(s)-c_0(s)\|\leq C {\varepsilon}\gamma(s)^{1+\frac{1}{p}}s\max\{1,s^\frac{1}{p}\},\\ \|\alpha_{\varepsilon}(s)-\alpha_0(s)\|&=s^{-1}\|c_{\varepsilon}(s)-c_0(s)\|\leq C {\varepsilon}\gamma(s)^{1+\frac{1}{p}}\max\{1,s^\frac{1}{p}\}\end{aligned}$$ as it was required. The one-dimensional Fu[č]{}[í]{}k problem {#sec.1d} ========================================= In this section we state some properties related with the following one-dimensional asymmetric equation in $\Omega=(a,b)$ $$\begin{aligned} \label{P10.1d} \bigg\{ \begin{array}{ll} -\Delta_p u=\alpha m(u^+ )^{p-1} - \beta n(u^- )^{p-1} &\quad \textrm{ in } \Omega\\ u(a)=u(b)=0. \end{array}\end{aligned}$$ As it was pointed in the introduction, Rynne [@RYN] shown that its spectrum is given by $$\Sigma(m,n):= \bigcup_{k\in {\mathbb N}} \mathcal{C}_{k},$$ where the curves ${\mathcal{C}}_k={\mathcal{C}}_k^+\cup {\mathcal{C}}_k^-$, $k\in {\mathbb N}_0$ are composed of pairs $(\alpha,\beta)\in{\mathbb R}^2$ whose corresponding eigenfunctions have $k$ internal zeros and positive (resp. negative) slote at $x=a$. In particular, ${\mathcal{C}}_0^+=\lambda_1(m)\times {\mathbb R}$ and ${\mathcal{C}}_0^-={\mathbb R}\times {\lambda}_1(n)$ have eigenfunctions which do not change signs in $\Omega$, being ${\lambda}_1(r)$ the first eigenvalue of $$\begin{aligned} \label{lap.1d} \bigg\{ \begin{array}{ll} -\Delta_p u={\lambda}r\|u\|^{p-2}u &\quad \textrm{ in } \Omega\\ u(a)=u(b)=0. \end{array}.\end{aligned}$$ For simplicity, when the $r=1$ we denote $\mu_k$ the $k-$th eigenvalue of . Sometimes, in order to empathize the dependence on the domain we write ${\lambda}_k(r,\Omega)$ and $\mu_k(\Omega)$ to denote the $k-$th eigenvalues of . Observe that, in contrast with the higher dimensional case, eigenvalues of the Dirichlet $p-$laplacian can be explicitly computed as $$\mu_k(I)=\pi_p^p k^p \|\Omega\|^{-p}$$ where $\pi_p=2(p-1)^{1/p}\int_0^1(1-s^p)^{-1/p} \, ds$, see [@dPDM]. Moreover, the sequence of variational eigenvalues of can be described as $$\label{variac.1d} {\lambda}_k = \inf_{U \in T_k} \sup_{u \in C} \frac{\int_a^b \|u'\|^p\, dx}{\int_a^b r(x)\|u\|^p\, dx}$$ where $$\begin{aligned} T_k & = \{ U \subset W^{1,p}_0(\Omega) \ : \ U \ \mbox{ is compact,} \ U=-U, \ \gamma(U) \ge k \},\end{aligned}$$ and $\gamma$ is the Krasnoselskii genus, see [@GAP] for details. From it follows that $$\label{eig.comp} \theta_+^{-1}\mu_k \leq {\lambda}_k \leq \theta_-^{-1} \mu_k$$ for any $k\geq 1$. The paper [@terra] characterizes the curves of $\Sigma(m,n)$ in terms of the first eigenvalue of weighted $p-$laplacian problems (see Theorem 1.3 and Remark 2.2). The description of the curves is made as follows. A couple $(\alpha,\beta)$ belonging to $\mathcal{C}_{1}$ has eigenfunctions with an internal zero, i.e., it has two nodal domains. Such couple can be written as $(s^{-1}c_{2}(s),c_{2}(s))$, where $$c_{2}(s)=\inf \max\{s {\lambda}_1(m,I_1),{\lambda}_2(n,I_2)\}$$ and $s$ is the slope of the line $\ell_s$ passing through the origin such that $(\alpha,\beta)=\mathcal{C}_{1} \cap \ell_s$. The infumum is taken over all the partitions $\mathcal{P}_2$ of $\Omega$ such that $a=t_0<t_1<t_2=b$, and $I_1=t_1-t_0$, $I_2=t_2-t_1$. Now, a couple belonging to $\mathcal{C}_{2}$ has associated eigenfunctions with three nodal domains. Such pair can be characterized as $(s^{-1}c_{3}(s),c_{3}(s))$, where $$c_{3}(s)=\inf \max\{s {\lambda}_1(m,I_1),{\lambda}_2(n,I_2), s{\lambda}_1(m,I_3)\}$$ and $s$ is the slope of the line $\ell_s$ passing through the origin such that $(\alpha,\beta)=\mathcal{C}_{2} \cap \ell_s$. Here the infumum is taken over all the partition $\mathcal{P}_3$ of $\Omega$ such that $a=t_0<t_1<t_2<t_3=b$, with $I_1=t_1-t_0$, $I_2=t_2-t_1$ and $I_3=t_3-t_2$. In order to state the general case we introduce the following notation: for $k\geq 0$ we denote $$\mathcal{P}_{k+1}\, : \{\,a=t_0<t_1<\ldots<t_{k+1}=b\}$$ a partition of $\Omega=(a,b)$, and we write $I_{i+1}=t_{i+1}-t_i$ for $0\leq i \leq k$. Although the result in [@terra] was proved for the case $p=2$ of and with constant weights, as the authors comment, by mixing Theorem 1.3 and Remark 2.2 from [@terra] it is straightforward to obtain the following result concerning to the spectrum of the weighted equation for any $p>2$. \[teo\_part\_1d\] Given $k\geq 1$ let us define $$\begin{aligned} \label{elcn1} \begin{split} c_{k+1}^+(s)=\inf_{\mathcal{P}_{k+1}} \max_{0\leq i\leq k} \{s {\lambda}_1(m,I_{2i+1}), {\lambda}_1(n,I_{2i+2})\},\\ c_{k+1}^-(s)=\inf_{\mathcal{P}_{k+1}} \max_{0\leq i\leq k} \{s {\lambda}_1(m,I_{2i+2}), {\lambda}_1(n,I_{2i+1})\} \end{split}\end{aligned}$$ for all $s>0$. Then the pair $(s^{-1}c_{k+1}^\pm(s),c_{k+1}^\pm,(s))$ belongs to a curve $\mathcal{C}_{k}^\pm$. Moreover, the infima above are attained for suitable optimal partitions $P^\pm\in\mathcal{P}_{k+1}$. Furthermore, there are eigenfunctions $u^\pm\in W^{1,p}_0(\Omega)$ of associated to $(\alpha=s^{-1}c_{k+1}^\pm(s),\beta=c_{k+1}^\pm,(s))$ whose nodal domains are given by $P^\pm$. From the definition of $c_{k+1}(s)$ it is easy to check that $s_2>s_1$ implies $c_{k+1}(s_2)>c_{k+1}(s_1)$. Moreover, it can be proved that $s_2^{-1}c_{k+1}(s_2)<s_1^{-1}c_{k+1}(s_1)$, from where the monotonicity of $\mathcal{C}_{k+1}$ follows: \[Theorem 21, [@RYN]\] The curve $\mathcal{C}_{k+1}$ is decreasing in the sense that if the points $(\alpha(s_1),\beta(s_1))$ and $(\alpha(s_2),\beta(s_2))$ belong to $\mathcal{C}_{k+1}$ then $$\alpha(s_1)> \alpha(s_2) \qquad \mbox{ and } \qquad \beta(s_2)>\beta(s_1)$$ whenever $s_2>s_1$. The following inequality relates $c_{k}^\pm (1)$ with the $k-$th eigenvalue of the Dirichlet $p-$laplacian. \[lema.4.3\] Let $k\geq 1$ and $c_{k+1}^\pm(\cdot )$ given in . It holds that $$c_{k+1}^\pm (1) \leq \theta_-^{-1} \mu_{k+1}(\Omega).$$ By using and we have that $$\begin{aligned} \label{eg1} \begin{split} c^+_{k+1}(1)&\leq \inf_{\mathcal{P}_{k+1}} \max_i \{ {\lambda}_1(\theta_-,I_{2i+1}), {\lambda}_1(\theta_-,I_{2i+2}) \}\\ &\leq \theta_-^{-1}\inf_{\mathcal{P}_{k+1}} \max_i \{ \mu_1(I_{i}) \} \end{split} \end{aligned}$$ In particular, if we take an uniform partition of $\Omega$, i.e., $\|I_i\|=\|\Omega\|/(k+1)$, it follows that $\mu_1(I_i)= \pi_p^p\|I_i\|^{-p}=\mu_{k+1}(\Omega)$ for each $0\leq i \leq k$ and the result follows. As a consequence of Lemma \[lema.4.3\], we obtain upper bounds for $\alpha(s)$ and $\beta(s)$. The following result is a one-dimensional version of Lemma \[lema.2\] for every curve in the spectrum of . \[lema.21d\] Let $(\alpha(s),\beta(s))\in \mathcal C_k(m,n)$. For each $s>0$ it holds that $$\alpha(s) \leq \theta_-^{-1} \mu_{k+1}(\Omega) \gamma(s), \qquad \beta(s) \leq \theta_-^{-1} \mu_{k+1}(\Omega) s \gamma(s)$$ with $\gamma$ defined by $$\label{gama.1} \gamma(s)= \bigg\{ \begin{array}{ll} 1 &\mbox{ if }\ s \geq 1 \\[0.05 cm] s^{-1} &\mbox{ if }\ s \leq 1. \end{array}$$ Let $s>0$ and $(\alpha(s),\beta(s))\in \mathcal{C}_{k}$. From Theorem \[teo\_part\_1d\] we can write $\alpha(s)=s^{-1}c_{k+1}(s)$ and $\beta(s)=c_{k+1}(s)$ (here $c_k$ denotes any of $c_k^\pm$). We empathize that $\mathcal{C}_{k}(m,n)$ is an decreasing curve. When $s\geq 1$, by using Lemma \[lema.4.3\] we can bound $$\begin{aligned} \label{rell1} \alpha(s)\leq \alpha(1)=c_{k+1}(1) \leq \theta_-^{-p} \mu_{k+1}(\Omega). \end{aligned}$$ When $s\leq 1$ we have that $\beta(s)\leq \beta(1)$, from where $ s^{-1}\beta(s)\leq s^{-1}\beta(1)$. Since $\beta(s)=s\alpha(s)$, we conclude that $$\begin{aligned} \label{rell3} \alpha(s)&=s^{-1}\beta(s) \leq s^{-1}\alpha(1) =s^{-1} c_{k+1}(1) \leq s^{-1} \theta_-^{-p}\mu_{k+1}(\Omega). \end{aligned}$$ By using and together with the relation $\beta=s\alpha$ the conclusion of the lemma follows. Finally, the following lemma allow us to estimate eigenvalues of the $p-$laplacian on nodal domains corresponding to eigenfunctions of . \[1d.lema.3\] Let $(\alpha(s),\beta(s))\in \mathcal{C}_k(m,n)$ with associated eigenfunction $u$. Let $I_+$ (resp. $I_-$) be a nodal domain of $u$ in which $u>0$ (resp. $u<0$). Then $$\mu_1(I_{+})\leq C\gamma(s) , \qquad \mu_1(I_{-})\leq C s\gamma(s)$$ where $C=\frac{\theta_+}{\theta_-}\mu_{k+1}(\Omega)$ and $\gamma(s)$ is given in . By arguing in the same way that in the proof of Lemma \[lema.3\] it is obtained that $$\mu_1(I_+) \leq \theta_+ \alpha(s), \qquad \mu_1(I_-) \leq \theta_+ \beta(s).$$ The result now follows by applying Lemma \[lema.21d\]. proof of the result in the one-dimensional case {#sect1d.p} =============================================== Aimed at proving our main result for one-dimensional Fu[č]{}[í]{}k spectrum, first we introduce the following notation we will use along this section. As we have pointed in the introduction, given the bounded interval $\Omega=(a,b)\subset {\mathbb R}$ and a function $r$ satisfying , we denote ${\lambda}_{k}(r_{\varepsilon},\Omega)$ the $k-$th eigenvalue of $$\begin{aligned} \label{Plap.sec} \begin{cases} -\Delta_p u_{\varepsilon}= {\lambda}_{\varepsilon}r_{\varepsilon}\|u_{\varepsilon}\|^{p-2}u_{\varepsilon}&\quad \textrm{in } \Omega\\ u_{\varepsilon}(a)=u_{\varepsilon}(b)=0. \end{cases}\end{aligned}$$ In the case in which $r\equiv 1$ we just put $\mu_k(\Omega)$. Observe that, since ${\mathcal{C}}_{k,{\varepsilon}} \to {\mathcal{C}}_{k,0}$ (see Theorem 1, [@FBPS-fucik]) in the sense that $$\label{ccon} \alpha_{k,{\varepsilon}}(s)\to \alpha_{k,0}(s) \quad \mbox{and} \quad \beta_{k,\varepsilon}(s)\to \beta_{k,0}(s)$$ where, for $s\in {\mathbb R}^+$, $$(\alpha_{\varepsilon}(s),\beta_{\varepsilon}(s))\in \mathcal{C}_{k\,{\varepsilon}}(m_{\varepsilon},n_{\varepsilon}) \quad \mbox{ and } \quad (\alpha_0(s),\beta_0(s))\in \mathcal{C}_{k,0}(\bar m,\bar n)$$ eigenfunctions corresponding to $(\alpha_0(s),\beta_0(s))$ have exactly $k$ nodal domains on $\Omega$. With the previous remarks and lemmas stated in Section \[sec.1d\] we are ready to prove the rates of the convergences . We consider the curve $\mathcal{C}_{k,{\varepsilon}}^+$. An eigenfunction corresponding to a pair over this curve has positive slope at $x=a$, therefore it is positive over odd nodal domains and negative over even nodal domains. The treatment for $\mathcal{C}_{k,{\varepsilon}}^-$ is analogous. Let $s>0$. According to Theorem \[teo\_part\_1d\] a pair $(\alpha_{\varepsilon}(s),\beta_\varepsilon(s))\in \mathcal{C}_{k,{\varepsilon}}^+$ can be written as $$(\alpha_{\varepsilon}(s),\beta_\varepsilon(s)) = (s^{-1}c_{k+1,{\varepsilon}}(s) , c_{k+1,{\varepsilon}}(s))$$ where $$\label{1d.el_c_eps} c_{k+1,{\varepsilon}}(s)=\inf_{\mathcal{P}_{k+1}} \max_i \{s {\lambda}_{1,{\varepsilon}}(m_{\varepsilon},I_{2i+1}), {\lambda}_{1,{\varepsilon}}(n_{\varepsilon},I_{2i+2})\}$$ and in a similar way, the limit pair $(\alpha_0(s),\beta_0(s))$ belonging to the limit curve $\mathcal{C}_{k,0}^+$ can be written as $$(\alpha_0(s),\beta_0(s)) = (s^{-1}c_{k+1,0}(s) , c_{k+1,0}(s))$$ where $$\label{1d.el_c_0} c_{k+1,0}(s)=\inf_{\mathcal{P}_{k+1}} \max_i \{s {\lambda}_1(\bar m,I_{2i+1}), {\lambda}_1(\bar n,I_{2i+2})\}.$$ Let $P_{k+1}\in \mathcal{P}_{k+1}$ a partition where the infimum is attained in . By considering $P_{k+1}$ in the expression we get $$\begin{aligned} \label{1d.eq.c.0} \begin{split} c_{k+1,{\varepsilon}}(s)&\leq \max_i \{s {\lambda}_{1}(m_{\varepsilon},I_{2i+1}), {\lambda}_{1}(n_{\varepsilon},I_{2i+2})\}. \end{split}\end{aligned}$$ Now, using Theorem \[teo\_1\] we can bound ${\lambda}_{1}(m_{\varepsilon},I_{2i+1})$ and ${\lambda}_{1}(n_{\varepsilon},I_{2i+2})$ in term of ${\lambda}_1(\bar m,I_{2i+1})$ and ${\lambda}_1(\bar n,I_{2i+2})$, from where we find an upper bound of as: $$\begin{aligned} \label{1d.eq.c.1} \begin{split} &\max_i\{s\big({\lambda}_1(\bar m ,I_{2i+1})+ C_m \mu_1(I_{2i+1})^{\frac{1}{p}+1} \varepsilon \big),{\lambda}_1(\bar n,I_{2i+2}) + C_n \mu_1(I_{2i+2})^{\frac{1}{p}+1} \varepsilon \}\\ &\leq \max\{s{\lambda}_1(\bar m ,I_{2i+1}) ,{\lambda}_1(\bar n,I_{2i+2})\}+C{\varepsilon}\max\{ s\mu_1(I_{2i+1})^{\frac{1}{p}+1},\mu_1(I_{2i+2})^{\frac{1}{p}+1} \} \\ &= c_{k+1,0}(s) +C_1{\varepsilon}\max\{ s\mu_1(I_{2i+1})^{\frac{1}{p}+1},\mu_1(I_{2i+2})^{\frac{1}{p}+1} \} \end{split} \end{aligned}$$ where $C_1=\max\{C_m,C_n\}$. On the other hand, by using Lemma \[1d.lema.3\] we obtain that $$\begin{aligned} \label{1d.eq.c.11} \begin{split} \max\{s\mu_1&(I_{2i+1})^{\frac{1}{p}+1},\mu_1(I_{2i+2})^{\frac{1}{p}+1} \} \\ &\leq C_2 \max\{s (\mu_{k+1}(I)\gamma(s))^{\frac{1}{p}+1},(s\mu_{k+1}(I)\gamma(s))^{\frac{1}{p}+1} \}\\ &\leq C_2 \mu_{k+1}(I)^{1+\frac{1}{p}}\gamma(s)^{1+\frac{1}{p}} s \max\{1,s^\frac{1}{p}\}. \end{split} \end{aligned}$$ where $C_2=\left(\frac{\theta_+}{\theta_-}\right)^{\frac{1}{p}+1}$. Collecting – we obtain that $$\begin{aligned} \label{1d.eq.c.2} \begin{split} c_{k+1,{\varepsilon}}(s)\leq c_{k+1,0}(s)+ C_1C_2 {\varepsilon}\mu_{k+1}(I)^{1+\frac{1}{p}} \gamma(s)^{1+\frac{1}{p}}s\max\{1,s^\frac{1}{p}\}. \end{split}\end{aligned}$$ Interchanging the roles of $c_{k+1,{\varepsilon}}(s)$ and $c_{k+1,0}(s)$ we similarly obtain that $$\begin{aligned} \label{1d.eq.c.21} c_{k+1,0}(s)\leq c_{k+1,{\varepsilon}}(s)+ C_1C_2 {\varepsilon}\mu_{k+1}(I)^{1+\frac{1}{p}} \gamma(s)^{1+\frac{1}{p}}s\max\{1,s^\frac{1}{p}\}.\end{aligned}$$ Mixing up and it follows that $$\|c_{\varepsilon}(s)-c_0(s)\|\leq C_1C_2 {\varepsilon}\mu_{k+1}(I)^{1+\frac{1}{p}} \gamma(s)^{1+\frac{1}{p}}s\max\{1,s^\frac{1}{p}\}.$$ Finally, since $\mu_k(I)=k^p \pi_p^p \|I\|^{-p}$, we get $$\begin{aligned} \|\beta_{\varepsilon}(s)-\beta_0(s)\|&=\|c_{k+1,{\varepsilon}}(s)-c_{k+1,0}(s)\|\leq C {\varepsilon}(k+1)^{p+1} \gamma(s)^{1+\frac{1}{p}}s\max\{1,s^\frac{1}{p}\},\\ \|\alpha_{\varepsilon}(s)-\alpha_0(s)\|&=s^{-1}\|c_{k+1,{\varepsilon}}(s)-c_{k+1,0}(s)\|\leq C {\varepsilon}(k+1)^{p+1} \gamma(s)^{1+\frac{1}{p}}\max\{1,s^\frac{1}{p}\}\end{aligned}$$ where $C=C_1C_2 \big(\frac{\pi_p}{b-a} \big)^{1+p}$, and the result is proved. Acknowledgements ================ This paper was mostly written during a visit at Universitá degli Studi di Torino. The author wishes to thank to Prof. Susanna Terracini for her useful discussions about this topic and help along the stay.
--- abstract: 'Dissipation and decoherence, and the evolution from pure to mixed states in quantum physics are handled through master equations for the density matrix. Master equations such as the Lindblad equation preserve the trace of this matrix. Viewing them as first-order time-dependent operator equations for the elements of the density matrix, a unitary integration procedure can be adapted to solve for these elements. A simple model for decoherence preserves the hermiticity of the density matrix. A single, classical Riccati equation is the only one requiring numerical handling to obtain a full solution of the quantum evolution. The procedure is general, valid for any number of levels, but is illustrated here for a three-level system with two driving fields. For various choices of the initial state, we study the evolution of the system as a function of the amplitudes, relative frequencies and phases of the driven fields, and of the strength of the decoherence. The monotonic growth of the entropy is followed as the system evolves from a pure to a mixed state. An example is provided by the $n=3$ states of the hydrogen atom in a time-dependent electric field, such degenerate manifolds affording an analytical solution.' author: - 'A. R. P. Rau$^{}$ and Weichang Zhao' title: 'Decoherence in a driven three-level system' --- Unitary integration procedure for master equations ================================================== Master equations, such as the Lindblad equation [@ref1], can describe dissipation and decoherence in quantum systems. In recent work [@ref2], one of us adapted a “unitary integration" procedure [@ref3; @ref4] for solving such equations while preserving desirable properties such as the hermiticity of the density matrix even in the presence of dissipation and decoherence. This permits keeping track of quantities such as the entropy while the system evolves from a possibly initial pure state to a final mixed one. The two-state illustration given in that initial work is extended now to a three-level system through suitable combinations of density matrix elements to preserve the hermiticity of the operators involved. Consider the master equation for the density matrix $\rho $ called the Liouville-von Neumann-Lindblad equation [@ref1; @ref2], $$\begin{aligned} i\dot{\rho} & = & [H,\rho ]+ \frac{1}{2}i\!\sum_{k}\left( [L_k\rho,L_k^{\dagger }]+ [L_k,\rho L_k^{\dagger }]\right) \nonumber \\ & = & [H,\rho ]-\frac{1}{2}i\!\sum_{k}\left( L_k^{\dagger }L_k\rho +\rho L_k^{\dagger }L_k-2L_k\rho L_k^{\dagger }\right)\!, \label{eqn1}\end{aligned}$$ where an over-dot denotes differentiation with respect to time and $\hbar$ has been set equal to unity. $H$ is a Hermitian Hamiltonian while the $L_k$ are operators in the system through which dissipation and decoherence are introduced. Even though this can result in non-unitary evolution, the form of the equation preserves Tr($\rho$) and positivity of probabilities. A more mathematical discussion of such “super-operators" and “dynamical semigroups" is given in [@ref5]. A commonly used form of $H$ is $$H(t)=\epsilon(t)A_z+2J(t)A_x, \label{eqn2}$$ with $$\begin{aligned} A_x=\left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array} \right)&,& A_y=\left( \begin{array}{ccc} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{array} \right),\, \nonumber \\ A_z &=& \left( \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right). \label{eqn3}\end{aligned}$$ The couplings indicated in Eq. (\[eqn2\]) between states 1 and 2 and between 2 and 3 of a three-state system are referred to as $\Lambda$ and $V$ depending on the relative energy positions of the three states, whether 2 lies above or below, respectively, relative to levels 1 and 3. The three operators in Eq. (\[eqn3\]) close under commutation according to the standard relations satisfied by angular momentum algebra: $[A_x,A_y]=iA_z$, and cyclic. Hioe and Eberly [@ref6] considered such a Hamiltonian for the Liouville version of Eq. (\[eqn1\]), that is, without the dissipative term, along with solutions for certain forms of $\epsilon$ and $J$. Population trapping and dispersion was also considered in [@ref7] with a similar Hamiltonian, and [@ref8] generalized to $n$-level systems. Our work presented here may be regarded as extending such studies to include also dissipation and decoherence. In general, with each of the three states having distinct energies $E_1,E_2,E_3$, and the driving fields having finite detunings from resonance, entries along the diagonal of $H$ in Eq. (\[eqn2\]) complete the Hamiltonian for such systems. Full treatment according to our formalism below then requires all the elements of the SU(3) algebra, namely five more linearly independent $3\times 3$ matrices to supplement those in Eq. (\[eqn3\]). We expect to return to this later but, in this paper, we restrict ourselves to the degenerate case of equal eigenvalues in which case the above three matrices suffice and the calculations reduce to solving a single equation just as in the two-state system considered in [@ref2]. Applications include three identical coupled pendula with nearest neighbor time-dependent couplings, driven systems on resonance, and the degenerate states of the $n=3$ manifold of hydrogen driven by time-dependent electric fields. Dissipation and decoherence are introduced through the $L_k$ matrices in Eq. (\[eqn1\]). Here again, as shown in [@ref2], a choice of all eight linearly independent matrices affords a simplification because of a sum rule that inserts the decoherence as a unit operator in such an eight-dimensional space. In this procedure, Eq. (\[eqn1\]) is recast into a set of eight equations for the elements of the density matrix (recall that the trace remains invariant). An appropriate linear combination of the elements such that the operators in Eq. (\[eqn3\]) map onto three Hermitian $8\times 8$ matrices obeying the same angular momentum commutators is given by the choice $$\begin{aligned} \eta(t)\!\! &=& \!\!(\rho_{11}\!-\!\rho_{33}, \!\frac{1}{\sqrt 3}(\rho_{11}\!+\!\rho_{33}\!-\!2\rho_{22}), \rho_{12}\!+\!\rho_{21}, \rho_{21} \nonumber \\ & &\!\! -\!\rho_{12}, \rho_{13}\!+\!\rho_{31}, \rho_{31}\!-\!\rho_{13}, \rho_{23}\!+\!\rho_{32}, \rho_{32}\!-\!\rho_{23}). \label{eqn4}\end{aligned}$$ Our choice differs only slightly from that in [@ref6; @ref8]where this set is called a “coherence" vector. The resulting equation for $\eta(t)$ takes the form $$i\dot{\eta}(t)=\mathcal{L}(t) \eta(t), \label{eqn5}$$ with $$\mathcal{L}(t)=-i\Gamma \mathcal{I} +\epsilon(t) B_z +2J(t) B_x, \label{eqn6}$$ where $\Gamma$ indexes the strength of the decoherence. The matrices $B$ take the form $$\begin{aligned} B_x &=& \left( \begin{array}{cccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt 3 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & -\sqrt 3 & 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right), \end{aligned}$$ $$\begin{aligned} B_y &=& \left( \begin{array}{cccccccc} 0 & 0 & 0 & 0 & -2i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -i & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & i \\ 2i & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & i & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -i & 0 & 0 & 0 & 0 \end{array} \right), \nonumber \\ B_z &=& \left( \begin{array}{cccccccc} 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \sqrt 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & \sqrt 3 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \end{array} \right). \label{eqn7}\end{aligned}$$ As per the unitary integration procedure [@ref2; @ref3], the solution of Eq. (\[eqn6\]) is written as a product of exponentials $$\begin{aligned} \eta (t)& = &\exp [-\Gamma t] \exp [-i\mu _{+}(t)B_{+}] \nonumber \\ && \times \exp [-i\mu_{-}(t)B_{-}] \exp [-i\mu (t)B_z]\eta (0), \label{eqn8}\end{aligned}$$ with $B_{\pm }\equiv B_x\pm iB_y$. Because our procedure depends only on the commutation relations which remain as in [@ref2], the classical functions $\mu$ in the exponents satisfy the same equations as before, $$\begin{aligned} \dot{\mu}_{+}-i\epsilon (t)\mu _{+}-J(t)(1+\mu _{+}^2) &=&0 , \label{subeq1} \\ \dot{\mu}=2iJ(t)\mu _{+}-\epsilon (t) , && \label{subeq2} \\ \dot{\mu}_{-}-i\dot{\mu}\mu _{-}=J(t) ,\; && \mu _i(0)=0. \label{subeq3}\end{aligned}$$ The first of these equations, involving $\mu _{+}(t)$ alone in Riccati form, is the only non-trivial member of this set. Once solved, $\mu_{-}$ and $\mu$ are obtained through simple integration of the remaining two equations. For given $\epsilon(t)$ and $J(t)$, a mathematica program solves the set of equations readily. Also, the subsequent algebra involved in evaluating the exponentials in Eq. (\[eqn8\]) and their product is easily carried out. Thereby, for any initial density matrix and its $\eta(0)$, we obtain $\eta(t)$ and thus $\rho(t)$ at any later time. Since our model for decoherence introduces its effect through the single real factor which is the first term on the right-hand side of Eq. (\[eqn8\]), the density matrix remains Hermitian throughout. This is an advantage, permitting evaluation of its eigenvalues and calculation of quantities such as the entropy of the system. It is also clear that for any finite $\Gamma$ all elements in $\eta(t)$ in Eq. (\[eqn4\]) vanish asymptotically with $t$ so that all off-diagonal elements of the density matrix so vanish while all diagonal elements become equal. With the trace invariant and chosen to be unity, the density matrix evolves to that of the so-called chaotically mixed state, $\frac{1}{3} \mathcal{I}$. Correspondingly, the entropy reaches asymptotically the value $\ln 3 = 1.0986$. These are aspects of the general result valid for all $n$-level systems [@ref2]. We note again that other models of decoherence and dissipation through other choices for the operators $L_k$ in Eq. (\[eqn1\]) than the one we made will, in general, lead to a larger set of exponential factors in Eq. (\[eqn8\]), making for more complicated algebra therein and in the coupled set of equations in Eq. (9). However, inclusion of a term involving also $A_y$ in Eq. (\[eqn2\]), that is, a coupling also between levels 1 and 3, causes no additional difficulty since it does not enlarge the number of $A$ or $B$ matrices in our procedure. Two different driving fields between neighboring states ======================================================= We present results for three degenerate states, such as of three identical pendula, with different nearest neighbor couplings between 1-2 and 2-3, that is, with $\epsilon(t)$ and $J(t)$ differing in amplitude and frequency, $$\epsilon(t)=A \cos \Omega t, \,\, J(t)=\frac{1}{2} B \cos (\omega t + \delta), \label{eqn10}$$ with $\delta$ a relative phase difference. A representative sample of the density matrix upon starting with all population in the state 1 and all other elements zero is shown in Figs. 1-4. Note the appearance of a complicated frequency spectrum beyond just the two introduced driving frequencies. The analytically solvable problem presented in the next section provides an understanding of the origin of these other frequencies. As shown in Fig. 3, the entire population can be transferred from level 1 to level 3 over certain time intervals. ![Time evolution of the elements of the density matrix of a $n=3$ system driven by the fields in Eq. (\[eqn10\]) and Hamiltonian in Eq. (\[eqn2\]), with $\Omega =0, \omega =1, \delta =0, A =0.05, B= 0.5, \Gamma =0.02$. Right hand panels show the off-diagonal elements, two of which are imaginary and one real.](WZFig01.eps){width="3in"} ![same as in Fig. 1, except that $A =0.15$.](WZFig02.eps){width="3in"} ![same as in Fig. 1, except that $A =0.5, B =1, \Omega =0.1, \omega =1$.](WZFig03.eps){width="3in"} ![same as in Fig. 3 except that $A =1, B =1/\sqrt{2}, \Gamma =0.08$ and longer times shown to illustrate asymptotic evolution.](WZFig04.eps){width="3in"} Specializing to equal driving frequencies with a fixed amplitude ratio, results for various phase differences between the two fields are shown in Figs. 5-8. Clearly, the density matrix elements depend on the relative phase. ![Elements of the density matrix with driving fields of same frequency, $\Omega =\omega =1$, and amplitudes $A =1, B =1/\sqrt{2}$, and $\Gamma =0.02, \delta =-\pi/6$. ](WZFig05.eps){width="3in"} ![same as in Fig. 5 except that $\delta =\pi/6$.](WZFig06.eps){width="3in"} ![same as in Fig. 5 except that $\delta =\pi/4$.](WZFig07.eps){width="3in"} ![same as in Fig. 5 except that $\delta =\pi/2$.](WZFig08.eps){width="3in"} To contrast with a different initial state, Figs. 9-10 show results when all population is in the state 2 at $t=0$. ![Evolution of density matrix elements, starting with an initial non-zero value only for $\rho_{22} =1$. Contrast with Figs. 1-4. The parameters are $\Omega =\omega =1, A=2, B=\sqrt{2}, \Gamma =0.02, \delta =0$.](WZFig09.eps){width="3in"} ![same as in Fig. 9 except that $A=10, B=5\sqrt{2}$. Note the appearance of more rapid oscillations with the harder driving fields.](WZFig10.eps){width="3in"} Fig. 11 presents the evolution of the entropy, $S=-{\rm Tr}\rho \ln \rho$, showing a monotonic rise independent of amplitudes and phases and of the initial pure state. Indeed, the eigenvalues of the density matrix are $\frac{1}{3}(1-e^{-\Gamma t}), \frac{1}{3}(1-e^{-\Gamma t})$, and $\frac{1}{3}(1+2e^{-\Gamma t})$, from which the entropy easily follows. ![Evolution of the entropy to accompany the results shown in previous figures. The rise is monotonic from $0$ to $\ln 3$, the rate of rise depending only on the value of $\Gamma$.](WZFig11.eps){width="3in"} $n=3$ states of the hydrogen atom in an oscillating electric field ================================================================== An example of a three-state degenerate system is provided by the $n=3, m=0$ states of the hydrogen atom. An oscillating electric field such as that of incident radiation couples $3s-3p$ and $3p-3d$ states, the dipole matrix elements being in the ratio $\sqrt 2:1$. Our results in this paper apply to this situation with the two frequencies in Eq. (\[eqn10\]) equal and $A/B=\sqrt 2$. These were the choices made in Figs. 5-8. We present in Figs. 12-15 a sample of results for initial population in $3s$ for different amplitudes of the driving field. ![Evolution of the density of states of the $n=3$ Stark field with initial population in the $3s$ state. The amplitude of the driving field is $A=1$ and $\Gamma =0.02$.](WZFig12.eps){width="3in"} ![same as in Fig. 12 except that $A=2$.](WZFig13.eps){width="3in"} ![same as in Fig. 12 except that $A=10$.](WZFig14.eps){width="3in"} ![same as in Fig. 13 except with a larger $\Gamma =0.08$.](WZFig15.eps){width="3in"} This problem is, of course, exactly solvable in terms of the parabolic eigenstates of hydrogen. With $H(t)$ in Eq. (\[eqn2\]) containing a single time dependence, the resulting Schrödinger equation, $$i\left( \begin{array}{c} \dot{s}(t) \\ \dot{p}(t) \\ \dot{d}(t) \end{array} \right) =\left( \begin{array}{ccc} 0 & -A & 0 \\ -A & 0 & -A/\sqrt 2 \\ 0 & -A/\sqrt 2 & 0 \end{array} \right) \left( \begin{array}{c} s(t) \\ p(t) \\ d(t) \end{array} \right) \cos \omega t, \label{eqn11}$$ can be solved after diagonalizing the matrix of constant coefficients to obtain the parabolic eigenstates $\{\frac{1}{\sqrt 3} s \pm \frac {1}{\sqrt 2} p+\frac{1}{\sqrt 6} d, \frac{1}{\sqrt 3}(s-\sqrt{2} d)\}$ and corresponding eigenvalues $-A\{\pm \sqrt {\frac{3}{2}},0\}$. The independent time evolution of each eigenstate is then easily followed. Thus, for initial population in the $s$ state, we have $$\begin{aligned} s(t)&=&\frac{1}{3}(1+2\cos[\sqrt {\frac{3}{2}} (A/\omega) \sin \omega t]) \label{subeq1} \\ p(t)&=&\sqrt{\frac{2}{3}} i \sin[\sqrt{\frac{3}{2}}(A/\omega)\sin \omega t] \label{subeq2} \\ d(t)&=&\frac{\sqrt 2}{3}(\cos[\sqrt{\frac{3}{2}}(A/\omega) \sin \omega t]-1). \label{subeq3}\end{aligned}$$ Together with the exponential decrease of the elements of $\eta(t)$, the density matrix can be constructed to reproduce the results in Figs. 12-15. It is also clear that when one of the parabolic states is used as a starting point, the density matrix will remain frozen for $\Gamma=0$ and decay monotonically for finite $\Gamma$ as shown in Figs. 16 and 17. ![Evolution of the $n=3$ states of hydrogen, starting with a Stark eigenstate, and $\Gamma =0$. Such eigenstates remain frozen in their time dependence, in contrast to other initial states as shown in the previous figures.](WZFig16.eps){width="3in"} ![same as in Fig. 16 except that $\Gamma =0.2$. Note the monotonic evolution from the initial Stark state to the mixed state described by the density matrix $\frac{1}{3}\mathcal{I}$.](WZFig17.eps){width="3in"} A recent paper has presented results similar to the above in Eqs. (12) for $n=2, 3$ [@ref9]. The results can be readily extended to any $n$ since the expansion of parabolic states in terms of spherical states of hydrogen are well known and given by $3j$-coefficients [@ref10]. The occurrence of the “Floquet form" in Eqs. (12), with trigonometric functions whose arguments are themselves a trigonometric function scaled by $A/\omega$, accounts for the appearance of higher frequencies than $\omega$ in Figs. 1-8 for stronger driving fields. We note that such studies of Rydberg atoms in an $n$ manifold under microwave ionization, sometimes with an additional static field, have been of considerable experimental and theoretical interest [@ref11; @ref12]. Summary ======= The method of unitary integration as extended to problems involving dissipation and decoherence affords a convenient and powerful way of treating $n$-state systems in time-dependent fields. Through the solution of a single, classical, Riccati equation (first-order in time and quadratically nonlinear), we can follow the evolution of the density matrix in time without any restrictions to infinitesimal steps or time orderings. Results have been presented for three-state systems with examples of coupled pendula and the $n=3$ states of hydrogen in a radiation field. This work has been supported by the U.S. Department of Energy under Grant No. DE-FG02-02ER46018. Email: arau@phys.lsu.edu G. Lindblad, Commun. Math. Phys. [48]{}, 119 (1976); V. Gorini, A. Kassokowski, and E. C. G. Sudarshan, J. Math. Phys. [17]{}, 821 (1976); D. F. Walls and G. J. Milburn, Quantum Optics* (Springer-Verlag, Berlin, 1994); M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge Univ. Pr., 1996); W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, Berlin, 2001); D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I. O. Stamatescu, and H. D. Zeh, Decoherence and the Appearance of a Classical World in Quantum Theory (Springer-Verlag, Berlin, 1996). A. R. P. Rau and R. A. Wendell, Phys. Rev. Lett. [89]{}, 220405 (2002). A. R. P. Rau and K. Unnikrishnan, Phys. Lett. A [222]{}, 304 (1996); A. R. P. Rau, Phys. Rev. Lett. [81]{}, 4785 (1998) and Phys. Rev. A [61]{}, 032301 (2000). J. Wei and E. Norman, J. Math. Phys. [4]{}, 575 (1963); G. Campolieti and B. C. Sanctuary, J. Chem. Phys. [91]{}, 2108 (1989); B. A. Shadwick and W. F. Buell, Phys. Rev. Lett. [79]{}, 5189 (1997) and J. Phys. A [34]{}, 4771 (2001). See, for instance, R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications (Springer-Verlag, Berlin, 1987); E. B. Davies, Quantum Theory of Open Systems (Academic Press, London, 1976). F. T. Hioe and J. H. Eberly, Phys. Rev. Lett. [47]{}, 838 (1981) and Phys. Rev. A [25]{}, 2168 (1982); F. T. Hioe, Phys. Rev. A [30]{}, 2100 (1984). P. M. Radmore and P. L. Knight, J. Phys. B [15]{}, 561 (1982). F. T. Hioe and J. H. Eberly, Phys. Rev. A [29]{}, 1164 (1984). Kh. Kh. Shakov and J. H. McGuire, Phys. Rev. A [67]{}, 033405 (2003); Kh. Kh. Shakov, J. H. McGuire, and Kh. Yu. Rakhimov, Bull. Am. Phys. Soc. [48]{}, No. 3, 33 (2003). See, for instance, L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon Press, Oxford, 1977), Third edition, Sec. 37. C. H. Cheng and T. F. Gallagher, Phys. Rev. A [61]{}, 063411 (2000). V. N. Ostrovsky and E. Horsdal-Pedersen, Phys. Rev. A [67]{}, 033408 (2003).
--- author: - 'Paul Kwiat, Joe Altepeter, David Branning, Evan Jeffrey, Nicholas Peters, and Tzu-Chieh Wei' title: \| To appear in the Proceedings of the 6th International Conference on\ [Quauntum Communication, Measurement and Computing]{}\ (QCMC 2002), Boston, Massachusetts, July 22-26, 2002.\ Taming Entanglement --- Central to the long-term future of quantum information processing is the capability of performing extremely accurate and reproducible gate operations. The restrictions for fault-tolerant quantum computation are extremely demanding: the tolerable error-per-gate operation should be less than $10^{-4}$ to $10^{-6}$. Implementing such precise gate operations and preparing the requisite input states is therefore one of the key milestones for quantum information processing. Using optical realizations of qubits, e.g., polarization states of photons, we have the potential to meet these demanding tolerances. Therefore, although large-scale quantum computers will perhaps never be constructed solely using optical qubits, these systems nevertheless form a unique and convenient testbed with which to experimentally investigate the issues surrounding state creation, manipulation, and characterization, and also ways of dealing with decoherence. Our primary tool for these investigations is a source of correlated photons produced via the process of spontaneous parametric down conversion: with small probability, a pump photon of appropriate polarization may split into two longer-wavelength daughter photons, subject to energy and momentum conservation (Fig. \[SinglePDC\]). By triggering on one of these photons, the other is prepared in a single-photon Fock state[@Mandel]. We can apply local unitary transformations to the polarizations of these photons using a birefringent half-waveplate (HWP) and quarter-waveplate (QWP). We can also introduce decoherence (either independently or collectively) by passing the photons through birefringent delay lines. Using these techniques for the single photon case, the initial pure horizontal state $\|H\rangle$ may be precisely converted into an arbitrary pure or mixed state (Fig. \[SingleQubit\]). We estimate that we can create and reliably distinguish (with fidelities of 0.998 or better) over 100,000 single-qubit states. Applying these single-qubit techniques to each output of a downconversion crystal, we can create arbitrary [product]{} states for the two photons. But these comprise only a very small part of the total two-qubit Hilbert space. To access the rest, we must create entangled states; this is done by adding a second downconverter with an orthogonal optic axis as shown in Fig. \[TwoQubit\]a. A given pair of signal and idler photons could have been born in the first crystal, with vertical polarizations, or in the second with horizontal polarizations. These two possibilities cannot be distinguished by any measurements other than polarization, so the quantum state for these photons is a superposition of $\|V\rangle\|V\rangle$ and $\|H\rangle\|H\rangle$. Because each crystal responds to only one pump polarization, the relative weights of the two downconversion processes can be controlled by adjusting the input pump polarization[@KwiatSource]. A birefringent phase plate is also added to one of the outputs to control the relative phase of the two contributions, so that we can create nonmaximally entangled states of the form: $$\| \psi \rangle \propto \|H\rangle\|H\rangle + \epsilon e^{i \phi}\|V\rangle\|V\rangle. %\end{equation}$$ Combined with the single-photon local unitary transformations, any pure 2-qubit state can be produced. In this way, we have prepared a variety of states (Figs. \[TwoQubit\]b, \[TangleEnt\]b ). The density matrices are tomographically determined by measuring the polarization correlations in 16 bases, and performing a maximum-likelihood analysis to find the legitimate density matrix most consistent with the experimental results[@James]. In order to improve the speed and accuracy of our tomographic measurements, we have implemented a fully automated system. In addition to reducing the total time for a measurement, and significantly decreasing the uncertainty in the measurement settings, this automated system will also enable the implementation of an adaptive tomography routine – by making an initial fast estimate of the state, one could spend most of the data collection time making an optimized set of measurements. With this sort of optimal quantum tomography, we hope to reach the ultimate limit in quantum state characterization. Our automated system has enabled the creation of a large number of states with widely varying degrees of purity and entanglement. A convenient way to display these states is the “Tangle-entropy” plane, shown in Fig. \[TangleEnt\]. Because it is impossible to have a state that is both completely mixed and completely entangled, there is an implied boundary between states that are physically possible and those that are not: this boundary is formed by the “maximally entangled mixed states” (MEMS), which possess the largest degree of entanglement possible for their entropies[@T.C.]. Finally, using the modification of our system shown in Fig. \[ProcessTomo\]a, we can realize several methods of quantum process tomography[@NielsenChuang], whose goal is to completely characterize some unknown process affecting a qubit. This process may be any combination of unitary transformations, state-dependent losses, and decoherence. One method is to send a variety of input states through the process, and tomographically determine the output states. Another technique, known as “entangelement-assisted” or “ancilla-assisted” process tomography[@Altepeter], exploits the two-photon correlations available at the source, and requires only a single, fixed input state to perform an entire process tomography. In the future, we will continue to expand our abilities to create an ever-widening range of quantum states and processes, and to push the level of precision with which they are created and characterized with adaptive tomography. Ultimately, this promising set of tools should be useful for implementing and testing various protocols in quantum information processing. This work was supported in part by the DCI Postdoctoral Research Fellowship Program and by ARDA, and in part under NSF Grant \#EIA-0121568. [99]{} C. K. Hong and L. Mandel, . P. G. Kwiat [et al.]{}, [Phys. Rev. A]{} [60]{}, R773 (1999). D. F. V. James [et al.]{}, [Phys. Rev. A]{} [64]{}, 052312 (2001); A. G. White [et al.]{}, . T. C. Wei [et al.]{}, in preparation; W. J. Munro [et al.]{}, [Phys. Rev. A]{} [64]{}, 030302(R) (2001). I. L. Chuang and M. A. Nielsen, [J. Mod. Opt.]{} [44]{}, 2455 (1997); J. F. Poyatos [et al.]{}, . J. B. Altepeter [et al.]{}, in preparation.
--- abstract: \| For [h=3]{} or [4]{}, Egyptian decompositions into [h]{} unit fractions, like 2/[D]{} = 1[/D[1]{}]{} + ... +1[/D[h]{}]{} , were given by using ([h-1]{}) divisors ([d[i]{}]{}) of [D[1]{}]{}. This ancient [modus operandi]{}, well recognized today, provides [D[i]{}=DD[1]{}/d[i]{}]{} for [i]{} greater than [1]{}. Decompositions selected (depending on [d[i]{}]{}) have generally been studied by modern researchers through the intrinsic features of [d[i]{}]{} itself. An unconventional method is presented here without considering the [d[i]{}]{} properties but just the differences [d]{}-[d]{}. In contrast to widespread ideas about the last denominator like ‘[D[h]{}]{} smaller than 1000’, it is more appropriate to adopt a global boundary of the form ‘[D[h]{}]{} smaller or equal to [10D]{}’, where [10]{} comes from the Egyptian decimal system. Singular case [2/53]{} (with [15]{} instead of [10]{}) is explained. The number of preliminary alternatives before the final decisions is found to be so low (71) for [h=3]{} or [4]{} that a detailed overview is possible. A simple additive method of trials, independent of any context, can be carried out, namely [2n+1= d[2]{}]{}+ ... + [d[h]{}]{}. Clearly the decisions fit with a minimal value of the differences [d]{}-[d]{}, independently of any [d[i]{}]{} values.\ [*[Subject:]{}* math.HO]{}\ [*[MSC:]{}* 01A16]{}\ [*[Keywords:]{}* Rhind Papyrus, 2/n table, Egyptian fractions]{} author: - \| Lionel Bréhamet\ Retired research scientist\ [brehamet.l@orange.fr]{} title: 'Remarks on the Egyptian 2/D table in favor of a global approach (D prime number)' --- Preamble {#preamble .unnumbered} ======== The recto of the Rhind Mathematical Papyrus (RMP) [[@Peet; @Chace; @Robins]]{} contains the so-called Egyptian $2/D$ table. The genesis of a project such as build this table will never really be apprehended. This is not a project as impressive as the construction of a pyramid or temple, however it has been well and truly succeeded. It is impossible to doubt that pyramid works have not been carried out without a hierarchy of teams well organized in various specialties. A perfectly organized hierarchy that included team leaders and supervisors.\ It is not hard to imagine that a structured similar organization was also used for the $2/D$ table. This table has not been an exercise in style. It is imperative to keep in mind that it can not be the work of a single scribe, but surely results of indefinite periods of trials and improvements done by an elite team of scribes talented for calculating. As it is well known through dialogues of Plato, the idea of a small number of scholars (philosophers) comes frequently. To these people only, was reserved the right to reflect on issues such as calculations or the study of numbers. He knew very well that this type of elite was present in the community of scribes of ancient Egypt. He was also aware of their very advanced knowledges in these areas, but without knowing all secrets. There is no reason today to reject the idea of an elite team or even a chief scribe empowered to decide the last.\ The time for carrying the table was perhaps over more than a generation [^1], in order to provide a satisfactory completed product. In such a product nothing should have been left to chance and everything has been deliberately chosen. This is not like a school exercise where one can use a decomposition rather than another to solve a given problem.\ Once found suitable methods for calculations, it becomes possible to take a look at “the preliminary draft" in its entirety. This look is necessary in order to preserve an overall coherence. Some difficulties thus may be highlighted and resolved by a minimum of general decisions, the simplest as possible. The number of potential solutions appears as considerably lower than [ab initio]{} unrealistic calculations published in the modern literature [[@Gillings; @BruckSalom]]{}, namely 22295 or around 28000. We find that it is enough to consider only $71+71$ possibilities, then results could be examined before making consistent decisions. This is realistic. A team spirit is very suitable to make obvious the need for a classification and successive resolutions of difficulties encountered during the project progress. Directives given by a leader are implied. All these ideas have put us on the track to a comprehensive approach. These ones are the filigree of our analysis.\ Data from the papyrus ===================== RMP is also well known by the name of his transcriber, the scribe Ahmes. This latter copied the document around 1650 BCE. The source, now lost, could date from XIIth dynasty, a golden age of the middle kingdom. RMP recto shows a table of $2$ divided by numbers $D$ from $5$ up to $101$ into “unit fractions”. Number $3$ may be considered as implicitly included, because its decomposition is used in the verso for some problems or it appears elsewhere in Papyrus Kahun [[@Imhausen]]{}. This fact has been commented pertinently by Abdulaziz [[@Abdulaziz] ]{}.\ For $D$ prime only (except number $101$), we present below a reordered excerpt from the $2/D$ table by using , that just show the multiplicity of a denominator with $D$. Please note that [they are not the red auxiliary numbers used by Ahmes]{}, [ie]{} those “decoded" by Gardner [[@GardnerMilo]]{}, but related with these latter by means of the divisors of the first denominator $D_1$.\ [c]{} [ccc]{} $\!\!\!\!\!$ $\scriptstyle 2/D=1/D_1+1/D_2 \;\, \sf [2-terms] $ ----------------------------------------------------- -- $2/3=1/2+1/6\,_ {\textcolor{red}{2}}$ $2/5=1/3+1/15\,_{\textcolor{red}{ 3}}$ $2/7=1/4+1/28\,_ {\textcolor{red}{4}}$ $2/11=1/6+1/66\,_{\textcolor{red}{ 6}}$ $2/23=1/12+1/276\,_{\textcolor{red}{ {12}}}$ : REORDERED $2/D$ TABLE FOR PRIME NUMBERS $D$ & $\!\!\!\!\!$ $\scriptstyle 2/D=1/D_1+1/D_2+1/D_3\;\, \sf [3-terms] $ ------------------------------------------------------------------------- -- $2/13=1/8+1/52\,_{\textcolor{red}{ 4}}+1/104\,_{\textcolor{red}{ 8}}$ $2/17=1/12+1/51\,_{\textcolor{red}{ 3}}+1/68\,_{\textcolor{red}{ 4}}$ $2/19=1/12+1/76\,_{\textcolor{red}{ 4}}+1/114\,_{\textcolor{red}{ 6}}$ $2/31=1/20+1/124\,_{\textcolor{red}{ 4}}+1/155\,_{\textcolor{red}{ 5}}$ $2/37=1/24+1/111\,_{\textcolor{red}{ 3}}+1/296\,_{\textcolor{red}{ 8}}$ $2/41=1/24+1/246\,_{\textcolor{red}{ 6}}+1/328\,_{\textcolor{red}{ 8}}$ $2/47=1/30+1/141\,_{\textcolor{red}{ 3}}+1/470\,_{\textcolor{red}{10}}$ $2/53=1/30+1/318\,_{\textcolor{red}{ 6}}+1/795\,_{\textcolor{red}{15}}$ $2/59=1/36+1/236\,_{\textcolor{red}{ 4}}+1/531\,_{\textcolor{red}{9}}$ $2/67=1/40+1/335\,_{\textcolor{red}{ 5}}+1/536\,_{\textcolor{red}{8}}$ $2/71=1/40+1/568\,_{\textcolor{red}{ 8}}+1/710\,_{\textcolor{red}{10}}$ $2/97=1/56+1/679\,_{\textcolor{red}{ 7}}+1/776\,_{\textcolor{red}{8}}$ : REORDERED $2/D$ TABLE FOR PRIME NUMBERS $D$ & $\!\!\!\!\!$ $\scriptstyle 2/D=1/D_1+1/D_2+1/D_3+1/D_4\;\, \sf[4-terms] $ -------------------------------------------------------------------------------------------------------- -- $2/29=1/24+1/58\,_{\textcolor{red}{ 2}}+1/174\,_{\textcolor{red}{ 6}}+1/232\,_{\textcolor{red}{ 8}}$ $2/43=1/42+1/86\,_{\textcolor{red}{ 2}}+1/129\,_{\textcolor{red}{ 3}}+1/301\,_{\textcolor{red}{ 7}}$ $2/61=1/40+1/244\,_{\textcolor{red}{ 4}}+1/488\,_{\textcolor{red}{ 8}}+1/610\,_{\textcolor{red}{ 10}}$ $2/73=1/60+1/219\,_{\textcolor{red}{ 3}}+1/292\,_{\textcolor{red}{ 4}}+1/365\,_{\textcolor{red}{ 5}}$ $2/79=1/60+1/237\,_{\textcolor{red}{ 3}}+1/316\,_{\textcolor{red}{ 4}}+1/790\,_{\textcolor{red}{ 10}}$ $2/83=1/60+1/332\,_{\textcolor{red}{ 4}}+1/415\,_{\textcolor{red}{ 5}}+1/498\,_{\textcolor{red}{ 6}}$ $2/89=1/60+1/356\,_{\textcolor{red}{ 4}}+1/534\,_{\textcolor{red}{ 6}}+1/890\,_{\textcolor{red}{ 10}}$ : REORDERED $2/D$ TABLE FOR PRIME NUMBERS $D$ \[Papyrus\] Outlines of a global approach {#PART 0} ============================== Actually the whole $2/D$ project can been viewed as a 3-component set (or 3-phases, if you like). FIRST: discovery of a unique \[2-terms\] solution, if $D$ is a prime number.\ SECOND: for a [sub-project \[composite numbers\] ]{}from $9$ up to $99$, realize that a mini-table, with just four numbers, enables to derive all the composite numbers by using a multiplicative operation [^2].\ Four numbers, 3, 5, 7, 11 are enough. For instance $99$ is reached with $ \times \mathit {33} $ or $ \times \mathit {9}$. This mini-table, a kind of ’Mother-table’, looks as follows:\ -------------------------------------------------------- $2/3=1/2+1/6\,_ {\textcolor{red}{2}}$ \[0.01in\] $2/5=1/3+1/15\,_{\textcolor{red}{ 3}}$ \[0.01in\] $2/7=1/4+1/28\,_ {\textcolor{red}{4}}$ \[0.01in\] $\cdots\cdots\cdots\cdots\cdots\cdots\cdot$ \[0.01in\] $2/11=1/6+1/66\,_{\textcolor{red}{ 6}}$ \[0.01in\] -------------------------------------------------------- : Basic Mother-Table \[MotherTable\] One sees the first four two-terms decompositions of $2/D$. $D$ being prime, the table is unique.\ In ‘theory’, [except if a better decision should be token,]{} any fraction $2/D$ ($D$ composite) could be decomposed from this table by dividing a given row by a convenient number. Consider an example: $2/65\!=\mbox{\sf [ (row 2 )/ (number 13) ]}=\!1/39+1/195\,_{\textcolor{red}{3 }} $, what is the solution adopted in the papyrus. As a matter of fact, all decompositions for the [sub-project]{} were given in two-terms (except for [2/95]{} as a logical consequence of the guidelines adopted by the scribes, that we will justify properly later) [^3] .\ As the ‘Mother-table’ has no need to higher value than $11$ for the [sub-project]{}, we can better understand that, from $13$, it could have been decided to leave decompositions into 2 terms.\ [THIRD:]{} nothing does more obstacle to start a main part of the whole project, namely decompositions into 3 (or 4 terms if necessary), for all prime numbers starting from 13 until 97.\ The study carried in this paper is devoted to the third phase. General presentation {#PART I} -------------------- We could have present the problems in the Egyptian manner, as did Abdulaziz [[@Abdulaziz]]{} like for example $47 \quad \overline{30}\quad \overline{141}\quad \overline{470}\quad \mbox{what means}\quad 2/47=1/30+1/141+1/470$, but we preferred a modern way, more easily understandable to us today. This is unrelated to the spirit in which we thought. Consider $D$ as given, $D_1$ is an unknown value to be found. Assume now that $d_2$, $d_3$, $d_4$ are distinct divisors of $D_1$, with $d_2> d_3> d_4$. These are also unknowns to find.\ In order to standardize the notations, $D$ is used for [D]{}enominators and $d$ for [d]{}ivisors.\ Look at the following (modern) equations that [decompose the ’unity’]{} in $3$ or $4$ parts: $$\mathbf{1}= \frac{D}{2D_1}+ \frac{d_2}{2D_1}+ \frac{d_3}{2D_1}.$$ $$\mathbf{1}= \frac{D}{2D_1}+ \frac{d_2}{2D_1}+ \frac{d_3}{2D_1}+ \frac{d_4}{2D_1}.$$ It can be viewed under another standpoint like additive operations on integers: $$2D_1={D}+ d_2+ d_3. \label{eq:additive3}$$ $$2D_1={D}+ d_2+ d_3+ d_4. \label{eq:additive4}$$ Since $d_2$, $d_3$, $d_4$ divide $D_1$ then we are sure to find Egyptian decompositions. Indeed, dividing by $DD_1$ we always get sums of unit fractions: $$\frac{2}{D}= \frac{1}{D_1}+ \frac{1}{(D_1/d_2)D}+ \frac{1}{(D_1/d_3)D}. \label{eq:FEgypt3}$$ $$\frac{2}{D}= \frac{1}{D_1}+ \frac{1}{(D_1/d_2)D}+ \frac{1}{(D_1/d_3)D}+ \frac{1}{(D_1/d_4)D}. \label{eq:FEgypt4}$$ This method was apparently followed [[@GardnerMilo]]{} in RMP table for prime numbers $D$ from $13$ up to $97$ .\ As can be seen, except $D_1$, all denominators of each equation appear as a multiple of $D$, namely $$D_i=m_i D, \mbox{\hspace{0.5em}where}\hspace{0.5em}m_i=(D_1/d_i). \label{eq:Relationmd}$$ Let us briefly summarize the possibilities as follows $$\frac{2}{D}= \frac{1}{D_1}+ \frac{1}{D_2}+ \frac{1}{D_3}. \label{eq:Egypt3}$$ $$\frac{2}{D}= \frac{1}{D_1}+ \frac{1}{D_2}+ \frac{1}{D_3}+ \frac{1}{D_4}. \label{eq:Egypt4}$$ The main task consists in the determination of $D_1$ and the convenient choice of $d_i$, from the additive equations (\[eq:additive3\]) or (\[eq:additive4\]). The $d_i$’s are the red auxiliary numbers used by the scribe Ahmes. $$d_i=\frac{D_1}{m_i}. \label{eq:Ahmesdi}$$ \[2-terms\] analysis {#TwoTerms} ==================== $$\frac{2}{D}= \frac{1}{D_1}+ \frac{1}{D_2}. \label{eq:Egypt2}$$ The only comment (admiring) on the subject is that the scribes actually found the right solution (unique) to the problem, namely $$D_1=\frac{D+1}{2}\quad \mbox{and}\quad D_2=\frac{D(D+1)}{2}.$$ \[3-terms\] analysis {#ThreeTerms} ==================== Right now consider the \[3-terms\] cases. Egyptians gave:\ [ccll]{} ------------------------------------------------------------------------------------- Ahmes’s selections \[3-terms\]\ \[0.01in\] $2/13=1/8+1/52\,_{\textcolor{red}{ 4}}+1/104\,_{\textcolor{red}{ 8}}$\ \[0.01in\] $2/17=1/12+1/51\,_{\textcolor{red}{ 3}}+1/68\,_{\textcolor{red}{ 4}}$\ \[0.01in\] $2/19=1/12+1/76\,_{\textcolor{red}{ 4}}+1/114\,_{\textcolor{red}{ 6}}$\ \[0.01in\] $2/31=1/20+1/124\,_{\textcolor{red}{ 4}}+1/155\,_{\textcolor{red}{ 5}}$\ \[0.01in\] $2/37=1/24+1/111\,_{\textcolor{red}{ 3}}+1/296\,_{\textcolor{red}{ 8}}$\ \[0.01in\] $2/41=1/24+1/246\,_{\textcolor{red}{ 6}}+1/328\,_{\textcolor{red}{ 8}}$\ \[0.01in\] $2/47=1/30+1/141\,_{\textcolor{red}{ 3}}+1/470\,_{\textcolor{red}{10}}$\ \[0.01in\] $2/53=1/30+1/318\,_{\textcolor{red}{ 6}}+1/795\,_{\textcolor{red}{15}}$\ \[0.01in\] $2/59=1/36+1/236\,_{\textcolor{red}{ 4}}+1/531\,_{\textcolor{red}{9}}$\ \[0.01in\] $2/67=1/40+1/335\,_{\textcolor{red}{ 5}}+1/536\,_{\textcolor{red}{8}}$\ \[0.01in\] $2/71=1/40+1/568\,_{\textcolor{red}{ 8}}+1/710\,_{\textcolor{red}{10}}$\ \[0.01in\] $2/97=1/56+1/679\,_{\textcolor{red}{ 7}}+1/776\,_{\textcolor{red}{8}}$\ \[0.01in\] ------------------------------------------------------------------------------------- & -------------- $\Leftarrow$ -------------- & ---------------------------------- Unity decomposition \[0.01in\] $16 = 13 + 2 + 1_{}$ \[0.01in\] $24 = 17 + 4 + 3_{}$ \[0.01in\] $24 = 19 + 3 + 2_{}$ \[0.01in\] $40= 31 + 5 + 4_{}$ \[0.01in\] $48 = 37 + 8 + 3_{}$ \[0.01in\] $48 = 41 + 4 + 3_{}$ \[0.01in\] $60 = 47 + 10 + 3_{}$ \[0.01in\] $60 = 53 + 5 + 2_{}$ \[0.01in\] $72 = 59 + 9 + 4_{}$ \[0.01in\] $80 = 67 + 8 + 5_{}$ \[0.01in\] $80 = 71 + 5 + 4_{}$ \[0.01in\] $112 = 97 + 8 + 7_{}$ \[0.01in\] ---------------------------------- & --- . --- \ The task of finding $D_1$ is rather simple, from the moment when one realizes that it is enough to establish a table of odd numbers $(2n+1)_{\|n\geq 1}$ as a sum of two numbers $ d_2 +d_3$, with $d_2>d_3$. This is easy to do and independent of any context. The table contains $n$ doublets {$d_2, d_3$} and $\sup(d_2)=2n$. One can start with the lowest values as follows: $d_3=1, d_2=2,4,6, \cdots; d_3=2, d_2=3,5,7, \cdots$ and so on.\ From Eq.(\[eq:additive3\]) the first candidate possible for $D_1$ starts at an initial value $D_1^0=(D+1 )/2$ as in Fibonnaci’s studies [[@Fibonacci]]{}. We can search for general solutions of the form $$D_1^n=D_1^0 + n,$$ whence $$2D_1^n-{D}= 2n+1 =d_2+ d_3. \label{eq:additive3bis}$$ Since one of the two $D_1$ divisors {$d_2,d_3$} is even, then $D_1$ can not be odd, it must be even. This was rightly stressed by Bruins [[@Bruins]]{}. From the first table of doublets, a new table (of trials) is built, where this time doublets are selected if $d_2,d_3$ divide $[(D+d_2+d_3)/2]$. This provides a $D_1^n$ possible. In this favorable case, first $D_3$ is calculated by $DD_1/d_3$, then $D_2$ by $DD_1/d_2$.\ For $D$ given, the table of trials defined by the equation just below $$\mathtt{2n+1=d_2+d_3}, \mbox{\hspace{0.5em} where $d_2$ and $d_3$ divide $D_1^n$ }, \label{eq:dividers3terms}$$ is bounded by a $n_{max}$ [^4]. By simplicity in our tables, $D_1^n$ will not be written as $D_1^n (d_2,d_3)$.\ Even by hand, a realization of this table takes few time. For example decompositions into 3 terms lead to a total of trials with only $71$ possibilities! From this low value, it is conceivable to present all results according to an appropriate parameter. Once found a $d_3$, a good idea would be select a $d_2$ the closest as possible of $d_3$. This provides a type of classification never glimpsed to our knowledge. Thus, a [key parameter]{} of our paper is defined as follows: $$\Delta_{d}= d_2 -d_3.$$ [Remarks: Clearly Eq. (\[eq:dividers3terms\]) is related to Bruins’s method of “parts” redistribution $d_2,d_3$ [[@Bruins]]{}. However our method is [‘artisanal’]{} and does not need to know the arithmetic properties of $D_1$. Once $D$ given, $D_1^n$ are found by trials, without calculations. Unlike to Bruins which sought some forms of $D_1$ for finding then possible $D$ values. The approach is quite different as well as the reasons justifying the Egyptian choices.]{}\ $$D_1 [R] = (2D_1 - D) = 2n+1 =d_2+ d_3,$$ or equivalently expressed $$[R] =\frac{1}{(D_1/ d_2)}+\frac{1}{(D_1/ d_3)}.$$ [When it is said “[... keeping the terms of $[R]$ less than $10$ was an essential part of determining how $2$:$n$ is to be decomposed.]{}”, this should be understood as $(D_1/ d_3)\leq 10$ and formulated for us as the condition (\[eq:ConditionD1\_3\]) with a Top-flag $\mbox{\boldmath $\top$ }\!\!_ f^{\;\;[3]}=10$. ]{}(See below for our Top-flag definition)\ [ However note that the ‘necessity’ of our Top-flag comes directly from the value of $D$, without constituting a check on $D_1$. That only follows from Eq. (\[eq:TopFlag3\]).\ In contrast, parameter $[Q]$, defined in Ref. [[@Abdulaziz]]{} by $[Q]=1-[R]$, does not appear to us and plays no role in our analyses. In addition, as the impact of closeness ($\Delta_{d}$) does not seem to have been apprehended, it is clear that our argumentation will generally be different. Even if, for some ’easy’ cases, we agree.]{} In short, for producing their final table, we assume that the scribes have analyzed all preliminary trial results before doing their choice among various alternatives, considered in their totality, not individually.\ Furthermore, [due to decimal numeration used by ancient Egyptians]{}, one can easily understand that a boundary with a Top-flag $\mbox{\boldmath $\top$ }\!\!_ f^{\;\;[3]}$ for the last denominator was chosen with a priority value equal to $\mathbf {10}$ (if possible according to the results given by trials).\ The idea of a Top-flag is far to be a [‘deus ex machina’]{}. It naturally arises if we try to solve the problem of decomposition in full generality. See [Appendix A]{} for more details.\ Chief scribe wisely decided to impose a upper bound to all the denominators $D_3$, such that $$D_3 \leq D \mbox{\boldmath $\top$ }\!\!_ f^{\;\;[3]} . \label{eq:TopFlag3}$$ This cut-off beyond $\mbox{\boldmath $\top$ }\!\!_ f^{\;\;[3]}$ is equivalent to a mathematical condition on $D_1$: $$D_1 \leq d_3\,\mbox{\boldmath $\top$ }\!\!_ f^{\;\;[3]} . \label{eq:ConditionD1_3}$$ Remark that this condition might be exploited from the beginning of the calculations for avoiding to handle too large denominators $D_3$. Simply find $d_3$, find $d_2$, then calculate $D_1$, if condition ( \[eq:ConditionD1\_3\])\ is not fulfilled then quit, do not calculate $D_3$, $D_2$ and go to next values for $d_3$, $d_2$, $D_1$ and so on.\ Actually, if we follow the method of trials for finding the good choices in the order $d_3 \rightarrow d_2 \rightarrow D_1$, we are naturally led to be careful of the closeness of $d_2$, $d_3$, measured by $\Delta_{d}$. This can suggest the idea of a classification according to increasing values of $\Delta_{d}$.\ Since this classification seriously enlightens many solutions chosen by the scribes, it is not impossible to imagine that this ‘[artisan method]{}’ was actually followed. This is a plausible hypothesis, valueless of evidence obviously. An advantage is also that a similar classification can be applied to the decompositions into 4 terms with the same success, see Sect. \[FourTerms\].\ The symbol $^{Eg}$ will be used for indicating Egyptian selections in our tables.\ Let us now display a preliminary table of trials, see Table \[Tble3terms71\].\ \[htp\] [\|l\|c\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $\textcolor{red}{\Delta_{d}}$ & $D_1^n$ & Possible \[3-terms\] decompositions\ \[0.01in\] $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $8$ & $ \mathbf {2/13=1/8+1/52\,_{\textcolor{red}{ 4}}+1/104\,_{\textcolor{red}{ 8}}}\;\, ^{Eg}$\ $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $10$ & $ \mathbf {2/17\mathit{_a}=1/10+1/85\,_{\textcolor{red}{ 5}}+1/170\,_{\textcolor{red}{ 10}}}$\ $3$ & $7$ & $4$ & $3$ & $\mathbf{\textcolor{red}{ 1}}$& $12$ & $ \mathbf {2/17\mathit{_b}=1/12+1/51\,_{\textcolor{red}{ 3}}+1/68\,_{\textcolor{red}{ 4}}}\;\, ^{Eg}$\ $2$ & $5$ & $3$ & $2$ & $\mathbf{\textcolor{red}{ 1}}$& $12$ & $\mathbf {2/19=1/12+1/76\,_{\textcolor{red}{ 4}}+1/114\,_{\textcolor{red}{ 6}}}\;\, ^{Eg}$\ $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $16$ & $ \mathbf {2/29=1/16+1/232\,_{\textcolor{red}{ 8}}+1/464\,_{\textcolor{red}{ 16}} }$\ $2$ & $5$ & $3$ & $2$ & $\mathbf{\textcolor{red}{ 1}}$& $18$ & $ \mathbf {2/31\mathit{_a}=1/18+1/186\,_{\textcolor{red}{ 6}}+1/279\,_{\textcolor{red}{ 9}}}$\ $4$ & $9$ & $5$ & $4$ & $\mathbf{\textcolor{red}{ 1}}$& $20$ & $ \mathbf {2/31\mathit{_b}=1/20+1/124\,_{\textcolor{red}{ 4}}+1/155\,_{\textcolor{red}{ 5}}}\;\, ^{Eg}$\ $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $20$ & $ \mathbf {2/37=1/20+1/370\,_{\textcolor{red}{ 10}}+1/740\,_{\textcolor{red}{ 20}}}$\ $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $22$ & $ \mathbf {2/41\mathit{_a}=1/22+1/451\,_{\textcolor{red}{ 11}}+1/902\,_{\textcolor{red}{ 22}}}$\ $3$ & $7$ & $4$ & $3$ & $\mathbf{\textcolor{red}{ 1}}$& $24$ & $ \mathbf {2/41\mathit{_b}=1/24+1/246\,_{\textcolor{red}{ 6}}+1/328\,_{\textcolor{red}{ 8}}}\;\, ^{Eg}$\ $2$ & $5$ & $3$ & $2$ & $\mathbf{\textcolor{red}{ 1}}$& $24$ & $ \mathbf {2/43=1/24+1/344\,_{\textcolor{red}{ 8}}+1/516\,_{\textcolor{red}{ 12}} }$\ $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $28$ & $ \mathbf {2/53=1/28+1/742\,_{\textcolor{red}{ 14}}+1/1484\,_{\textcolor{red}{ 28}}} $\ $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $32$ & $ \mathbf {2/61=1/32+1/976\,_{\textcolor{red}{ 16}}+1/1952\,_{\textcolor{red}{ 32}} }$\ $2$ & $5$ & $3$ & $2$ & $\mathbf{\textcolor{red}{ 1}}$& $36$ & $ \mathbf {2/67=1/36+1/804\,_{\textcolor{red}{ 12}}+1/1206\,_{\textcolor{red}{ 18}}} $\ $4$ & $9$ & $5$ & $4$ & $\mathbf{\textcolor{red}{ 1}}$& $40$ & $ \mathbf {2/71\mathit{_a}=1/40+1/568\,_{\textcolor{red}{ 8}}+1/710\,_{\textcolor{red}{ 10}} }\;\, ^{Eg}$\ $6$ & $13$ & $7$ & $6$ & $\mathbf{\textcolor{red}{ 1}}$& $42$ & $ \mathbf {2/71\mathit{_b}=1/42+1/426\,_{\textcolor{red}{ 6}}+1/497\,_{\textcolor{red}{ 7}}}$\ $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $38$ & $ \mathbf {2/73=1/38+1/1387\,_{\textcolor{red}{ 19}}+1/2274\,_{\textcolor{red}{ 38}} }$\ $2$ & $5$ & $3$ & $2$ & $\mathbf{\textcolor{red}{ 1}}$& $42$ & $ \mathbf {2/79=1/42+1/1106\,_{\textcolor{red}{ 14}}+1/1659\,_{\textcolor{red}{ 21}} }$\ $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $46$ & $ \mathbf {2/89\mathit{_a}=1/46+1/2047\,_{\textcolor{red}{ 23}}+1/4094\,_{\textcolor{red}{ 46}} }$\ $3$ & $7$ & $4$ & $3$ & $\mathbf{\textcolor{red}{ 1}}$& $48$ & $ \mathbf {2/89\mathit{_b}=1/48+1/1068\,_{\textcolor{red}{ 12}}+1/1424\,_{\textcolor{red}{ 16}} }$\ $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $50$ & $ \mathbf {2/97\mathit{_a}=1/50+1/2425\,_{\textcolor{red}{ 25}}+1/4850\,_{\textcolor{red}{ 50}}} $\ $7$ & $15$ & $8$ & $7$ & $\mathbf{\textcolor{red}{ 1}}$& $56$ & $ \mathbf {2/97\mathit{_b}=1/56+1/679\,_{\textcolor{red}{ 7}}+1/776\,_{\textcolor{red}{ 8}}}\;\, ^{Eg}$\ $3$ & $7$ & $5$ & $2$ & $\mathbf{\textcolor{red}{ 3}}$& $10$ & $ \mathbf {2/13=1/10+1/26\,_{\textcolor{red}{ 2}}+1/65\,_{\textcolor{red}{ 5}}}$\ $2$ & $5$ & $4$ & $1$ & $\mathbf{\textcolor{red}{ 3}}$& $12$ & $ \mathbf {2/19=1/12+1/57\,_{\textcolor{red}{ 3}}+1/228\,_{\textcolor{red}{ 12}}}$\ $2$ & $5$ & $4$ & $1$ & $\mathbf{\textcolor{red}{ 3}}$& $24$ & $ \mathbf {2/43=1/24+1/258\,_{\textcolor{red}{ 6}}+1/1032\,_{\textcolor{red}{ 24}} }$\ $3$ & $7$ & $5$ & $2$ & $\mathbf{\textcolor{red}{ 3}}$& $30$ & $ \mathbf {2/53=1/30+1/318\,_{\textcolor{red}{ 6}}+1/795\,_{\textcolor{red}{ 15}}}\;\, ^{Eg}$\ $2$ & $5$ & $4$ & $1$ & $\mathbf{\textcolor{red}{ 3}}$& $32$ & $ \mathbf {2/59=1/32+1/472\,_{\textcolor{red}{ 8}}+1/1888\,_{\textcolor{red}{ 32}}} $\ $2$ & $5$ & $4$ & $1$ & $\mathbf{\textcolor{red}{ 3}}$& $36$ & $ \mathbf {2/67\mathit{_a}=1/36+1/603\,_{\textcolor{red}{ 9}}+1/2412\,_{\textcolor{red}{ 36}}} $\ $6$ & $13$ & $8$ & $5$ & $\mathbf{\textcolor{red}{ 3}}$& $40$ & $ \mathbf {2/67\mathit{_b}=1/40+1/335\,_{\textcolor{red}{ 5}}+1/536\,_{\textcolor{red}{ 8}}}\;\, ^{Eg}$\ $3$ & $7$ & $5$ & $2$ & $\mathbf{\textcolor{red}{ 3}}$& $40$ & $ \mathbf {2/73=1/40+1/1584\,_{\textcolor{red}{ 8}}+1/1460\,_{\textcolor{red}{ 20}} }$\ $2$ & $5$ & $4$ & $1$ & $\mathbf{\textcolor{red}{ 3}}$& $44$ & $ \mathbf {2/83=1/44+1/913\,_{\textcolor{red}{ 11}}+1/3652\,_{\textcolor{red}{ 44}} }$\ $3$ & $7$ & $6$ & $1$ & $\mathbf{\textcolor{red}{ 5}}$& $12$ & $ \mathbf {2/17=1/12+1/34\,_{\textcolor{red}{ 2}}+1/204\,_{\textcolor{red}{ 12}}}$\ $4$ & $9$ & $7$ & $2$ & $\mathbf{\textcolor{red}{ 5}}$& $14$ & $ \mathbf {2/19=1/14+1/38\,_{\textcolor{red}{ 2}}+1/133\,_{\textcolor{red}{ 7}}}$\ $3$ & $7$ & $6$ & $1$ & $\mathbf{\textcolor{red}{ 5}}$& $18$ & $ \mathbf {2/29=1/18+1/87\,_{\textcolor{red}{ 3}}+1/522\,_{\textcolor{red}{ 18}} }$\ $5$ & $11$ & $8$ & $3$ & $\mathbf{\textcolor{red}{ 5}}$& $24$ & $ \mathbf {2/37=1/24+1/111\,_{\textcolor{red}{ 3}}+1/296\,_{\textcolor{red}{ 8}}}\;\, ^{Eg}$\ $3$ & $7$ & $6$ & $1$ & $\mathbf{\textcolor{red}{ 5}}$& $24$ & $ \mathbf {2/41=1/24+1/164\,_{\textcolor{red}{ 4}}+1/984\,_{\textcolor{red}{ 24}}}$\ $4$ & $9$ & $7$ & $2$ & $\mathbf{\textcolor{red}{ 5}}$& $28$ & $ \mathbf {2/47=1/28+1/188\,_{\textcolor{red}{ 4}}+1/658\,_{\textcolor{red}{ 14}}}$\ $3$ & $7$ & $6$ & $1$ & $\mathbf{\textcolor{red}{ 5}}$& $30$ & $ \mathbf {2/53=1/30+1/265\,_{\textcolor{red}{ 5}}+1/1590\,_{\textcolor{red}{ 30}}} $\ $6$ & $13$ & $9$ & $4$ & $\mathbf{\textcolor{red}{ 5}}$& $36$ & $ \mathbf {2/59=1/36+1/236\,_{\textcolor{red}{ 4}}+1/531\,_{\textcolor{red}{ 9}}}\;\, ^{Eg}$\ $3$ & $7$ & $6$ & $1$ & $\mathbf{\textcolor{red}{ 5}}$& $48$ & $ \mathbf {2/89=1/48+1/712\,_{\textcolor{red}{ 8}}+1/4272\,_{\textcolor{red}{ 48}} }$\ $4$ & $9$ & $8$ & $1$ & $\mathbf{\textcolor{red}{ 7}}$& $16$ & $ \mathbf {2/23=1/16+1/46\,_{\textcolor{red}{ 2}}+1/368\,_{\textcolor{red}{ 16}}}$\ $6$ & $13$ & $10$ & $3$ & $\mathbf{\textcolor{red}{ 7}}$& $30$ & $ \mathbf {2/47=1/30+1/141\,_{\textcolor{red}{ 3}}+1/470\,_{\textcolor{red}{ 10}}}\;\, ^{Eg}$\ $5$ & $11$ & $9$ & $2$ & $\mathbf{\textcolor{red}{ 7}}$& $36$ & $ \mathbf {2/61=1/36+1/244\,_{\textcolor{red}{ 4}}+1/1098\,_{\textcolor{red}{ 18}} }$\ $4$ & $9$ & $8$ & $1$ & $\mathbf{\textcolor{red}{ 7}}$& $40$ & $ \mathbf {2/71=1/40+1/355\,_{\textcolor{red}{ 5}}+1/2840\,_{\textcolor{red}{ 40}}} $\ $7$ & $15$ & $11$ & $4$ & $\mathbf{\textcolor{red}{ 7}}$& $44$ & $ \mathbf {2/73=1/44+1/292\,_{\textcolor{red}{ 4}}+1/803\,_{\textcolor{red}{ 11}} }$\ $5$ & $11$ & $9$ & $2$ & $\mathbf{\textcolor{red}{ 7}}$& $54$ & $ \mathbf {2/97=1/54+1/582\,_{\textcolor{red}{ 6}}+1/2619\,_{\textcolor{red}{ 27}}} $\ $5$ & $11$ & $10$ & $1$ & $\mathbf{\textcolor{red}{ 9}}$& $20$ & $ \mathbf {2/29=1/20+1/58\,_{\textcolor{red}{ 2}}+1/580\,_{\textcolor{red}{ 20}} }$\ $6$ & $13$ & $11$ & $2$ & $\mathbf{\textcolor{red}{ 9}}$& $22$ & $ \mathbf {2/31=1/22+1/62\,_{\textcolor{red}{ 2}}+1/341\,_{\textcolor{red}{ 11}}}$\ $5$ & $11$ & $10$ & $1$ & $\mathbf{\textcolor{red}{ 9}}$& $50$ & $ \mathbf {2/89=1/50+1/445\,_{\textcolor{red}{ 5}}+1/4450\,_{\textcolor{red}{ 50}} }$\ $7$ & $15$ & $13$ & $2$ & $\mathbf{\textcolor{red}{ 11}}$& $26$ & $ \mathbf {2/37=1/26+1/74\,_{\textcolor{red}{ 2}}+1/481\,_{\textcolor{red}{ 13}}}$\ $6$ & $13$ & $12$ & $1$ & $\mathbf{\textcolor{red}{ 11}}$& $36$ & $ \mathbf {2/59=1/36+1/177\,_{\textcolor{red}{ 3}}+1/2124\,_{\textcolor{red}{ 36}}} $\ $8$ & $17$ & $14$ & $3$ & $\mathbf{\textcolor{red}{ 11}}$& $42$ & $ \mathbf {2/67=1/42+1/201\,_{\textcolor{red}{ 3}}+1/938\,_{\textcolor{red}{ 14}}}$\ $6$ & $13$ & $12$ & $1$ & $\mathbf{\textcolor{red}{ 11}}$& $48$ & $ \mathbf {2/83=1/48+1/332\,_{\textcolor{red}{ 4}}+1/3984\,_{\textcolor{red}{ 48}} }$\ $7$ & $15$ & $13$ & $2$ & $\mathbf{\textcolor{red}{ 11}}$& $52$ & $ \mathbf {2/89=1/52+1/356\,_{\textcolor{red}{ 4}}+1/2314\,_{\textcolor{red}{ 26}} }$\ $7$ & $15$ & $14$ & $1$ & $\mathbf{\textcolor{red}{ 13}}$& $28$ & $ \mathbf {2/41=1/28+1/82\,_{\textcolor{red}{ 2}}+1/1148\,_{\textcolor{red}{ 28}}}$\ $8$ & $17$ & $15$ & $2$ & $\mathbf{\textcolor{red}{ 13}}$& $30$ & $ \mathbf {2/43=1/30+1/86\,_{\textcolor{red}{ 2}}+1/645\,_{\textcolor{red}{ 15}} }$\ $7$ & $15$ & $14$ & $1$ & $\mathbf{\textcolor{red}{ 13}}$& $56$ & $ \mathbf {2/97=1/56+1/388\,_{\textcolor{red}{ 4}}+1/5432\,_{\textcolor{red}{ 56}}} $\ $8$ & $17$ & $16$ & $1$ & $\mathbf{\textcolor{red}{ 15}}$& $32$ & $ \mathbf {2/47=1/32+1/94\,_{\textcolor{red}{ 2}}+1/1504\,_{\textcolor{red}{ 32}}} $\ $8$ & $17$ & $16$ & $1$ & $\mathbf{\textcolor{red}{ 15}}$& $48$ & $ \mathbf {2/79=1/48+1/237\,_{\textcolor{red}{ 3}}+1/3792\,_{\textcolor{red}{ 48}} }$\ $9$ & $19$ & $18$ & $1$ & $\mathbf{\textcolor{red}{ 17}}$& $36$ & $ \mathbf {2/53=1/36+1/106\,_{\textcolor{red}{ 2}}+1/1908\,_{\textcolor{red}{ 36}}}$\ $9$ & $19$ & $18$ & $1$ & $\mathbf{\textcolor{red}{ 17}}$& $54$ & $ \mathbf {2/89=1/54+1/267\,_{\textcolor{red}{ 3}}+1/4306\,_{\textcolor{red}{ 54}} }$\ $11$ & $23$ & $20$ & $3$ & $\mathbf{\textcolor{red}{ 17}}$& $60$ & $ \mathbf {2/97=1/60+1/291\,_{\textcolor{red}{ 3}}+1/1940\,_{\textcolor{red}{ 20}}}$\ $10$ & $21$ & $20$ &$1$ & $\mathbf{\textcolor{red}{ 19}}$& $40$ & $ \mathbf {2/59=1/40+1/118\,_{\textcolor{red}{ 2}}+1/2360\,_{\textcolor{red}{ 40}}}$\ $11$ & $23$ & $21$ & $2$ & $\mathbf{\textcolor{red}{ 19}}$& $42$ & $ \mathbf {2/61=1/42+1/122\,_{\textcolor{red}{ 2}}+1/1281\,_{\textcolor{red}{ 21}} }$\ $12$ & $25$ & $23$ & $2$ & $\mathbf{\textcolor{red}{ 21}}$& $46$ & $ \mathbf {2/67=1/46+1/134\,_{\textcolor{red}{ 2}}+1/1541\,_{\textcolor{red}{ 23}}}$\ $12$ & $25$ & $24$ & $1$ & $\mathbf{\textcolor{red}{ 23}}$& $48$ & $ \mathbf {2/71=1/48+1/142\,_{\textcolor{red}{ 2}}+1/3408\,_{\textcolor{red}{ 48}}}$\ $13$ & $27$ & $25$ & $2$ & $\mathbf{\textcolor{red}{ 23}}$& $50$ & $ \mathbf {2/73=1/50+1/146\,_{\textcolor{red}{ 2}}+1/1825\,_{\textcolor{red}{ 25}} }$\ $14$ & $29$ & $27$ & $2$ & $\mathbf{\textcolor{red}{ 25}}$& $54$ & $ \mathbf {2/79=1/54+1/158\,_{\textcolor{red}{ 2}}+1/2133\,_{\textcolor{red}{ 27}} }$\ $14$ & $29$ & $28$ & $1$ & $\mathbf{\textcolor{red}{ 27}}$& $56$ & $ \mathbf {2/83=1/56+1/166\,_{\textcolor{red}{ 2}}+1/4648\,_{\textcolor{red}{ 56}} }$\ $15$ & $31$ & $30$ & $1$ & $\mathbf{\textcolor{red}{ 29}}$& $60$ & $ \mathbf {2/89=1/60+1/178\,_{\textcolor{red}{ 2}}+1/5340\,_{\textcolor{red}{ 60}} }$\ $17$ & $35$ & $33$ & $2$ & $\mathbf{\textcolor{red}{ 31}}$& $66$ & $ \mathbf {2/97=1/66+1/194\,_{\textcolor{red}{ 2}}+1/3201\,_{\textcolor{red}{ 33}}} $\ \[Tble3terms71\] As it is clear from Table \[Tble3terms71\] an obvious preference for the smallest $\Delta_{d}$ seems to be well followed.\ After cut-off by $\mbox{\boldmath $\top$ }\!\!_ f^{\;\;[3]}= 10$ Table \[Tble3terms71\] is reduced and allows us to analyze the following options: \[htbp\] [\|l\|c\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $\textcolor{red}{\Delta_{d}}$ & $D_1^n$ & \[3-terms\] decompositions $\mathbf {\textcolor{red}{ m_3\leq 10}}$\ \[0.01in\] $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $8$ & $ \mathbf {2/13=1/8+1/52\,_{\textcolor{red}{ 4}}+1/104\,_{\textcolor{red}{ 8}}}\;\, ^{Eg}$\ $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $10$ & $ \mathbf {2/17\mathit{_a}=1/10+1/85\,_{\textcolor{red}{ 5}}+1/170\,_{\textcolor{red}{ 10}}}$\ $3$ & $7$ & $4$ & $3$ & $\mathbf{\textcolor{red}{ 1}}$& $12$ & $ \mathbf {2/17\mathit{_b}=1/12+1/51\,_{\textcolor{red}{ 3}}+1/68\,_{\textcolor{red}{ 4}}}\;\, ^{Eg{\textcolor{red}{{\star }}}}$\ $2$ & $5$ & $3$ & $2$ & $\mathbf{\textcolor{red}{ 1}}$& $12$ & $\mathbf {2/19=1/12+1/76\,_{\textcolor{red}{ 4}}+1/114\,_{\textcolor{red}{ 6}}}\;\, ^{Eg}$\ $2$ & $5$ & $3$ & $2$ & $\mathbf{\textcolor{red}{ 1}}$& $18$ & $ \mathbf {2/31\mathit{_a}=1/18+1/186\,_{\textcolor{red}{ 6}}+1/279\,_{\textcolor{red}{ 9}}}$\ $4$ & $9$ & $5$ & $4$ & $\mathbf{\textcolor{red}{ 1}}$& $20$ & $ \mathbf {2/31\mathit{_b}=1/20+1/124\,_{\textcolor{red}{ 4}}+1/155\,_{\textcolor{red}{ 5}}}\;\, ^{Eg{\textcolor{red}{{\star }}}}$\ $3$ & $7$ & $4$ & $3$ & $\mathbf{\textcolor{red}{ 1}}$& $24$ & $ \mathbf {2/41=1/24+1/246\,_{\textcolor{red}{ 6}}+1/328\,_{\textcolor{red}{ 8}}}\;\, ^{Eg}$\ $4$ & $9$ & $5$ & $4$ & $\mathbf{\textcolor{red}{ 1}}$& $40$ & $ \mathbf {2/71\mathit{_a}=1/40+1/568\,_{\textcolor{red}{ 8}}+1/710\,_{\textcolor{red}{ 10}} }\;\, ^{Eg}$\ $6$ & $13$ & $7$ & $6$ & $\mathbf{\textcolor{red}{ 1}}$& $42$ & $ \mathbf {2/71\mathit{_b}=1/42+1/426\,_{\textcolor{red}{ 6}}+1/497\,_{\textcolor{red}{ 7}}}$\ $7$ & $15$ & $8$ & $7$ & $\mathbf{\textcolor{red}{ 1}}$& $56$ & $ \mathbf {2/97=1/56+1/679\,_{\textcolor{red}{ 7}}+1/776\,_{\textcolor{red}{ 8}}}\;\, ^{Eg{\textcolor{red}{{\star }}}}$\ $3$ & $7$ & $5$ & $2$ & $\mathbf{\textcolor{red}{ 3}}$& $10$ & $ \mathbf {2/13=1/10+1/26\,_{\textcolor{red}{ 2}}+1/65\,_{\textcolor{red}{ 5}}}$\ $6$ & $13$ & $8$ & $5$ & $\mathbf{\textcolor{red}{ 3}}$& $40$ & $ \mathbf {2/67=1/40+1/335\,_{\textcolor{red}{ 5}}+1/536\,_{\textcolor{red}{ 8}}}\;\, ^{Eg}$\ $4$ & $9$ & $7$ & $2$ & $\mathbf{\textcolor{red}{ 5}}$& $14$ & $ \mathbf {2/19=1/14+1/38\,_{\textcolor{red}{ 2}}+1/133\,_{\textcolor{red}{ 7}}}$\ $5$ & $11$ & $8$ & $3$ & $\mathbf{\textcolor{red}{ 5}}$& $24$ & $ \mathbf {2/37=1/24+1/111\,_{\textcolor{red}{ 3}}+1/296\,_{\textcolor{red}{ 8}}}\;\, ^{Eg}$\ $6$ & $13$ & $9$ & $4$ & $\mathbf{\textcolor{red}{ 5}}$& $36$ & $ \mathbf {2/59=1/36+1/236\,_{\textcolor{red}{ 4}}+1/531\,_{\textcolor{red}{ 9}}}\;\, ^{Eg}$\ $6$ & $13$ & $10$ & $3$ & $\mathbf{\textcolor{red}{ 7}}$& $30$ & $ \mathbf {2/47=1/30+1/141\,_{\textcolor{red}{ 3}}+1/470\,_{\textcolor{red}{ 10}}}\;\, ^{Eg}$\ \ \[3TERMSOPT\] \ This table shows rare instances where multipliers $m_2$, $m_3$ are consecutive. It is always an interesting quality that does not require sophisticated mathematical justification. That will be denoted by a asterisk ${\textcolor{red}{^{\star }}}$. Two instances are found also in \[4-terms\] series with $m_2$, $m_3$, $m_4$, see Section \[FourTerms\].\ Just as an indication, we display below the cases dropped out of a \[3-terms\] decomposition:\ [\|l\|c\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $\textcolor{red}{\Delta_{d}}$ & $D_1^n$ & Possible \[3-terms\] decompositions\ $4$ & $9$ & $8$ & $1$ & $\mathbf{\textcolor{red}{ 7}}$& $16$ & $ \mathbf {2/23=1/16+1/46\,_{\textcolor{red}{ 2}}+1/368\,_{\textcolor{red}{ 16}}}$\ $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $16$ & $ \mathbf {2/29=1/16+1/232\,_{\textcolor{red}{ 8}}+1/464\,_{\textcolor{red}{ 16}} }$\ $3$ & $7$ & $6$ & $1$ & $\mathbf{\textcolor{red}{ 5}}$& $18$ & $ \mathbf {2/29=1/18+1/87\,_{\textcolor{red}{ 3}}+1/522\,_{\textcolor{red}{ 18}} }$\ $5$ & $11$ & $10$ & $1$ & $\mathbf{\textcolor{red}{ 9}}$& $20$ & $ \mathbf {2/29=1/20+1/58\,_{\textcolor{red}{ 2}}+1/580\,_{\textcolor{red}{ 20}} }$\ $2$ & $5$ & $4$ & $1$ & $\mathbf{\textcolor{red}{ 3}}$& $24$ & $ \mathbf {2/43=1/24+1/258\,_{\textcolor{red}{ 6}}+1/1032\,_{\textcolor{red}{ 24}} }$\ $2$ & $5$ & $3$ & $2$ & $\mathbf{\textcolor{red}{ 1}}$& $24$ & $ \mathbf {2/43=1/24+1/344\,_{\textcolor{red}{ 8}}+1/516\,_{\textcolor{red}{ 12}} }$\ $8$ & $17$ & $15$ & $2$ & $\mathbf{\textcolor{red}{ 13}}$& $30$ & $ \mathbf {2/43=1/30+1/86\,_{\textcolor{red}{ 2}}+1/645\,_{\textcolor{red}{ 15}} }$\ $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $32$ & $ \mathbf {2/61=1/32+1/976\,_{\textcolor{red}{ 16}}+1/1952\,_{\textcolor{red}{ 32}} }$\ $5$ & $11$ & $9$ & $2$ & $\mathbf{\textcolor{red}{ 7}}$& $36$ & $ \mathbf {2/61=1/36+1/244\,_{\textcolor{red}{ 4}}+1/1098\,_{\textcolor{red}{ 18}} }$\ $11$ & $23$ & $21$ & $2$ & $\mathbf{\textcolor{red}{ 19}}$& $42$ & $ \mathbf {2/61=1/42+1/122\,_{\textcolor{red}{ 2}}+1/1281\,_{\textcolor{red}{ 21}} }$\ $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $38$ & $ \mathbf {2/73=1/38+1/1387\,_{\textcolor{red}{ 19}}+1/2274\,_{\textcolor{red}{ 38}} }$\ $3$ & $7$ & $5$ & $2$ & $\mathbf{\textcolor{red}{ 3}}$& $40$ & $ \mathbf {2/73=1/40+1/1584\,_{\textcolor{red}{ 8}}+1/1460\,_{\textcolor{red}{ 20}} }$\ $7$ & $15$ & $11$ & $4$ & $\mathbf{\textcolor{red}{ 7}}$& $44$ & $ \mathbf {2/73=1/44+1/292\,_{\textcolor{red}{ 4}}+1/803\,_{\textcolor{red}{ 11}} }$\ $13$ & $27$ & $25$ & $2$ & $\mathbf{\textcolor{red}{ 23}}$& $50$ & $ \mathbf {2/73=1/50+1/146\,_{\textcolor{red}{ 2}}+1/1825\,_{\textcolor{red}{ 25}} }$\ $2$ & $5$ & $3$ & $2$ & $\mathbf{\textcolor{red}{ 1}}$& $42$ & $ \mathbf {2/79=1/42+1/1106\,_{\textcolor{red}{ 14}}+1/1659\,_{\textcolor{red}{ 21}} }$\ $8$ & $17$ & $16$ & $1$ & $\mathbf{\textcolor{red}{ 15}}$& $48$ & $ \mathbf {2/79=1/48+1/237\,_{\textcolor{red}{ 3}}+1/3792\,_{\textcolor{red}{ 48}} }$\ $14$ & $29$ & $27$ & $2$ & $\mathbf{\textcolor{red}{ 25}}$& $54$ & $ \mathbf {2/79=1/54+1/158\,_{\textcolor{red}{ 2}}+1/2133\,_{\textcolor{red}{ 27}} }$\ $2$ & $5$ & $4$ & $1$ & $\mathbf{\textcolor{red}{ 3}}$& $44$ & $ \mathbf {2/83=1/44+1/913\,_{\textcolor{red}{ 11}}+1/3652\,_{\textcolor{red}{ 44}} }$\ $6$ & $13$ & $12$ & $1$ & $\mathbf{\textcolor{red}{ 11}}$& $48$ & $ \mathbf {2/83=1/48+1/332\,_{\textcolor{red}{ 4}}+1/3984\,_{\textcolor{red}{ 48}} }$\ $14$ & $29$ & $28$ & $1$ & $\mathbf{\textcolor{red}{ 27}}$& $56$ & $ \mathbf {2/83=1/56+1/166\,_{\textcolor{red}{ 2}}+1/4648\,_{\textcolor{red}{ 56}} }$\ $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $46$ & $ \mathbf {2/89=1/46+1/2047\,_{\textcolor{red}{ 23}}+1/4094\,_{\textcolor{red}{ 46}} }$\ $3$ & $7$ & $6$ & $1$ & $\mathbf{\textcolor{red}{ 5}}$& $48$ & $ \mathbf {2/89=1/48+1/712\,_{\textcolor{red}{ 8}}+1/4272\,_{\textcolor{red}{ 48}} }$\ $3$ & $7$ & $4$ & $3$ & $\mathbf{\textcolor{red}{ 1}}$& $48$ & $ \mathbf {2/89=1/48+1/1068\,_{\textcolor{red}{ 12}}+1/1424\,_{\textcolor{red}{ 16}} }$\ $5$ & $11$ & $10$ & $1$ & $\mathbf{\textcolor{red}{ 9}}$& $50$ & $ \mathbf {2/89=1/50+1/445\,_{\textcolor{red}{ 5}}+1/4450\,_{\textcolor{red}{ 50}} }$\ $7$ & $15$ & $13$ & $2$ & $\mathbf{\textcolor{red}{ 11}}$& $52$ & $ \mathbf {2/89=1/52+1/356\,_{\textcolor{red}{ 4}}+1/2314\,_{\textcolor{red}{ 26}} }$\ $9$ & $19$ & $18$ & $1$ & $\mathbf{\textcolor{red}{ 17}}$& $54$ & $ \mathbf {2/89=1/54+1/267\,_{\textcolor{red}{ 3}}+1/4306\,_{\textcolor{red}{ 54}} }$\ $15$ & $31$ & $30$ & $1$ & $\mathbf{\textcolor{red}{ 29}}$& $60$ & $ \mathbf {2/89=1/60+1/178\,_{\textcolor{red}{ 2}}+1/5340\,_{\textcolor{red}{ 60}} }$\ \[Frac3become4\] Our definition of $\mbox{\boldmath $\top$ }\!\!_ f $ does not depend on a arbitrary value of $D_3$ fixed to $1000$ as often assumed in the literature. It depends only on the circumstances imposed by the current project. Subdivide now table \[3TERMSOPT\] into 3 sets according to the properties of each $D$. A first with a only one ${\Delta_{d}}$, a second with two different ${\Delta_{d}} $ and a third with two conflicting identical ${\Delta_{d}}$. That yields:\ \[htbp\] ----- -------- ------- ------- -------------------------------- --------- ------------------------------------------------------------------------------------------------------------------------- $n$ $2n+1$ $d_2$ $d_3$ $\textcolor{red}{\Delta_{d}}$ $D_1^n$ $\qquad$ \[3-terms\] decomposition $3$ $7$ $4$ $3$ $\mathbf{\textcolor{red}{ 1}}$ $24$ $ \mathbf {2/41=1/24+1/246\,_{\textcolor{red}{ 6}}+1/328\,_{\textcolor{red}{ 8}}}\;\, ^{Eg}$ $7$ $15$ $8$ $7$ $\mathbf{\textcolor{red}{ 1}}$ $56$ $ \mathbf {2/97=1/56+1/679\,_{\textcolor{red}{ 7}}+1/776\,_{\textcolor{red}{ 8}}}\;\, ^{Eg{\textcolor{red}{{\star }}}}$ $6$ $13$ $8$ $5$ $\mathbf{\textcolor{red}{ 3}}$ $40$ $ \mathbf {2/67=1/40+1/335\,_{\textcolor{red}{ 5}}+1/536\,_{\textcolor{red}{ 8}}}\;\, ^{Eg}$ $5$ $11$ $8$ $3$ $\mathbf{\textcolor{red}{ 5}}$ $24$ $ \mathbf {2/37=1/24+1/111\,_{\textcolor{red}{ 3}}+1/296\,_{\textcolor{red}{ 8}}}\;\, ^{Eg}$ $6$ $13$ $9$ $4$ $\mathbf{\textcolor{red}{ 5}}$ $36$ $ \mathbf {2/59=1/36+1/236\,_{\textcolor{red}{ 4}}+1/531\,_{\textcolor{red}{ 9}}}\;\, ^{Eg}$ $6$ $13$ $10$ $3$ $\mathbf{\textcolor{red}{ 7}}$ $30$ $ \mathbf {2/47=1/30+1/141\,_{\textcolor{red}{ 3}}+1/470\,_{\textcolor{red}{ 10}}}\;\, ^{Eg}$ ----- -------- ------- ------- -------------------------------- --------- ------------------------------------------------------------------------------------------------------------------------- : A single $\Delta_{d}$ \[3-terms\] \ \[1Delta3\] \[h\] [\|l\|c\|\|l\|\|l\|\|c\|l\|l\|]{}\ &\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $\textcolor{red}{\Delta_{d}}$ & $D_1^n$ & $\qquad$ \[3-terms\] decompositions\ $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $8$ & $ \mathbf {2/13=1/8+1/52\,_{\textcolor{red}{ 4}}+1/104\,_{\textcolor{red}{ 8}}}\;\, ^{Eg}$\ $3$ & $7$ & $5$ & $2$ & $\mathbf{\textcolor{red}{ 3}}$& $10$ & $ \mathbf {2/13=1/10+1/26\,_{\textcolor{red}{ 2}}+1/65\,_{\textcolor{red}{ 5}}}$\ $2$ & $5$ & $3$ & $2$ & $\mathbf{\textcolor{red}{ 1}}$& $12$ & $\mathbf {2/19=1/12+1/76\,_{\textcolor{red}{ 4}}+1/114\,_{\textcolor{red}{ 6}}}\;\, ^{Eg}$\ $4$ & $9$ & $7$ & $2$ & $\mathbf{\textcolor{red}{ 5}}$& $14$ & $ \mathbf {2/19=1/14+1/38\,_{\textcolor{red}{ 2}}+1/133\,_{\textcolor{red}{ 7}}}$\ \ \[2Delta3\] \[h\] [\|l\|c\|\|l\|\|l\|\|c\|l\|l\|]{}\ &\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $\textcolor{red}{\Delta_{d}}$ & $D_1^n$ & $\qquad$ \[3-terms\] decompositions\ $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $10$ & $ \mathbf {2/17\mathit{_a}=1/10+1/85\,_{\textcolor{red}{ 5}}+1/170\,_{\textcolor{red}{ 10}}}$\ $3$ & $7$ & $4$ & $3$ & $\mathbf{\textcolor{red}{ 1}}$& $12$ & $ \mathbf {2/17\mathit{_b}=1/12+1/51\,_{\textcolor{red}{ 3}}+1/68\,_{\textcolor{red}{ 4}}}\;\, ^{Eg{\textcolor{red}{{\star }}}}$\ $2$ & $5$ & $3$ & $2$ & $\mathbf{\textcolor{red}{ 1}}$& $18$ & $ \mathbf {2/31\mathit{_a}=1/18+1/186\,_{\textcolor{red}{ 6}}+1/279\,_{\textcolor{red}{ 9}}}$\ $4$ & $9$ & $5$ & $4$ & $\mathbf{\textcolor{red}{ 1}}$& $20$ & $ \mathbf {2/31\mathit{_b}=1/20+1/124\,_{\textcolor{red}{ 4}}+1/155\,_{\textcolor{red}{ 5}}}\;\, ^{Eg{\textcolor{red}{{\star }}}}$\ \ ----- -------- ------- ------- -------------------------------- --------- ----------------------------------------------------------------------------------------------------------- $n$ $2n+1$ $d_2$ $d_3$ $\textcolor{red}{\Delta_{d}}$ $D_1^n$ $\qquad$ \[3-terms\] decompositions $4$ $9$ $5$ $4$ $\mathbf{\textcolor{red}{ 1}}$ $40$ $ \mathbf {2/71\mathit{_a}=1/40+1/568\,_{\textcolor{red}{ 8}}+1/710\,_{\textcolor{red}{ 10}} }\;\, ^{Eg}$ $6$ $13$ $7$ $6$ $\mathbf{\textcolor{red}{ 1}}$ $42$ $ \mathbf {2/71\mathit{_b}=1/42+1/426\,_{\textcolor{red}{ 6}}+1/497\,_{\textcolor{red}{ 7}}} {\;\,^{\textcolor{red}{{\star }}}}$ ----- -------- ------- ------- -------------------------------- --------- ----------------------------------------------------------------------------------------------------------- : Two conflicting identical $\Delta_{d}$ \[3-terms\] \[11Delta3\] Remark: [in the cases involving options possible, and in these cases only,]{} the solutions for\ {2/D = 2/13, 2/19, 2/17, 2/31} were chosen respectively in the set {$n= 1, 2, 3, 4$}$_{\mathbf{\|}_\mathbf{{\textcolor{red}{2n\leq 10}}}}$.\ For ruling on [2/71]{} there is no convincing arithmetical argumentation, then the choice could\ have been the simplicity and direct observation: once again a boundary like $2n\leq {\textcolor{red}{ 10}}$ is used for picking $n=4$. That’s it. Too simple, but why not?\ it remains some cases to be examined, especially these with $\boxed{10 < m_3 \leq 16}$ because of the singular status of [2/23,]{} that the scribes will retain with a decomposition into 2 terms. We display below these cases. Of course [2/61, 2/83]{} are [ex officio]{} excluded from the analysis.\ (Anticipation is made on \[4-terms\] analysis and related decisions that follow, like $\sf \mbox{\boldmath $\top$ }\!\!_ f^{\;\;[4]}= \sf10$) ----------------------------------------------------------------------------- -- [Unique]{} \[2-terms\] solution \[0.01in\] $\mathbf {2/23=1/12+1/276\,_{\textcolor{red}{ {12}}}}\;\, ^{Eg}$ \[0.01in\] ----------------------------------------------------------------------------- -- : Dynamic comparison for transitions $\mathbf {3 \Rightarrow 4}$ \ [\|l\|c\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $\textcolor{red}{\Delta_{d}}$ & $D_1^n$ & [Unique]{} \[3-terms\] decomposition\ \[0.01in\] $4$ & $9$ & $8$ & $1$ & $\mathbf{\textcolor{red}{ 7}}$& $16$ & $ \mathbf {2/23=1/16+\boxed{1/46\,_{\textcolor{red}{ 2}}}+1/368\,_{\textcolor{red}{ 16}}}$\ [\|l\|c\|l\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $d_4$ &$\textcolor{red}{\Delta_{d}^{'}}$ &$D_1^n$ & \[4-terms\] decomposition $\mathbf {\textcolor{red}{ m_4\leq 10}}$\ \[0.01in\] $8$ & $17$ & $10$ & $5$ & $2$ & $\mathbf{\textcolor{red}{ 3}}$& $20$ & $\mathbf {2/23=1/20+\boxed{1/46\,_{\textcolor{red}{ 2}}}+1/92\,_{\textcolor{red}{ 4}}+1/230\,_{\textcolor{red}{10}}}$\ [\|l\|c\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $\textcolor{red}{\Delta_{d}}$ & $D_1^n$ & Possible \[3-terms\] decomposition\ \[0.01in\] $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $16$ & $ \mathbf {2/29=1/16+\boxed{1/232\,_{\textcolor{red}{ 8}}}+1/464\,_{\textcolor{red}{ 16}} }$\ \[0.01in\] [\|l\|c\|l\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $d_4$ &$\textcolor{red}{\Delta_{d}^{'}}$ &$D_1^n$ & Possible \[4-terms\] decompositions $\mathbf {\textcolor{red}{ m_4\leq 10}}$\ \[0.01in\] $9$ & $19$ & $12$ & $4 $ & $3 $ & $\mathbf{\textcolor{red}{ 1}}$& $24$ & $\mathbf {2/29=1/24+1/58\,_{\textcolor{red}{ 2}}+1/174\,_{\textcolor{red}{ 6}}+ \boxed{1/232\,_{\textcolor{red}{8}}}}\;\, ^{Eg}$\ $5$ & $11$ & $5$ & $4$ & $2$ & $\mathbf{\textcolor{red}{ 2}}$& $20$ & $ \mathbf {2/29=1/20+1/116\,_{\textcolor{red}{ 4}}+1/145\,_{\textcolor{red}{ 5}}+1/290\,_{\textcolor{red}{ 10}}}$\ $15$ & $31$ & $15$ & $10$ & $6$ & $\mathbf{\textcolor{red}{ 4}}$& $30$ & $ \mathbf {2/29=1/30+1/58\,_{\textcolor{red}{ 2}}+1/87\,_{\textcolor{red}{ 3}}+1/145\,_{\textcolor{red}{5}}}$\ [\|l\|c\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $\textcolor{red}{\Delta_{d}}$ & $D_1^n$ & Possible \[3-terms\] decomposition\ \[0.01in\] $3$ & $7$ & $4$ & $3$ & $\mathbf{\textcolor{red}{ 1}}$& $48$ & $ \mathbf {2/89\mathit{_b}=1/48+1/1068\,_{\textcolor{red}{ 12}}+1/1424\,_{\textcolor{red}{ 16}} }$\ [\|l\|c\|l\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $d_4$ &$\textcolor{red}{\Delta_{d}^{'}}$ &$D_1^n$ & Possible \[4-terms\] decompositions $\mathbf {\textcolor{red}{ m_4\leq 10}}$\ \[0.01in\] $15$ & $31$ & $15$ & $10$ & $6$ & $\mathbf{\textcolor{red}{ 4}}$& $60$ & $ \mathbf {2/89=1/60+1/356\,_{\textcolor{red}{ 4}}+1/534\,_{\textcolor{red}{ 6}}+1/890\,_{\textcolor{red}{10}}}\;\, ^{Eg}$\ [\|l\|c\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $\textcolor{red}{\Delta_{d}}$ & $D_1^n$ & Possible \[3-terms\] decomposition\ \[0.01in\] $3$ & $7$ & $5$ & $2$ & $\mathbf{\textcolor{red}{ 3}}$& $30$ & $ \mathbf {2/53=1/30+\boxed{1/318\,_{\textcolor{red}{ 6}}}+1/795\,_{\textcolor{red}{ 15}}}\;\, ^{Eg}$\ [\|l\|c\|l\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $d_4$ &$\textcolor{red}{\Delta_{d}^{'}}$ &$D_1^n$ & \[4-terms\] decomposition $\mathbf {\textcolor{red}{ m_4\leq 10}}$\ \[0.01in\] $9$ & $19$ & $9$ & $6 $ & $4 $ & $\mathbf{\textcolor{red}{ 2}}$& $36$ & $\mathbf {2/53=1/36+1/212\,_{\textcolor{red}{ 4}} +\boxed{1/318\,_{\textcolor{red}{ 6}}}+1/477\,_{\textcolor{red}{9}}}$\ [\|l\|c\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $\textcolor{red}{\Delta_{d}}$ & $D_1^n$ & Possible \[3-terms\] decompositions\ \[0.01in\] $2$ & $5$ & $3$ & $2$ & $\mathbf{\textcolor{red}{ 1}}$& $24$ & $ \mathbf {2/43=1/24+1/344\,_{\textcolor{red}{ 8}}+1/516\,_{\textcolor{red}{ 12}} }$\ \[0.01in\] $8$ & $17$ & $15$ & $2$ & $\mathbf{\textcolor{red}{ 13}}$ ------------------------------------------------------------------------ & $30$ & $ \mathbf {2/43=1/30+\boxed{1/86\,_{\textcolor{red}{ 2}}}+1/645\,_{\textcolor{red}{ 15}} }$ ------------------------------------------------------------------------ \ \[0.01in\] = [\|l\|c\|l\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $d_4$ &$\textcolor{red}{\Delta_{d}^{'}}$ &$D_1^n$ & \[4-terms\] decomposition $\mathbf {\textcolor{red}{ m_4\leq 10}}$\ \[0.01in\] $20$ & $41$ & $21$ & $14$ & $6$ & $\mathbf{\textcolor{red}{ 8}}$& $42$ & $ \mathbf {2/43=1/42+\boxed{1/86\,_{\textcolor{red}{ 2}}}+1/129\,_{\textcolor{red}{ 3}}+1/301\,_{\textcolor{red}{7}}}\;\, ^{Eg}$\ [\|l\|c\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $\textcolor{red}{\Delta_{d}}$ & $D_1^n$ & Possible \[3-terms\] decomposition\ \[0.01in\] $7$ & $15$ & $11$ & $4$ & $\mathbf{\textcolor{red}{ 7}}$& $44$ & $ \mathbf {2/73=1/44+\boxed{1/292\,_{\textcolor{red}{ 4}}}+1/803\,_{\textcolor{red}{ 11}} }$\ \[0.01in\] [\|l\|c\|l\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $d_4$ &$\textcolor{red}{\Delta_{d}^{'}}$ &$D_1^n$ & \[4-terms\] decomposition $\mathbf {\textcolor{red}{ m_4\leq 10}}$\ \[0.01in\] $23$ & $47$ & $20$ & $15$ & $12$ & $\mathbf{\textcolor{red}{ 3}}$& $60$ & $ \mathbf {2/73\mathit{_c}=1/60+1/219\,_{\textcolor{red}{ 3}}+\boxed{1/292\,_{\textcolor{red}{ 4}}}+1/365\,_{\textcolor{red}{5}}}\;\, ^{Eg{\textcolor{red}{{\star }}}}$\ We repeat that we are always in a logic of a construction site with difficulties arising in different parts of the project. Problems are processed case after case and do not interfere with another previous part. If not, all becomes incomprehensible. A overview supervised by a chief scribe can not be conflicted. The 6 cases presented above confront us with [a dynamic alternative: select the transition from 3 to 4 fractions, or reject it]{}. This exceptional situation is new in the table construction project, as well as the solution itself! It can be observed that 5 cases on 6 have in common the fact that a same denominator appears in \[3-terms\] and \[4-terms\] decompositions.\ [A priori]{}, this fact may be seen as not being an improvement to better decompose a \[3-terms\] fraction into \[4-terms\]. Unless we find a real improvement worthwhile.\ [$\boxed{2/89}$ :]{} sixth case, out of the category ‘same denominator’, is quickly ruled and \[4-terms\] decomposition is adopted. (Anyway it belonged to this table only because $m_3=16$).\ [$\boxed{2/43}$ :]{} once dropped out the option $m_3=15$, due to a too high gap $\Delta_{d}=13$, the same argument holds, then \[4-terms\] decomposition is adopted.\ [$\boxed{2/73}$ :]{} the \[4-terms\] expansion provides an improvement since that leads to three consecutive multipliers {$3,\,4,\,5$}, thus this solution is adopted.\ [Three cases (slightly reordered) remain to be solved, they are displayed in the following table.]{}\ ----------------------------------------------------------------------------- -- [Unique]{} \[2-terms\] solution \[0.01in\] $\mathbf {2/23=1/12+1/276\,_{\textcolor{red}{ {12}}}}\;\, ^{Eg}$ \[0.01in\] ----------------------------------------------------------------------------- -- \ [\|l\|c\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $\textcolor{red}{\Delta_{d}}$ & $D_1^n$ & [Unique]{} \[3-terms\] decomposition\ \[0.01in\] $4$ & $9$ & $8$ & $1$ & $\mathbf{\textcolor{red}{ 7}}$& $16$ & $ \mathbf {2/23=1/16+\boxed{1/46\,_{\textcolor{red}{ 2}}}+1/368\,_{\textcolor{red}{ 16}}}$\ [\|l\|c\|l\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $d_4$ &$\textcolor{red}{\Delta_{d}^{'}}$ &$D_1^n$ & \[4-terms\] decomposition $\mathbf {\textcolor{red}{ m_4\leq 10}}$\ \[0.01in\] $8$ & $17$ & $10$ & $5$ & $2$ & $\mathbf{\textcolor{red}{ 3}}$& $20$ & $\mathbf {2/23=1/20+\boxed{1/46\,_{\textcolor{red}{ 2}}}+1/92\,_{\textcolor{red}{ 4}}+1/230\,_{\textcolor{red}{10}}}$\ [\|l\|c\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $\textcolor{red}{\Delta_{d}}$ & $D_1^n$ & Possible \[3-terms\] decomposition\ \[0.01in\] $3$ & $7$ & $5$ & $2$ & $\mathbf{\textcolor{red}{ 3}}$& $30$ & $ \mathbf {2/53=1/30+\boxed{1/318\,_{\textcolor{red}{ 6}}}+1/795\,_{\textcolor{red}{ 15}}}\;\, ^{Eg}$\ [\|l\|c\|l\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $d_4$ &$\textcolor{red}{\Delta_{d}^{'}}$ &$D_1^n$ & \[4-terms\] decomposition $\mathbf {\textcolor{red}{ m_4\leq 10}}$\ \[0.01in\] $9$ & $19$ & $9$ & $6 $ & $4 $ & $\mathbf{\textcolor{red}{ 2}}$& $36$ & $\mathbf {2/53=1/36+1/212\,_{\textcolor{red}{ 4}} +\boxed{1/318\,_{\textcolor{red}{ 6}}}+1/477\,_{\textcolor{red}{9}}}$\ [\|l\|c\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $\textcolor{red}{\Delta_{d}}$ & $D_1^n$ & Possible \[3-terms\] decomposition\ \[0.01in\] $1$ & $3$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $16$ & $ \mathbf {2/29=1/16+\boxed{1/232\,_{\textcolor{red}{ 8}}}+1/464\,_{\textcolor{red}{ 16}} }$\ \[0.01in\] [\|l\|c\|l\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $d_4$ &$\textcolor{red}{\Delta_{d}^{'}}$ &$D_1^n$ & Possible \[4-terms\] decompositions $\mathbf {\textcolor{red}{ m_4\leq 10}}$\ \[0.01in\] $9$ & $19$ & $12$ & $4 $ & $3 $ & $\mathbf{\textcolor{red}{ 1}}$& $24$ & $\mathbf {2/29\mathit{_a}=1/24+1/58\,_{\textcolor{red}{ 2}}+1/174\,_{\textcolor{red}{ 6}}+ \boxed{1/232\,_{\textcolor{red}{8}}}}\;\, ^{Eg}$\ $5$ & $11$ & $5$ & $4$ & $2$ & $\mathbf{\textcolor{red}{ 2}}$& $20$ & $ \mathbf {2/29\mathit{_b}=1/20+1/116\,_{\textcolor{red}{ 4}}+1/145\,_{\textcolor{red}{ 5}}+1/290\,_{\textcolor{red}{ 10}}}$\ $15$ & $31$ & $15$ & $10$ & $6$ & $\mathbf{\textcolor{red}{ 4}}$& $30$ & $ \mathbf {2/29\mathit{_c}=1/30+1/58\,_{\textcolor{red}{ 2}}+1/87\,_{\textcolor{red}{ 3}}+1/145\,_{\textcolor{red}{5}}}$\ For each fraction the same denominators (inside a box) have a well defined position in a \[3-terms\] expansion and another in a \[4-terms\]. We denote respectively these positions by rank$^{[3]}$ [and]{} rank$^{[4]}$.\ Same denominators will be denoted by $\boxed{same D_i}$. The table below summarizes the situation.\ [\|c\|c\|c\|c\|l\|]{}Fraction & $\boxed{same D_i}$ & rank$^{[3]}$ & rank$^{[4]}$ & Appreciation on ranks\ 2/23 & $\boxed{46}$ & $\mathbf 2 $ & $\mathbf 2 $ & no interest\ 2/53 & $\boxed{318}$ & $\mathbf 2 $ & $\mathbf 3 $ & too near\ 2/29$\mathit{_a} $& $\boxed{232}$ & $\mathbf 2 $ & $\mathbf 4 $ & acceptable + smallest $\textcolor{red}{\Delta_{d}^{'}}$\ \ Some convenient rulings ensue, namely\ [2/23]{}; no solution; then come back to the only one solution in 2 terms.\ [2/53]{}; maintain \[3-terms\] solution; reject \[4-terms\] solution.\ [2/29$\mathit{_a} $]{}; adopt \[4-terms\] solution. \[4-terms\] analysis {#FourTerms} ==================== Right now consider the \[4-terms\] cases. Egyptians gave:\ [ccll]{} -------------------------------------------------------------------------------------------------------------------- Ahmes’s selections \[4-terms\]\ \[0.01in\] $2/29=1/24+1/58\,_{\textcolor{red}{ 2}}+1/174\,_{\textcolor{red}{ 6}}+1/232\,_{\textcolor{red}{ 8}}$\ \[0.01in\] $2/43=1/42+1/86\,_{\textcolor{red}{ 2}}+1/129\,_{\textcolor{red}{ 3}}+1/301\,_{\textcolor{red}{ 7}}$\ \[0.01in\] $2/61=1/40+1/244\,_{\textcolor{red}{ 4}}+1/488\,_{\textcolor{red}{ 8}}+1/610\,_{\textcolor{red}{ 10}}$\ \[0.01in\] $2/73=1/60+1/219\,_{\textcolor{red}{ 3}}+1/292\,_{\textcolor{red}{ 4}}+1/365\,_{\textcolor{red}{ 5}}$\ \[0.01in\] $2/79=1/60+1/237\,_{\textcolor{red}{ 3}}+1/316\,_{\textcolor{red}{ 4}}+1/790\,_{\textcolor{red}{ 10}}$\ \[0.01in\] $2/83=1/60+1/332\,_{\textcolor{red}{ 4}}+1/415\,_{\textcolor{red}{ 5}}+1/498\,_{\textcolor{red}{ 6}}$\ \[0.01in\] $2/89=1/60+1/356\,_{\textcolor{red}{ 4}}+1/534\,_{\textcolor{red}{ 6}}+1/890\,_{\textcolor{red}{ 10}}$\ \[0.01in\] -------------------------------------------------------------------------------------------------------------------- & -------------- $\Leftarrow$ -------------- & ----------------------------------------- Unity decomposition \[0.01in\] $48 = 29 + 12 + 4 + 3_{}$ \[0.01in\] $84 = 43 + 21 + 14 + 6_{}$ \[0.01in\] $80 = 61 + 10 + 5 + 4_{}$ \[0.01in\] $120 = 73 + 20 + 15 +12_{}$ \[0.01in\] $120 = 79 + 20 + 15 +6_{}$ \[0.01in\] $120 = 83 + 15 + 12 + 10_{}$ \[0.01in\] $120 = 89 + 15 + 10 + 6_{}$ \[0.01in\] ----------------------------------------- & --- . --- \ Square brackets here $ \boldsymbol [\;\boldsymbol]$ means ‘integral part of’. [One can start with the lowest values as follows: $d_4=1, d_3=2,3,4, \cdots, d_2=3,4,5, \cdots; d_4=2, d_3=3,4,5, \cdots, d_2=4,5,6, \cdots$ and so on, with the condition $d_3+d_2 \equiv d_4+1 \mod(2)$. ]{}\ From Eq.(\[eq:additive4\]) the first candidate possible for $D_1$ starts at the value $D_1^0=(D+1 )/2$. We can search for general solutions of the form $$D_1^n=D_1^0 + n,$$ whence $$2D_1^n-{D}= 2n+1 =d_2+ d_3+d_4. \label{eq:additive4bis}$$ From the first table of triplets, a new table (of trials) is built, where this time triplets are selected if $d_2,d_3,d_4$ divide $[(D+d_2+d_3+d_4)/2]$. This provides a $D_1^n$ possible. In this favorable case, first $D_4$ is calculated by $DD_1/d_4$, then $D_3$ by $DD_1/d_3$, and $D_2$ by $DD_1/d_2$.\ This table of trials, properly defined by the equation just below (included the constraints), ie $$\mathtt{2n+1=d_2+d_3+d_4}, \mbox{\hspace{0.5em} where $d_2$, $d_3$ and $d_4$ divide $D_1^n$ }, \label{eq:dividers4terms}$$ is obviously a bit longer to establish than for doublets. By simplicity $D_1^n$ will be not written as $D_1^n (d_2,d_3,d_4)$. For decompositions into 4 terms the total of trials yields only $71$ possibilities !\ Of course our remark previously made about doublets is still valid for triplets. Likewise, Abdulaziz’s parameter \[R\] takes the form $$[R] =\frac{1}{(D_1/ d_2)}+\frac{1}{(D_1/ d_3)}+\frac{1}{(D_1/ d_4)}.$$ The notation used in our tables will be $$\Delta_{d}^{'}= d_3 -d_4,$$ Chief scribe wisely decided to impose a upper bound to all the denominators $D_4$, such that $$D_4 \leq D \mbox{\boldmath $\top$ }\!\!_ f^{\;\;[4]} .$$ This cut-off beyond $\mbox{\boldmath $\top$ }\!\!_ f^{\;\;[4]}$ is equivalent to a mathematical condition on $D_1$: $$D_1 \leq d_4\,\mbox{\boldmath $\top$ }\!\!_ f^{\;\;[4]} . \label{eq:ConditionD1_4}$$ Here again, choosing $\mbox{\boldmath $\top$ }\!\!_ f^{\;\;[4]}=10$ is quite appropriate. Thus a general coherence is ensured throughout the project, since 11 out of 12 decompositions into 3 terms were solved with $\mbox{\boldmath $\top$ }\!\!_ f^{\;\;[3]}=10$.\ Remark that the condition (\[eq:ConditionD1\_4\]) might be exploited from the beginning of the calculations for avoiding to handle too large denominators $D_4$. Simply find $d_4$, find $d_3$, find $d_2$, calculate $D_1$, if (\[eq:ConditionD1\_4\]) is not fulfilled then quit, do not calculate $D_4$, $D_3$, $D_2$ and go to next values for $d_4$, $d_3$, $d_2$, $D_1$ etc. [\|l\|c\|l\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $d_4$ &$\textcolor{red}{\Delta_{d}^{'}}$ &$D_1^n$ & Possible \[4-terms\] decompositions\ \[0.01in\] $9$ & $19$ & $12$ & $4 $ & $3 $ & $\mathbf{\textcolor{red}{ 1}}$& $24$ & $\mathbf {2/29=1/24+1/58\,_{\textcolor{red}{ 2}}+1/174\,_{\textcolor{red}{ 6}}+1/232\,_{\textcolor{red}{8}}}\;\, ^{Eg}$\ $5$ & $11$ & $6$ & $3$ & $2$ & $\mathbf{\textcolor{red}{ 1}}$& $36$ & $\mathbf {2/61\mathit{_a}=1/36+1/366\,_{\textcolor{red}{ 6}}+1/732\,_{\textcolor{red}{ 12}}+1/1098\,_{\textcolor{red}{ 18}}}$\ $9$ & $19$ & $10$ & $5$ & $4$ & $\mathbf{\textcolor{red}{ 1}}$& $40$ & $\mathbf {2/61\mathit{_b}=1/40+1/244\,_{\textcolor{red}{ 4}}+1/488\,_{\textcolor{red}{ 8}}+1/610\,_{\textcolor{red}{ 10}}}\;\, ^{Eg}$\ $3$ & $7$ & $4$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $40$ & $ \mathbf {2/73\mathit{_a}=1/40+1/730\,_{\textcolor{red}{ 10}}+1/1460\,_{\textcolor{red}{ 20}}+1/2920\,_{\textcolor{red}{ 40}}}$\ $5$ & $11$ & $6$ & $3$ & $2$ & $\mathbf{\textcolor{red}{ 1}}$& $42$ & $ \mathbf {2/73\mathit{_b}=1/42+1/511\,_{\textcolor{red}{ 7}}+1/1022\,_{\textcolor{red}{ 14}}+1/1533\,_{\textcolor{red}{ 21}}}$\ $11$ & $23$ & $16$ & $4$ & $3$ & $\mathbf{\textcolor{red}{ 1}}$& $48$ & $ \mathbf {2/73\mathit{_c}=1/48+1/219\,_{\textcolor{red}{ 3}}+1/876\,_{\textcolor{red}{ 12}}+1/1168\,_{\textcolor{red}{ 16}}}$\ $8$ & $17$ & $12$ & $3$ & $2$ & $\mathbf{\textcolor{red}{ 1}}$& $48$ & $ \mathbf {2/79\mathit{_a}=1/48+1/316\,_{\textcolor{red}{ 4}}+1/1264\,_{\textcolor{red}{ 16}}+1/1896\,_{\textcolor{red}{ 24}}}$\ $20$ & $41$ & $30$ & $6$ & $5$ & $\mathbf{\textcolor{red}{ 1}}$& $60$ & $ \mathbf {2/79\mathit{_b}=1/60+1/158\,_{\textcolor{red}{ 2}}+1/790\,_{\textcolor{red}{ 10}}+1/948\,_{\textcolor{red}{ 12}}}$\ $6$ & $13$ & $8$ & $3$ & $2$ & $\mathbf{\textcolor{red}{ 1}}$& $48$ & $ \mathbf {2/83\mathit{_a}=1/48+1/498\,_{\textcolor{red}{ 6}}+1/1328\,_{\textcolor{red}{ 16}}+1/1992\,_{\textcolor{red}{ 24}}}$\ $6$ & $13$ & $6$ & $4$ & $3$ & $\mathbf{\textcolor{red}{ 1}}$& $48$ & $ \mathbf {2/83\mathit{_b}=1/48+1/664\,_{\textcolor{red}{ 8}}+1/996\,_{\textcolor{red}{ 12}}+1/1328\,_{\textcolor{red}{ 16}}}$\ $14$ & $29$ & $14$ & $8$ & $7$ & $\mathbf{\textcolor{red}{ 1}}$& $56$ & $ \mathbf {2/83\mathit{_c}=1/56+1/332\,_{\textcolor{red}{ 4}}+1/581\,_{\textcolor{red}{ 7}}+1/664\,_{\textcolor{red}{ 8}}}$\ $18$ & $37$ & $30$ & $4$ & $3$ & $\mathbf{\textcolor{red}{ 1}}$& $60$ & $ \mathbf {2/83\mathit{_d}=1/60+1/166\,_{\textcolor{red}{ 2}}+1/1245\,_{\textcolor{red}{ 15}}+1/1660\,_{\textcolor{red}{20}}}$\ $3$ & $7$ & $4$ & $2$ & $1$ & $\mathbf{\textcolor{red}{ 1}}$& $48$ & $ \mathbf {2/89\mathit{_a}=1/48+1/1068\,_{\textcolor{red}{ 12}}+1/2136\,_{\textcolor{red}{ 24}}+1/4272\,_{\textcolor{red}{ 48}}}$\ $15$ & $31$ & $20$ & $6$ & $5$ & $\mathbf{\textcolor{red}{ 1}}$& $60$ & $ \mathbf {2/89\mathit{_b}=1/60+1/267\,_{\textcolor{red}{ 3}}+1/890\,_{\textcolor{red}{ 5}}+1/1068\,_{\textcolor{red}{12}}}$\ $6$ & $13$ & $9$ & $3 $ & $1$ & $\mathbf{\textcolor{red}{ 2}}$& $18$ & $\mathbf {2/23=1/18+1/46\,_{\textcolor{red}{ 2}}+1/138\,_{\textcolor{red}{ 6}}+1/414\,_{\textcolor{red}{18}}}$\ $5$ & $11$ & $5$ & $4$ & $2$ & $\mathbf{\textcolor{red}{ 2}}$& $20$ & $ \mathbf {2/29=1/20+1/116\,_{\textcolor{red}{ 4}}+1/145\,_{\textcolor{red}{ 5}}+1/290\,_{\textcolor{red}{ 10}}}$\ $6$ & $13$ & $7$ & $4$ & $2$ & $\mathbf{\textcolor{red}{ 2}}$& $28$ & $ \mathbf {2/43=1/28+1/172\,_{\textcolor{red}{ 4}}+1/301\,_{\textcolor{red}{ 7}}+1/602\,_{\textcolor{red}{ 14}}}$\ $8$ & $17$ & $13$ & $3$ & $1$ & $\mathbf{\textcolor{red}{ 2}}$& $39$ & $\mathbf {2/61=1/39+1/183\,_{\textcolor{red}{ 3}}+1/793\,_{\textcolor{red}{ 13}}+1/2379\,_{\textcolor{red}{ 39}}}$\ $5$ & $11$ & $7$ & $3$ & $1$ & $\mathbf{\textcolor{red}{ 2}}$& $42$ & $ \mathbf {2/73\mathit{_a}=1/42+1/438\,_{\textcolor{red}{ 6}}+1/1022\,_{\textcolor{red}{ 14}}+1/3066\,_{\textcolor{red}{ 42}}}$\ $8$ & $17$ & $9$ & $5$ & $3$ & $\mathbf{\textcolor{red}{ 2}}$& $45$ & $ \mathbf {2/73\mathit{_b}=1/45+1/365\,_{\textcolor{red}{ 5}}+1/657\,_{\textcolor{red}{ 9}}+1/1095\,_{\textcolor{red}{ 15}}}$\ $18$ & $37$ & $15$ & $12$ & $10$ & $\mathbf{\textcolor{red}{ 2}}$& $60$ & $ \mathbf {2/83=1/60+1/332\,_{\textcolor{red}{ 4}}+1/415\,_{\textcolor{red}{ 5}}+1/498\,_{\textcolor{red}{6}}}\;\, ^{Eg}$\ $18$ & $37$ & $21$ & $9$ & $7$ & $\mathbf{\textcolor{red}{ 2}}$& $63$ & $ \mathbf {2/89=1/63+1/267\,_{\textcolor{red}{ 3}}+1/623\,_{\textcolor{red}{ 7}}+1/801\,_{\textcolor{red}{9}}}$\ $8$ & $17$ & $10$ & $5$ & $2$ & $\mathbf{\textcolor{red}{ 3}}$& $20$ & $\mathbf {2/23=1/20+1/46\,_{\textcolor{red}{ 2}}+1/92\,_{\textcolor{red}{ 4}}+1/230\,_{\textcolor{red}{10}}}$\ $8$ & $17$ & $10$ & $5$ & $2$ & $\mathbf{\textcolor{red}{ 3}}$& $30$ & $ \mathbf {2/43\mathit{_a}=1/30+1/129\,_{\textcolor{red}{ 3}}+1/258\,_{\textcolor{red}{ 6}}+1/645\,_{\textcolor{red}{15}}}$\ $10$ & $21$ & $16$ & $4$ & $1$ & $\mathbf{\textcolor{red}{ 3}}$& $32$ & $ \mathbf {2/43\mathit{_b}=1/32+1/86\,_{\textcolor{red}{ 2}}+1/344\,_{\textcolor{red}{ 8}}+1/1376\,_{\textcolor{red}{32}}}$\ $5$ & $11$ & $6$ & $4$ & $1$ & $\mathbf{\textcolor{red}{ 3}}$& $36$ & $\mathbf {2/61\mathit{_a}=1/36+1/366\,_{\textcolor{red}{ 6}}+1/549\,_{\textcolor{red}{ 9}}+1/2196\,_{\textcolor{red}{ 36}}}$\ $11$ & $23$ & $14$ & $6$ & $3$ & $\mathbf{\textcolor{red}{ 3}}$& $42$ & $ \mathbf {2/61\mathit{_b}=1/42+1/183\,_{\textcolor{red}{ 3}}+1/427\,_{\textcolor{red}{ 7}}+1/854\,_{\textcolor{red}{14}}}$\ $13$ & $27$ & $22$ & $4$ & $1$ & $\mathbf{\textcolor{red}{ 3}}$& $44$ & $ \mathbf {2/61\mathit{_c}=1/44+1/122\,_{\textcolor{red}{ 2}}+1/671\,_{\textcolor{red}{ 11}}+1/2684\,_{\textcolor{red}{44}}}$\ $15$ & $31$ & $26$ & $4$ & $1$ & $\mathbf{\textcolor{red}{ 3}}$& $52$ & $ \mathbf {2/73\mathit{_a}=1/52+1/146\,_{\textcolor{red}{ 2}}+1/949\,_{\textcolor{red}{ 13}}+1/3796\,_{\textcolor{red}{ 52}}}$\ $19$ & $39$ & $28$ & $7$ & $4$ & $\mathbf{\textcolor{red}{ 3}}$& $56$ & $ \mathbf {2/73\mathit{_b}=1/56+1/146\,_{\textcolor{red}{ 2}}+1/584\,_{\textcolor{red}{ 8}}+1/1022\,_{\textcolor{red}{14}}}$\ $23$ & $47$ & $20$ & $15$ & $12$ & $\mathbf{\textcolor{red}{ 3}}$& $60$ & $ \mathbf {2/73\mathit{_c}=1/60+1/219\,_{\textcolor{red}{ 3}}+1/292\,_{\textcolor{red}{ 4}}+1/365\,_{\textcolor{red}{5}}}\;\, ^{Eg}$\ $8$ & $17$ & $12$ & $4$ & $1$ & $\mathbf{\textcolor{red}{ 3}}$& $48$ & $ \mathbf {2/79\mathit{_a}=1/48+1/316\,_{\textcolor{red}{ 4}}+1/948\,_{\textcolor{red}{ 12}}+1/3792\,_{\textcolor{red}{ 48}}}$\ $8$ & $17$ & $8$ & $6$ & $3$ & $\mathbf{\textcolor{red}{ 3}}$& $48$ & $ \mathbf {2/79\mathit{_b}=1/48+1/474\,_{\textcolor{red}{ 6}}+1/632\,_{\textcolor{red}{ 8}}+1/1264\,_{\textcolor{red}{ 16}}}$\ $16$ & $33$ & $28$ & $4$ & $1$ & $\mathbf{\textcolor{red}{ 3}}$& $56$ & $ \mathbf {2/79\mathit{_c}=1/56+1/158\,_{\textcolor{red}{ 2}}+1/1106\,_{\textcolor{red}{ 14}}+1/4424\,_{\textcolor{red}{ 56}}}$\ $6$ & $13$ & $8$ & $4$ & $1$ & $\mathbf{\textcolor{red}{ 3}}$& $48$ & $ \mathbf {2/83\mathit{_a}=1/48+1/498\,_{\textcolor{red}{ 6}}+1/996\,_{\textcolor{red}{ 12}}+1/3984\,_{\textcolor{red}{ 48}}}$\ $8$ & $17$ & $10$ & $5$ & $2$ & $\mathbf{\textcolor{red}{ 3}}$& $50$ & $ \mathbf {2/83\mathit{_b}=1/50+1/415\,_{\textcolor{red}{ 5}}+1/830\,_{\textcolor{red}{ 10}}+1/2075\,_{\textcolor{red}{ 25}}}$\ $18$ & $37$ & $30$ & $5$ & $2$ & $\mathbf{\textcolor{red}{ 3}}$& $60$ & $ \mathbf {2/83\mathit{_c}=1/60+1/166\,_{\textcolor{red}{ 2}}+1/996\,_{\textcolor{red}{ 12}}+1/2490\,_{\textcolor{red}{30}}}$\ $15$ & $31$ & $15$ & $10$ & $6$ & $\mathbf{\textcolor{red}{ 4}}$& $30$ & $ \mathbf {2/29=1/30+1/58\,_{\textcolor{red}{ 2}}+1/87\,_{\textcolor{red}{ 3}}+1/145\,_{\textcolor{red}{5}}}$\ $14$ & $29$ & $15$ & $9$ & $5$ & $\mathbf{\textcolor{red}{ 4}}$& $45$ & $ \mathbf {2/61=1/45+1/183\,_{\textcolor{red}{ 3}}+1/305\,_{\textcolor{red}{ 5}}+1/549\,_{\textcolor{red}{9}}}$\ $17$ & $35$ & $27$ & $6$ & $2$ & $\mathbf{\textcolor{red}{ 4}}$& $54$ & $ \mathbf {2/73=1/54+1/146\,_{\textcolor{red}{ 2}}+1/657\,_{\textcolor{red}{ 9}}+1/1971\,_{\textcolor{red}{ 27}}}$\ $15$ & $31$ & $15$ & $10$ & $6$ & $\mathbf{\textcolor{red}{ 4}}$& $60$ & $ \mathbf {2/89=1/60+1/356\,_{\textcolor{red}{ 4}}+1/534\,_{\textcolor{red}{ 6}}+1/890\,_{\textcolor{red}{10}}}\;\, ^{Eg}$\ $9$ & $19$ & $12$ & $6 $ & $1 $ & $\mathbf{\textcolor{red}{ 5}}$& $24$ & $\mathbf {2/29=1/24+1/58\,_{\textcolor{red}{ 2}}+1/116\,_{\textcolor{red}{ 4}}+1/696\,_{\textcolor{red}{24}}}$\ $8$ & $17$ & $10$ & $6$ & $1$ & $\mathbf{\textcolor{red}{ 5}}$& $30$ & $ \mathbf {2/43=1/30+1/129\,_{\textcolor{red}{ 3}}+1/215\,_{\textcolor{red}{ 5}}+1/1290\,_{\textcolor{red}{30}}}$\ $11$ & $23$ & $14$ & $7$ & $2$ & $\mathbf{\textcolor{red}{ 5}}$& $42$ & $ \mathbf {2/61\mathit{_a}=1/42+1/183\,_{\textcolor{red}{ 3}}+1/366\,_{\textcolor{red}{ 6}}+1/1281\,_{\textcolor{red}{21}}}$\ $17$ & $35$ & $24$ & $8$ & $3$ & $\mathbf{\textcolor{red}{ 5}}$& $48$ & $ \mathbf {2/61\mathit{_b}=1/48+1/122\,_{\textcolor{red}{ 2}}+1/366\,_{\textcolor{red}{ 6}}+1/976\,_{\textcolor{red}{16}}}$\ $11$ & $23$ & $16$ & $6$ & $1$ & $\mathbf{\textcolor{red}{ 5}}$& $48$ & $ \mathbf {2/73\mathit{_a}=1/48+1/219\,_{\textcolor{red}{ 3}}+1/584\,_{\textcolor{red}{ 8}}+1/3504\,_{\textcolor{red}{ 48}}}$\ $11$ & $23$ & $12$ & $8$ & $3$ & $\mathbf{\textcolor{red}{ 5}}$& $48$ & $ \mathbf {2/73\mathit{_b}=1/48+1/292\,_{\textcolor{red}{ 4}}+1/438\,_{\textcolor{red}{ 6}}+1/1168\,_{\textcolor{red}{ 16}}}$\ $12$ & $25$ & $18$ & $6$ & $1$ & $\mathbf{\textcolor{red}{ 5}}$& $54$ & $ \mathbf {2/83\mathit{_a}=1/54+1/249\,_{\textcolor{red}{ 3}}+1/747\,_{\textcolor{red}{ 9}}+1/4482\,_{\textcolor{red}{ 54}}}$\ $18$ & $37$ & $30$ & $6$ & $1$ & $\mathbf{\textcolor{red}{ 5}}$& $60$ & $ \mathbf {2/83\mathit{_b}=1/60+1/166\,_{\textcolor{red}{ 2}}+1/830\,_{\textcolor{red}{ 10}}+1/4980\,_{\textcolor{red}{60}}}$\ $11$ & $23$ & $14$ & $7$ & $2$ & $\mathbf{\textcolor{red}{ 5}}$& $56$ & $ \mathbf {2/89=1/56+1/356\,_{\textcolor{red}{ 4}}+1/712\,_{\textcolor{red}{ 8}}+1/2492\,_{\textcolor{red}{ 28}}}$\ $14$ & $29$ & $18$ & $9$ & $2$ & $\mathbf{\textcolor{red}{ 7}}$& $36$ & $ \mathbf {2/43=1/36+1/86\,_{\textcolor{red}{ 2}}+1/172\,_{\textcolor{red}{ 4}}+1/774\,_{\textcolor{red}{18}}}$\ $9$ & $19$ & $10$ & $8$ & $1$ & $\mathbf{\textcolor{red}{ 7}}$& $40$ & $\mathbf {2/61=1/40+1/244\,_{\textcolor{red}{ 4}}+1/305\,_{\textcolor{red}{ 5}}+1/2440\,_{\textcolor{red}{ 40}}}$\ $23$ & $47$ & $30$ & $12$ & $5$ & $\mathbf{\textcolor{red}{ 7}}$& $60$ & $ \mathbf {2/73=1/60+1/146\,_{\textcolor{red}{ 2}}+1/365\,_{\textcolor{red}{ 5}}+1/876\,_{\textcolor{red}{12}}}$\ $14$ & $29$ & $18$ & $9$ & $2$ & $\mathbf{\textcolor{red}{ 7}}$& $54$ & $ \mathbf {2/79=1/54+1/237\,_{\textcolor{red}{ 3}}+1/474\,_{\textcolor{red}{ 6}}+1/2133\,_{\textcolor{red}{ 27}}}$\ $18$ & $37$ & $20$ & $12$ & $5$ & $\mathbf{\textcolor{red}{ 7}}$& $60$ & $ \mathbf {2/83=1/60+1/249\,_{\textcolor{red}{ 3}}+1/415\,_{\textcolor{red}{ 5}}+1/996\,_{\textcolor{red}{12}}}$\ $11$ & $23$ & $14$ & $8$ & $1$ & $\mathbf{\textcolor{red}{ 7}}$& $56$ & $ \mathbf {2/89=1/56+1/356\,_{\textcolor{red}{ 4}}+1/623\,_{\textcolor{red}{ 7}}+1/4984\,_{\textcolor{red}{ 56}}}$\ $20$ & $41$ & $21$ & $14$ & $6$ & $\mathbf{\textcolor{red}{ 8}}$& $42$ & $ \mathbf {2/43=1/42+1/86\,_{\textcolor{red}{ 2}}+1/129\,_{\textcolor{red}{ 3}}+1/301\,_{\textcolor{red}{7}}}\;\, ^{Eg}$\ $15$ & $31$ & $15$ & $12$ & $4$ & $\mathbf{\textcolor{red}{ 8}}$& $60$ & $ \mathbf {2/89=1/60+1/356\,_{\textcolor{red}{ 4}}+1/445\,_{\textcolor{red}{ 5}}+1/1335\,_{\textcolor{red}{ 15}}}$\ $21$ & $43$ & $26$ & $13$ & $4$ & $\mathbf{\textcolor{red}{ 9}}$& $52$ & $ \mathbf {2/61=1/52+1/122\,_{\textcolor{red}{ 2}}+1/244\,_{\textcolor{red}{ 4}}+1/793\,_{\textcolor{red}{13}}}$\ $20$ & $41$ & $30$ & $10$ & $1$ & $\mathbf{\textcolor{red}{ 9}}$& $60$ & $ \mathbf {2/79\mathit{_a}=1/60+1/158\,_{\textcolor{red}{ 2}}+1/474\,_{\textcolor{red}{ 6}}+1/4740\,_{\textcolor{red}{ 60}}}$\ $20$ & $41$ & $20$ & $15$ & $6$ & $\mathbf{\textcolor{red}{ 9}}$& $60$ & $ \mathbf {2/79\mathit{_b}=1/60+1/237\,_{\textcolor{red}{ 3}}+1/316\,_{\textcolor{red}{ 4}}+1/790\,_{\textcolor{red}{10}}}\;\, ^{Eg}$\ $15$ & $31$ & $20$ & $10$ & $1$ & $\mathbf{\textcolor{red}{ 9}}$& $60$ & $ \mathbf {2/89=1/60+1/267\,_{\textcolor{red}{ 3}}+1/534\,_{\textcolor{red}{ 6}}+1/5340\,_{\textcolor{red}{ 60}}}$\ $25$ & $51$ & $35$ & $14$ & $2$ & $\mathbf{\textcolor{red}{ 12}}$& $70$ & $ \mathbf {2/89=1/70+1/178\,_{\textcolor{red}{ 2}}+1/445\,_{\textcolor{red}{ 5}}+1/3115\,_{\textcolor{red}{35}}}$\ $23$ & $47$ & $30$ & $15$ & $2$ & $\mathbf{\textcolor{red}{13}}$& $60$ & $ \mathbf {2/73=1/60+1/146\,_{\textcolor{red}{ 2}}+1/292\,_{\textcolor{red}{ 4}}+1/2190\,_{\textcolor{red}{30}}}$\ $18$ & $37$ & $20$ & $15$ & $2$ & $\mathbf{\textcolor{red}{ 13}}$& $60$ & $ \mathbf {2/83=1/60+1/249\,_{\textcolor{red}{ 3}}+1/332\,_{\textcolor{red}{ 4}}+1/2490\,_{\textcolor{red}{30}}}$\ $24$ & $49$ & $32$ & $16$ & $1$ & $\mathbf{\textcolor{red}{ 15}}$& $64$ & $ \mathbf {2/79=1/64+1/158\,_{\textcolor{red}{ 2}}+1/316\,_{\textcolor{red}{ 4}}+1/5056\,_{\textcolor{red}{ 64}}}$\ $26$ & $53$ & $34$ & $17$ & $2$ & $\mathbf{\textcolor{red}{ 15}}$& $68$ & $ \mathbf {2/83=1/68+1/166\,_{\textcolor{red}{ 2}}+1/332\,_{\textcolor{red}{ 4}}+1/2822\,_{\textcolor{red}{34}}}$\ $23$ & $47$ & $27$ & $18$ & $2$ & $\mathbf{\textcolor{red}{ 16}}$& $54$ & $ \mathbf {2/61=1/54+1/122\,_{\textcolor{red}{ 2}}+1/183\,_{\textcolor{red}{ 3}}+1/1647\,_{\textcolor{red}{27}}}$\ $27$ & $55$ & $36$ & $18$ & $1$ & $\mathbf{\textcolor{red}{ 17}}$& $72$ & $ \mathbf {2/89=1/72+1/178\,_{\textcolor{red}{ 2}}+1/356\,_{\textcolor{red}{ 4}}+1/6408\,_{\textcolor{red}{72}}}$\ $30$ & $61$ & $36$ & $24$ & $1$ & $\mathbf{\textcolor{red}{ 23}}$& $72$ & $ \mathbf {2/83=1/72+1/166\,_{\textcolor{red}{ 2}}+1/249\,_{\textcolor{red}{ 3}}+1/5976\,_{\textcolor{red}{72}}}$\ $33$ & $67$ & $39$ & $26$ & $2$ & $\mathbf{\textcolor{red}{ 24}}$& $78$ & $ \mathbf {2/89=1/78+1/178\,_{\textcolor{red}{ 2}}+1/267\,_{\textcolor{red}{ 3}}+1/3471\,_{\textcolor{red}{39}}}$\ \[Complete4Terms\] Table \[Complete4Terms\] shown above is only as an indication for us and, certainly, was not calculated in its entirety. [2/23]{} has been reported only for memory because it was solved at the end of Sect. \[ThreeTerms\].\ With their experience related to 3-terms series, cut-off beyond $10$ has been applied by the scribes. Indeed all cases (here 7) may support this cut-off without any exception. Table \[Complete4Terms\] becomes:\ [\|l\|c\|l\|\|l\|\|l\|\|c\|l\|l\|]{}\ $n$ & $2n+1$ & $d_2$ & $d_3$ & $d_4$ &$\textcolor{red}{\Delta_{d}^{'}}$ &$D_1^n$ & Possible \[4-terms\] decompositions $\mathbf {\textcolor{red}{ m_4\leq 10}}$\ $9$ & $19$ & $12$ & $4 $ & $3 $ & $\mathbf{\textcolor{red}{ 1}}$& $24$ & $\mathbf {2/29=1/24+1/58\,_{\textcolor{red}{ 2}}+1/174\,_{\textcolor{red}{ 6}}+1/232\,_{\textcolor{red}{8}}}\;\, ^{Eg}$\ $9$ & $19$ & $10$ & $5$ & $4$ & $\mathbf{\textcolor{red}{ 1}}$& $40$ & $\mathbf {2/61=1/40+1/244\,_{\textcolor{red}{ 4}}+1/488\,_{\textcolor{red}{ 8}}+1/610\,_{\textcolor{red}{ 10}}}\;\, ^{Eg}$\ $14$ & $29$ & $14$ & $8$ & $7$ & $\mathbf{\textcolor{red}{ 1}}$& $56$ & $ \mathbf {2/83=1/56+1/332\,_{\textcolor{red}{ 4}}+1/581\,_{\textcolor{red}{ 7}}+1/664\,_{\textcolor{red}{ 8}}}$\ $5$ & $11$ & $5$ & $4$ & $2$ & $\mathbf{\textcolor{red}{ 2}}$& $20$ & $ \mathbf {2/29=1/20+1/116\,_{\textcolor{red}{ 4}}+1/145\,_{\textcolor{red}{ 5}}+1/290\,_{\textcolor{red}{ 10}}}$\ $18$ & $37$ & $15$ & $12$ & $10$ & $\mathbf{\textcolor{red}{ 2}}$& $60$ & $ \mathbf {2/83=1/60+1/332\,_{\textcolor{red}{ 4}}+1/415\,_{\textcolor{red}{ 5}}+1/498\,_{\textcolor{red}{6}}}\;\, ^{Eg{\textcolor{red}{{\star }}}}$\ $18$ & $37$ & $21$ & $9$ & $7$ & $\mathbf{\textcolor{red}{ 2}}$& $63$ & $ \mathbf {2/89=1/63+1/267\,_{\textcolor{red}{ 3}}+1/623\,_{\textcolor{red}{ 7}}+1/801\,_{\textcolor{red}{9}}}$\ $8$ & $17$ & $10$ & $5$ & $2$ & $\mathbf{\textcolor{red}{ 3}}$& $20$ & $\mathbf {\cancel{2/23}=1/20+1/46\,_{\textcolor{red}{ 2}}+1/92\,_{\textcolor{red}{ 4}}+1/230\,_{\textcolor{red}{10}}}$\ $23$ & $47$ & $20$ & $15$ & $12$ & $\mathbf{\textcolor{red}{ 3}}$& $60$ & $ \mathbf {2/73=1/60+1/219\,_{\textcolor{red}{ 3}}+1/292\,_{\textcolor{red}{ 4}}+1/365\,_{\textcolor{red}{5}}}\;\, ^{Eg{\textcolor{red}{{\star }}}}$\ $15$ & $31$ & $15$ & $10$ & $6$ & $\mathbf{\textcolor{red}{ 4}}$& $30$ & $ \mathbf {2/29=1/30+1/58\,_{\textcolor{red}{ 2}}+1/87\,_{\textcolor{red}{ 3}}+1/145\,_{\textcolor{red}{5}}}$\ $14$ & $29$ & $15$ & $9$ & $5$ & $\mathbf{\textcolor{red}{ 4}}$& $45$ & $ \mathbf {2/61=1/45+1/183\,_{\textcolor{red}{ 3}}+1/305\,_{\textcolor{red}{ 5}}+1/549\,_{\textcolor{red}{9}}}$\ $15$ & $31$ & $15$ & $10$ & $6$ & $\mathbf{\textcolor{red}{ 4}}$& $60$ & $ \mathbf {2/89=1/60+1/356\,_{\textcolor{red}{ 4}}+1/534\,_{\textcolor{red}{ 6}}+1/890\,_{\textcolor{red}{10}}}\;\, ^{Eg}$\ $20$ & $41$ & $21$ & $14$ & $6$ & $\mathbf{\textcolor{red}{ 8}}$& $42$ & $ \mathbf {2/43=1/42+1/86\,_{\textcolor{red}{ 2}}+1/129\,_{\textcolor{red}{ 3}}+1/301\,_{\textcolor{red}{7}}}\;\, ^{Eg}$\ $20$ & $41$ & $20$ & $15$ & $6$ & $\mathbf{\textcolor{red}{ 9}}$& $60$ & $ \mathbf {2/79=1/60+1/237\,_{\textcolor{red}{ 3}}+1/316\,_{\textcolor{red}{ 4}}+1/790\,_{\textcolor{red}{10}}}\;\, ^{Eg}$\ \[4TERMSOPT\] We follow the same way as for the \[3-terms\] series with slightly different subsets. That yields:\ ------ -------- ------- ------- ------- ----------------------------------- --------- ------------------------------------------------------------------------------------------------------------------------------------------------------- $n$ $2n+1$ $d_2$ $d_3$ $d_4$ $\textcolor{red}{\Delta_{d}^{'}}$ $D_1^n$ $\qquad \quad$\[4-terms\] decompositions $23$ $47$ $20$ $15$ $12$ $\mathbf{\textcolor{red}{ 3}}$ $60$ $ \mathbf {2/73=1/60+1/219\,_{\textcolor{red}{ 3}}+1/292\,_{\textcolor{red}{ 4}}+1/365\,_{\textcolor{red}{5}}}\;\, ^{Eg{\textcolor{red}{{\star }}}}$ $20$ $41$ $21$ $14$ $6$ $\mathbf{\textcolor{red}{ 8}}$ $42$ $ \mathbf {2/43=1/42+1/86\,_{\textcolor{red}{ 2}}+1/129\,_{\textcolor{red}{ 3}}+1/301\,_{\textcolor{red}{7}}}\;\, ^{Eg}$ $20$ $41$ $20$ $15$ $6$ $\mathbf{\textcolor{red}{ 9}}$ $60$ $ \mathbf {2/79=1/60+1/237\,_{\textcolor{red}{ 3}}+1/316\,_{\textcolor{red}{ 4}}+1/790\,_{\textcolor{red}{10}}}\;\, ^{Eg}$ ------ -------- ------- ------- ------- ----------------------------------- --------- ------------------------------------------------------------------------------------------------------------------------------------------------------- : A single or two different $\Delta_{d}^{'}$ \[4-terms\] \ ------ -------- ------- ------- ------- ----------------------------------- --------- ---------------------------------------------------------------------------------------------------------------------------- $n$ $2n+1$ $d_2$ $d_3$ $d_4$ $\textcolor{red}{\Delta_{d}^{'}}$ $D_1^n$ $\qquad \quad$\[4-terms\] decompositions $9$ $19$ $12$ $4 $ $3 $ $\mathbf{\textcolor{red}{ 1}}$ $24$ $\mathbf {2/29=1/24+1/58\,_{\textcolor{red}{ 2}}+1/174\,_{\textcolor{red}{ 6}}+1/232\,_{\textcolor{red}{8}}}\;\, ^{Eg}$ $5$ $11$ $5$ $4$ $2$ $\mathbf{\textcolor{red}{ 2}}$ $20$ $ \mathbf {2/29=1/20+1/116\,_{\textcolor{red}{ 4}}+1/145\,_{\textcolor{red}{ 5}}+1/290\,_{\textcolor{red}{ 10}}}$ $15$ $31$ $15$ $10$ $6$ $\mathbf{\textcolor{red}{ 4}}$ $30$ $ \mathbf {2/29=1/30+1/58\,_{\textcolor{red}{ 2}}+1/87\,_{\textcolor{red}{ 3}}+1/145\,_{\textcolor{red}{5}}}$ $9$ $19$ $10$ $5$ $4$ $\mathbf{\textcolor{red}{ 1}}$ $40$ $\mathbf {2/61=1/40+1/244\,_{\textcolor{red}{ 4}}+1/488\,_{\textcolor{red}{ 8}}+1/610\,_{\textcolor{red}{ 10}}}\;\, ^{Eg}$ $14$ $29$ $15$ $9$ $5$ $\mathbf{\textcolor{red}{ 4}}$ $45$ $ \mathbf {2/61=1/45+1/183\,_{\textcolor{red}{ 3}}+1/305\,_{\textcolor{red}{ 5}}+1/549\,_{\textcolor{red}{9}}}$ ------ -------- ------- ------- ------- ----------------------------------- --------- ---------------------------------------------------------------------------------------------------------------------------- : A single or two different $\Delta_{d}^{'}$ \[4-terms\] \ ------ -------- ------- ------- ------- ----------------------------------- --------- ------------------------------------------------------------------------------------------------------------------------------------------------------ $n$ $2n+1$ $d_2$ $d_3$ $d_4$ $\textcolor{red}{\Delta_{d}^{'}}$ $D_1^n$ $\qquad \quad$\[4-terms\] decompositions $14$ $29$ $14$ $8$ $7$ $\mathbf{\textcolor{red}{ 1}}$ $56$ $ \mathbf {2/83=1/56+1/332\,_{\textcolor{red}{ 4}}+1/581\,_{\textcolor{red}{ 7}}+1/664\,_{\textcolor{red}{ 8}}}$ $18$ $37$ $15$ $12$ $10$ $\mathbf{\textcolor{red}{ 2}}$ $60$ $ \mathbf {2/83=1/60+1/332\,_{\textcolor{red}{ 4}}+1/415\,_{\textcolor{red}{ 5}}+1/498\,_{\textcolor{red}{6}}}\;\, ^{Eg{\textcolor{red}{{\star }}}}$ ------ -------- ------- ------- ------- ----------------------------------- --------- ------------------------------------------------------------------------------------------------------------------------------------------------------ : A single or two different $\Delta_{d}^{'}$ \[4-terms\] \ ------ -------- ------- ------- ------- ----------------------------------- --------- ---------------------------------------------------------------------------------------------------------------------------- $n$ $2n+1$ $d_2$ $d_3$ $d_4$ $\textcolor{red}{\Delta_{d}^{'}}$ $D_1^n$ $\qquad \quad$\[4-terms\] decompositions $18$ $37$ $21$ $9$ $7$ $\mathbf{\textcolor{red}{ 2}}$ $63$ $ \mathbf {2/89=1/63+1/267\,_{\textcolor{red}{ 3}}+1/623\,_{\textcolor{red}{ 7}}+1/801\,_{\textcolor{red}{9}}}$ $15$ $31$ $15$ $10$ $6$ $\mathbf{\textcolor{red}{ 4}}$ $60$ $ \mathbf {2/89=1/60+1/356\,_{\textcolor{red}{ 4}}+1/534\,_{\textcolor{red}{ 6}}+1/890\,_{\textcolor{red}{10}}}\;\, ^{Eg}$ ------ -------- ------- ------- ------- ----------------------------------- --------- ---------------------------------------------------------------------------------------------------------------------------- : A single or two different $\Delta_{d}^{'}$ \[4-terms\] \[1Delta4\] We recall that any odd denominator $D_1$ could lead to a solution for \[3-terms\] decompositions as checked in tables \[3TERMSOPT\] or \[Frac3become4\]. Its occurrence arises only 2 times in table \[1Delta4\] \[4-terms\]. The first, for $2/61$, was dropped out because a ${\Delta_{d}^{'}}=4$ too high. The second one regards [2/89]{} (first row). Then, for a unifying sake and avoiding singularity, chief scribe decided to discard $D_1=63$ in this case.\ Remark that we are very far from assumptions of Gillings [[@Gillings]]{} about Egyptian preferences for even numbers instead of odd, regarding the denominators in general. Thus the [‘no odd precept’ ]{} was a low priority. At low ratio also (2 times only), this will be applied to the composite numbers $D$ [[@Brehamet]]{}. Conclusion ========== As we saw, the most recent analysis (2008) has been performed on the ‘$2/n$’ table by Abdulaziz [[@Abdulaziz]]{} (see his group $G_2$). It can be appreciated as a kind of mathematical anastylosis using materials issued from the RMP and other documentation. Ancient calculation procedure, using mainly fractions, is faithfully respected, but leads to arithmetical depth analyses of each divisor of $D_1$.\ Our global approach avoided the difficulties of sophisticated arithmetical studies. This provides the advantage of forgetting quickly some widespread ‘modern’ ideas about the topic.\ $\bullet$ No, the last denominator is not bounded by a fixed value of $1000$. It only depends on the ‘circumstances’ related to the value of $D$. For 3 or 4 terms, a limitation like $D_h \leq 10D$ is quite suitable, except only for [2/53]{} where $10$ is replaced by $15$. An observation well stressed in Ref. [[@Abdulaziz]]{}.\ $\bullet$ No requirement is found about the denominator $D_1$ as having to be the greatest if alternatives.\ $\bullet$ Once for all, a systematic predilection for even denominators does not need to be considered. Only once, we were forced to discard $D_1 = 63$ (odd) for deciding on [2/89]{}.\ $\bullet$ Of course, there is no theoretical formula that can give immediately the first denominator as a function of $D$. It must necessarily go through trials and few selection criteria. The simpler the better, like the $\Delta$-classification presented in this paper. Maybe is it this classification that induces the opportunity of a comprehensive approach ? Strictly speaking, there are no algorithms in the method, just tables and pertinent observation. This is how $2/23$, $2/29 $ or $2/53$ have found a logical explanation, more thorough than the arguments commonly supplied for these ‘singularities’.\ Find a simple logic according to which there is no singular case was the goal of the present paper.\ Perhaps, chronologically, the study of prime numbers has been elaborated before that of composite numbers. It is nothing more than an hypothesis consistent with the spirit of our study. Yes ancient scribes certainly have been able to calculate and analyze all the preliminary cases. Ultimately, our unconventional method allows to reconstruct the table fairly easily with weak mathematical assumptions, except maybe the new idea to consider as beneficial to have consecutive multipliers. Appendix A: why a boundary with a Top-flag? {#appendix-a-why-a-boundary-with-a-top-flag .unnumbered} =========================================== In this appendix, we continue to consider prime denominators $D$. For \[2-terms\] decompositions this concept of a Top-flag has no meaning since the last denominator is unique.\ Obviously, doubtless far from Egyptian concepts, there are another equations [more general]{} than\ Eqs. (\[eq:FEgypt3\]) or (\[eq:FEgypt4\]), namely $$\frac{2}{D}= \frac{1}{D_1}+ \frac{1}{m_2D}+ \frac{1}{m_3D}.$$ $$\frac{2}{D}= \frac{1}{D_1}+ \frac{1}{m_2D}+ \frac{1}{m_3D}+ \frac{1}{m_4D}.$$ We can imagine these as issued from another kind of unity decomposition like $$\mathbf{1}= \frac{D}{2D_1}+ \frac{1}{2m_2}+ \frac{1}{2m_3}.$$ $$\mathbf{1}= \frac{D}{2D_1}+ \frac{1}{2m_2}+ \frac{1}{2m_3}+ \frac{1}{2m_4}.$$ $D/2D_1$ remains in the lead of equality and $\mathbf{1}$ is a sum of terms, each with a even denominator .\ These (modern) equations have additional solutions of no use for the scribes .\ A priori the solutions are infinite, then for avoiding such a tedious research (today and in the past time), it is necessary to limit the highest denominator $D_h=m_h D$. How to do that ? Simply by defining a kind of ‘Top-flag’ $\mbox{\boldmath $\top$ }\!\!_ f^{\;\;[h]}$ such as $$D_h \leq D \mbox{\boldmath $\top$ }\!\!_ f^{\;\;[h]} .$$ Indeed, as soon as one decides to study a three-terms decomposition or more, it should be realized that an upper boundary for the last denominator has to be fixed. If not, the number of solutions becomes infinite \[countable\]. Recall that $m_2 < m_3 < m_4$ and $D_2 < D_3 < D_4$. Unfortunately (or not) the author of this paper has begun the calculations with a even more general problem, this of solving $$\frac{2}{D}= \sum _{i=1}^{h} \frac{1}{D_i}, \label{eq:EgyptGeneral}$$ without any criteria of multiplicity involving multipliers like $m_i$ ($i>2$).\ Certainly this was the reflex of Gillings [[[@Gillings]]{}]{} or Bruckheimer and Salomon [[[@BruckSalom]]{}]{}. The problem is solvable and the solutions available by means of a small computer. After a necessary arithmetical analysis, it can be found that $(h-1)$ sets of solutions exist. One with $(h-1)$ multipliers $m_i$, another with $(h-2)$ multipliers and so on. No solution exists if one searches for $D_i$ ($i\geq 2$) not multiple of $D$.\ Even a low-level programming code like [sb]{} can be used instead of [Fortran]{} to perform computations in a very acceptable speed. We quickly realized the necessity of stopping the calculations by using a limitation regarding the last highest denominator $D_h$. Whence the introduction of a Top-flag.\ Actually the Egyptian $2/D$ table shows a subset of more general solutions because the multipliers $m_i$ have a specific form involving $D_\mathbf1$ and some of its divisors $d_i$. For example out of this subset, you can find an unexpected \[4-terms\] solution for 2/23 with $\mbox{\boldmath $\top$ }\!\!_ f^{\;\;[4]}= 10$, namely -------------------------------------------------------------------------------------------------------- $2/23=1/15+1/115\,_{\textcolor{red}{ 5}}+1/138\,_{\textcolor{red}{ 6}}+1/230\,_{\textcolor{red}{ 10}}$ \[0.01in\] -------------------------------------------------------------------------------------------------------- .\ So, if we restrict ourself to retrieve Egyptian fractions given in the table, it naturally comes to mind to limit the highest denominator by an upper boundary: a convenient Top-flag.\ Excepted the Babylonian system example in base $60$, a numeration in base $10$ is rather universal, because of our two hands with each [5]{} fingers. It is of common sense that the selection was generally $\mbox{\boldmath $\top$ }\!\!_ f^{\;\;[h]}= 10 \;(=2\times \sf 5)$, not excluding a favorable appreciation for $\mbox{\boldmath $\top$ }\!\!_ f^{\;\;[3]}= 15 \;(=3\times \sf 5)$ as for [2/53]{}. [99]{} T. E. PEET: [The Rhind Mathematical Papyrus, British Museum 10057 and 10058]{},\ London: The University Press of Liverpool limited and Hodder - Stoughton limited (1923). A. B. CHACE; l. BULL; H. P. MANNING; and R. C. ARCHIBALD: [The Rhind Mathematical Papyrus,]{} Mathematical Association of America, [Vol.1 ]{}(1927), [Vol. 2 ]{}(1929), Oberlin, Ohio. G. ROBINS and C. SHUTE: [The Rhind Mathematical Papyrus: An Ancient Egyptian Text]{}, London: British Museum Publications Limited, (1987). \[A recent overview\]. R.J. GILLINGS: [Mathematics in the Time of Pharaohs]{}, MIT Press (1972), reprinted by Dover Publications (1982). M. BRUCKHEIMER and Y. SALOMON: [Some comments on R.J Gillings’s analysis of the 2/n table in the Rhind Papyrus]{}, Historia Mathematica, [Vol. 4]{}, pp. 445-452 (1977). A. IMHAUSEN and J. RITTER: [Mathematical fragments \[see fragment UC32159\]]{}. (2004).\ In: The UCL Lahun Papyri, [Vol. 2]{} , pp. 71-96. Archeopress, Oxford,\ Eds M. COLLIER, S. QUIRKE. A. ABDULAZIZ: [ On the Egyptian method of decomposing 2/n into unit fractions]{}, Historia Mathematica, [Vol. 35]{}, pp. 1-18 (2008). M. GARDNER: [Egyptian fractions:]{} Unit Fractions, Hekats and Wages - an Update (2013), available on the site of academia.edu. \[Herein can be found an historic of various researches about the subject\]. L. BREHAMET: [Remarks on the Egyptian 2/D table in favor of a global approach (D composite number)]{}, arXiv \[math.HO\], to be submitted. L. FIBONACCI: [Liber abaci]{} (1202). E.M. BRUINS: [The part in ancient Egyptian mathematics]{}, Centaurus, [Vol. 19]{}, pp. 241-251 (1975). [^1]: The creative flash of an inspired scholar (ancient or modern) is short. What is generally much longer is the development of the idea and achievement of tools (theoretical or practical) necessary for its application. Of course once the tools lapped their use takes little time! [^2]: Idea already suggested by Gillings [[@Gillings]]{} [^3]: All the Egyptian decompositions for composite numbers are analyzed in our second paper [[@Brehamet]]{} [^4]: It can be proved that no solution can be found beyond $n =(D-3)/2$.
--- abstract: 'We consider the dispersive Degasperis-Procesi equation $u_t-u_{x x t}-\mathtt{c} u_{xxx}+4 \mathtt{c} u_x-u u_{xxx}-3 u_x u_{xx}+4 u u_x=0$ with $\mathtt{c} \in \mathbb{R} \setminus \{0\}$. In [@Deg] the authors proved that this equation possesses infinitely many conserved quantities. We prove that there are infinitely many of such constants of motion which control the Sobolev norms and which are analytic in a neighborhood of the origin of the Sobolev space $H^s$ with $s \geq 2$, both on $\mathbb{R}$ and ${\mathbb{T}}$. By the analysis of these conserved quantities we deduce a result of global well-posedness for solutions with small initial data and we show that, on the circle, the formal Birkhoff normal form of the Degasperis-Procesi at any order is action-preserving.' author: - \| Roberto Feola$^{}$, Filippo Giuliani$^{\dag}$, Stefano Pasquali$^{\dag}$\ ${}^{}$ SISSA, Trieste, rfeola@sissa.it;\ ${}^\dag$ RomaTre, Roma, fgiuliani@mat.uniroma3.it, spasquali@mat.uniroma3.it[^1] title: \| On the integrability of Degasperis-Procesi equation:\ control of the Sobolev norms and Birkhoff resonances. --- Introduction ============ In $1999$ Degasperis and Procesi [@DegPro] applied a method of asymptotic integrability to the family of third-order dispersive PDE conservation laws $$\label{DPfamiglia} u_t+c_0 u_x+\gamma u_{xxx}-\alpha^2 u_{xx t}=(c_1 u^2+c_2 u_x^2+c_3 u u_{xx})_x,$$ where $\alpha, c_0, c_1, c_2, c_3$ are real constants and subindices denote the partial derivatives.\ Only three equations within this family result to satisfy the asymptotic integrability condition, the KdV equation ($\alpha=c_2=c_3=0$), the Camassa-Holm equation ($c_1=-3 c_3/2 \alpha^2, c_2=c_3/2$) and the Degaperis-Procesi equation $$\label{DPvera} u_t+c_0 u_x+\gamma u_{xxx}-\alpha^2 u_{xx t}=-\frac{4 c_2}{\alpha^2} u u_x+ 3 c_2 u_x u_{xx}+ c_2 u u_{xxx}.$$ In [@Deg] Degasperis-Holm-Hone showed that the equation is integrable by constructing its Lax pair. They also proposed a bi-Hamiltonian structure and a recursive method to generate infinitely many constants of motion (see Section $4$ in [@Deg]).\ Later Constantin and Lannes showed in [@ConstLannes] that the Degasperis-Procesi equation, as well as the Camassa-Holm equation, can be regarded as a model for nonlinear shallow water dynamics, and that it accomodates wave-breaking phenomena. We observe that the equation is a quasi-linear PDE, namely the linear and the nonlinear terms contain the same order of derivatives. We also remark that is not translation invariant and the linear dispersion (see for instance for the case $x\in{\mathbb{T}}$) is related to the chosen frame.\ By taking $c_0=-\gamma$ and $\alpha^2=1$ in the linearized equation at $u=0$ transforms into the following transport equation $$u_t=\gamma u_x.$$ Hence, in this case, all the solutions are travelling waves and there are no dispersive effects. In particular, by choosing $c_0=\gamma=0$, $c_2=1$ and $\alpha^2=1$, the equation can be transformed into the dispersionless form $$\label{dispersionless} u_t-u_{x x t}+ 4 u u_x=3 u_x u_{xx}+u u_{xxx}.$$ The family of equations is covariant under the group of transformations $$\label{boosts} u\mapsto \lambda u( t, \xi x+\eta t ) +\delta, \qquad \lambda, \xi, \eta, \delta\in\mathbb{R},$$ which are compositions of translations and Galilean boosts, and all the parametrized equations in can be obtained from by applying such changes of coordinates (see [@scolar]). As we said above, in order to consider the dispersive effects of we have to impose that $c_0\neq -\gamma$ and $c_0, \gamma\neq 0$; if we let the coefficient in front of $u_{xx t}$ be $-1$, we can obtain an equation with this feature from if and only if we apply a transformation of the form with $\delta\neq 0$, namely we have to consider translations of the variable $u$.\ We will consider the dispersive Degasperis-Procesi equation $$\label{DP} u_t-u_{x x t}-\mathtt{c} u_{xxx}+4 \mathtt{c} u_x- u u_{xxx}-3 u_x u_{xx}+4 u u_x=0,$$ obtained from by translating $u\mapsto u+\mathtt{c}$, for some real parameter $\mathtt{c}\neq 0$. We remark that the same equation can be obtained from by setting $ \alpha^2=1, \gamma=-\mathtt{c}, c_0=4 \mathtt{c}, c_2=c_3=1. $ We note that the mass $\int u \,dx$ is a constant of motion of the Degasperis-Procesi equation , hence the subsets of functions with fixed finite average $$\label{polpetta} \mathfrak{L}_{\mathtt{c}}:=\{ u : \int u\,dx=\mathtt{c}\}$$ are left invariant by the flow of . On the subspace $\mathfrak{L}_0$ it is possible to define a non-degenerate symplectic structure for which can be seen as a Hamiltonian PDE. Our purpose is to investigate the dynamics of on the invariant subsets $\mathfrak{L}_{\mathtt{c}}$ with $\mathtt{c}\neq 0$, which is equivalent to the one of when $u \in \mathfrak{L}_0$.\ The equation has been widely investigated by many authors since it presents wave breaking phenomena, peakon and soliton solutions and blow-up scenarios (see for instance [@ConstantinEscher], [@Escher], [@CH], [@stabpeak], [@Matsuno], [@Li], [@Liu], [@LiuYin], [@Deg] and [@globbreak]).\ Lundmark and Szmigielski [@Lund] presented an inverse scattering approach for computing n-peakon solutions to equation . Vakhnenko and Parkes [@Vak] investigated traveling wave solutions. Holm and Staley [@Ho] studied stability of solitons and peakons numerically.\ Regarding the well-posedness of the equation we cite the results of Yin [@YinR], [@YinT] on the local well-posedness with initial data $u_0\in H^s$, $s>3/2$, both on the line and on the circle, and the result by Himonas-Holliman [@HH], who showed that for the data-to-solution map is not uniformly continuous. We also mention the paper of Coclite-Karlsen [@Coclite] for the well-posedness in classes of discontinuous functions and Wu [@Wu] for the periodic generalized Degasperis-Procesi equation. Although the bi-Hamiltonian structure of equation provides an infinite number of conservation laws ([@Deg]), there is no way to find conservation laws controlling the $H^1$-norm of $u \in \mathfrak{L}_0$ ([@Escher]), which for instance represents an important difference with respect to the Camassa-Holm equation. We want to show that if we consider the dynamics on $\mathfrak{L}_{\mathtt{c}}$ with $\mathtt{c}\neq 0$, then the situation, close to the origin, is truly different. In this paper we construct infinitely many conserved quantities $K_n$ for the dispersive Degasperis-Procesi equation , starting from the ones proposed by Degasperis-Holm-Hone in [@Deg].\ We prove that the $K_n$’s are analytic functions in a ball centered at the origin of $H^{n+1}$ with radius depending on the parameter $\mathtt{c}$, but independent of $n$ (see ), such that - for any $s \geq 2$ there exists $n=n(s)\in\mathbb{N}$ such that $K_n$ controls the $H^s$-norm of solutions $u$ of with initial datum $u_0$ sufficiently close to $0$ in $H^s$, - the quadratic part of $K_n(u)$ is given by $$K_n^{(0)}=\int (\partial_x^{n-1} w)^2\,dx, \quad w:=u-u_{xx}$$ and so, if $x\in{\mathbb{T}}$, it reads $$K_n^{(0)}=\sum_{j\in\mathbb{Z}} (1+j^2)^2 \,j^{2 (n-1)}\,\lvert u_j \rvert^2.$$ These facts are actually proved in Theorem \[costantiMotoDP\], which is the main result of the paper.\ We provide also two applications of this result. The first one is a result of global in time well-posedness and stability in a neighborhood of the origin. - Fixed $s \geq 2$, there exists a ball centered at the origin $B_s$ of $H^s$ in which the equation is globally well-posed (see [@Tao]), namely any solution with initial datum belonging to $B_s$ exists for all times and, moreover, it remains inside a slightly bigger ball centered at the origin (hence the origin is a stable fixed point). This is the content of Theorem \[esistenzaglobale\], which in turn is due to $(F1)$.\ The use of the conserved quantities to prove existence results for solutions of integrable PDE’s is a classical argument. We mention for instance the celebrated result by Bona-Smith [@BonaSmith] on the initial value problem for the KdV equation.\ We also mention [@Luca], [@Visciglia], [@Visciglia2], in which the properties of conserved quantities of other integrable PDE’s, such as the DNLS and the Benjamin-Ono equation on ${\mathbb{T}}$, are used to construct functional Gibbs measures and study the long time existence of regular solutions. It would be interesting to investigate if similar results could be applied to equation .\ Another application of Theorem \[costantiMotoDP\] concerns the Birkhoff normal form analysis for the Degasperis-Procesi equation on the circle. First we briefly recall some facts and literature about Birkhoff normal form. The Birkhoff normal form theory for PDE’s has been widely developed since the 1990s, in order to extend to the infinite dimensional dynamical case the classical results which hold for finite dimensional “nearly” integrable Hamiltonian systems. Let us consider a Hamiltonian function $H(p,q)=H=H^{(0)}+P$, with $(p,q)\in {\mathbb{R}}^{2n}$ with $$\label{ham} H^{(0)}=\sum_{j=1}^{n}{\omega}_j\frac{p_j^{2}+q_{j}^{2}}{2},$$ and $P$ is a smooth function with a zero of order at least $3$ at the origin. The origin is an elliptic fixed point and the Hamiltonian system can be seen as a system of harmonic oscillators with frequencies $\omega_j$ coupled by the nonlinearities. Classical theory guarantees that, for any $N\geq1$, there is a real analytic and symplectic map $\Phi_{N}$ such that $$H\circ{\Phi_{N}}=H^{(0)}+{Z}_{N}+{R}_{N},$$ where ${R}_{N}$ is a function with a zero of order $N+3$ and ${Z}_{N}$ is a polynomial of degree $N+2$ which Poisson-commutes with $H^{(0)}$. In particular, if the frequencies ${\omega}_{j}$ are “non resonant”, namely $$\label{reso} \sum_{j=1}^{n}{\omega}_{j}k_{j}\neq 0, \quad \forall\; k\in \mathbb{Z}^{n}\setminus\{0\},$$ then ${Z}_{N}$ depends only on the actions $I_{j}=(p_{j}^{2}+q_{j}^{2})/2$. This implies that if the initial datum has size $\e\ll1$ then the solution remains in a neighborhood of radius $2\e$ for times of order $\e^{-(N+1)}$. The key idea to obtain such a result is to remove from the nonlinearity $P$ all the monomials which do not commutes with $H^{(0)}$. This can be done iteratively by means of symplectic transformations. More precisely, one uses a sequence of maps $\Phi_{p}$ generated as the Hamiltonian flow at time one of an auxiliary Hamiltonian $F_{p}$ of degree of homogeneity $p+2$. The $F_{p}$ is chosen in such a way the equation $$\label{homo} \{H^{(0)},F_p\}+G_{p}=Z$$ where $G_p$ is some homogeneous Hamiltonian of degree $p+2$ and $\{H^{(0)},Z\}=0$, is satisfied. It turns out that, in order to solve it , one needs some non-resonance conditions on the frequencies ${\omega}_{j}$, for instance the relation . In the case $n<\infty$ the number of monomials which have to be canceled out is finite. Many PDE’s (NLS, KdV, Klein-Gordon...) $u_t=L(u)+f(u)$, with $L$ possibly an unbounded linear operator and $f$ some nonlinear function, can be written, on compact manifolds, as Hamiltonian systems whose quadratic part has a form similar to with $n=\infty$.\ There are several difficulties in extending the theory to the infinite dimensional case: 1. one needs suitable non-resonance conditions which replace in infinite dimension; 2. one needs to cancel out an infinite number of monomials; 3. one needs to check that the normal form $Z_{N}$ is action-preserving; 4. the Birkhoff transformations are flows of possibly ill-posed PDEs. The first difficulty has been overcome in [@Bambu] and in [@BG]. To deal with the second one, a good definition of the class of formal polynomials on which one works is required. For instance, one at least needs that the Hamiltonian $F_p$ are continuos functions on the phase space. In the framework of Hamiltonian PDEs typical phase spaces are Sobolev spaces of functions (or weighted spaces of sequences). We mention [@DelortSzeft2] and [@DelortSzeft1], in which the authors introduce a suitable classes of multilinear forms.\ Concerning item $(iii)$, we say that the normal form $Z_k$ is action-preserving if it depends only on the actions $\lvert u_j \rvert^2$. This implies that the flow generated by the Hamiltonian function $Z_{k}$ leave the actions invariant. In the PDE context this means that the Sobolev norms $H^{s}$ are preserved for any $s>0$. Proving that the obtained normal form is action-preserving is a problem concerning the specific equation one is studying.\ Regarding item $(iv)$, if the nonlinearity depends upon some derivatives the Birkhoff transformations could be not well-defined. Especially for quasi-linear PDEs, the problem of constructing rigorous symplectic transformations in order to apply a Birkhoff normal form procedure is delicate.\ The Birkhoff normal form methods have been used by many authors to prove long time existence of solutions with data in a small neighborhood of a fixed point. We quote for instance the papers by Delort [@Delort-2009], [@Delort-sphere], in which the author studied quasi-linear perturbations of Klein-Gordon (K-G) and the recent paper by Berti-Delort [@BertiDelort], where the authors applied a suitable Birkhoff normal form procedure to the capillarity-gravity water waves (WW) equation. In the latter papers the linear frequencies depends on some “external” parameters (the “mass” for the K-G, the capillarity for the WW). This fact has been used in order to prove very strong non-resonance conditions (which hold for “most” values of parameters) and, as a consequence, that the normal forms are action-preserving.\ When the equation does not depend on physical parameters, the problem of showing the integrability of the normal form relies on an analysis of the algebraic structure of the resonances. We mention the paper by Craig-Worfolk [@CraigBF] in which the authors study (at a formal level) the Birkhoff normal form for pure-gravity water waves at order four. It is known that for this equation there are non trivial resonances (called “Benjamin-Feir”), which could prevent the normal forms to be action-preserving. Actually, they show that there are suitable cancellations in the coefficients of the Hamiltonian which allows to obtain an action preserving normal form.\ Our result focuses only on $(iii)$, by formulating the Birkhoff normal form procedure only at a formal level, and it is similar to the study performed in [@CraigBF], since we do not deal with $(i)$ , $(ii)$ and $(iv)$ (which concern convergence problem).\ The main differences are that: - we use the integrability of the equation in order to overcome the problem of non trivial resonances; - we are able to prove the integrability of the normal form at any order, since we exploit the algebraic structure of the resonances. More precisely we prove the following: - at a purely formal level, it is possible to put the Hamiltonian in a action-preserving Birkhoff normal form at any order. This result is achieved thanks to $(F2)$ and it is the content of Theorem \[teoBirk\]. One of the main applications of the Birkhoff normal form methods concerns the KAM theory for PDEs. It is well known that in order to apply perturbative arguments to construct periodic and quasi-periodic solutions for perturbed autonomous integrable equations one needs to control the frequencies of the expected invariant tori. In the infinite dimensional context, this requires the presence of parameters which modulate the frequencies, since the non-resonance conditions to be imposed are quite complicated. When the considered system does not present external parameters, one has to extract them from the equation itself: a way to do that is to perform a Birkhoff normal form procedure. Actually, the result presented in Theorem \[teoBirk\] has been motivated by the study of quasi-periodic solutions for perturbations of the Degasperis-Procesi equation [@FGP]. Preliminaries {#prelimi} ------------- In order to state the main result of the paper we introduce the Hamiltonian setting and the space of formal polynomials and power series. When there is no specification of the spatial domain we mean that the arguments hold for both cases $x\in {\mathbb{T}}$ or $x\in\mathbb{R}$. #### Hamiltonian setting. The equation can be formulated as a Hamiltonian PDE $u_t=J\,\nabla_{L^2} H(u)$, where $\nabla_{L^2} H$ is the $L^2$ gradient of the Hamiltonian $$\label{DPHamiltonian} H(u)=\int \frac{\mathtt{c}\,u^2}{2}-\frac{u^3}{6}\, dx.$$ The Hamiltonian is defined on the real phase space (recall ) $$H_0^1:=H^1\cap \mathfrak{L}_0$$ endowed with the non-degenerate symplectic form $$\label{SymplecticFormDP} \Omega(u, v):=\int (J^{-1} u)\,v\,dx, \quad \forall u, v\in H_0^1, \qquad J:=(1-\partial_{xx})^{-1}(4-\partial_{xx})\partial_x.$$ The Poisson bracket induced by $\Omega$ between two functions $F, G\colon H_0^1 {\rightarrow}\mathbb{R}$ is $$\label{PoissonBracketDP} \{ F(u), G(u) \}:=\Omega(X_F, X_G)=\int \nabla F(u)\,J \nabla G(u)\,dx,$$ where $X_F$ and $X_G$ are the vector fields associated to the Hamiltonians $F$ and $G$, respectively.\ On the circle the dispersion law of the Degasperis-Procesi equation is given by $$\label{dispersionLaw} j\mapsto\omega(j):=j\,\frac{4+j^2}{1+j^2}=j+\frac{3 j}{1+j^2}, \qquad j\in\mathbb{Z}\setminus\{0\},$$ where $\omega(j)$ are the linear frequencies of oscillations or the eigenvalues of the operator $J$ on the circle (see ). Let us define $$\label{emme} w:=(1-{\partial}_{xx})u, \quad m:=\mathtt{c}+u-u_{xx},\quad p=-m^{\frac{1}{3}}.$$ One can easily check that the functions $$\label{nonleho} \begin{aligned} &H(u)=\int \frac{\mathtt{c}\,u^2}{2}-\frac{u^3}{6}\, dx,\quad M_0(u)=\frac{1}{2}\int (J^{-1}u_x)u dx,\quad M_1(u)=\int m^{\frac{1}{3}}dx, \end{aligned}$$ are constant of motions for equation , i.e. if $u(t,x)$ solves then $$\label{pissonCOMMU} \frac{d}{dt}M_0(u)=\{ M_0, H\}(u)=0, \quad \frac{d}{dt}M_1(u)=\{ M_1, H\}(u)=0.$$ We will consider the Sobolev spaces $$\label{spazio} H^{s}(\mathbb{T};\mathbb{R}):= \big\{ u(x)\in H^1_0(\mathbb{T};\mathbb{R}) : \\|u\\|_{H^{s}}^2:=\sum_{j\in\mathbb{Z} \setminus \{0\} } \lvert u_{j} \rvert^2 \langle j \rangle^{2 s}<\infty, \,\,\overline{u}_j=u_{-j} \big\}$$ where $\langle j\rangle:=\sqrt{1+j^{2}}$ and $$\label{spazioR} H^{s}(\mathbb{R};\mathbb{R}):= \big\{ u(x)\in H^1_0(\mathbb{R};\mathbb{R}) : \\|u\\|_{H^{s}}^2:=\sum_{k=0}^s \int_{\mathbb{R}} (\partial_x^k u)^2\,dx \big\}.$$ We will denote both the spaces and with $H^s$ in Section \[costantiMotoPROOF\], since all the arguments hold independently by the $x$-space. We denote by $$\lvert u \rvert_{L^{\infty}}:=\sup_x \lvert u(x) \rvert$$ the $L^{\infty}$-norm either on $\mathbb{R}$ or on $\mathbb{T}$.\ Given a Banach space $(E, \lVert \cdot \rVert_E)$ and $r\geq 0$, we denote by $$B_{E}(v, r):=\{ u\in E : \lVert u-v \rVert_{E}< r \}$$ the open ball centered at $v\in E$ with radius $r$. #### Space of polynomials. When $x \in \mathbb{T}$ it is convenient to introduce a class of polynomials which describes the Hamiltonians in terms of their Fourier coefficients. These definitions will be used in Section \[proofteoBirk\].\ We use the multi-index notation $\alpha\in \mathbb{N}^{\mathbb{Z}}$, $\lvert \alpha \rvert:=\sum_j \alpha_j$. We define - the monomial associated to $\alpha$: $u^{\alpha}:=\prod_j u_j^{\alpha_j}.$ - the momentum associated to $\alpha$: $\mathcal{M}(\alpha)=\sum_j j\, \alpha_j$. - the divisor associated to $\alpha$ (recall the linear frequencies ): $\Omega(\alpha):=\sum_j \omega(j)\alpha_j$. We define the set of indices with zero momentum of order $n\in\mathbb{N}$ $$\mathcal{I}_n:=\{ \alpha\in\mathbb{N}^{\mathbb{Z}} : \lvert \alpha \rvert=n+2, \,\,\mathcal{M}(\alpha)=0 \}.$$ \[defpolinomi\] We say that $P:= \big( P_{\alpha} \big)_{\alpha\in\mathcal{I}_n}$, $P_{\alpha}\in\mathbb{C}$ for any $\alpha\in\mathcal{I}_n$, is a formal homogenous polynomial of degree $n+2$ and we write the (formal) expression $P(u)=\sum_{\alpha\in\mathcal{I}_n} P_{\alpha} u^{\alpha}$.\ We call $\mathscr{P}^{(n)}$ the space of the formal homogenous polynomial of degree $n$. \[defserie\] We define the space product $$\mathscr{F}:=\prod_{n\geq 0}\mathscr{P}^{(n)}.$$ If $P\in \mathscr{F}$ then we write the (formal) expression $P=\sum_{n=0}^{\infty} P^{(n)}$ where $P^{(n)}\in \mathscr{P}^{(n)}$. There exists a obvious inclusion of $\mathscr{P}^{(n)}$ into $\mathscr{F}$ given by $$(P_{\alpha})_{\alpha\in\mathcal{I}_n} \mapsto (\underbrace{0, \dots, 0}_{(n-1) \mbox{times}}, (P_{\alpha})_{\alpha\in\mathcal{I}_n}, 0, \dots)$$ and we denote by $\Pi^{(n)}\colon \mathscr{F}{\rightarrow}\mathscr{P}^{(n)}$ the projection $$(\dots, (P_{\alpha})_{\alpha\in\mathcal{I}_n}, \dots)\mapsto (P_{\alpha})_{\alpha\in\mathcal{I}_n}.$$ We call $\mathscr{P}^{(\le n)}:=\prod_{k=0}^n \mathscr{P}^{(k)}$ the space of the formal polynomials of degree (at most) $n+2$. As above, $\mathscr{P}^{(\le n)}$ can be embedded into the space of formal power series $\mathscr{F}$. We denote by $\Pi^{(\le n)}$ the projection of $\mathscr{F}$ onto $\mathscr{P}^{(\le n)}$.\ We define $\mathscr{F}^{\geq n}:=\prod_{k\geq n}\mathscr{P}^{(k)}$. We denote by $\Pi^{(\geq n)}$ the projection of $\mathscr{F}$ onto $\mathscr{P}^{(\geq n)}$. In particular, if $G$ is a formal power series we write $$\Pi^{(n)} G=G^{(n)}, \quad \Pi^{(\le n)} G=G^{(\le n)}, \quad \Pi^{(\geq n)} G=G^{(\geq n)}.$$ #### Birkhoff resonances. Now we introduce the notion of Birkhoff resonances. \[defrisonanza\] We say that $\alpha\in \mathcal{I}_n$ is resonant if its associated divisor $\Omega(\alpha)=0$ and we write $\alpha\in\mathcal{N}_n$.\ We say that $\alpha$ is trivially resonant if $\alpha_j=\alpha_{-j}$ for all $j$ and we write $\alpha\in\mathcal{N}^_n$. By the fact that $\omega(-j)=-\omega(j)$, if $\alpha$ is trivially resonant then it is resonant and its associated monomials depend only on the actions* $I_j:=\lvert u_j \rvert^2=u_j u_{-j}$ $$u^{\alpha}=\prod_{j>0} (\lvert u_{j} \rvert^2)^{\alpha_{j}}=\prod_{j>0} I_{j}^{\alpha_{j}}.$$ We say that a polynomial which depends only upon the actions $I_j$ is action-preserving. Main result and applications ---------------------------- The main result of the paper is the following. \[costantiMotoDP\] Let $\mathtt{c}\in\mathbb{R}\setminus\{ 0\}$. For any $n\geq1$ there exist a decreasing sequence of positive numbers $(r_n)_{n \geq 1}$, and a sequence of functions $K_{n}: H^{n+1} {\rightarrow}{\mathbb{R}}$ with the following properties: - [Involution:]{} if we set $K_0:=H$ (see ) then for any $n\geq0$ one has that $$\label{commutano} \{H(u),K_{n}(u)\}=0.$$ - [Analyticity:]{} the function $K_n$ is analytic on $B_{H^{n+1}}(0, \lvert \mathtt{c} \rvert /2)$; more precisely, there exists a function $\Psi_n\colon\mathbb{C}^{n+2}{\rightarrow}\mathbb{C}$ analytic on $B_{\mathbb{C}}(0, \lvert \mathtt{c} \rvert)\times\dots\times B_{\mathbb{C}}(0, \lvert \mathtt{c} \rvert)$ such that $$K_n(u)=\int \Psi_n(u, u_x, \dots, \partial_x^{n+1} u)\,dx.$$ Moreover, $\Psi_n$ admits the following Taylor expansion $$\label{formaaffine} \Psi_n(u, u_x, \dots, \partial_x^{n+1} u)=\sum_{k\geq 0}\,\, \sum_{\substack{\alpha\in \mathbb{N}^{\{0, \dots, n+1\}}, \\ \sum \alpha_i=k,\\ \sum i\alpha_i \le n+1 } } \Psi_{\alpha} \,\, u^{\alpha_0}( \partial_x u)^{\alpha_1}\dots ( \partial_x^{n+1} u)^{\alpha_{n+1}}, \qquad \Psi_{\alpha}\in \mathbb{C}.$$ - [Characterization of quadratic parts:]{} the Taylor polynomial of order $2$ of $K_n$ at $u=0$ has the form $$\label{formacostanti} \begin{aligned} &K_n^{(0)}(u)=\int_{\mathbb{R}} (\partial_x^{n-1} (u-u_{xx}))^2\,dx \quad x\in\mathbb{R}, \\ & K_{n}^{(0)}(u)= \sum_{j\in \mathbb{Z}\backslash\{0\}}\|j\|^{2(n-1)}(1+j^{2})^2\|u_{j}\|^{2} \quad x\in{\mathbb{T}}; \end{aligned}$$ - [Control of Sobolev norms:]{} there exist positive constants $C=C(n, \mathtt{c})$ and $\tilde{c}=\tilde{c}(n, \mathtt{c})$ such that for any $n\geq1$ $$\label{equivalenzaNorma} \|K_{n}^{(0)}(u)\|\leq \\|u\\|^2_{H^{n+1}}\leq \\|u\\|^2_{L^{2}} +\tilde{c} \|K_{n}^{(0)}(u)\| \quad \forall u\in B_{H^{n+1}}(0, \lvert \mathtt{c} \rvert/2)$$ and $$\label{stimecostanti} \|K_{n}^{(\geq1)}(u)\|\leq C\\|u\\|^{3}_{H^{n+1}} \quad \forall u\in B_{H^{n+1}}(0, r_n).$$ Let us make some comments. - and imply that $K_n$ is equivalent to the $H^{n+1}$-norm in a neighborhood of the origin, and this does not hold as the parameter $\mathtt{c}$ goes to zero (see for instance Remark \[clinprop\] and Remark \[displess\]). - We remark that the radius of analyticity of the $K_n$’s depends only on the parameter $\mathtt{c}$. - Our result is based on an explicit computation of the coefficients of the quadratic part of the constructed conserved quantities. We point out that the radii $r_n$ in decrease to zero as $n{\rightarrow}\infty$. It may be possible to improve by studying the higher order expansions of the constants of motion. - By the expression the function $\Psi_n$ is affine in the variable $\partial_x^{n+1} u$ (see Remark \[remarkaffine\]). This is a key point to prove the bounds in item $(iii)$. Let us discuss the applications we obtain by the result above. We prove the following stability result. \[esistenzaglobale\] Let $X$ be $\mathbb{R}$ or ${\mathbb{T}}$. For any $s \geq 2$ there is $r=r(s)>0$ such that for any $u_0\in B_{H^s}(0, r)$ there exists a unique solution $u(t,x)$ of , defined for all times, belonging to $C^{0}(\mathbb{R};H^{s}(X;\mathbb{R}))$ such that $$\sup_{t\in{\mathbb{R}}}\\|u(t,x)\\|_{H^{s}}\leq C' r,$$ for some constant $C'=C'(s, \mathtt{c})>0$. The above theorem is in turn based on a local well-posedness result for the equation (the proof, which follows [@HH], is deferred to the Appendix). The second application concerns the study of the Birkhoff normal form of the equation . \[teoBirk\] Let $H$ be the Hamiltonian with $x\in{\mathbb{T}}$. For any $N\in\mathbb{N}$ there exist, at least formally, a symplectic transformation $\Phi_N$ such that $$\label{BirkhoffFormN} H\circ \Phi_N=H^{(0)}+Z_N+R_N$$ where $Z_N\in \mathscr{P}^{(\le N)}$ (recall Definition \[defpolinomi\]) is action-preserving, hence it Poisson commutes with $H^{(0)}$ and depends only on the actions $I_j:=\lvert u_j \rvert^2$. The function $R_N\in\mathscr{F}^{(\geq N+1)}$ (recall Definition \[defserie\]). \[defBirkformN\] We say that a Hamiltonian $G\in\mathscr{F}$ is in a Birkhoff normal form of order $N$ if it has the form described in Theorem \[teoBirk\]. In the proof of such result will be fundamental the explicit form of the quadratic part of the constant of motion that we give in . The proof of Theorem \[teoBirk\] is based on the following classical result (see, for instance, [@KdVeKAM]): - two commuting Hamiltonians $H,K\in\mathscr{F}$ can be put in Birkhoff normal form, up to order $N$, by the same change of coordinates (at least at the formal level). This fact will be proved in Lemma \[lemmabellissimo\], which is a variation of Theorem $G. 2$ in [@KdVeKAM], since we do not assume that the linear frequencies are non resonant. #### Plan of the paper The paper is organized as follows. In Section \[costantiMotoPROOF\] we prove Theorem \[costantiMotoDP\]. In Section \[GWPDP\] we first state a local-well posedness result, which is proved in the Appendix \[appendA\], and then we prove \[esistenzaglobale\] by using the bounds , and a bootstrap argument. In Section \[proofteoBirk\] we give a proof of Theorem \[teoBirk\]. #### Acknowledgements We warmly thank Michela Procesi, Luca Biasco and Alberto Maspero for many useful suggestions and fruitful discussions. Constants of motion {#costantiMotoPROOF} =================== In Section $3$ of [@Deg] Degasperis-Holm-Hone give the Lax pair for the equation , which we write in the following with the choice of the parameters that leads to consider the equation (recall the definition of $m$ in ), $$\label{LaxPair} \begin{cases} (1-\partial_{xx})\Psi_x &= m\Psi \\ \Psi_t+\frac{1}{\lambda}\Psi_{xx}+(u+\mathtt{c})\Psi_{x}-u_x \Psi &= 0 \end{cases} ,$$ for a real parameter $\lambda\neq 0$. In Section $4$ of [@Deg] the authors derive many conservation laws by considering the following relations, $$\label{RELAZIONE} (1-{\partial}_{xx})\rho=3\rho \rho_x+\rho^{3}+\lambda m,$$ and $$\label{conservationlaw} \rho_t=j_x, \qquad j=u_x-\frac{1}{\lambda}(\rho_x+\rho^2)-(u+\mathtt{c})\rho,$$ for the quantity $$\rho:=\Big(\mbox{log} (p\Psi) \Big)_x,$$ where comes from the spatial part of the Lax pair and comes from the time part of the Lax pair . By , for any $u(t,x)$ solution of defined on some time interval $I\subseteq\mathbb{R}$, $$\label{costanteneltempo} \frac{d}{dt}\int \rho(u(t,x))\, dx=0, \quad t\in I.$$ In [@Deg] $\rho$ is written as a formal series in powers of the spectral parameter $\lambda=\zeta^{-3}$, $\zeta\in \mathbb{R}$, with the coefficients determined recursively from . One of the possible expansions is $$\label{RHO} \rho=p\zeta^{-1}+\sum_{n=0}^{\infty}\rho^{(n)}\zeta^{n},$$ and we are interesting in studying the constants of motion $$\label{seqcos} \Gamma^{(n)}:=\int \rho^{(n)}dx, \quad n\geq0,$$ given by the series in . By using we have that is equivalent to $$\label{ordbyord} \begin{aligned} &0=\rho^{(0)}p^2+pp_x , \\ &p-p_{xx}=\rho^{(1)}p^2+2p(\rho^{(0)})^{2}+3{\partial}_{x}(\rho^{(0)}p) , \\ \end{aligned}$$ $$\label{ordbyord1} \begin{aligned} \rho^{(n)}-\rho^{(n)}_{xx}&=\rho^{(n+2)}p^{2}+3\sum_{k_1+k_2=n+1}p\rho^{(k_1)}\rho^{(k_2)} +\sum_{k_1+k_2+k_3=n}\rho^{(k_1)}\rho^{(k_2)}\rho^{(k_3)}\\ &+3{\partial}_{x}(\rho^{(n+1)}p)+3\sum_{k_1+k_2=n}\rho^{(k_1)}\rho^{(k_2)}_x, \qquad n\geq0. \end{aligned}$$ From we get $$\label{iprimi} \rho^{(0)}=-\frac{p_x}{p}, \quad \rho^{(1)}=-\frac{p_{x}^{2}}{p^{3}}+\frac{2p_{xx}}{3p^{2}}+\frac{1}{3p}.$$ Now we want to prove that $\rho^{(n)}(w)$ can be expressed as a power series in the variables $w$ and its derivatives in a small neighborhood of the origin of some $H^s$ Sobolev space. We refer to the Appendix to recall some definitions and facts on analytic functions on Banach spaces. Analyticity of composition operators on Sobolev spaces ------------------------------------------------------ \[NemistkyLemma\] Let $n\geq 1$ and $f\colon \mathbb{C}^n{\rightarrow}\mathbb{C}$ be an analytic function on $B_{\mathbb{C}}(0, r)\times \dots \times B_{\mathbb{C}}(0, r)$ for some $r>0$. Then the composition operator $$\label{Nemitsky} T_{f}[u_1, \dots, u_n]=f( u_1, \dots, u_n)\colon B_{H^s({\mathbb{T}}, \mathbb{C})}(0, \rho)\times \dots \times B_{H^s({\mathbb{T}}, \mathbb{C})}(0, \rho){\rightarrow}H^s, \quad \forall\,\, 0<\rho<r,\quad s>1/2$$ is weakly analytic on $B_{H^{s}({\mathbb{T}}, \mathbb{C})}(0, r/2)\times \dots\times B_{H^{s}({\mathbb{T}}, \mathbb{C})}(0, r/2)$ for $s>1/2$. First we want to show that, given $0<\rho<r$, $T_f$ maps $B_{H^s({\mathbb{T}}, \mathbb{C})}(0, \rho)\times \dots \times B_{H^s({\mathbb{T}}, \mathbb{C})}(0, \rho)$ into $H^s$ for $s>1/2$. Since $f$ is analytic we can write $$f(z_1, \dots, z_n)=\sum_{k\geq 0}\sum_{\substack{\beta\in\mathbb{N}^{\mathbb{Z}},\\ \lvert \beta \rvert=k}} f_{\beta} z^{\beta}, \quad z^{\beta}:=\prod_{i=1}^n z_i^{\beta_i}$$ for some coefficients $f_{\beta}\in\mathbb{C}$ satisfying $$\sum_{k\geq 0}\sum_{\substack{\beta\in\mathbb{N}^{\mathbb{Z}},\\ \lvert \beta \rvert=k}} \lvert f_{\beta} \rvert \rho^{k}\le C, \qquad \forall 0<\rho<r$$ for some constant $C>0$ depending only on $\rho$. By using the algebra property of the Sobolev spaces $H^s$ with $s>1/2$, we have $$\lVert T_f[u_1, \dots, u_n] \rVert_{H^s({\mathbb{T}}, \mathbb{C})}\le \sum_{k\geq 0}\sum_{\substack{\beta\in\mathbb{N}^{\mathbb{Z}},\\ \lvert \beta \rvert=k}} \lvert f_{\beta} \rvert \lVert u_1 \rVert_{H^s({\mathbb{T}}, \mathbb{C})}^{\beta_1}\dots \lVert u_n \rVert_{H^s({\mathbb{T}}, \mathbb{C})}^{\beta_n}$$ and the claim follows. In order to prove the weak analyticity of the operator $T_f$, we have to show that for all $w_i\in B_{H^{s}({\mathbb{T}}, \mathbb{C})}(0, r/2)$, $h_i\in H^s({\mathbb{T}}, \mathbb{C})$, $i=1, \dots, n$ and $L\in (H^s({\mathbb{T}}, \mathbb{C}))^$ the function $(z_1, \dots, z_n)\mapsto L T_f(w_1+z_1 h_1, \dots, w_n+z_n h_n)$ is analytic in a neighborhood of the origin of $\mathbb{C}^n$. By Riesz theorem, for every $L\in (H^s({\mathbb{T}}, \mathbb{C}))^$ there exists a function $g\in H^s({\mathbb{T}}, \mathbb{C})$ such that $$\label{maledetto} \begin{aligned} & L T_f(w_1+z_1 h_1, \dots, w_n+z_n h_n) =\sum_{m=0}^s\int \partial_x^m\Big( f \big(w_1(x)+z_1 h_1(x), \dots, w_n(x)+z_n h_n(x) \big)\Big)\,\partial_x^m g(x)\,dx\\ &=\int f \big(w_1(x)+z_1 h_1(x), \dots, w_n(x)+z_n h_n(x) \big)\, g(x)\,dx\\ &+\sum_{m=1}^s \sum_{k=1}^m \sum_{\substack{ {\bf v}\in \mathcal{A}_{m, k},\\ {\bf p}\in \mathcal{B}_m({\bf v})}} C_{{\bf p}} \int \big( D^{ {\bf v}} f\big) \prod_{i=1}^n \prod_{j=1}^{\lvert {\bf v}_i \rvert} (\partial_x^{{\bf p}^{(i)}_{j}}w_{i}+z_{i} \partial_x^{{\bf p}^{(i)}_{j}} h_{i} )\,\partial_x^m g(x)\,dx \end{aligned}$$ where $C_{{\bf p}}$ are combinatorial factors, $$\mathcal{A}_{m, k}:=\{ {\bf v}\in \{0, \dots, m\}^n\, :\,\lvert {\bf v} \rvert:=\sum_{i=1}^n {\bf v}_i=k \},$$ $$\mathcal{B}_r({\bf v}):=\{ {\bf p}=({\bf p}^{(1)}, \dots, {\bf p}^{(n)}),\, {\bf p}^{(i)}\in\{1, \dots, m \}^{\lvert {\bf v}\rvert},\,\sum_{i=1}^n {\bf p}^{(i)}=r \} \qquad \mbox{for}\,\,\, {\bf v}\in \mathcal{A}_{m, k}$$ and $$D^{{\bf v}} f:= \partial_{ {\bf v}_1 \dots {\bf v}_n } f=\frac{\partial^k}{\partial_{z_1}^{{\bf v}_1}\dots \partial_{z_n}^{{\bf v}_n} }, \qquad \partial_{i} :=\partial_{z_i}.$$ The last summand in can be written as $$\label{maledetto2} \begin{aligned} &\sum_{m=1}^{s-1} \sum_{k=1}^m \sum_{\substack{ {\bf v}\in \mathcal{A}_{m, k},\\ {\bf p}\in \mathcal{B}_m({\bf v})}} C_{{\bf p}} \int \big( D^{ {\bf v} } f\big) \prod_{i=1}^n \prod_{j=1}^{\lvert {\bf v}_i \rvert} (\partial_x^{{\bf p}^{(i)}_{j}}w_{i}+z_{i} \partial_x^{{\bf p}^{(i)}_{j}} h_{i} )\,\partial_x^m g(x)\,dx\\ &+ \sum_{k=2}^s \sum_{\substack{ {\bf v}\in \mathcal{A}_{m, k},\\ {\bf p}\in \mathcal{B}_s({\bf v})}} C_{{\bf p}} \int \big( D^{ {\bf v}} f\big) \prod_{i=1}^n \prod_{j=1}^{\lvert {\bf v}_i \rvert} (\partial_x^{{\bf p}^{(i)}_{j}}w_{i}+z_{i} \partial_x^{{\bf p}^{(i)}_{j}} h_{i} )\,\partial_x^s g(x)\,dx\\ &+ \sum_{i=1}^n \int (\partial_{i} f)\,(\partial_x^{s}w_{i}+z_i \partial_x^{s}h_{i} )\,\partial_x^s g\,dx. \end{aligned}$$ We want to prove that there exist the derivatives in the complex variable $z_i$ of the function $L T_f(w_1+z_1 h_1, \dots, w_n+z_n h_n)$: to do that it is sufficient to prove that the derivative in $z_i$ of the integrands in is $L^1$ uniformly in the parameter $z$, since by dominated convergence we can pass the derivative inside the integral and use the analyticity of $f$. Hence we now show that the following sum $$\label{lucarelli} \begin{aligned} & \int \lvert (\partial_{z_{\xi}} f)h_{\xi}\, g(x)\, \rvert dx+\sum_{m=1}^{s-1} \sum_{k=1}^m \sum_{\substack{ {\bf v}\in \mathcal{A}_{m, k},\\ {\bf p}\in \mathcal{B}_m({\bf v})}} C_{{\bf p}} \int \lvert \big( D^{ {\bf v}+\mathtt{e}_{\xi} } f\big)\,h_{\xi}\, \prod_{i=1}^n \prod_{j=1}^{\lvert {\bf v}_i \rvert} (\partial_x^{{\bf p}^{(i)}_{j}}w_{i}+z_{i} \partial_x^{{\bf p}^{(i)}_{j}} h_{i} )\,\partial_x^m g(x)\,\rvert\,dx\\ &+\sum_{m=1}^{s-1} \sum_{k=1}^m \sum_{\substack{ {\bf v}\in \mathcal{A}_{m, k},\\ {\bf p}\in \mathcal{B}_m({\bf v})}} C_{{\bf p}} \int \lvert \big( D^{ {\bf v} } f\big) \prod_{i=1, i\neq {\xi}}^n \prod_{j=1}^{\lvert {\bf v}_i \rvert} (\partial_x^{{\bf p}^{(i)}_{j}}w_{i}+z_{i} \partial_x^{{\bf p}^{(i)}_{j}} h_{i} )\\ &\qquad\qquad\times\Big(\sum_{j=1}^{\lvert {\bf v}_{\xi} \rvert} \prod_{b\neq j} (\partial_x^{{\bf p}^{({\xi})}_{b}}w_{{\xi}}+z_{{\xi}} \partial_x^{{\bf p}^{({\xi})}_{b}} h_{{\xi}}) \partial_x^{{\bf p}_j^{({\xi})}} h_{\xi} \Big)\,\partial_x^m g(x)\,\rvert\,dx\\ &+ \sum_{k=2}^s \sum_{\substack{ {\bf v}\in \mathcal{A}_{m, k},\\ {\bf p}\in \mathcal{B}_s({\bf v})}} C_{{\bf p}} \int \lvert \big( D^{ {\bf v}+\mathtt{e}_{\xi}} f\big)\, h_{\xi}\, \prod_{i=1}^n \prod_{j=1}^{\lvert {\bf v}_i \rvert} (\partial_x^{{\bf p}^{(i)}_{j}}w_{i}+z_{i} \partial_x^{{\bf p}^{(i)}_{j}} h_{i} )\,\partial_x^s g(x)\,\rvert\,dx\\ &+ \sum_{k=2}^s \sum_{\substack{ {\bf v}\in \mathcal{A}_{m, k},\\ {\bf p}\in \mathcal{B}_s({\bf v})}} C_{{\bf p}} \int \lvert \big( D^{ {\bf v}} f\big)\, \prod_{i=1}^n \prod_{j=1}^{\lvert {\bf v}_i \rvert} (\partial_x^{{\bf p}^{(i)}_{j}}w_{i}+z_{i} \partial_x^{{\bf p}^{(i)}_{j}} h_{i} ) \\ &\qquad \qquad\times\Big(\sum_{j=1}^{\lvert {\bf v}_{\xi} \rvert} \prod_{b\neq j} (\partial_x^{{\bf p}^{({\xi})}_{b}}w_{{\xi}}+z_{{\xi}} \partial_x^{{\bf p}^{({\xi})}_{b}} h_{{\xi}}) \partial_x^{{\bf p}_j^{({\xi})}} h_{\xi} \Big) \,\partial_x^s g(x)\,\rvert\,dx\\ &+ \sum_{i=1}^n \int \lvert (\partial_{i\,{\xi}} f)\,h_{\xi}\,(\partial_x^{s}w_{i}+z_i \partial_x^{s}h_{i} )\,\partial_x^s g\,\rvert\,dx+ \int \lvert (\partial_{{\xi}} f)\partial_x^{s}h_{{\xi}}\,\partial_x^s g\, \rvert\,dx \end{aligned}$$ is finite for some ${\xi}\in\{ 1,\dots, n \}$. Fix $\lvert z_i \rvert<\min\{r/2\lVert h_i \rVert_{H^{s}({\mathbb{T}}, \mathbb{C})}, r/2\}$ for $i=1, \dots, n$. First we bound the derivatives of $f$ $$\lvert \big( D^{{\bf v}} f\big)\big(w_1+z_1 h_1, \dots, w_n+z_n h_n \big)\rvert_{L^{\infty}}<\infty \qquad \forall w_i\in B_{H^s({\mathbb{T}}, \mathbb{C})}(0, r/2), \,\,h_i\in H^s({\mathbb{T}}, \mathbb{C})$$ since the derivatives of $f$ are analytic on $B_{\mathbb{C}}(0, r)\times \dots \times B_{\mathbb{C}}(0, r)$ and $$\lvert w_i+z_i h_i \rvert_{L^{\infty}}<r, \qquad i=1, \dots, n.$$ Since $w_i$ and $h_i$ belong to $H^s({\mathbb{T}}, \mathbb{C})$ then, for $1\le k \le m$, $1\le m \le s$, ${\bf v}\in \mathcal{A}_{m, k}$, ${\bf p}\in\mathcal{B}_m({\bf v})$ we have by Cauchy-Schwarz and Sobolev embeddings $$\begin{aligned} &\int \lvert \prod_{i=1}^n \prod_{j=1}^{\lvert {\bf v}_i \rvert} (\partial_x^{{\bf p}^{(i)}_{j}}w_{i}+z_{i} \partial_x^{{\bf p}^{(i)}_{j}} h_{i} )\,h_{\xi}\,\partial_x^m g(x) \rvert\,dx\le \prod_{i=1}^n \prod_{j=1}^{\lvert {\bf v}_i \rvert}\lvert (\partial_x^{{\bf p}^{(i)}_{j}}w_{i}+z_{i} \partial_x^{{\bf p}^{(i)}_{j}} h_{i} ) \rvert_{L^{\infty}({\mathbb{T}}, \mathbb{C})} \lVert h_{\xi} \rVert_{L^2({\mathbb{T}}, \mathbb{C})} \lVert \partial_x^m g \rVert_{L^2({\mathbb{T}}, \mathbb{C})}\\ &\le \prod_{i=1}^n \prod_{j=1}^{\lvert {\bf v}_i \rvert} (\lVert \partial_x^{{\bf p}^{(i)}_{j}}w_{i} \rVert_{H^1({\mathbb{T}}, \mathbb{C})}+\lvert z_{i} \rvert \lVert \partial_x^{{\bf p}^{(i)}_{j}} h_{i}\rVert_{H^1({\mathbb{T}}, \mathbb{C})} ) \lVert h_{\xi} \rVert_{L^2({\mathbb{T}}, \mathbb{C})} \lVert \partial_x^m g \rVert_{L^2({\mathbb{T}}, \mathbb{C})}\le r^m \, \lVert h_{\xi} \rVert_{L^2({\mathbb{T}}, \mathbb{C})} \lVert g \rVert_{H^s({\mathbb{T}}, \mathbb{C})}\end{aligned}$$ and similarly $$\begin{aligned} \int \lvert \prod_{i=1, i\neq \xi}^n \prod_{j=1}^{\lvert {\bf v}_i \rvert} (\partial_x^{{\bf p}^{(i)}_{j}}w_{i}+z_{i} \partial_x^{{\bf p}^{(i)}_{j}} h_{i} )\,\Big(\sum_{j=1}^{\lvert {\bf v}_{\xi} \rvert} \prod_{b\neq j} (\partial_x^{{\bf p}^{(\xi)}_{b}}w_{\xi}+z_{\xi} \partial_x^{{\bf p}^{(\xi)}_{b}} h_{\xi}) & \partial_x^{{\bf p}_j^{(\xi)}} h_{\xi} \Big)\,\partial_x^m g(x) \rvert\,dx\\ &\le m r^{m-1}\, \lVert h_{\xi} \rVert_{L^2({\mathbb{T}}, \mathbb{C})} \lVert g \rVert_{H^s({\mathbb{T}}, \mathbb{C})}.\end{aligned}$$ We bounded the first three terms in . The fourth and the fifth terms in have similar bounds and the proof follows the arguments above; we remark only that the Cauchy-Schwarz inequality has to be applied to the $L^2$-product of $\partial_x^s g$ with $\partial_x^{{\bf p}_j^{(\xi)}} h_{\xi} $. For the last two summands of we have $$\begin{aligned} \int \lvert h_r (\partial_x^{s}w_{i}+z_i \partial_x^{s}h_{i} )\,\partial_x^s g \rvert\,dx &\le \lvert h_r \rvert_{L^{\infty}({\mathbb{T}}, \mathbb{C})} \lVert \partial_x^{s}w_{i}+z_i \partial_x^{s}h_{i} \rVert_{L^2({\mathbb{T}}, \mathbb{C})}\lVert \partial_x^s g \rVert_{L^2({\mathbb{T}}, \mathbb{C})}\\ &\le \lvert h_r \rvert_{L^{\infty}({\mathbb{T}}, \mathbb{C})} (\lVert w_{i} \rVert_{H^s({\mathbb{T}}, \mathbb{C})}+\lvert z_i \rvert\lVert h_{i} \rVert_{H^s({\mathbb{T}}, \mathbb{C})})\lVert g \rVert_{H^s({\mathbb{T}}, \mathbb{C})} \le r\, \lVert h_r \rVert_{H^1({\mathbb{T}}, \mathbb{C})} \lVert g \rVert_{H^s({\mathbb{T}}, \mathbb{C})}\end{aligned}$$ and $$\int \lvert \partial_x^{s}h_{\xi} \,\partial_x^s g \rvert\,dx\le \lVert h_{\xi}\rVert_{H^s({\mathbb{T}}, \mathbb{C})} \lVert g \rVert_{H^s({\mathbb{T}}, \mathbb{C})}.$$ \[secondlem\] Let $\sigma\in\mathbb{N}$ and $f\colon \mathbb{C}{\rightarrow}\mathbb{C}$ be analytic on a ball $B_{\mathbb{C}}(0, r)$. Then there exists a function $g\colon \mathbb{C}^{\sigma+1}{\rightarrow}\mathbb{C}$ analytic on $B_{\mathbb{C}}(0, r)\times \dots\times B_{\mathbb{C}}(0, r)$ such that $\big(\partial_x^{\sigma}\circ T_f \big)(w)$ is the restriction to $$w_0=w, \dots, w_{\sigma}=\partial_x^{\sigma} w$$ of the composition operator $T_g[w_0, \dots, w_{\sigma}]\colon H^s({\mathbb{T}}, \mathbb{C})\times\dots \times H^s({\mathbb{T}}, \mathbb{C}) {\rightarrow}H^s({\mathbb{T}}, \mathbb{C})$ for $s>1/2$. Moreover $T_g$ is analytic on $B_{H^s({\mathbb{T}}, \mathbb{C})}(0, r/2)\times \dots \times B_{H^{s}({\mathbb{T}}, \mathbb{C})}(0, r/2)$ for $s>1/2$. By the chain rule $$\label{faadibruno} \partial_x^{\sigma} f(w)=\sum_{k=1}^{\sigma} \sum_{p_1+\dots+p_k=\sigma} C_k f^{(k)}(w)(\partial_x^{p_1} w) \dots (\partial_x^{p_k} w),$$ hence the function $\partial_x^{\sigma}\circ T_f$ is the restriction of the composition operator $T_g$ on $w_0=w, \dots, w_{\sigma}=\partial_x^{\sigma} w$ for a function $g=g(z_0, \dots, z_{\sigma})\colon \mathbb{C}^{\sigma+1}{\rightarrow}\mathbb{C}$ which has the form $$\sum_{k=1}^{\sigma} \sum_{p_1+\dots+p_k=\sigma} C_k f^{(k)}(z_0)z_{p_1} \dots z_{p_k} =\sum_{k=1}^{\sigma} \tilde{C}_k f^{(k)}(z_0) z_1^{\alpha_1^{(k)}}\dots z_{\sigma}^{\alpha_{\sigma}^{(k)}}$$ for some $\alpha_i^{(k)}\in \mathbb{N}$ and some positive constants $\tilde{C}_k$.\ The function $g$ is clearly analytic on $B_{\mathbb{C}}(0, r)\times \dots\times B_{\mathbb{C}}(0, r)$ and we have the weakly analyticity of $T_g$ by using Lemma \[NemistkyLemma\]. The fact that $T_g$ is locally bounded as operator from $H^s({\mathbb{T}}, \mathbb{C})\times\dots \times H^s({\mathbb{T}}, \mathbb{C})$ to $H^s({\mathbb{T}}, \mathbb{C})$ follows trivially by the following estimate, obtained by exploiting the algebra property of $H^s({\mathbb{T}}, \mathbb{C})$ with $s>1/2$ and the analyticity of $f$, $$\lVert \partial_x^{\sigma} f(w) \rVert_{H^s({\mathbb{T}}, \mathbb{C})}\le \sum_{k=1}^{\sigma} \sum_{p_1+\dots+p_k=\sigma} C_k \lVert f^{(k)}(w_0) \rVert_{H^s({\mathbb{T}}, \mathbb{C})} \lVert w_{p_1}\rVert_{H^s({\mathbb{T}}, \mathbb{C})} \dots \lVert w_{p_k}\rVert_{H^s({\mathbb{T}}, \mathbb{C})}.$$ The function $p(y)=-(\mathtt{c}+y)^{1/3}$ is analytic in $\{ y\in \mathbb{C} : \lvert y \rvert< \lvert \mathtt{c} \rvert \}$, hence by Lemma \[NemistkyLemma\] the map $p(w)=T_p[w]=-(\mathtt{c}+w)^{1/3}$ defined in is weakly analytic in $B_{H^s({\mathbb{T}}, \mathbb{C})}(0, \lvert \mathtt{c} \rvert /2)$. Moreover $T_p$ is locally bounded, hence $p(w)$ is analytic in $B_{H^s({\mathbb{T}}, \mathbb{C})}(0, \lvert \mathtt{c} \rvert /2)$ and it can be represented by its Taylor expansion at the origin $$\label{taylor} p(w)=\sum_{n\geq 0} \frac{p^{(n)}(0)}{n!}\,w^n.$$ \[realta\] We note that the function $p(y)=-(\mathtt{c}+y)^{1/3}$ is real on real, namely it assumes real valued when it is restricted to the real line. Then its restriction to $\mathbb{R}$ is a real analytic function.\ As a consequence, it is easy to see that the composition operator $T_p$ is real on real and then it is analytic on $H^s:=H^s(X, \mathbb{R})$, $X={\mathbb{T}}, \mathbb{R}$. ### Class of differential polynomials {#diffpol} We introduce a class of differential polynomials to which the Taylor expansion of the $\rho^{(n)}$ belongs. The particular form of these polynomial results to be fundamental for the Sobolev estimates on the constants of motion which we construct. We define $$\label{definterval} \mathcal{J}_n^q:=\{ \alpha\in \mathbb{N}^{\{0, \dots, n\}} : \sum_{i=0}^n \alpha_i=q, \,\, \sum_{i=0}^n i \alpha_i \le n \}$$ and for $\alpha\in \mathcal{J}_n^q$, $w=(w_0, \dots, w_n)$, $w_i:=\partial_x^i w$ the monomial $$\label{defmonomial} w^{\alpha}=\prod_{i=0}^n w_i^{\alpha_i}=\prod_{i=0}^n (\partial_x^i w)^{\alpha_i}.$$ We denote by ${{\mathcal P}}_n^q$ the class of formal homogenous polynomials of degree $q$ and order $n$ of the form $$f=\sum_{\alpha\in \mathcal{J}_n^q} f_{\alpha} w^{\alpha}, \quad f_{\alpha}\in \mathbb{C}.$$ We denote by ${{\mathcal P}}_n^{\le q}$ the class of formal polynomials of degree at most $q$ and order $n$ of the form $$f=\sum_{k=0}^q f_k, \quad f_k\in\mathcal{P}_n^k.$$ We denote by $\Sigma_n^q$ the class of formal power series of degree at least $q$ of the form $$f=\sum_{k=q}^{\infty}f_{n, k}, \quad f_{n, k}\in {{\mathcal P}}_n^k.$$ The Taylor series is an element of $\Sigma_0^0$. \[polem\] Let $\sigma, n, m, q, r\in\mathbb{N}$. Then 1. If $f\in {{\mathcal P}}_n^q$, $g\in{{\mathcal P}}_m^r$ then $$f+g\in {{\mathcal P}}_{\max\{n, m \}}^{\le \max\{ q, r \}}, \quad f\,g\in{{\mathcal P}}_{\max\{n, m \}}^{\le q+r}.$$ 2. The operator $\partial_x^{\sigma}$ maps ${{\mathcal P}}_n^q$ into ${{\mathcal P}}_{n+\sigma}^q$. Proof of $(1)$: for the sum the proof is trivial. For the product, suppose that $m\geq n$, the claim follows by the fact that $$w_0^{\alpha_0}\dots w_n^{\alpha_n}\, w_0^{\beta_0}\dots w_m^{\beta_m}=\prod_{i=0}^{n} w_i^{\alpha_i+\beta_i}\,w_{n+1}^{\beta_{n+1}}\dots w_m^{\beta_m}$$ where $\lvert\alpha \rvert=q$ and $\lvert \beta \rvert=r$. Proof of $(2)$: clearly it is sufficient to look at the action of $\partial_x^{\sigma}$ on the monomials. Fixed $i\in\mathbb{N}$, we have that $$\partial_x^p w_i=w_{i+p} \quad \mbox{for}\,\,p\in\mathbb{N}$$ and by the chain rule $$\partial_x^{j} w_i^{\alpha_i}=\sum_{k=1}^j \sum_{p_1+\dots+p_k=j} C_k w_i^{\alpha_i-k} (\partial_x^{p_1} w_i)\dots (\partial_x^{p_k} w_i)=\sum_{k=1}^j \sum_{p_1+\dots+p_k=j} C_k w_i^{\alpha_i-k} w_{i+p_1}\dots w_{i+p_k}$$ is a function of variables $w_i, \dots, w_{i+j}$. It is easy to see that, in these variables, the degree of homogeneity has not been changed, namely it is already $\alpha_i$. Hence $$\partial_x^{\sigma} w^{\alpha}=\sum_{j_0+\dots+j_n=\sigma } C_{j_0 \dots j_n} (\partial_x^{j_0} w_0^{\alpha_0})\dots (\partial_x^{j_n} w_n^{\alpha_n})$$ is a function of the variables $w_0, \dots, w_{n+\sigma}$ with homogeneity degree $\alpha_0+\dots+\alpha_n= q$. The following remark is fundamental for getting bounds on the Sobolev norms of the constants of motion. \[remarkaffine\] Let $f\in \Sigma_n^q$ for some $n, q\geq 0$. We note that $f$ is necessarily affine in the variable $w_n=\partial_x^n w$, namely in the highest order derivative. Indeed, $\mathcal{M}(\alpha)=n=\sum_{i=0}^n i \alpha_i$ for $\alpha\in \mathcal{J}_n^q$, hence $$\alpha_0=q-1, \quad \alpha_i =0, \quad \alpha_n=1, \quad i=1, \dots, n-1, \qquad \mbox{or} \qquad \alpha_n=0.$$ So $$f=w_n\,\sum_{k\geq q} f_{(k-1, 0, \dots, 0, 1)} w^k+\sum_{k\geq q} \sum_{\substack{\alpha\in\mathcal{J}_n^q,\\ \alpha_n=0}} f_{\alpha} w^{\alpha}.$$ The structure of the conserved quantities $\rho^{(n)}$ ------------------------------------------------------ The functions $\rho^{(n)}$ are given by sums and products of $p$ and its $x$-derivatives. We want to show that the $\rho^{(n)}$ are composition operators for analytic and real on real functions of $w$ and its derivatives and that the Taylor expansions of these operators belong to some $\Sigma_n^q$ (recall the definitions given in Section \[diffpol\]). This fact will allow to prove the bound and consequently the estimates in Theorem \[costantiMotoDP\]-$(iii)$. Given two composition operators $T_f$ and $T_g$ we have that $T_f+T_g=T_{f+g}$ and $T_{f}T_g=T_{f g}$. Hence if $f, g$ are analytic we can apply to $T_f+T_g=T_{f+g}$ and $T_{f}T_g=T_{f g}$ Lemmata \[NemistkyLemma\], \[secondlem\]. \[lemmaron\] Fix $n\in\mathbb{N}$. Then there exists a function $f_n\colon \mathbb{C}^{n+2}{\rightarrow}\mathbb{C}$ real on real, analytic on $B_{\mathbb{C}}(0, \lvert \mathtt{c} \rvert)\times \dots\times B_{\mathbb{C}}(0, \lvert \mathtt{c} \rvert)$ such that $$\rho^{(n)}(w)=T_{f_n}[w, w_x, \dots, \partial_{x}^{n+1} w].$$ Moreover the Taylor series of $T_{f_n}$ restricted to $w_0=w, \dots, w_{n+1}=\partial_x^{n+1} w$ belongs to $\Sigma_{n+1}^{0}$. Let us start from $\rho^{(0)}$ and $\rho^{(1)}$, and then we argue by induction on $n$.\ Recalling that $p$ is analytic as function of the variable $w$, by Lemma \[secondlem\] $p_x$ is analytic as function of the two variables $w$, $w_x$. By , we have that $ \rho^{(0)} = \frac{1}{3} (-\mathtt{c}-w)^{-1} w_x, $ and since the function $f_0:\mathbb{C}^2 {\rightarrow}\mathbb{C}$ given by $$f_0(z_0,z_1):= \frac{1}{3} (-\mathtt{c}-z_0)^{-1} z_1$$ is real on real and analytic in $B_{H^s}(0,\|c\|) \times B_{H^s}(0,\|c\|)$, by Lemma \[NemistkyLemma\] and by local boundedness of the operator $T_{f_0}$ we get that $\rho^{(0)}$ is analytic in $B_{H^s}(0,\|\mathtt{c}\|/2) \times B_{H^s}(0,\|\mathtt{c}\|/2)$. From the explicit formula of $f_0$ one can deduce that the Taylor series of $T_{f_0}$ restricted to $w_0=w, w_1=w_x$ belongs to $\Sigma_{1}^{0}$.\ Similarly, we obtain that $\rho^{(1)}$ is real on real and analytic in the variables $w$, $w_x$ and $w_{xx}$, since it can be written as a composition operator for the following analytic function $$\begin{aligned} f_1(z_0,z_1,z_2) &:= - \frac{1}{9} (-\mathtt{c}-z_0)^{-7/3} z_1^2 + \frac{2}{3} \left( - \frac{2}{9} \frac{ z_1^2 }{ (-\mathtt{c}-z_0)^{5/3} }-\frac{1}{3}(-\mathtt{c}-z_0)^{-2/3} z_2 \right) + \frac{1}{3} (-\mathtt{c}-z_0)^{-1/3}. \end{aligned}$$ Furthermore, from the explicit formula of $f_1$ one can deduce that the Taylor series of $T_{f_1}$ restricted to $w_0=w, \dots, w_2=w_{xx}$ belongs to $\Sigma_{2}^{0}$. Now we assume that the thesis holds for $\rho^{(k)}$, $k \leq n+1$, and we only have to control that $\rho^{(n+2)}$ depends only on $w$, $w_x$, $\ldots$, ${\partial}_x^{n+3}w$. But by recalling , we just observe that - $\rho^{(n)}(w) = T_{f_n}[w,\ldots,{\partial}_x^{n+1}w]$ for some $f_n:\mathbb{C}^{n+2} {\rightarrow}\mathbb{C}$ analytic on $\times_{i=1}^{n+2} B_{\mathbb{C}}(0,\|\mathtt{c}\|)$, by inductive hypothesis; - $\rho_{xx}^{(n)}(w) = T_{g_n}[w,\ldots,{\partial}_x^{n+3}w]$ for some $g_n:\mathbb{C}^{n+4} {\rightarrow}\mathbb{C}$ analytic on $\times_{i=1}^{n+4} B_{\mathbb{C}}(0,\|\mathtt{c}\|)$, by Lemma \[secondlem\] and by inductive hypothesis; - $p(w) \rho^{(k_1)}(w) \rho^{(k_2)}(w) = T_{h_{k_1,k_2}}[w,\ldots,{\partial}_x^{\max(k_1,k_2)+1}w]$ (where $k_1+k_2=n+1$), for some $h_{k_1,k_2}:\mathbb{C}^{\max(k_1,k_2)+2} {\rightarrow}\mathbb{C}$ analytic on $\times_{i=1}^{\max(k_1,k_2)+2} B_{\mathbb{C}}(0,\|\mathtt{c}\|)$, by inductive hypothesis and by the above remark; - $\rho^{(k_1)}(w) \rho^{(k_2)}(w) \rho^{(k_3)}(w)= T_{l_{k_1,k_2,k_3}}[w,\ldots,{\partial}_x^{\max(k_1,k_2,k_3)+1}w]$ ($k_1+k_2+k_3=n$), for some $l_{k_1,k_2,k_3}:\mathbb{C}^{\max(k_1,k_2,k_3)+2} {\rightarrow}\mathbb{C}$ analytic on $\times_{i=1}^{\max(k_1,k_2,k_3)+2} B_{\mathbb{C}}(0,\|\mathtt{c}\|)$, by inductive hypothesis and by the above remark; - ${\partial}_x(\rho^{(n+1)}p)(w)=({\partial}_x\rho^{(n+1)})(w) p(w) + \rho^{(n+1)}(w) p_x(w) = T_{m_{n+1}}[w,\ldots,{\partial}_x^{n+3}w]$ for some $m_{n+1}:\mathbb{C}^{n+4} {\rightarrow}\mathbb{C}$ analytic on $\times_{i=1}^{n+4} B_{\mathbb{C}}(0,\|\mathtt{c}\|)$, by inductive hypothesis and by the above remark; - $\rho^{(k_1)}(w) \rho_x^{(k_2)}(w) = T_{v_{k_1,k_2}}[w,\ldots,{\partial}_x^{\max(k_1,k_2+1)+1}w]$ ($k_1+k_2=n$), for some $v_{k_1,k_2}:\mathbb{C}^{\max(k_1,k_2+1)+2} {\rightarrow}\mathbb{C}$ analytic on $\times_{i=1}^{\max(k_1,k_2+1)+2} B_{\mathbb{C}}(0,\|\mathtt{c}\|)$, by inductive hypothesis and by the above remark. Furthermore, again by using formula , we have that - the Taylor series of $T_{f_n}$ restricted to $w_0=w, \dots, w_{n+1}=\partial_x^{n+1} w$ belongs to $\Sigma_{n+1}^{0}$, by inductive hypothesis; - the Taylor series of $T_{g_n}$ restricted to $w_0=w, \dots, w_{n+3}=\partial_x^{n+3} w$ belongs to $\Sigma_{n+3}^{0}$, by inductive hypothesis and by Lemma \[polem\]; - the Taylor series of $T_{ h_{k_1,k_2} }$ ($k_1+k_2=n+1$) restricted to $w_0=w, \dots, w_{\max(k_1,k_2)+1}=\partial_x^{\max(k_1,k_2)+1} w$ belongs to $\Sigma_{n+2}^0$, by inductive hypothesis and by Lemma \[polem\]; - the Taylor series of $T_{ l_{k_1,k_2,k_3} }$ ($k_1+k_2+k_3=n$) when restricted to $w_0=w, \dots, w_{\max(k_1,k_2,k_3)+1}=\partial_x^{\max(k_1,k_2,k_3)+1} w$ belongs to $\Sigma_{n+1}^0$, by inductive hypothesis and by Lemma \[polem\]; - the Taylor series of $T_{ m_{n+1} }$ restricted to $w_0=w, \dots, w_{n+3}=\partial_x^{n+3} w$ belongs to $\Sigma_{n+3}^0$, by inductive hypothesis and by Lemma \[polem\]; - the Taylor series of $T_{ v_{k_1,k_2} }$ ($k_1+k_2=n$) restricted to $w_0=w, \dots, w_{\max(k_1,k_2+1)+1}=\partial_x^{\max(k_1,k_2+1)+1} w$ belongs to $\Sigma_{n+2}^0$, by inductive hypothesis and by Lemma \[polem\]. This implies that the Taylor series of $T_{f_{n+2}}$ restricted to $w_0=w, \dots,w_{n+3}={\partial}_x^{n+3}w$ belongs to $\Sigma_{n+3}^0$. By Remark \[realta\] the composition operators $\rho^{(n)}$ are real analytic and by Theorem \[TruboTeo\] $\rho^{(n)}(w)$ can be represented by their Taylor expansion at the origin as functions of $w_0:=w, \dots, w_n:=\partial_x^n w$ if $w$ belongs to a sufficiently small ball of $H^{s+n}$ centered at the origin. For instance we can write $$\begin{aligned} p &= -\mathtt{c}^{1/3} - \frac{1}{ 3 \mathtt{c}^{2/3} } w + \frac{1}{ 9 \mathtt{c}^{5/3} } w^{2}+g_0(w), \label{p} \\ \rho^{(0)}&=-\frac{w_x}{3\mathtt{c}}+\frac{ww_x}{3\mathtt{c}^2}+g_1(w,w_x), \label{rho0} \\ \rho^{(1)}&=-\frac{1}{3\mathtt{c}^{1/3}}+\frac{1}{9\mathtt{c}^{4/3}} w-\frac{2}{9\mathtt{c}^{4/3}} w_{xx} -\frac{2}{27\mathtt{c}^{7/3}} w^{2} +\frac{8}{27\mathtt{c}^{7/3}} ww_{xx} +\frac{7}{27\mathtt{c}^{7/3}} w_{x}^{2}+g_{2}(w,w_x,w_{xx}), \label{rho1}\end{aligned}$$ where $g_0$, $g_1$ and $g_2$ have a zero of order $3$ at the origin. We remark that $\Gamma^{(n)}$ defined in is the integral (on the torus ${\mathbb{T}}$ or on $\mathbb{R}$) of elements of $\Sigma_n^q$. In the following lemma we prove an estimate on Sobolev spaces for this kind of functions. \[LemmaStimeTaylor\] Fix $n\in\mathbb{N}$ and let $F(w):=\int f(w, \dots, \partial_x^n w)\,dx$, where $f\colon \mathbb{C}^{n+1}{\rightarrow}\mathbb{C}$ is real on real, analytic on $B_{\mathbb{C}}(0, r)\times \dots \times B_{\mathbb{C}}(0, r)$ for some $r>0$ and the Taylor expansion of $f(w, \dots, \partial_x^n w)$ at the origin belongs to $\Sigma_n^q$. Then $F\colon H^n{\rightarrow}\mathbb{C}$ is analytic on $B_{H^n}(0, r/2)$ and the following estimate holds $$\label{stimafondam} \lvert F(w) \rvert\le C(n, r)\lVert w \rVert^q_{H^n}$$ First we prove the bound , which implies also that $F$ is locally bounded on $B_{H^n}(0, r)$. We note that, since the Taylor series of $f(w, \dots, \partial_x^n w)$ belongs to $\Sigma_n^q$, we can write by Remark \[remarkaffine\] $$F(w)=\int \sum_{k\geq q }\sum_{\alpha\in\mathcal{J}_n^k} f_{\alpha} w^{\alpha}\,dx=\int_{{\mathbb{T}}}w_n\,\sum_{k\geq q} f_{(k-1, 0, \dots, 0, 1)} w^k\,dx+\int\sum_{k\geq q} \sum_{\substack{\alpha\in\mathcal{J}_n^k,\\ \alpha_n=0}} f_{\alpha} w^{\alpha}\,dx.$$ Hence by Cauchy-Schwarz and Sobolev embeddings $$\label{briscola} \begin{aligned} \lvert F(w) \rvert &\le\sum_{k\geq q} \lvert f_{(k-1, 0, \dots, 0, 1)} \rvert \int \lvert \partial_x^n w \rvert \lvert w^{k-1} \rvert\,dx+\sum_{k\geq q} \sum_{\substack{\alpha\in\mathcal{J}_n^k,\\ \alpha_n=0}}\lvert f_{\alpha} \rvert \int \lvert w^{\alpha} \rvert\,dx\\ &\le \sum_{k\geq q} \lvert f_{(k-1, 0, \dots, 0, 1)} \rvert \lvert w \rvert_{L^{\infty}}^{k-2} \int \lvert \partial_x^n w \rvert\lvert w \rvert\,dx+\sum_{k\geq q} \sum_{\substack{\alpha\in\mathcal{J}_n^k,\\ \alpha_n=0}}\lvert f_{\alpha} \rvert \prod_{i=0}^{k-2} \lVert w \rVert_{H^{i+1}} \lVert w \rVert_{H^{n-1}}^2\\ &\le \sum_{k\geq q} \lvert f_{(k-1, 0, \dots, 0, 1)} \rvert \lvert w \rvert_{L^{\infty}}^{k-2} \lVert w \rVert_{H^n}\lVert w \rVert_{L^2}+\sum_{k\geq q} \sum_{\substack{\alpha\in\mathcal{J}_n^k,\\ \alpha_n=0}}\lvert f_{\alpha} \rvert \lVert w \rVert_{H^{n}}^{k}\\ &\le \sum_{k\geq q} \lvert f_{(k-1, 0, \dots, 0, 1)} \rvert \lVert w \rVert_{H^1}^{k-1} \lVert w \rVert_{H^n}+\sum_{k\geq q} \sum_{\substack{\alpha\in\mathcal{J}_n^k,\\ \alpha_n=0}}\lvert f_{\alpha} \rvert \lVert w \rVert_{H^{n}}^{k}\\ &\le \lVert w \rVert_{H^n}^q \Big( \sum_{k\geq q} \lvert f_{(k-1, 0, \dots, 0, 1)} \rvert \lVert w \rVert^{k-q}_{H^n}+\sum_{k\geq q} \sum_{\substack{\alpha\in\mathcal{J}_n^k,\\ \alpha_n=0}}\lvert f_{\alpha} \rvert \lVert w \rVert_{H^{n}}^{k-q}\Big)\\ &\le \lVert w \rVert_{H^n}^q \sum_{k\geq q} \sum_{\substack{\alpha\in\mathcal{J}_n^k}}\lvert f_{\alpha} \rvert r^{k-q}\le r^{-q} \lVert w \rVert_{H^n}^q \sum_{k\geq q} \sum_{\substack{\alpha\in\mathcal{J}_n^k}}\lvert f_{\alpha} \rvert r^{k}\le \frac{C(f, n, r)}{r^q} \lVert w \rVert_{H^n}^q \end{aligned}$$ Now we prove the weakly analyticity of $F$. Since $\mathbb{C}^=\mathbb{C}$ it is sufficient to show that for every $w\in B_{H^s}(0, r/2)$ and $h\in H^s$ there exists $a=a(h)>0$ such that the function $z\mapsto F(w+z h)$ is analytic as function of a complex variable in $B_{\mathbb{C}}(0, a)$.\ If $\lvert z \rvert<\min\{ \frac{r}{2 \lVert h \rVert_{H^n}} , \frac{r}{2} \}$ then $$\label{sharp} \sup_{x}\, \big(\lvert \partial_x^i w \rvert+ \lvert z \partial_x^i h \rvert\big)<r, \quad i=0, \dots, n-1$$ since $w\in H^n$. The proof follows the strategy adopted in the proof of Lemma \[NemistkyLemma\], namely we isolate the terms with the pair of functions in the integrands with the highest order of derivatives and we apply the Cauchy-Schwarz inequality to their $L^2$-scalar product. We remark that is fundamental, as for obtaining the bound , that $\partial_x^n w$ appears linearly in the Taylor expansion of $F$. Indeed by this fact it is sufficient to require the condition for $i \le n-1$ and the loss of regularity due to the Sobolev embedding does not force us to require more smoothness on $w$ than $w\in H^n$. Eventually we have $$\begin{aligned} \frac{d}{d z} F(w+z h) &=\int (w_n+z h_n)\,\frac{d}{d z}\sum_{k\geq q} f_{(k-1, 0, \dots, 0, 1)} (w+z h)^k\,dx\\ &+\int h_n \,\sum_{k\geq q} f_{(k-1, 0, \dots, 0, 1)} (w+z h)^k\,dx\\ &+\int \frac{d}{dz}\sum_{k\geq q} \sum_{\substack{\alpha\in\mathcal{J}_n^k,\\ \alpha_n=0}} f_{\alpha} (w+z h)^{\alpha}\,dx $$ and these derivatives exist by the analyticity of $f$ on $B_{\mathbb{C}}(0, r)\times \dots \times B_{\mathbb{C}}(0, r)$. Now we prove two facts (recall for the definition of $\Gamma^{(n)}$): - the quadratic terms in the expansion of $\Gamma^{(n)}$ have a particular form ; - the cubic remainder of the expansion of $\Gamma^{(n)}$ does not contain derivatives of $w$ of order greater than the ones appearing in the quadratic part (see Section \[appsec\]). We will do that by showing that the coefficient of the quadratic part associated to the monomials containing the highest number of derivatives is non-zero. \[remarkone\] We point out the following properties of differential polynomial in ${{\mathcal P}}_{n}^{k}$. - Let $g_0\in {{\mathcal P}}_{n}^{1}$ for some $n$, and recall that $\int w\,dx=0$. Then one has $$\int g_0 \,dx=0;$$ - let $f$ be a polynomial of degree $2$ depending on the derivatives of $w$ of order exactly $n=2k+1$ for $k\in \N$, then $$\int f\,dx=0.$$ Indeed, by , we need to show that $$\int ({\partial}_{x}^{k_{1}}w)({\partial}_{x}^{k_{2}}w)dx=0,$$ when $k_1+k_2=2k+1$ and at least one between $k_1$ and $k_2$ is $\geq1$. Assume $k_2\geq1$. Hence by integrating by parts we have, for ${\sigma}=1$ or ${\sigma}=-1$ $$\int (-1)^{{\sigma}}({\partial}_{x}^{k}w)({\partial}_{x}^{k+1}w)dx= \int (-1)^{{\sigma}}{\partial}_{x}[({\partial}_{x}^{k}w)^{2}]dx=0;$$ - by the above computations for $n=2k$, $k\geq0$, $$\label{costpari} \Gamma^{(n)}(w)=\int g^{(n)}(w)dx,$$ for some $g$ belonging to the class $\Sigma_{n+1}^{3}$, namely $\Gamma^{(n)}$ has a zero of order three at the origin.\ On the other hand for $n=2k+1$, $k\geq0$, we simply have that $$\label{costdispari} \Gamma^{(n)}(w)=\int f_{2}^{(n)}+h^{(n)}(w)dx,\qquad h^{(n)}(w)\in\Sigma^{3}_{n+1}, \quad f_{2}^{(n)}\in {{\mathcal P}}_{n+1}^2.$$ More precisely $f_2^{(n)}$ has the form $$\label{geneform} f_{2}^{(n)}=\sum_{p=0}^{n+1}\sum_{k_1+k_2=p}({\partial}_{x}^{k_1}w)({\partial}_{x}^{k_2}w)c^{k_1,k_2}_{n}.$$ In the following Sections we analyse precisely the form of $c^{k_1,k_2}_{n}$. Computation of $\Gamma^{(n)}$ for $n$ odd {#appsec} ----------------------------------------- Now we want to derive some explicit expression for the coefficients of the quadratic part of the functions $\Gamma^{(n)}$ ($n \in \mathbb{N}$ is odd) introduced in ; more precisely, by recalling the definitions of Section \[diffpol\] and -, we want to compute the coefficients $c_{n}^{k_1,k_2}$ of $f_2^{(n)}$. ### Coefficients of the linear terms {#linsubsec} We begin by computing the coefficients $c_{n}:=c_{n}^{n+1}$ of the linear terms, since this will be useful for the computation of the coefficients of the quadratic terms. Consider the recursion relation between the $\rho^{(n)}$; since $p \in \Sigma^0_0$ and that $p=-\mathtt{c}^{1/3}+{{\mathcal O}}(w)$ for small $\|w\|$, the coefficients in front of the leading order linear term of $\rho^{(n+2)}p$ is proportional to the coefficient of maximal order of the linear term of $\rho^{(n+2)}$. Hence, if we write only the coefficients of maximal order for the linear terms, we get $$\begin{aligned} -c_{n} &= \mathtt{c}^{2/3} \, c_{n+2} -\mathtt{c}^{1/3} 3 c_{n+1}, \nonumber \\ c_{n+2} &= -\mathtt{c}^{-2/3} c_{n}+3 \mathtt{c}^{-1/3} c_{n+1}, n \geq 0. \label{lincoeff}\end{aligned}$$ Now, recall that by and we have that $c_0=-\frac{1}{3\mathtt{c}}$ and $c_1=-\frac{2}{9\mathtt{c}^{4/3}}$. One can check that $$\begin{aligned} c_m = d_1 a^{m} + d_2 b^{m}, \; & m \geq 0, \label{cm} \\ a:= a(\mathtt{c}) = \frac{3+\sqrt{5}}{2\mathtt{c}^{1/3} }, &\; \; b:= b(\mathtt{c}) = \frac{3-\sqrt{5}}{2\mathtt{c}^{1/3} }, \label{ab} \\ d_1 := d_1(\mathtt{c}) = \frac{ -3-\sqrt{5} }{18 \mathtt{c}}, &\; \; d_2 := d_2(\mathtt{c})=\frac{ -3+\sqrt{5} }{18 \mathtt{c}}. \label{d1d2}\end{aligned}$$ \[clinprop\] Notice that the following properties hold. 1. From one readily obtains that $$\begin{aligned} \lim_{\mathtt{c} {\rightarrow}0^\pm} d_1(\mathtt{c}) &= \mp \infty, \\ \lim_{\mathtt{c} {\rightarrow}0^\pm} d_2(\mathtt{c}) &= \mp \infty. \end{aligned}$$ The last two limits imply that in the dispersionless limit $$\begin{aligned} \lim_{\mathtt{c} {\rightarrow}0^\pm} c_m &= -(sgn( \mathtt{c} ))^{m} \infty, \; \; m \geq 1.\end{aligned}$$ 2. By direct computation one also obtains that: 1. for any $\mathtt{c}>0$ $$\begin{cases} c_m<0 &\text{for even} \; m \geq 2; \\ c_m<0 &\text{for odd} \; m \geq 2, \end{cases}$$ 2. for any $\mathtt{c}<0$ $$\begin{cases} c_m>0 &\text{for even} \; m \geq 2; \\ c_m<0 &\text{for odd} \; m \geq 2, \end{cases}$$ 3. $\lim_{m {\rightarrow}\infty} \|c_m\| = +\infty$ (respectively, $\lim_{m {\rightarrow}\infty} \|c_m\| = 0$) for $\|\mathtt{c}\| < \mathtt{c}^\ast:=\left(\frac{3+\sqrt{5}}{2}\right)^3$ (respectively, for $\|\mathtt{c}\| > \mathtt{c}^\ast$). ### Coefficients of the quadratic terms {#quadsubsec} To determine the coefficients in front of the quadratic terms containing the maximal number of derivatives in $f_2^{(n)}$, we integrate $$\begin{aligned} \int \rho^{(n+2)}p^{2} dx &=\int \rho^{(n)}-\rho^{(n)}_{xx}-3\sum_{k_1+k_2=n+1}p\rho^{(k_1)}\rho^{(k_2)} -\sum_{k_1+k_2+k_3=n}\rho^{(k_1)}\rho^{(k_2)}\rho^{(k_3)} dx \label{eqint1} \\ &-\int 3{\partial}_{x}(\rho^{(n+1)}p)-3\sum_{k_1+k_2=n}\rho^{(k_1)}\rho^{(k_2)}_x dx. \label{eqint2}\end{aligned}$$ Now, it is easy to see that the integral in vanishes, since the term ${\partial}_{x}(\rho^{(n+1)}p)$ is a total derivative, and since $$\begin{aligned} \sum_{k_1+k_2=n}\rho^{(k_1)}\rho^{(k_2)}_x &= \sum_{ \substack{k_1+k_2=n \\ k_1>k_2} } {\partial}_x(\rho^{(k_1)}\rho^{(k_2)}). \end{aligned}$$ Now, if write $\rho^{(n)} = f_2^{(n)}+h^{(n)}$ and we consider only the coefficients in front of the quadratic terms containing the maximal number of derivatives, the relation - reads as $$\begin{aligned} \label{inteq2} \mathtt{c}^{2/3} \int \sum_{k_1+k_2=n+3} c^{k_1,k_2}_{n+3} ({\partial}_x^{k_1}w)({\partial}_x^{k_2}w) dx &= 3\mathtt{c}^{1/3} \int \sum_{k_1+k_2=n+1} c_{k_1}c_{k_2} ({\partial}_x^{k_1+1}w)({\partial}_x^{k_2+1}w) dx,\end{aligned}$$ but since $$\begin{aligned} \int \sum_{k_1+k_2=n+3} c^{k_1,k_2}_{n+3} ({\partial}_x^{k_1}w)({\partial}_x^{k_2}w) dx &= \sum_{k=0}^{n+3} c^{k,n+3-k}_{n+3} (-1)^{q(k)} \int ({\partial}_x^{\frac{n+3}{2}} w)^2 dx, \\ & q(k):= \frac{n+3}{2}-k, \\ \int \sum_{k_1+k_2=n+1} c_{k_1}c_{k_2} ({\partial}_x^{k_1+1}w)({\partial}_x^{k_2+1}w) dx &= \sum_{k=0}^{n+1} c_{k} c_{n-k+1} (-1)^{\tilde{q}(k)} \int ({\partial}_x^{\frac{n+3}{2}} w)^2 dx, \\ & \tilde{q}(k):= \frac{n+1}{2}-k,\end{aligned}$$ we obtain $$\begin{aligned} \label{coeffquadeq} \sum_{k=0}^{n+3} (-1)^{q(k)} c^{k,n+3-k}_{n+3} &= 3\mathtt{c}^{-1/3} \sum_{k=0}^{n+1} (-1)^{\tilde{q}(k)} c_{k}c_{n-k+1}.\end{aligned}$$ Now we show that the coefficients in front of the quadratic terms containing the maximal number of derivatives do not vanish. We recall that by Remark \[remarkone\]-$(ii)$ we deal only with $n$ odd. \[coeffquadlemma\] Let $n$ be odd, then, recalling , we have $$\begin{aligned} S_n := \sum_{k=0}^{n+1} (-1)^{\tilde{q}(k)} c_{k}c_{n-k+1} &\neq 0.\end{aligned}$$ Consider the right-hand side of Eq. ; by simple calculations $$\begin{aligned} \label{coeffquadeq2} S_n &= 2 (-1)^{\frac{n+1}{2}} \left( \sum_{ \substack{ k=0,\ldots,\frac{n-1}{2} \\ k \; \text{even} } } c_{k} c_{n+1-k} - \sum_{ \substack{ k=0,\ldots,\frac{n-1}{2} \\ k \; \text{odd} } } c_{k} c_{n+1-k} \right) + c_{\frac{n+1}{2}}^2.\end{aligned}$$ First, one can verify explicitly that $$\begin{aligned} S_1 &= -2c_0c_2+c_1^2 = -2(d_1+d_2)(d_1a^2+d_2b^2)+(d_1a+d_2b)^2 \nonumber \\ &= -\frac{14}{81 \mathtt{c}^{8/3} } \neq 0.\end{aligned}$$ Now we distiguish the two cases $n=4l+3$ ($l \geq 0$) and $n=4l+1$ ($l>0$). Case $n=4l+3$:* first observe that $$\begin{aligned} c_{2s} c_{n+1-2s} - c_{2s+1} c_{n-2s} &= d_1 d_2 \left[ a^{2s} b^{n+1-2s} + b^{2s} a^{n+1-2s} - a^{2s+1} b^{n-2s} - b^{2s+1} a^{n-2s} \right] \nonumber \\ &= d_1 d_2 \left[ a^{2s} b^{n-2s} (b-a) + b^{2s} a^{n-2s} (a-b) \right] \nonumber \\ &= d_1 d_2 \left[ b^n \left(\frac{a}{b}\right)^{2s} (b-a) - a^n \left( \frac{b}{a} \right)^{2s} (b-a) \right] \nonumber \\ &= d_1 d_2 (b-a) \left[ b^n \left( \frac{a}{b} \right)^{2s} - a^n \left( \frac{b}{a} \right)^{2s} \right], \label{prodcoeff}\end{aligned}$$ so in this case we have $$\begin{aligned} \sum_{s=0}^{\frac{n-3}{4}} (c_{2s} c_{n+1-2s} - c_{2s+1} c_{n-2s}) &= d_1 d_2 (b-a) \left[ b^n \frac{ 1- \left( \frac{a^2}{b^2} \right)^{\frac{n+1}{4}} }{ 1-a^2/b^2 } - a^n \frac{ 1- \left( \frac{b^2}{a^2} \right)^{\frac{n+1}{4}} }{ 1-b^2/a^2 } \right] \\ &= d_1 d_2 (b-a) \left( b^{n+2-\frac{n+1}{2}} \frac{ b^{\frac{n+1}{2}} - a^{\frac{n+1}{2}} }{b^2-a^2} + a^{n+2-\frac{n+1}{2}} \frac{ a^{\frac{n+1}{2}}-b^{\frac{n+1}{2}} }{b^2-a^2} \right) \\ &=\frac{d_1 d_2}{a+b} ( b^{\frac{n+3}{2}} - a^{\frac{n+3}{2}} )( b^{\frac{n+1}{2}} - a^{\frac{n+1}{2}} ) \\ &\stackrel{n=4l+3}{=} \frac{d_1d_2}{a+b} (b^{2l+3}-a^{2l+3})(b^{2l+2}-a^{2l+2}),\end{aligned}$$ and the thesis is equivalent to $$\begin{aligned} \label{case3ineq} 2 \frac{d_1 d_2}{a+b} (a^{2l+3}-b^{2l+3})(a^{2l+2}-b^{2l+2}) + (d_1 a^{2l+2}+d_2 b^{2l+2})^2 &\neq 0\end{aligned}$$ In order to verify we observe that $$\begin{aligned} (a^{2l+3}-b^{2l+3})(a^{2l+2}-b^{2l+2}) &= a^{4l+5} \left[ 1 - \left(\frac{b}{a}\right)^{2l+2} - \left(\frac{b}{a}\right)^{2l+3} + \left(\frac{b}{a}\right)^{4l+5} \right] =: a^{4l+5} \alpha_l,\end{aligned}$$ where $(\alpha_l)_{l \in \mathbb{N}}$ is an increasing sequence of positive numbers (which do not depend on $\mathtt{c}$) satisfying $\lim_{l {\rightarrow}\infty} \alpha_l=1$; similarly, we have $$\begin{aligned} (d_1 a^{2l+2} + d_2 b^{2l+2})^2 &= a^{4l+4} d_1^2 \left[ 1 + \frac{d_2}{d_1} \left( \frac{b}{a} \right)^{2l+2} \right]^2 =: a^{4l+4} d_1^2 \beta_l,\end{aligned}$$ where $(\beta_l)_{l \in \mathbb{N}}$ is a decreasing sequence of positive numbers (which do not depend on $\mathtt{c}$) satisfying $\lim_{l {\rightarrow}\infty} \beta_l=1$. Since $2a \frac{d_1 d_2}{a+b}>0$ by and , we have that the left-hand side in is given by $$\begin{aligned} a^{4l+4} \left( 2a \frac{d_1d_2}{a+b} \alpha_l + d_1^2 \beta_l \right) &= \frac{1}{\mathtt{c}^{2+(4l+4)/3}} \left(\frac{3+\sqrt{5}}{2}\right)^{4l+4} \left( \frac{3+\sqrt{5}}{243} \alpha_l + \frac{7+3\sqrt{5}}{162} \beta_l \right) > 0.\end{aligned}$$ Case $n=4l+1$ ($l \geq 1$): by arguing as in , we get $$\begin{aligned} \sum_{s=0}^{l-1} (c_{2s} c_{n+1-2s} - c_{2s+1} c_{n-2s}) &= \frac{d_1 d_2}{a+b} (b^{2l+2}-a^{2l+2})(b^{2l+1}-a^{2l+1}),\end{aligned}$$ and the thesis is equivalent to the following inequality, $$\begin{aligned} \label{case1ineq} 2 \frac{d_1 d_2}{a+b} (b^{2l+2}-a^{2l+2})(b^{2l+1}-a^{2l+1}) + 2 (d_1 a^{2l}+d_2 b^{2l})(d_1 a^{2l+2} + d_2 b^{2l+2} ) - (d_1 a^{2l+1} + d_2 b^{2l+1})^2 &\neq 0.\end{aligned}$$ In order to verify we observe that $$\begin{aligned} & 2 \frac{d_1 d_2}{a+b} (a^{2l+2}-b^{2l+2})(a^{2l+3}-b^{2l+3}) \\ &= 2 \frac{d_1 d_2}{a+b} a^{4l+3} \left[ 1 - \left( \frac{b}{a} \right)^{2l+1} - \left( \frac{b}{a} \right)^{2l+2} + \left( \frac{b}{a} \right)^{4l+3} \right] =: 2 \frac{d_1 d_2}{a+b} a^{4l+3} \gamma_l,\end{aligned}$$ where $(\gamma_l)_{l \geq 1}$ is a increasing sequence of positive numbers (which do not depend on $\mathtt{c}$) satisfying $\lim_{l {\rightarrow}\infty} \gamma_l=1$; similarly, $$\begin{aligned} & 2(d_1 a^{2l} + d_2 b^{2l})(d_1 a^{2l+2} + d_2 b^{2l+2}) \\ &= 2d_1^2 a^{4l+2} \left[ \left(1 + \frac{d_2}{d_1} \left(\frac{b}{a}\right)^{2l} \right) \left(1 + \frac{d_2}{d_1} \left(\frac{b}{a}\right)^{2l+2} \right) \right] =: 2d_1^2 a^{4l+2} \delta_l,\end{aligned}$$ where $(\delta_l)_{l \geq 1}$ is an decreasing sequence of positive numbers (which do not depend on $\mathtt{c}$) satisfying $\lim_{l {\rightarrow}\infty} \delta_l=1$, and $$\begin{aligned} (d_1 a^{2l+1} + d_2 b^{2l+1})^2 &= a^{4l+2} d_1^2 \left( 1 + \frac{d_2}{d1} (b/a)^{2l+1} \right)^2 =: a^{4l+2} d_1^2 \epsilon_l,\end{aligned}$$ where $(\epsilon_l)_{l \geq 1}$ is a decreasing sequence of positive numbers (which do not depend on $\mathtt{c}$) such that $\lim_{l {\rightarrow}\infty} \epsilon_l=1$. Since $2a \frac{d_1 d_2}{a+b}>0$ by and , we have that the left-hand side in is given by $$\begin{aligned} a^{4l+2} \left( 2a \frac{d_1 d_2}{a+b} \gamma_l + 2 d_1^2 \delta_l - d_1^2\epsilon_l \right) &= \frac{1}{\mathtt{c}^{2+(4l+2)/3}} \left(\frac{3+\sqrt{5}}{2}\right)^{4l+2} \left(\frac{3+\sqrt{5}}{243} \gamma_l + \frac{7+3\sqrt{5}}{81} \delta_l - \frac{7+3\sqrt{5}}{162} \epsilon_l \right) \\ &\neq 0.\end{aligned}$$ \[displess\] By arguing as in the proof of the above proposition, one can also show that for any odd number $n$ $$\lim_{\mathtt{c} {\rightarrow}0} \|S_n\| = + \infty,$$ which reflects the fact that the present approach cannot be applied to the dispersionless DP equation . By Proposition \[coeffquadlemma\] we have that the number of derivatives appearing in the quadratic part is greater or equal than the one appearing in the cubic remainder. Proof of Theorem \[costantiMotoDP\] ----------------------------------- The proof of Theorem \[costantiMotoDP\] is based on the following reasoning. We recall that, by the discussion in Section \[appsec\], we have constructed sequence of constants of motion $\Gamma^{(n)}(w)$ with $n\geq0$ of the form $$\Gamma^{(n)}(w)=\int_{\mathbb{T}}\rho^{(n)}(w)dx,$$ where $\rho^{(n)}\in \Sigma^{0}_{n+1}$ (see Lemma \[lemmaron\]) are defined iteratively by , . We recall also (see , ) that, for $n$ even $$\label{costpari10} \Gamma^{(n)}(w)=\int_{\mathbb{T}}\rho^{(n)}(w)dx=\int_{\mathbb{T}}g^{(n)}(w)dx, \quad g^{(n)}\in \Sigma^{3}_{n+1},$$ while for $n$ odd $$\label{costdispari10} \Gamma^{(n)}(w)=\int_{\mathbb{T}}\rho^{(n)}(w)dx=\int_{\mathbb{T}}f_2^{(n)}(w)+h^{(n)}(w)dx, \quad h^{(n)}\in \Sigma^{3}_{n+1},$$ and $f_{2}^{(n)} \in {{\mathcal P}}_{n+1}^2$ as in . \[seqvera\] For any $N\in \N$, $N\geq 2$ there are $r=r(N)>0$, $c_1=c_1(N)>0$, $c_2=c_{2}(N)>0$ and a sequence of functions $$\label{funz} F_n : H^N {\rightarrow}{\mathbb{R}},\quad 1\leq n\leq N,$$ analytic on $B_{H^N}(0, \lvert \mathtt{c} \rvert /2)$ with the following properties. - For any $0\leq n\leq N$, we have $$\label{uffa1} F_n(w)=\int g^{(n)}dx, \quad g^{(n)}\in \Sigma_{n+1}^{3}, \;\;\; n=2k, \quad {\rm for \;\; some} \quad k\in \N,$$ and $$\label{uffa2} F_n(w)=\int \Big(q_2^{(n)}+q_3^{(n)} \Big) dx, \quad q_3^{(n)}\in \Sigma_{n+1}^{3}, \;\;\; n=2k+1, \quad {\rm for \;\; some} \quad k\in \N,$$ where $q_{2}^{(n)} \in {{\mathcal P}}^{2}_{n+1}$ and $$\label{uffa22} \int q_{2}^{(n)}(w)\,dx=\int ({\partial}_{x}^{k+1}w)^{2}\,dx.$$ - For $n=2k$, $k\in \N$, we have $$\label{uffa4} \|F_n(w)\|\leq c_1\\|w\\|_{H^{n+1}}^{3}, \quad \forall \; 0\leq n\leq N, \;\; w\in B_{H^{N}}(0,r),$$ - for $n=2k+1$, $k\in \N$, we have $$\label{uffa5} \|F_n(w)-\int q_2^{(n)}dx\|\leq c_{2} \\|w\\|_{H^{n+1}}^{3}, \quad \forall \; 0\leq n\leq N, \;\; w\in B_{H^{N}}(0,r).$$ Finally, let $u(t,x)$ be the solution of with $u(0,x)=u_0(x)\in H^{N+2}$ such that $\\|u_0\\|_{H^{N+2}}\leq \e$. Define $w(t,x)=(1-{\partial}_{xx})u(t,x)$. Then, as long as $\\|u(t,\cdot)\\|_{H^{N+2}}\leq r/2$, we have that $$\label{uffa3} \frac{d}{dt}F_n(w)=0, \quad 0\leq n\leq N.$$ First of all note that for $0\leq s\leq N$ $$\\|w\\|_{H^{s}}\leq \\|u\\|_{H^{s}}+\\|u\\|_{H^{s+2}}.$$ This implies that as long as $\\|u(t,\cdot)\\|_{H^{N+2}}\leq r/2$ one has $\\|w\\|_{H^{s}}\leq r$. For $n\in \N$ even we define $F_n(w):=\Gamma^{(n)}(w)$, hence holds by . The bound follows by Proposition \[LemmaStimeTaylor\].\ For $n$ odd we construct the functions $F_n$ iteratively, by putting the quadratic parts of the functions $\Gamma^{(n)}$ with $n$ odd in a triangular form.\ First we observe that for any $n$ odd (see ) $$\label{real} \int f_2^{(n)}\,dx= \sum_{\substack{0\le p\leq n+1,\\ p\equiv 0 (2)} } \sum_{k_1+k_2=p} c^{k_1 k_2}_n \,\int (\partial_x^{k_1} w)\,(\partial_x^{k_2} w)\,dx=\sum_{i=0}^{(n+1)/2} d_{n}^i\,\int (\partial_x^{i} w)^2\,dx$$ for some coefficients $d_n^i$ obtained after the integration by parts and by Lemma \[coeffquadlemma\] $$\label{REM} d_n^{(n+1)/2}\neq 0 \qquad \forall n.$$ Now let us show the first step of the triangularization. We write the quadratic part of $\Gamma^{(1)}$ as $$\int f_2^{(1)}\,dx=d_1^0 \int w^2\,dx+d_1^1\,\int w_x^2\,dx.$$ Recall the definition of the constants of motion $M_1$ in . Its quadratic part is, by , and Remark \[remarkone\], $$M_1^{(2)}=\frac{1}{9} \int w^2\,dx.$$ We remark that $\Gamma^{(1)}$, $M_1$ and their linear combinations are constants of motion. We define (recall ) $$F_1:=\frac{1}{d_1^1}\left(\Gamma^{(1)}-9 d_1^0 M_1\right)$$ so that its quadratic part reads as $$\label{passozero} \int q_2^{(1)}\,dx:=\int w_x^2\,dx $$ and for the cubic part we have $$\int q_3^{(1)}\,dx=\frac{1}{d_1^1}h^{(1)}-9 d_1^0 M_1^{(\geq 3)}.$$ Since by Lemma \[polem\] $q_3^{(1)}\in\Sigma_{2}^3$, by Proposition \[LemmaStimeTaylor\] the bound holds for $F^{(1)}$.\ We define iteratively for $n=2 k+1$, $k\geq 1$ $$\label{iteration} F_{2 k+1}:=\frac{1}{d_{2 k+1}^{k+1}}\left( \Gamma^{(2 k+1)}-9 d_{2k+1}^0 M_1-\sum_{i=1}^{k} d_{2k+1}^i F_{2 i+1} \right)$$ and follows easily by Lemma \[polem\]. Now we prove by induction on $k\geq 0$. We just showed the basis of the induction in ($k=0$ ). Now suppose that $$\int q_2^{(2 k+1)}\,dx=\int (\partial_x^{k+1} w)^2\,dx.$$ Then by $$\begin{aligned} d^{k+2}_{2k+3}\int q_2^{(2 k+3)}\,dx &=\int f_2^{(2k +3)}\,dx- d_{2k+3}^0 \int w^2\,dx-\sum_{i=1}^{k+1} d_{2k+3}^i \int (\partial_x^i w)^2\,dx\stackrel{(\ref{real})}{=}d^{k+2}_{2k+3}\int (\partial_x^{k+2} w)^2\,dx,\end{aligned}$$ which proved the claim. The bound follows trivially by triangle inequality and Proposition \[LemmaStimeTaylor\].\ Obviously the $F_{2k +1}$ defined in are constants of motion because they are linear combinations of conserved quantities. We set $K_1(u):=M_1(u)$ where $M_1$ is defined in . For any $n\geq2$ of the form $n=k+1$ we set $$K_{n}(u)=F_{2k-1}(w),$$ where $F_{2k-1}(w)$ is given in Proposition \[seqvera\].\ The item $(0)$ is verified since the functions $F_{n}$ constructed in Proposition satisfy .\ The item $(i)$ and $(ii)$ follows by item $(i)$ of Proposition .\ For the item $(iii)$, follows by and by , the definitions , and by considering the equivalent norm $\lVert u \rVert_{L^2}+\lVert \partial_x^s u \rVert_{L^2} \approx \lVert u \rVert_{H^s}$. Global well-posedness near the origin {#GWPDP} ===================================== In this Section we give the proof of Theorem \[esistenzaglobale\] by using Theorem \[costantiMotoDP\]. First we state the following local well-posedness result for Eq. . \[esistesol2\] For any $s > 3/2$ and for any $u_0\in H^s$ there exist $\bar{T}=\bar{T}(\\|u_0\\|_{H^s})$ and a unique solution $u(t,x)$ of with initial condition $u(0,x)=u_0(x)$ defined for $t\in [-\bar{T},\bar{T}]$ belonging to the space $C([-\bar{T},\bar{T}],{{H}}^{s})$, such that $$\begin{aligned} \\|u(t)\\|_{H^s} &\leq 2 \\|u_0\\|_{H^s}, \; \; \|t\| \leq \bar{T} \leq \frac{1}{ 2C_s \\|u_0\\|_{H^s} }, \label{estnormsol}\end{aligned}$$ for some constant $C_s>0$ depending only on $s$. The proof of the above result can be found in the Appendix \[appendA\]. It is based on a Galerkin-type approximation method, and follows closely the argument reported in [@HH] for the dispersionless DP equation . \[smalldata\] Proposition \[esistesol2\] implies that for any $s>3/2$ there exists $r(s)>0$ such that for any $u_0 \in B_{H^s}(0,r(s))$ Eq. with initial datum $u_0$ admits a solution $u(t,x)$ such that $\\|u(t)\\|_{H^s} \leq 2\\|u_0\\|_{H^s}$ for $\|t\| \leq \frac{1}{2C_s r(s)}$. The proof of Theorem \[esistenzaglobale\] is based on the following bootstrap argument. Consider the function $u(t,x)$ solution of , defined on some interval $t\in[-{T},{T}]$ with $0<T\leq \bar{T}$ with initial datum $\\|u(0;\cdot)\\|_{H^{s}}\leq r_0$ given by Proposition \[esistesol2\]. By choosing $n:=n(s)=[s]-1$, if $r_0$ small enough, we have by item $(iii)$ of Theorem \[costantiMotoDP\] and by Proposition \[esistesol2\] that $$\begin{aligned} \label{bound1} \\|u(t,\cdot)\\|_{H^{s}}^{2}\leq \\|u(\cdot)\\|_{L^{2}}^{2}+\tilde{c} \|K_{n}^{(0)}(u(t,\cdot))\| &\leq \\|u(\cdot)\\|_{L^{2}}^{2}+\tilde{c} \|K_{n}(u(t,\cdot))\|+ \tilde{c} C\\|u(t,\cdot)\\|_{H^{s}}\\|u(t,\cdot)\\|^{2}_{H^{s}},\end{aligned}$$ with $\tilde{c},C$ given by Theorem \[costantiMotoDP\]. Since $$\begin{aligned} K_1^{(0)}(u(t,\cdot)) &= \int (u-u_{xx})^2 dx \\ &= \int u^2 dx + 2 \int u_x^2 dx + \int u_{xx}^2 dx $$ by we have that for $r_0$ small enough $$\begin{aligned} \label{boundL2} \\|u(t,\cdot)\\|^2_{L^2} &\leq C \|K_1(u(t,\cdot))\|.\end{aligned}$$ Since $K_1(u),K_{n}(u)$ are constant of motion, we have $$\label{valesempre} \|K_{1}(u(t,\cdot))\|+\|K_{n}(u(t,\cdot))\|\leq \|K_{1}(u(0,\cdot))\|+\|K_{n}(u(,\cdot))\|\stackrel{(\ref{equivalenzaNorma}),(\ref{stimecostanti})}{\leq} {\kappa}(s)\\|u(0,\cdot)\\|^{2}_{H^{s}}\leq {\kappa}(s)r_0^{2},$$ for some ${\kappa}(s)>0$ depending only on $s$. The bound reads $$\label{bound1bis} \\|u(t,\cdot)\\|_{H^{s}}^{2}\leq\tilde{c}{\kappa}(s)r_0^{2}+\tilde{c} C\\|u(t,\cdot)\\|_{H^{s}}\\|u(t,\cdot)\\|^{2}_{H^{s}}.$$ Now let $\widehat{T}$ be the supremum of those $T$ such that the solution $u(t,x)$ is defined on $[-T,T]$ and $$\label{assurdo} \sup_{t\in [-T,T]}\\|u(t,\cdot)\\|^{2}_{H^{s}}\leq Q(s)r^2_0,$$ where $Q(s)\geq 4 {\kappa}(s)\tilde{c}$, with ${\kappa}(s)$ given in and $\tilde{c}$ given by Theorem \[costantiMotoDP\]. For $t\in [-\hat{T},\hat{T}]$ we deduce, by , that $$\label{bound1tris} \\|u(t,\cdot)\\|_{H^{s}}^{2}\leq \tilde{c}{\kappa}(s)r_0^{2}+\tilde{c}C\sqrt{Q(s)} r_0\\|u(t,\cdot)\\|^{2}_{H^{s}}.$$ Hence, if we take $r_0$ sufficiently small such that $\tilde{c}C\sqrt{Q(s)} r_0\leq 1/2$, we obtain $$\label{speriamobene} \\|u(t,\cdot)\\|_{H^{s}}^{2}\leq 2\tilde{c} r_0^{2}<Q(s)r_0^{2}.$$ Of course estimate leads to the contradiction of the fact that $\widehat{T}$ is the supremum. Since estimate does not depend on $\hat{T}$, we must have $\widehat{T}=+\infty$, which implies Theorem \[esistenzaglobale\]. Birkhoff resonances {#proofteoBirk} =================== In this Section we prove Theorem \[teoBirk\]. First we need some preliminaries definitions and results to show the formal Birkhoff normal form procedure. Recall the definitions given in Section \[prelimi\] for the space of formal polynomials. [(Poisson brackets)]{} Let $P=\sum_{\alpha\in\mathcal{I}_n} P_{\alpha} u^{\alpha}$ and $Q=\sum_{\beta\in\mathcal{I}_m} Q_{\beta} u^{\beta}$ (recall ) two formal homogenous polynomials. We define $\{\,\cdot\,,\, \cdot\, \}\colon \mathscr{P}^{(n)}\times \mathscr{P}^{(m)}{\rightarrow}\mathscr{F}$ $$\label{poisson} \{ P, Q\}:=\sum_{\alpha\in\mathcal{I}_n, \beta\in\mathcal{I}_m} P_{\alpha} Q_{\beta}\,\{ u^{\alpha}, u^{\beta}\}=\sum_{\alpha\in\mathcal{I}_n, \beta\in\mathcal{I}_m} \sum_{j} \big((-\mathrm{i}) \omega(j) \alpha_j \beta_{-j} \big) P_{\alpha} Q_{\beta} \,u^{\alpha+\beta-\mathtt{e}_j-\mathtt{e}_{-j}}$$ where $\mathtt{e}_k$ is the element of $\mathbb{N}^{\mathbb{Z}}$ with all components equal to zero except for the $k$-th one, which is equal to $1$. In the following lemma we prove that the above definition is well posed. We point out that the assumption of zero momentum (see ) is a key ingredient for the proof. Given $P\in\mathscr{P}^{(n)}$, $Q\in\mathscr{P}^{(m)}$ we have that - the sum $P+Q\in\mathscr{P}^{({\max\{m, n\}})}$. - the Poisson bracket $\{ P, Q\}\in\mathscr{P}^{(n+m)}$. The item $(i)$ is trivial. We prove item $(ii)$. We write $P=\sum_{\alpha\in\mathcal{I}_n} P_{\alpha} u^{\alpha}$ and $Q=\sum_{\beta\in\mathcal{I}_m} Q_{\beta} u^{\beta}$.\ Recalling , we have to prove the following claim: given $\gamma\in\mathcal{I}_{n+m-2}$ there is only a finite number of $\alpha, \beta, j$ such that $$\label{bob} \gamma=\alpha+\beta-\mathtt{e}_j-\mathtt{e}_{-j}, \quad \alpha_j \beta_{-j}\neq 0,$$ where we denoted by $\mathtt{e}_j$ the element of $\mathbb{N}^{\mathbb{Z}}$ with all components zero except for the $j$-th, which is $1$. Indeed, if this holds, there exists a sequence $(R_{\gamma})_{\gamma\in\mathcal{I}_{n+m-2}}$ of complex numbers such that $$\{ P, Q\}=\sum_{\gamma\in \mathcal{I}_{m+n-2}} R_{\gamma}\,u^{\gamma}.$$ First we observe the following: given $\gamma\in\mathcal{I}_{k}$, for some $k\geq 2$, there exist only finitely many couples $(a, b)\in\mathbb{N}^{\mathbb{Z}}\times \mathbb{N}^{\mathbb{Z}}$ such that we can decompose $\gamma=a+b$. Note that $\mathcal{M}(\gamma)=\mathcal{M}(a)+\mathcal{M}(b)=0$. We call $\mathtt{M}^{\gamma}:=\sum_{j>0} j \gamma_j$ and we observe that, for any choice of $a$ and $b$, we have $\lvert \mathcal{M}(a) \rvert\le \mathtt{M}^{\gamma}$.\ We can choose $a$ and $b$ such that $\mathcal{M}(a)=j$, for instance, $$a=\alpha-\mathtt{e}_{-j}, \quad b=\beta-\mathtt{e}_{j}.$$ hence $\lvert j \rvert\le \mathtt{M}^{\gamma}$. So given $\gamma$ there is only a finite number of $j$ for which holds.\ Now we prove that, given $\gamma$ and $j$, there is only a finite number of $\alpha$ and $\beta$ such that holds and so the claim is proved. We have $\gamma+\mathtt{e}_j+\mathtt{e}_{-j}=\alpha+\beta$. Hence we have to split the left-hand side in two elements of $\mathbb{N}^{\mathbb{Z}}$ with zero momentum. If $\gamma=a+b$ with $(a, b)\in\mathbb{N}^{\mathbb{Z}}\times \mathbb{N}^{\mathbb{Z}}$, then there is only a finite number of choices for $a$ and $b$, which are $$a=\alpha-\mathtt{e}_{\pm j}, \quad b=\beta-\mathtt{e}_{\mp j}.$$ #### Adjoint action and quadratic Hamiltonians. Let $G\in\mathscr{P}^{(m)}$, $m\geq 0$. We define the adjoint action of $G$ as $$\mathrm{ad}_G \colon \mathscr{P}^{(n)}{\rightarrow}\mathscr{P}^{(m+n)} \subset \mathscr{F}^{(\geq m)},\qquad \mathrm{ad}_G[P]:=\{ G, P\},$$ then we extend it to the entire $\mathscr{F}$ by setting $$\mathrm{ad}_G [P]:=\big(\mathrm{ad}_G [P^{(n)}] \big)_{n\geq 0}$$ with $$\Pi^{(d)} \mathrm{ad}_G[P]= \begin{cases} \sum_{n=d-m} \{ G^{(m)}, P^{(n)} \} \quad \mbox{if}\,\, d\geq m,\\ 0 \qquad \mbox{otherwise} \end{cases}$$ for $d\geq 0$. Note that $\Pi^{(d)} \mathrm{ad}_G[P]\in\mathscr{P}^{(d)}$ since the above sum is finite, so $\mathrm{ad}_G$ is well defined on $\mathscr{F}$.\ We define the kernel and range of $\mathrm{ad}_G$ as $$Ker(G):=\{ F\in\mathscr{F} : \{ F, G\}=0\}, \quad Rg(G):=\{ F\in\mathscr{F} : \{ F, G\}\neq 0\}.$$ Consider a quadratic Hamiltonian $G$ in diagonal form, $G=\sum_{j} \lambda(j) \lvert u_j \rvert^2$, $\lambda(j)\in\mathbb{C}$. Given $\alpha\in\mathbb{N}^{\mathbb{Z}}$ we define the associated $G$-divisor as (recall ) $$\label{Gdivisor} \Omega_G(\alpha)=\sum_j \omega(j) \lambda(j) \alpha_j$$ and we have $$\{ G(u), u^{\alpha}\}=\Big(-i \sum_{j} \omega(j)\lambda(j) \alpha_j\Big) u^{\alpha}=-i \Omega_{G}({\alpha})u^{{\alpha}}.$$ We define $\Pi_{{\rm Ker}(G)}$ as the projector on the kernel of the adjoint action ${\rm ad}_{G}[\cdot]$, i.e. $$\forall \,{\alpha}\in \mathcal{I}_{n}\quad \Pi_{{\rm Ker}(G)}(u^{{\alpha}}):=\left\{ \begin{aligned} &u^{{\alpha}}\;\;{\rm if}\;\; \Omega_G({\alpha})\neq0,\\ &0 \;\;\;\; {\rm if}\;\; \Omega_G({\alpha})=0. \end{aligned}\right.$$ We define the projector on the range of the adjoint action as $\Pi_{{\rm Rg}(G)}:=\mathrm{I}-\Pi_{{\rm Ker}(G)}$ where $\mathrm{I}$ is the identity. We define the action of $\Pi_{{\rm Rg}(G)}$ and $\Pi_{{\rm Ker}(G)}$ on any Hamiltonian $H\in \mathscr{P}^{(n)}$ by linearity. #### Exponential map and Lie transformation. Let $G\in\mathscr{P}^{(m)}$ with $m\geq 1$. We note that for $k\geq 0$ $$\mathrm{ad}^k_G\colon \mathscr{P}^{(n)}{\rightarrow}\mathscr{P}^{(n+k m)}\subset \mathscr{F}^{(\geq n+k m)}$$ We define the exponential map $e^{\{G, \cdot}\colon \mathscr{F}{\rightarrow}\mathscr{F}$ as $$e^{\{G, \cdot} P:=\Big(\sum_{k\geq 0} \frac{\mathrm{ad}^{k}_G[P^{(n)}]}{k!}\Big)_{n\geq 0}$$ and it is well-defined since $$\Pi^{(d)} e^{\{G, \cdot} P= \sum_{(n, k)\,:\,n+k m=d} \frac{\mathrm{ad}^k_G[P^{(n)}] }{k!}$$ but there is only a finite number of couples $(n, k)\in\mathbb{N}^2$ such that $n+m k=d$ ($d \geq 0$ and $m\geq 1$ are fixed). Hence $\Pi^{(d)} e^{\{G, \cdot} P\in\mathscr{P}^{(d)}$. Let $\chi\in\mathscr{P}^{(n)}$ and $H\in\mathscr{F}$ be two Hamiltonians. We call $\Phi_{\chi}^t$ the flow of $\chi$, namely $$\begin{cases} \dfrac{d}{d t} \Phi_{\chi}^t(u)=J \nabla \chi(u),\\[2mm] \Phi_{\chi}^0(u)=u. \end{cases}$$ We have $$\frac{d^k}{d t^k} \Big( H \circ \Phi_{\chi}^t\Big)=\mathrm{ad}_{\chi}^k[H]\circ \Phi_{\chi}^t$$ Then by expanding in the (formal) Taylor series at $t=0$ we get $$H\circ \Phi_{\chi}^t:=\sum_{k\geq 0} \frac{\mathrm{ad}^k_{\chi}[H]}{k!}\,t^k.$$ We call Lie transformation the map at time one $\Phi_{\chi}^{t=1}=:\Phi_{\chi}$ and we have $$H\circ \Phi_{\chi}:=\sum_{k\geq 0} \frac{\mathrm{ad}^k_{\chi}[H]}{k!}=e^{\{ \chi, \cdot} H.$$ Proof of Theorem \[teoBirk\] ---------------------------- Let $H\in\mathscr{P}^{(\le n)}$ be a Hamiltonian. The goal of the formal Birkhoff normal form procedure is to construct a change of coordinates $\Phi$ which puts the Hamiltonian $H$ in a Birkhoff normal form of some order (recall ). This algorithm consists of different steps. If we denote by $\chi_k\in\mathscr{P}^{(k)}$ the generators of the Birkhoff transformation at the step $k$ and with $$H_0:=H, \quad H_k:=e^{\{ \chi_k}\cdots e^{\{\chi_1, \cdot} H \quad k>0,$$ then we have to choose $\chi_{k+1}$ such that $$\label{beam} \Pi^{(k+1)}e^{\{ \chi_{k+1}, \cdot} H_k=\Pi_{Ker(H^{(0)})} \Pi^{(k+1)} H_k=\Pi_{Ker(H^{(0)})} H_k^{(k+1)}$$ The right-hand side of contributes to the normal form of $H_{k+1}$. We want to show that this homogenous polynomial of degree $k+3$ is supported on $\mathcal{N}_{k+1}^$ (recall Definition \[defrisonanza\]). First we prove that, given a finite set of Hamiltonians in involution (namely which pairwise commute) and fixed $N$, there exists a Birkhoff transformation $\Phi_N$ which puts all these Hamiltonians in Birkhoff normal form of order $N$ according to Definition \[defBirkformN\]. It is sufficient to prove that for two commuting Hamiltonians. \[lemmabellissimo\] Consider $H, K\in\mathscr{F}$ two commuting Hamiltonians. For any $N$ there exist, at least formally, a change of coordinates $\Phi_N$ such that $$H\circ \Phi_N=H^{(0)}+Z_{N}+R_N, \quad K\circ \Phi_N=K^{(0)}+W_{N}+Q_N$$ where $Z_N, W_N\in\mathcal{P}^{(N)}$ commuting with $H^{(0)}$ and $K^{(0)}$. $R_N, Q_N\in\mathscr{F}^{(\geq N+1)}$. We argue by induction on the number of steps $N$. For $N=0$ it is trivial since $\Phi_0$ is the identity map.\ Suppose that we have performed $N$ steps. By the fact that $\{ H, K \}=0$ then $\{ H, K\}\circ \Phi_N=0$ and so, at each order, we have $$\begin{aligned} &\{ H^{(0)}, K^{(0)}\}=0,\\ &\{ H^{(0)}, W_N\}+\{ Z_N, K^{(0)}\}+\Pi^{(\le N)}\{ Z_N, W_N\}=0,\\ &\Pi^{(N+1)} \{ Z_N, W_N\}+\{ H^{(0)}, Q_N^{(N+1)}\}+\{ R_N^{(N+1)}, K^{(0)} \}=0,\\ &\dots\end{aligned}$$ By the inductive hypothesis $W_N, Z_N\in Ker(H^{(0)})\cap Ker(K^{(0)})$, hence $\{ H^{(0)}, W_N\}=\{ Z_N, K^{(0)}\}=0$ and $$\label{identita} \{ H^{(0)}, Q_N^{(N+1)}\}+\{ R_N^{(N+1)}, K^{(0)} \}=0$$ since $\{ H^{(0)}, Q_N^{(N+1)}\}\in Rg(H^{(0)})$ and $\{ R_N^{(N+1)}, K^{(0)} \}\in Rg(K^{(0)})$. We note the following fact, which derives from the Jacobi identity: if $f\in Ker(H^{(0)})$ then $\{ f, K^{(0)}\}\in Ker(H^{(0)})$.\ Then we have that $\{ \Pi_{Ker(H^{(0)})} R_N^{(N+1)}, K^{(0)} \}\in Ker(H^{(0)})$ and by $$\{ \Pi_{Ker(H^{(0)})} R_N^{(N+1)}, K^{(0)} \}=-\{ \Pi_{Rg(H^{(0)})} R_N^{(N+1)}, K^{(0)} \}+\{ H^{(0)}, Q_N^{(N+1)}\}\in Rg(H^{(0)}).$$ Thus $\{ \Pi_{Ker(H^{(0)})} R_N^{(N+1)}, K^{(0)} \}=0$ and $$\Pi_{Ker(H^{(0)})} R_N^{(N+1)}=\Pi_{Ker(H^{(0)})} \Pi_{Ker(K^{(0)})} R_N^{(N+1)}.$$ By symmetry $\Pi_{Ker(K^{(0)})} Q_N^{(N+1)}=\Pi_{Ker(H^{(0)})} \Pi_{Ker(K^{(0)})} Q_N^{(N+1)}$. Hence $$\label{prisoner} \Pi_{Rg(H^{(0)})} \Pi_{Ker(K^{(0)})} Q_N^{(N+1)}=\Pi_{Rg(K^{(0)})} \Pi_{Ker(H^{(0)})} R_N^{(N+1)}=0.$$ In order to obtain the Birkhoff normal form at order $N+1$ we consider a Birkhoff transformation $\Phi_{\chi_{n+1}}$ with generator $\chi_{N+1}\in \mathscr{P}^{(N+1)}$ and we define $\Phi_{N+1}:=\Phi_N\circ \Phi_{\chi_{n+1}}$. The function $\chi_{N+1}$ is chosen in order to solve the homological equation $$\{ H^{(0)}, \chi_{N+1}\}=-\Pi_{Rg(H^{(0)})} R_N^{(N+1)} \stackrel{(\ref{prisoner})}{=}-\Pi_{Rg(K^{(0)})}\Pi_{Rg(H^{(0)})} R_N^{(N+1)}.$$ We now show that $\chi_{N+1}$ solves also the homological equation for $K_N$. Indeed, by the fact that $\mathrm{ad}_{H^{(0)}}^{-1}$ commutes with $\mathrm{ad}_{K^{(0)}}$ on the intersection $Rg(H^{(0)})\cap Rg(K^{(0)})$ $$\{ K^{(0)}, \chi_{N+1}\}=-\mathrm{ad}_{H^{(0)}}^{-1}\{ K^{(0)}, \Pi_{Rg(K^{(0)})}\Pi_{Rg(H^{(0)})} R_N^{(N+1)} \}$$ and by , we get $$\{ K^{(0)}, \Pi_{Rg(K^{(0)})}\Pi_{Rg(H^{(0)})} R_N^{(N+1)} \}=\{H^{(0)}, \Pi_{Rg(K^{(0)})}\Pi_{Rg(H^{(0)})} Q_N^{(N+1)} \}.$$ Suppose that we performed $N$ Birkhoff steps. Then the transformed Hamiltonian is $$H_N=H^{(0)}+Z_N+R_N, \quad Z_N\in\mathscr{P}^{(\le N)}, R_N\in\mathscr{F}^{(\geq N+1)}$$ where $Z_N$ is action-preserving.\ If we perform the $(N+1)$-th step then the term $\Pi_{Ker(H^{(0)})} R_N^{(N+1)}\in\mathscr{P}^{(N+1)}$ contributes to the normal form. We have to show the following claim - The resonant term $\Pi_{Ker(H^{(0)})} R_N^{(N+1)}$ is supported on $\mathcal{N}_{N+1}^$. The identity holds for any $N\geq 0$. Thus if we consider the set of commuting Hamiltonians $K_1,\dots, K_{N+2}$ (defined in and in Theorem \[costantiMotoDP\]), which are in the Birkhoff normal form after $N$ steps $$K_{m, N}=K_m^{(0)}+W_{m, N}+Q_{m, N} \qquad m=1, \dots, N+2,$$ we have that $R^{(N+1)}_N\in\mathscr{P}^{(N+1)}$ satisfies $$\{ H^{(0)}, Q^{(N+1)}_{m, N}\}=\{ K_m^{(0)}, R^{(N+1)}_N \}, \quad \forall m=1, \dots, N+2.$$ Thus by the above relation writes as (recall ) $$\label{base} \Omega(\alpha)\, K_{m, N, \alpha}^{(N+1)}=\Omega_{K_m^{(0)}}(\alpha)\,R_{N, \alpha}^{(N+1)}=\Big(\sum_j \,j^{2 (m-1)}(1+j^2)^2\,\omega(j)\,\alpha_j\Big) R_{N, \alpha}^{(N+1)}, $$ for any $\ \alpha\in\mathcal{I}_{N+1}$ and any $m=1, \dots, N+2$. If $\Omega(\alpha)\neq 0$ or $R_{N, \alpha}^{(N+1)}=0$ then the resonant term $\Pi_{Ker(H^{(0)})} R_N^{(N+1)}=0$.\ If $\Omega(\alpha)=0$ then $R_{N, \alpha}^{(N+1)}\neq 0$ if and only if $$\label{torrone} \sum_j \,j^{2 (m-1)}(1+j^2)^2\,\omega(j)\,\alpha_j=\sum_j \,j^{2 m-1}(1+j^2)\,(4+j^2)\,\alpha_j=0, \quad \forall \alpha \in\mathcal{I}_{N+1}, \quad \forall m=1, \dots, N+2.$$ We have to prove that the linear $(N+3)\times(N+3)$-dimensional system has no solutions, except for $\alpha\in \mathcal{N}_{N+1}^$.\ Finding a solution of is equivalent to prove that there are integers $j_1, \dots, j_{N+3}\in\mathbb{Z}\setminus\{0\}$ such that the following matrix $M$ has a non trivial kernel $$\label{torrone2} M:=\mathrm{diag}_{i=1, \dots, N+3} \Big( (1+j_i^2)\,(4+j_i^2) \Big) \,V, \quad V : =\begin{bmatrix} j_1 & \dots & j_N\\ j_1^3 & \dots & j_N^3\\ \vdots & \dots & \vdots\\ j_1^{2 (N+3)+1}& \dots & j_N^{2(N+3)+1} \end{bmatrix} .$$ Hence our goal is to prove that $\det M=0$ if and only if $(j_1, \dots, j_{N+3})=(i, -i, j, -j, \dots)$ or its permutations. Clearly $\det M=0$ if and only if $\det V=0$. Note that $$\det V=\Big(\prod_{i=1}^{N+3} j_i\Big)\,\,\,\det \begin{bmatrix} 1 & \dots & 1\\ j_1^2 & \dots & j_{N+3}^2\\ \vdots & \dots & \vdots\\ j_1^{2 N+6}& \dots & j_{N+3}^{2N+6} \end{bmatrix}$$ and by renaming $x_i=j_i^2$ and by using the well known formula for the determinant of the Vandermonde matrix we have that $$\det V =\sqrt{\prod_{i=1}^{N+3} x_i}\,\,\,\prod_{i< j} (x_i-x_j).$$ So it is clear that $\det V=0$ if and only if $$\Big((j_1+j_2)\dots (j_{N+2}+j_{N+3})\Big)\Big( (j_1-j_2)\dots (j_{N+2}-j_{N+3}) \Big)=0.$$ By the fact that the indices $j_i$ are all distincts we deduce the claim. Appendix ========= Proof of local existence {#appendA} ------------------------- Here we prove Proposition \[esistesol2\] about the local well-posedness of Eq. . The argument follows closely the proof of Theorem 1.1 in [@HH], which discusses the well-posedness of the dispersionless DP equation . We present the proof in the compact case; during the proof we point out the minor changes one has to make in order to adjust the proof to the noncompact case. First observe that if $u \in H^s(\mathbb{T};\mathbb{R})$, then $u {\partial}_x u + (1-{\partial}_{xx})^{-1} {\partial}_{xxx}u \in H^{s-1}(\mathbb{T};\mathbb{R})$. We handle this problem by considering the mollified version of : fix a Schwartz function $j \in \mathcal{S}({\mathbb{R}})$ satysfying $0 \leq \hat{j}(\xi) \leq 1$ for all $\xi \in \mathbb{R}$, and $\hat{j}(\xi)=1$ for $\|\xi\|\leq 1$. Then we define the periodic functions $j_\epsilon$, $0 < \epsilon \leq 1$ by the following formula, $$\begin{aligned} j_\epsilon(x) &:=\frac{1}{2\pi} \sum_{n \in \mathbb{Z}} \hat{j}(\epsilon n) e^{inx},\end{aligned}$$ and we define the mollifier by $$\begin{aligned} J_\epsilon f &:= j_\epsilon \ast f.\end{aligned}$$ By direct computation one has that $\hat{j}_\epsilon(k) =\hat{j}(\epsilon k)$; furthermore, for $0 < \sigma \leq s$ the map $Id-J_{\epsilon}:H^s(\mathbb{T};\mathbb{R}) {\rightarrow}H^\sigma(\mathbb{T};\mathbb{R})$ satisfies $$\begin{aligned} \label{opnorm} \\| Id-J_{\epsilon} \\|_{ L( H^s(\mathbb{T};\mathbb{R}) , H^\sigma(\mathbb{T};\mathbb{R}) ) } &= o(\epsilon^{s-\sigma}).\end{aligned}$$ Now recall that Eq. is obtained by its dispersionless version by applying the boost $u \mapsto \mathtt{c}+u$. This means that we can derive the mollified version of simply by translation form the mollified dispersionless DP equation (see Eq. (110) in [@HH]; in the rest of this section we denote by $D$ the operator $(1-{\partial}_{xx})^{1/2}$) $$\begin{aligned} \label{mollDP} u_t &= - J_\epsilon (J_\epsilon (\mathtt{c}+u) {\partial}_x J_\epsilon (\mathtt{c}+u)) -\frac{3}{2} {\partial}_x D^{-2} \left( (\mathtt{c}+u)^2 \right), \; 0 < \epsilon \leq 1,\end{aligned}$$ with initial datum $u(0,x)=u_0(x) \in H^s$. Now introduce the map $F_\epsilon: H^s {\rightarrow}H^s$, $$\begin{aligned} F_\epsilon(u) &= - J_\epsilon (J_\epsilon (\mathtt{c}+u) {\partial}_x J_\epsilon (\mathtt{c}+u)) -\frac{3}{2} {\partial}_x D^{-2} \left( (\mathtt{c}+u)^2 \right) \nonumber \\ &= -J_\epsilon (\mathtt{c} J_\epsilon {\partial}_x J_\epsilon u) - J_\epsilon (J_\epsilon u {\partial}_x J_\epsilon u) - \frac{3}{2} {\partial}_x D^{-2}(u^2) - 3\mathtt{c} {\partial}_x D^{-2} (u); \label{Feps}\end{aligned}$$ we can observe that for any $\epsilon$ the map $F_\epsilon$ is differentiable. Therefore with initial datum $u(0,\cdot)=u_0 \in H^s$, $s>3/2$, defines an ODE on $H^s$, which thus admits a unique solution $u_\epsilon$ with existence time $T_\epsilon>0$. We now prove Proposition \[esistesol2\] after some intermediate lemmata. \[LWPlemma1\] Let $s>3/2$, then there exists $C(s)>0$ such that the existence time $T_\epsilon$ for the solution of satisfies $$\begin{aligned} \label{timespan} T_\epsilon &\geq \bar{T}:=\bar{T}(\\|u_0\\|_{H^s})=\frac{1}{2C(s)\\|u_0\\|_{H^s}},\end{aligned}$$ while the solution $u_\epsilon$ satisfies $$\begin{aligned} \label{estsolmoll} \\|u_\epsilon(t)\\|_{H^s} \leq 2 \\|u_0\\|_{H^s}, \; \|t\| \leq \bar{T}.\end{aligned}$$ First apply the operator $D^{s}$ to both sides of , and multiply both sides by $D^{s} u_\epsilon$. By integration and we have $$\begin{aligned} \frac{1}{2} \frac{d}{dt} \\|u_\epsilon\\|^2_{H^s} &= - \int_0^{2\pi} D^s u_\epsilon \; D^s \left[ J_\epsilon ( J_\epsilon u_\epsilon {\partial}_x J_\epsilon u_\epsilon ) + \frac{3}{2} {\partial}_x D^{-2} \left( u^2 \right) \right] dx,\end{aligned}$$ where we exploited the fact that the first and the last term in are linear in $u$ and are given by the action of skew-adjoint Fourier multipliers which commute with $D^s$ (all these facts imply that the terms one would need to add to formula (117) in [@HH] vanish).\ In order to commute the operator $D^s$ with $J_\epsilon u_\epsilon$ we apply the following Kato-Ponce commutator estimate (see [@KatoPonce]). Let $s>0$, then there exists $C(s)>0$ such that $$\begin{aligned} \label{KPineq} \\|D^s(fg)-f D^s g\\|_{L^2} &\leq C(s) \left( \\|D^s f\\|_{L^2} \\|g\\|_{L^\infty} + \\|{\partial}_x f\\|_{L^\infty} \\|D^{s-1}g\\|_{L^2} \right).\end{aligned}$$ By exploiting and Sobolev embedding we have that there exists $C(s)>0$ such that $$\begin{aligned} \frac{d}{dt} \\|u_\epsilon\\|^2_{H^s} \leq 2 C(s) \\|u_\epsilon\\|^3_{H^s},\end{aligned}$$ which implies that $$\begin{aligned} \label{ineq} \\| u_\epsilon(t) \\|_{H^s} &\leq \frac{ \\|u_0\\|_{H^s} }{ 1- C(s) \\|u_0\\|_{H^s} t},\end{aligned}$$ and by setting $\bar{T}:= \frac{1}{ 2 C(s) \\|u_0\\|_{H^s} }$ we get the thesis. \[LWPexist\] Consider Eq. with initial datum $u_0 \in H^s$, $s>3/2$. Then there exists a solution $u \in C([-\bar T, \bar T]; H^s)$ with $\bar T$ as in such that $$\begin{aligned} \label{estsol} \\|u(t)\\|_{H^s} \leq 2 \\|u_0\\|_{H^s}, \; \|t\| \leq \bar{T}.\end{aligned}$$ To simplify the notation we set $I:=[-\bar T, \bar T]$. The proof is divided in several steps, whose purpose is to obtain the convergence of the family $(u_\epsilon)_{0 < \epsilon \leq 1}$ by extracting subsequences $(u_{\epsilon_\nu})_\nu$; after each such extraction, we assume that the resuling sequence is relabeled as $(u_\epsilon)_\epsilon$. [Step 1: weak$^\star$ convergence in $L^\infty(I;H^s)$.]{} The family $(u_\epsilon)_{0 < \epsilon \leq 1}$ is bounded in the space $C(I;H^s) \subset L^\infty(I;H^s)$. Since $L^\infty(I;H^s)$ is the dual of the space $L^1(I;H^s)$, Alaoglu’s Theorem implies that $(u_\epsilon)_\epsilon$ is precompact with respect to the weak$^\star$ topology. Hence there exists a subsequence $(u_{\epsilon_\nu})_\nu$ which converges to $u \in L^\infty(I;H^s)$ weakly$^\star$, and such that $u$ satisfies .\ [Step 2: convergence in $C(I;H^{s-1})$.]{} In order to show the strong convergence in $C(I;H^{s-1})$, we show that $(u_\epsilon)_\epsilon$ satisfies the hypotheses of Ascoli-Arzelá Theorem. Indeed, $(u_\epsilon)_\epsilon$ is equicontinuous, since for any $t_1, t_2 \in I$ $$\begin{aligned} \\|u_\epsilon(t_1) - u_\epsilon(t_2)\\|_{H^{s-1}} &\leq \sup_{t \in I} \\| {\partial}_t u_\epsilon \\|_{H^{s-1}} \, \|t_1-t_2\| \nonumber \\ &\stackrel{ \eqref{mollDP},\eqref{Feps}, \eqref{KPineq} }{\leq} 10 C(s) (\\|u_0\\|_{H^s}+\\|u_0\\|^2_{H^s}) \|t_1-t_2\|.\end{aligned}$$ Setting $U(t):=(u_\epsilon(t))_\epsilon$, we see that for any $t \in I$ the set $U(t) \subset H^s$ is bounded. On the other hand, since $\mathbb{T}$ is a compact manifold, we have that the inclusion $i:H^s {\rightarrow}H^{s-1}$ is compact. Therefore $U(t)$ is precompact in $H^{s-1}$.\ [Step 3: convergence in $C(I;H^{s-\sigma})$, $\sigma \in (0,1)$.]{} For each $\sigma \in (0,1)$ we have $$\begin{aligned} \\|u_\epsilon\\|_{ C^\sigma(I;H^{s-\sigma}) } &= \sup_{t \in I} \\|u_\epsilon\\|_{ H^{s-\sigma} } + \sup_{t \neq t'} \frac{ \\|u_\epsilon(t)-u_\epsilon(t')\\|_{H^{s-\sigma}} }{ \|t-t'\|^\sigma }.\end{aligned}$$ Now, the first term in the right-hand side of the above inequality is bounded, since $$\begin{aligned} \sup_{t \in I} \\|u_\epsilon(t)\\|_{H^{s-\sigma}} &\stackrel{ \eqref{estsolmoll} }{\leq} 2 \\|u_0\\|_{H^s},\end{aligned}$$ while the second term can be bounded by exploiting and . Putting these bounds together allows us to apply Ascoli-Arzelá Theorem, since the equicontinuity condition follows form $$\begin{aligned} \\|u_\epsilon(t_1)-u_\epsilon(t_2)\\|_{H^{s-\sigma}} &\leq \\|u_\epsilon\\|_{C^\sigma(I;H^{s-\sigma})} \|t_1-t_2\|^\sigma,\end{aligned}$$ while the precompactness condition can be verified as in the previous step.\ [Step 4: convergence in $C( I;C^1(\mathbb{T}) )$.]{} Now fix $\sigma \in (0,1)$ such that $s-\sigma>3/2$, then by Sobolev embedding implies that $u_\epsilon {\rightarrow}u$ in $C(I;C^1(\mathbb{T}))$. Now we need to study ${\partial}_t u_\epsilon$. Starting with the two non-local terms of , the continuity of the operator ${\partial}_x D^{-2}$ implies that ${\partial}_x D^{-2}(u_\epsilon^2) {\rightarrow}{\partial}_x D^{-2} (u^2)$ and ${\partial}_x D^{-2} u_\epsilon {\rightarrow}{\partial}_x D^{-2} u$ in $C(I;C(\mathbb{T}))$. To handle the first two terms, first observe that $$\begin{aligned} \label{estBurgterm} \\|J_\epsilon u_\epsilon - u \\|_{ C(I;C(\mathbb{T})) } &\leq \\|J_\epsilon u_\epsilon - u_\epsilon \\|_{ C(I;C(\mathbb{T})) } + \\| u_\epsilon - u \\|_{ C(I;C(\mathbb{T})) }. \end{aligned}$$ To estimate the first term in the right-hand side of , choose $r \in (1/2,s)$, and observe that for any $t \in I$ implies that there exists $C(r) >0$ such that $$\begin{aligned} \\|J_\epsilon u_\epsilon - u_\epsilon \\|_{ C(I;C(\mathbb{T})) } &\leq 2 C(r) \\|Id-J_\epsilon\\|_{L(H^s ; H^r)} \\|u_0\\|_{H^s} = o(\epsilon^{s-r}),\end{aligned}$$ from which we can deduce that $J_\epsilon u_\epsilon{\rightarrow}u$ in $C(I;C(\mathbb{T}))$. With a similar argument we can show that $J_\epsilon {\partial}_x u_\epsilon {\rightarrow}{\partial}_x u$ in $C(I;C(\mathbb{T}))$. Therefore one can conclude that $$\begin{aligned} {\partial}_t u_\epsilon &{\rightarrow}-(\mathtt{c}+u){\partial}_x(\mathtt{c}+u) -\frac{3}{2}{\partial}_x D^{-2}( (\mathtt{c}+u)^2 )\end{aligned}$$ in $C(I;C(\mathbb{T}))$. Recalling that also $u_\epsilon {\rightarrow}u$ in $C(I;C^1(\mathbb{T}))$, we can deduce that $t \mapsto u(t)$ is a differentiable map such that $$\begin{aligned} {\partial}_t u &= -(\mathtt{c}+u){\partial}_x(\mathtt{c}+u) -\frac{3}{2}{\partial}_x D^{-2}( (\mathtt{c}+u)^2 ).\end{aligned}$$ [Step 5: convergence in $C(I;H^s)$.]{} Fix $t \in I$ and take a sequence $(t_n)_{n \in \mathbb{N}} {\rightarrow}t$. Since $u \in L^\infty(I;H^s)$, we have that $t \mapsto u(t)$ is continuous with respect to the weak topology on $H^s$; thus, to verify the continuity we just need to check that the map $t \mapsto \\|u(t)\\|^2_{H^s}$ is continuous. We begin by introducing $$\begin{aligned} F(t) &:= \\|u(t)\\|^2_{H^s}, \\ F_\epsilon(t) &:= \\|J_\epsilon u(t)\\|^2_{H^s}.\end{aligned}$$ Now, implies that $F_\epsilon {\rightarrow}F$ pointwise as $\epsilon {\rightarrow}0$. Therefore it suffices to show that each $F_\epsilon$ is Lipschtiz and that the Lipschitz constants for this family are bounded. Since $$\begin{aligned} \label{HsFeps} \frac{1}{2} F'_\epsilon(t) &= - \int_0^{2\pi} D^s J_\epsilon u \; D^s J_\epsilon \left[(\mathtt{c}+u){\partial}_x(\mathtt{c}+u)\right] \, dx -\frac{3}{2} \int_0^{2\pi} D^s J_\epsilon u D^sJ_\epsilon {\partial}_x D^{-2}( (\mathtt{c}+u)^2 ) \, dx.\end{aligned}$$ To bound the first term on the right-hand side of we need to use the commutator estimate , in order to commute the operator $D^s$ with $u$; but since we also need to commute $J_\epsilon$ with $u$, we exploit the following result (see [@Taylor]): Let $f,g \in H^s$, then there exists $C>0$ such that $$\begin{aligned} \\| [f,J_\epsilon] {\partial}_x g\\|_{L^2} &\leq C \\|f\\|_{C^1(\mathbb{T})} \\|g\\|_{L^2(\mathbb{T})}.\end{aligned}$$ By applying the Cauchy-Schwarz inequality, the algebra property, and estimate on the size of the solution, we can conclude that there exists $C(s)>0$ such that $$\begin{aligned} \|F'_\epsilon(t)\| &= \left\| \frac{d}{dt} \\|J_\epsilon u(t)\\|^2_{H^s} \right\| \leq C(s).\end{aligned}$$ To adjust the proof for the noncompact case, we have to define the mollifiers $J_\epsilon$ in the following way: first we fix $j \in \mathcal{S}(\mathbb{R})$ such that $\hat{j}(\xi)=1$ for $\|\xi\|\leq 1$. Then we set $j_\epsilon(x):= \epsilon^{-1} j(x/\epsilon)$; this gives again that $\\|Id-J_\epsilon\\|_{L(H^s,H^r)} =o(\epsilon^{s-r})$.\ Moreover, we exploited the compactness of $\mathbb{T}$ in order to satisfy the hypotheses of Ascoli’s theorem; in the noncompact case the embedding $H^s {\rightarrow}H^{s'}$ for $s>s'$ does not define a compact operator. We handle this problem by first fixing $\phi \in \mathcal{S}(\mathbb{R})$ with $0 < \phi(x) \leq 1$. Then Rellich’s Theorem implies that the operator $u_\epsilon \mapsto \phi u_\epsilon$ is compact from $H^s$ to $H^{s'}$. Using this modification and by recalling that $\phi \neq 0$, we obtain again the existence of a solution $u$. \[LWPunique\] Consider Eq. with initial datum $u_0 \in H^s$, $s>3/2$. Then its solution $u \in C([-\bar T, \bar T]; H^s)$ with $\bar T$ as in is unique. Let $u_0 \in H^s$, and let $u$ and $w$ be two solutions to with $u(0,\cdot)=w(0,\cdot)=u_0$. Consider $v:=u-w$, then $$\begin{aligned} \label{eqdiff} {\partial}_t v &= -\frac{1}{2} {\partial}_x \left[ (2\mathtt{c}+u+w)v \right] -\frac{3}{2} {\partial}_x D^{-2} ((2\mathtt{c}+u+w)v).\end{aligned}$$ Fix $\sigma \in (1/2,s-1)$; then $$\begin{aligned} \label{normv} \frac{d}{dt} \\|v\\|^2_{H^\sigma} &= -\int_0^{2\pi} D^\sigma v \left[ D^\sigma {\partial}_x ((2\mathtt{c}+u+w)v) + 3 {\partial}_x D^{\sigma-2}((2\mathtt{c}+u+w)v) \right] \, dx.\end{aligned}$$ In order to bound the first term in the right-hand side of we commute $D^\sigma {\partial}_x$ with $u+w$ by exploiting the following Calderon-Coifman-Meyer estimate (see Proposition 4.2 in [@TayCommEst]) Let $\sigma \geq -1$, then for any $\rho>3/2$ such that $\sigma+1 \leq \rho$ there exists $C>0$ such that $$\begin{aligned} \\| [D^\sigma {\partial}_x,f]v \\|_{L^2} &\leq C \\|f\\|_{H^\rho} \\|v\\|_{H^\sigma}.\end{aligned}$$ The nonlocal term is bounded by Plancherel and Cauchy-Schwarz inequality. Hence there exists $c(s)>0$ such that $$\begin{aligned} \frac{d}{dt} \\|v(t)\\|^2_{H^\sigma} &\leq c(s) \\|v\\|^2_{H^\sigma}; \\ \\|v\\|_{H^\sigma} &\leq e^{c(s) \, T} \\|v(0)\\|_{H^\sigma}=0,\end{aligned}$$ and we can conclude that $u=w$. \[LWPcontdep\] Consider Eq. with initial datum $u_0 \in H^s$, $s>3/2$. Then the solution map from $H^s {\rightarrow}C(I;H^s)$ ($I=[-\bar T,\bar T]$, with $\bar T$ as in ) given by $u_0 \mapsto u$ is continuous. Fix $u_0 \in H^s$, and let $(u_{0,n})_n \subset H^s$ be a sequence such that $\lim_{n {\rightarrow}\infty} u_{0,n} = u_0$. Then, if $u_n$ is the solution of Eq. with initial datum $u_{0,n}$, we want to show that $$\begin{aligned} \label{contdepconv} \lim_{n {\rightarrow}\infty} u_n &= u \; \; \text{in} \; \; C(I;H^s);\end{aligned}$$ equivalently, let $\eta>0$, we want to show hat there exists $N>0$ such that $$\begin{aligned} \label{contdepconv2} \\|u-u_n\\|_{C(I;H^s)} &< \eta, \; \; \forall n > N.\end{aligned}$$ As before, we will ue the convolution operator to smooth out the initial data. Let $0 < \epsilon \leq 1$, let $u^\epsilon$ be the solution to with initial datum $J_\epsilon u_0 = j_\epsilon \ast u_0$ and let $u^\epsilon_n$ be the solution of with initial datum $J_\epsilon u_{0,n}$. Then $$\begin{aligned} \label{distsol} \\|u-u_n\\|_{C(I;H^s)} &\leq \\|u-u^\epsilon\\|_{C(I;H^s)} + \\|u^\epsilon-u_n^\epsilon\\|_{C(I;H^s)} + \\|u^\epsilon-u_n\\|_{C(I;H^s)}.\end{aligned}$$ We will prove that each of these terms can be bounded by $\eta/3$, for suitable choices of $\epsilon$ and $N$. We also point out that the quantity $\epsilon$ will be independent of $N$ and will only depend on $\eta$, while the choice of $N$ will depend on both $\eta$ and $\epsilon$. We start with $\\|u^\epsilon-u_n^\epsilon\\|_{C(I;H^s)}$. Set $v:=u^\epsilon-u_n^\epsilon$, then $v$ satisfies $$\begin{aligned} {\partial}_t v &= -\frac{1}{2} {\partial}_x \left[ (2\mathtt{c}+u^\epsilon+u_n^\epsilon)v \right] -\frac{3}{2} {\partial}_x D^{-2} ((2\mathtt{c}+u^\epsilon+u_n^\epsilon)v), \nonumber \\ v(0) &= u^\epsilon(0)-u_n^\epsilon(0) = J_\epsilon u_0 -J_\epsilon u_{0,n},\end{aligned}$$ and $$\begin{aligned} \label{Hsnormv} \frac{1}{2} \frac{d}{dt} \\|v\\|^2_{H^s} &= - \int_0^{2\pi} D^s v \, D^s \left[ \frac{1}{2} {\partial}_x \left[ (2\mathtt{c}+u^\epsilon+u_n^\epsilon)v \right] +\frac{3}{2} {\partial}_x D^{-2} ((2\mathtt{c}+u^\epsilon+u_n^\epsilon)v) \right] dx.\end{aligned}$$ Applying and the estimate $\\|u^\epsilon\\|_{H^{s+1}} \leq C/\epsilon$, implies that there exists $c_s>0$ such that $$\begin{aligned} \label{ineqdiff} \frac{1}{2} \frac{d}{dt} \\|v(t)\\|^2_{H^s} &\leq \frac{c_s}{\epsilon} \\|v(t)\\|^2_{H^s},\end{aligned}$$ which in turn leads to $$\begin{aligned} \label{ineqdiff2} \\|v(t)\\|_{H^s} &\leq e^{c_s T/\epsilon} \\|v(0)\\|_{H^s} \; \leq \; 2 e^{c_sT/\epsilon} \\|u_0-u_{0,n}\\|_{H^s}.\end{aligned}$$ Notice that does not imply any constraint on $\epsilon$; however, handling the first and the third term in the right-hand side of will require $\epsilon$ to be small. After chosing $\epsilon$, we will take $N$ so large that $\\|u_0-u_{0,n}\\|_{H^s} < \frac{\eta}{6} e^{-c_sT/\epsilon}$, which will imply that $\\|u^\epsilon-u_n^\epsilon\\|_{C(I;H^s)} < \eta/3$. Now we estimate $\\|u^\epsilon-u\\|_{C(I;H^s)}$ and $\\|u^\epsilon-u_n\\|_{C(I;H^s)}$. We set $v:=u^\epsilon-u$ and $v_n:=u_n^\epsilon-u_n$. Since $v$ and $v_n$ will satisfy the same energy estimates, we will write $v_{(n)}$ to mean that an equation holds both with and without the subscript. We observe that $v_{(n)}$ solves the Cauchy problem $$\begin{aligned} {\partial}_t v_{(n)} &= -\frac{1}{2} {\partial}_x \left[ (2\mathtt{c}+u^\epsilon+u_{(n)})v_{(n)} \right] -\frac{3}{2} {\partial}_x D^{-2} ((2\mathtt{c}+u^\epsilon+u_{(n)})v_{(n)} ) \\ &= -\frac{1}{2} {\partial}_x \left[ (2\mathtt{c}+2 u^\epsilon+v_{(n)})v_{(n)} \right] -\frac{3}{2} {\partial}_x D^{-2} ((2\mathtt{c}+2 u^\epsilon+v_{(n)})v_{(n)} ), \nonumber \\ v(0) &= j_\epsilon \ast u_{0,(n)}-u_{0,(n)}.\end{aligned}$$ By exploiting , the Cauchy-Schwarz inequality and Sobolev embedding we get $$\begin{aligned} \label{Hsnormvn} \frac{1}{2} \frac{d}{dt} \\|v_{(n)}(t)\\|_{H^s} &\leq c'_s \left[ \\|v_{(n)}\\|^3_{H^s} + (1+\\|u^\epsilon_{(n)}\\|_{H^s}) \\|v_{(n)}\\|^2_{H^s} + (1+\\|u^\epsilon_{(n)}\\|_{H^{s+1}}) \\|v_{(n)}\\|_{H^{s-1}} \\|v_{(n)}\\|_{H^s} \right]\end{aligned}$$ for some $c'_s>0$. Since $\\|u^\epsilon_{(n)}(t)\\|_{H^{s+1}} \leq c_1(s)/\epsilon$ and that $\\|v_{(n)}(t)\\|_{L^2}=o(\epsilon)$, gives $$\begin{aligned} \label{diffineq3} \frac{d y}{d t} &\leq c_2(s) \left( y^2+y+\delta \right),\end{aligned}$$ where $\delta=\delta(\epsilon) {\rightarrow}0$ as $\epsilon {\rightarrow}0$. The quadratic expression $y^2+y+\delta$ has roots $$\begin{aligned} \label{roots} r_{-1} = \frac{-1-\sqrt{1-4\delta}}{2}, \; &\; r_{0} = \frac{-1+\sqrt{1-4\delta}}{2}.\end{aligned}$$ Restricting $\epsilon$ so that the roots given in are real-valued, we observe that $r_0$ and $r_{-1}$ are negative and, as $\delta {\rightarrow}0$, we have $r_{-1} {\rightarrow}-1$ and $r_0 {\rightarrow}0$. Setting $R := \sqrt{1 - 4\delta}$ and taking into account the constant $c_s$, we solve via $$\begin{aligned} \label{diffineqsol} \frac{y(t)-r_0}{y(t)-r_{-1}} &\leq \gamma, \\ \gamma &:= e^{c_s RT} \frac{y(0)-r_0}{y(0)-r_{-1}}.\end{aligned}$$ From here, we will treat the cases $y = \\|v\\|_{H^s}$ and $y = \\|v_n\\|_{H^s}$ separately. Case $y = \\|v\\|_{H^s}$.* Using we have $y(0) {\rightarrow}0$ as $\epsilon {\rightarrow}0$. This implies that $\gamma {\rightarrow}0$ as $\epsilon {\rightarrow}0$. From , we then obtain $y(t) \leq y(t)-r_0 \leq \gamma [y(t)-r_{-1}]$. Solving for $y(t)$ gives us $$\begin{aligned} y(t) &\leq \frac{-r_{-1}}{1-\gamma} \gamma \\ &\stackrel{\gamma {\rightarrow}0}{ {\rightarrow}} 0.\end{aligned}$$ Therefore, for sufficiently small $\epsilon$ we can bound the first term of by $\eta/3$. Case $y = \\|v_n\\|_{H^s}$. We begin by bounding $y(0)$ by $$\begin{aligned} \\|j_\epsilon \ast u_{0,n}-u_{0,n} \\|_{H^s} &\leq 2 \\|u_{0,n}-u_0\\|_{H^s} + \\|j_\epsilon \ast u_0-u_0\\|_{H^s},\end{aligned}$$ which implies that $$\begin{aligned} \gamma &\leq \frac{e^{c_S RT}}{-r_{-1}} \left( 2 \\|u_{0,n}-u_0\\|_{H^s}+\\|j_\epsilon \ast u_0-u_0\\|_{H^s} \right) + \frac{r_0 e^{c_s RT}}{r_{-1}},\end{aligned}$$ where we may independently choose $\epsilon$ sufficiently small and $N$ sufficiently large so that $\gamma < 1/2$. Then, arguing as in the previous case we obtain $y(t) \leq 2\gamma$. We may now further refine the choice of $\epsilon$ and $N$ so that $y(t)<\eta/3$, completing this case. Collecting our results completes the proof. The proofs for uniqueness of the solution and for the continuous dependence on the initial datum do not rely on compactness properties, hence they do not nedd any adjustment in the noncompact case. Analyticity on Sobolev spaces ----------------------------- We recall some facts about analytic functions on Banach spaces following Appendix A of [@PoTr]. [([Weakly analyticity]{})]{} Let $E, F$ two complex Banach spaces and $U$ an open subset of $E$. A map $f\colon U{\rightarrow}F$ is said weakly analytic if for each $w\in U$, $h\in E$ and $L\in F^*$ the function $$z\mapsto L f(w+z h)$$ is analytic in some neighborhood of the origin in $\mathbb{C}$ in the usual sense of one complex variable. \[TruboTeo\] Let $f\colon U{\rightarrow}F$ be a map from an open subset $U$ of a complex Banach space $E$ into a complex Banach space $F$. Then the following three statements are equivalent. 1. $f$ is analytic in $U$. 2. $f$ is locally bounded and weakly analytic in $U$. 3. $f$ is infinitely often differentiable on $U$, and is represented by its Taylor series in a neighborhood of each point in $U$. [^1]: This research was supported by PRIN 2015 “Variational methods, with applications to problems in mathematical physics and geometry”. This research was supported by PRIN 2012 “Variational and perturbative aspects of nonlinear differential problems”. This research was supported by by ERC grant 306414 HamPDEs under FP7.
--- abstract: 'Bulk phase separation is responsible for the occurrence of stacks of different layers in sedimentation of colloidal mixtures. A recently proposed theory (de las Heras and Schmidt 2013 Soft Matter 9 8636) establishes a unique connection between the bulk phase behaviour and sedimentation-diffusion-equilibrium. The theory constructs a stacking diagram of all possible sequences of stacks under gravity in the limit of very high (infinite) sample heights. Here, we study the stacking diagrams of colloidal mixtures at finite sample height, $h$. We demonstrate that $h$ plays a vital role in sedimentation-diffusion-equilibrium of colloidal mixtures. The region of the stacking diagram occupied by a given sequence of stacks depends on $h$. Hence, two samples with different heights but identical colloidal concentrations can develop different stacking sequences. In addition, the stacking diagrams for different heights can be qualitatively different since some stacking sequences occur only in a given interval of sample heights. We use the theory to investigate the stacking diagrams of both model bulk systems and mixtures of patchy particles that differ either by the number or by the types of patches.' author: - Thomas Geigenfeind - Daniel de las Heras title: The role of sample height in the stacking diagram of colloidal mixtures under gravity --- Introduction \[introduction\] ============================= Since the pioneer work of Perrin [@perrin], sedimentation has become a central tool for investigating the phase behaviour in colloidal systems. The height-dependent colloidal concentration profile provides a direct measurement of the equation of state for monocomponent systems [@PhysRevLett.71.4267; @biben1993density]. Sedimentation experiments are also used to extract information from the bulk phase behaviour in binary colloidal mixtures, see e.g. Refs. [@Piazzareview; @C1SM06535A; @C5SM00615E]. However, thermal and gravitational energies are of the same order of magnitude for typical colloidal systems. This results in additional gravity-induced phenomenology not present in bulk systems. Examples are denser particles floating on top of lighter colloids [@piazza1], a nematic layer sandwiched by isotropic layers in mixtures of platelets and spheres [@floating] and mixtures of thin and thick rods [@C6SM00736H], and reentrant network formation in mixtures of patchy colloids [@Lucas]. It is also common to observe complex stacking sequences in sedimentation with three or more different layers, such as e.g., in mixtures of charged platelets and polymers [@doi:10.1021/la804023b], plate-plate binary systems [@C3SM52311J], mixtures of spheres of different sizes [@ahhh], and colloidal rod-plate mixtures [@JeR]. The relation between bulk phase behaviour and sedimentation-diffusion-equilibrium in mixtures is therefore intertwined with gravity-induced effects. From a theoretical viewpoint, a generalized Archimedes principle was formulated [@piazza1; @piazza2] for the case where one of the components is very diluted. Sedimentation was also studied by analyzing the macroscopic osmotic equilibrium conditions [@0295-5075-66-1-125; @PhysRevE.70.051401]. Recently, de las Heras and Schmidt have proposed a theory [@stack1; @stack2] for obtaining the stacking diagram, i.e., the set of all possible stacking sequences under gravity, from the bulk phase diagram of a given binary system. The theory is based on the concept of sedimentation paths. Each sedimentation path, which is a line in the plane of chemical potentials representation of the bulk phase diagram, describes the state of the mixture under gravity, in sedimentation-diffusion-equilibrium. Using this theory the stacking diagrams of mixtures of spheres and platelets [@stack1] and mixtures of platelets and nonadsorbing polymers [@stack2] were obtained. Also, very recently, van Roij and coworkers have obtained the stacking diagrams of mixtures of thin and thick colloidal rods [@C6SM00736H]. Although in all these cases the bulk phase diagrams of the colloidal systems are relatively simple, the resulting stacking diagrams are extremely rich and show complex stacking sequences. These works are focused on the limit of very high (infinite) sample heights. This idealized limit is very relevant in experimental work since the height of the test tube is typically larger than the gravitational lengths of the colloids. The interplay between micro confinement and colloidal sedimentation has been experimentally and theoretically investigated [@Royall]. However, little attention has been paid to the influence of the total (macroscopic) sample height in colloidal sedimentation. A remarkable exception is the experimental work of Jamie et al. [@B915788C], in which the properties of the gas-fluid interface of a polymer-colloid mixture are analyzed as a function of the overall height of the container. By systematically changing the total sample height while keeping the polymer-colloid concentrations fixed, the interfacial properties were found to move towards the critical point. Theoretically, it has been shown that varying the sample height might led to a change in the stacking sequence in mixtures of colloids and nonadsorbing polymers [@0295-5075-66-1-125; @stack2]. Here, we use the theory of Refs. [@stack1; @stack2] to study sedimentation-diffusion-equilibrium of colloidal mixtures for the case of finite height samples. We systematically investigate the role of sample height in the stacking diagrams of colloidal mixtures. We first apply the theory to model binary systems. That is, systems with generic bulk phase diagrams typical of model Hamiltonian which we however do no explicitly specify. As an application of current interest we also study sedimentation in patchy colloidal mixtures. Patchy colloids are functionalized colloids that interact via a directional and valence-limited potential [@in1; @in2]. Here, we use Wertheim’s theory [@wertheim] to obtain the bulk behaviour of the mixture. The two species of the mixture differ in either the number or in the types of patches. We show that the sample height is a crucial variable in sedimentation-diffusion-equilibrium of colloidal mixtures. The stacking diagrams for the same mixture but for different sample heights differ not only quantitatively but also qualitatively. For example, some stacking sequences occur only in a given range of sample heights. Theory \[theory\] ================= The sedimentation path ---------------------- Consider a colloidal mixture under gravity in a sedimentation vessel of sample height $h$. The gravitational potential for each species $i=1,2$ is $m_igz$, where $m_i$ is the buoyant mass of species $i$, $g$ is the acceleration due to gravity, and $z$ is the vertical coordinate (we set the origin of coordinates, $z=0$, at the bottom of the sample). Using a local density approximation [@schmidt04aog; @floating; @stack1; @stack2], we define a height-dependent local chemical potential for $0\le z\le h$ for each species $$\begin{aligned} \mu_1(z)=\mu_1^{\text{b}}-m_1gz,\nonumber\\ \mu_2(z)=\mu_2^{\text{b}}-m_2gz,\label{chemi}\end{aligned}$$ where $\mu_i^{\text{b}}$ is the bulk chemical potential, i.e. the chemical potential in absence of gravity. The local density approximation assumes that for each $z$ the state of the sample is analogous to a bulk system (no gravity) with chemical potentials given by . This constitutes a very good approximation provided that the correlation lengths are small compared to the gravitational lengths, $\xi_i=k_{\text{B}}T/m_ig$ with $k_{\text{B}}$ the Boltzmann constant, and $T$ the absolute temperature. Combining the expressions for the local chemical potentials, cf. , and eliminating the height variable $z$ we find $$\mu_2(\mu_1)=a+s\mu_1,\label{path}$$ where both $a$ and $s$ are constants given by $$\begin{aligned} a&=&\mu_2^b-s\mu_1^b,\nonumber\\ s&=&m_2/m_1=\xi_1/\xi_2.\label{constants}\end{aligned}$$ The finite size of the sample $0\le z\le h$ is translated into a range for the local chemical potentials $$0\le\frac{\mu_i^{\text{b}}-\mu_i}{m_igh}\le1,\text{ }i=1,2.~\label{limit}.$$ Eqs. and represent a line segment, which we refer to as the sedimentation path, in the plane of chemical potentials. The sedimentation path describes how the local chemical potentials vary along the height coordinate in the vessel. Each point in the sedimentation path corresponds to the state of the sample at a given $z$. The sedimentation path is directly related to the stacking sequence, i.e., the sequence of stacks of different phases that appear under gravity. If a path crosses a boundary between two phases in the phase diagram, e.g., a binodal, an interface appears in the vessel. The sedimentation path provides a direct link between the bulk phase diagram of the mixture and the stacking sequence. An example of a sedimentation path and its corresponding stacking sequence is shown in Fig. \[fig1\]. A sedimentation path is fully described by its (i) slope, (ii) location in the $\mu_1-\mu_2$ plane specified by a point on the path, (iii) direction and (iv) length. The slope is fixed by the ratio of the buoyant masses, cf. . The position is determined by the bulk chemical potentials in absence of gravity, and hence by the overall colloidal composition and concentration via a change of variables using the equation of state of the mixture. The direction is given by the signs of the buoyant masses (note $m_i$ can be negative if the mass density of the solvent is higher than the mass density of the colloids). Finally, the length of the path is proportional to the height of the vessel since the difference in chemical potentials between the top ($z=h$) and the bottom of the sample ($z=0$) is $$\Delta\mu_i=\mu_i(h)-\mu_i(0)=-m_igh.\label{length}$$ The stacking diagram {#theory} -------------------- We have shown how each sedimentation path is associated to a stacking sequence. The phase stacking diagram is the set of all possible stacking sequences for a given mixture. [Infinite height]{}. For standard colloidal particles in typical sedimentation vessels the length of the sedimentation path is of several $k_{\text{B}}T$ in the $\mu_1-\mu_2$ plane. That is, the path extends over a big region of the bulk phase diagram of the mixture. Hence, a very relevant idealization is to consider the limit of very high (infinite) sample heights. Within this limit [@stack1; @stack2] a sedimentation path is a straight line of infinite length (not a line segment) in the plane of chemical potentials. Hence, a sedimentation path can be fully described by using only the slope of the path $s$, and the crossing point between the path and the $\mu_2$ axis $a$, cf. . The stacking diagram can then be represented in the $s-a$ plane. There are three types of boundaries between different stacking sequences in the stacking diagram. Here we only describe each one briefly, see [@stack1; @stack2] for a full account: \(i) [Sedimentation binodals.]{} The set of all sedimentation paths tangent to a binodal in the bulk phase diagram is a boundary between two phases in the stacking diagram. The path labeled as (1) in Fig. \[fig1\] is an example. An infinitesimally small change in one or in both variables of the path, $a$ and $s$, can change the stacking sequence. \(ii) [Terminal lines.]{} The set of all paths crossing an ending point of a binodal in the bulk phase diagram is a boundary in the stacking diagram that we call the terminal line. The sedimentation path (2) in Fig. \[fig1\] is an example. An infinitesimal change of $a$ changes the stacking sequence. An ending point can be e.g. a critical point, triple point, critical end point, etc. \(iii) [Asymptotic terminal lines.]{} The third type of boundaries in the stacking diagram is formed by those paths that are parallel to the asymptotic behaviour of a binodal. See the path (3) in Fig. \[fig1\]. In this case an infinitesimal change of the slope $s$ alters the stacking sequence. A binodal is not the only possible boundary between two regions present in the bulk phase diagram. For example, a percolating line dividing the bulk phase diagram into percolated and nonpercolated states is another type of a boundary between phases. Any boundary present in the bulk phase diagram generates boundaries in the stacking diagram (sedimentation binodals, terminal lines and asymptotic terminal lines). For convenience we speak always of binodal but one should bear in mind that other lines also give rise to boundaries in the stacking diagram. The patchy colloid mixtures studied below feature such percolation lines. [Finite height]{}. In this paper we focus on the stacking diagrams for finite height samples. There exist several possibilities to represent the stacking diagram for finite heights. In an experimental work one typically varies the concentration and composition of the mixture, while keeping the solvent and the mass density of the colloids unchanged. The sample height is, in principle, easy to adjust [^1] and hence forms a useful control parameter. Under these circumstances the slope and the length of the path in the $\mu_1-\mu_2$ plane are fixed, cf. and , and its position in the $\mu_1-\mu_2$ plane varies. A sensible choice of variables for the stacking diagram is the plane of average local chemical potentials along the path $\bar\mu_1-\bar\mu_2$. As the sedimentation paths are straight lines, the average local chemical potentials are just the local chemical potential evaluated at the middle of the sample $\bar\mu_i=\mu_i(z=h/2)$. The stacking diagram for finite height samples in the $\bar\mu_1-\bar\mu_2$ plane contains three possible types of boundaries between different stacking sequences. Two of them are sedimentation binodals originated from coexisting lines in the bulk phase diagrams and one boundary is due to the ending points of the binodals. We describe each of them in detail in the following. \(i) [Sedimentation binodal type I]{} (SBI). The set of all sedimentation paths that either start or end at a binodal form a sedimentation binodal of type I. A path starts (ends) at a binodal if the bottom (top) of the sample is located at the binodal. The path labeled as (1) in Fig. \[fig2\] is an example. For each binodal in the bulk phase diagram there are two corresponding SBI in the stacking diagram. One SBI for those paths that end at the binodal and the other one SBI for those paths that start at the binodal. Both SBI lines have the same shape as the bulk binodal. This type of boundary is not present in the case of infinite height because the paths do not have starting and ending points. Let $\mu_{2,\text{C}}(\mu_1)$ be the parameterization of the chemical potential of species $2$ at a bulk coexistence line, such as a binodal, as a function of $\mu_1$. Then, the two corresponding sedimentation binodals of type I are given by $$\begin{aligned} \bar\mu_2(\bar\mu_1)=\mu_{2,\text{C}}(\mu_{1}^-)+ m_2g\frac h2,\nonumber\\ \bar\mu_2(\bar\mu_1)=\mu_{2,\text{C}}(\mu_{1}^+)- m_2g\frac h2,\end{aligned}$$ where $$\mu_1^{\pm}=\bar\mu_1\pm m_1g\frac h2.$$ Here, $\mu_1^+$ ($\mu_1^-$) is the local chemical potential of species $1$ at the bottom (top) of the sample. \(ii) [Sedimentation binodal type II]{} (SBII). The set of all paths tangent to a bulk binodal is also a boundary (sedimentation binodal type II) in the stacking diagram. See the path (2) in Fig. \[fig2\]. This boundary is analogous to the sedimentation binodals in the case of infinite height. The SBII boundaries are straight lines in the stacking diagram. A SBII line is present if and only if the slope of the path is the same as the slope of the binodal at same point(s). Each point of a binodal sharing the same slope as the path generates a SBII line. Let $(\mu_{1,\text{t}},\mu_{2,\text{t}})$ be the chemical potentials of a point on a bulk binodal. Let its local slope be that of the sedimentation path. That is $$\left.\frac{d\mu_{2,\text{C}}}{d\mu_1}\right\|_{\mu_{2,\text{t}}}=s.\label{mmm}$$ Then, the associated SBII line is given by $$\bar\mu_2(\bar\mu_1)=\mu_{2,\text{t}}+(\bar\mu_1-\mu_{1,\text{t}})s.$$ The finite size of the path limits the range of $\bar\mu_1$ to $$\left\|\bar\mu_1-\mu_{1,\text{t}}\right\|\le\left\|m_1g\frac h2\right\|.$$ \(iii) [Terminal lines]{} (TL). The terminal lines are, as in the infinite height case, the set of all paths that cross an ending point of a binodal. See path (3) in Fig. \[fig2\]. For each ending point in the bulk phase diagram there is one and only one TL in the stacking diagram. The TL is always a straight line. Let $(\mu_{1,\text{e}},\mu_{2,\text{e}})$ be the chemical potentials of an ending point in bulk, such as a critical point, a triple point, etc. The corresponding terminal line is $$\bar\mu_2(\bar\mu_1)=\mu_{2,\text{e}}+(\bar\mu_1-\mu_{1,\text{e}})s,$$ for $$\left\|\bar\mu_1-\mu_{1,\text{e}}\right\|\le\left\|m_1g\frac h2\right\|.$$ In the three cases (i),(ii), and (iii) any infinitesimal displacement of the path changes the stacking sequence (except for the special case in which the displacement is such that the path moves along the boundary of the stacking diagram). The asymptotic terminal lines that occur in the case of infinite sample height do not appear at finite height since the slope of the sedimentation path is fixed and the paths are of finite length. The three boundaries SBI, SBII, and TL divide the stacking diagram in different regions. Each region corresponds to a different stacking sequence. In order to identify each sequence we first select one point inside of the desired region. Next we plot the corresponding path in the bulk phase diagram such that we can determine the sequence by inspecting the crossings between the path and the binodals. Once the stacking diagram has been calculated in the $\bar\mu_1-\bar\mu_2$ plane, we can easily transform to any other set of variables provided that the equation of state of the mixture is known. In order to ease comparison to experimental work, a sensible choice of variables for the stacking diagram is the $\bar\eta_1-\bar\eta_2$ plane, where $\bar\eta_i$ is the average packing fraction of species $i$, $$\bar\eta_i=\frac1h\int_0^hdz\eta_i(z).$$ Here, $\eta_i(z)$ is the local packing fraction of species $i$ at a distance $z$ from the bottom of the sample. To obtain $\eta_i(z)$ we first compute the local chemical potentials at $z$ using Eq. , and then use the equation of state of the system $\eta_i=\eta_i(\mu_1,\mu_2)$. The phase diagram in the $\bar\eta_1-\bar\eta_2$ plane is then obtained by transforming the coordinates of the boundaries in the stacking diagram from $(\bar\mu_1,\bar\mu_2)$ to $(\bar\eta_1,\bar\eta_2)$. Other representations of the stacking diagram such as for example average osmotic pressure versus average composition are also possible, following a similar transformation procedure. Results ======= We first apply our theory to obtain the stacking diagrams at finite height of different bulk model phase diagrams (Sec. \[aaa\]). Although the bulk phase diagrams do not correspond to real microscopic models, they are representative of the behaviour of typical colloidal mixtures. The model bulk phase diagrams provide relevant examples of possible topologies of the stacking diagrams. In Sec. \[bbb\] we apply the theory to model binary mixtures of patchy colloids for which we use Wertheim’s theory to obtain the bulk phase diagram. Finally, in Sec. \[ccc\] we compare the stacking diagrams at finite and infinite sample heights. Model bulk phase diagrams {#aaa} ------------------------- One of the simplest possible bulk phase diagrams of a binary mixture is schematically represented in Fig. \[fig3\]a. There is a single binodal at which two phases A and B coexist. The binodal ends at a critical point. The species $2$ undergoes an A-B phase transition. Hence, the binodal has a horizontal asymptote and tends to the value of $\mu_2$ at the transition (chosen as $\lim_{\mu_2\rightarrow\infty}=0.89$). This phase diagram might be representative of e.g. a mixture of spherical colloids (species $1$) and anisotropic colloids undergoing an isotropic-nematic phase transition (species $2$). A small degree of polydispersity in the spherical colloids could prevent a liquid-solid phase transition in the pure system of species $1$ [^2]. The stacking diagram of the chosen model bulk phase diagram for the case of infinite height is shown in Fig. 5 of Ref. [@stack2]. Here we focus on the case of finite height. In Fig. \[fig3\]b we represent the stacking diagram ($\bar\mu_1-\bar\mu_2$ plane) for two different heights $h_1$ and $h_2$ with $h_2>h_1$. In both cases the slope of the paths are the same, $s=m_2/m_1=1$, and both buoyant masses are positive such that both local chemical potentials decrease from the bottom to the top of the sample. We show representative sedimentation paths in Fig. \[fig3\]a. Each sedimentation path in the $\mu_1-\mu_2$ plane in the bulk diagram is a point in the $\bar\mu_2-\bar\mu_1$ plane of the stacking diagram (the coordinates of the middle point of the path). The stacking diagrams contain two sedimentation binodals of type I (generated by paths starting and ending at the bulk binodal) and one terminal line (paths crossing the critical point). There are three possible stacking sequences, namely A, B, and AB. We label the sequences according to the order of different stacks from bottom to top. Next, we study the same model bulk phase diagram but for sedimentation paths with a different slope, $s=-1$. See representative paths in Fig. \[fig3\]c. Here $m_1>0$ and $m_2<0$ such that $\mu_1$ ($\mu_2$) decreases (increases) from the bottom to the top of the sample. The slope of the path is, in this case, compatible with the slope of the binodal in the sense that there is one point at the binodal whose derivative equals the slope of the path, cf. . Hence, the stacking diagram contains a sedimentation binodal of type II which is formed by the set of paths that are tangent to the binodal in bulk. The boundaries of the stacking diagrams (Fig. \[fig3\]d) are: two SDI lines, one SDII line, and one TL. These boundaries split the stacking diagram into five regions. The possible stacking sequences are A, B, AB, ABA, and BA [^3]. The ABA sequence appears when a path crosses the bulk binodal twice [@floating; @schmidt04aog]. This very simple example already shows the richness of the stacking diagram. It also suggests that the sample height plays a major role. The size of the area of the stacking diagrams occupied by each stacking sequence depends strongly on the height of the sample. For example, the AB region substantially increases with $h$, cf. Fig. \[fig3\]b. Two samples of different height and different stacking sequences might have the same composition and concentration of colloids (we will see examples in the next section). The height of the sample might have an even stronger influence on the stacking diagram, as we will demonstrate in the following. In Fig. \[fig4\]a we show a further model bulk phase diagram. There are three different phases: A, B and C. Three binodals for A-B, A-C, and B-C coexistence meet at a triple point. A phase diagram like this might represent a mixture in which the species $1$ represents spherical colloids and the species $2$ consists of e.g., elongated colloidal particles. The elongated particles can undergo isotropic-nematic and nematic-smectic phase transitions. The stacking diagrams for this mixture are depicted in Fig. \[fig4\]b for two different heights, $h_1$ and $h_2$, with $h_1<h_2$. In both cases the slope of the path is $s=1$ and both buoyant masses are positive. The boundaries in the stacking diagram are: six SDI lines (two for each of the three binodals), one SDII line (the slope of the path matches the slope of the B-C binodal at one point), and one TL line (originating from the triple point). The stacking diagrams for heights $h_1$ and $h_2$ differ substantially from each other, see left and right panels of Fig. \[fig4\]b, respectively. We observe two main differences between the diagrams for short and long samples: First, the sedimentation paths for the small system ($h_1$) fit in the space between the A-B and B-C binodals of the bulk phase diagram, see an example in Fig. \[fig4\]a. Consequently the stacking sequence B occurs in the stacking diagram, Fig. \[fig4\]b (left). In contrast, the stacking sequence B does not occur in the large samples ($h_2$). The B sequence is replaced by an ABC state, Fig. \[fig4\]b (right). The sedimentation paths in this case are long enough such that they do not fit in the region between the A-B and B-C binodals in bulk. Instead, the path must cross at least one of the binodals. Second, the sequence CABC is generated by paths crossing the three binodals in bulk. This sequence is present only in the long samples, Fig. \[fig4\]b (right). The paths corresponding to the short samples ($h_1$) are not long enough to cross the three binodals, and hence the CABC sequence is absent. These examples illustrate how the stacking diagrams for different heights might differ qualitatively. By changing the overall height of the sample some stacking sequences are replaced by others (e.g. the B sequence for $h_1$ is replaced by ABC for $h_2$) and one also observes the occurrence of new sequences, such as the CABC sequence for $h_2$. Mixtures of patchy colloids {#bbb} --------------------------- We next apply our theory to patchy colloidal binary mixtures. We study two cases in which the species differ either by the number or by the types of patches. ### Different number of patches {#23m} We model the colloids by hard spheres of diameter $\sigma$ with identical patches (spheres of size $\delta$) on the surface, see Fig. \[fig5\]a. If two patches overlap the internal energy of the system decreases by $\epsilon$. We use Wertheim’s first order perturbation theory [@wertheim] and a generalization of the Flory-Stockmayer theory of polymerization [@flory; @stock] to compute the bulk phase diagram of the mixture. We follow exactly the same implementation of the theory as in Ref. [@C0SM01493A]. Theory and Monte Carlo simulations for the bulk phase behaviour are in semi-quantitative agreement with each other [@emptyscio; @felix]. The species $1$ has two patches, and the species $2$ has three patches. The colloids with three patches undergo a phase transition between two fluids with different densities. With only two patches present the particles of species $1$ can form only chains. The absence of branching prevents phase separation and there is no fluid-fluid phase transition in the pure system of species $1$. In the mixture the transition between high and low density fluids ends at a critical point. See the binodal in the bulk phase diagram of the mixture shown in Fig. \[fig6\]a for scaled temperature $k_{\text{B}}T/\epsilon=0.09$. In addition to the binodal, the phase diagram contains a percolation line that divides percolated and non-percolated states. The system is percolated if the probability that a patch is bonded, $f_{\text{b}}$, is higher than the percolation threshold $p_{\text{T}}$. The percolation line intersects the binodal close to the critical point on the low density side. The high density phase (G) is an equilibrium gel or network fluid which is always percolated. The low density phase does not percolate (N) except for a very narrow region close to the critical point (G’). We refer the reader to Refs. [@emptyscio; @C0SM01493A] for further details about the bulk phase behaviour of this mixture. \[fig6\] To proceed and to obtain the stacking diagrams we need to set the slope of the sedimentation paths and the height of the sample. We fix the gravitational lengths of the colloids to $\xi_1=5\text{ mm}$ and $\xi_2=2\text{ mm}$ (typical values for colloidal particles). Hence, the slope of the path is fixed to $s=\xi_1/\xi_2=2.5$. The stacking diagrams in the $\bar\mu_1-\bar\mu_2$ plane for three different heights $h=1$ mm, $10$ mm, and $25$ mm are shown in Fig. \[fig6\]b. Each of them contains four SDI lines (two for the binodal and two for the percolation line) and two terminal lines (one for the critical point and one for the ending point of the percolation line). Six different stacking sequences are possible for this value of the slope: We use a dash between two stacks in the stacking sequence, like in the G-N sequence, to indicate that the sedimentation path crosses the binodal. The absence of a dash, e.g. in the GN sequence, indicates that the path crosses the percolation line. Once the stacking diagrams in the plane of average chemical potentials have been computed, we can transform the variables using the procedure described at the end of Section \[theory\]. In Fig. \[fig6\]d we show the resulting stacking diagrams in the plane of average packing fractions. The number and types of stacking sequences remain the same for the sample heights investigated here. However, the region of the phase space occupied by each sequence significantly depends on the value of the sample height. We show a specific example in Fig. \[fig7\] in which we plot the density profiles of two samples with the same average packing fractions ($\bar\eta_1=0.002$, and $\bar\eta_2=0.35$), but different heights ($h=25$ mm and $10$ mm). The corresponding state points are highlighted by green solid circles in the stacking diagrams of Fig. \[fig6\]. Despite the average colloidal concentrations being the same, the stacking sequences differ: G-N for the sample with $h=25$ mm and G for the case $h=10$ mm. Other values of the sample height and the gravitational lengths will result in identical phenomenology provided that the ratios $h/\xi_i$ with $i=1,2$, are unchanged. ### Different types of patches As a concluding example we study a binary mixture of patchy colloids with different types of patches. The species $1$ ($2$) possesses three patches of type A (B), see Fig. \[fig5\]b for an illustration. When two patches of type $\alpha$ and $\beta$ with $\alpha,\beta=\{A,B\}$ overlap, the energy of the system decreases by $\epsilon_{\alpha\beta}$. The bulk phase diagram of this model has been studied theoretically [@C1SM06948A] and by Monte Carlo simulations [@felix] for different values of the bonding energies $\epsilon_{\alpha\beta}$. The phenomenology that emerges is very rich as different types of gels can occur depending on the set of bonding energies. Here, we set $\epsilon_{\text{BB}}=\epsilon$ (energy scale), $\epsilon_{\text{AA}}=0.80\epsilon$, and $\epsilon_{\text{AB}}=0.85\epsilon$. We fix the scaled temperature, as in the previous case, to $k_{\text{B}}T/\epsilon=0.09$. The bulk phase diagrams in the planes of chemical potentials and of packing fractions are shown in Fig. \[fig8\]a and \[fig8\]b, respectively. At the value of temperature considered only the species $2$ (strongest bonds) undergoes a fluid-fluid phase transition. Hence, in the mixture there is only one binodal that ends at a critical point. In addition there are three percolation lines. One of these indicates whether the full mixture percolates, and the other two percolation lines indicate whether the individual species percolate. Although species $1$ does not undergo a first order fluid-fluid phase transition at this temperature, it still undergoes a percolation transition. The percolation lines and the binodal divide the bulk phase diagram into five different regions. At low chemical potentials (and hence low densities) the system is non-percolated (N). The other four states are equilibrium percolated gels: (i) a mixed gel (M) in which the mixture percolates but none of the species percolates independently, (ii) a bicontinuous gel or bigel (B) in which the mixture and both species percolate, (iii) two gels (G$_i$, $i=1,2$) in which the mixture and the species $i$ percolate. See [@C1SM06948A; @felix] for further details about the bulk behaviour. Here, we study sedimentation-diffusion-equilibrium. As in the example above we chose the gravitational lengths to be $\xi_1=5$ mm and $\xi_2=2$ mm, and study two different sample heights $h=1$ mm and $10$ mm. The resulting stacking diagrams are extremely rich, see Fig. \[fig8\]c and d, with more than $20$ distinct stacking sequences. Again, the regions occupied for each stacking sequence depend on the sample height. In some cases the same stacking sequence occurs in a completely different range of average packing fractions when varying the sample height, see for example the sequence 11 (G$_2$M) in Fig. \[fig8\]d. Even more important is the fact that the stacking diagrams for $h=1$ mm and $h=10$ mm are [qualitatively]{} different. There are several stacking sequences that are present only in one of the selected sample heights. For example, the sequence MG$_1$ (number $10$ in Fig. \[fig8\]) is only present in samples with $h=1$ mm, and the sequence G$_2$MG$_1$M (number $25$) occurs only for the case $h=10$ mm. Infinite vs finite height stacking diagrams {#ccc} ------------------------------------------- We conclude the section with several comments regarding the connection between the stacking diagrams for infinite and finite samples. The main effect is the occurrence of new sequences in the case of finite height samples. The new sequences are formed by the removal of one or more stacks of the sequences for $h\rightarrow\infty$. In general, a sequence observed at finite height might be a truncated sequence of the infinite system. This observation has strong implications for the correct interpretation of observed stacking sequences in finite height samples. In Fig. \[fig9\] we show the stacking diagram (infinite height) of the mixture of patchy colloids with two and three patches analyzed in Sec. \[23m\]. The stacking diagram is represented in the $s-a$ plane, cf . The diagram has been computed for $m_1>0$. Hence positive (negative) values of the slope $s=m_2/m_1$ correspond to positive (negative) values of $m_2$. There exists an analogous diagram for $m_1<0$ in which the only difference is that the stacking sequences have the reverse order. There are two sedimentation binodals (one for the binodal and one for the percolation line), two terminal lines (critical point of the binodal and ending point of the percolation point), and two asymptotic terminal lines (asymptotic behaviour of the binodal and the percolation lines). We have shown previously the diagrams at finite height for the slope $s=2.5$, see Fig. \[fig6\]. For this value of $s$ only three sequences are possible at infinite height, see Fig. \[fig9\]: . The finite height diagram is richer with up to six different sequences. These are the same three as for $h\rightarrow\infty$ and three new truncated sequences of the infinite case (G,N, and G-G’). As expected, by increasing the height of the sample, the regions occupied by the new truncated sequences in the stacking diagrams shrink (see Fig. \[fig6\]d). In this particular example, for $h=25$ mm, the stacking diagram is already dominated by the stacking sequences of the infinite height case. The infinite height stacking diagram provides the set of possible sequences for different values of $s$. Here, we have only analysed the value $s=2.5$ of the slope for finite height samples. The infinite height stacking diagram shows that for other values of $s$ further complex phenomenology occurs. For example, for negative values of the slope, i.e. $m_2<0$, it is possible to stabilize the sequence NGN-G which constitutes a reentrant percolation phenomena. This sequence also occurs in two-dimensional binary mixtures of patchy colloids [@Lucas]. Experimentally one can change the slope of the path via the synthesis of colloids with cores of different materials [@doi:10.1021/la0101548; @doi:10.1021/am400490h] or changing the mass density of the solvent. Hence, the full range of stacking sequences of a given colloidal mixture is, in principle, experimentally accessible. Discussion and conclusions ========================== Our theory is based on a local density approximation which assumes that for each $z$ the state of the sample can be approximated by a bulk state. Non-local effects might modify the stacking diagrams. In particular, the theory neglects the surface tension of the interfaces between stacks in the stacking sequence. If one of the stacks is very narrow the surface tension of the upper and lower interfaces might be higher than the gain in free energy due to the formation of the stack, as observed in colloid-polymer model mixtures [@schmidt04aog]. Under such circumstances, the final equilibrium stacking sequence might be different than that predicted by our local theory. This condition is analogue of capillary condensation/evaporation. Surface effects such as e.g., the occurrence of wetting and layering near the walls of the vessel might also modify the stacking diagrams. Our theory can be easily extended to multicomponent systems since the sedimentation paths remain lines in the phase space of chemical potentials. Also, the theory is directly applicable to molecular systems. There, the gravitational lengths are orders of magnitude higher than in colloidal systems. Hence, to observe similar phenomenology one needs containers of considerable size, such as for example geological deposits. We have obtained the stacking diagrams at constant sample height and fixed ratio of the buoyant masses. Other choices, such as for example keeping the colloidal concentrations fixed and varying the sample height, are also possible. A stacking diagram in which one of the variables is the height might be relevant to study the effects of slow solvent evaporation, which is a process that changes the total volume but keeps the particle number fixed. We have shown that two samples with the same colloidal concentrations but placed in vessels of different heights might have different stacking sequences. We have also shown that the stacking diagrams might be qualitatively different for different heights. Therefore, the sample height plays a major role in sedimentation-diffusion-equilibrium experiments. This role is as important as for example the average colloidal concentrations. We conclude that the sample height should be carefully measured and specified in any sedimentation experiment. We have focused on sedimentation-diffusion-equilibrium in colloidal mixtures. Future studies could consider the dynamics of sedimentation using dynamic density functional theory [@evans_dft; @ddft] and power functional theory [@pft]. We thank Matthias Schmidt for very useful discussions and a careful reading of the manuscript. This work was supported in part by the Portuguese Foundation for Science and Technology (FCT) through project EXCL/FIS-NAN/0083/2012. [39]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [](https://archive.org/details/atomsjean00perrrich) (, ) [, ()](\doibase 10.1103/PhysRevLett.71.4267) @noop [, ()]{} [, ()](http://stacks.iop.org/0034-4885/77/i=5/a=056602) [, ()](\doibase 10.1039/C1SM06535A) [, ()](\doibase 10.1039/C5SM00615E) [, ()](\doibase 10.1039/C2SM26120K) [, ()](\doibase 10.1038/srep00789) [, ()](\doibase 10.1039/C6SM00736H) [, ()](\doibase 10.1103/PhysRevE.93.030601) [, ()](\doibase 10.1021/la804023b) [, ()](\doibase 10.1039/C3SM52311J) [, ()](\doibase 10.1021/acsnano.5b01116) [, ()](\doibase 10.1021/acs.langmuir.5b02224) [, ()](http://stacks.iop.org/1478-3975/10/i=4/a=045005) [, ()](\doibase 10.1209/epl/i2003-10140-1) [, ()](\doibase 10.1103/PhysRevE.70.051401) [, ()](\doibase 10.1039/C3SM51491A) [, ()](http://stacks.iop.org/0953-8984/27/i=19/a=194115) [, ()](\doibase 10.1103/PhysRevLett.98.188304) [, ()](\doibase 10.1039/B915788C) [, ()](\doibase 10.1039/C0CP02296A) [, ()](\doibase 10.1038/nature11564) [, ()](\doibase 10.1007/BF01127721) [, ()](\doibase 10.1103/PhysRevLett.93.088303) [, ()](\doibase 0.1021/ja01856a061) [, ()](\doibase 10.1063/1.1723803) [, ()](\doibase 10.1039/C0SM01493A) [, ()](\doibase 10.1103/PhysRevLett.97.168301) [, ()](\doibase http://dx.doi.org/10.1063/1.4960808) [, ()](\doibase 10.1039/C1SM06948A) [, ()](\doibase 10.1021/la0101548) [, ()](\doibase 10.1021/am400490h) [, ()](\doibase 10.1080/00018737900101365) [, ()](\doibase http://dx.doi.org/10.1063/1.478705) [**, ()](\doibase http://dx.doi.org/10.1063/1.4807586) [^1]: Solvent evaporation might occur, changing the effective sample height and hence the concentration of colloids. [^2]: This ignores phase coexistence phenomena in polydisperse systems [^3]: Note that sequences with a single stack, such as A, are actually one phase-systems and not proper sequences made of different stacks
--- abstract: 'We present an extended analysis of deep [Chandra]{} HETG observations of the WR$+$OB binary system [WR 147 ]{}that was resolved into a double X-ray source [@zhp_10]. Our analysis of the profiles of strong emission lines shows that their centroids are blue-shifted in the spectrum of the northern X-ray source. We find no suppressed forbidden line in the He-like triplets which indicates that the X-ray emitting region is not located near enough to the stars in the binary system to be significantly affected by their UV radiation. The most likely physical picture that emerges from the entire set of HETG data suggests that the northern X-ray source can be associated with the colliding stellar wind region in the wide WR$+$OB binary system, while the X-rays of its southern counterpart, the WN8 star, are result from stellar wind shocking onto a close companion (a hypothesized third star in the system).' author: - 'Svetozar A. Zhekov and Sangwook Park' title: ' [Chandra]{} HETG Observations of the Colliding Stellar Wind System [WR 147 ]{} ' --- Introduction ============ WR$+$OB binaries are the brightest X-ray sources amongst the Wolf-Rayet (WR) stars [@po_87]. Their enhanced emission originates from the interaction region of the winds of the two massive stars (@pril_76; @cherep_76). Since the winds are highly supersonic, the interaction region is bounded by two shocks each compressing the stellar wind of the WR or OB star, and a contact discontinuity surface separating their shocked plasmas (for the first hydrodynamic models see @lm_90; @luo_90; @st_92; Myasnikov & Zhekov 1991, 1993). Given the wind velocities (typical values of 1,000-3,000[ km s$^{-1}$ ]{}), the postshock temperatures are in the keV range, and most of the plasma emission is thus in X-rays. If the shocks are adiabatic, the X-ray luminosity of the colliding stellar winds (CSW) is proportional to the square of the mass-loss rate ($\dot{M}$) and inversely to the binary separation ($D$): $L_X \propto \dot{M}^2 V_{wind}^{-3.2} D^{-1}$ (see the references above for discussion). From this follows: (i) the shocked WR wind dominates the X-ray emission from CSW in WR$+$OB binaries due to its much more massive wind ($\dot{M}$ of a WR star is about an order of magnitude higher than that of an O star); (ii) by the same argument, a WR$+$OB binary will be more luminous in X-rays than an OB$+$OB binary, with similar wind velocities and binary separation. On the other hand, CSWs in close and wide binaries have quite different behavior. While in the former, shocks are radiative and thus the interaction region is subject to instabilities (e.g., @st_92; @mzhb_98; @wa_00), the shocks are adiabatic in the latter and the effects of thermal conduction might also be important [@mzh_98]. Moreover, the effects of electron-ion temperature equilibration behind the shocks [@zhsk_00] and non-equilibrium ionization [@zh_07] can influence the X-ray emission from CSWs. A difference between CSWs in close and wide WR binaries is also seen in radio, and as a rule non-thermal radio (NTR) sources are associated with the wide binary systems [@do_00]. The strong shocks are the likely place for accelerating relativistic particles, and the CSWs in wide binaries offer good conditions for this mechanism to operate efficiently since the shocks are located relatively far from the optically bright sources (the stars) and thus the inverse Compton losses are minimal (for the first detailed models of NTR emission from CSWs see @do_03; @pitt_06). It is therefore seen that the CSWs in stellar binaries are an ideal laboratory for studying the wealth of physical processes related to strong shocks. With the launch of the modern X-ray observatories providing high spectral and spatial resolution observations ([Chandra]{}, [XMM-Newton]{}), a new window has opened for gathering detailed information about this exciting phenomenon. By confronting theoretical models of CSWs with observations, we can rigorously test them, and thus improve our understanding of the underlying physics. For such a goal, it is necessary to have an object which is bright enough in X-rays, it is a strong non-thermal radio source and it is relatively close to us. Thus, its radio-to-X-ray emission might be spatially resolved and provide us with detailed information about the CSW region and the stars in the binary system. Because the two components in the binary system can be separated, [WR 147 ]{}is unique amongst the massive WR$+$OB binaries and offers a rare opportunity for studying the CSW-binary phenomenon in its entirety. We report here results from our analysis of deep [Chandra]{} HETG observations of the CSW binary [WR 147 ]{}. @zhp_10 presented the first part of our study based on the zeroth-order HETG data whose most important result is that the X-ray emission from [WR 147 ]{}was resolved into two sources. This paper is organized as follows. We give basic information about the WR$+$OB binary [WR 147 ]{}in Section \[sec:thesystem\]. In Section \[sec:observations\], we briefly review the [Chandra]{} HETG observations. In Section \[sec:lines\], we present the results from analysis of strong X-ray emission lines. In Section \[sec:origin\], we discuss the origin of X-rays in the two X-ray components of [WR 147 ]{}. In Section \[sec:global\], we report results from the global spectral models. In Section \[sec:discussion\], we discuss our results and we list our conclusions in Section \[sec:conclusions\]. The Wolf-Rayet Binary [WR 147 ]{} {#sec:thesystem} ================================= The Wolf-Rayet star [WR 147 ]{}(WR$+$OB; @vdh_01) is a classical example of colliding wind binary at a distance of $630\pm70$ pc [@ch_92]. High-resolution radio observations resolved its emission into two components: a southern thermal source, [WR 147S ]{}(the WN8 star in the system), and a northern non-thermal source, [WR 147N ]{} (@ab_86; @mo_89; @ch_92; @con_96; @wi_97; @sk_99) with separation of $\sim 0\farcs57$. The binary system was spatially resolved both in infrared and optical to have a separation of $\sim 0\farcs64$ (@wi_97; @nie_98) which at the distance to this object corresponds to projected (or minimum) binary separation of $403\pm13$ au. While the spectral type of the WR star in the binary is well defined (WN8h; @vdh_01), that of the OB companion is not well constrained. From spatially resolved near-infrared and optical photometry, the spectral type of the latter was estimated correspondingly as a B0.5V [@wi_97] and O8-9 V-III [@nie_98]. And @lepine_01 classified it as a O5-7 I-II from spatially resolved spectra, but in the red optical domain. Due to the high extinction towards [WR 147 ]{}(A$_V=10.45$; A$_V=$ A$_v/1.11$; A$_v=11.6$; @vdh_01), the WR wind parameters, $\dot{M}=4\times10^{-5}$[ M$_{\odot}$ yr$^{-1}$ ]{}; $V_{wind}=950$[ km s$^{-1}$ ]{}, were derived from radio and NIR observations (@sk_99; @mo_00). For consistency with the previous works (@sk_99; @sk_07; @zh_07), we adopt $\dot{M}=6.6\times10^{-7}$[ M$_{\odot}$ yr$^{-1}$ ]{}; $V_{wind}=1600$[ km s$^{-1}$ ]{}for the stellar wind parameters of the OB companion. The previous X-ray observations of [WR 147 ]{}have revealed the presence of thermal emission from high temperature plasma: $\mbox{kT} \geq 0.5$ keV ([Einstein]{} observatory; @cai_85); $\mbox{kT} \approx 1$ keV ([ASCA]{}; @sk_99); $\mbox{kT} = 2.7$ keV ([XMM-Newton]{}; @sk_07) as the [XMM-Newton]{} observations detected the Fe K$_{\alpha}$ complex at 6.67 keV which is a clear sign of thermal X-rays even at high energies. Note that the temperature change simply reflects the improving quality of the X-ray data over the years. The observations with [Chandra]{} High-Resolution Camera (having low photon statistics, $\sim 148$ source counts, and no spectral information) found indications that the X-ray emitting region in [WR 147 ]{} might be extended and it peaks north from the WN8 star although a deeper X-ray image was needed to determine the degree of spatial extent [@pitt_02]. The recent high resolution [Chandra]{} observations resolved the X-ray emission from [WR 147 ]{}into two sources, [WR 147N ]{}and [WR 147S ]{}, with a spatial separation of $\approx 0\farcs60$ [@zhp_10]. The corresponding analysis of undispersed spectra showed that [WR 147N ]{}and [WR 147S ]{}have different global characteristics as the latter being more absorbed and having higher plasma temperature: N$_H = 2.28\,[2.08 - 2.57]\times10^{22}$ cm$^{-2}$; $\mbox{kT} = 1.78\,[1.52 - 1.98]$ keV for [WR 147N ]{}; N$_H = 3.83\,[3.51 - 4.20]\times10^{22}$ cm$^{-2}$; $\mbox{kT} = 2.36\,[2.12 - 2.56]$ keV for [WR 147S ]{}. It is worth noting that the absorption towards [WR 147N ]{}almost perfectly corresponds to the optical extinction of [WR 147 ]{}if the @go_75 conversion is used. Observations and Data Reduction {#sec:observations} =============================== [WR 147 ]{}was observed with [Chandra]{} in the configuration HETG-ACIS-S in eight consecutive runs ( ) in the period 2009 Mar 28 - Apr 10, providing a total effective exposure of 286 ksec. The roll angle was between $75^{\circ}$ and $82^{\circ}$: therefore, the dispersion axis was aligned approximately with the position angle of the binary system [WR 147 ]{}as derived in the optical (P.A. $=350^{\circ}\pm2$; @nie_98). The instrument configuration was such that the [negative]{} first-order MEG/HEG arms were pointing to north. By default, the pixel randomization is switched off in grating data. As reported in @zhp_10, the X-ray emission from [WR 147 ]{}was spatially resolved into a northern, [WR 147N ]{}, and a southern counterpart, [WR 147S ]{}(the WN8 star in the binary). Thus, following the Science Threads for Grating Spectroscopy in the CIAO 4.1.2 [^1] data analysis software, the positive and negative first-order MEG/HEG spectra for each of the eight observations were extracted centered on the [WR 147S ]{}position on the sky. The [Chandra]{} calibration database CALDB v.4.1.3 was used in this analysis. The resultant spectra were merged into one spectrum each for the positive and negative MEG/HEG arms with respective total counts of 2742 (MEG$+1$), 2396 (MEG$-1$), 1438 (HEG$+1$) and 1606 (HEG$-1$). The zeroth-order data were discussed in @zhp_10 and we only mention that the total number of zero order counts in the [WR 147N ]{}and [WR 147S ]{}spectra were 2158 and 5108, respectively. We note that terms [WR 147 ]{}and WR 147N$+$S will be used throughout the text to refer to the total X-ray emission (and spectra) of the studied WR$+$OB binary system. For the spectral analysis in this study, we made use of standard as well as custom models in version 11.3.2 of XSPEC [@Arnaud96]. Spectral Lines {#sec:lines} ============== The X-ray emission from [WR 147 ]{}is dominated by [WR 147S ]{}[@zhp_10] and the presence of another source ([WR 147N ]{}) at $\sim 0\farcs6-0\farcs64$ will cause the emission line profiles to be ‘blurred’ since the [WR 147S ]{}- [WR 147N ]{}orientation on the sky is approximately along the South-North direction, that is parallel to the dispersion axis. If there were no line shifts in the [WR 147N ]{}emission, the line centroids in the total [WR 147 ]{}spectrum would correspondingly show [blue shifts]{} in the positive and [red shifts]{} in the negative arms of the MEG/HEG spectra and their values will be the same ($\vert z(+1)\vert = \vert z(-1)\vert$). If the [WR 147N ]{}emission was intrinsically blue-shifted, this would result in larger line shifts in the MEG/HEG($+1$) than in the MEG/HEG($-1$) for each spectral line ($\vert z(+1)\vert > \vert z(-1)\vert$). The result would be just the opposite ($\vert z(+1)\vert < \vert z(-1)\vert$), if the [WR 147N ]{}spectrum was red-shifted. For each spectral line, we fitted all four spectra, MEG($+1/-1$) and HEG($+1/-1$), simultaneously as they were re-binned every two bins to improve the photon statistics. For the S XV and Si XIII He-like triplets, we fitted a sum of three Gaussians and a constant continuum. The centers of the triplet components were held fixed according to the [Chandra]{} atomic data base[^2] and all components share the same line width and line shifts. Similarly, we fitted the Si XIV and Mg XII H-like doublets but the component intensities were fixed through their atomic data values. Figure \[fig:shifts\] shows the results for the line shifts of prominent lines in the X-ray spectrum of [WR 147 ]{}. We see that all the lines are blue-shifted in the MEG/HEG($+1$) and red-shifted in the MEG/HEG($-1$) spectra and $\vert z(+1)\vert > \vert z(-1)\vert$. Thus, the results indicate [blue-shifted]{} X-ray emission from [WR 147N ]{}. And, we note that we do not find suppressed forbidden line in the He-like triplets. Motivated by these results, we developed a custom model for XSPEC to fit the line profiles that consist of line emission from two sources with an ‘offset’, $\Delta$, for the line center of the second one. We note that for the instrument configuration of our observations (§ \[sec:observations\]) and since the second source ([WR 147N ]{}) is located north from [WR 147S ]{}its spectral lines will get a red shift in the negative first-order spectra and a blue shift in the positive first-order spectra that correspond to the spatial offset $\Delta$. But for MEG and HEG this line shift will be different in units of wavelength due to their different spectral resolution ($\Delta_{MEG} = 2 \Delta_{HEG}$). For each source, the profiles of the spectral line doublets and triplets were treated as described above. We fitted this model to the line profiles of the H-like doublets of Mg XII and Si XIV as well as of the He-like triplets of Si XIII and S XV again simultaneously for the four first-order spectra. The fit results are given in Table \[tab:lines\] and individual line profiles are shown in Fig. \[fig:profiles\]. From the fit results we see that the line profiles in the spectrum of [WR 147S ]{}are broader than those in [WR 147N ]{}: bulk gas velocities (FWHM) with a typical value of $\approx 1000$[ km s$^{-1}$ ]{}are present in the former while the line widths indicate slower gas motion ($ < 1000$[ km s$^{-1}$ ]{}) for the latter. This may be a sign of different X-ray production mechanisms and we will return to this issue in § \[sec:origin\]. An important feature of these mechanisms is that we do not find suppressed forbidden line in the He-like triplets which means that the X-ray emission plasma has relatively low density and/or the hot plasma regions both in [WR 147S ]{}and [WR 147N ]{}are located far enough from strong UV sources. Perhaps the two most interesting fit results are related to [WR 147N ]{}alone. First, we see that despite their relatively large errors the centroids of [all]{} the studied lines are [blue-shifted]{} as anticipated from the preliminary analysis of the line profiles (see above). Second, the values for the spatial ‘offset’ between [WR 147N ]{} and [WR 147S ]{}are consistent between different spectral lines. Interestingly, the average value from all four spectral lines, $\bar{\Delta} = 0\farcs603^{+0.10}_{-0.08}$, is in a very good correspondence with the source separation measured directly from the deconvolved (1.0 - 2.0 keV) zeroth-order image of [WR 147 ]{}[@zhp_10]. This is a very important internal cross-check for the HETG results and another nice illustration of the superior spatial and spectral resolution of the [Chandra]{} observatory. Finally, for the analysis here we assumed that the spectral lines of [WR 147S ]{}are not shifted and the reason was that the fits are too complicated already, thus, an extra free parameter in the fits cannot be constrained well due to the quality of the data. But we note that we derived results generally consistent with zero line shifts if we let this parameter vary (the average value from the line shifts of the prominent line complexes in the spectrum of [WR 147S ]{}is -96$^{+75}_{-76}$[ km s$^{-1}$ ]{}). The Origin of X-ray Emission from [WR 147 ]{} {#sec:origin} ============================================= Thanks to its superior spatial resolution, [Chandra]{} resolved the CSW binary [WR 147 ]{}into a double X-ray source. The different spectral characteristics (plasma temperature and X-ray absorption) as deduced from the analysis of the undispersed spectra of both components (@zhp_10; see also § \[sec:thesystem\]) as well as the results from the fits to spectral line profiles in the first-order spectra (§ \[sec:lines\]) likely point to different emission mechanisms responsible for the X-ray emission from [WR 147N ]{}and [WR 147S ]{}. X-rays from [WR 147N ]{} {#subsec:wr147n} ------------------------ Since [WR 147 ]{}is a spatially resolved radio source with thermal and non-thermal components, all the analyses of its X-ray emission before the [Chandra]{} observations have assumed that the colliding stellar winds are responsible for the total X-ray emission from this WR$+$OB binary. We now know that the physical picture is more complex and at least two components are contributing to the X-ray emission of the [WR 147 ]{}system. Thus, only the northern source, [WR 147N ]{}, can be associated with the CSW region in this binary system [@zhp_10]. But, the presence of another massive star (the OB companion in the binary) in vicinity of [WR 147N ]{}requires a more careful look into such an identification. We recall that @zhp_10 found that the CSW model with nominal wind parameters for [WR 147 ]{}perfectly matches the shape of the X-ray spectrum of [WR 147N ]{}. They also reported a correspondence between the location of [WR 147N ]{}in X-rays and [WR 147N ]{}in the radio, and the latter is definitely associated with the CSW region in [WR 147 ]{}[@con_99]. We could prove that [WR 147N ]{}is located in the CSW region of the binary system if the X-ray separation between [WR 147N ]{}and [WR 147S ]{}is smaller than the value derived from the optical and NIR imaging. This would indicate that the northern X-ray source is detached from the surface of the OB companion. However, the values derived from the X-ray analysis do not provide solid evidence for that due to their uncertainties (see § \[sec:lines\] and Table \[tab:lines\]). Extended emission in [WR 147N ]{}would support that this source is the CSW region, but we see no extended X-ray emission in the HETG zeroth-order image that might have morphology similar to that of the extended non-thermal radio emission in [WR 147N ]{}[@con_99]. A reason for that could be the limited photon statistics in soft X-rays (only $\sim 700$ counts in the 1-2 keV band) and we note that soft X-rays originate downstream from the shocks that is in the ‘outskirts’ of the CSW region. Thus, future X-ray observations that provide images with higher quality may help us establish with certainty the morphology of [WR 147N ]{}which in turn will be very helpful in resolving the issue about the proper identification of this X-ray source. On the other hand, the massive OB stars are X-ray sources and could it be that the northern X-ray source, [WR 147N ]{}, is in fact the OB companion in this wide binary system? In general, this might well be the case but there are some pieces of indirect evidence that make such an identification more unlikely. For example, the plasma temperature in [WR 147N ]{}is relatively high ($\mbox{kT}= 1.78$ keV; § \[sec:thesystem\]) while OB stars are in general ’soft’ X-ray sources with temperatures below 1 keV (e.g., @woj_05; @zhp_07; see also §4.1.2 and §4.3 in the review paper of @gudel_naze_09 and the references therein). As a reference case, we ran XSPEC simulations having exposure of 286 ksec for a ‘soft’ X-ray source with $\mbox{kT} = 0.6$ keV, L$_X (0.5-10 \mbox{ keV}) = 10^{31}$ ergs s$^{-1}$ and N$_H = 2.3\times10^{22}$ cm$^{-2}$ (typical for [WR 147N ]{}), using the ancillary response functions from our zeroth-order HETG observation. We found 30-40 source counts or $< 2$% from the total zero order counts of [WR 147N ]{}. This indicates that a ‘soft’ X-ray source will likely have a small contribution to the total emission from [WR 147N ]{}. But, exceptions could be the rare hot magnetic objects like the massive O star $\theta^1$ Ori C and the B star $\tau$ Sco whose spectra show signs of considerably hotter plasma (@sch_03; @gagne_05; @cohen_03). The X-ray production mechanism that is believed to operate in such objects is that of magnetically confined wind shocks (MCWS, e.g., @babel_97; @ud_02). This model suggests that the X-ray emission mostly originates in regions very close to the stellar surface, thus, due to the strong stellar UV emission the forbidden line can likely be suppressed even in the He-like triplets as Si XIII and S XV. In fact, this is the case in the classical MCWS object $\theta^1$ Ori C [@gagne_05] and similar indications are found for $\tau$ Sco [@cohen_03]. Opposite to this, we do not find suppressed forbidden line in the Si XIII and S XV He-like triplets in the X-ray spectrum of [WR 147N ]{}(§ \[sec:lines\]) which indicates that these lines are formed in regions far from the surface of the OB companion. It is also worth noting that because of the expected decay of the magnetic field strength with the age of a massive star, the MCWS mechanism is associated only with young massive stars (age $\le 1$ Myr; @sch_03). But, the age of the OB star in the wide WR$+$OB binary system [WR 147 ]{}must be larger than 1-2 Myr as indicated by the presence of a WR star in the system. From all this, a physical picture in which [WR 147N ]{}resides in the CSW region seems more realistic than assuming that [WR 147N ]{}is associated with the OB companion in the massive binary system [WR 147 ]{}. However, we should keep in mind that the CSW model with nominal wind parameters for [WR 147 ]{}overestimates the observed [WR 147N ]{}luminosity by a factor of $\sim 16$ [@zhp_10], a discrepancy that must find reasonable explanation in the CSW scenario. We will discuss possible solutions to this problem in § \[sec:discussion\]. [WR 147S ]{}: Unusual X-ray Source {#subsec:wr147s} ---------------------------------- The X-ray detection of the WN8 star in [WR 147 ]{}and its brightness make it unusual amongst the WR stars of the same subtype. As shown by @sk_10, single WN7-9 stars are very weak X-ray sources. And, the X-ray emission of [WR 147S ]{}is quite hard as the analysis of its undispersed spectrum indicated ($\mbox{kT} = 2.36$ keV). Skinner et al. (2002a,b, 2010) discussed in some detail possible mechanisms for X-ray production in WN stars, which include: instability-driven wind shocks; magnetically confined wind shocks; wind accretion shocks; colliding wind shocks (including the case of stellar wind shocking onto a close companion); non-thermal X-ray emission. We note that none of these mechanisms finds solid observational support for the moment and each of them has its own limitations and caveats. We recall that instability-driven wind shocks are supposed to be soft X-ray emitters which is not the case with [WR 147S ]{}. Accretion wind shocks result in a relatively high X-ray luminosity ($\sim 10^{36-37}$ ergs s$^{-1}$) and the observed value is short by 3-4 orders of magnitude. As discussed above (§ \[subsec:wr147n\]), in the magnetically confined wind shock picture X-rays form close to the stellar surface which results in a suppressed forbidden line even in He-like triplets as Si XIII and S XV due to the strong stellar UV emission. Opposite to this, we find no indications for such a suppression in the spectrum of [WR 147S ]{}(§ \[sec:lines\]). And, the X-ray spectrum of [WR 147S ]{}definitely has a thermal origin as indicated by the presence of various spectral lines that originate in thermal plasma and this is also the case at high energies. @sk_07 detected a Fe XXV K-line at 6.67 keV in the [XMM-Newton]{} spectra of [WR 147 ]{}and the zeroth-order HETG data undoubtedly showed that this line is associated with [WR 147S ]{}, the WN8 star in the binary system [@zhp_10]. But, the case of X-rays from the stellar wind shocking onto a close companion seems to find some support from our deep [Chandra]{} observations that span an almost two-week period. Figure \[fig:LC\] shows the light curve of [WR 147 ]{}in different energy bands. Obviously, a long-term variability is present in the data with a tentative period of 15-20 days. This variability is well established at high energies, thus, it should be associated with [WR 147S ]{}which dominates the X-ray emission at $\mbox{E} > 3$ keV. We fitted the light curve with two models: a constant flux and a simple sinusoidal curve. The values for the reduced $\chi^2$ for various fits are: $\chi^2(0.3-10~keV) =$ 1.55 (sine) and 3.79 (const); $\chi^2(0.3-2~keV = $ 1.63 (sine) and 1.14 (const); $\chi^2(3-10~keV) = $ 0.49 (sine) and 4.49 (const). The formal goodness of fit is: 0.20 (sine) and 0.0009 (const) for the (0.3-10 keV) LC; 0.18 (sine) and 0.34 (const) for the (0.3-2 keV) LC; 0.69 (sine) and 0.0002 (const) for the (3-10 keV) LC. The corresponding periods are: P(0.3-10 keV)$= 17.61\pm3.74$ days; P(3-10 keV)$= 15.48\pm1.91$ days. The amplitudes of the variability with respect to the constant flux are: 7% in (0.3-10 keV); 13% in (3-10 keV). Thus, if the X-rays in [WR 147S ]{}originate from stellar wind shocking onto a close companion, the observed variability might simply correspond to the orbital period of the companion star. The changes in the X-ray emission could be a result from the ellipticity of the orbit (variable emission measure) and/or they may be due to a variable X-ray absorption depending on the inclination angle of the orbit. Unfortunately, the quality of the individual spectra in the HETG data set (zeroth and first order spectra) does not allow us address these issues. We only note that for an adopted value of 15.48 days for the orbital period and a total mass of the system of 20 M$_{\odot}$, the Kepler’s third law ($a = 0.01952 P^{2/3}_d [M/M_{\odot}]^{1/3}$ au; $P_d$ is the orbital period in days; $M$ is the total mass) gives $a = 0.33$ au for the semi-major axis of the orbit. For a distance of 630 pc to [WR 147 ]{}[@ch_92], the separation between the WN8 star and the normal star companion will be $\approx 0\farcs0005$. Being that close to a luminous hot star makes it highly unlikely for such an object be detected by imaging techniques. On the other hand, we can speculate that this variable X-ray source provides extra ionizing photons that are capable of changing the population in highly ionization stages of elements as carbon, nitrogen and oxygen. Given the spectral subtype of the WR star in [WR 147 ]{}(WN8), we can propose that variability in the profiles of NIV and NV lines can be expected in the optical with a periodicity as seen in X-rays, an effect similar to the Hatchett-McCray effect in massive X-ray binaries [@hat_77]. Global Spectral Models {#sec:global} ====================== A systematic approach to modeling the total X-ray spectrum of WR 147N$+$S would be to fit the entire observed spectrum using a reasonable physical model. We note that the HETG first-order spectra represent the total (combined) X-ray emission of the two X-ray sources [WR 147N ]{}and [WR 147S ]{}. As discussed in § \[subsec:wr147n\], the most realistic assumption for the X-rays in [WR 147N ]{}is that they originate from the CSW region in the binary system. Thus, we explore this picture in some detail. On the other hand, the origin of the X-ray emission in [WR 147S ]{}is not well constrained and we consider two limiting cases: a distribution of adiabatic shocks and a distribution of equilibrium plasma. For the spectrum of [WR 147N ]{}, we adopted the CSW model with non-equilibrium ionization by @zh_07 and we made a new version for XSPEC that explicitly takes into account the line broadening (bulk gas velocities) from the hydrodynamic CSW model. An important feature of the new version is that it includes a spatial ‘offset’ for the CSW source with respect to a near by source which is the case with the HETG first-order spectra of [WR 147N ]{}. Moreover, to get realistic line profiles we need to know the position of the observer (line of sight) with respect to the CSW region (its axis of symmetry). This requires two more model parameters: the inclination angle, $i$ (the angle between the line of sight and the orbital axis) and the azimuthal angle, $\omega$ (defining the position of the observer in the orbital plane). We recall that the CSW model by @zh_07 makes use of the hydrodynamic model of @mzh_93 which adopts a convention that the O star in the binary system is located at the origin of the coordinate system. Panel (a) in Fig. \[fig:grid\] shows a schematic diagram of the wind interaction of two spherically symmetric stellar winds in a WR$+$OB binary. Since the CSW spectral calculations require 3D integration, it is not feasible to fit for the values of the orbital inclination and azimuthal angle of the observer’s line of sight. Some information about these parameters comes from the analysis of the spectral line profiles (§ \[sec:lines\]). We recall that blue-shifted lines were found in the spectrum of [WR 147N ]{}. This indicates that the CSW region is inclined towards the observer. That is, the CSW region is located between the observer and the WN8 star ([WR 147S ]{}). To explore this in detail, we have run a grid of CSW models for $\omega \in [90, 180]$. One should keep in mind that due to the axial symmetry of the CSW region, the line shifts (and line profiles) from the CSW region are the same for $\omega \in [90, 180]$ and $\omega = 360 - \omega$, and a given value of $i$. To be as close as possible to the observational situation, we carried out the CSW spectral simulations in XSPEC using the ancillary response functions from our HETG observations. We fitted the spectral lines in simulated CSW spectra in a similar way as for the real data analysis: using a sum of two and three Gaussians for the lines of H-like doublets and He-like triplets, respectively. Panel (b) in Fig. \[fig:grid\] shows the isolines for the observed line shifts of S XV, Si XIV, Si XIII and Mg XII (from Table \[tab:lines\]) in the ($i, \omega$) parameter space from our model calculations. From this in conjunction with the result about the orientation of the WR-OB binary axis on the sky [@nie_98], we conclude that the orbital inclination is smaller than $60^\circ$ with an average value of $30^\circ$: the isolines cluster around $20^\circ-40^\circ$. The upper limit of the inclination angle comes from the largest possible value from the $1\sigma$-error on the line shift of acceptably well constrained data: in this case it is from the Si XIII triplet and is shown with a solid curve in panel (b) of Fig. \[fig:grid\]. We note that the value for the $1\sigma$-error of the S XV triplet is beyond this limit but this is likely due to the poorer quality of the data for this line and the fact that the contribution from the CSW region is small (Table \[tab:lines\]). This prevents having a tighter constraint (smaller uncertainties) on its line shift although the derived value is consistent with those for the other lines in our analysis. From the values of the observed position angle of the WR-OB star axis on the sky [@nie_98], we derive a value for the azimuthal angle $\omega = 170^\circ-174^\circ$ (see Fig. \[fig:grid\]). @con_99 derived a ($45^\circ \pm 15^\circ$) range of possible inclination angles for the [WR 147 ]{}system from the analysis of the morphology of the non-thermal radio source [WR 147N ]{}. Thus, in the framework of the CSW picture, the results from the HETG observations are generally consistent with those from the radio observations of [WR 147 ]{}. But, we now obtain an additional piece of information: the CSW region is inclined towards the observer. Blue-shifted lines were detected, so the CSW region lies between the observer and the WN8 star. Our global spectral model consists of two components: a CSW NEI model representing the X-ray emission from [WR 147N ]{}and a thermal plasma component for the X-ray spectrum of [WR 147S ]{}. In accord with the spectral analysis of @zhp_10, both spectra are subject to the same X-ray absorption from the interstellar matter (ISM; N$_H = 2.3\times10^{22}$ cm$^{-2}$) and there is an excess X-ray absorption of the [WR 147S ]{}spectrum which we assume is due to the WN8 wind. Since the value for the ISM absorption corresponds well to the optical extinction to [WR 147 ]{}, we keep its value fixed in this analysis. The WN-wind absorption is a free parameter and for that purpose we adopted the cold wind approximation by making use of the [vphabs]{} model in XSPEC. This allows us to impose the same set of chemical abundances on the hot X-ray emitting plasma and the wind absorber. We recall that the CSW model at hand takes into account that the shocked OB-star wind has solar abundances while the shocked WN wind has a different set of abundances that can be adjusted in the fitting process. Thus, the CSW model and the model for the X-ray emission from [WR 147S ]{}share their abundances. For consistency with the previous studies we adopted the same set of WN abundances (see @sk_07; @zh_07) as only the values for Ne, Mg, Si, S, Ar, Ca and Fe were allowed to vary while those for the other elements were held fixed in the fits. The CSW model uses nominal stellar wind parameters (V$_{WR} = 950$ km s$^{-1}$, $\dot{M}_{WR} = 4\times 10^{-5}$ M$_{\odot}$ yr$^{-1}$; V$_{O} = 1600$ km s$^{-1}$, $\dot{M}_{O} = 6.6\times 10^{-7}$ M$_{\odot}$ yr$^{-1}$; $[\dot{M}_{O} V_{O} / \dot{M}_{WR} V_{WR}] = 0.028$) and a value of 403 au for the projected binary separation (§ \[sec:thesystem\]). But the mass-loss values for the winds of both stars were reduced by a factor of 4 to correct for the CSW luminosity discrepancy revealed from the analysis of the undispersed spectra [@zhp_10]. The orbital inclination and azimuthal angle were held fixed to $i = 30^\circ$ and $\omega = 171^\circ$. Also, a spatial offset with a fixed value of $\Delta = 0\farcs603$ (§ \[sec:lines\]) was adopted for the CSW X-ray source ([WR 147N ]{}). As mentioned above, we adopted thermal plasma models for the X-ray spectrum of [WR 147S ]{}that consider two limiting cases. The first case is a distribution of adiabatic shocks with non-equilibrium ionization effects (NEI shocks) and we made use of our custom model for XSPEC which was successfully used in the analysis of the X-ray spectra of SNR 1987A (e.g. @zh_09 and references therein). The second one considers a distribution of emission measure of thermal plasma in ionization equilibrium (CIE plasma) and we adopted our custom model which is similar to $c6pvmkl$ in XSPEC but uses the $apec$ collisional plasma model for the X-ray spectrum at given plasma temperature. We note that the line broadening is self-consistently calculated in the CSW model and for the spectrum of [WR 147S ]{}we adopted Gaussian broadening using the $gsmooth$ model in XSPEC. We fitted simultaneously four HETG first-order spectra of [WR 147 ]{}: MEG($+1$), MEG($-1$), HEG($+1$), HEG($-1$); and two zeroth-order (undispersed) spectra: one for [WR 147N ]{}and [WR 147S ]{}, respectively. Table \[tab:fits\] and Figures \[fig:fits\] and \[fig:dem\] show the corresponding fit results. The following results are worth noting. We see that there is a good correspondence between the derived abundances from the models that assume CIE plasma or NEI shocks are responsible for the X-ray emission of [WR 147S ]{}. The total observed X-ray flux of [WR 147 ]{}from the [Chandra]{} HETG is $1.31\times10^{-12}$ ergs cm$^{-2}$ s$^{-1}$ which is $\sim12$% smaller than that from the [XMM-Newton]{} observation in November 2004 [@sk_07]. We believe that this difference is due to the calibration uncertainties between the two telescopes but the 15.48-day X-ray variability of [WR 147S ]{}(see Fig. \[fig:LC\]) may also contribute some part of it. Note that the flux value from HETG is the average X-ray flux over the proposed variability period while the [XMM-Newton]{} data provide a ‘snap-shot’ X-ray spectrum of [WR 147 ]{}. Despite the fact that we adopted an average line broadening for the [WR 147S ]{}spectrum, the derived value (FWHM) from the global fits is in agreement with the results from the fits to individual lines (see Table \[tab:lines\]). But perhaps the most interesting and robust result from the global fits is that all the adopted models require very hot plasma to be present in [WR 147S ]{}(Fig. \[fig:dem\]). We mention that the hot plasma (CIE or NEI) at temperatures kT$~> 3$ keV supplies $\sim 56-63$% of the total observed flux from [WR 147S ]{}. Finally, we note that we have run two more cases of the global spectral models that have orbital inclination and azimuthal angle fixed correspondingly to: (i) $i = 10^\circ$ and $\omega = 170^\circ$; (ii) $i = 50^\circ$ and $\omega = 174^\circ$. Their results are consistent with those in Table \[tab:fits\] and Figures \[fig:fits\] and \[fig:dem\] of our basic case with $i = 30^\circ$ and $\omega = 171^\circ$. Discussion {#sec:discussion} ========== Based on the zeroth-order data [@zhp_10] and the analysis of the first-order spectra presented in this study, we have argued that the northern X-ray source, [WR 147N ]{}, is likely associated with the CSW region of the [WR 147 ]{}binary system, while we speculated that the X-ray emission of its southern counterpart, [WR 147S ]{}, is likely due to stellar wind shocking onto a close unseen companion. Next, we discuss in some detail two of the most important results related to this physical picture. Reduced Mass-Loss Rates ----------------------- As described in § \[sec:global\], we reduced the mass-loss rates by a factor of 4 to calibrate the CSW X-ray luminosity to that of [WR 147N ]{}. On the other hand, the amount of absorption due to the stellar wind can provide us with another estimate of this parameter. From the radial column density of the assumed ‘cold’ stellar wind and adopted [WR 147 ]{}abundances, we derive the corresponding minimum value of the absorption column density for the X-ray source located in the wind at some distance from the star: N$_{H, wind} = 2.76\times10^{22} \dot{M}_5 r_{12}^{-1} V_{1000}$ cm$^{-2}$, where $\dot{M}_5$ is the WN8 mass loss in units of $10^{-5}$ [ M$_{\odot}$ yr$^{-1}$ ]{}; $r_{12}$ is the distance from the star in $10^{12}$ cm; $V_{1000}$ is the wind velocity in units of 1000[ km s$^{-1}$ ]{}. If the X-ray source in [WR 147S ]{}is located at distance equal to the semi-major axis of the orbit of the hypothesized close companion (§ \[subsec:wr147s\]; $a = 0.33~\mbox{au} \approx 5\times10^{12}$ cm), we see that the minimum column density of the stellar wind is $\sim 2-3$ times larger than the value derived from the global fits (Table \[tab:fits\]) even for the factor of 4 reduced mass-loss rate ($\dot{M}_5 = 1$). A likely explanation for such a discrepancy is that we adopted a ‘cold’ stellar wind absorption model in the global spectral fits, that is all the chemical elements are in their neutral state. Cold gas is an efficient X-ray absorber, so, this model underestimates the column density of the stellar wind. In reality, the stellar wind is ionized considerably, more so in vicinity of the WN8 star where the putative close companion is orbiting the primary star. This results in a higher X-ray transparency of the ionized wind. Thus, to have the same optical depth as in the case of the ‘cold’ wind, the ionized wind must have a larger effective column density. This can likely resolve the wind-absorption discrepancy. However, to test such an absorption model we need to know the exact ionization structure of the stellar wind. This requires a detailed modeling in the optical-UV spectral range which is beyond the scope of the present study. We only note that we find higher values for N$_{H, wind}$ from the fits, with no deterioration of their quality, if we enforce a smaller column density for H, He and N, thus mimicking an appreciable deviation from neutral state for these elements. On the other hand, the important conclusion is that both the CSW luminosity and the WN8 stellar wind absorption indicate a reduced mass loss of the WN8 star by a factor of $\sim 4$ compared to its nominal value for [WR 147 ]{}(§ \[sec:thesystem\]). From a detailed spectroscopy study of the optical-to-infrared spectrum of [WR 147 ]{}, @mo_00 concluded that the clumping factor in the stellar wind of the WN8 star is in the range 0.04 to 0.25, suggesting a reduction to the mass-loss rate of 0.2 - 0.5 compared to the values derived under assumptions of wind homogeneity. Thus, if the smaller mass loss of the WN star needed to explain the X-ray data is due to wind clumping, this would mean that a considerably inhomogeneous wind extends out to large distances from the star (e.g., to the thermal radio emission region of $\sim 1000$ stellar radii) and only further beyond does it become homogeneous and form the CSW region of [WR 147 ]{}. But an alternative and interesting explanation can be proposed as well. A homogeneous stellar wind can have a smaller mass-loss rate, provided the distance to [WR 147 ]{}is less than the currently adopted value. We note that for the mass-loss values derived from the radio observations $\dot{M} \propto d^{1.5}$ (@pf_75; @wb_75). It is worth noting that the distance to [WR 147 ]{}may bear some appreciable uncertainties since it was derived by @ch_92 from comparative NIR photometry with the only known galactic WN8 star (WR 105) which is a member of an association and WR 105 is now re-classified as a WN9h star [@vdh_01]. Moreover, due to the appreciable uncertainties in the spectral type of the OB companion, @lepine_01 pointed out another discrepancy related to the optical emission from [WR 147 ]{}: either the WR star is too bright (by 1.5 mag) or its OB companion is too faint (by 1.5 mag) for their respective spectral types. We believe that future spatially resolved observations of [WR 147 ]{}in the optical can help us obtain an accurate distance to this object based on a well-constrained spectral type and luminosity class of the OB companion in the binary system. In turn, the mass-loss rate of the WN8 star will be constrained better which could help us reveal the importance of clumpiness in the stellar wind at large distances from the star. Also, a well-constrained luminosity of the OB star will result in a better estimate of its X-ray luminosity, using the L$_X$ - L$_{bol}$ relation for hot massive stars. Consequently, we will have a more realistic value for the contribution from the OB star to the total X-ray emission of [WR 147N ]{}. Note that the higher the L$_X$ of the OB star, the higher its contribution to the X-ray emission of [WR 147N ]{}, thus, the higher the mass-loss reduction required. And all this may allow us build a self-consistent physical picture of the CSW binary [WR 147 ]{}. High Plasma Temperature in [WR 147S ]{} --------------------------------------- The presence of very hot plasma (kT$~\sim 4$ keV; Fig. \[fig:dem\]) in the X-ray emission region of the WN8 star ([WR 147S ]{}) is a result that does not depend on the accuracy of the distance to the observed object. We recall that in the framework of the CSW model the shape of the X-ray spectrum of [WR 147N ]{}is well matched using the currently accepted values for the wind velocities (§ \[sec:thesystem\]). But the high plasma temperature in [WR 147S ]{}poses some problems for the [WR 147S ]{}emission model: X-rays from stellar wind shocking onto a close companion. Given the chemical composition of the WN8 wind (Table \[tab:fits\]), the temperature of the shocked plasma is $\mbox{kT}_{sh} = 2.27 V_{1000}^2$ keV, where $V_{1000}$ is the shock velocity in units of 1000[ km s$^{-1}$ ]{}. Thus, the maximum temperature we can get for V$_{wind} = 950$[ km s$^{-1}$ ]{}is considerably smaller than that derived from the global spectral fits. We think that one way to resolve this problem could be the following. In the adopted scenario of X-rays from a stellar wind shocking onto a close (normal star) companion, apart from its own velocity the WN8 wind gets additionally accelerated by the gravitational field of the normal star. Qualitatively, we can assume that near the surface of the close companion the effective velocity of the WN wind is: V$_{eff}^2 = \mbox{V}_{wind}^2 + V_G^2$, where V$_{G}^2 = 2 G M_{CO}/ R_{CO}$; $G$ is the gravitational constant; $M_{CO}$ and $R_{CO}$ are the companion mass and radius, respectively. In such a case, a value of V$_{G} = 930$[ km s$^{-1}$ ]{}is required to match a plasma postshock temperature of $\sim 4$ keV. To model the X-ray emission from [WR 147S ]{}, we adopted two limiting cases of hot plasma distribution: (i) collisional plasma with equilibrium ionization; (ii) plasma in adiabatic shocks with non-equilibrium ionization (§ \[sec:global\]). @zh_07 discussed the NEI effects in colliding stellar wind shocks in some detail and introduced a dimensionless parameter $\Gamma_{NEI}$. This parameter is a measure of whether or not the NEI effects are important: if $\Gamma_{NEI} \leq 1$ they must be taken into account; and they can be neglected if $\Gamma_{NEI} \gg 1$. For the scenario of the X-rays from stellar wind shocking onto a close companion, adopted here for [WR 147S ]{}, we have $\Gamma_{NEI} \gg 1$ for the stellar wind parameters of [WR 147 ]{}(even with the reduced mass-loss rates) and the semi-major axis of the companion orbit of $a = 0.33~\mbox{au}$. Similarly, we can introduce a dimensionless parameter that can be an indicator of whether the X-ray emitting plasma is associated with adiabatic or radiative shocks. Namely, $\Gamma_{cool} = t_{flow}/t_{cool}$, where $t_{flow} = r/V_{wind}$ and $t_{cool} = \frac{3}{2}\frac{nkT}{\Lambda_{cool}}$, ($\Lambda_{cool}$ is the cooling function for collisionally ionized optically-thin plasma). Note that if $\Gamma_{cool} \ll 1$ the shocks are adiabatic while values of $\Gamma_{cool} \ge 1$ indicate radiative shocks. If for simplicity we assume that the cooling is due only to bremsstrahlung emission (which gives a lower limit to the cooling, thus, an upper limit to $t_{cool}$), for helium dominated plasma with stellar wind parameters $\dot{M} = 10^{-5}$[ M$_{\odot}$ yr$^{-1}$ ]{}, V$_{wind} = 950$[ km s$^{-1}$ ]{}and orbital radius of the close companion $a = 0.33~\mbox{au}$, the dimensionless parameter becomes: $\Gamma_{cool} = 3.64 T_{keV}^{-0.5}$, where $T_{keV}$ is the plasma temperature in keV. From this and Fig. \[fig:dem\] we see that $\Gamma_{cool} \ge 1$ for the X-ray plasma in [WR 147S ]{}. Thus, in the framework of the adopted X-ray production mechanism of stellar wind shocking onto a close companion, the values of $\Gamma_{NEI} \gg 1$ and $\Gamma_{cool} \ge 1$ indicate that X-ray emission from a distribution of CIE plasma is a more appropriate physical model for the X-ray spectra of [WR 147S ]{}. In such a case and since the shocks are radiative, we have an upper limit on the available energy (luminosity) that can be converted into X-ray emission: no more energy is emitted than the energy flux crossing the shock front per unit area. Taking into account the additional acceleration of the wind in the gravitational field of the close companion (see above), we have: L$_X = \frac{1}{2}\int \rho \left(V_{wind,\perp} + V_G \right)^3 dS$ ($\rho$ is the density of the wind in front of the shock; $V_{wind,\perp}$ is the wind velocity component perpendicular to the shock front; $S$ is the shock surface). For radiative shocks, the corresponding shock surface follows the shape of the stellar surface and we derive: L$_X = \frac{1}{8} \left(\frac{R_{CO}}{a}\right)^2 L_{wind} \left[1 + 3\left(\frac{V_G}{V_{wind}}\right) + 3\left(\frac{V_G}{V_{wind}}\right)^2\right]$, where $L_{wind} = \frac{1}{2}\dot{M} V_{wind}^2$ is the mechanical luminosity of the WN8 wind. Note that this is similar to eq.(80) in @usov_92 but with a correction factor for the gravity of the companion star. And, the simple considerations presented here allow us to further check the consistency of the adopted physical picture. For a distance of 630 pc to [WR 147 ]{}and the value of unabsorbed X-ray flux derived for [WR 147S ]{} (Table \[tab:fits\]), we have L$_X = 1.25\times10^{33}$ ergs s$^{-1}$. Adopting the values of $\dot{M} = 10^{-5}$, $V_{wind} = 950$[ km s$^{-1}$ ]{}, $a = 0.33$ au and $V_G = 930$[ km s$^{-1}$ ]{}(for the latter see above), we derive $R_{CO} = 1.6$ R$_{\odot}$ and in turn we use the value of $V_G$ to derive $M_{CO} = 3.6$ M$_{\odot}$. Thus, we see that the physical characteristics of the putative close companion of the WN8 star in [WR 147 ]{}are consistent with those of an AB main sequence star [@allen_73]. Note that the mass and radius of the companion star will be smaller, provided [WR 147 ]{}is located closer to us than it is currently assumed (it can be shown that in the picture described above $R_{CO}, M_{CO} \propto d^{0.25}$). It is important to emphasize that the quantitative estimates presented here serve only as a check on the global consistency of the physical picture we speculated about in the case of [WR 147S ]{}and they do not represent a detailed physical model. Nevertheless, two other results deserve a brief discussion in the framework of the adopted physical picture. If the X-rays from [WR 147S ]{}originate from a region close to the unseen and much cooler companion (therefore, [not a strong source]{} of far UV photons) of the main WN8 star, this may qualitatively explain why we do not find suppressed forbidden line in the helium-like triplets of Si XIII and S XV. We note that the forbidden and intercombination lines in these line complexes have intensities that are within $(1-2)\sigma$ of their nominal values (see Table \[tab:lines\] and compare with the [Chandra]{} ATOMDB). This fact seems to be in accord with the conclusion above that the emission of CEI plasma is a more appropriate model for the X-ray spectrum of [WR 147S ]{}. But, if much better grating data for this source become available in the future and they reveal enhanced emission for the forbidden line in Si XIII and/or S XV, then this will pose a problem for the physical picture adopted here for the X-ray emission from [WR 147S ]{}. Grating data with very high quality are also needed to study possible variations of the line centroids in the X-ray spectrum of [WR 147S ]{}. Such variations are expected with the proposed orbital period of the unseen companion if its orbit is not seen pole-on from an observer. For the data at hand, we can analyze in detail only the integrated over the orbital period X-ray spectrum of [WR 147S ]{}. Thus, we can speculate that the derived line centroids being consistent with zero line shifts (see §\[sec:lines\]) may simply indicate that the axis of the orbital plane of the unseen companion in [WR 147S ]{}(the hypothesized third star in the system) is almost along our line of sight. Finally, it is worth noting that the X-ray emission from [WR 147S ]{}(the WN8 star in the [WR 147 ]{}binary system) shares common characteristics with that of presumably single WN stars detected in X-rays [@sk_10]: their spectra are rather hard and a hot plasma with $kT > 2$ keV must be present in the X-ray emission region. This could be a sign that similar mechanism for X-ray production operates in all these objects. If we assume that this common mechanism is identified as a stellar wind shocking onto a close companion, then the simple considerations presented above would suggest that a $L_X \propto L_{wind}$ relation might exist for these WN stars as well. Interestingly, @sk_10 found such a trend which, despite the appreciable scatter in the data, is stronger than that for the commonly adopted relation between the X-ray and bolometric luminosity in massive stars (e.g., compare their Figs. 9 and 10). We note that such a scatter might simply indicate various companion radii and orbital separation in different WN stars. However, it should be kept in mind that we need additional pieces of evidence for the presence of a close companion (supposedly a normal star) that would come from future X-ray observations (e.g., revealing periodic variability) or from observations in other spectral domains (e.g, see the end of § \[subsec:wr147s\]). The physical picture will be on even more solid ground if objects with similar characteristics in the X-rays are found amongst supposedly single objects of the other types of Wolf-Rayet stars (WC and WO). And we note that an WO object with hard X-ray emission is already detected: a presumably single WO star WR 142 (@oskin_09; @kim_10). Conclusions {#sec:conclusions} =========== In this work, we presented the second part of our analysis of the [Chandra]{} HETG data of the WR$+$OB binary system [WR 147 ]{}which was resolved into a double X-ray source in the zeroth-order HETG images [@zhp_10]. The basic results and conclusions are as follows: 1. Profiles of the strong emission lines in the positive and negative first-order MEG and HEG spectra were modeled simultaneously. The results are consistent with the X-ray emission coming from two sources spatially separated by $0\farcs603^{+0.10}_{-0.08}$. This value is in very good correspondence with the results from the analysis of the zeroth-order image [@zhp_10]. 2. The line profile analysis showed that the centroids of the lines in the spectrum of [WR 147N ]{}are [blue-shifted]{} and we find no indication of suppressed forbidden lines in the He-like triplets either in the [WR 147N ]{}or in the [WR 147S ]{}spectra. The latter means that in both sources: (i) the X-ray plasma is not located very close to strong UV sources, and (ii) densities in the X-ray plasma do not significantly exceed the critical density. 3. Our deep [Chandra]{} observations that span a period of about two weeks show that the southern source in the binary, [WR 147S ]{}, exhibits X-ray variability with a period of $15.48\pm1.91$ days and a 13% amplitude with respect to its average flux. 4. The northern X-ray source, [WR 147N ]{}, is probably associated with the CSW region in the wide WR$+$OB binary system with orbital inclination $i = 30^\circ [0^\circ - 60^\circ]$ . The (variable ) X-rays of its southern counterpart, [WR 147S ]{}(the WN8 star), likely result from the WN8 star wind shocking onto a close companion (a hypothesized third star in the system). 5. Global spectral models having two components, X-ray emission from NEI CSW hydrodynamic models for [WR 147N ]{}and optically-thin plasma spectra of a distribution of hot plasma for [WR 147S ]{}, provide a good match to the entire set of HETG (first and zeroth-order) spectra. We note that, first, a factor of 4 reduction of the stellar mass-loss rate relative to currently accepted values is required to match the X-ray luminosity of [WR 147N ]{}. Second, very hot plasma (kT $ \sim 4$ keV) must be present in [WR 147S ]{}. 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The results for the positive, $z(+1)$, and negative, $z(-1)$, arm are correspondingly denoted by diamonds (with error bars in blue) and squares (with error bars in red). For each line, the MEG and HEG data were fitted simultaneously. []{data-label="fig:shifts"}](f1.eps){width="2.80in" height="2.0in"} ![The zeroth-order background-subtracted light curves of [WR 147 ]{}: the seven data points correspond to the average count rate for each observation (the data for ObsIDs 10897 and 10678 were combined, since this is one observation split into two parts). The constant-flux fit is presented by the dashed line and the fit with simple sinusoidal curve is given by the dotted line. The time (x-axis) is in days and is with respect to the start of the first observation in the HETG data set. []{data-label="fig:LC"}](f3a.eps "fig:"){width="2.in" height="1.5in"} ![The zeroth-order background-subtracted light curves of [WR 147 ]{}: the seven data points correspond to the average count rate for each observation (the data for ObsIDs 10897 and 10678 were combined, since this is one observation split into two parts). The constant-flux fit is presented by the dashed line and the fit with simple sinusoidal curve is given by the dotted line. The time (x-axis) is in days and is with respect to the start of the first observation in the HETG data set. []{data-label="fig:LC"}](f3b.eps "fig:"){width="2.in" height="1.5in"} ![The zeroth-order background-subtracted light curves of [WR 147 ]{}: the seven data points correspond to the average count rate for each observation (the data for ObsIDs 10897 and 10678 were combined, since this is one observation split into two parts). The constant-flux fit is presented by the dashed line and the fit with simple sinusoidal curve is given by the dotted line. The time (x-axis) is in days and is with respect to the start of the first observation in the HETG data set. []{data-label="fig:LC"}](f3c.eps "fig:"){width="2.in" height="1.5in"} ![ Line shifts of spectral lines that originate in the colliding stellar wind region. [Panel (a)]{} shows a schematic diagram of the stellar wind interaction in a WR$+$O binary system. The wind interaction cone is denoted by CSW (the axis Z is its axis of symmetry; the axis X is perpendicular to the orbital plane; the axis Y completes the right-handed coordinate system); the line of sight towards observer by l.o.s.; and the two related angles, $i$ (orbital inclination) and $\omega$ (azimuthal angle) are marked as well. [Panel (b)]{} shows the isolines for the observed line shifts for various strong spectral lines (Table \[tab:lines\]) on the grid of theoretical CSW models with the same stellar wind parameters but for different values of the inclination and azimuthal angles. The isoline shown with the solid line gives the upper limit to the line shifts. The three vertical solid lines correspond to the observed position angle of the WR-O star axis on the sky (the PA value $\pm1\sigma$ error; @nie_98). []{data-label="fig:grid"}](f4a.eps "fig:"){width="3.20in" height="2.286in"} ![ Line shifts of spectral lines that originate in the colliding stellar wind region. [Panel (a)]{} shows a schematic diagram of the stellar wind interaction in a WR$+$O binary system. The wind interaction cone is denoted by CSW (the axis Z is its axis of symmetry; the axis X is perpendicular to the orbital plane; the axis Y completes the right-handed coordinate system); the line of sight towards observer by l.o.s.; and the two related angles, $i$ (orbital inclination) and $\omega$ (azimuthal angle) are marked as well. [Panel (b)]{} shows the isolines for the observed line shifts for various strong spectral lines (Table \[tab:lines\]) on the grid of theoretical CSW models with the same stellar wind parameters but for different values of the inclination and azimuthal angles. The isoline shown with the solid line gives the upper limit to the line shifts. The three vertical solid lines correspond to the observed position angle of the WR-O star axis on the sky (the PA value $\pm1\sigma$ error; @nie_98). []{data-label="fig:grid"}](f4b.eps "fig:"){width="3.20in" height="2.286in"} ![ Emission measure (EM) of the [WR 147S ]{}distribution of thermal plasma in collisional ionization equilibrium (DEM of CIE plasma) and of adiabatic shock with non-equilibrium ionization effects taken into account (DEM of NEI shocks). The shaded area represents $1\sigma$ errors from the fits. The EM values are for adopted distance of 630 pc to [WR 147 ]{}. []{data-label="fig:dem"}](f6a.eps "fig:"){width="2.80in" height="2.0in"} ![ Emission measure (EM) of the [WR 147S ]{}distribution of thermal plasma in collisional ionization equilibrium (DEM of CIE plasma) and of adiabatic shock with non-equilibrium ionization effects taken into account (DEM of NEI shocks). The shaded area represents $1\sigma$ errors from the fits. The EM values are for adopted distance of 630 pc to [WR 147 ]{}. []{data-label="fig:dem"}](f6b.eps "fig:"){width="2.80in" height="2.0in"} [lcrrrrcc]{} S XV K$_{\alpha}$ & 5.0387 & 1300$^{+395}_{-265}$ & 15.36$^{+1.92}_{-1.92}$ & $< 500$ & 3.84$^{+0.48}_{-0.48}$ & -152$^{+16}_{-283}$ & $0\farcs54^{+0.13}_{-0.14}$\ (i/r)$^{f}$ & & & 0.32$^{+0.17}_{-0.14}$ & & 0.32 & &\ (f/r)$^{f}$ & & & 0.67$^{+0.20}_{-0.14}$ & & 0.67 & &\ Si XIV L$_{\alpha}$ & 6.1804 & 665$^{+450}_{-300}$ & 2.94$^{+0.51}_{-0.78}$ & $< 1050$ & 1.62$^{+0.65}_{-0.44}$ & -140$^{+180}_{-80}$ & $0\farcs66^{+0.22}_{-0.25}$\ Si XIII K$_{\alpha}$ & 6.6479 & 1200$^{+225}_{-185}$ & 11.43$^{+2.03}_{-2.66}$ & 670$^{+261}_{-254}$ & 7.84$^{+2.88}_{-1.80}$ & -246$^{+68}_{-68}$ & $0\farcs54^{+0.10}_{-0.13}$\ (i/r) & & & 0.04$^{+0.08}_{-0.04}$ & & 0.22$^{+0.17}_{-0.16}$ & &\ (f/r) & & & 0.75$^{+0.20}_{-0.17}$ & & 0.66$^{+0.28}_{-0.19}$ & &\ Mg XII L$_{\alpha}$ & 8.4192 & 1550$^{+680}_{-600}$ & 2.04$^{+0.84}_{-0.79}$ & $< 1500$ & 1.15$^{+0.50}_{-0.49}$ & -150$^{+167}_{-107}$ & $0\farcs67^{+0.27}_{-0.09}$ [lcc]{} $\chi^2$/dof & 561/993 & 526/992\ N$_{H, ISM}$$^a$ & 2.3 & 2.3\ N$_{H, wind}$$^a$ & 0.23$^{+0.01}_{-0.01}$ & 0.19$^{+0.01}_{-0.01}$\ Ne & 0.0$^{+2.7}_{-0.0}$ & 1.3$^{+1.6}_{-1.3}$\ Mg & 4.1$^{+0.9}_{-0.8}$ & 3.3$^{+0.5}_{-0.5}$\ Si & 3.8$^{+0.7}_{-0.3}$ & 3.5$^{+0.2}_{-0.2}$\ S & 6.1$^{+0.7}_{-0.7}$ & 5.5$^{+0.4}_{-0.4}$\ Ar & 11.6$^{+2.8}_{-2.8}$ & 8.7$^{+1.3}_{-1.3}$\ Ca & 19.4$^{+5.4}_{-5.3}$ & 12.6$^{+2.4}_{-2.4}$\ Fe & 6.6$^{+0.8}_{-0.9}$ & 7.7$^{+0.6}_{-0.6}$\ FWHM$^b$ & 840$^{+400}_{-220}$ & 1000$^{+100}_{-260}$\ $\tau^c$ & & 1.03$^{+0.17}_{-0.12}$\ F$_{X,~WR~147N}$$^c$ & 0.327 ( 9.5) & 0.326 (10.1)\ F$_{X,~WR~147S}$$^c$ & 0.975 (26.3) & 0.992 (28.1)\ [^1]: Chandra Interactive Analysis of Observations (CIAO), http://cxc.harvard.edu/ciao/ [^2]: For ATOMDB, see http://cxc.harvard.edu/atomdb/
--- abstract: 'In unsupervised learning, an unbiased uniform sampling strategy is typically used, in order that the learned features faithfully encode the statistical structure of the training data. In this work, we explore whether active example selection strategies — algorithms that select which examples to use, based on the current estimate of the features — can accelerate learning. Specifically, we investigate effects of heuristic and saliency-inspired selection algorithms on the dictionary learning task with sparse activations. We show that some selection algorithms do improve the speed of learning, and we speculate on why they might work.' author: - \| Tomoki Tsuchida & Garrison W. Cottrell\ Department of Computer Science and Engineering\ University of California, San Diego\ 9500 Gilman Drive, Mail Code 0404\ La Jolla, CA 92093-0404, USA\ `{ttsuchida,gary}@ucsd.edu` bibliography: - 'ExampleSelection.bib' title: Example Selection For Dictionary Learning --- Introduction ============ The efficient coding hypothesis, proposed by @Barlow:1961p1757, posits that the goal of perceptual system is to encode the sensory signal in such a way that it is efficiently represented. Based on this hypothesis, the past two decades have seen successful computational modeling of low-level perceptual features based on dictionary learning with sparse codes. The idea is to learn a set of dictionary elements that encode “naturalistic” signals efficiently; the learned dictionary might then model the features of early sensory processing. Starting with @Olshausen:1996p2797, the dictionary learning task has thus been used extensively to explain early perceptual features. Because the objective of such a learning task is to capture the statistical structure of the observed signals faithfully and efficiently, it is an instance of unsupervised learning. As such, the dictionary learning is usually performed using unbiased sampling: the set of data to be used for learning are sampled uniformly from the training dataset. At the same time, the world contains an overabundance of sensory information, requiring organisms with limited processing resources to select and process only information relevant for survival [@Tsotsos:1990vv]. This selection process can be expressed as perceptual action or attentional filtering mechanisms. This might at first appear at odds with the goal of the dictionary learning task, since the selection process necessarily biases the set of observed data for the organism. However, the converse is also true: as better (or different) features are learned over the course of learning, the mechanisms for selecting what is relevant may change, even if the selection objective stays the same. If a dictionary learning task is to serve as a realistic algorithmic model of the feature learning process in organisms capable of attentional filtering, this mutual dependency between the dictionary learning and attentional sample selection bias must be taken into consideration. In this work, we examine the effect of such sampling bias on the dictionary learning task. In particular, we explore interactions between learned dictionary elements and example selection algorithms. We investigate whether any selection algorithm can approach, or even improve upon, learning with unbiased sampling strategy. Some of the heuristics we examine also have close relationships to models of attention, suggesting that they can be plausibly implemented by organisms evolving to effectively encode stimuli from their environment. Dictionary Learning {#sec:dictionary_learning} =================== Assume that a training set consisting of $N$ $P$-dimensional signals ${\bf X}_{N} \triangleq \{ {\bf x}^{(i)} \}_{i=1}^N$ is generated from a $K$-element “ground-truth” dictionary set ${\bf A}^* = [{\bf a}_1 {\bf a}_2 \cdots {\bf a}_K]$ under the following model: $$\label{eqn:genmodel} \begin{aligned} {\bf x}^{(i)} &= {\bf A}^* {\bf s}^{(i)} + \epsilon^{(i)}, \\ \{s^{(i)}_j : s^{(i)}_j > 0\} &\sim Exp(\lambda) \quad \textrm{iid},\\ \epsilon^{(i)} &\sim \mathcal{N}(0, {\bf I} \sigma_\epsilon^2) \quad \textrm{iid}. \end{aligned}$$ Each signal column vector ${\bf x}^{(i)}$ is restricted to having exactly $k$ positive activations: ${\bf s}^{(i)} \in \mathcal{C}_s \triangleq \{ {\bf s} \in {I\!\!R}_{\ge 0}^P : \\|{\bf s}\\|_0 = k \}$, and each dictionary element is constrained to the unit-norm: ${\bf A}^* \in \mathcal{C}_{\bf A} \triangleq \{{\bf A} : \\|({\bf A})_j\\|_2 = 1 \; \forall j\}$. The goal of dictionary learning is to recover ${\bf A}^$ from ${\bf X}_N$, assuming $\lambda$ and $\sigma_\epsilon^2$ are known. To that end, we wish to calculate the maximum a posteriori estimate of ${\bf A}^$, $$\label{eqn:aml} \begin{aligned} \operatorname{arg\,min}_{{\bf A} \in \mathcal{C}_{\bf A}} \frac{1}{N} \sum_{i=1}^N \min_{{\bf s}^{(i)} \in \mathcal{C}_s} \left ( \frac{1}{2\sigma_\epsilon^2} \\| {\bf x}^{(i)} - {\bf A} {\bf s}^{(i)} \\|_2^2 + \lambda \\| {\bf s}^{(i)} \\|_1 \right ) . \end{aligned}$$ This is difficult to calculate, because ${\bf A}$ and $\{{\bf s}^{(i)}\}_{i=1}^N$ are simultaneously optimized. One practical scheme is to fix one variable and alternately optimize the other, leading to subproblems $$\begin{aligned} \label{eqn:encoding} {\bf {\hat S}} &= \left[ \operatorname{arg\,min}_{{\bf s}^{(i)} \in \mathcal{C}_s} \left ( \frac{1}{2\sigma_\epsilon^2} \\| {\bf x}^{(i)} - {\bf {\hat A}} {\bf s}^{(i)} \\|_2^2 + \lambda \\|{\bf s}^{(i)}\\|_1 \right ) \right]_{i=1}^N \textrm{,} \\ \label{eqn:updating} {\bf {\hat A}} &= \operatorname{arg\,min}_{{\bf A} \in \mathcal{C}_{\bf A}} \frac{1}{2N} \\| {\bf X}_N - {\bf A} {\bf {\hat S}} \\|_F^2.\end{aligned}$$ As in the Method of Optimal Directions (MOD) [@Engan:1999kg], this alternate optimization scheme is guaranteed to converge to a locally optimal solution for ${\bf {\hat A}}_{\textrm{MAP}}$ estimation problem . This scheme is also attractive as an algorithmic model of low-level feature learning, since each optimization process can be related to the “analysis” and “synthesis” phases of an autoencoder network [@Olshausen:1997uh]. In this paper, we henceforth refer to problems and as encoding* and updating stages, and their corresponding optimizers as $f_{enc}$ and $f_{upd}$. Encoding algorithms ------------------- The $L^0$-constrained encoding problem is NP-Hard [@elad], and various approximation methods have been extensively studied in the sparse coding literature. One approach is to ignore the $L^0$ constraint and solve the remaining nonnegative $L^1$-regularized least squares problem $$\begin{aligned} \textrm{\tt LARS}: {\hat {\bf s}}^{(i)} &= \operatorname{arg\,min}_{{\bf s} \ge 0} \left ( \frac{1}{2\sigma_\epsilon^2} \\| {\bf x}^{(i)} - {\bf {\hat A}} {\bf s} \\|_2^2 + \lambda' \\|{\bf s} \\|_1 \right ), \end{aligned}$$ with a larger sparsity penalty $\lambda' \triangleq \lambda P / k$ to compensate for the lack of the $L^0$ constraint. This works well in practice, since the distribution of $s_j^{(i)}$ (whose mean is $1/\lambda'$) is well approximated by $Exp(\lambda')$. For our simulations, we use the Least Angle Regression (LARS) algorithm [@lars] implemented by the SPAMS package [@Mairal:2010us] to solve this. Another approach is to greedily seek nonzero activations to minimize reconstruction errors. The matching pursuit family of algorithms operate on this idea, and they effectively approximate the encoding model $$\begin{aligned} \begin{split} \textrm{\tt OMP}: \quad {\hat {\bf s}}^{(i)} =\operatorname{arg\,min}_{{\bf s} \ge 0} \left ( \frac{1}{2\sigma_\epsilon^2} \\| {\bf x}^{(i)} - {\bf {\hat A}} {\bf s} \\|_2^2 \right) \\ \textrm{s.t.} \; \\|{\bf s}\\|_0 \le k . \end{split}\end{aligned}$$ This approximation ignores the $L^1$ penalty, but because nonzero activations are exponentially distributed and mostly small, this approximation is also effective. We use the Orthogonal Matching Pursuit (OMP) algorithm [@mallat1993matching], also implemented by the SPAMS package, for this problem. An even simpler variant of the pursuit-type algorithm is the thresholding [@elad] or the $k$-Sparse algorithm [@Makhzani:2013wn]. This algorithm takes the $k$ largest values of ${\bf {\hat A}}^\intercal {\bf x}^{(i)}$ and sets every other component to zero: $$\begin{aligned} \textrm{\tt k-Sparse}: \quad {\hat {\bf s}}^{(i)} &=\operatorname{supp}_k \{ {\bf {\hat A}}^\intercal {\bf x}^{(i)} \}\end{aligned}$$ This algorithm is plausibly implemented in a feedforward phase of an autoencoder with a hidden layer that competes horizontally and picks $k$ “winners”. The simplicity of this algorithm is important for our purposes, because we allow the training examples to be selected after* the encoding stage, and the encoding algorithm must operate on a much larger number of examples than the updating algorithm. This view also motivated the nonnegative constraint on ${\bf s}^{(i)}$, because the activations of the hidden layers are likely to be conveyed by nonnegative firing rates. Dictionary update algorithm --------------------------- For the updating stage, we only consider the stochastic gradient update, another simple algorithm for learning. For the reconstruction loss $L_{rec}({\bf A}) \triangleq \frac{1}{2N} \\|{\bf X}_N - {\bf A} {\bf {\hat S}}\\|_F^2$, the gradient is $\nabla L_{rec} = 2 ({\bf A} {\bf {\hat S}} - {\bf X}) {\bf {\hat S}}^\intercal / N$, yielding the update rule $$\begin{aligned} \label{eq:sgd} \quad {\bf {\hat A}} \leftarrow {\bf {\hat A}} - \eta_t ({\bf {\hat A}} {\bf {\hat S}}-{\bf X}_N) {\bf {\hat S}}^\intercal / N .\end{aligned}$$ Here, $\eta_t$ is a learning rate that decays inversely with the update epoch $t$: $\eta_t \in \Theta(1/t + c)$. After each update, ${\bf {\hat A}}$ is projected back to ${\mathcal C}_{\bf A}$ by normalizing each column. Given a set of training examples, this encoding and updating procedure is repeated a small number of times (10 times in our simulations). Activity equalization --------------------- One practical issue with this task is that a small number of dictionary elements tend to be assigned to a large number of activations. This produces “the rich get richer” effect: regularly used elements are more often used, and unused elements are left at their initial stages. To avoid this, an activity normalization procedure takes place after the encoding stage. The idea is to modulate all activities, so that the mean activity for each element is closer to the across-element mean of the mean activities; this is done at the cost of increasing the reconstruction error. The equalization is modulated by $\gamma$, with $\gamma = 0$ corresponding to no equalization and $\gamma = 1$ to fully egalitarian equalization (i.e. all elements would have equal mean activities). We use $\gamma=0.2$ for our simulations, which we found empirically to provide a good balance between equalization and reconstruction. Example Selection Algorithms ============================ (XN) [${\bf X}_N$\ All signals]{}; (encoding) [encoding]{}; (SN) [${\bf S}_N$\ All activations]{}; (selection) [Example\ selection]{}; (Sn) [${\bf X}_n, {\bf S}_n$\ Selected examples,\ activations]{}; (learning) [updating]{}; (Ahat) [${\bf {\hat A}}$\ Dictionary estimate]{}; (XN) – (encoding); (encoding) – (SN); (SN) – (selection); (selection) – (Sn); (Sn) – (learning); (learning) – (Ahat); (Ahat) – (encoding); (XN) to\[out=10,in=170\] (selection); To examine the effect of the example selection process on the learning, we extend the alternate optimization scheme in equations (\[eqn:encoding\], \[eqn:updating\]) to include an example selection stage. In this stage, a selection algorithm picks $n \ll N$ examples to use for the dictionary update (Figure \[fig:flow\_diagram\]). Ideally, the examples are to be chosen in such a way as to make learned dictionary ${\bf {\hat A}}$ closer to the ground-truth ${\bf A}^$ compared to the uniform sampling. In the following, we describe a number of heuristic selection algorithms that were inspired by models of attention. We characterize example selection algorithms in two parts. First, there is a choice of goodness measure* $g_j$, which is a function that maps $({\bf s}^{(i)}, {\bf x}^{(i)})$ to a number reflecting the “goodness” of the instance $i$ for the dictionary element $j$. Applying $g_j$ to $\{{\bf s}^{(i)}\}_{i=1}^N$ yields goodness values ${\bf G}_N$ for all $k$ dictionary elements and all $N$ examples. Second, there is a choice of selector function $f_{sel}$. This function dictates the way a subset of ${\bf X}_N$ is chosen using ${\bf G}_N$ values. Goodness measures ----------------- Of the various goodness measures, we first consider $$\begin{aligned} \textrm{\tt Err}: \quad g_j({\bf s}^{(i)}, {\bf x}^{(i)}) = \\| {\bf {\hat A}}{\bf s}^{(i)} - {\bf x}^{(i)} \\|_1 .\end{aligned}$$ [Err]{} is motivated by the idea of “critical examples” in [@Zhang:1994uy], and it favors examples with large reconstruction errors. In our paradigm, the criticality measured by [Err]{} may not correspond to ground-truth errors, since it is calculated using current estimate ${\bf {\hat A}}$ rather than ground-truth ${\bf A}^$. Another related idea is to select examples that would produce large gradients in the dictionary update equation , without regard to their directions. This results in $$\begin{aligned} \textrm{\tt Grad}: \quad g_j({\bf s}^{(i)}, {\bf x}^{(i)}) = \\| {\bf {\hat A}}{\bf s}^{(i)} - {\bf x}^{(i)} \\|_1 \cdot s^{(i)}_j.\end{aligned}$$ We note that [Grad]{} extends [Err]{} by multiplying the reconstruction errors by the activations $s^{(i)}_j$. It therefore prefers examples that are both critical and produce large activations. One observation is that the level of noise puts a fundamental limit on the recovery of true dictionary: better approximation bound is obtained when observation noise is low. It follows that, if we can somehow collect examples that happen to have low noise, learning from those examples might be beneficial. This motivated us to consider $$\begin{aligned} \textrm{\tt SNR}: \quad g_j({\bf s}^{(i)}, {\bf x}^{(i)}) = \frac{\\| {\bf x}^{(i)}\\|_2^2}{\\| {\bf {\hat A}}{\bf s}^{(i)} - {\bf x}^{(i)} \\|_2^2} \cdot s^{(i)}_j.\end{aligned}$$ This measure prefers examples with large estimated signal-to-noise ratio (SNR). Another idea focuses on the statistical property of activations ${\bf s}^{(i)}$, inspired by a model of visual saliency proposed by @Zhang2008. Their saliency model, called the SUN model, asserts that signals that result in rare feature activations are more salient. Specifically, the model defines the saliency of a particular visual location to be proportional the self-information of the feature activation, $-\log P(F = f)$. Because we assume nonzero activations are exponentially distributed, this corresponds to $$\begin{aligned} \textrm{\tt SUN}: \quad g_j({\bf s}^{(i)}, {\bf x}^{(i)}) = s^{(i)}_j \quad \left (\propto -\log P(s^{(i)}_j) \right ) .\end{aligned}$$ We note that this model is not only simple, but also does not depend on ${\bf x}^{(i)}$ directly. This makes [SUN]{} attractive as a neurally implementable goodness measure. Another saliency-based goodness measure is inspired by the visual saliency map model of @Itti:2002tq: $$\begin{aligned} \textrm{\tt SalMap}: \quad g_j({\bf s}^{(i)}, {\bf x}^{(i)}) = SaliencyMap({\bf x}^{(i)}).\end{aligned}$$ In contrast to the [SUN]{} measure, [SalMap]{} depends only on ${\bf x}^{(i)}$. Consequently, [SalMap]{} is impervious to changes in ${\bf {\hat A}}$. Since the signals in our simulations are small monochrome patches, the “saliency map” we use only has a single-scale intensity channel and an orientation channel with four directions. Selector functions ------------------ We consider two selector functions. The first function chooses top $n$ examples with high goodness values across dictionary elements: $$\begin{aligned} \textrm{\tt BySum}: \quad f_{sel}({\bf G}_N) = \textrm{top $n$ elements of} \sum_{j=1}^K {\bf G}_j^{(i)}.\end{aligned}$$ The second selector function,selects examples that are separately “good” for each dictionary element: $$\begin{aligned} \begin{split} \textrm{\tt ByElement}: \quad f_{sel}({\bf G}_N) = \\ \{ \textrm{top $n / K$ elements of } {\bf G}_j^{(i)} \;\|\; j \in 1 ... K \} . \end{split}\end{aligned}$$ This is done by first sorting ${\bf G}_j^{(i)}$ for each $j$ and then picking top examples in a round-robin fashion, until $N$ examples are selected. Barring duplicates, this yields a set consisting of top $n / k$ elements of ${\bf G}_j^{(i)}$ for each element $j$. Algorithm \[alg:main\] describes how these operations take place within each learning epoch. In our simulations, we consider all possible combinations of the goodness measures and selector functions for the example selection algorithm, except for [Err]{} and [SalMap]{}. Since these two goodness measures do not produce different values for different dictionary element activations $s_j^{(i)}$, [BySum]{} and [ByElement]{} functions select equivalent example sets. Initialize random ${\bf {\hat A}}_0 \in \mathcal{C}_{\bf A}$ from training examples\ For $t=1$ to max. epochs: 1. Obtain training set ${\bf X}_N = \{ {\bf x}^{(i)} \}_{i=1}^N$ 2. Encode ${\bf X}_N$: ${\bf S}_N = \{ f_{enc}({\bf x}^{(i)}; {\bf {\hat A}}) \}_{i=1}^N$ 3. Select $n$ “good” examples - Calculate ${\bf G}_N = \{[g_j({\bf s}^{(i)}, {\bf x}^{(i)})]_{j=1 ... k} \}_{i=1}^N$ - Select $n$ indices: $\Gamma = f_{sel}( {\bf G}_N )$ - ${\bf S}_n = \{{\bf s}^{(i)}\}_{i \in \Gamma}$, ${\bf X}_n = \{{\bf x}^{(i)}\}_{i \in \Gamma}$ 4. Loop 10 times: 1. Encode ${\bf X}_n$: ${\bf S}_n \leftarrow \{ f_{enc}({\bf x}^{(i)}; {\bf {\hat A}}) \}_{i=1}^n$ 2. Equalize ${\bf S}_n$: $\forall {\bf s}^{(i)} \in {\bf S}_n$,\ $s^{(i)}_j \leftarrow s^{(i)}_j \cdot (\frac{1}{K} \sum_{j=1}^K \sum_{i=1}^n s^{(i)}_j / \sum_{i=1}^n s^{(i)}_j )^{\gamma}$ 3. Update ${\bf {\hat A}}$: ${\bf {\hat A}} \leftarrow {\bf {\hat A}} - \eta_t ({\bf {\hat A}} {\bf S}_n-{\bf X}_n) {\bf S}_n^\intercal / n$ 4. Normalize columns of ${\bf {\hat A}}$. Simulations =========== In order to evaluate example selection algorithms, we present simulations across a variety of dictionaries and encoding algorithms. Specifically, we compare results using all three possible encoding models ([L0]{}, [L1]{}, and [k-Sparse]{}) with all eight selection algorithms. Because we generate the training examples from a known ground-truth dictionary ${\bf A}^$, we quantify the integrity of learned dictionary ${\bf {\hat A}}_t$ at each learning epoch $t$ using the minimal mean square distance $$\begin{aligned} D^({\bf {\hat A}}, {\bf A}^) \triangleq \min_{{\bf P}_\pi} \frac{1}{KP} \\| {\bf {\hat A}}_t {\bf P}_\pi - {\bf A}^* \\|_F^2 ,\end{aligned}$$ with ${\bf P}_\pi$ spanning all possible permutations. [0.5]{} ![image](nips-gabors--dictionary.png){width="40.00000%"} ![image](nips-gabors--sample_X0-for_presentation-only_lasso-for_talk.png){width="40.00000%"} [0.5]{} ![image](nips-letters--dictionary.png){width="40.00000%"} ![image](nips-letters--sample_X0-for_presentation-only_lasso-for_talk.png){width="40.00000%"} We also investigate the effect of ${\bf A}^$ on the learning. One way to characterize a dictionary set ${\bf A}$ is its mutual coherence $\mu({\bf A}) \triangleq \max_{i \neq j} \| {\bf a}_i^\intercal {\bf a}_j\|$ [@elad]. This measure is useful in theoretical analysis of recovery bounds [@donoho]. A more practical characterization is the average coherence ${\bar \mu}({\bf A}) \triangleq \frac{2}{K(K-1)} \sum_{i \neq j} \|{\bf a}_i^\intercal {\bf a}_j\|$. Regardless, exact recovery of the dictionary is more challenging when the coherence is high. The first dictionary set comprises $100$ $8$x$8$ Gabor patches (Figure \[fig:dic\_gabors\]). This dictionary set is inspired by the fact that dictionary learning of natural images leads to such a dictionary [@Olshausen:1996p2797], and they correspond to simple receptive fields in mammalian visual cortices [@Jones:1987ub]. With $\mu({\bf A}^) = 0.97$ but ${\bar \mu}({\bf A}^) = 0.13$, this dictionary set is relatively incoherent, and so the learning problem should be easier. The second dictionary set is composed of $64$ $8$x$8$ alphanumeric letters with alternating rotations and signs (Figure \[fig:dic\_letters\]). This artificial dictionary set has $\mu({\bf A}^) = 0.95$ with ${\bar \mu}({\bf A}^) = 0.34$[^1]. Within each epoch, 50,000 examples are generated with 5 nonzero activations per example ($k=5$), whose magnitudes are sampled from $Exp(1)$. $\sigma^2_\epsilon$ is set so that examples have SNR of $\approx 6$ dB. Each selection algorithm then picks 1% ($n=500$) of the training set for the learning. For each experiment, ${\bf {\hat A}}$ is initialized with random examples from the training set. Results ------- Figure \[fig:res\_dist\] shows the average distance of ${\bf {\hat A}}$ from ${\bf A}^$ for each learning epoch. We observe that [ByElement]{} selection policies generally work well, especially in conjunction with [Grad]{} and [SUN]{} goodness measures. This trend is especially noticeable for the alphanumeric dictionary case, where most of the [BySum]{}-selectors perform worse than the baseline selector that chooses examples randomly ([Uniform]{}). The ranking of the selector algorithms is roughly consistent across the learning epochs (Figure \[fig:res\_dist\], left column), and it is also robust with the choice of the encoding algorithms (Figure \[fig:res\_dist\], right column). In particular, good selector algorithms are beneficial even at the relatively early stages of learning ($< 100$ epochs, for instance), in contrast to the simulation in [@Amiri:2014ct]. This is surprising, because at early stages of learning, poor ${\bf {\hat A}}$ estimates result in bad activation estimates as well. Nevertheless, good selector algorithms soon establish a positive feedback loop for both dictionary and activation estimates. One interesting exception is the [SalMap]{} selector. It works relatively well for Gabor dictionary (and closely tracks the [SUNBySum]{} selector), but not for the alphanumeric dictionary. This is presumably due to the design of the [SalMap]{} model: because the model uses oriented Gabor filters as one of its feature maps, the overall effect is similar to the [SUNBySum]{} algorithm when the signals are generated from Gabor dictionaries. [rl]{} &\ \ &\ \ Robustness ---------- In order to assess the robustness of the example selection algorithms, we repeated the Gabor dictionary simulation across a range of parameter values. Specifically, we experimented with modifying the following parameters one at a time, starting from the original parameter values: - The signal-to-noise ratio ($10 \log_{10} (2 \lambda^2 / \sigma_\epsilon^2)$ \[dB\]) - The number of nonzero elements in the generated examples ($k$) - The ratio of selected examples to the original training set ($n / N$) - The number of dictionary elements ($K$) Figure \[fig:params\] shows the result of these simulations. These results show that good selector algorithms improve learning across a wide range of parameter values. Of note is the number of dictionary elements $K$, whose results suggest that the improvement is greatest for the “complete” dictionary learning cases; the advantage of selection appears to diminish for extremely over-complete (or under-complete) dictionary learning tasks. -- -- -- -- Discussion ========== In this work, we examined the effect of selection algorithms on the dictionary learning based on stochastic gradient descent. Simulations using training examples generated from known dictionaries revealed that some selection algorithms do indeed improve learning, in the sense that the learned dictionaries are closer to the known dictionaries throughout the learning epochs. Of special note is the success of [SUN]{} selectors; since these selectors are very simple, they hold promise for more general learning applications. Few studies have so far investigated example selection strategies for the dictionary learning task, although some learning algorithms contain such procedures implicitly. For instance, K-SVD [@Aharon:2006kn] relies upon identifying a group of examples that use a particular dictionary element during its update stage. The algorithm in [@Arora:2013vq] also makes use of a sophisticated example grouping procedure to provably recover dictionaries. In both cases, though, the focus is on breaking the inter-dependency between ${\bf {\hat A}}$ and ${\bf {\hat S}}$, instead of characterizing how some algorithms – notably those of the perceptual systems – might improve learning despite this inter-dependency. One recent paper that does consider example selection on its own is [@Amiri:2014ct], whose [cognit]{} algorithm is explicitly related to perceptual attention. The point that differentiates this work lies in the generative assumption: [cognit]{} relies on having additional information available to the learner, in their case the temporal contiguity of the generative process. With a spatially and temporally independent generation process, the generative model we considered here is simpler but more difficult to solve. Why do selection algorithms improve learning at all? At first glance, one may assume that any non-uniform sampling would skew the apparent distribution $\mathcal{D}({\bf X}_n)$ from the true distribution of the training set $\mathcal{D}({\bf X}_N)$, and thus lead to learning of an incorrect dictionary. However, as we have empirically shown, this is not the case. One intuitive reason – one that also underlies the design of the [SNR]{} selectors – is that “good” selection algorithms picks samples with high information content. For instance, samples with close to zero activation content provide little information about the dictionary elements that compose them, even though such samples abound under our generative model with exponentially-distributed activations. It follows that such samples provide little benefit to the inference of the statistical structure of the training set, and the learner would be well-advised to discard them. To validate this, we calculated the (true) SNR of ${\bf X}_n$ at the last epoch of the learning for each selection algorithm (Figure \[fig:snr\], left columns). This shows that all selection algorithms picked ${\bf X}_n$ with much higher SNR than [Uniform]{}. However, the correlation between the overall performance ranking and SNR is weak, suggesting that this is not the only factor driving good example selection. Another factor that contributes to good learning is the spread of examples within ${\bf X}_n$. Casual observation revealed that the [BySum]{} selector is prone to picking similar examples, whereas [ByElement]{} selects a larger variety of examples and thus retains the distribution of ${\bf X}_N$ more faithfully. To quantify this, we measured the distance of the distribution of selected examples, $\mathcal{D}({\bf X}_n)$, from that of all training examples, $\mathcal{D}({\bf X}_N)$, using the histogram intersection distance[@Rubner:2000uj]. The right columns of Figure \[fig:snr\] shows that this distance, $D(\mathcal{D}({\bf X}_n) \|\| \mathcal{D}({\bf X}_N))$, tends to be lower for [ByElement]{} selectors (solid lines) than [BySum]{} selectors (dashed lines). Like the SNR measure, however, this quantity itself is only weakly predictive of the overall performance, suggesting that it is important to pick a large variety of high-SNR examples for the dictionary learning task. -- -- -- -- -- -- -- -- There are several directions to which we plan to extend this work. One is the theoretical analysis of the selection algorithms. For instance, we did not explore under what conditions learning with example selection leads to the same solutions as an unbiased learning, although empirically we observed that to be the case. As in the curriculum learning paradigm [@bengio2009curriculum], it is also possible that different selection algorithms are better suited at different stages of learning. Another is to apply the active example selection processes to hierarchical architectures such as stacked autoencoders and Restricted Boltzmann Machines. In these cases, an interesting question arises as to how information from each layer should be combined to make the selection decision. We intend to explore some of these questions in the future using learning tasks similar to this work. [^1]: Both dictionaries violate the recovery bound described in [@donoho]. @Amiri:2014ct notes that this bound is prone to be violated in practice; as such, we explicitly chose “realistic” parameters that violate the bounds in our simulations.
--- --- [Christian Salas]{} The Open University, Walton Hall, Milton Keynes, MK7 6AA, United Kingdom e-mail: c.p.h.salas@open.ac.uk \[section\] \[Theorem\][Definition]{} \[Theorem\][Corollary]{} \[Theorem\][Lemma]{} \[Theorem\][Example]{} [Abstract]{} Cantor primes are primes $p$ such that $1/p$ belongs to the middle-third Cantor set. One way to look at them is as containing the base-3 analogues of the famous Mersenne primes, which encompass all base-2 repunit* primes, i.e., primes consisting of a contiguous sequence of 1’s in base 2 and satisfying an equation of the form $p + 1 = 2^q$. The Cantor primes encompass all base-3 repunit primes satisfying an equation of the form $2p + 1 = 3^q$, and I show that in general all Cantor primes $> 3$ satisfy a closely related equation of the form $2pK + 1 = 3^q$, with the base-3 repunits being the special case $K = 1$. I use this to prove that the Cantor primes $> 3$ are exactly the prime-valued cyclotomic polynomials of the form $\Phi_s(3^{s^j}) \equiv 1$ (mod 4). Significant open problems concern the infinitude of these, making Cantor primes perhaps more interesting than previously realised.* [Keywords:]{} Cantor set, prime numbers, cyclotomic polynomials [Mathematics Subject Classification:]{} 11A41, 11R09 Introduction ============ Any base-$N$ repunit prime $p$ is a cyclotomic polynomial evaluated at N, $\Phi_q(N)$, with $q$ also prime, i.e., $$p = \Phi_q(N) = \frac{N^q - 1}{N - 1} = \sum_{k=0}^{q-1} N^k$$ It is therefore expressible as a contiguous sequence of 1’s in base $N$. For example, $p = 31$ satisfies (1) for $N = 2$ and $q = 5$ and can be expressed as 11111 in base 2. The term repunit was coined by A. H. Beiler [@BEIL] to indicate that numbers like these consist of repeated units. The case $N = 2$ corresponds to the famous Mersenne primes on which there is a vast literature [@GUY]. They are sequence number A000668 in The Online Encyclopedia of Integer Sequences [@SLOANE] and are exactly the prime-valued cyclotomic polynomials of the form $\Phi_s(2) \equiv 3$ (mod 4). In this note I show that Cantor primes can be characterised in a similar way as being exactly the prime-valued cyclotomic polynomials of the form $\Phi_s(3^{s^j}) \equiv 1$ (mod 4). They are primes whose reciprocals belong to the middle-third Cantor set $\mathcal{C}_3$. It is easily shown that $\mathcal{C}_3$ contains the reciprocals of all base-3 repunit primes, i.e., those primes $p$ which satisfy an equation of the form $2p + 1 = 3^q$ with $q$ prime. $\mathcal{C}_3$ is a fractal consisting of all the points in $[0, 1]$ which have non-terminating base-3 representations involving only the digits 0 and 2. Rerranging (1) to get the infinite series $$\frac{1}{p} = \frac{N - 1}{N^q - 1} = \sum_{k=1}^{\infty} \frac{N-1}{N^{qk}}$$ and putting $N = 3$ shows that those primes $p$ which satisfy $2p + 1 = 3^q$ are such that $\frac{1}{p}$ can be expressed in base $3$ using only zeros and the digit $2$. This single digit $2$ will appear periodically in the base-$3$ representation of $\frac{1}{p}$ at positions which are multiples of $q$. Since only zeros and the digit $2$ appear in the ternary representation of $\frac{1}{p}$, $\frac{1}{p}$ is never removed in the construction of $\mathcal{C}_3$, so $\frac{1}{p}$ must belong to $\mathcal{C}_3$. Base-3 repunit primes are sequence number A076481 in The Online Encyclopedia of Integer Sequences and the exact analogues of the Mersenne primes, i.e., they are the case $N = 3$ in (1). In the next section I show that Cantor primes $> 3$ more generally satisfy a closely related equation of the form $2pK + 1 = 3^q$, with the base-3 repunits being the special case $K = 1$. A subsequent section proves that the Cantor primes $> 3$ are exactly the prime-valued cyclotomic polynomials of the form $\Phi_s(3^{s^j}) \equiv 1$ (mod 4), and a final section considers related open problems. An Exponential Equation Characterising All Cantor Primes ======================================================== A prime number $p > 3$ is a Cantor prime if and only if it satisfies an equation of the form $2pK + 1 = 3^q$ where $q$ is the order of 3 modulo $p$ and $K$ is a sum of non-negative powers of $3$ each smaller than $3^q$. [Comment.]{} The base-3 repunit primes are then the special case in which $K = 3^0 = 1$. An example is 13, which satisfies $2p + 1 = 3^3$. A counterexample which shows that not all Cantor primes are base-3 repunit primes is 757, which satisfies $26p + 1 = 3^9$ with $K = 3^0 + 3^1 + 3^2 = 13$ and $q = 9$. [Proof.]{} Each $x \in \mathcal{C}_3$ can be expressed in ternary form as $$x = \sum_{k=1}^{\infty} \frac{a_k}{3^k} = 0.a_1a_2\ldots$$ where all the $a_k$ are equal to 0 or 2. The construction of $\mathcal{C}_3$ amounts to systematically removing all the points in $[0, 1]$ which cannot be expressed in ternary form with only 0’s and 2’s, i.e., the removed points all have $a_k = 1$ for one or more $k \in \mathbb{N}$ [@OLM]. The construction of the Cantor set suggests some simple conditions which a prime number must satisfy in order to be a Cantor prime. If a prime number $p > 3$ is to be a Cantor prime, the first non-zero digit $a_{k_1}$ in the ternary expansion of $\frac{1}{p}$ must be 2. This means that for some $k_1 \in \mathbb{N}$, $p$ must satisfy $$\frac{2}{3^{k_1}} < \frac{1}{p} < \frac{1}{3^{k_1-1}}$$ or equivalently $$3^{k_1} \in (2p, 3p)$$ Prime numbers for which there is no power of 3 in the interval $(2p, 3p)$, e.g., 5, 7, 17, 19, 23, 41, 43, 47, …, can therefore be excluded immediately from further consideration. Note that there cannot be any other power of $3$ in the interval (2p, 3p) since $3^{k_1 - 1}$ and $3^{k_1 + 1}$ lie completely to the left and completely to the right of $(2p, 3p)$ respectively. If the next non-zero digit after $a_{k_1}$ is to be another 2 rather than a 1, it must be the case for some $k_2 \in \mathbb{N}$ that $$\frac{2}{3^{k_1 + k_2}} < \frac{1}{p} - \frac{2}{3^{k_1}} < \frac{1}{3^{k_1 + k_2 - 1}}$$ or equivalently $$3^{k_2} \in \bigg(\frac{2p}{3^{k_1} - 2p}, \frac{3p}{3^{k_1} - 2p}\bigg)$$ Thus, any prime numbers for which there is a power of 3 in the interval $(2p, 3p)$ but for which there is no power of 3 in the interval $(\frac{2p}{3^{k_1} - 2p}, \frac{3p}{3^{k_1} - 2p})$ can again be excluded, e.g., 37, 113, 331, 337, 353, 991, 997, 1009. Continuing in this way, the condition for the third non-zero digit to be a 2 is $$3^{k_3} \in \bigg(\frac{2p}{3^{k_2}(3^{k_1} - 2p) - 2p}, \frac{3p}{3^{k_2}(3^{k_1} - 2p) - 2p}\bigg)$$ and the condition for the $n$th non-zero digit to be a 2 is $$3^{k_n} \in \\ \\ \bigg(\frac{2p}{3^{k_{n-1}}(\cdots(3^{k_2}(3^{k_1} - 2p) - 2p)\cdots) - 2p}, \\ \\ \frac{3p}{3^{k_{n-1}}(\cdots(3^{k_2}(3^{k_1} - 2p) - 2p)\cdots) - 2p}\bigg)$$ The ternary expansions under consideration are all non-terminating, so at first sight it seems as if an endless sequence of tests like these would have to be applied to ensure that $a_k \neq 1$ for any $k \in \mathbb{N}$. However, this is not the case. Let $p$ be a Cantor prime and let $3^{k_1}$ be the smallest power of 3 that exceeds $2p$. Since $p$ is a Cantor prime, both (5) and (9) must be satisfied for all $n$. Multiplying (9) through by $3^{k_1-k_n}$ we get $$3^{k_1} \in \bigg(\frac{3^{k_1-k_n}\cdot2p}{3^{k_{n-1}}(\cdots(3^{k_2}(3^{k_1} - 2p) - 2p)\cdots) - 2p}, \frac{3^{k_1-k_n}\cdot3p}{3^{k_{n-1}}(\cdots(3^{k_2}(3^{k_1} - 2p) - 2p)\cdots) - 2p}\bigg)$$ Since all ternary representations of prime reciprocals $\frac{1}{p}$ for $p > 3$ have a repeating cycle which begins immediately after the point, it must be the case that $k_n = k_1$ for some $n$ in (10). Setting $k_n = k_1$ in (10) we can therefore deduce from the fact that $3^{k_1} \in (2p, 3p)$ and the fact that (10) must be consistent with this for all values of $n$, that all Cantor primes must satisfy an equation of the form $$3^{k_{n-1}}(\cdots(3^{k_2}(3^{k_1} - 2p) - 2p)\cdots) - 2p = 1$$ where $k_1 + k_2 + \cdots + k_{n-1} = q$ is the cycle length in the ternary representation of $\frac{1}{p}$. In other words, $q$ is the order of 3 modulo $p$. By successively considering the cases in which there is only one non-zero term in the repeating cycle, two non-zero terms, three non-zero terms, etc., in (11), and defining $$\begin{aligned} d_1 &= q - k_1 \\ d_2 &= q - k_1 - k_2 \\ d_3 &= q - k_1 - k_2 - k_3 \\ \vdots \\ d_n &= q - k_1 - k_2 - \cdots - k_n = 0 \end{aligned}$$ it is easy to see that (11) can be rearranged as $$2p\sum_{i=1}^n 3^{d_i} + 1 = 3^q$$ Setting $K = \sum_{i=1}^n 3^{d_i}$, we conclude that every Cantor prime must satisfy an equation of the form $2pK + 1 = 3^q$ as claimed. Conversely, every prime which satisfies an equation of this form must be a Cantor prime. To see this, note that we can rearrange (12) to get $$\frac{1}{p} = \frac{2\sum_{i=1}^n 3^{d_i}}{3^q - 1} = 2\sum_{i=1}^n 3^{d_i}\bigg\{\frac{1}{3^q} + \frac{1}{3^{2q}} + \frac{1}{3^{3q}} + \cdots \bigg\}$$ Since $2\sum_{i=1}^n 3^{d_i}$ involves only products of $2$ with powers of $3$ which are each less than $3^q$, (13) is an expression for $\frac{1}{p}$ which corresponds to a ternary representation involving only 2s. Thus, $\frac{1}{p}$ must be in the Cantor set if $2pK + 1 = 3^q$. Cantor Primes as Cyclotomic Polynomials ======================================= Let $n$ be a positive integer and let $\zeta_n$ be the complex number $e^{2 \pi i/n}$. The $n^{\text{th}}$ cyclotomic polynomial is defined as $$\Phi_n(x) = \prod_{\substack{1 \leq k < n \\ \text{gcd}(k, n) = 1}} (x - \zeta_n^k)$$ The degree of $\Phi_n(x)$ is $\varphi(n)$ where $\varphi$ is the Euler totient function. There is now a powerful body of theory relating to cyclotomic polynomials and discussions of their basic properties can be found in any textbook on abstract algebra. $x^{(n-1)a} + x^{(n-2)a} + \cdots + x^{2a} + x^a + 1$ is irreducible in $\mathbb{Z}[x]$ if and only if $n = p$ and $a = p^k$ for some prime $p$ and non-negative integer $k$. [Proof.]{} This is proved as Theorem 4 in [@GRASSL]. A prime number $p > 3$ is a Cantor prime if and only if $p = \Phi_s(3^{s^j}) \equiv 1$ (mod 4) where $s$ is an odd prime and $j$ is a non-negative integer. [Proof.]{} Assume $p$ is a Cantor prime. By Theorem 2.1 we then have $$pK = \frac{3^q - 1}{2} = R_q^{(3)}$$ where $R_q^{(3)}$ denotes the base-3 repunit consisting of $q$ contiguous units, and $q$ and $K$ are as defined in that theorem. If $q$ is composite, say $q = rs$, we obtain the factorisation $$R_q^{(3)} = R_r^{(3)}\cdot (3^{(s-1)r} + 3^{(s-2)r} + \cdots + 3^{2r} + 3^r + 1)$$ If $q$ is prime we can take $r = 1$. Therefore in both cases at least one factor of $pK$ must be a base-3 repunit. If $K = 1$ then $p = R_q^{(3)} = \Phi_s(3)$, since $q$ must be prime in this case. ($R_q^{(3)}$ is composite if $q$ is). If $K > 1$, $p$ is not a base-3 repunit and by Theorem 2.1 K is a sum of powers of 3, so $p$ must be of the general form $$p = 3^{(s-1)r} + 3^{(s-2)r} + \cdots + 3^{2r} + 3^r + 1$$ for some $s$ and $r$, and $K$ must be a corresponding base-3 repunit $R_r^{(3)}$, otherwise their product could not be $R_{rs}^{(3)}$. But the polynomial in (16) can only be prime if it is irreducible in $\mathbb{Z}[x]$. By Lemma 3.1, this requires $s$ to be a prime number and $r = s^j$ for some non-negative integer $j$, and we therefore have $p = \Phi_s(3^{s^j})$ in this case. We conclude that in all cases we must have $p = \Phi_s(3^{s^j})$ if $p$ is a Cantor prime. Note that $s$ must be an odd prime as $\Phi_s(3^{s^j})$ is even for $s = 2$. Conversely, suppose that $p = \Phi_s(3^{s^j})$ is a prime number. Then we can multiply it by the base-3 repunit $R_r^{(3)}$ where $r = s^j$ to get the repunit $R_q^{(3)}$ as in (15). Thus, $p$ must satisfy (14) and must therefore be a Cantor prime. Base-3 repunits are congruent to 0 modulo 4 when they consist of an even number of digits, and to 1 modulo 4 otherwise. Therefore if $p > 3$ is a base-3 repunit prime it must be of the form $4k + 1$. If $p$ is prime but not a base-3 repunit, both $r = s^j$ and $q = rs$ in (15) are odd, so both $R_q^{(3)}$ and $R_r^{(3)}$ are base-3 repunits with odd numbers of digits, and thus of the form $4k + 1$. It follows that $p$ is also of the form $4k + 1$ in this case. Open Problems ============= The infinitude of Cantor primes is currently an open problem shown to be significant in this paper because of the equivalence of Cantor primes and prime-valued cyclotomic polynomials of the form $\Phi_s(3^{s^j})$. In the case $j = 0$, it is known that $\Phi_s(3)$ is prime for $s =$ 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, and 877843. It seems plausible that there are infinitely many such values of $s$ but this remains to be proved. The Cantor prime $757 = \Phi_3(3^3)$ is an example with $j > 0$. It is again an open problem to prove there are infinitely many integers $j > 0$ for which $\Phi_s(3^{s^j})$ is prime given a prime $s$, though all such cyclotomic polynomials must be irreducible. Previous studies have considered the infinitude of prime-valued cyclotomic polynomials of other types. For example, primes of the form $\Phi_s(1)$ and $\Phi_s(2)$ are studied in [@GALLOT], and other cases are discussed in [@DAMIANOU]. [ACKNOWLEDGEMENTS.]{} I would like to express my gratitude to anonymous reviewers of this manuscript. [6]{} A. Beiler, Recreations in the theory of numbers, Dover, 1964. P. A. Damianou, On prime values of cyclotomic polynomials, [Int. Math. Forum]{} [6]{} (2011), 1445–1456. Y. Gallot, Cyclotomic polynomials and prime numbers, http://perso.orange.fr/yves.gallot/papers/cyclotomic.pdf, 2001. B. R. Gelbaum and J. M. H. Olmsted, Counterexamples in analysis, Holden-Day, 1964. R. Grassl and T. Mingus, Cyclotomic polynomial factors, [Math. Gaz.]{} [89]{} (2005), 195–201. R. K. Guy, Unsolved problems in number theory, Springer, 2004. N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, http://oeis.org/, 2012.
--- abstract: 'We propose an adaptive diffusion mechanism to optimize a global cost function in a distributed manner over a network of nodes. The cost function is assumed to consist of a collection of individual components. Diffusion adaptation allows the nodes to cooperate and diffuse information in real-time; it also helps alleviate the effects of stochastic gradient noise and measurement noise through a continuous learning process. We analyze the mean-square-error performance of the algorithm in some detail, including its transient and steady-state behavior. We also apply the diffusion algorithm to two problems: distributed estimation with sparse parameters and distributed localization. Compared to well-studied incremental methods, diffusion methods do not require the use of a cyclic path over the nodes and are robust to node and link failure. Diffusion methods also endow networks with adaptation abilities that enable the individual nodes to continue learning even when the cost function changes with time. Examples involving such dynamic cost functions with moving targets are common in the context of biological networks.' author: - 'Jianshu Chen, and Ali H. Sayed, [^1][^2]' bibliography: - 'DistOpt.bib' title: Diffusion Adaptation Strategies for Distributed Optimization and Learning over Networks --- Distributed optimization, diffusion adaptation, incremental techniques, learning, energy conservation, biological networks, mean-square performance, convergence, stability. EDICS Category: SEN–DIST, SEN–COLB, ASD–ANAL Introduction {#sec:intro} ============ consider the problem of optimizing a global cost function in a distributed manner. The cost function is assumed to consist of the sum of individual components, and spatially distributed nodes are used to seek the common minimizer (or maximizer) through local interactions. [ Such problems abound in the context of biological networks, where agents collaborate with each other via local interactions for a common objective, such as locating food sources or evading predators[@tu2011fish]. Similar problems are common in distributed resource allocation applications and in online machine learning procedures. In the latter case, data that are generated by the same underlying distribution are processed in a distributed manner over a network of learners in order to recover the model parameters (e.g., [@dekel2011optimal; @zaid2011collaborativeGMM]). ]{} There are already a few of useful techniques for the solution of optimization problems in a distributed manner [@bertsekas1997new; @nedic2001incremental; @rabbat2005quantized; @lopes2007incremental; @bertsekasparallel; @tsitsiklis1984convergence; @tsitsiklis1986distributed; @barbarossa2007bio; @nedic2009bookchapter; @nedic2009distributed; @schizas2008consensus1; @kar2008sensor; @kar2011converegence; @dimakis2010gossip; @olfati2004consensus; @aysal2009broadcast; @sardellitti2010fast; @xiao2005scheme; @eksin2011asilomar]. Most notable among these methods is the incremental approach [@bertsekas1997new; @nedic2001incremental; @rabbat2005quantized; @lopes2007incremental] and the consensus approach [@bertsekasparallel; @tsitsiklis1984convergence; @tsitsiklis1986distributed; @barbarossa2007bio; @nedic2009bookchapter; @nedic2009distributed; @schizas2008consensus1; @kar2008sensor; @kar2011converegence; @dimakis2010gossip; @olfati2004consensus; @aysal2009broadcast; @sardellitti2010fast; @xiao2005scheme]. In the incremental approach, a cyclic path is defined over the nodes and data are processed in a cyclic manner through the network until optimization is achieved. However, determining a cyclic path that covers all nodes is known to be an NP-hard problem[@karp1972reducibility] and, in addition, cyclic trajectories are prone to link and node failures. When any of the edges along the path fails, the sharing of data through the cyclic trajectory is interrupted and the algorithm stops performing. In the consensus approach, vanishing step-sizes are used to ensure that nodes reach consensus and converge to the same optimizer in steady-state. However, in time-varying environments, diminishing step-sizes prevent the network from continuous learning and optimization; when the step-sizes die out, the network stops learning. In earlier publications [@lopesdistributed; @lopes2007diffusion; @Sayed07; @lopes2008diffusion; @cattivelli2008diffusion; @Cattivelli10; @cattivelli2007diffusionRLS; @cattivelli2008TSPdiffusionRLS; @cattivelli2010TACdiffusionKalman; @takahashi2010diffusion], and motivated by our work on adaptation and learning over networks, we introduced the concept of diffusion adaptation and showed how this technique can be used to solve global minimum mean-square-error estimation problems efficiently both in real-time and in a distributed manner. In the diffusion approach, information is processed locally and simultaneously at all nodes and the processed data are diffused through a real-time sharing mechanism that ripples through the network continuously. Diffusion adaptation was applied to model complex patterns of behavior encountered in biological networks, such as bird flight formations [@cattivelli2011modeling] and fish schooling [@tu2011fish]. Diffusion adaptation was also applied to solve dynamic resource allocation problems in cognitive radios[@di2011bio], to perform robust system identification[@chouvardas2011adaptive], and to implement distributed online learning in pattern recognition applications[@zaid2011collaborativeGMM]. This paper generalizes the diffusive learning process and applies it to the distributed optimization of a wide class of cost functions. The diffusion approach will be shown to alleviate the effect of gradient noise on convergence. Most other studies on distributed optimization tend to focus on the almost-sure convergence of the algorithms under diminishing step-size conditions[@bertsekas1997new; @nedic2001incremental; @ram2010distributed; @bianchi2011convergence; @bertsekas2010incremental; @borkar2000ode; @srivastava2011distributed], or on convergence under deterministic conditions on the data [@bertsekas1997new; @nedic2001incremental; @rabbat2005quantized; @nedic2009distributed]. In this article we instead examine the distributed algorithms from a mean-square-error perspective at constant step-sizes. This is because constant step-sizes are necessary for continuous adaptation, learning, and tracking, which in turn enable the resulting algorithms to perform well even under data that exhibit statistical variations, measurement noise, and gradient noise. [ This paper is organized as follows. In Sec. \[Sec:ProblemFormulation\], we introduce the global cost function and approximate it by a distributed optimization problem through the use of a second-order Taylor series expansion. In Sec. \[Sec:Diffusion\], we show that optimizing the localized alternative cost at each node $k$ leads naturally to diffusion adaptation strategies. In Sec. \[Sec:ConvergenceAnalysis\], we analyze the mean-square performance of the diffusion algorithms under statistical perturbations when stochastic gradients are used. In Sec. \[Sec:Simulation\], we apply the diffusion algorithms to two application problems: sparse distributed estimation and distributed localization. Finally, in Sec. \[Sec:Conclusion\], we conclude the paper. ]{} [Notation]{}. Throughout the paper, all vectors are column vectors except for the regressors $\{\bm{u}_{k,i}\}$, which are taken to be row vectors for simplicity of notation. We use boldface letters to denote random quantities (such as $\bm{u}_{k,i}$) and regular font letters to denote their realizations or deterministic variables (such as $u_{k,i}$). We write $\E$ to denote the expectation operator. We use $\mathrm{diag}\{x_1,\ldots,x_N\}$ to denote a diagonal matrix consisting of diagonal entries $x_1,\ldots,x_N$, and use $\mathrm{col}\{x_1,\ldots,x_N\}$ to denote a column vector formed by stacking $x_1,\ldots,x_N$ on top of each other. For symmetric matrices $X$ and $Y$, the notation $X \le Y$ denotes $Y-X \ge 0$, namely, that the matrix difference $Y-X$ is positive semi-definite. Problem Formulation {#Sec:ProblemFormulation} =================== The objective is to determine, in a collaborative and distributed manner, the $M\!\times\! 1$ column vector $w^o$ that minimizes a global cost of the form: $$\begin{aligned} \label{Equ:ProblemFormulation:J_glob_original} \boxed{ J^{\mathrm{glob}}(w) = \sum_{l=1}^N J_l(w) } \end{aligned}$$ where $J_l(w)$, $l=1,2,\ldots,N$, are individual real-valued functions, defined over $w \in \mathds{R}^M$ and assumed to be differentiable and strictly convex. Then, $J^{\mathrm{glob}}(w)$ in is also strictly convex so that the minimizer $w^o$ is unique[@poliak1987introduction]. In this article we study the important case where the component functions $\{J_l(w)\}$ are minimized at the same $w^o$. This case is common in practice; situations abound where nodes in a network need to work cooperatively to attain a common objective (such as tracking a target, locating the source of chemical leak, estimating a physical model, or identifying a statistical distribution). This scenario is also frequent in the context of biological networks. For example, during the foraging behavior of an animal group, each agent in the group is interested in determining the same vector $w^o$ that corresponds to the location of the food source or the location of the predator [@tu2011fish]. This scenario is equally common in online distributed machine learning problems, where data samples are often generated from the same underlying distribution and they are processed in a distributed manner by different nodes (e.g., [@dekel2011optimal; @zaid2011collaborativeGMM]). The case where the $\{J_l(w)\}$ have different individual minimizers is studied in [@chen2012ssp]; this situation is more challenging to study. Nevertheless, it is shown in [@chen2012ssp] that the same diffusion strategies – of this paper are still applicable and nodes would converge instead to a Pareto-optimal solution. Our strategy to optimize the global cost $J^{\mathrm{glob}}(w)$ in a distributed manner is based on three steps. First, using a second-order Taylor series expansion, we argue that $J^{\mathrm{glob}}(w)$ can be approximated by an alternative localized cost that is amenable to distributed optimization — see . Second, each individual node optimizes this alternative cost via a steepest-descent procedure that relies solely on interactions within the neighborhood of the node. Finally, the local estimates for $w^o$ are spatially combined by each node and the procedure repeats itself in real-time. To motivate the approach, we start by introducing a set of nonnegative coefficients $\{c_{l,k}\}$ that satisfy: $$\begin{aligned} \label{Equ:ProblemFormulation:C_Condition} \boxed{ \displaystyle \sum_{k=1}^N c_{l,k} = 1,\quad c_{l,k}=0~\mathrm{if}~l \notin \mathcal{N}_k, \quad l=1,2,\ldots,N } \end{aligned}$$ where $\mathcal{N}_k$ denotes the neighborhood of node $k$ (including node $k$ itself); the neighbors of node $k$ consist of all nodes with which node $k$ can share information. Each $c_{l,k}$ represents a weight value that node $k$ assigns to information arriving from its neighbor $l$. Condition states that the sum of all weights leaving each node $l$ should be one. Using the coefficients $\{c_{l,k}\}$, we can express $J^{\mathrm{glob}}(w)$ from as $$\begin{aligned} \label{Equ:ProblemFormulation:J_glob} J^{\mathrm{glob}}(w) &= \displaystyle J_k^{\mathrm{loc}}(w) + \sum_{l \neq k}^N J_l^{\mathrm{loc}}(w) \end{aligned}$$ where $$\begin{aligned} \label{Equ:ProblemFormulation:J_k_loc0} J_k^{\mathrm{loc}}(w) \triangleq \sum_{l \in \mathcal{N}_k} c_{l,k} J_l(w) \end{aligned}$$ In other words, for each node $k$, we are introducing a new local cost function, $J_k^{\mathrm{loc}}(w)$, which corresponds to a weighted combination of the costs of its neighbors. Since the $\{c_{l,k}\}$ are all nonnegative and each $J_l(w)$ is convex, then $J_k^{\mathrm{loc}}(w)$ is also a convex function (actually, the $J_k^{\mathrm{loc}}(w)$ will be guaranteed to be strongly convex in our treatment in view of Assumption \[Assumption:Hessian\] further ahead). Now, each $J_l^{\mathrm{loc}}(w)$ in the second term of can be approximated via a second-order Taylor series expansion as: $$\begin{aligned} J_l^{\mathrm{loc}}(w) % \approx& % J_l^{\mathrm{loc}}(w^o) % + % (w-w^o)^* % \nabla^2 J_l^{\mathrm{loc}}(w^o) % (w-w^o) % \nonumber\\ \label{Equ:ProblemFormulation:J_k_loc} \approx\; & J_l^{\mathrm{loc}}(w^o) + \\|w-w^o\\|_{\Gamma_l}^2 \end{aligned}$$ where $\Gamma_l\!=\!\frac{1}{2}\nabla_w^2 J_l^{\mathrm{loc}}(w^o)$ is the (scaled) Hessian matrix relative to $w$ and evaluated at $w\!=\!w^o$, and the notation $\\|a\\|_\Sigma^2$ denotes $a^T \Sigma a$ for any weighting matrix $\Sigma$. The analysis in the subsequent sections will show that the second-order approximation is sufficient to ensure mean-square convergence of the resulting diffusion algorithm. Now, substituting into the right-hand side of gives: $$\begin{aligned} \label{Equ:ProblemFormulation:J_glob_approx} J^{\mathrm{glob}}(w) \approx \displaystyle J_k^{\mathrm{loc}}(w) \!+\! \sum_{l \neq k} \\|w\!-\!w^o\\|_{\Gamma_l}^2 \!+\! \sum_{l \neq k} J_l^{\mathrm{loc}} (w^o) \end{aligned}$$ The last term in the above expression does not depend on the unknown $w$. Therefore, we can ignore it so that optimizing $J^{\mathrm{glob}}(w)$ is approximately equivalent to optimizing the following alternative cost: $$\begin{aligned} \label{Equ:ProblemFormulation:J_glob_prime} J^{\mathrm{glob}'}(w)&\triangleq \displaystyle J_k^{\mathrm{loc}}(w) + \sum_{l \neq k} \\|w-w^o\\|_{\Gamma_l}^2 \end{aligned}$$ Iterative Diffusion Solution {#Sec:Diffusion} ============================ Expression relates the original global cost to the newly-defined local cost function $J_{k}^{\mathrm{loc}}(w)$. The relation is through the second term on the right-hand side of , which corresponds to a sum of quadratic terms involving the minimizer $w^o$. Obviously, $w^o$ is not available at node $k$ since the nodes wish to estimate $w^o$. Likewise, not all Hessian matrices $\Gamma_l$ are available to node $k$. Nevertheless, expression suggests a useful approximation that leads to a powerful distributed solution, as we proceed to explain. ![A network with $N$ nodes; a cost function $J_k(w)$ is associated with each node $k$. The set of neighbors of node $k$ is denoted by ${\cal N}_k$; this set consists of all nodes with which node $k$ can share information.[]{data-label="Fig:Network"}](Fig_Network){width="36.00000%"} Our first step is to replace the global cost $J^{\mathrm{glob}'}(w)$ by a reasonable localized approximation for it at every node $k$. Thus, initially we limit the summation on the right-hand side of to the neighbors of node $k$ and introduce the cost function: $$\begin{aligned} \label{Equ:DiffusionAdaptation:J_glob_prime_local} J_k^{\mathrm{glob}'}(w) \triangleq \displaystyle J_k^{\mathrm{loc}}(w) + \sum_{l\in{\mathcal{N}}_k\backslash\{k\}} \\|w-w^o\\|^2_{\Gamma_l} \end{aligned}$$ Compared with , the last term in involves only quantities that are available in the neighborhood of node $k$. The argument involving steps – therefore shows us one way by which we can adjust the earlier local cost function $J_k^\mathrm{loc}(w)$ defined in by adding to it the last term that appears in . [ Doing so, we end up replacing $J_k^\mathrm{loc}(w)$ by $J_k^{\mathrm{glob}'}(w)$, and this new localized cost function preserves the second term in up to a second-order approximation. This correction will help lead to a diffusion step (see –). ]{} Now, observe that the cost in includes the quantities $\{\Gamma_l\}$, which belong to the neighbors of node $k$. These quantities may or may not be available. If they are known, then we can proceed with and rely on the use of the Hessian matrices $\Gamma_l$ in the subsequent development. Nevertheless, the more interesting situation in practice is when these Hessian matrices are not known beforehand (especially since they depend on the unknown $w^o$). For this reason, in this article, we approximate each $\Gamma_l$ in by a multiple of the identity matrix, say, $$\begin{aligned} \label{Equ:DiffusionAdaptation:Gamma_l_approx} \Gamma_l \approx b_{l,k} I_M \end{aligned}$$ for some nonnegative coefficients $\{b_{l,k}\}$; observe that we are allowing the coefficient $b_{l,k}$ to vary with the node index $k$. Such approximations are common in stochastic approximation theory and help reduce the complexity of the resulting algorithms — see [@poliak1987introduction pp.20–28] and [@Sayed08 pp.142–147]. Approximation is reasonable since, in view of the Rayleigh-Ritz characterization of eigenvalues [@golub1996matrix], we can always bound the weighted squared norm $\\|w-w^o\\|^2_{\Gamma_{l}}$ by the unweighted squared norm as follows $$\begin{aligned} \lambda_{\min}(\Gamma_{l})\cdot\\|w\!-\!w^o\\|^2\leq \\|w\!-\!w^o\\|^2_{\Gamma_{l}}\leq \lambda_{\max}(\Gamma_{l})\cdot \\|w\!-\!w^o\\|^2 \nonumber \end{aligned}$$ Thus, we replace by $$\begin{aligned} \label{Equ:DiffusionAdaptation:J_glob_prime_prime} J_k^{\mathrm{glob}''}(w) \triangleq \displaystyle J_k^{\mathrm{loc}}(w) \;+\; \sum_{l\in{\mathcal N}_k\backslash\{k\}} b_{l,k} \\|w-w^o\\|^2 \end{aligned}$$ As the derivation will show, we do not need to worry at this stage about how the scalars $\{b_{l,k}\}$ are selected; they will be embedded into other combination weights that the designer selects. If we replace $J_k^\mathrm{loc}(w)$ by its definition , we can rewrite as $$\begin{aligned} \label{Equ:DiffusionAdaptation:J_glob_prime_prime_final} \boxed{ J_k^{\mathrm{glob}''}\!(w) = \displaystyle \sum_{l\in{\cal N}_k} \! c_{l,k}J_l(w)+\!\! \sum_{l\in{\cal N}_k\backslash\{k\}} b_{l,k}\\|w\!\!-\!\!w^o\\|^2 } \end{aligned}$$ Observe that cost is different for different nodes; this is because the choices of the weighting scalars $\{c_{l,k},b_{l,k}\}$ vary across nodes $k$; moreover, the neighborhoods vary with $k$. Nevertheless, these localized cost functions now constitute the important starting point for the development of diffusion strategies for the online and distributed optimization of . Each node $k$ can apply a steepest-descent iteration to minimize $J_k^{\mathrm{glob}''}(w)$ by moving along the negative direction of the gradient (column) vector of the cost function, namely, $$\begin{aligned} w_{k,i} =\;& \displaystyle w_{k,i-1} - \mu_k \sum_{l \in \mathcal{N}_k} c_{l,k} \nabla_w J_l(w_{k,i-1}) \nonumber\\ \label{Equ:DiffusionAdaptation:GradientDescent} \;& - \displaystyle\mu_k \sum_{l \in \mathcal{N}_k\backslash \{k\}} 2b_{l,k}(w_{k,i-1}-w^o), \qquad i \ge 0 \end{aligned}$$ where $w_{k,i}$ denotes the estimate for $w^o$ at node $k$ at time $i$, and $\mu_k$ denotes a small constant positive step-size parameter. [ While vanishing step-sizes, such as $\mu_{k}(i)=1/i$, can be used in , we consider in this paper the case of constant step-sizes. This is because we are interested in distributed strategies that are able to continue adapting and learning. An important question to address therefore is how close each of the $w_{k,i}$ gets to the optimal solution $w^o$; we answer this question later in the paper by means of a mean-square-error convergence analysis (see expression ). It will be seen then that the mean-square-error (MSE) of the algorithm will be of the order of the step-size; hence, sufficiently small step-sizes will lead to sufficiently small MSEs. ]{} Expression adds two correction terms to the previous estimate, $w_{k,i-1}$, in order to update it to $w_{k,i}$. The correction terms can be added one at a time in a succession of two steps, for example, as: $$\begin{aligned} \label{Equ:DiffusionAdaptation:Adaptation_intermedieate} \psi_{k,i} &= \displaystyle w_{k,i-1} - \mu_k \sum_{l \in \mathcal{N}_k} c_{l,k} \nabla_w J_l(w_{k,i-1}) \\ \label{Equ:DiffusionAdaptation:Combination_intermediate} w_{k,i} &= \displaystyle \psi_{k,i} - \mu_k \sum_{l \in \mathcal{N}_k\backslash\{k\}} 2b_{l,k} (w_{k,i-1}-w^o) \end{aligned}$$ Step updates $w_{k,i-1}$ to an intermediate value $\psi_{k,i}$ by using a combination of local gradient vectors. Step further updates $\psi_{k,i}$ to $w_{k,i}$ by using a combination of local estimates. However, two issues arise while examining : 1. First, iteration requires knowledge of the optimizer $w^o$. However, all nodes are running similar updates to estimate the $w^o$. By the time node $k$ wishes to apply , each of its neighbors would have performed its own update similar to and would have available their intermediate estimates, $\{\psi_{l,i}\}$. Therefore, we replace $w^o$ in by $\psi_{l,i}$. This step helps diffuse information over the network and brings into node $k$ information that exists beyond its immediate neighborhood; this is because each $\psi_{l,i}$ is influenced by data from the neighbors of node $l$. [ We observe that this diffusive term arises from the quadratic approximation we have made to the second term in . ]{} 2. Second, the intermediate value $\psi_{k,i}$ in is generally a better estimate for $w^o$ than $w_{k,i-1}$ since it is obtained by incorporating information from the neighbors through . Therefore, we further replace $w_{k,i-1}$ in by $\psi_{k,i}$. This step is reminiscent of incremental-type approaches to optimization, which have been widely studied in the literature [@bertsekas1997new; @nedic2001incremental; @rabbat2005quantized; @lopes2007incremental]. Performing the substitutions described in items (a) and (b) into , we obtain: $$\begin{aligned} % \label{Equ:DiffusionAdaptation:Adaptation_intermedieate2} % \psi_{k,i} &= \displaystyle w_{k,i-1} - \mu_k \sum_{l \in \mathcal{N}_k} c_{l,k} \nabla_w J_l(w_{k,i-1}) \\ \label{Equ:DiffusionAdaptation:Combination_intermediate2} w_{k,i} &= \displaystyle \psi_{k,i} - \mu_k \sum_{l \in \mathcal{N}_k\backslash\{k\}} 2b_{l,k} (\psi_{k,i}-\psi_{l,i}) \end{aligned}$$ Now introduce the coefficients $$\begin{aligned} \label{Equ:DiffusionAdaptation:a_lk_def} a_{l,k} \triangleq 2 \mu_k b_{l,k} \quad (l \! \neq \! k), \quad a_{k,k} \triangleq 1 \!-\! \mu_k \!\! \sum_{l \in \mathcal{N}_k\backslash\{k\}} 2b_{l,k} \end{aligned}$$ Note that the $\{a_{l,k}\}$ are nonnegative for $l\neq k$ and $a_{k,k}\geq 0$ for sufficiently small step-sizes. Moreover, the coefficients $\{a_{l,k}\}$ satisfy $$\begin{aligned} \label{Equ:AdaptiveDiffusion:Condition_a} \displaystyle \sum_{l=1}^N a_{l,k} = 1, \quad a_{l,k} = 0~\mathrm{if}~l \notin \mathcal{N}_{k} \end{aligned}$$ Using in , we arrive at the following Adapt-then-Combine (ATC) diffusion strategy (whose structure is the same as the ATC algorithm originally proposed in [@lopes2008diffusion; @cattivelli2008diffusion; @Cattivelli10] for mean-square-error estimation): $$\begin{aligned} % \mathrm{(ATC)~} \boxed{ \label{Equ:DiffusionAdaptation:ATC0} \begin{array}{l} \psi_{k,i} = \displaystyle w_{k,i-1} - \mu_k \sum_{l \in \mathcal{N}_k} c_{l,k} \nabla_w J_l(w_{k,i-1}) \\ w_{k,i} = \displaystyle \sum_{l \in \mathcal{N}_k} a_{l,k} \psi_{l,i} \end{array} } \end{aligned}$$ To run algorithm , we only need to select combination coefficients $\{a_{l,k},c_{l,k}\}$ satisfying and , respectively; there is no need to worry about the intermediate coefficients $\{b_{l,k}\}$ any more, since they have been blended into the $\{a_{l,k}\}$. The ATC algorithm involves two steps. In the first step, node $k$ receives gradient vector information from its neighbors and uses it to update its estimate $w_{k,i-1}$ to an intermediate value $\psi_{k,i}$. All other nodes in the network are performing a similar step and generating their intermediate estimate $\psi_{l,i}$. In the second step, node $k$ aggregates the estimates $\{\psi_{l,i}\}$ of its neighbors and generates $w_{k,i}$. Again, all other nodes are performing a similar step. Similarly, if we reverse the order of steps and to implement , we can motivate the following alternative Combine-then-Adapt (CTA) diffusion strategy (whose structure is similar to the CTA algorithm originally proposed in [@lopesdistributed; @lopes2007diffusion; @cattivelli2007diffusionRLS; @Sayed07; @lopes2008diffusion; @cattivelli2008diffusion; @Cattivelli10] for mean-square-error estimation): $$\begin{aligned} % \mathrm{(CTA)}~ \boxed{ \label{Equ:DiffusionAdaptation:CTA0} \begin{array}{l} \psi_{k,i\!-\!1} = \displaystyle \sum_{l \in \mathcal{N}_k} a_{l,k} w_{l,i-1} \\ w_{k,i} = \displaystyle\psi_{k,i-1} - \mu_k \sum_{l \in \mathcal{N}_k} c_{l,k} \nabla_w J_l(\psi_{k,i-1}) \end{array} } \end{aligned}$$ Adaptive diffusion strategies of the above ATC and CTA types were first proposed and extended in [@lopesdistributed; @lopes2007diffusion; @cattivelli2007diffusionRLS; @Sayed07; @lopes2008diffusion; @cattivelli2008diffusion; @cattivelli2008TSPdiffusionRLS; @Cattivelli10; @cattivelli2010TACdiffusionKalman] for the solution of distributed mean-square-error, least-squares, and state-space estimation problems over networks. [ The special form of ATC strategy for minimum-mean-square-error estimation is listed further ahead as Eq. in Example \[Example:ATC\_MMSE\]; the same strategy as also appeared in [@stankovic2011decentralized] albeit with a vanishing step-size sequence to ensure convergence towards consensus. ]{}A special case of the diffusion strategy (corresponding to choosing $c_{l,k}=0$ for $l\neq k$ and $c_{k,k}=1$, i.e., without sharing gradient information) was used in the works [@ram2010distributed; @bianchi2011convergence; @srivastava2011distributed] to solve distributed optimization problems that require all nodes to reach agreement about $w^o$ by relying on step-sizes that decay to zero with time. Diffusion recursions of the forms and are more general than these earlier investigations in a couple of respects. First, they do not only diffuse the local estimates, but they can also diffuse the local gradient vectors. In other words, two sets of combination coefficients $\{a_{l,k},c_{l,k}\}$ are used. Second, the combination weights $\{a_{l,k}\}$ are not required to be doubly stochastic (which would require both the rows and columns of the weighting matrix $A=[a_{l,k}]$ to add up to one; as seen from , we only require the entries on the columns of $A$ to add up to one). Finally, and most importantly, the step-size parameters $\{\mu_k\}$ in and are not required to depend on the time index $i$ and are not required to vanish as $i\rightarrow\infty$. Instead, they can assume constant values, which is critical to endow the network with continuous adaptation and learning abilities (otherwise, when step-sizes die out, the network stops learning). Constant step-sizes also endow networks with tracking abilities, in which case the algorithms can track time changes in the optimal $w^o$. Constant step-sizes will be shown further ahead to be sufficient to guarantee agreement among the nodes when there is no noise in the data. However, when measurement noise and gradient noise are present, using constant step-sizes does not force the nodes to attain agreement about $w^o$ (i.e., to converge to the same $w^o$). Instead, the nodes will be shown to tend to individual estimates for $w^o$ that are within a small mean-square-error (MSE) bound from the optimal solution; the bound will be proportional to the step-size so that sufficiently small step-sizes lead to small MSE values. Multi-agent systems in nature behave in this manner; they do not require exact agreement among their agents but allow for fluctuations due to individual noise levels (see [@cattivelli2011modeling; @tu2011fish]). Giving individual nodes this flexibility, rather than forcing them to operate in agreement with the remaining nodes, ends up leading to nodes with enhanced learning abilities. [ Before proceeding to a detailed analysis of the performance of the diffusion algorithms –, we note that these strategies differ in important ways from traditional consensus-based distributed solutions, which are of the following form [@nedic2009distributed; @bertsekasparallel; @nedic2009bookchapter; @kar2011converegence]: $$\begin{aligned} \label{Equ:DiffusionAdaptation:ConsensusTypeAlgorithm} w_{k,i} = \sum_{l \in \mathcal{N}_k} a_{l,k} w_{k,i-1} - \mu_k(i) \cdot \nabla_w J_l(w_{k,i-1}) \end{aligned}$$ usually with a time-variant step-size sequence, $\mu_k(i)$, that decays to zero. For example, if we set $C \triangleq [c_{l,k}] = I$ in the CTA algorithm and substitute the combination step into the adaptation step, we obtain: $$\begin{aligned} \label{Equ:DiffusionAdaptation:CTA_oneline} w_{k,i} = \sum_{l \in \mathcal{N}_k} a_{l,k} w_{k,i-1} - \mu_k \nabla_w J_l\Big(\sum_{l \in \mathcal{N}_k} a_{l,k} w_{k,i-1}\Big) \end{aligned}$$ Thus, note that the gradient vector in is evaluated at $\psi_{k,i-1}$, while in it is evaluated at $w_{k,i-1}$. Since $\psi_{k,i-1}$ already incorporates information from neighbors, we would expect the diffusion algorithm to perform better. Actually, it is shown in [@Tu2012diffcons] that, for mean-square-error estimation problems, diffusion strategies achieve higher convergence rate and lower mean-square-error than consensus strategies due to these differences in the dynamics of the algorithms. ]{} Mean-Square Performance Analysis {#Sec:ConvergenceAnalysis} ================================ The diffusion algorithms and depend on sharing local gradient vectors $\nabla_{w} J_l(\cdot)$. In many cases of practical relevance, the exact gradient vectors are not available and approximations are instead used. We model the inaccuracy in the gradient vectors as some [random]{} additive noise component, say, of the form: $$\begin{aligned} \label{Equ:Diffusion:NoisyGradient} \widehat{\nabla}_w J_l({w}) = \nabla_w J_l({w}) + \bm{v}_{l,i}({w}) \end{aligned}$$ where $\bm{v}_{l,i}(\cdot)$ denotes the perturbation and is often referred to as gradient noise. Note that we are using a boldface symbol $\bm{v}$ to refer to the gradient noise since it is generally stochastic in nature. \[FN:GradientNoiseExample\] [Assume the individual cost $J_l(w)$ at node $l$ can be expressed as the expected value of a certain loss function $Q_l(\cdot,\cdot)$, i.e., $J_l(w)=\E\{Q_l(w,\bm{x}_{l,i})\}$, where the expectation is with respect to the randomness in the data samples $\{\bm{x}_{l,i}\}$ that are collected at node $l$ at time $i$. Then, if we replace the true gradient $\nabla_w J_l(w)$ with its stochastic gradient approximation $\widehat{\nabla}_wJ_l(w)=\nabla_w Q_l(w,\bm{x}_{l,i})$, we find that the gradient noise in this case can be expressed as $$\begin{aligned} \label{Equ:PerformanceAnalysis:GradientNoise_ExpectedLoss} \bm{v}_{l,i}(w) = \nabla_w Q_l(w,\bm{x}_{l,i}) - \nabla_w \E\{Q_l(w,\bm{x}_{l,i})\} \end{aligned}$$ ]{} Using the perturbed gradient vectors , the diffusion algorithms – become the following: $$\begin{aligned} (\mathrm{ATC})~ \boxed{ \label{Equ:DiffusionAdaptation:ATC} \begin{array}{l} \bm{\psi}_{k,i} = \displaystyle \bm{w}_{k,i-1} \!-\! \mu_k \sum_{l \in \mathcal{N}_k} c_{l,k} \widehat{\nabla}_w J_l(\bm{w}_{k,i-1}) \\ \bm{w}_{k,i} = \displaystyle \sum_{l \in \mathcal{N}_k} a_{l,k} \bm{\psi}_{l,i} \end{array} } \end{aligned}$$ $$\begin{aligned} (\mathrm{CTA})~ \boxed{ \label{Equ:DiffusionAdaptation:CTA} \begin{array}{l} \bm{\psi}_{k,i-1} = \displaystyle \sum_{l \in \mathcal{N}_k} a_{l,k} \bm{w}_{l,i-1} \\ \bm{w}_{k,i} = \displaystyle \bm{\psi}_{k,i-1} \!-\! \mu_k \sum_{l \in \mathcal{N}_k} c_{l,k} \widehat{\nabla}_w J_l(\bm{\psi}_{k,i-1}) \end{array} } \end{aligned}$$ Observe that, starting with –, we will be using boldface letters to refer to the various estimate quantities in order to highlight the fact that they are also stochastic in nature due to the presence of the gradient noise. Given the above algorithms, it is necessary to examine their performance in light of the approximation steps – that were employed to arrive at them, and in light of the gradient noise that seeps into the recursions. A convenient framework to carry out this analysis is mean-square analysis. In this framework, we assess how close the individual estimates $\bm{w}_{k,i}$ get to the minimizer $w^o$ in the mean-square-error (MSE) sense. [ In practice, it is not necessary to force the individual agents to reach agreement and to converge to the same $w^o$ using diminishing step-sizes. It is sufficient for the nodes to converge within acceptable MSE bounds from $w^o$. This flexibility is beneficial and is common in biological networks; it allows nodes to learn and adapt in time-varying environments without the forced requirement of having to agree with neighbors. ]{} The main results that we derive in this section are summarized as follows. First, we derive conditions on the constant step-sizes to ensure boundedness and convergence of the mean-square-error for sufficiently small step-sizes — see and further ahead. Second, despite the fact that nodes influence each other’s behavior, we are able to quantify the performance of every node in the network and to derive closed-form expressions for the mean-square performance at small step-sizes — see –. Finally, as a special case, we are able to show that constant step-sizes can still ensure that the estimates across all nodes converge to the optimal $w^o$ and reach agreement in the absence of noise — see Theorem \[Corollary:ConvergenceAnalysis:Convergence\_NoiseFreeCase\]. Motivated by [@Cattivelli10], we address the mean-square-error performance of the adaptive ATC and CTA diffusion strategies – by treating them as special cases of a general diffusion structure of the following form: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:Diffusion_General} \bm{\phi}_{k,i-1} &= \displaystyle \sum_{l=1}^N p_{1,l,k} \bm{w}_{l,i-1} \\ \label{Equ:ConvergenceAnalysis:Diffusion_General1} \bm{\psi}_{k,i} &= \displaystyle \bm{\phi}_{k,i-1} - \mu_k \sum_{l=1}^N s_{l,k} \big[ \nabla_w J_l(\bm{\phi}_{k,i-1}) + \bm{v}_{l,i}(\bm{\phi}_{k,i-1}) \big] \\ \label{Equ:ConvergenceAnalysis:Diffusion_General2} \bm{w}_{k,i} &= \displaystyle \sum_{l=1}^N p_{2,l,k} \bm{\psi}_{l,i} \end{aligned}$$ The coefficients $\{p_{1,l,k}\}$, $\{s_{l,k}\}$, and $\{p_{2,l,k}\}$ are nonnegative real coefficients corresponding to the $\{l,k\}$-th entries of three matrices $P_1$, $S$, and $P_2$, respectively. Different choices for $\{P_1,P_2,S\}$ correspond to different cooperation modes. For example, the choice $P_1 = I$, $P_2 = I$ and $S=I$ corresponds to the non-cooperative case where nodes do not interact. On the other hand, the choice $P_1 = I$, $P_2=A=[a_{l,k}]$ and $S=C=[c_{l,k}]$ corresponds to ATC [@lopes2008diffusion; @cattivelli2008diffusion; @Cattivelli10], while the choice $P_1 = A$, $P_2 = I$ and $S=C$ corresponds to CTA [@lopesdistributed; @lopes2007diffusion; @Sayed07; @lopes2008diffusion; @cattivelli2008diffusion; @Cattivelli10]. We can also set $S=I$ in ATC and CTA to derive simplified versions that have no gradient exchange[@lopes2008diffusion]. Furthermore, if in CTA ($P_2=I$), we enforce $P_1=A$ to be doubly stochastic, set $S=I$, and use a time-decaying step-size parameter ($\mu_k(i)\rightarrow 0$), then we obtain the unconstrained version used by [@ram2010distributed; @srivastava2011distributed]. The matrices $\{P_1,P_2,S\}$ are required to satisfy: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:ConditionCombinationWeights} \boxed{ P_1^T \mathds{1} = \mathds{1}, \; P_2^T \mathds{1} = \mathds{1}, \; S \mathds{1} = \mathds{1} } \end{aligned}$$ where the notation $\mathds{1}$ denotes a vector whose entries are all equal to one. Error Recursions ---------------- We first derive the error recursions corresponding to the general diffusion formulation in –. Introduce the error vectors: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:Def_errorQuantities} \tilde{\bm{\phi}}_{k,i} \triangleq w^o\!-\!\bm{\phi}_{k,i}, \; \tilde{\bm{\psi}}_{k,i} \triangleq w^o\!-\!\bm{\psi}_{k,i}, \; \tilde{\bm{w}}_{k,i} \triangleq w^o\!-\!\bm{w}_{k,i} \end{aligned}$$ Then, subtracting both sides of – from $w^o$ gives: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:phi_ki} \tilde{\bm{\phi}}_{k,i-1} &= \displaystyle\sum_{l=1}^N p_{1,l,k} \tilde{\bm{w}}_{l,i-1} \\ \label{Equ:ConvergenceAnalysis:psi_ki} \tilde{\bm{\psi}}_{k,i} &= \displaystyle \tilde{\bm{\phi}}_{k,i-1} + \mu_k \sum_{l=1}^N s_{l,k} \big[ \nabla_w J_l(\bm{\phi}_{k,i-1}) + \bm{v}_{l,i}(\bm{\phi}_{k,i-1}) \big] \\ \label{Equ:ConvergenceAnalysis:w_ki} \tilde{\bm{w}}_{k,i} &= \displaystyle\sum_{l=1}^N p_{2,l,k} \tilde{\bm{\psi}}_{l,i} \end{aligned}$$ Expression still includes terms that depend on $\bm{\phi}_{k,i-1}$ and not on the error quantity, $\tilde{\bm{\phi}}_{k,i-1}$. We can find a relation in terms of $\tilde{\bm{\phi}}_{k,i-1}$ by calling upon the following result from [@poliak1987introduction p.24] for any twice-differentiable function $f(\cdot)$: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:MeanValueTheorem_VecVariant} \nabla f(y) = \nabla f(x) + \left[\int_{0}^{1} \nabla^2 f\big(x\!+\!t(y\!-\!x)\big)dt\right] (y-x) \end{aligned}$$ where $\nabla^2 f(\cdot)$ denotes the Hessian matrix of $f(\cdot)$ and is symmetric. Now, since each component function $J_l(w)$ has a minimizer at $w^o$, then, $\nabla_w J_l(w^o)= 0$ for $l=1,2,\ldots,N$. Applying to $J_l(w)$ using $x=w^o$ and $y=\bm{\phi}_{k,i-1}$, we get $$\begin{aligned} &\nabla_w J_l(\bm{\phi}_{k,i-1}) \nonumber\\ &\quad= \;\nabla_w J_l(w^o) - \left[ \int_{0}^{1} \nabla_w^2 J_l\big(w^o - t\tilde{\bm{\phi}}_{k,i-1}\big) dt \right] \tilde{\bm{\phi}}_{k,i-1} \nonumber\\ \label{Equ:ConvergenceAnalysis:Gradient_LinearRepresentation} &\quad\triangleq - \bm{H}_{l,k,i-1} \tilde{\bm{\phi}}_{k,i-1} \end{aligned}$$ where we are introducing the symmetric random matrix $$\begin{aligned} \label{Equ:ConvergenceAnalysis:H_kim1} \boxed{ \bm{H}_{l,k,i-1} \triangleq \int_{0}^{1} \nabla_w^2 J_l\big(w^o - t\tilde{\bm{\phi}}_{k,i-1}\big) dt } \end{aligned}$$ Observe that one such matrix is associated with every edge linking two nodes $(l,k)$; observe further that this matrix changes with time since it depends on the estimate at node $k$. Substituting – into leads to: $$\begin{aligned} % \begin{cases} % \tilde{\bm{\phi}}_{k,i-1} = \displaystyle\sum_{l=1}^N p_{1,l,k} \tilde{\bm{w}}_{l,i-1} \\ \tilde{\bm{\psi}}_{k,i} =\;& \displaystyle \Big[ I_M - \mu_k \sum_{l=1}^N s_{l,k} \bm{H}_{l,k,i-1} \Big] \tilde{\bm{\phi}}_{k,i-1} \nonumber\\ \label{Equ:ConvergenceAnalysis:ErrorRecursions} \;&+ \mu_k \sum_{l=1}^N s_{l,k} \bm{v}_{l,i}(\bm{\phi}_{k,i-1}) \end{aligned}$$ We introduce the network error vectors, which collect the error quantities across all nodes: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:global_error_vector} \tilde{\bm{\phi}}_{i} \triangleq \begin{bmatrix} \tilde{\bm\phi}_{1,i} \\ \vdots \\ \tilde{\bm\phi}_{N,i} \end{bmatrix}, \qquad \tilde{\bm\psi}_{i} \triangleq \begin{bmatrix} \tilde{\bm\psi}_{1,i} \\ \vdots \\ \tilde{\bm\psi}_{N,i} \end{bmatrix}, \qquad \tilde{\bm w}_{i} \triangleq \begin{bmatrix} \tilde{\bm w}_{1,i} \\ \vdots \\ \tilde{\bm w}_{N,i} \end{bmatrix} \end{aligned}$$ and the following block matrices: [ $$\begin{aligned} \label{Equ:ConvergenceAnalysis:P1_P2} \mathcal{P}_1 \;=\;& P_1 \otimes I_M,\; \mathcal{P}_2 \;=\; P_2 \otimes I_M \\ \label{Equ:ConvergenceAnalysis:S_M} \mathcal{S} \;=\;& S \otimes I_M, \; \mathcal{M} \;=\; \Omega \otimes I_M \\ \label{Equ:ConvergenceAnalysis:Xi} \Omega \;=\;& \mathrm{diag} \left\{ \mu_1, \; \ldots, \; \mu_N \right\} \\ \label{Equ:ConvergenceAnalysis:D_i_minus_1} \!\!\!\bm{\mathcal{D}}_{i-1} \;=\;& \sum_{l=1}^N \mathrm{diag} \big\{ s_{l,1} \bm{H}_{l,1,i-1}, \cdots, s_{l,N} \bm{H}_{l,N,i-1} \big\} \\ \label{Equ:ConvergenceAnalysis:G_i} \bm{g}_i \;=\;& \sum_{l=1}^N\! \mathrm{col} \big\{ s_{l,1}\bm{v}_{l,i}(\bm{\phi}_{1,i\!-\!1}), \cdots,\! s_{l,N}\bm{v}_{l,i}(\bm{\phi}_{N,i\!-\!1}) \big\} \end{aligned}$$ ]{}where the symbol $\otimes$ denotes Kronecker products [@horn1990matrix]. Then, recursions , and lead to: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:ErrorRecursion_final} \boxed{ \tilde{\bm w}_i = \mathcal{P}_2^T [I_{MN}-\mathcal{M}\bm{\mathcal{D}}_{i-1}] \mathcal{P}_1^T \tilde{\bm w}_{i-1} + \mathcal{P}_2^T \mathcal{M} \bm{g}_i } \end{aligned}$$ To proceed with the analysis, we introduce the following assumption on the cost functions and gradient noise, followed by a lemma on $\bm{H}_{l,k,i-1}$. \[Assumption:Hessian\] Each component cost function $J_l(w)$ has a bounded Hessian matrix, i.e., there exist nonnegative real numbers $\lambda_{l,\min}$ and $\lambda_{l,\max}$ such that $\lambda_{l,\min} \le \lambda_{l,\max}$ and that for all $w$: $$\begin{aligned} \label{Assumption:StrongConvexity} \lambda_{l,\min} I_M \le \nabla_w^2 J_l(w) \le \lambda_{l,\max} I_M \end{aligned}$$ Furthermore, the $\{\lambda_{l,\min}\}_{l=1}^N$ satisfy $$\begin{aligned} \label{Assumption:StrongConvexity1} \sum_{l=1}^N s_{l,k} \lambda_{l,\min} >0, k=1,2,\ldots,N \end{aligned}$$ Condition ensures that the local cost functions $\{J_{k}^{\mathrm{loc}} (w)\}$ defined earlier in are strongly convex and, hence, have a unique minimizer at $w^o$. \[Assumption:GradientNoise\] There exist $\alpha \ge 0$ and $\sigma_{v}^2 \ge 0$ such that, for all $\bm{w} \in \mc{F}_{i-1}$ and for all $i$, $l$: [ $$\begin{aligned} \label{Assumption:GradientNoise:ZeroMean_Uncorrelated} &\mathbb{E}\left\{\bm{v}_{l,i}(\bm{w}) \;\|\; \mc{F}_{i-1} \right\} = 0 \\ \label{Assumption:GradientNoise:Norm} &\mathbb{E}\left\{ \\|\bm{v}_{l,i}(\bm{w}) \\|^2 \right\} \le \alpha \E\\|w^o-\bm{w}\\|^2 + \sigma_{v}^2 \end{aligned}$$ ]{} where $\mc{F}_{i-1}$ denotes the past history ($\sigma-$field) of estimates $\{\bm{w}_{k,j}\}$ for $j \le i-1$ and all $k$. \[Lemma:H\_lki\_bounds\] Under Assumption \[Assumption:Hessian\], the matrix $\bm{H}_{l,k,i-1}$ defined in is a nonnegative-definite matrix that satisfies: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:H_lki_Bounds} \lambda_{l,\min} I_M \le \bm{H}_{l,k,i-1} \le \lambda_{l,\max} I_M \end{aligned}$$ It suffices to prove that $\lambda_{l,\min} \le x^T \bm{H}_{l,k,i-1} x \le \lambda_{l,\max}$ for arbitrary $M \times 1$ unit Euclidean norm vectors $x$. By and , we have $$\begin{aligned} x^T \bm{H}_{l,k,i-1} x % &= x^T % \left[ % \int_{0}^{1}\nabla_w^2 J_l\left(w^o - t\tilde{\bm{\phi}}_{k,i-1}\right)dt % \right] % x &= \int_{0}^{1} x^T\nabla_w^2 J_l\left(w^o - t\tilde{\bm{\phi}}_{k,i-1}\right) x \; dt \nonumber\\ &\le \int_0^1 \lambda_{l,\max} dt = \lambda_{l,\max} \nonumber \end{aligned}$$ In a similar way, we can verify that $x^T \bm{H}_{l,k,i-1} x \ge \lambda_{l,\min}$. [ In distributed subgradient methods (e.g., [@nedic2009distributed; @ram2010distributed; @srivastava2011distributed]), the norms of the subgradients are usually required to be uniformly bounded. Such assumption is restrictive in the unconstrained optimization of differentiable functions. ]{} Assumption \[Assumption:Hessian\] is more relaxed in that it allows the gradient vector $\nabla_w J_l(w)$ to have unbounded norm (e.g., quadratic costs). Furthermore, condition allows the variance of the gradient noise to grow no faster than $\E\\|w^o-\bm{w}\\|^2$. This condition is also more general than the uniform bounded assumption used in [@ram2010distributed] (Assumptions 5.1 and 6.1), which requires instead: $$\begin{aligned} \label{Equ:AbsoluteRandomNoise_GradientNoise} \!\!\!\mathbb{E} \\| \bm{v}_{l,i}(\bm{w}) \\|^2 \!\le\! \sigma_v^2, \quad \mathbb{E} \left\{ \\|\bm{v}_{l,i}(\bm{w}) \\|^2 \| \mathcal{F}_{i\!-\!1} \right\} \!\le\! \sigma_v^2 \end{aligned}$$ Furthermore, condition is similar to condition (4.3) in [@bertsekas2000gradient p.635]: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:GradientNoise_Bertsekas} \mathbb{E}\left\{ \\| \bm{v}_{l,i}(\bm{w}) \\|^2 \| \mathcal{F}_{i-1} \right\} \le \alpha \big[\\|\nabla_w J_l(\bm{w})\\|^2 + 1\big] \end{aligned}$$ which is a combination of the “relative random noise” and the “absolute random noise” conditions defined in [@poliak1987introduction pp.100–102]. Indeed, we can derive by substituting into , taking expectation with respect to $\mathcal{F}_{i-1}$, and then using . Such a mix of “relative random noise” and “absolute random noise” is of practical importance. For instance, consider an example in which the loss function at node $l$ is chosen to be of the following quadratic form: $$\begin{aligned} Q_l(w,\{\bm{u}_{l,i},\bm{d}_l(i)\})=\|\bm{d}_l(i)-\bm{u}_{l,i}w\|^2 \nonumber \end{aligned}$$ for some scalars $\{\bm{d}_l(i)\}$ and $1\times M$ regression vectors $\{\bm{u}_{l,i}\}$. The corresponding cost function is then: $$\begin{aligned} \label{Equ:PerformanceAnalysis:QuadraticCost_LMS} J_l(w) &= \E \|\bm{d}_l(i)-\bm{u}_{l,i}w\|^2 \end{aligned}$$ Assume further that the data $\{\bm{u}_{l,i},\bm{d}_l(i)\}$ satisfy the linear regression model $$\begin{aligned} \label{Equ:PerformanceAnalysis:gradientNoise_linearmodel} \bm{d}_l(i) = \bm{u}_{l,i}w^o + \bm{z}_l(i) \end{aligned}$$ where the regressors $\{\bm{u}_{l,i}\}$ are zero mean and independent over time with covariance matrix $R_{u,l}=\E \{\bm{u}_{l,i}^T\bm{u}_{l,i}\}$, and the noise sequence $\{\bm{z}_k(j)\}$ is also zero mean, white, with variance $\sigma_{z,k}^2$, and independent of the regressors $\{\bm{u}_{l,i}\}$ for all $l,k,i,j$. Then, using and , the gradient noise in this case can be expressed as: $$\begin{aligned} \label{Equ:PerformanceAnalysis:GradientNoise_LMS} \bm{v}_{l,i}(\bm{w}) = 2(R_{u,l}-\bm{u}_{l,i}^T\bm{u}_{l,i})(w^o-\bm{w}) - 2\bm{u}_{l,i}^T \bm{z}_l(i) \end{aligned}$$ It can easily be verified that this noise satisfies both conditions stated in Assumption \[Assumption:GradientNoise\], namely, and also: $$\begin{aligned} &\mathbb{E}\left\{ \\|\bm{v}_{l,i}(\bm{w})\\|^2\right\} \nonumber\\ \label{Equ:PerformanceAnalysis:GradientNoiseNorm_LMS} &\le 4 \E\\|R_{u,l}\!-\!\bm{u}_{l,i}^T\bm{u}_{l,i}\\|^2 \cdot \E\\|w^o\!-\!\bm{w}\\|^2 \!+\! 4 \sigma_{z,l}^2 \mathrm{Tr}(R_{u,l}) \end{aligned}$$ for all $\bm{w} \in \mc{F}_{i-1}$. Note that both relative random noise and absolute random noise components appear in and are necessary to model the statistical gradient perturbation even for quadratic costs. Such costs, and linear regression models of the form , arise frequently in the context of adaptive filters — see, e.g., [@Sayed08; @haykin2002adaptive; @lopes2007incremental; @lopesdistributed; @lopes2007diffusion; @Sayed07; @lopes2008diffusion; @cattivelli2008diffusion; @Cattivelli10; @cattivelli2007diffusionRLS; @cattivelli2008TSPdiffusionRLS; @cattivelli2011modeling; @arenas2006plant; @silva2008improving; @theodoridis2011adaptive]. \[Example:ATC\_MMSE\] Quadratic costs of the form are common in mean-square-error estimation for linear regression models of the type . If we use the instantaneous approximations as is common in the context of stochastic approximation and adaptive filtering [@Sayed08; @haykin2002adaptive; @poliak1987introduction], then the actual gradient $\nabla_w J_l(w)$ can be approximated by $$\begin{aligned} \widehat{\nabla}_{w} J_{l}(w) &= \nabla_wQ_l(w,\{\bm{u}_{l,i},\bm{d}_l(i)\}) \nonumber\\ &= -2\bm{u}_{l,i}^T[\bm{d}_{l}(i)-\bm{u}_{l,i} w] \end{aligned}$$ Substituting into –, and assuming $C=I$ for illustration purposes only, we arrive at the following ATC and CTA diffusion strategies originally proposed and extended in [@lopesdistributed; @lopes2007diffusion; @Sayed07; @lopes2008diffusion; @cattivelli2008diffusion; @Cattivelli10] for the solution of distributed mean-square-error estimation problems: $$\begin{aligned} \!\!(\mathrm{ATC})~\boxed{ \label{Equ:DiffusionAdaptation:ATC_MMSE} \begin{array}{l} \bm{\psi}_{k,i} = \displaystyle \bm{w}_{k,i\!-\!1} \!+\! 2\mu_k \bm{u}_{k,i}^T[\bm{d}_{k}(i)\!-\!\bm{u}_{k,i} \bm{w}_{k,i\!-\!1}] \\ \bm{w}_{k,i} = \displaystyle \sum_{l \in \mathcal{N}_k} a_{l,k} \bm{\psi}_{l,i} \end{array} } \end{aligned}$$ $$\begin{aligned} \!\!(\mathrm{CTA})~\boxed{ \label{Equ:DiffusionAdaptation:CTA_MMSE} \begin{array}{l} \bm{\psi}_{k,i-1} = \displaystyle \sum_{l \in \mathcal{N}_k} a_{l,k} \bm{w}_{l,i-1} \\ \bm{w}_{k,i} = \displaystyle \bm{\psi}_{k,i\!-\!1} \!+\! 2\mu_k \bm{u}_{k,i}^T[\bm{d}_{k}(i) \!-\! \bm{u}_{k,i} \bm{\psi}_{k,i\!-\!1}] \end{array} } \end{aligned}$$ Variance Relations {#Sec:ConvergenceAnalysis:VarianceRelation} ------------------ The purpose of the mean-square analysis in the sequel is to answer two questions in the presence of gradient perturbations. First, how small the mean-square error, $\mathbb{E}\\|\tilde{\bm{w}}_{k,i}\\|^2$, gets as $i\rightarrow\infty$ for any of the nodes $k$. Second, how fast this error variance tends towards its steady-state value. The first question pertains to steady-state performance and the second question pertains to transient/convergence rate performance. Answering such questions for a distributed algorithm over a network is a challenging task largely because the nodes influence each other’s behavior: performance at one node diffuses through the network to the other nodes as a result of the topological constraints linking the nodes. The approach we take to examine the mean-square performance of the diffusion algorithms is by studying how the variance $\mathbb{E}\\|\tilde{\bm{w}}_{k,i}\\|^2$, or a weighted version of it, evolves over time. As the derivation will show, the evolution of this variance satisfies a nonlinear relation. Under some reasonable assumptions on the noise profile, and the local cost functions, we will be able to bound these error variances as well as estimate their steady-state values for sufficiently small step-sizes. We will also derive closed-form expressions that characterize the network performance. The details are as follows. Equating the squared weighted Euclidean norm of both sides of , applying the expectation operator and using using , we can show that the following variance relation holds: $$\label{Equ:ConvergenceAnalysis:WeightedEnergyConservation_Relation} \boxed{ \begin{split} &\mathbb{E}\\|\tilde{\bm{w}}_{i}\\|_\Sigma^2 = \mathbb{E}\big\{\\|\tilde{\bm{w}}_{i-1}\\|_{\bm{\Sigma}'}^2\big\} + \mathbb{E}\\|\mathcal{P}_2^T \mathcal{M} \bm{g}_i\\|_\Sigma^2 \\ &\bm{\Sigma}' = \mathcal{P}_1 [I_{MN}\!-\!\mathcal{M}\bm{\mathcal{D}}_{i\!-\!1}] \mathcal{P}_2 \Sigma \mathcal{P}_2^T [I_{MN}\!-\!\mathcal{M}\bm{\mathcal{D}}_{i\!-\!1}] \mathcal{P}_1^T \end{split} }$$ where $\Sigma$ is a positive semi-definite weighting matrix that we are free to choose. The variance expression shows how the quantity $\mathbb{E}\\|\tilde{\bm{w}}_i\\|^2_{\Sigma}$ evolves with time. Observe, however, that the weighting matrix on $\tilde{\bm{w}}_{i-1}$ on the right-hand side of is a different matrix, denoted by $\bm{\Sigma}'$, and this matrix is actually random in nature (while $\Sigma$ is deterministic). As such, result is not truly a recursion. Nevertheless, it is possible, under a small step-size approximation, to rework variance relations such as into a recursion by following certain steps that are characteristic of the energy conservation approach to mean-square analysis [@Sayed08]. The first step in this regard would be to replace $\bm{\Sigma}'$ by its mean $\mathbb{E}\bm{\Sigma}'$. However, the matrix $\bm{\Sigma}'$ depends on the $\{\bm{H}_{l,k,i-1}\}$ via $\bm{\mathcal{D}}_{i-1}$ (see ). It follows from the definition of $\bm{H}_{l,k,i-1}$ in that $\bm{\Sigma}'$ is dependent on $\tilde{\bm{\phi}}_{k,i-1}$ as well, which in turn is a linear combination of the $\{\tilde{\bm{w}}_{l,i-1}\}$. Therefore, the main challenge to continue from is that $\bm{\Sigma}'$ depends on $\tilde{\bm{w}}_{i-1}$. For this reason, we cannot apply directly the traditional step of replacing $\bm{\Sigma}'$ in the first equation of by $\mathbb{E}\bm{\Sigma}'$ as is typically done in the study of stand-alone adaptive filters to analyze their transient behavior [@Sayed08 p.345]; in the case of conventional adaptive filters, the matrix $\bm{\Sigma}'$ is independent of $\tilde{\bm{w}}_{i-1}$. To address this difficulty, we shall adjust the argument to rely on a set of inequality recursions that will enable us to bound the steady-state mean-square-error at each node — see Theorem \[Thm:ConvergenceAnalysis:MS\_Theorem\] further ahead. The procedure is as follows. First, we note that $\\| x \\|^2$ is a convex function of $x$, and that expressions and are convex combinations of $\{\tilde{\bm{w}}_{l,i-1}\}$ and $\{\tilde{\bm{\psi}}_{l,i}\}$, respectively. Then, by Jensen’s inequality[@boyd2004convex p.77] and taking expectations, we obtain [ $$\begin{aligned} \label{Equ:ConvergenceAnalysis:Combination1_bound} \mathbb{E}\\|\tilde{\bm{\phi}}_{k,i-1}\\|^2 &\le \sum_{l=1}^N p_{1,l,k}\mathbb{E}\\| \tilde{\bm{w}}_{l,i-1}\\|^2 \\ \label{Equ:ConvergenceAnalysis:Combination2_bound} \mathbb{E}\\|\tilde{\bm{w}}_{k,i}\\|^2 &\le \sum_{l=1}^N p_{2,l,k}\mathbb{E}\\| \tilde{\bm{\psi}}_{l,i}\\|^2 \end{aligned}$$ ]{}for $k=1,\ldots,N$. Next, we derive a variance relation for . Equating the squared Euclidean norms of both sides of , applying the expectation operator, and using from Assumption \[Assumption:GradientNoise\], we get $$\begin{aligned} \label{Equ:ConvergenceAnalysis:VarianceRelation_Original} \!\!\mathbb{E}\\|\tilde{\bm{\psi}}_{k,i}\\|^2 &\!= \mathbb{E}\big\{\\|\tilde{\bm{\phi}}_{k,i-1}\\|_{\bm{\Sigma}_{k,i-1}}^2\big\} \!+\! \mu_k^2 \mathbb{E} \Big\\| \sum_{l=1}^N\! s_{l,k} \bm{v}_{l,i}(\bm{\phi}_{k,i-1}) \Big\\|^2 \end{aligned}$$ where $$\begin{aligned} \bm{\Sigma}_{k,i-1} %\triangleq&\; % \Big[ % I_M \!-\! \mu_k \sum_{l=1}^N s_{l,k} \bm{H}_{l,k,i-1} % \Big]^T % \nonumber\\ % &\;\times % \Big[ % I_M \!-\! \mu_k \sum_{l=1}^N s_{l,k} \bm{H}_{l,k,i-1} % \Big] % \nonumber\\ \label{Equ:ConvergenceAnalysis:VarianceRelation_Sigma_kim1} =&\; \Big[ I_M \!-\! \mu_k \sum_{l=1}^N s_{l,k} \bm{H}_{l,k,i-1} \Big]^2 \end{aligned}$$ We call upon the following two lemmas to bound . \[Lemma:Sigma\_k\_im1\] The weighting matrix $\bm{\Sigma}_{k,i-1}$ defined in is a symmetric, positive semi-definite matrix, and satisfies: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:Lemma:Sigma_kim1_Bound} 0 \le \bm{\Sigma}_{k,i-1} \le \gamma_k^2 I_M \end{aligned}$$ where $$\begin{aligned} \label{Equ:ConvergenceAnalysis:gamma_k} \boxed{ \gamma_k \triangleq \max \Big\{ \Big\|1\!-\!\mu_k \sum_{l=1}^N s_{l,k} \lambda_{l,\max} \Big\|,\; \Big\|1\!-\!\mu_k \sum_{l=1}^N s_{l,k} \lambda_{l,\min}\Big\| \Big\} } \end{aligned}$$ By definition and the fact that $\bm{H}_{l,k,i-1}$ is symmetric — see definition , the matrix $I_M \!-\! \mu_k \sum_{l=1}^N s_{l,k} \bm{H}_{l,k,i-1}$ is also symmetric. Hence, its square, $\bm{\Sigma}_{k,i-1}$, is symmetric and also nonnegative-definite. To establish , we first use to note that: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:Proof_Lemma_SimgaBound_Ineq1} &I_M \!-\! \mu_k \sum_{l=1}^N s_{l,k} \bm{H}_{l,k,i-1} \ge \Big(1\!-\!\mu_k \sum_{l=1}^N s_{l,k} \lambda_{l,\max}\Big) I_M \\ \label{Equ:ConvergenceAnalysis:Proof_Lemma_SimgaBound_Ineq2} &I_M \!-\! \mu_k \sum_{l=1}^N s_{l,k} \bm{H}_{l,k,i-1} \le \Big(1\!-\!\mu_k \sum_{l=1}^N s_{l,k} \lambda_{l,\min}\Big) I_M \end{aligned}$$ The matrix $I_M - \mu_k \sum_{l=1}^N s_{l,k} \bm{H}_{l,k,i-1}$ may not be positive semi-definite because we have not specified a range for $\mu_k$ yet; the expressions on the right-hand side of – may still be negative. However, inequalities – imply that the eigenvalues of $I_M - \mu_k \sum_{l=1}^N s_{l,k} \bm{H}_{l,k,i-1}$ are bounded as: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:Proof_Lemma_SimgaBound_Ineq3} &\lambda\Big(I_M - \mu_k \sum_{l=1}^N s_{l,k} \bm{H}_{l,k,i-1}\Big) \ge 1-\mu_k \sum_{l=1}^N s_{l,k} \lambda_{l,\max} \\ \label{Equ:ConvergenceAnalysis:Proof_Lemma_SimgaBound_Ineq3_2} &\lambda\Big(I_M - \mu_k \sum_{l=1}^N s_{l,k} \bm{H}_{l,k,i-1}\Big) \le 1-\mu_k \sum_{l=1}^N s_{l,k} \lambda_{l,\min} \end{aligned}$$ By definition , $\bm{\Sigma}_{k,i-1}$ is the square of the symmetric matrix $I_M \!-\! \mu_k \sum_{l=1}^N s_{l,k} \bm{H}_{l,k,i-1}$, meaning that $$\begin{aligned} \label{Equ:ConvergenceAnalysis:Proof_Lemma_SimgaBound_eq1} \lambda\left(\bm{\Sigma}_{k,i-1}\right) = \left[\lambda\left(I_M - \mu_k \sum_{l=1}^N s_{l,k} \bm{H}_{l,k,i-1}\right)\right]^2 \ge 0 \end{aligned}$$ Substituting – into leads to $$\begin{aligned} &\lambda\left(\bm{\Sigma}_{k,i-1}\right) \nonumber\\ \label{Equ:ConvergenceAnalysis:Proof_Lemma_SimgaBound_Ineq4} &\le \max \Big\{ \Big\| 1 - \mu_k \sum_{l=1}^N s_{l,k} \lambda_{l,\max} \Big\|^2 , \Big\| 1 - \mu_k \sum_{l=1}^N s_{l,k} \lambda_{l,\min} \Big\|^2 \Big\} \end{aligned}$$ which is equivalent to . The second term on the right-hand-side of satisfies: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:NoiseTerm_bound_final} \mathbb{E} \Big\\| \! &\sum_{l=1}^N s_{l,k} \bm{v}_{l,i}(\bm{\phi}_{k,i\!-\!1}) \Big\\|^2 \le \\|S\\|_1^2 \cdot \left[ \alpha \mathbb{E}\\|\tilde{\bm{\phi}}_{k,i-1}\\|^2 \!+\! \sigma_{v}^2 \right] \end{aligned}$$ where $\\|S\\|_1$ denotes the $1$-norm of the matrix $S$ (i.e., the maximum absolute column sum). Applying Jensen’s inequality[@boyd2004convex p.77], it holds that $$\begin{aligned} &\mathbb{E} \Big\\| \! \sum_{l=1}^N s_{l,k} \bm{v}_{l,i}(\bm{\phi}_{k,i-1}) \Big\\|^2 \nonumber\\ &\qquad= \big(\sum_{l=1}^N s_{l,k}\big)^2 \cdot \mathbb{E} \Big\\| \! \sum_{l=1}^N \frac{s_{l,k}}{\sum_{l=1}^N s_{l,k}} \bm{v}_{l,i}(\bm{\phi}_{k,i-1}) \Big\\|^2 \nonumber\\ &\qquad\le \big(\sum_{l=1}^N s_{l,k}\big)^2 \cdot \sum_{l=1}^N \frac{s_{l,k}}{\sum_{l=1}^N s_{l,k}} \mathbb{E} \\| \bm{v}_{l,i}(\bm{\phi}_{k,i-1}) \\|^2 \nonumber\\ &\qquad= \big(\sum_{l=1}^N s_{l,k}\big) \cdot \sum_{l=1}^N s_{l,k} \mathbb{E} \\| \bm{v}_{l,i}(\bm{\phi}_{k,i-1}) \\|^2 \nonumber\\ \label{Equ:ConvergenceAnalysis:NoiseTerm_bound} &\qquad\le \big( \sum_{l=1}^N s_{l,k} \big)^2 \cdot \big[ \alpha \mathbb{E}\\|\tilde{\bm{\phi}}_{k,i-1}\\|^2 + \sigma_{v}^2 \big] \\ \label{Equ:ConvergenceAnalysis:NoiseTerm_bound2} &\qquad\le \\|S\\|_1^2 \cdot \left[\alpha \mathbb{E}\\|\tilde{\bm{\phi}}_{k,i-1}\\|^2 + \sigma_{v}^2\right] \end{aligned}$$ where inequality follows by substituting , and is obtained using the fact that $\\|S\\|_1$ is the maximum absolute column sum and that the entries $\{s_{l,k}\}$ are nonnegative. Substituting and into , we obtain: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:Adaptation_bound} \!\!\mathbb{E}\\|\tilde{\bm{\psi}}_{k,i}\\|^2 % &\le \gamma_k^2 % \cdot % \mathbb{E}\\|\tilde{\bm{\phi}}_{k,i-1}\\|^2 % + % \mu_k^2 % \; % \\|S\\|_1^2 % \left[\alpha \mathbb{E}\\|\tilde{\bm{\phi}}_{k,i-1}\\|^2 + \sigma_{v}^2\right] % \nonumber\\ &\le (\gamma_k^2 \!+\! \mu_k^2 \alpha\\|S\\|_1^2 ) \!\cdot\! \mathbb{E}\\|\tilde{\bm{\phi}}_{k,i-1}\\|^2 \!+\! \mu_k^2 \;\\|S\\|_1^2 \; \sigma_v^2 \end{aligned}$$ for $k=1,\ldots,N$. Finally, introduce the following network mean-square-error vectors (compare with ): $$\begin{aligned} \mathcal{X}_{i} = \begin{bmatrix} \mathbb{E}\\|\tilde{\bm\phi}_{1,i}\\|^2\\ \vdots\\ \mathbb{E}\\|\tilde{\bm\phi}_{N,i}\\|^2 \end{bmatrix}, \; \mathcal{Y}_{i} = \begin{bmatrix} \mathbb{E}\\|\tilde{\bm\psi}_{1,i}\\|^2\\ \vdots\\ \mathbb{E}\\|\tilde{\bm\psi}_{N,i}\\|^2 \end{bmatrix}, \; \mathcal{W}_{i} = \begin{bmatrix} \mathbb{E}\\|\tilde{\bm w}_{1,i}\\|^2\\ \vdots\\ \mathbb{E}\\|\tilde{\bm w}_{N,i}\\|^2 \end{bmatrix} \nonumber \end{aligned}$$ and the matrix $$\begin{aligned} \label{Equ:ConvergenceAnalysis:Gamma} \Gamma \;=&\; \mathrm{diag} \left\{ \gamma_1^2 + \mu_1^2\alpha\\|S\\|_1^2, \;\ldots\;, \gamma_N^2 + \mu_N^2\alpha\\|S\\|_1^2 \right\} \end{aligned}$$ Then, – and can be written as $$\begin{aligned} \label{Equ:ConvergenceAnalysis:VarianceRecursion_Inequality_Separate} \begin{cases} \mathcal{X}_{i-1} \preceq P_1^T \mathcal{W}_{i-1}\\ \mathcal{Y}_i \preceq \Gamma \mathcal{X}_{i-1} + \sigma_v^2\\|S\\|_1^2\Omega^2 \mathds{1}\\ \mathcal{W}_i \preceq P_2^T \mathcal{Y}_i \end{cases} \end{aligned}$$ where the notation $x \preceq y$ denotes that the components of vector $x$ are less than or equal to the corresponding components of vector $y$. We now recall the following useful fact that for any matrix $F$ with nonnegative entries, $$\begin{aligned} \label{Equ:ConvergenceAnalysis:x_y_Fx_Fy} x \preceq y \Rightarrow Fx \preceq Fy \end{aligned}$$ This is because each entry of the vector $Fy-Fx = F(y-x)$ is nonnegative. Then, combining all three inequalities in leads to: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:VarianceRecursion_Inequality_Final} \boxed{ \mathcal{W}_i \preceq P_2^T \Gamma P_1^T \mathcal{W}_{i-1} + \sigma_v^2 \\|S\\|_1^2 \cdot P_2^T \Omega^2 \mathds{1} } \end{aligned}$$ Mean-Square Stability --------------------- Based on , we can now prove that, under certain conditions on the step-size parameters $\{\mu_k\}$, the mean-square-error vector $\mathcal{W}_i$ is bounded as $i\rightarrow \infty$, and we use this result in the next subsection to evaluate the steady-state MSE for sufficiently small step-sizes. \[Thm:ConvergenceAnalysis:MS\_Theorem\] If the step-sizes $\{\mu_k\}$ satisfy the following condition: $$\label{Equ:ConvergenceAnalysis:MS_Theorem:Stepsizes} \boxed{ 0 < \mu_k < \min\left\{ \frac{2\sigma_{k,\max}} {\sigma_{k,\max}^2\!+\!\alpha\\|S\\|_1^2}, \frac{2\sigma_{k,\min}} {\sigma_{k,\min}^2\!+\!\alpha\\|S\\|_1^2} \right\} }$$ for $k=1,\ldots,N$, where $\sigma_{k,\max}$ and $\sigma_{k,\min}$ are defined as $$\begin{aligned} \label{Equ:ConvergenceAnalysis:MS_Theorem:sigma_min_max} &\sigma_{k,\max} \triangleq \sum_{l=1}^N s_{l,k} \lambda_{l,\max}, \quad \sigma_{k,\min} \triangleq \sum_{l=1}^N s_{l,k} \lambda_{l,\min} \end{aligned}$$ then, as $i\rightarrow \infty$, $$\begin{aligned} \label{Equ:ConvergenceAnalysis:MS_Theorem:W_infty_Bound} \boxed{ \limsup_{i \rightarrow \infty}\\| \mathcal{W}_{i}\\|_{\infty} \le \frac{ \displaystyle \big( \max_{1\le k \le N} \mu_k^2 \big) \cdot \\|S\\|_1^2 \sigma_v^2 } { \displaystyle 1 - \max_{1 \le k \le N} (\gamma_k^2 + \mu_k^2 \alpha\\|S\\|_1^2) } } \end{aligned}$$ where $\\| x \\|_{\infty}$ denotes the maximum absolute entry of vector $x$. See Appendix \[Appendix:Proof\_MeanSquaredStability\]. [ If we let $\alpha\!\!=\!\!0$ and $\sigma_v^2\!\!=\!\!0$ in Theorem \[Thm:ConvergenceAnalysis:MS\_Theorem\], and examine the arguments leading to it, we conclude the validity of the following result, which establishes the convergence of the diffusion strategies – in the absence of gradient noise (i.e., using the true gradient rather than stochastic gradient — see and ). ]{} \[Corollary:ConvergenceAnalysis:Convergence\_NoiseFreeCase\] If there is no gradient noise, i.e., $\alpha=0$ and $\sigma_v^2=0$, then the mean-square-error vector becomes the deterministic vector $\mathcal{W}_i = \mathrm{col}\{\\|\tilde{ w}_{1,i}\\|^2,\cdots,\\|\tilde{ w}_{N,i}\\|^2\}$, and its entries converge to zero if the step-sizes $\{\mu_k\}$ satisfy the following condition: $$\label{Equ:ConvergenceAnalysis:NoiseFree_Corollary:Stepsizes} \boxed{ 0 < \mu_k < \frac{2}{\sigma_{k,\max}} }$$ for $k=1,\ldots,N$, where $\sigma_{k,\max}$ was defined in . We observe that, in the absence of noise, the deterministic error vectors, $\tilde{w}_{k,i}$, will tend to zero as $i\rightarrow \infty$ even with constant (i.e., non-vanishing) step-sizes. This result implies the interesting fact that, in the noise-free case, the nodes can reach agreement without the need to impose diminishing step-sizes. Steady-State Performance {#Sec:ConvergenceAnalysis:SteadyState} ------------------------ Expression provides a condition on the step-size parameters $\{\mu_k\}$ to ensure the mean-square stability of the diffusion strategies –. At the same time, expression gives an upper bound on how large $\mathcal{W}_i$ can be at steady-state. Since the $\infty$-norm of a vector is defined as the largest absolute value of its entries, then bounds the MSE of the worst-performing node in the network. We can derive closed-form expressions for MSEs when the step-sizes are assumed to be sufficiently small. Indeed, we first conclude from that for step-sizes that are sufficiently small, each $\bm{w}_{k,i}$ will get closer to $w^o$ at steady-state. To verify this fact, assume the step-sizes are small enough so that the nonnegative factor $\gamma_k$ that was defined earlier in becomes $$\begin{aligned} \label{Equ:ConvergenceAnalysis:gamma_k_SmallStepsize} \gamma_k = 1 - \mu_k \sum_{l=1}^N s_{l,k} \lambda_{l,\min} = 1 - \mu_k \sigma_{k,\min} \end{aligned}$$ where $\sigma_{k,\min}$ was given by . Substituting into , we obtain: $$\begin{aligned} &\limsup_{i\rightarrow\infty} \\|\mathcal{W}_{i}\\|_{\infty} \nonumber\\ &\quad \le \frac{ \displaystyle \Big(\max_{1\le k \le N} \mu_k^2\Big) \cdot \\|S\\|_1^2 \sigma_v^2 } { \displaystyle 1 \!-\! \max_{1 \le k \le N} \Big\{ (1\!-\!\mu_k\sigma_{k,\min})^2 \!+\! \mu_k^2 \alpha \\|S\\|_1^2 \Big\} } \nonumber\\ &\quad \le \frac{ \displaystyle\Big(\max_{1\le k \le N} \mu_k^2\Big) \cdot \\|S\\|_1^2 \sigma_v^2 } {\displaystyle \min_{1\le k \le N} \Big\{ \mu_{k} \Big[ 2\sigma_{k,\min} - \mu_k (\sigma_{k,\min}^2+\alpha \\|S\\|_1^2) \Big] \Big\} } \nonumber\\ \label{Equ:ConvergenceAnalysis:MSD_Vanish_SmallStepSize} &\quad \le \frac{ \\|S\\|_1^2 \sigma_v^2 } {\displaystyle \min_{1\le k \le N} \Big\{ 2\sigma_{k,\min} \!-\! \mu_k (\sigma_{k,\min}^2\!+\!\alpha \\|S\\|_1^2) \Big\} } \cdot \frac{\mu_{\max}^2}{\mu_{\min}} \end{aligned}$$ where $$\begin{aligned} \label{Equ:PerformanceAnalysis:mu_max_min_def} \mu_{\max} \!\triangleq \!\displaystyle\max_{1\le k\le N} \mu_k, \qquad \mu_{\min} \!\triangleq \!\displaystyle\min_{1\le k\le N}\mu_k \end{aligned}$$ For sufficiently small step-sizes, the denominator in can be approximated as $$\begin{aligned} \label{Equ:PerformanceAnalysis:W_inf_bound_intermediate_approx} 2\sigma_{k,\min} \!-\! \mu_k (\sigma_{k,\min}^2\!+\!\alpha \\|S\\|_1^2) \approx 2\sigma_{k,\min} \end{aligned}$$ Substituting into , we get $$\begin{aligned} \label{Equ:ConvergenceAnalysis:MSD_Vanish_SmallStepSize_final} \boxed{ \limsup_{i\rightarrow\infty} \\|{\mathcal{W}}_{i}\\|_{\infty} \;\le\; \frac{\\|S\\|_1^2 \sigma_v^2} {\displaystyle2\min_{1\le k \le N} \sigma_{k,\min}} \cdot \frac{\mu_{\max}^2}{\mu_{\min}} } \end{aligned}$$ Therefore, if the step-sizes are sufficiently small, the MSE of each node becomes small as well. This result is clear when all nodes use the same step-sizes such that $\mu_{\max}=\mu_{\min}=\mu$. Then, the right-hand side of is on the order of $O(\mu)$, as indicated. It follows that $\{\tilde{\bm{w}}_{k,i}\}$ are small in the mean-square-error sense at small step-sizes, which also means that the mean-square value of $\tilde{\bm{\phi}}_{k,i-1}$ is small because it is a convex combination of $\{\tilde{\bm{w}}_{k,i}\}$ (recall ). Then, by definition , [ in steady-state (for large enough $i$), ]{} the matrix $\bm{H}_{l,k,i-1}$ can be approximated by: $$\begin{aligned} \label{Equ:PerformanceAnalysis:H_lkim1_approx_steadystate} \bm{H}_{l,k,i-1} \approx \int_{0}^{1} \nabla^2 J_l(w^o) dt = \nabla^2 J_l(w^o) \end{aligned}$$ In this case, the matrix $\bm{H}_{l,k,i-1}$ is not random anymore and is not dependent on the error vector $\tilde{\bm{\phi}}_{k,,i-1}$. Accordingly, in steady-state, the matrix $\bm{\mathcal{D}}_{i-1}$ that was defined in is not random anymore and it becomes $$\begin{aligned} \label{Equ:ConvergenceAnalysis:D_inf} \boxed{ \bm{\mathcal{D}}_{i-1} \!\approx\! \mathcal{D}_{\infty} \!\triangleq\! \sum_{l=1}^N \mathrm{diag} \big\{ s_{l,1} \nabla_w^2 J_l(w^o), \cdots,\! s_{l,N} \nabla_w^2 J_l(w^o) \big\} } \end{aligned}$$ [ As a result, in steady-state, the original error recursion can be approximated by $$\begin{aligned} \label{Equ:ConvergenceAnalysis:ErrorRecursion_final_approx} \boxed{ \tilde{\bm w}_i = \mathcal{P}_2^T [I_{MN}-\mathcal{M}{\mathcal{D}}_{\infty}] \mathcal{P}_1^T \tilde{\bm w}_{i-1} + \mathcal{P}_2^T \mathcal{M} \bm{g}_i } \end{aligned}$$ Taking expectations of both sides of , we obtain the following mean-error recursion $$\begin{aligned} \label{Equ:PerformanceAnalysis:MeanErrorRecursion_Approx} \E\tilde{\bm w}_i = \mathcal{P}_2^T [I_{MN}-\mathcal{M}{\mathcal{D}}_{\infty}] \mathcal{P}_1^T \cdot \E\tilde{\bm w}_{i-1}, \quad i \rightarrow\infty \end{aligned}$$ which converges to zero if the matrix $$\begin{aligned} \label{Equ:PerformanceAnalysis:B_cal_def} \mathcal{B} \triangleq \mathcal{P}_2^T[I_{MN}-\mathcal{M}{\mathcal{D}}_{\infty}]\mathcal{P}_1^T \end{aligned}$$ is stable. The stability of $\mathcal{B}$ can be guaranteed when the step-sizes are sufficiently small (or chosen according to ) — see the proof in Appendix \[Appendix:Stability\_F\]. Therefore, in steady-state, we have $$\begin{aligned} \label{Equ:PerformanceAnalysis:Ew_inf_zero} \boxed{ \lim_{i \rightarrow \infty} \E\tilde{\bm{w}}_i = 0 } \end{aligned}$$ ]{}Next, we determine an expression (rather than a bound) for the MSE. To do this, we need to evaluate the covariance matrix of the gradient noise vector $\bm{g}_i$. Recall from that $\bm{g}_i$ depends on $\{{\bm{\phi}}_{k,i-1}\}$, which is close to $w^o$ at steady-state for small step-sizes. Therefore, it is sufficient to determine the covariance matrix of $\bm{g}_i$ at $w^o$. We denote this covariance matrix by: $$\begin{aligned} R_v \;\triangleq\;& \mathbb{E}\{\bm{g}_i\bm{g}_i^T\}\big\|_{\phi_{k,i-1}=w^o} \nonumber\\ \;=\;& \mathbb{E} \bigg\{\! \Big[\! \sum_{l=1}^N \mathrm{col} \big\{ s_{l,1}\bm{v}_{l,i}(w^o), \cdots, s_{l,N}\bm{v}_{l,i}(w^o) \big\}\! \Big] \nonumber\\ \label{Equ:ConvergenceAnalysis:GradientNoise_PreciseModel} &\times \Big[\! \sum_{l=1}^N \mathrm{col} \big\{ s_{l,1}\bm{v}_{l,i}(w^o), \cdots, s_{l,N}\bm{v}_{l,i}(w^o) \big\}\! \Big]^T\! \bigg\} \end{aligned}$$ In practice, we can evaluate $R_v$ from the expressions of $\{\bm{v}_{l,i}(w^o)\}$. For example, for the case of the quadratic cost , we can substitute into to evaluate $R_v$. Returning to the last term in the first equation of , we can evaluate it as follows: $$\begin{aligned} \mathbb{E}\\|\mathcal{P}_2^T \mathcal{M} \bm{g}_i\\|_\Sigma^2 \;=&\; \mathbb{E} \bm{g}_i^T\mathcal{M} \mathcal{P}_2 \Sigma \mathcal{P}_2^T \mathcal{M} \bm{g}_i \nonumber\\ \;=&\; \mathrm{Tr} \left( \Sigma \mathcal{P}_2^T \mathcal{M} \mathbb{E}\{ \bm{g}_i \bm{g}_i^T \} \mathcal{M} \mathcal{P}_2 \right) \nonumber\\ \;=&\; \label{Equ:ConvergenceAnalysis:NoiseTerm_PreciseModel} \mathrm{Tr} \left( \Sigma \mathcal{P}_2^T \mathcal{M} R_v \mathcal{M} \mathcal{P}_2 \right) \end{aligned}$$ Using , the matrix $\bm{\Sigma}'$ in becomes a deterministic quantity as well, and is given by: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:Sigma_Prime_Approx_SmallStepSize} {\Sigma}' \approx \mathcal{P}_1 [I_{MN}-\mathcal{M}{\mathcal{D}}_{\infty}] \mathcal{P}_2 \Sigma \mathcal{P}_2^T [I_{MN}-\mathcal{M}{\mathcal{D}}_{\infty}] \mathcal{P}_1^T \end{aligned}$$ Substituting and into , an approximate variance relation is obtained for small step-sizes: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:WeightedEnergyConservation_Relation_Approx_SmallStepSize} \!\!\!\!\mathbb{E}\\|\tilde{\bm{w}}_{i}\\|_\Sigma^2 &\approx \mathbb{E}\\|\tilde{\bm{w}}_{i-1}\\|_{{\Sigma}'}^2 +\mathrm{Tr} \left( \Sigma \mathcal{P}_2^T \mathcal{M} R_v \mathcal{M} \mathcal{P}_2 \right) \\ \label{Equ:ConvergenceAnalysis:Sigma_PrimePrime} \!\!\!\!{\Sigma}' &\approx \mathcal{P}_1 [I_{MN}\!\!-\!\!\mathcal{M}{\mathcal{D}}_{\infty}] \mathcal{P}_2 \Sigma \mathcal{P}_2^T [I_{MN}\!\!-\!\!\mathcal{M}{\mathcal{D}}_{\infty}] \mathcal{P}_1^T \end{aligned}$$ Let $\sigma=\mathrm{vec}(\Sigma)$ denote the vectorization operation that stacks the columns of a matrix $\Sigma$ on top of each other. We shall use the notation $\\|x\\|_{\sigma}^2$ and $\\|x\\|_{\Sigma}^2$ interchangeably to denote the weighted squared Euclidean norm of a vector. Using the Kronecker product property[@laub2005matrix p.147]: $\mathrm{vec}(U\Sigma V) = (V^T \otimes U ) \mathrm{vec}(\Sigma)$, we can vectorize ${\Sigma}'$ in and find that its vector form is related to $\Sigma$ via the following linear relation: $\sigma' \triangleq \mathrm{vec}(\Sigma') \approx \mc{F}\sigma$, where, for sufficiently small steps-sizes (so that higher powers of the step-sizes can be ignored), the matrix $\mc{F}$ is given by $$\begin{aligned} \label{Equ:ConvergenceAnalysis:F} \!\!\!\boxed{ \mc{F} \!\triangleq\! \big( \mathcal{P}_1 [I_{MN}\!-\!\mathcal{M}{\mathcal{D}}_{\infty}] \mathcal{P}_2 \big)\! \otimes\! \big( \mathcal{P}_1 [I_{MN}\!-\!\mathcal{M}{\mathcal{D}}_{\infty}] \mathcal{P}_2 \big) } \end{aligned}$$ Here, we used the fact that $\mathcal{M}$ and $\mathcal{D}_\infty$ are block diagonal and symmetric. Furthermore, using the property $\mathrm{Tr}(\Sigma X) = \mathrm{vec}(X^T)^T \sigma$, we can rewrite as $$\begin{aligned} \label{Equ:ConvergenceAnalysis:WeightedEnergyConservation_final} \mathbb{E}\\|\tilde{\bm{w}}_{i}\\|_\sigma^2 \;\approx\;& \mathbb{E}\\|\tilde{\bm{w}}_{i-1}\\|_{\mc{F}\sigma}^2 + \left[ \mathrm{vec} \left( \mathcal{P}_2^T \mathcal{M} R_v \mathcal{M} \mathcal{P}_2 \right) \right]^T \!\! \sigma \end{aligned}$$ It is shown in [@Sayed08 pp.344–346] that recursion converges to a steady-state value if the matrix $\mc{F}$ is stable. This condition is guaranteed when the step-sizes are sufficiently small (or chosen according to ) — see Appendix \[Appendix:Stability\_F\]. Finally, denoting $$\begin{aligned} \mathbb{E}\\|\tilde{\bm{w}}_{\infty}\\|_{\sigma}^2 \triangleq \lim_{i\rightarrow\infty} \mathbb{E}\\|\tilde{\bm{w}}_{i}\\|_{\sigma}^2 \end{aligned}$$ and letting $i \rightarrow \infty$, expression becomes $$\begin{aligned} % \label{Equ:ConvergenceAnalysis:Limit_WeightedEnergyConservation_final} \mathbb{E}\\|\tilde{\bm{w}}_{\infty}\\|_\sigma^2 \;\approx\;& \mathbb{E}\\|\tilde{\bm{w}}_{\infty}\\|_{\mc{F}\sigma}^2 + \left[ \mathrm{vec} \left( \mathcal{P}_2^T \mathcal{M} R_v \mathcal{M} \mathcal{P}_2 \right) \right]^T \sigma \nonumber \end{aligned}$$ so that $$\begin{aligned} \label{Equ:ConvergenceAnalysis:SteadyStatePerformance_final} \boxed{ \mathbb{E}\\|\tilde{\bm{w}}_{\infty}\\|_{(I-\mc{F})\sigma}^2 \approx \left[ \mathrm{vec} \left( \mathcal{P}_2^T \mathcal{M} R_v \mathcal{M} \mathcal{P}_2 \right) \right]^T \sigma } \end{aligned}$$ Expression is a useful result: it allows us to derive several performance metrics through the proper selection of the free weighting parameter $\sigma$ (or $\Sigma$). First, to be able to evaluate steady-state performance metrics from , we need $(I-\mc{F})$ to be invertible, which is guaranteed by the stability of matrix $\mc{F}$ — see Appendix \[Appendix:Stability\_F\]. Given that $(I-\mc{F})$ is a stable matrix, we can now resort to and use it to evaluate various performance metrics by choosing proper weighting matrices $\Sigma$ (or $\sigma$), as it was done in [@Cattivelli10] for the mean-square-error estimation problem. For example, the MSE of any node $k$ can be obtained by computing $\mathbb{E}\\|\tilde{\bm{w}}_{\infty}\\|_{T}^2$ with a block weighting matrix $T$ that has an identity matrix at block $(k,k)$ and zeros elsewhere: $$\begin{aligned} \mathbb{E}\\|\tilde{\bm{w}}_{k,\infty}\\|^2 = \mathbb{E}\\|\tilde{\bm{w}}_{\infty}\\|_{T}^2 \end{aligned}$$ Denote the vectorized version of this matrix by $t_k$, i.e., $$\begin{aligned} t_k \triangleq \mathrm{vec}(\mathrm{diag}(e_k)\otimes I_M) \end{aligned}$$ where $e_k$ is a vector whose $k$th entry is one and zeros elsewhere. Then, if we select $\sigma$ in as $\sigma = (I-\mc{F})^{-1}t_k$, the term on the left-hand side becomes the desired $\mathbb{E}\\|\tilde{\bm{w}}_{k,\infty}\\|^2$ and MSE for node $k$ is therefore given by: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:MSD_k} \mathrm{MSE}_k \approx \left[ \mathrm{vec} \left( \mathcal{P}_2^T \mathcal{M} R_v \mathcal{M} \mathcal{P}_2 \right) \right]^T (I-\mc{F})^{-1} t_k \end{aligned}$$ This value for $\mathrm{MSE}_k$ is actually the $k$th entry of $\mathcal{W}_{\infty}$ defined as $$\begin{aligned} \mathcal{W}_\infty \triangleq \lim_{i\rightarrow\infty} \mathcal{W}_i \end{aligned}$$ Then, we arrive at an expression for $\mathcal{W}_{\infty}$ (as opposed to the bound for it in , as was explained earlier; expression is derived under the assumption of sufficiently small step-sizes): $$\begin{aligned} \label{Equ:ConvergenceAnalysis:W_infty} \!\!\boxed{ \mathcal{W}_{\infty} \!\approx\! \left\{\! I_N \!\otimes\! \left( \left[ \mathrm{vec} \left( \mathcal{P}_2^T \mathcal{M} R_v \mathcal{M} \mathcal{P}_2 \right) \right]^T (I\!\!-\!\!\mc{F})^{-1} \right)\! \right\} t } \end{aligned}$$ where $t=\mathrm{col}\{t_1,\ldots,t_N\}$. If we are interested in the network MSE, then the weighting matrix of $\mathbb{E}\\|\tilde{\bm{w}}_{\infty}\\|_{T}^2$ should be chosen as $T=I_{MN}/N$. Let $q$ denote the vectorized version of $I_{MN}$, i.e., $$\begin{aligned} q \triangleq \mathrm{vec}(I_{MN}) \end{aligned}$$ and select $\sigma$ in as $\sigma = (I\!-\!\mc{F})^{-1} q/N$. The network MSE is then given by $$\label{Equ:ConvergenceAnalysis:MSD_Network} \boxed{ \begin{split} \overline{\mathrm{MSE}} &\triangleq \frac{1}{N} \sum_{k=1}^N \mathrm{MSE}_k\\ &\approx \frac{1}{N} \left[ \mathrm{vec} \left( \mathcal{P}_2^T \mathcal{M} R_v \mathcal{M} \mathcal{P}_2 \right) \right]^T (I-\mc{F})^{-1} q \end{split} }$$ [ The approximate expressions and hold when the step-sizes are small enough so that holds. In the next section, we will see that they are consistent with the simulation results. ]{} Simulation Results {#Sec:Simulation} ================== In this section we illustrate the performance of the diffusion strategies – by considering two applications. We consider a randomly generated connected network topology with a cyclic path. There are a total of $N=10$ nodes in the network, and nodes are assumed connected when they are close enough geographically. In the simulations, we consider two applications: a regularized least-mean-squares estimation problem with sparse parameters, and a collaborative localization problem. Distributed Estimation with Sparse Data {#Sec:Simulation:L1} --------------------------------------- Assume each node $k$ has access to data $\{ \bm{U}_{k,i}, \bm{d}_{k,i}\}$, generated according to the following model: $$\begin{aligned} \label{Equ:Simulation:RegularizedLMS:LinearModel} \bm{d}_{k,i} = \bm{U}_{k,i} {w}^o + \bm{z}_{k,i} \end{aligned}$$ where $\{\bm{U}_{k,i}\}$ is a sequence of $K \times M$ i.i.d. Gaussian random matrices. The entries of each $\bm{U}_{k,i}$ have zero mean and unit variance, and $ \bm{z}_{k,i} \sim \mathcal{N}(0,\sigma_z^2I_K)$ is the measurement noise that is temporally and spatially white and is independent of $ \bm{U}_{l,j}$ for all $k,l,i,j$. Our objective is to estimate ${w}^o$ from the data set $\{ \bm{U}_{k,i}, \bm{d}_{k,i}\}$ in a distributed manner. In many applications, the vector ${w}^o$ is sparse such as $${w}^o = [ 1 \; 0 \; \ldots \; 0 \; 1 ]^T \in \mathbb{R}^M$$ One way to search for sparse solutions is to consider a global cost function of the following form [@diLorenzo2012icassp]: $$\begin{aligned} \label{Equ:Simulation:J_glob_L1RLS} J^{\mathrm{glob}}(w) = \sum_{l=1}^N \mathbb{E}\\| \bm{d}_{l,i} - \bm{U}_{l,i} w \\|_2^2 + \rho R(w) \end{aligned}$$ where $R(w)$ and $\rho$ are the regularization function and regularization factor, respectively. A popular choice is $R(w)=\\|w\\|_1$, which helps enforce sparsity and is convex [@tibshirani1996regression; @baraniuk2007compressive; @candes2008enhancing; @mateos2010distributed; @kopsinis2011online; @diLorenzo2012icassp]. However, this choice is non-differentiable, and we would need to apply sub-gradient methods [@poliak1987introduction pp.138–144] for a proper implementation. Instead, we use the following twice-differentiable approximation for $\\|w\\|_1$: $$\begin{aligned} \label{Equ:Simulation:R_w} R(w) = \sum_{m=1}^M \sqrt{[w]_m^2 + \epsilon^2} \end{aligned}$$ where $[w]_m$ denotes the $m$-th entry of $w$, and $\epsilon$ is a small number. We see that, as $\epsilon$ goes to zero, $R(w) \approx \\|w\\|_1$. Obviously, $R(w)$ is convex, and we can apply the diffusion algorithms to minimize in a distributed manner. To do so, we decompose the global cost into a sum of $N$ individual costs: $$\begin{aligned} \label{Equ:Simulation:J_k_L1RLS} J_l(w) = \mathbb{E}\\| \bm{d}_{l,i} - \bm{U}_{l,i} w \\|_2^2 + \frac{\rho}{N} R(w) \end{aligned}$$ for $l=1,\ldots,N$. Then, using algorithms and , each node $k$ would update its estimate of ${w}^o$ by using the gradient vectors of $\{J_l(w)\}_{l \in \mathcal{N}_k}$, which are given by: $$\begin{aligned} \nabla_w J_l(w) \;=\;& 2\mathbb{E}\left(\bm{U}_{l,i}^T\bm{U}_{l,i}\right) {w} - 2\mathbb{E}\left( \bm{U}_{l,i}^T \bm{d}_{l,i}\right) \nonumber\\ \;&+ \frac{\rho}{N} \nabla_w R(w) \end{aligned}$$ However, the nodes are assumed to have access to measurements $\{{U}_{l,i},{d}_{l,k}\}$ and not to the second-order moments $\mathbb{E}\big(\bm{U}_{l,i}^T \bm{U}_{l,i}\big)$ and $\mathbb{E}\big(\bm{U}_{l,i}^T\bm{d}_{l,i}\big)$. In this case, nodes can use the available measurements to approximate the gradient vectors in and as: $$\begin{aligned} \widehat{\nabla}_w J_l(w) = 2{U}_{l,i}^T \left[ {U}_{l,i} w \!-\! {d}_{l,i} \right] \!+\! \frac{\rho}{N} \nabla_w R(w) \end{aligned}$$ where $$\begin{aligned} \nabla_w R(w) = \begin{bmatrix} \displaystyle\frac{[w]_1}{\sqrt{[w]_1^2 + \epsilon^2}} & \cdots & \displaystyle\frac{[w]_M}{\sqrt{[w]_M^2 + \epsilon^2}} \end{bmatrix}^T \end{aligned}$$ In the simulation, we set $M=50$, $K=5$, $\sigma_v^2=1$, and ${w}^o=[1 \; 0 \; \ldots \; 0 \; 1]^T$. We apply both diffusion and incremental methods to solve the distributed learning problem, where the incremental approach [@bertsekas1997new; @nedic2001incremental; @rabbat2005quantized; @lopes2007incremental] uses the following construction to determine $\bm{w}_i$: - Start with $\bm{\psi}_{0,i} = \bm{w}_{i-1}$ at the node at the beginning of the incremental cycle. - Cycle through the nodes $k=1,\ldots,N$: $$\begin{aligned} \bm{\psi}_{k,i} = \bm{\psi}_{k-1,i} - \mu\widehat{\nabla}_w J_{k}(\bm{\psi}_{k-1,i}) \end{aligned}$$ - Set $\bm{w}_i \leftarrow \bm{\psi}_{N,i}$. - Repeat. The results are averaged over $100$ trials. The step-sizes for ATC, CTA and non-cooperative algorithms are set to $\mu=10^{-3}$, and the step-size for the incremental algorithm is set to $\mu=10^{-3}/N$. This is because the incremental algorithm cycles through all $N$ nodes every iteration. We therefore need to ensure the same convergence rate for both algorithms for a fair comparison [@takahashi2010diffusion]. For ATC and CTA strategies, we use simple averaging weights for the combination step, and for ATC and CTA with gradient exchange, we use Metropolis weights for $\{c_{l,k}\}$ to combine the gradients (see Table III in [@Cattivelli10] for the definitions of averaging weights and Metropolis weights). We use expression to evaluate the theoretical performance of the diffusion strategies. [ As a remark, expression gives the MSE with respect to the minimizer of the cost $J^{\mathrm{glob}}(w)$ in . In this example, the minimizer of the cost , denoted as $\hat{w}^o$, is biased away from the model parameter $w^o$ in when the regularization factor $\gamma \neq 0$. To evaluate the theoretical MSE with respect to $w^o$, we use $$\begin{aligned} \overline{\mathrm{MSD}} &= \lim_{i \rightarrow \infty} \frac{1}{N} \sum_{k=1}^N \E \\| w^o-\bm{w}_{k,i} \\|^2 \nonumber\\ \label{Equ:Simulation:L1_MSD_modified} &= \E \\| w^o - \hat{w}^o \\|^2 + \lim_{i \rightarrow \infty} \frac{1}{N} \sum_{k=1}^N \E \\| \hat{w}^o-\bm{w}_{k,i} \\|^2 \end{aligned}$$ where the second term in can be evaluated by expression with $w^o$ replaced by $\hat{w}^o$. Moreover, in the derivation of , we used the fact that $\lim_{i\rightarrow \infty}\E(\hat{w}^o-\bm{w}_{k,i})=0$ to eliminate the cross term, and this result is due to with $w^o$ there replaced by $\hat{w}^o$. ]{}Fig. \[fig:Simulation:LearningCurve\] shows the learning curves for different algorithms for $\gamma=2$ and $\epsilon=10^{-3}$. We see that the diffusion and incremental schemes have similar performance, and both of them have about $10$ dB gain over the non-cooperation case. To examine the impact of the parameter $\epsilon$ and the regularization factor $\gamma$, we show the steady-state MSE for different values of $\gamma$ and $\epsilon$ in Fig. \[fig:Simulation:MSDvsGamma\]. When $\epsilon$ is small ($\epsilon=10^{-2}$), adding a reasonable regularization ($\gamma=1\sim4$) decreases the steady-state MSE. However, when $\epsilon$ is large ($\epsilon=1$), expression is no longer a good approximation for $\\|w\\|_1$, and regularization does not improve the MSE. Distributed Collaborative Localization {#Sec:Simulation:Localization} -------------------------------------- The previous example deals with a convex cost . Now, we consider a localization problem that has a non-convex cost function and apply the same diffusion strategies to its solution. Assume each node is interested in locating a common target located at $w^o=[0 \; 0]^T$. Each node $k$ knows its position $x_k$ and has a noisy measurement of the squared distance to the target: $$\begin{aligned} \label{Equ:Simulation:Localization_datamodel} \bm{d}_{k}(i) = \\|w^o - x_k\\|^2 + \bm{z}_k(i), \quad k=1,2,\ldots,N \nonumber \end{aligned}$$ where $\bm{z}_k(i) \sim \mathcal{N}(0,\sigma_{z,k}^2)$ is the measurement noise of node $k$ at time $i$. The component cost function $J_k(w)$ at node $k$ is chosen as $$\begin{aligned} J_k(w) = \frac{1}{4}\mathbb{E}\left\|\bm{d}_k(i) - \\|w - x_k\\|^2 \right\|^2 \end{aligned}$$ where we multiply by $1/4$ here to eliminate a factor of $4$ that will otherwise appear in the gradient. If each node $k$ minimizes $J_k(w)$ individually, it is not possible to solve for $w^o$. Therefore, we use information from other nodes, and instead seek to minimize the following global cost: $$\begin{aligned} \label{Equ:Simulation:Example1:J_glob} J^{\mathrm{glob}}(w) = \frac{1}{4} \sum_{k=1}^N \mathbb{E}\left\|\bm{d}_k(i) - \\|w - x_k\\|^2 \right\|^2 \end{aligned}$$ This problem arises, for example, in cellular communication systems, where multiple base-stations are interested in locating users using the measured distances between themselves and the user. Diffusion algorithms and can be applied to solve the problem in a distributed manner. Each node $k$ would update its estimate of ${w}^o$ by using the gradient vectors of $\{J_l(w)\}_{l \in \mathcal{N}_k}$, which are given by: $$\begin{aligned} \nabla_w J_l(w) = -\mathbb{E}\bm{d}_l(i) \; (w-x_l) +\\|w-x_l\\|^2 (w-x_l) \end{aligned}$$ However, the nodes are assumed to have access to measurements $\{{d}_{l}(i),x_l\}$ and not to $\mathbb{E}\bm{d}_l(i)$. In this case, nodes can use the available measurements to approximate the gradient vectors in and as: $$\begin{aligned} \widehat{\nabla}_w J_l(w) = -{d}_l(i) (w-x_l) + \\|w-x_l\\|^2 (w-x_l) \end{aligned}$$ If we do not exchange the local gradients with neighbors, i.e., if we set $S=I$, then the base-stations only share the local estimates of the target position $w^o$ with their neighbors (no exchange of $\{x_l\}_{l \in \mathcal{N}_k}$). We first simulate the stationary case, where the target stays at $w^o$. In Fig. \[fig:Fig\_Localization\_LearningCurve\_MSD\_Truewo\], we show the MSE curves for non-cooperative, ATC, CTA, and incremental algorithms. The noise variance is set to $\sigma_{z,k}^2=1$. We set the step-sizes to $\mu=0.0025/N$ for the incremental algorithm, and $\mu= 0.0025$ for other algorithms. For ATC and CTA strategies, we use simple averaging for the combination step $\{a_{l,k}\}$, and for ATC and CTA with gradient exchange, we use Metropolis weights for $\{c_{l,k}\}$ to combine the gradients. The performance of CTA and ATC algorithms are close to each other, and both of them are close to the incremental scheme. In Fig. \[fig:Fig\_Localization\_MSDvsMU\_Truewo\], we show the steady state MSE with respect to different values of $\mu$. [ We also use expression to evaluate the theoretical performance of the diffusion strategies. ]{} As the step-size becomes small, the performances of diffusion and incremental algorithms are close, and the MSE decreases as $\mu$ decreases. Furthermore, we see that exchanging only local estimates ($S=I$) is enough for localization, compared to the case of exchanging both local estimates and gradients ($S=C$). Next, we apply the algorithms to a non-stationary scenario, where the target moves along a trajectory, as shown in Fig. \[fig:Fig\_Localization\_TrackingTrajectory\]. The step-size is set to $\mu=0.01$ for diffusion algorithms, and to $\mu=0.01/N$ for the incremental approach. To see the advantage of using a constant step-size for continuous tracking, we also simulate the vanishing step-size version of the algorithm from [@ram2010distributed; @srivastava2011distributed] ($\mu_{k,i} = 0.01/i$). The diffusion algorithms track well the target but not the non-cooperative algorithm and the algorithm from [@ram2010distributed; @srivastava2011distributed], because a decaying step-size is not helpful for tracking. The tracking performance is shown in Fig. \[fig:Fig\_Localization\_TrackingMSD\]. Conclusion {#Sec:Conclusion} ========== This paper proposed diffusion adaptation strategies to optimize global cost functions over a network of nodes, where the cost consists of several components. Diffusion adaptation allows the nodes to solve the distributed optimization problem via local interaction and online learning. We used gradient approximations and constant step-sizes to endow the networks with continuous learning and tracking abilities. We analyzed the mean-square-error performance of the algorithms in some detail, including their transient and steady-state behavior. Finally, we applied the scheme to two examples: distributed sparse parameter estimation and distributed localization. Compared to incremental methods, diffusion strategies do not require a cyclic path over the nodes, which makes them more robust to node and link failure. Proof of Mean-Square Stability {#Appendix:Proof_MeanSquaredStability} ============================== Taking the $\infty-$norm of both sides of , we obtain [ $$\begin{aligned} \\|\mathcal{W}_i\\|_{\infty} \;\le&\; \\|P_2^T \Gamma P_1^T\\|_{\infty} \!\cdot\! \\|\mathcal{W}_{i-1}\\|_{\infty} \!+\! \sigma_v^2 \\|S\\|_1^2 \!\cdot\! \\|P_2^T\\|_{\infty} \!\cdot\! \\|\Omega\\|_{\infty}^2 \nonumber\\ \;\le&\; \\|P_2^T\\|_{\infty} \cdot \\|\Gamma \\|_{\infty} \cdot \\|P_1^T\\|_{\infty} \cdot \\|\mathcal{W}_{i-1}\\|_{\infty} \nonumber\\ &\;+ \sigma_v^2 \\|S\\|_1^2 \cdot \\|P_2^T\\|_{\infty} \cdot \\|\Omega\\|_{\infty}^2 \nonumber\\ \label{Equ:Appendix:W_i_norm_inf_ineq} \;=&\; \\|\Gamma \\|_{\infty} \!\cdot\! \\|\mathcal{W}_{i-1}\\|_{\infty} \!+\! \big(\max_{1 \le k \le N} \mu_k^2\big) \!\cdot\! \sigma_v^2 \\|S\\|_1^2 \end{aligned}$$ ]{}where we used the fact that $\\|P_1^T\\|_{\infty}=\\|P_2^T\\|_{\infty}=1$ because each row of $P_1^T$ and $P_2^T$ sums up to one. Moreover, from , we have $$\begin{aligned} \label{Equ:ConvergenceAnalysis:gamma} \\|\Gamma\\|_{\infty} = \max_{1 \le k \le N} (\gamma_k^2+\mu_k^2\alpha\\|S\\|_1^2) \end{aligned}$$ Iterating , we obtain $$\begin{aligned} \\|\mathcal{W}_i\\|_{\infty} \;\le\;& \\|\Gamma\\|_{\infty}^i \cdot \\|\mathcal{W}_0\\|_{\infty} \nonumber\\ \label{Equ:ConvergenceAnalysis:TheoremProof:W_i_Expr} \;&+ \big(\max_{1 \le k \le N} \mu_k^2\big) \cdot \sigma_v^2 \\|S\\|_1^2 \sum_{j=0}^{i-1} \\|\Gamma\\|_{\infty}^j \end{aligned}$$ We are going to show further ahead that condition guarantees $\\|\Gamma\\|_{\infty}<1$. Now, given that $\\|\Gamma\\|_{\infty}<1$, the first term on the right hand side of converges to zero as $i \rightarrow \infty$, and the second term on the right-hand side of converges to: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:Convergence_SecondTerm_Entry_k} \lim_{i \rightarrow \infty} \sigma_v^2 \\|S\\|_1^2 \sum_{j=0}^{i-1} \\|\Gamma\\|_{\infty}^j = \frac{\sigma_v^2\\|S\\|_1^2}{1-\\|\Gamma\\|_{\infty}} \end{aligned}$$ Therefore, we establish as follows: $$\begin{aligned} \limsup_{i\rightarrow \infty} \\|\mathcal{W}_{i}\\|_{\infty} &\le \big(\max_{1 \le k \le N} \mu_k^2\big) \cdot \frac{\sigma_v^2\\|S\\|_1^2}{1-\\|\Gamma\\|_{\infty}} \nonumber\\ \label{Equ:Appendix:lisup_W_i_inf_norm_bound} &= \frac{ \displaystyle \Big(\max_{1\le k \le N} \mu_k^2\Big) \cdot \\|S\\|_1^2\sigma_v^2 } { \displaystyle 1 - \max_{1 \le k \le N} (\gamma_k^2 + \mu_k^2 \alpha\\|S\\|_1^2) } \end{aligned}$$ The only fact that remains to prove is to show that ensures $\\|\Gamma\\|_{\infty}<1$. From , we see that the condition $\\|\Gamma\\|_{\infty}<1$ is equivalent to requiring: $$\begin{aligned} \label{Equ:ConvergenceAnalysis:gamma_bound1} & \gamma_k^2 + \mu_k^2 \alpha\\|S\\|_1^2 < 1, \qquad k=1,\ldots,N. \end{aligned}$$ Then, using , this is equivalent to: $$\begin{aligned} \label{Equ:Appendix:Condition_Stepsize_intermediate_original1} &\big(1-\mu_k \sum_{l=1}^N s_{l,k} \lambda_{l,\max}\big)^2 + \mu_k^2 \alpha\\|S\\|_1^2 < 1 \qquad \\ \label{Equ:Appendix:Condition_Stepsize_intermediate_original2} & \big(1-\mu_k \sum_{l=1}^N s_{l,k} \lambda_{l,\min}\big)^2 + \mu_k^2 \alpha \\|S\\|_1^2< 1 \end{aligned}$$ for $k=1,\ldots,N$. Recalling the definitions for $\sigma_{k,\max}$ and $\sigma_{k,\min}$ in and solving these two quadratic inequalities with respect to $\mu_k$, we arrive at: $$\begin{aligned} &0 < \mu_k < \frac{2\sigma_{k,\max}}{\sigma_{k,\max}^2 + \alpha\\|S\\|_1^2}, \qquad 0 < \mu_k < \frac{2\sigma_{k,\min}}{\sigma_{k,\min}^2 + \alpha\\|S\\|_1^2} \nonumber \end{aligned}$$ and we are led to . Block Maximum Norm of a Matrix {#Appendix:Proof_Lemma_BMNorm_BlockDiagonalMatrix} ============================== Consider a block matrix $X$ with blocks of size $M\times M$ each. Its block maximum norm is defined as[@takahashi2010diffusion]: $$\begin{aligned} \label{Equ:Appendix:BMNorm:Def_BMNorm_Matrix} \\|X\\|_{b,\infty} \triangleq \max_{x \neq 0} \frac{\\|X x\\|_{b,\infty}}{\\|x\\|_{b,\infty}} \end{aligned}$$ where the block maximum norm of a vector $x \triangleq \mathrm{col}\{x_1,\ldots,x_N\}$, formed by stacking $N$ vectors of size $M$ each on top of each other, is defined as[@takahashi2010diffusion]: $$\begin{aligned} \label{Equ:Appendix:BMNorm:Def_BMNorm_Vector} \\|x\\|_{b,\infty} \triangleq \max_{1\le k \le N} \\|x_k\\| \end{aligned}$$ where $\\|\cdot\\|$ denotes the Euclidean norm of its vector argument. \[Lemma:BMNorm\_BlockDiagonalMatrix\] If a block diagonal matrix $X \triangleq \mathrm{diag}\{X_1,\ldots,X_N\}\in \mathbb{R}^{NM \times NM}$ consists of $N$ blocks along the diagonal with dimension $M \times M$ each, then the block maximum norm of $X$ is bounded as $$\begin{aligned} \label{Equ:ConvergenceAnalysis:Lemma:BMNorm_BlockDiagonalMatrix_Bound} \\|X\\|_{b,\infty} \le \max_{1\le k \le N} \\|X_k\\| \end{aligned}$$ in terms of the $2$-induced norms of $\{X_k\}$ (largest singular values). Moreover, if $X$ is symmetric, then equality holds in . Note that $X x \!=\! \mathrm{col}\{X_1 x_1,\!\ldots,\!X_Nx_N\}$. Evaluating the block maximum norm of vector $X x$ leads to $$\begin{aligned} \\|X x\\|_{b,\infty} &= \max_{1 \le k \le N} \\| X_k x_k \\| \nonumber\\ &\le \max_{1 \le k \le N} \\| X_k \\| \cdot \\|x_k \\| \nonumber\\ \label{Equ:Appendix:BMNorm:BMNorm_Xx} &\le \max_{1 \le k \le N} \\| X_k \\| \cdot \max_{1 \le k \le N}\\|x_k \\| \end{aligned}$$ Substituting and into , we establish as $$\begin{aligned} \\|X\\|_{b,\infty} &\triangleq \max_{x \neq 0} \frac{\\|X x\\|_{b,\infty}}{\\|x\\|_{b,\infty}} \nonumber\\ &\le \max_{x \neq 0} \frac{\max_{1 \le k \le N} \\| X_k \\| \cdot \max_{1 \le k \le N}\\|x_k \\|} {\max_{1\le k \le N} \\|x_k\\|} \nonumber\\ \label{Equ:Appendix:BMNorm:BMNorm_X_bound} &= \max_{1 \le k \le N} \\| X_k \\| \end{aligned}$$ Next, we prove that, if all the diagonal blocks of $X$ are symmetric, then equality should hold in . To do this, we only need to show that there exists an $x_0 \neq 0$, such that $$\begin{aligned} \label{Equ:Appendix:BMNorm:BMNorm_X_attainablepoint} \frac{\\|X x_0\\|_{b,\infty}}{\\|x_0\\|_{b,\infty}} = \max_{1 \le k \le N} \\| X_k \\| \end{aligned}$$ which would mean that $$\begin{aligned} \\|X\\|_{b,\infty} &\triangleq \max_{x \neq 0} \frac{\\|X x\\|_{b,\infty}}{\\|x\\|_{b,\infty}} \nonumber\\ &\ge \frac{\\|X x_0\\|_{b,\infty}}{\\|x_0\\|_{b,\infty}} \nonumber\\ \label{Equ:Appendix:BMNorm:BMNorm_X_lowerbound} &= \max_{1 \le k \le N} \\| X_k \\| \end{aligned}$$ Then, combining inequalities and , we would obtain desired equality that $$\begin{aligned} \label{Equ:Appendix:X_Xb_bmaxNorm} \\|X\\|_{b,\infty} = \max_{1 \le k \le N} \\|X_k\\| \end{aligned}$$ when $X$ is block diagonal and symmetric. Thus, without loss of generality, assume the maximum in is achieved by $X_1$, i.e., $$\begin{aligned} \displaystyle\max_{1 \le k \le N} \\|X_k\\| = \\|X_1\\| \nonumber \end{aligned}$$ For a symmetric $X_k$, its 2-induced norm $\\|X_k\\|$ (defined as the largest singular value of $X_k$) coincides with the spectral radius of $X_k$. Let $\lambda_{0}$ denote the eigenvalue of $X_1$ of largest magnitude, with the corresponding right eigenvector given by $z_0$. Then, $$\begin{aligned} \max_{1 \le k \le N} \\|X_k\\| = \|\lambda_0\|, \qquad X_1 z_0 = \lambda_0 z_0 \nonumber \end{aligned}$$ We select $x_0=\mathrm{col}\{z_0,0,\ldots,0\}$. Then, we establish by: $$\begin{aligned} \frac{\\|X x_0\\|_{b,\infty}}{\\|x_0\\|_{b,\infty}} &= \frac{\\|\mathrm{col}\{X_1 z_0,0,\ldots,0\}\\|_{b,\infty}} {\\|\mathrm{col}\{z_0,0,\ldots,0\}\\|_{b,\infty}} \nonumber\\ &= \frac{\\|X_1 z_0\\|} {\\|z_0\\|} = \frac{\\|\lambda_0 z_0\\|} {\\|z_0\\|} = \|\lambda_0\| = \max_{1 \le k \le N} \\|X_k\\| \nonumber \end{aligned}$$ Stability of $\mathcal{B}$ and $\mc{F}$ {#Appendix:Stability_F} ======================================= Recall the definitions of the matrices $\mc{B}$ and $\mc{F}$ from and : $$\begin{aligned} \label{Equ:ConvergenceAnalysis:SpectralRadius_B} \mc{B} &= \mc{P}_2^T [I_{MN} - \mc{M}\mc{D}_{\infty}]\mc{P}_1^T \\ \mc{F} &= \big(\mc{P}_1 [I_{MN} - \mc{M}\mc{D}_{\infty}]\mc{P}_2\big) \otimes \big(\mc{P}_1 [I_{MN} - \mc{M}\mc{D}_{\infty}]\mc{P}_2\big) \nonumber\\ \label{Equ:ConvergenceAnalysis:SpectralRadius_F} &= \mc{B}^T \otimes \mc{B}^T \end{aligned}$$ From –, we obtain (see Theorem 13.12 from [@laub2005matrix p.141]): $$\begin{aligned} \rho(\mc{F}) = \rho(\mc{B}^T \otimes \mc{B}^T) = [\rho(\mc{B}^T)]^2 = [\rho(\mc{B})]^2 \end{aligned}$$ where $\rho(\cdot)$ denotes the spectral radius of its matrix argument. Therefore, the stability of the matrix $\mc{F}$ is equivalent to the stability of the matrix $\mc{B}$, and we only need to examine the stability of $\mc{B}$. Now note that the block maximum norm (see the definition in Appendix \[Appendix:Proof\_Lemma\_BMNorm\_BlockDiagonalMatrix\]) of the matrix $\mc{B}$ satisfies $$\begin{aligned} \\|\mc{B}\\|_{b,\infty} \le \\|I_{MN} - \mc{M}\mc{D}_{\infty}\\|_{b,\infty} \end{aligned}$$ since the block maximum norms of $\mc{P}_1$ and $\mc{P}_2$ are one (see [@takahashi2010diffusion p.4801]): $$\begin{aligned} &\left\\|\mathcal{P}_1^T\right\\|_{b,\infty} = 1, \qquad \left\\|\mathcal{P}_2^T\right\\|_{b,\infty} = 1 \end{aligned}$$ Moreover, by noting that the spectral radius of a matrix is upper bounded by any matrix norm (Theorem 5.6.9, [@horn1990matrix p.297]) and that $I_{MN} - \mc{M}\mc{D}_{\infty}$ is symmetric and block diagonal, we have $$\begin{aligned} \label{Equ:Appendix:rho_B_bound} \rho(\mc{B}) \le \\|I_{MN} - \mc{M}\mc{D}_{\infty}\\|_{b,\infty} = \rho(I_{MN} - \mc{M}\mc{D}_{\infty}) \end{aligned}$$ Therefore, the stability of $\mc{B}$ is guaranteed by the stability of $I_{MN} - \mc{M}\mc{D}_{\infty}$. Next, we call upon the following lemma to bound $\left\\|I_{MN}\!-\!\mathcal{M}{\mathcal{D}}_{\infty}\right\\|_{b,\infty}$. \[Lemma:I\_MDinf\_Bound\] It holds that the matrix $\mathcal{D}_{\infty}$ defined in satisfies $$\begin{aligned} \label{Equ:ConvergenceAnalysis:I_MDinf_Bound} \left\\| I_{MN}\!-\!\mathcal{M}{\mathcal{D}}_{\infty} \right\\|_{b,\infty} \le \max_{1 \le k \le N} \gamma_k \end{aligned}$$ where $\gamma_k$ is defined in . Since $\mathcal{D}_{\infty}$ is block diagonal and symmetric, $I_{MN}-\mathcal{M}\mathcal{D}_{\infty}$ is also block diagonal with blocks $\{I_{M}\!-\!\mu_k{\mathcal{D}}_{k,\infty}\}$, where ${\mathcal{D}}_{k,\infty}$ denotes the $k$th diagonal block of $\mathcal{D}_{\infty}$. Then, from in Lemma \[Lemma:BMNorm\_BlockDiagonalMatrix\] in Appendix \[Appendix:Proof\_Lemma\_BMNorm\_BlockDiagonalMatrix\], it holds that $$\begin{aligned} \label{Equ:ConvergenceAnalysis:I_MDinf_NoriyukiFact} \left\\| I_{MN}\!-\!\mathcal{M}{\mathcal{D}}_{\infty} \right\\|_{b,\infty} &= \max_{1 \le k \le N} \left\\| I_{M}\!-\!\mu_k{\mathcal{D}}_{k,\infty} \right\\| \end{aligned}$$ By the definition of $\mathcal{D}_{\infty}$ in , and using condition from Assumption \[Assumption:Hessian\], we have $$\begin{aligned} \Big(\sum_{l=1}^N s_{l,k} \lambda_{l,\min}\Big) \cdot I_M \le {\mathcal{D}}_{k,\infty} \le \Big(\sum_{l=1}^N s_{l,k} \lambda_{l,\max}\Big) \cdot I_M \nonumber \end{aligned}$$ which implies that $$\begin{aligned} \label{Equ:Appendix:Bound_I_muD_k_Lower} &I_M - \mu_k \mc{D}_{k,\infty} \ge \Big(1-\mu_k\sum_{l=1}^N s_{l,k} \lambda_{l,\max}\Big) \cdot I_M \\ \label{Equ:Appendix:Bound_I_muD_k_Upper} &I_M - \mu_k \mc{D}_{k,\infty} \le \Big(1- \mu_k\sum_{l=1}^N s_{l,k} \lambda_{l,\min}\Big) \cdot I_M \end{aligned}$$ Thus, $\\| I_M \!-\! \mu_k \mathcal{D}_{k,\infty} \\| \!\le\! \gamma_k$. Substituting into , we get . Substituting into , we get: $$\begin{aligned} \rho(\mc{B}) \le \max_{1 \le k \le N} \gamma_k \end{aligned}$$ As long as $\displaystyle\max_{1 \le k \le N} \gamma_k < 1$, then all the eigenvalues of $\mc{B}$ will lie within the unit circle. By the definition of $\gamma_k$ in , this is equivalent to requiring $$\begin{aligned} \|1-\mu_k \sigma_{k,\max}\| < 1, \qquad \|1-\mu_k \sigma_{k,\min}\| < 1 \nonumber \end{aligned}$$ for $k=1,\ldots,N$, where $\sigma_{k,\max}$ and $\sigma_{k,\min}$ are defined in . These conditions are satisfied if we choose $\mu_k$ such that $$\begin{aligned} 0 < \mu_k < {2}/{\sigma_{k,\max}}, \qquad k=1,\ldots,N \end{aligned}$$ which is obviously guaranteed for sufficiently small step-sizes (and also by condition ). [^1]: Manuscript received October 30, 2011; revised March 15, 2012. This work was supported in part by NSF grants CCF-1011918 and CCF-0942936. Preliminary results related to this work are reported in the conference presentations[@chen2011diffusionOpt] and [@chen2011MSE]. [^2]: The authors are with Department of Electrical Engineering, University of California, Los Angeles, CA 90095. Email: {jshchen, sayed}@ee.ucla.edu.
--- abstract: 'We extend the notions of conditioned and controlled invariant spaces to linear dynamical systems over the max-plus or tropical semiring. We establish a duality theorem relating both notions, which we use to construct dynamic observers. These are useful in situations in which some of the system coefficients may vary within certain intervals. The results are illustrated by an application to a manufacturing system.' address: - 'Laboratoire Ampère, UMR CNRS 5005, INSA de Lyon, 20 Avenue Albert Einstein, 69621 Villeurbanne Cedex, France.' - 'INRIA and Centre de Mathématiques Appliquées, École Polytechnique. Postal address: CMAP, École Polytechnique, 91128 Palaiseau Cédex, France. Tèl: +33 1 69 33 46 13, Fax: +33 1 39 63 57 86' - 'CONICET. Postal address: Instituto de Matemática “Beppo Levi”, Universidad Nacional de Rosario, Avenida Pellegrini 250, 2000 Rosario, Argentina.' - 'Institut de Recherche en Communications et en Cybernétique de Nantes (IRCCyN), UMR CNRS 6597, 1 rue de la Noe BP 92 101, 44 321 Nantes Cedex 3, France.' author: - Michael Di Loreto - Stéphane Gaubert - 'Ricardo D. Katz$^$' - 'Jean-Jacques Loiseau' title: 'Duality between invariant spaces for max-plus linear discrete event systems' --- Introduction ============ The use of geometric-type techniques when dealing with linear dynamical systems, following a line of work initiated by Basile and Marro [@BasMar69] and Morse and Wonham [@MorWon70; @MorWon71], has provided important insights to system-theoretic and control-synthesis problems. In particular, this kind of techniques lead to elegant solutions to many control problems, such as the disturbance decoupling problem and the model matching problem, to quote but a few. To achieve this, a geometric approach, using certain linear spaces known as conditioned and controlled invariant spaces, has been developed (see [@wonham; @BasMar91] and the references therein). In the classical geometric approach to the theory of linear dynamical systems, the scalars belong to a field, or at least to a ring. However, the case where the scalars belong to a semiring is also of practical interest. In particular, linear dynamical systems with coefficients in the max-plus or tropical semiring, and other similar algebraic structures sometimes referred to as “dioids” or “idempotent semirings”, arise in the modeling and analysis of some manufacturing systems following the approach initiated by Cohen, Dubois, Quadrat and Viot [@cohen85a] (a systematic account can be found in the book by Baccelli, Cohen, Quadrat and Olsder [@bcoq]). More recent developments of the max-plus approach include the “network calculus” of Le Boudec and Thiran [@leboudec], which can be used to assess certain issues concerning the quality of service in telecommunication networks, or an application to train networks by Heidergott, Olsder and van der Woude [@how06]. All these works provide important examples of discrete event dynamical systems subject to synchronization constraints that can be described by max-plus linear dynamical systems. Several results from linear system theory have been extended to the max-plus algebra framework, such as transfer series methods or the connection between spectral theory and stability questions (see [@cohen89a]). In view of the potentiality of the theory of linear dynamical systems over the max-plus semiring, it is also tempting to generalize the geometric approach to these systems, a problem which was raised by Cohen, Gaubert and Quadrat in [@ccggq99]. However, this generalization is not straightforward, because many concepts and results must be properly redefined and adapted. Similar difficulties were already met in the case of linear dynamical systems over rings (for which we refer to the works of Hautus [@hautus82] and of Conte and Perdon [@conte94; @conte95]), and in the case of linear systems of infinite dimensions on Hilbert spaces (see for instance Curtain [@Cu86]). The difficulties are in two directions. In the first place, there are algorithmic issues. The concepts of conditioned and controlled invariant subspaces (or submodules) are no longer dual, and the convergence of the algorithms of the geometric approach is not guaranteed. Then, the computation of these spaces may be difficult or impossible in general. In the second place, the connection between invariance and control or estimation problems is more difficult to establish. Hypothesis must be added to overcome these problems. In this paper, we show that some of the main results of the geometric approach do carry over to the max-plus case. A first work in this direction was developed by Katz [@katz07], who studied the $(A,B)$-invariant spaces of max-plus linear systems providing solutions to some control problems. The max-plus analogue of the disturbance decoupling problem has been studied by Lhommeau et al. [@LHC03; @Lhommeau] making use of invariant sets in the spirit of the classical geometric approach. More precisely, principal ideal invariant sets were considered, which is an elegant solution to the algorithmic issues, leading to effective algorithms at the price of a restrictive assumption. However, these works differ from [@katz07] in the fact that they are based on residuation theory and transfer series techniques. The present paper is devoted to studying the max-plus analogues of conditioned and controlled invariance and the duality between them. In the classical linear system theory, conditioned invariant spaces are defined in terms of the kernel of the output matrix. In the semiring case, the usual definition of the kernel of a matrix is not pertinent because it is usually trivial. In their places, we consider a natural extension of kernels, the congruences, which are equivalence relations with a semimodule structure (see [@CGQ96a; @CGQ97a; @gk08u]). Instead of considering, for instance, situations in which the perturbed state $x'$ of the system is the sum of the unperturbed state $x$ and of a noise $w$, we require the states $x$ and $x'$ to belong to the same equivalence class modulo a relation (congruence) which represents the perturbation. Indeed, in the max-plus setting, considering only additive perturbations would be an important restriction, because adding only means delaying events, whereas congruences allow us to model situations in which the perturbation drives some events to occur at an earlier time. By a systematic application of these ideas, we generalize the main notions of the classical geometric approach. However, this generalization raises new theoretical as well as algorithmic issues, because we have to work with congruences (sets of pairs of vectors), rather than with linear spaces (sets of vectors), leading to a general “doubling” of the dimension. Considering max-plus linear systems subject to perturbations modeled by congruences actually leads to an extension of the modeling power of the max-plus approach, allowing one to take certain classes of constraints or uncertainties into account. For instance, we show that max-plus linear dynamical systems with uncertain holding times can be modeled in this way, if these times are assumed to belong to certain intervals. For this kind of perturbed systems, the minimal conditioned invariant space containing the perturbation can be interpreted as the “best information” that can be learned on the state of the system from a given observation and initial state. Our final result (Theorem \[TheoObserver\]) shows that this “optimal information” on the perturbed state of the system can be reconstructed from the output by means of a dynamic observer. In order to compute this dynamic observer, we extend to the max-plus algebra framework the classical fixed point algorithms used to compute the minimal conditioned invariant and the maximal controlled invariant spaces containing and contained, respectively, in a given space. Our main result, Theorem \[TheoCondContr\], establishes a duality between conditioned and controlled invariant spaces. This allows us to reduce the computation of minimal conditioned invariant spaces to the computation of maximal controlled invariant spaces, and in this way to reduce algorithmic problems concerning congruences to algorithmic problems concerning semimodules, which are easier to handle. Thus, this duality theorem solves the previously mentioned algorithmic difficulties related to the “doubling” of the dimension. Then, Proposition \[PropFiniteVolume\] identifies conditions which guarantee that the fixed point algorithm used to compute the minimal conditioned invariant congruence containing a given congruence terminates in a finite number of steps. Its proof is based on a finite chain condition, which is valid thanks to the finiteness and integrity assumptions made in the proposition. Under more general circumstances, the max-plus case shows difficulties which seem somehow reminiscent of the ones encountered in the theory of invariant spaces for linear systems over non-Noetherian rings. Recall that the computation of such spaces is still an open problem in the case of general rings. This paper is organized as follows. The next section is devoted to recalling basic definitions and results on max-plus algebra which will be used throughout this paper. In Section \[SCondCont\] we introduce the max-plus analogues of conditioned and controlled invariant spaces and extend to the max-plus algebra framework the classical fixed point algorithms used for their computation. Duality between conditioned and controlled invariance is investigated in Section \[SDuality\] but previously, in Section \[SOrthogonal\], it is convenient to introduce the notions of orthogonal of semimodules and congruences and study their properties. Finally, in Section \[SApplication\], we illustrate the results presented here with their application to a manufacturing system. Preliminaries {#SPreliminaries} ============= The max-plus semiring, ${{\mathbb{R}}_{\max}}$, is the set ${\mathbb{R}}\cup\{-\infty\}$ equipped with the addition $(a,b)\mapsto \max(a,b)$ and the multiplication $(a,b)\mapsto a+b$. To emphasize the semiring structure, we write $a\oplus b:=\max(a,b)$ and $ab:=a+b$. For $p,q \in {\mathbb{N}}$, we denote by ${{\mathbb{R}}_{\max}}^{p\times q}$ the set of all $p$ times $q$ matrices over the max-plus semiring. As usual, if $E\in {{\mathbb{R}}_{\max}}^{p\times q}$, $E_{ij}$ denotes the element of $E$ in its $i$-th row and $j$-th column, and $E^t\in {{\mathbb{R}}_{\max}}^{q\times p}$ the transposed of $E$. The semiring operations are extended in the natural way to matrices over the max-plus semiring: $(E\oplus F)_{ij}:=E_{ij}\oplus F_{ij}$, $(EF)_{ij}:=\oplus_k E_{ik} F_{kj}$ and $(\lambda E)_{ij}:=\lambda E_{ij}$ for all $i,j$, where $E$ and $F$ are matrices of compatible dimension and $\lambda \in {{\mathbb{R}}_{\max}}$. For $E\in {{\mathbb{R}}_{\max}}^{p\times q}$, we denote by ${\mbox{\rm Im}\,}E:={\{Ex\mid\,x\in {{\mathbb{R}}_{\max}}^q\}}$ the image of $E$. We usually denote by ${\varepsilon}:=-\infty$ the neutral element for addition as well as the null matrix of any dimension. We equip ${{\mathbb{R}}_{\max}}$ with the usual topology which can be defined by the metric: $d(a,b):=\|\exp(a)-\exp(b)\|$. The Cartesian product ${{\mathbb{R}}_{\max}}^n$ is equipped with the product topology. Note that the semiring operations are continuous with respect to this topology. The analogues of vector spaces or modules obtained by replacing the field or ring of scalars by an idempotent semiring are called [semimodules]{} or [idempotent spaces]{}. They have been studied by several authors with different motivations (see for example [@zimmerman77; @maslov92; @GargKumar95; @litvinov00; @cgq02]). Here, we will only consider subsemimodules of the Cartesian product ${{\mathbb{R}}_{\max}}^n$, also known as [max-plus cones]{}, which are subsets ${\mathcal{K}}$ of ${{\mathbb{R}}_{\max}}^n$ stable by max-plus linear combinations, meaning that $$\begin{aligned} \lambda x\oplus \mu y \in {\mathcal{K}}\label{e-stable}\end{aligned}$$ for all $x,y\in {\mathcal{K}}$ and $\lambda,\mu \in {{\mathbb{R}}_{\max}}$. We denote by ${{\text{\rm span}}\,}{\mathcal{S}}$ the smallest semimodule containing a subset ${\mathcal{S}}$ of ${{\mathbb{R}}_{\max}}^n$. Therefore, ${{\text{\rm span}}\,}{\mathcal{S}}$ is the set of all max-plus linear combinations of finitely many elements of ${\mathcal{S}}$. A semimodule ${\mathcal{K}}$ is said to be finitely generated, if there exists a finite set ${\mathcal{S}}$ such that ${\mathcal{K}}={{\text{\rm span}}\,}{\mathcal{S}}$, which also means that ${\mathcal{K}}={\mbox{\rm Im}\,}E$ for some matrix $E$. We shall need the following lemma. \[FGClosed\] Finitely generated subsemimodules of ${{\mathbb{R}}_{\max}}^n$ are closed. A [congruence]{} on ${{\mathbb{R}}_{\max}}^n$ is an equivalence relation ${\mathcal{W}}\subset ({{\mathbb{R}}_{\max}}^n)^2$ on ${{\mathbb{R}}_{\max}}^n$ which has a semimodule structure when it is thought of as a subset of $({{\mathbb{R}}_{\max}}^n)^2$. Congruences can be seen as the max-plus analogues of kernels of the classical theory: due to the absence of minus sign, given a matrix $E\in {{\mathbb{R}}_{\max}}^{p\times n}$, it is natural to define the [kernel]{} of $E$ (see [@CGQ96a; @CGQ97a]) as the congruence $$\ker E:={\{(x,y)\in ({{\mathbb{R}}_{\max}}^n)^2\mid\,E x = E y\}} \; .$$ The usual definition $\ker E:={\{x\in {{\mathbb{R}}_{\max}}^n\mid\,E x = {\varepsilon}\}}$ is not convenient in the max-plus algebra case, because this semimodule is usually trivial even if $E$ is not injective, so it carries little information. If $A\in {{\mathbb{R}}_{\max}}^{n\times n}$ is a matrix and ${\mathcal{W}}\subset ({{\mathbb{R}}_{\max}}^n)^2$ is a congruence, we define $$A{\mathcal{W}}:={\{(Ax,Ay)\in ({{\mathbb{R}}_{\max}}^n)^2\mid\,(x,y)\in {\mathcal{W}}\}}$$ and $$A^{-1}{\mathcal{W}}:={\{(x,y)\in ({{\mathbb{R}}_{\max}}^n)^2\mid\,(Ax,Ay)\in {\mathcal{W}}\}} \; .$$ Observe that $A{\mathcal{W}}$ is not necessarily a congruence even if ${\mathcal{W}}$ is. For ${\mathcal{S}}\subset {{\mathbb{R}}_{\max}}^n$, we define as usual $$A{\mathcal{S}}:={\{Ax\in {{\mathbb{R}}_{\max}}^n\mid\,x\in {\mathcal{S}}\}} \; \makebox{ and} \; A^{-1}{\mathcal{S}}:={\{x\in {{\mathbb{R}}_{\max}}^n\mid\,Ax\in {\mathcal{S}}\}} \; .$$ In the sequel, if ${\mathcal{W}}$ is a congruence, we write $x\sim_{\mathcal{W}}y$ for $(x,y)\in {\mathcal{W}}$ and denote by $[x]_{{\mathcal{W}}}$ the equivalence class of $x$ modulo ${\mathcal{W}}$. Max-plus conditioned and controlled invariance {#SCondCont} ============================================== We consider max-plus dynamical systems of the form $$\label{e-fond} \left\{ \begin{array}{l} x(k+1) \sim_{\mathcal{V}}Ax(k) \\ y(k)=Cx(k) \\ x(0)=x \end{array} \right.$$ where $A\in {{\mathbb{R}}_{\max}}^{n\times n}$, $C\in {{\mathbb{R}}_{\max}}^{q\times n}$, $x\in {{\mathbb{R}}_{\max}}^n$ is the initial state, $x(k)\in {{\mathbb{R}}_{\max}}^n$ is the state, $y(k)\in {{\mathbb{R}}_{\max}}^q$ is the output and ${\mathcal{V}}\subset ({{\mathbb{R}}_{\max}}^n)^2$ is a congruence which represents unobservable perturbations of the max-plus linear system $x(k+1)=Ax(k)$. Hence, the dynamics in is multi-valued, meaning that several values of $x(k+1)$ are compatible with a given $x(k)$. We assume that the output $y(k)$ is observed. If ${\mathcal{V}}=\ker E$ for some matrix $E$, then $x(k+1)\sim_{{\mathcal{V}}} A x(k)$ is equivalent to $E x(k+1)=E A x(k)$, so $x\mapsto E x$ may be interpreted as an invariant which must be preserved by the perturbation. Hence, system might be viewed as an implicit linear system. In classical system theory, implicit systems are often used to represent systems subject to disturbances. In Section \[SApplication\] we will show that dynamical systems of the form can be used, for instance, to model max-plus linear dynamical systems of the form $$x(k+1)=\bar{A} x(k)\; ,$$ when some entries of $\bar{A}$ are unknown but belong to certain intervals. The analogy between congruences and classical kernels leads us to the following definition. Given $A\in {{\mathbb{R}}_{\max}}^{n\times n}$ and $C\in {{\mathbb{R}}_{\max}}^{q\times n}$, a congruence ${\mathcal{W}}\subset ({{\mathbb{R}}_{\max}}^n)^2$ is said to be $(C,A)$-conditioned invariant if $$A({\mathcal{W}}\cap {\mathcal{C}})\subset {\mathcal{W}}\; ,$$ where ${\mathcal{C}}:=\ker C$. The following proposition establishes the connection between conditioned invariants and the observation problem for dynamical systems of the form . \[CondInvInterp\] Let ${\mathcal{W}}$ be a congruence containing the perturbation ${\mathcal{V}}$. Then, ${\mathcal{W}}$ is $(C,A)$-conditioned invariant if, and only if, for any trajectory $\left\{ x(k)\right\}_{k\geq 0}$ of system and any $m\in {\mathbb{N}}$, the equivalence class of $x(m)$ modulo ${\mathcal{W}}$ is uniquely determined by the equivalence class of $x(0)$ modulo ${\mathcal{W}}$ and by the observations $y(0),\ldots,y(m-1)$. Assume that ${\mathcal{W}}$ is $(C,A)$-conditioned invariant. Let $x(k+1)\sim_{\mathcal{V}}Ax(k)$ and $x'(k+1)\sim_{\mathcal{V}}Ax'(k)$. Then, if $x(k)\sim_{\mathcal{W}}x'(k)$ and $y(k):=Cx(k)=y'(k):=Cx'(k)$, we have $(x(k),x'(k))\in {\mathcal{W}}\cap {\mathcal{C}}$, and so $(Ax(k),Ax'(k))\in A({\mathcal{W}}\cap {\mathcal{C}})\subset {\mathcal{W}}$ because ${\mathcal{W}}$ is $(C,A)$-conditioned invariant and ${\mathcal{C}}=\ker C$. Therefore, $Ax(k)\sim_{\mathcal{W}}Ax'(k)$, and since ${\mathcal{W}}\supset {\mathcal{V}}$, we deduce that $x(k+1)\sim_{\mathcal{W}}x'(k+1)$. The “only if” part of the proposition follows from an immediate induction. Conversely, assume that for any trajectory $\left\{ x(k)\right\}_{k\geq 0}$ of system the equivalence class of $x(m)$ modulo ${\mathcal{W}}$ is uniquely determined by the equivalence class of $x(0)$ modulo ${\mathcal{W}}$ and by the observations. Let $(x(0),x'(0))\in {\mathcal{W}}\cap {\mathcal{C}}$. Then, if $x(1)\sim_{\mathcal{V}}Ax(0)$ and $x'(1)\sim_{\mathcal{V}}Ax'(0)$, we have $x(1)\sim_{\mathcal{W}}x'(1)$ because $x(0)\sim_{\mathcal{W}}x'(0)$ and $y(0):=Cx(0)=y'(0):=Cx'(0)$. Therefore, it follows that $Ax(0) \sim_{\mathcal{W}}Ax'(0)$ because ${\mathcal{V}}\subset {\mathcal{W}}$. Since this holds for any $(x(0),x'(0))\in {\mathcal{W}}\cap {\mathcal{C}}$, we conclude that $A ({\mathcal{W}}\cap {\mathcal{C}}) \subset {\mathcal{W}}$, which proves the “if” part of the proposition. This proposition raises, for observation purpose, the question of the existence, and the computation when it exists, of the minimal $(C,A)$-conditioned invariant congruence containing ${\mathcal{V}}$. Like in the case of coefficients in a field, the following lemma can be easily proved. \[InterConditioned\] The intersection of $(C,A)$-conditioned invariant congruences is a $(C,A)$-conditioned invariant congruence. If we denote by ${\mathscr{L}}(C,A,{\mathcal{V}})$ the set of all $(C,A)$-conditioned invariant congruences containing a given congruence ${\mathcal{V}}$, then, as a consequence of the previous lemma, it follows that ${\mathscr{L}}(C,A,{\mathcal{V}})$ is a lower semilattice with respect to $\subset$ and $\cap$. Moreover, Lemma \[InterConditioned\] also implies that ${\mathscr{L}}(C,A,{\mathcal{V}})$ admits a smallest element, the minimal $(C,A)$-conditioned invariant congruence containing ${\mathcal{V}}$, which will be denoted by ${\mathcal{V}}_(C,A)$. In order to compute ${\mathcal{V}}_(C,A)$, we extend the classical fixed point algorithm (see [@BasMar69; @BasMar91; @wonham]) to the max-plus algebra framework. With this purpose in mind, consider the self-map $\psi$ of the set of congruences given by $$\label{DefPsi} \psi ({\mathcal{W}}):={\langle {\mathcal{V}}\oplus A({\mathcal{W}}\cap {\mathcal{C}}) \rangle} \enspace,$$ where ${\langle {\mathcal{U}}\rangle}$ denotes the smallest congruence containing the set ${\mathcal{U}}\subset ({{\mathbb{R}}_{\max}}^n)^2$. Note that ${\mathcal{W}}\subset {\mathcal{U}}$ implies $\psi ({\mathcal{W}})\subset \psi ({\mathcal{U}})$. Define the sequence of congruences $\left\{{\mathcal{W}}_k\right\}_{k\in {\mathbb{N}}}$ by: $$\label{DefSequenceW} {\mathcal{W}}_1 := {\mathcal{V}}\; \makebox{ and } \; {\mathcal{W}}_{k+1}:=\psi ({\mathcal{W}}_{k}) \; \makebox{ for } \; k\in {\mathbb{N}}\; .$$ Then, this sequence is (weakly) increasing, that is, ${\mathcal{W}}_k\subset {\mathcal{W}}_{k+1}$ for all $k\in {\mathbb{N}}$. As a matter of fact, ${\mathcal{W}}_1={\mathcal{V}}\subset {\mathcal{V}}\oplus A({\mathcal{W}}_1 \cap {\mathcal{C}})\subset {\langle {\mathcal{V}}\oplus A({\mathcal{W}}_1 \cap {\mathcal{C}}) \rangle}={\mathcal{W}}_2$ and if ${\mathcal{W}}_r\subset {\mathcal{W}}_{r+1}$, then ${\mathcal{W}}_{r+1}=\psi ({\mathcal{W}}_r)\subset \psi ({\mathcal{W}}_{r+1})={\mathcal{W}}_{r+2}$. We define ${\mathcal{V}}_\infty$ as the limit of the sequence $\left\{{\mathcal{W}}_k\right\}_{k\in {\mathbb{N}}}$, that is, ${\mathcal{V}}_\infty=\cup_{k\in {\mathbb{N}}}{\mathcal{W}}_k$. Note that ${\mathcal{V}}_\infty$ is a congruence, because $\left\{{\mathcal{W}}_k\right\}_{k\in {\mathbb{N}}}$ is an increasing sequence of congruences. \[FixedPointCong\] Let ${\mathcal{V}}\subset ({{\mathbb{R}}_{\max}}^n)^2$ be a congruence. Then, ${\mathcal{V}}_\infty$ is the minimal $(C,A)$-conditioned invariant congruence containing ${\mathcal{V}}$. Let ${\mathcal{W}}\subset ({{\mathbb{R}}_{\max}}^n)^2$ be a $(C,A)$-conditioned invariant congruence containing ${\mathcal{V}}$. We next show that ${\mathcal{W}}_k\subset {\mathcal{W}}$ for all $k\in {\mathbb{N}}$, and therefore ${\mathcal{V}}_\infty\subset {\mathcal{W}}$. In the first place, note that ${\mathcal{W}}_1={\mathcal{V}}\subset {\mathcal{W}}$. Assume now that ${\mathcal{W}}_r \subset {\mathcal{W}}$. Then, as $A({\mathcal{W}}\cap {\mathcal{C}})\subset {\mathcal{W}}$ and ${\mathcal{V}}\subset {\mathcal{W}}$, it follows that ${\mathcal{W}}_{r+1}=\psi({\mathcal{W}}_r)\subset \psi({\mathcal{W}})={\langle {\mathcal{V}}\oplus A({\mathcal{W}}\cap {\mathcal{C}}) \rangle} \subset {\langle {\mathcal{W}}\rangle}= {\mathcal{W}}$. To prove that ${\mathcal{V}}_(C,A)={\mathcal{V}}_\infty$, it only remains to show that ${\mathcal{V}}_\infty$ is $(C,A)$-conditioned invariant. Since $$A({\mathcal{V}}_\infty\cap {\mathcal{C}})=A((\cup_k{\mathcal{W}}_k)\cap {\mathcal{C}})=A(\cup_k ({\mathcal{W}}_k\cap {\mathcal{C}}))= \cup_k(A({\mathcal{W}}_k\cap {\mathcal{C}}))\subset \cup_k{\mathcal{W}}_{k+1}={\mathcal{V}}_\infty \; ,$$ it follows that ${\mathcal{V}}_\infty$ is a $(C,A)$-conditioned invariant congruence. Concerning the computation of ${\mathcal{V}}_(C,A)$, Proposition \[FixedPointCong\] presents two drawbacks in relation to the classical theory. In the first place, for linear systems over fields, the sequence $\left\{{\mathcal{W}}_k\right\}_{k\in {\mathbb{N}}}$ always converges in at most $n$ steps because it is an increasing sequence of subspaces of a vector space of dimension $n$. However, in the max-plus case, this sequence does not necessarily converge in a finite number of steps (see the example below). This difficulty is mainly due to the fact that $({{\mathbb{R}}_{\max}}^n)^2$ is not Noetherian, meaning that there exist infinite increasing sequences of subsemimodules of $({{\mathbb{R}}_{\max}}^n)^2$. The second difficulty comes from the fact the ${\mathcal{V}}\oplus A({\mathcal{W}}_k \cap {\mathcal{C}})$ need not be a congruence, so it is necessary to compute ${\langle {\mathcal{V}}\oplus A({\mathcal{W}}_k \cap {\mathcal{C}}) \rangle}$. However, the duality results established in the present paper will allow us to dispense with this operation. \[ExampleSeqW\] Consider the matrices $$A = \begin{pmatrix} {0}& {\varepsilon}\cr {\varepsilon}& 1 \cr \end{pmatrix} \; \makebox{ and }\; C = \begin{pmatrix} {\varepsilon}& {\varepsilon}\end{pmatrix} \; ,$$ and the congruence ${\mathcal{V}}\subset ({{\mathbb{R}}_{\max}}^2)^2$ defined by: $x\sim_{\mathcal{V}}y$ if, and only if, $x_1=y_1$ and $x_1\oplus x_2 = y_1\oplus y_2$. In order to determine the minimal $(C,A)$-conditioned invariant congruence containing ${\mathcal{V}}$, we next compute the sequence of congruences $\left\{{\mathcal{W}}_k\right\}_{k\in {\mathbb{N}}}$ defined in and . We claim that ${\mathcal{W}}_k$ is defined as follows: $x\sim_{{\mathcal{W}}_k} y$ if, and only if, $x_1=y_1$ and $(k-1) x_1\oplus x_2 =(k-1) y_1\oplus y_2$. In the first place, note that this property is satisfied by definition for $k=1$. Assume now that it holds for $k=m$. Note that $$x_1=y_1 \; \makebox{ and } \; (m-1) x_1\oplus x_2 =(m-1) y_1\oplus y_2$$ is equivalent to $$\left( x_1=y_1 , (m-1) x_1\geq x_2 , (m-1) y_1\geq y_2 \right) \; \makebox{ or } \; \left( x_1=y_1 , x_2=y_2 \right) \; .$$ Then, in this particular case $A{\mathcal{W}}_m$ is a congruence which is defined by $$x\sim_{A{\mathcal{W}}_m} y \iff \left( x_1=y_1 , m x_1\geq x_2 , m y_1\geq y_2 \right) \; \makebox{ or } \; \left( x_1=y_1 , x_2=y_2 \right)$$ and thus $${\mathcal{W}}_{m+1}=\psi ({\mathcal{W}}_{m})= {\langle {\mathcal{V}}\oplus A({\mathcal{W}}_m \cap {\mathcal{C}}) \rangle} ={\langle {\mathcal{V}}\oplus A {\mathcal{W}}_m \rangle} ={\langle A {\mathcal{W}}_m \rangle} = A {\mathcal{W}}_m$$ because ${\mathcal{C}}=\ker C=({{\mathbb{R}}_{\max}}^2)^2$ and ${\mathcal{V}}\subset A{\mathcal{W}}_m$. This proves our claim. Therefore, ${\mathcal{V}}_(C,A)={\mathcal{V}}_\infty$ is the congruence defined as follows: $$x\sim_{{\mathcal{V}}_\infty } y \iff \left( x_1 = y_1\neq {\varepsilon}\right) \; \makebox{ or } \; \left( x_1 = y_1 = {\varepsilon}, x_2 = y_2 \right) \; .$$ Note that ${\mathcal{V}}_(C,A)={\mathcal{V}}_\infty$ is not closed even if ${\mathcal{V}}$ is closed. For instance, if $\lambda_1\neq \lambda_2$, we have $$(-k,\lambda_1)^t\sim_{{\mathcal{V}}_\infty } (-k,\lambda_2)^t$$ for all $k\in {\mathbb{N}}$, but $({\varepsilon},\lambda_1)^t {\not \sim}_{{\mathcal{V}}_\infty } ({\varepsilon},\lambda_2)^t$. For linear systems over fields, the minimal $(C,A)$-conditioned invariant space containing a given space can be alternatively computed through the notion of controlled invariance, which is dual of the notion of conditioned invariance. In the max-plus case, this dual notion can be defined as follows. \[DefControlled\] Given $A\in {{\mathbb{R}}_{\max}}^{n\times n}$ and $B\in {{\mathbb{R}}_{\max}}^{n\times q}$, a semimodule ${\mathcal{X}}\subset {{\mathbb{R}}_{\max}}^n$ is said to be $(A,B)$-controlled invariant if $$A{\mathcal{X}}\subset {\mathcal{X}}\oplus {\mathcal{B}}\; ,$$ where ${\mathcal{B}}:={\mbox{\rm Im}\,}B$ and ${\mathcal{X}}\oplus {\mathcal{B}}:={\{x\oplus b\mid\,x\in {\mathcal{X}},b\in {\mathcal{B}}\}}$. From a dynamical point of view, the interpretation of $(A,B)$-controlled invariance differs from the classical one. For linear dynamical systems over fields of the form $$\label{ABSystem} x(k+1)=A x(k) + B u(k) \; ,$$ where $x(k)$ is the state, $u(k)$ is the control, and $A$ and $B$ are matrices of suitable dimension, it can be shown (see [@BasMar91; @wonham]) that ${\mathcal{X}}$ is $(A,B)$-controlled invariant if, and only if, any trajectory of starting in ${\mathcal{X}}$ can be kept inside ${\mathcal{X}}$ by a suitable choice of the control. However, due to the non-invertibility of addition, this is no longer true in the max-plus case. For this property to hold true, in Definition \[DefControlled\] the semimodule ${\mathcal{X}}\oplus {\mathcal{B}}$ must be replaced by ${\mathcal{X}}\ominus {\mathcal{B}}:= {\{z\in {{\mathbb{R}}_{\max}}^n\mid\,\exists b\in {\mathcal{B}}, z\oplus b\in {\mathcal{X}}\}}$ (see [@katz07] for details). The proof of the following simple lemma, which is dual of Lemma \[InterConditioned\], is left to the reader. \[SumControlled\] The (max-plus) sum of $(A,B)$-controlled invariant semimodules is $(A,B)$-controlled invariant. By Lemma \[SumControlled\] the set of all $(A,B)$-controlled invariant semimodules contained in a given semimodule ${\mathcal{K}}\subset {{\mathbb{R}}_{\max}}^n$, which will be denoted by ${\mathscr{M}}(A,B,{\mathcal{K}})$, is an upper semilattice with respect to $\subset$ and $\oplus$. In this case, ${\mathscr{M}}(A,B,{\mathcal{K}})$ admits a biggest element, the maximal $(A,B)$-controlled invariant semimodule contained in ${\mathcal{K}}$, which will be denoted by ${\mathcal{K}}^(A,B)$. In order to compute ${\mathcal{K}}^(A,B)$, consider the self-map $\phi$ of the set of semimodules defined by: $$\label{DefPhiX} \phi ({\mathcal{X}}) := {\mathcal{K}}\cap A^{-1}({\mathcal{X}}\oplus {\mathcal{B}}) \; .$$ Define the sequence of semimodules $\left\{{\mathcal{X}}_k\right\}_{k\in {\mathbb{N}}}$ as follows: $$\label{DefSequenceX} {\mathcal{X}}_1 := {\mathcal{K}}\; \makebox{ and } \; {\mathcal{X}}_{k+1} := \phi({\mathcal{X}}_{k}) \; \makebox{ for }\; k\in {\mathbb{N}}\; .$$ Note that $\left\{{\mathcal{X}}_k\right\}_{k\in {\mathbb{N}}}$ is (weakly) decreasing, that is, ${\mathcal{X}}_{k+1}\subset {\mathcal{X}}_k$ for all $k\in {\mathbb{N}}$. As a matter of fact, ${\mathcal{X}}_2= \phi({\mathcal{X}}_1)= {\mathcal{K}}\cap A^{-1}({\mathcal{X}}_1 \oplus {\mathcal{B}})\subset {\mathcal{K}}={\mathcal{X}}_1$ and if ${\mathcal{X}}_{r+1}\subset {\mathcal{X}}_r$, then $ {\mathcal{X}}_{r+2}=\phi({\mathcal{X}}_{r+1})\subset \phi({\mathcal{X}}_r)={\mathcal{X}}_{r+1}$, since $\phi({\mathcal{Z}})\subset \phi({\mathcal{Y}})$ whenever ${\mathcal{Z}}\subset {\mathcal{Y}}$. We define the semimodule ${\mathcal{K}}^\infty$ as the limit of the sequence $\left\{{\mathcal{X}}_k\right\}_{k\in {\mathbb{N}}}$, that is, ${\mathcal{K}}^\infty =\cap_{k\in {\mathbb{N}}}{\mathcal{X}}_{k}$. \[ABsimple\] Any $(A,B)$-controlled invariant semimodule contained in ${\mathcal{K}}$ is contained in ${\mathcal{K}}^\infty$. In particular, ${\mathcal{K}}^(A,B)\subset {\mathcal{K}}^\infty$. Let ${\mathcal{X}}$ be an $(A,B)$-controlled invariant semimodule contained in ${\mathcal{K}}$. We next prove (by induction on $k$) that ${\mathcal{X}}\subset {\mathcal{X}}_k$ for all $k\in {\mathbb{N}}$, and thus ${\mathcal{X}}\subset {\mathcal{K}}^\infty$. In the first place, note that ${\mathcal{X}}\subset {\mathcal{K}}={\mathcal{X}}_{1}$. Assume now that ${\mathcal{X}}\subset {\mathcal{X}}_r$. Then, as $A{\mathcal{X}}\subset {\mathcal{X}}\oplus {\mathcal{B}}$ and ${\mathcal{X}}\subset {\mathcal{K}}$, it follows that ${\mathcal{X}}\subset {\mathcal{K}}\cap A^{-1}({\mathcal{X}}\oplus {\mathcal{B}}) = \phi({\mathcal{X}})\subset \phi({\mathcal{X}}_r)= {\mathcal{X}}_{r+1}$. In the sequel, we will repeatedly use the following elementary observation. \[SumClosed\] If ${\mathcal{X}}$ and ${\mathcal{Y}}$ are closed subsemimodules of ${{\mathbb{R}}_{\max}}^n$, then so is ${\mathcal{X}}\oplus {\mathcal{Y}}$. Let $\left\{z_k\right\}_{k\in {\mathbb{N}}}$ denote a sequence of elements of ${\mathcal{X}}\oplus {\mathcal{Y}}$ converging to some $z\in {{\mathbb{R}}_{\max}}^n$. Then, we can write $z_k=x_k\oplus y_k$ with $x_k\in {\mathcal{X}}$ and $y_k\in {\mathcal{Y}}$ for $k\in {\mathbb{N}}$. Since $\left\{z_k\right\}_{k\in {\mathbb{N}}}$ is bounded, $\left\{x_k\right\}_{k\in {\mathbb{N}}}$ and $\left\{y_k\right\}_{k\in {\mathbb{N}}}$ must be bounded, and so, by taking subsequences if necessary, we may assume that $\left\{x_k\right\}_{k\in {\mathbb{N}}}$ and $\left\{y_k\right\}_{k\in {\mathbb{N}}}$ converge to some vectors $x$ and $y$, respectively. Since ${\mathcal{X}}$ and ${\mathcal{Y}}$ are closed, we have $x\in {\mathcal{X}}$ and $y\in {\mathcal{Y}}$, and so, $z=x\oplus y\in {\mathcal{X}}\oplus {\mathcal{Y}}$. In order to state a dual of Proposition \[FixedPointCong\], we shall need a topological assumption. \[Kclosed\] Let ${\mathcal{K}}\subset {{\mathbb{R}}_{\max}}^n$ be a closed semimodule. Then, ${\mathcal{K}}^\infty$ is the maximal $(A,B)$-controlled invariant semimodule contained in ${\mathcal{K}}$. By Lemma \[ABsimple\], it suffices to show that ${\mathcal{K}}^\infty$ is $(A,B)$-controlled invariant, that is, $A{\mathcal{K}}^\infty\subset {\mathcal{K}}^\infty\oplus {\mathcal{B}}$. With this aim, as $A{\mathcal{K}}^\infty = A(\cap_k{\mathcal{X}}_{k+1}) \subset \cap_k A{\mathcal{X}}_{k+1}\subset \cap_k ({\mathcal{X}}_k\oplus {\mathcal{B}})$, it is enough to prove that $\cap_k ({\mathcal{X}}_k\oplus {\mathcal{B}})\subset (\cap_k{\mathcal{X}}_k)\oplus {\mathcal{B}}= {\mathcal{K}}^\infty\oplus {\mathcal{B}}$. In the first place, note that by Lemma \[SumClosed\], $\phi({\mathcal{X}})$ is closed whenever ${\mathcal{X}}$ and ${\mathcal{K}}$ are closed, because ${\mathcal{B}}$ is closed by Lemma \[FGClosed\]. Then, the semimodules ${\mathcal{X}}_k$ are all closed since ${\mathcal{K}}$ is closed. If $x\in \cap_k ({\mathcal{X}}_k\oplus {\mathcal{B}})$, there exist sequences $\left\{b_k\right\}_{k\in {\mathbb{N}}}$ and $\left\{x_k\right\}_{k\in {\mathbb{N}}}$ such that $x=x_k\oplus b_k$, $x_k\in {\mathcal{X}}_k$ and $b_k\in {\mathcal{B}}$ for all $k\in {\mathbb{N}}$. As these sequences are bounded by $x$, we may assume, by taking subsequences if necessary, that there exist $y\in {{\mathbb{R}}_{\max}}^n$ and $b\in {\mathcal{B}}$ such that $\lim_{k\rightarrow\infty}x_k=y$ and $\lim_{k\rightarrow\infty}b_k=b$ (recall that ${\mathcal{B}}$ is closed by Lemma \[FGClosed\]). Then, as the sequence $\left\{{\mathcal{X}}_k\right\}_{k\in {\mathbb{N}}}$ is decreasing and the semimodules ${\mathcal{X}}_k$ are all closed, it follows that $y=\lim_{k\rightarrow\infty}x_k\in {\mathcal{X}}_r$ for all $r\in {\mathbb{N}}$. Therefore, $y\in \cap_k {\mathcal{X}}_k={\mathcal{K}}^\infty$ and $x=\lim_{k\rightarrow\infty}(x_k\oplus b_k)= (\lim_{k\rightarrow\infty}x_k)\oplus (\lim_{k\rightarrow\infty}b_k)= y\oplus b\in {\mathcal{K}}^\infty\oplus {\mathcal{B}}$. Observe that, by Lemma \[FGClosed\], the condition of the previous proposition is in particular satisfied when ${\mathcal{K}}$ is finitely generated. Note also that ${\mathcal{K}}^(A,B)={\mathcal{K}}^\infty$ is closed if ${\mathcal{K}}$ is closed, because in that case ${\mathcal{K}}^\infty$ is an intersection of closed semimodules (recall that in the previous proof we showed that the semimodules ${\mathcal{X}}_k$ are all closed when ${\mathcal{K}}$ is closed). Like in the case of the sequence of congruences $\left\{{\mathcal{W}}_k\right\}_{k\in {\mathbb{N}}}$, and unlike the case of coefficients in a field in which it converges in at most $n$ steps (see [@BasMar69; @BasMar91; @wonham]), the sequence of semimodules $\left\{{\mathcal{X}}_k\right\}_{k\in {\mathbb{N}}}$ does not necessarily converge in a finite number of steps (see the example below). This is in part a consequence of the fact that ${{\mathbb{R}}_{\max}}^n$ is not Artinian, meaning that there exist infinite decreasing sequences of subsemimodules of ${{\mathbb{R}}_{\max}}^n$. However, in Section \[SDuality\] we will give a condition which ensures the convergence of this sequence in a finite number of steps. This difficulty is also found when the coefficients belong to a ring, where except for Principal Ideal Domains, the computation of the maximal $(A,B)$-controlled invariant module is still under investigation (see [@conte94; @conte95]). \[ExampleSeqX\] Consider the matrices $$A = \begin{pmatrix} {0}& {\varepsilon}\cr {\varepsilon}& 1 \cr \end{pmatrix} \; \makebox{ and }\; B = \begin{pmatrix} {\varepsilon}\cr {\varepsilon}\end{pmatrix} \; ,$$ and the semimodule ${\mathcal{K}}={\{x\in {{\mathbb{R}}_{\max}}^2\mid\,x_1\geq x_2\}}$. Since ${\mathcal{K}}$ is clearly closed, we can apply Proposition \[Kclosed\] in order to compute ${\mathcal{K}}^(A,B)$. If we define the sequence of semimodules $\left\{{\mathcal{X}}_k\right\}_{k\in {\mathbb{N}}}$ by and , then, using the fact that in this particular case $A$ is invertible and that $$A^{-1} = \begin{pmatrix} {0}& {\varepsilon}\cr {\varepsilon}& -1 \cr \end{pmatrix} \; ,$$ it can be easily seen that ${\mathcal{X}}_k={\{x\in {{\mathbb{R}}_{\max}}^2\mid\,x_1\geq (k-1)x_2\}}$ for all $k\in {\mathbb{N}}$. Therefore, ${\mathcal{K}}^(A,B)={\mathcal{K}}^\infty={\{x\in {{\mathbb{R}}_{\max}}^2\mid\,x_2={\varepsilon}\}}$. Orthogonal semimodules and congruences {#SOrthogonal} ====================================== Before studying the duality between controlled and conditioned invariance, it is convenient to introduce the notions of orthogonal of semimodules and congruences, and study their properties. The orthogonal of a semimodule ${\mathcal{X}}\subset {{\mathbb{R}}_{\max}}^n$ is the congruence ${\mathcal{X}}^\bot ={\{(x,y)\in ({{\mathbb{R}}_{\max}}^n)^2\mid\,x^tz=y^tz,\forall z\in {\mathcal{X}}\}}$. Analogously, the orthogonal of a congruence (or more generally a semimodule) ${\mathcal{W}}\subset ({{\mathbb{R}}_{\max}}^n)^2$ is the semimodule ${\mathcal{W}}^\top ={\{z\in {{\mathbb{R}}_{\max}}^n\mid\,x^tz=y^tz,\forall (x,y)\in {\mathcal{W}}\}}$. Note that the orthogonal, being the intersection of closed sets, is always closed. We shall need the following duality theorem. \[SepTheo\] If ${\mathcal{X}}\subset {{\mathbb{R}}_{\max}}^n$ is a closed semimodule, and if ${\mathcal{W}}\subset ({{\mathbb{R}}_{\max}}^n)^2$ is a closed congruence, then: $${\mathcal{X}}=({\mathcal{X}}^\bot)^\top,\; \makebox{ and } \; {\mathcal{W}}=({\mathcal{W}}^\top)^\bot \; .$$ The first equality follows from the separation theorem for closed semimodules, see [@zimmerman77 Th. 4], [@shpiz], see also [@cgqs04 Th. 3.14] for recent improvements. The second equality is proved in [@gk08u] as a consequence of a new separation theorem, which applies to closed congruences. The orthogonal has the following properties. \[propiedad\] Let $A\in {{\mathbb{R}}_{\max}}^{n\times n}$ be a matrix, ${\mathcal{W}},{\mathcal{W}}_1,{\mathcal{W}}_2 \subset ({{\mathbb{R}}_{\max}}^n)^2$ be congruences and ${\mathcal{X}},{\mathcal{X}}_1,{\mathcal{X}}_2\subset {{\mathbb{R}}_{\max}}^n$ be semimodules. Then, - \[ph1\] $({\mathcal{W}}_1\oplus {\mathcal{W}}_2)^\top= {\mathcal{W}}_1^\top \cap {\mathcal{W}}_2^\top$ $({\mathcal{X}}_1\oplus {\mathcal{X}}_2)^\bot= {\mathcal{X}}_1^\bot \cap {\mathcal{X}}_2^\bot$ , - \[ph2\] $(A{\mathcal{W}})^\top= (A^t)^{-1}{\mathcal{W}}^\top$ $(A{\mathcal{X}})^\bot= (A^t)^{-1}{\mathcal{X}}^\bot$ . Moreover, if ${\mathcal{W}}_1$, ${\mathcal{W}}_2$, ${\mathcal{X}}_1$ and ${\mathcal{X}}_2$ are closed, then - \[ph3\] $({\mathcal{W}}_1\cap {\mathcal{W}}_2)^\top= {\mathcal{W}}_1^\top \oplus {\mathcal{W}}_2^\top$ $({\mathcal{X}}_1\cap {\mathcal{X}}_2)^\bot= {\mathcal{X}}_1^\bot \oplus {\mathcal{X}}_2^\bot$ . We next prove these properties for congruences. In the case of semimodules, these properties can be proved along the same lines. \(i) As ${\mathcal{W}}_r\subset {\mathcal{W}}_1\oplus {\mathcal{W}}_2$ for $r=1,2$, we have $({\mathcal{W}}_1\oplus {\mathcal{W}}_2)^\top\subset {\mathcal{W}}_r^\top$ for $r=1,2$ and thus $({\mathcal{W}}_1\oplus {\mathcal{W}}_2)^\top\subset {\mathcal{W}}_1^\top \cap {\mathcal{W}}_2^\top$. Let $z\in {\mathcal{W}}_1^\top \cap {\mathcal{W}}_2^\top$. Since $$(x_1\oplus x_2)^t z = x_1^t z\oplus x_2^t z = y_1^t z \oplus y_2^t z = (y_1\oplus y_2)^t z$$ for all $(x_1,y_1)\in {\mathcal{W}}_1$ and $(x_2,y_2)\in {\mathcal{W}}_2$, it follows that $z\in ({\mathcal{W}}_1\oplus {\mathcal{W}}_2)^\top$. Therefore, ${\mathcal{W}}_1^\top \cap {\mathcal{W}}_2^\top \subset ({\mathcal{W}}_1\oplus {\mathcal{W}}_2)^\top$. \(ii) We have $$\begin{aligned} (A{\mathcal{W}})^\top & = & {\{z\in {{\mathbb{R}}_{\max}}^n\mid\,x^tz=y^tz,\forall (x,y)\in A{\mathcal{W}}\}} \\ & = & {\{z\in {{\mathbb{R}}_{\max}}^n\mid\,(Ax)^tz=(Ay)^tz,\forall (x,y)\in {\mathcal{W}}\}} \\ & = & {\{z\in {{\mathbb{R}}_{\max}}^n\mid\,x^tA^tz=y^tA^tz,\forall (x,y)\in {\mathcal{W}}\}} \\ & = & {\{z\in {{\mathbb{R}}_{\max}}^n\mid\,A^tz\in {\mathcal{W}}^\top\}}=(A^t)^{-1}{\mathcal{W}}^\top\; .\end{aligned}$$ \(iii) Since ${\mathcal{W}}_1$ and ${\mathcal{W}}_2$ are closed, by Theorem \[SepTheo\] we have ${\mathcal{W}}_1=({\mathcal{W}}_1^\top )^\bot $ and ${\mathcal{W}}_2=({\mathcal{W}}_2^\top)^\bot$. Then, from (i) and Theorem \[SepTheo\], it follows that $$({\mathcal{W}}_1\cap {\mathcal{W}}_2)^\top= (({\mathcal{W}}_1^\top)^\bot \cap ({\mathcal{W}}_2^\top)^\bot)^\top= (({\mathcal{W}}_1^\top \oplus {\mathcal{W}}_2^\top)^\bot)^\top= {\mathcal{W}}_1^\top \oplus {\mathcal{W}}_2^\top \; ,$$ because ${\mathcal{W}}_1^\top \oplus {\mathcal{W}}_2^\top$ is closed by Lemma \[SumClosed\]. In Property (iii) above, when the semimodules ${\mathcal{X}}_1$ and ${\mathcal{X}}_2$ are not closed, the only thing that can be said is that $${\mathcal{X}}_1^\bot \oplus {\mathcal{X}}_2^\bot \subset ({\mathcal{X}}_1\cap {\mathcal{X}}_2)^\bot \; .$$ As a matter of fact, since ${\mathcal{X}}_1\cap {\mathcal{X}}_2\subset {\mathcal{X}}_r$ for $r=1,2$, it follows that ${\mathcal{X}}_r^\bot \subset ({\mathcal{X}}_1\cap {\mathcal{X}}_2)^\bot$ for $r=1,2$ and so ${\mathcal{X}}_1^\bot \oplus {\mathcal{X}}_2^\bot \subset ({\mathcal{X}}_1\cap {\mathcal{X}}_2)^\bot$. To see that the other inclusion does not necessarily hold, consider the semimodules ${\mathcal{X}}_1={\{x\in {{\mathbb{R}}_{\max}}^2\mid\,x_1=x_2\}}$ and ${\mathcal{X}}_2={\{x\in {{\mathbb{R}}_{\max}}^2\mid\,x_1>x_2\}}\cup \left\{ ({\varepsilon}, {\varepsilon})^t\right\}$. Then, ${\mathcal{X}}_1^\bot={\{(x,y)\in ({{\mathbb{R}}_{\max}}^2)^2\mid\,x_1\oplus x_2=y_1\oplus y_2\}}$ and ${\mathcal{X}}_2^\bot={\{(x,y)\in ({{\mathbb{R}}_{\max}}^2)^2\mid\,x_1=y_1,x_1\oplus x_2=y_1\oplus y_2\}}$, thus ${\mathcal{X}}_1^\bot \oplus {\mathcal{X}}_2^\bot={\mathcal{X}}_1^\bot \varsubsetneq ({{\mathbb{R}}_{\max}}^2)^2$ because ${\mathcal{X}}_2^\bot \subset {\mathcal{X}}_1^\bot $. However, $({\mathcal{X}}_1\cap {\mathcal{X}}_2)^\bot=\left\{ ({\varepsilon}, {\varepsilon})^t\right\}^\bot =({{\mathbb{R}}_{\max}}^2)^2$. \[LemmaFG\] For any matrix $E$ we have $({\mbox{\rm Im}\,}E)^\bot=\ker E^t$ and $(\ker E)^\top ={\mbox{\rm Im}\,}E^t$. Note that $(x,y)\in ({\mbox{\rm Im}\,}E)^\bot \iff x^tz=y^tz, \forall z\in {\mbox{\rm Im}\,}E \iff x^tEv=y^tEv, \forall v \iff E^tx=E^ty \iff (x,y)\in \ker E^t$. Therefore, $({\mbox{\rm Im}\,}E)^\bot =\ker E^t$. Since by Lemma \[FGClosed\] ${\mbox{\rm Im}\,}E^t$ is closed, we have $(({\mbox{\rm Im}\,}E^t)^\bot)^\top ={\mbox{\rm Im}\,}E^t$. Then, as $({\mbox{\rm Im}\,}E^t)^\bot =\ker E$, it follows that $(\ker E)^\top ={\mbox{\rm Im}\,}E^t$. Duality between conditioned and controlled invariance {#SDuality} ===================================================== In this section we investigate the duality between controlled and conditioned invariants in max-plus algebra. \[conditionedcontrolled\] If ${\mathcal{X}}\subset {{\mathbb{R}}_{\max}}^n$ is $(A,B)$-controlled invariant, then ${\mathcal{X}}^\bot$ is $(B^t,A^t)$-conditioned invariant. Moreover, if a closed congruence ${\mathcal{W}}\subset ({{\mathbb{R}}_{\max}}^n)^2$ is $(C,A)$-conditioned invariant, then ${\mathcal{W}}^\top$ is $(A^t,C^t)$-controlled invariant. If ${\mathcal{X}}$ is $(A,B)$-controlled invariant, then $A{\mathcal{X}}\subset {\mathcal{X}}\oplus {\mbox{\rm Im}\,}B$. Since $$\begin{aligned} A{\mathcal{X}}\subset {\mathcal{X}}\oplus {\mbox{\rm Im}\,}B &\implies & ({\mathcal{X}}\oplus {\mbox{\rm Im}\,}B)^\bot \subset (A{\mathcal{X}})^\bot \\ &\implies & {\mathcal{X}}^\bot \cap ({\mbox{\rm Im}\,}B)^\bot \subset (A^t)^{-1}{\mathcal{X}}^\bot \\ &\implies & A^t ({\mathcal{X}}^\bot \cap \ker B^t)\subset {\mathcal{X}}^\bot, \end{aligned}$$ it follows that ${\mathcal{X}}^\bot$ is $(B^t,A^t)$-conditioned invariant. Assume that a closed congruence ${\mathcal{W}}$ is $(C,A)$-conditioned invariant, that is $A ({\mathcal{W}}\cap \ker C)\subset {\mathcal{W}}$. Then, by Lemma \[propiedad\] we have $({\mathcal{W}}\cap \ker C)^\top ={\mathcal{W}}^\top \oplus (\ker C)^\top$ and thus $$\begin{aligned} A ({\mathcal{W}}\cap \ker C)\subset {\mathcal{W}}&\implies & {\mathcal{W}}^\top \subset (A({\mathcal{W}}\cap \ker C))^\top \\ & \implies & {\mathcal{W}}^\top \subset (A^t)^{-1} ({\mathcal{W}}\cap \ker C)^\top\\ &\implies & A^t{\mathcal{W}}^\top \subset ({\mathcal{W}}\cap \ker C)^\top ={\mathcal{W}}^\top \oplus {\mbox{\rm Im}\,}C^t \enspace .\end{aligned}$$ Therefore, ${\mathcal{W}}^\top$ is $(A^t,C^t)$-controlled invariant. The following duality theorem establishes a bijective correspondence between closed controlled invariant semimodules and closed conditioned invariant congruences. This is the basis of the algorithmic results which follow, since dealing with invariant semimodules is technically simpler than dealing with invariant congruences (because, in particular, the later objects show a “doubling” of the dimension). It is worth mentioning that for systems with coefficients in a ring the duality between controlled and conditioned invariant modules does not hold in general [@DLLL08]. For systems in infinite dimensions on a Hilbert space as well, closeness is instrumental for obtaining duality results [@Cu86]. In this case other hypothesis are necessary concerning the domain of the operators and their boundedness. \[TheoCondContr\] Let ${\mathcal{V}}\subset ({{\mathbb{R}}_{\max}}^ n)^2$ be a congruence. If ${\mathcal{W}}$ is a closed congruence, then $${\mathcal{W}}\in {\mathscr{L}}(C,A,{\mathcal{V}})\iff {\mathcal{W}}^\top\in {\mathscr{M}}(A^t,C^t,{\mathcal{V}}^\top) \enspace .$$ If the congruence ${\mathcal{W}}$ is $(C,A)$-conditioned invariant and closed, then, by Lemma \[conditionedcontrolled\], ${\mathcal{W}}^\top$ is $(A^t,C^t)$-controlled invariant. Moreover, if ${\mathcal{W}}\supset {\mathcal{V}}$, it follows that ${\mathcal{W}}^\top\subset {\mathcal{V}}^\top$. This shows the “only if” part of the theorem. Conversely, if ${\mathcal{X}}:={\mathcal{W}}^\top$ is $(A^t,C^t)$-controlled invariant, then, by Lemma \[conditionedcontrolled\], ${\mathcal{X}}^\bot$ is $(C,A)$-conditioned invariant. Moreover, if ${\mathcal{W}}$ is closed and ${\mathcal{X}}\subset {\mathcal{V}}^\top$, then ${\mathcal{W}}=({\mathcal{W}}^\top)^\bot={\mathcal{X}}^\bot\supset ({\mathcal{V}}^\top)^\bot\supset {\mathcal{V}}$, which shows the “if” part of the theorem. As a consequence, we have. \[controcondi\] Let ${\mathcal{V}}\subset ({{\mathbb{R}}_{\max}}^ n)^2$ be a congruence. If we define ${\mathcal{K}}={\mathcal{V}}^\top$, then ${{\mathcal{K}}^(A^t,C^t)}^\bot$ is the minimal closed $(C,A)$-conditioned invariant congruence containing ${\mathcal{V}}$. Therefore, ${\mathcal{V}}_(C,A)\subset {{\mathcal{K}}^(A^t,C^t)}^\bot$ and ${\mathcal{V}}_(C,A)= {{\mathcal{K}}^(A^t,C^t)}^\bot$ if ${\mathcal{V}}_(C,A)$ is closed. By Lemma \[conditionedcontrolled\] we know that ${{\mathcal{K}}^(A^t,C^t)}^\bot$ is $(C,A)$-conditioned invariant. Moreover, since ${\mathcal{K}}^(A^t,C^t)\subset {\mathcal{K}}$, we have ${\mathcal{V}}\subset ({\mathcal{V}}^\top)^\bot ={\mathcal{K}}^\bot \subset {{\mathcal{K}}^(A^t,C^t)}^\bot$. Let ${\mathcal{W}}$ be a closed $(C,A)$-conditioned invariant congruence containing ${\mathcal{V}}$. Then, by Theorem \[TheoCondContr\], we have ${\mathcal{W}}^\top\in {\mathscr{M}}(A^t,C^t,{\mathcal{K}})$ and thus ${\mathcal{W}}^\top \subset {\mathcal{K}}^(A^t,C^t)$. Therefore, ${{\mathcal{K}}^(A^t,C^t)}^\bot \subset ({\mathcal{W}}^\top)^\bot={\mathcal{W}}$. Since in the previous proposition ${\mathcal{K}}={\mathcal{V}}^\top$ is closed, we can apply Proposition \[Kclosed\] in order to compute ${\mathcal{K}}^(A^t,C^t)$. This means that in and we have to take ${\mathcal{K}}={\mathcal{V}}^\top$, $B=C^t$ and $A^t$ instead of $A$. Consider again the matrices $A$ and $C$ and the congruence ${\mathcal{V}}$ of Example \[ExampleSeqW\]. We have seen that in this case ${\mathcal{V}}_(C,A)$ is not closed. Taking ${\mathcal{K}}={\mathcal{V}}^\top$, by Proposition \[controcondi\], we know that ${{\mathcal{K}}^(A^t,C^t)}^\bot$ is the minimal closed $(C,A)$-conditioned invariant congruence containing ${\mathcal{V}}$. Note that ${\mathcal{V}}=\ker E$, where $$E = \begin{pmatrix} {0}& {\varepsilon}\cr {0}& {0}\cr \end{pmatrix} \; ,$$ so that ${\mathcal{K}}={\mbox{\rm Im}\,}E^t$ is the semimodule considered in Example \[ExampleSeqX\]. Since $A=A^t$ and the matrix $B$ of Example \[ExampleSeqX\] is equal to $C^t$, ${\mathcal{K}}^(A^t,C^t)$ is the semimodule ${\mathcal{K}}^(A,B)={\{x\in {{\mathbb{R}}_{\max}}^2\mid\,x_2={\varepsilon}\}}$ computed in Example \[ExampleSeqX\]. Therefore, we conclude that the minimal closed $(C,A)$-conditioned invariant congruence containing ${\mathcal{V}}$ is ${{\mathcal{K}}^(A^t,C^t)}^\bot = {\{(x,y)\in ({{\mathbb{R}}_{\max}}^2)^2\mid\,x_1=y_1\}}$. We say that a congruence ${\mathcal{W}}$ is cofinitely generated if ${\mathcal{W}}=\ker E$ for some matrix $E$. Since a congruence ${\mathcal{W}}$ on ${{\mathbb{R}}_{\max}}^n$ is in particular a subsemimodule of $({{\mathbb{R}}_{\max}}^n)^2$, we say that ${\mathcal{W}}$ is finitely generated if it is finitely generated as a semimodule, that is, if there exists a finite family $\left\{(x_i,y_i)\right\}_{i\in I}\subset ({{\mathbb{R}}_{\max}}^n)^2$ such that ${\mathcal{W}}$ is the set of elements of the form $\bigoplus_{i\in I}\lambda_i (x_i,y_i)$, with $\lambda_i\in {{\mathbb{R}}_{\max}}$. We next show that the class of cofinitely generated congruences coincides with the class of finitely generated congruences. With this aim, we shall need the following lemma, which tells us that the solution sets of homogeneous max-plus linear systems of equations are finitely generated semimodules. If $F$ and $G$ are two rectangular matrices of the same dimension, then, ${\{z\mid\,Fz=Gz\}}$ is a finitely generated semimodule. \[th-C\] A congruence ${\mathcal{W}}$ is cofinitely generated if, and only if, it is finitely generated as a semimodule. If ${\mathcal{W}}=\ker E$, then, it is finitely generated as a semimodule, because ${\mathcal{W}}={\{(x,y)\mid\,Ex=Ey\}}$ is the solution set of an homogeneous max-plus linear system of equations, which is finitely generated as a semimodule by the previous lemma. Conversely, if ${\mathcal{W}}$ is finitely generated as a semimodule, then it is closed, and so ${\mathcal{W}}=({\mathcal{W}}^\top)^{\bot}$ by Theorem \[SepTheo\]. Using again the previous lemma, we deduce that ${\mathcal{W}}^\top$ is a finitely generated semimodule, so ${\mathcal{W}}^\top ={\mbox{\rm Im}\,}G$ for some matrix $G$. It follows that ${\mathcal{W}}=({\mbox{\rm Im}\,}G)^\bot=\ker G^t$ is cofinitely generated. In practice, the objects of interest are usually finitely generated congruences and semimodules. This is why, in the sequel, we focus our attention to them and consider the following sets $${\mathscr{L}_{\rm fg}}(C,A,{\mathcal{V}})= {\{{\mathcal{W}}\in {\mathscr{L}}(C,A,{\mathcal{V}})\mid\,{\mathcal{W}}=\ker D \; \text{\rm for some matrix } D\}}$$ and $${\mathscr{M}_{\rm fg}}(A,B,{\mathcal{K}})= {\{{\mathcal{X}}\in {\mathscr{M}}(A,B,{\mathcal{K}})\mid\,{\mathcal{X}}={\mbox{\rm Im}\,}D \; \text{\rm for some matrix } D\}} \; .$$ The following theorem relates ${\mathscr{L}_{\rm fg}}(C,A,{\mathcal{V}})$ with ${\mathscr{M}_{\rm fg}}(A^t,C^t,{\mathcal{V}}^\top)$. \[FGDuality\] Let ${\mathcal{V}}\subset ({{\mathbb{R}}_{\max}}^ n)^2$ be a congruence. If we define ${\mathcal{K}}={\mathcal{V}}^\top$, then ${\mathscr{L}_{\rm fg}}(C,A,{\mathcal{V}})$ admits a minimal element ${\mathcal{V}_{\rm fg}}(C,A)$ if, and only if, ${\mathscr{M}_{\rm fg}}(A^t,C^t,{\mathcal{K}})$ admits a maximal element ${\mathcal{K}^{\rm fg}}(A^t,C^t)$. Moreover, when these elements exist, they satisfy ${\mathcal{V}_{\rm fg}}(C,A)= {\mathcal{K}^{\rm fg}}(A^t,C^t)^\bot$. In the first place, note that $$\ker D_1\subset \ker D_2 \iff {\mbox{\rm Im}\,}D_2^t \subset {\mbox{\rm Im}\,}D_1^t\; ,$$ since $\ker D_1\subset \ker D_2 \implies (\ker D_2)^\top\subset (\ker D_1)^\top \implies {\mbox{\rm Im}\,}D_2^t \subset {\mbox{\rm Im}\,}D_1^t$ and ${\mbox{\rm Im}\,}D_2^t \subset {\mbox{\rm Im}\,}D_1^t \implies ({\mbox{\rm Im}\,}D_1^t)^\bot \subset ({\mbox{\rm Im}\,}D_2^t)^\bot \implies \ker D_1\subset \ker D_2$ by Lemma \[LemmaFG\]. In addition, by Theorem \[TheoCondContr\] we know that ${\mathcal{W}}=\ker D\in {\mathscr{L}_{\rm fg}}(C,A,{\mathcal{V}})$ if, and only if, ${\mathcal{X}}={\mbox{\rm Im}\,}D^t\in {\mathscr{M}_{\rm fg}}(A^t,C^t,{\mathcal{K}})$. Therefore, ${\mathscr{L}_{\rm fg}}(C,A,{\mathcal{V}})$ admits a minimal element ${\mathcal{V}_{\rm fg}}(C,A)$ if, and only if, ${\mathscr{M}_{\rm fg}}(A^t,C^t,{\mathcal{K}})$ admits a maximal element ${\mathcal{K}^{\rm fg}}(A^t,C^t)$. Moreover, if ${\mathcal{V}_{\rm fg}}(C,A)$ exists and ${\mathcal{V}_{\rm fg}}(C,A)=\ker D$, then ${\mathcal{K}^{\rm fg}}(A^t,C^t)={\mbox{\rm Im}\,}D^t$ and thus ${\mathcal{V}_{\rm fg}}(C,A)={\mathcal{K}^{\rm fg}}(A^t,C^t)^\bot$. We next give a condition which ensures the existence of ${\mathcal{V}_{\rm fg}}(C,A)$. With this aim, it is convenient to restrict ourselves to the subsemiring ${{\mathbb{Z}}_{\max}}=({{\mathbb{Z}}\cup\{-\infty\}},\max,+)$ of ${{\mathbb{R}}_{\max}}$ and introduce the notion of volume of a subsemimodule of ${{\mathbb{Z}}_{\max}}^n$. \[defvolumen\] Let ${\mathcal{K}}\subset {{\mathbb{Z}}_{\max}}^n$ be a semimodule. We call [volume]{} of ${\mathcal{K}}$, and we represent it with ${{\text{\rm vol}}\,}({\mathcal{K}})$, the cardinality of the set ${\{z\in {\mathcal{K}}\mid\,z_1\oplus \cdots \oplus z_n=0\}}$. Moreover, if $E\in {{\mathbb{Z}}_{\max}}^{n\times p}$, we represent with ${{\text{\rm vol}}\,}(E)$ the volume of the semimodule ${\mathcal{K}}={\mbox{\rm Im}\,}E$, that is, ${{\text{\rm vol}}\,}(E)={{\text{\rm vol}}\,}({\mbox{\rm Im}\,}E)$. Note that a semimodule ${\mathcal{K}}\subset {{\mathbb{Z}}_{\max}}^n$ with finite volume is necessarily finitely generated because clearly ${\mathcal{K}}= {{\text{\rm span}}\,}{\{z\in {\mathcal{K}}\mid\,z_1\oplus \cdots \oplus z_n=0\}}$. We shall need the following properties. \[lemapropvol\] Let $E\in {{\mathbb{Z}}_{\max}}^{n\times p}$ be a matrix and ${\mathcal{Z}},{\mathcal{Y}}\subset {{\mathbb{Z}}_{\max}}^n$ be semimodules. Then, we have 1. \[pv1\] ${\mathcal{Y}}\subset {\mathcal{Z}}\Rightarrow {{\text{\rm vol}}\,}({\mathcal{Y}}) \leq {{\text{\rm vol}}\,}({\mathcal{Z}}) \;$, 2. \[pv2\] if ${{\text{\rm vol}}\,}({\mathcal{Y}})<\infty$, then ${\mathcal{Y}}\varsubsetneq {\mathcal{Z}}\Rightarrow {{\text{\rm vol}}\,}({\mathcal{Y}}) <{{\text{\rm vol}}\,}({\mathcal{Z}}) \;$, 3. \[pv3\] ${{\text{\rm vol}}\,}(E) ={{\text{\rm vol}}\,}(E^t)\; $. \[PropFiniteVolume\] Let $A\in {{\mathbb{Z}}_{\max}}^{n\times n}$, $C\in {{\mathbb{Z}}_{\max}}^{q\times n}$ and ${\mathcal{V}}=\ker E$, where $E\in {{\mathbb{Z}}_{\max}}^{p\times n}$ is a matrix with finite volume. Then, ${\mathcal{V}_{\rm fg}}(C,A)$ exists. Moreover, if we define the sequence $\left\{{\mathcal{X}}_k\right\}_{k\in {\mathbb{N}}}$ by , with ${\mathcal{K}}={\mathcal{V}}^\top$, $B=C^t$ and $A^t$ instead of $A$ in , then ${\mathcal{V}_{\rm fg}}(C,A)={{\mathcal{X}}_r}^\bot$ for some $r\leq {{\text{\rm vol}}\,}(E)+1$. Firstly, note that by Property (iii) of Lemma \[lemapropvol\] we have ${{\text{\rm vol}}\,}({\mathcal{X}}_1)={{\text{\rm vol}}\,}({\mathcal{K}})= {{\text{\rm vol}}\,}(E^t)={{\text{\rm vol}}\,}(E)<\infty $. Then, since $\left\{{\mathcal{X}}_k\right\}_{k\in {\mathbb{N}}}$ is a decreasing sequence of semimodules, from Property (i) of Lemma \[lemapropvol\] we deduce that $\left\{{{\text{\rm vol}}\,}({\mathcal{X}}_k)\right\}_{k\in {\mathbb{N}}}$ is a decreasing sequence of non-negative integers. Therefore, there exists $r\leq {{\text{\rm vol}}\,}({\mathcal{X}}_1)+1={{\text{\rm vol}}\,}(E)+1$ such that ${{\text{\rm vol}}\,}({\mathcal{X}}_{r+1})={{\text{\rm vol}}\,}({\mathcal{X}}_r)$, so by Property (ii) of Lemma \[lemapropvol\], we have ${\mathcal{X}}_{r+1}={\mathcal{X}}_r$. It follows that ${\mathcal{X}}_k={\mathcal{X}}_r$ for all $k\geq r$ and thus ${\mathcal{K}}^(A^t,C^t)={\mathcal{K}}^\infty={\mathcal{X}}_r$ by Proposition \[Kclosed\]. Finally, since ${\mathcal{X}}_r$ has finite volume, and so it is finitely generated, we conclude that ${\mathcal{K}^{\rm fg}}(A^t,C^t)={\mathcal{K}}^(A^t,C^t)={\mathcal{X}}_r$ and then, by Theorem \[FGDuality\] we have ${\mathcal{V}_{\rm fg}}(C,A)= {\mathcal{X}}_r^\bot$. Note that in the previous proof we in particular showed that when ${\mathcal{K}}$ has finite volume, the sequence $\left\{{\mathcal{X}}_k\right\}_{k\in {\mathbb{N}}}$ defined in converges in at most ${{\text{\rm vol}}\,}({\mathcal{K}})+1$ steps to ${\mathcal{K}}^(A,B)$, for any pair of matrices $A$ and $B$. For sufficient conditions for ${\mathcal{K}}={\mbox{\rm Im}\,}E$ to have finite volume, and a bound for ${{\text{\rm vol}}\,}(E)$ in terms of the additive version of Hilbert’s projective metric when $E$ only has finite entries, we refer the reader to [@katz07]. To end this section, observe that Proposition \[CondInvInterp\] raises the question of constructing a dynamic observer for system , allowing us to compute $[x(m)]_{{\mathcal{W}}}$ as a function of $[x(0)]_{{\mathcal{W}}}$ and $y(0),\ldots, y(m-1)$. To do so, we shall assume that the congruence ${\mathcal{W}}$ is cofinitely generated, so that ${\mathcal{W}}=\ker F$ for some matrix $F$. Then, we must compute $Fx(m)$ in terms of $Fx(0)$ and $y(0),\ldots,y(m-1)$. \[TheoObserver\] Assume that the minimal $(C,A)$-conditioned invariant congruence ${\mathcal{W}}$ containing ${\mathcal{V}}$ is cofinitely generated, so that ${\mathcal{W}}=\ker F$ for some matrix $F$. Then, there exist two matrices $U$ and $V$ such that $$\begin{aligned} \label{e-transpose} FA=UF\oplus VC \enspace ,\end{aligned}$$ and for any choice of these matrices, if we define $z(k):=F x(k)$, we have $$\begin{aligned} \label{e-do} z(k+1)=Uz(k)\oplus Vy(k) \; \makebox{ for }\; k\geq 0 \; ,\end{aligned}$$ where $\left\{ x(k)\right\}_{k\geq 0}$ is any trajectory of system and $\left\{ y(k)\right\}_{k\geq 0}$ is the corresponding output trajectory. By Theorem \[TheoCondContr\] we know that ${\mathcal{W}}^\top$ is $(A^t,C^t)$-controlled invariant and thus $$A^t({\mathcal{W}}^\top)\subset {\mathcal{W}}^\top\oplus {\mbox{\rm Im}\,}C^t \; .$$ Since ${\mathcal{W}}^\top ={\mbox{\rm Im}\,}F^t$ by Lemma \[LemmaFG\], we deduce that $$A^tF^t=F^tU^t\oplus C^tV^t$$ for some matrices $U$ and $V$. After transposing, we obtain . Since ${\mathcal{V}}\subset \ker F$, we have $z(k+1)=Fx(k+1)=FAx(k)$, and using , we get $$F A x(k)=U F x(k)\oplus V C x(k)= U z(k)\oplus V y(k)\; ,$$ which shows . Suppose that, given a matrix $G$, we want to construct an observer for reconstructing, from the observation and initial condition, the linear functional of the state of system $$w(k)=Gx(k) \; ,$$ where $k\geq 0$. With this aim, assume that the minimal $(C,A)$-conditioned invariant congruence containing ${\mathcal{V}}$ is contained in $\ker G$ and cofinitely generated, so that ${\mathcal{V}}_(C,A)=\ker F\subset \ker G$ for some matrix $F$. Then, if we define $z(k):=Fx(k)$ like in Theorem \[TheoObserver\], we have $w(k)=G (-F^t) z(k)$ for all $k\geq 0$, where the product by $-F^t$ is performed in the semiring $({\mathbb{R}}\cup\{-\infty,+\infty\},\min ,+)$ with the convention $(+\infty)+a=+\infty$ for all $a\in{\mathbb{R}}\cup\{-\infty,+\infty\}$. As a matter of fact, from the equality $F (-F^t) F=F$ (see for example [@bcoq; @BlythJan72]), it follows that $$F (-F^t) z(k) = F (-F^t) F x(k) = F x(k) \; ,$$ and since $\ker F \subset \ker G$, we get $G (-F^t) z(k) = G x(k) =w(k)$. Therefore, by Theorem \[TheoObserver\], we conclude that it is possible to effectively reconstruct the linear functional of the state $w(k)=Gx(k)$ from the observation and initial state if the minimal $(C,A)$-conditioned invariant congruence containing ${\mathcal{V}}$ is contained in $\ker G$ and cofinitely generated. Application to a manufacturing system {#SApplication} ===================================== Dynamical systems of the form can be used, for instance, to model max-plus linear dynamical systems of the form $$x(k+1)=\bar{A} x(k)\; ,$$ where some entries of $\bar{A}$ are unknown but belong to certain intervals, at the price of adding new variables. To see this, assume for example that in the max-plus linear dynamical system $$\label{persystem} \begin{pmatrix} x_1(k+1) \cr x_2(k+1)\cr \end{pmatrix}= \begin{pmatrix} a_{11} & a_{12} \cr a_{21}(k) & a_{22} \cr \end{pmatrix} \begin{pmatrix} x_1(k) \cr x_2(k)\cr \end{pmatrix} \; ,$$ $a_{21}(k)$ can take any value in the interval $[a,b]$ for each $k\in {\mathbb{N}}$, but $a_{11}$, $a_{12}$ and $a_{22}$ are fixed. Consider the “extended state” vector $x(k)=\left( x_1(k), x_2(k), x_3(k), x_4(k) \right)^t$ and define $$A= \begin{pmatrix} a_{11} & a_{12} & {\varepsilon}& {\varepsilon}\cr {\varepsilon}& a_{22} & a & b \cr a_{11} & a_{12} & {\varepsilon}& {\varepsilon}\cr a_{11} & a_{12} & {\varepsilon}& {\varepsilon}\cr \end{pmatrix} \; \mbox{ and } \; E= \begin{pmatrix} 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}\cr {\varepsilon}& 0 & {\varepsilon}& {\varepsilon}\cr {\varepsilon}& {\varepsilon}& 0 & 0 \cr \end{pmatrix} \; .$$ Then, as $$Ax(k)= \begin{pmatrix} a_{11}x_1(k)\oplus a_{12}x_2(k) \cr a_{22}x_2(k)\oplus ax_3(k)\oplus bx_4(k) \cr a_{11}x_1(k)\oplus a_{12}x_2(k) \cr a_{11}x_1(k)\oplus a_{12}x_2(k) \cr \end{pmatrix} \; ,$$ we have $$Ex(k+1) = EAx(k)\iff \left\{ \begin{array}{l} x_1(k+1) = a_{11}x_1(k)\oplus a_{12}x_2(k) \\ x_2(k+1) = a_{22}x_2(k)\oplus ax_3(k)\oplus bx_4(k) \\ x_3(k+1)\oplus x_4(k+1)= a_{11}x_1(k)\oplus a_{12}x_2(k) \end{array}\right. \; .$$ Since $$x_3(k+1)\oplus x_4(k+1)=a_{11}x_1(k)\oplus a_{12} x_2(k)= x_1(k+1)$$ it follows that $$a x_3(k+1)\oplus b x_4(k+1)=a_{21}(k+1) x_1(k+1)$$ for some $a_{21}(k+1)\in [a,b]$. Thus, if we assume that the initial state $x(0)$ satisfies the condition $x_3(0)\oplus x_4(0)= x_1(0)$, for all $k\geq 0$ we have $$x_2(k+1) = a_{22}x_2(k)\oplus ax_3(k)\oplus bx_4(k) = a_{22}x_2(k)\oplus a_{21}(k) x_1(k)$$ for some $a_{21}(k)\in [a,b]$. Therefore, the first two entries of the state vector of the dynamical system $$Ex(k+1) = EAx(k)$$ describe the evolution of system , in the sense that they are equal to the state vector of system corresponding to some choice of $a_{21}(k)$ in $[a,b]$ for each $k\in {\mathbb{N}}$, and vice versa. This idea can be generalized to the case of more than one uncertain holding time by adding two auxiliary variables for each of them. In particular, this method can be used to model manufacturing systems and transportation networks in which some (processing or traveling) times are unknown but bounded. When applying it, in order to satisfy the previous condition on the initial state of the extended state vector, for simplicity we will assume that $x_i(0)={0}$ for all $i$. (0,0)![A manufacturing system (flow-shop)[]{data-label="figure1"}](Figure1.eps "fig:") \#1\#2\#3\#4\#5[ @font ]{} (12765,11052)(376,-10282) (5326,-5236)[(0,0)\[lb\]]{} (6376,-5986)[(0,0)\[lb\]]{} (2776,-5986)[(0,0)\[lb\]]{} (5926,-3811)[(0,0)\[lb\]]{} (9526,-3811)[(0,0)\[lb\]]{} (9526,-511)[(0,0)\[lb\]]{} (7726,-1486)[(0,0)\[lb\]]{} (2326,-3811)[(0,0)\[lb\]]{} (5926,-7111)[(0,0)\[lb\]]{} (2326,-7111)[(0,0)\[lb\]]{} (9526,-7111)[(0,0)\[lb\]]{} (4126,-8086)[(0,0)\[lb\]]{} (5926,-9286)[(0,0)\[lb\]]{} (7726,-4786)[(0,0)\[lb\]]{} (9976,-5986)[(0,0)\[lb\]]{} (9976,-2686)[(0,0)\[lb\]]{} (6376,-2686)[(0,0)\[lb\]]{} (2851,-2686)[(0,0)\[lb\]]{} (6526,-3886)[(0,0)\[lb\]]{} (2326,-511)[(0,0)\[lb\]]{} (12451,-2011)[(0,0)\[lb\]]{} (7651,-9886)[(0,0)\[lb\]]{} (11626,-6286)[(0,0)\[lb\]]{} (11626,-2986)[(0,0)\[lb\]]{} (8626,-9886)[(0,0)\[lb\]]{} (12526,-5311)[(0,0)\[lb\]]{} (12151,-7636)[(0,0)\[lb\]]{} (12151,-4336)[(0,0)\[lb\]]{} (12151,-1036)[(0,0)\[lb\]]{} (10576,-4336)[(0,0)\[lb\]]{} (376,-4336)[(0,0)\[lb\]]{} (5926,614)[(0,0)\[lb\]]{} (5926,-511)[(0,0)\[lb\]]{} (7726,-8086)[(0,0)\[lb\]]{} (2251,-10111)[(0,0)\[lb\]]{} (5851,-10111)[(0,0)\[lb\]]{} (9451,-10111)[(0,0)\[lb\]]{} (3901,-4786)[(0,0)\[lb\]]{} (3901,-1486)[(0,0)\[lb\]]{} As an example, consider the Timed Event Graph of Figure \[figure1\]. This figure represents a manufacturing system (flow-shop) composed of three machines, denoted by $M_1$, $M_2$ and $M_3$, which is supposed to produce three kinds of parts, denoted by $P_1$, $P_2$ and $P_3$ (we refer the reader to [@bcoq] for background on the modeling of Timed Event Graphs using max-plus algebra). We assume that each machine processes each part exactly once, that all parts follow the same sequence of machines: $M_1$, $M_2$ and finally $M_3$, and that the sequencing of part types on each machine is the same: $P_1$, $P_2$ and finally $P_3$. Parts are carried on pallets form one machine to the next one. When a part has been processed by the three machines, it is removed from the pallet, which returns to the staring point for a new part. In Figure \[figure1\], each of the nine transitions corresponds to a combination of a machine and a part type. For instance, the transition labeled $x_6$ corresponds to the combination of machine $M_2$ processing part $P_3$. To each transition corresponds a variable $x_i(k)$ which denotes the earliest time at which the transition can be fired for $k$-th time, that is, the earliest time at which a specific machine can start processing a specific part type for $k$-th time. Places between transitions express the precedence constrains between operations due to the sequencing of operations on the machines. For instance, $x_6$ depends on $x_3$, which corresponds to $M_1$ processing $P_3$, and on $x_5$, which corresponds to $M_2$ processing $P_2$. The holding time assigned to each place is determined, for instance, as a function of some or all of the following variables: the processing time of machines on parts, the transportation time between machines and the set up time on machines when switching from one part type to another. These times are given in Figure \[figure1\]. Note that the times $x_1\rightarrow x_2$, $x_2\rightarrow x_5$ and $x_4\rightarrow x_5$ are not fixed but are assumed to belong to the intervals $\left[ 1,7 \right]$, $\left[ 3,5 \right]$ and $\left[ 1,3 \right]$ respectively. This variation could be due, for instance, to possible breakdowns. All the other times are supposed to be fixed. For simplicity, we assume that in the initial state there is a token in each place. This physically means that each machine can process at most three parts at the same time, and that there are three pallets carrying each part type. The evolution of this flow-shop can be described by a max-plus linear dynamical system of the form $x(k+1)=\bar{A} x(k)$, where $x(k)\in {{\mathbb{R}}_{\max}}^9$ is the vector of $k$-th firing times of the nine transitions and $\bar{A}_{ij}$ is the holding time of the place in the arc that goes from $x_j$ to $x_i$ ($\bar{A}_{ij}={\varepsilon}$ if there is no such an arc, see [@bcoq] for details). Due to the presence of uncertain holding times, three entries of $\bar{A}$ may vary with $k$, so we next use the method described above to model this system. After adding six auxiliary variables $x_i$, $i=10,\dots ,15$ (two for each uncertain holding time), the evolution of the flow-shop of Figure \[figure1\] can be described by the following dynamical system $$\label{EqEvSystem} Ex(k+1) = EAx(k) \; ,$$ where $$E= \left( \begin{array}{ccccccccccccccc} 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 0 & 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 0 & 0 & {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 0 & 0 \end{array} \right)$$ and $$A= \left( \begin{array}{ccccccccccccccc} {\varepsilon}& {\varepsilon}& 4 & {\varepsilon}& {\varepsilon}& {\varepsilon}& 2 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 3 & {\varepsilon}& 1 & 7 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& 5 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 1 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ 4 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 3 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 3 & 5 & 1 & 3 \\ {\varepsilon}& {\varepsilon}& 5 & {\varepsilon}& 4 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& 4 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 3 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 3 & {\varepsilon}& 5 &{\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 2 & {\varepsilon}& 4 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& 4 & {\varepsilon}& {\varepsilon}& {\varepsilon}& 2 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& 4 & {\varepsilon}& {\varepsilon}& {\varepsilon}& 2 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 3 & {\varepsilon}& 1 & 7 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 3 & {\varepsilon}& 1 & 7 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ 4 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 3 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ 4 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 3 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\end{array} \right) \; .$$ Now, assume that we observe the firing times of transitions $x_3$, $x_6$ and $x_8$, that is, we define $y(k)=Cx(k)$, where $$C= \left( \begin{array}{ccccccccccccccc} {\varepsilon}& {\varepsilon}& 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\end{array} \right) \; .$$ Note that these times have a physical meaning. For instance, $x_8$ represents the time at which $M_3$ starts processing $P_2$. Taking into account Theorem \[FGDuality\], in order to determine if there exists a minimal cofinitely generated $(C,A)$-conditioned invariant congruence ${\mathcal{V}_{\rm fg}}(C,A)$ containing ${\mathcal{V}}=\ker E$, in the first place we compute the maximal $(A^t,C^t)$-controlled invariant semimodule ${\mathcal{K}}^(A^t,C^t)$ contained in ${\mathcal{K}}={\mathcal{V}}^\top={\mbox{\rm Im}\,}E^t$. With this aim, we apply Proposition \[Kclosed\] and compute the sequence of semimodules $${\mathcal{X}}_1={\mathcal{K}}\; \makebox{ and }\; {\mathcal{X}}_{k+1}=\phi({\mathcal{X}}_{k}) \; \makebox{ for }\; k\in {\mathbb{N}}\; ,$$ where $$\phi ({\mathcal{X}})= {\mathcal{K}}\cap (A^t)^{-1}({\mathcal{X}}\oplus {\mbox{\rm Im}\,}C^t) \; .$$ This can be done with the help of the max-plus toolbox of Scilab. To be more precise, this is performed expressing the intersection and inverse image of finitely generated semimodules as the solution sets of appropriate homogeneous max-plus linear systems of equations (see [@gaubert98n]), and by solving these systems using the function [mpsolve]{} of this toolbox (see [@AGG08] for a discussion of the complexity of the algorithm involved). In this way, we obtain ${\mathcal{X}}_4={\mathcal{X}}_3\varsubsetneq {\mathcal{X}}_2\varsubsetneq {\mathcal{X}}_1$ and thus ${\mathcal{K}}^(A^t,C^t)={\mathcal{X}}_3$, where ${\mathcal{X}}_3$ is the semimodule generated by the rows of the following matrix $$F= \left( \begin{array}{ccccccccccccccc} 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 0 & 0 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& 0 & 0 \end{array} \right) \; .$$ Therefore, by Theorem \[FGDuality\] we conclude that ${\mathcal{V}_{\rm fg}}(A,C)=\ker F$. By Theorem \[TheoObserver\] we know that it is possible to reconstruct the functional of the state $z(k):=Fx(k)$ in terms of the initial condition $x(0)$ and the observations $y(0),\ldots ,y(m-1)$. More precisely, we have the following dynamic observer allowing us to compute $z(k)$: $$\label{EqObserver} \left\{ \begin{array}{l} z(k+1)=Uz(k)\oplus Vy(k) \\ z(0)=Fx(0) \end{array} \right.$$ where $$U= \left( \begin{array}{cccccc} {\varepsilon}& {\varepsilon}& 2 & {\varepsilon}& {\varepsilon}& {\varepsilon}\\ 4 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& 4 & {\varepsilon}& 3 & {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& {\varepsilon}& 2 & {\varepsilon}& {\varepsilon}& {\varepsilon}\\ 4 & {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}& {\varepsilon}\end{array} \right) \; \makebox{ and } \; V= \left( \begin{array}{ccc} 4 & {\varepsilon}& {\varepsilon}\\ {\varepsilon}& 3 & {\varepsilon}\\ {\varepsilon}& {\varepsilon}& {\varepsilon}\\ {\varepsilon}& 2 & 4 \\ 4 & {\varepsilon}& {\varepsilon}\\ {\varepsilon}& 3 & {\varepsilon}\end{array} \right) \; .$$ The matrices $U$ and $V$ are obtained by solving the equation $FA=UF\oplus VC$ (this kind of one sided max-plus linear systems of equations can be efficiently solved with the help of residuation theory, see [@BlythJan72; @bcoq; @gaubert98n; @cuning]). Observe that, due to the form of $F$, this in particular means that we can determine $x_1(m)$, $x_4(m)$, $x_7(m)$ and $x_9(m)$ in terms of the initial condition and the observations. [cc]{} (0,0)![Random holding times and the corresponding output trajectory[]{data-label="figure2"}](Figure2a.eps "fig:") \#1\#2\#3\#4\#5[ @font ]{} (8278,6298)(901,-5459) (1156,-5410)[(0,0)\[lb\]]{} (1876,-5410)[(0,0)\[lb\]]{} (2596,-5410)[(0,0)\[lb\]]{} (3316,-5410)[(0,0)\[lb\]]{} (4036,-5410)[(0,0)\[lb\]]{} (4710,-5410)[(0,0)\[lb\]]{} (5430,-5410)[(0,0)\[lb\]]{} (6150,-5410)[(0,0)\[lb\]]{} (6870,-5410)[(0,0)\[lb\]]{} (7590,-5410)[(0,0)\[lb\]]{} (8310,-5410)[(0,0)\[lb\]]{} (901,-4466)[(0,0)\[lb\]]{} (901,-3012)[(0,0)\[lb\]]{} (901,-2286)[(0,0)\[lb\]]{} (901,-1559)[(0,0)\[lb\]]{} (901,-832)[(0,0)\[lb\]]{} (901,-105)[(0,0)\[lb\]]{} (901,-5161)[(0,0)\[lb\]]{} (901,-3736)[(0,0)\[lb\]]{} (8626,-4936)[(0,0)\[lb\]]{} (1651,-61)[(0,0)\[lb\]]{} (1651,239)[(0,0)\[lb\]]{} (1651,539)[(0,0)\[lb\]]{} & (0,0)![Random holding times and the corresponding output trajectory[]{data-label="figure2"}](Figure2b.eps "fig:") \#1\#2\#3\#4\#5[ @font ]{} (8458,5952)(721,-5459) (1156,-5410)[(0,0)\[lb\]]{} (1876,-5410)[(0,0)\[lb\]]{} (2596,-5410)[(0,0)\[lb\]]{} (3316,-5410)[(0,0)\[lb\]]{} (4036,-5410)[(0,0)\[lb\]]{} (4710,-5410)[(0,0)\[lb\]]{} (5430,-5410)[(0,0)\[lb\]]{} (6150,-5410)[(0,0)\[lb\]]{} (6870,-5410)[(0,0)\[lb\]]{} (7590,-5410)[(0,0)\[lb\]]{} (8310,-5410)[(0,0)\[lb\]]{} (721,-4628)[(0,0)\[lb\]]{} (721,-4062)[(0,0)\[lb\]]{} (721,-3497)[(0,0)\[lb\]]{} (721,-2932)[(0,0)\[lb\]]{} (721,-2366)[(0,0)\[lb\]]{} (721,-1801)[(0,0)\[lb\]]{} (721,-1236)[(0,0)\[lb\]]{} (721,-670)[(0,0)\[lb\]]{} (721,-136)[(0,0)\[lb\]]{} (901,-5193)[(0,0)\[lb\]]{} (1351, 89)[(0,0)\[lb\]]{} (8626,-4936)[(0,0)\[lb\]]{} \ (0,0)![Random holding times and the corresponding output trajectory[]{data-label="figure2"}](Figure2c.eps "fig:") \#1\#2\#3\#4\#5[ @font ]{} (8638,5952)(541,-5459) (1156,-5410)[(0,0)\[lb\]]{} (1876,-5410)[(0,0)\[lb\]]{} (2596,-5410)[(0,0)\[lb\]]{} (3316,-5410)[(0,0)\[lb\]]{} (4036,-5410)[(0,0)\[lb\]]{} (4710,-5410)[(0,0)\[lb\]]{} (5430,-5410)[(0,0)\[lb\]]{} (6150,-5410)[(0,0)\[lb\]]{} (6870,-5410)[(0,0)\[lb\]]{} (7590,-5410)[(0,0)\[lb\]]{} (8310,-5410)[(0,0)\[lb\]]{} (901,-5193)[(0,0)\[lb\]]{} (721,-4684)[(0,0)\[lb\]]{} (721,-4175)[(0,0)\[lb\]]{} (721,-3667)[(0,0)\[lb\]]{} (721,-3158)[(0,0)\[lb\]]{} (721,-2649)[(0,0)\[lb\]]{} (721,-2140)[(0,0)\[lb\]]{} (721,-1631)[(0,0)\[lb\]]{} (721,-1123)[(0,0)\[lb\]]{} (721,-614)[(0,0)\[lb\]]{} (541,-105)[(0,0)\[lb\]]{} (8626,-4936)[(0,0)\[lb\]]{} (1351, 89)[(0,0)\[lb\]]{} & (0,0)![Random holding times and the corresponding output trajectory[]{data-label="figure2"}](Figure2d.eps "fig:") \#1\#2\#3\#4\#5[ @font ]{} (8563,5960)(541,-5459) (1156,-5410)[(0,0)\[lb\]]{} (1876,-5410)[(0,0)\[lb\]]{} (2596,-5410)[(0,0)\[lb\]]{} (3316,-5410)[(0,0)\[lb\]]{} (4036,-5410)[(0,0)\[lb\]]{} (4710,-5410)[(0,0)\[lb\]]{} (5430,-5410)[(0,0)\[lb\]]{} (6150,-5410)[(0,0)\[lb\]]{} (6870,-5410)[(0,0)\[lb\]]{} (7590,-5410)[(0,0)\[lb\]]{} (8310,-5410)[(0,0)\[lb\]]{} (901,-5193)[(0,0)\[lb\]]{} (721,-4684)[(0,0)\[lb\]]{} (721,-4175)[(0,0)\[lb\]]{} (721,-3667)[(0,0)\[lb\]]{} (721,-3158)[(0,0)\[lb\]]{} (721,-2649)[(0,0)\[lb\]]{} (721,-2140)[(0,0)\[lb\]]{} (721,-1631)[(0,0)\[lb\]]{} (721,-1123)[(0,0)\[lb\]]{} (721,-614)[(0,0)\[lb\]]{} (541,-105)[(0,0)\[lb\]]{} (8551,-4936)[(0,0)\[lb\]]{} (1351, 89)[(0,0)\[lb\]]{} In Figure \[figure2\] we represented the output trajectory (corresponding to the initial condition $x_i(0)={0}$, for $i=1,\dots ,9$) of the usual description, that is through a max-plus linear dynamical system of the form $$\left\{ \begin{array}{l} x(k+1)=\bar{A} x(k) \\ y(k)=C x(k) \end{array} \right. \; ,$$ of the flow-shop of Figure \[figure1\], when the uncertain holding times take the values given on the upper left-hand side of Figure \[figure2\]. These times have been generated at random, in their respective intervals, using Scilab. With this output trajectory and the initial condition, we computed the sequence $\left\{z(k)\right\}_{k\in {\mathbb{N}}}$ given by the dynamic observer . In particular, in Figure \[figure3\] we represented the sequence $\left\{z_3(k)\right\}_{k\in {\mathbb{N}}}$ which is equal to the sequence $\left\{x_7(k)\right\}_{k\in {\mathbb{N}}}$ of firing times of the seventh transition $x_7$ because $z_3(k)=x_7(k)$ by the form of $F$. (0,0)![The (exact) state trajectory reconstructed by the observer[]{data-label="figure3"}](Figure3.eps "fig:") \#1\#2\#3\#4\#5[ @font ]{} (8458,5952)(721,-5459) (1156,-5410)[(0,0)\[lb\]]{} (1876,-5410)[(0,0)\[lb\]]{} (2596,-5410)[(0,0)\[lb\]]{} (3316,-5410)[(0,0)\[lb\]]{} (4036,-5410)[(0,0)\[lb\]]{} (4710,-5410)[(0,0)\[lb\]]{} (5430,-5410)[(0,0)\[lb\]]{} (6150,-5410)[(0,0)\[lb\]]{} (6870,-5410)[(0,0)\[lb\]]{} (7590,-5410)[(0,0)\[lb\]]{} (8310,-5410)[(0,0)\[lb\]]{} (901,-5193)[(0,0)\[lb\]]{} (721,-4628)[(0,0)\[lb\]]{} (721,-4062)[(0,0)\[lb\]]{} (721,-3497)[(0,0)\[lb\]]{} (721,-2932)[(0,0)\[lb\]]{} (721,-2366)[(0,0)\[lb\]]{} (721,-1801)[(0,0)\[lb\]]{} (721,-1236)[(0,0)\[lb\]]{} (721,-670)[(0,0)\[lb\]]{} (721,-105)[(0,0)\[lb\]]{} (1351,164)[(0,0)\[lb\]]{} (8626,-4936)[(0,0)\[lb\]]{} Indeed, it is possible to directly check the dynamic observer . Assume that $z(k)=F x(k)$. Then, for instance, by we have $$z_3(k+1) = 4 z_2(k)\oplus 3 z_4(k)= 4 x_4(k)\oplus 3 x_9(k) \; ,$$ because $z_2(k)= x_4(k)$ and $z_4(k)=x_9(k)$, and by we know that $$x_7(k+1) = 4 x_4(k)\oplus 3 x_9(k) \; .$$ Therefore, $z_3(k+1)=x_7(k+1)$. Let us finally mention that Timed Event Graphs in which the number of initial tokens and holding times are only known to belong to certain intervals have been considered in [@LHCJ04]. This work addresses the existence and computation of a robust control set in order to guarantee that the output of the controlled system is contained in a set of reference outputs. In contrast to the present paper, it is based on interval analysis in dioids, residuation theory and transfer series methods, and does not address any observation problem. [HOvdW06]{} X. Allamigeon, S. Gaubert, and É. Goubault. Inferring min and max invariants using max-plus polyhedra. 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--- abstract: 'The generalized Zakharov–Shabat systems with complex-valued [non-regular]{} Cartan elements and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their gauge equivalent are studied. This study includes: the properties of fundamental analytical solutions (FAS) for the gauge-equivalent to CBC systems and the minimal set of scattering data; the description of the class of nonlinear evolutionary equations, solvable by the inverse scattering method, and the recursion operator, related to such systems; the hierarchies of Hamiltonian structures. The results are illustrated on the example of the multi-component nonlinear Schrödinger (MNLS) equations and the corresponding gauge-equivalent multi-component Heisenberg ferromagnetic (MHF) type models, related to $so(5,\bbbc)$ algebra.' --- [The Generalised Zakharov-Shabat System and the Gauge Group Action]{} [Georgi G. Grahovski$^{1,2}$]{} [$^{1}$ School of Mathematical Sciences, Dublin Institute of Technology,\ Kevin Street, Dublin 8, Ireland]{}\ [$^2$ Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences,\ 72 Tsarigradsko chausee, Sofia 1784, Bulgaria ]{}\ [E-mail: [georgi.grahovski@dit.ie]{} $\quad$ [grah@inrne.bas.bg]{}]{} Introduction {#sec:1} ============ The multi-component Zakharov-Shabat (ZS) system leads to such important systems as the multi-component non-linear Schrödinger equation (NLS), the $N$-wave type equations, etc. All of these systems are integrable via the inverse scattering method. In the class of nonlinear evolution equations (NLEE) related to the Zakharov–Shabat (ZS) system [@ZaSh; @1], the Lax operator belonging to $sl(2,\bbbc) $ algebra is studied. This class of NLEE contains physically important equations as the nonlinear Schrödinger equation (NLS), the sine-Gordon, Korteweg–de-Vriez (KdV) and the modified Korteweg–de-Vriez (mKdV) equations. In the recent years, the gauge equivalent systems to various versions of the generalised Zakharov-Shabat have been systematically studied [@Ferap; @vg1; @vg2; @vgn; @vgn1; @ours; @ours2; @ours3; @ours4; @ours5; @yan1; @yan2; @yan3; @wu]. Recently, it was also shown [@cil2010] that the spectral problem for the Degasperis-Procesi equation can be cast into Zakharov-Shabat form with an $sl(3, \bbbc)$ Lax pair with additional $\bbbz_3$ and $\bbbz_2$ symmetries. Here, we consider the $n \times n $ system [@BS; @Caud80; @ForGib80b]: $$\begin{aligned} \label{eq:1.5} L\Psi (x,t,\lambda )= \left({\rm i}{{\rm d} \over {\rm d} x}+Q(x,t)-\lambda J\right) \Psi (x,t,\lambda ),\end{aligned}$$ where the potential $Q(x,t) $ takes values in the semi-simple Lie algebra ${\frak g} $ [@MiOlPer81; @G86; @ZaMi; @ForKu83]: $$\begin{aligned} \label{eq:1.6} Q(x,t) = \sum_{\alpha \in \Delta _+} \left(q_{\alpha }(x,t)E_{\alpha }+ q_{-\alpha }(x,t)E_{-\alpha } \right)\in {\frak g}_J \qquad J = \sum_{j=1}^{r}a_jH_j \in {\frak h}. \nonumber\end{aligned}$$ For the case of complex $J$, we refer to this system as the Caudrey-Beals-Coifman (CBC) system. Here, $J $ is a [non-regular]{} element in the Cartan subalgebra ${\frak h} $ of ${\frak g} $, ${\frak g}_J $ is the image of $\mbox{ad}_J $, $\{E_{\alpha }, H_i\} $ form the Cartan–Weyl Basis in ${\frak g} $, $\Delta _+ $ is the set of positive roots of the algebra, $r= \mbox{rank}\, {\frak g}= \mbox{dim}\, {\frak h}$. For more details, see section 2 below. The [non-regularity]{} of the Cartan elements means that ${\frak g}_J $ is [not]{} spanned by all root vectors $E_{\alpha } $ of ${\frak g} $, i.e. $\alpha (J) \neq 0 $ for any root $\alpha $ of ${\frak g} $. The given NLEE, as well as the other members of its hierarchy, possess a Lax representation of the form (according to (\[eq:1.5\])): $ [L(\lambda ), M_P(\lambda )]=0, $ where $$\begin{aligned} \label{eq:1.7} M_P\Psi (x,t,\lambda ) = \left({\rm i}{{\rm d} \over {\rm d} t}+\sum_{k=-S}^{P-1} V_k(x,t)-\lambda ^Pf_PI \right) \Psi (x,t,\lambda )=0, \qquad I\in {\frak h},\end{aligned}$$ which must hold identically with respect to $\lambda $. A standard procedure generalising the one proposed by Ablowitz, Kaup, Newell and Segur (AKNS) [@AKNS] allows us to evaluate $V_k(x,t) $ in terms of $q(x,t) $ and its $x$-derivatives. Here and below, we consider only the class of potentials $q(x,t) $ vanishing fast enough for $\|x\|\rightarrow \infty $. Then, one may also check that the asymptotic value of the potential in $M_P(\lambda )$, namely $f^{(P)}(\lambda )=f_P\lambda ^PI$, may be understood as the dispersion law of the corresponding NLEE. Another important step in the development of the Inverse Scattering Method (ISM) is the introduction of the reduction group by A. V. Mikhailov [@2], and further developed in [@ForGib80b; @ForKu83; @ZaMi; @MiOlPer81]. This allows one to prove that some of the well known models in field theory [@2], and also a number of new interesting NLEE [@2; @ForGib80b; @MiOlPer81; @GEI], are integrable by the ISM and possess special symmetry properties. As a result, its potential $q(x,t) $ has a very special form and $J$ [can no-longer be chosen real]{}. This problem of constructing the spectral theory for (\[eq:1.5\]) in the most general case, when $J $ has an arbitrary complex eigenvalues, was initialized by Beals, Coifman and Caudrey [@BC; @BC2; @BC3; @Caud80], and continued by Zhou [@Zhou] in the case when the algebra ${\frak g} $ is $ sl(n) $, $q(x,t) $ vanishing fast enough for $\|x\| \rightarrow \infty$, and no [a priori]{} symmetry conditions are imposed on $q(x,t) $. This was done later for any semi-simple Lie algebras by Gerdjikov and Yanovski [@VYa]. The applications of the differential geometric and Lie algebraic methods to soliton type equations lead to the discovery of a close relationship between the multi-component (matrix) NLS equations and the symmetric and homogeneous spaces [@ForKu83]. In [@ForKu83] it was shown that the integrable MNLS systems have a Lax representation of the form (\[eq:1.5\]), where $J $ is a constant element of the Cartan subalgebra $\mathfrak{h} \subset \mathfrak{g} $ of the simple Lie algebra $\mathfrak{g} $ and $Q(x,t) \equiv [J,\widetilde{Q}(x,t)] \in \mathfrak{g}/\mathfrak{h}$. In other words, $Q(x,t) $ belongs to the co-adjoint orbit $\mathcal{M}_J $ of $\mathfrak{g} $ passing through $J$. Later on, this approach was extended to other types of multi-component integrable models, like the derivative NLS, Korteweg-de-Vriez and modified Korteweg-de-Vriez, $N$-wave, Davey-Stewartson, Kadomtsev-Petviashvili equations [@AthFor; @For]. The choice of $J $ determines the dimension of $\mathcal{M}_J $ which can be viewed as the phase space of the relevant nonlinear evolution equations (NLEE). It is is equal to the number of roots of $ \mathfrak{g} $ such that $\alpha (J)\neq 0 $. Taking into account that if $\alpha $ is a root, then $-\alpha $ is also a root of $\mathfrak{g} $, then $\dim \mathcal{M}_J $ is always even [@loos]. The degeneracy of $J $ means that the subalgebra $\mathfrak{g}_J \subset \mathfrak{g} $ of elements commuting with $J $ (i.e., the kernel of the operator $\ad_J $) is non-commutative. This makes more difficult the derivation of the fundamental analytic solutions (FAS) of the Lax operator (\[eq:1.5\]) and the construction of the corresponding (generating) recursion operator $\Lambda$. The explicit construction of the recursion operator related to (\[eq:1.5\]), using the gauge covariant approach [@G86; @VSG94], is outlined in[@vgn2]. As we mentioned above the Lax operator for the MNLS equations formally has the form (\[eq:1.5\]) but now $J $ is no longer a regular element of $\mathfrak{h} $. This means that the subalgebra $\mathfrak{g}_J\subset \mathfrak{g} $ of elements commuting with $J $ (i.e., the kernel of the operator $\ad_J $) is a non-commutative one. The dispersion law of the MNLS eqn. is quadratic in $\lambda $: $f_{\rm MNLS}=2\lambda ^2 J $. The general form of the MNLS equations and their $M $-operators is: $$\begin{aligned} \label{eq:MNLS} && i {dq \over dt } + 2\ad_J^{-1}{d^2q \over dx^2 } +[q, \pi_0[q, \ad_J^{-1}q]] -2i(\openone - \pi _0)[q, \ad_J^{-1}q_x] =0, \\\label{eq:M-NLS} && M(\lambda)\psi \equiv \left(i{d\over dt} - V_0^{\rm d} + 2i \mbox{ad}_J^{-1}q_x(x,t) +2\lambda q(x,t) -2\lambda ^2 J \right) \psi (x,t,\lambda )=0,\end{aligned}$$ where $V_0^d=\pi_0 \left( [q, \ad_J^{-1}q_x]\right)$ and $\pi_0$ is the projector onto $\mathfrak{g}_J $; (see also Section 4.1 below). The zero-curvature condition $ [L(\lambda ), M_P(\lambda )]=0, $ is invariant under the action of the group of gauge transformations [@ZaTa]. Therefore, the gauge equivalent systems are again completely integrable, possess a hierarchy of Hamiltonian structures, etc, [@FaTa; @1; @VYa; @ZaTa]. The structure of this paper is as follows: In Section 2 we summarize some basic facts about the reduction group and Lie algebraic details. The construction of the fundamental analytic solutions (FAS) is sketched in Section 3 which is done separately for the case of real Cartan elements (Section 3.1) and for complex ones (Section 3.2). The gauge equivalent MHF’s to the MNLS systems are described in Section 4. In Section 5 we present an example of a MNLS type system, related to the $so(5)$ Lie algebra and its corresponding MHF one. The present article presents an extension of our results [@gc]. Preliminaries {#sec:2} ============= Simple Lie Algebras {#ssec:2.1} ------------------- Here, we fix up the notations and the normalization conditions for the Cartan-Weyl generators of ${\frak g} $ [@Helg]. We introduce $h_k\in {\frak h} $, $k=1,\dots,r $ and $E_\alpha $, $\alpha \in \Delta$, where $\{h_k\} $ are the Cartan elements dual to the orthonormal basis $\{e_k\}$ in the root space ${\mathbb E}^r $. Along with $h_k $, we introduce also $$\label{eq:31.1} H_\alpha = {2 \over (\alpha ,\alpha ) } \sum_{k=1}^{r} (\alpha ,e_k) h_k, \quad \alpha \in \Delta ,$$ where $(\alpha ,e_k) $ is the scalar product in the root space ${\mathbb E}^r $ between the root $\alpha $ and $e_k $. The commutation relations are given by [@LA]: $$\begin{aligned} \label{eq:31.2} [h_k,E_\alpha ] = (\alpha ,e_k) E_\alpha , \quad [E_\alpha ,E_{-\alpha }]=H_\alpha , \quad [E_\alpha ,E_\beta ] = \left\{ \begin{array}{ll} N_{\alpha ,\beta } E_{\alpha +\beta } \quad & \mbox{for}\; \alpha +\beta \in \Delta \\ 0 & \mbox{for}\; \alpha +\beta \not\in \Delta \cup\{0\}. \end{array} \right. \end{aligned}$$ We will denote by $\vec{a}=\sum_{k=1}^{r}a_k e_k $ the $r $-dimensional vector dual to $J\in {\frak h} $; obviously $J=\sum_{k=1}^{r}a_k h_k $. If $ J $ is a regular real element in ${\frak h}$, then without restrictions we may use it to introduce an ordering in $\Delta $. Namely, we will say that the root $\alpha \in\Delta _+ $ is positive (negative) if $(\alpha ,\vec{a})>0 $ ($(\alpha ,\vec{a})<0 $ respectively). The normalization of the basis is determined by: $$\begin{aligned} \label{eq:32.1} E_{-\alpha } =E_\alpha ^T, \quad \langle E_{-\alpha },E_\alpha \rangle ={2 \over (\alpha ,\alpha ) }, \quad N_{-\alpha ,-\beta } = -N_{\alpha ,\beta }, \quad N_{\alpha ,\beta } = \pm (p+1),\end{aligned}$$ where the integer $p\geq 0 $ is such that $\alpha +s\beta \in\Delta $ for all $s=1,\dots,p $ $ \alpha +(p+1)\beta \not\in\Delta $ and $\langle \cdot,\cdot \rangle $ is the Killing form of ${\frak g}$ [@GoGr; @Helg]. The root system $\Delta $ of ${\frak g} $ is invariant with respect to the Weyl reflections $A^_\alpha $; on the vectors $\vec{y}\in {\mathbb E}^r $ they act as $ A^_\alpha \vec{y} = \vec{y} - {2(\alpha ,\vec{y}) \over (\alpha ,\alpha )} \alpha , \quad \alpha \in \Delta$. All Weyl reflections $A^_\alpha $ form a finite group $W_{{\frak g}} $ known as the Weyl group. One may introduce, in a natural way, an action of the Weyl group on the Cartan-Weyl basis, namely [@FG; @Hu]: $$\begin{aligned} \label{eq:32.3} A^_\alpha (H_\beta ) \equiv A_\alpha H_\beta A^{-1}_{\alpha } = H_{A^_\alpha \beta }, \qquad A^_\alpha (E_\beta ) \equiv A_\alpha E_\beta A^{-1}_{\alpha } = n_{\alpha ,\beta } E_{A^_\alpha \beta }, \quad n_{\alpha ,\beta }=\pm 1.\end{aligned}$$ It is also well known that the matrices $A_\alpha $ are given (up to a factor from the Cartan subgroup) by $ A_\alpha ={\rm e}^{E_\alpha } {\rm e}^{-E_{-\alpha }} {\rm e}^{E_\alpha } H_A, $ where $H_A $ is a conveniently chosen element from the Cartan subgroup such that $H_A^2=\openone $. As we already mentioned in the Introduction, the MNLS equations correspond to the Lax operator (\[eq:1.5\]) with non-regular (constant) Cartan elements $J\in \mathfrak{h}$. If $J$ is a regular element of the Cartan subalgebra of $\mathfrak{g}$, then $\ad_J$ has as many different eigenvalues as is the number of the roots of the algebra and they are given by $a_j=\alpha_j(J)$, $\alpha_j\in \Delta$. Such $J $’s can be used to introduce ordering in the root system by assuming that $\alpha >0 $ if $\alpha (J)>0 $. In what follows, we will assume that all roots for which $\alpha (J)>0 $ are positive. Obviously, we can consider the eigensubspaces of $\ad_J $ as a grading of the algebra $\mathfrak{g} $. In what follows, we will consider symmetric spaces related to maximally degenerated $J $, i.e. $\ad_J $ has only four non-vanishing eigenvalues: $ \pm a $ and $ \pm 2a $. Then $\mathfrak{g}$ is split into a direct sum of the subalgebra $\mathfrak{g}_0 $ and the linear subspaces $\mathfrak{g}_{\pm } $: $$\begin{aligned} \label{eq:KF14.1} \mathfrak{ g} &=& \mathfrak{ g}_0 \oplus \mathfrak{ g}_+ \oplus \mathfrak{g}_{-},\qquad \mathfrak{ g}_{\pm } = \mathop{\oplus}\limits_{k=1}^2 \mathfrak{ g}_{\pm k} \qquad \mathfrak{ g}_{\pm } = \mbox{l.c.} \left\{ X_{\pm j}\,\|\, [J, X_{\pm }]=\pm ka X_{\pm } \right\}, \qquad k=1,2. \nonumber\end{aligned}$$ The subalgebra $\mathfrak{ g}_0 $ contains the Cartan subalgebra $\mathfrak{h} $ and also all root vectors $E_{\pm \alpha }\in \mathfrak{g} $ corresponding to the roots $\alpha $ such that $\alpha (J)=(\vec{a},\alpha )=0 $. The root system $\Delta $ is split into subsets of roots $\Delta = \theta _0\cup \theta _+\cup (-\theta _+)$, where: $$\begin{aligned} \label{eq:KF14.3} \theta _0 &=& \left\{ \alpha \in \Delta \,\|\, \alpha (J) =0 \right\}, \quad \theta _+ = \left\{ \alpha \in \Delta \,\|\, \alpha (J) >0 \right\}.\end{aligned}$$ The Reduction Group {#ssec:2.2} ------------------- The principal idea underlying Mikhailov’s reduction group [@2] is to impose algebraic restrictions on the Lax operators $L $ and $M$, which will be automatically compatible with the corresponding equations of motion. Due to the purely Lie-algebraic nature of the Lax representation, this is most naturally done by embedding the reduction group as a subgroup of $\mbox{Aut}\, {\frak g} $ – the group of automorphisms of ${\frak g} $. Obviously, to each reduction imposed on $L $ and $M$, there will correspond a reduction of the space of fundamental solutions ${\bf S}_\Psi \equiv \{\Psi (x,t,\lambda )\} $ of (\[eq:1.5\]). Some of the simplest ${\mathbb Z}_2 $-reductions of Zakharov–Shabat systems have been known for a long time (see [@2]) and are related to outer automorphisms of ${\frak g} $ and ${\frak G} $, namely: $$\begin{aligned} \label{eq:C-1} C_1\left( \Psi (x,t,\lambda ) \right) = A_1 \Psi^\dag (x,t,\kappa (\lambda )) A_1^{-1} = \widetilde{\Psi}^{-1}(x,t,\lambda ), \qquad \kappa (\lambda )=\pm \lambda ^, \\ C_2\left( \Psi (x,t,\lambda ) \right) = A_3 \Psi^* (x,t,\kappa (\lambda )) A_3^{-1} = \widetilde{\Psi}(x,t,\lambda ),\end{aligned}$$ where $A_1$ and $A_3$ are elements of the group of authomorphisms $\mbox{Aut}\,{\frak g}$ of the algebra ${\frak g}$. Since our aim is to preserve the form of the Lax pair, we limit ourselves to automorphisms preserving the Cartan subalgebra ${\frak h} $. The reduction group, $G_R$, is a finite group which preserves the Lax representation, i.e. it ensures that the reduction constraints are automatically compatible with the evolution. $G_R $ must have two realizations: i\) $G_R \subset {\rm Aut}{\frak g} $ ii\) $G_R \subset {\rm Conf}\, \Bbb C $, i.e. as conformal mappings of the complex $\lambda $-plane. To each $g_k\in G_R $, we relate a reduction condition for the Lax pair as follows [@2]: $$\label{eq:2.1} C_k(U(\Gamma _k(\lambda ))) = \eta_k U(\lambda ),$$ where $U(x,\lambda )=q(x)-\lambda J $, $C_k\in \mbox{Aut}\; {\frak g} $ and $\Gamma _k(\lambda )$ are the images of $g_k $ and $\eta_k =1 $ or $-1 $ depending on the choice of $C_k $. Since $G_R $ is a finite group, then for each $g_k$, there exists an integer $N_k $ such that $g_k^{N_k} =\openone $. It is well known that $\Aut {\frak g} \equiv V\otimes \Aut_0 {\frak g}$, where $V $ is the group of outer automorphisms (the symmetry group of the Dynkin diagram) and $\Aut_0 {\frak g} $ is the group of inner automorphisms. Since we start with $I,J\in {\frak h}$, it is natural to consider only those inner automorphisms that preserve the Cartan subalgebra ${\frak h} $. Then $\Aut_0 {\frak g} \simeq \Ad_H \otimes W $ where $\Ad_H $ is the group of similarity transformations with elements from the Cartan subgroup and $W $ is the Weyl group of ${\frak g} $. Generically, each element $g_k\in G $ maps $\lambda $ into a fraction-linear function of $\lambda $. Such action however is appropriate for a more general class of Lax operators which are fraction linear functions of $\lambda $. The Caudrey–Beals–Coifman systems {#sec:3} ================================= Fundamental analytical solutions and scattering data for real $J $. {#3.1} ---------------------------------------------------------------- The direct scattering problem for the Lax operator (\[eq:1.5\]) is based on the Jost solutions: $$\begin{aligned} \label{eq:3.1.1} \lim_{x \to \infty }\psi (x,\lambda ){\rm e}^{{\rm i}\lambda Jx}=\openone , \qquad \lim_{x \to -\infty }\phi (x,\lambda ){\rm e}^{{\rm i}\lambda Jx}=\openone ,\end{aligned}$$ and the scattering matrix $$\begin{aligned} \label{eq:3.1.2} T(\lambda )=(\psi (x,\lambda ))^{-1}\phi (x,\lambda ).\end{aligned}$$ The fundamental analytic solutions (FAS) $\chi^{\pm} (x,\lambda ) $ of $L(\lambda ) $ are analytic functions of $\lambda $ for $\mbox{Im}\,\lambda \gtrless 0$ and are related to the Jost solutions by [@G86; @GVY08] $$\begin{aligned} \label{eq:3.1.3} \chi ^{\pm}(x,\lambda )=\phi (x,\lambda )S_J^{\pm}(\lambda )= \psi ^{\pm}(x,\lambda )T_J^{\mp}(\lambda )D_J^{\pm}(\lambda ),\end{aligned}$$ where $T_J^{\pm}(\lambda ) $, $S_J^{\pm}(\lambda ) $ and $D_J^{\pm}(\lambda ) $ are the factors of the Gauss decomposition of the scattering matrix: $$\begin{aligned} \label{eq:3.1.4} &&T(\lambda )=T_J^-(\lambda )D_J^+(\lambda )\hat{S}_J^{+}(\lambda )= T_J^+(\lambda )D_J^-(\lambda )\hat{S}_J^-(\lambda ) \\ &&T_J^{\pm}(\lambda )=\exp \left(\sum_{\alpha >0}t^{\pm}_{\pm \alpha ,J}(\lambda )E_{\alpha } \right), \qquad S_J^{\pm}(\lambda )=\exp \left(\sum_{\alpha >0}s^{\pm}_{\pm \alpha,J }(\lambda )E_{\alpha } \right), \nonumber\\ &&D_J^{+}(\lambda )=I\exp \left(\sum_{j=1}^{r}{2d_J^+(\lambda ) \over (\alpha _j,\alpha _j)}H_j \right), \qquad D_J^{-}(\lambda )=I\exp \left(\sum_{j=1}^{r}{2d_J^-(\lambda ) \over (\alpha _j,\alpha _j)}H_j^- \right). \nonumber\end{aligned}$$ Here, $H_j=H_{\alpha _j} $, $H_j^-=w_0(H_j) $, $\hat{S}\equiv S^{-1} $, $I$ is an element from the universal center of the corresponding Lie group ${\frak G} $ and the superscript $+ $ (or $- $) in the Gauss factors means upper- (or lower-) block-triangularity for $T_J^{\pm} (\lambda ) $, $S_J^{\pm} (\lambda ) $ and shows that $D_J^{+} (\lambda ) $ (or $D_J^{-} (\lambda ) $) are analytic functions with respect to $\lambda $ for $\mbox{Im}\, \lambda >0 $ (or $\mbox{Im}\, \lambda <0$, respectively). On the real axis $\chi ^+(x,\lambda )$ and $\chi ^-(x,\lambda ) $ are linearly related by: $$\begin{aligned} \label{eq:3.1.6} \chi ^+(x,\lambda )=\chi ^-(x,\lambda )G_{J,0}(\lambda ), \qquad G_{J,0}(\lambda )=S_J^+(\lambda )\hat{S}_J^-(\lambda ),\end{aligned}$$ and the sewing function $G_{J,0}(\lambda ) $ may be considered as a minimal system of scattering data provided that the Lax operator (\[eq:1.5\]) has no discrete eigenvalues [@G86]. The CBC Construction for Semisimple Lie Algebras {#3.2} ------------------------------------------------ Here, we will sketch the construction of the FAS for the case of complex-valued regular Cartan elements $J$: $\alpha (\psi )\neq 0$, following the general ideas of Beals and Coifmal [@BC; @BDT] for the $sl(n) $ algebras and [@VYa] for the orthogonal and symplectic algebras. These ideas consist of the following: 1. For potentials $q(x) $ with small norm $\|\|q(x)\|\|_{L^1} <1$, one can divide the complex $\lambda $–plane into sectors and then construct an unique FAS $m_{\nu }(x,\lambda ) $ which is analytic in each of these sectors $\Omega _{\nu } $; 2. For these FAS in each sector, there is a certain Gauss decomposition problem for the scattering matrix $T(\lambda ) $ which has a unique solution in the case of absence of discrete eigenvalues. The main difference between the cases of real-valued and complex-valued $ J $ lies in the fact that for complex $J $ the Jost solutions and the scattering data exist only for the potentials on compact support. We define the regions (sectors) $\Omega _{\nu } $ as consisting of those $ \lambda $’s for which $\mbox{Im}\, (\lambda \alpha (J))\neq 0 $ for any $ \alpha \in \Delta $. Thus, the boundaries of the $\Omega _{\nu } $’s consist of the set of straight lines: $$\begin{aligned} \label{eq:3.2.1} l_{\alpha }\equiv \{ \lambda : \mbox{Im}\, \lambda \alpha (J)=0, \qquad \alpha \in \Delta \},\end{aligned}$$ and to each root $\alpha$, we can associate a certain line $l_{\alpha } $; different roots may define coinciding lines. Note that with the change from $\lambda $ to $\lambda {\rm e}^{{\rm i}\eta } $ and $ J $ to $J{\rm e}^{-{\rm i}\eta } $ (this leads the product $\lambda \alpha (J) $ invariant), we can always choose $l_1 $ to be along the positive real $\lambda $ axis. To introduce an ordering in each sector $\Omega _{\nu }$, we choose the vector $\vec{a}_{\nu }(\lambda )\in {\mathbb E}^r $ to be dual to the element $\mbox{Im}\, \lambda J \in {\frak h} $. Then, in each sector we split $\Delta $ into $$\begin{aligned} \label{eq:3.2.2} \Delta =\Delta _{\nu }^+ \cup \Delta _{\nu }^-, \qquad \Delta _{\nu }^{\pm}=\{\alpha \in \Delta : \mbox{Im}\, \lambda \alpha (J)\gtrless 0,\, \lambda \in \Omega _{\nu }\}.\end{aligned}$$ If $\lambda \in \Omega _{\nu } $ then $-\lambda \in \Omega _{M+\nu } $ (if the lines $l_{\alpha } $ split the complex $\lambda $-plane into $2M $ sectors). We also need the subset of roots: $$\begin{aligned} \label{eq:3.2.2b} \delta _{\nu }=\{\alpha \in \Delta \, : \, \mbox{Im}\, \lambda \alpha (J)=0, \, \lambda \in l_{\nu }\}\end{aligned}$$ which will be a root system of some subalgebra ${\frak g}_{\nu }\subset {\frak g} $. Then, we can write that $$\begin{aligned} \label{eq:3.2.3} {\frak g}= \mathop{\oplus}\limits_{\nu =1}^{M} {\frak g}_{\nu } \qquad \Delta = \mathop{\cup}\limits_{\nu =1}^{M}\delta _\nu \qquad \delta _{\nu }=\delta _{\nu }^+ \cup \delta _{\nu }^-, \qquad \delta _{\nu }^{\pm}=\delta _{\nu }\cap \Delta _{\nu }^\pm \nonumber .\end{aligned}$$ Thus, we can describe in more detail the sets $\Delta _{\nu }^{\pm} $: $$\begin{aligned} \label{eq:3.2.4} \Delta _k^+=\delta _1^+ \cup \delta _2^+ \cup \dots \cup \delta _k^+ \cup \delta _{k+1}^-\cup \dots \cup \delta _M^-, \quad \Delta _{k+M}^+=\Delta _k^-, \quad k=1,\dots , M.\end{aligned}$$ Note that each ordering in $\Delta $ can be obtained from the “canonical” one by an action of a properly chosen element of the weyl group ${\frak W}({\frak g}) $. Now, in each sector $\Omega _{\nu }$, we introduce the FAS $\chi _{\nu }(x,\lambda ) $ and $m_{\nu }(x,\lambda )=\chi _{\nu }(x,\lambda ){\rm e}^{{\rm i}\lambda Jx} $ satisfying the equivalent equation: $$\begin{aligned} \label{eq:3.2.5} {\rm i}{{\rm d} m_{\nu } \over {\rm d} x} + q(x)m_{\nu }(x,\lambda )-\lambda [J,m_{\nu }(x,\lambda )]=0, \qquad \lambda \in \Omega _{\nu }.\end{aligned}$$ If $q(x) $ is a potential on compact support, then the FAS $m_{\nu }(x,\lambda ) $ are related to the Jost solutions by $$\begin{aligned} \label{eq:3.2.6} &&m_{\nu }(x,\lambda )=\phi (x,\lambda )S_{J,\nu }^+(\lambda ){\rm e}^{{\rm i} \lambda Jx} =\psi (x,\lambda )T_{J,\nu }^-(x,\lambda )D_{J,\nu }^+(\lambda ){\rm e}^{{\rm i}\lambda Jx}, \\ &&m_{\nu -1}(x,\lambda )=\phi (x,\lambda )S_{J,\nu }^-(\lambda ){\rm e}^{{\rm i}\lambda Jx} =\psi (x,\lambda )T_{J,\nu }^+(x,\lambda )D_{J,\nu }^-(\lambda ){\rm e}^{{\rm i}\lambda Jx}, \qquad \lambda \in l_{\nu }. \nonumber\end{aligned}$$ From the definitions of $m_{\nu }(x,\lambda ) $ and the scattering matrix $T(\lambda )$, we have $$\begin{aligned} \label{eq:3.2.7} T(\lambda )=T_{J,\nu }^-(\lambda )D_{J,\nu }^+(\lambda )\hat{S}_{J,\nu }^+ (\lambda )= T_{J,\nu }^+(\lambda )D_{J,\nu }^-(\lambda )\hat{S}_{J,\nu }^- (\lambda ) , \quad \lambda \in l_{\nu }\end{aligned}$$ where, in the first equality, we take $\lambda =\mu {\rm e}^{{\rm i}0} $ and for the second– $\lambda =\mu {\rm e}^{-{\rm i}0} $ with $\mu \in l_{\nu } $. The corresponding expressions for the Gauss factors have the form: $$\begin{aligned} \label{eq:3.2.8} &&S_{J,\nu }^+(\lambda )=\exp \left(\sum_{\alpha \in \Delta _{\nu }^+}s_{\nu ,\alpha }^+(\lambda )E_{\alpha } \right), \qquad S_{J,\nu }^-(\lambda )=\exp \left(\sum_{\alpha \in \Delta _{\nu -1}^+}s_{\nu ,\alpha }^-(\lambda )E_{-\alpha } \right), \nonumber\\ &&T_{J,\nu }^+(\lambda )=\exp \left(\sum_{\alpha \in \Delta _{\nu -1}^+}t_{\nu ,\alpha }^+(\lambda )E_{\alpha } \right), \qquad T_{J,\nu }^-(\lambda )=\exp \left(\sum_{\alpha \in \Delta _{\nu }^+}t_{\nu ,\alpha }^-(\lambda )E_{-\alpha } \right), \nonumber\\ &&D_{J,\nu }^+(\lambda )=\exp ({\bf d}_{\nu }^+(\lambda )\cdot {\bf H}_{\nu }), \qquad D_{J,\nu }^-(\lambda )=\exp ({\bf d}_{\nu }^-(\lambda )\cdot {\bf H}_{\nu -1}).\end{aligned}$$ Here ${\bf d}_{\nu }^{\pm}(\lambda )=(d_{\nu ,1}^{\pm},\dots ,d_{\nu ,r}^{\pm}) $ is a vector in the root space and $$\begin{aligned} \label{eq:3.2.9} {\bf H}_{\eta}= \left( {2H_{\eta ,1} \over (\alpha _{\eta,1}, \alpha _{\eta,1}) }, \dots , {2H_{\eta ,r} \over (\alpha _{\eta,r}, \alpha_{\eta,r}) } \right), \qquad ({\bf d}_{\nu }^{\pm}(\lambda ), {\bf H}_{\eta})= \sum_{k=1}^{r}{2d_{\nu ,k}^{\pm}(\lambda )H_{\eta ,k} \over (\alpha _{\eta ,k}, \alpha _{\eta ,k}) },\end{aligned}$$ where $\alpha _{\eta ,k} $ is the $k$-th simple root of ${\frak g} $ with respect to the ordering $\Delta _{\eta}^+$ and $H_{\eta ,k} $ are their dual elements in the Cartan subalgebra ${\frak h} $. The Gauge Group Action {#sec:4} ====================== Before proceeding with the study of the gauge-equivalent systems, the following remark is in order: We can use the gauge transformation commuting with $J $ to simplify $Q $; in particular, we can remove all components of $Q $ in $\mathfrak{ g}_0 $; effectively, this means that our $Q(x,t)= Q_+(x,t) + Q_{-}(x,t)\in \mathfrak{g}_+\cup \mathfrak{g}_- $ can be viewed as a local coordinate in the co-adjoint orbit $\mathcal{M}_J\simeq \mathfrak{g}\backslash \mathfrak{g}_0 $: $$\begin{aligned} \label{eq:KF14.4} Q_+(x,t) = \sum_{\alpha \in\theta _+}^{} q_\alpha(x,t) E_{\alpha }, \qquad Q_{-}(x,t) =\sum_{\alpha \in \theta _-}^{} p_\alpha(x,t) E_{-\alpha }.\end{aligned}$$ The class of the gauge equivalent NLEE’s {#ssec:4.1} ---------------------------------------- The notion of gauge equivalence allows one to associate to any Lax pair of the type (\[eq:1.5\]), (\[eq:1.7\]) an equivalent one [@VYa], solvable by the inverse scattering method for the gauge equivalent linear problem: $$\begin{aligned} \label{eq:2.3} \widetilde{L}\widetilde{\psi }(x,t,\lambda )\equiv \left({\rm i} {{\rm d} \over {\rm d} x}-\lambda S(x,t) \right) \widetilde{\psi }(x,t,\lambda )=0, \nonumber\\ \widetilde{M}\widetilde{\psi }(x,t,\lambda )\equiv \left(i{d \over dt} -2i\lambda \mbox{ad}_{{\cal S}}^{-1}{\cal S}_x -2\lambda ^2 {\cal S} \right) \widetilde{\psi }(x,t,\lambda)=0,\end{aligned}$$ where $\widetilde{\psi }(x,t,\lambda ) = g^{-1}(x,t)\psi (x,t,\lambda )$, $ S = \mbox{Ad}_{g}\cdot J \equiv g^{-1}(x,t)Jg(x,t), $ and $g(x,t)=m_\nu (x,t,0) $ is FAS at $\lambda =0 $. The functions $m_\nu (x,t,\lambda)$ are analytic with respect to $\lambda$ in each sector $\Omega_\nu$ and do not lose their analyticity for $\lambda=0$ (in the case of potential on compact support). From the integral representation for the FAS $m_\nu (x,t,\lambda)$ at $\lambda=0$, it follows that $$m_1(x,t,0)= \cdots = m_\nu(x,t,0)=\cdots = m_{2M}(x,t,0).$$ Therefore, the gauge group action is well defined. The zero-curvature condition $[\widetilde{L},\widetilde{M}]=0 $ gives: $$\begin{aligned} \label{eq:2.5} i {d {\cal S} \over dt } + 2 { d\over dx } \left(\mbox{ad}_{{\cal S}}^{-1} {d {\cal S}\over dx }\right) =0.\end{aligned}$$ Both Lax operators $L(\lambda)$ and $\widetilde{L}(\lambda)$ have equivalent spectral properties and spectral data and therefore, the classes of NLEE’s related to them are equivalent. Following [@AKNS], one can consider more general $\widetilde{M} $-operators of the form: $$\label{eq:M-op} \widetilde{M}(\lambda)\widetilde{\Psi}\equiv {\rm i}{d\widetilde{\Psi} \over dt}+\left( \sum_{k=1}^{N}\widetilde{V}_k(x,t)\lambda^k \right)\widetilde{\Psi}(x,t,\lambda)=0, \qquad f(\lambda ) = \lim_{x\to\pm\infty } \widetilde{V}(x,t,\lambda ),$$ where $\widetilde{V}(x,t,\lambda )=\sum_{k=1}^{N}\widetilde{V}_k(x,t)\lambda^k$. The Lax representation $[\widetilde{L}(\lambda), \widetilde{M}(\lambda)]=0$ leads to recurrent relations between $\widetilde{V}_k(x,t)=\widetilde{V}_{k}^{\rm f}+\widetilde{V}_{k}^{\rm d}$ $$\begin{aligned} \label{eq:RecLax} && \widetilde{V}_{k+1}^{\rm f}(x,t)\equiv \pi_{\cal S}(\widetilde{V}_{k+1}) =\widetilde{\Lambda}_\pm \widetilde{V}_k^{\rm f}(x,t)+{\rm i} \ad_{\cal S}^{-1} [C_k,\ad_{\cal S}^{-1}{\cal S}_x(x,t)], \\ && \widetilde{V}_{k}^{\rm d}(x,t)\equiv (\openone -\pi_{\cal S})(\widetilde{V}_{k}) =\widetilde{C}_k + \int_{\pm \infty}^x dy\,[ \ad_{\cal S}^{-1}{\cal S}_x(y,t), \widetilde{V}_k^{\rm f}(y,t)], \qquad k=1,...,N; \nonumber\end{aligned}$$ where $ \pi_{\cal S}=\ad_{\cal S}^{-1}\circ\ad_{\cal S}$ and $\widetilde{C}_k=(\openone -\pi_{\cal S})\widetilde{C}_k$ are block-diagonal integration constants, for details see, e.g. [@AKNS; @ForKu83]. These relations are resolved by the recursion operators (\[eq:Lamb\]): $$\begin{aligned} \label{eq:Lamb} \widetilde{\Lambda}=\mbox{Ad}_{g}\cdot \Lambda={1\over 2}\left(\widetilde{\Lambda}_+ + \widetilde{\Lambda}_-\right), \qquad \widetilde{\Lambda}_\pm =\mbox{Ad}_{g} \cdot\Lambda _\pm\end{aligned}$$ where $$\begin{aligned} \label{eq:Lambd} \Lambda _\pm Z &=& \mbox{ad}_J^{-1} (\openone -\pi_0) \left( i {dZ\over dx } + [q(x), Z(x)] + i \left[ q(x) , \int_{\pm\infty }^{x} dy \; \pi_0 [q(y), Z(y)] \right] \right),\end{aligned}$$ and we assume that $Z \equiv \pi_{0} Z \in \mathcal{M}_{\cal S}$, where $\pi_{0}=\mbox{ad}_J^{-1} \circ \mbox{ad}_J$ is the projector onto the off-diagonal part. As a result, we obtain that the class of (generically nonlocal) multi-component Heisenberg feromagnet (MHF) type models, solvable by the ISM, have the form: $$\label{eq:NLEE} i\ad_{\cal S}^{-2} {d{\cal S} \over dt } = \sum_{k=0}^{N}\widetilde{\Lambda}_\pm^{N-k} \left[\widetilde{C}_k,\ad_{\cal S}^{-2}{\cal S}(x,t)\right], \qquad f(\lambda ) =\left( \begin{array}{cc} f^+(\lambda ) & 0 \\ 0& f^-(\lambda )\end{array}\right),$$ where $f(\lambda ) =\sum_{k=0}^{N}\widetilde{C}_k\lambda ^{N-k} $ determines their dispersion law. The NLEE (\[eq:2.5\]) become local if $f(\lambda )=f_0(\lambda ){\cal S} $, where $f_0(\lambda ) $ is a scalar function. In particular, if $f(\lambda )=-2\lambda ^2{\cal S} $ we get the MHF eqn. (\[eq:2.5\]). The Minimal Set of Scattering Data for $L(\lambda)$ and $\widetilde{L}(\lambda)$ {#ssec:4.2} -------------------------------------------------------------------------------- We skip the details about CBC construction which can be found in [@VYa] and go to the minimal set of scattering data for the case of complex $J $ which are defined by the sets ${\cal F}_1 $ and ${\cal F}_2 $ as follows: $$\begin{aligned} \label{eq:3.2.10} {\cal F}_1= \mathop{\cup}\limits_{\nu =1}^{2M}{\cal F}_{1,\nu }, \qquad {\cal F}_2= \mathop{\cup}\limits_{\nu =1}^{2M}{\cal F}_{2,\nu }, \nonumber\\ {\cal F}_{J,1,\nu }= \{\rho _{J,B,\nu ,\alpha }^{\pm }(\lambda ), \, \alpha \in \delta _{\nu }^+, \, \lambda \in l_{\nu }\} \qquad {\cal F}_{J,2,\nu }= \{\tau _{J,B,\nu ,\alpha }^{\pm }(\lambda ), \, \alpha \in \delta _{\nu }^+, \, \lambda \in l_{\nu }\},\end{aligned}$$ where $$\begin{aligned} \label{eq:3.2.12} \rho _{J,B,\nu ,\alpha }^{\pm }(\lambda )=\langle S_{J,\nu }^{\pm}(\lambda )B\hat{S}_{J,\nu }^{\pm}(\lambda ), E_{\mp \alpha }\rangle , \qquad \tau _{J,B,\nu ,\alpha }^{\pm }(\lambda )=\langle T_{J,\nu }^{\pm}(\lambda )B\hat{T}_{J,\nu }^{\pm}(\lambda ), E_{\mp \alpha }\rangle ,\end{aligned}$$ with $\alpha \in \delta _{\nu }^+ $, $\lambda \in l_{\nu } $ and $B $ is a properly chosen regular element of the Cartan subalgebra ${\frak h} $. Without loss of generality, we can take in (\[eq:3.2.12\]) $B=H_{\alpha } $. Note that the functions $\rho _{J,B,\nu ,\alpha }^{\pm}(\lambda ) $ and $\tau _{J,B,\nu ,\alpha }^{\pm}(\lambda ) $ are continuous functions of $\lambda $ for $\lambda \in l_{\nu } $. If we choose $J $ in such way that $2M=\|\Delta \| $– the number of the roots of ${\frak g} $, then to each pair of roots $\{\alpha ,-\alpha \} $ one can relate a separate pair of rays $\{l_{\alpha }, l_{\alpha +M}\} $, and $l_{\alpha }\neq l_{\beta } $ if $\alpha \neq \pm \beta $. In this case, each of the subalgebras ${\frak g}_{\alpha } $ will be isomorphic to $sl(2) $. In order to determine the scattering data for the gauge equivalent equations, we need to start with the FAS for these systems: $$\begin{aligned} \label{eq:3.1} \widetilde{m }_\nu^{\pm}(x,\lambda )=g^{-1}(x,t)m_\nu^{\pm}(x,\lambda )g_-,\end{aligned}$$ where $g_-= \lim_{x \to -\infty }g (x,t) $ and due to (\[eq:1.7\]) and $g_-=\hat{T}(0)$. In order to ensure that the functions $\widetilde{\xi }^{\pm}(x,\lambda ) $ are analytic with respect to $\lambda $, the scattering matrix $T(0) $ at $ \lambda =0 $ must belong to ${\frak H}\otimes {\frak G}_0$, where ${\frak H} $ is the corresponding Cartan subgroup and ${\frak G}_0 $ is the subgroup, that corresponds to the subalgebra ${\frak g}_0$. Then, equation (\[eq:3.1\]) provides the fundamental analytic solutions of $\widetilde{L} $. We can calculate their asymptotics for $x\to\pm\infty $ and thus establish the relations between the scattering matrices of the two systems: $$\begin{aligned} \label{eq:3.2} \lim_{x \to -\infty }\widetilde{\xi }^+(x,\lambda )= e^{-{\rm i} \lambda Jx}T(0)S^+_J(\lambda )\hat{T}(0) \qquad \lim_{x \to \infty }\widetilde{\xi }^+(x,\lambda )= e^{-{\rm i} \lambda Jx}T^-_J(\lambda )D^+_J(\lambda )\hat{T}(0)\end{aligned}$$ with the result: $ \widetilde{T}(\lambda )= T(\lambda )\hat{T}(0)$. The factors in the corresponding Gauss decompositions are related by: $$\begin{aligned} \label{eq:3.4} \widetilde{S}_J^{\pm}(\lambda )= T(0)S_J^{\pm}(\lambda )\hat{T}(0), \qquad \widetilde{T}_J^{\pm}(\lambda )=T_J^{\pm}(\lambda ) \nonumber\qquad \widetilde{D}_J^{\pm}(\lambda )=D_J^{\pm}(\lambda )\hat{T}(0).\end{aligned}$$ On the real axis, again, the FAS $\widetilde{\xi }^+(x,\lambda ) $ and $\widetilde{\xi }^-(x,\lambda ) $ are related by $ \widetilde{\xi }^+(x,\lambda )=\widetilde{\xi }^-(x,\lambda )\widetilde{G}_{J,0}(\lambda ) $ with the normalization condition $\widetilde{\xi }(x,\lambda =0)=\openone $ and $\widetilde{G}_{J,0}(\lambda )=\widetilde{S}_J^+(\lambda )\hat{\widetilde{S}}_J^-(\lambda ) $ again can be considered as a minimal set of scattering data. The minimal set of scattering data for the gauge-equivalent CBC systems are defined by the sets $\widetilde{\cal F}_1 $ and $\widetilde{\cal F}_2 $ as follows: $$\begin{aligned} \label{eq:3.2.10a} \widetilde{\cal F}_1= \mathop{\cup}\limits_{\nu =1}^{2M}\widetilde{\cal F}_{1,\nu }, \qquad \widetilde{\cal F}_2= \mathop{\cup}\limits_{\nu =1}^{2M}\widetilde{\cal F}_{2,\nu }, \nonumber\\ \widetilde{\cal F}_{1,\nu }= \{\widetilde{\rho }_{B,\nu ,\alpha }^{\pm }(\lambda ), \, \alpha \in \delta _{\nu }^+, \, \lambda \in l_{\nu }\} \qquad \widetilde{\cal F}_{2,\nu }= \{\widetilde{\tau }_{B,\nu ,\alpha }^{\pm }(\lambda ), \, \alpha \in \delta _{\nu }^+, \, \lambda \in l_{\nu }\},\end{aligned}$$ where $$\begin{aligned} \label{eq:3.2.12a} \widetilde{\rho }_{J,B,\nu ,\alpha }^{\pm }(\lambda )=\langle T_J(0)S_{J,\nu }^{\pm}(\lambda )B\hat{S}_{J,\nu }^{\pm}(\lambda )\hat{T}_J(0), E_{\mp \alpha }\rangle , \quad \widetilde{\tau }_{J,B,\nu ,\alpha }^{\pm }(\lambda )=\langle T_{J,\nu }^{\pm}(\lambda )B\hat{T}_{J,\nu }^{\pm}(\lambda ), E_{\mp \alpha }\rangle ,\end{aligned}$$ with $\alpha \in \delta _{\nu }^+ $, $\lambda \in l_{\nu } $ and $B $ is again a properly chosen regular element of the Cartan subalgebra ${\frak h} $. Without loss of generality, we can take in (\[eq:3.2.12a\]) $B=H_{\alpha } $ (as in (\[eq:3.2.12\])). The functions $\widetilde{\rho }_{B,\nu ,\alpha }^{\pm}(\lambda ) $ and $\widetilde{\tau }_{B,\nu ,\alpha }^{\pm}(\lambda ) $ are continuous functions of $\lambda $ for $\lambda \in l_{\nu }$, and have the same analyticity properties as the functions $\rho _{B,\nu ,\alpha }^{\pm}(\lambda ) $ and $\tau _{B,\nu ,\alpha }^{\pm}(\lambda ) $. Integrals of Motion and Hierarchies of Hamiltonian Structures {#ssec:4.3} ------------------------------------------------------------- If $q(x,t) $ evolves according to the MNLS (\[eq:NLS-so5\]), then $$\label{eq:dS_J} i {dS_{J}^{\pm} \over dt } -2\lambda ^2 [J, S_{J}^{\pm} (t,\lambda )] =0, \qquad i {dT_{J}^{\pm} \over dt } -2\lambda ^2 [J, T_{J}^{\pm} (t,\lambda )] =0, \qquad {dD_{J}^{\pm} \over dt }=0.$$ This means that the MNLS eq. (\[eq:NLS-so5\]) has four series of integrals of motion. This is due to the special (degenerate) choice of the dispersion law $f_{\rm MNLS}=-2\lambda ^2J $. We have to remember, however, that only two of these four series are in involution, which in turn is related to the non-commutativity of the subalgebra $\mathfrak{g}_J $. Both classes of NLEE’s are infinite dimensional, completely integrable Hamiltonian systems and possess hierarchies of Hamiltonian structures. The phase space ${\cal M}_{\rm MNLS} $ is the linear space of all off-diagonal matrices $q(x,t) $ tending fast enough to zero for $x\to\pm\infty $. The hierarchy of pair-wise compatible symplectic structures on ${\cal M}_{\rm MNLS} $ is provided by the $2 $-forms: $$\label{eq:ome-nls} \Omega _{\rm MNLS}^{(k)} = {\rm i } \int_{-\infty }^{\infty } dx \tr \left( \delta q(x,t) \wedge \Lambda ^k [J, \delta q(x,t) ] \right),$$ where $\Lambda= (\Lambda_+ + \Lambda_-)/2 $ is the generating (recursion) operator for (\[eq:1.5\]) defined in (\[eq:Lambd\]). The symplectic forms $\Omega _{\rm MNLS}^{(k)}$ can be expressed in terms of the scattering data for $L(\lambda)$: $$\begin{aligned} \label{eq:3.2.a} && \Omega _{\rm MNLS}^{(k)} = {c_k \over 2\pi }\sum_{\nu =1}^{M} \int_{\lambda \in l_{\nu }\cup l_{M+\nu } } d\lambda \lambda ^k \left( \Omega _{0,\nu }^+(\lambda ) - \Omega _{0,\nu }^-(\lambda )\right), \nonumber\\ && \Omega _{0,\nu }^\pm(\lambda ) = \left\langle \hat{D}_{J,\nu } ^\pm(\lambda ) \hat{T}_{J,\nu }^\mp(\lambda ) \delta T_{J,\nu }^\mp(\lambda ) D_{J,\nu }^\pm(\lambda ) \wedge \hat{S}_{J,\nu }^\pm(\lambda ) \delta S_{J,\nu }^\pm(\lambda ) \right\rangle .\end{aligned}$$ Note that the kernels of $\Omega _{\rm MNLS}^{(k)}$ differ only by the factor $\lambda^k$ so all of them can be cast into canonical form simultaneously. The phase space ${\cal M}_{\rm MHF} $ of the gauge equivalent to the MNLS systems is the manifold of all ${\cal S}(x,t) $, satisfying appropriate boundary conditions. The family of compatible $2 $-forms is: $$\label{eq:ome-hf} \widetilde{\Omega} _{\rm MHF}^{(k)} = {i \over 4} \int_{-\infty }^{\infty } dx \tr \left( \delta S^{(0)} \wedge \widetilde{\Lambda} ^k [S^{(0)}, \delta S^{(0)}(x,t) ] \right).$$ Again, like for the “canonical” MNLS models, the symplectic forms $\Omega _{\rm MHF}^{(k)}$ for their gauge-equivalent MHF’s can be expressed in terms of the scattering data for $\widetilde{L}(\lambda)$: $$\begin{aligned} \label{eq:3.2.b} && \widetilde{\Omega} _{\rm MHF}^{(k)} = {c_k \over 2\pi }\sum_{\nu =1}^{M} \int_{\lambda \in l_{\nu }\cup l_{M+\nu } } d\lambda \lambda ^k \left( \widetilde{\Omega} _{0,\nu }^+(\lambda ) - \widetilde{\Omega }_{0,\nu }^-(\lambda )\right), \nonumber\\ && \widetilde{\Omega} _{0,\nu }^\pm(\lambda ) = \left\langle \hat{\widetilde{D}}_{J,\nu } ^\pm(\lambda ) \hat{\widetilde{T}}_{J,\nu }^\mp(\lambda ) \delta \widetilde{T}_{J,\nu }^\mp(\lambda ) \widetilde{D}_{J,\nu }^\pm(\lambda ) \wedge \hat{\widetilde{S}}_{J,\nu }^\pm(\lambda ) \delta \widetilde{S}_{J,\nu }^\pm(\lambda ) \right\rangle .\end{aligned}$$ The spectral theory of these two operators $\Lambda$ and $\widetilde{\Lambda }$ underlie all the fundamental properties of these two classes of gauge equivalent NLEE, for details see [@VYa]. Note that the gauge transformation relates, in a nontrivial manner, the symplectic structures, i.e. $\Omega _{\rm MNLS}^{(k)} \simeq\widetilde{\Omega} _{\rm MHF}^{(k+2)} $ [@RST; @VYa]. Example ${\frak g}\simeq so(5,\bbbc)$ {#sec:5} ===================================== Here, we consider a MNLS model with the Lax operators $L(\lambda)$ and $M(\lambda)$ belonging to $so(5,\bbbc)$ Lie algebra. \[fig:1\] (95.11,94.77) (80.00,50.00)[(1,0)[0.2]{}]{} (50.00,80.00)[(0,1)[0.2]{}]{} (80.33,57.33)[(0,0)\[cc\][$\Omega_1$]{}]{} (65.00,71.00)[(0,0)\[cc\][$\Omega_2$]{}]{} (34.67,71.00)[(0,0)\[cc\][$\Omega_3$]{}]{} (19.00,57.33)[(0,0)\[cc\][$\Omega_4$]{}]{} (19.33,39.00)[(0,0)\[cc\][$\Omega_5$]{}]{} (34.67,26.33)[(0,0)\[cc\][$\Omega_6$]{}]{} (64.67,26.33)[(0,0)\[cc\][$\Omega_7$]{}]{} (80.33,39.00)[(0,0)\[cc\][$\Omega_8$]{}]{} (97.33,49.67)[(0,0)\[cc\][$l_1$ (${\frak g}_0$)]{}]{} (92.00,69.33)[(0,0)\[cc\][$l_2$]{}]{} (92.33,26.00)[(0,0)\[cc\][$l_8$]{}]{} (51.00,4.00)[(0,0)\[cc\][$l_7$]{}]{} (51.67,91.00)[(0,0)\[cc\][$l_3$]{}]{} (9.00,70.00)[(0,0)\[cc\][$l_4$]{}]{} (6.67,49.67)[(0,0)\[cc\][$l_5$]{}]{} (8.33,27.00)[(0,0)\[cc\][$l_6$]{}]{} (90.33,90.00) (90.33,90.00)[(0,0)\[cc\][$\lambda$]{}]{} This algebra has 4 positive roots: $\alpha_1=e_1-e_2$, $\alpha_2=e_2$, $\alpha_3=\alpha_1+\alpha_2$ and $\alpha_4=\alpha_1+2\alpha_2$. Let us choose also $J $ to be a degenerate Cartan element ($\alpha _1(J)=0 $) so that the set of roots $\Delta _1^+=\{ \alpha _2,\alpha_3,\alpha _4\}$ of $so(5)$, for which $\alpha (J)\neq 0 $ labels the coefficients of the potential $q(x,t)$: $$\begin{aligned} \label{grah:ex-nls} q(x,t)&\equiv & \sum_{\alpha \in \Delta _1^+}^{} (q_{\alpha }E_{\alpha } + p_{\alpha }E_{-\alpha }) = \left(\begin{array}{ccccc} 0 & 0 & q_{11} & q_{12} & 0\\ 0 & 0 & q_1 & 0 & q_{12} \\ p_{11} & p_1 & 0 & q_1 & -q_{11}\\ p_{12} & 0 & p_1 & 0 & 0\\ 0 & p_{12} & -p_{11} & 0 & 0\\ \end{array} \right); \qquad J=\mbox{diag}\,(a,a,0,-a,-a)\nonumber.\end{aligned}$$ Here, $q_1$ and $p_1$ are related to the root $\alpha_2$; the labels $mn$ in $q_{mn}(x,t)$ and $p_{mn}(x,t)$ refer to the roots $(mn)\leftrightarrow m\alpha_1+n\alpha_2$. The continuous spectrum of the Lax operator $L(\lambda)$ related to (\[eq:6MNLS\]) is depicted on fig.1. Then, the corresponding MNLS type system is of the form: $$\begin{aligned} \label{eq:6MNLS} \hspace{-7mm} i{\partial q_{12}\over \partial t}+ {1\over 2a}{\partial ^2q_{12}\over \partial x^2}+{1 \over a}q_{12}(q_1p_1+q_{11}p_{11}+q_{12}p_{12}) + {i\over a}q_{1}q_{11,x}-{i\over a}q_{11}q_{1,x}&=&0, \nonumber\\ \hspace{-7mm} i{\partial q_{11}\over \partial t}+ {1\over a}{\partial ^2q_{11}\over \partial x^2}+{1 \over a}q_{11}(q_1p_1+q_{11}p_{11}+{1\over 2}q_{12}p_{12}) + {i\over a}q_{12}p_{1,x}+{i\over 2a}q_{12,x}p_{1}&=&0, \nonumber\\ \hspace{-7mm} i{\partial q_{1}\over \partial t}+ {1\over a}{\partial ^2q_{1}\over \partial x^2}+{1 \over a}q_{1}(q_1p_1+q_{11}p_{11}+{1\over 2}q_{12}p_{12}) - {i\over a}q_{12}p_{11,x}-{i\over 2a}q_{12,x}p_{11}&=&0, \\ \hspace{-7mm} i{\partial p_{1}\over \partial t}- {1\over a}{\partial ^2p_{1}\over \partial x^2}-{1 \over a}p_{1}(q_1p_1+q_{11}p_{11}+{1\over 2}q_{12}p_{12}) - {i\over a}p_{12}q_{11,x}-{i\over 2a}p_{12,x}q_{11}&=&0, \nonumber\\ \hspace{-7mm} i{\partial p_{11}\over \partial t} -{1\over a}{\partial ^2p_{11}\over \partial x^2}-{1 \over a}p_{11}(q_1p_1+q_{11}p_{11}+{1\over 2}q_{12}p_{12}) + {i\over a}p_{12}q_{1,x}+ {i\over 2a}p_{12,x}q_{1}&=&0, \nonumber\\ \hspace{-7mm} i{\partial p_{12}\over \partial t}- {1\over 2a}{\partial ^2p_{12}\over \partial x^2}-{1 \over a}p_{12}(q_1p_1+q_{11}p_{11}+q_{12}p_{12}) + {i\over a}p_{1}p_{11,x}-{i\over a}p_{11}p_{1,x}&=&0. \nonumber\end{aligned}$$ In order to evaluate the gauge equivalent recursion operator, we use the gauge covariant approach. First, we need to express any function $f(K)$ of the the operator $K=\mbox{ad}_J$ through the projectors onto its eigensubspaces. Our choice of $J $ means that $K=\ad_J $ has five different eigenvalues: $-2a $, $-a $, $0 $, $a $ and $2a $. Then, the minimal characteristic polynomial for $K $ is $ K(K^2-a^2)(K^2-4a^2) =0$. Let us also introduce the projectors onto the eigensubspaces of $K $ as follows: $$\begin{aligned} \label{eq:1_5)} \pi_{\pm 2} = {K(K^2-a^2) (K\pm 2a)\over 24a^4 }, \quad \pi_{\pm 1} = -{K(K^2-4a^2) (K\pm a)\over 6a^4 }, \quad \pi_0 = {(K^2-a^2) (K^2-4a^2) \over 4a^4 }. \end{aligned}$$ Using the characteristic equation $ K(K^2-a^2)(K^2-4a^2) =0$, it is easy to check that $\pi_{j} $ are orthogonal projectors; i.e. they satisfy: $ \pi_{j} \pi_{k} = \delta _{jk}\pi_{j}$ for all $j,k=\pm 2 $, $\pm 1 $, $0 $ and that $ K\pi_{\pm 2} = \pm 2a \pi_{\pm 2}$, $K\pi_{\pm 1} = \pm a \pi_{\pm 1}$, $K\pi_0 = 0$. Thus any function $f(K) $ can be expressed in terms of these projectors: $ f(K) = f(2a)\pi_{2} + f(a)\pi_{1} + f(0)\pi_{0}+ f(-a)\pi_{-1} f(-2a)\pi_{-2} $ provided $f(\lambda ) $ is regular for $\lambda =\pm 2a $, $\pm a $ and $0 $. Note also that $\ad_J=K $ introduces a grading on $\mathfrak{g} =\mathop{\oplus}\limits_{j=-2}^2 \mathfrak{g}_j $ and the projectors $\pi_j $ project precisely onto $\mathfrak{g}_j $. Obviously, $\mathfrak{g}_j=\pi_j \mathfrak{g} $, $\mathfrak{g}_0\equiv \mathfrak{g}_J $ and $\mathcal{M}_J \simeq \mathfrak{g}\backslash \mathfrak{g}_J $. Then, applying a gauge transformation, one can recalculate easily the projectors on the eigensubspaces of $\ad_{S(x)}\equiv \widetilde{K}(x) = g_0^{-1}Kg_0(x,t) $: $$\begin{aligned} \label{eq:1_5} \widetilde{\pi}_{\pm 2} = {\widetilde{K}(\widetilde{K}^2-a^2) ( \widetilde{K}\pm 2a)\over 24a^4 }, \quad \widetilde{\pi}_{\pm 1} = -{\widetilde{K}(\widetilde{K}^2-4a^2) (\widetilde{K}\pm a)\over 6a^4 }, \quad \widetilde{\pi}_0 = {(\widetilde{K}^2-a^2) (\widetilde{K}^2-4a^2) \over 4a^4 }. \end{aligned}$$ Using these formulae, one can cast also, the gauge-equivalent MHF-type system (\[eq:2.5\]) in the form: $$\label{eq:2.5a} iS_t - {5 \over 4a^2 } [ S, S_{xx}] + {1 \over 4a^4 } \left( (\ad_S)^3 S_x \right)_x =0,$$ where ${\cal S}$ is constrained by ${\cal S}({\cal S}^2-a^2)^2=0$. In addition, the operator $\widetilde{K}(x,t) $ satisfies the equation $ \widetilde{K}(\widetilde{K}^2-a^2)(\widetilde{K}^2-4a^2) =0$. Now, let us apply to the system (\[eq:6MNLS\]) the following reduction: $L(\lambda)=-L(\lambda^)^\dag$. This implies, that the potential matrix $Q(x,t)$ is hermitian, i.e. that $p_\alpha =q_\alpha^$, and, that the matrix elements of the Cartan element $J$ are real. As a result, we get the following 3-component MNLS system for the complex-valued fields $q_{1}(x,t)$, $q_{11}(x,t)$ and $q_{12}(x,t)$: $$\begin{aligned} &&i{dq_{12}\over dt}+ {1\over 2a}{d^2q_{12}\over dx^2}-{1 \over a}q_{12} (\|q_1\|^2 +\|q_{11}\|^2 + \|q_{12}\|^2) + {i\over a} q_{1}q_{11,x}- {i\over a}q_{11}q_{1,x}=0 \nonumber\\ &&i{dq_{11}\over dt}+ {1\over a}{d^2q_{11}\over dx^2}-{1 \over a} q_{11}(\|q_1\|^2 + \|q_{11}\|^2 + {1\over 2} \|q_{12}\|^2 ) + {i\over a} q_{12} q^_{1,x}+ {i\over 2a}q_{12,x}q^_{1}=0 \\ \label{eq:NLS-so5} &&i{dq_{1}\over dt}+ {1\over a}{d^2q_{1}\over dx^2}-{1 \over a} q_{1} (\|q_1\|^2 + \|q_{11}\|^2 + {1\over 2} \|q_{12}\|^2 ) - {i\over a} q_{12}q^_{11,x} - {i\over 2a} q_{12,x}q^_{11}=0,\nonumber \end{aligned}$$ Using the well-known isomorphism between the algebras $so(5,\bbbc)$ and $sp(4,\bbbc)$ [@Helg], due to the purely Lie-algebraic nature of the Lax representation, one can convert this 3-component MNLS system in the typical representation of $sp(4,\bbbc)$[^1]. This system is equivalent to the 3-component one, describing $F=1$ spinor Bose-Einstein condensate [@IMW04; @kagg] in one dimensional approximation. Finally, applying the reduction $\widetilde{L}(\lambda)=-\widetilde{L}(\lambda^)^\dag$ we obtain that the reduced model, that corresponds to (\[eq:2.5a\]) will be constrained by the condition, that the matrix ${\cal S}$ must be hermitian: ${\cal S}={\cal S}^\dag$. Conclusions {#sec:6} =========== We will finish this article with several concluding remarks. In order to obtain the soliton solutions for the gauge equivalent MHF systems, one needs to apply the Zakharov–Shabat dressing method to a regular FAS $\widetilde{\chi}_{(0)}^{\pm}(x,\lambda ) $ of $\widetilde{L} $ with potential ${\cal S}_{(0)} $. Thus, one gets a new singular solution $\widetilde{\chi }_{(1)}^{\pm}(x, \lambda ) $ of the Riemann–Hilbert problem with singularities located at prescribed positions $\lambda _1^{\pm} $. It is related to the regular one by the dressing factors $\widetilde{u}(x,\lambda)$. The dressing method for the generalised Zakharov-Shabat systems (related to semi-simple Lie algebras) is developed in [@ZaMi; @G86] , [@VSG87], [@HSAS] and [@Ivanov]. To MNLS systems and their gauge equivalent MHF ones, one can apply the analysis [@VYa] and derive the completeness relations for the corresponding system of ”squared” solutions. Such analysis will allow one to prove the pair-wise compatibility of the Hamiltonian structures and eventually, to derive their action-angle variables, see e.g. [@ZM] and [@BS] for the ${\bf A}_r$-series. The approach presented here allows one to consider CBC systems with more general $\lambda$- dependence, like the Principal Chiral field models and other relativistic invariant field theories [@ZaMi]. 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--- abstract: \| We report on 25 sub-arcsecond binaries, detected for the first time by means of lunar occultations in the near-infrared as part of a long-term program using the ISAAC instrument at the ESO Very Large Telescope. The primaries have magnitudes in the range K=3.8 to 10.4, and the companions in the range K=6.4 to 12.1. The magnitude differences have a median value of 2.8, with the largest being 5.4. The projected separations are in the range 6 to 748 milliarcseconds and with a median of 18 milliarcseconds, or about 3 times less than the diffraction limit of the telescope. Among our binary detections are a pre-main sequence star and an enigmatic Mira-like variable previously suspected to have a companion. Additionally, we quote an accurate first-time near-IR detection of a previously known wider binary. We discuss our findings on an individual basis as far as made possible by the available literature, and we examine them from a statistical point of view. We derive a typical frequency of binarity among field stars of $\approx 10$%, in the resolution and sensitivity range afforded by the technique ($\approx 0\farcs003$ to $\approx 0\farcs5$, and K$\approx 12$mag, respectively). This is in line with previous results by the same technique but we point out interesting differences that we can trace up to sensitivity, time sampling, and average distance of the targets. Finally, we discuss the prospects for further follow-up studies. author: - 'A. Richichi, O. Fors, F. Cusano, and M. Moerchen' title: 'Twenty-Five Sub-Arcsecond Binaries Discovered By Lunar Occultations [$^$ ]{} ' --- Introduction {#section:introduction} ============ Lunar occultations (LOs) can efficiently yield high angular resolution observations from the analysis of the diffraction light curves generated when background sources are covered by the lunar limb. The technique has been employed to measure hundreds of stellar angular diameters, binary stars, and sources with extended circumstellar emission [see CHARM2 catalog, @CHARM2]. In the past few years, a program to observe LOs in the near-infrared at the ESO Very Large Telescope (VLT) has been very successful both in quantity, with over a thousand events recorded, and in quality with a combination of angular resolution far exceeding the diffraction limit of a single telescope ($\approx 0\farcs001$) and a sensitivity significantly better than that currently achieved by long-baseline interferometry (K $\approx 12$mag). The drawbacks are that LOs are fixed-time events, yielding mainly a one-dimensional scan of the source, and that the source cannot be chosen at will. Details on the LOs program at the VLT can be found in @apjssur, and references therein. Here, we report on 25 sources discovered to be binary with projected separations below one arcsecond, and in fact mostly below the 57 milliarcseconds (mas) diffraction limit of the telescope at the given wavelength. We also report on one previously known system. In Sect. \[section:data\] we describe the observational procedure, the sample composition, and the data analysis. In Sect. \[section:results\] we report the individual results, and provide some context from previous bibliography when available. Some considerations on the statistics of binary detections from our VLT LO program and on the prospects of follow-up of selected systems are given in Sect. \[sec:conclusions\]. Observations and data analysis {#section:data} ============================== The observations were carried out between April 2010 and October 2011, using the 8.2-m UT3 Melipal telescope of the VLT and the ISAAC instrument operated in burst mode. Most of the observations were carried out in service mode, based on a strategy of profiting from short slots that might become available depending on the atmospheric conditions and execution status of other service programs. Consequently, the targets were inherently random. A few observations were parts of isolated nights dedicated to LO observations in visitor mode, and in this case the sources were selected on the basis of their colors and brightness in very extincted regions of the Galactic Bulge. The sources observed in service mode have the field “Our ID” in Table \[tab:list\] beginning with P85 to P88, which are the ESO Periods under consideration. The sources without this prefix were observed in visitor mode. Table \[tab:list\] provides a list of the observations and of the characteristics of the sources, ordered by time. A sequential number is included, for ease of cross-reference. Our predictions were generated from the 2MASS Catalogue, and so is the listed near-infrared photometry. We did not attempt to derive proper K-band photometry from our light curves, due to the lack of calibration sources. However we notice that in some cases differences between our counts and the 2MASS magnitudes of up to a factor 2 are present, pointing to possible variability (Fig. \[fig:counts\]). Further identifications in Table \[tab:list\], as well as visual photometry and spectral types, are extracted from the [Simbad]{} database. Each observation consisted of 7500 or 5000 frames (in service or visitor runs, respectively) in a 32x32-pixel ($4\farcs7 \times 4\farcs7$) sub-window, with a time sampling of 3.2ms. This was also the effective integration time. A broad-band K$_{\rm s}$ filter was employed for all events, except in the case of the brighter sources Stars 4 and 22, for which a narrow-band filter centered at 2.07$\mu$m was employed to avoid possible non-linearities. The events were disappearances, with lunar phases ranging from 38% to 96% (median 66%). Airmass ranged from 1.1 to 2.0, while seeing ranged from $0\farcs6$ to $2\farcs9$ (median $1\farcs1$). The LO light curves are generated in vacuum at the lunar limb and the diffraction fringes span a range of few 0.1s, so that the technique is in any case largely insensitive to atmospheric perturbations. The data cubes were converted to light curves by extracting the signal at each frame within a mask tailored to the seeing and image quality. The light curves were then analyzed using both a model-dependent [@richichi96] and a model-independent method (CAL, @CAL). This latter is well suited to derive brightness profiles in the case of faint binaries, from which the initial values of the model can be inferred. The least-squares fits are driven by changes in normalized $\chi^2$, with a noise model defined for each light curve from the data before and after the occultation. More details on the instrumentation and the method can be found in @richichi2011 and references therein. It should be noted that only restricted portions of the light curves around the main disappearance, corresponding to angular extensions of $\approx 0\farcs5$, were considered. In general, companions with projected separations larger than this would not appear in our list. [rllllrrrrrrrl]{} 1 & P85-06 & 07283985+2151062 & BD+22 1693 & 21-Apr-10 & 00:03:25 & 9.52 & 9.12 & 8.50 & 8.36 & 8.30 & F2\ 2 & P85-23 & 19224512-2046033 & BD-21 5366 & 21-Aug-10 & 05:39:03 & 11.18 & 10.07 & 8.09 & 7.59 & 7.44 &\ 3 & P85-26 & 19240606-2103008 & BD-21 5373 & 21-Aug-10 & 05:59:25 & 11.63 & 10.28 & 6.73 & 5.93 & 5.65 &\ 4 & s033 & 18115684-2330488 & AKARI-IRC-V1 & 16-Sep-10 & 03:08:18 & 16.50 & 15.60$_{\rm R}$ & 5.70 & 4.51 & 4.02 &\ 5 & P86-02 & 22275268-0418586 & HD 212913 & 15-Nov-10 & 00:52:55 & 10.61 & 8.92 & 5.42 & 4.28 & 4.07 & M...\ 6 & P86-06 & 22282898-0417248 & & 15-Nov-10 & 01:33:46 & & & 10.32 & 9.76 & 9.63 &\ 7 & P86-13 & 22304733-0348326 & HD 213344 & 15-Nov-10 & 03:24:31 & 10.82 & 9.75 & 7.77 & 7.23 & 7.11 & K0\ 8 & P86-21 & 23551978+0532182 & TYC 593-1067-1 & 17-Nov-10 & 00:35:44 & 10.44 & 9.92 & 8.94 & 8.74 & 8.66 &\ 9 & P86-27 & 23581573+0612341 & TYC 593-1360-1 & 17-Nov-10 & 03:37:07 & 10.85 & 9.88 & 8.44 & 8.04 & 7.95 &\ 10 & P86-31 & 23592250+0614580 & TYC 593-1337-1 & 17-Nov-10 & 04:20:09 & 11.62 & 11.11 & 9.98 & 9.72 & 9.66 &\ 11 & P87-014 & 09463209+0913340 & & 13-Apr-11 & 23:11:43 & & & 10.21 & 9.67 & 9.55 &\ 12 & P87-026 & 10220366+0515554 & TYC 252-1217-1 & 11-May-11 & 23:03:59 & 12.03 & 11.33 & 9.86 & 9.48 & 9.42 &\ 13 & P87-041 & 10070867+0628201 & TYC 250-82-1 & 07-Jun-11 & 22:55:04 & 11.23 & 10.55 & 9.53 & 9.27 & 9.15 &\ 14 & vmj-023 & 17350423-2319491 & & 13-Jul-11 & 04:49:05 & & & 9.75 & 8.60 & 8.08 &\ 15 & vmj-056 & 17395562-2303408 & HD 160257 & 13-Jul-11 & 07:26:09 & 9.18 & 8.58 & 7.47 & 7.20 & 7.13 & G2V\ 16 & vma-049 & 18142147-2207047 & IRAS 18113-2208 & 10-Aug-11 & 03:14:03 & & & 7.48 & 6.20 & 5.65 &\ 17 & vma-055 & 18151114-2213294 & & 10-Aug-11 & 03:33:19 & & & 7.80 & 6.55 & 5.85 &\ 18 & vma-062 & 18154557-2220045 & & 10-Aug-11 & 03:57:43 & & & 8.45 & 7.32 & 6.71 &\ 19 & vma-083 & 18172695-2147313 & & 10-Aug-11 & 05:27:34 & & & 9.25 & 8.05 & 7.37 &\ 20 & P88-003 & 17370333-2239165 & HD 159700 & 02-Oct-11 & 23:33:40 & 11.28 & 9.67 & 5.81 & 5.04 & 4.60 & K7\ 21 & P88-008 & 17371972-2215045 & IRAS 17343-2213 & 03-Oct-11 & 00:02:24 & & & 8.89 & 7.86 & 7.10 &\ 22 & P88-021 & 17401076-2208280 & IRAS 17371-2207 & 03-Oct-11 & 01:30:13 & 12.16 & 10.67 & 5.32 & 4.11 & 3.75 &\ 23 & P88-046 & 18395035-2052114 & BD-20 5222 & 04-Oct-11 & 02:02:01 & 11.00 & 9.79 & 7.30 & 6.70 & 6.53 &\ 24 & P88-050 & 18402840-2040072 & KO Sgr & 04-Oct-11 & 02:31:16 & 12.70 & 14.50 & 6.84 & 5.91 & 5.36 &\ 25 & P88-052 & 18403433-2034059 & IRAS 18375-2036 & 04-Oct-11 & 02:45:15 & & & 7.89 & 6.85 & 6.31 &\ Results {#section:results} ======= Table \[tab:results\] lists our results, following closely the format already used in previous papers. The same sequential number used in Table \[tab:list\] is included, followed by the 2MASS identification. The next two columns are the observed rate of the event and its deviation from the predicted value. The difference is due to the local limb slope $\psi$, from which the actual observed position angle PA and contact angle CA are derived. We then list the binary fit results, namely the signal-to-noise ratio (SNR) of the light curve, the separation and brightness ratio, and the two individual magnitudes based on the total magnitude listed in the 2MASS. As mentioned in Sect. \[section:data\], for Stars 4 and 22 narrow-band filters were used but we do not expect significant effects on K$_1$, K$_2$. [rlrrrrrrccrr]{} 1 & 07283985+2151062 & 0.6326 & -12.0% & -26 & 94 & -23 & 26.5 & 190.29 $\pm$ 0.06 & 2.335 $\pm$ 0.004 & 8.69 & 9.61\ 2 & 19224512-2046033 & 0.4199 & -8.5% & -4 & 7 & -58 & 64.2 & 55.5 $\pm$ 0.4 & 7.98 $\pm$ 0.02 & 7.56 & 9.82\ 3 & 19240606-2103008 & 0.4715 & -12.2% & -6 & 107 & 43 & 91.6 & 8.4 $\pm$ 0.2 & 12.06 $\pm$ 0.04 & 5.74 & 8.44\ 4 & 18115684-2330488 & 0.6350 & -5.8% & -5 & 33 & -38 & 70.3 & 8.8 $\pm$ 0.3 & 18.9 $\pm$ 0.2 & 4.07 & 7.27\ 5 & 22275268-0418586 & 0.4576 & -7.9% & -6 & 83 & 33 & 116.8 & 6.29 $\pm$ 0.05 & 7.53 $\pm$ 0.01 & 4.20 & 6.39\ 6 & 22282898-0417248 & 0.2254 & -21.5% & -6 & 110 & 58 & 10.4 & 6.9 $\pm$ 2.6 & 4.5 $\pm$ 0.1 & 9.85 & 11.48\ 7 & 22304733-0348326 & 0.6890 & 10.9% & 9 & 283 & 46 & 47.7 & 42.5 $\pm$ 0.4 & 29.2 $\pm$ 0.3 & 7.15 & 10.81\ 8 & 23551978+0532182 & 0.5415 & -7.0% & -10 & 58 & 8 & 39.4 & 15.3 $\pm$ 0.7 & 12.7 $\pm$ 0.2 & 8.74 & 11.50\ 9 & 23581573+0612341 & 0.6639 & 0.4% & 1 & 47 & -10 & 46.1 & 19.6 $\pm$ 1.0 & 45.1 $\pm$ 1.6 & 7.97 & 12.11\ 10 & 23592250+0614580 & 0.6767 & -4.2% & -9 & 62 & 2 & 2.5 & 40.1 $\pm$ 0.3 & 1.03 $\pm$ 0.01 & 10.40 & 10.43\ 11 & 09463209+0913340 & 0.4553 & -22.3% & -16 & 141 & 20 & 7.5 & 47.9 $\pm$ 0.2 & 1.50 $\pm$ 0.01 & 10.11 & 10.54\ 12 & 10220366+0515554 & 0.6121 & -3.2% & -4 & 146 & 20 & 9.0 & 29.4 $\pm$ 3.5 & 2.34 $\pm$ 0.02 & 9.81 & 10.73\ 13 & 10070867+0628201 & 0.4769 & 5.3% & 3 & 179 & 53 & 10.3 & 371.7 $\pm$ 4.1 & 1.447 $\pm$ 0.007 & 9.72 & 10.12\ 14 & 17350423-2319491 & 0.3648 & -0.3% & 0 & 137 & 58 & 31.6 & 8.3 $\pm$ 0.8 & 9.7 $\pm$ 0.1 & 8.19 & 10.66\ 15 & 17395562-2303408 & 0.6512 & 26.4% & 12 & 324 & 67 & 29.3 & 41.96 $\pm$ 0.07 & 1.137 $\pm$ 0.003 & 7.81 & 7.95\ 16 & 18142147-2207047 & 0.5564 & 0.1% & 0 & 222 & -32 & 202.4 & 32.4 $\pm$ 0.3 & 144.4 $\pm$ 1.6 & 5.65 & 11.05\ 17 & 18151114-2213294 & 0.6585 & -1.7% & -7 & 71 & -3 & 209.2 & 15.1 $\pm$ 0.2 & 100.1 $\pm$ 1.3 & 5.86 & 10.86\ 18 & 18154557-2220045 & 0.3918 & -28.0% & -17 & 95 & 22 & 72.9 & 9.9 $\pm$ 0.4 & 24.7 $\pm$ 0.3 & 6.75 & 10.24\ 19 & 18172695-2147313 & 0.4303 & 9.8% & 3 & 194 & -58 & 22.5 & 27.8 $\pm$ 0.9 & 12.2 $\pm$ 0.1 & 7.45 & 10.17\ 20 & 17370333-2239165 & 0.3548 & -33.8% & -20 & 98 & 19 & 99.8 & 8.8 $\pm$ 0.1 & 25.60 $\pm$ 0.09 & 4.64 & 8.16\ 21 & 17371972-2215045 & 0.4863 & 10.0% & 5 & 31 & -47 & 124.9 & 7.30 $\pm$ 0.07 & 18.9 $\pm$ 0.1 & 7.16 & 10.35\ 22 & 17401076-2208280 & 0.6415 & -1.0% & -1 & 36 & -40 & 135.5 & 6.2 $\pm$ 0.3 & 36.5 $\pm$ 0.5 & 3.78 & 7.69\ 23 & 18395035-2052114 & 0.7922 & 0.4% & 1 & 262 & 12 & 109.3 & 748.4 $\pm$ 0.2 & 23.13 $\pm$ 0.04 & 6.58 & 9.99\ 24 & 18402840-2040072 & 0.7780 & -2.1% & -3 & 227 & -23 & 217.6 & 18.1 $\pm$ 0.2 & 78.3 $\pm$ 0.6 & 5.37 & 10.11\ 25 & 18403433-2034059 & 0.5391 & -15.4% & -9 & 19 & -52 & 111.9 & 8.1 $\pm$ 0.3 & 18.5 $\pm$ 0.1 & 6.37 & 9.54\ Many of our sources are in the direction of the Galactic Bulge, having 17$^{\rm h}$ $\la$ RA $\la$ 19$^{\rm h}$ and Dec $\approx$ $-20\degr$. They have generally very red colors; however a color-color diagram shows that these are mostly consistent with significant amounts of interstellar extinction (Fig. \[fig:colors\]), with a possible notable exception to be discussed later. In line with this extinction, many of our sources have relatively faint optical counterparts, or none. Few of them have been studied in detail previously, and spectral information is correspondingly scarce. In the reminder of this section we discuss individual cases in the context of available literature, when possible. Star 5: is and , for which no literature exists except a generic late M spectral type. The Tycho-2 parallax values seem to place this star at $\approx$300pc, thus hinting at a giant star. The star is not in the Bulge and is $49\degr$ below the galactic disk, yet its location in Fig. \[fig:colors\] is peculiar and indicative of substantial reddening. We attempted fits to our light curve with a binary, a resolved diameter, and a diameter plus companion models. The first one gave the best normalized $\chi^2$ and is the one we adopt in our results. Star 12: was included in Tycho-2 as . However, no binarity was detected, including in the subsequent dedicated re-analysis of the Tycho Double Star Catalogue . No literature references were found for this source. Star 13: is . In spite of the relatively large measured projected separation, no binarity was detected in Tycho 2. No literature references are present for this source. Star 15: is , a bright, nearby, G2V-type star. It was classified as a pre-main sequence star by , on the basis of X-Ray emission and high resolution spectra. We find it to to be binary with a rather large projected separation (see Fig. \[fig:vmj-056\]). It was observed by Hipparcos as , but no binarity was found. As for the recent case of new binaries in the Pleiades , this shows the potential of LO to further extend the statistics of binarity among young stars in the context of multiple star formation mechanisms. Star 16: is the infrared source . No literature references were found for this source. It is also the binary with the largest magnitude difference in our sample, $\Delta$K=5.4mag against a dynamic range of 5.8mag. Star 21: is the infrared source , also measured by the AKARI satellite. There are no bibliographical references. It is one of the reddest sources in our sample, having J-K=1.8mag from 2MASS. Our recorded flux shows an increase of $\approx$80% above the 2MASS K$_{\rm s}$ magnitude (cfr. Fig. \[fig:counts\]), pointing to possible variability. The detection of the 3.2mag fainter companion with 7mas projected separation is shown in Fig. \[fig:p88-08\]. In the lower panel of the figure, note that the CAL algorithm preserves the integrated flux ratios, not the peaks in the brightness profile. Star 23: is the star with the largest projected separation in our list, $0\farcs75$, however it does not seem to have been detected before in spite of being relatively bright. The 1:23 brightness ratio in the K-band, and possibly more at visual wavelengths, could be one reason. Concerning projection effects of a nearby field star, we have investigated images from DSS, HST and 2MASS without evidence of other stars within the ISAAC subwindow. Star 24: , also identified as , was found to have a regular period of 312 days by @Hof60 and thus tentatively classified as a Mira-type star. The same author, however, noted the peculiarity of rather steep increases in luminosity, and the relatively short-lived maxima. She suggested the possibility of a companion in a cataclysmic system, or just a visual association. In this scenario, the light at minimum would come mostly from the secondary. The photographic magnitude difference between minima and maxima is about 3.1mag. Using the Hoffleit photometric period, our observation would have occurred at phase 0.8 or just before the onset of the outburst-like maximum. Uncertainties are possible due to the $\approx$100 cycles intervened since Hoffleit’s first determination, but are not easy to estimate. Data in the AAVSO database are also not very complete. Under the above assumptions, the $\Delta$K=4.7mag would point to a primary much redder than the secondary, consistent with the scenario outlined in @Hof60. We also recorded a LO light curve of = on April 13, 2011 (star P87-010 in our database), detecting a well-separated companion with $385.5 \pm 0.9$mas projected separation along PA=$161\degr$. The 2MASS-based magnitudes of the two components are K=9.04 and K=10.31. This is listed as the wide double [@2001AJ....122.3466M], without orbital parameters and with a separation of $1\farcs4$. Given the wide nature of this binary, we did not include it in our tables and figures, and we mention our results only as a complement to the previous observations. Discussion and Conclusions {#sec:conclusions} ========================== @apjssur list the sources observed in the same program and found to be unresolved, covering also the period under consideration here. From the first to the last night considered in Table \[tab:list\], a total of 403 LO light curves were observed at the VLT with ISAAC, therefore our serendipitous binary detection fraction is (26/403) $\approx 6.5$%. Restricting ourselves to the purely random service mode observations, the number is (19/231) $\approx 8.2$%. We note that the visitor mode targets were in crowded and extincted regions in, and consequently mostly deep into, the Galactic Bulge, with a median K=6.4mag. By contrast, the service mode targets were randomly scattered and with median K=8.0mag. One possible reason for the higher incidence of binaries among the service mode targets is that they are statistically closer to us, therefore providing a better spatial scale for the same angular resolution. ![Distribution of the K magnitudes of the primaries and secondaries in our sample, as computed from the flux ratio in our light curves and the 2MASS K$_{\rm s}$ values. []{data-label="fig:k1k2"}](richichi_fig05.eps){width="7.5cm"} In a previous work based on a comparable volume of LO observed also in the near-IR in subarray fast mode from Calar Alto , it was found that the serendipitous binary detection frequency was significantly smaller than the present one, $\approx 4$%. The difference can be justified in principle with the lower sensitivity (limiting K $\approx 8$mag then, and $\approx 12$mag here) and the slower time sampling ($\approx 8.5$ms then, against 3.2ms here). However, both cases appear to have an inferior yield of binary stars than observations by a fast photometer: e.g., @richichi96 using the TIRGO 1.5m telescope found (26/157) $\approx 16$% binary detection frequency. The explanation in this case is that the targets were brighter, being selected prior to the availability of the 2MASS catalog, with average K $\approx 4$mag. Hence, the sources were generally significantly closer to us than in the present study. The flux ratios of the binaries in our list range from $\Delta$K=0.0 to 5.4mag, with a median of 2.8mag. From Fig. \[fig:k1k2\] it can be seen that half of the sample falls in the range 2$\le \Delta{\rm K} \le$4mag. As for the separations, LO can measure only projected values, and the median for our sample is 18mas. Considering a random projection correction of 2/$\pi$, about half of our binaries would remain inaccessible to techniques with less than $0\farcs03$ resolution. This corresponds, e.g., to the K-band diffraction limit of a 12m telescope. Long-baseline interferometry at some of the largest facilities can provide the necessary angular resolution, but it suffers from a sensitivity which is significantly more limited than for LO and in general from a reduced dynamic range. We outline schematically the situation in Fig. \[fig:brsep\]. From the figure, it can be estimated that only about 2/3 of the sources in our list might be effectively followed up by independent methods. ![Distribution of brightness ratios against projected separations for the stars in our sample. The shaded areas mark the approximate regions where near-infrared long-baseline interferometry (left) and adaptive optics at an 8m telescope (right) are mostly effective. []{data-label="fig:brsep"}](richichi_fig06.eps){width="7.5cm"} In conclusion, we find from the present work as well as from previous similar research that a routine program of random LO with an IR detector operated in subarray mode at an 8m-class telescope with time resolution of order 3ms can detect companions to stars in the general field down to a sensitivity K $\approx 12$mag, with separations as small as a few milliarcseconds (at least 10 times better than the diffraction limit of the telescope). The expected detection frequency can range from $\approx 6$% in distant regions such as deep in the Galactic Bulge, to $\approx 15$% in areas closer to the solar neighbourhood. Most of these binaries or multiples turn out to be new detections, with limited cross-identifications, spectral determinations and previous literature. Due to the small separations and high brightness ratios, only up to $\approx 50$% of these LO-detected binaries can be followed up by other more flexible methods such as adaptive optics and interferometry. A significant fraction of such new binary detections will remain isolated until significantly larger telescopes or more sensitive interferometers become available. We are grateful to the ESO staff in Europe and Chile for their continued support especially of the service observations. OF acknowledges financial support from MINECO through a [Juan de la Cierva]{} fellowship and from MCYT-SEPCYT Plan Nacional I+D+I AYA\#2008-01225*. This research made use of the Simbad database, operated at the CDS, Strasbourg, France, and of data products from the Two Micron All Sky Survey (2MASS), which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This research has made use of the Washington Double Star Catalog maintained at the U.S. Naval Observatory. Bessell, M. S., & Brett, J. M. 1988, , 100, 1134 Fabricius, C., H[ø]{}g, E., Makarov, V. V., et al. 2002, , 384, 180 Hoffleit, D. 1960, , 65, 100 H[ø]{}g, E., Fabricius, C., Makarov, V. V., et al. 2000, , 357, 367 Mason, B. D., Wycoff, G. L., Hartkopf, W. I., Douglass, G. G., & Worley, C. E. 2001, , 122, 3466 Richichi, A. 1989, , 226, 366 Richichi, A., Baffa, C., Calamai, G., & Lisi, F. 1996, , 112, 2786 Richichi, A., Percheron, I., & Khristoforova, M. 2005, , 431, 773 Richichi, A., Fors, O., Merino, M., et al. 2006, , 445, 1081 Richichi, A., Chen, W. P., Fors, O., & Wang, P. F. 2011, , 532, A101 Richichi, A., Chen, W. P., Cusano, F., et al. 2012a, , 541, A96 Richichi, A., Cusano, F., Fors, O., Moerchen, M.2012b, , 203, 33 Rieke, G. H., & Lebofsky, M. J. 1985, , 288, 618 Torres, C. A. O., Quast, G. R., da Silva, L., et al. 2006, , 460, 695
--- abstract: 'The risk premium is one of main concepts in mathematical finance. It is a measure of the trade-offs investors make between return and risk and is defined by the excess return relative to the risk-free interest rate that is earned from an asset per one unit of risk. The purpose of this article is to determine upper and lower bounds on the risk premium of an asset based on the market prices of options. One of the key assumptions to achieve this goal is that the market is Markovian. Under this assumption, we can transform the problem of finding the bounds into a second-order differential equation. We then obtain upper and lower bounds on the risk premium by analyzing the differential equation.' author: - \| Jihun Han[^1] and Hyungbin Park[^2]\ \ Courant Institute of Mathematical Sciences,\ New York University, New York, USA\ \ [First version: Nov 09, 2014]{}\ [Final version: Mar 03, 2015]{} title: The Intrinsic Bounds on the Risk Premium of Markovian Pricing Kernels --- Introduction {#sec:intro} ============ The [risk premium]{} or [market price of risk]{} is one of main concepts in mathematical finance. The risk premium is a measure of the trade-offs investors make between return and risk and is defined by the excess return relative to the risk-free interest rate earned from an asset per one unit of risk. The risk premium determines the relation between an objective measure and a risk-neutral measure. An objective measure describes the actual stochastic dynamics of markets, and a risk-neutral measure determines the prices of options. Recently, many authors have suggested that the risk premium (or, equivalently, objective measure) can be determined from a risk-neutral measure. Ross [@Ross13] demonstrated that the risk premium can be uniquely determined by a risk-neutral measure. His model assumes that there is a finite-state Markov process $X_{t}$ that drives the economy in discrete time $t\in\mathbb{N}.$ Many authors have extended his model to a continuous-time setting using a Markov diffusion process $X_{t}$ with state space $\mathbb{R}$; see, e.g., [@Borovicka14],[@Carr12],[@Dubynskiy13],[@Goodman14],[@Park14b],[@Qin14b] and [@Walden13]. Unfortunately, in the continuous-time model, the risk premium is not uniquely determined from a risk-neutral measure [@Goodman14], [@Park14b]. To determine the risk premium uniquely, all of the aforementioned authors assumed that some information about the objective measure was known or restricted the process $X_t$ to some class. Borovicka, Hansen and Scheinkman [@Borovicka14] made the assumption that the process $X_t$ is [stochastically stable]{} under the objective measure. In [@Carr12], Carr and Yu assumed that the process $X_{t}$ is a [bounded]{} process. Dubynskiy and Goldstein [@Dubynskiy13] explored Markov diffusion models with [ reflecting boundary]{} conditions. In [@Park14b], Park assumed that $X_t$ is non-attracted to the left (or right) boundary under the objective measure. Qin and Linetsky [@Qin14b] and Walden [@Walden13] assumed that the process $X_{t}$ is [recurrent]{} under the objective measure. Without these assumptions, one cannot determine the risk premium uniquely. The purpose of this article is to investigate the bounds of the risk premium. As mentioned above, without further assumptions, the risk premium is not uniquely determined, but one can determine upper and lower bounds on the risk premium. To determine these bounds, we need to consider how the risk premium of an asset is determined in a financial market. A key assumption of this article is that the reciprocal of the pricing kernel is expressed in the form $e^{\beta t}\,\phi(X_{t})$ for some positive constant $\beta$ and positive function $\phi(\cdot).$ For example, in the [consumption-based capital asset model]{} [@Campbell99], [@Karatzas98], the pricing kernel is expressed in the above form. We will see that in this case the risk premium $\theta_t$ is given by $$\label{eqn:theta} \theta_t=(\sigma\phi'\phi^{-1})(X_t)\;,$$ where $\sigma(X_t)$ is the volatility of $X_t.$ The problem of determining the bounds of the risk premium can be transformed into a second-order differential equation. We will demonstrate that $\phi(\cdot)$ satisfies the following differential equation: $$\mathcal{L}\phi(x):=\frac{1}{2}\sigma^2(x){\phi ''(x)}+k(x)\phi '(x) -r(x)\phi (x) =-\beta \,\phi (x)$$ for some unknown positive number $\beta.$ Thus, we can determine the bounds of the risk premium by investigating the bounds of $(\sigma\phi'\phi^{-1})(\cdot)$ for a positive solution $\phi(\cdot).$ It will be demonstrated that two special solutions of $\mathcal{L}h=0$ play an important role for the bounds of the risk premium $\theta_t.$ The following provides an overview of this article. In Section \[sec:Markovian\_pricing\], we state the notion of Markovian pricing kernels. In Section \[sec:risk\_premium\], we investigate the risk premium of an asset and see how the problem of determining the bounds of the risk premium is transformed into a second-order differential equation. In Section \[sec:intrinsic\_bounds\], we find upper and lower bounds on the risk premium of an asset, which is the main result of this article. In Section \[sec:appli\], we see how this result can be applied to determine the range of return of an asset. Finally, Section \[sec:conclusion\] summarizes this article. Markovian pricing kernels {#sec:Markovian_pricing} ========================= A financial market is defined as a probability space $(\Omega,\mathcal{F},\mathbb{P})$ having a Brownian motion $B_{t}$ with the filtration $\mathcal{F}=(\mathcal{F}_{t})_{t=0}^{\infty}$ generated by $B_{t}$. All the processes in this article are assumed to be adapted to the filtration $\mathcal{F}$. $\mathbb{P}$ is the objective measure of this market. In the financial market, there are two assets. One is a [money market account]{} $e^{\int_0^t\,r_s\,ds}$ with an [interest rate]{} process $r_{t}$ and the other is a risky asset $S_{t}$ satisfying $$dS_t=\mu_tS_t\,dt+v_tS_t\,dB_t\;.$$ Throughout this article, the stochastic discount factor is the money market account. Let $\mathbb{Q}$ be a risk-neutral measure in the market $(\Omega,\mathcal{F},\mathbb{P})$ such that $S_t\,e^{-\int_0^tr_s\,ds}$ is a local martingale under $\mathbb{Q}.$ Put the Radon-Nikodym derivative $$\Sigma_{t}=\left.\frac{d \mathbb{Q}}{d \mathbb{P}} \right\|_{\mathcal{F}_{t}} \; ,$$ which is known to be a martingale process on $(\Omega,\mathcal{F},\mathbb{P}).$ We can write in the SDE form $$d\Sigma_{t}=-\theta_{t}\Sigma_{t}\, dB_{t}$$ where $$\label{eqn:rho} \theta_{t}:=\frac{\mu_t-r_t}{v_t}$$ is the [risk-premium]{} or [market price of risk.]{} It is well-known that $W_{t}$ defined by $$\label{eqn:Girsanov} dW_{t}=\theta_{t}dt+dB_{t}$$ is a Brownian motion under $\mathbb{Q}.$ We define the reciprocal of the [pricing kernel]{} by $L_{t}=e^{\int_0^tr_s\,ds}/\Sigma_{t}.$ Using the Ito formula, $$\label{eqn:RN_SDE} \begin{aligned} dL_t&=(r_{t}+\theta_{t}^2)\,L_t\,dt +\theta_{t}L_t\, dB_{t}\\ &=r_{t}L_t\,dt + \theta_{t}L_t\, dW_{t} \end{aligned}$$ is obtained. \[assume:Markovian\] Assume that (the reciprocal of) the pricing kernel $L_t$ is Markovian in the sense that there are a positive function $\phi\in C^{2}(\mathbb{R}),$ a positive number $\beta$ and a state variable $X_t$ such that $$\label{eqn:Markovian} L_{t}=e^{\beta t}\,\phi(X_{t})\,\phi^{-1}(X_{0})\; .$$ In this case, we say $(\beta,\phi)$ is a [principal pair]{} of $X_{t}.$ We imposed a special structure on the pricing kernel. This specific form is commonly assumed in the recovery literature as in [@Borovicka14],[@Carr12],[@Dubynskiy13],[@Goodman14],[@Park14b],[@Qin14b],[@Ross13] and [@Walden13]. In general, $L_t$ can be expressed as $$L_{t}=e^{\beta t}\,\phi(X_{t})\,\phi^{-1}(X_{0})\,M_t$$ where $M_t$ is a $\mathbb{Q}$-martingale. Refer to [@Hansen09] for this general expression. Assumption \[assume:Markovian\] has an implication that the martingale term $M_t$ is equal to $1.$ We now shift our attention to the assumption that $\beta>0.$ In lots of literature on asset pricing theory, $\beta$ is the discount rate of the representative agent, which is typically a positive number. For example, in the [consumption-based capital asset model]{} [@Campbell99], [@Karatzas98], (the reciprocal of) the pricing kernel is expressed by $$e^{\beta t}\frac{U'(c_0)}{U'(c_t)}$$ where $U$ is the utility of the representative agent, $c_t$ is the aggregate consumption process and $\beta$ is the discount rate of the agent. \[assume:X\] The state variable $X_{t}$ is a time-homogeneous Markov diffusion process satisfying the following SDE. $$dX_{t}=k(X_{t})\,dt+\sigma(X_{t})\,dW_{t}\,,\;X_{0}=\xi\;.$$ $k(\cdot)$ and $\sigma(\cdot)$ are assumed to be known ex ante. The process $X_t$ takes values in some interval $I$ with endpoints $c$ and $d,\,-\infty\leq c<d\leq\infty.$ It is assumed that $b(\cdot)$ and $\sigma(\cdot)$ are continuous on $I$ and continuously differentiable on $(c,d)$ and that $\sigma(x)>0$ for $x\in(c,d).$ \[assume:interest\_rate\] The short interest rate $r_t$ is determined by $X_{t}.$ More precisely, there is a continuous positive function $r(\cdot)$ such that $r_{t}=r(X_{t}).$ Under these assumptions, the next section demonstrates how to transform the problem of determining the bounds of the risk premium into a second-order differential equation. We will also describe the properties of positive solutions of the differential equation. Risk premium {#sec:risk_premium} ============ The purpose of this article is to determine upper and lower bounds on the risk premium $\theta_t.$ First, we investigate how the risk premium $\theta_t$ is determined with the Markovian pricing kernel. Applying the Ito formula to , we have $$dL_{t}=\left(\beta+\frac{1}{2}(\sigma^{2}\phi''\phi^{-1})(X_{t})+(k\phi'\phi^{-1})(X_{t})\right)L_{t} \,dt+(\sigma\phi'\phi^{-1})(X_{t})\,L_{t}\,dW_{t}$$ and by , we know $dL_{t}=r(X_{t})\,L_{t}\,dt + \theta_{t}L_{t}\, dW_{t}. $ By comparing these two equations, we obtain $$\frac{1}{2}\sigma^2(x){\phi ''(x)}+k(x)\phi '(x) -r(x)\phi (x) =-\beta \,\phi (x)$$ and $$\label{eqn:Markovian_theta} \theta_{t}=(\sigma\phi'\phi^{-1})(X_{t})\;.$$ Define a infinitesimal operator $\mathcal{L}$ by $$\mathcal{L}\phi(x)=\frac{1}{2}\sigma^2(x){\phi ''(x)}+k(x)\phi '(x) -r(x)\phi (x)\;.$$ Under Assumption 1-\[assume:interest\_rate\], let $(\beta,\phi)$ be a principal pair of $X_{t}.$ Then, $(\beta,\phi)$ satisfies $\mathcal{L}\phi=-\beta\phi\;.$ We also have the following theorem by and . \[thm:theta\_phi\] The risk premium is given by $\theta_{t}=\theta(X_{t})$ where $\theta(\cdot):=(\sigma\phi'\phi^{-1})(\cdot).$ We thus have that $dB_{t}=-\theta(X_{t})\,dt+dW_{t}.$ This theorem explains the relation between the risk premium and the pricing kernel $L_t.$ The purpose of this article is to determine upper and lower bounds on $\theta(\cdot)$ based on $k(\cdot),\,\sigma(\cdot)$ and $r(\cdot).$ The positive function $\phi(\cdot)$ and the positive number $\beta$ are assumed to be unknown. The main idea is to determine the properties of all of the possible $\phi(\cdot)$’s and $\beta$’s and then to obtain upper and lower bounds on the possible $(\phi'\phi^{-1})(\cdot)$ values. From , we can determine the bounds of the risk premium $\theta_t.$ Intrinsic bounds {#sec:intrinsic_bounds} ================ We are interested in a solution pair $(\lambda,h)$ of $\mathcal{L}h=-\lambda h$ with positive function $h.$ There are two possibilities. - there is no positive solution $h$ for any $\lambda\in\mathbb{R}$, or - there exists a number $\overline{\beta}$ such that it has two linearly independent positive solutions for $\lambda<\overline{\beta},$ has no positive solution for $\lambda>\overline{\beta}$ and has one or two linearly independent solutions for $\lambda=\overline{\beta}.$ Refer to page 146 and 149 in [@Pinsky]. In this article, we implicitly assumed the second case by Assumption \[assume:Markovian\]. For each $\lambda$ with $\lambda\leq\overline{\beta},$ we say $(\lambda,h)$ is a candidate pair if $(\lambda,h)$ is a solution pair of $\mathcal{L}h=-\lambda h$ and if $h(\xi)=1$ (i.e., $h$ is normalized). We define the candidate set by $$\mathcal{C}_\lambda:=\{\,h'(\xi)\in\mathbb{R}\,\|\,\mathcal{L}h=- \lambda h,\, h(\xi)=1,\,h(\cdot)>0 \,\}\;.$$ It is known that $\mathcal{C}_\lambda$ is a connected compact set. Refer to [@Park14b] or [@Pinsky]. Denote the functions corresponding to $\max\mathcal{C}_\lambda$ and $\min\mathcal{C}_\lambda$ by $H_\lambda$ and $h_\lambda,$ respectively. It is assumed that $H_\lambda(\xi)=h_\lambda(\xi)=1.$ For a solution pair $(\lambda,h)$ with $h>0,$ it is easily checked that $$e^{\lambda t-\int_{0}^{t} r(X_{s})ds}\,h(X_{t})\,h^{-1}(\xi)$$ is a local martingale under $\mathbb{Q}.$ To be a Radon-Nikodym derivative, this should be a martingale. Thus, we are interested in solution pairs that induces martingales. Let $(\lambda,h)$ be a candidate pair. We say $(\lambda,h)$ is a admissible pair if $$e^{\lambda t-\int_{0}^{t} r(X_{s})ds}\,h(X_{t})\,h^{-1}(\xi)$$ is a martingale under $\mathbb{Q}.$ In this case, a measure obtained from the risk-neutral measure $\mathbb{Q}$ by the Radon-Nikodym derivative $$\left.\frac{\,d\,\cdot\,}{d\mathbb{Q}}\right\|_{\mathcal{F}_{t}}=e^{\lambda t-\int_{0}^{t} r(X_{s})ds}\,h(X_{t})\,h^{-1}(\xi)$$ is called [the transformed measure]{} with respect to the pair $(\lambda,h).$ We now investigate the bounds of the risk premium. We set $$\begin{aligned} \ell:&=\inf\{0\leq\lambda\leq\overline{\beta}\,\|\,(\lambda,h_\lambda)\text{ is an admissible pair}\} \\ L:&=\inf\{0\leq\lambda\leq\overline{\beta}\,\|\,(\lambda,H_\lambda)\text{ is an admissible pair}\}\;. \end{aligned}$$ Two functions $h_\ell$ and $H_L$ will play a crucial role in determining the bounds of the risk premium $\theta_t.$ To see this, we need the following proposition. \[prop:relation\] Let $\alpha<\lambda$ and let $(\lambda,h)$ be a candidate. Then, $$\begin{aligned} (h_\alpha'h_\alpha^{-1})(x)\leq(h'h^{-1})(x)\leq(H_\alpha'H_\alpha^{-1})(x)\;. \end{aligned}$$ See Appendix \[app:pf\_relation\] for proof. This proposition says that if a candidate pair $(\lambda,h)$ with $\lambda>0$ is an admissible pair, then $$\begin{aligned} (h_\ell'h_\ell^{-1})(x)\leq(h'h^{-1})(x)\leq(H_L'H_L^{-1})(x)\;. \end{aligned}$$ This equation gives upper and lower bounds on the risk premium. The only information that we know about the principal pair $(\beta,\phi)$ is that $\beta>0$ and that $(\beta,\phi)$ is an admissible pair. Thus, we can conclude that $$(h_\ell'h_\ell^{-1})(x)\leq(\phi'\phi^{-1})(x)\leq(H_L'H_L^{-1})(x)\;.$$ By Theorem \[thm:theta\_phi\], we obtain the main theorem of this article. \[thm:intrinsic\_bounds\][(Intrinsic Bounds of Risk Premium)]{.nodecor} Let $\theta_t$ be the risk premium. Then, $$\begin{aligned} (\sigma h_\ell'h_\ell^{-1})(X_t)\leq\theta_t\leq(\sigma H_L'H_L^{-1})(X_t)\;. \end{aligned}$$ This theorem implies that we can determine the range of the risk premium when a risk-neutral measure is given. Upper and lower bounds can then be calculated using option prices. In the next section, as an example, we show that in the classical Black-Scholes model, the risk premium satisfies $$-\frac{2r}{v}\leq\theta_t\leq v$$ where $r$ is the interest rate and $v$ is the volatility of the stock. Let $\theta_t$ be the risk premium. Then, $$\begin{aligned} (\sigma h_0'h_0^{-1})(X_t)\leq\theta_t\leq(\sigma H_0'H_0^{-1})(X_t)\;. \end{aligned}$$ This gives rough bounds on the risk premium. In general, two solutions $h_0$ and $H_0$ are more straightforward to find than $h_\ell$ and $H_L.$ We can find better bounds if some information of $\mathbb{P}$-dynamics of $X_t$ is known. As mentioned in Section \[sec:intro\], if $X_t$ is recurrent under the objective measure, then we can find the exact risk premium. Thus, the bounds is not informative. For example, if the state variable is an interest rate, which is usually recurrent under the objective measure, then we can find the precise risk premium. For more details, see [@Borovicka14],[@Park14b],[@Qin14b] and [@Walden13]. We can find better bounds when it is known that the state variable is non-attracted to the left (or right) boundary under the objective measure. This is a reasonable assumption under some situations. For example, a stock price process is usually not attracted to zero boundary. The process $X_t$ is non-attracted to the left boundary under the transformed measure with respect to $(\lambda,h)$ if only if $h=H_\lambda.$ See [@Park14b] for proof. Similarly, the process $X_t$ is non-attracted to the right boundary under the transformed measure with respect to $(\lambda,h)$ if only if $h=h_\lambda.$ This proposition says the following theorem. \[thm:non-attracted\] If the state process $X_t$ is non-attracted to the left boundary, then the risk premium $\theta_t$ satisfies $$(\sigma H_{\overline{\beta}}'H_{\overline{\beta}}^{-1})(X_t)\leq\theta_t\leq(\sigma H_L'H_L^{-1})(X_t)\;.$$ Similarly, if the state process $X_t$ is non-attracted to the right boundary, then the risk premium $\theta_t$ satisfies $$(\sigma h_\ell'h_\ell^{-1})(X_t)\leq\theta_t\leq(\sigma h_{\overline{\beta}}'h_{\overline{\beta}}^{-1})(X_t)\;.$$ Returns of Stock {#sec:appli} ================ In this section, we investigate the bounds of the risk premium when the state variable $X_t$ is the stock price process $S_t.$ In practice, the S$\&$P 500 index process, which can theoretically be regarded as a stock price process, is used as a state variable [@Audrino14]. In this case, we can determine upper and lower bounds on the return of the stock process $S_t.$ Suppose that $S_t=X_t$ and that the interest rate is constant $r.$ Under the risk-neutral measure, the dynamics of $X_t$ is $$dX_t=rX_t\,dt+\sigma(X_t)\,X_t\,dW_t\;.$$ By Theorem \[thm:intrinsic\_bounds\], the risk premium satisfies $$(\sigma h_\ell'h_\ell^{-1})(x)\leq\theta(x)\leq(\sigma H_L'H_L^{-1})(x)$$ where $h_\ell$ and $H_L$ are the corresponding solutions. We obtain upper and lower bounds on the return $\mu_t$ by using equation . $$r+(\sigma^2h_\ell'h_\ell^{-1})(X_t)\leq\mu_t\leq r+(\sigma^2H_L'H_L^{-1})(X_t)\;.$$ As an example, we explore the classical Black-Scholes model for stock price $X_t.$ $$dX_t=rX_t\,dt+vX_t\,dW_t\,,\;X_0=1$$ for $v>0.$ The infinitesimal operator is $$\mathcal{L}h(x)=\frac{1}{2}v^2x^2h''(x)+rxh'(x)-rh(x)\;.$$ It can be easily shown that every solution pair of $\mathcal{L}h=-\lambda h$ induces a martingale, that is, every candidate pair is an admissible pair. Thus it is obtained that $\ell=L=0.$ We want to find the positive solutions of $\mathcal{L}h=0$ with $h(0)=1.$ The solutions are given by $h(x)=c\,x^{-\frac{2r}{v^2}}+(1-c)x$ for $0\leq c\leq 1.$ Thus $$h_0(x)=x^{-\frac{2r}{v^2}},\; H_0(x)=x\;.$$ The risk premium $\theta_t$ satisfies $-\frac{2r}{v}\leq\theta_t\leq v.$ The upper and lower bounds of the return $\mu_t$ of $X_t$ is given by $-r\leq\mu_t\leq r+v^2.$ We can find better bounds if we know that the stock price is non-attracted to zero boundary. By direct calculation, we have that $\overline{\beta}=\frac{(r+\frac{1}{2}v^2)^2}{2v^2}$ and that $$H_{\overline{\beta}}(x)=x^{\frac{1}{2}-\frac{r}{v^2}},\; H_0(x)=e^x\;.$$ By Theorem \[thm:non-attracted\], the risk premium $\theta_t$ satisfies $\frac{1}{2}-\frac{r}{v}\leq\theta_t\leq v.$ The upper and lower bounds of the return $\mu_t$ of $X_t$ is given by $\frac{v}{2}\leq\mu_t\leq r+v^2.$ Conclusion {#sec:conclusion} ========== This article determined the possible range of the risk premium of the market using the market prices of options. One of the key assumptions to achieve this result is that the market is Markovian driven by a state variable $X_t.$ Under this assumption, we can transform the problem of determining the bounds into a second-order differential equation. We then obtain the upper and lower bounds of the risk premium by analyzing the differential equation. We illuminated how the problem of the risk premium is transformed into a problem described by a second-order differential equation. The risk premium is determined by $\theta_t=(\sigma\phi'\phi^{-1})(X_t)$ with a positive function $\phi(\cdot).$ We demonstrated that $\phi(\cdot)$ satisfies $\mathcal{L}\phi=-\beta\phi$ for some positive number $\beta$, where $\mathcal{L}$ is a second-order operator that is determined by option prices. We demonstrated that two special solutions $h_\ell$ and $H_L$ play a crucial role for determining upper and lower bounds on the risk premium. The risk premium $\theta_t$ satisfies $$\begin{aligned} (\sigma h_\ell'h_\ell^{-1})(X_t)\leq\theta_t\leq(\sigma H_L'H_L^{-1})(X_t)\;. \end{aligned}$$ We also discussed better bounds when it is known that the state variable is non-attracted to the left (or right) boundary. It is stated how this result can be applied to determine the range of return of an asset. The following extensions for future research are suggested. First, it would be interesting to extend the process $X_t$ to a multidimensional process or a process with jump. Second, it would be interesting to determine the bounds of the risk premium when the process $X_t$ is a non-Markov process. In this article, we discussed only a time-homogeneous Markov process. Third, it would be interesting to explore more general forms of the pricing kernel. We discussed only the case in which (the reciprocal of) the pricing kernel has the form $e^{\beta t}\phi(X_{t}).$ Acknowledgement {#acknowledgement .unnumbered} =============== The authors highly appreciate the detailed valuable comments of the referee. Proof of Proposition \[prop:relation\] {#app:pf_relation} ====================================== Hereafter, without loss of generality, we assume that $X_0=0,$ the left boundary of the range of $X_t$ is $-\infty$ and the right boundary of the range of $X_t$ is $\infty$. \[lem:finite\] Let $\alpha<\lambda$ and let $(\alpha,g)$ and $(\lambda,h)$ be candidate pairs. Then, $$\begin{aligned} \int_{-\infty}^{0}v^{-2}(y)\,dy\textnormal{ is finite if } h'(0)\geq g'(0)\;,\\ \int_{0}^{\infty}v^{-2}(y)\,dy\textnormal{ is finite if } h'(0)\leq g'(0)\;. \end{aligned}$$ where $g=vq$ and $q(x)=e^{-\int_{0}^{x}\frac{k(y)}{\sigma^{2}(y)}\,dy}.$ Write $h=uq.$ Assume that $h'(0)\geq g'(0),$ equivalently $u'(0)\geq v'(0).$ Define $\Gamma=\frac{u'}{u}-\frac{v'}{v}.$ Then $$\Gamma'=-\Gamma^{2}-\frac{2v'}{v}\Gamma-\frac{2(\lambda-\alpha)}{\sigma^{2}}\;.$$ Because $\Gamma(0)\geq0,$ we have that $\Gamma(x)>0$ for $x<0,$ since if $\Gamma$ ever gets close to $0,$ then term $-\frac{2(\lambda-\alpha)}{\sigma^{2}}$ dominates the right hand side of the equation. Choose $x_{0}$ with $x_{0}<0.$ For $x<x_{0},$ we have $$-\frac{2v'(x)}{v(x)}=\frac{\Gamma'(x)}{\Gamma(x)}+\Gamma(x)+\frac{2(\lambda-\alpha)} {\sigma^{2}(x)}\cdot\frac{1}{\Gamma(x)}\;.$$ Integrating from $x_{0}$ to $x,$ $$-2\ln\frac{v(\,x\,)}{v(x_{0})}=\ln \frac{\Gamma(\,x\,)}{\Gamma(x_{0})}+\int_{x_{0}}^{x} \Gamma(y)\,dy+\int_{x_{0}}^{x}\frac{2(\lambda-\alpha)}{\sigma^{2}(y)}\cdot\frac{1}{\Gamma(y)} \,dy$$ which leads to $$\frac{v^{2}(x_{0})}{v^{2}(\,x\,)}\leq \frac{\Gamma(\,x\,)}{\Gamma(x_{0})}\,e^{\int_{x_{0}}^{x} \Gamma(y)\,dy}$$ for $x<x_{0}.$ Thus, $$\begin{aligned} \int_{-\infty}^{x_{0}}\frac{1}{v^{2}(y)}\,dy &\leq (\text{constant})\cdot\int_{-\infty}^{x_{0}}\Gamma(y)\,e^{\int_{x_{0}}^{y}\Gamma(w)dw}\,dy \\ &=(\text{constant})\cdot\left(1-e^{-\int_{-\infty}^{x_{0}}\Gamma(w)dw}\right)\\ &\leq (\text{constant})\\ &< \infty \;. \end{aligned}$$ This implies that $\int_{-\infty}^{0}v^{-2}(y)\,dy$ is finite. Similarly, we can show that $\int_{0}^{\infty}v^{-2}(y)\,dy$ is finite if $h'(0)\leq g'(0).$ \[lem:infinite\] Write $H_\lambda=Vq$ and $h_\lambda=vq$, where $q(x):=e^{-\int_{0}^{x}\frac{k(y)}{\sigma^{2}(y)}\,dy}.$ Then, $$\int_{-\infty}^{0}V^{-2}(y)\,dy=\infty\quad\text{ and }\quad \int_{0}^{\infty}v^{-2}(y)\,dy=\infty\;.$$ We only prove that $\int_{-\infty}^{0}V^{-2}(y)\,dy=\infty.$ A general (normalized to $h(0;c)=1$) solution of $\mathcal{L}h=-\lambda h$ is expressed by $$h(x;c):=H_\lambda(x)\left(1+c\cdot \int_{0}^{x}V^{-2}(y)\,dy\right)\;,$$ thus we have $$h'(0;c)=H_\lambda'(0)+c.$$ Since $H_\lambda'(0)$ is the maximum value by definition, we have that $c\leq 0.$ Suppose that $\int_{-\infty}^{0}V^{-2}(y)\,dy<\infty$ then we can choose a small positive number $c$ such that $h(x;c)$ is a positive function. This is a contradiction. We now prove Proposition \[prop:relation\]. We only show the inequality for $x<0.$ First, we prove $(h_\alpha'h_\alpha^{-1})(x)<(h'h^{-1})(x)$ for $x<0.$ Let $q(x):=e^{-\int_{0}^{x}\frac{k(y)}{\sigma^{2}(y)}\,dy}.$ Write $h=uq$ and $h_\alpha=vq.$ Then we have $h'(0)>h_\alpha'(0),$ equivalently $u'(0)>v'(0).$ It is because, if not, by Lemma \[lem:finite\], we have $$\int_{0}^{\infty}v^{-2}(y)\,dy<\infty\;,$$ and this contradicts to Lemma \[lem:infinite\]. Now we define $\gamma=\frac{u'}{u}-\frac{v'}{v}.$ Then $$\gamma'=-\gamma^{2}-\frac{2v'}{v}\gamma-\frac{2(\lambda-\alpha)}{\sigma^{2}}\;.$$ Because $\gamma(0)>0,$ we have that $\gamma(x)>0$ for $x<0,$ since if $\gamma$ ever gets close to $0,$ then term $-\frac{2(\lambda-\alpha)}{\sigma^{2}}$ dominates the right hand side of the equation. Thus for $x<0,$ it is obtained that $(v'v^{-1})(x)<(u'u^{-1})(x),$ which implies that $(h_\alpha'h_\alpha^{-1})(x)<(h'h^{-1})(x).$ We now prove $(h'h^{-1})(x)<(H_\alpha'H^{-1}_\alpha)(x)$ Write $H_\lambda=Vq.$ Then we have $h'(0)<H_\alpha'(0),$ equivalently $u'(0)<V'(0).$ Define $\Gamma:=\frac{u'}{u}-\frac{V'}{V}.$ Then $$\Gamma'=-\Gamma^{2}-\frac{2V'}{V}\Gamma-\frac{2(\lambda-\alpha)}{\sigma^{2}}\;.$$ We claim that $\Gamma(x)<0$ for $x<0.$ Suppose there exists $x_0<0$ such that $\Gamma(x_0)\geq0.$ Then for all $z<x_0,$ it is obtained that $\Gamma(z)>0$ since if $\Gamma$ ever gets close to $0,$ then term $-\frac{2(\lambda-\alpha)}{\sigma^{2}}$ dominates the right hand side of the equation. For $z<x_0,$ we have $$-\frac{2V'(z)}{V(z)}=\frac{\Gamma'(z)}{\Gamma(z)}+\Gamma(z)+\frac{2(\beta-\lambda)} {\sigma^{2}(z)}\cdot\frac{1}{\Gamma(z)}\;.$$ Integrating from $x_{0}$ to $y,$ $$-2\ln\frac{V(\,y\,)}{V(x_{0})}=\ln \frac{\Gamma(\,y\,)}{\Gamma(x_{0})}+\int_{x_{0}}^{y} \Gamma(z)\,dz+\int_{x_{0}}^{y}\frac{2(\lambda-\alpha)}{\sigma^{2}(z)}\cdot\frac{1}{\Gamma(z)} \,dz$$ which leads to $$\frac{V^{2}(x_{0})}{V^{2}(\,y\,)}\leq \frac{\Gamma(\,y\,)}{\Gamma(x_{0})}\,e^{\int_{x_{0}}^{y} \Gamma(z)\,dz}$$ for $y<x_{0}.$ Thus, $$\begin{aligned} \int_{-\infty}^{x_{0}}V^{-2}(y)\,dy &\leq (\text{constant})\cdot\int_{-\infty}^{x_{0}}\Gamma(y)\,e^{\int_{x_{0}}^{y}\Gamma(w)dw}\,dy \\ &=(\text{constant})\cdot\left(1-e^{-\int_{-\infty}^{x_{0}}\Gamma(w)dw}\right)\\ &\leq (\text{constant})\\ &< \infty \;. \end{aligned}$$ This implies that $\int_{-\infty}^{0}V^{-2}(y)\,dy$ is finite. This contradicts to Lemma \[lem:infinite\]. [plain]{} Amrein, W.O., A.M. Hinz, and D.B. Pearson. 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--- abstract: 'Fully symmetric learning rules for principal component analysis can be derived from a novel objective function suggested in our previous work. We observed that these learning rules suffer from slow convergence for covariance matrices where some principal eigenvalues are close to each other. Here we describe a modified objective function with an additional term which mitigates this convergence problem. We show that the learning rule derived from the modified objective function inherits all fixed points from the original learning rule (but may introduce additional ones). Also the stability of the inherited fixed points remains unchanged. Only the steepness of the objective function is increased in some directions. Simulations confirm that the convergence speed can be noticeably improved, depending on the weight factor of the additional term.' author: - \| Ralf Möller\ Computer Engineering Group, Faculty of Technology\ Bielefeld University, Bielefeld, Germany\ [www.ti.uni-bielefeld.de](www.ti.uni-bielefeld.de) title: ' [Improved Convergence Speed of Fully Symmetric Learning Rules for Principal Component Analysis]{}' --- \#1 [ ]{}[ ]{} mathcmds-1-4.tex Introduction ============ In our previous work [@own_Moeller20a], we derived several fully symmetric learning rules[^1] for principal component analysis (PCA), starting from a novel objective function (in this paper referred to as “original” objective function). We analyzed the fixed points of these learning rules and (indirectly via the objective function) their stability. We could show that the learning rules have stable, desired fixed points in the eigenvectors of the covariance matrix, but exhibit additional undesired fixed points; however, the latter are unstable. Preliminary simulations confirmed that the learning rules converge towards the desired fixed points, but also revealed a disadvantage: If some principal eigenvalues of the covariance matrix are close to each other, the learning rules operate close the undesired fixed points which noticeably slows down convergence. In this continuation of our work, we introduce an additional term into our objective function which mitigates the convergence problem. We derive a learning rule from this modified objective function.[^2] We determine the fixed points of the learning rule and show that the modified learning rule shares the fixed points of the original one, but may introduce additional fixed points. Using the same indirect method as in our previous work, we study the stability at the shared fixed points and show that it is unchanged compared to the original objective function. Simulations confirm both the theoretical results and the improved convergence speed of the novel learning rule. We recapitulate the notation in section \[sec\_notation\] and our Lagrange-multiplier approach in section \[sec\_lagrange\]. The original objective function and the corresponding (“short”) learning rule are recapitulated in \[sec\_objfct\_orig\] together with insights on the fixed-point structure which motivate the modifications introduced here. Section \[sec\_objfct\_mod\_1\] introduces the modified objective function from which we derive a (“short”) learning rule in section \[sec\_lr\_mod\]. The fixed points of this modified learning rule are analyzed in section \[sec\_M2S\_fp\]. The stability of the fixed points is analyzed indirectly from the modified objective function in section \[sec\_stability\_mod\]. Simulations are presented in section \[sec\_simulations\]. The report ends with a discussion (section \[sec\_discussion\]) and conclusions (section \[sec\_conclusion\]). Notation {#sec_notation} ======== We use the same notation as in our previous work [@own_Moeller20a]. Table \[tab\_matrices\] shows the names of widely used matrices. Column vector $i$ of a matrix $\matX$ is written as $\vecx_i$. Fixed-point variables are marked by a bar (e.g. $\matWnull$). Sometimes, matrix and vector sizes are indicated by suffixes; for vectors and symmetric matrices, only one suffix is provided. Table \[tab\_operators\] shows the operators used. ------------------ -------------- ------------------------------------------------------------------------------ $\matC$ $n \times n$ covariance matrix $\matV$ $n \times n$ matrix of eigenvectors $\vecv_i$ (columns) of $\matC$ $\matLambda$ $n \times n$ diagonal matrix of eigenvalues $\lambda_i$ (distinct, descending) of $\matC$ $\matW$ $n \times m$ matrix of principal eigenvector estimates $\vecw_i$ (columns) of $\matC$ $\matA$ $n \times m$ projection of $\matW$ onto the eigenvectors $\matV$ $\matQ$ $n \times n$ orthogonal matrix: $\matQ^T \matQ = \matQ \matQ^T = \matI_n$ $\matB$ $m \times m$ matrix used to form the Lagrange multipliers $\matI_n$ $n \times n$ unit matrix $\matNull_{n,m}$ $n \times m$ null matrix ------------------ -------------- ------------------------------------------------------------------------------ : Notation: matrices[]{data-label="tab_matrices"} --------------------------------------- ----------------------------------------------------- $\delta_{ij}$ Kronecker’s delta $\dg\{\matX\}$ diagonal matrix with diagonal elements from $\matX$ $\diag_{i=1}^{n}\{x_i\}$ diagonal matrix with $n$ diagonal elements $x_i$ $\blkdiag_{l=1}^{k}\{\matX_l\}$ block-diagonal matrix with $k$ blocks $\matX_l$ $\\|\matX\\|^2_F = \tr\{\matX^T\matX\}$ squared Frobenius norm of $\matX$ --------------------------------------- ----------------------------------------------------- : Notation: operators[]{data-label="tab_operators"} References to equations and lemmata from our previous work [@own_Moeller20a] are printed in bold font. Lagrange-Multiplier Approach {#sec_lagrange} ============================ We use the Lagrange-multiplier approach from our previous work [@own_Moeller20a]. For a given objective function $J$, we write the extended objective function $J^$ as $$\begin{aligned} J^(\matB,\matW) = J(\matW) + C(\matB,\matW)\end{aligned}$$ where $C$ is the constraint term which includes the matrix $\matB$ (elements $\beta_{jk}$) which forms the Lagrange multipliers. We use the same symmetric construction for the Lagrange multipliers as in equation [(39)]{}, here with $\Omega_j = 1$ (i.e. operating on a Stiefel manifold): $$\begin{aligned} C(\matB,\matW) = \half \summe{j=1}{m}\summe{k=1}{m} \half (\beta_{jk}+\beta_{kj}) \left(\vecw_j^T\vecw_k-\delta_{jk}\right).\end{aligned}$$ Original Objective Function {#sec_objfct_orig} =========================== The original “novel” objective function from equation [(23)]{} is $$\begin{aligned} \label{eq_objfct_orig} J(\matW) = \quarter \summe{j=1}{m} \left(\vecw_j^T\matC\vecw_j\right)^2.\end{aligned}$$ We are interested in the local maxima of this function. From (\[eq\_objfct\_orig\]) we derived a fully symmetric learning rule “N2S”, either from our “short” form derivation [(450)]{} or from the canonical metric on the Stiefel manifold [(486)]{}: $$\begin{aligned} \label{eq_N2S} \tau \matWdot = \matC\matW\matD - \matW\matD\matW^T\matC\matW\end{aligned}$$ where $\tau$ is a time constant and $$\begin{aligned} \matD = \Diag{j=1}{m}\{\vecw_j^T\matC\vecw_j\}.\end{aligned}$$ The fixed-point structure of this equation is relatively complex. If all diagonal elements of $\matDnull$ are pairwise different, we obtain the special solution [(270)]{} $$\begin{aligned} \label{eq_N2S_fp_special} \matWnull = \matV \matP \pmat{\matI_m\\ \matNull}\end{aligned}$$ where $\matP$ is an arbitrary $n\times n$ permutation matrix. If some diagonal elements of $\matDnull$ may coincide, we obtain the general solution for the fixed points $$\begin{aligned} \label{eq_N2S_fp_general} \matWnull = \matV \matP \pmat{\matU^{T} \matP^\\ \matNull}.\end{aligned}$$ Here $\matU^$ is an orthogonal block-diagonal matrix (where the size of each block depends on the number of identical diagonal elements in $\matDnull$), and $\matP^$ another permutation matrix (which is chosen such that identical diagonal elements in $\matDnull$ are contiguous in a rearranged matrix $\matDnull^$, see [(249)]{}). To motivate our modified objective function below, we look at the term $\matWnull^T \matC \matWnull$. From equations [(282)]{} and [(284)]{} we know that, in the fixed points, we have $$\begin{aligned} \label{eq_N2S_WTCW} \matWnull^T \matC \matWnull &= \matP^{T} \Blkdiag{l=1}{k}\{\matU^{'}_l\hat{\matLambda}^_l\matU^{'T}_l\} \matP^\end{aligned}$$ under the constraint [(295)]{} $$\begin{aligned} \label{eq_N2S_constraint} \dg\{\matU^{'}_l\hat{\matLambda}^_l\matU^{'T}_l\} = \overline{d}^{'}_l \matI_l.\end{aligned}$$ Each diagonal matrix $\hat{\matLambda}^_l$ is a block of the upper-left $m \times m$ part of a permuted version of the eigenvalue matrix $\matLambda$. It is not clear which matrices $\matU_l^{'}$ fulfill constraint (\[eq\_N2S\_constraint\]). However, we can say that if the constraint is fulfilled and $\matU^{'}_l$ is not of size $1 \times 1$, the matrix $\matU^{'}_l\hat{\matLambda}^_l\matU^{'T}_l$ cannot be diagonal [(Lemma 5)]{}. Therefore the block-diagonal matrix in (\[eq\_N2S\_WTCW\]) has non-zero off-diagonal elements. A permutation transformation of a matrix as in (\[eq\_N2S\_WTCW\]) permutes the positions of the diagonal elements (see [(516)]{}), which entails that off-diagonal elements remain at off-diagonal positions. Therefore $\matWnull^T \matC \matWnull$ from (\[eq\_N2S\_WTCW\]) has non-zero off-diagonal elements if we are at an [undesired]{} fixed point. In contrast, if we look at the [desired]{} fixed points from (\[eq\_N2S\_fp\_special\]), we see that $\matWnull^T \matC \matWnull$ is diagonal:[^3] $$\begin{aligned} \matWnull^T \matC \matWnull &= \pmat{\matI_m & \matNull} \matP^T \matV^T \matC \matV \matP \pmat{\matI_m\\ \matNull}\\ &= \pmat{\matI_m & \matNull} \matP^T \matLambda \matP \pmat{\matI_m\\ \matNull}\\ &= \pmat{\matI_m & \matNull} \matLambda^* \pmat{\matI_m\\ \matNull}\\ &= \hat{\matLambda}^.\end{aligned}$$ Therefore we introduce a term into the objective function where off-diagonal elements in $\matWnull^T \matC \matWnull$ are pushed towards zero. Modified Objective Function {#sec_objfct_mod} =========================== In the following, we suggest a modified objective function by introducing an additional term, derive a learning rule, analyze its fixed points, and study the stability of the fixed points indirectly through the behavior of the objective function. Modified Objective Function {#sec_objfct_mod_1} --------------------------- We suggest the following modified objective function: $$\begin{aligned} \label{eq_objfct_mod} J(\matW) = \quarter \left[ (1 + \alpha) \summe{j=1}{m} \left(\vecw_j^T\matC\vecw_j\right)^2 - \alpha \summe{j=1}{m}\summe{k=1}{m}\left(\vecw_j^T\matC\vecw_k\right)^2 \right].\end{aligned}$$ Again, we are interested in the local maxima of this function. The first term of (\[eq\_objfct\_mod\]) coincides with the original objective function (\[eq\_objfct\_orig\]). A second term with negative sign is added which penalizes non-zero off-diagonal elements in $\matW^T \matC \matW$ as motivated in section \[sec\_objfct\_orig\]. A weight factor $\alpha$ is introduced which expresses the influence of the second term. The factors of the two terms are chosen such that terms $(\vecw_j^T \matC \vecw_k)^2$ with $j = k$ are weighted with $1$, and terms with $j \neq k$ are weighted with $-\alpha$. By writing the equation in this way we can avoid the use of Kronecker’s delta. Due to the negative sign and the squared expressions, the terms with $j \neq k$ are maximized if they are zero. Derivation of Modified Learning Rule {#sec_lr_mod} ------------------------------------ To derive a learning rule from the modified objective function (\[eq\_objfct\_mod\]), we first determine its derivative with respect to a single weight vector $\vecw_l$: $$\begin{aligned} \nonumber &\ddf{J}{\vecw_l}\\ &= \half \left[ (1 + \alpha) \summe{j=1}{m} \left(\vecw_j^T\matC\vecw_j\right) \ddf{\vecw_j^T\matC\vecw_j}{\vecw_l} - \alpha \summe{j=1}{m}\summe{k=1}{m}\left(\vecw_j^T\matC\vecw_k\right) \ddf{\vecw_j^T\matC\vecw_k}{\vecw_l} \right]\\ &= (1 + \alpha) \summe{j=1}{m} \left(\vecw_j^T\matC\vecw_j\right) \matC \vecw_j \delta_{jl} - \half \alpha \summe{j=1}{m}\summe{k=1}{m}\left(\vecw_j^T\matC\vecw_k\right) \left(\matC \vecw_k \delta_{jl} + \matC \vecw_j \delta_{kl}\right)\\ &= (1 + \alpha) \left(\vecw_l^T\matC\vecw_l\right) \matC \vecw_l - \half \alpha \left[ \summe{k=1}{m}\left(\vecw_l^T\matC\vecw_k\right) \matC \vecw_k + \summe{j=1}{m}\left(\vecw_j^T\matC\vecw_l\right) \matC \vecw_j \right]\\ &= (1 + \alpha) \left(\vecw_l^T\matC\vecw_l\right) \matC \vecw_l - \half \alpha \left[ \summe{j=1}{m}\left(\vecw_l^T\matC\vecw_j\right) \matC \vecw_j + \summe{j=1}{m}\left(\vecw_j^T\matC\vecw_l\right) \matC \vecw_j \right]\\ &= (1 + \alpha) \left(\vecw_l^T\matC\vecw_l\right) \matC \vecw_l - \alpha \summe{j=1}{m}\left(\vecw_j^T\matC\vecw_l\right) \matC \vecw_j\\ &= (1 + \alpha) \left(\vecw_l^T\matC\vecw_l\right) \matC \vecw_l - \alpha \summe{j=1}{m}\matC \vecw_j\vecw_j^T\matC\vecw_l\\ &= (1 + \alpha) \left(\vecw_l^T\matC\vecw_l\right) \matC \vecw_l - \alpha \matC \left(\summe{j=1}{m} \vecw_j\vecw_j^T\right)\matC\vecw_l\\ &= (1 + \alpha) \left(\vecw_l^T\matC\vecw_l\right) \matC \vecw_l - \alpha \matC \matW \matW^T \matC\vecw_l.\end{aligned}$$ Now we combine the expression above into a derivative with respect to the entire matrix $\matW$ (with $m$ columns $\vecw_l$, $l = 1,\ldots,m$): $$\begin{aligned} \matM \coloneqq \ddf{J}{\matW} &= (1 + \alpha) \matC \matW \underbrace{\Diag{j=1}{m}\{\vecw_j^T \matC \vecw_j\}}_{\matD} - \alpha \matC \matW \matW^T \matC \matW\\ &= (1 + \alpha) \matC \matW \matD - \alpha \matC \matW \matW^T \matC \matW.\end{aligned}$$ In our previous work [@own_Moeller20a] we found that there are two variants to eliminate the Lagrange multipliers, the first leading to “uninteresting” principal subspace rules, the second to “interesting” PCA rules. We use the second variant and our “short” form derivation and obtain the following “modified” learning rule which we henceforth refer to as “M2S”: $$\begin{aligned} \nonumber \tau \matWdot &= \matM\\ &- \matW \matM^T \matW\\[5mm] \nonumber &= (1 + \alpha) \matC \matW \matD - \alpha \matC \matW \matW^T \matC \matW\\ &- \matW \left[ (1 + \alpha) \matD \matW^T \matC - \alpha \matW^T \matC \matW \matW^T \matC \right] \matW\\[5mm] \nonumber &= (1 + \alpha) \matC \matW \matD - \alpha \matC \matW (\matW^T \matC \matW)\\ \label{eq_M2S_form0} &- (1 + \alpha) \matW \matD (\matW^T \matC \matW) + \alpha \matW (\matW^T \matC \matW) (\matW^T \matC \matW).\end{aligned}$$ We can arrange equation (\[eq\_M2S\_form0\]) in two ways. In the first arrangement, we sort the terms according to the common factors $(1+\alpha)$ and $-\alpha$: $$\begin{aligned} \label{eq_M2S_form1} \tau \matWdot = (1 + \alpha) (\matC\matW\matD-\matW\matD\matW^T\matC\matW) - \alpha (\matC\matW-\matW\matW^T\matC\matW)(\matW^T\matC\matW).\end{aligned}$$ This arrangement leads to an interesting insight on the fixed-point structure of “M2S” which is elaborated in section \[sec\_M2S\_fp\]. The second arrangement is obtained from (\[eq\_M2S\_form0\]) by combining the first with the second and the third with the forth term, and factoring out common terms: $$\begin{aligned} \tau \matWdot &= \matC \matW \left[(1 + \alpha) \matD - \alpha \matW^T\matC\matW\right] - \matW \left[(1 + \alpha) \matD - \alpha \matW^T\matC\matW\right] \matW^T\matC\matW\\ \label{eq_M2S_form2} &= \matC \matW \matD'_\alpha - \matW \matD'_\alpha \matW^T\matC\matW.\end{aligned}$$ We see that we obtain the same form as in “N2S” (\[eq\_N2S\]), but with a matrix $$\begin{aligned} \label{eq_Dp_alpha} \matD'_\alpha = (1 + \alpha) \matD - \alpha \matW^T\matC\matW\end{aligned}$$ instead of $\matD$. Note that $\matD'_\alpha$ is not generally diagonal (but $\matDnull'_\alpha$ would be diagonal if the rule actually converges to the principal eigenvectors). Fixed Points of Modified Learning Rule {#sec_M2S_fp} -------------------------------------- We can gain an interesting insight on the fixed-point structure of “M2S” from an analysis of the first arrangement of terms in (\[eq\_M2S\_form1\]). We see that the first term coincides with the original learning rule “N2S” from (\[eq\_N2S\]). The second term contains the right-hand side of Oja’s subspace rule [(110)]{} $$\begin{aligned} \label{eq_Oja_subspace} \tau \matWdot = \matC\matW-\matW\matW^T\matC\matW\end{aligned}$$ as the first factor [@nn_Oja89]. We know from [(113)]{} that the fixed points of (\[eq\_Oja\_subspace\]) are $$\begin{aligned} \label{eq_Oja_subspace_fp} \matWnull = \matV \matP \pmat{\matR\\ \matNull}\end{aligned}$$ where $\matR$ is an arbitrary orthogonal matrix, thus the subspace factor in the second term of (\[eq\_M2S\_form1\]) will disappear as soon as the eigenvector estimates span the same subspace as an arbitrary selection of $m$ eigenvectors of $\matC$. The general fixed-point solution of “N2S” (\[eq\_N2S\_fp\_general\]) always fulfills (\[eq\_Oja\_subspace\_fp\]) with $\matR = \matU^{T}\matP^$ ($\matU^$ and $\matP^$ are orthogonal, as is their product), thus the second term of (\[eq\_M2S\_form1\]) disappears in the fixed points of “N2S”. This leads to the insight that all fixed points of “N2S” are also present in “M2S”. The additional term in the modified objective function apparently only shapes the landscape outside the fixed points. Note, however, that learning rule “M2S” may have additional fixed points compared to “N2S”. Aside from this observation, the interpretation of the second term is difficult. The entire second term may also disappear for other values of $\matW$, depending on the interplay between first and second factor. Moreover, the negative sign of the second term implies that this term will probably not push $\matW$ towards the subspace described above. For the fixed-point analysis of “M2S”, we proceed as in our previous work [@own_Moeller20a]. We express $\matWnull$ through the projections $\matAnull$ onto the eigenvectors by $\matWnull = \matV \matAnull$, apply $\matV^T \matC \matV = \matLambda$, insert the ansatz [(84)]{} $$\begin{aligned} \matAnull = \matQ \pmat{\matI_m\\ \matNull}\end{aligned}$$ where $\matQ$ is an orthogonal matrix and therefore $\matAnull$ is semi-orthogonal (located on a Stiefel manifold defined by $\matAnull^T \matAnull = \matI_m$), and define $$\begin{aligned} \matM \coloneqq \matQ^T \matLambda \matQ = \pmat{\matS & \matT^T\\ \matT & \matU}.\end{aligned}$$ In appendix \[app\_constraints\_S\_T\] we describe two attempts — starting from either (\[eq\_M2S\_form1\]) or (\[eq\_M2S\_form2\]) — at deriving constraints on $\matS$ and $\matT$ which lead to the same result, namely $$\begin{aligned} \label{eq_M2S_constraint_S} \matS\matDnull &= \matDnull\matS\\ \label{eq_M2S_constraint_T} \matT \left[(1+\alpha)\matDnull-\alpha\matS\right] &= \matNull.\end{aligned}$$ While the constraint on $\matS$ (\[eq\_M2S\_constraint\_S\]) coincides with the one for “N2S”, the constraint on $\matT$ (\[eq\_M2S\_constraint\_T\]) differs from the one for “N2S” (where it is $\matT\matDnull=\matNull$ with the only solution $\matT = \matNull$). The constraint (\[eq\_M2S\_constraint\_T\]) also has the solution $\matT = \matNull$, but can have additional, non-zero solutions if the factor $\matD'_\alpha=(1+\alpha)\matDnull-\alpha\matS$ is singular. A simulation shows that $\det\{\matD'_{\alpha}\}$ can actually be zero, see figure \[fig\_det\_Dp\_alpha\] in appendix \[app\_add\_fp\]. We will focus on the case $\matT = \matNull$ which coincides with “N2S”. For this case, the derivation completely coincides with the one of “N2S” (from equation [(248)]{} onward) and leads to the special solution (\[eq\_N2S\_fp\_special\]) and the general solution (\[eq\_N2S\_fp\_general\]). Stability Analysis {#sec_stability_mod} ------------------ The stability analysis uses the same indirect approach as in our previous work [@own_Moeller20a Sec. 8]. We can use the the following expressions from [(330)]{}, [(335)]{}, and [(338)]{}: $$\begin{aligned} \matWnull^T \matC \matWnull &= \matU_m^T \hat{\matLambda}^ \matU_m \eqqcolon \matH\\ \matW^T \matC \matW &= \matF^T \matH \matF + \matB^T \check{\matLambda}^* \matB\\ \nonumber \matF^T \matH \matF &\approx \matH + \matA^T \matH + \matH \matA + \matA^T \matH \matA\\ &- \half (\matA^T \matA + \matB^T \matB) \matH - \half \matH (\matA^T \matA + \matB^T \matB).\end{aligned}$$ We compute the change in the objective function under a small step from fixed point $\matWnull$ (on the Stiefel manifold) to point $\matW$ obtained by an approximated back-projection onto the Stiefel manifold. The step is parametrized by a skew-symmetric $m \times m$ matrix $\matA$ and an $(n-m)\times m$ matrix $\matB$. For the modified objective function from equation (\[eq\_objfct\_mod\]) we get $$\begin{aligned} \Delta J &= J(\matW) - J(\matWnull)\\ &= (1+\alpha) \underbrace{\quarter \bigg[ \summe{j=1}{m} \left(\vecw_j^T\matC\vecw_j\right)^2 - \summe{j=1}{m} \left(\vecwnull_j^T\matC\vecwnull_j\right)^2 \bigg]}_{\Delta J_1}\\ &+ \alpha \underbrace{\quarter \bigg[ \summe{j=1}{m}\summe{k=1}{m}\left(\vecwnull_j^T\matC\vecwnull_k\right)^2 - \summe{j=1}{m}\summe{k=1}{m}\left(\vecw_j^T\matC\vecw_k\right)^2 \bigg]}_{\Delta J_2}\end{aligned}$$ where the negative sign was incorporated into $\Delta J_2$. We see that $\Delta J_1$ describes the change of the original objective function (\[eq\_objfct\_orig\]) for which we derived [(413)]{} $$\begin{aligned} \nonumber &\Delta J_1\approx\\ & \half \summe{j=1}{m} \matH_{jj} \left\{ (\matA^T \matH \matA)_{jj} - [(\matA^T \matA + \matB^T \matB) \matH]_{jj} + (\matB^T \check{\matLambda}^* \matB)_{jj} \right\} + \summe{j=1}{m} [(\matA^T \matH)_{jj}]^2.\end{aligned}$$ For $\Delta J_2$ we obtain $$\begin{aligned} \nonumber \Delta J_2 &= \quarter \summe{j=1}{m} \summe{k=1}{m} \left[ (\vecwnull_j^T\matC\vecwnull_k)^2 - (\vecw_j^T\matC\vecw_k)^2 \right]\\ &= \quarter \summe{j=1}{m} \summe{k=1}{m} \left[ (\vece_j^T\matWnull^T\matC\matWnull\vece_k)^2 - (\vece_j^T\matW^T\matC\matW\vece_k)^2 \right]\\ &= \quarter \summe{j=1}{m} \summe{k=1}{m} \left[ (\vece_j^T\matH\vece_k)^2 - (\vece_j^T\{\matF^T\matH\matF + \matB^T\check{\matLambda}^\matB\}\vece_k)^2 \right]\\ &= \quarter \left[ \\|\matH\\|^2_F - \\|\matF^T\matH\matF + \matB^T\check{\matLambda}^\matB\\|^2_F\right]\\ &= \quarter \left[ \\|\matU_m^T \hat{\matLambda}^* \matU_m \\|^2_F - \tr\{(\matF^T\matH\matF + \matB^T\check{\matLambda}^\matB)^2\}\right]\\ &= \quarter \left[ \\|\hat{\matLambda}^\\|^2_F - \tr\{(\matF^T\matH\matF + \matB^T\check{\matLambda}^\matB)^2\}\right]\\ &\approx \quarter \left[ \\|\hat{\matLambda}^\\|^2_F - \tr\{ (\matF^T\matH\matF)^2 + 2 (\matF^T\matH\matF)(\matB^T\check{\matLambda}^\matB) \}\right]\\ \label{eq_deltaJ2} &= \quarter \left[ \\|\hat{\matLambda}^\\|^2_F - \tr\{(\matF^T\matH\matF)^2\} - 2 \tr\{(\matF^T\matH\matF)(\matB^T\check{\matLambda}^\matB)\}\right]\end{aligned}$$ where we omitted terms above second order in the approximation. Note that for symmetric $\matX$ we have $\\|\matX\\|^2_F = \tr\{\matX^T\matX\} = \tr\{\matX^2\}$. We further process the second term of (\[eq\_deltaJ2\]), using the invariance of the trace to cyclic permutation, exploiting skew-symmetry $\matA^T = -\matA$ and symmetry $\matH^T = \matH$, and omitting terms above second order in $\matA$ and $\matB$: $$\begin{aligned} \nonumber & \tr\{(\matF^T\matH\matF)^2\}\\[5mm] \nonumber &\approx \tr\bigg\{ \Big[\matH + \matA^T \matH + \matH \matA + \matA^T \matH \matA\\ &- \half (\matA^T \matA + \matB^T \matB) \matH - \half \matH (\matA^T \matA + \matB^T \matB) \Big]^2\bigg\}\\[5mm] \nonumber &\approx \tr\bigg\{ \matH^2 + \matH\matA^T\matH + \matH^2\matA + \matH\matA^T\matH\matA\\ \nonumber &-\half\matH(\matA^T\matA+\matB^T\matB)\matH -\half\matH^2(\matA^T\matA+\matB^T\matB)\\ \nonumber &+\matA^T\matH^2 + \matA^T\matH\matA^T\matH + \matA^T\matH\matH\matA\\ \nonumber &+\matH\matA\matH + \matH\matA\matA^T\matH + \matH\matA\matH\matA\\ \nonumber &+\matA^T\matH\matA\matH\\ &-\half(\matA^T\matA+\matB^T\matB)\matH^2 -\half \matH(\matA^T\matA+\matB^T\matB)\matH \bigg\}\\[5mm] \nonumber &= \tr\bigg\{ \matH^2 + \matH^2\matA^T + \matH^2\matA + \matH\matA^T\matH\matA\\ \nonumber &-\half\matH^2(\matA^T\matA+\matB^T\matB) -\half\matH^2(\matA^T\matA+\matB^T\matB)\\ \nonumber &+\matH^2\matA^T + \matH\matA^T\matH\matA^T + \matH^2\matA\matA^T\\ \nonumber &+\matH^2\matA + \matH^2\matA\matA^T + \matH\matA\matH\matA\\ \nonumber &+\matH\matA^T\matH\matA\\ &-\half\matH^2(\matA^T\matA+\matB^T\matB) -\half\matH^2(\matA^T\matA+\matB^T\matB)\bigg\}\\[5mm] &= \tr\left\{ \matH^2 -2\matH^2(\matA^T\matA+\matB^T\matB) +2\matH^2\matA\matA^T\right\}\\[5mm] &= \tr\left\{ \matH^2 -2\matH^2(\matA^T\matA+\matB^T\matB) +2\matH^2\matA^T\matA\right\}\\[5mm] &= \tr\left\{ \matH^2 -2\matH^2\matB^T\matB\right\}\\[5mm] &= \tr\left\{\matH^2\right\} -2\tr\left\{\matH^T\matB^T\matB\matH\right\}\\[5mm] &= \\|\hat{\matLambda}^\\|^2_F -2\\|\matB\matH\\|^2_F.\end{aligned}$$ The third term of (\[eq\_deltaJ2\]) only has terms of second order (or below) by taking $\matH$ from the first factor: $$\begin{aligned} \tr\left\{(\matF^T\matH\matF)(\matB^T\check{\matLambda}^\matB)\right\} \approx \tr\left\{\matH\matB^T\check{\matLambda}^\matB\right\}.\end{aligned}$$ We summarize: $$\begin{aligned} \label{eq_deltaJ2_final} \Delta J_2 \approx \half\left[ \tr\{\matH^2\matB^T\matB\} - \tr\{\matH\matB^T\check{\matLambda}^\matB\} \right].\end{aligned}$$ For the special case with pairwise different elements in $\matDnull$ we have $\matU_m = \matI_m$ and thus $\matH = \hat{\matLambda}^$. We apply [(576)]{} and obtain $$\begin{aligned} \Delta J_2 &\approx \half\left[ \tr\{\hat{\matLambda}^{2}\matB^T\matB\} - \tr\{\hat{\matLambda}^\matB^T\check{\matLambda}^\matB\} \right]\\ &= \half\left[ \summe{j=1}{m} \hat{\lambda}_j^{2} (\matB^T\matB)_{jj} - \summe{j=1}{m} \hat{\lambda}_j^* (\matB^T\check{\matLambda}^\matB)_{jj} \right]\\ &= -\half\left[ \summe{j=1}{m} \hat{\lambda}_j^ (\matB^T\check{\matLambda}^\matB)_{jj} - \summe{j=1}{m} \hat{\lambda}_j^{2} (\matB^T\matB)_{jj} \right]\\ \label{eq_deltaJ2_final_mod} &= -\half\summe{j=1}{m}\left[ \hat{\lambda}_j^* (\matB^T\check{\matLambda}^\matB)_{jj} - \hat{\lambda}_j^{2} (\matB^T\matB)_{jj} \right].\end{aligned}$$ For the special case we also have with [(376)]{} $$\begin{aligned} \nonumber \Delta J_1 &\approx \half\summe{j=1}{m}\left[ \hat{\lambda}^_j (\matA^T \hat{\matLambda}^ \matA)_{jj} - \hat{\lambda}^{^2}_j (\matA^T \matA)_{jj}\right]\\ &+ \half\summe{j=1}{m}\left[ \hat{\lambda}^_j (\matB^T \check{\matLambda}^* \matB)_{jj} - \hat{\lambda}^{^2}_j (\matB^T \matB)_{jj}\right],\end{aligned}$$ thus by combining the two expressions we obtain $$\begin{aligned} \nonumber \Delta J &\approx \half (1 + \alpha) \summe{j=1}{m}\left[ \hat{\lambda}^_j (\matA^T \hat{\matLambda}^* \matA)_{jj} - \hat{\lambda}^{^2}_j (\matA^T \matA)_{jj}\right]\\ \nonumber &+ \half (1 + \alpha) \summe{j=1}{m}\left[ \hat{\lambda}^_j (\matB^T \check{\matLambda}^* \matB)_{jj} - \hat{\lambda}^{^2}_j (\matB^T \matB)_{jj}\right]\\ &- \half \alpha \summe{j=1}{m}\left[ \hat{\lambda}_j^ (\matB^T\check{\matLambda}^\matB)_{jj} - \hat{\lambda}_j^{2} (\matB^T\matB)_{jj} \right]\\ \nonumber &= \half (1 + \alpha) \summe{j=1}{m}\left[ \hat{\lambda}^_j (\matA^T \hat{\matLambda}^ \matA)_{jj} - \hat{\lambda}^{^2}_j (\matA^T \matA)_{jj}\right]\\ &+ \half \summe{j=1}{m}\left[ \hat{\lambda}^_j (\matB^T \check{\matLambda}^* \matB)_{jj} - \hat{\lambda}^{^2}_j (\matB^T \matB)_{jj}\right].\end{aligned}$$ As in the original objective function (\[eq\_objfct\_orig\]), we can demonstrate the existence of a maximum ($\Delta J < 0$) if the first $m$ eigenvectors are associated with the $m$ largest eigenvalues [(section 8.4.1)]{}; these are the “desired” fixed points. Otherwise we obtain a saddle point or a minimum ($\Delta J > 0$ in some directions). For the general case where diagonal elements in $\matDnull$ may coincide and where we have $\matH = \matU_m^T \hat{\matLambda}^ \matU_m$, we could show that $\Delta J > 0$ for $\matB = \matNull$ and a specific choice of $\matA$ [(section 8.4.2)]{}. Since $\Delta J_2$ only depends on $\matB$ and disappears for $\matB = \matNull$, we can demonstrate that the “undesired” fixed points are either saddle points or minima. We conclude that the additional term introduced in the modified objective function (\[eq\_objfct\_mod\]) leaves the stability of the fixed points unchanged. We also see that the factor $(1+\alpha)$ leads to a steeper shape of the objective function in the vicinity of the fixed points, at least in some directions (determined by step parameter $\matA$). Simulations {#sec_simulations} =========== As in our previous work, we restrict our simulations to averaged learning rules operating on the covariance matrix $\matC = E\{\vecx\vecx^T\}$ (in contrast, online learning rules operate on individual data vectors $\vecx$). Methods {#sec_simulations_methods} ------- We explore the behavior of the following learning rules: “TwJ2S” : from [(449)]{}, which is the same as rule (15a) from [@nn_Xu93], with $\matTheta = \diag_{j=1}^m\{j/m\}$, “N2S” : from (\[eq\_N2S\]), which is the same as “M2S” with $\alpha=0$, and “M2S” : from (\[eq\_M2S\_form2\]) for $\alpha \in \{1.0, 2.0, 5.0, 10.0, 20.0\}$. We determine eigenvector estimates $\matW$ with $n = 10$ and $m = 4$. We start from a random initial $\matW$ located on the Stiefel manifold ($\matW^T \matW = \matI_m$) which is the same for all learning rules and all figures. We generate a $n \times n$ covariance matrix $\matC$ from a random orthogonal $\matV$ and a diagonal eigenvalue matrix $\matLambda$ through $\matC = \matV \matLambda \matV^T$. The matrix $\matLambda$ is generated from one of the following eigenvalue sets, either “nearby eigenvalues” : $\{0.91, 0.9, 0.8, \ldots, 0.1\}$ or “evenly spaced eigenvalues” : $\{1.0, 0.9, 0.8, \ldots, 0.1\}$ through $\matLambda = \diag_{i=1}^n \lambda_i$. The simulation uses an Euler step $\matW'_{t+1} = \matW_t + \matWdot_t$ where $\matWdot_t$ contains the parameter $\gamma = 1/\tau$. Three different subsequent back-projection modes are tested: “exact back-projection to Stiefel manifold” : $$\begin{aligned} \matW_{t+1} = \matW'_{t+1} \left( \matW'^T_{t+1} \matW'_{t+1} \right)^{-\half}, \end{aligned}$$ “approximated back-projection to Stiefel manifold” : from [(630)]{} $$\begin{aligned} \matW_{t+1} = \matW'_{t+1} - \half \matW_t \matWdot_t^T \matWdot_t, \end{aligned}$$ “no back-projection” : $$\begin{aligned} \matW_{t+1} = \matW'_{t+1}. \end{aligned}$$ To evaluate the deviation of $\matW$ from semi-orthogonality (“orthonormality” for short) and the deviation of $\matW$ from the true principal eigenvectors (in arbitrary order), we define three error measures $e_1$, $e_2$, and $e'_2$ on square matrices of size $m$: $$\begin{aligned} e_1(\matX) &= \frac{1}{m^2} \summe{i=1}{m}\summe{j=1}{m} \left\|X_{ij} - \delta_{ij}\right\|\\ e_2(\matX) &= \frac{1}{m} \summe{j=1}{m} \left\|\max_{i=1}^m\{\|(\vecx_j)_i\|\} - 1\right\|\\ e'_2(\matX) &= \half\left(e_2(\matX) + e_2(\matX^T)\right)\end{aligned}$$ Error measure $e_1$ is zero if $\matX$ coincides with the identity matrix of the same size. Error measure $e_2$ is zero if the maximal absolute element in each column of $\matX$ is $1$. Error measure $e'_2$ considers $e_2$ in both columns and rows. We define the orthonormality error $e_o$ and the error of the projection to the eigenvectors $e_p$ as $$\begin{aligned} e_o(\matW) &= e_1(\matW^T\matW)\\ e_p(\matW,\hat{\matV}) &= e'_2(\hat{\matV}^T\matW)\end{aligned}$$ where $\hat{\matV}$ (size $n \times m$) contains the $m$ principal eigenvectors in its columns. Error measure $e_o$ is zero for a semi-orthogonal $\matW$. Error measure $e_p$ is zero if each eigenvector estimate $\vecw_j$ corresponds to a true eigenvector $\pm\vecv_i$ (arbitrary sign) in a one-to-one mapping. To motivate the error measure $e_p$, we show two examples of final values of $\hat{\matV}^T \matW$ which lead to $e_p \approx 0$. The first is from learning rule “TwJ2S” where the ordering of the estimated eigenvectors with respect to the eigenvalues is determined by the fixed matrix $\matTheta$: $$\begin{aligned} \hat{\matV}^T\matW = \begin{pmatrix}[r] 0.00 & 0.00 & 0.00 & -1.00\\ -0.00 & -0.00 & 1.00 & 0.00\\ 0.00 & -1.00 & -0.00 & -0.00\\ 1.00 & 0.00 & 0.00 & 0.00 \end{pmatrix}.\end{aligned}$$ The corresponding eigenvalue estimates $\vecw_j^T\matC\vecw_j$ are, in the same order: $0.70,\, 0.80,\, 0.90,\, 1.00$. The second example is from learning rule “N2S” where the approached ordering is arbitrary: $$\begin{aligned} \hat{\matV}^T\matW = \begin{pmatrix}[r] -0.00 & -1.00 & 0.00 & 0.00\\ 0.00 & 0.00 & -0.00 & 1.00\\ 1.00 & -0.00 & 0.00 & -0.00\\ -0.00 & 0.00 & 1.00 & 0.00\\ \end{pmatrix}.\end{aligned}$$ The eigenvalue estimates are, in the same order: $0.80,\, 1.00,\, 0.70,\, 0.90$. Results {#sec_simulations_results} ------- Figure \[fig\_spaced\] shows the simulation results for the evenly spaced eigenvector set for the three back-projection methods (note the reduced number of simulation steps). Looking at the projection error $e_p$ (right diagrams), we see fast convergence for all learning rules, particularly for the exact back-projection where the learning rate $\gamma$ can be higher than in the other two back-projection methods. “N2S” converges more slowly than “TwJ2S”, but is in the same convergence range. “M2S” converges faster with increasing $\alpha$ and even surpasses “TwJ2S” (but see section \[sec\_discussion\]), but the gain decreases for the highest values of $\alpha$. The orthonormality error $e_o$ (left diagrams) stays small for exact back-projection, reduces very fast for approximated back-projection, and reduces somewhat slower for no back-projection. In the latter two cases, increasing $\alpha$ accelerates the convergence of the orthonormality error. The improved convergence of the orthonormality error from no back-projection to approximated back-projection is not reflected in faster reduction of the projection error, though. Figure \[fig\_nearby\] shows the simulation results for nearby eigenvalues $\lambda_1 \approx \lambda_2$. Looking at the projection error (right diagrams), both “N2S” and “TwJ2S” show slower convergence than for evenly spaced eigenvalues (note the larger number of simulation steps), but we see that “N2S” converges considerably slower than “TwJ2S” which confirms the observation reported before [@own_Moeller20a]. However, with increasing $\alpha$ in “M2S”, the time course of the projection error approaches that of “TwJ2S”. Looking at the orthonormality error (left diagrams), we see small values for exact back-projection, fast convergence for approximated back-projection, and much slower convergence with no back-projection. In the latter case, there is a tendency for faster convergence with increasing $\alpha$ in “M2S”, approaching “TwJ2S” for $\alpha=20$. Again, the projection error does not differ between no back-projection and approximated back-projection, even though the latter shows a noticeably faster reduction of the orthonormality error. All learning rules seem to approach a lower limit in both $e_o$ and $e_p$ which can probably be explained by numerical effects. ![Orthonormality error $e_o$ (left) and error of projection to eigenvectors $e_p$ (right) for evenly spaced eigenvalues (logarithmic, $20.000$ steps, subsampling $100$).[]{data-label="fig_spaced"}](orthomeas_n_10_m_4_sm_3_g_1_l1_1_l2_0.9_r_20000.eps "fig:"){width="74mm"} ![Orthonormality error $e_o$ (left) and error of projection to eigenvectors $e_p$ (right) for evenly spaced eigenvalues (logarithmic, $20.000$ steps, subsampling $100$).[]{data-label="fig_spaced"}](projmeas_n_10_m_4_sm_3_g_1_l1_1_l2_0.9_r_20000.eps "fig:"){width="74mm"}\ ![Orthonormality error $e_o$ (left) and error of projection to eigenvectors $e_p$ (right) for evenly spaced eigenvalues (logarithmic, $20.000$ steps, subsampling $100$).[]{data-label="fig_spaced"}](orthomeas_n_10_m_4_sm_2_g_0.1_l1_1_l2_0.9_r_20000.eps "fig:"){width="74mm"} ![Orthonormality error $e_o$ (left) and error of projection to eigenvectors $e_p$ (right) for evenly spaced eigenvalues (logarithmic, $20.000$ steps, subsampling $100$).[]{data-label="fig_spaced"}](projmeas_n_10_m_4_sm_2_g_0.1_l1_1_l2_0.9_r_20000.eps "fig:"){width="74mm"}\ ![Orthonormality error $e_o$ (left) and error of projection to eigenvectors $e_p$ (right) for evenly spaced eigenvalues (logarithmic, $20.000$ steps, subsampling $100$).[]{data-label="fig_spaced"}](orthomeas_n_10_m_4_sm_1_g_0.1_l1_1_l2_0.9_r_20000.eps "fig:"){width="74mm"} ![Orthonormality error $e_o$ (left) and error of projection to eigenvectors $e_p$ (right) for evenly spaced eigenvalues (logarithmic, $20.000$ steps, subsampling $100$).[]{data-label="fig_spaced"}](projmeas_n_10_m_4_sm_1_g_0.1_l1_1_l2_0.9_r_20000.eps "fig:"){width="74mm"}\ ![Orthonormality error $e_o$ (left) and error of projection to eigenvectors $e_p$ (right) for nearby eigenvalues $\lambda_1 \approx \lambda_2$ (logarithmic, $50.000$ steps, subsampling $100$).[]{data-label="fig_nearby"}](orthomeas_n_10_m_4_sm_3_g_1_l1_0.91_l2_0.9_r_50000.eps "fig:"){width="74mm"} ![Orthonormality error $e_o$ (left) and error of projection to eigenvectors $e_p$ (right) for nearby eigenvalues $\lambda_1 \approx \lambda_2$ (logarithmic, $50.000$ steps, subsampling $100$).[]{data-label="fig_nearby"}](projmeas_n_10_m_4_sm_3_g_1_l1_0.91_l2_0.9_r_50000.eps "fig:"){width="74mm"}\ ![Orthonormality error $e_o$ (left) and error of projection to eigenvectors $e_p$ (right) for nearby eigenvalues $\lambda_1 \approx \lambda_2$ (logarithmic, $50.000$ steps, subsampling $100$).[]{data-label="fig_nearby"}](orthomeas_n_10_m_4_sm_2_g_0.1_l1_0.91_l2_0.9_r_50000.eps "fig:"){width="74mm"} ![Orthonormality error $e_o$ (left) and error of projection to eigenvectors $e_p$ (right) for nearby eigenvalues $\lambda_1 \approx \lambda_2$ (logarithmic, $50.000$ steps, subsampling $100$).[]{data-label="fig_nearby"}](projmeas_n_10_m_4_sm_2_g_0.1_l1_0.91_l2_0.9_r_50000.eps "fig:"){width="74mm"}\ ![Orthonormality error $e_o$ (left) and error of projection to eigenvectors $e_p$ (right) for nearby eigenvalues $\lambda_1 \approx \lambda_2$ (logarithmic, $50.000$ steps, subsampling $100$).[]{data-label="fig_nearby"}](orthomeas_n_10_m_4_sm_1_g_0.1_l1_0.91_l2_0.9_r_50000.eps "fig:"){width="74mm"} ![Orthonormality error $e_o$ (left) and error of projection to eigenvectors $e_p$ (right) for nearby eigenvalues $\lambda_1 \approx \lambda_2$ (logarithmic, $50.000$ steps, subsampling $100$).[]{data-label="fig_nearby"}](projmeas_n_10_m_4_sm_1_g_0.1_l1_0.91_l2_0.9_r_50000.eps "fig:"){width="74mm"}\ Discussion {#sec_discussion} ========== The simulations show a marked improvement of the convergence speed of the modified learning rule “M2S” for increasing $\alpha$, particularly if some principal eigenvalues are close to each other. Nearby principal eigenvalues slow down both the “N2S” and the “M2S” rule, but the latter is affected more strongly for which we can provide the explanation that the “symmetry-breaking” effect of $\matD$ is reduced if the eigenvalue estimates on its diagonal are close to each other [@own_Moeller20a]. Introducing the additional terms in the modified objective function mitigates this effect. However, we originally expected that the additional terms will also modify or suppress the “undesired” fixed points, but our analysis shows that all fixed points of “N2S” are also present in “M2S”. We assume that the contributions by the different terms of the additional sum cancel out in the fixed points which therefore remain unchanged. Only the steepness of the landscape outside of the fixed points is increased. Additional fixed points may be present in “M2S”, but this was not analyzed here (particularly the different constraint on $\matT$ may lead to additional fixed points). It is always unfortunate if an additional parameter (in this case $\alpha$) has to be introduced. We currently cannot provide a universal guideline on how $\alpha$ has to be adjusted for different eigenvalue spectra and dimensions. We observed that higher values of $\alpha$ than the ones tested in the simulations sometimes lead to divergence. We could imagine that learning rules can be designed where $\alpha$ is suitably chosen depending on $\matW$. Note that also the suitable range for the learning rate $\gamma$ is not clear. With exact back-projection, it can be higher than with approximated back-projection (since the approximation is based on the assumption of small steps) or without back-projection, but suitable values may depend on the eigenvalue spectrum and the dimensions. We compare “N2S” and “M2S” with the learning rule “TwJ2S” where we have a fixed diagonal weight-factor matrix $\matTheta$ with distinct elements in the place of $\matD$ or $\matD'_\alpha$. The influence of the choice of the elements of $\matTheta$ on the convergence speed has to be studied. An absolute statement like ‘rule “M2S” performs better than “TwJ2S” for a certain $\alpha$’ is therefore debatable. The time course of the projection error of “TwJ2S” should therefore only be taken as a coarse reference. The stability analysis in section \[sec\_stability\_mod\] revealed that the additional second term in the modified objective function (\[eq\_objfct\_mod\]) leads to a term depending only on the step parameter $\matB$, see equations (\[eq\_deltaJ2\_final\]) and (\[eq\_deltaJ2\_final\_mod\]). With inverted sign, this term alone would be sufficient to explain PCA behavior. One could therefore assume that the additional term in objective function alone (with inverted sign) could lead to a PCA rule. However, we have also shown that the corresponding terms in the learning rule “M2S” are characteristic for subspace behavior, thus the fixed-point structure is completely different without the original first term. We did not explore the difference between the “short” learning rules studied here and the alternative of learning rules derived from the “embedded” metric on the Stiefel manifold. Conclusion {#sec_conclusion} ========== We introduced an additional term into the objective function which improves the convergence speed of the corresponding learning rule, particularly in the case of nearby principal eigenvalues. The modified learning rule “M2S” is structurally similar to the original rule “N2S”, with a different matrix $\matD'_\alpha$ in place of $\matD$. Our analysis shows that the modified learning rule has all fixed points of the original rule but may introduce new fixed points (which was not studied further). Also the stability of the fixed points shared with the original rule is unaffected by the modification. [ symmpca2.bbl ]{} Changes {#sec_changes .unnumbered} ======= 6 June 2020: Started report.\ 18 July 2020: Submission to arXiv. Terms of Learning Rules Close to the Stiefel Manifold {#app_near_st} ===================================================== In our previous work, we derived different learning rules, either from a derivation in “short” or “long” form or from two different metrics on the Stiefel manifold, canonical and embedded [(section 9)]{}. The “short” rules coincide with the “canonical” rules, so we have three groups: “short”, “long”, and “embedded”. In this report we focus on “short” learning rules. In simulations (data not shown) comparing rules from the three groups for the “original” objective function (N2S, NL, NSE), the time course of the projection error $e_p$ was not markedly different, regardless of the back-projection method used. In the following we explore how the different terms can be approximated if the learning rule operates in the vicinity of the Stiefel manifold where $\matW^T\matW \approx \matI_m$. We start from learning rule NL which contains all types of terms known so far [(476)]{}: $$\begin{aligned} \tau \matWdot &= 5 \matC \matW \matD\nonumber\\ &- \matW \matW^T \matC \matW \matD - \matW \matD \matW^T \matC \matW \nonumber\\ &- \matC \matW \matD \matW^T \matW - \matC \matW \matD^* - \matC \matW \matW^T \matW \matD\end{aligned}$$ where $$\begin{aligned} \matD &= \Diag{j=1}{m}\{\vecw_j^T \matC \vecw_j\} = \dg\{\matW^T\matC\matW\}\\ \matD^* &= \Diag{j=1}{m}\{\vecw_j^T \matC \matW \matW^T \vecw_j\} = \dg\{\matW^T\matC\matW\matW^T\matW\}.\end{aligned}$$ Close to the Stiefel manifold, we have $\matD^* \approx \dg\{\matW^T\matC\matW\} = \matD$. We can approximate the different terms as $$\begin{aligned} \tau \matWdot &= 5 \matC \matW \matD\nonumber\\ &- \matW \matW^T \matC \matW \matD - \matW \matD \matW^T \matC \matW \nonumber\\ &- \matC \matW \matD - \matC \matW \matD - \matC \matW \matD\end{aligned}$$ which leads to the rule called NSE [(485)]{}: $$\begin{aligned} \tau \matWdot = 2 \matC \matW \matD\nonumber - \matW \matW^T \matC \matW \matD - \matW \matD \matW^T \matC \matW.\end{aligned}$$ There is no obvious approximation which leads from here to N2S, so we assume that in the vicinity of the Stiefel manifold, there are essentially just the two forms N2S and NSE, which correspond to the gradient in the canonical or embedded metric [(section 9.3)]{}, respectively. Even these two rules show very similar behavior. It is obvious that exact back-projection keeps $\matW$ on the Stiefel manifold, and it is also clear that the approximated back-projection almost achieves the same, at least for small learning rates $\gamma = 1 / \tau$. Why the rules return to the Stiefel manifold after each learning step [without]{} back-projection remains to be explored. Additional Fixed Points of “M2S” {#app_add_fp} ================================ We analyze whether the second factor in equation (\[eq\_M2S\_constraint\_T\]) can become singular: $$\begin{aligned} \matDnull'_\alpha &= (1+\alpha) \dg\{\matWnull^T\matC\matWnull\} - \alpha \matWnull^T\matC\matWnull\\ &= (1+\alpha) \dg\{\matAnull^T\matV^T\matC\matV\matAnull\} - \alpha \matAnull^T\matV^T\matC\matV\matAnull\\ &= (1+\alpha) \dg\{\matAnull^T\matLambda\matAnull\} - \alpha \matAnull^T\matLambda\matAnull.\end{aligned}$$ In a simulation, we generate a random semi-orthogonal $\matA$ of size $n \times m$ (with $n=10$, $m=4$) and use eigenvalues $\{n, n-1, \ldots, 1\}$ to form $\matLambda$. We vary $\alpha$ and plot $\det\{\matD'_{\alpha}\}$ in steps of $0.1$ from $0.0$ to $20.0$ in figure \[fig\_det\_Dp\_alpha\]. We often see two zero-crossings as shown in the figure, but curves with other shapes appear as well, depending on the random initialization of $\matA$. ![Determinant $\det\{\matD'_{\alpha}\}$ over $\alpha$ for a random, semi-orthogonal $\matW$ of size $10 \times 4$ and eigenvalues descending from $10.0$ to $1.0$ in steps of $1.0$.[]{data-label="fig_det_Dp_alpha"}](det.eps "fig:"){width="10cm"}\ Fixed-Point Constraints of “M2S” {#app_constraints_S_T} ================================ We describe two attempts at deriving the constraints on matrices $\matS$ and $\matT$ which lead to the same result. Attempt 1 --------- The first attempt starts from (\[eq\_M2S\_form1\]): $$\begin{aligned} \nonumber \matNull &= (1 + \alpha) (\matC\matWnull\matDnull -\matWnull\matDnull\matWnull^T\matC\matWnull)\\ &- \alpha (\matC\matWnull -\matWnull\matWnull^T\matC\matWnull) (\matWnull^T\matC\matWnull)\\[5mm] \nonumber \matNull &= (1 + \alpha) (\matC\matV\matAnull\matDnull -\matV\matAnull\matDnull\matAnull^T\matV^T\matC \matV\matAnull)\\ &- \alpha (\matC\matV\matAnull -\matV\matAnull\matAnull^T\matV^T\matC\matV\matAnull) (\matAnull^T\matV^T\matC\matV\matAnull)\\[5mm] \nonumber \matNull &= (1 + \alpha) (\matC\matV\matAnull\matDnull -\matV\matAnull\matDnull\matAnull^T\matLambda\matAnull)\\ &- \alpha (\matC\matV\matAnull -\matV\matAnull\matAnull^T\matLambda\matAnull) (\matAnull^T\matLambda\matAnull)\\[5mm] \nonumber \matNull &= (1 + \alpha) (\matV^T\matC\matV\matAnull\matDnull -\matV^T\matV\matAnull\matDnull\matAnull^T\matLambda\matAnull)\\ &- \alpha (\matV^T\matC\matV\matAnull -\matV^T\matV\matAnull\matAnull^T\matLambda\matAnull) (\matAnull^T\matLambda\matAnull)\\[5mm] \nonumber \matNull &= (1 + \alpha) (\matLambda\matAnull\matDnull -\matAnull\matDnull\matAnull^T\matLambda\matAnull)\\ &- \alpha (\matLambda\matAnull -\matAnull\matAnull^T\matLambda\matAnull) (\matAnull^T\matLambda\matAnull)\\[5mm] \nonumber \matNull &= (1 + \alpha) \left[\matLambda\matQ\pmat{\matI_m\\ \matNull}\matDnull -\matQ\pmat{\matI_m\\ \matNull}\matDnull\pmat{\matI_m & \matNull^T}\matQ^T \matLambda\matQ\pmat{\matI_m\\ \matNull}\right]\\ &- \alpha \left[\matLambda\matQ\pmat{\matI_m\\ \matNull} -\matQ\pmat{\matI_m\\ \matNull}\pmat{\matI_m & \matNull^T}\matQ^T \matLambda\matQ\pmat{\matI_m\\ \matNull}\right] \left[\pmat{\matI_m & \matNull^T}\matQ^T \matLambda \matQ\pmat{\matI_m\\ \matNull}\right]\\[5mm] \nonumber \matNull &= (1 + \alpha) \left[\matQ^T\matLambda\matQ\pmat{\matI_m\\ \matNull}\matDnull -\matQ^T\matQ\pmat{\matI_m\\ \matNull}\matDnull\pmat{\matI_m & \matNull^T} \matQ^T\matLambda\matQ\pmat{\matI_m\\ \matNull}\right]\\ \nonumber &- \alpha \left[\matQ^T\matLambda\matQ\pmat{\matI_m\\ \matNull} -\matQ^T\matQ\pmat{\matI_m\\ \matNull}\pmat{\matI_m & \matNull^T} \matQ^T\matLambda\matQ \pmat{\matI_m\\ \matNull}\right]\\ &\cdot \left[\pmat{\matI_m & \matNull^T} \matQ^T\matLambda\matQ \pmat{\matI_m\\ \matNull}\right]\\[5mm] \nonumber \matNull &= (1 + \alpha) \left[\matM\pmat{\matI_m\\ \matNull}\matDnull -\pmat{\matI_m\\ \matNull}\matDnull\pmat{\matI_m & \matNull^T} \matM\pmat{\matI_m\\ \matNull}\right]\\ &- \alpha \left[\matM\pmat{\matI_m\\ \matNull} -\pmat{\matI_m\\ \matNull}\pmat{\matI_m & \matNull^T} \matM \pmat{\matI_m\\ \matNull}\right] \left[\pmat{\matI_m & \matNull^T} \matM\pmat{\matI_m\\ \matNull}\right]\\[5mm] \nonumber \matNull &= (1 + \alpha) \left[\pmat{\matS\\ \matT}\matDnull -\pmat{\matI_m\\ \matNull}\matDnull \matS\right]\\ &- \alpha \left[\pmat{\matS\\ \matT} -\pmat{\matI_m\\ \matNull}\matS\right] \matS\\[5mm] \nonumber \matNull &= (1 + \alpha) \left[\pmat{\matS\matDnull\\ \matT\matDnull} -\pmat{\matDnull\matS\\ \matNull}\right]\\ &- \alpha \left[\pmat{\matS^2\\ \matT\matS} -\pmat{\matS^2\\ \matNull}\right]\\[5mm] \label{eq_M2S_constraint_1_matrix} \matNull &= \pmat{ (1+\alpha)[\matS\matDnull-\matDnull\matS]\\ \matT[(1+\alpha)\matDnull-\alpha\matS]}.\end{aligned}$$ The upper part of equation (\[eq\_M2S\_constraint\_1\_matrix\]) gives $$\begin{aligned} \label{eq_M2S_constraint_1} \matS\matDnull &= \matDnull\matS.\end{aligned}$$ Equation (\[eq\_M2S\_constraint\_1\]) coincides with the constraint [(247)]{} derived for the fixed points of learning rule “N2S”. The lower part of equation (\[eq\_M2S\_constraint\_1\_matrix\]) gives $$\begin{aligned} \matT[(1+\alpha)\matDnull-\alpha\matS] = \matNull\end{aligned}$$ which differs from the equation $\matT\matDnull = \matNull$ derived for “N2S”. Attempt 2 --------- The second attempt starts from (\[eq\_M2S\_form2\]) and proceeds in the same way as in section 7.8 of our previous work [@own_Moeller20a], from equation [(239)]{} onward, except with $\matDnull'_\alpha$ instead of $\matDnull$: $$\begin{aligned} \matNull &= \matC \matWnull \matDnull'_\alpha - \matWnull \matDnull'_\alpha \matWnull^T \matC \matWnull\\ \matNull &= \matC \matV \matAnull \matDnull'_\alpha - \matV \matAnull \matDnull'_\alpha \matAnull^T \matV^T \matC \matV \matAnull\\ \matNull &= \matV^T \matC \matV \matAnull \matDnull'_\alpha - \matV^T\matV \matAnull \matDnull'_\alpha \matAnull^T \matV^T \matC \matV \matAnull\\ \label{eq_M2S_intermediate} \matNull &= \matLambda \matAnull \matDnull'_\alpha - \matAnull \matDnull'_\alpha \matAnull^T \matLambda \matAnull\\ \matNull &= \matLambda \matQ \pmat{\matI_m\\ \matNull} \matDnull'_\alpha - \matQ \pmat{\matI_m\\ \matNull} \matDnull'_\alpha \pmat{\matI_m & \matNull^T} \matQ^T \matLambda \matQ \pmat{\matI_m\\ \matNull}\\ \matNull &= \matQ^T \matLambda \matQ \pmat{\matI_m\\ \matNull} \matDnull'_\alpha - \pmat{\matI_m\\ \matNull} \matDnull'_\alpha \pmat{\matI_m & \matNull^T} \matQ^T \matLambda \matQ \pmat{\matI_m\\ \matNull}\\ \matNull &= \pmat{\matS & \matT^T\\\matT & \matU} \pmat{\matI_m\\ \matNull} \matDnull'_\alpha - \pmat{\matI_m\\ \matNull} \matDnull'_\alpha \pmat{\matI_m & \matNull^T} \pmat{\matS & \matT^T\\\matT & \matU} \pmat{\matI_m\\ \matNull}\\ \label{eq_M2S_constraint_2} \pmat{\matNull\\ \matNull} &= \pmat{\matS \matDnull'_\alpha \\ \matT \matDnull'_\alpha} - \pmat{\matDnull'_\alpha \matS\\ \matNull}.\end{aligned}$$ To analyze the constraint on $\matS$ in the upper equation of (\[eq\_M2S\_constraint\_2\]), we look at $$\begin{aligned} \matDnull'_\alpha = (1 + \alpha) \matDnull - \alpha \matWnull^T\matC\matWnull\end{aligned}$$ and see that $$\begin{aligned} \matWnull^T\matC\matWnull &= \matAnull^T\matV^T\matC\matV\matAnull\\ &= \matAnull^T\matLambda\matAnull\\ &= \pmat{\matI_m & \matNull^T} \underbrace{\matQ^T\matLambda\matQ}_{\matM} \pmat{\matI_m\\ \matNull}\\ &= \pmat{\matI_m & \matNull^T} \pmat{\matS & \matT^T\\ \matT & \matU} \pmat{\matI_m\\ \matNull}\\ &= \matS.\end{aligned}$$ Note that we also have $\matDnull = \dg\{\matWnull^T\matC\matWnull\} = \dg\{\matS\}$. We can therefore write $\matDnull'_\alpha$ as $$\begin{aligned} \matDnull'_\alpha = (1 + \alpha) \matDnull - \alpha \matS.\end{aligned}$$ We proceed with the upper equation of (\[eq\_M2S\_constraint\_2\]): $$\begin{aligned} \matS\matDnull'_\alpha &= \matDnull'_\alpha \matS\\ \matS\left[(1+\alpha)\matDnull-\alpha\matS\right] &= \left[(1+\alpha)\matDnull-\alpha\matS\right]\matS\\ (1+\alpha)\matS\matDnull - \alpha\matS^2 &= (1+\alpha)\matDnull\matS - \alpha\matS^2\\ \matS\matDnull &= \matDnull\matS.\end{aligned}$$ This constraint is the same as [(247)]{} which was derived for the fixed points of the “N2S” learning rule. We now look at the lower equation of (\[eq\_M2S\_constraint\_2\]): $$\begin{aligned} \matT \matDnull'_\alpha &= \matNull\\ \matT \left[(1+\alpha)\matDnull-\alpha\matS\right] &= \matNull.\end{aligned}$$ The corresponding equation for “N2S” was $\matT\matDnull=\matNull$. [^1]: In a fully symmetric learning rule, all units see the same input and perform exactly the same computations. Earlier symmetric learning rules required a distinct weight factor in each unit to ensure convergence to the principal eigenvectors and not just to the principal subspace. [^2]: Our analysis focuses on the simplest (“short”) learning rule from our previous work, since our simulations show that the more complex (“long”) learning rules differ only marginally in their behavior, probably since terms coincide in the vicinity of the Stiefel manifold of the eigenvector estimates; see appendix \[app\_near\_st\]. [^3]: Note that $\hat{\matLambda}^*$ in this derivation may differ from the one in (\[eq\_N2S\_WTCW\]) and (\[eq\_N2S\_constraint\]).
--- abstract: 'Given an elliptic surface $\mathcal{E}\to\mathcal{C}$ over a field $k$ of characteristic zero equipped with zero section $O$ and another section $P$ of infinite order, we give a simple and explicit upper bound on the number of points where $O$ is tangent to a multiple of $P$.' address: - \| Department of Mathematics\ University of Arizona\ Tucson, AZ 85721 USA - \| Facultad de Matemáticas\ Pontificia Universidad Católica de Chile\ Santiago, Chile author: - Douglas Ulmer - Giancarlo Urzúa bibliography: - 'database.bib' title: Bounding tangencies of sections on elliptic surfaces --- Introduction ============ Let $k=\C$ be the complex numbers[^1], let $\CC$ be an irreducible, smooth, projective curve of genus $g$ over $k$, and let $\EE\to\CC$ be an elliptic surface over $\CC$ equipped with a section of $\pi$ denoted $O:\CC\to\EE$ which will play the role of a zero section. Let $P:\CC\to\EE$ be another section of $\pi$ which is of infinite order in the group law with $O$ as origin. Write $\EE[n]$ for the union of the points of order $n$ in each fiber of $\pi$. It is known that $\EE[n]$ is a smooth, locally closed subset of $\EE$ which is quasi-finite over $\CC$ of generic degree $n^2$ (See [@UUpp19 Sections 2.1 and 2.2] for more details.) In [@UUpp19], we proved that the set $$\begin{aligned} T_{tor}&:=\bigcup_{n>0}\left\{t\in\CC\left\|nP\text{ is tangent to }O\text{ over }t\right.\right\}\\ &\phantom{:}=\bigcup_{n>0}\left\{t\in\CC\left\|P\text{ is tangent to }\EE[n]\text{ over }t\right.\right\}\end{aligned}$$ is finite. Our goal in this paper is to give an explicit upper bound for $\|T_{tor}\|$, the cardinality of $T_{tor}$. We say that $\EE$ is constant if there is an elliptic curve $E$ over $k$ such that $\EE\cong\CC\times_kE$ and $\pi$ is the projection to $\CC$. If $\EE$ is constant, we say $P$ is constant if there is point $p\in E$ such that $P(t)=(t,p)$ for all $t\in\CC$. \[thm:main\] Let $\omega=O^(\Omega^1_{\EE/\CC})$, let $d=\deg(\omega)$, and let $\delta$ be the number of singular fibers of $\pi:\EE\to\CC$. Suppose that $\EE$ is not constant, or that $\EE$ is constant and $P$ is not constant. Then $$\left\|T_{tor}\right\|\le 2g-2-d+\delta.$$ This is proved as Corollary \[cor:bound\] below. In fact, we will prove a more precise result (Theorem \[thm:total-int\]) which gives an exact formula* for the cardinality, with multiplicities, of a more general set of tangencies. The constant case ----------------- The constant case of our result is very transparent and gives a hint of how to proceed in general, so we discuss it here. Suppose that $\EE\cong\CC\times E$ is constant. Then a section $P:\CC\to\EE$ may be identified with a morphism $f:\CC\to E$, and $P$ is constant if and only if $f$ is constant. We assume that $P$ is non-constant. The torsion subset $\EE[n]$ consists of the $n^2$ constant sections $\CC\times\{p\}$ where $p$ is an $n$-torsion point of $E$. It is of interest to consider tangencies with general constant sections $\CC\times\{p\}$ for any $p\in E$. Let $$T_{const}:=\bigcup_{p\in E}\left\{t\in\CC\left\|P\text{ is tangent to }\CC\times\{p\}\text{ over }t\right.\right\}$$ i.e., the set of points of $\CC$ where $P$ is tangent to a constant section. Obviously $T_{tor}\subset T_{const}$. To take into account multiplicities, suppose $P(t)=(t,p)$ and let $I(P,t)$ be the intersection number of $\CC\times\{p\}$ and $P$ at $(t,p)$. By definition, $I(P,t)\ge1$ and it is $\ge2$ if and only if $P$ is tangent to $\CC\times\{p\}$ over $t$. On the other hand, it is clear that $I(P,t)$ is $e_t(f)$, the ramification index of $f$ at $t$. Let $\eta_P$ be the pull-back under $f$ of a non-zero invariant differential on $E$. Since $f$ is non-constant, $\eta_P$ is a non-zero section of $\Omega^1_\CC$, and the order of vanishing of $\eta_P$ at $t$ is $$\ord_t(\eta_P)=e_f(t)-1=I(P,t)-1.$$ Thus we have $$\left\|T_{tor}\right\|\le\left\|T_{const}\right\| \le\sum_{t\in\CC}\left(I(P,t)-1\right) =\sum_{t\in\CC}\ord_t(\eta_P)=2g-2.$$ Since $d=\delta=0$ when $\EE$ is constant, this proves the Theorem \[thm:main\] in the constant case. Sketch of the general case -------------------------- In the general case, we will define a “Betti foliation” on an open subset of $\EE$ which generalizes the foliation of $\CC\times E$ by the leaves $\CC\times\{p\}$ and which has the subsets $\EE[n]$ among its closed leaves. This leads to a set of tangencies $T_{Betti}\subset\CC$ with $T_{tor}\subset T_{Betti}$ and intersection multiplicities $I(P,t)$ which measure the order of contact between $P$ and the Betti foliation. We will also define a certain twisted real-analytic 1-form $\eta_P$ on an open subset of $\CC$ whose local indices $J(\eta_P,t)$ satisfy $J(\eta_P,t)=I(P,t)-1$ at all places $t$ of good reduction. Summing over all points of $\CC$ will lead to a formula $$\sum_{t\in\CC}\left(I(P,t)-1\right)=\sum_{t\in\CC}J(\eta_P,t)=2g-2-d,$$ and taking into account what happens at the bad fibers leads to the upper bound $$\left\|T_{tor}\right\|\le\left\|T_{Betti}\right\|\le2g-2-d+\delta.$$ A trivialization essentially equivalent to the Betti foliation was used in the first version of [@UUpp19], and we later adopted the Betti terminology, following [@CorvajaMasserZannier18]. The form $\eta_P$ appears implicitly in the first version of [@UUpp19]. A more general version of it is discussed at some length in [@ACZpp18 §4], and their account inspired our use of it here to count tangencies. The finiteness of $T_{tor}$ was proved independently in [@CorvajaDemeioMasserZannierpp]. Plan of the paper ----------------- In Section \[s:analytic\] we review certain aspects of Kodaira’s construction of $\EE$ as an analytic surface. In Sections \[s:Betti\] and \[s:intersections\], we define the Betti foliation and local intersection numbers $I(P,t)$ measuring the order of contact between a section $P$ and the Betti foliation. In Section \[s:1-form\], we attach to $P$ a real-analytic section $\eta_P$ of $\Omega^1_\CC\tensor\omega^{-1}$ over a Zariski open subset of $\CC$ and with isolated zeroes, define local indices $J(\eta_P,t)$, and calculate their sum. In Section \[s:zeros\], we relate the local indices $I(P,t)$ and $J(\eta_P,t)$. This leads to the proof, in Section \[s:proofs\], of the main theorem. Finally, in Section \[s:examples\] we give examples illustrating edges cases and the sharpness of the main theorem. Acknowledgements ---------------- The first-named author thanks the Simons Foundation for partial support in the form of Collaboration Grant 359573 and Doug Pickrell for a pointer to the topology literature. The second-named author thanks FONDECYT for support from grant 1190066. Both authors thank Brian Lawrence for drawing their attention to [@ACZpp18]. $\EE$ as an analytic surface {#s:analytic} ============================ For the rest of the paper, we consider $\EE$ as an analytic surface (a 2-dimensional complex manifold) and $\CC$ as a Riemann surface. Let $\CC^0\subset\CC$ be the open set over which $\pi$ is smooth and let $\EE^0=\pi^{-1}(\CC^0)$. Let $j:\CC\to\P^1$ be the meromorphic function which on $\CC^0$ sends $t$ to the $j$-invariant of $\pi^{-1}(t)$. Our goal in this section is to review aspects of the analytic description of $\EE$ due to Kodaira. In [@Kodaira63a §7], Kodaira attaches to $\EE$ a period map from $\CC^0$ to the upper half plane and a monodromy representation from the fundamental group of $\CC^0$ to $\SL_2(\Z)$. We assume the reader is familiar with these invariants. In [@Kodaira63a §8], Kodaira reconstructs $\EE$ from this data, and in [@Kodaira63b §11], he describes the group law on (a subset of) $\EE$ in sheaf theoretic terms. We will use these ideas in the rest of the paper to define a foliation on $\EE$, study its intersections with sections of $\EE$, and relate them to a certain real-analytic 1-form. Uniformization {#ss:unif} -------------- We review the well-known construction of $\EE^0$ as a quotient space. Let $\widetilde{\CC^0}$ be the universal cover of $\CC^0$, choose a point $\tilde b\in\widetilde{\CC^0}$, let $b$ be the image of $\tilde b$ in $\CC^0$, and let $\Gamma=\pi_1(\CC^0,b)$. Let $\HH$ denote the upper half plane. Choosing an oriented basis of $H_1(\pi^{-1}(b),\Z)$, we get a period morphism $\tau:\widetilde{\CC^0}\to\HH$ and a monodromy representation $\rho:\Gamma\to\SL_2(\Z)$. We write $$\rho(\gamma)=\pmat{a_\gamma&b_\gamma\\c_\gamma&d_\gamma}.$$ The period and monodromy data satisfy the following compatibility: if $\gamma\in\Gamma$ and $\tilde t\in\widetilde{\CC^0}$, then $$\tau(\gamma\tilde t)=\rho(\gamma)\left(\tau(\tilde t)\right)$$ where $\rho(\gamma)$ acts as a linear fractional transformation on $\HH$. Form the semi-direct product $\Gamma\ltimes\Z^2$ by using the monodromy representation and the right action of $\SL_2(\Z)$ on $\Z^2$: $$\begin{aligned} \left(\gamma_1,m_1,n_1\right) \left(\gamma_2,m_2,n_2\right) &=\left(\gamma_1\gamma_2,(m_1,n_1)\gamma_2+(m_2,n_2)\right)\\ &=\left(\gamma_1\gamma_2, a_{\gamma_2}m_1+c_{\gamma_2}n_1+m_2,b_{\gamma_2}m_1+d_{\gamma_2}n_1+n_2\right).\end{aligned}$$ For $\tilde t\in\widetilde{\CC^0}$ and $\gamma\in\Gamma$, let $$f_\gamma(\tilde t)=\left(c_\gamma\tau(\tilde t)+d_\gamma\right)^{-1}.$$ One checks that $f$ satisfies the cocycle relation $f_{\gamma_1\gamma_2}(\tilde t) =f_{\gamma_1}(\gamma_2\tilde t)f_{\gamma_2}(\tilde t)$. Now let $\Gamma\ltimes\Z^2$ act on $\widetilde{\CC^0}\times\C$ by $$\left(\gamma,m,n\right)(\tilde t,w) =\left(\gamma\tilde t,f_{\gamma}(\tilde t)(w+m\tau(\tilde t)+n)\right).$$ This action is properly discontinuous, and we have isomorphisms $$\EE^0\cong\left(\widetilde{\CC^0}\times\C\right)/(\Gamma\ltimes\Z^2)$$ and $$\CC^0\cong\widetilde{\CC^0}/\Gamma.$$ We will also consider the quotient $$\FF^0:=\left(\widetilde{\CC^0}\times\C\right)/\Z^2.$$ With these isomorphisms and definition, we may identify the diagram of complex manifolds $$\xymatrix{ \left(\widetilde{\CC^0}\times\C\right)/\Z^2\ar[r]\ar[d] &\left(\widetilde{\CC^0}\times\C\right)/(\Gamma\ltimes\Z^2)\ar[d]\\ \widetilde{\CC^0}\ar[r]&\widetilde{\CC^0}/\Gamma}$$ with the Cartesian diagram $$\xymatrix{\FF^0\ar[r]\ar[d]&\EE^0\ar[d]\\ \widetilde{\CC^0}\ar[r]&{\CC^0}.}$$ In the introduction, we defined $\omega$ as the line bundle $O^(\Omega^1_{\EE/\CC})$. Let $\omega^{-1}$ be the dual line bundle. It is clear from the definitions in this section that a section of $\omega^{-1}$ over an open set $U\subset\CC^0$ can be identified with a function $w:\tilde U\to\C$ where $\tilde U$ is the inverse image of $U$ in $\widetilde{\CC^0}$ and $w$ satisfies $$\label{eq:omega-1} w(\gamma\tilde t)=f_\gamma(\tilde t)w(\tilde t).$$ Global monodromy {#ss:g-mono} ---------------- We recall three well-known results about the monodromy group $\rho(\Gamma)\subset\SL_2(\Z)$: 1. $j:\CC\to\P^1$ is non-constant if and only if $\rho(\Gamma)$ is infinite, in which case it has finite index in $\SL_2(\Z)$. 2. $j:\CC\to\P^1$ is constant if and only if $\rho(\Gamma)$ is finite. 3. $\EE$ is constant if and only if $\rho(\Gamma)$ is trivial. Indeed, if $j$ is non-constant, the period $\tau$ induces a factorization $$\CC^0\cong\widetilde{\CC^0}/\Gamma\to\HH/\rho(\Gamma)\to \HH/\PSL_2(\Z)\cong\A^1$$ where the composed map $\CC^0\to\A^1$ is the $j$-invariant. Since $j$ has finite degree, the index of the image of $\Gamma$ in $\PSL_2(\Z)$ is at most the degree of $j$. If $j$ is constant, all fibers of $\EE^0\to\CC^0$ are isomorphic to $E_b:=\pi^{-1}(b)$, and analytic continuation of local sections induces an inclusion $\rho(\Gamma)\subset\aut(E_b)$. Since the latter has order $2$, $4$, or $6$, this shows that $\rho(\Gamma)$ is finite. If $\EE$ is constant, it is clear that $\rho(\Gamma)$ is trivial. Conversely, if the monodromy is trivial, [@Kodaira63a p. 585, $1_1$] shows that all fibers of $\pi:\EE\to\CC$ are smooth elliptic curves, so the $j$-function defines a morphism $\CC\to\A^1$. Since $\CC$ is projective, $j$ must be constant, so all fibers of $\pi$ are isomorphic to a fixed elliptic curve $E$. Since $\pi$ has a section, we find that $\EE\cong\CC\times E$, as desired. We say that $\EE\to\CC$ is isotrivial* (resp. non-isotrivial) if $j$ is constant (resp. non-constant). Obviously, if $\EE$ is constant, it is isotrivial, but not conversely. Local invariants {#ss:l-mono} ---------------- In this section, we recall from [@Kodaira63a §8] the local monodromy, a branch of the period map, and the line bundle $\omega^{-1}$ in a neighborhood of each point $t\in\CC$. We use Kodaira’s notation ($I_0$, $I_0^$, …) for the reduction type of each fiber to label the rows of the table at the end of the section. For each $t\in\CC$, let $\Delta_t$ be a neighborhood of $t$ biholomorphic to a disk such that $\Delta'_t=\Delta_t\setminus\{t\}\subset\CC^0$, and let $z$ be a coordinate on $\Delta_t$ such that $z=0$ at $t$. To define the local monodromy, choose a path $p$ from $b$ to a point of $s\in\Delta'_t$, and let $\gamma$ be a positively oriented loop in $\Delta'_t$ based at $s$. Then $\rho$ applied to the class of $p^{-1}\gamma p$ is an element $g_t\in\rho(\Gamma)\subset\SL_2(\Z)$ which is well defined up to conjugation by $\rho(\Gamma)$. We say that $g_t$ is a generator of the local monodromy at $t$. In the table below, the column “monodromy” gives a representative for the local monodromy for fibers of each type. If $t\in\CC^0$, the local monodromy is trivial, and the period map $\tau$ is holomorphic on $\Delta'_t$ and extends to a holomorphic function on $\Delta_t$. If $t\in\CC\setminus\CC^0$, the period map is well-defined on the universal cover $\widetilde\Delta'_t$ of $\Delta'_t$ and often on a subcover. In the table below, the column “domain” gives a subcover of $\widetilde\Delta'_t\to\Delta'_t$ over which the monodromy becomes trivial, and thus over which a branch of $\tau$ becomes a well-defined function. The column “period” describes this function for a suitable choice of a branch of the period map. We described $\omega^{-1}$ over $\CC^0$ in the last paragraph of Section \[ss:unif\] above. For $t\in\CC\setminus\CC^0$, we may specify $\omega^{-1}$ restricted to $\Delta_t$ by giving a section of $\omega^{-1}$ over $\Delta'_t$ which extends to a generating section over $\Delta_t$. Since the monodromy is trivial on the domain, so is the cocycle $f_\gamma$, and a section of $\omega^{-1}$ on $\Delta'_t$ is a function on the domain. The column “generator of $\omega^{-1}$” describes this function. Fiber Monodromy Domain Period Generator of $\omega^{-1}$ ---------------- ----------------------- ---------------------- --------------------------------------------------------------------- ---------------------------------- $I_0$ $\psmat{1&0\\0&1}$ $z\in\Delta'_t$ $\tau=holo(z)$ $w=1$ $I_b$, $b>0$ $\psmat{1&b\\0&1}$ $e^{2\pi i\zeta}= z$ $\tau=b\zeta$ $w=1$ $I_b^$, $b>0$ $\psmat{-1&-b\\0&-1}$ $e^{2\pi i\zeta}= z$ $\tau=b\zeta$ $w=e^{\pi i\zeta}$ $I_0^$ $\psmat{-1&0\\0&-1}$ $\zeta^2= z$ $\tau=holo(z)$ $w=\zeta$ $II$ $\psmat{1&1\\-1&0}$ $\zeta^6=z$ $\tau=\frac{\eta-\eta^2\zeta^{2h}}{1-\zeta^{2h}}$, $h\equiv1\pmod3$ $w=\frac{\zeta}{1-\zeta^{2h}}$ $III$ $\psmat{0&1\\-1&0}$ $\zeta^4=z$ $\tau=\frac{i+i\zeta^{2h}}{1-\zeta^{2h}}$, $h\equiv1\pmod2$ $w=\frac{\zeta}{1-\zeta^{2h}}$ $IV$ $\psmat{0&1\\-1&-1}$ $\zeta^3=z$ $\tau=\frac{\eta-\eta^2\zeta^{h}}{1-\zeta^{h}}$, $h\equiv2\pmod3$ $w=\frac{\zeta}{1-\zeta^{h}}$ $IV^$ $\psmat{-1&-1\\1&0}$ $\zeta^3=z$ $\tau=\frac{\eta-\eta^2\zeta^{h}}{1-\zeta^{h}}$, $h\equiv1\pmod3$ $w=\frac{\zeta^2}{1-\zeta^{h}}$ $III^$ $\psmat{0&-1\\1&0}$ $\zeta^4=z$ $\tau=\frac{i+i\zeta^{2h}}{1-\zeta^{2h}}$, $h\equiv1\pmod2$ $w=\frac{\zeta^3}{1-\zeta^{2h}}$ $II^$ $\psmat{0&-1\\1&1}$ $\zeta^6=z$ $\tau=\frac{\eta-\eta^2\zeta^{2h}}{1-\zeta^{2h}}$, $h\equiv2\pmod3$ $w=\frac{\zeta^5}{1-\zeta^{2h}}$ In the table, we write $\eta$ for $e^{2\pi i/3}$ and $holo(z)$ for a holomorphic function on $\Delta'_t$ which extends holomorphically to $\Delta_t$. Global group structure ---------------------- Let $\EE^{sm}$ be the open subset of $\EE$ where $\pi:\EE\to\CC$ is smooth, and let $\EE^{id}$ be the union over all $t\in\CC$ of the identity component of the fiber of $\EE^{sm}$ over $t$. We may view $\EE^{id}$ as the sheaf of abelian groups over $\CC$ which assigns to $U\subset\CC$ the group of holomorphic sections of $\EE^{id}\to\CC$ over $U$. In [@Kodaira63b §11], Kodaira gives a description of $\EE^{id}$ in terms of two other sheaves which we now review. The monodromy representation $\rho$ gives rise to a locally constant sheaf $\GG_0$ on $\CC^0$ with stalks $\Z^2$. Taking the direct image of $\GG_0$ along along the inclusion $\CC^0\subset\CC$ yields a sheaf $\GG$. Using the description of the local monodromy in the preceding section, we see that the stalk of $\GG$ at points of multiplicative reduction ($I_b$, $b\ge1$) is $\Z$, and the stalk at points of additive reduction ($I_b^$, $II$, ...) is $0$. Using the period morphism $\tau$, we define an inclusion $\GG\to\omega^{-1}$. On $\widetilde{\CC}^0$ it sends $\Z^2$ to $\C$ via $(m,n)\mapsto m\tau(\tilde t)+n$, and at points of multiplicative reduction it sends $\Z\to\C$ via $n\mapsto n$. Kodaira [@Kodaira63b Thm 11.2] showed that there is an exact sequence $$\label{eq:EE-id} 0\to\GG\to\omega^{-1}\to\EE^{id}\to 0$$ of sheaves of abelian groups on $\CC$. We will use this sequence to work with sections of $\EE$ near bad fibers. The Betti foliation {#s:Betti} =================== In this section, we will define a foliation on $\EE^0$ which has the torsion multisections $\EE^0\cap\EE[n]$ among its closed leaves. The global Betti foliation -------------------------- Given $(r,s)\in\R^2$, consider the set $$\left\{(\tilde t,r\tau(\tilde t)+s)\left\|\ \tilde t\in\widetilde{\CC^0}\right.\right\} \subset \widetilde{\CC^0}\times\C,$$ and define $\FF_{r,s}$ to be its image in $\FF^0$. Then $\FF_{r,s}$ is a section of the projection $\FF^0\to\widetilde{\CC^0}$ which depends only on the class of $(r,s)\in(\R/\Z)^2$, and we have an isomorphism of real analytic manifolds $$\FF^0=\bigcup_{(r,s)\in(\R/\Z)^2}\FF_{r,s}\cong \widetilde{\CC^0}\times(\R/\Z)^2.$$ We define the (global) Betti leaf* $\GG_{r,s}$ attached to $(r,s)\in(\R/\Z)^2$ to be the image of $\FF_{r,s}$ in $\EE^0\cong\FF^0/\Gamma$. (This terminology is inspired by [@CorvajaMasserZannier18], where $r$ and $s$ are called “Betti coordinates”. ) The collection of leaves $\GG_{r,s}$ gives a foliation of $\EE^0$ by (not necessarily closed) analytic submanifolds. A straightforward calculation shows that $\GG_{r,s}=\GG_{r',s'}$ if and only if $$(r,s)=(r',s')\rho(\gamma)=(a_\gamma r'+c_\gamma s',b_\gamma r'+d_\gamma s')$$ in $(\R/\Z)^2$ for some $\gamma\in\Gamma$. In particular, the leaves $\GG_{r,s}$ are in bijection with the orbits of $\Gamma$ acting on $(\R/\Z)^2$ from the right. The local Betti foliation {#ss:l-Betti} ------------------------- We define local Betti leaves as in [@UUpp19]. Let $V\subset\CC^0$ be non-empty, connected, and simply connected open subset and choose a lifting $V\to\widetilde{\CC^0}$, $t\mapsto\tilde t$. Then we get a branch of the period $\tau:V\to\HH$, $t\mapsto\tau(\tilde t)$, and we foliate $\pi^{-1}(V)\subset\EE^0$ by leaves $\LL_{r,s}$ where $\LL_{r,s}$ is the image of the section of $\EE^0\to\CC^0$ given by $$t\mapsto\text{ the class of }(\tilde t,r\tau(\tilde t)+s)\in \left(\widetilde{\CC^0}\times\C\right)/(\Gamma\ltimes\Z^2)\cong\EE^0.$$ With this definition we have a trivialization $$\pi^{-1}(V)\cong V\times(\R/\Z)^2.$$ The following relation between the local and global leaves follows immediately from the definitions: for $(r,s)\in(\R/\Z)^2$, $$\pi^{-1}(V)\cap\GG_{r,s}=\bigcup_{(r',s')\in(r,s)\rho(\Gamma)}\LL_{r',s'}.$$ In other words, over $V$, a global leaf $\GG_{r,s}$ decomposes into the disjoint union of local leaves, where the union is indexed by the orbit of the monodromy group on $(\R/\Z)^2$ through $(r,s)$. From this we deduce a criterion for a leaf $\GG_{r,s}$ to be closed in $\EE^0$. \[prop:closed-leaves\] 1. If $\EE$ is isotrivial, every leaf $\GG_{r,s}$ is closed. 2. If $\EE$ is non-isotrivial, $\GG_{r,s}$ is closed if and only if $(r,s)\in(\Q/\Z)^2$ if and only if every point of $\GG_{r,s}$ is a torsion point in its fiber. 3. If $\EE$ is not constant, then a section $P$ has image lying in a leaf $\GG_{r,s}$ if and only if $P$ is a torsion section. From the local description above, it is clear that $\GG_{r,s}$ is closed in $\EE^0$ if the orbit of $\rho(\Gamma)$ through $(r,s)$ is finite. Since $\rho(\Gamma)$ is finite when $\EE$ is isotrivial, this establishes part (1). For part (2), suppose that $\EE$ is non-isotrivial. Then as noted in Section \[ss:g-mono\], $\rho(\Gamma)$ has finite index in $\SL_2(\Z)$. If $(r,s)\in(\Q/\Z)^2$ it is clear that the orbit through $(r,s)$ is finite and that $\GG_{r,s}$ consists of points which are torsion in their fiber. Suppose then that $(r,s)\in(\R/\Z)^2\setminus(\Q/\Z)^2$. It is clear that the points of $\GG_{r,s}$ are not torsion in their fiber. Since $\rho(\Gamma)$ has finite index in $\SL_2(\Z)$, there is an integer $b$ such that $\psmat{1&b\\0&1}\in\rho(\Gamma)$. If $r\not\in\Q$, then the orbit contains $$(r,s)\pmat{1&b\\0&1}^n=(r,nbr+s)$$ and thus $\GG_{r,s}$ is not closed by Weyl equidistribution. If $s\not\in\Q$, a similar argument shows that $\GG_{r,s}$ is not closed. This completes the proof of part (2). For part (3), assume that $\EE$ is not constant and that $P$ is a section. If $P$ is torsion, then in every fiber its “Betti coordinates” $(r,s)$ are rational. Since $\Q$ is totally disconnected, these coordinates must be the same in every fiber, so $P$ lies in $\GG_{r,s}$ for some rational pair $(r,s)$. Conversely, if $P$ lies in $\GG_{r,s}$ then $(r,s)$ must be invariant under the monodromy group $\rho(\Gamma)$. Similarly for the multiples $nP$. But $\EE$ is non-constant, and this implies that the monodromy group is non-trivial and either finite or of finite index in $\SL_2(\Z)$ (as noted in Section \[ss:g-mono\]). In both cases, it has elements with only finitely many fixed points on $(\R/\Z)^2$, so the set $\{nP\|n\in\Z\}$ is finite, i.e., $P$ is torsion. This completes the proof of part (3). Behavior at infinity -------------------- We consider the local geometry of Betti leaves near a singular fiber. Suppose $t\in\CC\setminus\CC^0$ and, as in Section \[ss:l-mono\], let $\Delta_t\subset\CC$ be a neighborhood of $t$ biholomorphic to a disk with $\Delta'_t:=\Delta_t\setminus\{t\}\subset\CC^0$. Let $V\subset\Delta'_t$ be a non-empty, connected, and simply connected open set, and define the local monodromy $g_t\in\SL_2(\Z)$ as in Section \[ss:l-mono\] and local Betti leaves $\LL_{r,s}$ as in Section \[ss:l-Betti\]. We say that a local leaf $\LL_{r,s}$ is an invariant leaf (with respect to $t$) if $(r,s)\in(\R/\Z)^2$ is fixed by $g_t$ (acting on the right), and we say it is a vanishing leaf (with respect to $t$) if $(r,s)$ is not invariant under $g_t$. The latter terminology is motivated by part (4) of the following result. \[prop:inv-van\] 1. If $\LL_{r,s}$ is an invariant leaf, then it extends to a section of $\pi:\EE\to\CC$ over $\Delta_t$, and this section meets the special fiber $\pi^{-1}(t)$ in a smooth point. 2. If $\EE$ has multiplicative reduction at $t$ type $I_b$, $b\ge1$, let $S\cong(\Z/b\Z)\times S^1$ be the closure of the set of points of finite order in the special fiber. The invariant leaves extend to sections meeting the special fiber at points of $S$, and every point of $S$ is met by the extension of a unique invariant leaf $\LL_{r,s}$. 3. If $\EE$ has additive reduction at $t$ types $I_b^, b\ge0$, $II$, $II^$, $III$, $III^$, $IV$, $IV^$, the invariant leaves extend to sections meeting the special fiber at one of its finitely many torsion points, and each such point is met by the extension of a unique invariant leaf $\LL_{r,s}$. 4. If $\LL_{r,s}$ is a vanishing leaf, then it extends to a connected multisection of $\EE\to\CC$ over $\Delta_t$ of degree $>1$ possibly infinite, and this multisection meets the special fiber $\pi^{-1}(t)$ in one singular point. Suppose $\LL_{r,s}$ is an invariant leaf. Then by analytic continuation, $\LL_{r,s}$ extends to a section of $\pi$ over $\Delta'_t$. The closure of of this section in $\pi^{-1}(\Delta_t)$ is proper over $\Delta_t$ (since $\pi$ is proper) and by invariance of the intersection number, it meets the special fiber with intersection number 1, and thus must meet it at a smooth point. This establishes part (1). Now assume that $\EE$ has reduction type $I_1$ at $t$. Let $\XX=\pi^{-1}(\Delta_t)$ and $\XX'=\pi^{-1}(\Delta'_t)$. Then Kodaira showed that $$\XX\setminus\XX' = \text{nodal cubic} \cong \C^\times\cup \{q\}$$ where $q$ is the node of the cubic, and that, with a suitable choice of coordinates, $\XX'$ has the form $$\XX'\cong \left(\Delta'_t\times\C^\times\right)/\Z$$ where the action of $\Z$ on $\Delta'_t\times\C^\times$ is $$m\cdot(u,v)=(u,u^mv).$$ Moreover, there is a holomorphic map $$\phi:\Delta_t\times\C^\times \to \XX$$ such that $\{t\}\times\C^\times$ maps biholomorphically to the complement of $q$ in the special fiber, and $\Delta'\times\C^\times\to\XX'\subset\XX$ is the natural quotient map. In terms of a suitable basis, the local monodromy map is $$g_t=\pmat{1&1\\0&1}.$$ It is then straightforward to calculate that the invariant leaves are those of the form $\LL_{0,s}$ for $s\in\R/\Z$. The corresponding extended section is $$u\mapsto\text{ the class of }(u,e^{2\pi i s}),$$ these sections specialize to points on the unit circle $S=S^1\subset\C^\times$, and we get the asserted bijection between the invariant leaves and points on $S$. The establishes the case $b=1$ of part (2). The case of $I_b$ reduction for general $b$ is very similar, with additional notational complexities. In suitable coordinates, the local monodromy is $$g_t=\pmat{1&b\\0&1}$$ and the invariant leaves are those of the form $\LL_{r,s}$ where $r\in(1/b)\Z/\Z$ and $s\in\R/\Z$. The smooth part of $\XX=\pi^{-1}(\Delta_t)$ is covered by open subsets as follows: For $i\in\Z/b\Z$, let $$W_i=W_i'\cup\C^\times_i,\qquad W_i'=\left(\Delta'_t\times\C^\times\right)/\Z$$ where the action of $\Z$ on $\Delta_t'\times\C^\times$ is $$m\cdot(u,v)=(u,u^{bm}v).$$ For $u\in\Delta'_t$ and $v\in\C^\times$, write $(u,v)_i$ for the class of $(u,v)$ in $W_i'$. Then $\XX^{sm}$ is obtained by gluing the $W_i$ according to the rule $$(u,v)_i=(u,u^{j-i}v)_j$$ for all $u\in\Delta'_t$, $v\in\C^\times$, and $i,j\in\Z/b\Z$. The invariant leaf $\LL_{i/b,s}$ lies in the open corresponding to $i$ and extends to the section $$u\mapsto\text{ the class of }(u,e^{2\pi i s})_i,$$ and we find that the specializations of extensions of invariant leaves are in bijection with $$(1/b)\Z/\Z\times S^1\subset \pi^{-1}(t),$$ as required. This establishes part (2) in the general case. For part (3), recall the explicit generators for the local monodromy groups in the table at the end of Section \[ss:l-mono\] . Using these, one computes the invariant leaves, which are as follows: $$\begin{aligned} I_b^, b\text{ odd}:&\LL_{0,0}, \LL_{1/2,1/4},\LL_{0,1/2},\LL_{1/2,3/4}\\ I_b^, b\text{ even}:&\LL_{0,0}, \LL_{1/2,1/2},\LL_{0,1/2},\LL_{1/2,0}\\ II, II^:&\LL_{0,0}\\ III, III^:&\LL_{0,0},\LL_{1/2,1/2}\\ IV, IV^:&\LL_{0,0},\LL_{1/3,2/3},\LL_{2/3,1/3}.\end{aligned}$$ It is then straightforward to see that each of the corresponding sections specializes to a torsion point and that all torsion points on the special fiber are met by the extension of a unique invariant leaf. For part (4), it is clear that analytic continuation of a vanishing leaf $\LL_{r,s}$ yields a multisection over $\Delta'_t$ whose degree is the order of the orbit of the monodromy group through $(r,s)$, which by assumption is $>1$. That its closure in $\pi^{-1}(\Delta_t)$ adds a single point over $t$ which is singular in the special fiber requires a tedious analysis of cases. Since we will not use this result elsewhere in the paper, we omit the details. Intersections with the Betti foliation {#s:intersections} ====================================== For the rest of the paper, we assume that $\pi:\EE\to\CC$ is non-constant and that $P$ is not torsion. In this section, we will quantify tangencies between $P$ and the Betti foliation in terms of intersection numbers. Local intersection numbers {#ss:local-int} -------------------------- Suppose first that $t\in\CC^0$, i.e., that $\EE$ has good reduction at $t$. Over a neighborhood of $t$, there is a unique local Betti leaf $\LL$ passing through $P(t)$. Since $P$ is not torsion, Proposition \[prop:closed-leaves\](3) implies that this intersection is isolated, i.e., by shrinking the neighborhood, we may assume $P$ and $\LL$ meet only over $t$. We define $I(P,t)$ to be the intersection multiplicity of $P$ and $\LL$ at $P(t)$. (This is the local intersection number of two holomorphic 1-manifolds meeting at an isolated point of a holomorphic 2-manifold. We will make it explicit in terms of the order of vanishing of a holomorphic function below.) Note that the intersection in question satisfies $I(P,t)\ge1$, and $I(P,t)\ge2$ if and only if $P$ is tangent to $\LL$ at $t$, i.e., if and only if $t\in T_{Betti}$. Now assume that $t\in\CC\setminus\CC^0$. Let $S\subset\pi^{-1}(t)$ be the closure of the set of torsion points in the special fiber. As noted in Proposition \[prop:inv-van\], $S\cong(\Z/b\Z)\times S^1$ if $\EE$ has reduction type $I_b$ at $t$, and it is a finite group in the other cases. If $P(t)\not\in S$, we define $I(P,t)=0$. If $P(t)\in S$, then by Proposition \[prop:inv-van\], there is a unique invariant local leaf $\LL$ extending over a neighborhood of $t$ and meeting $P$ over $t$. We define $I(P,t)$ to be the intersection number of $P$ and this extended leaf at $t$. \[lemma:nP\] For all integers $n>0$ and all points $t\in\CC$, $$I(P,t)=I(nP,t).$$ Indeed, since the multiplication by $n$ map $\EE^{sm}\to\EE^{sm}$ is étale, for every $(r,s)\in(\R/\Z)^2$, the intersection number of $P$ with the local leaf $\LL_{r,s}$ at $t$ is the same as the intersection number of $nP$ with $\LL_{nr,ns}$ at $t$. Explicit intersection numbers {#ss:explicit} ----------------------------- In this section, we make the intersection number $I(P,t)$ more explicit by using the exact sequence . By Lemma \[lemma:nP\], we may replace $P$ with a multiple and thereby assume that $P$ passes through the identity component of each fiber. Fix $t\in\CC$ and choose a small enough neighborhood $\Delta_t$ of $t$ in $\CC$ such that the restricted section $P:\Delta_t\to\EE$ lifts to a section of $\omega^{-1}$ over $\Delta_t$ and such that $\omega^{-1}$ is trivial over $\Delta_t$. Let $z$ be a coordinate on $\Delta_t$ such that $t$ corresponds to $z=0$. We may then identify $P$ with a product $w=hw_0$ where $h$ is a holomorphic function on $\Delta_t$ and $w_0$ is a generating section of $\omega^{-1}$ over $\Delta_t$ as specified in the table at the end of Section \[ss:l-mono\]. The local multiplicity $I(P,t)$ is by definition the intersection number of $P$ and an invariant local Betti leaf $\LL_{r,s}$. Since the leaf is invariant, the map $z\mapsto r\tau(z)+s$ defines a section of $\omega^{-1}$ over $\Delta_t$, and the intersection multiplicity is the same as the intersection number between the graphs of the functions $z\mapsto h(z)$ and $z\mapsto (r\tau(z)+s)/w_0$ If $t\in\CC^0$, then $w_0=1$. If $h(t)=r\tau(t)+s$, $I(P,t)$ is the intersection number between the graph of $z\mapsto h(z)$ and the graph of $z\mapsto r\tau(z)+s$. Therefore, $$\label{eq:I-good} I(P,t)=\ord_{z=0}\left(h(z)-r\tau(z)-s\right).$$ If $\EE$ has multiplicative reduction ($I_b$, $b>0$) at $t$, then $w_0=1$ and $I(P,t)=0$ if $h(t)\not\in\R$. If $h(t)=s\in\R$, then $I(P,t)$ is the intersection number between the graph of $z\mapsto h(z)$ and the graph of the constant function $z\mapsto s$. Therefore, $$\label{eq:I-mult} I(P,t)= \begin{cases} 0&\text{if $h(t)\not\in\R$}\\ \ord_{z=0}\left(h(z)-s\right)&\text{if $h(t)=s\in\R$.} \end{cases}$$ If $\EE$ has additive reduction at $t$ (types $I_b^$, $II$, ...), then $I(P,t)=0$ unless $h(t)=0$, and if $h(t)=0$, then $I(P,t)$ is the intersection number between the graph of $z\mapsto h(z)$ and the graph of $z\mapsto 0$. Therefore, $$\label{eq:I-add} I(P,t)=\ord_{z=0}\left(h(z)\right).$$ A real analytic 1-form {#s:1-form} ====================== In this section, we review a connection between local and global degrees of smooth sections of a line bundle. We then construct a real analytic 1-form whose zeroes will turn out to control tangencies between a section $P$ and the Betti foliation. Local and global indices ------------------------ The number of zeroes and poles of a meromorphic section of a line bundle (counted with multiplicities) is the degree of the line bundle. This familiar result from basic algebraic geometry is in fact purely topological. In this section, we state and sketch the proof of the result in the smooth category. Our Proposition \[prop:zeroes-degree\] is in substance equivalent to [@BottTuDFIAT Thm. 11.17], but the language there is rather different than ours, so for the convenience of the reader, we review the main lines of the argument adapted to our situation. ### Winding numbers {#ss:winding} Let $\Delta$ be the unit disk in $\C$, and let $\Delta'=\Delta\setminus\{0\}$. Suppose that $f$ is a smooth, nowhere vanishing, complex-valued function on $\Delta'$. We define the winding number of $f$ to be $$W(f):=\frac1{2\pi i}\oint d\log f$$ where the path of integration is any positively oriented loop around 0. Equivalently $$W(f)=\frac1{2\pi i}\int_0^1\frac{g'(t)}{g(t)}dt$$ where $g(t)=f(re^{2\pi it})$ for some $0<r<1$. The following properties of $W(f)$ are well known. See, for example, [@FultonAT Ch. 3]. 1. $W(f)$ is an integer and is independent of the choice of path of integration. 2. If $f$ extends to a smooth nowhere vanishing function on $\Delta$, then $W(f)=0$. 3. $W(f_1f_2)=W(f_1)+W(f_2)$. 4. If $f$ is the restriction of a meromorphic function on $\Delta$, then $W(f)=\ord_{z=0}f(z)$. 5. If $F(\sigma,z)$ is a smooth, nowhere vanishing function on $[0,1]\times\Delta'$ and $f_\sigma(z)=F(\sigma,z)$, then $W(f_0)=W(f_1)$. The following is essentially the “dog on a leash” theorem, see [@FultonAT Thm 3.11]. \[lemma:W-sum\] 1. Suppose that $f_1$ and $f_2$ are smooth functions on the punctured disk $\Delta'$, and let $f=f_1-f_2$. Suppose also that there exist real numbers $m_1<m_2$ and positive real numbers $C_1$, and $C_2$ such that $$\begin{aligned} \|f_1(z)\|&\ge C_1\|z\|^{m_1}\\ \noalign{and} \|f_2(z)\|&\le C_2\|z\|^{m_2}\end{aligned}$$ for all $\in\Delta'$. Then $W(f)=W(f_1)$. 2. The same conclusion holds when $m_1=m_2$ provided that $C_1>C_2$. Define $F$ on $[0,1]\times\Delta'$ by $F(\sigma,z)=f_1(z)-\sigma f_2(z)$, so that $F(1,z)=f(z)$ and $F(0,z)=f_1(z)$. The displayed inequalities show that $F(\sigma,z)\neq0$ for all sufficiently small $z$, so we may shrink $\Delta'$ and have that $F(\sigma,z)$ is nowhere vanishing on $[0,1]\times\Delta'$. The winding numbers $W(f)$ and $W(f_1)$ are then well defined, and property (5) of winding numbers shows that $W(f)=W(f_1)$. ### Local indices Let $L$ be a holomorphic line bundle on $\CC$ and suppose that $s$ is a smooth, nowhere vanishing section of $L$ over an open subset of the form $U=\CC\setminus\{t_1,\dots,t_m\}$. We define a local index $J(s,t)$ for all $t\in\CC$ as follows: Given $t$, choose a neighborhood $U_t$ of $t$ diffeomorphic to a disk and such that $s$ is defined and non-zero on $U'_t:=U_t\setminus \{t\}$. Choose a trivializing section $s_t$ of $L$ (as a complex line bundle) over $U_t$, and write $s=f(z)s_t$ for $z\in U'_t$. Then $$J(s,t):=W(f)$$ where we identify $f$ with a function on $\Delta'$ via a diffeomorphism $\Delta\cong U_t$ sending $0$ to $t$. The properties of $W$ recalled above imply that $J(s,t)$ is an integer and is independent of the various choices. The also imply that if $s$ is a meromorphic section of $L$ near $t$, then $J(s,t)$ is exactly the order of zero or pole of $s$ at $t$ in the usual sense. The following global result generalizes the statement that the sum of the orders of zero or pole of a meromorphic section of a line bundle is the degree of the line bundle. Recall that $H^2(\CC,\Z)$ is canonically isomorphic to $\Z$. We define $\deg(L)$ to be the first Chern class $c_1(L)\in H^2(\CC,\Z)=\Z$. \[prop:zeroes-degree\] Suppose $s$ is a smooth, nowhere vanishing section of $L$ over $U=\CC\setminus\{t_1,\dots,t_m\}$. Then $$\sum_{t\in\CC}J(s,t)=\sum_{i=1}^mJ(s,t_i)=\deg(L).$$ Property (2) of winding numbers recalled above implies that $J(s,t)=0$ unless $t$ is in $\{t_1,\dots,t_m\}$, so the sum over $t\in\CC$ is well defined and equal to the sum over the $t_i$. To prove the equality with the degree of $L$ we will compare Cech and de Rham cohomologies. For $i=1,\dots,m$, let $U_i$ be a neighborhood of $t_i$ diffeomorphic to the disk $\Delta$ with $0$ corresponding to $t_i$ and such that the closures of the $U_i$ in $\CC$ are disjoint. Choose simply connected open sets $U_{m+1},\dots,U_n\subset U$ such that $U_1,\dots,U_n$ covers $\CC$ and such that $U_{ij}:=U_i\cap U_j$ is simply connected for all pairs of indices $1\le i,j\le n$. For $i=1,\dots,m$, choose generating sections $s_i$ of $L$ over $U_i$, and for $i=m+1,\dots,n$, let $s_i$ be the restriction of $s$ to $U_i$. Then there are smooth, nowhere vanishing functions $g_{ij}$ defined on $U_{ij}$ by $$s_i=g_{ij}s_j,$$ and the $g_{ij}$ form a 1-cocycle with values in $\AA^\times$, the sheaf of nowhere vanishing smooth functions on $\CC$. The class of this cocycle in Cech cohomology is $[L]\in H^1(\CC,\AA^\times)$. We have an exact sequence $$0\to\Z\to\AA\to\AA^\times\to0$$ where $\AA$ is the sheaf of smooth functions on $\CC$ and the map $\AA\to\AA^\times$ is $f\mapsto e^{2\pi i f}$. Taking the coboundary of $[L]$ in the long exact sequence of cohomology, we find that $c_1(L)\in H^2(\CC,\Z)$ is represented by the 2-cocycle $$\eta_{ijk}=\frac1{2\pi i}\left(\log g_{ij}-\log g_{ik}+\log g_{jk}\right).$$ Next, we write down a 2-form representing the image of $[L]$ under $$H^2(\CC,\Z)\to H^2(\CC,\C)\cong H^2_{dR}(\CC)\tensor\C.$$ Since $\eta_{ijk}$ is $\Z$-valued, we have $d\eta_{ijk}=0$. This implies that $$h_{ij}=\frac1{2\pi i}d\log g_{ij}$$ is a 1-cocycle with values in $\AA^1$, the sheaf of smooth 1-forms on $\CC$. Now choose a partition of unity $\rho_i$ subordinate to the cover $U_i$ of $\CC$. Shrinking $U_i$ for $i=m+1,\dots,n$ if necessary, we may assume that for $i=1,\dots,m$, there are closed disks of positive radius $K_{i,1}\subset K_{i,2}\subset U_i$ such that $\rho_i$ is identically 1 on $K_{i,1}$ and identically zero on the complement of $K_{i,2}$. Setting $$\theta_i=\frac1{2\pi i}\sum_{\ell=1}^n \rho_\ell \,d\log g_{i\ell}\in\AA^1(U_{i})$$ we see that $\theta_i-\theta_j=h_{ij}$. Since $h_{ij}$ is $d$-closed for all $ij$, we find that $d\theta_i=d\theta_j$ on $U_{ij}$ and so we may define a global 2-form $\Omega$ on $\CC$ by requiring that $$\begin{aligned} \Omega&=-d\theta_i \\ &=\frac{-1\phantom{-}}{2\pi i}\sum_{\ell=1}^nd\rho_\ell\,d\log g_{i\ell}\end{aligned}$$ on $U_i$. It follows from the “generalized Mayer-Vietoris principle” [@BottTuDFIAT §8] (also known fondly to some as the “Cech-de Rham shuffle”), that $\Omega$ represents the class of $L$ in de Rham cohomology. More formally $$c_1(L)=\int_\CC\Omega.$$ (The point is that the $\Z$-valued 2-cocycle $\eta_{ijk}$ and the $\AA^2$-valued 0-cocycle $\Omega$ represent the same class in the cohomology of the total complex of the Cech-deRham double complex because, by construction, they differ by a coboundary.) To finish the proof, we will relate the displayed integral to winding numbers. Let $U_i'=U_i\setminus\{t_i\}$, and let $g_i\in\AA^\times(U_i')$ be defined by $$s_i=g_is.$$ Then examining the definitions shows that $$\begin{aligned} g_{ij}=1&\quad\text{if $i,j>m$}\\ g_{ij}=g_i&\quad\text{if $i\le m$ and $j>m$}\\ U_i\cap U_j=\emptyset&\quad\text{if $i,j\le m$.}\end{aligned}$$ It follows that $\Omega$ vanishes identically on the complement of $\cup_{i=1}^m U_i$, and so $$\label{eq:deg-sum} \deg(L)=\sum_{i=1}^m\int_{U_i}\Omega.$$ On $U_i$ we have $$\begin{aligned} \Omega_{\|U_i} &=\frac{-1\phantom{-}}{2\pi i} \sum_{\ell=m+1}^nd\rho_\ell\,d\log g_i\\ &=\frac{1}{2\pi i} d\rho_i\,d\log g_i.\end{aligned}$$ Now $d\rho_i$ is identically zero on $K_{i,1}$ and on the complement of $K_{i,2}$, so we have $$\begin{aligned} \int_{U_i}\Omega &=\frac{1}{2\pi i}\int_{K_{i,2}\setminus K_{i,1}}d\rho_i\,d\log g_i\\ &=\frac{1}{2\pi i}\int_{\partial K_{i,2}}\rho_i\,d\log g_i -\frac{1}{2\pi i}\int_{\partial K_{i,1}}\rho_i\,d\log g_i\end{aligned}$$ by Stokes’ theorem. Since $\rho_i$ vanishes on $\partial K_{i,2}$ and is 1 on $\partial K_{i,1}$, we find $$\int_{U_i}\Omega=-W(g_i)=W(g_i^{-1})=J(s,t_i)$$ where the last equality follows from the definition of $J$. Combining this with Equation , we find that $$\deg(L)=\sum_{i=1}^mJ(s,t_i)$$ as desired. This completes the proof of the proposition. Constructing $\eta$ ------------------- We use the uniformization of Section \[ss:unif\]. Let $w$ be the standard coordinate on $\C$ and recall the period function $\tau:\widetilde{\CC^0}\to\HH$. Consider the real-analytic 1-form on $\widetilde{\CC^0}\times\C$ given by $$\tilde\eta=dw-\frac{\im w}{\im \tau}d\tau.$$ Under the action of $\Gamma\ltimes\Z^2$, straightforward calculation shows that $$(id,m,n)^(\tilde\eta) =\tilde\eta$$ and $$(\gamma,0,0)^(\tilde\eta)=f_\gamma\tilde\eta =\frac{\tilde\eta}{c_\gamma\tau+d_\gamma}$$ where as usual $\rho(\gamma)=\psmat{a_\gamma&b_\gamma\\c_\gamma&d_\gamma}$. These formulas show that $\tilde\eta$ descends to a real-analytic section $\eta$ of $$\Omega^1_{\EE^0}\tensor\pi^(\omega)^{-1} =\Omega^1_{\EE^0}\tensor\left(\Omega^1_{\EE^0/\CC^0}\right)^{-1}.$$ It is immediate from the definition of the Betti foliation in terms of the uniformization $\widetilde{\CC^0}\times\C\to\EE^0$ that at every point $x\in\EE^0$, the kernel of $\eta$ as a functional on the holomorphic tangent space of $\EE^0$ at $x$ is precisely the tangent space to the leaf of the Betti foliation passing through $x$. We will thus be able to use $\eta$ to quantify the tangencies between sections $P$ and the Betti foliation. Definition of $\eta_P$ {#ss:etaP-def} ---------------------- Now assume that $P$ is a non-torsion section of $\EE\to\CC$ and recall that the latter is assumed to be non-constant. Let $\eta_P:=P^(\eta)$. This is a real analytic section of $\Omega^1_\CC\tensor\omega^{-1}$ over $\CC^0$. Since the kernel of $\eta$ at a point of $\EE^0$ is the tangent space to the leaf of the Betti foliation through that point, we see that $\eta_P$ vanishes at a point of $\CC^0$ if and only if that point lies in $T_{Betti}\cap\CC^0$. Since $T_{Betti}$ is finite by [@UUpp19 §3], it follows that $\eta_P$ has only finitely many zeroes. Thus, Proposition \[prop:zeroes-degree\] applies, and we have the following key result. \[prop:EtaP-degree\] $$\sum_{t\in\CC}J(\eta_P,t)= \deg(\Omega^1_\CC\tensor\omega^{-1})= 2g-2-d.$$ We end this section with a lemma parallel to Lemma \[lemma:nP\]. \[lemma:nP’\] For all integers $n>0$ and all points $t\in\CC$, $$J(\eta_P,t) =J(\eta_{nP},t).$$ It is clear from the local expression for $\eta$ as $$dw-\frac{\im w}{\im\tau}d\tau$$ the $\eta_{nP}=n\eta_P$. The equality of local indices then follows from properties (3) and (4) of the winding number $W$. Zeroes and intersection numbers {#s:zeros} =============================== In this section, we relate the intersection number $I(P,t)$ to the local index $J(\eta_P,t)$. \[prop:int-van\] For all $t\in\CC$ $$J(\eta_P,t)=I(P,t)-1.$$ By Lemmas \[lemma:nP\] and \[lemma:nP’\], we may replace $P$ with a multiple and reduce to the case where $P$ passes through the identity component of every fiber of $\EE\to\CC$. Fix $t\in\CC$ and let $\Delta_t$ be a neighborhood of $t$ biholomorphic to a disk and such that $\Delta'_t:=\Delta_t\setminus\{t\}$ lies in $\CC^0$. Let $z$ be a coordinate on $\Delta_t$ such that $t$ corresponds to $z=0$. Recall from Section \[ss:explicit\] that shrinking $\Delta_t$ if necessary, we may lift $P$ to $\omega^{-1}$ in the exact sequence of Equation and identify the lift with a product $w=hw_0$ where $w_0$ is a generating section of $\omega^{-1}$ (as specified in the table at the end of Section \[ss:l-mono\]) and $h$ is a holomorphic function on $\Delta_t$. In terms of this data, we have $$\eta_P=d(hw_0)-\frac{\im hw_0}{\im \tau}d\tau.$$ The winding number that defines $J(\eta_P,t)$ is then $W(f)$ where $$f=\frac1{w_0}\left(\frac{d(hw_0)}{dz} -\frac{\im (hw_0)}{\im \tau}\frac{d\tau}{dz}\right).$$ To lighten notation, let $n=I(P,t)$, so that our goal is to prove that $W(f)=n-1$. We will complete the proof of the proposition in the next four sections, dividing into cases according to the reduction of $\EE$ at $t$. Points of good reduction ------------------------ If $\EE$ has good reduction at $t$, then we saw in Equation that $$n:=I(P,t)=\ord_{z=0}\left(h(z)-r\tau(z)-s\right),$$ where $h(t)=r\tau(t)+s$. Since $w_0=1$, we have $J(\eta_P,t)=W(f)$ where $$f=\frac{dh}{dz}-\frac{\im h}{\im \tau}\frac{d\tau}{dz}.$$ Let $$f_1(z)=\frac{dh}{dz}-r\frac{d\tau}{dz} \quad\text{and}\quad f_2(z)=\left(\frac{\im h}{\im \tau}-r\right)\frac{d\tau}{dz}.$$ Since $h-r\tau-s$ is holomorphic and vanishes to order $n\ge1$ at $t$, we have $$\|f_1(z)\|\ge C_1\|z\|^{n-1}$$ for some positive constant $C_1$ and all sufficiently small $z$. On the other hand, $$\im h-r\im \tau=\frac12\left((h-r\tau-s)-(\overline{h-r\tau-s})\right),$$ $\im\tau(t)>0$, and $\tau$ is holomorphic on $\Delta$, so $$\|f_2(z)\|\le C_2\|z\|^n$$ for some positive constant $C_2$ and all sufficiently small $z$. Applying Lemma \[lemma:W-sum\], we have $W(f)=W(f_1)$, and since $f_1$ is holomorphic on $\Delta$ and vanishes to order $n-1$ at $z=0$, we have $W(f)=W(f_1)=n-1$. This establishes that $J(\eta_P,t)=I(P,t)-1$ for all $t\in\CC^0$. Points of multiplicative reduction ---------------------------------- Next assume that $\EE$ has reduction type $I_b$ ($b\ge1$) at $t$. According to Equation , $$n:=I(P,t)= \begin{cases} 0&\text{if $h(t)\not\in\R$}\\ \ord_{z=0}\left(h(z)-s\right)&\text{if $h(t)=s\in\R$} \end{cases}$$ and by the first part of the proof, $J(\eta_P,t)=W(f)$ where $$f=\frac1{w_0}\left(\frac{d(hw_0)}{dz} -\frac{\im (hw_0)}{\im \tau}\frac{d\tau}{dz}\right).$$ According to the table at the end of Section \[ss:l-mono\], $w_0=1$ and $\tau=(b/2\pi i)\log z$, so $$f=\frac{dh}{dz}-\frac{i\im h}{z\log\|z\|}.$$ Suppose that $h(t)\not\in\R$. Then after shrinking $\Delta_t$, we may assume that $\|h(z)\|$ and $\|(\im h(z))\|$ are bounded above and below on $\Delta_t$ by positive constants. Since $dh/dz$ is holomorphic, we have that $\|dh/dz\|$ is bounded above on $\Delta_t$ as well. On the other hand $\|1/(z\log\|z\|)\|>C\|z^{-1+\epsilon}\|$ for all $\epsilon>0$. Thus, setting $f_1=(i\im h)/(z\log\|z\|)$ and $f_2=dh/dz$, Lemma \[lemma:W-sum\] implies that $W(f)=W(f_1)$. Finally, $$W\left(\frac{i\im h}{z\log\|z\|}\right) =W(i\im h)-W(z)-W(\log\|z\|)=0-1-0=-1.$$ Thus we find that $W(f)=-1=n-1$, establishing that $J(\eta_P,t)=I(P,t)-1$ when $h(t)\not\in\R$. To finish the multiplicative case, assume that $h(t)=s\in\R$. Then $\|dh/dz(z)\|\ge C_1\|z\|^{n-1}$ for some positive $C_1$ where $n=\ord_{z=0}(h(z)-s)$. Also, $\im h\le C_2\|z\|^n$, and we find that $\|(i\im h)/(z\log\|z\|)\|\le C_2\|z\|^{n-1}/\|\log\|z\|\|$. Shrinking $\Delta'$, we may assume that $C_2<C_1$ and $\|(i\im h)/(z\log\|z\|)\|\le C_2\|z\|^{n-1}$. Setting $f_1=dh/dz$ and $f_2=(i\im h)/(z\log\|z\|)$, Lemma \[lemma:W-sum\] implies that $W(f)=W(f_1)$. Since $f_1$ is holomorphic with $\ord_{z=0}(f_1)=n-1$, we conclude that $W(f)=n-1$. This establishes that $J(\eta_P,t)=I(P,t)-1$ when $h(t)\in\R$ and completes the proof for places $t$ of multiplicative reduction. Points of potentially multiplicative reduction ---------------------------------------------- Now assume that $\EE$ has reduction type $I_b^$ ($b>0$) at $t$. According to Equation , $$n:=I(P,t)=\ord_{z=0}\left(h(z)\right)$$ and by the first part of the proof, $J(\eta_P,t)=W(f)$ where $$f=\frac1{w_0}\left(\frac{d(hw_0)}{dz} -\frac{\im (hw_0)}{\im \tau}\frac{d\tau}{dz}\right).$$ According to the table at the end of Section \[ss:l-mono\], $w_0=z^{1/2}$ and $\tau=(b/2\pi i)\log z$, so $$f=\frac{dh}{dz}+\frac12\frac hz -\frac{\im(hz^{1/2})}{z^{1/2}}\frac{i}{z\log\|z\|}.$$ Note that $$\left\|\frac{dh}{dz}+\frac12\frac hz\right\|\ge C_1\|z\|^{n-1}$$ on $\Delta'_t$ for some positive constant $C_1$. On the other hand, $\|\im(hz^{1/2})/z^{1/2}\|\le C\|z\|^n$ on $\Delta'_t$ for some positive constant $C$. Thus $$\left\|\frac{\im(hz^{1/2})}{z^{1/2}}\frac{i}{z\log\|z\|}\right\|\le C\frac{\|z\|^{n-1}}{\|log\|z\|\|}.$$ Shrinking $\Delta$, we may ensure that $$\left\|\frac{\im(hz^{1/2})}{z^{1/2}}\frac{i}{z\log\|z\|}\right\|\le C_2\|z\|^{n-1}$$ for some positive $C_2<C_1$. Setting $$f_1=\frac{dh}{dz}+\frac12\frac hz \quad\text{and}\quad f_2=\frac{\im(hz^{1/2})}{z^{1/2}}\frac{i}{z\log\|z\|},$$ Lemma \[lemma:W-sum\] implies that $W(f)=W(f_1)$, and since $f_1$ is holomorphic on $\Delta_t$ and vanishes to order $n-1$ at $z=0$, we have $W(f_1)=n-1$. This establishes that $J(\eta_P,t)=I(P,t)-1$ when $\EE$ has reduction type $I_b^$ at $t$, completing the proof for places $t$ of potentially multiplicative reduction. Points of potentially good reduction ------------------------------------ Now assume that the reduction type of $\EE$ at $t$ is one of those of additive, potentially good reduction, namely $I_0^$, $II$, $III$, $IV$, $IV^$ $III^$, or $II^$. According to Equation , $$n:=I(P,t)=\ord_{z=0}\left(h(z)\right)$$ and by the first part of the proof, $J(\eta_P,t)=W(f)$ where $$f=\frac1{w_0}\left(\frac{d(hw_0)}{dz} -\frac{\im (hw_0)}{\im \tau}\frac{d\tau}{dz}\right).$$ Using the table at the end of Section \[ss:l-mono\], we see that $w_0$ is a fractional power of $z$ times a non-vanishing, holomorphic function of $\zeta$ on the domain listed in the table. From this we calculate that $$f_1:=\frac1{w_0}\left(\frac{d(hw_0)}{dz}\right) =\frac{dh}{dz}+\alpha\frac{h}{z}+g$$ where $\alpha\in\{1/6,1/4, 1/3,1/2,2/3,3/4,5/6\}$ and $g$ is holomorphic and vanishes to order $\ge n$. Thus $f_1$ is holomorphic on $\Delta_t$ and $$\|f_1(z)\|\ge C_1\|z\|^{n-1}$$ for some positive constant $C_1$. Now consider $$f_2=\frac{\im (hw_0)}{w_0\im \tau}\frac{d\tau}{dz}.$$ Since $\ord_{z=0}(h(z))=n$, and $\im \tau$ is bounded away from zero, we find that $$\left\|\frac{\im (hw_0)}{w_0\im \tau}\right\|\le C\|z\|^n$$ on $\Delta'_t$ for some positive constant $C$. On the other hand, since $\tau$ is a holomorphic function of $\zeta$, and $z=\zeta^b$ with $b\in\{1,3,4,6\}$, we see that $$\left\|\frac{d\tau}{dz}(z)\right\|\le C'\|z\|^{-\beta}$$ with $\beta\in\{0,2/3,3/4,5/6\}$ for some positive $C'$. It follows that $$\|f_2(z)\|\le C_2\|z\|^{n-\beta}$$ with $n-\beta>n-1$. Applying Lemma \[lemma:W-sum\] we find that $W(f)=W(f_1)=n-1$. This establishes that $J(\eta_P,t)=I(P,t)-1$ when $\EE$ has additive and potentially good reduction type at $t$. This completes the proof of Proposition \[prop:int-van\] in all cases. Proof of main theorems {#s:proofs} ====================== The key result of this paper is the following equality. \[thm:total-int\] Suppose that $\pi:\EE\to\CC$ is non-constant and that $P$ is a section of $\pi$ of infinite order. Let $g$ be the genus of $\CC$ and let $d$ be the degree of the line bundle $\omega=O^(\Omega^1_{\EE/\CC})$. Let $I(P,t)$ be the local intersection indices defined in Section \[ss:local-int\]. Then $$\sum_{t\in\CC}\left(I(P,t)-1\right)=2g-2-d.$$ Let $\eta_P$ be the 1-form attached to $P$ in Section \[ss:etaP-def\]. Then according to Proposition \[prop:EtaP-degree\], we have $$2g-2-d=\sum_{t\in\CC}J(\eta_P,t)$$ and according to Proposition \[prop:int-van\] $$J(\eta_P,t)=I(P,t)-1$$ for all $t\in\CC$. \[cor:bound\] Let $T_{Betti}$ be the set of points $t\in\CC$ where $I(P,t)\ge2$. Then $$\left\|T_{tor}\right\|\le\left\|T_{Betti}\right\|\le 2g-2-d+\delta,$$ where $\delta$ is the number of singular fibers of $\pi:\EE\to\CC$. From the definitions, $T_{tor}$ is the subset of $T_{Betti}$ where $P(t)$ is a torsion point in its fiber, so $\|T_{tor}\|\le\|T_{Betti}\|$. Let $S$ be the set of points of $\CC$ where $\EE$ has bad reduction. Then $I(P,t)\ge0$ for all $t\in\CC$, $I(P,t)\geq1$ for all $t\not\in S$, and $I(P,t)\ge2$ if and only $t\in T_{Betti}$. Thus by the Theorem, $$2g-2-d\ge\left\|T_{Betti}\right\|-\left\|S\right\|$$ and the corollary follows immediately. Examples {#s:examples} ======== We consider some explicit families illustrating various aspects of the main theorem. Suppose as usual that $\pi:\EE\to\CC$ is a Jacobian elliptic surface with zero section $O$, $g$ is the genus of $\CC$, $d=\deg\left(O^(\Omega^1_{\EE/\CC})\right)$, and $\delta$ is the number of singular fibers of $\pi$. \[prop:degen\] If $2g-2-d+\delta<0$, then the group of sections of $\pi$ is finite. Suppose $P$ is a section of $\pi$ of infinite order, and let $T_{tor}$ be the corresponding set of tangencies between $P$ and torsion multisections. Then by Corollary \[cor:bound\] the cardinality of $T_{tor}$ would be negative, a contradiction. Thus, there are no sections of infinite order. It would be interesting to have a more direct proof of the proposition. We note that the proposition is sharp in the sense that we give examples below of elliptic surfaces with $2g-2-d+\delta=0$ and with a section of infinite order. Degenerate cases ---------------- Next, we give two examples where $2g-2-d+\delta<0$, one isotrivial, one non-isotrivial. In both cases, it is straightforward to check that the group of sections is torsion, in agreement with Proposition \[prop:degen\]. Let $E$ be any elliptic curve over $\C$ with a given Weierstrass model $y^2=x^3+ax+b$ where $a,b\in\C$. Let $E$ be the twisted elliptic curve $$E:\quad y^2=x^3+at^2x+bt^3$$ over $\C(t)$, and let $\EE\to\P^1$ be the regular minimal model of $E/\C(t)$. Then one verifies easily that $d=1$ and $\delta=2$ ($E$ has $ I_0^$ reduction at $t=0$ and $t=\infty$ and good reduction elsewhere), so that $2g-2-d+\delta=-1$. For a non-isotrivial example, consider $$E:\quad y^2=x^3-3t^4(t^2-1)^2x+2t^5(t^2-1)^3$$ over $\C(t)$, and let $\EE\to\P^1$ be the regular minimal model. Then one verifies that $d=2$ and $\delta=3$, so $2g-2-d+\delta=-1$. Moreover, the $j$-invariant of $E$ is $1728t^2/(t^2-1)$, so $\EE\to\P^1$ is non-isotrivial. (Thanks to Rick Miranda for pointing out how to construct an example like this.) We refer to [@Schmickler-Hirzebruch85] and [@Nguyen99] for the complete list of Jacobian elliptic surfaces over $\P^1$ with three singular fibers. We have the following general result. Suppose that $\pi:\EE\to\CC$ is a Jacobian elliptic fibration with zero section $O$. Suppose $\pi$ has everywhere semi-stable reduction i.e., the bad fibers are of type $I_b$ and is non-constant. Then $$2g-2-d+\delta>0.$$ Here, as usual, $g$ is the genus of $\CC$, $d=\deg\left(O^(\Omega^1_{\EE/\CC})\right)$, and $\delta$ is the number of singular fibers of $\pi$. Note that since $\pi$ is everywhere semi-stable and non-constant, it is in fact non-isotrivial. Let $D$ be the divisor $$D=O+\sum_{\text{bad }t}\pi^{-1}(t)$$ where the sum is over the set of points where the fiber of $\pi$ is singular. By hypothesis, each $\pi^{-1}(t)$ appearing in the sum is a chain of $\P^1$s meeting in nodes, and the divisor $D$ thus has normal crossings. Consider the logarithmic Chern classes $\overline{c}_1(\EE,D)$ and $\overline{c}_2(\EE,D)$ as in [@Urzua11]. By [@Urzua11 Thm. 9.2], we have $$\overline{c}_1(\EE,D)^2<3\overline{c}_2(\EE,D).$$ From the definitions, one computes that $$\overline{c}_1(\EE,D)^2=4g-4+d+2\delta \quad\text{and}\quad \overline{c}_2(\EE,D)=2g-2+\delta,$$ so we find that $$0<3\overline{c}_2(\EE,D)-\overline{c}_1(\EE,D)^2=2g-2-d+\delta$$ as desired. Optimality in the constant case ------------------------------- Fix an elliptic curve $E$ over $\C$ and a positive integer $g$. Let $B=2g-2$ and let $A$ be any integer with $0\le A\le B$. We will produce a constant elliptic surface $\EE=\CC\times E$ and a section whose corresponding $T_{tor}$ satisfies $$\|T_{tor}\|=A\le B=2g-2.$$ Indeed, by the Riemann existence theorem, there exists a branched cover $f:\CC\to\EE$ where $\CC$ is a curve of genus $g$ and $f$ has simple ramification. I.e., $f$ is ramified over $2g-2$ points with distinct images in $\CC$, and the ramification indices are all $2$. Moreover, we may choose the points in $E$ where $f$ is ramified freely. Let $\EE=\CC\times E$ and let $P$ be the section corresponding to the map $f$. Then $T_{Betti}$ is exactly the set of branch points, and it has cardinality $2g-2$. Moreover, by suitable choice of those points, we may arrange for $\|T_{tor}\|$ to take any value between $0$ and $2g-2$. This shows that Corollary \[cor:bound\] is sharp in the constant case. Optimality in the non-constant cases ------------------------------------ A similar idea works in the non-constant cases once we have suitable starting data. To that end, we construct $\pi:\EE_1\to\P^1$ with $$2g-2-d+\delta=-2-d+\delta=0,$$ and with a section $P_1$ of infinite order. By Corollary \[cor:bound\], there are no tangencies bewteen $P_1$ and a torsion multisection. In the isotrivial case, we may take the example considered in [@UUpp19 §7], namely the quotient of the square of an elliptic curve by the diagonal map $(t,t)\mapsto(-t,-t)$. The minimal regular model $\EE_1\to\P^1$ has $d=2$ and $\delta=4$ bad fibers and a section $P_1$ of infinite order (namely the image of the graph of the identity map). The set of tangencies between $P_1$ is thus empty by Corollary \[cor:bound\]. For a non-isotrivial example, consider the elliptic curve $$E_1: \quad y^2=x^3-tx+t$$ and let $\EE_1\to\P^1$ be the regular minimal model. One computes that $\EE_1$ has $\deg\left(O^*(\Omega^1_{\EE_1/\P^1})\right)=1$, and good reduction away from $t=0$, $t=27/4$, and $t=\infty$, and has bad reduction at these points. Thus for any non-torsion section, the corresponding set of torsion tangencies $T_{tor}$ is empty. Let $P_1$ be the section corresponding to the rational point $(x,y)=(1,1)$. Straightforward calculation shows that $P_1$ is of infinite order (and in fact generates the group of sections of $\EE_1$). By Corollary \[cor:bound\], the corresponding et $T_{tor}$ is empty. Now fix a positive, even integer $B$ and let $f:\CC\to\P^1$ be a branched cover with exactly $B$ ramification points and simple branching over each one. (We could also insist that $\CC\to\P^1$ have low degree, say degree 2, but this is not relevant for what follows.) Let $\EE\to\CC$ be the regular minimal model of the pull back of $\EE_1\to\P^1$ to $\CC$ (where $\EE_1$ is either of the examples above), and let $P$ be the section induced by $P_1$. Assuming that the branch points of $\CC\to\P^1$ are distinct from the points where $\EE_1$ has bad reduction, we have $I(P,t)=e_f(t)$ where $e_f(t)$ is the ramification index of $f$ at $t\in\CC$. The Riemann-Hurwitz formula yields $$2g_{\CC}-2-d+\delta=B$$ where $d$ and $\delta$ are the usual invariants attached to $\EE$. Thus, Corollary \[cor:bound\] implies that $\|T_{tor}\|\le B$. By choosing some of the branch points of $f$ to be among the points of $\P^1$ where $P$ takes a torsion value, we may arrange for $\|T_{tor}\|$ to take any value between 0 and $B$. This shows that Corollary \[cor:bound\] is sharp. [^1]: We will work over $\C$ for simplicity. By a standard reduction given in [@UUpp19], our results also hold when $k$ is any field of characteristic zero.
--- abstract: 'The free energy of a static quark-antiquark pair is obtained in an interacting dyon ensemble near the deconfinement temperature. Comparing the results with the noninteracting case, we observe that the string tension between the quark-antiquark pair increases for the interacting ensemble. As a result, the confinement temperature decreases.' author: - \| Motahareh Kiamari$^{1}$, Sedigheh Deldar$^{1}$\ $^1$Department of Physics, University of Tehran,\ P.O. Box 14395/547, Tehran 1439955961, Iran. title: 'Interacting dyon ensemble and confinement by particle mesh Ewald’s method' --- Introduction ============ Calorons - as one of the candidates of QCD vacuum structure - were first introduced in a set of papers by Diakonov and Petrov [@1][@2][@3] to describe quark confinement. They studied the noninteracting ensemble of calorons to calculate the Polyakov loop correlator and obtained the free energy of static quark-antiquark pairs. They also found the temperature of the confinement-deconfinement phase transition by considering the Polyakov loop as an order parameter. However, since the interaction of calorons inside the region of their cores are complex, the interacting ensemble of calorons remained unstudied. This is basically because the core structure of calorons are nonlinear and they are neutral objects without any interactions outside the core. On the other hand, Bruckmann et al. [@4] showed that the metric introduced by Diakonov and Petrov [@1] for the noninteracting calorons, is only positive definite for dyons of different charges or for dyons of the same charge at separations larger than the $ \frac{2}{\pi T} $ in the SU(2) gauge group. They [@5] used a numerical method called Ewald’s method [@6] to solve the problem. For interacting ensembles, they suggested the particle mesh Ewald’s (PME) method which is more efficient from the point of view of running time cost. The main idea of Ewald’s method is to split the interaction into a converging short-range term and a smooth long-range term which is convergent in the Fourier space. Applying this method, Bruckmann et al. [@5] obtained the free energy of static quark-antiquark pairs versus their separations by calculating the Polyakov loop correlator of a noninteracting dyon gas. They also showed that the finite-size volume effects were under control in their calculations. In Ref. [@12], we applied the particle mesh Ewald’s method to noninteracting ensemble of dyons and showed that this method also works very well for calculating the free energy between a static quark-antiquark pair. We got a linear rising potential with a well-behaved string tension decreases with increasing temperature. In this paper we apply the PME method to an interacting dyon ensemble and compare the results with the noninteracting case. For a noninteracting dyon ensemble, the Polyakov loop correlator is calculated by the temporal gauge field of each dyon whereas the dyons themselves do not interact with each other. For the interacting case, we consider some Coulomb-like interaction between dyons. Our results show that the free energy of the static quark-antiquark pair is also linear for the interacting dyon ensemble, as expected. Comparing the results obtained from the noninteracting and interacting ensembles, we show that by adding the dyonic interactions, the string tension between the quark-antiquark pair increases and therefore the confinement temperature decreases. The paper is organized as follows. In Sec. \[sec:dyon\], some features of dyons are introduced and the Polyakov loop correlator and the action are derived. Ewald’s method and the particle mesh Ewald’s method are described briefly in Sec. \[sec:em\]. The setup of our simulations and the numerical results are presented in Sec. \[sec:results\]. The conclusion and discussions are given in Sec. \[sec:Conclusions\]. Interacting Dyon ensemble for SU(2) Yang-Mills theory {#sec:dyon} ===================================================== KvBLL calorons - found by Kraan and van Baal [@7], as well as Lee and Lu [@8] - are the periodic solutions of the finite-temperature Yang-Mills theory. These neutral objects consist of $N$ dyons in the $SU(N)$ group and have non-Abelian and nonlinear cores which makes it difficult to study their interactions. Dyons are basically non-Abelian objects, but in the far-field limit, they can be considered as $U(1)$ objects with Coulombic electric and magnetic fields. Using the Abelian temporal gauge field in the third direction of color space in $SU(2)$, $$A_{4}\rightarrow 2\pi\omega T \sigma_{3}, \label{A4}$$ $$\pm B=E\rightarrow \frac{q}{r^{2}} \sigma_{3}, \label{ebfield}$$ where T is the temperature, $\sigma _{3}$ is the third Pauli matrix and $\omega $ is the holonomy which specifies the confinement and deconfinement phases. The Polyakov loop, $$P(\textbf{r})=\frac{1}{2}Tr\left(\exp\left(i\int_{0}^{1/T} dx_{4}A_{4}\left(x_{4},\textbf{r}\right) \right) \right) \label{polyakovloop}$$ is related to the holonomy in the far-field limit, $$P(\textbf{r})\rightarrow \frac{1}{2}Tr\left(\exp\left(2\pi i\omega\sigma _{3}\right)\right)=\cos\left(2\pi\omega\right).$$ The free energy versus Polyakov loop is defined as $$\begin{aligned} F_{\bar{Q}Q}(d)=-T\ln \left\langle P(\textbf{r})P^{\dag} (\textbf{r}')\right\rangle , d\equiv \lvert \textbf{r}-\textbf{r}' \rvert , \label{free-energy}\end{aligned}$$ where d is the distance between a quark located in $\textbf{r}$ and an antiquark in $\textbf{r}'$. Hence, for maximally nontrivial holonomy, where $ \omega=\frac{1}{4} $, the system is in the confinement phase and $ P(\textbf{r})\rightarrow 0 $. For trivial holonomy, the system is in the deconfinement phase and $ P(\textbf{r})\rightarrow \pm 1 $. To find the Polyakov loop of Eq. (\[polyakovloop\]), $A_{4}$ of the dyon ensemble has to be found. The long-range gauge fields of a dyon are Coulombic and Abelian in the third direction of color space, $$\begin{aligned} a_{4}\left(\textbf{r};q\right)=\frac{q}{r}, a_{1}\left(\textbf{r};q\right)=-\frac{qy}{r\left(r-z\right)}, a_{2}\left(\textbf{r};q\right)=+\frac{qx}{r\left(r-z\right)}, a_{3}\left(\textbf{r};q\right)=0. \label{dyonpotential}\end{aligned}$$ There are two self-dual dyons in $SU(2)$, with electric and magnetic charges equal to $(+1,+1)$ and $(-1,-1)$ corresponding to the plus sign in Eq. (\[ebfield\]) and two anti-self-dual dyons with charges $(+1,-1)$ and $(-1,+1)$ corresponding to the minus sign in Eq. (\[ebfield\]). Since we study self-dual dyons and their electric and magnetic charges are equal, these dyons can be considered as objects with one charge $q=\pm 1$. Using $a_{4}$ of Eq. (\[dyonpotential\]), the Polyakov loop of the dyon ensemble in confinement phase is obtained from Eq. (\[polyakovloop\]), $$\begin{aligned} P(\textbf{r})=\cos\left(2\pi\omega+\frac{1}{2T}\Phi(\textbf{r})\right), P(\textbf{r})\| _{\omega=1/4 } =-\sin\left(\frac{1}{2T}\Phi(\textbf{r})\right), \end{aligned}$$ $$\Phi(\textbf{r})\equiv \sum_{i=1}^{2K}\frac{q_{i}}{\lvert \textbf{r}-\textbf{r}_{i}\rvert }. \label{phi}$$ Keeping in mind that the original system we study is the ensemble of $K$ calorons, we consider the neutral system of $2K$ dyons: $K$ dyons with charge $q=+1$ and $K$ dyons with charge $q=-1$. To obtain the free energy of Eq. (\[free-energy\]), the Polyakov loop correlator should be computed. The expectation value of an observable O, $$\langle O \rangle=\frac{1}{Z} \int \left( \prod_{k=1}^{n_{D}} d^{3}r_{k} \right) O\left( \left\lbrace \textbf{r}_{k}\right\rbrace \right) \exp \left[S\left( \left\lbrace \textbf{r}_{k}\right\rbrace \right)\right] \label{evofO}$$ where Z is the partition function, $n_{D}$ is the number of dyons in the system, $$Z=\int \left( \prod_{k=1}^{n_{D}} d^{3}r_{k} \right) \exp \left[S\left( \left\lbrace \textbf{r}_{k}\right\rbrace \right)\right]. \label{partitionfun}$$ and S is the effective action of the ensemble. For noninteracting dyon gas the effective action is constant for all simulations. For the interacting ensemble, the integration measure should be rewritten as $$\left(\prod _{k=1}^{n_{D}}d^{3}r_{k}\right)\det(G), \label{measure}$$ where G is the moduli space metric. This metric is exactly known for two dyons with different charges or a caloron [@7], but for two dyons with the same charge the metric is approximate [@1]. Thus, the moduli-space metric for the two-body interaction is $$G_{(i,j)}= \begin{pmatrix} 2\pi -\frac{2q_{i}q_{j}}{T{\lvert \textbf{r}_{i}-\textbf{r}_{j}\rvert }}& \frac{2q_{i}q_{j}}{T{\lvert \textbf{r}_{i}-\textbf{r}_{j}\rvert }} \\ \frac{2q_{i}q_{j}}{T{\lvert \textbf{r}_{i}-\textbf{r}_{j}\rvert }} & 2\pi -\frac{2q_{i}q_{j}}{T{\lvert \textbf{r}_{i}-\textbf{r}_{j}\rvert }}, \end{pmatrix} \label{metric}$$ with the eigenvalues, $$\begin{aligned} \lambda _{1}=2\pi, \lambda _{2}=2\pi -\frac{4q_{i}q_{j}}{T{\lvert \textbf{r}_{i}-\textbf{r}_{j}\rvert }}. \end{aligned}$$ To have a positive-definite metric, the distance between dyons of the same charge should not be less than $ \frac{2q_{i}q_{j}}{\pi T} $. The determinant of the moduli-space metric is $$\resizebox{0.99\textwidth}{!}{$\prod _{(i,j)} det(G_{(i,j)}) = \prod _{(i,j)} 4\pi^{2} \left( 1- \frac{2q_{i}q_{j}}{\pi T \lvert \textbf{r}_{i}-\textbf{r}_{j}\rvert} \right) = (4\pi^{2})^{n_{D}^{2}} \exp \left[ \sum _{(i,j)} \ln \left( 1- \frac{2q_{i}q_{j}}{\pi T \lvert \textbf{r}_{i}-\textbf{r}_{j}\rvert} \right) \right]. $}$$ Now one can rewrite the expectation value (\[evofO\]) and the partition function (\[partitionfun\]), $$\langle O \rangle=\frac{1}{Z} \int \left( \prod_{k=1}^{n_{D}} d^{3}r_{k} \right) O\left( \left\lbrace \textbf{r}_{k}\right\rbrace \right) \exp \left[S_{eff}\left( \left\lbrace \textbf{r}_{k}\right\rbrace \right)\right]$$ $$Z=\int \left( \prod_{k=1}^{n_{D}} d^{3}r_{k} \right) \exp \left[S_{eff}\left( \left\lbrace \textbf{r}_{k}\right\rbrace \right)\right].$$ where the effective action is, $$S_{eff}\left( \left\lbrace \textbf{r}_{k}\right\rbrace \right)=\frac{1}{2}\sum _{i=1}^{n_{D}}\sum _{j=1,j\neq i}^{n_{D}} \ln \left(1 -\frac{2q_{i}q_{j}}{\pi T{\lvert \textbf{r}_{i}-\textbf{r}_{j}\rvert }} \right). \label{action}$$\ To include the contribution of anti-self-dual dyons, one should modify the metric of Eq. (\[metric\]) to a (44) matrix [@1]. The diagonal (22) blocks of the new metric describe the same-duality dyons, while the off-diagonal (22) blocks represent the interactions of different-duality dyons. Thus, we should calculate the determinant of this metric with the nonzero off-diagonal elements. All terms in the modified metric are Coulombic and we should apply all steps of Ewald’s method to the anti-self-dual dyons, as well. This modification makes the calculations very difficult and cumbersome. However, in Ref. [@1] Diakonov showed that adding anti-self-dual dyons only changes the string tension to $\sqrt{2}$ of its value when we do not use them and the physics of the quark-antiquark potential does not change. Therefore, we trust Diakonov’s calculations and study the ensemble of $K$ calorons as he did. Our main goal - which is to study the linearity of the free energy and to observe the increasing of the string tension due to the dyonic interactions - will not be affected. In the next section we calculate the Polyakov correlator with Ewald’s method using the partition function and the action we obtained in this section. Ewald’s method {#sec:em} ============== The first step is applying Ewald’s method is to mimic the space with a basic cell called a super cell, and copy it in all three directions and put the particles in the super cell. The copies contain the copies of the particles. This is how the periodic boundary condition is applied. Therefore we put $n_{D}$ dyons randomly in the super cell. The second and main step is to split the long-range term $\frac{1}{r}$ into an exponentially short-range part and a smooth long-range part, $$\Phi(\textbf{r}) =\Phi ^{\texttt{short}}(\textbf{r})+\Phi ^{\texttt{long}}(\textbf{r}),$$ $$\Phi ^{S}(\textbf{r})\equiv \sum_{\textbf{n}\in \mathbb{Z} ^{3} } \sum_{j=1}^{n_{D}}\left( 1-\texttt{erf}\left( \frac{\lvert \textbf{r}-\textbf{r}_{j} -\textbf{n}L\rvert}{\sqrt{2}\lambda}\right) \right) \frac{q_{j}}{\lvert \textbf{r}-\textbf{r}_{j} -\textbf{n}L \rvert }, \label{short}$$ $$\Phi ^{L}(\textbf{r})\equiv \sum_{\textbf{n}\in \mathbb{Z} ^{3} } \sum_{j=1}^{n_{D}}\texttt{erf}\left( \frac{\lvert \textbf{r}-\textbf{r}_{j} -\textbf{n}L\rvert}{\sqrt{2}\lambda}\right) \frac{q_{j}}{\lvert \textbf{r}-\textbf{r}_{j} - \textbf{n}L\rvert }, \label{long}$$ where $ \lambda $ is an arbitrary parameter and `erf` is the error function. The vector n specifies the copies of the super cell and $L^{3}$ is the spatial volume of the super cell. $\Phi ^{S}$ is convergent for a finite cutoff but $\Phi ^{L}$ is a divergent smooth function. Thus, its Fourier transformation is convergent for finite cutoff, $$\begin{aligned} \Phi ^{L}(\textbf{r})=\frac{4\pi}{L^{3}}\sum_{\textbf{n}\in \mathbb{Z} ^{3} \setminus \vec{0}} \frac{e^{-\lambda ^{2}\textbf{k}(\textbf{n})^{2}/2}}{\textbf{k}(\textbf{n})^{2}}Re\left( \sum_{j=1}^{n_{D}} q_{j}e^{+i\textbf{k}(\textbf{n})\textbf{r}}e^{-i\textbf{k}\textbf{}(\textbf{n})\textbf{r}_{j}}\right) , \textbf{k}(\textbf{n})\equiv \frac{2\pi }{L}\textbf{n} . \label{ploop}\end{aligned}$$ where $$S(k)= \sum_{j=1}^{n_{D}} q_{j}e^{-i\textbf{k}\textbf{}(\textbf{n})\textbf{r}_{j}} \label{sfactor}$$ is called the structure factor. To reduce the operating costs, one needs $\Phi ^{S}(\textbf{r}) $ to be convergent in the original super cell. However, the arbitrary parameter $\lambda $ should be chosen such that $\Phi ^{S}(\textbf{r}) $ converges in a sphere with a maximum radius $r_{max}<L/2$ within an appropriate error [@9]. The center of the sphere is located at position r. Consider $ J(\textbf{r})$ which indicates all dyons and copies of them in this sphere, $$\Phi ^{S}(\textbf{r})\equiv \sum_{j\in J(\textbf{r})} \texttt{erfc}\left( \frac{\lvert \textbf{r}-\textbf{r}_{j} \rvert}{\sqrt{2}\lambda}\right) \frac{q_{j}}{\lvert \textbf{r}-\textbf{r}_{j} \rvert }, \label{short2}$$ where $ \texttt{erfc}(x) = 1-\texttt{erf}(x)$. As mentioned before, the long-range part $\Phi ^{L}(\textbf{r})$ converges in Fourier space. Hence, one can consider a sphere with radius $ n_{max}$, in which $\Phi ^{L}(\textbf{r})$ has to converge [@9]. For large dyon separations $r_{ij}$, the action in Eq. (\[action\]) can be expanded in powers of $ \frac{1}{r} $, $$S^{N}=\frac{1}{2} \sum_{i\neq j}\left(-\frac{2q_{i}q_{j}}{\pi Tr_{ij}} -\frac{2\left( q_{i}q_{j}\right) ^{2}}{\left( \pi Tr_{ij}\right) ^{2}} - \frac{8\left( q_{i}q_{j}\right) ^{3}}{3\left( \pi Tr_{ij}\right) ^{3}} + O\left( \frac{1}{r_{ij}^{4}}\right) \right), \label{expansion}$$ where $ r_{ij}=\lvert \textbf{r}_{i}-\textbf{r}_{j} \rvert $ and the superscript $N$ denotes the nonperiodic summation of the action. The action of Eq. (\[expansion\]) has the $\frac{1}{r^{p}}$, $ p\in \mathbb{R}$ terms, so to apply Ewald’s method to calculate these terms we should generalize the above procedure to the $\frac{1}{r^{p}}$ terms. With the definition of the Euler gamma function and the Fourier integral expansion of the three-dimensional Gaussian distribution, one can obtain the $ \frac{1}{r^{p}} $ term, $$\frac{1}{r^{p}}=\frac{\pi ^{3/2}}{\left(\sqrt{2}\lambda\right)^{p-3}} \int d^{3}\textbf{u}f_{p}\left(\sqrt{2}\lambda\pi \lvert \textbf{u}\rvert \right) \exp\left(-2i\pi\textbf{u}.\textbf{r}\right)+\frac{g_{p}\left(r/\sqrt{2}\lambda\right)}{r^{p}}, \label{rinv}$$ where $$g_{p}(x)=\frac{2}{\Gamma\left(p/2\right)}\int_{x}^{\infty } s^{p-1}\exp (-s^{2})ds \label{gp}$$ $$f_{p}(x)=\frac{2x^{p-3}}{\Gamma\left(p/2\right)}\int_{x}^{\infty } s^{2-p}\exp (-s^{2})ds. \label{fp}$$ The first and the second terms of Eq. (\[rinv\]) express the long-range part and the short-range part, respectively. This is because $ \lim _{x\rightarrow \infty} g_{p}(x) = 0 $ while $ \lim _{x\rightarrow \infty} f_{p}(x) \neq 0 $. Using Eq. (\[rinv\]) for each term of Eq. (\[expansion\]) and using periodic boundary conditions, one can split the terms of the action into the short-range term, long-range term, and self-energy term, $$S_{p}^{P}=\sum_{l=1}^{p}\left(S^{S}_{(l)}+S^{L}_{(l)}-S^{self}_{(l)}\right), \label{periodicS}$$ where the superscript P denotes the periodic summation of the action that consists of the copies of dyons in copies of the super cell. We should modify the formula in Ref. [@9] since the charges of the dyons in that reference are $\pm 1$ and therefore the multiplication of charges in the numerators of equations like (\[expansion\]) is equal to 1 for the even power of the charges. As a result, the only odd power of the charges is 1. While in our case, we are dealing with the interpolated charges with different values which depend on the positions of the randomly located dyons for each configuration. The interpolated charges are introduced in the next section. We should also add the self-energy part to the action. This is because the self-energy is a function of the power of the charges \[Eq. (\[self-s\])\]. These terms have different and important values for our dyons, while for dyons with $\pm 1$ charges the self-energy terms are constant for the simulations with a fixed number of dyons and thus they do not affect the correlation function of Eq. (\[evofO\]): $$S^{S}_{(l)} = c(l)\frac{1}{2} \sum_{\textbf{n}\in \mathbb{Z} ^{3} }\sum_{i\neq j} \frac{q_{i}^{l}q_{j}^{l}}{\lvert \textbf{r}_{i}-\textbf{r}_{j} -\textbf{n}L\rvert^{l}} g_{l} \left( \frac{\lvert \textbf{r}_{i}-\textbf{r}_{j} -\textbf{n}L\rvert}{\sqrt{2}\lambda } \right), \label{SS}$$ and $c(l)$ is the coefficient of the $l$th term in Eq. (\[expansion\]), $$S^{L}_{(l)} = c(l) \frac{\pi ^{3/2}}{2V (\sqrt{2}\lambda )^{l-3}} \sum_{\textbf{k}_{sym}} f_{l} \left( \frac{\lambda k}{\sqrt{2}} \right) \left( 2\lvert S(\textbf{k},l)\rvert^{2} \right). \label{LS}$$ $ S(\textbf{k},l)=\sum_{i=1}^{n_D}q^{l}_{i}e^{-i\textbf{k}.\textbf{r}_{i}} $ and k is symmetric with respect to $\textbf{k}=0$, and the summation on $ \textbf{n}$ is done by the term $ \exp\left(-2i\pi\textbf{u}.\textbf{n}L \right) $ of Eq. (\[rinv\]), $$\sum _{\textbf{n}} \exp(-2\pi i \textbf{u}\textbf{n}L ) = \frac{1}{V} \sum _{m}^{\infty} \delta \left( \textbf{u} - \frac{\textbf{m}}{L} \right),$$ since u is the reciprocal vector, and the integral on u in Eq. (\[rinv\]) changes all u to $ \frac{m}{L}$, where $ \textbf{k} = 2\pi \frac{\textbf{m}}{L}$. The self-energy part of the short-range term can be canceled by omitting the $ i=j $ term, but the self-energy part of the long-range term should be separated. This term is the long-range part of the energy when $ \textbf{r}_{j}-\textbf{r}_{i}\rightarrow 0 $. In general, this term can be obtained by subtracting the short-range part in Eq. (\[gp\]) from the total term $ \frac{1}{r^{p}} $, $$\lim_{r\rightarrow 0} \left( \frac{1}{r^{p}}-\frac{g_{p}\left( r/\sqrt{2}\lambda\right) }{r^{p}}\right) = \frac{2\left( \sqrt{2}\lambda\right)^{p} }{p \Gamma \left( p/2\right) }, \label{self}$$ which gives $$S^{self}_{(l)} = \frac{2(1/\sqrt{2}\lambda)^{p}}{p\Gamma (p/2)} c(l) \sum_{i=1}^{n_{D}} q_{i}^{l}. \label{self-s}$$ Now, for $l=1,2,3$, $$S^{S}_{(1)}=-\frac{1}{\pi}\sum_{i=1}^{n_{D}} \sum_{j\in J(\textbf{r}_{i})} \frac{q_{i}q_{j}}{Tr_{ij}} \texttt{erfc}\left(\frac{r_{ij}}{\sqrt{2}\lambda } \right), \label{S1S}$$ $$S^{S}_{(2)}=-\frac{1}{\pi^{2}}\sum_{i=1}^{n_{D}} \sum_{j\in J(\textbf{r}_{i})} \frac{q^{2}_{i}q^{2}_{j} }{T^{2}r_{ij}^{2}}\exp \left(-\frac{r_{ij}^{2}}{2\lambda ^{2}}\right), \label{S2S}$$ $$S^{S}_{(3)}=-\frac{4}{3\pi^{3}}\sum_{i=1}^{n_{D}} \sum_{j\in J(\textbf{r}_{i})} q^{3}_{i}q^{3}_{j}\left( \frac{\texttt{erfc}\left( \frac{r_{ij}}{\sqrt{2}\lambda }\right)}{T^{3}r_{ij}^{3}}+ \sqrt{\frac{2}{\pi }} \frac{\exp \left(-\frac{r_{ij}^{2}}{2\lambda ^{2}}\right)}{T^{3}\lambda r_{ij}^{2}}\right) , \label{S3S}$$ $$S^{L}_{(1)}=-\frac{8}{TV}\sum_{\textbf{k}sym} \lvert S(\textbf{k},1)\rvert^{2} \frac{\exp \left( -\frac{\lambda ^{2}\textbf{k}^{2}}{2} \right)}{\textbf{k}^{2}}, \label{S1L}$$ $$S^{L}_{(2)}=-\frac{4}{T^{2}V}\sum_{\textbf{k}sym} \lvert S(\textbf{k},2)\rvert^{2} \frac{\texttt{erfc} \left( \frac{\lambda k}{\sqrt{2}} \right)}{k},$$ $$S^{L}_{(3)}=-\frac{16}{3\pi ^{2}T^{3}V}\sum_{\textbf{k}sym} \lvert S(\textbf{k},3)\rvert^{2} \left(-\texttt{Ei}\left(-\frac{k^{2}\lambda ^{2} }{2}\right)\right), \label{S3L}$$ where Ei is the exponential integral $ \texttt{Ei}(x)=-\int _{-x}^{\infty }\frac{e^{-t}}{t}dt $, and $$S^{self}_{(1)}=\frac{-2}{\sqrt{2}\lambda \pi ^{3/2}}\sum^{n_{D}} _{i=1} q^{2}_{i}, \label{S1self}$$ $$S^{self}_{(2)}=\frac{-1}{2\lambda^{2} \pi ^{2}}\sum^{n_{D}} _{i=1} q^{4}_{i}, \label{S2self}$$ $$S^{self}_{(3)}=\frac{-8}{9\sqrt{2}\lambda^{3} \pi ^{7/2}}\sum^{n_{D}} _{i=1} q^{6}_{i}. \label{S3self}$$ As mentioned before, the expansion in Eq. (\[expansion\]) is only appropriate for large dyon separations, and thus for small dyon separations a correction term should be added to the periodic action in Eq. (\[periodicS\]), $$S=S_{p}^{P}-S_{p}^{Corr}. \label{Self}$$ To have a continuous action on the boundary of small and large dyon separations, $ r_{Corr} $, we should subtract the expansion of the action in Eq. (\[expansion\]) from the periodic result in Eq. (\[periodicS\]) and add S from Eq. (\[action\]), $$S_{p}^{Corr}=\sum_{j=1}^{n_D}\sum_{i\in I(\textbf{r}_{j})} \left[\sum_{l=1}^{p} S_{(l)}^{N}(q_{i}q_{j},r_{ij})-\frac{1}{2}\ln \left( 1-\frac{2q_{i}q_{j}}{\pi Tr_{ij}}\right) \right], \label{correct}$$ because for small $ r_{ij} $, $ S^{P} $ and $ S^{N} $ are approximately equal and for large $ r_{ij} $, $ S^{N} $ and the action in Eq. (\[action\]) are equal [@9]. Here, $ I(\textbf{r}_{j}) $ is the set of dyons and their copies and their separations from the ith dyon are less than $ r_{Corr} $, and $ S_{(l)}^{N} $ stands for the lth-order term of $ S^{N} $ in Eq. (\[expansion\]). By expanding Eq. (\[correct\]), $ S_{p}^{Corr} $ for different values of p is found, $$\begin{split} S^{corr}_{1} =& \frac{1}{2}\sum_{j=1}^{n_D}\sum_{i\in I(\textbf{r}_{j})}\left[ -\frac{2q\left(\textbf{r}_{i} \right)q\left(\textbf{r}_{j} \right) }{\pi Tr_{ij}}-\left(-\frac{2q\left(\textbf{r}_{i} \right)q\left(\textbf{r}_{j} \right) }{\pi Tr_{ij}} -\frac{2q^{2}\left(\textbf{r}_{i} \right)q^{2}\left(\textbf{r}_{j} \right)}{\pi^{2} T^{2}r^{2}_{ij}} + O \left( \frac{1}{r_{ij}^{3}} \right) \right)\right] {}\\ =& \frac{1}{2}\sum_{j=1}^{n_D}\sum_{i\in I(\textbf{r}_{j})}\frac{2q^{2}\left(\textbf{r}_{i} \right)q^{2}\left(\textbf{r}_{j} \right)}{\pi^{2} T^{2}r^{2}_{ij}} + O \left( \frac{1}{r_{ij}^{3}} \right). \end{split} \label{S1}$$ Performing the same procedure, $$S^{corr}_{2} = \frac{1}{2}\sum_{j=1}^{n_D}\sum_{i\in I(\textbf{r}_{j})}\frac{8q^{3}\left(\textbf{r}_{i} \right)q^{3}\left(\textbf{r}_{j} \right)}{3\pi^{3} T^{3}r^{3}_{ij}} + O \left( \frac{1}{r_{ij}^{4}} \right), \label{S2}$$ $$S^{corr}_{3} = \frac{1}{2}\sum_{j=1}^{n_D}\sum_{i\in I(\textbf{r}_{j})}\frac{4q^{4}\left(\textbf{r}_{i} \right)q^{4}\left(\textbf{r}_{j} \right)}{\pi^{4} T^{4}r^{4}_{ij}} + O \left( \frac{1}{r_{ij}^{5}} \right). \label{S3}$$ Since we approximate the action terms of Eq. (\[expansion\]) up to order $ O(r^{3}) $, the correction terms up to $ O(r^{4}) $ are good enough. To summarize this section, we have obtained the following action for an interacting dyonic system: $$S = \sum _{l=1}^{p} \left(S^{S}_{(l)}+S^{L}_{(l)}-S^{self}_{(l)}\right)-S_{p}^{Corr}, \label{finalS}$$ where $S^{S}_{(l)}$, $S^{L}_{(l)}$, and $S^{self}_{(l)}$ were introduced in Eqs. (\[S1S\]) (\[S3self\]), respectively. $S_{p}^{Corr}$ in Eq. (\[S3\]) is added to the action which represents a correction term corresponding to the small dyon separations. We calculated the action for $p=3$ in the above Eq. (\[finalS\]) and we have discussed that it is a good approximation. Particle mesh Ewald’s method {#subsec:pme} ---------------------------- The main idea of the particle mesh Ewald’s method [@10] is to grid the super cell in reciprocal space and interpolate the charge of each particle to the nearest neighboring mesh points. This method was first introduced by Hockney and Eastwood [@11] within a computer simulation and is more efficient for interacting dyon gas. Consider $n_{D}$ dyons distributed randomly in a super cell at positions $ \textbf{r}_{1},\textbf{r}_{2},...,\textbf{r}_{n_{D}} $. Each dyon at position $\textbf{r}_{i}$ in real space has fractional coordinates $s_{\alpha i}=\textbf{a}_{\alpha }^{}.\textbf{r}_{i}$ in reciprocal space. Then, we grid the super cell by the points $K_{i}$ for each direction. The new scaled fractional coordinates $u_{1}$,$u_{2}$,$u_{3}$ are defined as $ u_{\alpha }=K_{\alpha }\textbf{a}_{\alpha }^{}.\textbf{r} $, $ \alpha=1,2,3 $, and $ 0\leq u_{\alpha } < K_{\alpha } $ due to the periodic boundary condition. Then, the terms of the structure factor of Eq. (\[sfactor\]) can be rewritten with these new coordinates. m is the reciprocal vector, $ \textbf{m}=m_{1}\textbf{a}_{1}^{}+m_{2}\textbf{a}_{2}^{}+m_{3}\textbf{a}_{3}^{*} $, $$\exp \left( -i\textbf{m}.\textbf{r}\right)=\exp \left(-i\frac{m_{1}u_{1}}{K_{1}} \right).\exp \left(-i\frac{m_{2}u_{2}}{K_{2}} \right).\exp \left(-i\frac{m_{3}u_{3}}{K_{3}} \right). \label{sfterms}$$ In Ref. [@10] both piecewise Lagrangian and cardinal B-Spline interpolations were introduced, but the latter was applied to calculate the energy of the molecular system. This is because the coefficients of this interpolation are $n-2$ times continuously differentiable. $n$ is the number of neighbor mesh points used for interpolation, and the authors needed differentiability to calculate the forces between molecules, while the coefficients of piecewise Lagrangian interpolation are only piecewise differentiable. Since we do not need to calculate the force and therefore differentiability, we apply piecewise Lagrangian interpolation. By this interpolation, these exponentials can be approximated for $p> 1$, $$\exp \left(-i\frac{m_{\alpha}}{K_{\alpha}}u_{\alpha}\right) \approx \sum _{k=-\infty}^{\infty}W_{2p}(u_{\alpha}-k). \exp\left(-i\frac{m_{\alpha}}{K_{\alpha}}k\right), \label{exp}$$ where $W_{2p}(u^{'})=0$ for $\|u^{'}\| > p$ and for $-p\leq u^{'} \leq p$ the coefficient $W_{2p}(u^{'})$ is $$W_{2p}(u^{'}) = \frac{\prod _{j=-p,j\neq k^{'}}^{p-1} (u^{'}+j-k^{'})}{\prod _{j=-p,j\neq k^{'}}^{p-1} (j-k^{'})}, k^{'}\leq u^{'}\leq k^{'}+1, k^{'}=-p,-p+1,...,p-1. \label{W2p}$$ The subscript $2p$ is the order of interpolation and specifies the number of mesh points used to interpolate the $\exp(-i mu/K)$ in each direction. These points are $[u]-p+1$, $[u]-p+2$ , ..., $[u]+p$, which are the $2p$ nearest neighbor mesh points to the point $ u $. Using Eq. (\[exp\]), one can approximate the structure factor in Eq. (\[sfactor\]), $$\begin{split} S(\textbf{m})\approx & \widetilde{S}(\textbf{m})=\sum _{i=1}^{n_{D}}q_{i} \sum _{k_{1}=-\infty}^{\infty}\sum _{k_{2}=-\infty}^{\infty}\sum _{k_{3}=-\infty}^{\infty} W_{2p}(u_{1i}-k_{1})W_{2p}(u_{2i}-k_{2}) {}\\ &.W_{2p}(u_{3i}-k_{3})\exp \left(-i\frac{m_{1}}{K_{1}}k_{1}\right)\exp \left(-i\frac{m_{2}}{K_{2}}k_{2}\right)\exp \left(-i\frac{m_{3}}{K_{3}}k_{3}\right). \end{split} \label{stilda}$$ Comparing the new structure factor of Eq. (\[stilda\]) with the structure factor of Eq. (\[sfactor\]), the new charges assigned to the mesh points are $$\begin{split} Q(k_{1},k_{2},k_{3})= \sum_{i=1}^{n_D}\sum_{n_{1},n_{2}n_{3}}q_{i} & W_{2p}(u_{1i}-k_{1}-n_{1}K_{1})W_{2p}(u_{2i}-k_{2}-n_{2}K_{2}) {}\\ & .W_{2p}(u_{3i}-k_{3}-n_{3}K_{3}). \end{split} \label{Q}$$ The new structure factor is $$S(\textbf{m})\approx \sum_{k_{1}=0 }^{K_{1}-1 }\sum_{k_{2}=0 }^{K_{2}-1 }\sum_{k_{3}=0 }^{K_{3}-1 } Q(k_{1},k_{2},k_{3}) \exp \left[-i\left(\frac{m_{1}k_{1}}{K_{1}}+\frac{m_{2}k_{2}}{K_{2}}+\frac{m_{3}k_{3}}{K_{3}} \right) \right]. \label{newstructurefac}$$ The structure factor of Eq. (\[newstructurefac\]) describes the new system with new $K_{1}K_{2}K_{3}$ charges $Q(k_{1},k_{2},k_{3})$ introduced in Eq. (\[Q\]) which are located on mesh points $(k_{1},k_{2},k_{3})$ on a three-dimensional ($3D$) lattice. We use this system instead of the system with $n_{D}$ dyons located randomly on $r_{i}$. Now, we apply the simple Ewald’s method to this new system. The advantage of this new system is the constant number of charges, $K_{1}K_{2}K_{3}$, which are the same in all simulations, in contrast to the number of dyons $n_{D}$ of the original system which are different for each individual simulation. Simulation results {#sec:results} ================== As mentioned in the Introduction, studying quark confinement with dyons as the constituents of the QCD vacuum is the main purpose of this article. Using the Polyakov loop correlator of Sec. \[sec:dyon\], the free energy of a static quark-antiquark pair is calculated for both non interacting and interacting dyon ensembles. Ewald’s method (introduced in Sec. \[sec:em\]) is applied to the system of the charges obtained with the PME method in Sec. \[subsec:pme\], for dyons located randomly on 3D lattice. Before applying the particle mesh Ewald’s method, we need to put some dyons randomly in a super cell on the lattice. To make sure that dyons are sitting randomly in the super cell, we also use a Metropolis algorithm to make sure the system is in a stable energy. We do this procedure for each configuration before applying Ewald’s method and the dyonic interaction. $n_{D}$ dyons are assumed to be located randomly in a super cell in the following procedure:\ 1. Fill the super cell with $N$ dyons with random 3D coordinates.\ 2. Displace one dyon slightly.\ 3. Compute the change of the action due to this displacement, $\Delta S$.\ 4. If $\Delta S < 0 $, accept the new configuration.\ 5. If $\Delta S > 0 $, accept the new configuration with the conditional probability: pick a random number $0<x<1$; if $\exp(-\Delta S) > x $, accept the new configuration; if $\exp(-\Delta S) < x $, reject the new configuration.\ We should mention that in this procedure we calculate only the part of the action related to the dyon which is displaced. For each configuration, we perform steps 2 to 5 for all $N$ dyons. We interpolate the charges of these dyons to the 3D lattice with $K_{i}=16$ as described in Sec. \[subsec:pme\]. This interpolation leads the system to a new setup with charges located on the mesh points according to Eq. (\[Q\]). Since the structure factors of these old and new systems are approximately equal, the two systems are equivalent and we use the new system of interpolated charges instead of the original old system of dyons. We apply Ewald’s method to this new system to calculate both the short-range and long-range parts of the Polyakov loop and also the action introduced in Sec. \[sec:dyon\], while in Ref. [@10] the PME method was only applied to calculate the long-range part of the action. Using this method, we do not have to increase the number of mesh points even for a large number of dyons, since for any number of dyons we can interpolate them to a constant number of mesh points. This saves on operating costs, in contrast to the case where one puts dyons directly on a lattice and increases the lattice points as the number of dyons increases [@5]. We fix the dyon density $ \rho $ and temperature $ T $ to $ \rho /T^{3} = 1 $ which scales the separations by $ \rho ^{1/3} $ or $T$, as done in Ref. [@5]. Various lattice sizes, the number of configurations, the number of dyons and other parameters of our simulations are listed in Table \[tab:input\]. For both noninteracting and interacting ensembles, the simulations are done for maximally nontrivial holonomy corresponding to the confinement phase, as described in Sec. \[sec:dyon\]. Therefore, we expect that the potential grows linearly by increasing the quark-antiquark separation. As an example, Fig. \[fig:2030\] illustrates this linear dependence for $LT = 20$ and 30 for noninteracting and interacting simulations. $ n_{D} $ $ LT $ configurations ----------- -------- ---------------- -- 1000 10 1600 8000 20 800 27000 30 120 125000 50 60 : Number of dyon configurations, number of dyons, $n_D$, and $LT$ for each simulation. $L^3$ indicates the spatial volume of the super cell and $T$ is the temperature.[]{data-label="tab:input"} [.45]{} ![The linear dependence of the free energy on the quark-antiquark separation for noninteracting and interacting dyon ensembles for $LT=20$ and 30. $\rho /T^{3}$ is fixed to one. The free energy grows linearly as the quark-antiquark separation increases. We are very close to the deconfinement temperature, $T=312 $ MeV.[]{data-label="fig:2030"}](2020.eps "fig:"){width="1\linewidth" height="1\linewidth"} [.45]{} ![The linear dependence of the free energy on the quark-antiquark separation for noninteracting and interacting dyon ensembles for $LT=20$ and 30. $\rho /T^{3}$ is fixed to one. The free energy grows linearly as the quark-antiquark separation increases. We are very close to the deconfinement temperature, $T=312 $ MeV.[]{data-label="fig:2030"}](3030.eps "fig:"){width="1\linewidth" height="1\linewidth"} To be able to compare the results of different simulations, we scale the data by the ansatz $$\frac{\sigma }{T^{2}} = \frac{\sigma(T=0)}{T_{c}^{2}}\left(\frac{T_{c}}{T}\right)^{2}A\left(1-\frac{T}{T_{c}}\right)^{0.63}\left(1+B\left(1-\frac{T}{T_{c}}\right)^{1/2}\right), \label{scale}$$ where $B = 1 - 1/A$, $A=1.39$ [@5], and $\sigma (T=0)=\left(440 MeV \right)^{2}$ corresponding to $T_{c}=312 $ MeV. Here, $\sigma$ indicates the string tension between the static quark and antiquark, and $T_c$ is the critical temperature. $\frac{\sigma }{T^{2}}$ (obtained from the plots like Fig. \[fig:2030\]) is inserted into Eq. (\[scale\]) and the corresponding temperature is obtained. Then, using the information in Table \[tab:input\], the lattice spacings are found for each simulation. As represented in Table \[tab:alldata\], the temperatures of our simulations are very close to the deconfinement temperature, $T=312 $ MeV, for both noninteracting and interacting simulations. The spatial lattice spacings and string tensions for each simulation are listed in Table \[tab:alldata\].\ Since we use the interpolated original charges on the lattice, we should show that this approximation and the space discretization do not affect our results. In fact, we should show that the string tensions obtained from the lattices with different lattice spacings are equal at the same temperature. For both interacting and noninteracting ensembles, one can learn from Table \[tab:alldata\] that the string tensions of the lattices with the same temperature agree very well within the errors. For example, for a noninteracting ensemble, for $LT=20$ and $30$ for which the temperatures are almost equal, the string tensions agree within the errors. Thus, our lattice spacings are small enough to not encounter discretization error. ![The scaled results of a noninteracting dyonic ensemble for different volumes.[]{data-label="fig:sall"}](sall.eps){width=".55\linewidth"} Figures \[fig:sall\] and \[fig:isall\] illustrate the results of noninteracting and interacting simulations for different $LT$, after scaling. In general, as the temperature increases the string tension decreases, as one expects from the ansatz (\[scale\]). To get the interacting results, we add dyonic interactions to the lattice of the noninteracting ensemble for the same $LT$. Therefore, we can compare the noninteracting and interacting results for each $LT$. As indicated in Table \[tab:alldata\], by adding the Coulombic interaction to the dyon ensemble the confinement temperature decreases slightly. The string tension of the quark-antiquark pair increases for the interacting ensemble. This is a nice result. The interpretation is as follows: the interaction between dyons increases the free energy between the quark antiquark-pair, as the plots show. This means that the quark-antiquark pair system is more stable now and is further from the deconfinement phase compared with the noninteracting dyonic ensemble. In other words, it seems that the interaction between dyons increases the gluonic field strength compared with the noninteracting dyons. This explains the decrease in temperature for the same lattice when we just add the dyonic interaction to the noninteracting ensembles. Figure \[fig:bothdata\] shows the results of interacting and noninteracting ensembles in one plot. Since the free energy is scaled, the slopes of the same “$LT$” simulations which show the string tensions between the static quark and antiquark can be compared easily between the interacting and noninteracting dyonic ensembles. A quantitative comparison is shown in Table \[tab:alldata\]. Our simulation results are fitted to the plot of Eq. (\[scale\]) in Fig. \[fig:sigma\]. For all noninteracting diagrams the order of interpolation $2p$ \[in (\[exp\]) of Sec. \[subsec:pme\]\] is equal to 4. This means that the charge of each dyon is interpolated to the four nearest neighbor points of the dyon location. But it seems that the $2p = 4$ is not enough for interacting simulations because of correlations between the dyon charges. Hence, we use $2p = 8$ for interacting dyons. We tried $2p=6$ and $2p=8$ for the noninteracting case and $2p=6$ for the interacting case and the results did not changed. To show how good our choice $K_{i}=16$ is, we tried $K_{i}=8$ and $K_{i}=10$ for $LT=30$ for the noninteracting dyonic system. The errors on $\sigma/T^{2}$ are $21$% and $8$% for $K_{i}=8$ and $K_{i}=10$, respectively. Therefore, it seems that $K_{i}=16$ is a good choice. Increasing the parameter $K$ to the higher values does not give us a better estimation of the string tension, but the operating time increases drastically. By increasing the number of dyons, the effective charge becomes more efficient and a better result is expected. However, since we fix the parameter $\rho / T^{3} = 1$ in our simulations, the volume of the lattice would be increasing without increasing the number of lattice points, and therefore we get larger lattice spacings and larger errors. Therefore, there is a compromise between increasing the number of dyons and not getting a larger lattice spacing error. Table \[tab:alldata\] shows that we are on the safe side. As mentioned in Sec. \[sec:dyon\], adding antidyons changes the string tensions by a constant factor from physical results, $\sigma \rightarrow \sqrt{2}\sigma$ [@1]. This affects the value of the temperature, although the system remains close to the deconfinement phase. However, our main results - the linearity of the free energy and the increase of the string tension due to the interaction - do not change. Conclusion {#sec:Conclusions} ========== We have computed the free energy of a static quark-antiquark pair as a function of their separation by studying the Polyakov loop correlator for noninteracting and interacting dyon ensembles. We first applied the PME method to the dyons located randomly in different volumes to interpolate their charges on a 3D lattice with fixed dimensions, and then applied Ewald’s method to this new system. As one expects, the free energy grows linearly as the separation increases. However, the string tension between the static quark-antiquark pair increases for the interacting dyonic ensemble. It seems that the dyonic interaction increases the gluonic strength, as expected. Acknowledgement {#acknowledgement .unnumbered} =============== We are grateful to the research council of the University of Tehran for supporting this study. ![The same as Fig. \[fig:sall\] but for the interacting dyonic ensemble.[]{data-label="fig:isall"}](isall.eps){width=".55\linewidth"} ![The scaled results of noninteracting and interacting simulations for different volumes. Comparison between the same values of $LT$ shows that when using interacting dyons, the string tension of the quark-antiquark pair increases.[]{data-label="fig:bothdata"}](all.eps){width=".55\linewidth"} ![Our results fitted to the results of lattice gauge theory \[Eq. (\[scale\])\]. []{data-label="fig:sigma"}](sigma.eps){width=".55\linewidth"} [99]{}
--- abstract: 'Phase diagram of the spin-1 quantum Heisenberg model with both exchange as well as single-ion anisotropy is constructed within the framework of pair approximation formulated as a variational procedure based on the Gibbs-Bogoliubov inequality. In this form adapted variational approach is used to obtain the results equivalent with the Oguchi’s pair approximation. It is shown that the single-ion anisotropy induces a tricritical behaviour in the considered model system and a location of tricritical points is found in dependence on the exchange anisotropy strength.' author: - Jozef Strečka - Ján Dely - Lucia Čanová title: 'Phase diagram of the spin-1 XXZ Heisenberg ferromagnet with a single-ion anisotropy' --- Introduction {#intro} ============ Phase transitions and critical phenomena of quantum spin systems currently attract a great deal of interest [@sach99]. As usual, the quantum Heisenberg model is used as a basic generating model which should be appropriate for investigating quantum properties of insulating magnetic materials [@gat85; @car86; @kah93; @gat99]. However, a rigorous proof known as the Mermin-Wagner theorem [@mer66] prohibits a spontaneous long-range order for the isotropic spin-1/2 Heisenberg model on the one- and two-dimensional lattices and hence, the spontaneous ordering might in principle appear either if a three-dimensional magnetic structure is considered [@dys76] or a non-zero magnetic anisotropy is involved in the studied model Hamiltonian [@fro77]. On the other hand, it is currently well established that obvious quantum manifestations usually arise from a mutual combination of several factors, especially, when the low-dimensional magnetic structure is combined with as low coordination number as possible and low quantum spin number. Apparently, these opposite trends make hard to find a long-range ordered system that simultaneously exhibits evident quantum effects. Investigation of quantum spin systems, which can exhibit a non-trivial criticality, thus remains among the most challenging tasks in the statistical and solid-state physics. Over the last few decades, there has been increasing interest in the study of the effect of different anisotropies (single-ion, Dzyaloshinskii-Moriya, exchange) on the critical behaviour of the spin-1 quantum Heisenberg ferromagnet. The main interest to study this model system arises since Stanley and Kaplan [@sta66; @sta67] proved the existence of a phase transition in the two- and three-dimensional Heisenberg ferromagnets. In addition, the ferromagnetic quantum Heisenberg model with the spin-1 has relevant connection with several nickel-based coordination compounds, which provide excellent experimental realization of this model system [@def; @djm]. Up to now, the spin-1 quantum Heisenberg model has been explored within the standard mean-field approximation [@tag74; @buz88], random phase approximation [@mic77] or linked-cluster expansion [@pan93; @pan95]. By making use of the pair approximation [@ury80; @iwa97; @lu06; @sun06], several further studies have been concerned with the critical behaviour of the anisotropic spin-1 XXZ Heisenberg ferromagnet with bilinear and biquadratic interactions [@ury80; @iwa97], the isotropic spin-1 Heisenberg ferromagnet with the bilinear and biquadratic interactions and the single-ion anisotropy [@lu06], as well as, the anisotropic spin-1 XXZ Heisenberg ferromagnet with an antisymmetric Dzyaloshinskii-Moriya interaction [@sun06]. To the best of our knowledge, the critical properties of the spin-1 XXZ Heisenberg ferromagnet with the uniaxial single-ion anisotropy have not been dealt with in the literature yet. Therefore, the primary goal of present work is to examine this model system which represents another eligible candidate for displaying an interesting criticality affected by quantum fluctuations. The rest of this paper is organized as follows. In Section \[model\], we briefly describe the model system and basic steps of the variational procedure which gives results equivalent to the Oguchi’s pair approximation [@ogu55]. Section \[result\] deals with the most interesting numerical results obtained for the ground-state and finite-temperature phase diagrams. Magnetization dependences on the temperature, for several values of exchange and single-ion anisotropies, are also displayed in Section \[result\]. Finally, some concluding remarks are drawn in Section \[conclusion\]. Model and method {#model} ================ Let us consider the Hamiltonian of the spin-1 quantum Heisenberg model: $$\begin{aligned} {\cal H} = - J \sum_{(i,j)}^{Nq/2} [\Delta (S_i^x S_j^x + S_i^y S_j^y) + S_i^z S_j^z] - D \sum_{i=1}^{N} (S_i^z)^2 - H \sum_{i=1}^{N} S_i^z, \label{ham}\end{aligned}$$ where $S_i^{\alpha}$ ($\alpha=x,y,z$) denotes spatial components of the spin-1 operator at the lattice site $i$, the first summation runs over nearest-neighbour pairs on a lattice with a coordination number $q$ and the other two summations are carried out over all $N$ lattice sites. The first term in Hamiltonian (\[ham\]) labels the ferromagnetic XXZ Heisenberg exchange interaction with the coupling constant $J>0$, $\Delta$ is the exchange anisotropy in this interaction, the parameter $D$ stands for the uniaxial single-ion anisotropy and the last term incorporates the effect of external magnetic field $H$. The model system described by means of the Hamiltonian (\[ham\]) will be treated within the pair approximation formulated as a variational procedure based on the Gibbs-Bogoliubov inequality [@bog47; @fey55; @bog62; @fal70]: $$\begin{aligned} G \leq G_0 + \langle {\cal H} - {\cal H}_0 \rangle_0. \label{gbf}\end{aligned}$$ Above, $G$ is the Gibbs free energy of the system described by the Hamiltonian (\[ham\]), $G_0$ is the Gibbs free energy of a simplified model system given by a trial Hamiltonian ${\cal H}_0$, and $\langle \ldots \rangle_0$ indicates a canonical ensemble averaging performed within this simplified model system. Notice that the choice of the trial Hamiltonian ${\cal H}_0$ is arbitrary, however, its form directly determines an accuracy of the obtained results. If only single-site interaction terms are included in the trial Hamiltonian, i.e. single-spin cluster terms are used as the trial Hamiltonian, then, one obtains results equivalent to the mean-field approximation. Similarly, if a two-spin cluster Hamiltonian is chosen as the trial Hamiltonian, the obtained results will be equivalent to the Oguchi’s pair approximation, which is superior to the mean-field approach. In the present work, we shall employ the two-spin cluster approach for the considered model system in order to obtain results equivalent to the Oguchi’s pair approximation [@ogu55]. The two-spin cluster trial Hamiltonian can be written in this compact form: $$\begin{aligned} {\cal H}_0 &=& \sum_{k=1}^{N/2} {\cal H}_k, \label{trial1} \\ {\cal H}_k &=& - \lambda [\delta (S_{k1}^x S_{k2}^x + S_{k1}^y S_{k2}^y) + S_{k1}^z S_{k2}^z] \nonumber\\ &&- \eta [(S_{k1}^z)^2 + (S_{k2}^z)^2] - \gamma (S_{k1}^z + S_{k2}^z), \label{trial2}\end{aligned}$$ where the first summation is carried out over $N/2$ spin pairs and $\lambda$, $\delta$, $\eta$, and $\gamma$ denote variational parameters which have obvious physical meaning. It is noteworthy that an explicit expression of the variational parameters can be obtained by minimizing the right-hand-side of Eq. (\[gbf\]), i.e. by obtaining the best estimate of the true Gibbs free energy. Following the standard procedure one easily derives: $$\begin{aligned} \lambda = J, \quad \delta = \Delta, \quad \eta = D, \quad \gamma = (q - 1) J m_0 + H, \label{para}\end{aligned}$$ where $m_0 \equiv \langle S_i^z \rangle_0$ denotes the magnetization per one site of the set of independent spin-1 dimers described by means of the Hamiltonian ${\cal H}_0$. By substituting optimized values of the variational parameters (\[para\]) into the inequality (\[gbf\]) one consequently yields the best upper estimate of the true Gibbs free energy within the pair-approximation method: $$\begin{aligned} G = \frac{N}{2} G_k + \frac{N J}{2} (q-1) m_0^2. \label{gfe}\end{aligned}$$ Above, $G_k$ labels the Gibbs free energy of the spin-1 Heisenberg dimer given by the Hamiltonian (\[trial2\]). With the help of Eq. (\[gfe\]), one can straightforwardly verify that the magnetization of the original model directly equals to the magnetization of the corresponding dimer model, i.e. $m \equiv \langle S_i^z \rangle = \langle S_i^z \rangle_0 \equiv m_0$. Of course, similar relations can be established for another quantities, as well. To complete solution of the model under investigation, it is further necessary to calculate the Gibbs free energy, magnetization and other relevant quantities of the corresponding spin-1 dimer model given by the Hamiltonian (\[trial2\]). Fortunately, an explicit form of all relevant quantities (Gibbs free energy, magnetization, correlation functions, quadrupolar moment) can be found for this model system elsewhere [@str05]. Referring to these results, the solution of the considered model system is formally completed. For the sake of brevity, we just merely quote final expressions for the Gibbs free energy $G_k$ and the magnetization $m_0$, both entering into Eq. (\[gfe\]): $$\begin{aligned} G_k &=& - \beta^{-1} \ln Z_k, \label{fin1} \\ Z_k &=& 2 \exp[\beta (\lambda + 2 \eta)] \cosh(2 \beta \gamma) + 4 \exp(\beta \eta) \cosh(\beta \gamma) \cosh(\beta \lambda \delta) \nonumber\\ &+& \exp[\beta (2\eta - \lambda)] + 2 \exp[\beta (\eta- \lambda / 2)] \cosh(\beta W), \label{fin2} \\ m_0 &=& \frac{1}{Z_d} \{ 2 \exp[\beta(\lambda + 2 \eta)] \sinh(2 \beta \gamma) + 2 \exp(\beta \eta) \sinh(\beta \gamma) \cosh(\beta \lambda \delta) \}, \label{fin3} \end{aligned}$$ where $W = \sqrt{(\eta- \lambda/2)^2 + 2 (\lambda \delta)^2}$, $\beta = 1/(k_{\rm B} T)$, $k_{\rm B}$ is the Boltzmann’s constant, $T$ labels the absolute temperature, and the variational parameters $\lambda$, $\delta$, $\eta$, and $\gamma$ take their optimized values (\[para\]). It is quite evident that the magnetization $m_0$ must obey the self-consistent transcendental Eq. (\[fin3\]) (recall that it enters into the variational parameter $\gamma$ given by Eq. (\[para\])), which might possibly have more than one solution. Accordingly, the stable solution for the magnetization $m_0$ is the one that minimizes the overall Gibbs free energy (\[gfe\]). In an absence of the external magnetic field ($H=0$), the magnetization tends gradually to zero in the vicinity of a continuous (second-order) phase transition from the ordered phase ($m=m_0 \neq 0$) towards the disordered phase ($m=m_0 = 0$). According to this, the magnetization (\[fin3\]) close to the second-order phase transition can be expanded into the series: $$\begin{aligned} m = a m + b m^3 + c m^5 + \ldots.\end{aligned}$$ Notice that the coefficients $a$, $b$, and $c$ depend on the temperature and all parameters involved in the model Hamiltonian (\[ham\]). Then, the power expansion of the magnetization $m$ can be straightforwardly used to locate second-order transition lines and tricritical points by following the standard procedure described in several previous works [@ben85; @kan86; @tuc89; @cha92; @jia93]. The critical temperatures corresponding to the second-order transitions must obey the condition $a = 1$, $b < 0$, while the tricritical points can be located from the constraint $a = 1$, $b = 0$, and $c < 0$. Finally, the critical temperatures of discontinuous (first-order) transitions must be obtained from a comparison of Gibbs free energy of the lowest energy ordered phase with the Gibbs free energy of the disordered phase. Results and discussion {#result} ====================== Before proceeding to a discussion of the most interesting numerical results, let us firstly mention that some particular results for the considered model system have already been reported on by the present authors elsewhere [@del06]. Note that in the former preliminary report we have used an alternate approach based on the original Oguchi’s pair approximation to study a particular case with the coordination number $q=4$ corresponding to the square and diamond lattices. In the present article, we shall further focus our attention to other particular case with the coordination number $q=6$, which corresponds to the case of the triangular and simple-cubic lattices. A brief comparison with the results obtained previously will be made in conclusion. Now, let us take a closer look at the ground-state behaviour. A detailed analysis of our numerical results shows that the ground-state phase boundary between the ferromagnetically ordered and the disordered phases can be allocated with the aid of following condition: $$\begin{aligned} \frac{D_{\rm b}}{J} = - \frac{q}{2} + \frac{\Delta^2}{q+1}. \label{gs}\end{aligned}$$ It is quite obvious from the Eq. (\[gs\]) that the ground-state phase boundary between the ordered and disordered phases shifts to the more positive (weaker) single-ion anisotropies when the parameter $\Delta$ is raised from zero. As a matter of fact, the order-disorder transition moves towards the weaker single-ion anisotropies for any $\Delta \neq0 $ in comparison with the result $D/J = - q/2$ attained in the semi-classical Ising limit ($\Delta = 0$). This result is taken to mean that a destabilization of the ferromagnetic order originates from raising quantum fluctuations, which work in conjunction with the single-ion anisotropy in the view of destroying of the ferromagnetic long-range order at zero temperature. It is worthwhile to remark that an appearance of the planar (XY) long-range ordering cannot be definitely ruled out in the parameter space with predominant easy-plane interactions ($D<0$ and/or $\Delta>1$), where we have found the disordered phase only. It should be stressed, however, that the present form of two-spin cluster mean-field treatment cannot resolve a presence of the ferromagnetic long-range order inherent to XY-type models [@lie62; @dys78; @ken88; @kub88] unlike the conventional Ising-like ferromagnetic long-range order with only one non-zero component of the spontaneous magnetization. Next, let us turn our attention to the finite-temperature phase diagram, which is shown in Fig. 1 in the reduced units $d = D/J$ and $t = k_{\rm B}T/J$ for the simple-cubic (triangular) lattice and different values of the exchange anisotropy $\Delta$. In this figure, the solid and dashed lines represent second- and first-order phase transitions between the ferromagnetic and paramagnetic phases, respectively, while the black circles denote positions of the tricritical points. It is quite obvious from this figure that the considered model system exhibits the highest values of critical temperature ![The phase diagram of the spin-1 Heisenberg model for the simple-cubic (triangular) lattice and several values of the exchange anisotropy $\Delta$. The solid and dashed lines represent second- and first-order phase transitions, respectively. The black circles denote positions of the tricritical points.](fig1.eps){width="100mm"} in the Ising limit ($\Delta = 0$). The gradual increase of the exchange anisotropy $\Delta$ reduces the transition temperature as a result of raising quantum fluctuations. It is worthwhile to remark that all the lines of second-order phase transitions, for arbitrary but finite $\Delta$, have the same asymptotic behaviour in the limit $d \to \infty$. Actually, the critical temperature of the continuous transitions does not depend on the exchange anisotropy in this limiting case and it is equal to $t^* = 5.847$. Moreover, it should be also mentioned that our approach yields for the Ising case without the single-ion anisotropy ($\Delta = 0$, $d = 0$) the critical temperature $t_c = 3.922$, which is consistent with the result of other pair-approximation methods [@sun06] and is simultaneously superior to the result $t_c = 4.0$ obtained from the standard mean-field approximation [@fit92]. In addition, it can be clearly seen from Fig. 1 that the transition temperature of the continuous phase transition monotonically decreases by decreasing the single-ion anisotropy $d$ until the tricritical point (TCP) is reached. Further decrease of the anisotropy parameter $d$ changes the second-order phase transitions towards the first-order ones. It should be realized, nevertheless, that the first-order phase transitions occur merely in a narrow region of single-ion anisotropies close to the boundary value (\[gs\]) at which both completely ordered phases with $m = \pm 1$ have the identical energy (coexist together) with the disordered phase with $m=0$ and one asymptotically reaches the first-order phase transition between them in the zero-temperature limit. An origin of discontinuous phase transitions could be therefore related to the fact that the ordered and disordered phases have very close energies near the boundary single-ion anisotropy (\[gs\]) (the former ones are being slightly lower in energy) and the small temperature change might possibly induce a phase coexistence (energy equivalence) between them, what consequently leads to the discontinuous phase transition. In Fig. 2, we depict more clearly the position of TCPs in dependence on the single-ion and exchange anisotropies by the ![The phase diagram of the spin-1 Heisenberg model for $q = 6$ and $\Delta = 0.0, 1.0, 2.0, 3.0,$ and $4.0$. The solid and dashed lines represent second- and first-order phase transitions, respectively. The black circles denote positions of the tricritical points. The dot-and-dash line represents the location of tricritical points in dependence on the exchange anisotropy $\Delta$.](fig2.eps){width="100mm"} dot-and-dash line in order to clarify how the type of phase transition changes with the anisotropy parameters. As one can see from this figure, the $d$-coordinate of TCPs ($d_t$) shifts to more positive values upon strengthening of $\Delta$, while the $t$-coordinate of TCPs behaves as a non-monotonic function of the exchange anisotropy $\Delta$ with a minimum at $\Delta_{min} = 3.459$. To illustrate the effect of uniaxial single-ion anisotropy on the phase transitions, the thermal variation of the magnetization $m$ is shown in Fig. 3 for the case of isotropic spin-1 Heisenberg model ($\Delta = 1.0$) and several values of $d$. It can be clearly seen from this figure that the reduction of the single-ion anisotropy causes lowering of the critical temperature $t_c$. Furthermore, ![The temperature dependence of the magnetization $m$ for the isotropic spin-1 Heisenberg model ($\Delta$ = 1.0) on the simple-cubic (triangular) lattice, when the value of the single-ion anisotropy parameter $d$ changes. The dashed lines represent the discontinuities of the magnetization at the first-order phase transitions.](fig3.eps){width="100mm"} it is also evident that the magnetization varies smoothly to zero for $d = 0.0$, $-2.0$, and $-2.6$ until the temperature reaches its critical value. This behaviour of magnetization, which is typical for the second-order (continuous) phase transitions, persists until $d > d_t$ ($d_t = -2.656$ for $\Delta = 1.0$ and $q=6$). On the other hand, the magnetization jumps discontinuously to zero for $d<d_t$ (e.g. see the curves for $d=-2.7$ and $-2.8$), what is characteristic feature of the first-order (discontinuous) phase transitions. As one can see, this discontinuity in the magnetization increases rather abruptly as the single-ion anisotropy moves to more negative values with respect to the $d_t$ value. Finally, it should be pointed out that the similar variations of magnetization curves occur for any value of the exchange anisotropy $\Delta$. Conclusion ========== In the present paper, the phase diagram of the anisotropic spin-1 XXZ Heisenberg model with the uniaxial single-ion anisotropy is examined within the variational procedure based on the Gibbs-Bogoliubov inequality, which gives results equivalent to the Oguchi’s pair approximation [@ogu55]. A comparison between the results obtained in the present study and those attained within the standard Oguchi approximation actually implies an equivalence between both the methods. The most important benefit of using the variational approach based on the Gibbs-Bogoliubov inequality is that in this way adapted method enables obtaining of all thermodynamic quantities in a self-consistent manner and moreover, it is also well suited to discern the continuous phase transitions from the discontinuous ones by distinguishing of the stable, metastable and unstable solutions inherent to the approximation used. In the spirit of the applied pair-approximation method we have demonstrated that the single-ion anisotropy as well as the exchange anisotropy have a significant influence on the critical behaviour and both these anisotropy parameters can cause a tricritical phenomenon, i.e. the change of the continuous phase transition to the discontinuous one. Our results can serve in evidence that the tricritical phenomenon may occur in the investigated model system if at least one of the anisotropy parameters provides a sufficiently strong source of the easy-plane anisotropy. Note furthermore that the obtained results are rather general in that they are qualitatively independent of the lattice coordination number. The comparison between the results to be presented in this work with those reported on previously for other particular case [@del06] actually implies that the model under investigation shows qualitatively the same features irrespective of the lattice coordination number. 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--- abstract: 'Speech enhancement is a crucial task for several applications. Among the most explored techniques are the Wiener filter and the LogMMSE, but approaches exploring deep learning adapted to this task, such as SEGAN, have presented relevant results. This study compared the performance of the mentioned techniques in 85 noise conditions regarding quality, intelligibility, and distortion; and concluded that classical techniques continue to exhibit superior results for most scenarios, but, in severe noise scenarios, SEGAN performed better and with lower variance.' author: - \| Tito Spadini[^1]\ CECS\ Universidade Federal do ABC\ Santo André, SP\ `tito.caco@ufabc.edu.br`\ Ricardo Suyama\ CECS\ Universidade Federal do ABC\ Santo André, SP\ `ricardo.suyama@ufabc.edu.br`\ bibliography: - 'main.bib' title: Comparative Study between Adversarial Networks and Classical Techniques for Speech Enhancement --- Introduction ============ Since the 1980s, speech enhancement and denoising researches exploit neural networks’ ability to work as non-linear filters [@DNNSPEECHI; @DNNSPEECHII; @DNNSPEECHIII], but it’s performance was often unsatisfactory — usually due to the reduced amount of training data or even the limited flexibility of the networks, caused by the inefficiency of the training algorithm for more extensive networks, with more neurons and layers. A new perspective for the use of neural networks arose after [@HINTON2006], which has exhibited that employing an unsupervised pre-training of the network for layers can bypass the limitation found by gradient-based algorithms. This new possibility, coupled with the rise of new tools for training neural nets using GPU (Graphics Processing Unit) [@scherer2010evaluation], has rekindled interest in neural networks. Implementations of neural networks with large numbers of neurons and layers now include the term “Deep” in their nomenclature, and therefore the term Deep Learning has become popular when referring to Deep Neural Networks (DNN). The flexibility and power of this type of neural network have shown promising results and have attracted a growing number of researchers in the field. However, there is a wide variety of DNN architectures; each with its respective characteristics and particularities, which may be more convenient for specific applications and therefore less suitable for others. For the applications discussed in this paper, an architecture known as convolutional autoencoder will be explored. A recent approach called Generative Adversarial Networks (GAN) [@NIPS2014_5423] is a structure composed of individuals — usually, two neural networks — competing against each other and exploring concepts of Game Theory and Deep Learning. In this competitive two-player game, there is a well-prepared dataset, composed of samples of the same type, appropriately chosen, but with different attribute values. The Discriminator player, $D$, has the purpose of discriminating whether a sample came from the original dataset or not; the Generator, $G$, must capture the distribution of the original dataset and use it to generate entirely new samples. Thus, while one of the players intends to generate the perfect imitation of the original data, the other player tries to be the best possible counterfeit identifier. For GAN-based methodes, the network training should occur gradually and concomitantly for both players, otherwise, an evolutionary imbalance may occur in favor of one of the actors, therefore, instead of achieving a good evolution for both, only one network will evolve minimally compared to the other, which does not even guarantee good results for at least one of the players. The GAN adjusting criterion is given by: $$\min_{G} \max_{D} V\left(D, G\right) = \mathbb{E}_{\pmb{x}\thicksim p_{\text{data}}\left(\pmb{x}\right)}\left[\log D\left(\pmb{x}\right)\right] + \mathbb{E}_{z\thicksim p_{z}\left(z\right)}\left[\log \left(1 - D\left(G\left(z\right)\right)\right)\right], \label{eq:minimax}$$ where $V$ is the value function; $D$ is the Discriminator (player), a multi-layer perceptron that generates the probability that $\pmb{x}$ has originated from the legitimate data instead of the distribution $p_{G}$; $G$ is the Generator (player); $p_{\text{data}}$ is the data distribution; and $p_{\text{\pmb{z}}}$ is a prior in the input noise variables. In the same way that GAN can learn a generative model for training data, conditional GAN (cGAN) [@CGAN], as its name may suggest, provides a conditional generative model for the data. The data generation is based on a prior distribution and also on an additional input $\pmb{x}_c$, thus conditioning the distribution of the generated data to the additional information provided by $\pmb{x}_c$. The cost function of cGAN is given by: $$\min_{G} \max_{D} V\left(D, G\right) = \mathbb{E}_{\pmb{x}, \pmb{x}_c\thicksim p_{\text{data}}\left(\pmb{x}, \pmb{x}_c\right)}\left[\log D\left(\pmb{x},\pmb{x}_c\right)\right] + \mathbb{E}_{z\thicksim p_{z}\left(z\right), \pmb{x}_c\thicksim p_{data}\left(\pmb{x}_c\right)}\left[\log \left(1 - D\left(G\left(z, \pmb{x}_c\right),\pmb{x}_c\right)\right)\right], \label{eq:minimax_cGAN}$$ Despite the advancement of the original GAN’s and cGANs based on minimizing and , in some cases the training may converge to solutions with performance below desired. For this reason, in [@LSGAN], an alternative proposal was presented, called Least Squares GAN (LSGAN), which seeks to adapt the discriminator and the generator according to the following criteria: $$\min_{D} V_{LSGAN} \left(D\right) = \frac{1}{2}\mathbb{E}_{\pmb{x}, \pmb{x}_c\thicksim p_{\text{data}}\left(\pmb{x}, \pmb{x}_c\right)}\left[\left( D\left(\pmb{x},\pmb{x}_c\right)-1\right)^2\right] + \frac{1}{2}\mathbb{E}_{z\thicksim p_{z}\left(z\right), \pmb{x}_c\thicksim p_{data}\left(\pmb{x}_c\right)}\left[ D\left(G\left(z, \pmb{x}_c\right),\pmb{x}_c\right)^2\right], \label{eq:LSGAN_D}$$ $$\min_{G} V_{LSGAN} \left(D\right) = \frac{1}{2}\mathbb{E}_{z\thicksim p_{z}\left(z\right), \pmb{x}_c\thicksim p_{data}\left(\pmb{x}_c\right)}\left[ \left(D\left(G\left(z, \pmb{x}_c\right),\pmb{x}_c\right)-1\right)^2\right], \label{eq:LSGAN_G}$$ A few years later, a GAN-based approach called Speech Enhancement GAN (SEGAN) was introduced in [@SEGAN], exploring autoencoders neural networks adapted using RMSProp. Also, it has presented promising results. Further details on SEGAN will be provided in Section II. One of the most decisive aspects regarding the adoption of an autoencoder, which in this case is fully convolutional, is its infundibuliform architecture capable of preserving the signal structure, ensuring that the obtained output will respect the same form used as network input. Also, the autoencoder can discard dispensable parts of the signals, i.e., noise, which causes the preservation of signal information to occur. However, due to the connections between encoding layers and decoding layers, the denoising effect caused by the network does not occur aggressively to the point of destroying the signal in terms of quality and intelligibility only for if noise reduction is achieved. This work’s objective is to compare SEGAN’s performance against Wiener filter and Log Minimum Mean Square Error (LogMMSE), concerning quality and intelligibility through objective and perceptual metrics, for several different noise scenarios. However, different from [@SEGAN], this work considered LogMMSE in its comparisons, in addition to Wiener’s filter and SEGAN itself. Moreover, in addition to the Perceptual Evaluation of Speech Quality (PESQ) metric, explored in both studies, this work includes the Short-Time Objective Intelligibility (STOI) and Signal-to-Distortion Ratio (SDR) metrics. This work considered 5 different SNR scenarios (0 dB, 5 dB, 10 dB, 15 dB and 20 dB), which includes a new SNR value (20 dB) compared to those used in [@SEGAN]. The scenarios used in this paper considered 17 different types of noise, while [@SEGAN] used 5 types of noise. In order to present the results of this study, the article adopts the following structure: section II presents a review on the main concepts related to SEGAN; section III covers the simulation scenarios and the metrics; section IV discuss the results and offers the final comments on the study; and section V exhibits the conclusions of the work. Speech Enhancement GAN - SEGAN ============================== Speech Enhancement Generative Adversarial Network (SEGAN) [@SEGAN] is based on the idea introduced by cGAN, discussed in the previous section. The proposed structure for the generator, in this case, resembles an autoencoder, shown in Figure \[fig:autoencoder\]. The successive convolutional layers provide, at the end of the coding process, a vector $\mathbf{c}$ which corresponds to a condensed representation of the input signal. In this process, strided convolutions layers are used, since this type of layer has been shown to be more stable in GAN training. However, unlike the structure of a traditional autoencoder, the decoding process is done from the vector $\mathbf{c}$ concatenated with the vector of latent variables $\mathbf{z} $. This new vector is then subjected to a sequence of layers that seek to reverse the coding process by means of fractional-strided transposed convolutions [@SEGAN]. The network structure also includes skip connections connecting the outputs of the layers in the encoding process directly to corresponding layers in the decoding process. The reason for this was to try to maintain the underlying structure of the observed (noisy) data in the data being generated by GAN. ![\[fig:autoencoder\]Autoencoder architecture for speech enhancement. Based on [@SEGAN].](autoencoder.pdf){width=".375\linewidth"} Although the presented LSGAN criterion may help with some known problems of structure adaptation, the authors propose, based on preliminary simulations, that the cost function includes an additional term in order to favor solutions that minimize the distance between generated data and authentic examples. The distance, however, is measured with the norm $\ell_1$, so that the adopted criterion is defined by: $$\min_{G} V_{LSGAN} \left(G\right) = \frac{1}{2}\mathbb{E}_{\pmb{z}\thicksim p_{\pmb{z}}\left(\pmb{z}\right),\ \pmb{\tilde{x}}\thicksim p_{data}\left(\pmb{\tilde{x}}\right)} \left[\left(D\left(G\left(\pmb{z},\ \pmb{\tilde{x}}\right),\ \pmb{\tilde{x}}\right)-1\right)^{2}\right] + \lambda\ \norm{G\left(\pmb{z},\ \pmb{\tilde{x}}\right)-\pmb{x}}_{1}, \label{eq:LSGAN}$$ where $\pmb{\tilde{x}}$ represents the input (noisy) signals. Simulation Setups ================= To evaluate the performance, several testing scenarios were created from the combination of 20 voices of different people from the dataset VCTK-Corpus [@veaux2017cstr] reading three different sentences; (0 dB, 5 dB, 10 dB, 15 dB and 20 dB) for each of the 17 types of noise, all coming from the DEMAND [@thiemann2013demand] dataset, resulting in 85 different noise conditions for each sentence read by each person. In order to preserve a variability that would assist in the quest for broader generalizability in the SEGAN model training, the selected corpus was chosen to maintain a uniform gender distribution and to ensure different accents. Although the datasets were used in SEGAN’s work, the selected individuals were not the same and more types of noise were used, including an extra SNR level (20 dB) that was not explored in the mentioned work. With all different voice and noise scenarios previously detailed, 5100 different mixtures were obtained to be processed by the three speech enhancement techniques (Wiener filter, LogMMSE, and SEGAN). Using SoX, pre-processing was performed to ensure that all input signals conform to the 16 kHz, 16-bit, and mono configuration in WAV format. The selected Wiener filter belongs to the Scipy library; LogMMSE [@LOGMMSE; @LOGMMSE_Python], is also available as a Python package of the same name; and the SEGAN (pre-trained) model is the same as the original work [@SEGAN], which is openly distributed by the SEGAN authors themselves in their GitHub repository. Such a model had been trained for 86 times using RMSprop [@tieleman2012lecture] and learning rate of 0.0002 in batches of size 400 [@SEGAN]. For each technique, after processing, an enhanced signal was obtained for each noisy signal used as input; and, based on the improved signal and the reference clean voice signal (directly from the VCTK-Corpus dataset), the (PESQ) [@PESQ], a perceptual quality metric with values from -0.5 to 4.5; the STOI [@STOI], which measures the improvement of intelligibility with values from 0 to 1; and the SDR [@SDR], which quantifies the rate between the speech signal and the distorting effects of improved speech signal, were calculated to perform the improvement evaluation. Results ======= The average values of PESQ can be seen on Figures \[fig:pesq\_avg\] and \[fig:pesq\]. It can be noted that the Wiener filter maintained a near linear behavior as the SNR was increased; although it was the technique with the worst performance for SNR 0 dB, it reached the best result of PESQ observed in this work for the case in 20 dB. LogMMSE has proven to be very effective from the start, going through all the scenarios as one of the best in terms of quality. SEGAN, on the other hand, showed a subtle superiority in the 0 dB scenario, but showed little improvement for higher SNR values, being on average much lower than the other two quality techniques analyzed during the enhancement process. Nevertheless, it is important to highlight the fact that the variance of SEGAN was lower in all scenarios. ![\[fig:pesq\]PESQ for different noise levels.](pesq_avg.pdf){width="\linewidth"} ![\[fig:pesq\]PESQ for different noise levels.](pesq.pdf){width="\linewidth"} Figures \[fig:stoi\_avg\] and \[fig:stoi\] show STOI averages. The intelligibility is shown to be higher for the Wiener filter in all scenarios, which means that such a technique resulted in lower degeneration of speech comprehension. The LogMMSE showed a much lower performance than the other techniques for low SNR scenarios; improved slightly for 15 dB and 20 dB, but still got much worse than the Wiener filter. SEGAN showed a performance similar to that of the Wiener filter for the 0 dB scenario and remained superior to LogMMSE for this relation between the signal and distortions for almost all scenarios of different SNR values, except for 20 dB. ![\[fig:stoi\]STOI for different noise levels.](stoi_avg.pdf){width="\linewidth"} ![\[fig:stoi\]STOI for different noise levels.](stoi.pdf){width="\linewidth"} The performance of each technique in terms of SDR can be seen in Figures \[fig:sdr\_avg\] and \[fig:sdr\]. Notwithstanding the poor performance of Wiener filter for cases of lower SNR, it proved to be quite effective for higher SNR scenarios. The LogMMSE approach presented a similar performance to the Wiener filter. And, although it was the technique with the best performance for low SNR cases, SEGAN showed little improvement for cases with higher SNR; in the case with SNR 20 dB, its performance was well below that obtained by the other techniques; however, as with the PESQ metric, the variance of this technique was much lower than the other techniques. ![\[fig:sdr\]SDR for different noise levels.](sdr_avg.pdf){width="\linewidth"} ![\[fig:sdr\]SDR for different noise levels.](sdr.pdf){width="\linewidth"} Discussion ========== Although it was not the focus of this paper, there are some considerations to be made regarding the performance in terms of resources required for the speech enhancement process to be performed. While both classical methods adopted are based on an unsupervised approach, the GAN-based method is supervised, which requires a fundamental training step to be performed based on a pre-selected data set; and this step is computationally expensive as it took several hours to complete, even though it was performed on a GPU. Still, the application of the enhancement process itself through SEGAN’s trained-model is not fast either, and it may take several seconds to complete the application over a single audio track of a few seconds. By contrast, classical methods performed the process almost instantaneously for each audio track and required no prior training. Regarding speech quality and intelligibility, even considering the respectable, effective and efficient existing objective metrics, such as PESQ and STOI, respectively, if the enhanced signal is aimed for direct use by people, the use of metrics that still have the opinion of people, like Mean Opinion Score (MOS) [@streijl2016mean], may continue to be utilized, even if it has a lesser weight; after all, for various purposes, the human sense to evaluate and perceive distinct levels of quality may not yet have been well-enough designed in computational algorithms and metrics. The use of before-mentioned metrics would introduce factors of subjectivity into the process, which can be understood as something to be avoided; yet, if their influence is carefully managed in the appraisal and weighting, perhaps the results may be more satisfactory. It is appropriate to indicate that Figure \[fig:stoi\] also shows two critical details respecting the STOI results: an enormous variance and a colossal amount of outliers. The results show a considerable decrease in variance accompanied by a noticeable increase in already evident outliers. Such peculiarities described in this paragraph are worrisome regarding speech perception issues in noisy scenarios. This result may have been negatively affected by an innocently naive choice of complex audios, or by a training step that underwent from severe data frugality. Given the observations indicated in the preceding paragraph, which emphasize certain undesired peculiarities about part of the results, especially regarding intelligibility, improvements may be perceived if meaningful arrangements are implemented to the assembled speech corpus. Perhaps using more numerous personalities of diverse ages, with different accents and more notable distinction in their vocal characteristics may enhance the results in future work. About SDR, despite its relevance in this work, also because it is a usual objective metric, which tends to reduce human-related failures, it may be a less robust metric for some scenarios of speech enhancement or source separation, mainly for monoaural signals, which are of the type discussed in this paper. In order to address the problems associated with this metric, work [@le2019sdr] proposed an alternative metric called SI-SDR. Thus, in a future continuation of this work, this new metric proposal can be explored. Conclusions =========== The results show that, although it is a classic technique confronted by more advanced ones, at least for the scenarios covered in this particular paper, the Wiener filter is still able to perform speech enhancement tasks for several scenarios, and remains a proper method for quality, intelligibility and distorting effects on speech signals. Despite its subtly inferior performance for some considered scenarios, SEGAN did well at 0 dB SNR scenarios, which are much more complicated, as well as exhibiting substantially lower variances. Although results obtained in [@SEGAN] indicate superiority over the Wiener filter, the divergence of results in relation to this work may be due to the wider variety of scenarios considered in this work. It is noteworthy, however, the need for a more detailed analysis of specific scenarios and also a more in-depth investigation into SEGAN. Acknowledgment {#acknowledgment .unnumbered} ============== The authors are grateful for the support received from CAPES and from the Conselho Nacional de Desenvolvimento Científico e Tecnológico – CNPq. [^1]: https://spadini.info
--- abstract: 'In this paper, we consider the use of UAVs to provide wireless connectivity services, for example after failures of wireless network components or to simply provide additional bandwidth on demand, and introduce the concept of UAVs as a service (UaaS). To facilitate UaaS, we introduce a novel framework, dubbed $D\mathrm{^3}S$, which consists of four phases: demand, decision, deployment and service. The main objectives of this framework is to develop efficient and realistic solutions to implement these four phases. The technical problems include determining the type and number of UAVs to be deployed, and also their final locations (e.g., hovering or on-ground), which is important for serving certain applications. These questions will be part of the decision phase. They also include trajectory planning of UAVs when they have to travel between charging stations and deployment locations, and may have to do this several times. These questions will be part of the deployment phase. The service phase includes the implementation of the backbone communication and data routing between UAVs, and between UAVs and ground control stations.' author: - bibliography: - './bibliographies/main-d3s-uav-commag.bib' title: 'D$^3$S: A Framework for Enabling Unmanned Aerial Vehicles as a Service' --- Unmanned Aerial Vehicles (UAVs), Wireless Networks, UaaS. Introduction {#sec:framework} ============ Demand Forecasting and Characterization ======================================= Decision and Dimensioning Phase =============================== Deployment and Trajectory Planning Phase ======================================== The information from the demand and decision phases play significant role in the deployment and trajectory planning phase. Information from the decision phase such as the rate requirements for certain users can affect the UAVs trajectory. For example, obtaining good channel gains between the UAVs and targeted users require, in general, the UAVs to move closer to the targeted users to obtain better channel, which in turn expect increasing the achievable rate. On the other hand, the information from the decision phase such as number of UAVs and bandwidth limitation will directly affect the deployment and trajectory design by limiting the available resources to use. By exploiting a careful trajectory design of the UAVs, significant performance gains can be achieved compared to traditional static wireless systems. However, several limitation factors need to be considered. The first one is the instantaneous battery levels of the UAVs, where each UAV determines its battery level periodically to make sure it has enough battery for both hovering and communications. The second factor is the nearby available charging stations. We assume that each charging station can accommodate a maximum number of UAVs at a time instance. Thus, each UAV needs to pre-define the available charging stations in order to land for charging when needed. The third factor is recharging period, i.e., how much time the UAV needs to stay in the charging station. This depends on the decision of the central unit which is based on the user’s demand. Finally, the last main factor is the safely path planning, for example, the UAVs are required to avoid flying over some restricted regions, such as airports and military regions. Also, they are required to respect the obstacles on the way, such as high buildings and avoid collisions with other UAVs. Let us assume that we have certain number of charging station locations with maximum UAVs that can be accommodated in each charging station. Thus, two constraints need to be respected. First, the maximum number of UAVs that can be charged during each time slot at each charging station.Second, no more than one UAV can be at the same location during each time slot. Therefore the possible scenarios can be summarized as follows: 1) when the UAV is located not in the charging station;2) when the UAV stays at the same serving location but not at charging stations; 3) when the UAV moves to return to the charging station while it was located at the serving location; and 4) when the UAV decides to remain in the charging station. We categorize the ground users into stationary and mobile users. The only difference between these two types of users is that the speed of stationary users is equal to 0. We assume that the total time period is discretized into equal sub-slots, where the communication channel is approximately unchanged during the sub-slot. Furthermore, We assume that each user can be associated to one UAV at most during each short time slot. On the other hand, based on the moving speed of the ground user and UAV, we assume that the maximum distances the ground user and UAV can travel in each sub-slot are limitted by their speeds during the sub-slot. Service Phase ============= In this service phase, the proper coverage service to achieve end-to-end connectivity will be provided. This includes communication of UAVs with ground users (stationary or mobile), routing of data between UAVs, and routing of data to and from access points to the core network. In order to provide UaaS for end-to-end connectivity, it is critically important to establish a reliable backbone network between UAVs to allow reliable, low-latency data delivery either from a UAV to a base station, or from a base station to one or more UAVs, or from a base station to another base station through a network of UAVs. Existing works on routing in UAV networks have typically used or adapted classic MANET (Mobile Ad-hoc Networks) protocols. These protocols are classified as either proactive or reactive, depending on whether they maintain routes a priori or build routes on demand. Hybrid protocols, such as the Hybrid Wireless Mesh Protocol (HWMP) of the IEEE 802.11s standard, also exist. However, these classic protocols generally perform poorly in UAV networks where nodes are moving fast. Some works have adapted MANET protocols for use in UAV networks [@8255752]. Although these excellent efforts were successful in handling some of the scenarios for UAV networks, more innovations are needed to improve the reliability and latency performances of the routing protocols. In the following, we describe three methods that may be more suitable for routing in the dynamic and challenging environment of UAV networks: (1) proactive routing based on cohesive swarming and machine learning; (2) fast-converging reactive routing based on back-pressure; and (3) opportunistic routing based on anycast.\ Proactive Routing based on Cohesive Swarming and Machine Learning: Unlike conventional wireless ad hoc networks, designing optimal multi-path routing and congestion control algorithms for UAV networks is particularly challenging due to the highly dynamic energy-aware UAV flight maneuvers, which yields constantly changing network topology and fluctuating channel qualities. Classic proactive routing methods are known to perform poorly in such an environment. One possible way to enhance proactive routing for UAV networks is to combine it with cohesive swarming, which coordinates UAVs to form a swarm or shape that suits best the underlying proactive routing method, as well as the events or users of interest. In addition, machine learning techniques can be used for more accurate traffic prediction and thus to enhance in-routing functions among UAVs.\ Fast-Converging Reactive Routing: In addition to accurate predictive proactive routing, designing fast-converging reactive routing methods also plays a critical role in UAV networks. In classic reactive methods, queue-length changes are often used as weights in making dynamic routing decisions. Such methods are known to converge slowly. One possible way to improve the convergence speed is to couple queue-length changes with route update from the previous time slot (called momentum). Momentum-based reactive routing methods such as the one proposed in [@Liu16:HeavyBall_INFOCOM] could be a good candidate for routing in UAV networks, due to its low-complexity, and its strong performance guarantees in terms of throughput-optimality, delay reduction, and convergence speed.\ Opportunistic Routing based on Anycast: Opportunistic routing refers to the practice of making routing decisions dynamically (instead of following pre-determined routes) based on network events and conditions, such as link availability and quality. The opportunistic approach gives nodes multiple options for forwarding a packet and, thus, is particularly suited to UAV networks where the set of a node’s neighbors can be constantly changing. The method proposed in [@wymore:2015ab] could be a good candidate for opportunistic routing in UAV networks, which is a cross-layer approach that merges information from both network and link layers to make dynamic routing decisions based on the available links. Moreover, the opportunistic approach may be integrated with proactive or reactive routing methods to further improve the system performance.\ Case Study: UaaS for Self-healing ================================= Conclusion ==========
--- abstract: 'We consider the cases of the self-adjoint and skew-self-adjoint discrete Dirac systems, obtain explicit expressions for reflection coefficients and show that rational reflection coefficients and Weyl functions coincide.' author: - 'B. Fritzsche, B. Kirstein, I.Ya. Roitberg and A.L. Sakhnovich' title: 'Discrete Dirac systems on the semiaxis: rational reflection coefficients and Weyl functions' --- [MSC(2010): 39A10, 39A12, 47A40]{} [Keywords:]{} Discrete self-adjoint Dirac system, discrete skew-self-adjoint Dirac system, Weyl function, reflection coefficient, Bäcklund-Darboux transformation. Introduction {#Intro} ============ Discrete self-adjoint and skew-self-adjoint Dirac systems play an essential role in the study of Toeplitz matrices (and corresponding measures), of discrete integrable nonlinear equations (including isotropic Heisenberg magnet model) and of spectral theory of difference equations (see, e.g., [@DerSi; @FKKS; @KaS; @ALSJFA; @ALSverb] and references therein). Weyl-Titchmarsh theory of discrete systems is actively studied (see, e.g., [@CGe; @Jo; @SZ; @ZeC] and various references therein). In particular, Weyl–Titchmarsh theory of discrete self-adjoint and skew-self-adjoint Dirac systems was studied in [@FKKS; @FKRS14; @FKRS-LAA; @KaS; @RoS; @SaSaR] (see also references therein). It is known that Weyl–Titchmarsh (or simply Weyl) functions of continuous Dirac systems on the semi-axis are closely related to the scattering data. Some particular results for the self-adjoint systems are contained, for instance, in [@BeM; @GKS6] and the general cases of continuous self-adjoint and skew-self-adjoint systems were treated in the recent paper [@ALS-Scat]. The present article may be considered as the continuation of the paper [@ALS-Scat], where the important discrete case is dealt with. We consider the cases of the self-adjoint and skew-self-adjoint discrete Dirac systems, obtain explicit expressions for reflection coefficients and show that rational reflection coefficients and Weyl functions coincide. General-type discrete self-adjoint Dirac system has the form: $$\label{0.1} y_{k+1}(z)=(I_m+ {\mathrm{i}}z j C_k) y_k(z) \quad \left( k \in \BN_0 \right),$$ where $\BN_0$ stands for the set of non-negative integers, $I_m$ is the $m \times m$ identity matrix, $``{\mathrm{i}}"$ is the imaginary unit (${\mathrm{i}}^2=-1$) and the $m \times m$ matrices $C_k$ are positive and $j$-unitary: $$\label{0.2} C_k>0, \quad C_k j C_k=j, \quad j: = \left[ \begin{array}{cc} I_{m_1} & 0 \\ 0 & -I_{m_2} \end{array} \right] \quad (m_1+m_2=m; \, \, m_1, \, m_2 > 0).$$ We introduce the Jost solution and reflection coefficient of the system , in a similar to the continuous case way. Namely, the Jost solution $\{F_k(z) \}$ ($z\in \BR$) of the Dirac system , is defined via its asymptotics $$\begin{aligned} & \label{R!} F_k(z)=\big(I_m +{\mathrm{i}}z j \big)^{k}\big(I_m+o(1)\big), \quad k \to \infty.\end{aligned}$$ The reflection coefficient $\clr(z)$ is introduced via the blocks of $F_0(z)$: $$\begin{aligned} & \label{R1!} \clr(z)= \begin{bmatrix} I_{m_1} & 0 \end{bmatrix} F_0(z)\begin{bmatrix} 0 \\ I_{m_2} \end{bmatrix}\left(\begin{bmatrix} 0 & I_{m_2} \end{bmatrix}F_0(z) \begin{bmatrix} 0 \\ I_{m_2} \end{bmatrix}\right)^{-1}.\end{aligned}$$ Discrete skew-self-adjoint Dirac system (SkDDS) is given (see [@FKRS-LAA; @KaS]) by the formula: $$\label{d1} y_{k+1}(z)=\left(I_m+ \frac{{\mathrm{i}}}{ z} C_k\right) y_k(z), \quad C_k=U_k^jU_k \quad \left( k \in \BN_0 \right),$$ where the matrices $U_k$ are unitary and $j$ is defined in . Direct and inverse problems (in terms of Weyl functions) were solved for systems , in [@FKRS14; @RoS] and for systems in [@FKKS; @FKRS-LAA]. In particular, in the case of rational Weyl matrix functions, direct and inverse problems were solved explicitly using our GBDT version [@ALS94; @ALS-JMAA; @SaSaR] of the Bäcklund-Darboux transformation. For various versions of Bäcklund-Darboux transformations and related commutation methods see, for instance, [@Ci; @D; @GeT; @Gu; @KoSaTe; @Mar; @MS] and references therein. The results of the paper imply that the procedures to recover systems from the Weyl functions enable us to recover systems from the reflection coefficients as well. In the next section, we give some preliminary definitions and results in order to make the paper self-sufficient. Two subsections of Section \[Reflect\] are dedicated to the reflection coefficients in the self-adjoint and skew-self-adjoint cases. In the paper, $\BN$ denotes the set of natural numbers, $\BR$ denotes the real axis, $\BC$ stands for the complex plane, and $\BC_+$ ($\BC_-$) stands for the open upper (lower) half-plane. The spectrum of a square matrix $A$ is denoted by $\s(A)$. Preliminaries {#Prel} ============== Self-adjoint case ----------------- #### 1. Self-adjoint discrete Dirac system and stability of the explicit procedure to recover it from the Weyl function was studied in our recent paper [@RoS]. We refer to [@RoS] for the preliminary definitions and results in this subsection. The fundamental $m \times m$ solution $\{W_k \}$ of is normalized by $$\begin{aligned} \label{0.3}& W_0(z)=I_m.\end{aligned}$$ \[defWeyl\] The Weyl function of the Dirac system $($which is given on the semi-axis $0\leq k < \infty$ and satisfies $)$ is an $m_1\times m_2$ matrix function $\vp(z)$ in the lower half-plane, such that the following inequalities hold $:$ $$\begin{aligned} \label{0.4}& \sum_{k=0}^\infty q(z)^k \begin{bmatrix} \vp(z)^ &I_{m_2} \end{bmatrix} W_k(z)^C_k W_k (z) \begin{bmatrix} \vp(z) \\ I_{m_2} \end{bmatrix}<\infty \quad (z\in \BC_-), \\ & \label{0.5} q(z):=(1+\|z\|^2)^{-1}.\end{aligned}$$ (For the case $z \in \BC_+$, the definition of the Weyl function $\vp(z)$ of Dirac system , was given in [@FKRS14].) #### 2. In order to consider the case of rational Weyl functions, we introduce generalized Bäcklund-Darboux transformation (GBDT) of the discrete self-adjoint Dirac systems. Each GBDT of the initial discrete Dirac system is determined by a triple $\{A, S_0, \Pi_0\}$ of parameter matrices. Here, we take a trivial initial system and choose $n\in \BN$, two $n \times n$ parameter matrices $A$ ($\det A \not=0$) and $S_0>0$, and an $n \times m$ parameter matrix $\Pi_0$ such that $$\label{0.6} A S_0-S_0 A^={\mathrm{i}}\Pi_0 j \Pi_0^.$$ Define the sequences $\{\Pi_r\}$ and $\{S_r\}$ $(r \geq 0)$ using the triple $\{A, S_0, \Pi_0\}$ and recursive relations $$\begin{aligned} & \label{0.7} \Pi_{k+1}=\Pi_k+{\mathrm{i}}A^{-1}\Pi_k j \quad (k\geq 0), \\ & \label{0.8} S_{k+1}=S_k+A^{-1}S_k (A^)^{-1}+A^{-1}\Pi_k \Pi_k^(A^)^{-1} \quad (k\geq 0).\end{aligned}$$ From –, the validity of the matrix identity $$\label{0.9} A S_{r}-S_{r} A^={\mathrm{i}}\Pi_{r} j \Pi_{r}^ \quad (r \geq 0)$$ follows by induction. In the self-adjoint case, we introduce [admissible]{} triples $\{A, S_0, \Pi_0\}$ in the following way. \[adm\] The triple $\{A, S_0, \Pi_0\}$, where $\det A \not=0$, $S_0>0$ and holds, is called admissible. In view of , for the admissible triple we have $S_r>0$ $(r\geq 0)$. Thus, the sequence (potential) $\{C_k\}$ $(k\geq 0)$ is well-defined by the equality $$\label{0.10} C_k:=I_m+\Pi_k^S_k^{-1}\Pi_k-\Pi_{k+1}^S_{k+1}^{-1}\Pi_{k+1} .$$ Moreover, from [@RoS Theorem 2.5] we see that the matrices $C_k$ satisfy . We say that the potential $\{C_k\}$ is [determined]{} by the admissible triple. The potential determined by an admissible triple $\{A, S_0, \Pi_0\}$ is called [pseudo-exponential]{}. We note that the notion of the pseudo-exponential (and [strictly pseudo-exponential]{}) potentials for the self-adjoint continuous case was introduced first in [@GKS1] (see also [@GKS6]). In the discrete case, some additional requirements on the [admissible and strongly admissible triples]{} (which determine pseudo-exponential and strictly pseudo-exponential, respectively, potentials) appear. All Weyl functions $\vp(z)$ are contractive in $\BC_-$ and all the potentials $\{C_k\}$, such that $\vp(z)$ (for the corresponding systems) are contractive and $\vp(-1/z)$ are strictly proper rational, are determined by some admissible triples [@RoS]. Strongly admissible triples for the self-adjoint case are considered in Subsection \[Refl1\]. We will need also the matrix function $w_A$, which for each $k \geq 0$ is a so called transfer matrix function in Lev Sakhnovich form [@SaL1; @SaL3; @SaSaR] and is defined by the relation $$\label{0.11} w_A(k,\lambda):=I_m-{\mathrm{i}}j \Pi_k^S_k^{-1}(A-\lambda I_n)^{-1}\Pi_k.$$ The fundamental solution $\{W_{k}\}$ of the Dirac system admits the representation $$\label{0.12} W_{k}(z)=w_A(k,\,-1/z)\big(I_m +{\mathrm{i}}z j \big)^{k}w_A(0,\,-1/z)^{-1} \quad (k \geq 0),$$ where $w_A$ is defined in . Now, we partition $\Pi_k$ and write it down in the form $$\begin{aligned} \label{v7} & \Pi_k=\begin{bmatrix}(I_n+{\mathrm{i}}A^{-1})^k\vt_1 & (I_n-{\mathrm{i}}A^{-1})^k \vt_2 \end{bmatrix},\end{aligned}$$ where $\vt_1$ and $\vt_2$ are $n\times m_1$ and $n\times m_2$, respectively, blocks of $\Pi_0$. Assume further in this subsection that $$\begin{aligned} \label{R0} & \pm {\mathrm{i}}\not\in \s(A).\end{aligned}$$ In view of and , setting $$\begin{aligned} \label{v8} & R_{r}:=(I_n+{\mathrm{i}}A^{-1})^{-r}S_r\big(I_n-{\mathrm{i}}(A^)^{-1}\big)^{-r}\end{aligned}$$ we have $$\begin{aligned} \nn R_{k+1}-R_k=& 2(I_n+{\mathrm{i}}A^{-1})^{-k-1}A^{-1}(I_n-{\mathrm{i}}A^{-1})^k \vt_2\vt_2^* \big((I_n-{\mathrm{i}}A^{-1})^k\big)^\big(A^{-1}\big)^ \\ \label{v9} & \times \big((I_n+{\mathrm{i}}A^{-1})^{-k-1}\big)^\geq 0.\end{aligned}$$ Since $R_0=S_0>0$, relations imply that there is a limit $$\begin{aligned} \label{v10} & \lim_{k\to \infty}R_k^{-1}=\vk_R\geq 0.\end{aligned}$$ In a similar way we introduce the matrices $$\begin{aligned} \label{v8'} & Q_{r}:=(I_n-{\mathrm{i}}A^{-1})^{-r}S_r\big(I_n+{\mathrm{i}}(A^)^{-1}\big)^{-r} ,\end{aligned}$$ and show that $$\begin{aligned} \label{v9'} & Q_{k+1}-Q_k \geq 0.\end{aligned}$$ Since $Q_0=S_0>0$, relations imply that there is a limit $$\begin{aligned} \label{v10'} & \lim_{k\to \infty}Q_k^{-1}=\vk_Q\geq 0.\end{aligned}$$ Skew-self-adjoint case ---------------------- The preliminary definitions and results on the skew-self-adjoint discrete Dirac systems (SkDDS) we take from [@FKKS] and sometimes from [@FKRS-LAA]. \[RkNn\] The notations here slightly differ from the notations in [@FKKS; @FKRS-LAA]. In particular, we introduce the matrices $R_k$ and $Q_k$ in the both self-adjoint and skew-self-adjoint cases via formulas and , respectively, but in [@FKRS-LAA] $R_k$ stands for $Q_k$ in the the present notations and $Q_k$ stands for $R_k$. \[DnD0\] The Weyl function of SkDDS is an $m_1 \times m_2$ matrix function $\vp(z)$ in $$\BC_M=\{z\in \BC: \, \Im(z)>M\} \quad {\mathrm{for \,\, some}} \quad M>0,$$ which satisfies the inequality $$\label{k1} \sum_{k=0}^{\infty}\begin{bmatrix} \vp(z)^* & I_{m_2}\end{bmatrix}w_k(z)^* w_k(z) \left[ \begin{array}{c} \vp(z) \\ I_{m_2} \end{array} \right] < \infty,$$ where $w_k(z)$ is the [fundamental solution]{} of SkDDS normalized by $w_0(z) \equiv I_m$. Let us fix again an integer $n>0$, and consider an $n \times n$ matrix $A$ with $\det \, A \not= 0$, an $n \times n$ matrix $S_0>0$ and an $n \times m$ matrix $\Pi_0$. These matrices should satisfy the identity $$\label{k2} A S_0-S_0 A^= {\mathrm{i}}\Pi_0 \Pi_0^.$$ The sequences $\{\Pi_k\}$, $\{S_k\}$ and $\{C_k\}$ $(k \geq 0)$ are introduced using the triple $\{A, \, S_0, \, \Pi_0\}$ and relations $$\begin{aligned} \label{k3} &\Pi_{k+1}= \Pi_k+{\mathrm{i}}A^{-1} \Pi_k j, \\ & S_{k+1}=S_k+ A^{-1} S_k (A^)^{-1}+ A^{-1} \Pi_k j \Pi_k^ (A^)^{-1},\label{k4} \\ \label{k5} & C_k=j+ \Pi_k^ S_k^{-1} \Pi_k - \Pi_{k+1}^* S_{k+1}^{-1} \Pi_{k+1}.\end{aligned}$$ Similar to the self-adjoint case we write down $\Pi_k$ in the form $$\Pi_k=\begin{bmatrix}(I_n+{\mathrm{i}}A^{-1})^k\vt_1 & (I_n-{\mathrm{i}}A^{-1})^k \vt_2 \end{bmatrix}.$$ If $S_0>0$, the identity holds and the pair $\{A, \, \vt_1\}$ is controllable, then according to [@FKKS Lemma 3.2] and [@FKKS Proposition 3.6] we have $\det \, A \not= 0$, $S_k>0$ and the matrices $C_k$ admit representation $C_k=U_k^jU_k$ from . That is, the sequence $\{C_k\}$ is well-defined and the corresponding system is a skew-self-adjoint Dirac system. In the skew-self-adjoint case, the triple $\{A, \, S_0, \, \Pi_0\}$, such that $S_0>0$, the identity holds and the pair $\{A, \, \vt_1\}$ is controllable, is called [admissible]{}. The potential determined by this triple is called [pseudo-exponential]{}. Moreover, if $\vp(z)$ is a strictly proper rational $m_1\times m_2$ matrix function then it is the Weyl function of some skew-self-adjoint Dirac system with the pseudo-exponential potential (see [@FKRS-LAA Theorem 4.2]). We will require additionally that ${\mathrm{i}}\not\in \s(A)$. In the skew-self-adjoint case, the triple $\{A, \, S_0, \, \Pi_0\}$, where $S_0>0$, the identity is valid, the pair $\{A, \, \vt_1\}$ is controllable and ${\mathrm{i}}\not\in \s(A)$, is called strongly admissible. The potentials determined by the strongly admissible triples are called strictly pseudo-exponential. Note that [@FKRS-LAA Proposition 4.8] implies that if $S_0>0$, holds and $0,\, {\mathrm{i}}\not\in \s(A)$ then $S_k>0$, the matrices $C_k$ are well-defined and there is a strongly admissible triple which determines the same potential $\{C_k\}$ as $\{A, \, S_0, \, \Pi_0\}$. The fundamental solution $w_k$ of SkDDS determined by the strongly admissible triple $\{A, \, S_0, \, \Pi_0\}$ has the form $$\begin{aligned} & \label{k6} w_k( z )=w_{ A }(k, -z ) \left( I_{m}+\frac{{\mathrm{i}}}{z}j \right)^{k} w_{ A }(0, - z )^{-1},\end{aligned}$$ whereas $w_A$ in the skew-self-adjoint case is given by $$\begin{aligned} & \label{k7} w_{ A }(k, \la ):=I_{m}- {\mathrm{i}}\Pi_k^{} S_k^{-1} ( A- \lambda I_{n} )^{-1} \Pi_k.\end{aligned}$$ Taking into account Remark \[RkNn\], we see that [@FKRS-LAA Proposition 4.10] and [@FKRS-LAA (4.34)] imply that $$\begin{aligned} \label{k8} & \lim_{k\to \infty}Q_k^{-1}=0; \quad \lim_{k\to \infty}\big(Q_k^{-1}\wt G(A)^k\vt_1\big)=0, \quad \wt G(A):=(A-{\mathrm{i}}I_n)^{-1}(A+{\mathrm{i}}I_n)\end{aligned}$$ in the case of a strongly admissible triple $\{A, \, S_0, \, \Pi_0\}$. Reflection coefficients {#Reflect} ======================= Reflection coefficients: self-adjoint case {#Refl1} ------------------------------------------- In this subsection, we express (via the triple $\{A, \, S_0, \, \Pi_0\}$) the Jost solution and reflection coefficient, which are the analogs of the corresponding functions in the continuous case. Uniqueness of the solution of the inverse problem to recover system from the Weyl function (see [@RoS Theorem 2.3]) together with Theorems 2.6 and 2.8 and Proposition 2.7 (all from [@RoS]) imply that without loss of generality one can require that $\s(A )\subset (\BC_+\cup \BR)$. Further we use a stronger requirement $$\begin{aligned} \label{R1} & \s(A )\subset \BC_+, \quad {\mathrm{i}}\not\in \s(A).\end{aligned}$$ Following [@FKKS; @GKS1], we call the admissible triple satisfying and we introduce the class of the strictly pseudo-exponential potentials $\{C_k\}$. \[sps\] The potentials $\{C_k\}$ of the Dirac systems , , which are determined by the strongly admissible triples, are called strictly pseudo-exponential. In view of , , and , we have a representation $$\begin{aligned} \label{R2} & w_{A}(k,-1/ z ) \\ \nn &= I_m-{\mathrm{i}}zj \begin{bmatrix} \vt_1^R_k^{-1}(I_n+zA)^{-1}\vt_1 & \vt_1^R_k^{-1}(I_n+zA)^{-1} G(A)^k\vt_2 \\ \vt_2^(G(A)^k)^R_k^{-1} (I_n+zA)^{-1}\vt_1 & \vt_2^Q_k^{-1}(I_n+zA )^{-1}\vt_2 \end{bmatrix},\end{aligned}$$ where $$\begin{aligned} \label{R3} & G(A)=(I_n+{\mathrm{i}}A^{-1})^{-1}(I_n-{\mathrm{i}}A^{-1}).\end{aligned}$$ Relations and yield $$\begin{aligned} \label{R4} & \s\big(G(A)\big) \subset \{\la: \, \|\la \|<1 \}.\end{aligned}$$ Hence, from , and we derive $$\begin{aligned} \label{R5} & \lim_{k\to \infty} w_{A}(k,-1/ z ) = \begin{bmatrix} \chi_1(z) & 0 \\ 0 & \chi_2(z) \end{bmatrix}, \\ \label{R6} & \chi_1(z):=I_{m_1}-{\mathrm{i}}z \vt_1^\vk_R (I_n+zA)^{-1}\vt_1, \,\, \chi_2(z):=I_{m_2}+{\mathrm{i}}z \vt_2^\vk_Q (I_n+zA)^{-1}\vt_2.\end{aligned}$$ According to , and , the Jost solution $\{F_k\}$ is given by the equalities $$\begin{aligned} \label{R7} & F_k(z)=W_k(z)w_{A}(0,-1/ z )\begin{bmatrix} \chi_1(z)^{-1} & 0 \\ 0 & \chi_2(z)^{-1} \end{bmatrix}.\end{aligned}$$ Partition $w_A$ into the blocks corresponding to the partitioning of $j$: $$\begin{aligned} \label{R8}& w_A(0,\lambda)=\left[ \begin{array}{lr} a(\lambda) & b(\lambda) \\ c(\lambda) & d(\lambda) \end{array} \right].\end{aligned}$$ It was shown in the proof of [@RoS Theorem 2.6] (see [@RoS (2.29)]) that the Weyl function $\vp(z)$ of the system , (in $\BC_-$) is given by the formula $$\begin{aligned} \label{R9}& \vp(z)=b(-1/z)d(-1/z)^{-1}. \end{aligned}$$ Relations , and – imply the following theorem. \[Tm1\] Let Dirac system , be a system with the strictly pseudo-exponential potential $\{C_k\}$. Then the Weyl function $\vp(z)$ is the unique analytic continuation of the reflection coefficient $\clr(z)$ of this system. That is, the reflection coefficient and the Weyl function are given by the same rational matrix function. From [@RoS Theorem 2.6] and Theorem \[Tm1\] we derive the following corollary. \[Cy2\] Let the potential $\{C_k\}$ be determined by a strongly admissible triple $\{A, \, S_0, \, \Pi_0\}$. Then, the reflection coefficient of the Dirac system , is given by the formula $$\label{R10} \clr(z)=-{\mathrm{i}}z\vt_1^S_0^{-1}(I_n+zA^{\times})^{-1}\vt_2, \quad A^{\times}=A+{\mathrm{i}}\vt_2 \vt_2^S_0^{-1}.$$ Reflection coefficients: skew-self-adjoint case {#Refl2} ------------------------------------------------ In the skew-self-adjoint case, we define the reflection coefficient $\clr(z)$ in a slightly more general way than in the self-adjoint case. That is, we consider the matrix valued $m_2 \times m$ solution $Y$ of the system : $$\label{R11} Y_k(z)=\big(1-\frac{{\mathrm{i}}}{z}\big)^k\left(\begin{bmatrix} 0 \\ I_{m_2}\end{bmatrix}+o(1)\right), \quad k\to \infty ,$$ and set $$\begin{aligned} & \label{R12} \clr(z)= \begin{bmatrix} I_{m_1} & 0 \end{bmatrix} Y_0(z)\left(\begin{bmatrix} 0 & I_{m_2} \end{bmatrix}Y_0(z)\right)^{-1}.\end{aligned}$$ In order to express $\clr(z)$ via a strongly admissible triple $\{A, \, S_0, \, \Pi_0\}$, we derive from , , and the representation $$\begin{aligned} & \label{R13} w_{A}(k,-z )\begin{bmatrix} 0 \\ I_{m_2}\end{bmatrix}= \begin{bmatrix} 0 \\ I_{m_2}\end{bmatrix} -{\mathrm{i}}\begin{bmatrix} \vt_1^\big(\wt G(A)^k\big)^Q_k^{-1}(z I_n+A)^{-1}\vt_2 \\ \vt_2^Q_k^{-1}(z I_n+A )^{-1}\vt_2 \end{bmatrix},\end{aligned}$$ where $\wt G$ is introduced in . Formulas and imply that $$\begin{aligned} & \label{R14} \lim_{k \to \infty} w_{A}(k,-z )\begin{bmatrix} 0 \\ I_{m_2}\end{bmatrix}=\begin{bmatrix} 0 \\ I_{m_2}\end{bmatrix}.\end{aligned}$$ It follows from , and that $$\begin{aligned} & \label{R15} Y_k(z)=\big(1-\frac{{\mathrm{i}}}{z}\big)^k w_{A}(k,-z )\begin{bmatrix} 0 \\ I_{m_2}\end{bmatrix}. \end{aligned}$$ Hence, after we take into account and (similar to the self-adjoint case) partition $w_A$ (as in ), we obtain $$\begin{aligned} & \label{R16} \clr(z)=b(-z)d(-z)^{-1}. \end{aligned}$$ On the other hand, according to [@FKKS (3.24)] the Weyl function $\vp(z)$ of the system is also given by the right-hand side of . Thus, the following theorem is proved. \[Tm2\] Let Dirac system be a system with the strictly pseudo-exponential potential $\{C_k\}$. Then the Weyl function $\vp(z)$ is the analytic continuation of the reflection coefficient $\clr(z)$ of this system. 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--- abstract: 'We establish Manin’s conjecture for a quartic del Pezzo surface split over $\mathbb{Q}$ and having a singularity of type $\mathbf{A}_3$ and containing exactly four lines. It is the first example of split singular quartic del Pezzo surface whose universal torsor is not a hypersurface for which Manin’s conjecture is proved.' address: \| Université Denis Diderot (Paris VII)\ Institut de Mathématiques de Jussieu\ UMR 7586\ Case $7012$ - Bâtiment Chevaleret\ Bureau $7$C$14$\ $75205$ Paris Cedex 13, France author: - Pierre Le Boudec bibliography: - 'biblio.bib' title: 'Manin’s conjecture for a quartic del Pezzo surface with $\mathbf{A}_3$ singularity and four lines' --- \#1[10mu([@font mod]{}\#1)]{} Introduction ============ Manin’s conjecture (see [@MR974910]) gives a precise description of the distribution of rational points of bounded height on singular del Pezzo surfaces. More precisely, let $V \subset \mathbb{P}^n$ be such a surface defined over $\mathbb{Q}$ and anticanonically embedded and $U$ be the open subset formed by deleting the lines from $V$. We set $$\begin{aligned} N_{U,H}(B) & = & \# \{x \in U(\mathbb{Q}), H(x) \leq B \} \textrm{,}\end{aligned}$$ where $H : \mathbb{P}^n(\mathbb{Q}) \to \mathbb{R}_{> 0}$ is the exponential height defined by $$\begin{aligned} H(x_0: \dots :x_n) & = & \max \{ \|x_i\|, 0 \leq i \leq n \}\textrm{,}\end{aligned}$$ for $(x_0, \dots, x_n) \in \mathbb{Z}^{n+1}$ satisfying the condition $\gcd(x_0, \dots, x_n) = 1$. If $\widetilde{V}$ denotes the minimal desingularization of $V$ and $\rho = \rho_{\widetilde{V}}$ the rank of the Picard group of $\widetilde{V}$, then it is expected that $$\begin{aligned} N_{U,H}(B) & = & c_{V,H} B \log(B)^{\rho -1} (1+o(1)) \textrm{,}\end{aligned}$$ where $c_{V,H}$ is a constant which is expected to follow Peyre’s prediction [@MR1340296]. We are only interested here in singular del Pezzo surfaces of degree four. Their classification is rather classical and can be found in the work of Coray and Tsfasman [@MR940430]. Up to isomorphism over $\overline{\mathbb{Q}}$, there are fifteen types of such surfaces and they are categorized by their extended Dynkin diagrams which are the diagrams describing the intersection behaviour of the negative curves on the minimal desingularizations (see [@D-hyper Table $4$]). Here is a quick overview of the available results concerning Manin’s conjecture for singular quartic del Pezzo surfaces split over $\mathbb{Q}$. The conjecture is already known to hold for nine surfaces of different types. Using harmonic analysis techniques on adelic groups and studying the height Zeta function $$\begin{aligned} Z_{U,H}(s) & = & \sum_{x \in U(\mathbb{Q})} H(x)^{-s} \textrm{,}\end{aligned}$$ Batyrev and Tschinkel have proved it for toric varieties [@MR1620682] (which covers the three types $4 \mathbf{A}_1$, $2 \mathbf{A}_1 + \mathbf{A}_2$ and $2 \mathbf{A}_1 + \mathbf{A}_3$) and Chambert-Loir and Tschinkel have proved it for equivariant compactifications of vector groups [@MR1906155] (which covers the type $\mathbf{D}_5$). Note that for a certain surface of type $\mathbf{D}_5$, la Bretèche and Browning have proved the conjecture independently [@MR2320172]. Finally, the conjecture has been obtained for five other surfaces, a surface of type $\mathbf{D}_4$ by Derenthal and Tschinkel [@MR2290499], a surface of type $\mathbf{A}_1 + \mathbf{A}_3$ by Derenthal [@MR2520770], a surface of type $\mathbf{A}_4$ by Browning and Derenthal [@MR2543667] and two surfaces of respective types $3 \mathbf{A}_1$ and $\mathbf{A}_1 + \mathbf{A}_2$ by the author [@3A1]. These proofs are very different from those using the fact that the varieties considered are equivariant compactifications of algebraic groups. They all use a lift to universal torsors. This consists in defining a bijection between the set of rational points to be counted on $U$ and a certain set of integral points on an affine variety of higher dimension (which is equal to eight for quartic surfaces). Note that Derenthal has determined the equations of the universal torsors for most of the singular quartic del Pezzo surfaces in his doctoral thesis [@Der-th]. This can also be achieved using only elementary techniques, see section \[torsor section\] for an example. Our aim is to prove Manin’s conjecture for another surface split over $\mathbb{Q}$, having singularity type $\mathbf{A}_3$ and containing exactly four lines. This surface $V \subset \mathbb{P}^4$ is defined as the intersection of the two following quadrics, $$\begin{aligned} x_0 x_1 - x_2^2 & = & 0 \textrm{,} \\ (x_0 + x_1 + x_3) x_3 - x_2 x_4 & = & 0 \textrm{.}\end{aligned}$$ The lines on $V$ are given by $x_i = x_2 = x_3 = 0$ and $x_i = x_2 = x_0 + x_1 + x_3 = 0$ for $i \in \{0,1\}$ and the unique singularity is $(0:0:0:0:1)$. We see that $V$ is actually split over $\mathbb{Q}$ and thus, if $\widetilde{V}$ denotes the minimal desingularization of $V$, the Picard group of $\widetilde{V}$ has rank $\rho = 6$. Define the open subset $U$ and the quantity $N_{U,H}(B)$ as explained above. In section \[torsor section\], we define a bijection between the set of the points to be counted on $U$ and a certain set of integral points of an open subset of the affine variety embedded in $\mathbb{A}^{10} \simeq {\operatorname{Spec}}\left( \mathbb{Q}[\eta_1, \dots, \eta_7, \alpha_1, \alpha_2, \alpha_4] \right)$ and defined by $$\begin{aligned} \eta_1^2\eta_2\eta_4^2\eta_7 + \eta_5 \alpha_1 - \eta_6 \alpha_2 & = & 0 \textrm{,} \\ \eta_2 \eta_3^2 \eta_5^2 \eta_6 + \eta_7 \alpha_2 - \eta_4 \alpha_4 & = & 0 \textrm{.}\end{aligned}$$ The universal torsor corresponding to our present problem actually has five equations and can be embedded in $\mathbb{A}^{11} \simeq {\operatorname{Spec}}\left( \mathbb{Q}[\eta_1, \dots, \eta_7, \alpha_1, \alpha_2, \alpha_3, \alpha_4] \right)$ but we will neither use these three other equations nor the variable $\alpha_3$. Let us emphasize the fact that it is the first time that Manin’s conjecture is proved for a split singular quartic del Pezzo surface whose universal torsor has several equations. This obstacle is overcome in section \[First steps\] by turning the two equations into a single congruence in order to apply the usual techniques. Our result is the following. \[Manin\] As $B$ tends to $+ \infty$, we have the estimate $$\begin{aligned} N_{U,H}(B) & = & c_{V,H} B \log(B)^{5} \left( 1 + O \left( \frac1{\log(B)} \right) \right) \textrm{,}\end{aligned}$$ where $c_{V,H}$ agrees with Peyre’s prediction. Since $\rho = 6$, this estimate proves that $V$ satisfies Manin’s conjecture. Let us note here that Derenthal has proved that $V$ is not toric [@D-hyper Proposition 12] and Derenthal and Loughran have proved that it is not an equivariant compactification of $\mathbb{G}_a^2$ [@DL-equi], so theorem \[Manin\] does not follow from the general results [@MR1620682] and [@MR1906155]. In view of this result, it only remains to deal with five types of split singular quartic del Pezzo surfaces among the list of fifteen. In the following section, we prove several lemmas about summations of arithmetic functions. The next two sections are respectively devoted to the calculations of the universal torsor and of Peyre’s constant. Finally, the last section is dedicated to the proof of theorem \[Manin\]. It is a great pleasure for the author to thank his supervisor Professor de la Bretèche both for his encouragement and his advice during this work. This work has received the financial support of the ANR PEPR (Points Entiers Points Rationnels). Arithmetic functions ==================== We need to introduce the following collection of arithmetic functions, $$\begin{aligned} \varphi^{\ast}(n) = \prod_{p\|n} \left( 1 - \frac1{p} \right) \textrm{,} & \ & \varphi^{\circ}(n) = \prod_{\substack{p\|n \\ p \neq 2}} \left( 1 - \frac1{p-1} \right) \textrm{,} \\ \varphi^{\dag}(n) = \prod_{p\|n} \left( 1 - \frac1{p^2} \right) \textrm{,} & \ & \varphi^{\flat}(n) = \prod_{\substack{p\|n \\ p \neq 2}} \left( 1 + \frac1{p(p-2)} \right) \textrm{.}\end{aligned}$$ We can note here that if $n$ is odd then $\varphi^{\circ}(n) \varphi^{\flat}(n) = \varphi^{\ast}(n)$ and if $n$ is even then $\varphi^{\circ}(n) \varphi^{\flat}(n) = 2 \varphi^{\ast}(n)$. Moreover, for $a,b \geq 1$, we define $$\begin{aligned} \psi_{a,b}(n) & = & \begin{cases} \varphi^{\circ}(\gcd(a,n))^{-1} & \textrm{ if } \gcd(n,b) = 1 \textrm{,} \\ 0 & \textrm{ otherwise,} \end{cases}\end{aligned}$$ and $$\begin{aligned} \psi_{a,b}'(n) & = & \begin{cases} \varphi^{\circ}(\gcd(a,n))^{-1} \varphi^{\ast}(n) \varphi^{\ast}(\gcd(a,n))^{-1} & \textrm{ if } \gcd(n,b) = 1 \textrm{,} \\ 0 & \textrm{ otherwise.} \end{cases}\end{aligned}$$ Finally, for $\delta > 0$, we set $$\begin{aligned} \sigma_{- \delta}(n) & = & \sum_{k\|n} k^{- \delta} \textrm{.}\end{aligned}$$ \[arithmetic preliminary 0\] Let $0 < \delta \leq 1$ be fixed. We have the estimate $$\begin{aligned} \sum_{n \leq X} \psi_{a,b}(n)& = & \Psi(a,b) X + O_{\delta} \left( \sigma_{- \delta}(ab) X^{\delta} \right) \textrm{,}\end{aligned}$$ where $$\begin{aligned} \Psi(a,b) & = & \varphi^{\ast}(b) \frac{\varphi^{\flat}(a)}{\varphi^{\flat}(\gcd(a,b))} \textrm{.}\end{aligned}$$ We start by calculating the Dirichlet convolution of $\psi_{a,b}$ with the Möbius function $\mu$. We have $$\begin{aligned} (\psi_{a,b} \ast \mu)(n)& = & \sum_{d\|n} \psi_{a,b} \left( \frac{n}{d} \right) \mu(d) \\ & = & \prod_{p^\nu \parallel n} \left( \psi_{a,b} \left( p^\nu \right) - \psi_{a,b} \left( p^{\nu - 1} \right) \right) \textrm{.}\end{aligned}$$ Moreover $\psi_{a,b}(1) = 1$ and for all $\nu \geq 1$, we have $$\begin{aligned} \psi_{a,b} \left( p^\nu \right) = \psi_{a,b}(p) = \begin{cases} \left( 1 - 1/(p-1) \right)^{-1} & \textrm{ if } p\|a, p \neq 2 \textrm{ and } p \nmid b \textrm{,} \\ 1 & \textrm{ if } p \neq 2, p \nmid ab \textrm{,} \\ 1 & \textrm{ if } p = 2, 2 \nmid b \textrm{,} \\ 0 & \textrm{ if } p\|b \textrm{.} \end{cases}\end{aligned}$$ Thus, we easily obtain $$\begin{aligned} (\psi_{a,b} \ast \mu)(n) & = & \mu(n) \prod_{p\|\gcd(a,n), p\nmid b} \frac{-1}{p-2} \textrm{,}\end{aligned}$$ if $n\|ab$ and $2 \nmid n$ or $2\|b$ and $(\psi_{a,b} \ast \mu)(n) = 0$ otherwise. Writing $\psi_{a,b} = (\psi_{a,b} \ast \mu) \ast 1$, we get $$\begin{aligned} \sum_{n \leq X} \psi_{a,b}(n) & = & \sum_{n \leq X} \sum_{d\|n} (\psi_{a,b} \ast \mu)(d) \\ & = & \sum_{d = 1}^{+ \infty} (\psi_{a,b} \ast \mu)(d) \left[ \frac{X}{d} \right] \textrm{.}\end{aligned}$$ Let $0 < \delta \leq 1$ be fixed. Let us use the elementary estimate $[t] = t + O \left( t^{\delta} \right)$ for $t = X/d$. Since $\|(\psi_{a,b} \ast \mu)(n)\| \leq 1$, we get $$\begin{aligned} \sum_{d = 1}^{+ \infty} \frac{\| (\psi_{a,b} \ast \mu)(d) \|}{d^{\delta}} & \leq & \sigma_{- \delta}(ab) \textrm{,}\end{aligned}$$ and we have thus proved that $$\begin{aligned} \sum_{n \leq X} \psi_{a,b}(n) & = & X \sum_{d = 1}^{+ \infty} \frac{(\psi_{a,b} \ast \mu)(d)}{d} + O \left( \sigma_{- \delta}(ab) X^{\delta} \right) \textrm{.}\end{aligned}$$ Finally, a straigthforward calculation gives $$\begin{aligned} \sum_{d = 1}^{+ \infty} \frac{(\psi_{a,b} \ast \mu)(d)}{d} & = & \prod_{p\|b} \left( 1 - \frac1{p} \right) \prod_{\substack{p\|a, p \nmid b \\ p \neq 2}} \left( 1 + \frac1{p(p-2)} \right) \textrm{,}\end{aligned}$$ which concludes the proof. \[arithmetic preliminary 0’\] Let $0 < \delta \leq 1$ be fixed. We have the estimate $$\begin{aligned} \sum_{n \leq X} \psi_{a,b}'(n)& = & \Psi'(a,b) X + O_{\delta} \left( \sigma_{- \delta}(b) X^{\delta} \right) \textrm{,}\end{aligned}$$ where $$\begin{aligned} \Psi'(a,b) & = & \varphi^{\ast}(b) \frac{\varphi^{\flat}(a)}{\varphi^{\flat}(\gcd(a,b))} \frac{\zeta(2)^{-1}}{\varphi^{\dag}(ab)} \textrm{.}\end{aligned}$$ We proceed exactly as for the proof of lemma \[arithmetic preliminary 0\]. Let $$\begin{aligned} f(n) & = & \mu(n) \prod_{\substack{p\|n, p\nmid ab \\ p \neq 2}} \frac1{p} \prod_{\substack{p\|\gcd(a,n), p\nmid b \\ p \neq 2}} \frac{-1}{p-2} \textrm{.}\end{aligned}$$ A calculation provides $$\begin{aligned} (\psi_{a,b}' \ast \mu)(n) & = & \begin{cases} f(n) & \textrm{ if } 2 \nmid n \textrm{ or } 2\|b \textrm{,} \\ f(n)/2 & \textrm{ if } 2\|n \textrm{ and } 2 \nmid ab \textrm{,} \\ 0 & \textrm{ otherwise.} \end{cases}\end{aligned}$$ Now we see that $\| (\psi_{a,b}' \ast \mu)(n) \| \ll \gcd(b,n)/n$, which easily yields $$\begin{aligned} \sum_{d = 1}^{+ \infty} \frac{\| (\psi_{a,b}' \ast \mu)(d) \|}{d^{\delta}} & \ll & \sigma_{- \delta}(b) \textrm{.}\end{aligned}$$ Another straightforward calculation gives $$\begin{aligned} \sum_{d = 1}^{+ \infty} \frac{(\psi_{a,b}' \ast \mu)(d)}{d} & = & \Psi'(a,b) \textrm{,}\end{aligned}$$ which completes the proof. Using partial summation and the estimates of lemmas \[arithmetic preliminary 0\] and \[arithmetic preliminary 0’\] as in the proof of [@3A1 Lemma $6$], we see that we have the following result. \[arithmetic preliminary\] Let $0 < \delta \leq 1$ be fixed. Let $0 \leq t_1 < t_2$ and $I=[t_1,t_2]$. Let also $g : \mathbb{R}_{> 0} \to \mathbb{R}$ be a function having a piecewise continuous derivative on $I$ whose sign changes at most $R_g(I)$ times on $I$. We have $$\begin{aligned} \sum_{n \in I \cap \mathbb{Z}_{>0}} \psi_{a,b}(n) g(n) & = & \Psi(a,b) \int_I g(t) {\mathrm{d}}t + O_{\delta} \left( \sigma_{- \delta}(ab) t_2^{\delta} M_I(g) \right) \textrm{,}\end{aligned}$$ and $$\begin{aligned} \sum_{n \in I \cap \mathbb{Z}_{>0}} \psi_{a,b}'(n) g(n) & = & \Psi'(a,b) \int_I g(t) {\mathrm{d}}t + O_{\delta} \left( \sigma_{- \delta}(b) t_2^{\delta} M_I(g) \right) \textrm{,}\end{aligned}$$ where $M_I(g) = (1 + R_g(I)) \sup_{t \in I \cap \mathbb{R}_{> 0}} \|g(t)\|$. We also have the following estimation. \[arithmetic preliminary 2\] With the same notations, if $2 \nmid b$ then $$\begin{aligned} \sum_{\substack{n \in I \cap \mathbb{Z}_{>0} \\ n \equiv 0 \imod{2}}} \psi_{a,b}(n) g(n) & = & \frac1{2} \Psi(a,b) \int_I g(t) {\mathrm{d}}t + O_{\delta} \left( \sigma_{- \delta}(ab) t_2^{\delta} M_I(g) \right) \textrm{.}\end{aligned}$$ In a similar way, if $2\|a$ and $2 \nmid b$ then $$\begin{aligned} \sum_{\substack{n \in I \cap \mathbb{Z}_{>0} \\ n \equiv 0 \imod{2}}} \psi_{a,b}'(n) g(n) & = & \frac1{2} \Psi'(a,b) \int_I g(t) {\mathrm{d}}t + O_{\delta} \left( \sigma_{- \delta}(b) t_2^{\delta} M_I(g) \right) \textrm{.}\end{aligned}$$ Let us prove the statement for $\psi_{a,b}$, it suffices to notice that $$\begin{aligned} \sum_{\substack{n \leq X \\ n \equiv 0 \imod{2}}} \psi_{a,b}(n) & = & \sum_{d = 1}^{+ \infty} (\psi_{a,b} \ast \mu)(d) \sum_{\substack{k \leq X/d \\ k \equiv 0 \imod{2}}} 1 \\ & & + \sum_{\substack{d = 1 \\ d \equiv 0 \imod{2}}}^{+ \infty} (\psi_{a,b} \ast \mu)(d) \sum_{\substack{k \leq X/d \\ k \equiv 1 \imod{2}}} 1 \textrm{,}\end{aligned}$$ and $(\psi_{a,b} \ast \mu)(d) = 0$ for all $d \equiv 0 \imod{2}$ since $2 \nmid b$ and therefore $$\begin{aligned} \sum_{\substack{n \leq X \\ n \equiv 0 \imod{2}}} \psi_{a,b}(n) & = & \sum_{d = 1}^{+ \infty} (\psi_{a,b} \ast \mu)(d) \left( \frac{X}{2 d} + O \left( \frac{X^{\delta}}{d^{\delta}} \right) \right) \textrm{.}\end{aligned}$$ We can conclude exactly as in the proof of lemma \[arithmetic preliminary 0\] and finally, as for lemma \[arithmetic preliminary\], use partial summation to complete the proof. The proof for $\psi_{a,b}'$ is strictly identical, it only uses the fact that $(\psi_{a,b}' \ast \mu)(d) = 0$ for all $d \equiv 0 \imod{2}$ since $2\|a$ and $2 \nmid b$. The universal torsor {#torsor section} ==================== We now proceed to define a bijection between the set of rational points we want to count on $U$ and a certain set of integral points on the affine variety defined in the introduction. As explained in the introduction, the universal torsor of our problem is an open subset of an affine variety of dimension $8$ embedded in $\mathbb{A}^{11}$. It has five equations but we will only deal with ten of the eleven variables and will only make use of two equations among these five. Our choice of notation might be surprising but it is guided by our wish to adopt the notation used by Derenthal in [@Der-th Chapter $6$]. Note that if $(x_0:x_1:x_2:x_3:x_4) \in V(\mathbb{Q})$ then we have $(x_0:x_1:x_2:x_3:x_4) \in U(\mathbb{Q})$ if and only if $x_0x_1x_2x_3 \neq 0$. Let $(x_0,x_1,x_2,x_3,x_4) \in \mathbb{Z}_{\neq 0}^4 \times \mathbb{Z}$ be such that $$\begin{aligned} x_0 x_1 - x_2^2 & = & 0 \textrm{,} \\ (x_0 + x_1 + x_3) x_3 - x_2 x_4 & = & 0 \textrm{,}\end{aligned}$$ and $\max \{ \|x_i\|, 0 \leq i \leq 4 \} \leq B$ and $\gcd(x_0,x_1,x_2,x_3,x_4) = 1$. Since $\mathbf{x} = - \mathbf{x}$ in $\mathbb{P}^4$, we can assume that $x_0 > 0$, which implies $x_1 > 0$. Moreover, the symmetry given by $(x_2,x_4) \mapsto (-x_2,-x_4)$ shows that we can also assume that $x_2 > 0$ keeping in mind that we need to multiply our future result by $2$. The first equation shows that there is a unique way to write $x_0 = y_{01} x_0'^2$, $x_1 = y_{01} x_1'^2$ and $x_2 = y_{01} x_0' x_1'$ for some $x_0',x_1',y_{01} > 0$ such that $\gcd(x_0',x_1') = 1$. The second equation therefore gives $$\begin{aligned} \left( y_{01} x_0'^2+ y_{01} x_1'^2 + x_3 \right) x_3 - y_{01} x_0' x_1' x_4 & = & 0 \textrm{.}\end{aligned}$$ We define $y_{01}' = \gcd(y_{01}, x_3) > 0$ and write $y_{01} = y_{01}' \eta_2$ and $x_3 = y_{01}' x_3'$ with $\eta_2>0$ and $\gcd(\eta_2, x_3') = 1$. We obtain $$\begin{aligned} \left( \eta_2 x_0'^2+ \eta_2 x_1'^2 + x_3' \right) y_{01}' x_3' - \eta_2 x_0' x_1' x_4 & = & 0 \textrm{,}\end{aligned}$$ and thus $\eta_2\|y_{01}' x_3'^2$ and it follows $\eta_2\|y_{01}'$ since $\gcd(\eta_2,x_3') = 1$. We can therefore write $y_{01}' = \eta_2 y_{01}''$ for some $y_{01}'' > 0$. The equation becomes $$\begin{aligned} \left( \eta_2 x_0'^2+ \eta_2 x_1'^2 + x_3' \right) y_{01}'' x_3' - x_0' x_1' x_4 & = & 0 \textrm{.}\end{aligned}$$ We now see that $\gcd(x_0,x_1,x_2,x_3,x_4) = 1$ implies $\gcd(y_{01}'',x_4) = 1$ and thus $y_{01}''\|x_0' x_1'$ and $x_0'$, $x_1'$ being coprime, we can write $y_{01}'' = \eta_1 \eta_3$, $x_0' = \eta_3 x_0''$ and $x_1' = \eta_1 x_1''$ for some $\eta_1, \eta_3, x_0'', x_1'' > 0$. Now we set $x_3' = \alpha_1 x_3''$, $x_4 = \alpha_1 \alpha_4$ with $x_3'' > 0$ and $\gcd(x_3'',\alpha_4) = 1$ (we do not prescribe the sign of $\alpha_1=\pm \gcd(x_3',x_4)$). We finally get $$\begin{aligned} \left( \eta_2 \eta_3^2 x_0''^2 + \eta_2 \eta_1^2 x_1''^2 + \alpha_1 x_3'' \right) x_3'' - x_0'' x_1'' \alpha_4 & = & 0 \textrm{.}\end{aligned}$$ We observe that since $\gcd(x_3'',\alpha_4) = 1$, we have $x_3''\|x_0''x_1''$ and we can write $x_3'' = \eta_5 \eta_7$, $x_0'' = \eta_5 \eta_6$ and $x_1'' = \eta_4 \eta_7$, for some $\eta_4,\eta_5,\eta_6,\eta_7 > 0$. We have finally obtained $$\begin{aligned} x_0 & = & \eta_1 \eta_2^2 \eta_3^3 \eta_5^2 \eta_6^2 \textrm{,} \\ x_1 & = & \eta_1^3 \eta_2^2 \eta_3 \eta_4^2 \eta_7^2 \textrm{,} \\ x_2 & = & \eta_1^2 \eta_2^2 \eta_3^2 \eta_4 \eta_5 \eta_6 \eta_7 \textrm{,} \\ x_3 & = & \eta_1 \eta_2 \eta_3 \eta_5 \eta_7 \alpha_1 \textrm{,} \\ x_4 & = & \alpha_1 \alpha_4 \textrm{,}\end{aligned}$$ and the equation is $$\begin{aligned} \eta_2 \eta_3^2 \eta_5^2 \eta_6^2 + \eta_1^2 \eta_2 \eta_4^2 \eta_7^2 + \eta_5 \eta_7 \alpha_1 - \eta_4 \eta_6 \alpha_4 & = & 0 \textrm{.}\end{aligned}$$ Furthermore, it is easy to see that the coprimality conditions can be summed up by $$\begin{aligned} \label{coprim1} & & \gcd(\eta_3\eta_5\eta_6,\eta_1\eta_4\eta_7) = 1 \textrm{,} \\ \label{coprim2} & & \gcd(\eta_5\eta_7,\eta_2\alpha_4) = 1 \textrm{,} \\ \label{coprim3} & & \gcd(\eta_1\eta_2\eta_3, \alpha_1\alpha_4) = 1 \textrm{.}\end{aligned}$$ Since $\eta_6$ and $\eta_7$ are coprime, we see that the equation is equivalent to the existence of $\alpha_2 \in \mathbb{Z}$ such that $$\begin{aligned} \label{torsor 1} \eta_1^2 \eta_2 \eta_4^2 \eta_7 + \eta_5 \alpha_1 - \eta_6 \alpha_2 & = & 0 \textrm{,} \\ \label{torsor 2} \eta_2 \eta_3^2 \eta_5^2 \eta_6 + \eta_7 \alpha_2 - \eta_4 \alpha_4 & = & 0 \textrm{.}\end{aligned}$$ In a similar way, since $\eta_4$ and $\eta_5$ are coprime, we can derive the existence of $\alpha_3 \in \mathbb{Z}$ such that $$\begin{aligned} \eta_2 \eta_3^2 \eta_5 \eta_6^2 + \eta_7 \alpha_1 - \eta_4 \alpha_3 & = & 0 \textrm{,} \\ \eta_1^2 \eta_2 \eta_4 \eta_7^2 + \eta_5 \alpha_3 - \eta_6 \alpha_4 & = & 0 \textrm{,} \\ \eta_1^2 \eta_2^2 \eta_3^2 \eta_4 \eta_5 \eta_6 \eta_7 + \alpha_1 \alpha_4 - \alpha_2 \alpha_3 & = & 0 \textrm{.}\end{aligned}$$ As explained above, we will not use these three equations. We define $\mathcal{T}(B)$ as the set of $(\eta_1,\eta_2,\eta_3,\eta_4,\eta_5,\eta_6,\eta_7,\alpha_1,\alpha_2,\alpha_4) \in \mathbb{Z}_{>0}^7 \times \mathbb{Z}^3$ satisfying the coprimality conditions , , , the two equations and and finally the height conditions $$\begin{aligned} \label{condition1} \eta_1 \eta_2^2 \eta_3^3 \eta_5^2 \eta_6^2 & \leq & B \textrm{,} \\ \label{condition2} \eta_1^3 \eta_2^2 \eta_3 \eta_4^2 \eta_7^2 & \leq & B \textrm{,} \\ \label{condition3} \eta_1 \eta_2 \eta_3 \eta_5 \eta_7 \|\alpha_1\| & \leq & B \textrm{,} \\ \label{condition4} \|\alpha_1 \alpha_4\| & \leq & B \textrm{.}\end{aligned}$$ We have proved the following lemma. \[T\] We have the equality $$\begin{aligned} N_{U,H}(B) & = & 2 \# \mathcal{T}(B) \textrm{.}\end{aligned}$$ Calculation of Peyre’s constant =============================== We calculate the value of the constant $c_{V,H}$ predicted by Peyre. It is defined by $$\begin{aligned} c_{V,H} & = & \alpha(\widetilde{V}) \beta(\widetilde{V}) \omega_H(\widetilde{V}) \textrm{,}\end{aligned}$$ where $\alpha(\widetilde{V}) \in \mathbb{Q}$ is the volume of a certain polytope in the dual of the effective cone of $\widetilde{V}$ with respect to the intersection form, $\beta(\widetilde{V}) = \# H^1({\operatorname{Gal}}(\overline{\mathbb{Q}}/\mathbb{Q}), {\operatorname{Pic}}_{\overline{\mathbb{Q}}}(\widetilde{V})) = 1$ since $V$ is split over $\mathbb{Q}$ and finally $$\begin{aligned} \omega_H(\widetilde{V}) & = & \omega_{\infty} \prod_p \left( 1 - \frac1{p} \right)^6 \omega_p \textrm{,}\end{aligned}$$ where $\omega_{\infty}$ and $\omega_p$ are respectively the archimedean and $p$-adic densities. The work of Derenthal [@MR2318651] provides the value $$\begin{aligned} \alpha(\widetilde{V}) & = & \frac1{4320} \textrm{.}\end{aligned}$$ Furthermore, using [@Loughran Lemma 2.3], we get $$\begin{aligned} \omega_p & = & 1 + \frac{6}{p} + \frac1{p^2} \textrm{.}\end{aligned}$$ To calculate $\omega_{\infty}$, we set $f_1(x) = x_0 x_1 - x_2^2$, $f_2(x) = (x_0 + x_1 + x_3) x_3 - x_2 x_4$ and we parametrize the points of $V$ by $x_0$, $x_2$ and $x_3$. We have $$\begin{aligned} \det \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_4} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_4} \end{pmatrix} & = & \begin{vmatrix} x_0 & 0 \\ x_3 & -x_2 \end{vmatrix} \\ & = & - x_0 x_2 \textrm{.}\end{aligned}$$ Moreover, $x_1 = x_2^2/x_0$ and $x_4 = (x_0^2 + x_2^2 + x_0x_3) x_3 / ( x_0 x_2 )$. Since $\mathbf{x} = - \mathbf{x}$ in $\mathbb{P}^4$, we have $$\begin{aligned} \omega_{\infty} & = & 2 \int \int \int_{x_0, x_2 > 0, x_0, x_2^2/x_0, \|x_3\|, \left\|x_0^2 + x_2^2 + x_0x_3 \right\| \left\| x_3 \right\| / x_0 x_2 \leq 1} \frac{{\mathrm{d}}x_0 {\mathrm{d}}x_2 {\mathrm{d}}x_3}{x_0x_2} \textrm{.}\end{aligned}$$ Define the function $$\begin{aligned} \label{equation h} h & : & (u_2,t_7,t_6) \mapsto \max \{t_6,t_7, t_7 \|t_7 - t_6 u_2\|, \|t_7 - t_6 u_2\| \|t_6 + t_7 u_2\| \} \textrm{.}\end{aligned}$$ The change of variables given by $x_0 = t_6^2$, $x_2 = t_6 t_7$ and $x_3 = - t_7 (t_7 - t_6 u_2)$ yields $$\begin{aligned} \label{omega} \omega_{\infty} & = & 4 \int \int \int_{t_6,t_7>0, h(u_2,t_7,t_6) \leq 1} {\mathrm{d}}u_2 {\mathrm{d}}t_7 {\mathrm{d}}t_6 \textrm{.}\end{aligned}$$ Proof of the main theorem ========================= First steps of the proof {#First steps} ------------------------ The idea of the proof is to see the equations and as congruences respectively modulo $\eta_5$ and $\eta_4$ and then to count the number of $\alpha_2$ satisfying these two congruences. In order to do so, we replace the height conditions and by $$\begin{aligned} \eta_1 \eta_2 \eta_3 \eta_7 \left\| \eta_1^2 \eta_2 \eta_4^2 \eta_7 - \eta_6 \alpha_2 \right\| & \leq & B \textrm{,} \\ \eta_4^{-1} \eta_5^{-1} \left\| \eta_1^2 \eta_2 \eta_4^2 \eta_7 - \eta_6 \alpha_2 \right\| \left\| \eta_2 \eta_3^2 \eta_5^2 \eta_6 + \eta_7 \alpha_2 \right\| & \leq & B \textrm{,}\end{aligned}$$ and we carry on denoting them the same way. We note that the equation proves that we necessarily have $\gcd(\eta_1\eta_2,\eta_6\alpha_2) = 1$ since we also have $\gcd(\eta_1\eta_2,\eta_5\alpha_1)~=~1$. Exactly the same way we get $\gcd(\alpha_2,\eta_3\eta_5) = 1$ thanks to the equation and $\gcd(\eta_3\eta_5,\eta_4\alpha_4) = 1$. The equation and $\gcd(\eta_2,\eta_7\alpha_2) = 1$ also imply $\gcd(\eta_2,\eta_4) = 1$. This new coprimality condition together with the equation yield $\gcd(\eta_4,\alpha_2) = 1$ since we have $\gcd(\eta_4,\eta_2\eta_3\eta_5\eta_6) = 1$. In a similar way, we finally obtain $\gcd(\alpha_1,\eta_4\eta_6) = 1$, $\gcd(\eta_4,\eta_7) = 1$ and $\gcd(\eta_5,\eta_6) = 1$. We can therefore rewrite the coprimality conditions , , and all these new conditions as $$\begin{aligned} \label{gcd1} & & \gcd(\alpha_1,\eta_1\eta_2\eta_3\eta_4\eta_6) = 1 \textrm{,} \\ \label{gcd2} & & \gcd(\alpha_4,\eta_1\eta_2\eta_3\eta_5\eta_7) = 1 \textrm{,} \\ \label{gcd3} & & \gcd(\alpha_2,\eta_1\eta_2\eta_3\eta_4\eta_5) = 1 \textrm{,} \\ \label{gcd4} & & \gcd(\eta_7,\eta_2\eta_3\eta_4\eta_5\eta_6) = 1 \textrm{,} \\ \label{gcd5} & & \gcd(\eta_6,\eta_1\eta_2\eta_4\eta_5) = 1 \textrm{,} \\ \label{gcd6} & & \gcd(\eta_1\eta_4,\eta_3\eta_5) = 1 \textrm{,} \\ \label{gcd7} & & \gcd(\eta_2,\eta_4\eta_5) = 1 \textrm{.}\end{aligned}$$ From now on, we set $\boldsymbol{\eta} = (\eta_1, \eta_2, \eta_3, \eta_4, \eta_5) \in \mathbb{Z}_{>0}^5$ and $\boldsymbol{\eta}' = (\boldsymbol{\eta},\eta_6,\eta_7) \in \mathbb{Z}_{>0}^7$. Consider that $\boldsymbol{\eta}' \in \mathbb{Z}_{>0}^7$ is fixed and is subject to the height conditions , and to the coprimality conditions , , and . Let $N(\boldsymbol{\eta}',B)$ be the number of $(\alpha_1,\alpha_2,\alpha_4) \in \mathbb{Z}$ satisfying the equations , , the height conditions and and finally the coprimality conditions , and . For $(r_1,r_2,r_3,r_4,r_5) \in \mathbb{Q}^5$, we define $$\begin{aligned} \boldsymbol{\eta}^{(r_1,r_2,r_3,r_4,r_5)} & = & \eta_1^{r_1} \eta_2^{r_2} \eta_3^{r_3} \eta_4^{r_4} \eta_5^{r_5} \textrm{,}\end{aligned}$$ and we adopt the following notations in order to help in the understanding of the height conditions, $$\begin{aligned} A_2 & = & \boldsymbol{\eta}^{(1,1,1,1,1)} \textrm{,} \\ Y_6 & = & \frac{B^{1/2}}{\boldsymbol{\eta}^{(1/2,1,3/2,0,1)}} \textrm{,} \\ Y_7 & = & \frac{B^{1/2}}{\boldsymbol{\eta}^{(3/2,1,1/2,1,0)}} \textrm{,}\end{aligned}$$ and recalling the definition of the function $h$, we can sum up the height conditions , , and as $$\begin{aligned} h \left( \frac{\alpha_2}{A_2}, \frac{\eta_7}{Y_7}, \frac{\eta_6}{Y_6} \right) & \leq & 1 \textrm{.}\end{aligned}$$ We also introduce the real-valued functions $$\begin{aligned} g_1 & : & (t_7,t_6) \mapsto \int_{h(u_2,t_7,t_6) \leq 1} {\mathrm{d}}u_2 \textrm{,} \\ g_2 & : & (t_6;\boldsymbol{\eta},B) \mapsto \int_{t_7 Y_7 \geq 1} g_1(t_7,t_6) {\mathrm{d}}t_7 \textrm{,} \\ g_3 & : & (\boldsymbol{\eta},B) \mapsto \int_{t_6 Y_6 \geq 1} g_2(t_6;\boldsymbol{\eta},B) {\mathrm{d}}t_6 \textrm{.}\end{aligned}$$ We obviously have $$\begin{aligned} \label{g_3} g_3(\boldsymbol{\eta},B) & = & \int \int \int_{t_6 Y_6 \geq 1, t_7 Y_7 \geq 1, h(u_2,t_7,t_6) \leq 1} {\mathrm{d}}u_2 {\mathrm{d}}t_7 {\mathrm{d}}t_6 \textrm{.}\end{aligned}$$ \[bounds\] We have the bounds $$\begin{aligned} g_1(t_7,t_6) & \ll & t_6^{-1/2} t_7^{-1/2} \textrm{,} \\ g_2(t_6;\boldsymbol{\eta},B) & \ll & t_6^{-1/2} \textrm{.}\end{aligned}$$ Recall the definition of the function $h$. A little thought reveals that the condition $\|t_7 - t_6 u_2\| \|t_6 + t_7 u_2\| \leq 1$ implies that $u_2$ runs over a set whose measure is $\ll t_6^{-1/2} t_7^{-1/2}$ which gives the first bound. The second bound is an immediate consequence of the first since $t_7 \leq 1$. We have the following result. \[lemma inter\] The following estimate holds $$\begin{aligned} N(\boldsymbol{\eta}',B) & = & \frac{A_2}{\eta_4 \eta_5} g_1 \left( \frac{\eta_7}{Y_7}, \frac{\eta_6}{Y_6} \right) \theta(\boldsymbol{\eta}') + R( \boldsymbol{\eta}', B) \textrm{,}\end{aligned}$$ where $\theta(\boldsymbol{\eta}')$ is a certain arithmetic function given in and $$\begin{aligned} \sum_{\boldsymbol{\eta}'} R( \boldsymbol{\eta}', B) & \ll & B \log(B)^2 \textrm{.}\end{aligned}$$ Let us remove the coprimality conditions and employing two Möbius inversions, we get $$\begin{aligned} N(\boldsymbol{\eta}',B) & = & \sum_{k_1\|\eta_1\eta_2\eta_3\eta_4\eta_6} \mu(k_1) \sum_{k_4\|\eta_1\eta_2\eta_3\eta_5\eta_7} \mu(k_4) S_{k_1,k_4} \textrm{,}\end{aligned}$$ where, with the notations $\alpha_1 = k_1 \alpha_1'$ and $\alpha_4 = k_4 \alpha_4'$, $$\begin{aligned} S_{k_1,k_4} & = & \# \left\{ (\alpha_1',\alpha_4',\alpha_2) \in \mathbb{Z}^3, \begin{array}{l} \eta_1^2 \eta_2 \eta_4^2 \eta_7 + \eta_5 k_1 \alpha_1' - \eta_6 \alpha_2 = 0 \\ \eta_2 \eta_3^2 \eta_5^2 \eta_6 + \eta_7 \alpha_2 - \eta_4 k_4 \alpha_4' = 0 \\ \eqref{condition3}, \eqref{condition4}, \eqref{gcd3} \end{array} \right\} \\ & = & \# \left\{ \alpha_2 \in \mathbb{Z}, \begin{array}{l} \eta_6 \alpha_2 \equiv \eta_1^2 \eta_2 \eta_4^2 \eta_7 \imod{k_1 \eta_5} \\ \eta_7 \alpha_2 \equiv - \eta_2 \eta_3^2 \eta_5^2 \eta_6 \imod{k_4 \eta_4} \\ \eqref{condition3}, \eqref{condition4}, \eqref{gcd3} \end{array} \right\} \textrm{.}\end{aligned}$$ We note that we necessarily have $\gcd(k_1,\eta_6) = 1$ since $\gcd(\eta_6,\eta_1\eta_2\eta_4\eta_7) = 1$ and $\gcd(k_1,\eta_1\eta_2\eta_4) = 1$ since $\gcd(\eta_1\eta_2\eta_4,\eta_6\alpha_2) = 1$. In a similar way, we also have $\gcd(k_4,\eta_2\eta_3\eta_5\eta_7)=1$. In particular, we see that $\eta_6$ and $\eta_7$ are respectively invertible modulo $k_1 \eta_5$ and $k_4 \eta_4$. We therefore get $$\begin{aligned} N(\boldsymbol{\eta}',B) & = & \sum_{\substack{k_1\|\eta_3 \\ \gcd(k_1,\eta_1\eta_2\eta_4\eta_6) = 1}} \mu(k_1) \sum_{\substack{k_4\|\eta_1 \\ \gcd(k_4,\eta_2\eta_3\eta_5\eta_7) = 1}} \mu(k_4) S_{k_1,k_4} \textrm{,}\end{aligned}$$ and $$\begin{aligned} S_{k_1,k_4} & = & \# \left\{ \alpha_2 \in \mathbb{Z}, \begin{array}{l} \alpha_2 \equiv \eta_6^{-1} \eta_1^2 \eta_2 \eta_4^2 \eta_7 \imod{k_1 \eta_5} \\ \alpha_2 \equiv - \eta_7^{-1} \eta_2 \eta_3^2 \eta_5^2 \eta_6 \imod{k_4 \eta_4} \\ \eqref{condition3}, \eqref{condition4}, \eqref{gcd3} \end{array} \right\} \textrm{.}\end{aligned}$$ Furthermore, $k_1\eta_5$ and $k_4\eta_4$ are coprime since $\eta_3\eta_5$ and $\eta_1\eta_4$ are coprime thus the Chinese remainder theorem gives $$\begin{aligned} S_{k_1,k_4} & = & \# \left\{ \alpha_2 \in \mathbb{Z}, \begin{array}{l} \alpha_2 \equiv a \imod{k_1 k_4 \eta_4 \eta_5} \\ \eqref{condition3}, \eqref{condition4}, \eqref{gcd3} \end{array} \right\} \textrm{,}\end{aligned}$$ for a certain integer $a$ coprime to $k_1 k_4 \eta_4 \eta_5$ since $\gcd(k_1k_4\eta_4\eta_5,\alpha_2) = 1$. A Möbius inversion yields $$\begin{aligned} S_{k_1,k_4} & = & \sum_{k_2\|\eta_1\eta_2\eta_3\eta_4\eta_5} \mu(k_2) \# \left\{ \alpha_2' \in \mathbb{Z}, \begin{array}{l} k_2 \alpha_2' \equiv a \imod{k_1 k_4 \eta_4 \eta_5} \\ \eqref{condition3}, \eqref{condition4} \end{array} \right\} \\ & = & \sum_{\substack{k_2\|\eta_1\eta_2\eta_3 \\ \gcd(k_2,k_1k_4\eta_4\eta_5) = 1}} \mu(k_2) \# \left\{ \alpha_2' \in \mathbb{Z}, \begin{array}{l} \alpha_2' \equiv k_2^{-1} a \imod{k_1 k_4 \eta_4 \eta_5} \\ \eqref{condition3}, \eqref{condition4} \end{array} \right\}\textrm{,}\end{aligned}$$ since $\gcd(k_1 k_4 \eta_4 \eta_5,a) = 1$. Using the elementary estimate $$\begin{aligned} \# \left\{ n \in \mathbb{Z} \cap [t_1,t_2], n \equiv a \imod{q} \right\} & = & \frac{t_2 - t_1}{q} + O(1) \textrm{,}\end{aligned}$$ and the change of variable $u_2 \mapsto u_2 A_2/k_2$, we get $$\begin{aligned} \# \left\{ \alpha_2' \in \mathbb{Z}, \begin{array}{l} \alpha_2' \equiv k_2^{-1} a \imod{k_1 k_4 \eta_4 \eta_5} \\ \eqref{condition3}, \eqref{condition4} \end{array} \right\} & = & \frac{A_2}{k_2 k_1 k_4 \eta_4 \eta_5} g_1 \left( \frac{\eta_7}{Y_7},\frac{\eta_6}{Y_6} \right) + O(1) \textrm{.}\end{aligned}$$ We see that the main term of $N(\boldsymbol{\eta}',B)$ is equal to $$\begin{aligned} \frac{A_2}{\eta_4 \eta_5} g_1 \left( \frac{\eta_7}{Y_7},\frac{\eta_6}{Y_6} \right) \theta(\boldsymbol{\eta}') \textrm{,}\end{aligned}$$ where $$\begin{aligned} \theta(\boldsymbol{\eta}') & = & \sum_{\substack{k_1\|\eta_3 \\ \gcd(k_1,\eta_1\eta_2\eta_4\eta_6) = 1}} \frac{\mu(k_1)}{k_1} \sum_{\substack{k_4\|\eta_1 \\ \gcd(k_4,\eta_2\eta_3\eta_5\eta_7) = 1}} \frac{\mu(k_4)}{k_4} \sum_{\substack{k_2\|\eta_1\eta_2\eta_3 \\ \gcd(k_2,k_1k_4\eta_4\eta_5) = 1}} \frac{\mu(k_2)}{k_2} \\ & = & \varphi^{\ast}(\eta_1\eta_2\eta_3\eta_4\eta_5) \sum_{\substack{k_1\|\eta_3 \\ \gcd(k_1,\eta_2\eta_6) = 1}} \frac{\mu(k_1)}{k_1\varphi^{\ast}(k_1\eta_5)} \sum_{\substack{k_4\|\eta_1 \\ \gcd(k_4,\eta_2\eta_7) = 1}} \frac{\mu(k_4)}{k_4\varphi^{\ast}(k_4\eta_4)} \textrm{.}\end{aligned}$$ We have removed $\eta_1\eta_4$ from the condition over $k_1$ and $\eta_3\eta_5$ from the condition over $k_4$ respectively because $\gcd(\eta_3,\eta_1\eta_4) = 1$ and $\gcd(\eta_1,\eta_3\eta_5) = 1$. A straightforward calculation yields, for $a,b,c \geq 1$, $$\begin{aligned} \sum_{\substack{k\|a \\ \gcd(k,c) = 1}} \frac{\mu(k)}{k\varphi^{\ast}(kb)} & = & \frac{\varphi^{\ast}(\gcd(a,b))}{\varphi^{\ast}(b)\varphi^{\ast}(\gcd(a,b,c))} \prod_{p\|a, p \nmid bc} \left( 1 - \frac1{p-1} \right) \textrm{.}\end{aligned}$$ Therefore, we have obtained $$\begin{aligned} \label{theta} \theta(\boldsymbol{\eta}') & = & \theta_1 (\boldsymbol{\eta},\eta_6) \prod_{p\|\eta_1, p \nmid \eta_2\eta_4\eta_7} \left( 1 - \frac1{p-1} \right) \textrm{,}\end{aligned}$$ where $\theta_1 (\boldsymbol{\eta},\eta_6)$ denotes $$\begin{aligned} \varphi^{\ast}(\eta_1\eta_2\eta_3\eta_4\eta_5) \frac{\varphi^{\ast}(\gcd(\eta_1,\eta_4))}{\varphi^{\ast}(\eta_4)} \frac{\varphi^{\ast}(\gcd(\eta_3,\eta_5))}{\varphi^{\ast}(\eta_5)} \prod_{p\|\eta_3, p \nmid \eta_2\eta_5\eta_6} \left( 1 - \frac1{p-1} \right) \textrm{.}\end{aligned}$$ In addition, we see that the overall contribution of the error term is $$\begin{aligned} \sum_{\boldsymbol{\eta},\eta_6,\eta_7} 2^{\omega(\eta_3)} 2^{\omega(\eta_1)} 2^{\omega(\eta_1\eta_2\eta_3)} & \ll & \sum_{\boldsymbol{\eta}} 2^{\omega(\eta_3)} 2^{\omega(\eta_1)} 2^{\omega(\eta_1\eta_2\eta_3)} Y_6 Y_7 \\ & = & \sum_{\boldsymbol{\eta}} 2^{\omega(\eta_3)} 2^{\omega(\eta_1)} 2^{\omega(\eta_1\eta_2\eta_3)} \frac{B}{\boldsymbol{\eta}^{(2,2,2,1,1)}} \\ & \ll & B \log(B)^2 \textrm{,}\end{aligned}$$ where we have summed over $\eta_6$ and $\eta_7$ using respectively the height conditions and . This completes the proof of lemma \[lemma inter\]. Summation over $\eta_7$ ----------------------- To carry out the summations over $\eta_7$ and $\eta_6$, we let $$\begin{aligned} \label{V} \mathcal{V} & = & \left\{\boldsymbol{\eta} \in \mathbb{Z}_{>0}^5, Y_6 \geq 1, Y_7 \geq 1 \right\} \textrm{,}\end{aligned}$$ and we assume that $\boldsymbol{\eta} \in \mathcal{V}$ is fixed and is subject to the coprimality conditions and . Our next task is to sum over $\eta_7$, that is why we have isolated $\eta_7$ in $\theta(\boldsymbol{\eta}')$. Let us define $$\begin{aligned} \mathcal{N} & = & \{ (\eta_1,\eta_2,\eta_4) \in \mathbb{Z}_{>0}^3, 2 \nmid \eta_1 \textrm{ or } 2\|\eta_2\eta_4 \} \textrm{.}\end{aligned}$$ It is plain to see that if $(\eta_1,\eta_2,\eta_4) \in \mathcal{N}$ or $2\|\eta_7$ then $$\begin{aligned} \prod_{p\|\eta_1, p \nmid \eta_2\eta_4\eta_7} \left( 1 - \frac1{p-1} \right) & = & \prod_{\substack{p\|\eta_1, p \nmid \eta_2\eta_4\eta_7 \\ p \neq 2}} \left( 1 - \frac1{p-1} \right) \textrm{,}\end{aligned}$$ and this product is equal to $0$ otherwise. Furthermore, since $\eta_2 \eta_4$ and $\eta_7$ are coprime, we see that $$\begin{aligned} \prod_{\substack{p\|\eta_1, p \nmid \eta_2\eta_4\eta_7 \\ p \neq 2}} \left( 1 - \frac1{p-1} \right) & = & \frac{\varphi^{\circ}(\eta_1)}{\varphi^{\circ}(\gcd(\eta_1,\eta_2\eta_4))\varphi^{\circ}(\gcd(\eta_1,\eta_7))} \textrm{.}\end{aligned}$$ We need to treat two cases separately depending on whether $(\eta_1,\eta_2,\eta_4) \in \mathcal{N}$ or $(\eta_1,\eta_2,\eta_4) \notin \mathcal{N}$ (note that, in the latter case, the main term of $N(\boldsymbol{\eta}',B)$ vanishes if $2 \nmid \eta_7$). For fixed $\eta_6$ satisfying the height condition and the coprimality condition , we call $N(\boldsymbol{\eta},\eta_6,B)$ the sum of the main term of $N(\boldsymbol{\eta}',B)$ over $\eta_7$, $\eta_7$ being subject to the height condition and to the coprimality condition . We also use $N_1(\boldsymbol{\eta},\eta_6,B)$ and $N_2(\boldsymbol{\eta},\eta_6,B)$ to denote the sums over $\eta_7$ respectively for $(\eta_1,\eta_2,\eta_4) \in \mathcal{N}$ and $(\eta_1,\eta_2,\eta_4) \notin \mathcal{N}$. We now proceed to prove the following lemma. \[sum eta\_7\] We have the estimate $$\begin{aligned} N(\boldsymbol{\eta},\eta_6,B) & = & \frac{A_2Y_7}{\eta_4 \eta_5} g_2 \left( \frac{\eta_6}{Y_6}; \boldsymbol{\eta},B \right) \theta_1'(\boldsymbol{\eta}) \theta_2'(\boldsymbol{\eta},\eta_6) + R(\boldsymbol{\eta},\eta_6,B) \textrm{,}\end{aligned}$$ where $\theta_1'(\boldsymbol{\eta})$ and $\theta_2'(\boldsymbol{\eta},\eta_6)$ are arithmetic functions defined in and and $$\begin{aligned} \sum_{\boldsymbol{\eta},\eta_6} R(\boldsymbol{\eta},\eta_6,B) & \ll & B \log(B)^4 \textrm{.}\end{aligned}$$ First, we estimate the contribution of $N_1(\boldsymbol{\eta},\eta_6,B)$. For this, we make use of the first estimate of lemma \[arithmetic preliminary\] to deduce that for any fixed $0 < \delta \leq 1$, we have $$\begin{aligned} N_1(\boldsymbol{\eta},\eta_6,B) & = & \frac{A_2Y_7}{\eta_4 \eta_5} g_2 \left( \frac{\eta_6}{Y_6}; \boldsymbol{\eta},B \right) \theta_1 (\boldsymbol{\eta},\eta_6) \frac{\varphi^{\circ}(\eta_1)}{\varphi^{\circ}(\gcd(\eta_1,\eta_2\eta_4))} \Psi(\eta_1,\eta_2\eta_3\eta_4\eta_5\eta_6) \\ & & + O \left( \frac{A_2}{\eta_4 \eta_5} Y_7^{\delta} \sigma_{- \delta}(\eta_1\eta_2\eta_3\eta_4\eta_5\eta_6) \sup_{t_7 Y_7 \geq 1} g_1 \left( t_7, \frac{\eta_6}{Y_6} \right) \right) \textrm{.}\end{aligned}$$ To estimate the overall contribution of this error term, we use the bound of lemma \[bounds\] for $g_1$ and we choose $\delta = 1/4$. The average order of $\sigma_{- 1/4}$ is $O(1)$ so we see that this contribution is $$\begin{aligned} \sum_{\boldsymbol{\eta},\eta_6} \sigma_{- 1/4}(\eta_1\eta_2\eta_3\eta_4\eta_5\eta_6) \frac{A_2 Y_6^{1/2} Y_7^{3/4}}{\eta_4 \eta_5 \eta_6^{1/2}} & \ll & \sum_{\boldsymbol{\eta}} \sigma_{- 1/4}(\eta_1\eta_2\eta_3\eta_4\eta_5) \frac{A_2 Y_6 Y_7^{3/4}}{\eta_4 \eta_5} \\ & \ll & \sum_{\eta_1,\eta_2,\eta_3,\eta_5} \sigma_{- 1/4}(\eta_1\eta_2\eta_3\eta_5) \frac{B}{\boldsymbol{\eta}^{(1,1,1,0,1)}} \\ & \ll & B \log(B)^4 \textrm{,}\end{aligned}$$ where we have summed over $\eta_6$ and $\eta_4$ using respectively the conditions and $Y_7 \geq 1$. Concerning the main term, we have $$\begin{aligned} \Psi(\eta_1,\eta_2\eta_3\eta_4\eta_5\eta_6) & = & \varphi^{\ast}(\eta_2\eta_3\eta_4\eta_5\eta_6) \frac{\varphi^{\flat}(\eta_1)}{\varphi^{\flat}(\gcd(\eta_1,\eta_2\eta_4))} \textrm{,}\end{aligned}$$ and since $(\eta_1,\eta_2,\eta_4) \in \mathcal{N}$, we also have $$\begin{aligned} \frac{\varphi^{\circ}(\eta_1)}{\varphi^{\circ}(\gcd(\eta_1,\eta_2\eta_4))} \frac{\varphi^{\flat}(\eta_1)}{\varphi^{\flat}(\gcd(\eta_1,\eta_2\eta_4))} & = & \frac{\varphi^{\ast}(\eta_1)}{\varphi^{\ast}(\gcd(\eta_1,\eta_2\eta_4))} \textrm{.}\end{aligned}$$ These equalities and a short calculation prove that $$\begin{aligned} & & \theta_1(\boldsymbol{\eta},\eta_6) \frac{\varphi^{\circ}(\eta_1)}{\varphi^{\circ}(\gcd(\eta_1,\eta_2\eta_4))} \Psi(\eta_1,\eta_2\eta_3\eta_4\eta_5\eta_6)\end{aligned}$$ can be rewritten as $\theta_1'(\boldsymbol{\eta}) \theta_2'(\boldsymbol{\eta},\eta_6)$ for $$\begin{aligned} \label{theta_1'} \ \ \ \ \theta_1'(\boldsymbol{\eta}) & = & \varphi^{\ast}(\eta_1\eta_2\eta_3\eta_4\eta_5) \varphi^{\ast}(\eta_2\eta_3\eta_4\eta_5) \frac{\varphi^{\ast}(\eta_1\eta_2)}{\varphi^{\ast}(\eta_2\eta_4)} \frac{\varphi^{\ast}(\gcd(\eta_3,\eta_5))}{\varphi^{\ast}(\eta_5)} \textrm{,} \\ \label{theta_2'} \ \ \ \ \theta_2'(\boldsymbol{\eta},\eta_6) & =& \frac{\varphi^{\ast}(\eta_6)}{\varphi^{\ast}(\gcd(\eta_6,\eta_3))} \prod_{p\|\eta_3, p \nmid \eta_2\eta_5\eta_6} \left( 1 - \frac1{p-1} \right) \textrm{.}\end{aligned}$$ We now turn to the estimation of $N_2(\boldsymbol{\eta},\eta_6,B)$. We only need to sum on the even $\eta_7$ and so, given the coprimality condition , $\eta_2\eta_3\eta_4\eta_5\eta_6$ is odd and thus we can make use of the first estimate of lemma \[arithmetic preliminary 2\]. The error term is the same as the previous one and, in the main term, there are exactly two differences with the case of $N_1(\boldsymbol{\eta},\eta_6,B)$. The first is the factor $1/2$ and the second is the fact that here, since $(\eta_1,\eta_2,\eta_4) \notin \mathcal{N}$, $$\begin{aligned} \frac{\varphi^{\circ}(\eta_1)}{\varphi^{\circ}(\gcd(\eta_1,\eta_2\eta_4))} \frac{\varphi^{\flat}(\eta_1)}{\varphi^{\flat}(\gcd(\eta_1,\eta_2\eta_4))} & = & 2 \frac{\varphi^{\ast}(\eta_1)}{\varphi^{\ast}(\gcd(\eta_1,\eta_2\eta_4))} \textrm{,}\end{aligned}$$ and thus we find exactly the same main term, which completes the proof of lemma \[sum eta\_7\]. Summation over $\eta_6$ ----------------------- We now proceed to sum over $\eta_6$. We set $$\begin{aligned} \mathcal{M} & = & \{ (\eta_3,\eta_2,\eta_5) \in \mathbb{Z}_{>0}^3, 2 \nmid \eta_3 \textrm{ or } 2\|\eta_2\eta_5 \} \textrm{.}\end{aligned}$$ As for the summation over $\eta_7$, it is clear that if $(\eta_3,\eta_2,\eta_5) \in \mathcal{M}$ or $2\|\eta_6$ then $$\begin{aligned} \prod_{p\|\eta_3, p \nmid \eta_2\eta_5\eta_6} \left( 1 - \frac1{p-1} \right) & = & \prod_{\substack{p\|\eta_3, p \nmid \eta_2\eta_5\eta_6 \\ p \neq 2}} \left( 1 - \frac1{p-1} \right) \textrm{,}\end{aligned}$$ and this product is equal to $0$ otherwise. Furthermore, since $\eta_2 \eta_5$ and $\eta_6$ are coprime, we have $$\begin{aligned} \prod_{\substack{p\|\eta_3, p \nmid \eta_2\eta_5\eta_6 \\ p \neq 2}} \left( 1 - \frac1{p-1} \right) & = & \frac{\varphi^{\circ}(\eta_3)}{\varphi^{\circ}(\gcd(\eta_3,\eta_2\eta_5))\varphi^{\circ}(\gcd(\eta_3,\eta_6))} \textrm{.}\end{aligned}$$ We need to treat two cases separately depending on whether $(\eta_3,\eta_2,\eta_5) \in \mathcal{M}$ or $(\eta_3,\eta_2,\eta_5) \notin \mathcal{M}$ (note that, in the latter case, the main term of $N(\boldsymbol{\eta},\eta_6,B)$ vanishes if $2 \nmid \eta_6$). Let $\mathbf{N}(\boldsymbol{\eta},B)$ be the sum of the main term of $N(\boldsymbol{\eta},\eta_6,B)$ over $\eta_6$, $\eta_6$ satisfying the height condition and the coprimality condition and let also $\mathbf{N}_1(\boldsymbol{\eta},B)$ and $\mathbf{N}_2(\boldsymbol{\eta},B)$ be the sums over $\eta_6$ respectively for $(\eta_3,\eta_2,\eta_5) \in \mathcal{M}$ and $(\eta_3,\eta_2,\eta_5) \notin \mathcal{M}$. \[sum eta\_6\] We have the estimate $$\begin{aligned} \mathbf{N}(\boldsymbol{\eta},B) & = & \zeta(2)^{-1} \frac{B}{\boldsymbol{\eta}^{(1,1,1,1,1)}} g_3(\boldsymbol{\eta},B) \Theta(\boldsymbol{\eta}) + \mathbf{R}(\boldsymbol{\eta},B) \textrm{,}\end{aligned}$$ where $$\begin{aligned} \Theta(\boldsymbol{\eta}) & = & \frac{\varphi^{\ast}(\eta_1\eta_2\eta_3\eta_4\eta_5)}{\varphi^{\dag}(\eta_1\eta_2\eta_3\eta_4\eta_5)} \varphi^{\ast}(\eta_2\eta_3\eta_4\eta_5) \varphi^{\ast}(\eta_1\eta_2\eta_4\eta_5) \frac{\varphi^{\ast}(\eta_1\eta_2)}{\varphi^{\ast}(\eta_2\eta_4)} \frac{\varphi^{\ast}(\eta_2\eta_3)}{\varphi^{\ast}(\eta_2\eta_5)} \textrm{,}\end{aligned}$$ and $$\begin{aligned} \sum_{\boldsymbol{\eta}} \mathbf{R}(\boldsymbol{\eta},B) & \ll & B \log(B)^4 \textrm{.}\end{aligned}$$ We first treat the contribution of $\mathbf{N}_1(\boldsymbol{\eta},B)$. For this, we make use of the second estimate of lemma \[arithmetic preliminary\] to deduce that for any fixed $0 < \delta \leq 1$, we have $$\begin{aligned} \mathbf{N}_1(\boldsymbol{\eta},B) & = & \frac{A_2Y_7Y_6}{\eta_4 \eta_5} g_3 \left( \boldsymbol{\eta},B \right) \theta_1' (\boldsymbol{\eta}) \frac{\varphi^{\circ}(\eta_3)}{\varphi^{\circ}(\gcd(\eta_3,\eta_2\eta_5))} \Psi'(\eta_3,\eta_1\eta_2\eta_4\eta_5) \\ & & + O \left( \frac{A_2Y_7}{\eta_4 \eta_5} Y_6^{\delta} \sigma_{- \delta}(\eta_1\eta_2\eta_4\eta_5) \sup_{t_6 Y_6 \geq 1} g_2 \left( t_6; \boldsymbol{\eta},B \right) \right) \textrm{.}\end{aligned}$$ To estimate the overall contribution of the error term, we use the bound of lemma \[bounds\] for $g_2$ and we choose $\delta = 1/4$. Since the average order of $\sigma_{- 1/4}$ is $O(1)$, we obtain that this contribution is $$\begin{aligned} \sum_{\boldsymbol{\eta}} \sigma_{- 1/4}(\eta_1\eta_2\eta_4\eta_5) \frac{A_2 Y_7 Y_6^{3/4}}{\eta_4 \eta_5} & \ll & \sum_{\eta_1,\eta_2,\eta_3,\eta_4} \sigma_{- 1/4}(\eta_1\eta_2\eta_4) \frac{B}{\boldsymbol{\eta}^{(1,1,1,1,0)}} \\ & \ll & B \log(B)^4 \textrm{,}\end{aligned}$$ where we have summed over $\eta_5$ using the condition $Y_6 \geq 1$. Let us turn to the main term. First, note that $$\begin{aligned} \frac{A_2Y_7Y_6}{\eta_4 \eta_5} & = & \frac{B}{\boldsymbol{\eta}^{(1,1,1,1,1)}} \textrm{.}\end{aligned}$$ In addition, we have $$\begin{aligned} \Psi'(\eta_3,\eta_1\eta_2\eta_4\eta_5) & = & \varphi^{\ast}(\eta_1\eta_2\eta_4\eta_5) \frac{\varphi^{\flat}(\eta_3)}{\varphi^{\flat}(\gcd(\eta_3,\eta_2\eta_5))} \frac{\zeta(2)^{-1}}{\varphi^{\dag}(\eta_1\eta_2\eta_3\eta_4\eta_5)} \textrm{,}\end{aligned}$$ and since $(\eta_3,\eta_2,\eta_5) \in \mathcal{M}$, we also have $$\begin{aligned} \frac{\varphi^{\circ}(\eta_3)}{\varphi^{\circ}(\gcd(\eta_3,\eta_2\eta_5))} \frac{\varphi^{\flat}(\eta_3)}{\varphi^{\flat}(\gcd(\eta_3,\eta_2\eta_5))} & = & \frac{\varphi^{\ast}(\eta_3)}{\varphi^{\ast}(\gcd(\eta_3,\eta_2\eta_5))} \textrm{.}\end{aligned}$$ An easy calculation finally yields $$\begin{aligned} \theta_1' (\boldsymbol{\eta}) \frac{\varphi^{\circ}(\eta_3)}{\varphi^{\circ}(\gcd(\eta_3,\eta_2\eta_5))} \Psi'(\eta_3,\eta_1\eta_2\eta_4\eta_5) & = & \zeta(2)^{-1} \Theta(\boldsymbol{\eta}) \textrm{.}\end{aligned}$$ We now deal with the estimation of $\mathbf{N}_2(\boldsymbol{\eta},B)$. We only need to sum on the even $\eta_6$ and so, given the coprimality condition , $\eta_1\eta_2\eta_4\eta_5$ is odd and moreover since $(\eta_3,\eta_2,\eta_5) \notin \mathcal{M}$, we have $2\|\eta_3$ and thus we can make use of the second estimate of lemma \[arithmetic preliminary 2\]. The error term is the same as the previous one and, in the main term, there are exactly two differences with the case of $\mathbf{N}_1(\boldsymbol{\eta},B)$. The first is the factor $1/2$ and the second is that here, since $(\eta_3,\eta_2,\eta_5) \notin \mathcal{N}$, $$\begin{aligned} \frac{\varphi^{\circ}(\eta_3)}{\varphi^{\circ}(\gcd(\eta_3,\eta_2\eta_5))} \frac{\varphi^{\flat}(\eta_3)}{\varphi^{\flat}(\gcd(\eta_3,\eta_2\eta_5))} & = & 2 \frac{\varphi^{\ast}(\eta_3)}{\varphi^{\ast}(\gcd(\eta_3,\eta_2\eta_5))} \textrm{,}\end{aligned}$$ and we finally obtain the same main term, which concludes the proof of lemma \[sum eta\_6\]. Conclusion ---------- The aim of the following lemma is to replace the conditions $t_6 Y_6 \geq 1$ and $t_7 Y_7 \geq 1$ in the integral defining $g_3$ in the main term of $\mathbf{N}(\boldsymbol{\eta},B)$ in lemma \[sum eta\_6\] respectively by $t_6 > 0$ and $t_7 > 0$. For short, we introduce the notation $$\begin{aligned} D_h & = & \left\{ (u_2, t_7, t_6) \in \mathbb{R}^3, t_6,t_7>0, h(u_2,t_7,t_6) \leq 1 \right\} \textrm{.}\end{aligned}$$ For $Z_6, Z_7 > 0$, we have $$\begin{aligned} \label{1} {\operatorname{meas}}\{ (u_2, t_7, t_6) \in D_h, t_6 Z_6 < 1 \} & \ll & Z_6^{-1/2} \textrm{,} \\ \label{2} {\operatorname{meas}}\{ (u_2, t_7, t_6) \in D_h, t_7 Z_7 < 1 \} & \ll & Z_7^{-1/2} \textrm{.}\end{aligned}$$ These two bounds follow from the bound of lemma \[bounds\] for $g_1$ and the fact that $h(u_2,t_7,t_6) \leq 1$ implies $t_6,t_7 \leq 1$. Making use of the bound , we see that replacing the condition $t_6 Y_6 \geq 1$ in the integral defining $g_3$ in the main term of $\mathbf{N}(\boldsymbol{\eta},B)$ in lemma \[sum eta\_6\] by the condition $t_6 > 0$ creates an error term whose overall contribution is $$\begin{aligned} \sum_{\boldsymbol{\eta}} \frac{A_2 Y_7 Y_6^{1/2}}{\eta_4 \eta_5} & \ll & \sum_{\eta_1,\eta_2,\eta_3,\eta_4} \frac{B}{\boldsymbol{\eta}^{(1,1,1,1,0)}} \\ & \ll & B \log(B)^4 \textrm{,}\end{aligned}$$ where we have summed over $\eta_5$ using the condition $Y_6 \geq 1$. The bound shows that the same conclusion holds for the condition $t_7 Y_7 \geq 1$. Recalling the equality , we finally see that we can replace $g_3(\boldsymbol{\eta},B)$ in the main term of $\mathbf{N}(\boldsymbol{\eta},B)$ in lemma \[sum eta\_6\] by $$\begin{aligned} \int \int \int_{t_6,t_7>0, h(u_2,t_7,t_6) \leq 1} {\mathrm{d}}u_2 {\mathrm{d}}t_7 {\mathrm{d}}t_6 & = & \frac{\omega_{\infty}}{4} \textrm{.}\end{aligned}$$ Redefine $\Theta$ as being equal to zero if the remaining coprimality conditions and are not satisfied. Using lemma \[T\], we obtain the following result. \[final lemma\] We have the estimate $$\begin{aligned} N_{U,H}(B) & = & \zeta(2)^{-1} \frac{\omega_{\infty}}{2} B \sum_{\boldsymbol{\eta} \in \mathcal{V}} \frac{\Theta(\boldsymbol{\eta})}{\boldsymbol{\eta}^{(1,1,1,1,1)}} + O \left( B \log(B)^4 \right) \textrm{,}\end{aligned}$$ where $\mathcal{V}$ is defined in . Let us introduce the generalized Möbius function $\boldsymbol{\mu}$ defined for $(n_1, \dots, n_5) \in \mathbb{Z}_{>0}^5$ by $\boldsymbol{\mu}(n_1, \dots, n_5) = \mu(n_1) \cdots \mu(n_5)$. We set $\mathbf{k} = (k_1,k_2,k_3,k_4,k_5)$ and we define, for $s \in \mathbb{C}$ such that $\Re(s) > 1$, $$\begin{aligned} F(s) & = & \sum_{\boldsymbol{\eta} \in \mathbb{Z}_{>0}^5} \frac{\left\|(\Theta \ast \boldsymbol{\mu})(\boldsymbol{\eta})\right\|}{\eta_1^s \eta_2^s \eta_3^s \eta_4^s \eta_5^s} \\ & = & \prod_p \left( \sum_{\mathbf{k} \in \mathbb{Z}_{\geq 0}^5} \frac{\left\|(\Theta \ast \boldsymbol{\mu}) \left( p^{k_1},p^{k_2},p^{k_3},p^{k_4},p^{k_5} \right)\right\|} {p^{k_1 s}p^{k_2 s}p^{k_3 s}p^{k_4 s}p^{k_5 s}} \right) \textrm{.}\end{aligned}$$ It is easy to see that if $\mathbf{k} \notin \{0,1\}^5$ then $(\Theta \ast \boldsymbol{\mu}) \left( p^{k_1},p^{k_2},p^{k_3},p^{k_4},p^{k_5} \right) = 0$ and moreover if exactly one of the $k_i$ is equal to $1$, then $(\Theta \ast \boldsymbol{\mu}) \left( p^{k_1},p^{k_2},p^{k_3},p^{k_4},p^{k_5} \right) \ll 1/p$, so the local factors $F_p$ of $F$ satisfy $$\begin{aligned} F_p(s) & = & 1 + O \left( \frac1{p^{ \min \left( \Re(s)+1, 2 \Re(s) \right)}} \right) \textrm{.}\end{aligned}$$ This proves that $F$ actually converges in the half-plane $\Re(s) > 1/2$, which implies that $\Theta$ satifies the assumption of [@3A1 Lemma $8$]. Applying this lemma, we get $$\begin{aligned} \label{sum1} \ \ \ \ \ \sum_{\boldsymbol{\eta} \in \mathcal{V}} \frac{\Theta(\boldsymbol{\eta})}{\boldsymbol{\eta}^{(1,1,1,1,1)}} & = & \alpha \left( \sum_{\boldsymbol{\eta} \in \mathbb{Z}_{>0}^5} \frac{(\Theta \ast \boldsymbol{\mu})(\boldsymbol{\eta})}{\boldsymbol{\eta}^{(1,1,1,1,1)}} \right) \log(B)^5 + O \left( \log(B)^4 \right) \textrm{,}\end{aligned}$$ where $\alpha$ is the volume of the polytope defined in $\mathbb{R}^5$ by $t_1,t_2,t_3,t_4,t_5 \geq 0$ and $$\begin{aligned} t_1 + 2 t_2 + 3 t_3 + 2 t_5 & \leq & 1 \textrm{,} \\ 3 t_1 + 2 t_2 + t_3 + 2 t_4 & \leq & 1 \textrm{.}\end{aligned}$$ A computation using Franz’s additional Maple package [@Convex] provides $\alpha = 1/2160$, that is to say $$\begin{aligned} \label{alpha} \alpha & = & 2 \alpha(\widetilde{V}) \textrm{.}\end{aligned}$$ Moreover, $$\begin{aligned} \sum_{\boldsymbol{\eta} \in \mathbb{Z}_{>0}^5} \frac{(\Theta \ast \boldsymbol{\mu}) (\boldsymbol{\eta})}{\boldsymbol{\eta}^{(1,1,1,1,1)}} & = & \prod_p \left( \sum_{\mathbf{k} \in \mathbb{Z}_{\geq 0}^5} \frac{(\Theta \ast \boldsymbol{\mu}) \left( p^{k_1},p^{k_2},p^{k_3},p^{k_4},p^{k_5} \right)}{p^{k_1}p^{k_2}p^{k_3}p^{k_4}p^{k_5}} \right) \\ & = & \prod_p \left( 1 - \frac1{p} \right)^5 \left( \sum_{\mathbf{k} \in \mathbb{Z}_{\geq 0}^5} \frac{\Theta \left( p^{k_1},p^{k_2},p^{k_3},p^{k_4},p^{k_5} \right)}{p^{k_1}p^{k_2}p^{k_3}p^{k_4}p^{k_5}} \right) \textrm{.}\end{aligned}$$ The remaining coprimality conditions greatly simplify the calculation and we obtain $$\begin{aligned} \sum_{\mathbf{k} \in \mathbb{Z}_{\geq 0}^5} \frac{\Theta \left( p^{k_1},p^{k_2},p^{k_3},p^{k_4},p^{k_5} \right)}{p^{k_1}p^{k_2}p^{k_3}p^{k_4}p^{k_5}} & = & \left( 1 - \frac1{p^2} \right)^{-1} \left( 1 - \frac1{p} \right) \left( 1 + \frac{6}{p} + \frac1{p^2} \right) \textrm{,}\end{aligned}$$ which gives $$\begin{aligned} \label{sum2} \sum_{\boldsymbol{\eta} \in \mathbb{Z}_{>0}^5} \frac{(\Theta \ast \boldsymbol{\mu}) (\boldsymbol{\eta})}{\boldsymbol{\eta}^{(1,1,1,1,1)}} & = & \zeta(2) \prod_p \left( 1 - \frac1{p} \right)^6 \omega_p \textrm{.}\end{aligned}$$ We complete the proof of theorem \[Manin\] putting together the equalities , , and lemma \[final lemma\].
--- abstract: \| Caputo q-fractional derivatives are introduced and studied. A Caputo -type q-fractional initial value problem is solved and its solution is expressed by means of a new introduced q-Mittag-Leffler function. Some open problems about q-fractional integrals are proposed as well. [AMS Subject Classification:]{} 26A33; 60G05; 60G07; 60G012; 60GH05,41A05, 33D60, 34G10. [Key Words and Phrases:]{} Left q-fractional integral, right q-fractional integral, Caputo left and right q-fractional derivatives, Q-operator,q-Mittag-Leffler function, time scale. author: - \| Thabet Abdeljawad and Dumitru Baleanu[^1]\ Department of Mathematics and Computer Science\ Çankaya University, 06530 Ankara, Turkey title: ' Caputo q-Fractional Initial Value Problems and a q-Analogue Mittag-Leffler Function ' --- Introduction {#s:1} ============= The concept of fractional calculus is not new. It is believed to have stemmed from a question raised in 1695. However, it has gained considerable popularity and importance during the last three decades or so. This is due to its distinguished applications in numerous diverse fields of science and engineering ([@Samko], [@Podlubny], [@Kilbas]). The q-calculus is also not of recent appearance. It was initiated in twenties of the last century. As a survey about this calculus we refer to [@history]. Starting from the q-analogue of Cauchy formula [@Al-salam2], Al-Salam started the fitting of the concept of q-fractional calculus. After that he ([@Alsalaml], [@Alsalam]) and Agarwal R. [@Agarwal] continued on by studying certain q-fractional integrals and derivatives, where they proved the semigroup properties for left and right (Riemann)type fractional integrals but without variable lower limit and variable upper limit, respectively. Recently, the authors in [@Pred] generalized the notion of the (left)fractional q-integral and q-derivative by introducing variable lower limit and proved the semigroup properties. However, the case of the (right) q-fractional integral by introducing a variable upper limit is still open. This open problem will be stated clearly in this article. Very recently and after the appearance of time scale calculus (see for example [@Boh]), some authors started to pay attention and apply the techniques of time scale to discrete fractional calculus ([@Ferd],[@Feri],[@Ferq], [@Th]) benefitting from the results announced before in [@Miller]. All of these results are mainly about fractional calculus on the time scales $T_q=\{q^n:n \in \mathbb{Z}\}\cup \{0\}$ and $h\mathbb{Z}$ [@Bastos]. However, the study of fractional calculus on time scales combining the previously mentioned time scales is still unknown. Continuing in this direction and being motivated by all above, in this article we define and study Caputo type q-fractional derivatives. This manuscript is organized as follows: Section 2 contains essential definitions and results about fractional q-integrals and q-derivatives, where we present an open problem about the semigroup property. Section 3 is devoted to define and study left and right Caputo q-fractional derivatives. In Section 4, we solve a Caputo q-fractional nonhomogenuous linear dynamic equation, where the solution is expressed by a q-analogue of Mittag-Leffler function. Preliminaries and Essential Results about q-Calculus, Fractional q-Integrals and q-Derivatives {#s:3} ============================================================================================== For the theory of q-calculus we refer the reader to the survey [@history] and for the basic definitions and results for the q-fractional calculus we refer to [@Ferq]. Here we shall summarize some of those basics. For $0<q<1$, let $T_q$ be the time scale $$T_q=\{q^n:n \in \mathbb{Z}\}\cup \{0\}.$$ where $Z$ is the set of integers. More generally, if $\alpha$ is a nonnegative real number then we define the time scale $$T_q^\alpha=\{q^{n+\alpha}:n \in Z\}\cup \{0\},$$ we write $T_q^0=T_q.$ For a function $f:T_q\rightarrow \mathbb{R}$, the nabla q-derivative of $f$ is given by $$\label{qd} \nabla_q f(t)=\frac{f(t)-f(qt)}{(1-q)t},~~t \in T_q-\{0\}$$ The nabla q-integral of $f$ is given by $$\label{qi} \int_0^t f(s)\nabla_q s=(1-q)t\sum_{i=0}^\infty q^if(tq^i)$$ and for $0\leq a \in T_q$ $$\int_a^t f(s)\nabla_q s=\int_0^t f(s)\nabla_q s - \int_0^a f(s)\nabla_q s$$ On the other hand $$\label{r1} \int_t^\infty f(s)\nabla_q s=(1-q)t\sum_{i=1}^\infty q^{-i} f(tq^{-i})$$ and for $0<b<\infty$ in $T_q$ $$\label{r2} \int_t^b f(s)\nabla_q s=\int_t^\infty f(s)\nabla_q s - \int_b^\infty f(s)\nabla_q s$$ By the fundamental theorem in q-calculus we have $$\label{fq} \nabla_q \int_0^t f(s)\nabla_q s=f(t)$$ and if $f$ is continuous at $0$, then $$\label{cfq} \int_0^t \nabla_qf(s)\nabla_q s=f(t)-f(0)$$ Also the following identity will be helpful $$\label{help} \nabla_q\int_a^t f(t,s)\nabla_q s=\int_a^t \nabla_qf(t,s)\nabla_q s+f(qt,t)$$ Similarly the following identity will be useful as well $$\label{help1} \nabla_q\int_t^b f(t,s)\nabla_q s=\int_{qt}^b \nabla_qf(t,s)\nabla_q s-f(t,t)$$ The q-derivative in (\[help\]) and (\[help1\]) is applied with respect to t. From the theory of q-calculus and the theory of time scale more generally, the following product rule is valid $$\label{qproduct} \nabla_q (f(t)g(t))=f(qt)\nabla_q g(t)+\nabla_q f(t)g(t)$$ The q-factorial function for $n\in \mathbb{N}$ is defined by $$\label{qfact} (t-s)_q^n=\prod_{i=0}^{n-1}(t-q^is)$$ When $\alpha$ is a non positive integer, the q-factorial function is defined by $$\label{qfactg} (t-s)_q^\alpha=t^\alpha\prod_{i=0}^\infty \frac{1- \frac{s}{t} q^i} {1- \frac{s}{t} q^{i+\alpha}}$$ We summarize some of the properties of q-factorial functions, which can be found mainly in [@Ferq], in the following lemma \[qproperties\] (i)$ (t-s)_q^{\beta+\gamma}=(t-s)_q^\beta (t-q^\beta s)_q^\gamma$ (ii)$(at-as)_q^\beta=a^\beta (t-s)_q^\beta$ \(iii) The nabla q-derivative of the q-factorial function with respect to $t$ is $$\nabla_q (t-s)_q^\alpha =\frac{1-q^\alpha}{1-q}(t-s)_q^{\alpha-1}$$ (iv)The nabla q-derivative of the q-factorial function with respect to $s$ is $$\nabla_q (t-s)_q^\alpha =-\frac{1-q^\alpha}{1-q}(t-qs)_q^{\alpha-1}$$ where $\alpha,\gamma,\beta \in \mathbb{R}.$ For the q-gamma function, $\Gamma_q(\alpha)$, we refer the reader to [@Ferq] and the references therein. We just mention here the identity $$\label{qid} \Gamma_q(\alpha+1)=\frac{1-q^\alpha}{1-q}\Gamma_q(\alpha),~~\Gamma_q(1)=1,~\alpha >0.$$ The authors in [@Ferq] following [@Agarwal] defines the left fractional q-integral of order $\alpha\neq 0,-1,-2,...$ by $$\label{ag} _{q}I^\alpha f(t)=\frac{1}{\Gamma_q(\alpha)}\int_0^t (t-qs)_q^{\alpha-1}f(s)\nabla_qs$$ In [@Agarwal] it was proved that the left q-fractional integral obeys the identity $$\label{t1} _{q}I^\beta ~ _{q}I^\alpha f(t)= _{q}I^{\alpha+\beta}f(t),~~~~\alpha~,\beta>0$$ The left q-fractional integral $ _{q}I_a^\alpha$ starting from $0<a\in T_q$ is to be defined by $$\label{t3} _{q}I_a^\alpha f(t)=\frac{1}{\Gamma_q(\alpha)}\int_a^t (t-qs)_q^{\alpha-1}f(s)\nabla_qs$$ It is clear, from the q-analogue of Cauchy’s formula [@Al-salam2], that $$\label{t11} \nabla_q^n ~_{q}I_a^n f(t)=f(t)$$ where $n$ is a positive integer and $0\leq a \in T_q$ Recently, in Theorem 5 of [@Pred], the authors there have proved that $$\label{t4} _{q}I_a^\beta ~ _{q}I_a^\alpha f(t)= _{q}I_a^{\alpha+\beta}f(t),~~~~\alpha~,\beta>0$$ The right q-fractional integral of order $\alpha$ is defined by [@Agarwal] $$\label{t55} I_q^ \alpha f(t)=\frac{q^{-(1/2)\alpha(\alpha-1)}}{\Gamma_q(\alpha)}\int_t^\infty (s-t)_q^{\alpha-1}f(sq^{1-\alpha})\nabla_qs$$ and the right q-fractional integral of order $\alpha$ ending at $b$ for some $b \in T_q$ is defined by $$\label{t555} _{b}I_q^ \alpha f(t)=\frac{q^{-(1/2)\alpha(\alpha-1)}}{\Gamma_q(\alpha)}\int_t^b (s-t)_q^{\alpha-1}f(sq^{1-\alpha})\nabla_qs$$ Note that, while the left q-fractional integral $_{q}I_a^\alpha$ maps functions defined $T_q$ to functions defined on $T_q$, the right q-fractional integral $_{b}I_q^\alpha$, $0<b\leq\infty$, maps functions defined on $T_q^{1-\alpha}$ to functions defined on $T_q$. It is clear, from the q-analogue of Cauchy’s formula [@Al-salam2], that $$\label{s1} \nabla_q^n I_q^n f(t)=(-1)^nf(t)$$ In [@Alsalam] it was proved that the right q-fractional integral obeys the identity $$\label{s2} I_q^\beta ~ I_q^\alpha f(t)= I_q^{\alpha+\beta}f(t),~~~~\alpha~,\beta>0$$ Taking into account the domain and the range of the right q-fractional integral, as mentioned above, we note that the formula (\[s2\]) is valid under the condition that $f$ must be at least defined on $T_q$, $T_q^{1-\beta}$, $T_q^{1-\alpha}$ and $T_q^{1-(\alpha+\beta)}$. A particular case of the identity (\[s2\]) is $$\label{s22} I_q^{n-\alpha} ~ I_q^\alpha f(t)= I_q^nf(t),~~~~\alpha~>0.$$ \[remb1\] For $\alpha,\beta > 0$ and a function $f$ fitting suitable domains, we have $$\int_b^\infty (t-x)_q^{\beta-1}~_{b}I_q^\alpha f(tq^{1-\beta})\nabla_qt=0$$ From (\[r1\]) we can write $$\int_b^\infty (t-x)_q^{\beta-1}~_{b}I_q^\alpha f(tq^{1-\beta})\nabla_qt=$$ $$\label{eq1} (1-q)b \sum_{i=0}^\infty q^{-i}(bq^{-i}-x)_q^{\beta-1}~_{b}I_q^\alpha f(q^{1-\beta}bq^{-i})$$ From the fact that $(t-r)_q^{\beta-1}=0$, when $t<r$ we conclude that $_{b}I_q^\alpha f(q^{1-\beta}bq^{-i})=0$ and hence the result follows. Problem 1: Can we use Lemma \[remb1\] and following similar ideas to that in [@Pred] to prove that $$\label{s222} _{b}I_q^\beta ~ _{b}I_q^\alpha f(t)=~ _{b}I_q^{\alpha+\beta}f(t),~~~~\alpha~,\beta>0, ~0<b\in T_q$$ Alternatively, can we define the q-analogue of the Q-operator and prove that $Q ~_{q}I_a^\alpha f(t) =~ _{b}I_q^\alpha Q f(t)$? Then apply the Q-operator to the identity $$\label{t44} _{q}I_a^\beta ~ _{q}I_a^\alpha g(t)= _{q}I_a^{\alpha+\beta}g(t),~~~~\alpha~,\beta>0$$ with $g(t)=Qf(t)$ to obtain (\[s222\]). Recall that in the continuous case $Qf(t)=f(a+b-t)$. In connection to Problem 1, the following open problem is also raised Problem 2: Is it possible to obtain a by-part formula for q-fractional derivatives when the lower limit $a$ and the upper limit $b$ both exist. That is on the interval $[a,b]_q$. As for the $(0,\infty)$ case there is a formula was early obtained by Agarwal in [@Agarwal]. As for the left and right (Riemann) q-fractional derivatives of order $\alpha > 0$,as traditionally done in fractional calculus, they are defined respectively by $$\label{lrqd} _{q}\nabla_a^\alpha f(t)\triangleq \nabla_q^n~_{q}I_a^{n-\alpha}f(t)~\texttt{and}~_{b}\nabla_q^\alpha f(t)\triangleq (-1)^n \nabla_q^n~_{b}I_q^{n-\alpha}f(t)$$ where $n=[\alpha]+1$ and $a,b\in T_q\cup \{\infty\}$ with $0\leq a < b\leq \infty$. We usually remove the endpoints in the notation when $a=0$ or $b=\infty$. Here, we point that the operator $_{q}\nabla_a^\alpha$ maps functions defined on $T_q$ to functions defined on $T_q$, while the operator $ _{b}\nabla_q^\alpha$ maps functions defined on $T_q^{1-(n-\alpha)}$ to functions defined on $T_q$. Also, particularly, one has to note that $$\label{lrqd} _{q}\nabla_a^n f(t)=\nabla_q^n f(t)~\texttt{and}~_{b}\nabla_q^n f(t)=(-1)^n \nabla_q^n f(t)$$ where $\nabla_q^n$ always denotes the $n-th$ q-derivative (i.e. the q-derivative applied n times). Caputo q-Fractional Derivative {#s:3} =============================== In this section, before defining Caputo-type q-fractional derivatives and relating them to Riemann ones, we first state and prove some essential preparatory lemmas. \[m\] For any $\alpha>0$, the following equality holds: $$\label{m1} _{q}I_a ^\alpha\nabla_q f(t)= \nabla_q ~ _{q}I_a ^\alpha f(t)-\frac{(t-a)_q^{\alpha-1}}{\Gamma_q(\alpha)}f(a)$$ From (\[qproduct\]) and (iv) of Lemma \[qproperties\], we obtain the following q-integration by parts: $$\label{mm1} \nabla_q ((t-s)_q^{\alpha-1}f(s))= (t-qs)_q^{\alpha-1}\nabla_q f(s)- \frac{1-q^{\alpha-1}}{1-q}(t-qs)_q^{\alpha-2}f(s)$$ Applying (\[mm1\]) leads to $$\label{m2} _{q}I_a ^\alpha\nabla_q f(t)=\frac{(t-s)_q ^{\alpha-1}}{\Gamma_q(\alpha)}f(s)\|_a^t +\frac{1-q^{\alpha-1}}{1-q}\int_a^t (t-qs)_q^{\alpha-2}f(s)\nabla_qs$$ or $$_{q}I_a ^\alpha\nabla_q f(t)= - \frac{(t-a)_q ^{\alpha-1} }{\Gamma_q (\alpha)}f(a)+ \frac{1-q^{\alpha-1}}{1-q}\int_a^t (t-qs)_q^{\alpha-2}f(s)\nabla_qs$$ On the other hand, and by the help of (iii) of Lemma \[qproperties\], (\[help\]) and the identity (\[qid\]), we find that $$\label{m4} \nabla_q ~ _{q}I_a ^\alpha f(t)=\frac{1-q^{\alpha-1}}{1-q}\int_a^t (t-qs)_q^{\alpha-2}f(s)\nabla_qs,$$ which completes the proof. \[qinter\] For any real $\alpha>0$ and any positive integer $p$ such that $\alpha -p+1$ is not negative integer or 0, in particular $\alpha >p-1$, the following equality holds: $$\label{qinter1} _{q}I_a ^\alpha \nabla_q^p f(t)= \nabla_q^p~ _{q}I_a ^\alpha f(t)-\sum_{k=0}^{p-1}\frac{(t-a)_q^{\alpha-p+k}}{\Gamma_q(\alpha+k-p+1)}\nabla_q^k f(a)$$ The proof can be achieved by following inductively on $p~$ and making use of Lemma \[m\], (iii) of Lemma \[qproperties\] and (\[qid\]). Now we obtain an analogue to Lemma \[m\] for the right q-integrals. \[rm\] For any $\alpha > 0$, the following equality holds: $$\label{th} _{q^{-1}b}I_q^\alpha~_{b}\nabla_q f(t)=~_{b}\nabla_q~_{b}I_q^\alpha f(t)-\frac{r(\alpha)}{\Gamma_q(\alpha)}(b-qt)_q^{\alpha-1}f(q^{1-\alpha}q^{-1}b)$$ where $$r(\alpha)= q^{(-1/2)\alpha(\alpha-1)}$$ and $$_{b}\nabla_q f(t)=-\nabla_q f(t)$$. First, by the help of (iii) of Lemma \[qproperties\] and (\[qproduct\]), the following q-calculus by-parts version is valid: $$(s-t)_q^{\alpha-1}\nabla_qf(sq^{1-\alpha})q^{1-\alpha}=$$ $$\label{rid} \nabla_q((s-t)_q^{\alpha-1}f(sq^{1-\alpha}))-\frac{1-q^{\alpha-1}}{1-q}(s-t)_q^{\alpha-2} f(sq^{2-\alpha})$$ where the q-derivative is applied with respect to $s$. Using (\[rid\]) we obtain $$_{q^{-1}b}I_q^\alpha~_{b}\nabla_q f(t)=$$ $$\label{m1} \frac{q^{\alpha-1}r(\alpha)}{\Gamma_q(\alpha)}(\frac{1-q^{\alpha-1}}{1-q}\int_t^{q^{-1}b} (s-t)_q^{\alpha-2}f(q^{2-\alpha}s)\nabla_qs-(s-t)_q^{\alpha-1}f(q^{1-\alpha}s)\|_t^{q^{-1}b})$$ $$\label{m2} =\frac{ q^{\alpha-1}r(\alpha) } {\Gamma_q(\alpha)}(\frac{1-q^{\alpha-1}}{1-q}\int_t^{q^{-1}b} (s-t)_q^{\alpha-2}f(q^{2-\alpha}s)\nabla_qs-(q^{-1}b-t)_q^{\alpha-1}f(q^{1-\alpha}q^{-1}b))$$ On the other hand (\[help1\]) and (iv) of Lemma \[qproperties\] imply $$\label{m3} ~_{b}\nabla_q~ _{b}I_q^\alpha f(t)=\frac{ q^{\alpha-1}r(\alpha) } {\Gamma_q(\alpha)}\frac{1-q^{\alpha-1}}{1-q}\int_t^{q^{-1}b} (s-t)_q^{\alpha-2}f(q^{2-\alpha}s)\nabla_qs$$ Taking into account (\[m2\]) and (\[m3\]), identity (\[th\]) will follow and the proof is complete. One has to note that the above formula (\[th\]) holds under the request that $f$ must be at least defined on $T_q$ and $T_q^{1-\alpha}$. \[qcd\] Let $\alpha>0$. If $\alpha \notin \mathbb{N}$ , then the $\alpha-$order Caputo left q-fractional and right q-fractional derivatives of a function $f$ are, respectively, defined by $$\label{qrd} _{q}C_a^\alpha f(t)\triangleq~ _{q}I_a ^{(n-\alpha)}\nabla_q ^nf(t)=\frac{1}{\Gamma(n-\alpha)} \int_a^t(t-qs)_q^{n-\alpha-1} \nabla_q^nf(s)\nabla_q s$$ and $$\label{qld} _{b}C_q^\alpha f(t)\triangleq ~_{b}I_q ^{(n-\alpha)}\nabla_b ^nf(t)=\frac{q^{(-1/2)\alpha(\alpha-1)}}{\Gamma_q(n-\alpha)} \int_t^b(s-t)_q^{n-\alpha-1}~_{b}\nabla_q^nf(sq^{1-\alpha })\nabla_q s$$ where $n=[\alpha]+1$. If $\alpha \in \mathbb{N}$, then $_{q}C_a^\alpha f(t)\triangleq \nabla_q^n f(t)$ and $_{b}C_q^\alpha f(t)\triangleq~ _{b}\nabla_q^n= (-1)^n \nabla_q^n$ Also, it is clear that $_{q}C_a^\alpha $ maps functions defined on $T_q$ to functions defined on $T_q$, and that $_{b}C_q^\alpha $ maps functions defined on $T_q^{1-\alpha}$ to functions defined on $T_q$ If, in Lemma \[m\] and Lemma \[rm\] we replace $\alpha$ by $1-\alpha$ . Then, we can relate the left and right Riemann and Caputo q-fractional derivatives. Namely, we state \[qrelate\] For any $0<\alpha<1$, we have $$\label{relate1} _{q}C_a^\alpha f(t)=~_{q}\nabla_a^\alpha f(t)-\frac{ (t-a)_q^{-\alpha}}{\Gamma_q(1-\alpha)} f(a)$$ and $$\label{qrelate2} _{q^{-1}b}C_q^\alpha f(t)=~_{b}\nabla_q^\alpha f(t)-\frac{r(1-\alpha)}{\Gamma_q(1-\alpha)} (b-qt)_q^{-\alpha}f(q^\alpha q^{-1}b)$$ A Caputo q-fractional Initial Value Problem and q-Mittag-Leffler Function {#s:4} ========================================================================= The following identity which is useful to transform Caputo q-fractional difference equations into q-fractional integrals, will be our key in this section. \[qtrans\] Assume $\alpha>0$ and $f$ is defined in suitable domains. Then $$\label{qtrans1} _{q}I_a^\alpha~ _{q}C_a^\alpha f(t)=f(t)-\sum_{k=0}^{n-1}\frac{(t-a)_q^{k}}{\Gamma_q(k+1)}\nabla_q^kf(a)$$ and if $0<\alpha\leq1$ then $$\label{qtrans3} _{q}I_a^\alpha~ _{q}C_a^\alpha f(t)= f(t)-f(a)$$ The proof followed by definition of Caputo q-fractional derivatives, (\[t11\]), Lemma \[m\] and Theorem \[qinter\]. The following identity [@Pred] is essential to solve linear q-fractional equations $$\label{qpower} _{q}I_a^\alpha (x-a)_q^\mu =\frac{\Gamma_q(\mu+1)}{\Gamma_q(\alpha+\mu+1)}(x-a)_q^{\mu+\alpha}~~(0<a<x<b)$$ where $\alpha \in \mathbb{R}^+$ and $\mu \in (-1,\infty)$. \[qlinear\] Let $0<\alpha\leq 1$ and consider the left Caputo q-fractional difference equation $$\label{lfractional} _{q}C^ \alpha _a y(t)= \lambda y(t)+f(t),~~y(a)=a_0,~t\in T_q.$$ if we apply $_{q}I_a^\alpha$ on the equation (\[lfractional\]) then by the help of (\[qtrans3\]) we see that $$y(t)= a_0+ \lambda~ _{q}I_a^\alpha y(t)+~_{q}I_a^\alpha f(t).$$ To obtain an explicit clear solution, we apply the method of successive approximation. Set $y_0(t)=a_0$ and $$y_m(t)=a_0+\lambda~ _{q}I_a^\alpha y_{m-1}(t)+ _{q}I_a^\alpha f(t), m=1,2,3,....$$ For $m=1$, we have by the power formula (\[qpower\]) $$y_1(t)=a_0[1+\frac{\lambda ( t-a)_q^{(\alpha)}}{\Gamma_q(\alpha+1)}]+~ _{q}I_a^\alpha f(t).$$ For $m=2$, we also see that $$y_2(t)= a_0+ \lambda a_0~_{q}I_a^\alpha[1+ \frac{(t-a)_q^\alpha}{\Gamma_q(\alpha+1)}]+~_{q}I_a^{\alpha} f(t) +\lambda~ _{q}I_a^{2\alpha} f(t)$$ $$=a_0 [1+\frac{\lambda (t-a)_q^\alpha}{\Gamma_q(\alpha+1)}+\frac{\lambda^2 (t-a)_q^{2\alpha}}{\Gamma_q(2\alpha+1)}]+~_{q}I_a^{\alpha} f(t) +\lambda~ _{q}I_a^{2\alpha} f(t)$$ If we proceed inductively and let $m\rightarrow\infty$ we obtain the solution $$y(t)=a_0[1+\sum_{k=1}^\infty\frac{\lambda^k (t-a)_q^{k\alpha}} {\Gamma_q(k\alpha+1)}]+\int_a^t [\sum_{k=1}^{\infty}\frac{\lambda^{k-1}}{\Gamma_q(\alpha k)}(t-qs)_q^{\alpha k-1}]f(s)\nabla_qs$$ $$= a_0[1+\sum_{k=1}^\infty\frac{\lambda^k (t-a)_q^{k\alpha}} {\Gamma_q(k\alpha+1)}]+\int_a^t [\sum_{k=0}^{\infty}\frac{\lambda^{k}}{\Gamma_q(\alpha k+\alpha)}(t-qs)_q^{\alpha k+(\alpha-1)}]f(s)\nabla_qs$$ $$= a_0[1+\sum_{k=1}^\infty\frac{\lambda^k (t-a)_q^{k\alpha}} {\Gamma_q(k\alpha+1)}]+\int_a^t (t-qs)_q^{(\alpha-1)}[\sum_{k=0}^{\infty}\frac{\lambda^{k}}{\Gamma_q(\alpha k+\alpha)}(t-q^\alpha s)_q^{(\alpha k)}]f(s)\nabla_qs$$ If we set $\alpha=1$, $\lambda=1$, $a=0$ and $f(t)=0$ we come to a q-exponential formula $e_q(t)= \sum_{k=0}^\infty \frac{t^k}{\Gamma_q(k+1)}$ on the time scale $T_q$, where $\Gamma_q(k+1)=[k]_q!=[1]_q[2]_q...[k]_q$ with $[r]_q=\frac{1-q^r}{1-q}$. It is known that $e_q(t)=E_q((1-q)t)$, where $E_q(t)$ is a special case of the basic hypergeormetric series, given by $$E_q(t)=~_{1}\phi_0(0;q,t)=\Pi_{n=0}^\infty (1-q^nt)^{-1}=\sum_{n=0}^\infty \frac{t^n}{(q)_n},$$ where $(q)_n=(1-q)(1-q^2)...(1-q^n)$ is the q-Pochhammer symbol. If we compare with the classical case, then the above example suggests the following q-analogue of Mittag-Leffler function \[Mitt\] For $z ,z_0\in \mathbf{C}$ and $\mathfrak{R}(\alpha)> 0$, the q-Mittag-Leffler function is defined by $$\label{Mit} _{q}E_{\alpha,\beta}(\lambda,z-z_0)=\sum_{k=0}^\infty \lambda^k \frac{(z-z_0)_q^{\alpha k}}{\Gamma_q(\alpha k+\beta)}.$$ When $\beta=1$ we simply use $~_{q}E_{\alpha}(\lambda,z-z_0):=~_{q}E_{\alpha,1}(\lambda,z-z_0)$. According to Definition \[Mitt\] above, the solution of the q-Caputo-fractional equation in Example \[qlinear\] is expressed by $$y(t)=a_0 ~_{q}E_\alpha (\lambda,t-a)+ \int_a^t (t-qs)_q^{\alpha-1}~_{q}E_{\alpha,\alpha} (\lambda,t-q^\alpha s) f(s)\nabla_q s.$$ \[last\] 1) Note that the above proposed definition of the q-analogue of Mittag-Leffler function agrees with time scale definition of exponential functions. As it depends on the three parameters other than $\alpha$ and $\beta$. 2\) The power term of the q-Mittag-Leffler function contains $\alpha$ (the term $(z-z_0)_q^{\alpha k}$). We include this $\alpha$ in order to express the solution of q-Caputo initial value problem explicitly by means of the q-Mittag-Leffler function. This is due to that in general it is not true for the q-factorial function to satisfy the power formula $(z-z_0)_q^{\alpha k}= [(z-z_0)_q^{\alpha }]^k$. But for example the latter power formula is true when $z_0=0$. Therefore, for the case $z_0=0$, we may drop $\alpha$ from the power so that the q-Mittag-Leffler function will tend to the classical one when $q\rightarrow 1$. 3\) Once Problem 1 raised in section 2 is solved an analogue result to Proposition \[qtrans\] can be obtained for right Caputo q-fractional derivatives. [99]{} Abdeljawad T. and Dumitru B. Fractional differences and integration by parts, Journal of computational Analysis and Applications (to appear 2011 early). Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres, Discrete-Time Variational Problems, Signal Processing, Volume 91 Issue 3, March, 2011. At[i]{}c[i]{} F.M. and Eloe P. W., A Transform method in discrete fractional calculus , International Journal of Difference Equations, vol 2, no 2, (2007), 165–176. At[i]{}c[i]{} F.M. and Eloe P. W., Initial value problems in discrete fractional calculus, Proceedings of the American Mathematical Society, to appear. At[i]{}c[i]{} F.M. and Eloe P. W., Fractional q-calculus on a time scale , Journal of Nonlinear Mathematical Physics 14, 3, (2007), 333–344. Bohner M. and A. Peterson, Dynamic equations on Time Scales, Birkhäuser, Boston, 2001. Miller K. S. and Ross B.,Fractional difference calculus, Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan, (1989), 139-152. Ernst T., The history of q-calculus and new method (Licentiate Thesis), U.U.D.M. Report 2000: http://math.uu.se/thomas/Lics.pdf. Agrawal R. P., Certain fractional q-integrals and q-derivatives, Proc. Camb. Phil. Soc. (1969), 66,365, 365-370. Al-Salam W. A., Some fractional q-integrals and q-derivatives, Proc. Edin. Math. Soc., vol 15 (1969), 135-140. Al-Salam W. A. and Verma A., A fractional Leibniz q-formula, Pacific Journal of Mathematics, vol 60, (1975), 1-9. Al-Salam W. A., q-Analogues of Cauchy’s formula, Proc. Amer. Math. Soc. 17,182-184,(1952-1953). Predrag M.R., Sladana D. M. and Miomir S. S., Fractional Integrals and Derivatives in q-calculus, Applicable Analysis and Discrete Mathematics, 1, 311-323, (2007). Podlubny I., Fractional Differential Equations, Academic Press, 1999. Samko G. Kilbas A. A., Marichev, Fractional Integrals and Derivatives‘: Theory and Applications, Gordon and Breach, Yverdon, 1993. Kilbas A., Srivastava M. H.,and Trujillo J. J. Theory and Application of Fractional Differential Equations, North Holland Mathematics Studies 204, 2006. [^1]: On leave of absence from Institute of Space Sciences, P.O.BOX, MAG-23, R 76900,Maturely-Bucharest, Romania, Emails: dumitru@cankaya.edu.tr, baleanu@venus.nipne.ro
--- abstract: 'Several recent papers have examined a rational polyhedron $P_m$ whose integer points are in bijection with the numerical semigroups (cofinite, additively closed substets of the non-negative integers) containing $m$. A combinatorial description of the faces of $P_m$ was recently introduced, one that can be obtained from the divisibility posets of the numerical semigroups a given face contains. In this paper, we study the faces of $P_m$ containing arithmetical numerical semigroups and those containing certain glued numerical semigroups, as an initial step towards better understanding the full face structure of $P_m$. In most cases, such faces only contain semigroups from these families, yielding a tight connection to the geometry of $P_m$.' address: - \| Mathematics and Statistics Department\ San Diego State University\ San Diego, CA 92182 - \| Mathematics and Computer Science Department\ Colorado College\ Colorado Springs, CO 80903 - \| Mathematics and Statistics Department\ San Diego State University\ San Diego, CA 92182 - \| Mathematics and Statistics Department\ San Diego State University\ San Diego, CA 92182 - \| Mathematics Department\ University of Washington Tacoma\ Tacoma, WA 98402 - \| Mathematics and Statistics Department\ Swarthmore College\ Swarthmore, PA 19081 - \| Mathematics and Statistics Department\ San Diego State University\ San Diego, CA 92182 author: - Jackson Autry - Abigail Ezell - Tara Gomes - 'Christopher O’Neill' - Christopher Preuss - Tarang Saluja - Eduardo Torres Davila title: \| Numerical semigroups, polyhedra, and posets II: \ locating certain families of semigroups --- Introduction {#sec:intro} ============ A numerical semigroup is a cofinite subset $S \subseteq {{\ensuremath{\mathbb{Z}}}}_{\ge 0}$ of the non-negative integers that is closed under addition. Numerical semigroups are often specified using a set of generators $n_1 < \cdots < n_k$, i.e., $$S = {\langle}n_1, \ldots, n_k{\rangle}= \{a_1n_1 + \cdots + a_kn_k : a_i \in {{\ensuremath{\mathbb{Z}}}}_{\ge 0}\}.$$ The Apéry set of $m \in S$ is the set $$\operatorname{Ap}(S;m) = \{n \in S : n - m \notin S\}$$ of the minimal elements of $S$ within each equivalence class modulo $m$. Since $S$ is cofinite, we are guaranteed $\|\!\operatorname{Ap}(S;m)\| = m$, and that $\operatorname{Ap}(S;m)$ contains exactly one element in each equivalence class modulo $m$. The elements of $\operatorname{Ap}(S;m)$ are partially ordered by divisibility, that is, $a \preceq a'$ whenever $a' - a \in \operatorname{Ap}(S;m)$; we call this the Apéry poset of $m$ in $S$. A family of rational polyhedra whose integer points are in bijection with certain numerical semigroups, first introduced by Kunz [@kunz] and independently in [@kunzcoords], has received a flurry of recent attention [@alhajjarkunz; @wilfmultiplicity; @oversemigroupcone; @kaplanwilfconj; @kunzfaces1; @kunznew; @kunzcoords]. More specifically, given $m \ge 2$, the Kunz polyhedron $P_m$ is a pointed rational cone, translated from the origin, whose integer points are in bijection with the numerical semigroups containing $m$ (we defer precise definitions to Section \[sec:polyhedra\]). One of the primary goals of studying these polyhedra is to utilize tools from lattice point geometry (e.g., Ehrhart’s theorem) to approach some long-standing enumerative questions involving numerical semigroups [@wilfsurvey; @kaplanwilfconj]. Recent developments have produced a combinatorial description of the faces of $P_m$. Given a numerical semigroup $S$ containing $m$, the Kunz poset is the partially ordered set with ground set ${{\ensuremath{\mathbb{Z}}}}_m$ obtained by replacing each element of the Apéry poset $\operatorname{Ap}(S;m)$ with its equivalence class in ${{\ensuremath{\mathbb{Z}}}}_m$. In [@wilfmultiplicity], it was shown that two numerical semigroups lie in the interior of the same face of $P_m$ if and only if they have identical Kunz posets, thereby providing a natural combinatorial object that indexes the faces of $P_m$ containing numerical semigroups. This combinatorial description was extended in [@kunzfaces1] to include every face of $P_m$, even those that do not contain any numerical semigroups. When studying questions that are difficult to answer for general numerical semigroups, it is common to restrict to certain families with some additional structure. In this paper, we examine two such families. The first (one of the most common in the literature) are arithmetical numerical semigroups, whose minimal generating sets are arithmetic sequences. Thanks to a particularly powerful membership criterion, these semigroups admit closed forms for many quantities that are difficult to obtain in general [@setoflengthsets; @omidali; @numerical]. The second family is comporised of numerical semigroups obtained by scaling every element of a given semigroup $S$ by some common factor $\beta$ and then adding one new generator $\alpha$ to obtain ${\langle}\alpha{\rangle}+ \beta S$ (this process is known as gluing). The result is a broad class of semigroups (which we call monoscopic numerical semigroups (Definition \[d:monoscopic\])) that includes several other families of independent interest, such as supersymmetric [@supersymmetric] and telescopic [@telescopic] numerical semigroups, as well as numerical semigroups on compound sequences [@compseqs]. The combinatorial interpretation of the faces of $P_m$ in terms of posets is still young, and many basic questions are still unanswered. The goal of this paper is to describe geometrically the faces of $P_m$ containing numerical semigroups from the families described above, as an initial step towards better understanding the full face structure of $P_m$. To this end, we give a formula for the dimension of every such face, and in most cases characterize their extremal rays (both of which are still not well understood in general for $P_m$). Our results yield two particularly notable geometric insights. - We provide a collection of combinatorial embeddings of the form $P_m \hookrightarrow P_{\beta m}$, which we call monoscopic embeddings, whose images contain precisely those semigroups obtained from monoscopic gluings of semigroups in $P_m$ with scaling factor $\beta$. This provides a complete characterization of the faces containing monoscopic numerical semigroups, as well as all faces they contain. One interpretation of this construction is that monoscopic gluing of numerical semigroups can be realized as a geometric operation on Kunz polyhedra. - The posets associated to low-dimensional faces, such as rays, possess the most relations, making them more difficult to classify in general. Moreover, many rays do not contain numerical semigroups. When describing a ray $\vec r$ of a face $F$ containing arithmetical numerical semigroups, we do so by examining the effect adding $\vec r$ to each integer point in $F$ has on the minimal generators of the corresponding semigroup. In addition to providing a glimpse of the face structure of $P_m$, our results have several consequences for numerical semigroups outside the realm of geometry. - The elements in the Apéry sets of arithmetical and monoscopic numerical semigroups are well understood (indeed, this is one of the reasons these families are considered especially “nice”). We extend these classical results to include a description of the divisibility poset structure of the Apéry set, both of which have elegant combinatorial structure; see Figure \[f:previews\] for examples. [0.45]{} ![Apéry posets of two numerical semigroups. The first, given by $S = {\langle}15,17,19,21,23{\rangle}$ (left), is arithmetical, and the second, given by $S' = {\langle}15, 18, 20, 27{\rangle}$ (right), is monoscopic.[]{data-label="f:previews"}](posets-mainarith.pdf){height="2.5in"} [0.45]{} ![Apéry posets of two numerical semigroups. The first, given by $S = {\langle}15,17,19,21,23{\rangle}$ (left), is arithmetical, and the second, given by $S' = {\langle}15, 18, 20, 27{\rangle}$ (right), is monoscopic.[]{data-label="f:previews"}](posets-mainscopic.pdf){height="2.5in"} - In most cases, the membership criterion for generalized arithmetical numerical semigroups can be extended to all semigroups lying on their same face. This gives rise to a new family of semigroups, which we call extra-generalized arithmetical numerical semigroups, possessing most of the desirable properties of arithmetical numerical semigroups. We develop this new family, independent of the geometry of $P_m$, including a membership criterion (Proposition \[p:arithmembercrit\]) and a formula for their Frobenius number (Corollary \[c:extraarithfrob\]). The paper is organized as follows. After reviewing the necessary terminology in Section \[sec:polyhedra\], we introduce extra-generalized arithmetical numerical semigroups in Section \[sec:arithposets\], providing a membership criterion (Proposition \[p:arithmembercrit\]), a characterization of their Apéry posets (Theorem \[t:arithposet\]), and a formula for their Frobenius numbers (Corollary \[c:extraarithfrob\]). We then examine the faces of $P_m$ containing extra-generalized arithmetical numerical semigroups in Section \[sec:arithfaces\], giving a formula for their dimension (Theorem \[t:arithfacedim\]) and, in most cases, their extremal rays (Theorem \[t:arithrays\]). In the final two sections of the paper, we turn attention to monoscopic numerical semigroups, characterizing their Apéry poset structure (Theorem \[t:monoscopicposet\]) and the complete structure of the faces containing them (Theorems \[t:augmonoscopicfaces\] and \[t:monoscopicfaces\]) via monoscopic embeddings (Definition \[d:monoscopicembedding\]). The group cone and its faces {#sec:polyhedra} ============================ After recalling basic definitions from polyhedral geometry (see [@ziegler] for a thorough introduction), we define the Kunz polyhedron $P_m$ and a related polyhedron from [@kunzfaces1]. A rational polyhedron $P \subset {{\ensuremath{\mathbb{R}}}}^d$ is the set of solutions to a finite list of linear inequalities with rational coefficients, that is, $$P = \{x \in {{\ensuremath{\mathbb{R}}}}^d : Ax \le b\}$$ for some matrix $A$ and vector $b$. If none of the inequalities can be omitted without altering $P$, we call this list the $H$-description or facet description of $P$ (such a list of inequalities is unique up to reordering and scaling by positive constants). The inequalities appearing in the H-description of $P$ are called facet inequalities of $P$. Given a facet inequality $a_1x_1 + \cdots + a_dx_d \le b$ of $P$, the intersection of $P$ with the equation $a_1x_1 + \cdots + a_dx_d = b$ is called a facet of $P$. A face $F$ of $P$ is a subset of $P$ equal to the intersection of some collection facets of $P$. The set of facets containing $F$ is called the H-description or facet description of $F$. The dimension of a face $F$ is the dimension $\dim(F)$ of the affine linear span of $F$. The relative interior of a face $F$ is the set of points in $F$ that do not also lie in a face of dimension strictly smaller than $F$ (or, equivalently, do not lie in a proper face of $F$). We say $F$ is a vertex if $\dim(F) = 0$, and edge if $\dim(F) = 1$ and $F$ is bounded, a ray if $\dim(F) = 1$ and $F$ is unbounded, and a ridge if $\dim(F) = d - 2$. If there is a unique point $v$ satisfying every inequality in the H-description of $P$ with equality, then we call $P$ a cone with vertex $v$. If, additionally, $b = 0$ above, we call $P$ a pointed cone. Separately, we say $P$ is a polytope if $P$ is bounded. If $P$ is a pointed cone, then any face $F$ equals the non-negative span of the rays of $P$ it contains, and if $P$ is a polytope, then any face $F$ equals the convex hull of the set of vertices of $P$ it contains; in each case, we call this the V-description of $F$. A partially ordered set (or poset) is a set $Q$ equipped with a partial order $\preceq$ that is reflexive, antisymmetric, and transitive. We say $q$ covers $q'$ if $q' \prec q$ and and there is no intermediate element $q''$ with $q' \prec q'' \prec q$. If $(Q, \preceq)$ has a unique minimal element $0 \in Q$, the atoms of $Q$ are the elements that cover $0$. The set of faces of a polyhedron $P$ forms a poset under containment that is a lattice (i.e., every element has a unique greatest common divisor and least common multiple) and is graded, where the height function is given by dimension. If $P$ is a cone, then every face of $P$ equals the sum of some collection of extremal rays and the intersection of some collection of facets, meaning the face lattice of $P$ is both atomic and coatomic. \[d:kunzcoords\] Fix $m \in {{\ensuremath{\mathbb{Z}}}}_{\ge 2}$, and a numerical semigroup $S$ containing $m$. Write $$\operatorname{Ap}(S;m) = \{0, a_1, \ldots, a_{m-1}\},$$ where $a_i = mz_i + i$ for each $i = 1, \ldots, m-1$. We refer to the tuples $(a_1, \ldots, a_{m-1})$ and $(z_1, \ldots, z_{m-1})$ as the Apéry tuple/Apéry coordinates and the Kunz tuple/Kunz coordinates of $S$, respectively. \[d:kunzandgroupcone\] Fix a finite Abelian group $G$, and let $m = \|G\|$. The group cone $\mathcal C(G) \subset {{\ensuremath{\mathbb{R}}}}^{m-1}$ is the pointed cone with facet inequalities $$\begin{array}{r@{}c@{}l@{\qquad}l} x_i + x_j &{}\ge{}& x_{i+j} & \text{for } i, j \in G \setminus \{0\} \text{ with } i + j \ne 0, \end{array}$$ where the coordinates of ${{\ensuremath{\mathbb{R}}}}^{m-1}$ are indexed by $G \setminus \{0\}$. Additionally, for each integer $m \ge 2$, let $P_m$ denote the translation of $\mathcal C({{\ensuremath{\mathbb{Z}}}}_m)$ with vertex $(-\tfrac{1}{m}, \ldots, -\tfrac{m-1}{m})$, whose facets are given by $$\begin{array}{r@{}c@{}l@{\qquad}l} x_i + x_j &{}\ge{}& x_{i+j} & \text{for } 1 \le i \le j \le m - 1 \text{ with } i + j < m, \text{ and} \\ x_i + x_j + 1 &{}\ge{}& x_{i+j-m} & \text{for } 1 \le i \le j \le m - 1 \text{ with } i + j > m. \end{array}$$ We refer to $P_m$ as the Kunz polyhedron. Parts (a) and (b) of the following theorem appear in [@kunz] and [@kunzfaces1], respectively. \[t:kunzlatticepts\] Fix an integer $m \ge 2$. (a) The set of all Kunz tuples of numerical semigroups containing $m$ coincides with the set of integer points in $P_m$. (b) The set of all Apéry tuples of numerical semigroups containing $m$ coincides with the set of integer points $(a_1, \ldots, a_{m-1})$ in $\mathcal C({{\ensuremath{\mathbb{Z}}}}_m)$ with $a_i \equiv i \bmod m$ for every $i$. In view of Theorem \[t:kunzlatticepts\], given a face $F \subset \mathcal C({{\ensuremath{\mathbb{Z}}}}_m)$, we say $F$ contains a numerical semigroup $S$ if the Apéry tuple lies in the relative interior of $F$. Analogously, we say a face $F' \subset P_m$ contains $S$ if the Kunz tuple of $S$ lies in the relative interior of $F'$. \[t:groupconefacelattice\] Fix a finite Abelian group $G$ and a face $F \subset \mathcal C(G)$. (a) The set $H = \{h \in G : x_g = 0 \text{ for all } x \in F\}$ is a subgroup of $G$ (called the Kunz subgroup of $F$), and the relation $P = (G/H, \preceq)$ with unique minimal element $\ol 0$ and $\ol a \preceq_P \ol b$ whenever $x_a + x_{b-a} = x_b$ for distinct $a, b \in G$ is a well defined partial order (called the Kunz poset of $F$). (b) If $G = {{\ensuremath{\mathbb{Z}}}}_m$ with $m \ge 2$ and $F$ contains a numerical semigroup $S$, then the Kunz subgroup of $F$ is trivial and the Kunz poset of $F$ equals the Kunz poset of $S$. (c) In the Kunz poset $P$ of $F$, $\ol b$ covers $\ol a$ if and only if $\ol b - \ol a$ is an atom of $P$. \[r:automorphisms\] The automorphism group of a finite Abelian group $G$ acts on the face lattice of $\mathcal C(G)$ by permuting coordinates in the natural way. Each such automorphsim induces a group isomorphism between the corresponding Kunz subgroups, as well as on the corresponding Kunz posets. Extra-generalized arithmetical numerical semigroups {#sec:arithposets} =================================================== Arithmetical numerical semigroups, which are numerical semigroups whose minimal generating set is an arithmetic sequence, are a common focal point in the literature. It turns out the polyhedral faces that contain arithmetical numerical semigroups also contain semigroups from two related families: generalized arithmetical numerical semigroups (previously studied in [@omidali; @omidalirahmati]) and a new family (Definition \[d:extragenarith\]). In this section, we provide a characterization of the Kunz posets for this new family (Corollary \[c:arithposet\]) using an adapted membership criterion (Proposition \[p:arithmembercrit\]), as well as a formula for their Frobenius numbers (Corollary \[c:extraarithfrob\]). \[d:extragenarith\] An extra-generalized arithmetical numerical semigroup has the form $$S = {\langle}a, ah + d, ah + 2d, \ldots, ah + kd{\rangle}$$ for $a, h, k \in {{\ensuremath{\mathbb{Z}}}}_{\ge 1}$ and $d \in {{\ensuremath{\mathbb{Z}}}}$ with $k < a$, $\gcd(a,d) = 1$ and $ah + kd > a$. If $d \ge 1$, then we call $S$ a generalized arithmetical numerical semigroup, and if $d < 0$, then we call $S$ a pessimistic arithmetical numerical semigroup. \[e:extragenarith\] Consider the semigroup $S = {\langle}11,12,14,16,18,20{\rangle}$, whose Kunz poset is depicted in Figure \[f:pessimisticposet\]. Reordering the generators as $S = {\langle}11,20,18,16,14,12{\rangle}$ reveals that $S$ is extra-generalized arithmetical with $d = -2$. This has the same Kunz poset as $S' = {\langle}11,20,29,38,47,56{\rangle}$, whose common difference $d' = 9$ satisfies $d' \equiv d \bmod a$. It is important to note the extra requirement that $ah + kd > a$ in Definition \[d:extragenarith\] is in place to ensure the given generating set is minimal. Indeed, the semigroup ${\langle}11,20,18,16,14,12,10{\rangle}$ demonstrates this need not be the case if the assumption is dropped. As it turns out, this assumption also forces $a$ to be the multiplicity of $S$. [0.30]{} ![Apéry posets of the semigroups $S = {\langle}11,12,14,16,18,20{\rangle}$ (left) and $S' = {\langle}11,20,29,38,47,56{\rangle}$ (middle) from Example \[e:extragenarith\], along with their shared Kunz poset (right).](posets-arith-pess.pdf){height="1.5in"} [0.30]{} ![Apéry posets of the semigroups $S = {\langle}11,12,14,16,18,20{\rangle}$ (left) and $S' = {\langle}11,20,29,38,47,56{\rangle}$ (middle) from Example \[e:extragenarith\], along with their shared Kunz poset (right).](posets-arith-npess.pdf){height="1.5in"} [0.30]{} ![Apéry posets of the semigroups $S = {\langle}11,12,14,16,18,20{\rangle}$ (left) and $S' = {\langle}11,20,29,38,47,56{\rangle}$ (middle) from Example \[e:extragenarith\], along with their shared Kunz poset (right).](posets-arith-pesskunz.pdf){height="1.5in"} We begin by generalizing a membership criterion for generalized arithmetical numerical semigroups to the extra-generalized family. \[p:arithmembercrit\] Fix an extra-generalized arithmetical numerical semigroup $$S = {\langle}a, ah + d, ah + 2d, \ldots, ah + kd{\rangle}.$$ Fix $n \in {{\ensuremath{\mathbb{Z}}}}$, and let $n = qa + rd$ for $q, r \in {{\ensuremath{\mathbb{Z}}}}_{\ge 0}$ with $0 \le r \le a - 1$. We have (a) $n \in S$ if and only if $\lceil \tfrac{r}{k} \rceil h \le q$, and (b) $n \in \operatorname{Ap}(S;a)$ if and only if $\lceil \tfrac{r}{k} \rceil h = q$. First, suppose $n \in S$, so for some $z_0, z_1, \ldots, z_k \in {{\ensuremath{\mathbb{Z}}}}_{\ge 0}$, we have $$n = z_0 a + \sum_{i=1}^k z_i(ah + id).$$ Write $\sum_{i=1}^k z_i i = q'a + r$ for $q', r \in {{\ensuremath{\mathbb{Z}}}}_{\ge 0}$ with $r < a$. Letting $$q = z_0 + h\sum\limits_{i=1}^k z_i + q' d,$$ we obtain $$qa + rd = z_0 a + ah\sum_{i=1}^k z_i + q'ad + rd = z_0 a + \sum_{i=1}^k z_i (ah + id) =n$$ and $$\bigg\lceil \frac{r}{k} \bigg\rceil h \le \bigg\lceil \frac{1}{k}\sum_{i=1}^k z_i i \bigg\rceil h \le \bigg\lceil \sum_{i=1}^k z_i \bigg\rceil h = h \sum_{i=1}^k z_i \le q.$$ Conversely, assume $\lceil \frac{r}{k} \rceil h \le q$ for some $n = aq + rd$ with $q, r \in {{\ensuremath{\mathbb{Z}}}}_{\ge 0}$ and $0 \le r \le a - 1$. If $r = 0$, then $n = aq$ clearly implies $n \in S$, and if $0 < r \le k$, then the bounds on $r$ imply $\lceil \frac{r}{k} \rceil = 1$, so $h \le q$ and $$n = aq + rd = (q - h)a + (ha + rd) \in S.$$ Lastly, if $k < r$, then $$n = qa + rd = (q - h + h)a + (r - k + k)d = (ha + kd) + (q - h)a + (r - k)d,$$ so it suffices to show $(q - h)a + (r - k)d \in S$. Since $0 \le r - k \le a - 1$ and $$\bigg\lceil \frac{r-k}{k} \bigg\rceil h = \bigg\lceil \frac{r}{k} \bigg\rceil h - h \le q - h,$$ we conclude by induction on $r$ that $n \in S$. This completes the proof of part (a). For part (b), $n \in \operatorname{Ap}(S;a)$ occurs when $n \in S$ and $n - a \notin S$. Writing $n = qa + rd$ as above, we check by part (a) that this happens if and only if $q - 1 < \lceil \frac{r}{k} \rceil h \le q$, which is equivalent to the desired equality. \[t:arithposet\] Fix an extra-generalized arithmetical numerical semigroup $$S = {\langle}a, ah + d, ah + 2d, \ldots, ah + kd{\rangle},$$ and write $a - 1 = qk + r$ for $q, r \in {{\ensuremath{\mathbb{Z}}}}_{\ge 0}$ with $r < k$. (a) Each nonzero element $a_i \in \operatorname{Ap}(S;a)$ has the form $$a_i = x_iah + ((x_i - 1)k + y_i)d$$ for either $x_i \in [1, q]$ and $y_i \in [1,k]$, or $x_i = q + 1$ and $y_i \in [1,r]$. (b) We have $a_i \prec a_j$ in $\operatorname{Ap}(S;a)$ if and only if $x_i < x_j$ and $y_i \ge y_j$. (c) An element $a_j$ covers $a_i$ in $\operatorname{Ap}(S;a)$ if and only if $x_j = x_i + 1$ and $y_i \ge y_j$. Part (a) follows from Proposition \[p:arithmembercrit\](b). For part (c), if $x_j = x_i + 1$ and $y_i \ge y_j$, then $a_j - a_i = ah + (k + y_j - y_i)d$ is a minimal generator of $S$, which by Theorem \[t:groupconefacelattice\](c) implies $a_j$ covers $a_i$. Conversely, suppose $a_j$ covers $a_i$, which by Theorem \[t:groupconefacelattice\](c) means $a_j - a_i = ah + md$ with $1 \le m \le k$. With $a_i$ and $a_j$ written as in part (a), we see $x_i - x_j = 1$ and $y_i - y_j = k - m \ge 0$, as desired. Lastly, for part (b), we cannot have $a_i \prec a_j$ unless $x_i < x_j$ by part (c). In this case, $$a_j - a_i = (x_j - x_i)ah + ((x_j - x_i)k + y_j - y_i)d$$ with $0 \le (x_j - x_i)k + y_j - y_i \le a - 1$, so $a_j - a_i \in \operatorname{Ap}(S;a)$ by Proposition \[p:arithmembercrit\](b) when $$\big\lceil \tfrac{1}{k}((x_j - x_i)k + y_j - y_i) \big\rceil = x_j - x_i + \big\lceil \tfrac{1}{k}(y_j - y_i) \big\rceil = x_j - x_i,$$ which happens precisely when $y_i \ge y_j$. \[c:arithposet\] Resume notation from Theorem \[t:arithposet\], and suppose $S$ has Kunz poset $P$. (a) The elements of $P$ have the form $[a_i] = [md]$, where $m = ((x_i - 1)k + y_i)$ takes each integer value in $[0, a - 1]$. (b) The Kunz poset $P$ is graded, with each $[a_i]$ occuring with height $x_i$. (c) The Kunz poset $P$ depends only on $a$, $k$, and the residue of $d$ modulo $a$. Part (a) follows from Theorem \[t:arithposet\](a) and the division algorithm, and part (b) follows from Theorem \[t:arithposet\](c). Lastly, part (c) follows by examining the statements of parts (a) and (b). \[r:arithposetdrawing\] Again resuming notation from Theorem \[t:arithposet\], the poset $P$ can be drawn so that the nonzero elements form a grid with $k$ columns where $y_j$ specifies the position from the left within each row, and cover relations between sequential rows are drawn precisely when they do not have positive slope. Additionally, reading elements left to right and bottom to top yields $[0], [d], [2d], \ldots, [(a-1)d]$, and all rows have $k$ elements except possibly the top row (where $x_i = q + 1$), which has $r$ elements. We close this section with a formula for the Frobenius number of an extra-generalized arithmetical numerical semigroup, extending [@omidalirahmati Theorem 2.8] to the case when $d < 0$. Our proof, including for positive $d$, utilizes the poset structure from Theorem \[t:arithposet\]. \[c:extraarithfrob\] For a given extra-generalized arithmetical numerical semigroup $$S = {\langle}a, ah + d, ah + 2d, \ldots, ah + kd{\rangle},$$ we have $$F(S) = \begin{cases} \lceil \tfrac{a-1}{k} \rceil ah + (a - 1)d - a & \text{if } d > 0; \\ \lceil \tfrac{a-1}{k} \rceil (ah + kd) + (1-k)d - a & \text{if } d < 0. \end{cases}$$ Write $a - 1 = qk + r$ for $q, r \in {{\ensuremath{\mathbb{Z}}}}_{\ge 0}$ with $r < k$. Regardless of whether $d$ is positive or negative, by Theorem \[t:arithposet\] and Remark \[r:arithposetdrawing\], $a_j = \max(\operatorname{Ap}(S;a))$ must occur in the top row of the Apéry poset. If $d > 0$, then $a_j$ is the last element of the top row, which means either $x_j = q + 1$ and $y_j = r$, or $x_j = q$ and $y_j = k$, both of which yield $$F(S) = a_j - a = x_jah + ((x_j - 1)k + y_j)d - a = \big\lceil \tfrac{a-1}{k} \big\rceil ah + (a - 1)d - a.$$ On the other hand, if $d < 0$, then $a_j$ is the first element in the top row, meaning $y = 1$ and either $x_j = q+1$ or $x_j = q$. In either case, $x_j = \lceil \tfrac{a-1}{k} \rceil$ and $$F(S) = a_j - a = x_jah + ((x_j - 1)k + y_j)d - a = \big\lceil \tfrac{a-1}{k} \big\rceil (ah + kd) + (1-k)d - a,$$ as desired. Polyhedra faces containing arithmetical numerical semigroups {#sec:arithfaces} ============================================================ Having now characterized the Kunz posets of extra-generalized arithmetical numerical semigroups, we examine the geometric properties of faces containing such semigroups. In particular, we characterize their dimension (Theorem \[t:arithfacedim\]) and, for some, their defining rays (Theorem \[t:arithrays\]). Additionally, we prove that in most cases, the faces contain only extra-generalized numerical semigroups (Theorem \[t:onlyarithfaces\]) We begin by describing the orbits (in the sense of Remark \[r:automorphisms\]) of faces containing extra-generalized arithmetical numerical semigroups, allowing us to restrict some arguments to the case when $d \equiv 1 \bmod a$. \[l:arithautomorphisms\] Fix an extra-generalized arithmetical numerical semigroup $$S = {\langle}a, ah + d, ah + 2d, \ldots, ah + kd{\rangle}$$ with containing face $F \subset \mathcal C({{\ensuremath{\mathbb{Z}}}}_a)$. Applying an automorphism of $C({{\ensuremath{\mathbb{Z}}}}_a)$ induces by multiplication by some $u \in {{\ensuremath{\mathbb{Z}}}}_a^$ to $F$ yields the face $F'$ containing the semigroup $$S' = {\langle}a, ah + d', ah + 2d', \ldots, ah + kd'{\rangle}$$ for any positive $d'$ with $d' \equiv ud \bmod a$. Multiply each element of the Kunz poset of $S$ by $u$ and apply Corollary \[c:arithposet\]. \[t:arithfacedim\] Given an extra-generalized arithmetical numerical semigroup $$S = {\langle}a, ah + d, ah + 2d, \ldots, ah + kd{\rangle}$$ with containing face $F \subset C({{\ensuremath{\mathbb{Z}}}}_a)$, we have $$\dim F = \begin{cases} a - 1 & \text{if } k = a - 1; \\ \lfloor \tfrac{a}{2} \rfloor & \text{if } k = a - 2; \\ 2 & \text{if } 1 < k < a - 2; \\ 1 & \text{if } k = 1. \end{cases}$$ By Lemma \[l:arithautomorphisms\], it suffices to assume $d \equiv 1 \bmod a$. First, $k = 1$ implies $\mathsf e(S) = 2$, so $S$ lies on the ray $(1, 2, \ldots, a - 1)$, and if $k = a - 1$, then $S$ has maximal embedding dimension, meaning $\dim F = a - 1$. Next, suppose $k = a - 2$. By Corollary \[c:arithposet\], the facet equations of $F$ each have the form $x_i + x_{a-1-i} = x_{a-1}$ for $i = 1, 2, \ldots, \lfloor \tfrac{a - 1}{2} \rfloor$. Since the matrices $$\begin{bmatrix} 1 & 0 & \cdots & 0 & 0 & \cdots & 0 & 1 & -1 \\ 0 & 1 & \cdots & 0 & 0 & \cdots & 1 & 0 & -1 \\ \vdots & \vdots & \ddots & \vdots & \vdots & \iddots & \vdots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & 1 & \cdots & 0 & 0 & -1 \\ \end{bmatrix} \qquad \text{and} \qquad \begin{bmatrix} 1 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 1 & -1 \\ 0 & 1 & \cdots & 0 & 0 & 0 & \cdots & 1 & 0 & -1 \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \iddots & \vdots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & 0 & 1 & \cdots & 0 & 0 & -1 \\ 0 & 0 & \cdots & 0 & 2 & 0 & \cdots & 0 & 0 & -1 \\ \end{bmatrix}$$ both have full rank $\lfloor \tfrac{a-1}{2} \rfloor$, we conclude $\dim F = a - 1 - \lfloor \tfrac{a-1}{2} \rfloor = \lfloor \tfrac{a}{2} \rfloor$. Lastly, suppose $k \le a - 2$. Again applying Corollary \[c:arithposet\], we obtain facet equations of the form $$x_{k+1} = x_1 + x_k = x_2 + x_{k-1} = \cdots \qquad \text{and} \qquad x_{k+2} = x_2 + x_k = x_3 + x_{k-1} = \cdots.$$ Subtracting corresponding equations above yields $$x_2 - x_1 = x_3 - x_2 = \cdots = x_{k-1} - x_{k-2} = x_k - x_{k-1},$$ meaning that the values of $x_1$ and $x_2$ determine the values for $x_3 \ldots, x_k$, and thus for the remainder of the coordinates as well. This proves $\dim F \le 2$, and since the coordinates in $C({{\ensuremath{\mathbb{Z}}}}_a)$ of the semigroups $S$, $$S' = {\langle}a, a(h+1) + d, \ldots, a(h+1) + kd{\rangle}, \quad \text{and} \quad S'' = {\langle}a, ah + (d + a), \ldots, ah + k(d + a){\rangle},$$ are affine independent and lie in $F$ by Corollary \[c:arithposet\](c), we conclude $\dim F = 2$. Our next result implies that, outside of the first 2 cases in Theorem \[t:arithfacedim\], faces containing extra-generalized arithmetical numerical semigroups contain exclusively such semigroups. \[t:onlyarithfaces\] Fix an extra-generalized arithmetical numerical semigroup $$S = {\langle}a, ah + d, ah + 2d, \ldots, ah + kd{\rangle}.$$ If $k < a - 2$, then any numerical semigroup $S'$ with identical Kunz poset to $S$ is also an extra-generalized arithmetical numerical semigroup. In particular, the face of $C({{\ensuremath{\mathbb{Z}}}}_a)$ containing $S$ contains only extra-generalized arithmetical numerical semigroups. If $S$ and $S'$ have identical Kunz poset $P$, then $m(S') = a$ is the number of elements of $P$ and $e(S') = k+1$ is one more than the number of atoms of $P$ by Theorem \[t:groupconefacelattice\](c), so write $S' = {\langle}a, n_1, \ldots, n_k{\rangle}$ with each $n_i \equiv di \bmod a$. Since $k < a - 2$, $\operatorname{Ap}(S';a)$ has at least 2 more elements $a_{k+1} \equiv (k+1)d \bmod a$ and $a_{k+2} \equiv (k+2)d \bmod a$. By Theorem \[t:arithposet\](c), $$a_{k+1} = n_1 + n_k = n_2 + n_{k-1} = \cdots \qquad \text{and} \qquad a_{k+2} = n_2 + n_k = n_3 + n_{k-1} = \cdots$$ must all hold. Let $d' = a_{k+2} - a_{k+1}$. Subtracting corresponding equations above yields $$d' = a_{k+2} - a_{k+1} = n_2 - n_1 = n_3 - n_2 = \cdots$$ as well as $$d' = a_{k+2} - a_{k+1} = n_k - n_{k-1} = n_{k-1} - n_{k-2} = \cdots.$$ If $k = 2j$ is even, then $a_{k+2} = 2n_{j+1} = n_j + n_{j+2}$ and $a_{k+1} = n_j + n_{j+1}$, so $d' = n_j - n_{j-1}$ and $d' = n_{j+1} - n_j$ both appear above. If $k = 2j - 1$ is odd, then $a_{k+2} = n_j + n_{j+1}$ and $a_{k+1} = 2n_j = n_{j-1} + n_{j+1}$, so again $d' = n_j - n_{j-1}$ and $d' = n_{j+1} - n_j$ both appear above. Putting everything together, we must have $d \equiv d' \bmod a$, so we can write $n_1 = ah' + d'$ for some $h' \ge 1$, and thus each $n_i = ah' + id'$, as desired. \[e:arithonly\] The hypothesis $k < a - 2$ is necessary in Theorem \[t:onlyarithfaces\]. If $k = a - 1$, such as for the semigroup $S = {\langle}6, 7, 8, 9, 10, 11{\rangle}$, then $S$ is max embedding dimension, so there are ample examples of other numerical semigroups (e.g., $S_1' = {\langle}6,8,10,13,15,17{\rangle}$) with identical Kunz poset (depicted in Figure \[f:nonarith1\]). If $k = a - 2$, such as for the numerical semigroup $S_2 = {\langle}6,13,14,15,26{\rangle}$, then $S$ shares its Kunz poset (depicted in Figure \[f:nonarith2\]) with the semigroup $S_2' = {\langle}6,15,16,19,20{\rangle}$. [0.30]{} ![Kunz posets for $S_1$ and $S_1'$ (left) and for $S_2$ and $S_2'$ (middle) from Example \[e:arithonly\], along with the Kunz poset of $S_3 = {\langle}7,23,25,27,29{\rangle}$ to which Theorem \[t:onlyarithfaces\] does apply.](posets-arith-embdimex1.pdf){width="1.2in"} [0.30]{} ![Kunz posets for $S_1$ and $S_1'$ (left) and for $S_2$ and $S_2'$ (middle) from Example \[e:arithonly\], along with the Kunz poset of $S_3 = {\langle}7,23,25,27,29{\rangle}$ to which Theorem \[t:onlyarithfaces\] does apply.](posets-arith-embdimex2.pdf){width="1.2in"} [0.30]{} ![Kunz posets for $S_1$ and $S_1'$ (left) and for $S_2$ and $S_2'$ (middle) from Example \[e:arithonly\], along with the Kunz poset of $S_3 = {\langle}7,23,25,27,29{\rangle}$ to which Theorem \[t:onlyarithfaces\] does apply.](posets-arith-embdimex3.pdf){width="1.2in"} \[c:arithalsofaces\] Any face with an extra-generalized arithmetical numerical semigroup $$S = {\langle}a, ah + d, ah + 2d, \ldots, ah + kd{\rangle}$$ in its interior also contains infinitely many arithmetical numerical semigroups. This follows from applying Theorem \[t:arithposet\] to $S$ and $S' = {\langle}a, a + d', \ldots, a + kd'{\rangle}$ where $d'$ is any positive integer with $d' \equiv d \bmod a$. There is still no known classification of the extremal rays of group cones. The machinery developed in [@kunzfaces1] yields a method to prove that a given ray is indeed a ray, but proving that a given list of rays is complete, even for a particular face, is a much more difficult task. However, it is a fact from polyhedral geometry (see [@ziegler], for instance) that any 2-dimensional face of a pointed cone has exactly 2 extremal rays. As our final result in this section, we characterize both bounding rays of the 2-dimensional faces of the group cone containing extra-generalized arithmetical numerical semigroups. \[t:arithrays\] Fix an extra-generalized arithmetical numerical semigroup $$S = {\langle}a, ah + d, ah + 2d, \ldots, ah + kd{\rangle},$$ and suppose $1 < k < a - 2$. The extremal rays of the face $F \subset \mathcal C({{\ensuremath{\mathbb{Z}}}}_a)$ containing $S$ are spanned by the following: (i) the vector $\vec r$ that, when added to the Kunz coordinates of $S$ in $P_a$, yields $${\langle}a, ah + (d + a), \ldots, ah + k(d + a){\rangle};$$ and (ii) the vector $\vec t$ that, when added to the Kunz coordinates of $S$ in $P_a$, yields $${\langle}a, a(h + a) + (d - a\lfloor a/k \rfloor), \ldots, a(h + a) + k(d - a\lfloor a/k \rfloor){\rangle}.$$ The ray $\vec r$ contains all of the numerical semigroups ${\langle}a, b{\rangle}$ for positive $b \equiv d \bmod a$, and the ray $\vec t$ contains numerical semigroups if $k \mid (a - 1)$, in which case those semigroups have the form ${\langle}a, b{\rangle}$ for some positive $b \equiv kd \bmod a$. By Lemma \[l:arithautomorphisms\], it suffices to assume $d = 1$. Under this assumption, it is easy to check that $\vec r = (1, 2, \dots, a - 1)$, and that $\vec t$ is given by $$t_i = x_ia - ((x_i - 1)k + y_i) \lfloor a/k \rfloor,$$ with $x_i$ and $y_i$ defined as in Theorem \[t:arithposet\](a). Note each $t_i$ is surely non-negative, as $$((x_i - 1)k + y_i) \lfloor a/k \rfloor \le x_ik \lfloor a/k \rfloor \le x_ia.$$ Since every integer point in $F$ is the Kunz tuple of some extra-generalized arithmetical numerical semigroup by Theorem \[t:onlyarithfaces\], adding $\vec r$ or $\vec t$ to any integer point in $F$ yields another point in $F$, so the rays $\vec r$ and $\vec t$ both lie in $F$. Now, the coordinates of $\vec r$ satisfy the equation $2r_1 = r_2$, which is not satisfied by the Apéry coordinates of $S$ since $k > 1$. To prove the same for $\vec t$, we consider two cases. First, if $k \mid a$, then $$t_k = a - k \lfloor a/k \rfloor = 0,$$ so $\vec t$ lies in a face with nontrivial Kunz subgroup. Otherwise, we can write $a - 1 = qk + r$ with $q = \lfloor a/k \rfloor$ and $0 \le r \le k - 2$ and obtain $$t_{(q+1)k} = t_{k-(r+1)} = a - (k - (r + 1))q = a - (k - (a - qk))q = (a - kq) + (qa - q^2k) = t_k + t_{qk}$$ as a facet equation not satisfied by the Apéry coordinates of $S$. As such, we conclude $\vec r$ and $\vec t$ both lie in proper faces of $F$, which necessarily are rays since $\dim F = 2$. For the final claims, it is clear that $\vec r = (1, 2, \ldots, a - 1)$ contains semigroups of the form ${\langle}a, b{\rangle}$ for $b \equiv 1 \bmod a$ since the Kunz poset is a total ordering. Likewise, if $a - 1 = qk$ for $q \in {{\ensuremath{\mathbb{Z}}}}_{\ge 0}$, then $t_k = 1$ and $$t_{k-i} = a - (k - i)q = 1 + iq$$ for each $i = 1, \ldots, k - 1$, so the Kunz poset of $\vec t$ is also a total ordering with unique atom $k$. This completes the proof. \[e:arithrays\] The Kunz poset of $S = {\langle}13,14,15,16,17{\rangle}$ is depicted in Figure \[f:arithraysboth\], along with the posets of its bounding rays in Figures \[f:arithray1\] and \[f:arithray2\]. By Theorem \[t:arithrays\], the Kunz poset of the first ray will always be a total order, obtained by “reading across the rows” of the Kunz poset for $S$. For this particular semigroup $k \mid (a - 1)$, so the latter ray is also a total order, obtained by reading “bottom to top and right to left” in the Kunz poset of $S$. The condition $k \mid (a - 1)$ holds whenever the top row of the Kunz poset has the full $k$ elements. \[e:arithbadray\] When $k \nmid (a-1)$, the Kunz poset of the ray $\vec t$ in Theorem \[t:arithrays\] can be substantially more complicated. One such example, for $S = {\langle}16,23,30,37,44,51,58{\rangle}$, is depicted in Figure \[f:arithraybad\]. Generally speaking, the ray $\vec t$ often does not contain numerical semigroups, and sometimes corresponds to a nontrivial subgroup (for instance, it is not hard to show using Theorem \[t:arithrays\] that this happens if $k \mid a$). [0.15]{} ![The Kunz poset of $S = {\langle}13,14,15,16,17{\rangle}$ ((b) above) and its two bounding rays ((a) and (c) above), as in Example \[e:arithrays\]. The final Kunz poset is that of a bounding ray for $S = {\langle}16,23,30,37,44,51,58{\rangle}$ discussed in Example \[e:arithbadray\].](posets-arith-ray1.pdf){height="2.5in"} [0.20]{} ![The Kunz poset of $S = {\langle}13,14,15,16,17{\rangle}$ ((b) above) and its two bounding rays ((a) and (c) above), as in Example \[e:arithrays\]. The final Kunz poset is that of a bounding ray for $S = {\langle}16,23,30,37,44,51,58{\rangle}$ discussed in Example \[e:arithbadray\].](posets-arith-raysboth.pdf){height="2.5in"} [0.15]{} ![The Kunz poset of $S = {\langle}13,14,15,16,17{\rangle}$ ((b) above) and its two bounding rays ((a) and (c) above), as in Example \[e:arithrays\]. The final Kunz poset is that of a bounding ray for $S = {\langle}16,23,30,37,44,51,58{\rangle}$ discussed in Example \[e:arithbadray\].](posets-arith-ray2.pdf){height="2.5in"} [0.30]{} ![The Kunz poset of $S = {\langle}13,14,15,16,17{\rangle}$ ((b) above) and its two bounding rays ((a) and (c) above), as in Example \[e:arithrays\]. The final Kunz poset is that of a bounding ray for $S = {\langle}16,23,30,37,44,51,58{\rangle}$ discussed in Example \[e:arithbadray\].](posets-arith-raybad.pdf){height="2.5in"} \[r:remainingarithrays\] The remaining faces of $C({{\ensuremath{\mathbb{Z}}}}_a)$ containing extra-generalized arithmetical numerical semigroups have substantially more extremal rays than those described in Theorem \[t:arithrays\]. Indeed, if $a = 19$ and $k = 17$, then each such face has $726$ rays, while $P_{19}$ itself has a grand total of $11$,$665$,$781$ rays [@wilfmultiplicity]. Posets of monoscopic numerical semigroups {#sec:monoscopic} ========================================= In this section, we introduce monoscopic numerical semigroups (Definition \[d:monoscopic\]), and extend a known characterization of the Apéry set of monoscopic numerical semigroups to a characterization of their Apéry posets (Theorem \[t:monoscopicposet\]) and Kunz posets (Corollary \[c:monoscopicposet\]). \[d:monoscopic\] Fix a numerical semigroup $S = {\langle}n_1, \ldots, n_k{\rangle}$, an integer $\beta \in {{\ensuremath{\mathbb{Z}}}}_{\ge 2}$, and an element $\alpha \in S \setminus \{n_1, \ldots, n_k\}$ with $\gcd(\alpha, \beta) = 1$. The semigroup $$T = {\langle}\alpha{\rangle}+ \beta S = {\langle}\alpha, \beta n_1, \ldots, \beta n_k{\rangle}$$ is called a monoscopic gluing* of $S$, or simply monoscopic. It is well known [@numerical Lemma 8.8] that under the given conditions, the generating set for $T$ given in Definition \[d:monoscopic\] is minimal. \[t:monoscopicaperyset\] Suppose $S = {\langle}m, n_2, \ldots, n_k{\rangle}$ and that $T = {\langle}\alpha{\rangle}+ \beta S$ is a monoscopic gluing. We have $$\operatorname{Ap}(T;\beta m) = \{b\alpha + a\beta : a \in \operatorname{Ap}(S;m) \text{ and } 0 \le b \le \beta - 1\}.$$ \[e:monoscopicextensions\] Let $S = {\langle}4, 13, 18{\rangle}$, and consider the monoscopic gluings $$T_1 = {\langle}43{\rangle}+ 3S = {\langle}12, 39, 43, 54{\rangle}\qquad \text{and} \qquad T_2 = {\langle}31{\rangle}+ 3S = {\langle}12, 31, 39, 54{\rangle}$$ of $S$. Although the Apéry sets of $T_1$ and $T_2$ have identical structure by Theorem \[t:monoscopicaperyset\], and have nearly identical Kunz posets, as depicted in Figure \[f:monoscopicextensions\], there is a subtle distinction. The key turns out to be that $31 \in \operatorname{Ap}(S;4)$, meaning $\alpha = 31$ is the smallest possible value of $\alpha$ that take in its equivalence class modulo $12$. When we examine where monoscopic semigroups lie in the Kunz polyhedron in Section \[sec:monoscopicembeddings\], we will see that varying $\alpha$ within its equivalence class modulo $\beta m$ yields semigroups within the same face, unless $\alpha \in \operatorname{Ap}(S;m)$, which places the resulting semigroup on a boundary face. ![The Kunz poset for $T_1 = {\langle}12, 39, 43, 54{\rangle}$ (without the dashed edge) and $T_2 = {\langle}12, 31, 39, 54{\rangle}$ (with the dashed edge) from Example \[e:monoscopicextensions\], both of which are monoscopic gluings of $S = {\langle}4, 13, 18{\rangle}$.[]{data-label="f:monoscopicextensions"}](posets-scopic-extaug.pdf){width="2.0in"} \[t:monoscopicposet\] Let $S = {\langle}m, n_2, \ldots, n_k{\rangle}$. Suppose $T = {\langle}\alpha{\rangle}+ \beta S$ is a monoscopic gluing. (a) If $\alpha \notin \operatorname{Ap}(S;m)$, then $b\alpha + a\beta \preceq_T b'\alpha + a'\beta$ if and only if $a \preceq_S a'$ and $b \le b'$. (b) If $\alpha \in \operatorname{Ap}(S;m)$, then $b\alpha + a\beta \preceq_T b'\alpha + a'\beta$ if and only if $a \preceq_S a'$ and either $b \le b'$ or $\alpha \preceq_S a' - a$. It is clear, in either case, that if $a \preceq_S a'$ and $b \le b'$, then $b\alpha + a\beta \preceq_T b'\alpha + a'\beta$. On the other hand, if $b > b'$, but $\alpha \in \operatorname{Ap}(S;m)$ and $\beta\alpha \preceq_S a' - a$, then $$(b'\alpha + a'\beta) - (b\alpha + a\beta) = (b' - b)\alpha + (a' - a)\beta = (b' - b + \beta)\alpha + (a' - a - \alpha)\beta \in T,$$ so again $b\alpha + a\beta \preceq_T b'\alpha + a'\beta$. Conversely, suppose $b\alpha + a\beta \preceq_T b'\alpha + a'\beta$, meaning $(b' - b)\alpha + (a' - a)\beta \in \operatorname{Ap}(T;\beta m)$. By Theorem \[t:monoscopicaperyset\], this means $$(b' - b)\alpha + (a' - a)\beta = b''\alpha + a''\beta$$ with $a'' \in \operatorname{Ap}(S;m)$ and $0 \le b'' \le \beta - 1$. Rearranging this equality yields $$(b' - b - b'')\alpha + (a' - a - a'')\beta = 0,$$ and since $\gcd(\alpha, \beta) = 1$, we must have $\beta \mid (b' - b - b'')$ and $\alpha \mid (a' - a - a'')$. Given the bounds on $b$, $b'$, and $b''$, this is only possible if $b + b'' - b' = 0$ or $b + b'' - b' = \beta$. If the former holds, then $b' - b = b'' \ge 0$ and $a' - a = a'' \in S$, so we are done. If the latter holds, then $a' - a - \alpha = a'' \in S$, which is only possible if $\alpha \in \operatorname{Ap}(S;m)$ as well. \[c:monoscopicposet\] If $S = {\langle}m, n_2, \ldots, n_k{\rangle}$ and $T = {\langle}\alpha{\rangle}+ \beta S$ is a monoscopic gluing, then $b'\alpha + a'\beta$ covers $b\alpha + a\beta$ in $\operatorname{Ap}(T;\beta m)$ if and only if one of the following holds: (i) $a = a'$ and $b' - b = 1$; (ii) $b = b'$ and $a' - a = n_j$ for some $j \ge 2$; or (iii) $b = \beta - 1$, $b' = 0$, and $a' - a = \alpha$. Moreover, condition (iii) occurs in $\operatorname{Ap}(T;\beta m)$ if and only if $\alpha \in \operatorname{Ap}(S;m)$. By Theorem \[t:groupconefacelattice\](c), $b'\alpha + a'\beta$ covers $b\alpha + a\beta$ in $\operatorname{Ap}(T;\beta m)$ if and only if their difference equals either $\alpha$ or $\beta n_j$ for some $j \ge 2$. By Theorem \[t:monoscopicposet\], the former case forces (ii) to hold, and the latter case holds precisely when either (i) or (iii) holds. Monoscopic embeddings of polyhedra {#sec:monoscopicembeddings} ================================== In this section, we characterize the faces of the Kunz polyhedron $P_m$ containing monoscopic numerical semigroups, as well as all lower dimensional faces contained therein. We do so by constructing a family of combinatorial embeddings (Definition \[d:monoscopicembedding\]) of group cones, which we show in Theorem \[t:augmonoscopicfaces\] induces an injection of face lattices. The faces of $P_m$ corresponding to these faces, together with those described in Theorem \[t:monoscopicfaces\], contain all of the monoscopic numerical semigroups, and only monoscopic numerical semigroups (Theorem \[t:onlymonoscopicfaces\]). Figure \[f:monoscopicfacelattice\] depicts the portion of the face lattice of $\mathcal C({{\ensuremath{\mathbb{Z}}}}_{12})$ described by Theorems \[t:augmonoscopicfaces\] and \[t:monoscopicfaces\], including all posets therein. \[n:phantomzero\] In order to simplify numerous expressions in the remainder of the paper, we adopt the convention of prepending a “$0$” entry to each point in $\mathcal C(G)$, indexed by the identity element of $G$. More precisely, we write each $(x_1, x_2, \ldots) \in \mathcal C(G)$ in the form $(x_0, x_1, x_2, \ldots)$ with $x_0 = 0$, effectively replacing $\mathcal C(G)$ with $\{0\} \times \mathcal C(G)$. \[d:monoscopicembedding\] Fix an Abelian group $G$, a subgroup $H \subset G$ so that $G/H$ is cyclic, and an element $\rho \in G$ whose image in $G/H$ is a generator. Letting $\beta = \|G/H\|$, define $$\begin{array}{r@{}c@{}l} \Phi_\rho:\mathcal C(H) &{}\longrightarrow{}& \mathcal C(G) \\ w &{}\longmapsto{}& x \end{array}$$ where $x_{a+b\rho} = \beta w_a + b w_{\beta\rho}$ for each $a \in H$ and $0 \le b < \beta$. We call $\Phi_\rho$ a monoscopic embedding of $\mathcal C(H)$ into $\mathcal C(G)$ along $\rho$. ![image](m12embedding.pdf){width="7.9in"} \[l:phimap\] The monoscopic embedding $\Phi_\rho$ is well-defined and injective. Under the given assumptions, every element of $G$ can be written uniquely in the form $a + k\rho$ for some $a \in H$ and $b \in {{\ensuremath{\mathbb{Z}}}}$ with $0 \le b \le \beta - 1$. Moreover, if $w \in \mathcal C(H)$ and $x = \Phi_\rho(w)$, it is easy to check that $x_0 = 0$, and for any $a, a' \in H$ and $0 \le b, b' < \beta$, $$\begin{aligned} x_{a+b\rho} + x_{(a'-a)+(b'-b)\rho} &= (\beta w_a + b w_{\beta\rho}) + (\beta w_{a'-a} + (b' - b) w_{\beta\rho}) = \beta(w_a + w_{a'-a}) + b' w_{\beta\rho} \\ &\ge \beta w_j + b' w_{\beta\rho} = x_{b+b'\rho}\end{aligned}$$ so the image of $\Phi_\rho$ lies in $\mathcal C(G)$. This proves $\Phi_\rho$ is well-defined. From here, it is clear that $\Phi_\rho$ is linear, and projecting the image of $\Phi_\rho$ onto the coordinates indexed by $H$ yields a linear map that simply scales each vector by $\beta$, so $\Phi_\rho$ is injective as well. \[d:monoscopicextension\] Fix a poset $P = (H/H', \preceq_P)$, where $H' \subset H$ is some subgroup, and suppose $a, b \in H/H'$ and $0 \le b, b' \le \beta - 1$. (a) The monoscopic extension of $P$ along $\rho$ is the poset $Q = (G/H', \preceq_Q)$ satisfying $a + b \rho \preceq_Q a' + b' \rho$ whenever $a \preceq_P a'$ and $b \le b'$. (b) The augmented monoscopic extension of $P$ along $\rho$ is the poset $Q$ defined as follows. (i) If $\beta\rho \ne 0$, then $Q = (G/H', \preceq_Q)$ with $a + b \rho \preceq_Q a' + b' \rho$ whenever $a \preceq_P a'$ and either $b \le b'$ or $\beta\rho \preceq_P a' - a$. (ii) If $\beta\rho = 0$, then $Q$ is the poset on $G/(H' + {\langle}\rho{\rangle})$ that is identical to $P$ under the natural group isomorphism $G/(H' + {\langle}\rho{\rangle}) {\cong}H/H'$. \[r:monoscopicextension\] The (non-augmented) monoscopic extension of a poset $P$ is isomorphic to the Cartesian product of $P$ with a total ordering. Also, any augmented monoscopic extension is a refinement of the corresponding non-augmented monoscopic extension. \[c:monoscopicextensiongluing\] If $S = {\langle}m, n_2, \ldots, n_k{\rangle}$, and $T = {\langle}\alpha{\rangle}+ \beta S$ is a monoscopic gluing, then the Kunz poset of $T$ is the monoscopic extension of the Kunz poset of $S$ along $\rho = \ol \alpha$, one that is augmented if and only if $\alpha \in \operatorname{Ap}(S)$. Note that if $\beta\rho = 0$, then $\alpha \in \operatorname{Ap}(S)$ is impossible. As such, the claim follows from Theorem \[t:monoscopicposet\] upon unraveling definitions. \[t:augmonoscopicfaces\] The image of $\Phi_\rho$ is a face of $\mathcal C(G)$. More precisely, given any face $F \subset \mathcal C(H)$ with Kunz poset $P = (H/H', \preceq)$, the image $\Phi_\rho(F)$ is a face of $\mathcal C(G)$ whose Kunz poset is the augmented monoscopic extension $Q$ of $P$ along $\rho$. First, fix $w \in F \subset \mathcal G(H)$, let $x = \Phi_\rho(w)$, let $F'$ denote the face containing $x$, and let $Q$ denote the corresponding Kunz poset of $F'$. If $\beta\rho = 0$, then $x_{a + b\rho} = 0$ whenever $w_a = 0$ and $bw_{\beta\rho} = 0$. If $\beta\rho \ne 0$, then this occurs when $a \in H'$ and $b = 0$. On the other hand, if $\beta\rho = 0$, then this occurs whenever $a \in H'$, independent of $b$. In either case, $Q$ has the claimed ground set. Now, suppose $a, b \in H$ and $0 \le b, b' \le \beta - 1$. If $\beta\rho = 0$, then $$x_{a + b\rho} + x_{(a' - a) + (b' - b)\rho} = \beta w_a + \beta w_{a' - a} \ge \beta w_{a + a'} = x_{a' + b'\rho}$$ with equality precisely when $a \preceq_P a'$, so assume $\beta\rho \ne 0$. If $b \le b'$, then $$\begin{aligned} x_{a + b\rho} + x_{(a' - a) + (b' - b)\rho} &= (\beta w_a + bw_{\beta\rho}) + (\beta w_{a' - a} + (b' - b)w_{\beta\rho}) = \beta(w_a + w_{a' - a}) + b'w_{\beta\rho} \\ &\ge \beta w_{a'} + b'w_{\beta\rho} = x_{a' + b'\rho},\end{aligned}$$ with equality precisely when $a \preceq_P a'$. Alternatively, if $b > b'$, then $$(a' + b'\rho) - (a + b\rho) = (a' - a - \beta\rho) + (b' - b + \beta)\rho$$ with $0 \le b' - b + \beta \le \beta - 1$, so we have $$\begin{aligned} x_{a + b\rho} + x_{(a' - a - \beta\rho) + (b' - b + \beta)\rho} &= (\beta w_a + bw_{\beta\rho}) + (\beta w_{a' - a - \beta\rho} + (b' - b + \beta)w_{\beta\rho}) \\ &= \beta(w_a + w_{a' - a - \beta\rho} + w_{\beta\rho}) + b'w_{\beta\rho} \\ &\ge \beta(w_a + w_{a' - a}) + b'w_{\beta\rho} \\ &\ge \beta w_{a'} + b'w_{\beta\rho} = x_{a' + b'\rho},\end{aligned}$$ with equality precisely when $\beta\rho \preceq_P a' - a$ and $a \preceq_P a'$. In either case, we obtain $a + b\rho \preceq_Q a' + b'\rho$ in the exact cases required by Definition \[d:monoscopicextension\], thereby proving $\Phi_\rho(w)$ lies in the interior of the claimed face. Conversely, let $F' \subset \mathcal C(G)$ denote the face whose Kunz poset is the augmented monoscopic extension $Q$ of $P$ (which must exist by the above argument), and fix $x \in F'$. Defining $w_a = \tfrac{1}{\beta}x_a$ for $a \in H$, we see $$\begin{aligned} x_{a + b\rho} = x_a + bx_\rho = x_a + \tfrac{1}{\beta}b(x_\rho + x_{(\beta-1)\rho}) = x_a + \tfrac{1}{\beta}bx_{\beta\rho} = \beta w_a + bw_{\beta\rho} = \Phi_\rho(w),\end{aligned}$$ where the first and second equalities hold since $Q$ is an augmented monoscopic extension. This proves set equality $\Phi_\rho(F) = F'$, thereby completing the proof. \[d:betaray\] The beta ray $\vec s$ of a monoscopic embedding $\Phi_\rho$ is defined $$s_{a+b\rho} = b$$ for each $a \in H$ and $0 \le b \le \beta - 1$. Notice that $s_a = 0$ precisely when $a \in H$, so $\vec s$ must lie in a face whose Kunz subgroup under is $H$. \[l:betarayind\] The beta ray $\vec s$ (a) lies in a face of $\mathcal C(G)$ whose corresponding subgroup is $H$, and (b) is linearly independent to each vector in the image of $\Phi_\rho$. The first claim is easy to verify. For the second claim, since $\Phi_\rho$ is linear and $\mathcal C(H)$ is full-dimensional, it suffices to prove $\vec s$ lies outside of the image of $\Phi_\rho$. Indeed, projecting the image of $\Phi_\rho$ onto the coordinates indexed by $H$ is injective by the proof of Lemma \[l:phimap\], while applying the same projection to $\vec s$ yields $0$. \[t:monoscopicfaces\] For any face $F \subset \mathcal C(H)$, the set ${{\ensuremath{\mathbb{R}}}}_{\ge 0}\vec s + \Phi_\rho(F)$ is a face of $\mathcal C(G)$ whose Kunz poset is the monoscopic extension of the Kunz poset of $F$. Let $P = (H/H', \preceq_P)$ denote the Kunz poset of $F$, let $F'$ denote the smallest face containing ${{\ensuremath{\mathbb{R}}}}_{\ge 0}\vec s + \Phi_\rho(F)$, and let $Q = (G/H'', \preceq_Q)$ denote the Kunz poset of $F'$. Since $\Phi_\rho(F) \subset F'$, Theorem \[t:augmonoscopicfaces\] implies $H'' \subset H'$, and the coordinates in which $s$ is zero are precisely those indexed by $H$, so we must have $H'' = H'$. Next, fix $x \in {{\ensuremath{\mathbb{R}}}}_{\ge 0}\vec s + \Phi_\rho(F)$, and write $x$ in the form $x = y + cs$ for $y \in \Phi_\rho(F)$ and $c \ge 0$. If $a + b\rho, a' + b'\rho \in G$ satisfy $a \preceq_P a'$ and $b \le b'$, then by Theorem \[t:augmonoscopicfaces\], $$\begin{aligned} x_{a + b\rho} + x_{(a' - a) + (b' - b)\rho} = y_{a + b\rho} + cb + y_{(a' - a) + (b' - b)\rho} + c(b' - b) = y_{a' + b'\rho} + cb' = x_{a' + b'\rho}.\end{aligned}$$ Additionally, among the facet equations satisfied by $\Phi_\rho(F)$, these are the only ones satisfied by $s$. As such, we conclude $Q$ equals the monoscopic extension of $P$. Lastly, fix $x \in F'$. Let $c = x_\rho - \tfrac{1}{\beta}x_{\beta\rho}$, and write $y = x - cs$. To complete the proof, we must show $y \in \Phi_\rho(F)$. Fix $a + b\rho, a' + b'\rho \in G$. If $b \le b'$, then $$\begin{aligned} y_{a + b\rho} + y_{(a' - a) + (b' - b)\rho} = x_{a + b\rho} - cb + x_{(a' - a) + (b' - b)\rho} - c(b' - b) \ge x_{a' + b'\rho} - cb' = y_{a' + b'\rho}\end{aligned}$$ with equality precisely when $a \preceq_P a'$. On the other hand, if $b > b'$, then $$\begin{aligned} y_{a + b\rho} + y_{(a' - a - \beta\rho) + (b' - b + \beta)\rho} &= x_{a + b\rho} - cb + x_{(a' - a - \beta\rho) + (b' - b + \beta)\rho} - c(b' - b + \beta) \\ &= x_{a + b\rho} + x_{(a' - a - \beta\rho) + (b' - b + \beta)\rho} - c\beta - cb' \\ &= x_{a + b\rho} + x_{(a' - a - \beta\rho) + (b' - b + \beta)\rho} - \beta x_\rho + x_{\beta\rho} - cb' \\ &\ge x_{a + b\rho} + x_{(a' - a) + (b' - b + \beta)\rho} - \beta x_\rho - cb' \\ &= x_{a + b\rho} + x_{(a' - a)} - (b - b') x_\rho - cb' \\ &\ge x_{a' + b\rho} - (b - b') x_\rho - cb' \\ &= x_{a' + b'\rho} - cb' = y_{a' + b'\rho}\end{aligned}$$ with equality whenever $\beta\rho \preceq_P a' - a$ and $a \preceq_P a'$. We conclude $y \in \Phi_\rho(F)$. Our final result of this section is a converse of sorts to Corollary \[c:monoscopicextensiongluing\], namely that the faces of the group cone containing monoscopic numerical semigroups contain only monoscopic numerical semigroups. \[t:onlymonoscopicfaces\] Let $S = {\langle}m, n_2, \ldots, n_k{\rangle}$, suppose $T = {\langle}\alpha{\rangle}+ \beta S$ is a monoscopic gluing, and let $F \subset \mathcal C({{\ensuremath{\mathbb{Z}}}}_{\beta m})$ denote the face containing $T$. Any numerical semigroup $T'$ in $F$ can be expressed as a monoscopic gluing $T' = {\langle}\alpha'{\rangle}+ \beta S'$, where $\alpha' \equiv \alpha \bmod \beta m$ and $S'$ is a numerical semigroup on the same face of $\mathcal C({{\ensuremath{\mathbb{Z}}}}_m)$ as $S$. Let $\alpha' \in T'$ denote the minimal generator of $T'$ satisfying $\alpha' \equiv \alpha \bmod \beta m$, and let $\rho \in {{\ensuremath{\mathbb{Z}}}}_{\beta m}$ denote the equivalence class containing $\alpha$ and $\alpha'$. The remaining generators of $T'$ must each be divisible by $\beta$, so we can write $T' = {\langle}\alpha', \beta m, \beta n_2', \ldots, \beta n_k'{\rangle}$. Letting $S' = {\langle}m, n_2', \ldots, n_k'{\rangle}$, we claim $T' = {\langle}\alpha'{\rangle}+ \beta S'$ is a gluing. Indeed, it is clear $\gcd(\alpha, \beta) = 1$ since the above generating set for $T'$ is minimal, so we must show $\alpha' \in S' \setminus \{m, n_2', \ldots, n_k'\}$. However, this follows from Corollary \[c:monoscopicextensiongluing\] since $\beta\alpha'$ can be factored using $\beta n_2', \ldots, \beta n_k'$. This proves the claim. It remains to show that $S'$ lies in the same face of $\mathcal C({{\ensuremath{\mathbb{Z}}}}_m)$ as $S$. Again, Corollary \[c:monoscopicextensiongluing\] implies every element of $\operatorname{Ap}(T';\beta m)$ divisible by $\beta$ can be factored using $\beta n_2', \ldots, \beta n_k'$. 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--- abstract: 'Tunneling between opposite surfaces of topological insulator thin film populated by electrons and holes is considered. We predict considerable enhancement of tunneling conductivity by Cooper electron-hole pair fluctuations that are precursor of their Cooper pairing. Cooper pair fluctuations lead to the critical behavior of tunneling conductivity in vicinity of critical temperature with critical index $\nu=2$. If the pairing is suppressed by disorder the behavior of tunneling conductivity in vicinity of quantum phase transition is also critical with the index $\mu=2$. The effect can be interpreted as fluctuational internal Josephson effect and it is general phenomenon for electron-hole bilayers. The peculiarities of the effect in other realizations of electron-hole bilayer are discussed.' author: - 'D.K. Efimkin' - 'Yu.E. Lozovik' title: Fluctuational internal Josephson effect in topological insulator film --- Introduction ============ Cooper pairing of spatially separated electrons and holes was predicted in the system of semiconductor quantum wells more then thirty years ago[@LozovikYudson]. Later it was observed in quantum Hall bilayer at total filling factor $\nu_{\mathrm{T}}=1$ that can be presented as the system of spatially separated composite electrons and composite holes (see [@EisensteinMacDonald] and references therein). After graphene discovery Cooper pairing of Dirac electrons and holes in the structure of independently gated graphene layers has been proposed [@LozovikSokolik; @MinBistrizerSuMacDonald; @KharitonovEfetov]. Recently possibility of Cooper pairing of Dirac electrons and holes was predicted[@EfimkinLozovikSokolik; @SeradjehMooreFranz] in thin film of topological insulator (TI), new unique class of solids that has topologically protected Dirac surface states [@HasanKane; @QiZhang]. The electron-hole pairing in that system is the realization of topological superfluidity and hosts Majorana fermions on edges and vortices [@SeradjehMooreFranz; @Seradjeh] that is the topic of extraordinary interest due to possibility to use them in quantum computation [@Alicea; @Beenakker]. Also the Cooper pairing can lead to a number of interesting physical effects including superfluidity [@LozovikYudson], anomalous drag effect[@VignaleMacDonald], nonlocal Andreev reflection [@PesinMacDonald]. The most prominent manifestation of electron-hole pairing is internal Josephson effect [@LozovikPoushnov]. The coherence between electron and hole states leads to a tunnel current $j_{\mathrm{T}}=j_0 \sin(\phi-\phi_{\mathrm{T}})$ that depends on the phase $\phi$ of electron-hole condensate. Here $j_0$ is maximal value of the current carried by the condensate and $\phi_{\mathrm{T}}$ is the phase of the tunnel matrix element. Dynamic of the phase on the macroscopic scale is described by the action with Lagrangian analogous to the one for the superconducting Josephson junctions. But current-voltage characteristics of the electron-hole bilayer drastically differs from ones of the latter. In equilibrium state the phase of the order parameter is fixated $\phi=\phi_{\mathrm{T}}$ and the tunnel current is zero hence in electron-hole system there is no analog of DC Josephson effect. The coherent tunnel current flows in the non equilibrium state driven by a voltage bias between the electron and hole layers. It leads to a colossal enhancement of tunneling conductivity at zero voltage bias. The effect has been observed in quantum Hall bilayer[@JosehsonExp1; @JosehsonExp2] and its microscopical and macroscopical description were addressed in a number of interesting theoretical papers [@SternGirvinMacDonaldMa; @JoglekarMacDonald; @RosLee; @FoglerWilczek; @BezuglyjShevchenko]. Cooper electron-hole pairing can appear above critical temperature as thermodynamic fluctuations. Particulary they lead to the logarithmic divergence of a drag conductivity as a function of a temperature[@Hu; @Mink] and a pseudogap formation in single-particle density of states of electrons and holes[@Rist]. Manifestations of Cooper pair fluctuations in tunneling have not been consider previously. Since tunneling conductivity is colossally enhanced in the paired state one can anticipate its strong enhancement by Cooper pair fluctuations above critical temperature in analogy with strong contribution of electron-electron Copper pair fluctuations in superconductor to its diamagnetic susceptibility and electric conductivity [@LarkinVarlamov]. Indeed, tunneling current in electron-hole bilayer can be transferred by Cooper pair fluctuations. Since the amplitude of pairing fluctuations increases in a vicinity of critical temperature, as fluctuations of an ordered state do for different phase transitions, one can expect the significant enhancement of tunneling conductivity and its critical behavior. The described effect can be called fluctuational internal Josephson effect and it is rather general phenomenon for electron-hole bilayers. Here we develop the microscopic theory of the effect and its macroscopic theory will be published elsewhere. We have considered the effect in topological insulator thin film and its peculiarities in other realizations of electron-hole bilayer are discussed in Conclusions. The rest of the paper is organized as follows. In Section 2 we briefly discuss the model used for the description of interacting electrons and holes in TI film. In Section 3 the microscopical description of Cooper pair fluctuations is introduced. In Section 4 the tunneling conductivity between the opposite surfaces of topological insulator film is calculated and Section 5 is devoted to the analysis of results and conclusions. \[Fig1\] ![(Color online) a) Experimental setup for measurements of tunneling conductivity between opposite surfaces of TI populated by electrons and holes. b) Dispersion laws of the electrons and holes in equilibrium. Dashed line denotes electrochemical potential of TI surfaces. c) Dispersion law of the electrons and holes in nonequilibrium state induced by voltage bias V. ](Fig1.eps){width="8.5"} The model ========= The setup for the experimental investigation of tunneling conductivity between spatially separated electrons and holes in TI film is presented on Fig.1. Voltage $V_{\mathrm{eh}}$ between the external gates induces equilibrium concentrations of electrons and holes on the opposite surfaces. Voltage $V$ drives the system from equilibrium and induces charge current between the layers. If the side surfaces of the film are gapped, for example, by ordered magnetic impurities introduced to TI surface the charge can be transferred only via interlayer tunneling and the tunneling resistance can be measured. Also charge transport through TI side surfaces is unimportant if the area of the tunneling junction is large enough. Possibility and peculiarities of electron-hole Cooper pairing in the TI film in realistic model that takes into account screening, disorder and interlayer tunneling has been considered in our paper [@EfimkinLozovikSokolik]. Here we focus on investigation of Cooper pair fluctuations and their role in tunneling. Hamiltonian of the system $H=H_{\mathrm{eh}}+H_{\mathrm{d}}+H_{\mathrm{T}}$ includes kinetic and electron-hole interaction energies $H_\mathrm{eh}$, interaction with disorder $H_{\mathrm{d}}$ and the part describing tunneling $H_{\mathrm{T}}$. The first part in single-band approximation that ignores valence (conduction) band on the surface with excess of electrons (holes) is given by $$\begin{split} H_{\mathrm{eh}}=&\sum_{{\mathbf}{p}}\xi_{{\mathbf}{p}} a_{\mathbf}{p}^+a_{\mathbf}{p} - \sum_{{\mathbf}{p}}\xi_{{\mathbf}{p}} b_{\mathbf}{p}^+b_{\mathbf}{p} + \\ &+\sum_{{\mathbf}{p}{\mathbf}{p}'{\mathbf}{q}} U({\mathbf}{q})\Lambda_{{\mathbf}{p}'-{\mathbf}{q},{\mathbf}{p}'}^{{\mathbf}{p}+{\mathbf}{q},{\mathbf}{p}}a_{{\mathbf}{p}+{\mathbf}{q}}^+b^+_{{\mathbf}{p}\prime-{\mathbf}{q}} b_{{\mathbf}{p}\prime}^{\mathstrut} a_{{\mathbf}{p}}^{\mathstrut}. \end{split}$$ Here $a_{{\mathbf}{p}}$ is annihilation operator for a electron on the surface with excess of electrons and $b_{{\mathbf}{p}}$ is annihilation operator for a electron on the surface with excess of holes [@Comment1]; $\xi_{{\mathbf}{p}}=v_{\mathrm{F}}p-E_{\mathrm{F}}$ is Dirac dispersion law in which $v_{\mathrm{F}}$ and $E_{\mathrm{F}}$ are velocity and Fermi energy of electrons and holes. We consider the balanced case since it is favorable for Cooper pairing and the pairing is sensitive to concentration mismatch of electrons and holes. $U({\mathbf}{q})$ is screened Coulomb interaction between electrons and holes (see [@EfimkinLozovikSokolik] for its explicit value) and $\Lambda_{{\mathbf}{p}'-{\mathbf}{q},{\mathbf}{p}'}^{{\mathbf}{p}+{\mathbf}{q},{\mathbf}{p}} = \cos{\small (\phi_{{\mathbf}{p},{\mathbf}{p}+{\mathbf}{q}}/2)} \cos{(\phi_{{\mathbf}{p}',{\mathbf}{p}'+{\mathbf}{q}}/2)}$ is angle factor that comes from the overlap of spinor wave functions of two-dimensional Dirac fermions. Critical temperature of pairing in Bardeen-Cooper-Schrieffer (BCS) theory that ignores disorder and tunneling is given by $$T_0=\frac{2\gamma'E_{\mathrm{F}}}{\pi} \exp^{-1/\nu_\mathrm{F}U'}$$ where $U'$ is Coulomb coupling constant [@Comment2] ; $\gamma'=e^C$ where $C\approx0,577$ is the Euler constant. \[Fig2\] ![a) Diagrammatic representation of the Bethe-Salpeter equation for the Cooper propogator $\Gamma_\mathrm{c}$; b) Feynman diagrams for a tunnel conductivity. Solid (dashed) line corresponds to electrons on the surface of TI film with excess of electrons (holes).](Fig2.eps){width="8.7"} We do not specify explicitly the interaction Hamiltonian with disorder $H_\mathrm{d}$ since both short-range and long-range Coulomb impurities lead to pairbreaking and can suppress Cooper pairing. Short-range disorder scatters only one component of Cooper pair since they are spatially separated and long-range Coulomb impurities acts differently on components of Cooper pair since they have different charge. Below we introduce phenomenological decays of electrons $\gamma_\mathrm{a}$ and holes $\gamma_\mathrm{b}$. The tunneling of electrons between the opposite surfaces of topological insulator thin film with conserving momentum can be described by the following Hamiltonian $$H_{\mathrm{T}}=T+T^+=\sum_{{\mathbf}{p}} \left(t b_{{\mathbf}{p}}^+a_{{\mathbf}{p}}+t^* a_{{\mathbf}{p}}^+b_{{\mathbf}{p}}\right),$$ where $t$ is the tunneling amplitude. We consider influence of tunneling on pairing to be weak and treat it below as perturbation. The described model is applicable for description of tunneling in TI films which width is larger than value at which $t\approx T_0$. If $t \gg T_0$ tunneling strongly influences electron-hole pairing and it can not be treated as perturbation. Particularly it induces electron-hole condensate with fixated phase and smears critical temperature to the paired state. Our calculation [@EfimkinLozovikSokolik] shows that the described model is applicable for thin films of $\hbox{Bi}_2 \hbox{Se}_3$ at $d> 10\;\hbox{nm}$. In that case critical temperature without disorder can achieve $T_0\approx0.1\;\hbox{K}$ at $E_{\mathrm{F}}=5 \;\hbox{meV}$ and the pairing is not suppress by disorder if electrons and holes have exceptional hight mobilities of order $\mu\sim 10^6 \;\hbox{sm}^2/\hbox{V}\hbox{s}$. It should be noted that single-band and static screening approximations used for calculation of critical temperature[@EfimkinLozovikSokolik] usually underestimate critical temperature of Cooper pairing between Dirac particles [@LozovikOgarkovSokolik; @LozovikSokolikMultiband; @SodemannPesinMacDonald]. Cooper pair fluctuations ======================== For the microscopical description of Cooper pair fluctuations we introduce Cooper propagator $\Gamma_\mathrm{c}^{\mathrm{R}}(\omega)$. It corresponds to the two-particle vertex function in the Cooper channel [@LarkinVarlamov] and satisfies the Bethe-Salpeter equation depicted on Fig. 2 (a). In Bardeen-Cooper-Schrieffer (BCS) approximation its solution can be presented in the form $$\label{CooperPropagator} \Gamma_\mathrm{c}^{\mathrm{R}}(\omega)=\frac{U'}{1-U'\Pi_{\mathrm{c}}^{\mathrm{R}}(\omega)},$$ where $\Pi_{\mathrm{c}}^{\mathrm{R}}(\omega)$ corresponds to electron-hole bubble diagram that can be interpreted as Cooper susceptibility of the system. After direct calculation it can be presented in the following form $$\label{Bubble} \begin{split} \Pi_\mathrm{c}^\mathrm{R}(\omega)=\frac{1}{U'}-\nu_\mathrm{F}\ln\frac{T}{T_0} - \nu_\mathrm{F} \Psi\left(\frac{1}{2}\right) - \\ - \nu_\mathrm{F}\Psi\left(\frac{1}{2}-\frac{i\omega}{4 \pi T}+\frac{\gamma}{2\pi T}\right). \end{split}$$ Here $\gamma=(\gamma_\mathrm{a}+\gamma_\mathrm{b})/2$ is disorder caused Copper pair decay rate equals to the half-sum of phenomenological introduced decay rates of electrons and holes[@Comment3]; $\nu_{\mathrm{F}}$ is the density of states of electrons and holes on the Fermi level; $\Psi(x)$ is the digamma function. Cooper pair propagator acquires the following form $$\label{CooperPropagator2} \Gamma_\mathrm{c}^{\mathrm{R}}(\omega)=\frac{1}{\nu_{\mathrm{F}}}\frac{1}{\ln\frac{T}{T_\mathrm{0}}+\Psi\left(\frac{1}{2}-i\frac{\omega}{4 \pi T}+\frac{\gamma}{2\pi T}\right)-\Psi\left(\frac{1}{2}\right)}.$$ In the absence of disorder $\Gamma_\mathrm{c}^{\mathrm{R}}(\omega)=0$ at the critical temperature $T_0$ indicating Cooper instability of the system against Cooper pairing. Critical temperature for disordered system $T_\mathrm{d}$ at which $\Gamma_\mathrm{c}^{\mathrm{R}}(\omega)=0$ satisfies the following equation $$\ln\frac{T_\mathrm{d}}{T_\mathrm{0}}+\Psi\left(\frac{1}{2}+\frac{\gamma}{2\pi T}\right)-\Psi \left(\frac{1}{2}\right)=0.$$ This equation has nontrivial solution if $\gamma<\gamma_0$, where $\gamma_{0}=0.89 T_0$ is the critical Cooper pair decay value. In opposite case the pairing is suppressed by disorder. The value $\gamma_0$ corresponds to quantum critical point at zero temperature. Above critical temperature $T_{\mathrm{d}}$ the expression for Cooper pair propagator (\[CooperPropagator2\]) at $\omega\rightarrow 0$ can be approximated in the following way $$\Gamma_\mathrm{c}^{\mathrm{R}}(\omega)=\frac{1}{\nu_{\mathrm{F}}}\frac{4\pi T_{\mathrm{d}}}{4\pi T_{\mathrm{d}}\ln \frac{T}{T_{\mathrm{d}}}-i\omega\Psi^\prime(\frac{1}{2}+\frac{\gamma}{2\pi T_{\mathrm{d}}})}.$$ If the pairing is suppressed by the disorder Cooper pair propagator at zero temperature and at $\omega\rightarrow 0$ is given by $$\Gamma_\mathrm{c}^{\mathrm{R}}(\omega)=\frac{1}{\nu_{\mathrm{F}}}\frac{2 \gamma}{2\gamma\ln\frac{\gamma}{\gamma_0} - i \omega}.$$ \[Fig3\] ![(Color online) Tunneling conductivity $\sigma$ as a function of bias voltage $V$ for noninteracting (dashed line) and interacting (solid line) electrons and holes for $\gamma=0.2\;\hbox{K}$ and $T=0.2\;\hbox{K}$. Inset: The phase diagram of the system in which BCS denotes the pairing state. The arrows on the phase diagram correspond to the dependencies on Fig. 4-7.](Fig3.eps){width="8.5"} Tunneling conductivity ====================== For a calculation of the tunneling conductivity we use linear response theory in which the tunneling conductivity $\sigma(V)$ at a finite voltage bias $V$ can be presented in the form of Kubo formula[@Mahan] $$\sigma(V)=\frac{e^2}{h} \frac{4 \pi}{eV} {\mathop{\mathrm{Im}}\nolimits}[\chi^\mathrm{R}(eV)],$$ where the retarded response function $\chi^\mathrm{R}(\omega)$ can be obtained by the analytical continuation $i\Omega_n\rightarrow \omega+i\delta$ of $\chi^\mathrm{M}(i \Omega_n)$ that is given by $$\label{TunnelDef} \chi^\mathrm{M}(i \Omega_n)=-\frac{1}{2\beta}\int_{-\beta}^{\beta}d\tau\;e^{i\Omega_n\tau}\langle T_{\mathrm{M}}T(\tau)T^+(0)\rangle.$$ Here $T_{\mathrm{M}}$ is the time-ordering symbol for a imaginary time $\tau$ and $\Omega_n=2\pi n T$ is a bosonic Matsubara frequency. In the system of noninteracting electrons and holes the $\chi^\mathrm{M}(i \Omega_n)$ corresponds to the first diagram on the Fig. 1 (b) leading to $\chi^\mathrm{R}(\omega)=\|t\|^2\Pi_{\mathrm{c}}^{\mathrm{R}}(\omega)$. Hence the tunneling conductivity for noninteracting electrons and holes $\sigma_{0}$ is given by $$\label{TunnelBare} \sigma_{0}(V)=\frac{e^2}{h} \frac{4 \pi \|t\|^2}{eV} {\mathop{\mathrm{Im}}\nolimits}[\Pi_{\mathrm{c}}^\mathrm{R}(eV)].$$ Its value $\sigma_{0}^{\mathrm{m}}(\gamma,T)$ at zero bias is given by $$\label{SigmaBare} \sigma_{0}^{\mathrm{m}}(\gamma,T)=\frac{2 \pi e^2}{h} \frac{\nu_{\mathrm{F}}\|t\|^2}{2\pi T} \Psi'\left(\frac{1}{2}+\frac{\gamma}{2\pi T}\right),$$ and at low temperatures $T\ll\gamma$ it transforms to $$\label{igmaBareZero} \sigma_{0}^{\mathrm{m}}(\gamma,0)=\frac{2 \pi e^2}{h} \frac{\nu_{\mathrm{F}}\|t\|^2}{\gamma}.$$ \[Fig4\] ![(Color online) The height of the tunnel conductivity peak $\sigma_{0}^{\mathrm{m}}$ for noninteracting electrons and holes on temperature $T$ for different values of Cooper pair decay $\gamma$.](Fig4.eps){width="8.7"} Introduction of the electron-hole Coulomb interaction in the ladder approximation leads to three additional terms for the tunneling conductivity. The first one corresponds to second diagram on Fig.2 (b). It is singular in a vicinity of critical temperature and cannot be reduced to the tunneling conductivity of noninteracting quasiparticles with a renormalized spectrum due to Cooper pair fluctuations. The other two terms correspond to renormalization of single-particle Green functions of electrons and holes. They are not singular in a vicinity of critical temperature and can be neglected. The tunneling conductivity for interacting electrons and holes $\sigma_{\mathrm{c}}$ is given by $$\label{TunnelInt} \sigma_{\mathrm{c}}(V)=\frac{e^2}{h} \frac{4 \pi \|t\|^2}{eV} {\mathop{\mathrm{Im}}\nolimits}\left[\frac{\Pi_\mathrm{c}^\mathrm{R}(eV)}{1-U'\Pi_{\mathrm{c}}^\mathrm{R}(eV)}\right].$$ The denominator of (\[TunnelInt\]) coincides with that in the Cooper propagator (\[CooperPropagator\]) and tends to zero in a vicinity of critical temperature $T_{\mathrm{d}}$. Hence the Cooper pair fluctuations lead to the critical behavior of tunneling conductivity in vicinity of the critical temperature and the quantum critical point. Above critical temperature $T_{\mathrm{d}}$ the tunneling conductivity at zero bias is given by $$\label{SigmaFluctClassical} \sigma_\mathrm{c}^{\mathrm{m}}(\gamma,T)=\sigma_\mathrm{0}^{\mathrm{m}}(\gamma,T)\frac{1}{(\nu_\mathrm{F}U^\prime)^2}\frac{1}{\ln^2 \frac{T}{T_\mathrm{d}}}.$$ In vicinity of the critical temperature it diverges as $\sigma_\mathrm{c}^{\mathrm{m}}(\gamma,T)\sim(T-T_\mathrm{d})^{-\nu}$ with the critical index $\nu=2$. At zero temperature tunneling conductivity is given by $$\label{SigmaFluctQuantum} \sigma_{\mathrm{c}}^{\mathrm{m}}(\gamma,0)=\sigma_{0}^\mathrm{m}(\gamma,0)\frac{1}{(\nu_{\mathrm{F}}U')^2} \frac{1}{\ln^2\frac{\gamma}{\gamma_0}}.$$ It diverges in the vicinity of the quantum phase transition at $\gamma=\gamma_0$ as $\sigma_\mathrm{c}^{\mathrm{m}}(\gamma,0) \sim(\gamma -\gamma_0)^{-\mu}$ with the critical index $\mu=2$. The formulas (\[SigmaFluctClassical\]) and (\[SigmaFluctQuantum\]) are the main result of the article. The calculated contribution of Cooper pair fluctuations to the tunneling conductivity cannot be reduced to the one of noninteracting quasiparticles with renormalized spectrum due to Cooper pair fluctuations. It can be interpreted as the direct contribution of Cooper pair fluctuations. Hence the predicted effect of the enhancement of tunneling conductivity in a vicinity of the critical temperature and the quantum critical point is the collective effect that can be interpreted as fluctuational internal Josephson effect. \[Fig5\] ![(Color online) Half-width of the tunnel conductivity peak $V_\mathrm{HW}$ for noninteracting electrons and holes as function of temperature $T$ for different values of Cooper pair decay $\gamma$.](Fig5.eps){width="8.7"} Analysis and Discussion ======================= Tunneling conductivity at finite voltage bias and in full-range of temperature and Cooper pair decay was calculated numerically according to formulas (\[Bubble\]), (\[TunnelBare\]) and (\[TunnelInt\]). The following set[@EfimkinLozovikSokolik] of the parameters $T_0=0.1\;\hbox{K}$, $E_{\mathrm{F}}=5\;\hbox{meV}$, $\nu_{\mathrm{}F}U'=0.16$, $t=10\;\mu\hbox{eV}$ was used. The set corresponds to $\hbox{Bi}_2 \hbox{Se}_3$ TI film with width $10 \;\hbox{nm}$. The phase diagram of the system is presented on the inset of Fig. 3. If Cooper pair decay rate exceeds critical value $\gamma_0=0.09\;\hbox{K}$ then the electron-hole pairing is suppressed by a disorder. Calculated tunneling conductivity both for noninteracting electrons and holes and interacting ones for $\gamma=0.2 \; \hbox{K}$ and $T=0.2 \; \hbox{K}$ is presented on Fig. 3. The dependence has prominent peak and is qualitatively the same for all points of the phase diagram $(\gamma,T)$. The peak appears due to restrictions connected with energy and momentum conservation for tunneling electrons. Such peak was predicted and observed also in electron-electron bilayers [@Tunneling2D2DExp; @ZhengMacDonald] and it is the peculiarity of a tunneling between two two-dimensional systems. Coulomb interaction between electrons and holes considerably enhances the tunneling conductivity but does not change qualitatively its dependence on external bias. \[Fig6\] ![(Color online) The height of the tunnel conductivity peak $\sigma_{\mathrm{c}}^{\mathrm{m}}$ for interacting electrons and holes on temperature $T$ for different values of Cooper pair decay $\gamma$. ](Fig6.eps){width="8.7"} Height and half-width of the peak for noninteracting electrons are presented on Figs. 4 and 5. The peak becomes more prominent with decreasing of Cooper pair decay and temperature. The peaks width is determined by $\max\{\gamma,T\}$. The peaks height is determined as $1/\max\{\gamma,T\}$ and it is decreasing as $1/T$ at $T\gg\gamma$. The height and half-width of the peak for interacting electrons and holes are presented on Fig. 6 and Fig. 7, respectively. For $\gamma<\gamma_0$ Coulomb interaction leads to the critical behavior of the tunneling conductivity in a vicinity of the critical temperature that we interpret as fluctuational internal Josephson effect. In vicinity of the critical temperature height of the peak diverges with the critical index $\nu=2$ that agrees with the analytic results and the width of the peak linearly tends to zero. At high temperatures peaks height decreases as $1/(T\log^2 (T/T_\mathrm{d}))$. The critical region in which tunneling conductivity is considerably enhanced is of order $\Delta T\approx T_d$. If the Cooper pair decay exceeds the critical value $\gamma>\gamma_0$ Coulomb interaction considerably enhances the height and leads to reduction of the width but does not lead to any singularities. The peak becomes more prominent with decreasing of decay and temperature as it does in the model of noninteracting electrons and holes. The peaks height smoothly depends on temperature but its maximal value at zero temperature diverges as function of Cooper pair decay $\gamma$ in vicinity of quantum critical point at $\gamma_0$. The width of the critical region is of order $\Delta \gamma \approx \gamma_0$. The peaks width and height smoothly depend on the parameters of the system used for the calculation and listed above. But a satisfaction of the number of assumptions is important for observation of fluctuational internal Josephson effect . The model we use here is well applicable in the regime of weak hybridization $t \ll T_0$ in which influence of tunneling on Cooper pairing can be neglected. In ultrathin TI film the regime of strong hybridization $t\gg T_0$ can be realized. In that regime tunneling induces the gap $t$ in the spectrum of electrons and holes which is considerable larger than the one due to their Cooper pairing. Critical temperature and Cooper instability are considerably smoothed in that case. So in that regime we do not expect critical behavior of the tunneling conductivity due to Cooper pair fluctuations. Our calculations for $\hbox{Bi}_2 \hbox{Se}_3$ shows that regime of strong hybridization can be realized in films which width is less then $10\; \hbox{nm}$. \[Fig7\] ![(Color online) Half-width of the tunnel conductivity peak $V_\mathrm{HW}$ for interacting electrons and holes as function of temperature $T$ for different values of Cooper pair decay rate $\gamma$.](Fig7.eps){width="8.9"} The mean field theory we use here for the description of fluctuational internal Josephson effect does not account large scale fluctuation of phase of Cooper pair condensate. In two-dimensional superfluids phase fluctuations destroy long-range coherence and the transition to paired state at critical temperature $T_{\mathrm{d}}$ calculated within mean field theory is smoothed. Moreover the transition to superfluid state is Berezinskii-Kosterlitz-Thouless transition [@Berezinskii; @KosterlizThouless] that corresponds to dissociation of vortex-antivortex pairs and which temperature is below $T_{\mathrm{d}}$. Hence the large scale phase fluctuations of Cooper pair condensate can smooth the critical behavior of tunnel conductivity we predict here. But if the size of the system is comparable with coherence length of Cooper pair fluctuations $l_{\mathrm{c}}\approx\hbar v_{\mathrm{F}}/T_0$ the phase fluctuations are unimportant and mean field theory is well applicable. For $T_0=0.1 \; \hbox{K}$ the coherence length of Cooper pair fluctuations is of order $l_{\mathrm{c}}\sim 10 \; \hbox{mkm}$ and we conclude that the developed microscopical theory is applicable for samples of the corresponding size. The model we use here implies conservation of the momentum of tunneling electron. If the momentum is not conserved the tunneling process creates electron-hole pair with nonzero total momentum of order $l_{\mathrm{T}}^{-1}$. Here $l_{\mathrm{T}}$ is character length at which tunneling matrix matrix element $t$ can be considered as constant. Cooper pair is formed by electron and hole with opposite momenta and the Cooper instability is smoothed if $l_{\mathrm{c}}\gg l_{\mathrm{T}}$. For tunneling between opposite surfaces of topological insulator thin film of high crystalline quality momentum conservation can be achieved to remarkable degree. We have shown that electron-hole Coulomb interaction considerably enhances the tunneling conductivity in electron-hole bilayer even when the Cooper pairing is suppressed by disorder. The opposite situation takes place in electron-electron bilayer that also can be realized in semiconductor quantum well structure, in graphene double layer system and in a film of topological insulator. Coulomb interaction gives contribution to decay of electrons that was analyzed in [@JungwirthMacDonald] and to additional series of diagrams for the tunneling conductivity. We treated the additional diagrams in the ladder approximation (See Fig.2-b). If they are omitted the tunneling conductivity at zero temperature and at a finite bias $V$ is given by [@JungwirthMacDonald; @ZhengMacDonald] $$\sigma_{0}^{\mathrm{ee}}=gt_{\mathrm{A}}\frac{2\pi e^2}{h}\frac{\nu_{\mathrm{F}}\|t\|^2}{\gamma} \frac{4\gamma^2}{4\gamma^2+(eV)^2},$$ where $g$ is the degeneracy factor of electrons and $t_{\mathrm{A}}$ is additional factor that depends on internal nature of electrons[@Comment4]. The dependence of tunnel conductivity on external bias contains prominent peak which becomes more prominent with decreasing of decay rate $\gamma$. If the Coulomb interaction is treated in ladder approximation the tunneling conductivity is given by $$\sigma_{\mathrm{c}}^{\mathrm{ee}}=gt_{\mathrm{A}}\frac{2\pi e^2}{h}\frac{\nu_{\mathrm{F}}\|t\|^2}{\gamma} \frac{4\gamma^2}{4\gamma^2+(1+(\nu_{\mathrm{F}}U')^2)(eV)^2}.$$ For electron-electron bilayer Coulomb interaction does not influences the height of the peak and leads to decreasing of the width which is insignificant even in the case of strong interaction $\nu_{\mathrm{F}}U'\sim1$. The roles of the interlayer Coulomb interaction in electron-electron bilayer and electron-hole bilayer are drastically different because the correction to the tunneling conductivity of the primer is caused by the scattering diagrams in particle-antiparticle channel and the correction to the one of the latter is caused by the diagrams in particle-particle Cooper channel that contains instability. We have investigated the manifestations of Cooper electron-hole pairing fluctuations in thin film of topological insulator on tunneling between its opposite surfaces. The internal fluctuational Josephson effect is general phenomenon but each realization of electron-hole bilayer has its own peculiarities. Dirac points in graphene are situated in corners of first Brillouin zone. Electron-hole pairing was predicted[@LozovikSokolik; @MinBistrizerSuMacDonald; @KharitonovEfetov] in the system of two independently gated graphene layers separated by dielectric film. In that case orientations of the graphene lattices are uncorrelated. The distance between Dirac points of different layers in momentum space is of order $a_0^{-1}$, where $a_0$ is lattice constant of graphene. The tunneling of electrons between Dirac points is possible if $l_{\mathrm{T}}\sim a_0$ that corresponds to tunneling through impurity states or other defects. The condition $l_{\mathrm{c}}\gg l_{\mathrm{T}}$ is well satisfied and the critical behavior of tunneling conductivity in double layer graphene system is considerably smoothed. But if the mutual orientation of graphene layers can be controlled in experiment the presented here theory is well applicable for that system. The formulas (\[SigmaFluctClassical\]),(\[SigmaFluctQuantum\]) are reasonable and can be easily generalized to additional spin and valley degree of freedom of electrons and holes. So the fluctuational internal Josephson effect can also be experimentally investigates in that system. Recently anomalies in drag effect in semiconductor double well structure that contains spatially separated electrons and holes were observed [@CroxallExp; @MorathExp]. The analysis of results shows that the observed anomalies can be caused by electron-hole pairing not in BCS regime but rather in regime of BCS-BEC crossover[@Leggett; @PieriNeilsonStrinati]. In that regime electron-hole pairing fluctuations also should increase tunneling conductivity of the system but quantitative theory of the effect is interesting and challenging problem. The developed here microscopical theory of fluctuation internal Josephson effect is applicable for that system if the pairing is realized in BCS regime. Moreover the formulas (\[SigmaFluctClassical\]),(\[SigmaFluctQuantum\]) are reasonable in that case. In the semiconductor double well structures[@CroxallExp; @MorathExp] the concentrations of electrons and holes can be independently controlled and separate contacts to the layers have been made. So in that system tunneling conductivity between electron and hole layers can be measured and the predictions of our work can be also addressed. Internal Josephson effect has been observed in quantum Hall bilayer at total occupation factor $\nu_{\mathrm{T}}=1$. But above critical temperature the dependence of tunneling conductivity on external bias does not contain any peak due to non Fermi-liquid behavior of composite electrons and holes [@QHFTunnelingExp1; @QHFTunnelingExp2; @QHFTunnelingTheor1; @QHFTunnelingTheor2]. Thus fluctuational effects in that system are more complicated ones and need separate investigation. It should be noted that critical behavior of tunneling conductivity has not been observed yet in experiments in that system. We have considered influence of electron-hole Cooper pair fluctuations that are precursor of their Cooper pairing in topological insulator film on tunneling between its opposite surface. Cooper pair fluctuations lead to critical behavior of tunneling conductivity in vicinity of critical temperature with critical index $\nu=2$. If pairing is suppressed by disorder the behavior of tunneling conductivity in vicinity of quantum critical point at zero temperature is also critical with critical index $\mu=2$. 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--- abstract: 'In connection with the discussion and the measurements fulfilled in Ref. [@experiment], the full identity is demonstrated between the Feynman formula for the field of a moving charge and the Liénard-Wiechert potentials.' author: - 'D. M. Gitman$^{1,2,3}$' - 'A. E. Shabad$^{1,2}$' - 'A.A. Shishmarev $^{2,3}$' title: 'A note on “Measuring propagation speed of Coulomb fields” by R. de Sangro, G. Finocchiaro, P. Patteri, M. Piccolo, G. Pizzella ' --- In Ref. [@experiment] measurements were performed to decide between two approaches to the field of a moving charge: one based on the Liénard-Wiechert potentials and the other on the Feynman interpretation. The aim of the present note is to demonstrate that although apparently different physical ideas are layed into the Feynman formula [@Feynman], it is as a matter of fact mathematically identical to that found in standard text books, e. g. [@Landau2; @Jackson], for the Liénard-Wiechert potentials both for accelerated motion of the charge and its motion with a constant speed. We believe that this observation should be taken into account and prove to be useful in the general discussion on the matter [@Field]. The Feynmann formula for the electric field $\mathbf{E}$ of a moving accelerated charge $q$ is [@Feynman] $$\mathbf{E}=\frac{q}{4\pi\epsilon_{0}}\left[ \frac{\mathbf{e}_{r^{\prime}}}{r^{\prime2}}+\frac{r^{\prime}}{c}\frac{d}{dt}\left( \frac{\mathbf{e}_{r^{\prime}}}{r^{\prime2}}\right) +\frac{1}{c^{2}}\frac{d^{2}}{dt^{2}}\mathbf{e}_{r^{\prime}}\right] ,\label{A3_Feynman}$$ We set $c=1$ for the sake of simplicity and keep to the notations in [@Feynman]. In (\[A3\_Feynman\]) the function$$r^{\prime}=\sqrt{x_{1}^{\prime}+x_{2}^{\prime2}+x_{3}^{\prime2}}\label{1}$$ is the modulus of vector $\mathbf{x}^{\prime},$ directed from the position $\mathbf{\tilde{x}}$ of the moving charge to the observation point $\mathbf{x}$:$$\mathbf{x}^{\prime}=\mathbf{x}-\mathbf{\tilde{x},}\label{2}$$ and $\mathbf{e}_{r^{\prime}}$ is the unit vector in the direction of $\mathbf{x}^{\prime}.$ The charge trajectory is given as $\mathbf{\tilde {x}=\tilde{x}(}t^{\prime}),$ where $t^{\prime}$ is the time coordinate of the charge. Therefore, with the location of the observation point fixed, $\mathbf{x=const.,}$ we see that $\mathbf{x}^{\prime},$ as well as $r^{\prime },$ is a function solely of $t^{\prime}.$ Once the influence of the charge propagates exactly with the speed of light $c=1,$ the relation$$r^{\prime}(t^{\prime})=t-t^{\prime}\label{r'}$$ holds, where $t$ is the time of observation. With Eqs. (\[2\]) and (\[r’\]) the length (\[1\]) is just the distance between the position of the charge and the observation point at the moment of emission. Relation (\[r’\]) defines $t$ as a function of $t^{\prime}.$Then, according to the rule of differentiation of an inverse function, one has for any function $a(t^{\prime})$$$\frac{da\left( t^{\prime}\right) }{dt}=\frac{da\left( t^{\prime}\right) }{dt^{\prime}}\frac{dt^{\prime}}{dt}=\frac{da\left( t^{\prime}\right) }{dt^{\prime}}\left( \frac{dt}{dt^{\prime}}\right) ^{-1},$$ where $\frac{dt}{dt^{\prime}}$ follows from (\[r’\]) and (\[2\]) to be$$\frac{dt}{dt^{\prime}}=1+\frac{dr^{\prime}\left( t^{\prime}\right) }{dt^{\prime}}=1+\frac{d\|\mathbf{x}-\mathbf{\tilde{x}}(t^{\prime})\|}{dt^{\prime}}=1-\frac{\left( x_{i}-\tilde{x}_{i}(t^{\prime})\right) }{r^{\prime}}\frac{d\tilde{x}_{i}\left( t^{\prime}\right) }{dt^{\prime}}=1-\frac{\left( \mathbf{v\cdot x}^{\prime}\right) }{r^{\prime}}=1-\left( \mathbf{v\cdot e}_{r^{\prime}}\right) .$$ We have used here that $\frac{d\mathbf{\tilde{x}}\left( t^{\prime}\right) }{dt^{\prime}}=\mathbf{v}(t^{\prime})$ is the instantaneous speed of the charge. Referring to the designation $\left( \mathbf{e}_{r^{\prime}}\bar{v}\right) =\kappa$ used for brevity we can now rewrite Eq. (\[A3\_Feynman\]) as $$\mathbf{E}=\frac{q}{4\pi\epsilon_{0}}\left[ \frac{\mathbf{e}_{r^{\prime}}}{r^{\prime2}}+\frac{r^{\prime}}{\left( 1-\kappa\right) }\frac{d}{dt^{\prime}}\left( \frac{\mathbf{e}_{r^{\prime}}}{r^{\prime2}}\right) +\frac{1}{\left( 1-\kappa\right) }\frac{d}{dt^{\prime}}\left( \frac {1}{\left( 1-\kappa\right) }\frac{d\mathbf{e}_{r^{\prime}}}{dt^{\prime}}\right) \right] . \label{A3_5}$$ Taking into account that $$\frac{d\mathbf{e}_{r^{\prime}}}{dt^{\prime}}=\frac{d}{dt^{\prime}}\left( \frac{\mathbf{x}^{\prime}}{r^{\prime}}\right) =-\frac{\mathbf{v}}{r^{\prime}}+\frac{\kappa\mathbf{x}^{\prime}}{r^{\prime2}}=\frac{\kappa\mathbf{e}_{r^{\prime}}-\mathbf{v}}{r^{\prime}} \label{A3_6}$$ we calculate the second term in (\[A3\_5\]):$$\frac{r^{\prime}}{\left( 1-\kappa\right) }\frac{d}{dt^{\prime}}\left( \frac{\mathbf{e}_{r^{\prime}}}{r^{\prime2}}\right) =\frac{3\kappa \mathbf{e}_{r^{\prime}}-\mathbf{v}}{\left( 1-\kappa\right) r^{\prime2}}. \label{A3_7}$$ The third term in (\[A3\_5\]) is $$\frac{1}{\left( 1-\kappa\right) }\frac{d}{dt^{\prime}}\left( \frac {1}{\left( 1-\kappa\right) }\frac{d\mathbf{e}_{r^{\prime}}}{dt^{\prime}}\right) =\frac{1}{\left( 1-\kappa\right) }\frac{d}{dt^{\prime}}\left( \frac{\kappa\mathbf{e}_{r^{\prime}}-\mathbf{v}}{\left( 1-\kappa\right) r^{\prime}}\right) . \label{A3_8}$$ Let us calculate the derivative of $\left( 1-\kappa\right) ^{-1}$:$$\frac{d\left( 1-\kappa\right) ^{-1}}{dt^{\prime}}=\left( 1-\kappa\right) ^{-2}\left( \frac{d\mathbf{e}_{r^{\prime}}}{dt^{\prime}}\cdot\mathbf{v}+\mathbf{e}_{r^{\prime}}\cdot\frac{d\mathbf{v}}{dt^{\prime}}\right) =\left( 1-\kappa\right) ^{-2}\left[ \frac{\kappa^{2}-v^{2}}{r^{\prime}}+\left( \mathbf{e}_{r^{\prime}}\cdot\mathbf{v}\right) \right] , \label{A3_9}$$ where $\overset{\cdot}{\mathbf{v}}$ is the acceleration of the charge. Then the third term in (\[A3\_5\]) becomes $$\begin{aligned} \frac{1}{\left( 1-\kappa\right) }\frac{d}{dt^{\prime}}\left( \frac {1}{\left( 1-\kappa\right) }\frac{d\mathbf{e}_{r^{\prime}}}{dt^{\prime}}\right) & =\frac{\left( \kappa^{2}-v^{2}\right) \left( \kappa \mathbf{e}_{r^{\prime}}-\mathbf{v}\right) +\left( 1-\kappa\right) \left[ 2\kappa\left( \kappa\mathbf{e}_{r^{\prime}}-\mathbf{v}\right) +\mathbf{e}_{r^{\prime}}\left( \kappa^{2}-v^{2}\right) \right] }{\left( 1-\kappa\right) ^{3}r^{\prime2}}+\nonumber\\ & +\frac{\left( \kappa\mathbf{e}_{r^{\prime}}-\mathbf{v}\right) \left( \mathbf{e}_{r^{\prime}}\cdot\mathbf{v}\right) +\left[ \mathbf{e}_{r^{\prime }}\left( \mathbf{e}_{r^{\prime}}\cdot\overset{\cdot}{\mathbf{v}}\right) -\overset{\cdot}{\mathbf{v}}\right] \left( 1-\kappa\right) }{\left( 1-\kappa\right) ^{3}r^{\prime}}. \label{A3_10}$$ Finally, substituting (\[A3\_7\]) and (\[A3\_10\]) in (\[A3\_5\]) and separating the factor $\left( 1-\kappa\right) ^{3}r^{\prime2}$ , we get$$\begin{aligned} \mathbf{E} & =\frac{q}{4\pi\epsilon_{0}\left( 1-\kappa\right) ^{3}r^{\prime2}}\left[ \mathbf{e}_{r^{\prime}}\left( 1-\kappa\right) ^{3}+\left( 3\kappa\mathbf{e}_{r^{\prime}}-\mathbf{v}\right) \left( 1-\kappa\right) ^{2}+\right. \nonumber\\ & +\left( \kappa^{2}-v^{2}\right) \left( \kappa\mathbf{e}_{r^{\prime}}-\mathbf{v}\right) +2\kappa\left( \kappa\mathbf{e}_{r^{\prime}}-\mathbf{v}\right) \left( 1-\kappa\right) +\mathbf{e}_{r^{\prime}}\left( \kappa^{2}-v^{2}\right) \left( 1-\kappa\right) +\nonumber\\ & +\left. r^{\prime}\left( \kappa\mathbf{e}_{r^{\prime}}-\mathbf{v}\right) \left( \mathbf{e}_{r^{\prime}}\cdot\overset{\cdot}{\mathbf{v}}\right) +r^{\prime}\left[ \mathbf{e}_{r^{\prime}}\left( \mathbf{e}_{r^{\prime}}\cdot\overset{\cdot}{\mathbf{v}}\right) -\overset{\cdot}{\mathbf{v}}\right] \left( 1-\kappa\right) \right] . \label{A3_11}$$ It is easy to show that this is reduced to $$\begin{aligned} \mathbf{E} & =\frac{q}{4\pi\epsilon_{0}\left( 1-\kappa\right) ^{3}r^{\prime2}}\left[ \mathbf{e}_{r^{\prime}}\left( 1-v^{2}\right) -\mathbf{v}\left( 1-v^{2}\right) \right. \nonumber \\ & \left. +r^{\prime}\left( \kappa\mathbf{e}_{r^{\prime}}-\mathbf{v}\right) \left( \mathbf{e}_{r^{\prime}}\cdot\overset{\cdot}{\mathbf{v}}\right) +r^{\prime}\left( \mathbf{e}_{r^{\prime}}\left( \mathbf{e}_{r^{\prime}}\cdot\overset{\cdot}{\mathbf{v}}\right) -\overset{\cdot}{\mathbf{v}}\right) \left( 1-\kappa\right) \right] \nonumber \\ & =\frac{q}{4\pi\epsilon_{0}}\frac{\left( 1-v^{2}\right) \left( \mathbf{x}^{\prime}-\mathbf{v}r^{\prime}\right) }{\left( r^{\prime }-\mathbf{x}^{\prime}\cdot\mathbf{v}\right) ^{3}}+\frac{q}{4\pi\epsilon_{0}}\frac{\left( \mathbf{x}^{\prime}-\mathbf{v}r^{\prime}\right) \left( \mathbf{x}^{\prime}\cdot\overset{\cdot}{\mathbf{v}}\right) -\overset{\cdot }{\mathbf{v}}r^{\prime}\left( r^{\prime}-\mathbf{x}^{\prime}\cdot \mathbf{v}\right) }{\left( r^{\prime}-\mathbf{x}^{\prime}\cdot \mathbf{v}\right) ^{3}}. \label{A3_12}$$ Taking into account that the numerator in the second term in the latter expression can be rewritten as the double vector product$$\left( \mathbf{x}^{\prime}-\mathbf{v}r^{\prime}\right) \left( \mathbf{x}^{\prime}\cdot\overset{\cdot}{\mathbf{v}}\right) -\overset{\cdot }{\mathbf{v}}r^{\prime}\left( r^{\prime}-\mathbf{x}^{\prime}\cdot \mathbf{v}\right) =\left[ \mathbf{x}^{\prime}\times\left[ \left( \mathbf{x}^{\prime}-r^{\prime}\mathbf{v}\right) \times\overset{\cdot }{\mathbf{v}}\right] \right] , \label{A3_13}$$ expression (\[A3\_12\]) can be recognized (with the identification $q/4\pi\epsilon_{0}=e$, $\mathbf{x}^{\prime}=\mathbf{R}$, $r^{\prime }=R$) as the expression ($63.8)$ for electric field in Ref. [@Landau2]. Thus, expressions in Refs. [@Landau2] and [@Feynman] are the same. Acknowledgements {#acknowledgements .unnumbered} ================ Supported by FAPESP under grants 2013/00840-9, 2013/16592-4 and 2014/08970-1, by RFBR under Project 15-02-00293a, and by the TSU Competitiveness Improvement Program, by a grant from The Tomsk State University D.I. Mendeleev Foundation Program. A.A.S. thanks also CAPES for support. A.E.S. thanks the University of São Paulo for hospitality extended to him during the period when this work was being performed. [9]{} R. de Sangro, G. Finocchiaro, P. Patteri, M. Piccollo, and G. Pizzella, Eur. Phys. J. C 75, 137 (2015). L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields. (GIF-ML, Moscow, 1962; Pergamon Press, Oxford, 1971). J.D. Jackson, Classical Electrodynamics (Wiley, New York, 1962 – 1975, pp. 654 – 658). ** R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures in Physics (Addison-Wesley, Reading Massachusetts, 1963), “Electromagnetism II” Ch 21-1. See also http://www.feynmanlectures.caltech.edu/II\_21.html\#Ch21-S3. . J.H. Field, Int. J. Mod. Phys. A 23, 327 (2008); Retarded electric and magnetic fields of a moving charge: Feynman’s derivation of Liénard-Wiechert potentials revisited, arXiv:0704.1574\[physics.class-ph\](2007); Comment on “Measuring propagation speed of Coulomb fields”, arXiv:1506.06630\[physics.gen-ph\] (2015) (or arXiv:1506.06630v2 \[physics.gen-ph\] (2016) for completed version) .
--- abstract: 'In organic bulk heterojunction solar cells, the donor/acceptor interfacial energy offset ($\Delta E$) is found to provide the driving force for efficient charge separation which gives rise to high short circuit current density ($J_\mathrm{sc}$), but a high $\Delta E$ inevitably undermines the open circuit voltage ($V_\mathrm{oc}$). In this paper, employing the device model method we calculated the steady state current density-voltage ($J-V$) and the $J_\mathrm{sc}-\Delta E$ curves under two different charge separation mechanisms to investigate the optimum driving force required for achieving sizable $V_\mathrm{oc}$ and $J_\mathrm{sc}$ simultaneously. Under the Marcus charge transfer mechanism, with the increased $\Delta E$ the Jsc increases rapidly for $\Delta E\leq 0.2$ eV, and then maintains a nearly constant value before decreasing at the Marcus inverted region, which is due to the accumulation of undissociated excitons within their lifetime and is beneficial for obtaining a sizable $J_\mathrm{sc}$ under a $\Delta E$ much smaller than the reorganization energy $\lambda$. With inclusion of both the electron and the hole transfer pathways of different respective $\lambda$’s into the device model, the experimentally measured $J-V$ curves for donor/acceptor blend with different $\Delta E$’s can be reproduced. For the coherent charge transfer mechanism in which the driving force act as the energy window of accessible charge separated states, with two typical types of density of states for the charge transfer excitons, it is shown that the highest $J_\mathrm{sc}$ can also be achieved under a small $\Delta E$ of 0.2eV if the high-lying delocalized states are harvested in high proportion. This work demonstrates the existence of the optimum driving force of 0.2eV and provides some guidelines for engineering the interfacial energetics to achieve the high balanced $J_\mathrm{sc}$ and $V_\mathrm{oc}$.' author: - Wenchao Yang bibliography: - 'dfpaper.bib' title: 'Achieving balanced open circuit voltage and short circuit current by tuning the interfacial energetics in organic bulk heterojunction solar cells: A drift-diffusion simulation' --- Introduction ============ In organic bulk heterojunction solar cells, the randomly oriented donor/acceptor (D/A) interfaces are generally employed for converting the photogenerated excitons into free charge carriers[@deibel; @hains; @clark; @gunes; @hedley]. There have been plenty of experimental and theoretical works devoted to the investigation of the various interfacial properties, which demonstrated that the performance and stability of the devices are largely determined by the interfacial donor/acceptor morphology[@bavel; @huang; @szarko; @nuzzo], the interfacial molecular orientation and aggregation[@barry; @miller; @fu; @chen; @ryno; @ran; @ndjawa], since they have significant impacts on the interfacial energetics[@poelking; @guo; @fazzi]. In particular, the interfacial energetics plays the central role on the exciton dissociation and charge generation[@clark; @hedley], and thus attracted most of the research attention among all the interfacial properties. It is expected that through investigating the working principles of the interfacial energetics people can optimize it and fabricate high efficiency photovoltaic devices. However, the interfacial energetics is rather complicated, which not only involves the molecular frontier orbitals of the donor and acceptor materials, but also the tightly-bounded singlet excitonic states, the loosely-bounded charge transfer (CT) states and the charge separated (CS) states upon photo-excitation[@ohkita; @tvingstedt; @deibel2; @arndt; @grancini; @jailaubekov; @nuzzo]. Moreover, each type of these excited states consists of many energy levels forming a manifold[@clark; @barker; @gerhard; @ti]. Up to now, the intricate interactions among the states and their effects on the charge separation processes remain hotly debated. Macroscopically, the interfacial energetics is revealed to have direct impacts on both the short circuit current density ($J_{sc}$) and the open circuit voltage ($V_{oc}$) of the devices. For the $V_{oc}$, its upper limit is basically determined by the CT states energy and their disordered effect[@veldman; @vandewal; @burke; @collins; @zou; @guan]; while for the $J_{sc}$, it is experimentally demonstrated that a finite lowest unoccupied molecular orbital (LUMO) or highest occupied molecular orbital (HOMO) level offset across the interface is indispensable for obtaining sizable photocurrent[@ohkita; @dnuzzo; @hoke; @hendriks]. The measured current density-voltage ($J-V$) characteristics for devices with a fixed donor material and different fullerene acceptor materials suggest that, as the acceptor with higher LUMO level is employed, the $V_{oc}$ becomes larger due to the increased effective band gap, whereas the $J_{sc}$ decreases significantly. Especially for the recently popular fullerene-based acceptor of ICBA, when it is blended with P3HT as the photoactive layer, the interfacial LUMO offset is smaller than 0.05eV, and the corresponding $J-V$ curve exhibits a $V_{oc}$ of over 1V but an extremely small $J_{sc}$, representing poor charge generation efficiency[@dnuzzo; @hoke]. For the hole transfer processes, the required HOMO energy offset is found to be even 0.3eV higher than the driving force for electron transfer[@hendriks]. Thus, the interfacial LUMO (HOMO) offset is believed to play important roles on charge separation and is usually referred as the driving force for charge transfer and separation in literature[@rand; @clark; @ohkita; @shoaee; @ward; @coffey; @dimitrov; @wright; @jakowetz]. More rigorously, the driving force can be defined as the difference between the effective band gap and the CT state energy[@jakowetz]. According to the different scenarios proposed to explain how the exciton dissociation and charge separation process proceed at the donor/acceptor interface, the possible roles of the driving force could be the following three folds. First of all, the charge transfer at the interface may be a non-adiabatic process which involves a relatively large reorganization energy. The energy level offset provides the free energy for carriers to reach the intersection of potential energy surfaces and achieve resonant charge transfer, as described by the traditional Marcus theory[@ljakoster; @zhang; @wright; @volpi]. This mechanism has been demonstrated by the measurement of photo-carrier yield for a series of acceptors[@coffey]. Secondly, the driving force may provide the kinetic energy required for the electron-hole polaron pairs to escape from their mutual Coulomb attractive potential, which is the so-called hot CT state dissociation[@clark; @ohkita; @shoaee; @murthy; @grancini; @jailaubekov; @schulze; @fuzzi]. Thirdly, employing the pump-push-photocurrent measurements on the free carrier generation efficiency, Bakulin et al found that there exists a band of delocalized high-lying CT states or some vibrational modes which can facilitate the coherent transport of charge carriers on these states to achieve full separation, while those charge carriers on the low-lying CT states generally recombine geminately and do not contribute to the photocurrent[@bakulin1; @bakulin2; @jakowetz]. Due to this coherent charge separation mechanism the finite driving force seems unnecessary[@chner; @kaake; @whaley]. However, with the increased energy level offset, more delocalized states become accessible for the ballistic or coherent transfer of charge carriers, so that the thus measured charge generation rate still exhibits a weak dependence on the LUMO energy level offset[@jakowetz]. On the other hand, the impacts of donor: acceptor ratio is much stronger in this case, because the high proportion of fullerene acceptor material will spontaneously aggregates and forms crystalline phases, which give rise to much more delocalized electronic states[@jakowetz; @savoire; @tamura; @nan]. Each of the charge generation mechanisms can partly explain the experimental phenomena and it is highly controversial that which one is the dominant. Since most of the measurements are done under the transient pulsed luminescence conditions, it remains unclear to what extent the steady state performance of devices is limited by the charge generation rates under the incoherent (Marcus) or coherent mechanisms. Moreover, as mentioned above, there is always a tradeoff between the $J_{sc}$ and $V_{oc}$ under a specific value of the driving force, and increasing the driving force to boost the free carrier generation will inevitably lead to the decreased $V_{oc}$[@rand; @ohkita]. Thus people need to find the minimum driving force required for efficient charge separation to avoid sacrificing the $V_{oc}$ too much. Actually this has been realized in some non-fullerene acceptor solar cells, where the high and balanced $J_{sc}$ and $V_{oc}$ can be reached simultaneously[@liu; @baran]. But the theoretical explanation for this desirable effect is still lacking. In this paper, we employ the macroscopic device model simulation to investigate the effect of the driving force on the final device performance, especially the $J_{sc}$ which was less intensively studied than the properties of $V_{oc}$ in literature. The interfacial energy offset are incorporated into the device model both through its impacts on the effective band gap and on the exciton dissociation rate or proportion to calculate the $J-V$ curves under different driving forces. The investigations are done on the theoretical frameworks of the incoherent and the coherent charge separation mechanisms. It is found that both of them can give rise to $J-V$ curves similar to the experimentally measured ones. Moreover, there indeed exists an optimum driving force of 0.2eV or so for obtaining balanced $J_{sc}$’s and $V_{oc}$’s. Under the steady state, with the incoherent dissociation mechanism the relatively large $J_{sc}$ can be achieved in a broad range of the interfacial LUMO offset so that it could be restricted to a much less value than the reorganization energy; while with the coherent mechanism, the denser is the distribution of the delocalized CT states above the acceptor LUMO level, the higher is the $J_{sc}$ under small driving forces. The results are consistent with the finding that the efficient charge separation can be achieved under small driving forces[@lee; @heeger], and may also provide clues for the design and preparation of the organic donor/acceptor with optimized interfacial energetics to fabricate devices of high power conversion efficiency. In Sec.\[method\], we describe the model we used in simulation, and in Sec.\[results\], the simulated J-V curves for the incoherent and the coherent dissociation mechanisms are shown, respectively, and the variation of $J_{sc}$ under different driving forces is discussed in detail. Finally, we give the conclusions in Sec.\[conclusion\]. Theoretical device modeling method {#method} ================================== The one dimensional device model provide a straightforward method to calculate the device operating parameters under the influences of various microscopic electronic processes[@smith; @blom]. For the bulk heterojunction devices, the active layer in which the donor and the acceptor phases interpenetrate with each other and form percolating pathways for charge transport is considered as a homogeneous medium. Although the interfacial morphology cannot be taken into account in the model, for finely-mixed donor/acceptor phases this assumption is valid from a macroscopic point of view. In order to produce free charge carriers, the photo-generated singlet excitons must experience two successive dissociation steps. In the first one, the exciton diffuses to the donor/acceptor interfaces and transfer their electrons from the donor phase to the acceptor phase while leaving the holes in the donor phase, forming CT states on the interfaces[@clark]. The exciton dynamics is described by the following continuity equation $$\label{exciton} \frac{\partial X}{\partial t}=D_X\frac{\partial^2 X}{\partial x^2}-\frac{X}{\tau}-k_\mathrm{PET} X + G,$$ where $X$ is the exciton concentration. The terms on the right-hand side of Eq. (\[exciton\]) represents the diffusion, the radiative and non-radiative decay, the dissociation and the photo-generation processes, respectively, with $D_X$ the diffusion coefficient, $\tau$ the lifetime, $k_\mathrm{PET}$ the photo-electron transfer rate and $G$ the optical generation rate. This ultrafast electron transfer process is nonadiabatic and its rate is given by the Marcus theory:[@clark] $$\label{ket} k_\mathrm{PET}=\frac{2\pi}{\hbar\sqrt{4\pi \lambda kT}}V^2\exp\left(-\frac{(\triangle G+\lambda)^2}{4\lambda kT}\right),$$ in which the $V$ stands for the electronic coupling between the donor and acceptor molecules; the $\lambda$ represents the reorganization energy; and the $\triangle G$ is the free energy. In the context of charge transfer at the donor/acceptor heterojunction, the $\triangle G$ is actually equal to the interfacial energy offset. The value of $k_\mathrm{PET}$ is mainly dominated by the exponential factor on the right side of Eq. (\[ket\]), and the corresponding prefactor is assumed to be a constant of $k_0$, which may also represents the coherent (ballistic) charge transfer rate. The coherent charge transfer mechanism arises from the delocalized CT states and its modeling method is postponed to the Sec.\[results\] for compactness. In the incoherent charge separation mechanism, only certain proportion of the thus produced CT states can dissociate and generate free charge carriers, while the others recombine geminately to the ground state. According to the Onsager-Braun theory, the proportion of the successfully dissociated CT states $P(E)$ is mainly dependent on temperature, the electric field strength, and the CT states binding energy. With the approximate form of[@clark] $$\label{pe} P(E)=\exp\left(-\frac{e^2}{4\pi\varepsilon_0\varepsilon kT a}\right)\left(1+\frac{e^3}{8\pi\varepsilon_0\varepsilon (kT)^2}E\right),$$ it is incorporated into the free carrier generation rate. Now the continuity equations for electrons and holes can be written as: $$\begin{aligned} % \nonumber to remove numbering (before each equation) \frac{\partial p}{\partial t} &=& -\frac{1}{e}\frac{\partial J_p}{\partial x}+P(E)k_\mathrm{PET} X-R, \label{pt} \\ \frac{\partial n}{\partial t} &=& \frac{1}{e}\frac{\partial J_n}{\partial x}+P(E)k_\mathrm{PET} X-R. \label{nt} \end{aligned}$$ The electron (hole) current $J_n$ ($J_p$) has the common drift-diffusion form[@blom], with the Einstein’s relation being assumed. At the two ends of the device, the $J_n$ and $J_p$ are defined as the respective net surface recombination currents[@sandberg], which consist of the boundary conditions for Eqs (\[pt\],\[nt\]). The bimolecular recombination rate $$R=\zeta \frac{e(\mu_n+\mu_p)}{\varepsilon_0\varepsilon}(np-n_i^2)$$ where $\zeta$ is the reduction factor with respect to the Langevin bimolecular recombination rate[@burke]. The internal electric field $E(x)$ obeys the ordinary Poisson’s equation $$\label{poisson} \frac{\partial E}{\partial x}=\frac{e}{\varepsilon_0\varepsilon}(p-n)$$ with the constraint that $$\label{cons} \int_0^L E(x)dx=V_\mathrm{ext}-(E_g-\phi_p-\phi_n)/e ,$$ in which $V_\mathrm{ext}$ is the externally applied bias voltage, $E_g$ is the effective band gap, and $\phi_p(\phi_n)$ is the hole (electron) injection barrier at the anode (cathode), namely the effective voltage drop across the device is equal to the applied voltage subtracted by the built-in voltage. Using the equilibrium concentrations of $n(x), p(x)$ and the equilibrium internal field strength $E(x)$ as the initial conditions, the continuity equations and the Poisson’s equation are evolved together under the constant illumination condition to reach the steady state solutions, from which the $J-V$ curves are plotted. The simulation parameters are presented in Table. \[para\] except being noted otherwise. Parameter Symbol Value -------------------------------------------- ------------------ ----------------------------------------------- Donor (Acceptor) band gap $E_g$ 1.8eV Injection barriers $\phi_n, \phi_p$ 0.2eV Relative permitivity $\varepsilon$ 3.5 Active layer thickness $L$ 200nm Effective density of states $N_C, N_V$ $10^{21} \mbox{cm}^{-3}$ Charge carrier Mobilities $\mu_n, \mu_p$ $0.1\,\mbox{cm}^2/\mbox{Vs}$ CT generation rate $G$ $3\times 10^{21} \mbox{cm}^{-3}\mbox{s}^{-1}$ CT state lifetime $\tau$ 100ns CT state radius $a$ 2.25nm Coherent CT rate $k_0$ 0.1$\mbox{ns}^{-1}$ Bimolecular recombination reduction factor $\zeta$ 0.1 Reorganization energy $\lambda$ 0.5eV : The parameters used in the device model simulation\[para\] Results and discussion {#results} ====================== Charge separation through the incoherent Marcus mechanism --------------------------------------------------------- Based on the assumption that the major role of the driving force is to predominantly determine the charge transfer rate as described by the Marcus theory, we calculated the $J-V$ curves under a set of LUMO level offsets $\Delta E_\mathrm{L}$’s to examine their effects on the device performance, which are shown in Fig. \[jv1\]. It is observed that the calculated curves reflect well some features of the experimentally measured $J-V$ curves for the polymer/fullerene bulk heterojunction solar cells with a fixed donor material and varied acceptor materials, that is if a $J-V$ curve shows a high $V_\mathrm{oc}$ the corresponding $J_\mathrm{sc}$ is relatively small, and vice versa[@dnuzzo; @hoke]. Therefore, it is demonstrated that the excess free energy required for achieving a sufficiently high nonadiabatic charge transfer rate $k_\mathrm{PET}$ could be the probable origin of the observed tradeoff between the $J_\mathrm{sc}$ and $V_\mathrm{oc}$ in these devices. Generally, as the driving force increases by 0.1eV each time, the $V_\mathrm{oc}$ decreases exactly by 0.1V, which is simply due to the consequent decreasing of the effective band gap. In the following we will mainly focus on the behavior of $J_\mathrm{sc}$ with the varying driving forces. It can be observed that the $J_\mathrm{sc}$ increases significantly with the increasing $\Delta E_\mathrm{L}$ when $\Delta E_\mathrm{L}$ is as small as 0.1 or 0.2eV. In this case the enhanced electron transfer rate $k_\mathrm{PET}$ produces high concentration of CT states, whose subsequent dissociation leads to the increased photocurrent. As the $\Delta E_\mathrm{L}$ approaches 0.5eV, the $J_\mathrm{sc}$ reaches its maximum and then decreases. ![The calculated $J-V$ curves for different interfacial LUMO level offsets (the driving forces) under the Marcus charge separation mechanisms. The reorganization energy $\lambda$ is set to 0.5eV. In the calculation the Onsager-Braun theory for CT state dissociation is taken into account. []{data-label="jv1"}](JVcurve1){width="8cm"} In order to reveal quantitatively the relationship between the steady state photocurrent and the driving force, we calculated more $J_\mathrm{sc}$’s under different $\Delta E_\mathrm{L}$’s and plotted them in Fig. \[jsc\](a), where the effect of temperature is also examined considering the strong temperature-dependence of $k_\mathrm{PET}$ and $P(E)$. At the room temperature (RT) of 300K, as the $\Delta E_\mathrm{L}$ increases the $J_\mathrm{sc}$ quickly rises to over $8\,\mbox{mA}/\mbox{cm}^2$, and keeps this high and approximately constant value in the wide range from 0.2 and 0.7eV, beyond which the Marcus inverted region emerges and the $J_\mathrm{sc}$ becomes smaller. With the decreasing temperature, the $J_\mathrm{sc}$ reduces greatly due to the reduction of the dissociation proportion $P(E)$ of the CT states. Moreover, for the curves of low temperature, the high-and-flat region shown in the RT curve disappears, and the $J_\mathrm{sc}$ begins to decrease slowly as soon as it reaches its maximum at $\Delta E_\mathrm{L}=0.3$eV. This is because with the increasing $\Delta E_\mathrm{L}$, the built-in field is greatly weakened so that the $P(E)$ decreases as the result of the reduced internal field, leading to inefficient charge extraction and smaller $J_\mathrm{sc}$’s. Therefore, at low temperatures the free charge generation is strongly restricted by the small field dependent CT state dissociation rate. The sole effect of the driving force can be observed in Fig. \[jsc\](b), where we plotted the $J_\mathrm{sc}-\Delta E_\mathrm{L}$ curves calculated by setting $P(E)=1$. In this case all of the $J_\mathrm{sc}-\Delta E_\mathrm{L}$ curves have the high-and-flat region, even though the region gradually shrinks with the decreasing temperatures. In addition, the curves are basically symmetric with respect to the vertical line of $\Delta E_\mathrm{L}=0.5$eV, which is the feature of the Marcus charge transfer rate. Nevertheless, they display large discrepancy with the corresponding $k_\mathrm{PET}-\Delta E_\mathrm{L}$ curves, for the latter have prominent peaks when the $\Delta E_\mathrm{L}=\lambda$ and reduces much more rapidly when the $\Delta E_\mathrm{L}$ deviates from the $\lambda$. This result is in contradictory with Coffey et al’s finding that the photo-carrier relative yield data measured with the time-resolved microwave photoconductivity (TRMC) method for blends of fixed acceptor and different donors can be well fitted by the $k_\mathrm{PET}-\Delta E_\mathrm{L}$ curve[@coffey]. To find the underlying reason of the discrepancy, we calculated the steady state exciton concentration $X$ under the short circuit condition for different $\Delta E_\mathrm{L}$’s and temperatures, which are averaged over the whole active layer thickness, as shown in Fig. \[exdis\]. It is seen that the variation of the exciton concentration $X$ with respect to $\Delta E_\mathrm{L}$ is just opposite to that of $k_\mathrm{PET}$, i.e. the curve has a deep valley precisely at the point of the reorganization energy $\lambda$ of 0.5eV, and this feature does not change at different temperatures. Therefore, the product of $k_\mathrm{PET}X$ which is the CT states generation rate is approximately a constant for a wide range of $\Delta E_L$, which is the origin of the high-and-flat region for the $J_\mathrm{sc}-\Delta E_\mathrm{L}$ curves. Especially for small $\Delta E_\mathrm{L}$’s, under the constant illumination condition in steady state the induced small $k_\mathrm{PET}$ may give rise to high concentration of unquenched excitons, and many of which can dissociate within their lifetime to contribute to the photocurrent. On the contrary, under the pulsed illumination condition in the TRMC experiments, since the supply of excitons is limited by the light-pulse duration for each type of polymer:fullerene blend, only the $k_\mathrm{PET}$ basically governs the variational tendency of the photo-carrier yield with respect to the $\Delta E_\mathrm{L}$, such that a bell-like curve for the photo-carrier yield emerges[@coffey]. The occurrence of high $J_\mathrm{sc}$ under small $\Delta E_\mathrm{L}$ in steady state suggests that it is unnecessary to employ pairs of donor and acceptor materials with their $\Delta E_\mathrm{L}$ approaching the reorganization energy $\lambda$ to achieve the maximum photocurrent. According to Fig. 2, at the RT with $\lambda=0.5$eV, a moderate $\Delta E_\mathrm{L}$ of 0.2-0.3eV can provide sufficient driving force for charge separation at the D/A interface. Thus in principle given a donor material, much $V_\mathrm{oc}$ loss due to the interfacial energy level offset could be saved by employing acceptors with higher LUMO levels to form blend with the donor. However, in some polymer/fullerene blended systems the required driving force for achieving sizable photocurrent is still as high as 0.5eV[@wright; @hendriks]. This may be due to the fact that both of the electron transfer and the hole transfer processes contribute to the photocurrent, and the driving force for the latter is experimentally revealed to be 0.3eV higher than the former[@hendriks]. Consequently, if a significant proportion of the excitons are dissociated through transferring their holes from the acceptor to the donor, a relatively large HOMO offset $\Delta E_\mathrm{H}$ is essential for achieving a sufficiently high hole transfer rate $k_\mathrm{HT}$ and thus the high $J_\mathrm{sc}$. The situation may occur in devices made of non-fullerene acceptors, in which the photo-absorption of acceptors contribute greatly to the exciton formation[@stoltzfus]. Moreover, it is reported that even for fullerene-based acceptors such as ICBA, a large proportion of excitons generated in the polymers can diffuse into them through the F$\ddot{\mbox{o}}$rster resonant energy transfer process, and these excitons can only dissociate through the hole transfer pathway, and the inefficient hole transfer process in polymer-ICBA devices should be responsible for their low $J_\mathrm{sc}$[@dnuzzo; @hoke]. Here we denoted the respective proportions of excitons dissociating through the two pathways as $P_e$ and $P_h=1-P_e$, and calculated the $J_\mathrm{sc}-\Delta E$ curves with the varied $P_e$ to evaluate the combined roles of the electron and hole transfer pathways on the $J_\mathrm{sc}$. Since we assumed the same band gaps for the donor and acceptor materials, the HOMO offset $\Delta E_\mathrm{H}$ is equal to the LUMO offset $\Delta E_\mathrm{L}$; and the reorganization energy for hole transfer is 0.3eV higher than that of the electron transfer. The calculated results are shown in Fig. \[twopathway\]. It is observed that as more excitons are generated in or transferred into the acceptor, the high-and-flat region for $J_\mathrm{sc}$ is maintained, but its onset is shifted to the higher $\Delta E$, which suggests that the driving force required for achieving sizable photocurrent becomes larger if the hole transfer reaction, being of a higher reorganization energy, plays a significant role on exciton dissociation. In addition, the slow decreasing of $J_\mathrm{sc}$’s in the high $\Delta E$ regime is mainly caused by the reduced built-in field rather than the Marcus inverted effect, which is different from the sole electron transfer case (the red dashed line). Based on the above understanding, we calculated the RT $J-V$ curves with the $\Delta E$’s derived from real materials, which are some types of fullerene-based acceptors blended with the donor of PF10TBT, as shown in Fig. \[jvicba\]. The effective energy gap is set to 1.66eV, and the electron transfer driving forces $\Delta E_\mathrm{L}$ are set to 0.18, 0.08, 0.05, 0.03 and 0.01eV, corresponding to the acceptors of PCBM, $\mbox{t}_2$-bis-PCBM, bis-PCBM, si-bis-PCBM and ICBA, respectively. Without taking into account the hole transfer pathway, the high $J_\mathrm{sc}$ exhibited by the curve of PCBM further suggest that only a small driving force of about 0.2eV is required for obtaining sufficiently high photocurrent. Compared with the experimentally measured curves in Ref.[@dnuzzo], the curve of ICBA exhibits a much larger $J_\mathrm{sc}$ of 3.56$\mbox{mA/cm}^2$. So we speculate that when examining the performance of the ICBA-acceptor devices, it is important to incorporate the effects of the high proportion of excitons diffusing into the ICBA and the inefficient hole transfer pathway. With the $P_e=0.2$ and the hole transfer reorganization energy $\lambda_\mathrm{H}=0.8$eV, the recalculated $J-V$ curve (the dashed line) of ICBA gives rise to a relatively realistic $J_\mathrm{sc}$ of 1.25$\mbox{mA/cm}^2$. It is noticed that in this case the $V_\mathrm{oc}$ also becomes smaller because of the reduced photo-carrier density under the open circuit condition. ![The calculated short circuit current density versus the interfacial LUMO offset ($J_\mathrm{sc}-\Delta E_\mathrm{L}$) under the Marcus charge transfer mechanism at different temperatures, with the field and temperature dependent Onsager-Braun CT state dissociation probability $P(E)$ being taken into account (a) or neglected (b) in the calculation. For comparison, the corresponding $k_\mathrm{PET}-\Delta E_\mathrm{L}$ curves with their maximum being scaled to the value of $J_\mathrm{sc}$ at $\Delta E_\mathrm{L}=\lambda=0.5$eV are also plotted in (b). []{data-label="jsc"}](JscDF){width="16cm"} ![The calculated exciton density versus the interfacial LUMO offset under the Marcus charge transfer mechanism at different temperatures. Each point for density corresponds to the average value over the whole active layer in the device. The reorganization energy $\lambda$ is set to 0.5eV[]{data-label="exdis"}](excitondensity){width="8cm"} ![The calculated $J_\mathrm{sc}-\Delta E_\mathrm{L}$ curves under the Marcus charge transfer mechanism for a set of different proportions $P_e/(1-P_e)$ of the electron/hole transfer pathways. The reorganization energy $\lambda_\mathrm{L}, \lambda_\mathrm{H}$ for the electron and hole transfer pathways are set to 0.5 and 0.8eV, respectively. []{data-label="twopathway"}](twopath){width="8cm"} ![The calculated $J-V$ curves for devices with the donor of PF10TBT blended with different acceptors. The effective energy gap is set to 1.66eV; the reorganization energy is set to 0.5eV; and the electron transfer driving forces $\Delta E_\mathrm{L}$ are set to 0.18, 0.08, 0.05, 0.03 and 0.01eV, corresponding to the acceptors of PCBM, $\mbox{t}_2$-bis-PCBM, bis-PCBM, si-bis-PCBM and ICBA, respectively. The dashed line is calculated for the ICBA acceptor by taking into account the hole transfer pathway with $P_e=0.2$ and $\lambda_\mathrm{H}=0.8$eV, which can better fit the experimentally measured $J-V$ curve for the ICBA. []{data-label="jvicba"}](jvicba){width="8cm"} Charge separation through the coherent/ballistic dynamics --------------------------------------------------------- Next we examine the role of the driving force in coherent mechanism for charge separation. In contrast to the Onsager-Braun theory, for the coherent mechanism the charge transfer process is ballistic and band-like rather than diffusive, and results in complete charge separation. The relevant exciton dissociation rate can be assumed to be of a constant rate of $k_0$ which is independent of the ambient temperature and the electric field. But as have been demonstrated by lots of experiments like the PPP, only the proportion of excitons that are energetically resonant with the delocalized CT states (thus they can be deemed as the same entity) in the CT states manifold can participate the ballistic transfer process[@bakulin1; @bakulin2]. To make the picture of the involved energy levels simple, it is assumed that the delocalized CT states accessible for charge separation are those states which are energetically higher than the LUMO level of the acceptor, and the CT states below the acceptor LUMO level are relaxed and localized so that they cannot dissociate successfully but decay to the ground state through geminate recombination. Since the coherent charge transfer process is ultrafast, which occurs within 100fs upon photo-excitation and is prior to the relaxation of the hot CT states[@grancini; @jailaubekov; @whaley; @yao], initially the population in each level of the CT states manifolds does not obey the equilibrium distribution and is considered to be evenly distributed. In particular, the population of the in-gap CT states arises from the low energy such as the near infrared photo-absorption[@troisi], as has been verified by some external quantum efficiency (EQE) measurements[@vandewal]. According to the above considerations, if the total number of states in the intermediate CT states manifold is fixed to be $N$, the corresponding density of states (DOS) $g(E)$ solely determines the proportion of the delocalized CT states $P_\mathrm{band}$ in the whole manifold, thus we have $$\label{pband} P_\mathrm{band}(\Delta)=\frac{1}{N}\int_{E^\prime-\Delta}^{E^\prime} g(E) dE,$$ where $E^\prime$ is the upper limit of the CT states manifold and $\Delta$ is the width of the energy window of the delocalized electronic states, i.e. the interfacial LUMO level offset. To investigate the dynamics of free charge carrier, only the delocalized CT states should be taken into account. Whereas the low-lying ones do not contribute to the photocurrent and are just wasted. Thus the continuity equations of excitons should be modified into the form of $$\label{exciton2} \frac{\partial X}{\partial t}=D_X\frac{\partial^2 X}{\partial x^2}-\frac{X}{\tau}-k_0 X + P_\mathrm{band}G.$$ Solving Eq. (\[exciton2\]) together with other device model equations, the $J-V$ curves for the coherent charge separation mechanism can be obtained. To obtain the actual $P_\mathrm{band}$, the $g(E)$ must be explicitly given. However, for real materials there are many complicated effects impacting the CT DOS, such as the energetic disorder[@mcmahon], the entropic effect due to the different dimensionality of the donor and acceptor molecules[@bregg], the aggregation effects of the fullerene-based acceptors and the image charge effect at the donor/acceptor interface[@savoire; @xyz]. Currently the $g(E)$ can only be calculated using first principle or molecular dynamics methods for specific donor-acceptor material systems[@tamura; @chner; @nan; @savoire; @belj], and there lacks an analytical expression for it. For the convenience of the present phenomenological investigation, we assumed two simplified analytical expressions of $g(E)$ which may approximate the real DOS of the CT states manifold to some extent. ### Hydrogen-atom-like density of states Firstly, considering the CT excitons as bounded polaron pairs, their energy levels resemble those of a hydrogen atom so that the $g(E)$ has the hydrogen-atom DOS like expression[@xyz]. In Fig. \[schematic\](a) we schematically depicted the interfacial energetics, where the zero point of the CT state energy is set to be the acceptor LUMO level, i.e. $E_\mathrm{ct}$ is transformed to $E_\mathrm{ct}-E_g$. A cutoff energy level $E_c$ slightly above the acceptor LUMO level separates the high-lying continuous energy spectrum from the low-lying discrete one. Then for $E\geq E_c$, $g(E)$ equals to a constant of $\alpha (E_a-E_c)^{-3/2}$; while for $E<E_c$, $g(E)=\alpha (E_a-E)^{-3/2}$, in which the parameter $E_a$ is a small positive energy used to avoid the singularity in $g(E)$, and the prefactor $\alpha$ is determined by the normalization condition of $$\label{normalization} N=\int_{E^\prime-W}^{E^\prime} g(E) dE$$ with $W$ the width of the CT state manifold. Based on the Eqs. (\[pband\],\[normalization\]), the proportion $P_\mathrm{band}$ is deduced to be of the form of $$\label{pbandhy} P_\mathrm{band}(\Delta)=\frac{(\Delta-E_c)/(E_a-E_c)+2\left[1-\sqrt{(E_a-E_c)/E_c}\right]}{(\Delta-E_c)/(E_a-E_c)+2\left[1-\sqrt{(E_a-E_c)/(E_c+W-\Delta)}\right]}.$$ Substituting the $P_\mathrm{band}$ into Eq. (\[exciton2\]) with $E_c=0.05$eV, $E_a=0.1$eV and $W=1$eV, we calculated the $J-V$ curves with a set of different driving force $\Delta$’s, which are plotted in Fig. \[jvpband\]. It is observed that the curves exhibit the exactly same behavior as those calculated for the Marcus incoherent charge transfer mechanism (Fig. \[jv1\]). That is with the increased $\Delta$, the $J_\mathrm{sc}$ increases whereas the $V_\mathrm{oc}$ decreases evenly. Thus it is not plausible to gain some clues concerning which charge separation mechanism is the dominant one just from the variation of $J-V$ curves with respect to the driving force. For $\Delta=0.2$eV, the optimum device performance is obtained with $J_\mathrm{sc}=7.14\,\mbox{mA/cm}^2$ and $V_\mathrm{oc}=0.94$V, such that a driving force of 0.2eV is sufficient for achieving balanced $J_\mathrm{sc}$ and $V_\mathrm{oc}$ in devices where the coherent exciton dissociation mechanism plays the major role on charge generation. The quantitative relationship between the $J_\mathrm{sc}$ and $\Delta$ are calculated and presented with the varying cutoff energy $E_c$ and the fixed $E_a=0.1$eV, as shown in Fig. 8(a). For all the curves, the $J_\mathrm{sc}$ increases very rapidly with $\Delta$ when the latter is smaller than 0.2eV, because in the energy window of 0-0.2eV the CT states form a quasi-continuum band with high DOS $g(E)$, and a small increase of $\Delta$ can induce high extra population of the CT states to participate the coherent (ballistic) charge transfer and separation. For $E_c=0.08$eV, the $J_\mathrm{sc}$ reaches an approximately constant high value beyond $\Delta=0.2$eV, being similar to the high-and-flat region appearing in the $J_\mathrm{sc}-\Delta E_L$ curve under the incoherent charge transfer mechanism. On the other hand, with the reduced $E_c$ the $J_\mathrm{sc}$ becomes smaller and remain increases slowly in the relatively high $\Delta$ regime, which is due to the fact that the in-gap CT states close to the acceptor LUMO level are of a high DOS and thus need to be harvested by enhancing the $\Delta$ in order to reach a sufficiently high $J_\mathrm{sc}$. The decreasing of $J_\mathrm{sc}$ around $\Delta=0.8$eV is only caused by the reduction of the built-in electric field and the lowered charge extraction efficiency, rather than the Marcus inverted region in Fig. 2. We also included a curve with the varied $E_a$ of 0.2eV and $E_c=0.08$eV(the dashed line), where compared to the corresponding curve with $E_a=0.1$eV (the red line), the $J_\mathrm{sc}$ reduces significantly as a result of the reduced DOS for the high-lying levels in CT state manifold. The CT states lying in the energy window $\Delta$ can also possibly consist of purely discrete spectrum, namely the $E_c$ level is equal to or above the donor LUMO level, as shown in Fig. \[schematic\](b). In this case we assumed the $E_c$ and $E_a$ to be the same. Then the delocalized CT states proportion $P_{band}$ is modified to $$\label{pband2} P^{\prime}_\mathrm{band}(\Delta)=\frac{1-(1+\Delta/E_c)^{-1/2}}{1-(1+W/E_c)^{-1/2}}.$$ Inserting the $P'_\mathrm{band}$ into the device model, the $J_\mathrm{sc}-\Delta$ curves are calculated and presented in Fig. 8(b). The shape of the curves does not exhibit observable change as compared to that in Fig. 8(a), but it becomes more difficult to achieve a sizable $J_\mathrm{sc}$ for the small $\Delta$. Generally the $J_\mathrm{sc}$ increases with the decreasing $E_c$. Therefore the $E_c$ level should be tuned as close to the donor LUMO level as possible. Combined with the results in Fig. 8(a), we conclude that the criterion for good interfacial energetics being able to facilitate significant coherent exciton dissociation is that, the CT states manifold is low and the energy window $\Delta$ is resonant with at least part of the continuous spectrum and the high-lying discrete spectrum, so that plenty of hot excitons can be harvested. ![The schematic illustration of the donor/acceptor interfacial energetics which contains the delocalized CT states and may facilitate the coherent (ballistic) charge separation. The energy levels of donor are on the left side and those of acceptor are on the right side. The Charge transfer (CT) states manifold (marked in red) is of the width $W$, in which the levels above the acceptor LUMO level are delocalized and resonant with the charge separated (CS) states (marked in green), forming an energy window of $\Delta$ in which the ballistic charge transfer can take place. The energy parameter $E_c$ represents a critical energy level on which the continuous and the discrete spectra in the CT states manifold meet. The $E_c$ level may be in the energy window (a), or above it (b). The case of (b) may occur in the hydrogen-atom-like DOS of the CT states manifold. The specific forms of the CT states DOS are described in the text. []{data-label="schematic"}](schematic){width="16cm"} ![The calculated $J-V$ curves for different driving force $\Delta$’s under the coherent charge separation mechanism. The CT states manifold is of a hydrogen-atom-like DOS, with the energy parameter $E_c=0.05$eV, $E_a=0.1$eV and $W=1$eV. []{data-label="jvpband"}](jvpband){width="8cm"} ![The calculated short circuit current density versus the driving force ($J_\mathrm{sc}-\Delta$) curves for different energy parameters of $E_c,E_a$ of the hydrogen-atom-like CT states DOS. (a) The CT states manifold may consist of the continuous and discrete spectra, corresponding to the DOS schematically illustrated in Fig. \[schematic\](a); (b) or the manifold is of the purely discrete energy levels below the LUMO level of the donor, corresponding to the DOS schematically illustrated in Fig. \[schematic\](b). []{data-label="jscecea"}](jscecea){width="16cm"} ### Exponential density of states Secondly, motivated by the mobility edge model in organic semiconductors[@neher], we considered CT states manifold to be of the exponential type of DOS. The interfacial energetic levels can be still basically schematically illustrated by the Fig. \[schematic\](a), with $E_c$ representing the cutoff energy level separating the continuous band from the discrete levels. For $E\geq E_c$, the DOS $g(E)=\alpha/E_a$; while for $E<E_c$, $g(E)=\alpha/E_a \exp[(E-E_c)]/E_a$, where $E_a$ is a parameter characterizing the width of the exponential states, and the prefactor $\alpha$ is also determined by the normalization condition Eq. (\[normalization\]). Now it can be deduced that the proportion of the CT states lying in the energy window of the driving force $\Delta$ is $$\label{pbandexp} P^{\prime\prime}_\mathrm{band}(\Delta)=\frac{(\Delta-E_c)+E_a[1-\exp(-E_c/E_a)]}{(\Delta-E_c)+E_a\left[1-\exp\left(\frac{\Delta-W-E_c}{E_a}\right)\right]}.$$ With the energy parameters $E_c, E_a$ and $W$ being set to 0.05, 0.1 and 1eV, we substituted the above $P^{\prime\prime}_\mathrm{band}(\Delta)$ into Eq. (\[exciton2\]) and calculated the $J-V$ curves for different driving force $\Delta$’s, as shown in Fig. \[expjv\](a). It is obvious that the curves show almost the same features with respect to those for hydrogen-atom like CT state DOS (see Fig.\[jvpband\]), suggesting that the device performance is not very sensitive to the specific form of the DOS as long as the number of in-gap levels decreases quickly with the decreasing energy. In Fig. \[expjv\](b) we present the calculated $J_\mathrm{sc}-\Delta$ curves for the varying $E_c$ and $E_a$. Similar to the behaviors exhibited by the curves for hydrogen-atom like DOS, the relatively larger $E_c$ more or less give rises to the higher $J_\mathrm{sc}$, because of the inclusion of the dense high-lying discrete levels in the energy window for ballistic charge transfer. On the other hand, the $E_a$ plays a much more important role on determining the photocurrent. With the increasing of $E_a$ from 0.1eV to 0.2eV, the $J_\mathrm{sc}$ decreases by nearly 2$\mbox{mA/cm}^2$ under the $\Delta$ of 0.2eV. In order to obtain a sizable $J_\mathrm{sc}$ under a small driving force, the width of the in-gap states in the CT manifold should be restricted to a value at least being smaller than 0.2eV. Therefore, the optimization of the DOS for the CT sates manifold is important, which could be realized by changing donor:acceptor ratio of the blend to enhance the fullerene aggregation and crystallization so that more delocalized CT states may be formed. ![(a) The calculated calculated $J-V$ curves for different driving force $\Delta$’s under the coherent charge separation mechanism. The CT states manifold is of an exponential DOS, with the energy parameter $E_c=0.05$eV, $E_a=0.1$eV and $W=1$eV. (b) The calculated $J_\mathrm{sc}-\Delta$ curves for different energy parameters of $E_c,E_a$ of the exponential CT states DOS, as schematically illustrated in Fig. \[schematic\](a). []{data-label="expjv"}](exponential){width="16cm"} Conclusion and outlook {#conclusion} ====================== In this work, employing the phenomenological device model method we investigated the impacts of the charge separation driving force, which is defined as the donor/acceptor interfacial energy level offsets on the device performance of organic bulk heterojunction solar cells. The driving force $\Delta$ may either provide the free energy required for the incoherent Marcus charge transfer processes to happen or form an energy window where the delocalized CT states reside and facilitate the coherent charge transfer processes. Both of the two kinds of charge separation mechanisms probably play important roles and thus were studied independently by calculating the corresponding $J-V$ and $J_\mathrm{sc}-\Delta$ curves. Generally the $V_\mathrm{oc}$ reduces evenly with the increased $\Delta$, forming a significant $V_\mathrm{oc}$ loss pathway. For the Marcus charge transfer mechanism, with the increasing of $\Delta$ from 0eV, the $J_\mathrm{sc}$ initially increases extremely rapidly and begin to saturate under a small delta of 0.2eV or so; then the $J_\mathrm{sc}$ maintains a high and nearly constant value until the Marcus inverted effect emerges under too high $\Delta$’s, exhibiting a behavior which is largely different from that of the Marcus charge transfer rate $k_\mathrm{PET}$. The underlying reason is found that the reduced $k_\mathrm{PET}$ under a $\Delta$ deviating from the reorganization energy $\lambda$ is precisely compensated by the enhanced density of the accumulated exciton within their lifetime, such that the overall free charge generation rate changes very slowly. When the hole transfer pathway plays innegligible roles on charge separation, the required $\Delta$ for obtaining a sizable $J_\mathrm{sc}$ may become higher due to the relatively larger reorganization energy on the acceptor side, such as the case for the ICBA acceptor based devices. For the coherent mechanism, when calculating the $J-V$ and $J_\mathrm{sc}-\Delta$ curves we assumed the hydrogen-atom-like DOS and the exponential DOS for the interfacial CT states manifold, respectively. The results show similar behaviors and suggest that as long as the energy window formed by the interfacial energy offset (or the driving force) contains part of the continuous spectrum and the dense high-lying discrete levels in the CT state manifold while the low-lying in-gap levels are rare, a great proportion of the CT states can be converted into the fully separated charge carriers and consequently the high $J_\mathrm{sc}$ is obtained under a small $\Delta$ of about 0.2eV, which is consistent with the behavior of $J_\mathrm{sc}$ calculated for the incoherent mechanism. Therefore, regardless of the charge separation mechanism, people can obtain the relatively high $J_\mathrm{sc}$ and $V_\mathrm{oc}$ simultaneously without sacrificing one for the other, which may be hopefully realized in the recently popular non-fullerene acceptor solar cells. In addition, concerning the concrete charge separation mechanism in the actual donor/acceptor blended systems, the coherent and incoherent mechanisms may coexist, which is probably the reason that up to now, in different experiments people have observed that the photocurrent generation follows both the Marcus-type behavior with respect to the driving force and the composition dependence on the donor:acceptor blend ratio. It is demonstrated in our simulation that with a moderate driving force, there is no obvious feature on the $J-V$ curves and the $J_\mathrm{sc}-\Delta$ curves that can identify which one is the dominant mechanism. However, the incoherent mechanism induces strongly temperature-dependent effects for the photocurrent and thus can be singled out through observing the behavior of $J_\mathrm{sc}$ at the lowered ambient temperature. Also, future works on the DOS of CT states may be helpful for acquiring the high $J_\mathrm{sc}$ under the smaller driving forces. The authors would like to thank Professor R. $\ddot{\mbox{O}}$sterbacka for the fruitful discussion and his insightful comments. This work is supported by the National Natural Science Foundation of China under the Contract No. 11604280 and 51602276.
--- abstract: 'We present an integrated microsimulation framework to estimate the pedestrian movement over time and space with limited data on directional counts. Using the activity-based approach, simulation can compute the overall demand and trajectory of each agent, which are in accordance with the available partial observations and are in response to the initial and evolving supply conditions and schedules. This simulation contains a chain of processes including: activities generation, decision point choices, and assignment. They are considered in an iteratively updating loop so that the simulation can dynamically correct its estimates of demand. A Markov chain is constructed for this loop. These considerations transform the problem into a convergence problem. A Metropolitan Hasting algorithm is then adapted to identify the optimal solution. This framework can be used to fill the lack of data or to model the reactions of demand to exogenous changes in the scenario. Finally, we present a case study on Montréal Central Station, on which we tested the developed framework and calibrated the models. We then applied it to a possible future scenario for the same station.' author: - 'Alexis Pibrac [^1]' - 'Bilal Farooq[^2]' bibliography: - 'PibracFarooq\_PedDynamics.bib' title: 'Integrated Microsimulation Framework for Dynamic Pedestrian Movement Estimation in Mobility Hub [^3]' --- Introduction ============ With the constant increase in the population of urban areas around the world, transportation and logistic is facing more organizational problems in order to deal with complex networks, mixing new technologies, and modern modes of transport. Never in the history, society has offered such a number of different possibilities, from the traditional individual modes (such as cars or bikes) to new concepts born in the growing market of sharing economy. Public transit systems such as metro, bus or tramway are now available in all sufficiently big cities. Moreover, these cities are well interconnected, thanks to various long distance modes of transportation. Thus rendering the current network of transportation facilities highly efficient as well as highly complex. Despite the improvements in transportation technologies and increasing demand, the mode that has always remained central is the walking mode. Mobility hubs (e.g. train stations, terminals, etc.) within which walking is the only mode, are the key connections in the dominantly prevalent inter-modal urban travel patterns. They have a high risk of overcrowdedness and thus playing more and more prominent role in the fluidity and efficiency of the whole transportation network.\ Despite significant advances in the individual level microscopic models to describe and reproduce pedestrians movement, the main limitation for simulations remains the lack of data for such situations. Indeed, with enough data, one particular case can be reproduced in a consistent manner (not exact but at least representative). But problems may arise when the information is incomplete; when it comes to validation (where additional data are needed for another time period); or extension of the scenario to future situations (where data are impossible to get).\ That is why here, we develop a novel framework for pedestrian dynamics in which the demand part is no more a static estimation directly obtained from the data. The demand which is technically a time dependent Origin-Destination matrix, will for sure be based on the available data, but will also be influenced by other kinds of information, such as the schedule of transportation systems, infrastructure in which the agents are moving, estimates of the transfer times for each trajectory, etc. By bringing in new processes and dynamic supply information, we aim to account for incomplete data when it comes to generating the exact demand using a microsimulation. Furthermore, we will be able to estimate changes in this demand induced by the changes in exogenous inputs. For instance if the design of a train station has changed, we need to adapt the departure time of each individual following what would be their reaction in real life. In such a case, the demand still depends on the observations already gathered, but the link between them is no more direct. Some of these observations may not be exactly satisfied, but adapted depending on the changes we made in the scenario. Thereby now that the demand description is adapted depending on the situations, we are making a step forward in terms of realism, when it comes to simulating non-existent scenarios i.e. testing potential future changes.\ Traditionally, the demand part of a scenario has been the starting point of a simulation–especially in case of pedestrian simulations [@Abdelghany2016; @sahaleh2012scenario]. For example, in the four-step model, after the generation and distribution steps, the demand is completely described. Then comes the modal choice and assignment that are using the so-called demand and models like discrete choice theory, model of transport modes, etc. that describe the behavior of each agent. In this approach, the simulation is divided into two phases: first we create the demand, and then we use it into successive behavioral models. In fact, we create an agent and its characteristics, then we describe its movement thanks to a description of its behavior. And this behavior is simulated with a chain of models that are successively going deeper in term of information (first only the mode of transport is chosen, then the global itinerary is computed etc. until we obtain the complete time dependent description of the movement). Here the demand is no more considered completely exogenous or known a priori, but dependent on other parts of the scenario, that can also be partial results of behavioral models. We can’t consider its generation into a separate phase. We have no longer a clear chain of objects to generate in a simple order thanks to deterministic models. But we have to find an equilibrium between all the different parts of the state, verifying all dependencies settled between them. The behaviors depend on the demand, for example the transfer times of each travel or the occupancy of each transport mode is directly influenced by the number of pedestrian in the station and their temporal distribution. And the demand depends on the behavioral simulation results, for example, the departure time is impacted by the time agents need to transfer or availability of modes.\ The problem of finding such an equilibrium is analogous to the one in Dynamic Traffic Assignment for vehicular traffic. However, due to the presence of a well-defined network and clear constraints, the search process for equilibrium in vehicular network is relatively trivial. Due to the complex movement of pedestrians and the high number of external factors that influence it (for example, arrival or departure time of a bus or a train is such an external factor that does not exist in vehicular traffic), the resolution for the case of pedestrian is of a higher complexity. To solve our equilibrium search problem, we are using a similar solution: a looping process running several times the same models until the convergence is reached. The classic behavior models of pedestrians simulations will be looped and computed as long as an equilibrium has not been found (see Figure \[markov\]) i.e. until the simulated demand is consistent with the state generated from the partially observed demand. The purpose here is to present a novel microsimulation framework that controls the generation of the demand, intended movement patterns, and assignment in order to search for the equilibrium. In next section we present the existing work, after which the core methodology is presented. The case study of Montréal Central Station is developed as an implementation of the proposed methodology. The results of base case and future scenario are discussed in details. In the end we present the conclusions and future direction. ![ \[markov\]Organization of the proposed framework.](generalF) Literature Review ================= Extensive research on various aspects of pedestrians simulation can be found in the literature. This has resulted in variety of tools to model the problem [@daamen2004modelling]. Past research has either focused on a specific operation within a train station [@zhang2008modeling] or the whole station [@sahaleh2012scenario]. The classical way to describe the pedestrian behavior is divided into three levels [@daamen2004modelling]: strategic, tactical and operational.\ The generation of OD matrix, that contains complete information about departure location, arrival location, and departure time for all agents, is a classical but tough problem in transportation research. It has been extensively studied in different contexts, e.g. for vehicles at urban area level for planning purposes [@national2012travel], as well as at a smaller spatial scale like ours. Various available datasets have been used, beginning with traffic counts on the network in order to directly generate the matrix [@cascetta1988unified] or more recently with a Bayesian resolution [@cheng2014bayesian]. The schedule can also be used for this step [@hanseler2015schedule]. These different information can be mixed in order to generate the matrix with a crucial time dependency [@ashok1996estimation]. Depending on the type of specific problem, a wide range of algorithms have been developed and tested, [@antoniou2014framework] provide an extensive literature review and propose a framework to compare them.\ The tactical level is the process that affects for each pedestrian their global route, depending on the OD matrix. In this step, we consider that all agents think in a graph-styled simplified network that represents the practical space and decision points. The classical formulation of this problem is the search of a Nash equilibrium [@wardrop1952road]. For vehicular simulation, the tactical level proceeds to the route choice of all agents [@bovy1990route]. Similar works have been developed for pedestrians [@hoogendoorn2004pedestrian]. However, in case of pedestrians we are of the view that it is behaviorally more consistent to consider this step as selection of decision points. The pedestrian choose their way through a succession of crucial decision points at a rather aggregate and abstract level. For example: which door, or coffee stand, etc. The process can be similar to the way finding algorithms for urban navigation that often use a graph representation of the network [@gaisbauer2008wayfinding]. Contrary to a continuous simulation of the trajectories where the space of possible solutions is also continuous, this level is characterized by discrete choices and so a finite number of possible configurations. The discrete choice theories have played a crucial role in the transportation research [@ben1985discrete] since they are related to different levels, such as modal choice [@hausman1978conditional]. Finally the proposed process strongly depends on a route cost function that should take into account the main phenomena, such as the travel time [@avineri2006impact] or even the perception of the facilities [@sisiopiku2003pedestrian].\ The operational level goes one step further in terms of precision. Using the high level paths generated from previous process, it computes trajectories of all agents. Variety of models have been developed in this context. The more efficient are often aggregate model, where agents are gathered in order to consider the whole crowd like a flow [@hughes2002continuum]. This kind of approach can also be solved with a Cell Transmission Model, that discretize the space into cells [@daganzo1994cell]. [@hanseler2014macroscopic] have developed the cell transmission based model for pedestrians. The main advantages of these approaches are a quick simulation/enumeration time and a relatively good aggregate level precision for real crowd despite overly simplified assumptions. But in our case, we are interested in precise results with information on each pedestrian. As all the other levels are individual level, we want to maintain the consistency and disaggregation at operation level as well. We are interested in a microscopic scale. Several models have been developed at micro-scale, such as the use of discrete choices to model the next step of pedestrians[@robin2009specification] or an analogy with physical forces called the social force model [@helbing1995social]. In these models, it is always possible to go deeper in description to have better precision. Some studies have developed even more complicated description of agents taking into account for example the social or natural effects such as the use of field of view [@turner2002encoding]. These kind of agent-based model are now efficient on complex networks [@batty2003agent] and bring depth to the analysis.\ Once the estimation of all trajectories have been done, our goal is to authenticate the previous departure time and to correct them if needed. In the literature, algorithms have been developed that include a choice in the departure time generation [@de2002real]. The clear advantage is that it coincides more easily with real traffic conditions. Other algorithms try to deal with a real time correction of OD matrices [@bierlaire2004efficient]. But the new problem we are facing is that previous state estimated by the simulation step doesn’t match any more with the new departure times. These simulations need to be recomputed. We now have a loop and need to find a convergence (Figure \[markov\]). This problem is known as the Dynamic Traffic Assignment [@peeta2001foundations]. Some recent works proposed processes in order to solve this kind of problem [@nagel2012agent].\ We propose a stochastic approach to solve this convergence problem. The output of each process will no longer be deterministic, but subject to probabilities as it has been proposed in [@daganzo1977stochastic]. The outputs we will now consider are probability distributions over the possible states space. In such case, a Bayesian resolution can be used [@maher1983inferences]. Specifically, we propose to consider the series of processes as a Markov Chain, using only the previous estimation of the state and giving back a new one, following a stochastic rule. The Monte Carlo algorithms therefore be used in order to identify the most probable states. One of such algorithm, Metropolis-Hasting algorithm [@hastings1970monte] has already be used for route choice set generation in a complex traffic network with high numbers of alternatives [@flotterod2011bayesian]. Methodology =========== Problem Statement ----------------- Given the infrastructure $I$, schedule of all modes of transportation $C$ and the location of different considered activities $A$, we are interested in estimating the state $S$ of the station that matches as good as possible to a set of incomplete observations $D$. A state $S$ contains the complete information of each pedestrian i.e. their activity chain $A^i$, their start and end location $(l_s^i,l_e^i)$, their starting time $t_{dep}^i$ and their exact trajectory $T^i:t\mapsto l$.\ $$(I,C,A,D) \mapsto S=(A^i,l_s^i,l_e^i,t_{dep}^i,T^i)_i$$ Indeed, if all these information were contained in $D$, the proposed framework is obsolete. However, in reality $D$ is not sufficient to directly extract $S$: $D\neq \subset S$. Moreover, we may have to confront cases where $D$ was collected in a different scenario than the one in which $I$, $C$ and $A$ are defined. This happens when $(I,C,A)$ represents a non-existent scenario (for example possible perturbations of the reality or extensive changes that may occurs in the future). $D$ always corresponds to a scenario that has already happened i.e. base case. In such case, $D$ is still bringing a necessary amount of information, but they will not be directly considered as constraints for $S$. A necessary level of abstraction have to be brought to these observations: for example if a pedestrian $j$ is observed at a certain point of the time and space (this information $D_j$ is contained in $D$) it will not necessary be the case in $S$, the information could be transformed into $\overline{D_j}=$“$j$ $b$”. In $S$, $\overline{D_j}$ can bring pedestrian $j$ to have another trajectory if bus $b$ has a different departure time, $D_j$ is not satisfied. By calling $I_D$, $C_D$ and $A_D$ the respective infrastructure, schedule and activity of the scenario where $D$ was observed, we can write: $$(I=I_D,C=C_D,A=A_D) \Rightarrow D \subset S(I,C,A,D)$$ \[ovD\] By calling $\overline{D}$ the abstract information of $D$. This set is defined, when $D\not\subset S(I,C,A,D)$, such as it verifies: $$\overline{D} \subset S(I_D,C_D,A_D,D)$$ $$\overline{D} \subset S(I,C,A,D)$$ In cases where $D\neq S$, it means that one or more estimated states may represent D. These estimated states will only differ on the part where we have the lack of information. Note that we are assuming that there always exists at least one state in the search space that can verify all our constraints. This assumption is reasonable in the case where the search space is well-defined and D has enough information. The goal of the simulation is to fill in the exact amount of information needed and thus choose one final state S\. We can’t assure that there will be a unique state to which the convergence can bring us. Since it would mean that we have perfectly described all human phenomena that come into effect in the station. In fact we only want a representative of what could happen in reality, just a consistent case that allows us to understand the main phenomena in the station. We will be able to converge to several different and completely consistent solutions. But if we don’t bring enough constraints, this space of possible final solutions will be oversized, we need to restrain it enough to have usable results. This is why constraints such as schedule dependence and behavioral models’ consistency will be added in these cases where we don’t have enough data. Inputs ------ Different kind of inputs will be considered. The four main ones are the infrastructure, schedule, activity list, and observed data: $(I, C,A, D)$. - Infrastructure $I$: Spatial description of the infrastructure. Mainly composed of a CAD design model of the studied station, available facilities, and the main entrances. - Schedule $C$: List of all transportation modes, with their arrival or departure time, location and capacity. - Considered activities $A$: Description of all activities available, inside as well as outside the station, for considered agent. It should contain the type, location and possibly the time at which it is available. In case of mobility hubs, the prime activity is to go from one mode to another, so in this paper we will model a unique activity for every pedestrian. However, the proposed methodology can easily be extended to include full activity chain modeling. - Observations $D$: These data can be of various form. The less precise are aggregated counts on different point of the station, for example the number of people entering/exiting it per unit minute of the scenario. More precise data can be incorporated if they are available, for example observation of the exact time each pedestrian entered the station (or cross a specific point); information on the origin and destination of each travel; or even some local trajectories observed within the field of view of cameras in the station directly. A detailed discussion on the types of data commonly available on pedestrians in public spaces can be found in [@Farooq2016]. The different behaviorial models used in the simulation are also inputs: different results can be obtained depending on the accuracy of each model and their consistency with reality. As for the observation $D$, the models can be considered as constraints. Indeed there are constraints on the kind of behavior agents can have. The final state $S$ will have to verify these constraints to be consistent. We can bring more constraints with more restrictive models: where possible agents movement are more precisely defined. Simulation Processes -------------------- Here we use the chosen behavioral models, and the inputs $(I,C,A)$ in order to simulate pedestrian agents moving in the station and obtain a description of state $S$. The three levels of simulation (strategic, tactical and operational) are respectively implemented with the activities generation, decision points choice, and assignment models. ### Activities Generation {#gene} The generation phase aims to estimate the demand. At the end of this process we obtain a part of $S$: the number of pedestrians, the activity chain of each one, their start and end locations, and their starting time: $$(A^i,l_s^i,l_e^i,t_{dep}^i)_i$$ Since we only consider one type of activity (work), the model we choose here for the generation is Location Choice Model (LCM) that assign a destination for each pedestrian. This model, coupled with an estimation of the occupancy for every transportation mode and a description of the variability upon time, is sufficient to generate the demand. Estimating demand of a new scenario exactly corresponds to the calibration of location choice model. Thanks to the information contained in $D$, our framework corrects the demand until it is consistent with all parts of the scenario by calibrating this model. We can then use it for other scenarios, where at least one of $(I,C,A)$ is changed. Such calibrated model contains exactly $\overline{D}$ (see Section \[ovD\]). The information in $D$ is absorbed in the form of a model to have the abstraction necessary to be generic for several different scenarios. ### Decision Points Choice The decision point level generates a global movement pattern for each pedestrian depending on their origin and destination. In this phase, the station is viewed as a simplified network representing all different paths. At each node or decision point, a pedestrian is confronted to a choice scenario. Pedestrian chooses one of the possible direction towards the destination. At this level there is no description of time, and the pedestrian is not considering other agents or obstacle. But several kind of information can be brought, from a simple estimation of the different transfer times on each link to real information of perception: signs, sized of corridor, light etc. In our case we are using a basic model for the sake of simplification in the simulation i.e. shortest path model, but random utility based choice models can be used. ### Assignment Finally the assignment uses all information generated in the previous phases to compute the exact time-dependent trajectory of each involved agent. It depends on their global routes defined by decision points; on their interactions with other agents and obstacles; and on different personal characteristics that may change their behavior in order to represent the diversity. Here we used the social force model [@helbing1995social] in the simulations. Convergence ----------- After the three previous processes have been executed, a state is obtained. But, like traditional simulations, the demand was generated before the state of the station was estimated. This demand could not have used some crucial information on the station’s load such as transfer times or occupancy, because they were not observable yet. However, this demand may be in total adequacy with the obtained results. The first step is to observe this adequacy or not and measure what is not coherent. Then this information can be used in a correction process that will correct the previous estimation of the demand, now that more information are available. This correction process close a loop that we can be represented as a Markov process. We will then use the Simulated Annealing algorithm, a special case of Metropolis-Hasting sampler [@ross2013], to make it converge to the desired state. ### Corrections {#twoscenarios} The correction process is required to correct the estimated demand. As we saw in Section \[gene\], a set of rules formulated in a model and applied in the generation step results in obtaining the estimated demand for a scenario. The correction process has two kind of possible actions that directly define the kind of simulation direction we are interested in: - Calibration of the demand.* First application of the simulation is to calibrate our model used in the generation step. This simulation is based on the available observation $D$ and generates the exact demand to satisfy it. It corresponds to the creation and calibration of $\overline{D}$ that absorbs all the information of $D$. - Simulation of unknown scenario. Second application supposes that the first one has already found the equilibrium point in order to calibrate the demand generation step. It means that $\overline{D}$ has been created. This second application uses it to simulate a new scenario without the availability of $D$. The correction process for the first application is used to correct the location choice model: based on the difference observed between the current estimated state and $D$, it changes its rules $\overline{D}$ to try a different search point. Concretely, in our case, the probabilities of LCM are changed. After the observations are made, a correction is chosen depending on the lack or excess of people going into each kind of location. This correction can be focused on a specific location trying to increase or reduce the number of agent interested on it, or can be a mix of different changes. At each application of the correction process, since the choice is random any correction can be applied, but the probabilities are made set such that the correction has better chance to correct the observed difference in a right manner.\ The correction process for the second application is simpler: the estimated state is analyzed and the abstract conditions of $\overline{D}$ (that are gathered in the generation step) are tested. If some are not satisfied, the behavior of corresponding agents are changed with a probability according to the correction they need. For example if some agents miss their bus or train (that they should take according to the generation model) their starting time is corrected. The same principle can be applied if certain occupancy of a transportation mode needs to be reached by adding or deleting agents in the simulation. ### Markov Process #### Construction of the chain. The correction we just defined closes the loop. When applied on an estimated state, it gives us a new potential state with some probability of being chosen. This loop can be considered as the transition of a Markov chain (see figure \[markov\]). In order to use the powerful properties of Markov chains, we have to prove that the one we defined is one. This is done here by proving the two properties: irreducibly and aperiodicity.\ These properties are easily proved in the case of chains applied in finite space of state. This is definitely not the case here: the probabilities in LCM, that define the current position of the state in search space, are continuous (from 0 to 1). The space is not finite, neither discrete. In the case of such continuous space of state, it is common to use distribution probability (on which the chain is applied) in order to find the same properties than discrete spaces. In our case, it is not possible since the transition probabilities are not obtained with a formal function that we can easily integrate or apply in a region of states. Our transition is a function we can compute on only one state at a time. And since it is a whole simulation, we can’t apply to a consequent number of state at each transition. \[consideration\] In order to prove Markov chain properties, we are using another kind of consideration i.e. the location choice model can be continuous, but it is always applied on a finite number of pedestrians. There are infinite scenarios and demand that can be generated thanks to LCM, but for any particular scenario $(I,C,A)$, there is a maximum number of pedestrians that can be generated. In such case, the proportions of LCM may be continuous, but since they will be applied on a finite number, their effect is discrete. More precisely, around each LCM configuration of parameters, there is a small interval in which all other configuration of parameters have the same effect in a scenario. All these set of parameters correspond, in fact, to the same state. Finally we find that there is a finite number of different LCM in the particular scenario we are considering for the simulation. We have to prove that there exists $N$ for which from any set of parameters $P$ we can reach any other one $P'$ in $N$ iterations with a non zero probability. A set has finite number of parameters in our LCM, so we can write $P=(p^j)_{j\in [1,J]}$, $P'=(p'^j)_{j\in [1,J]}$. In each iteration, at least one of the parameters is changed. The amplitude of this change has a maximum, let’s call it $A$. $p^j\in [0,1]$ can reach any other parameter $p'^j$ in $\|p^j-p'^j\|/A < 1/A$ steps. The probability to jump from $p^j$ to $p'^j$ in $1/A$ steps is not zero since there is a finite number of values that can be reached–we just need to reach a value close enough to $p'^j$. There is $J$ parameters to change, each has a non-zero probability to be changed in any other value in $1/A$ steps. Moreover, each has a non zero probability to be chosen and changed at each iteration. It means from any state $P$, we can reach any other $P'$ in $J\times 1/A$ step with a non zero probability: $$N=\frac{J}{A}$$ Corollary \[consideration\] ensures that two “very close” sets of parameters can have the same effect in the generation process for one particular scenario. In fact it ensures that around one state $P$ there is a small open set (not empty) of parameter configurations that define the same state. Since a change at any iteration have a maximum amplitude of $A$ and minumum of $0$, the change made to a parameter can be so small that the new configuration of parameters is still in the open set of the same states. For any state, at any iteration, there is a non zero probability that we stay in the same state. The periodicity of all states can’t be higher than 1. Our chain is aperiodic. ### Search Algorithm A Markov Chain Monte Carlo (MCMC) simulation process can be used to sample from the developed Markov chain. In particular Simulated Annealing algorithm is used to converge to an optimal state. Transition of the process is already defined, the algorithm need an objective function and a temperature to decide whether or not each new state will be kept. ### Objective Function The objective function drives the choice of search towards the optimal state. This state have to be consistent with all inputs we had: $(I,C,A,D)$ and the behavioral models. Even if we can integrate the behavioral models consistency in the objective function by implementing a rating system, we don’t have to in our case. This can be easily done in future works. For example, it is possible to integrate the comfort (or security) appreciation for each pedestrian following a behavioral model that measure the perceived comfort of everyone. Integrating it in the objective function would lead to states where people tend to choose the travel by maximizing their evaluation of comfort.\ The behavioral models are already used to generate the processes $I$ and $A$. Elements that the objective function should integrate to assure their impact on the simulation are $D$ and $C$: the conformity to external observation and the consistency with schedule, respectively. - Observations $D$: We use the comparison between these observation with the exact same information taken from the estimated state. $D$ is a set of values $(D_i)$ that correspond to a list of observation function $(O_i)$ applied on the real life station $S_{rl}$. These functions can be, for example, the number of pedestrians going through a particular door between two points in time. $$D=(D_i)=(O_i(S_{rl}))$$ We evaluate these functions on the current estimated state $S$ to obtain $(s_i)$ the values to be compared with $D$. A rate is settled using the residual sum of square: $$OF_1(S,D)=RSS((s_i),(O_i(S_{rl})))=\sum_i (O_i(S)-D_i)^2$$ - Schedule $C$: the coherence of schedule measures the embarking and disembarking pattern for each train or any other mode of transport. It uses the list of pedestrians $(p_i^{(m)})$ taking each mode $m$. This list is obtained from the state $S$ thanks to the list of origin and destination of each pedestrian (time and space are considered) and the list of arrival and departure of each mode (time and platform also) in $C$. We assign a pedestrian going to or coming from a platform to a consistent bus or train.\ From this list of pedestrians we compute the arrival pattern $f(p_i^{(m)})$ of each mode for the state $S$. This pattern is compared to our embarking and disembarking pattern model $C_m$ (see Figure \[UNLDM\]) and a rate is given with the residual sum of square to measure the consistency: $$OF_2(S,C)=\prod_m RSS(f(p_i^{(m)}),C_m)$$ Finally, we may be unable to assign some pedestrians to a mode of transportation (origin or departure) if they are created before a mode arrives in the station or if they arrive too late to take the mode corresponding to their platform. We strongly penalize states with these incoherent observations. By denoting $Y(S,C)$ the number of such incoherent pedestrians in state $S$ with the schedule $C$, we have: $$OF_3(S,C)=e^{-Y(S,C)}$$ The relative importance of the three functions can be settled with two parameters $\alpha$ and $\beta$. We obtain the objective function: $$OF(S,C,D)=OF_1(S,D) \quad OF_2(S,C)^\alpha \quad OF_3(S,C)^\beta$$ ![\[UNLDM\]Unloading model for trains: the bottleneck model is used due to the form of the connecting stairs between platforms and main hall.](goulotCurve.png "fig:") ![\[UNLDM\]Unloading model for trains: the bottleneck model is used due to the form of the connecting stairs between platforms and main hall.](goulotMM.png "fig:") Implementation ============== Since the general algorithm, various behavioral models, and types of data are separate entities, we implement them in a way that each component can be independently and easily plugged in or replaced. We use an object oriented paradigm to implement in Java programming language. The implementation is available upon direct request to the corresponding author. Figure \[imple\] shows the UML diagram of the framework. Different kind of scenarios and type of states can be plugged to the corresponding classes. The implemented code can be used in many different scenarios and is able to take various kind of models. Please also note that a commercial software called MassMotion by Oasis Software is used for running the Assignment process. ![ \[imple\]UML diagram of the framework.](UML) Case Study ========== As a case study we explored our framework on the Montréal Central Train Station. Here 14 tracks are exploited by several national and local railways companies. The station is also linked to two metro stations, 16 bus lines, encloses an active underground mall, and is directly connected to several buildings. It is an important part of the Montréal city centre since it is located downtown, and is a central part of the Montreal’s Underground City, the biggest pedestrian indoor network in the world. Presentation ------------ ### Simulation Setup We will study a fixed part of the space and time of the scenario i.e. we will model the main hall of the central station (see Figure \[station\]). Pedestrians will be able to enter and leave through different portals that model the entrance/exit of boundaries of the station. These portals are the different corridors (1 to 8) arriving into the main hall of the station, and also the stairs connecting the platforms just under the hall (RA to RG), where trains arrive. The time of day that we are interested in is when the station is the most crowded i.e. the peak period. Since the afternoon peak period is more spread, it is less intense (see Figure \[data\]). We have chosen the morning peak period. Agent based simulations are computationally very demanding and because we are running several simulations in a single iteration, we began with a short window of time (i.e. 15 minutes of highest demand) to minimize the computational time. So our simulation concerns the duration between 8:30am and 8:45am, the most crowded quarter, during which several trains arrive and leave the station. ![\[station\] Representation of the station. Usable space is in light blue and obstacles in dark blue. Portal with which people can enter and leave the simulation are in green.](stationNumbered) The schedule $C$ gathers all departure and arrival trains of a normal day, including their capacity information. During the time window we are studying, several of them are unloading and others are taking passengers for the suburb. ### Behavioral Models {#models} For this first simulation we selected basic behavioral models. The three levels have to be implemented with one model: strategical, tactical and operational. In the strategical level, we should model the activity chain. Because we are simulating the morning peak hour, we assume that the main purpose of the displacements is work. The only dimension to generate here is the location of this activity. That is why we use an activity location choice model. This is particularly consistent because the studied space (the hall of central station) is small and doesn’t host too many different activities. There are still some coffees and restaurants. In future simulations the model could integrate them and propose a full activity chain modeling.\ The operational level, where we model decision points, is also impacted by the size of the station i.e. when there are not too many different ways to go from one point to another, its importance is diminished. We choose the simplest model there is, the shortest path. It is still particularly consistent since everyone is going to work at that time, people may mainly choose their trajectory to go as fast as possible. Finally the operational level is very important in term of realism. We used the social force model that gives a good description of real behavior. The parameter values calibrated in [@sahaleh2012scenario] were used. ### Scenario Development As explained in Section \[twoscenarios\], two kinds of simulation are possible: one using observations $D$ on existing use so as to calibrate the demand generator model. The second kind of simulation use the calibrated model to estimate the station’s load in a scenario for which $D$ does not exist. For this case study we are executing both kind of simulations. First, thanks to the data we collected on the real scenario, we will calibrate the behavioral models. Then it will be used in a possible future scenario for which no data could be collected. Base Case Scenario ------------------ $I$, $C$ and $A$ are already detailed, as well as the behavioral models. Only $D$ is needed to launch the simulation. ### Inputs We gathered observations by installing magnetic sensors on all entrances of the station’s main hall i.e. portal 1-8. Unfortunately, logistical issues bared us to measure the flow at access points to platforms. We also did not have access to the occupancy data of trains. The commercial sensors were provided by the manufacturer, Eco-Counter. Due to the data collection rate of these sensors, pedestrian counts were only recoded for 15 minute intervals during one typical day. Note that the pedestrian loading at portals in the simulation was done at 1 minute interval. So for this purpose the initial departures at entrance portals 1-8 were assigned based on Poisson Arrival Process for every 15 minutes of sensor counts. The departures from portals connected to platform were based on the arrival times of the trains and unloading curve. ![\[data\]Data recorded on October 1, 2014, at portal 2. Pedestrians using the portal as entrance are in green and leaving through it are in blue.](data) ### Optimal Solution Search After 500 iterations, the simulation converged to an optimal solution. Figure \[c500\] shows values of the objective function for each iteration, and for the selected states. We can observe that there is a gradual and steady progression towards search regions with better values and thus the selected value constantly improved towards optimal solution. The convergence is very slow[—]{}it took several hundreds of iteration to obtain an acceptable result. The reason is that, for this particular simulation, we begin with a particularly incoherent set of parameters for the location choice model. The goal here is to show the robustness of the method i.e. that it converges, though slowly, to a proper solution. For sure, when the goal of a simulation is only to have consistent results, we can begin with a more coherent set of parameters, simply generated with the common sense of the analyst. Final solution is an estimation of the station’s load with the trajectory of all pedestrians. For illustration purposes, we can see a 3D representation of the pedestrian movement in Figure \[screen\]. ![\[c500\]Left: $log(OF(S_i))$ for the state generated at each iteration $i$. Right: $log(OF(S^))$, value of the objective function at step $i$ i.e. its value for the best known state.](curve1 "fig:") ![\[c500\]Left: $log(OF(S_i))$ for the state generated at each iteration $i$. Right: $log(OF(S^))$, value of the objective function at step $i$ i.e. its value for the best known state.](curve2 "fig:") ![\[screen\]3D representation of a simulation. Two trains just arrived. The disembarking passengers mix with a continuous flow of pedestrians crossing the station.](screen) ### Validation In order to validate the results, we need real observations that have not been used in the convergence process. The problem is that the lack of data is exactly what we try to solve here. So, instead, we used real train occupancy information in order to validate the coherence with the real life. Since people disembarking trains can take several other exits than through the main hall, we particularly compare the count of pedestrian leaving through the ones we considered. In the simulation, we found that an average of 760 people using these exits after the arrival of a train. In the real data we have an average of 850 people disembarking from the trains, which is higher. This difference can be explained because some exits from platform to the hall were not considered, so the flow is limited, in the simulation, to the principal exits only.\ In order to validate the convergence, we can also analyze at what point the observed demand is satisfied by the solution. Figure \[table\] shows fit of the optimal solution with the observed data. We can clearly see that most of conditions are satisfied with only one exception. According to the observation, more people should be leaving the hall through portal 4, but this error is less than 10 %.\ ![\[table\]Comparison between the real life observations ($D_i$) and the same observation on the final state ($O_i(S_f)$) for the inflow and outflow of all portals of the station. The percentage of error is written when it is not 0%.](table2) Future Scenario --------------- After calibrating the simulation, we used it in a future scenario for which we did not have any real observations, but the scenario close enough to base case. Thus the utilization of base case calibrated models was consistent. We simulated here the same station, with the exact same facilities and people using it, but with an increase in the population by 50%. This is a possible and very realistic scenario, if the infrastructure at station is not updated in the near future. ### Inputs The inputs are almost the same: $I$, $A$ and $C$ are unchanged, as well as the behavior models. Only $D$ is no more used. The LCM is now simply used instead of being calibrated. The total number of agents involved in the station is multiplied by 150%. ### Simulation Even if the LCM is now static, we still need to make the Markov Chain converge. The demand still has to find an equilibrium with the estimation of the station. For example, an estimation of the transfer time is used in the generation step so that, after using the LCM, a departure time is assigned for each pedestrian. This is after several iterations that these estimations are consistent with the scenario so that the demand is generated in a consistent way. ### Results From the converged state, we can extract information on pedestrian trajectories over space and time. Figure \[flowR\] shows the principal paths used by pedestrians during the simulated time window. We can see what parts of the station are overcrowded and may present a risk of traffic congestion. The results also provide information on each agent. For example, a criterion could be used to measure the safety or satisfaction of each agent. The general OD matrix over time in the station can also be obtained from the pedestrian trajectories in the simulation. We represented it in Figure \[flowL\]. Each strip represent a flow from an origin (to which it is attached) to a destination. Its thickness is proportional to the number of agent using it. We can identify that the major flow is from portal 2 to 3. The two most loaded trains are arriving in platform B and G. Passengers arriving with the first one are mainly going to portal 2 and 3, while those arriving with second are oriented to portal 5 and 7. Only one platform is considered as a destination by pedestrians i.e. platform E. This is consistent since it is from this platform that the only train leaving in our simulation window time departs.\ The comparison between base case and possible future scenario can bring a detailed picture of the evolution of station. Figure \[density\] shows the densities over time in both scenarios. Densities are represented according to the standard [@fluin1971] and IATA (International Air Transport Association) level of service mappings. We can clearly see how an augmentation of the population in the station does not linearly increase the measured densities. With the 50% augmentation, the presence of higher densities explodes. Indeed, when serious congestion appears, pedestrians get blocked and stay longer in the station. This leads to even more pedestrians in the station and so a higher danger of congestion–it a vicious circle phenomenon. Also we can identify some details of what parts could require some improvements. We can see in Figure \[congested\] the different intersections where high levels of congestion appear.\ ![\[flowR\] Densities of pedestrians on each path](flux) ![\[flowL\] Flow between each portal of the main hall.](circos) ![\[density\]Representation of densities over time in the station for base case (left) and future scenario (right).](densityLOS "fig:") ![\[density\]Representation of densities over time in the station for base case (left) and future scenario (right).](density150 "fig:") ![\[congested\]Spatial representation of densities for base case (right) and future scenario (left).](densities "fig:") ![\[congested\]Spatial representation of densities for base case (right) and future scenario (left).](densities150 "fig:") Discussion ---------- The first point that we would like to discuss is the 8.8% error with respect to outflow observations at portal 4 (see Figure \[table\]). This error means that our model did not manage to send enough people to this portal. The Location Choice Model is responsible for this assignment. The error corresponds to a default aspect of our model, which defined a type of attraction for each portal: city, metro, train, etc. The probability for pedestrians to choose one of these attractions was calibrated in the base scenario. Once an attraction was set for an agent, a destination was assigned by selecting the nearest portal that proposes this attraction. The lack of people going to portal 4 means that the attraction we assigned to this portal put it in competition with other portals that were surely closer to the major inflows. We can see in Figure \[flowL\] that pedestrians leaving through portal 4 were essentially coming from platform D. And the flow created by this platform is limited.\ The error means that the function of portal 4 was not properly assessed. In order to improve the model, we can use the random utility theory and define a utility function for the attractivity of each portal based on their attributes. We could also imagine to simply extend the location choice model by adding a type of attraction, just for portal 4. But we have to be careful with these options, since they mean more parameters to calibrate in the model. The search algorithm will be faced with a higher degree of complexity in search space to explore. More information will be needed for the algorithm to be able to determine an optimal solution.\ We can identify in Figure \[flowR\] another limitation of the implementation: some path used are not coherent with the reality. They cross an area of shops that is not attractive for pedestrians, in reality they try to avoid it and mostly take the wider corridor just next to the area. This difference between the real observation and the simulation is also coming from the model at tactical level i.e. the shortest path model. We observed that in the simulation, pedestrians are taking the path through shops because it is shorter and so it corresponds to the model. But in real life, the choice of path is more complex than a simple shortest path choice. When it comes to choosing between the two directions, people tend to take the corridor because it seems more attractive. Such phenomena in the choice are not described by the model. In future, we can suggest the use of a random utility based decision points model–especially the dynamic mixed logit model, which fits very well with the choice scenario. Such model could describe some human behavior such as the impact of the perceived environment in the choice, same person making successive decisions, and correlation between the decisions. People may tend to be more attracted by bigger corridors, shown by direction signs, and presenting less obstacles in the sides (such as tables or shops’ advertising).\ Conclusion ========== We presented an agent based microsimulation framework for pedestrian movement in moblity hubs and public spaces. The problem is formulated as a Markov chain of activity generation, decision points choice, and assignment processes. Thanks to behavioral considerations of the demand and dependence on public transit schedules, the resulting framework is truly dynamic and can fill the lack of complete observations. We propose MCMC process that converges to an optimal solution depending on the type of behvaioral models, infrastructure data, public transit schedules, and incomplete observed demand. As a result the framework is able to predict the activities, location, start time, duration, and detailed trajectory of individual pedestrian.\ A case study of Montréal Central Station has been implemented for the base case and a future scenario with demand augmentation of 50%. The validation of the base case shows a good fit. We also observed several differences between the result and the data in station. They were all explained by the choices of model: a too simplistic description of the infrastructure and of the possible activities in the station; a calibrated location choice model not perfectly adaptable; and a shortest path model at tactical level that needs to be more representative of behavior and dynamic conditions. These are the dimensions where improvements can be made to the current implementation of the case study. The general algorithm is computed in a way that these changes can be easily integrated.\ Finally, there is a great potential and applicability of the proposed microsimulation framework. Once behaviorally richer models are implemented, the simulation will be able to render how the station will be affected if specific changes are made to the design or schedule, with a dynamic demand that effectively reacts to these changes. For example a change in train’s departure time will force pedestrians to leave at a different time in order to have a coherent behavior in the simulation. If the demand were not dynamic, these pedestrians’ departure time could not be changed and we could observe absurd situations where the arrival time of the pedestrian is completely not coherent with his/her train. Such a framework will be very useful in the network-level optimization of the schedule for various modes of transportation, in order to have perfect connections between them following what the population needs, and avoiding high densities that could lead to unstable situations. Acknowledgements {#acknowledgements .unnumbered} ================ We would like to thank Natural Sciences and Engineering Research Council of Canada (NSERC) and Fonds de recherche du Québec Nature et technologies (FRQNT) for funding this research. We would also like to thank Société de transport de Montréal (STM), Agence métropolitaine de transport (AMT), Oasis Software, and Eco-Counter for providing us the critical support to make the research possible in this paper. [^1]: Laboratory of Innovations in Transportation (LITrans), Department of Civil, Geotechnical, and Mining Engineering, Polytechnique Montréal, Montréal, Canada, Email: [alexis.pibrac@polymtl.ca]{} [^2]: Laboratory of Innovations in Transportation (LITrans), Department of Civil, Geotechnical, and Mining Engineering, Polytechnique Montréal, Montréal, Canada, Email: [bilal.farooq@polymtl.ca]{} [^3]: An extended abstract appeared in IATBR 2015
--- abstract: 'We report the relative frequency stabilization of an intracavity frequency doubled singly resonant optical parametric oscillator on a Fabry-Perot étalon. The red/orange radiation produced by the frequency doubling of the intracavity resonant idler is stabilized using the Pound-Drever-Hall locking technique. The relative frequency noise of this orange light, when integrated from 1 Hz to 50 kHz, corresponds to a standard deviation of 700 Hz. The frequency noise of the pump laser is shown experimentally to be transferred to the non resonant signal beam.' address: 'Laboratoire Aimé Cotton, CNRS-Université Paris Sud 11, 91405 Orsay Cedex, France' author: - 'O. Mhibik, D. Pabœuf, C. Drag, and F. Bretenaker' title: 'Sub-kHz-level relative stabilization of an intracavity doubled continuous wave optical parametric oscillator using Pound-Drever-Hall scheme' --- [99]{} H. de Riedmatten, M. Afzelius, M. U. Staudt, C. Simon, and N. 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Musser,“Raman-excited spin coherences in nitrogen-vacancy color centers in diamond,” , 361-363 (2001). Introduction ============ The recent demonstration of solid-state quantum memories [@Riedmatten2008; @Hedges2010; @Clausen2011; @Saglamyurek2011] has raised the possibility to implement solid-state quantum repeaters in quantum key distribution systems. This is an important step towards the increase of the distance over which quantum communications can be achieved. Such solid-state quantum memories are based on the use of the long-lived quantum coherences which exist in rare earth ions cooled down at cryogenic temperatures. However, among the possible rare earths, those which exhibit the longest hyperfine coherence lifetimes (i. e., the longest possible storage times) and the longest optical coherence lifetimes (i. e., the longest available durations for pulse sequences aiming at their coherent manipulation) are europium and praseodymium. For example, Eu$^{3+}$ has been shown to exhibit a dephasing time of 2.6 ms for its optical transition at 580 nm, leading to optical homogeneous linewidths as narrow as 122 Hz [@Equall1994]. Similarly, Pr$^{3+}$ has been shown to exhibit homogeneous linewidths of the order of 1 kHz for its optical transition at 606 nm [@Equall1995]. These long-lived coherences also constitute an interesting resource for applications of rare earth ions to quantum computing [@Longdell2004-1; @Longdell2004-2; @Rippe2005]. However, up to now, all these demonstrations based on Pr$^{3+}$ and Eu$^{3+}$ ions use frequency stabilized dye lasers. Indeed, the coherent manipulation of optical coherences exhibiting lifetimes in the ms range relies on the use of a laser source with a linewidth in the kHz range over durations of the order or longer than 1 ms. In the orange part of the visible spectrum, dye lasers are the only available sources which meet this specification. This is a serious drawback to the practical implementation of quantum memories using these two ions because dye lasers are relatively cumbersome systems. Moreover, their stabilization at the kHz level requires the use of an intracavity modulator to compensate for the high frequency noise induced by the fast fluctuations of the dye jet thickness [@Helmcke1982; @Hough1984; @Kallenbach1989; @Julsgaard2007]. In recent years, we have investigated several all-solid-state alternatives to dye lasers [@Melkonian2007; @My2008-1; @My2008-2; @Paboeuf2011]. Among these different attempts, the intracavity frequency-doubled singly resonant optical parametric oscillator (SHG-SROPO) has given the most promising results [@My2008-2]. However, when dealing with the frequency noise of optical parametric oscillators, one must be aware of a fundamental difference between these sources and usual lasers based on stimulated emission: the parametric gain process is a coherent process. Consequently, due to energy conservation at the microscopic level, the frequency fluctuations of the pump laser are transferred to the signal and idler beams, and must be shared between these two beams. In the case of the SHG-SROPO, since the pump laser is a commercial frequency-doubled Nd:YVO$_4$ laser exhibiting frequency fluctuations in the MHz range, the pump frequency fluctuations must not contaminate the useful beam, i. e., the idler beam in the present implementation. This is why we use a singly resonant architecture in which the signal beam frequency is free to fluctuate and to take away the major part of the pump frequency noise. However, the question remains of how the useful beam can be stabilized down to the kHz level. The relatively large frequency noise of the green pump that we use forbids the implementation of a stabilization of the OPO cavity on the pump frequency, which is commonly implemented by making the pump resonate in the OPO cavity [@Breitenbach1995; @Schneider1997; @Strossner1999; @Petelski2001; @Kovalchuk2001; @Strossner2002; @Kovalchuk2005; @Lenhard2011]. Consequently, the frequency of the field resonant in a SROPO can be stabilized only by comparison with an optical or atomic reference and by taking effect on the cavity length. For example, different SROPOs have been stabilized at the MHz level by comparison with an interferometric [@Melkonian2007] or atomic [@Zaske2010] reference. More recently, stabilizations at the kHz level have been achieved by locking the SROPO either on the side [@Mhibik2010] or the peak [@Andrieux2011] of the transmission fringe of a Fabry-Perot étalon. However, in the case of lasers, it is well known that the best results are often achieved using the Pound-Drever-Hall (PDH) stabilization scheme [@Drever1983]. In the case of SROPOs, this scheme is interesting because it avoids the transfer of intensity to frequency noise inherent to the side of transmission fringe locking scheme [@Mhibik2010] and it does not require any modulation of the length of the reference étalon as in Ref. [@Andrieux2011]. This constitutes an important potential progress towards the use of non-tunable ultrastable reference resonators in order to improve the long-term absolute stability of the system [@Helmcke1987]. Consequently, the aim of the present paper is to implement a PDH locking scheme with a high finesse ($\sim 3000$) reference cavity in the case of our SHG-SROPO and to compare its performances with the side-of-fringe locking scheme implemented earlier with a much lower finesse ($\sim 100$) cavity [@Mhibik2010]. Furthermore, we use this opportunity to experimentally analyze how the pump noise is transferred to the non resonant wavelength. Experimental set-up =================== Our experimental setup is schematized in Fig.\[fig1\]. The OPO is pumped at 532 nm by a 10 W single-frequency Verdi laser and is based on a 30-mm long MgO-doped periodically poled stoichiometric lithium tantalate (PPSLT) crystal ($d_{\mathrm{eff}}\simeq11~\mathrm{pm/V}$) manufactured and coated by HC Photonics. This crystal contains a single grating with a period of 7.97 $\mu$m, designed to lead to quasi-phasematching conditions for an idler wavelength in the 1200-1400 nm range, leading to a signal wavelength in the 960-860 nm range. The crystal is anti-reflection coated for the pump, the signal, and the idler. The OPO cavity is a 1.15-m long ring cavity and consists in four mirrors. The two mirrors sandwiching the nonlinear crystal have a 150 mm radius of curvature. The two other mirrors are planar. This cavity is resonant for the idler wavelength only. The estimated waist of the idler beam at the middle of the PPSLT crystal is 37 $\mu$m. The pump beam is focused to a 53 $\mu$m waist inside the PPSLT crystal. All mirrors are designed to exhibit a reflectivity larger than 99.8 % between 1.2 $\mu$m and 1.4 $\mu$m and a transmission larger than 95 % at 532 nm and between 850 nm and 950 nm. ![Experimental set-up. PZT: Piezoelectric transducer; HVA: High-Voltage Amplifier; DM: Dichroic Mirror; HWP: Half Wave Plate; LO: Local Oscillator; f$_i$, $1\le i\le 4$: Focusing lenses; EOM: Electro Optic Modulator; FC: Fiber Coupler; OI: Optical Isolator; QWP: Quarter Wave Plate; PBS: Polarization Beamsplitter Cube; FP: Fabry-Perot cavity; FPD: Fast Photodiode; PD: Photodiode.[]{data-label="fig1"}](setupV4.eps){width="60.00000%"} In order to generate the second harmonic of the resonating idler, a 25-mm long BBO crystal is inserted between the two plane mirrors, i. e., at the second waist of the cavity. It is anti-reflection coated for the idler and the second harmonic wavelengths. A 1.5-mm thick solid étalon with a finesse $F=3$ is also inserted in this leg of the cavity. The transmission of the plane mirrors is larger than 95 % for the orange radiation. In these conditions and with the PPSLT crystal heated at $T=103\;^{\circ}\mathrm{C}$, the system oscillates at an idler wavelength of 1208 nm with a pump threshold equal to 800 mW. At a pump power equal to 3.4 W, the SHG-SROPO emits a single-frequency orange beam at 604 nm with a 30 mW output power. The frequency stabilization loop that we describe here is also schematized in Fig. \[fig1\]. It is based on a 11.5-cm-long Fabry-Perot cavity (free spectral range equal to 1.3 GHz) with two mirrors of radii of curvature equal to 50 cm and with a finesse equal to 3000. The orange beam, which is spatially filtered and carried by a single-mode fiber, is matched to the TEM$_{00}$ Gaussian mode of this reference cavity, leading to a 78% contrast for the reflection dip at resonance. In order to implement the PDH locking scheme, the phase of the orange beam is modulated at 25 MHz using a resonant electro-optic modulator (New Focus model 4001). The beam reflected from the reference cavity is detected on a fast photodiode (EOT model ET-2030A) and demodulated using a frequency mixer (Mini-Circuits model ZFM-3). When the phase of the demodulation is correctly chosen, one obtains the typical PDH signal [@Drever1983] of Fig. \[fig2\](a) when the reference cavity length is scanned thanks to a piezoelectric transducer carrying one of its mirrors. The corresponding intensity transmitted by the reference cavity is reproduced in Fig. \[fig2\](b). The two sidebands, separated by the 25 MHz modulation frequency from the carrier, are clearly seen. ![\[signals\] (a): Pound-Drever-Hall signal. (b): Signal transmitted by the reference cavity, exhibiting the carrier frequency and the two side bands. Both signals are recorded while the length of the reference cavity is scanned.[]{data-label="fig2"}](signals.eps){width="65.00000%"} Experimental results and discussion =================================== Before describing the results obtained when the frequency of the second harmonic of the idler is locked, let us first discuss the free-running behavior of this frequency. Since the parametric gain is a coherent process, it is worth measuring the frequency noise spectrum of the pump laser. To perform this measurement, we send a small part of the pump beam to a confocal FP cavity with a 750 MHz free spectral range and a finesse equal to 16 at 532 nm. We manually maintain the cavity length so that the Verdi laser frequency is on the side of the transmission peak of the cavity. Then the average transmission of the cavity is one half of its transmission at resonance. In these conditions, we can record the intensity transmitted through the cavity during 1 s, which corresponds to the evolution of the frequency of the Verdi laser during 1 s. The Fourier transform of this signal gives the power spectral density (PSD) of the frequency noise [@Wassen1990] of the Verdi laser which is reproduced in Fig. \[fig3\](a). ![\[comparaison\] Spectrum of the relative frequency noise of (a) the pump laser and (b) the free-running SHG-SROPO.[]{data-label="fig3"}](Figure3-2-bon.eps){width="99.00000%"} The same technique is used to measure the frequency noise of the free running SHG-SROPO. A small part of the orange beam is sent to the same FP cavity, whose finesse is equal to 60 for this orange wavelength. The corresponding frequency noise spectrum is reproduced in Fig. \[fig3\](b). One can notice that the frequency noise for the orange beam is larger than for the pump, and is particularly large below 1 kHz. Notice however that for free-running lasers, when the frequency noise is large, this measurement technique cannot be considered to lead to a quantitative measurement but more to a qualitative indication of the noise bandwidth. Notice also that the noise floors at high frequency (for example above 2 kHz in Fig.\[fig3\](b)) correspond to the measurement noise and that the OPO frequency noise is actually below this floor. To reduce this noise, we use the servo-loop schematized in Fig. \[fig1\]. The PDH signal of Fig. \[fig2\](a) is filtered by a first order low-pass filter with a bandwidth of 70 kHz before being amplified using a variable gain amplifier based on an Analog Device OP-27 operational amplifier (gain bandwidth equal to 8 MHz). It is then filtered by an adjustable proportional-integrator (PI) filter (New Focus model LB1005) before being amplified by the high voltage amplifier HVA (Piezomechanik model SVR150-3) and applied to a piezoelectric transducer (Piezomechanik model PSt 150/10x10/2) carrying mirror M$_3$ of the cavity. This transducer has been specially chosen because its resonance frequencies lie above 20 kHz. We experimentally checked using an interferometer that the combination of this HVA and this transducer with the mirror attached on it behaves like a first-order low-pass filter with a 3-dB bandwidth equal to 300 Hz. ![Spectrum of the relative frequency noise of the servo-locked SHG-SROPO using the PDH technique. Notice the oscillation peaks between 10 kHz and 20 kHz.[]{data-label="fig4"}](instability.eps){width="70.00000%"} With the corner frequency of the proportional integrator filter tuned at 10 kHz, we obtain the frequency noise spectrum of Fig. \[fig4\] for the orange beam when the OPO is locked. In these conditions, we expect the 0 dB gain of the open-loop transfer function to lie about 5 kHz. By comparison with the free-running spectrum of Fig. \[fig3\](b), we can indeed see that the low-frequency noise has been reduced by many orders of magnitude. However, we also notice the appearance of some new peaks between 10 kHz and 20 kHz. These noise peaks correspond to the onset of oscillations in our servo-loop. These oscillations come from the phase shift accumulated through the low-pass filter, the amplifier, the proportional-integral filter, the high voltage amplifier, and the piezoelectric transducer, probably combined with a residual resonance of the piezoelectric transducer, which lead to the occurrence of an overall $\pi$ phase shift at these frequencies. In order to get rid of this spurious oscillation, we introduced a phase-lead compensation network in the servo-loop filter. This circuit adds a $\pi/5$ phase advance at 10 kHz and induces a 15 dB gain reduction at low frequencies. It thus permits to increase the phase margin for frequencies close to 10 kHz and allows one to increase the proportional gain of the proportional integral filter by 5 dB and the corner frequency of this filter from 10 to 100 kHz while keeping the loop stable. ![Same as Fig. \[fig4\] with a phase-lead compensator added in the loop.[]{data-label="fig5"}](PDHbest.eps){width="70.00000%"} In these conditions, the OPO remains locked to the reference étalon during several tens of minutes. One can even scan the resonance frequency of the reference étalon by several tens of MHz by tuning its length using a piezoelectric transducer carrying one of its mirrors while keeping the SHG-SROPO locked to this reference cavity. Analysis of the servo-loop signal leads to the relative frequency noise (i. e., frequency deviation with respect to the cavity resonance frequency) spectrum of Fig. \[fig5\]. By comparison with Fig. \[fig4\], one can clearly see that the spurious resonances close to 10 kHz have disappeared. The noise is almost flat below 10 Hz$^2$/Hz, except for stronger noise components located between 100 and 1000 Hz. These noise peaks are probably due to mechanical resonances in the OPO mirror mounts, some residual oscillations of the oven containing the nonlinear crystal, and some harmonics of the 50 Hz mains frequency. When one integrates the power spectral density of the relative frequency fluctuations of the SHG-SROPO of Fig. \[fig5\] from 1 Hz to 50 kHz, one obtains a rms value of 700 Hz. This corresponds to an improvement by a factor of 6 with respect to the results of ref. [@Mhibik2010]. It is also worth noticing that this corresponds to a rms frequency noise of 350 Hz for the idler frequency, which is the one that is actually resonating in the OPO cavity. ![\[nonresonate\] Spectrum of the relative frequency noise of the non-resonating signal beam while the servo-loop control of the idler frequency is on.[]{data-label="fig6"}](signalnr.eps){width="70.00000%"} This experiment also allowed us to clearly illustrate the transfer of frequency noise from the pump beam to the non-resonating signal beam. Indeed, comparison of the spectra of Figs. \[fig3\](a) and \[fig5\] shows that the noise of the locked OPO is lower than the pump noise by many orders of magnitude (up to 10 at low frequencies). Since the parametric process must respect the conservation of energy $\omega_{\mathrm{p}}=\omega_{\mathrm{s}}+\omega_{\mathrm{i}}$, where $\omega_{\mathrm{p}}$, $\omega_{\mathrm{s}}$, and $\omega_{\mathrm{i}}$ are the angular frequencies of the pump, signal, and idler beam, respectively, this means that the pump frequency noise which is absent from the idler beam must be transferred to the signal beam. This is evidenced in Fig. \[fig6\], which has been obtained by analyzing the non-resonating signal beam using a confocal cavity of free spectral range equal to 1 GHz with a finesse equal to 110. By comparison with the spectrum of Fig. \[fig3\](a), one can clearly see that the pump frequency noise, which is restrained from propagating to the idler beam by the servo-loop, is faithfully transferred to the frequency of the signal beam. Actually, the similarity of these two spectra (Fig. \[fig3\](a) and Fig. \[fig6\]) is a striking illustration of the conservation of the pump photon energy in the down conversion process. Moreover, the comparison of the spectra of Fig. \[fig3\](a) and Fig. \[fig5\], which present a huge difference, shows how the control of the idler frequency permits us to allow the transfer of the pump frequency noise only in the direction we are interested in (to the signal and not to the idler). All these features illustrate the power of the SROPO concept with respect to the DROPO concept in order to obtain an output beam with a frequency noise much lower than the pump laser and to allow the use of a relatively noisy pump laser even for applications requiring ultrahigh spectral purity. Conclusion ========== In conclusion, we have been able to stabilize the frequency of an SHG-SROPO to levels lower than 1 kHz rms relative frequency noise. This has been made possible by using the Pound-Drever-Hall scheme to lock the OPO frequency to a relatively high finesse (3000) reference cavity. The pump laser noise has been experimentally shown to be almost completely transferred to the frequency of the non-resonating beam, i. e., the signal beam in the present case. This opens the way to the use of an all solid-state OPO source in rare-earth based quantum memories. Moreover, this source could also be very useful for experiments using spin coherences in nitrogen vacancy centers in diamond [@Hemmer2001]. Finally, such a source could be modified in order to be usable for metrology and high-resolution spectroscopy experiments in the visible, provided the reference étalon was made more stable on a longer time scale [@Helmcke1987]. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by the Triangle de la Physique and the Agence Nationale de la Recherche.
--- abstract: 'We generalize the stochastic block model to the important case in which edges are annotated with weights drawn from an exponential family distribution. This generalization introduces several technical difficulties for model estimation, which we solve using a Bayesian approach. We introduce a variational algorithm that efficiently approximates the model’s posterior distribution for dense graphs. In specific numerical experiments on edge-weighted networks, this weighted stochastic block model outperforms the common approach of first applying a single threshold to all weights and then applying the classic stochastic block model, which can obscure latent block structure in networks. This model will enable the recovery of latent structure in a broader range of network data than was previously possible.' bibliography: - 'WSBM\_05.bib' --- Introduction ============ In social and biological networks, vertices often play distinct functional roles in the large-scale structure of the graph. The automatic detection of these latent roles, by identifying the induced “community” or block structures from connectivity data alone, is a fundamental problem in network analysis and many approaches have been proposed [@fortunato:2010; @porter_communities_2009]. The stochastic block model (SBM) is a popular generative model that solves this problem in an unsupervised fashion [@holland:laskey:leinhardt:1983; @wang:wong:1987]. In its classic form, the SBM is a probabilistic model of pairwise interactions among $n$ vertices. Each vertex belongs to one of $k$ latent groups, and each undirected edge exists or does not with a probability that depends only on the block memberships of the connecting vertices. The model is thus defined by a vector $z$ containing the block assignment of each vertex and a $k\times k$ matrix $p$, where $p_{uv}$ gives the probability that a vertex of block $u$ connects to a vertex of block $v$. This model can capture a wide variety of large-scale organizational patterns of network connectivity, depending on the choices of $p$ and $z$. If $p$’s diagonal elements are greater than its off-diagonal elements, the block structure is assortative, with communities exhibiting greater edge densities within than between them, as is often found in social networks. Other choices of $p$ can generate hierarchical, multi-partite, or core-periphery patterns, among others. This flexibility, and the principled probabilistic statements it produces, has made the SBM a popular tool for unsupervised network analysis, in which we seek to infer the latent block labels from the observed graph structure alone. There is broad interest in machine learning, physics, and computational social science to develop and apply generalizations of the classic SBM. Generalizations have been made to allow degree heterogeneity within blocks [@karrer_stochastic_2011], probabilistic or mixed block membership [@airoldi_mixed_2008; @ball:karrer:newman:2011], infinite number of blocks [@kemp_irm_2006], or hierarchical (nested) relationships among blocks [@clauset:moore:newman:2008]. Several efficient techniques exist for estimating latent block structures from data. Of particular relevance to our weighted generalization of the SBM are the variational algorithms, both Bayesian and frequentist. Scalability is typically achieved by constraining the parameter space or using modern optimization techniques. Examples include variational expectation-maximization (EM) for the classic SBM [@daudin_mixture_2008; @park_dynamic_2010], variational Bayes EM for a restricted, two-parameter $p$ matrix [@hofman_bayesian_2008], nested variational EM for the classic mixed membership SBM [@airoldi_mixed_2008], and stochastic variational inference for assortative mixed membership SBM [@gopalan_scalable_2012]. In most of these efforts, the SBM is restricted to binary or Bernoulli networks, in which edges are unweighted. The one exception has been block models with Poisson distributed edge weights [@mariadassou_variationalweight_2010; @karrer_stochastic_2011; @ball:karrer:newman:2011], which can be fitted to multigraphs. In practice, however, most binary networks are produced after applying a threshold to a weighted relationship [@thomas_valued_2011], and this practice clearly destroys potentially valuable information. To apply the SBM on weighted data without thresholding, we introduce a generalization of the SBM to the important case in which edges are annotated with weights drawn from an exponential family distribution. This weighted stochastic block model (WSBM) includes as special cases most standard distributional forms, and thus allows us to use weighted relations directly in recovering latent block structure, preventing the information loss caused by thresholding. Handling these general weight distributions presents several technical difficulties for model estimation, which we solve using a Bayesian approach. We first give the WSBM’s form and derive a variational Bayes algorithm for fitting to dense graphs. We then present synthetic examples that illustrate the type of behavior the WSBM captures that is overlooked by thresholding. We close with a brief discussion of extensions of the model. Weighted Stochastic Block Models ================================ The weighted stochastic block model is a generative model for weighted pairwise interactions among $n$ vertices, and is composed of an exponential family distribution ${\mathcal{F}}$ and a block structure ${\mathcal{R}}$. The block structure defines a set of vertex labels, denoted $z = \{z_1,\ldots,z_n\}$ where $z_i \in K = \{1,\ldots,k\}$. The block structure $\mathcal{R}$ defines a partition on the edges into $R$ disjoint bundles, one for each pair of blocks. Edges weights in some bundle are modeled by a distribution in ${\mathcal{F}}$, parameterized by $\theta_r \in \theta = \{\theta_1, \ldots, \theta_R\}$. That is, each bundle has its own set of distribution parameters. The choice of ${\mathcal{R}}$ determines the large-scale structure of the network, just as $p$ and $z$ do for the classic SBM. When ${\mathcal{F}}$ is a Bernoulli trial, we cover this classic case. Although constraining ${\mathcal{R}}$, or the variation of its parameters across edge bundles, can be used to create specific types of large-scale structure, here we focus on the general case of blocks with independent parameters. In principle, the form of ${\mathcal{R}}$ could be learned directly from data, but we do not explore this topic. We denote a WSBM with edge distribution family ${\mathcal{F}}$ and block structure ${\mathcal{R}}$ by ${\mathcal{M}_{{\mathcal{F}},{\mathcal{R}}}}$, whose parameters are the vertex labels $z$ and the matrix of edge bundle parameters $\theta$. The likelihood of observing a graph $A$, given distribution $f \in {\mathcal{F}}$, is then $$\Pr(A \, \| \, z,\theta, {\mathcal{M}_{{\mathcal{F}},{\mathcal{R}}}}) = \prod_{i<j} f(A_{i,j} \, \| \, \theta_{{\mathcal{R}}(z_i,z_j)}) \enspace .$$ Restricting ${\mathcal{F}}$ to exponential family distribution makes the mathematics tractable while covering a broad range of models of edge weights, including many common distributions produced by classic stochastic processes. A distribution $f$ belongs to an exponential family ${\mathcal{F}}$ if it can be written as $$f(x\,\|\,\phi) = h(x) \exp\left( T(x) \cdot \eta(\phi) \right) \text{ for } x \in \mathcal{X} \enspace$$ where $h$, $T$, $\eta$ are fixed mappings, $\phi$ is the distribution’s parameter, and $\mathcal{X}$ is the distribution’s support. Under these assumptions, the log-likelihood becomes $$\mathcal{L} = \sum_{i < j} \log h(A_{i,j}) + \sum_{r = 1}^R T_r \cdot \eta(\theta_r) \enspace$$ where $T_r = \sum_{i,j : {\mathcal{R}}(z_i,z_j) = r} T(A_{i,j})$ is the sufficient statistic for the weights in edge bundle $r$. For some choices of ${\mathcal{M}_{{\mathcal{F}},{\mathcal{R}}}}$, the likelihood function contains degeneracies that prevent the direct estimation of parameters $z$ and $\theta$. For instance, when weights are real-valued and ${\mathcal{F}}$ is a Normal distribution. An edge bundle with all-equal weights will have zero variance, which creates a degeneracy in the likelihood calculation. Another technical problem is that non-edges in a sparse graph (a zero in the adjacency matrix) may represent a pair of non-interacting vertices, an interaction with zero weight, or an interaction we have not yet observed. The classic SBM does not exhibit these problems because edge weights are Bernoulli random variables, whose sufficient statistics are always well defined. To regularize the degeneracy problem, we take a Bayesian approach and assign an appropriate prior distribution $\pi$ to our parameters $\theta$. Now, the posterior distribution $\pi^{}$ will exhibit no degeneracies and estimation can proceed smoothly. Estimating the posterior distribution $\pi^{}(z,\theta\,\|\,A)$ given the observed edge weights $A$ and prior $\pi$ is generally difficult, and so we approximate $\pi^{}$ by a factorizable distribution $q(z,\theta) = q(z)q(\theta)$. How we estimate $\pi^{}$ also depends on whether the graph $A$ is dense or sparse, and our interpretation of non-edges. Here, we present the solution for dense graphs. In a separate paper, we will present a belief propagation algorithm for sparse graphs that correctly handles non-edges. Variational Bayes ================= For a dense graph, we construct a variational Bayes (VB) expectation-maximization algorithm to estimate $\pi^{}$. We approximate the posterior distribution $\pi^{}(z,\theta\|A)$ by a product of marginals $q(z,\theta) = \prod_i q_i(z_i)\prod_r q(\theta_r)$. We then select $q$ by minimizing the Kullback-Leibler (KL) divergence between our approximation and the posterior $D_{\text{KL}}(q\, \|\| \,\pi^{})$. It can be shown that $$\log \Pr(A\, \|\,{\mathcal{M}_{{\mathcal{F}},{\mathcal{R}}}}) = {\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}(q) + D_{\text{KL}}\left( q \,\|\|\, \pi^{}\right) \enspace ,$$ where ${\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}(q)$ is a functional lower bound on the constant $\log \Pr(A \ \| \ {\mathcal{M}_{{\mathcal{F}},{\mathcal{R}}}})$, calculated as $${\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}(q) = \mathbb{E}_{q}\left(\mathcal{L}\right) + \mathbb{E}_{q}\!\left(\log \frac{\pi( z , \theta)}{q( z , \theta)}\right) \enspace .$$ The first term is the expected log-likelihood under the approximation $q$ and the second term is the KL-divergence of the approximation $q$ from the prior $\pi$. As the likelihood $\log \Pr(A\, \|\, {\mathcal{M}_{{\mathcal{F}},{\mathcal{R}}}})$ is constant, minimizing the KL divergence $D_{\text{KL}}(q\, \|\| \,\pi^{})$ is equivalent to maximizing ${\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}(q)$. To maximize ${\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}$, we maximize the expected log-likelihood of the data and weakly constrain the approximation to be close to the prior. This regularizer prevents over fitting and eliminates the aforementioned likelihood degeneracies. In practice, the first term overwhelms the second term given sufficient data. #### Conjugate priors. For mathematical convenience, we restrict the prior $\pi$ to a product of parameterized conjugate distributions. The conjugate prior for the parameter $\theta$ of an exponential family has the form $$\pi(\theta) = Z^{-1}(\tau)\exp\left(\tau \cdot \eta(\theta)\right) \enspace ,$$ where $\tau$ parameterizes the prior and $Z(\tau)$ is a normalizing constant. When we update the prior based on the observed weights in a given edge bundle $r$, the posterior’s parameter becomes $\tau^ = \tau + T_r$, and $\tau$ can be viewed as a set of pseudo-observations. This prevents the posterior from becoming degenerate since every edge bundle, no matter how small or uniform, produces a parameter estimate. The conjugate prior for a vertex label $z$ is a categorical distribution with parameter $\mu \in \mathbb{R}^k$, where $\mu_i(\kappa)$ is the probability that node $i$ belongs to group $\kappa$. We fit $\mu_i$ directly, with a flat prior $\mu_0(\kappa) = 1/k$. The form of our prior is thus $$\pi(z, \theta) = \prod_i \mu_0(z_i) \prod_{r} Z^{-1}(\tau_0)\exp\left(\tau_0 \cdot \eta(\theta_r)\right) \enspace ,$$ where $\mu_0$, $\tau_0$ are the parameters for the priors $\pi_i$, $\pi_r$. With conjugate priors for $\pi$, our approximation $q$ takes the form $$q(z, \theta) = \prod_i \mu_{i}(z_i) \prod_{r} Z^{-1}(\tau_r)\exp\left(\tau_r \cdot \eta(\theta_r)\right)\enspace .$$ Now, maximizing ${\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}$ is equivalent to maximizing ${\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}$ over $q$’s parameters $\mu_i$, $\tau_r$. #### Optimizing ${\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}$. These choices of $\pi$ and $q$ yield $$\begin{aligned} {\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}&= \sum_{i,j} \log h(A_{i,j}) + \sum_r \left(\left\langle T \right\rangle_r + \tau_0 - \tau_r \right) \cdot \left\langle \eta \right\rangle_r \\ &\quad + \sum_r \log \frac{Z(\tau_r)}{Z(\tau_0)} + \sum_i \sum_{z_i} \mu_i (z_i) \log \frac{\mu_0 (z_i)}{\mu_i (z_i)} \enspace , \end{aligned}$$ where $\left\langle T \right\rangle_r$, $\left\langle \eta \right\rangle_r$ are expectations of $T_r$, $\eta_r$ under the approximation $q$; for exponential families they are, $$\begin{aligned} \left\langle T \right\rangle_r & := \sum_{i,j} \sum_{R(z_i,z_j) = r} \mu_i(z_i) \, \mu_j(z_j) \, T(A_{i,j}) \nonumber \\ \left\langle \eta \right\rangle_r & := \frac{\partial \log Z(\tau_r)}{\partial \tau_r} \enspace . \nonumber\end{aligned}$$ To optimize ${\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}$ we take derivatives with respect to $q$’s parameters $\mu$, $\tau$ and set them to zero. We iteratively solve for the maximum by updating $\mu$ and $\tau$ independently. For $\tau$, this yields $$\begin{aligned} \frac{\partial {\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}}{\partial \tau_r} &= \left[\left\langle T \right\rangle_r + \tau_0 - \tau_r\right] \frac{\partial \left\langle \eta \right\rangle_r}{\partial \tau_r} - \left\langle \eta \right\rangle_r + \frac{\partial \log Z_r}{\partial \tau_r} \\ &\propto \left\langle T \right\rangle_r + \tau_0 - \tau_r \enspace ,\end{aligned}$$ and the update equation for each edge-bundle parameter is $\tau_r = \tau_0+ \left\langle T \right\rangle_r $. For $\mu$, we use Lagrange multipliers $\lambda_i$ to enforce $\sum_{z} \mu_i(z) = 1$. Setting the derivative of ${\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}$ with respect to $\mu_i$ equal to $\lambda_i$ yields $$\frac{\partial {\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}}{\partial \mu_i(z)} = \sum_r \left[ \frac{\partial \left\langle T \right\rangle_r}{\partial \mu_i(z)} \cdot \left\langle \eta \right\rangle_r \right] - \log \mu_i(z) = \lambda_i \enspace ,$$ where $$\frac{\partial \left\langle T \right\rangle_r}{\partial \mu_i(z)} := \sum_{z' : R(z,z') = r} \sum_{j \neq i} T(A_{i,j}) \mu_j(z') \enspace .$$ Solving for $\mu_i(z)$ produces the update equation $$\mu_i(z) \propto \exp\! \left( \sum_r \frac{\partial \left\langle T \right\rangle_r}{\partial \mu_i(z)} \cdot \left\langle \eta \right\rangle_r \right) \enspace ,$$ where each $\mu_i$ is normalized to a probability distribution. To calculate the $\mu_i$ values, we iteratively update each $\mu_i$ from some initial guess until convergence to within some tolerance. Data $A$, Model ${\mathcal{M}_{{\mathcal{F}},{\mathcal{R}}}}$ Initialize $\mu$ Set $\left\langle T \right\rangle_r := \sum_{i,j} \sum_{\mathcal{R}(z_i,z_j) = r} \mu_i(z_i) \mu_j(z_j) T(A_{i,j})$ Set $\tau_r := \tau_0 + \left\langle T \right\rangle_r$ Set $\left\langle \eta \right\rangle_r := \left. \frac{\partial}{\partial \tau} \log Z(\tau) \right\|_{\tau=\tau_r}$ $\frac{\partial \left\langle T \right\rangle_r}{\partial \mu_i(z)} := \sum_{\mathcal{R}(z,z') = r} \sum_{j \neq i} T(A_{i,j}) \mu_j(z') $ $ \mu_i(z) \propto \exp \left( \sum_r \frac{\partial \left\langle T \right\rangle_r}{\partial \mu_i(z)} \cdot \left\langle \eta \right\rangle_r \right) $ $\mu,\tau$ Algorithm \[alg:VB\] gives pseudocode for the full variational Bayes algorithm, which alternates between updating the edge-bundle parameters and the vertex label parameters using the update equations derived above. Because every pairwise interaction contributes to the estimation of some parameter, the algorithm takes $O(n^2)$ time, assuming fast convergence on $\theta$ and $\mu$. Like all VB approaches, only convergence to a local optima of ${\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}$ is guaranteed. In practical contexts, multiple trials with a variety of initial conditions are used, and the best overall model selected. Model Selection =============== An important intermediate step toward applying the WSBM to some graph is the selection of a class of distributions ${\mathcal{F}}$ or the number of blocks $k$. Any of a number of principled approaches could be employed, including maximum likelihood, possibly with cross-validation [@airoldi_mixed_2008], Bayes factors [@hofman_bayesian_2008], approximations thereof [@mariadassou_variationalweight_2010; @daudin_mixture_2008], or minimum description length [@peixoto_2012_parsimonious]. In our experiments below, we use Bayes factors, which assume a uniform prior and are equivalent to selecting the model with the largest model-likelihood, $$\log B(\mathcal{M}_1,\mathcal{M}_2) = \log\frac{\Pr(A \,\|\, \mathcal{M}_1)}{\Pr(A \,\|\, \mathcal{M}_2)} \approx {{\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}}_1 - {{\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}}_2 \enspace ,$$ where we approximate $\log\Pr(A \,\|\, {\mathcal{M}_{{\mathcal{F}},{\mathcal{R}}}})$ with ${\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}$. Although Bayes factors assign a uniform prior on a set of nested models, they have a built-in penalty for complex models. Recall that ${\mathcal{G}_{{\mathcal{F}},{\mathcal{R}}}}$ is penalized for large divergence from the prior and since the vertex-label prior is uniform on all $k$ groups, there is a penalty if an increase in $k$ does not sufficiently reduce the entropy or correspondingly increase the expected log-likelihood. Experimental results ==================== We compare the WSBM against several alternative methods for recovering latent block structure. Our goal is to demonstrate that the classic SBM after applying a single threshold to all edge weights may miss important structure and that the WSBM can be used to explicitly evaluate the accuracy of inferring latent block via thresholding. We also include k-means clustering and hierarchical clustering to show that the weighted behavior the WSBM captures is different. To demonstrate how the WSBM can find structure other methods may miss, we use synthetically generated dense graphs with $n$ vertices divided into $k^{}= 5$ heterogenous blocks; the weights of each edge bundle are Normally distributed with bundle-specific parameters (see Fig. \[fig:1\]). This $5$-block model is a weighted variation of Newman’s four-group test for unweighted graphs [@newman_finding_2004]. We then vary three model parameters—graph size $n$, variance of the edge weight distributions, and number of blocks we fit to the data—and measure the accuracy of the inferred block structure. Varying the graph size corresponds to consistency, varying the variance shows the performance in high-noise settings, and varying the number of blocks corresponds to robustness. We characterize the accuracy of the recovered block structures using the variation of information (VI) [@meila_comparing_2007], a standard metric for such tasks. The VI is a mathematically principled, information theoretic metric for the distance between the inferred and true assignment (vertex labels). Let $P$ denote the true block structure and $Q$ be our estimate. Then $\textrm{VI}(P,Q) = H(P\,\|\,Q) + H(Q\,\|\,P)$, with $H(P\,\|\,Q)$ being the conditional entropy. When $Q=P$ and we recover the true structure exactly, $\textrm{VI}(P,Q)=0$. One nice property of VI is that it increases only modestly when $Q$ differs from $P$ mainly by splitting or dividing blocks. Under all test settings, the WSBM outperforms the alternatives (Fig. \[Fig:Normal\]b–d). As edge-weight variance increases, all methods have decreased performance, but the WSBM fails most gracefully. As the graph size $n$ increases, all methods perform better, with the WSBM performing best by far. And, when varying the number of blocks we infer, all methods perform better when $k\approx k^{}$, but only the WSBM correctly recovers the latent structure at $k = 5 = k^$, which is the value selected under model selection using Bayes factors. Additionally, the WSBM fails gracefully when $k > k^$. Thresholding with the SBM performs poorly in all tests, because choosing a universal weight threshold destroys information about the latent block structure. Thresholding converts the original weights into a Bernoulli distribution with parameter equal to the probability of exceeding the threshold. This effect is substantial whenever distinct blocks exhibit similar weight distributions. If the two blocks’ distributions are similar (Fig. \[fig:v\]), the SBM with thresholding typically finds only one block because the probabilities of exceeding the threshold are too similar. In this case, thresholding confuses latent differences with Bernoulli sampling noise, and the SBM merges blocks that are distinct. With well-separated weight distributions and an optimal threshold, the SBM may find correct structure. However, selecting the ‘optimal’ threshold is a challenging problem itself. Because a threshold will impact different edge bundles differently, a single ‘optimal’ threshold may not, in fact, exist. As a result, when $k > k^$, the SBM with thresholding tends to under-fit the data, leading to very poor results. In contrast, the WSBM, having no thresholds, utilizes the complete weight information and performs well even when given more flexibility than the underlying data require. The performance of k-means and hierarchical clustering is particularly poor for increasing edge-weight variance, when the signal-to-noise ratio is low. These methods over fit the data less than the classic SBM when given $k>k^{}$, but they still perform more poorly than the WSBM. The reason for this difference is our particular choice example. The k-means algorithm uses principle component analysis, which suffers in high variance settings. Similarly, hierarchical clustering focuses on only intra-block behavior (the blocks on the diagonal) and misses out on inter-block behavior. Discussion ========== The weighted stochastic block model we introduce here generalizes the classic stochastic block model to the important case of edges with weights drawn from an exponential family distribution. This generalization presented several technical challenges, which we solved using a Bayesian approach to develop a variational Bayes algorithm for dense graphs. This model accurately recovers latent block structure under a wide variety of conditions, and performs substantially better than simple alternatives. These results demonstrate that applying a threshold to edges weights before applying the unweighted SBM is generally unreliable. The WSBM can be naturally generalized in several potentially useful ways. For sparse graphs, we have developed a scalable belief-propagation algorithm, to be presented in future work. It could also be extended to mixed membership [@airoldi_mixed_2008] or, in the sparse case, to allow degree heterogeneity [@karrer_stochastic_2011]. Stochastic variational inference has shown promising results for scaling in the mixed-membership SBM, and this technique could also be adapted to the WSBM [@gopalan_scalable_2012]. Finally, an interesting question is the extent to which utilizing weight information modifies the phase transition in the detectability of latent block structure, which is known to exist in the classic SBM [@decelle_phasetransition_2011]. Acknowledgements {#acknowledgements .unnumbered} ---------------- We thank D. Larremore for helpful conversations. We acknowledge financial support from Grant \#FA9550-12-1-0432 from the U.S. Air Force Office of Scientific Research (AFOSR) and the Defense Advanced Research Projects Agency (DARPA).
--- author: - Tommi Markkanen title: Renormalization of the inflationary perturbations revisited --- IMPERIAL/TP/2017/TM/02 Introduction ============ The earliest inflationary models were proposed over 30 years ago [@Guth:1980zm; @Linde:1981mu; @Albrecht:1982wi; @Starobinsky:1979ty; @Starobinsky:1980te] and to this day the framework of cosmological inflation is perhaps the most successful proposal for explaining several non-trivial features of the observed Universe [@Ade:2015lrj]. In the simplest realisation of inflation an epoch of quasi-exponential expansion of space results from a classical field $\varphi$ slowly rolling down its potential during which its quantum fluctuations $\hat{\phi}$ get stretched to cosmological scales providing the necessary inhomogeneities that seed the formation of structure in the Universe. The inflationary perturbations are quantified by the spectrum of the co-moving curvature perturbation [@Sasaki:1986hm; @Mukhanov:1988jd], $\mathcal{R}$, which in the spatially flat gauge is sourced by fluctuations from the scalar field as [$$\mathcal{P}_{\mathcal{R}}={\frac{{H}}{{\dot{\varphi}}}}\mathcal{P}_{\phi}\,,\label{eq:R}$$]{} where the power spectrum of a generic quantized variable is defined by a momentum space integral [$$\langle\hat{f}\,^2\rangle\equiv \int _0^\infty {\frac{{dk}}{{k}}}\,\mathcal{P}_{f}\,,\label{eq:PS}$$]{} with $\|\mathbf{k}\|\equiv k$. The current state-of-the-art measurements give for the observable long wavelength part of the spectrum [@Ade:2015lrj] [$$\mathcal{P}_{\mathcal{R}}=2.2\times 10^{-9}\,.$$]{} Although rarely discussed in works focussing on Early Universe phenomenology, according to general field theory principles the spectrum of perturbations $\mathcal{P}_{\mathcal{R}}$ can potentially be modified by the renormalization counter terms one must unavoidably introduce to render the theory finite. For example, in general the variance of the field fluctuations $\langle\hat{\phi}^2\rangle$ is a divergent quantity and any physical result containing it will also have to contain counter terms, $\delta\phi^2$, removing the divergences. By way of (\[eq:PS\]) a renormalized variance provides a definition for the renormalization of the power spectrum $\hat{\phi}$ [$$\langle\hat{\phi}^2\rangle_R\equiv\langle\hat{\phi}^2\rangle-\delta\phi^2\equiv\int _0^\infty {\frac{{dk}}{{k}}}\,\big(\mathcal{P}_{\phi}-\delta\mathcal{P}_{\phi}\big)\equiv \int _0^\infty {\frac{{dk}}{{k}}}\,(\mathcal{P}_{\phi})_R\,,\label{eq:Rp0}$$]{} and trivially via (\[eq:R\]) for $(\mathcal{P}_\mathcal{R})_R\equiv\mathcal{P}_\mathcal{R}-\delta\mathcal{P}_\mathcal{R}$. As the above shows, in general renomalization can have an effect on the amplitude of the inflationary perturbations, it is however widely assumed that renormalization may be performed in such a way that only the very ultraviolet contribution of the spectrum is modified with no effect on the observable long wavelength modes and robust inflationary predictions can be made without addressing proper renormalization of the theory. However, it has also been claimed that renormalization will unavoidably introduce a significant change on the amplitude of the spectrum perturbations from inflation, including the long wavelength modes thus resulting in completely different predictions when compared with standard results [@Parker:2007]. The results of [@Parker:2007] have been the subject of a considerable amount of debate [@Agullo:2008ka; @Agullo:2009vq; @Agullo:2009zza; @Agullo:2009zi; @Agullo:2010ui; @Agullo:2010hg; @Agullo:2011qg; @Bastero-Gil:2013; @Glenz:2009zn; @Seery:2010kh; @Durrer:2009ii; @Finelli:2007fr; @Marozzi:2011da; @Urakawa:2009xaa] and as far as we are aware, this issue is still viewed as unsolved by many [@Woodard:2014jba], see also the recent works [@Alinea:2015pza; @Alinea:2016qlf; @Alinea:2017ncx]. The approach that implies a significant modification of $\mathcal{P}_\phi$ from renormalization was laid out already in [@Parker:1974qw1; @Parker:1974qw; @Fulling:1974zr] and is generally referred to as adiabatic regularization or adiabatic subtraction. This technique has since been established as one of the most popular approaches for renormalization of quantum fields in curved spaces, for examples see [@Bunch:1980vc; @Brandenberger:2004kx; @Anderson:2005hi; @Kaya:2015hka; @MolinaParis:2000zz; @Habib:1999cs; @Anderson:1987yt]. It relies on an adiabatic expansion of modes or more concretely a series in increasing number of derivatives which provides a counter term with the identical divergences to the ones generated by the full quantum correlations, which upon subtraction leads to quantities with the divergences removed. Despite the great calculational advantages of adiabatic subtraction it does lack features that in some applications would be desirable. The adiabatic subtraction terms are not in any obvious manner related to redefinitions of (bare) constants as introduced by the action, despite the fact that at the most elementary level that is the essence of renormalizing a quantum theory [@Collins:1984xc]. The specific physical conditions, in particular having a handle on the renormalization scale and the finite terms in the subtractions that in a strict sense are required for providing physical definitions for the renormalized parameters are similarly only implicitly defined in the procedure. If one wished to make use of a set of counter terms with different finite contributions to the adiabatic prescription and hence a different physical interpretation in terms of matching to observations one would have to go beyond adiabatic subtraction. An approach allowing one to achieve this was recently presented in [@Markkanen:2013nwa]. Furthermore, in the works [@Cooper:1994ji; @Cooper:1996ib; @Lampert:1996qw] as summarized in section 14.2.3 of [@Calzetta:2008iqa] the adiabatic subtraction terms were converted into redefinitions of couplings in the framework of heavy ion collisions. Similar issues were addressed recently in the context of gravitational waves in [@Granese:2017gfb]. In this work by making use of the techniques of [@Markkanen:2013nwa] we set out to clarify the non-trivial issues related to the renormalization of the inflationary perturbations and in particular investigate if a modification to the observable spectrum can consistently be kept small even when proper renormalization is implemented. We will perform our analysis for the simple $m^2\varphi^2$-model given by the matter action [$$S_m=-\int d^4x\,\sqrt{-g}\bigg[{\frac{{1}}{{2}}}\nabla_\mu\varphi\nabla^\mu\varphi+{\frac{{1}}{{2}}}m^2\varphi^2\bigg]\,.\label{eq:act}$$]{} Even though this model is under tension from observations [@Ade:2015lrj] it serves as a useful toy model with which to illustrate our calculation in an accessible form. Following [@Parker:2007], we will perform the derivation for $\langle\hat{\phi}^2\rangle$ in the framework of quantum field theory on a curved space time, where the metric is assumed to be a classical background field [@Birrell:1982ix; @Parker:2009uva]. Although it is strictly speaking not correct to neglect the metric perturbations it can be shown that this approximation will only introduce a small error for the power spectrum. Our conventions are (+,+,+) [@Misner:1974qy] and $c\equiv\hbar\equiv1$. Adiabatic subtraction {#sec:subtraction} ===================== The adiabatic subtraction prescription is described at length in [@Birrell:1982ix; @Parker:2009uva] where we refer the reader for more information. In this section we will keep our discussion general by considering a Friedmann–Lemaître–Robertson–Walker (FLRW) type metric with [$$g_{\mu\nu}dx^\mu dx^\nu=-dt^2+a(t)d\mathbf{x}^2\,,$$]{} with from now on $a(t)\equiv a$, in section \[sec:renorm\] we will restrict our analysis to de Sitter space exclusively. Adiabatic subtraction was first discussed in [@Parker:1974qw1; @Parker:1974qw; @Fulling:1974zr] and in practical calculations is probably the most frequently utilized method used for renormalization on a curved background. In this approach a renormalized quantity is defined by subtracting an $A$th order derivative approximation from the full divergent expression. The order of the expansion $A$ depends on the adiabatic order of divergences in the bare quantity and $A$ should only be as high as to include the generated divergences. Thus in adiabatic subtraction all divergences are removed by a single subtraction. Counter terms in the usual sense of redefining constants as introduced by the action are only implicitly defined by the form of the adiabatic subtraction term. Although this brings about a practical advantage it does make the connection to traditional renormalization framework, discussed at length for example in [@Collins:1984xc], less clear. It is however possible also in curved spaces to apply the standard renormalization techniques relying on counter terms derived by redefining bare constants [@Markkanen:2013nwa], which we will discuss in section \[sec:renorm\]. In this work we have deliberately chosen to denote this prescription adiabatic subtraction instead of adiabatic regularization despite the fact that it is generally viewed as a regularization method instead of a complete renormalization prescription [@Birrell:1982ix; @Parker:2009uva]. This is to highlight an important property possessed by it that other regularization methods such as introducing a cut-off or dimensional regularization generally do not: it is not only a method for rendering formally infinite expressions finite, but it by definition contains a prescription for subtraction. When using cut-off or dimensional regularization, in order to complete the renormalization of a quantity one must also define what precisely is to be subtracted from a regularized expression in order to render it physically meaningful. This step one must perform and define [in addition to regularization]{} and the prescription one chooses defines the finite pieces contained in the subtraction. As an example one can think of the differences in finite terms introduced in the ${\rm MS}$ and $\overline{\rm MS}$ subtraction prescriptions used in particle physics [@Collins:1984xc]. This is in fact very important for our purposes: although adiabatic subtraction is usually viewed as a technique with which to regularize divergences it nonetheless implicitly also defines a subtraction prescription including the finite terms in the subtraction. For this reason quantities defined via this technique should be viewed more as renomalized instead of regularized expressions and this distinction will be reflected by our choice of language calling finite quantities defined via adiabatic subtraction renormalized. We can illustrate the adiabatic prescription with an example: the renormalized variance of the fluctuation of a scalar field we can symbolically define as [$$\begin{aligned} \label{eq:adsub1}\langle \hat{\phi}^2\rangle_R\equiv\langle \hat{\phi}^2\rangle-(\delta {\phi}^2)^{\rm ad}=\langle \psi\vert \hat{\phi}^2\vert\psi\rangle-\langle 0^{(A)}\vert \hat{\phi}^2\vert 0^{(A)}\rangle\big\vert_{A=2}\end{aligned}$$]{} where $\langle \hat{\phi}^2\rangle$ denotes the bare quantity calculated in some yet undefined state $\vert\psi\rangle$, $(\delta {\phi}^2)^{\rm ad}$ the adiabatic counter term, $\vert0^{(A)}\rangle$ the $A$th order adiabatic vacuum to be defined shortly and we have split the quantized field $\hat{\varphi}$ into a mean field $\langle\hat{\varphi}\rangle$ and a quantum fluctuation [$$\hat{\varphi}-\langle\hat{\varphi}\rangle\equiv\hat{\varphi}-{\varphi}\equiv\hat{\phi}\,.\label{eq:fluc}$$]{} The adiabatic subtraction term $(\delta{\phi}^2)^{\rm ad}$ can be derived as follows: first one solves the equation of motion for the fluctuation obtained from (\[eq:act\]) [$$\label{eq:eom}\big(-\square+m^2\big){\hat\phi}=0\,,$$]{} up to the appropriate adiabatic order. By using the standard mode decomposition [$$\begin{aligned} \hat{\phi}=\int d^{n-1}k\,e^{i\mathbf{k}\cdot\mathbf{x}} \big [\hat{a}_\mathbf{k} u_\mathbf{k}+\hat{a}^\dagger_{-\mathbf{k}} u_\mathbf{k}\big],\quad u_\mathbf{k}={\frac{{h_\mathbf{k}(t)}}{{\sqrt{(2\pi)^{n-1}a^{n-1}}}}}\label{eq:ans0}\,,\end{aligned}$$]{} the adiabatic solutions come by way of the ansatz [$$h^{\rm ad}_\mathbf{k}(t)={\frac{{1}}{{\sqrt{2W}}}}e^{-i\int^{t}Wdt'}\label{eq:ans}\,,$$]{} where $W$ is expressed as an adiabatic expansion, [$$W=c_0+c_1{\frac{{{\ensuremath{\dot{a}}}}}{{a}}}+c_2{\frac{{{\ensuremath{\dot{a}}}^2}}{{a^2}}}+c_3{\frac{{{\ensuremath{\ddot{a}}}}}{{a}}}+\cdots,$$]{} with the $c$’s being functions on $a$, $\mathbf{k}$ and $m$ and where we have normalized our modes such that [$$=[{\hat{a}}^\dagger_\mathbf{k},{\hat{a}}^\dagger_{\mathbf{k}'}]=0,\quad[{\hat{a}}_\mathbf{k},{\hat{a}}^\dagger_{\mathbf{k}'}]=\delta(\mathbf{k}-\mathbf{k}'),$$]{} and for completeness analytically continued our dimensions to $n$. Explicitly, equation (\[eq:eom\]) reads [$$\bigg[\partial_t\partial_t +(n-1){\frac{{{\ensuremath{\dot{a}}}}}{{a}}}\partial_t-a^{-2}\partial_i\partial^i+m^2\bigg]\hat{\phi}=0\,,$$]{} and using the ansatz from (\[eq:ans\]) gives rise to the equation [$$\label{eq:W}W^2=\underbrace{{\frac{{{k}^2}}{{a^2}}}+m^2}_{\displaystyle \equiv\omega^2}-{\frac{{{\ensuremath{\ddot{a}}}}}{{a}}}\bigg[{\frac{{1}}{{2}}}(n-1)\bigg]-\bigg({\frac{{{\ensuremath{\dot{a}}}}}{{a}}}\bigg)^2\bigg[{\frac{{1}}{{4}}}(n-1)(n-3)\bigg]+{\frac{{3\dot{W}^2}}{{4W^2}}}-{\frac{{\ddot{W}}}{{2W}}}\,.$$]{} From the above we trivially find the leading adiabatic term to be $(W^2)^{(0)}=\omega^2$ and upon straightforward iteration $(W^2)^{(1)}=0$ and [@Markkanen:2013nwa] [$$\begin{aligned} (W^2)^{(2)}&=\frac{{\ensuremath{\ddot{a}}}}{a}\left[\frac{1}{2}\big(2-n-m^2/\omega^2\big)\right]-\frac{\dot{a}^2}{a^2}\bigg[{\frac{{1}}{{4}}}\Big((n-2)^2+4m^2/\omega^2-5\big(m^2/\omega^2\big)^2\Big) \bigg]\label{eq:W2}\,,\end{aligned}$$]{} with which the adiabatic counter term for the variance in (\[eq:adsub1\]) reads [$$\begin{aligned} (\delta \phi^2)^{\rm ad}&=\int {\frac{{d^{n-1}k}}{{(2\pi a)^{n-1}}}}\|h^{\rm ad}_\mathbf{k}(t)\|^2_{A=2}=\int {\frac{{d^{n-1}k}}{{2(2\pi a)^{n-1}}}}\bigg[{\frac{{1}}{{W}}}\bigg]_{A=2}\nonumber \\&=\int {\frac{{d^{n-1}k}}{{2(2\pi a)^{n-1}\omega}}}\bigg\{1-{\frac{{{\ensuremath{\dot{a}}}^2}}{{a^2}}}\bigg[\frac{5 m^4-4 m^2 \omega ^2-(n-2)^2 \omega ^4}{8 \omega ^6}\bigg]+{\frac{{{\ensuremath{\ddot{a}}}}}{{a}}}\bigg[\frac{m^2+(n-2) \omega^2}{4 \omega^4}\bigg]\bigg\}\,.\label{eq:CTa}\end{aligned}$$]{} According to the adiabatic prescription the above includes terms with up to two time derivatives only since beyond this order no divergences are generated in four dimensions, which in the above is denoted with $A=2$. For inflation in the approximation of strict de Sitter space $a= e^{H t}$ with $H$ almost a constant the equation of motion (\[eq:eom\]) can be solved exactly. Making the standard and well-motivated choice of the Bunch-Davies (BD) vacuum as the boundary condition for the mode [@Chernikov:1968zm; @BD] and focussing on the four-dimensional case the solution is given as [$$h_{\mathbf{k}}(t)=\sqrt{{\frac{{\pi}}{{4H}}}}H^{(1)}_{{\nu}}\big(k/(aH)\big)\,; \qquad\nu^2={\frac{{9}}{{4}}}-{\frac{{m^2}}{{H^2}}}\label{eq:sol}\,.$$]{} Using the definitions (\[eq:Rp0\]) and (\[eq:adsub1\]) for the BD mode as well as the adiabatic counter term (\[eq:CTa\]) one may easily find an expression for the adiabatically renormalized power spectrum for the field to be [$$\begin{aligned} \label{eq:Rp}\mathcal{P}_\phi-\delta \mathcal{P}^{\rm ad}_\phi&\equiv(\mathcal{P}_\phi)_R={\frac{{(k/a)^3}}{{2\pi^2}}}\Big(\|h_\mathbf{k}(t)\|^2-\|h^{\rm ad}_\mathbf{k}(t)\|_{A=2}^2\Big)\nonumber \\&=\bigg({\frac{{k}}{{a H}}}\bigg)^3{\frac{{H^2}}{{8\pi}}}\bigg\{\big\|H^{(1)}_{{\nu}}\big(k/(aH)\big)\big\|^2-\bigg[\frac{2 H}{\pi \omega}-\frac{H^3 \left(5 m^4-6 m^2 \omega ^2-8 \omega^4\right)}{4 \pi \omega^7}\bigg]\bigg\}\,,\end{aligned}$$]{} which for long wavelengths, ${k}/({aH})\longrightarrow0$, and small masses, $\nu\longrightarrow3/2$, gives [$$(\mathcal{P}_\phi)_R\longrightarrow \bigg({\frac{{H}}{{2\pi}}}\bigg)^2\bigg\{1-\bigg({\frac{{k}}{{a H}}}\bigg)^3\bigg[\frac{H}{\omega}-\frac{H^3 \left(5 m^4-6 m^2 \omega^2-8 \omega^4\right)}{8 \omega ^7}\bigg]\bigg\}\,.$$]{} As discovered in [@Parker:2007] for small masses (\[eq:Rp\]) deviates from the unrenormalized expression by several orders of magnitude. For example for $m^2/H^2\sim 10^{-2}$ one gets [$${\frac{{(\mathcal{P}_\phi)_R}}{{\mathcal{P}_\phi}}}\sim 10^{-4} \,,\label{eq:mods}$$]{} for scales exiting the horizon, $k/(aH)\sim1$. This obviously implies a significant modification to the standard predictions. An important point to keep in mind however is that even if (\[eq:mods\]) is true in the adiabatic prescription, it does not imply that a large correction from renormalization is always unavoidable: it is well-known from the context of particle theory [@Peskin:1995ev] that the counter terms rendering a theory finite have universal divergencies but in general may have different finite contributions depending on the physical boundary conditions one imposes and as discussed in the beginning of this section, adiabatic subtraction is simply one prescription for defining the finite parts of the counter terms. In principle, nothing prevents the existence of other renormalization prescriptions where the long wavelength portion of the spectrum is only mildly modified. Also, the renormalization scale imposed in the adiabatic subtraction terms cannot be easily determined thus rendering the physical interpretation of the subtraction non-trivial. The above assertion that there must exist other prescriptions beyond adiabatic subtraction is also implied by the axiomatic approach laid out in [@Wald:1977up; @Wald:1978pj; @Wald:1978ce] by Wald: under a few physically motivated criteria the renormalized energy-momentum tensor is unique only up to a covariantly conserved local tensor. For more discussion of this axiomatic approach, see for example chapter 6.6 of [@Birrell:1982ix]. Furthermore, although for a non-interacting theory the terms introduced by adiabatic subtraction can be reduced to a redefinition of the bare constants of the action, which in principle is ultimately what one does when renormalizing, in the interacting case this no longer applies [@Markkanen:2013nwa]. Hence, when interactions are included adiabatic subtraction cannot, in a strict sense, be considered a consistent renormalization prescription. Let us then turn to the technique introduced in [@Markkanen:2013nwa] where these issues are not present and where one may easily choose arbitrary finite pieces for the counter terms. Counter terms in curved space {#sec:renorm} ============================= In this section we will throughout make use of the de Sitter approximation with $a=e^{H t}$ and $\dot{H}=0$. One of the main difficulties in performing consistent renormalization on a curved background is respecting coordinate invariance: the renormalized energy-momentum tensor is required to be covariantly conserved which directly implies the same for the respective counter term [$$\nabla^\mu\langle\hat{T}_{\mu\nu}\rangle_R\equiv\nabla^\mu\big(\langle\hat{T}_{\mu\nu}\rangle-\delta T_{\mu\nu}\big)=0\qquad\Leftrightarrow\qquad \nabla^\mu\delta T_{\mu\nu}=0\,.\label{eq:covCons1}$$]{} Trying to a priori devise a counter term $\delta T_{\mu\nu}$ that removes all generated divergences while at the same time respects the above requirement and does not subtract physically relevant contributions is generally quite non-trivial as (\[eq:covCons1\]) imposes constraints on the form of the subtractions as well as the regularization used [@Birrell:1982ix; @Parker:2009uva]. In adiabatic subtraction no issues arise since the adiabatic modes (\[eq:ans\]) satisfy the equation of motion up to the required truncation which implies that covariant conservation must be respected. Furthermore, since the subtraction term is given implicitly by an integral as in (\[eq:CTa\]) one may perform the subtraction procedure by simply combining two formally divergent expressions under the same integral and the issues related to properly regulating the integrals do not arise. This is true also when interactions are present [@MolinaParis:2000zz]. However, as discussed in the previous section adiabatic subtraction has two features that in some applications render it less ideal: first, as demonstrated in [@Markkanen:2013nwa] in the interacting case it does not correspond to redefining the parameters of the theory into counter terms and finite physical contributions. Second, it does not posses a handle with which to control the finite pieces of the subtraction. These issues are evaded when implementing the technique of [@Markkanen:2013nwa]. Any renormalization prescription in a strict sense should in the end be traced back to a redefinition of constants in the original action i.e. all necessary subtractions should be obtainable from counter terms resulting from the appropriate redefinition of the bare constants, $c_i$, into finite physical pieces, $c_{i,R}$, and divergent counter terms, $\delta c_i$, as [$$c_i\longrightarrow c_{i,R}+\delta c_i\,,\label{eq:cts}$$]{} which once determined for example from the renormalization of the energy-momentum tensor must remove all divergences that appear in physical quantities. Importantly, when renormalization is performed by introducing counter terms via (\[eq:cts\]) the covariant conservation requirement (\[eq:covCons1\]) is automatically satisfied given that general coordinate invariance is not broken by an improper choice of regularization, such as a cut-off [@Maggiore:2010wr2; @Akhmedov:2002ts]. Specifically, analytic continuation to $n$ dimensions allows one to write closed expressions for the counter terms resulting from (\[eq:cts\]) while respecting (\[eq:covCons1\]), as for example implemented in [@Markkanen:2013nwa]. In this work we will also highlight this feature with an example calculation (see equations (\[eq:l1\] – \[eq:l2\])). In a first principle approach to renormalization in curved space relying on (\[eq:cts\]) it is well-known that in order to have all necessary counter terms one must also introduce the following gravitational action[^1] [@Birrell:1982ix; @Parker:2009uva] [$$\label{eq:actg}S_g= {\frac{{1}}{{2}}}\int d^4x\sqrt{-g}~\bigg[-2\Lambda +{M_{\rm P}^2} R+\beta R^2+\epsilon_{1}R_{\alpha\beta}R^{\alpha\beta}+ \epsilon_{2}R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}\bigg] \,,$$]{} in addition to the matter piece in (\[eq:act\]) i.e. [$$S=S_m+S_g\,.$$]{} The first two terms in (\[eq:actg\]) are the familiar cosmological constant and Einstein-Hilbert term with the reduced Planck mass, $M_{\rm P}^2\equiv (8\pi G)^{-1}$, where however the parameters are not yet physical as the redefinitions (\[eq:cts\]) have not yet been made. In the semi-classical approach with quantum matter and a classical metric $S_g$ provides counter terms on the matter or energy-momentum side of the Einstein equation, despite the fact that the tree-level contributions from $S_g$ only appear in on the geometric side of the semi-classical Einstein equation [$$\begin{aligned} \label{eq:renomE0}{\frac{{2}}{{\sqrt{-g}}}}{\frac{{\delta S_g}}{{\delta g^{\mu\nu}}}}&=\langle\hat{T}_{\mu\nu}\rangle_R\,;\\\langle\hat{T}_{\mu\nu}\rangle_R&\equiv\langle\hat{T}_{\mu\nu}\rangle-\delta T_{\mu\nu}=-\bigg\langle{\frac{{2}}{{\sqrt{-g}}}}{\frac{{\delta S_m}}{{\delta g^{\mu\nu}}}}\bigg\rangle-{\frac{{2}}{{\sqrt{-g}}}}{\frac{{\delta S_g}}{{\delta g^{\mu\nu}}}}\bigg\vert_{c_i\rightarrow\delta c_i}\,,\end{aligned}$$]{} where for a non-interacting theory the only counter terms required come from $S_g$. In an interacting theory counter terms from $S_m$ also need to be included [@Markkanen:2013nwa]. The energy-momentum tensor can be conveniently split into a sum of the classical, quantum and counter term pieces, $\langle\hat{T}_{\mu\nu}\rangle_R\equiv T_{\mu\nu}^C+\langle\hat{T}^Q_{\mu\nu}\rangle-\delta T_{\mu\nu}$, which for our theory with (\[eq:act\]) and (\[eq:actg\]) are [$$\begin{aligned} \langle\hat{T}_{\mu\nu}\rangle_R&=-{\frac{{g_{\mu\nu}}}{{2}}}\Big[\partial_\rho\varphi\partial^\rho\varphi+m^2_R\varphi^2\Big]+\partial_\mu\varphi\partial_\nu\varphi\\&+\label{eq:renomE1}\bigg\langle-{\frac{{g_{\mu\nu}}}{{2}}}\Big[\partial_\rho\hat{\phi}\partial^\rho\hat{\phi} +m^2_R\hat{\phi}^2\Big]+\partial_\mu\hat{\phi}\partial_\nu\hat{\phi}\bigg\rangle\\ &-\Big[g_{\mu\nu}\delta\Lambda+\delta M_{\rm P}^2G_{\mu\nu} +\delta\beta~^{(1)}H_{\mu\nu}+\delta\epsilon_1~^{(2)}H_{\mu\nu}+\delta\epsilon_2H_{\mu\nu}\Big]\,,\label{eq:renomE2}\end{aligned}$$]{} where in $n$-dimensional de Sitter space we have [@Markkanen:2013nwa] $$\begin{aligned} G_{\mu\nu}&\equiv{\frac{{1}}{{\sqrt{-g}}}}{\frac{{\delta}}{{\delta g^{\mu\nu}}}}\int d^nx\sqrt{-g}~R^{\phantom{2}}=g_{\mu\nu}\frac{1}{2}(2-n) (n-1) H^2 \,,\\ ~^{(1)}H_{\mu\nu}&\equiv{\frac{{1}}{{\sqrt{-g}}}}{\frac{{\delta}}{{\delta g^{\mu\nu}}}}\int d^nx\sqrt{-g}~R^2= g_{\mu\nu}\frac{n}{2} (4-n) (n-1)^2 H^4\,,\label{eq:h4} \\ ~^{(2)}H_{\mu\nu}&\equiv{\frac{{1}}{{\sqrt{-g}}}}{\frac{{\delta}}{{\delta g^{\mu\nu}}}}\int d^nx\sqrt{-g}~R_{\mu\nu}R^{\mu\nu}= g_{\mu\nu}\frac{1}{2} (4-n) (n-1)^2 H^4\,, \\ H_{\mu\nu}&\equiv{\frac{{1}}{{\sqrt{-g}}}}{\frac{{\delta}}{{\delta g^{\mu\nu}}}}\int d^nx\sqrt{-g}~R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}= g_{\mu\nu} (4-n) (n-1) H^4 \label{eq:h42}\,.\end{aligned}$$ As one may see from (\[eq:h4\] - \[eq:h42\]), in (\[eq:renomE2\]) in de Sitter space one only needs to include one of the $\mathcal{O}(R^2)$ tensors in order to cancel all divergences $\propto H^4$ and from now on we will neglect the counter terms from the contributions $R_{\alpha\beta}R^{\alpha\beta}$ and $R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}$ by simply setting $\delta\epsilon_1=\delta\epsilon_2=0$[^2]. Before discussing the specifics of our method, we comment on an important and non-trivial feature of renormalization on a curved background that we already briefly mentioned at the beginning of this section. In order to verify that the divergences in the energy-momentum tensor do in fact organize themselves as a linear combination of local and conserved curvature tensors as in (\[eq:renomE2\]) a proper regularization method must be introduced. On a curved background some regulators such as a cut-off are problematic as they lead to explicit violation of covariance and hence render any renormalization approach based on redefinition of constants more involved. Specifically, when using a cut-off some non-covariant divergences must simply by removed by hand [@Maggiore:2010wr2; @Akhmedov:2002ts]. Fortunately, in dimensional regularization this problem does not surface which is the reason why throughout we have maintained $n$ dimensions in our expressions. In dimensional regularization covariance is often maintained in a very non-trivial manner related to the special properties of the analytically continued integrals. One may clarify the above discussion by calculating the vacuum energy and pressure densities in flat space, which come easily with the mode (\[eq:ans0\]) inserted into (\[eq:renomE1\]) with simply $W^2 = {k}^2+m^2_R$ [$$\begin{aligned} \langle\hat{T}^Q_{00}\rangle_{H=0}&={\frac{{1}}{{2}}}\int \frac{d^{n-1 }{k}}{(2\pi)^{n-1}}{\sqrt{{k}^2+m^2_R}}={\frac{{1}}{{2(4\pi)^{{\frac{{n-1}}{{2}}}}}}}{\frac{{\Gamma[-{\frac{{n}}{{2}}}]}}{{\Gamma[-{\frac{{1}}{{2}}}]}}}m^n_R\nonumber \\&=\frac{m^4_R}{32 \pi ^2 (n-4)}+\frac{m^4_R}{128 \pi ^2} \left(2 \log \left(\frac{m^2_R}{4 \pi }\right)+2 \gamma -3\right)+\mathcal{O}(n-4)\,,\label{eq:l1}\end{aligned}$$]{} for the energy density and for the pressure [$$\begin{aligned} \langle\hat{T}^Q_{ii}\rangle_{H=0}&={\frac{{1}}{{2}}}\int \frac{d^{n-1 }{k}}{(2\pi)^{n-1}}\frac{{k}^2}{(n-1)\sqrt{{k}^2+m^2_R}}={\frac{{1/2}}{{2(4\pi)^{{\frac{{n-1}}{{2}}}}}}}{\frac{{\Gamma[-{\frac{{n}}{{2}}}]}}{{\Gamma[{\frac{{1}}{{2}}}]}}}m_R^{n}\nonumber \\&=-\frac{m^4_R}{32 \pi ^2 (n-4)}-\frac{m^4_R}{128 \pi ^2} \left(2 \log \left(\frac{m^2_R}{4 \pi }\right)+2 \gamma -3\right)+\mathcal{O}(n-4)\,,\label{eq:l2}\end{aligned}$$]{} where the final expressions in (\[eq:l1\] – \[eq:l2\]) can be obtained with the standard formulae of dimensional regularization [@Peskin:1995ev]. Quite obviously (\[eq:l1\] – \[eq:l2\]) satisfy [$$\langle\hat{T}^Q_{\mu\nu}\rangle_{H=0}\propto g_{\mu\nu}\label{eq:porp}$$]{} as required by (\[eq:renomE1\] – \[eq:renomE2\]). The fact that all divergences generated for the energy-momentum tensor when dimensionally regularized organize themselves in the form (\[eq:renomE2\]) was verified in [@Markkanen:2013nwa], in a general FLRW background[^3]. What is also apparent is that for a cut-off (\[eq:porp\]) fails since, as we already mentioned, a cut-off violates covariance. This also implies that in a precise sense the integrals are valid as $n$-dimensional expressions and only if the integrand for some expression is convergent in four dimensions can one set $n=4$. In the adiabatic prescription the quantum piece of the energy-momentum tensor can be renormalized as was done in (\[eq:adsub1\]) for the variance [$$\begin{aligned} \label{eq:adsub2}\langle \hat{T}^Q_{\mu\nu}\rangle_R&= \langle\hat{T}^Q_{\mu\nu}\rangle-\delta T^{\rm ad}_{\mu\nu}\equiv\langle\psi\| \hat{T}^Q_{\mu\nu}\|\psi\rangle-\langle 0^{(A)}\vert \hat{T}^Q_{\mu\nu}\vert 0^{(A)}\rangle\big\vert_{A=4}\,,\end{aligned}$$]{} where in four dimensions the adiabatic expansion in general has to be performed up to fourth order in derivatives in order to remove all divergences. As we discussed in the previous section adiabatic subtraction is just one choice for the finite parts of the counter terms $\delta\Lambda$, $\delta M_{\rm P}^2$ and $\delta \beta$. However, since the divergencies are universal adiabatic subtraction can be used to determine the divergent parts of the counter terms. This leads to the following modified approach with which one may introduce arbitrary finite contributions in the counter terms: if we match the adiabatic counter term in (\[eq:adsub2\]) with the general form in given in (\[eq:renomE2\]) as [$$\langle 0^{(A)}\vert \hat{T}^Q_{\mu\nu}\vert 0^{(A)}\rangle\big\vert_{A=4}\equiv \delta T^{\rm ad}_{\mu\nu}=g_{\mu\nu}\delta\Lambda^{\rm ad}+\delta (M_{\rm P}^{2})^{\rm ad}G_{\mu\nu} +\delta\beta^{\rm ad}~^{(1)}H_{\mu\nu}\label{eq:match}\,,$$]{} we may extract the specific choice made implicitly for the counter terms in the adiabatic prescription. As is also apparent in the results of the previous section, adiabatic subtraction results in counter terms that are given in terms of momentum space integrals and once we have explicit expressions for $\delta\Lambda^{\rm ad}$, $\delta (M_{\rm P}^{2})^{\rm ad}$ and $\delta\beta^{\rm ad}$ we may straightforwardly find other sets of counter terms with identical divergences but with differing finite parts by changing the parts that remain finite when $k/a\rightarrow\infty$. We also note that it has been shown that adiabatic subtraction, including the generated finite pieces, does admit an expression of the form (\[eq:match\]) [@Markkanen:2013nwa]. The expression for energy-momentum can be read from (\[eq:renomE1\]) with which we can solve the expressions for the energy and pressure density adiabatic counter terms as [$$\delta\hat{T}^{\rm ad}_{00}={\frac{{1}}{{2}}}\int d^{n-1} {k}\bigg[\vert\dot{u}^{\rm ad}_\mathbf{k}\vert^2+\bigg( {\frac{{\mathbf{k}^2}}{{a^2}}}+m_R^2\bigg)\vert u^{\rm ad}_\mathbf{k}\vert^2\bigg]\label{eq:T00}$$]{} and [$$a^{-2}\delta\hat{T}^{\rm ad}_{ii}={\frac{{1}}{{2}}}\int d^{n-1} k\bigg[\vert\dot{u}^{\rm ad}_\mathbf{k}\vert^2-\bigg({\frac{{3-n}}{{1-n}}}{\frac{{\mathbf{k}^2}}{{a^2}}}+m_R^2\bigg)\vert u^{\rm ad}_\mathbf{k}\vert^2\bigg]\label{eq:Tii}\,,$$]{} again for completeness in $n$ dimensions. Making use of (\[eq:ans\]) and (\[eq:match\]) along with (\[eq:T00\]) and (\[eq:Tii\]) gives [$$\begin{aligned} \delta\hat{T}^{\rm ad}_{00}-a^{-2}\delta\hat{T}^{\rm ad}_{ii}&=-2\delta\Lambda^{\rm ad}+\delta (M_{\rm P}^{2})^{\rm ad}\big(G_{00}-a^{-2}G_{ii}\big) +\delta\beta^{\rm ad}\big(^{(1)}H_{00}-a^{-2}~^{(1)}H_{ii}\big)\label{eq:je} \\ &=\int {\frac{{d^{n-1}k}}{{2(2\pi a)^{n-1}}}}\bigg[{\frac{{1}}{{W}}}\bigg({\frac{{2-n}}{{1-n}}}{\frac{{\mathbf{k}^2}}{{a^2}}}+m_R^2\bigg) \bigg]_{A=4}\,,\end{aligned}$$]{} which now requires one to find a solution for $W$ to fourth adiabatic order, which comes via a straightforward but tedious iteration of equation (\[eq:W\]). With the help of the appendixes of [@Markkanen:2013nwa] and [@Markkanen:2016aes], after some work, we can solve the counter terms as defined in the adiabatic prescription to be [$$\begin{aligned} \label{eq:della}\delta\Lambda^{\rm ad}&\equiv\int{\frac{{d^{n-1}k}}{{(2\pi a)^{n-1}}}}\delta\Lambda^{\rm ad}_{\mathbf{k}}=\int{\frac{{d^{n-1}k}}{{(2\pi a)^{n-1}}}}\frac{m_R^2+(n-2) \omega ^2}{4 (1-n) \omega }\,,\\\label{eq:dellab}\delta (M_{\rm P}^{2})^{\rm ad}&\equiv\int{\frac{{d^{n-1}k}}{{(2\pi a)^{n-1}}}}\delta(M_{\rm P}^{2})^{\rm ad}_{\mathbf{k}} =\int {\frac{{d^{n-1}k}}{{(2\pi a)^{n-1}}}}\frac{\left(m_R^2+(n-2) \omega ^2\right)}{16 (n-2) (n-1)^2 \omega ^7} \bigg\{-5 m_R^4\\ \nonumber&+6 m_R^2 \omega ^2+(n-2) n \omega ^4\bigg\}\,,\end{aligned}$$]{} and [$$\begin{aligned} \delta\beta^{\rm ad}&\equiv\int{\frac{{d^{n-1}k}}{{(2\pi a)^{n-1}}}}\delta\beta^{\rm ad}_{\mathbf{k}}=\int{\frac{{d^{n-1}k}}{{(2\pi a)^{n-1}}}}\frac{m_R^2+(n-2) \omega^2}{256 (n-4) (n-1)^3 n \omega^{13}}\bigg\{1155 m_R^8-2772 m_R^6 \omega ^2\nonumber \\&-70 m_R^4 ((n-2) n-30) \omega ^4+20 m_R^2 (5 (n-2) n-24) \omega ^6+3 (n-4) (n^2-4) n\omega ^8\bigg\}\,.\label{eq:delbe}\end{aligned}$$]{} We note that all dependence on the scale factor in (\[eq:della\] - \[eq:delbe\]) is of the form $k/a$ as it only comes from the $a$-dependence of the integration measure and $\omega^2=(k/a)^2+m^2$. Hence, by redefining the integration variable [$${\frac{{{k}}}{{a}}}\equiv {q}\qquad \Rightarrow\qquad {\frac{{d^{n-1}k}}{{(2\pi a)^{n-1}}}}\equiv{\frac{{d^{n-1}q}}{{(2\pi )^{n-1}}}}\label{eq:intv}\,,$$]{} (\[eq:della\] - \[eq:delbe\]) can be seen to be strictly constant. By again using standard formulae of dimensional regularization [@Peskin:1995ev] one may check that the divergencies of $\delta\Lambda^{\rm ad}$ and $\delta (M_{\rm P}^{2})^{\rm ad}$ match the well-known results, for example as calculated by other means in section 6.2 of [@Birrell:1982ix][^4]. In (\[eq:della\] - \[eq:delbe\]) we have now successfully extracted the constant counter terms that are implicitly defined by adiabatic subtraction. We emphasize once more that only the divergent parts in (\[eq:della\] - \[eq:delbe\]) are universal and finite contributions, or terms in the integrands that vanish faster than $\sim q^{-4}$ at the high ultraviolet when $n=4$, should be determined according to the dedicated renormalization prescription of one’s choosing. Note that since we are interested in the four-dimensional results, the last term in (\[eq:je\]) only generates a finite contribution, since from (\[eq:h4\]) one has$~^{(1)}H_{\mu\nu}\sim (4-n)H^4$ and by explicitly performing the integral in (\[eq:delbe\]) one can show that the leading term in $\delta\beta^{\rm ad}$ when approaching $n\rightarrow4$ scales as $(4-n)^{-1}$. With (\[eq:della\] - \[eq:delbe\]) we can now address the main focus of this article, which is finding renormalization prescriptions where the long wavelength limit of the renormalized power spectrum $(\mathcal{P}_\phi)_R$ is much less affected than in the adiabatic approach in (\[eq:Rp\]). Renormalization of the spectrum of perturbations {#sec:modp} ================================================ In order to derive a renormalized expression for the spectrum of perturbations from inflation as defined in (\[eq:R\]) we need to find a physical quantity that is a function of $\langle\hat{\phi}^2\rangle$. After all, a physical expression involving $\langle\hat{\phi}^2\rangle$ must also include the appropriate counter terms providing a way of defining $\delta\mathcal{P}_\phi$. We need to look no further than the Einstein equations, since by making use of expressions (\[eq:T00\]) and (\[eq:Tii\]), but for the full mode $u_\mathbf{k}^{\rm ad}\rightarrow u_\mathbf{k}^{\phantom{\rm ad}}$ and with the definition for the spectrum (\[eq:PS\]) we get [$$\langle \hat{T}^Q_{00}\rangle-a^{-2}\langle \hat{T}^Q_{ii}\rangle=\int_0^\infty {\frac{{k^{n-2}dk}}{{(2\pi a)^{n-1}}}}\bigg({\frac{{2-n}}{{1-n}}}{\frac{{{k}^2}}{{a^2}}}+m_R^2\bigg){\frac{{\mathcal{P_\phi}}}{{(k/a)^{n-1}}}}\,.\label{eq:toi}$$]{} Equation (\[eq:toi\]) after the appropriate renormalization we can then use as the definition of the renormalized power spectrum [$$\begin{aligned} \langle\hat{T}^Q_{00}\rangle_R-a^{-2}\langle \hat{T}^Q_{ii}\rangle_R&=\langle \hat{T}^Q_{00}\rangle-a^{-2}\langle \hat{T}^Q_{ii}\rangle\nonumber \\&-\big[-2\delta\Lambda+\delta M_{\rm P}^{2}\big(G_{00}-a^{-2}G_{ii}\big) +\delta\beta\big(^{(1)}H_{00}-a^{-2}~^{(1)}H_{ii}\big)\big]\nonumber \\ &\equiv \int_0^\infty {\frac{{k^{n-2}dk}}{{(2\pi a)^{n-1}}}}\bigg({\frac{{2-n}}{{1-n}}}{\frac{{{k}^2}}{{a^2}}}+m_R^2\bigg){\frac{{\mathcal{P_\phi}-\delta \mathcal{P_\phi}}}{{(k/a)^{n-1}}}}\,.\label{eq:physP}\end{aligned}$$]{} As an example we can use the momentum space expressions for the adiabatic counter terms for $\delta\Lambda^{\rm ad}$ and $\delta(M_{\rm P}^2)^{\rm ad}$, furthermore setting $\delta\beta=0$ and finally solve $(\mathcal{P}_\phi)_R$ from (\[eq:physP\]) in $n=4$ to be [$$\begin{aligned} \mathcal{P_\phi}-\delta \mathcal{P}_\phi^{\rm ad}\equiv(\mathcal{P}_\phi)_R&=\bigg({\frac{{k}}{{a H}}}\bigg)^3{\frac{{H^2}}{{8\pi}}}\bigg\{\big\|H^{(1)}_{{\nu}}\big(k/(aH)\big)\big\|^2 \nonumber \\&-H(8\pi)^2\bigg({\frac{{2}}{{3}}}{\frac{{{k}^2}}{{a^2}}}+m_R^2\bigg)^{-1}\Big[-\delta\Lambda^{\rm ad}_\mathbf{k}+\delta (M_{\rm P}^{2})^{\rm ad}_\mathbf{k} 3H^2\Big]\bigg\}\,,\label{eq:adpi}\end{aligned}$$]{} which after plugging in the explicit results for $\delta\Lambda^{\rm ad}_\mathbf{k}$ and $\delta(M_{\rm P}^2)^{\rm ad}_\mathbf{k}$ from (\[eq:della\]) and (\[eq:dellab\]) precisely coincides with the result of the derivation of section \[sec:subtraction\] given in equation (\[eq:Rp\]). The reason why we needed to choose $\delta\beta=0$ in order to find agreement with adiabatic subtraction is that the variance generates divergences only to second adiabatic order and in the adiabatic prescription the subtraction term is truncated to neglect the orders beyond the divergences. However, the energy-momentum tensor generically has divergences up to fourth order and since we used the energy-momentum tensor to define the physical $(\mathcal{P}_\phi)_R$ in our approach a dependence on counter terms of $\mathcal{O}(H^4)$ is introduced, although $\delta\beta$ is strictly not required to cancel any divergences in $\mathcal{P}_\phi$. Before proceeding let us first discuss a few key elements of our approach and the definition for the renormalization of the power spectrum $(\mathcal{P}_\phi)_R$ we introduced in (\[eq:physP\]). Comparing the adiabatic prescription of section \[sec:subtraction\] and the result (\[eq:adpi\]) it seems we have only been able to derive the identical results in a different form, but in a seemingly more cumbersome manner. The significance of our approach however lies in the fact that throughout we maintain a connection with counter terms in the traditional sense as the renormalized spectrum depends explicitly on coupling constants introduced by the tree-level action. Namely, we succesfully mapped the renormalization of the power spectrum into redefinitions of the cosmological constant and the Planck mass, with the simplifying choice of $\delta\beta=0$. By deciphering the counter term content implicitly defined by adiabatic subtraction in (\[eq:della\] - \[eq:delbe\]) then allows us to make a leap forward: any set of counter terms that coincides with (\[eq:della\] - \[eq:delbe\]) in terms of the divergences represents in principle a valid renormalization prescription. As the explicit expressions (\[eq:della\] - \[eq:delbe\]) are given as momentum space integrals modifying the finite terms is straightforward, as we will now show. Suppose that instead of the adiabatic counter terms (\[eq:della\] - \[eq:delbe\]) we made use of the following choices [$$\begin{aligned} \label{eq:della0}\delta\underline{\Lambda}\equiv\int{\frac{{d^{n-1}k}}{{(2\pi a)^{n-1}}}}e^{-(aM/k)^5}\delta\Lambda^{\rm ad}_{\mathbf{k}}\,,\qquad\delta \underline{M}_{\rm P}^{2}\equiv\int{\frac{{d^{n-1}k}}{{(2\pi a)^{n-1}}}}e^{-(aM/k)^5}\delta(M_{\rm P}^{2})^{\rm ad}_{\mathbf{k}}\,,\qquad \delta\underline{\beta}=0\,,\end{aligned}$$]{} where $M$ is some large mass scale. For $n=4$ the leading infinities in (\[eq:della\] - \[eq:delbe\]) come from terms $\sim\int^\infty dq\, q^3$ where again the time dependence has disappeared by way of (\[eq:intv\]) and hence the above counter terms generate precisely the same divergences as (\[eq:della\] - \[eq:delbe\]). However, they become quickly suppressed for $q<M$ and the observable modes in $\mathcal{P}_\phi$ are unaltered in this renormalization prescription. Since (\[eq:della0\]) correspond to constant redefinitions of the cosmological constant and the Planck mass and furthermore we have not introduced a cut-off but give the counter terms as formally divergent integrals covariant conservation of the renormalized energy-momentum tensor (\[eq:covCons1\]) must hold. Furthermore, one may calculate the difference in renormalization prescriptions for the energy-momentum tensor between the adiabatic and modified approach by making use of (\[eq:match\]), (\[eq:della\] - \[eq:delbe\]) and (\[eq:della0\]) and subtracting the adiabatic result from the modified one, which results in a finite contribution formed as a linear combination of only local and conserved curvature tensors, $g_{\mu\nu},\, G_{\mu\nu}$ and ${\,}^{(1)}H_{\mu\nu}$. This is in accord with the ambiguity allowed by the axiomatic approach, as discussed at the end of section \[sec:subtraction\]. Finally, in regards to our discussion after equation (\[eq:porp\]) since the prescription (\[eq:della0\]) was defined by modifying the adiabatic expressions, there is no danger of generating new divergences that require the $n$-dimensional forms of the integrals and one may consistently set $n=4$. Making the choices in (\[eq:della0\]) will via (\[eq:physP\]) give rise to a renormalized power spectrum which explicitly reads [$$\begin{aligned} \mathcal{P}_\phi-\delta \underline{\mathcal{P}}_\phi&\equiv (\mathcal{P}_\phi)_R=\bigg({\frac{{k}}{{a H}}}\bigg)^3{\frac{{H^2}}{{8\pi}}}\bigg\{\big\|H^{(1)}_{{\nu}}\big(k/(aH)\big)\big\|^2\nonumber \\&-e^{-(aM/k)^5}\bigg[\frac{2 H}{\pi \omega}-\frac{H^3 \left(5 m_R^4-6 m_R^2 \omega ^2-8 \omega^4\right)}{4 \pi \omega^7}\bigg]\bigg\}\,;\qquad \omega^2\equiv {\frac{{k^2}}{{a^2}}}+m_R^2\,,\label{eq:Rpr}\end{aligned}$$]{} and which trivially coincides with the bare results for the observable modes, in particular [$$(\mathcal{P}_\phi)_R\longrightarrow \bigg({\frac{{H}}{{2\pi}}}\bigg)^2\,;\quad \text{for}\quad {\frac{{k}}{{aH}}}\longrightarrow0~~\text{and}~~\nu\longrightarrow3/2\,,$$]{} without compromising the successful removal of the ultraviolet divergences or coordinate invariance of the theory. As a final comment, we stress that the counter terms in (\[eq:della0\]) serve merely to illustrate that there exist consistent renormalization prescriptions where the observable modes in the spectrum of inflationary perturbations are not affected, contrary to the claim of [@Parker:2007]. A different question all together is providing a set of renormalization conditions based on observations corresponding to clear physical definitions for the cosmological constant and the Planck mass. However, this question can also be approached with the techniques presented here and in [@Markkanen:2013nwa]. Discussion ========== In works investigating Early Universe phenomenology, in particular the physics and implications of cosmological inflation, renormalization in any form is rarely included in the discussion which stems perhaps from the implicit assumption that observable predictions from the bare theory are not significantly altered by it. Although renormalizing a quantum theory on a curved background is known to be a non-trivial endeavour, for any theory containing divergent correlators some consistent procedure for removing the infinities must in principle be applied in order to obtain robust results. If the subtractions involved in devising a well-defined finite theory in curved space unavoidably gave rise to a significant change over the bare results as suggested in [@Parker:2007] it would force the community to completely overhaul the standard framework with which observable predictions from inflation are derived. This compels a closer look on the process of renormalization in curved space and specifically its effect on the amplitude of perturbations from inflation which was the primary source of the claim in [@Parker:2007] and the debate that followed. The specific prescription which when implemented on the renormalization of inflationary perturbations significantly modifies the bare results is adiabatic subtraction or regularization, a technique which does not immediately connect to the standard approaches used in the context of particle physics: it does not rely on explicitly redefining the coupling constants of the action to produce the necessary counter terms with which the infinities are removed. It also does not seem to require any information about the specific conditions one needs to choose for defining the physical couplings. This leaves the choices made for finite pieces in the subtraction, which in any subtraction prescription are unavoidable, completely implicit. A technique that can overcome these issues was laid out recently in [@Markkanen:2013nwa]. In this work by making use of the approach of [@Markkanen:2013nwa] we reduced the procedure of adiabatic subtraction into a redefinition of constants allowing one to easily connect it to the more traditional approaches of renormalization. From the renormalized energy-momentum we were able to define the renormalized power spectrum in a way where the subtraction terms were directly mapped to redefinitions of the bare cosmological constant and Planck mass. Disentangling the implicit definitions for the counter terms made in the adiabatic prescription allowed us to find other prescriptions with relative ease, which differ from one another by only local and conserved tensors and hence fall within the ambiguity allowed by the axiomatic approach of [@Wald:1977up; @Wald:1978pj; @Wald:1978ce]. Finally, we showed how one may renormalize the power spectrum such that it is observationally indistinguishable from the prediction of the bare theory. Our main result is likely to be viewed favourably by the majority of the community working on Early Universe physics as it implies that the observable spectrum of perturbations is not necessarily modified by renormalization, which is the mainstream assumption. However, making any stronger claims at this stage would be somewhat premature: we have showed that prescriptions exist where this is true, but it is a different question to define a well-motivated physical condition for renormalization where the same continues to hold. It is also non-trivial to generalize our approach to the case of an interacting theory as well as to a case where the background is not approximated to be strictly de Sitter, which also would require the inclusion of the metric fluctuations into the calculation. To conclude, we mention a potentially fruitful future line of research implied by our results. 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Torrado and D. Wands, “The stochastic spectator,” JCAP [1710]{} (2017) 018 doi:10.1088/1475-7516/2017/10/018 \[arXiv:1701.06473 \[astro-ph.CO\]\]. [^1]: Often the cosmological constant is defined to have mass dimension two, which is obtained from our definition via $\Lambda\rightarrow M_{\rm P}^2\Lambda$ [^2]: In a general FLRW background this cannot be done since$~^{(1)}H_{\mu\nu}$,$~^{(2)}H_{\mu\nu}$ and $H_{\mu\nu}$ all have distinct expressions in terms of ${\ensuremath{\dot{a}}}/a$, ${\ensuremath{\ddot{a}}}/a$, ${\ensuremath{\dddot{a}}}/a$ and $a^{(4)}/a$ with dimension $H^4$ [@Markkanen:2013nwa]. [^3]: See in particular equation (2.14), Appendix B and the discussion in section 5.2 of [@Markkanen:2013nwa]. [^4]: [$$\delta\Lambda^{\rm ad}=\frac{m_R^4}{32 \pi ^2 (4-n)}+\cdots\,;\qquad\delta (M_{\rm P}^{2})^{\rm ad}=\frac{m_R^2}{48 \pi ^2 (4-n)}+\cdots$$]{} As we have made use of simplifications arising in strict de Sitter space a direct correspondence with the results of [@Birrell:1982ix] and $\delta\beta^{\rm ad}$ is less obvious.
--- address: 'Department of Mathematics, University of California, San Diego, USA' author: - Jonathan Novak title: On the Complex Asymptotics of the HCIZ and BGW Integrals --- > There seems to be a connection between large $N$ and the permutation groups. — Stuart Samuel, 1980 Introduction ============ Objective --------- The purpose of this paper is to prove a longstanding conjecture on the $N \to \infty$ asymptotic behavior of the Harish-Chandra/Itzykson-Zuber (HCIZ) integral, $$\label{eqn:HCIZ} I_N = \int_{\mathrm{U}(N)} e^{zN \mathrm{Tr} AUBU^{-1}} \mathrm{d}U,$$ and its additive counterpart, the Brézin-Gross-Witten (BGW) integral, $$\label{eqn:BGW} J_N = \int_{\mathrm{U}(N)} e^{zN \mathrm{Tr}(AU + BU^{-1})} \mathrm{d}U.$$ These are integrals over $N \times N$ unitary matrices against unit mass Haar measure, the integrands of which depend on a complex parameter $z$ and a pair of $N \times N$ complex matrices $A$ and $B$. The conjecture we prove emerged from a cluster of 1980 theoretical physics papers on the large $N$ limit of ${\mathrm{U}}(N)$ lattice gauge theory [@Bars; @BG; @GW; @IZ; @Samuel; @Wadia], and has been of perennial interest in physics ever since; see the reviews [@BBMP; @Morozov; @ZZ]. It entered mathematics in the early 2000s along with growing interest in random matrices, and was precisely formulated in work of Collins [@Collins Section 5], Guionnet [@Guionnet:PS Section 4.3], and Zelditch [@Z Section 4]. The conjecture has since attained the status of an outstanding open problem in asymptotic analysis, and has become perhaps the most prominent question at the confluence of random matrix theory and representation theory; see e.g. [@BGH] for a recent perspective. It may be stated as follows. Given a Young diagram $\alpha$ with $d$ cells, $\ell(\alpha)$ rows, and $\alpha_i$ cells in the $i$th row, let $$p_\alpha(x_1,\dots,x_N) = \prod_{i=1}^{\ell(\alpha)} \sum_{j=1}^N x_j^{\alpha_i}$$ be the corresponding Newton power sum symmetric polynomial in $N$ variables. \[conj:Main\] Given any $M \geq 0,$ there exists a corresponding $\varepsilon_M>0$ such that, for any integer $k \geq 0$, $$I_N = e^{\sum_{g=0}^k N^{2-2g}F_N^{(g)}+ o(N^{2-2k})} \quad\text{ and }\quad J_N = e^{\sum_{g=0}^k N^{2-2g}G_N^{(g)} + o(N^{2-2k})}$$ as $N \to \infty$, where the error term is uniform over complex numbers $z$ of modulus at most $\varepsilon_M$ and complex matrices $A,B$ of spectral radius at most $M$, and $$\begin{split} F_N^{(g)} &= \sum_{d=1}^\infty \frac{z^d}{d!} \sum_{\alpha,\beta \vdash d} \frac{p_\alpha(a_1,\dots,a_N)}{N^{\ell(\alpha)}} \frac{p_\beta(b_1,\dots,b_N)}{N^{\ell(\beta)}} F_g(\alpha,\beta), \\ G_N^{(g)} &= \sum_{d=1}^\infty \frac{z^{2d}}{d!} \sum_{\beta \vdash d} \frac{p_\beta(c_1,\dots,c_N)}{N^{\ell(\beta)}} G_g(\beta) , \end{split}$$ are analytic functions of $z$, the eigenvalues $a_1,\dots,a_N$ of $A$, the eigenvalues $b_1,\dots,b_N$ of $B$, and the eigenvalues $c_1,\dots,c_N$ of $C=AB$. Moreover, the coefficients $F_g(\alpha,\beta)$ and $G_g(\beta)$ are integers. The main result of this paper is a proof of Conjecture \[conj:Main\]. Before outlining our argument, let us briefly unpack the conjecture’s meaning. Its salient feature is the claim that $I_N$ and $J_N$ admit what physicists call “strong coupling expansions” — their logarithms have complete $N \to \infty$ asymptotic expansions on the scale $N^{2-2g}$ provided the “coupling constant” $z$ is sufficiently small and the “external fields” $A$ and $B$ are uniformly bounded (our parameter $z$ is inversely proportional to the physical coupling constant, so that small $\|z\|$ corresponds to strong coupling). Without loss in generality, we may take $M=1$ as the uniform bound on the spectral radii of $A$ and $B$. The conjecture then asserts the existence of $\varepsilon>0$ such that, for any given $k \geq 0,\kappa > 0$, there is a corresponding $N(k,\kappa)$ with $N \geq N(k,\kappa)$ implying $$\left\| \log I_N - \sum_{g=0}^k N^{2-2g} F_N^{(g)} \right\| \leq \kappa N^{2-2k} \quad\text{ and }\quad \left\| \log J_N - \sum_{g=0}^k N^{2-2g} G_N^{(g)} \right\| \leq \kappa N^{2-2k}$$ for all complex numbers $z$ of modulus at most $\varepsilon$ and all complex matrices $A,B$ with eigenvalues of modulus at most $1,$ where “$\log$” denotes the principal branch of the complex logarithm. The coefficients of these purported asymptotic expansions — the “free energies” $F_N^{(g)}$ and $G_N^{(g)}$ — are themselves dependent on $N,$ and hence could conceivably interact with the asymptotic scale. The conjecture addresses this by further claiming that $F_N^{(g)}$ and $G_N^{(g)}$ are analytically determined by the data $(z,A,B)$ in a manner which precludes this possibility: it implies the bounds $$\|F_N^{(g)}\| \leq \sum_{d=1}^\infty \frac{\varepsilon^d}{d!} \sum_{\alpha,\beta \vdash d} \|F_g(\alpha,\beta)\| \quad\text{ and }\quad \|G_N^{(g)}\| \leq \sum_{d=1}^\infty \frac{\varepsilon^{2d}}{d!} \sum_{\beta \vdash d} \|G_g(\beta)\|,$$ which are finite and depend only on $\varepsilon$ and $g.$ Finally, the conjecture asserts that the universal coefficients $F_g(\alpha,\beta)$ and $G_g(\beta),$ which determine $F_N^{(g)}$ and $G_N^{(g)}$ but do not depend on the data $(z,A,B),$ are integers. This claim is rooted in the notion of “topological expansion,” a fundamental but analytically non-rigorous principle in quantum field theory which generalizes the apparatus of Feynman diagrams to matrix integrals [@tHooft1; @BIZ; @BIPZ; @DGZ; @IZ; @Witten], and beyond [@EO; @KS]. This principle predicts that the structure constants $F_g(\alpha,\beta)$ and $G_g(\beta)$ are combinatorial invariants of compact connected genus $g$ Riemann surfaces. Results ------- The main result of this paper is a proof of Conjecture \[conj:Main\]. Our argument proeeds in three stages: exact formulas, stable asymptotics, and functional asymptotics. ### Exact formulas Our point of departure is a pair of novel absolutely convergent series expansions of $I_N$ and $J_N$ which are amenable to large $N$ analysis. \[thm:StringExpansions\] For any $N \in \mathbb{N}$, we have $$\begin{aligned} I_N &= 1 + \sum_{d=1}^\infty \frac{z^d}{d!} {\mathbb{P}}({\mathrm{LIS}}_d \leq N) \sum_{\alpha,\beta \vdash d} p_\alpha(a_1,\dots,a_N) p_\beta(b_1,\dots,b_N) \langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle,\\ J_N &= 1 + \sum_{d=1}^\infty \frac{z^{2d}}{d!} N^d{\mathbb{P}}({\mathrm{LIS}}_d \leq N) \sum_{\beta \vdash d} p_\beta(c_1,\dots,c_N) \langle \Omega_N^{-1} \omega_\beta \rangle, \end{aligned}$$ where ${\mathbb{P}}({\mathrm{LIS}}_d \leq N)$ is the probability that a uniformly random permutation from the symmetric group ${\mathrm{S}}(d)$ has no increasing subsequence of length $N+1$, and $\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle$ is the Plancherel expectation of a certain natural observable of Young diagrams with $d$ cells and at most $N$ rows. These series converge absolutely and uniformly on compact subsets of ${\mathbb{C}}^{2N+1}$ and ${\mathbb{C}}^{N+1}$, respectively. We call these series the “string expansions” of $I_N$ and $J_N;$ this terminology is explained in Section \[sec:Exact\] below. In the absence of external fields, the string expansion of the BGW integral reduces to the beautiful formula $$\int_{\mathrm{U}(N)} e^{zN \mathrm{Tr}(U + U^{-1})} \mathrm{d}U = 1+ \sum_{d=1}^\infty \frac{z^{2d}}{d!} N^{2d}{\mathbb{P}}({\mathrm{LIS}}_d \leq N),$$ which is due to Rains [@Rains] and equivalent to a result of Gessel [@Gessel]. The Gessel-Rains identity was the starting point of Baik, Deift, and Johansson [@BDJ] in their seminal work showing that the $d \to \infty$ fluctuations of ${\mathrm{LIS}}_d$ around its asymptotic mean value of $2\sqrt{d}$ are governed by the Tracy-Widom distribution. Informative expositions of this landmark result may be found in [@AD; @Romik; @Stanley:ICM]. The existence of a connection between the HCIZ integral and increasing subsequences appears to have been previously unknown. Since the Fourier transform of any unitarily invariant random matrix is a mixture of HCIZ integrals, the HCIZ-LIS connection exposes a new and very direct link between random matrices and random permutations. ### Stable asymptotics In Section \[sec:Stable\], we analyze the $N \to \infty$ asymptotics of each fixed string coefficient of $I_N$ and $J_N,$ i.e. the large $N$ asymptotics of the Plancherel expectation $\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle$ with fixed $\alpha,\beta \vdash d.$ We show that $\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle$ admits a convergent asymptotic expansion on the scale $1/N,$ and that this expansion is a generating function for “monotone” walks on the Cayley graph of the symmetric group ${\mathrm{S}}(d)$ with boundary conditions $\alpha,\beta.$ Monotone walks are self-interacting trajectories: the future of a monotone walk depends on its past. It is a fundamental fact, discovered in [@Novak:Banach] and further developed in [@MN], that these trajectories play the role of Feynman diagrams for integration against Haar measure on the unitary group. For any fixed $N \in \mathbb{N},$ one can replace the first $N$ string coefficients of $I_N$ and $J_N$ with their $1/N$ expansions, but not so for higher terms. The issue is conceptually similar to that faced when studying the homotopy groups of ${\mathrm{U}}(N)$, which behave regularly at first but eventually become wild. Topologists see past this by studying the stable unitary group ${\mathrm{U}}$, an $N=\infty$ version of ${\mathrm{U}}(N)$ which does not suffer from this defect [@Bott]. The price paid is that ${\mathrm{U}}$ is not a Lie group, but an infinite-dimensional manifold which is not locally compact. In Section \[sec:Stable\], we introduce the stable HCIZ and BGW integrals, $I$ and $J$, which are $N=\infty$ versions of $I_N$ and $J_N$. Conceptually, these objects are the integrals $$I = \int_{{\mathrm{U}}} e^{\frac{z}{\hbar} {\operatorname{Tr}}AUBU^{-1}} \mathrm{d}U \quad\text{ and }\quad J = \int_{{\mathrm{U}}} e^{\frac{z}{\hbar} {\operatorname{Tr}}(AU+BU^{-1})} \mathrm{d}U,$$ with $\hbar$ an infinitely small parameter, $A$ and $B$ infinitely large matrices, and $\mathrm{d}U$ the non-existent Haar measure on the stable unitary group ${\mathrm{U}}.$ Like the homotopy groups of ${\mathrm{U}},$ the topological expansions of $I$ and $J$ can be completely understood; the price paid is that $I$ and $J$ are not analytic functions, but formal power series in infinitely many variables which are not convergent. \[thm:MainStable\] We have $$I = e^{\sum_{g=0}^\infty \hbar^{2g-2} F^{(g)}} \quad\text{ and} \quad J = e^{\sum_{g=0}^\infty \hbar^{2g-2}G^{(g)}},$$ where the stable free energies $F^{(g)}$ and $G^{(g)}$ are generating functions for the genus $g$ monotone double and single Hurwitz numbers, respectively. Hurwitz theory, familiar to algebraic geometers as the prototypical enumerative theory of maps from curves to curves, plays a prominent role in contemporary enumerative geometry; see [@ELSV; @GJV; @KL; @OP], and [@DYZ] for a recent overview. Monotone Hurwitz theory [@GGN1; @GGN2; @GGN3; @GGN4; @GGN5] is a desymmetrized version of classical Hurwitz theory which, rather surprisingly, is exactly solvable to exactly the same extent. Just as there are explicit formulas for classical Hurwitz numbers in genus zero and one [@Hurwitz; @Vakil], there are explicit formulas for monotone Hurwitz numbers in genus zero and one [@GGN1; @GGN2], and the two sets of formulas are structurally analogous. Monotone Hurwitz numbers manifest versions of polynomiality [@GGN2] and integrability [@GGN4] which mirror the polynomiality [@ELSV] and integrability [@Okounkov:MRL] of their classical counterparts. The consonance between the classical and monotone theories is to some extent explained by the fact that both are governed by the Eynard-Orantin topological recursion formalism — the two theories are structurally identical, but are generated by different spectral curves [@BEMS; @DDM]. Monotone Hurwitz theory has proved to be a useful tool with diverse applications [@BK; @CDO; @GGR; @Montanaro; @Novak:JSP], and its discovery has sparked a surge of interest in combinatorial deformations of classical Hurwitz numbers [@ACEH; @ALS; @CD; @DDM; @DK; @DKPS; @HKL]. Although the subject has taken on a life of its own, monotone Hurwtz numbers were originally summoned from the void as a weapon with which to attack Conjecture \[conj:Main\]. In this paper, they fulfill their initial purpose. ### Functional asymptotics In order to prove Conjecture \[conj:Main\], we must descend from the stable world of $N =\infty$ the unstable world of finite $N.$ To navigate this passage, we must address the questions of convergence and approximation which are the analytic substance of Conjecture \[conj:Main\]. Prior knowledge of the stable limit, which comprises the combinatorial substance of Conjecture \[conj:Main\], is extremely useful in this regard — since we know what the answer is supposed to be, the analysis becomes a task of verification rather than discovery. More precisely, if Conjecture \[conj:Main\] is true then the free energies $F_N^{(g)}$ and $G_N^{(g)}$ must be generating functions for monotone Hurwitz numbers of genus $g.$ Remarkably, monotone Hurwitz theory guarantees that the stable free energies $F^{(g)}$ and $G^{(g)}$ remain stable at finite $N$: replacing the formal parameter $\hbar$ with $N^{-1}$ and the formal alphabets $A,B,C$ with the spectra of uniformly bounded $N \times N$ complex matrices yields absolutely summable power series. Even better, the radius of convergence of these series is bounded below by a positive constant $\delta$ independent of both $N$ and $g.$ We thus have explicit analytic candidates for $F_N^{(g)}$ and $G_N^{(g)},$ with a stable domain of holomorphy. The stable analyticity of $F_N^{(g)}$ and $G_N^{(g)}$ does not mean that one can deduce Conjecture \[conj:Main\] from Theorem \[thm:MainStable\] simply by replacing $\hbar$ with $N^{-1}$ — this fails because the series $$F_N = \sum_{g=0}^\infty N^{2-2g}F_N^{(g)} \quad\text{ and }\quad G_N = \sum_{g=0}^\infty N^{2-2g}G_N^{(g)}$$ are not uniformly convergent on any nondegenerate polydisc for any finite $N.$ This is typical of generating functions associated with 2D quantum gravity [@DGZ; @Witten], and one sees similar phenomena in the world of maps on surfaces and Hermitian matrix integrals [@EM; @Maurel]. The divergence of these series forces the introduction of a cutoff at fixed genus $g=k$, and an ensuing analysis of the holomorphic discrepancy functions $$1- \frac{I_N}{e^{\sum_{g=0}^k N^{2-2g}F_N^{(g)}}} \quad\text{ and }\quad 1-\frac{J_N}{e^{\sum_{g=0}^k N^{2-2g}G_N^{(g)}}}.$$ It is here that knowledge of the full string expansions of $I_N$ and $J_N$ at finite $N$ is essential: it leads to a “topological bound” which controls the moduli of the discrepancy functions on small polydiscs by a quantity of order $N^{2-2k}.$ Complex analytic tools may then be utilized to convert the topological bound into a topological approximation, replacing a uniform $O$-term with a uniform $o$-term at the logarithmic scale. The upshot of this analysis is our main theorem, which proves Conjecture \[conj:Main\]. \[thm:Main\] Conjecture \[conj:Main\] is true, and the structure constants $F_g(\alpha,\beta)$ and $G_g(\beta)$ are given by $$F_g(\alpha,\beta) = (-1)^{\ell(\alpha)+\ell(\beta)} {\vec{H}}_g(\alpha,\beta) \quad\text{ and }\quad G_g(\beta) = (-1)^{d+\ell(\beta)} {\vec{H}}_g(\beta),$$ where ${\vec{H}}_g(\alpha,\beta)$ and ${\vec{H}}_g(\beta)$ are the monotone double and single Hurwitz numbers of genus $g$. Context ------- Conjecture \[conj:Main\] is the subject of a large literature, and many powerful and impressive results have previously been obtained. For the HCIZ integral, the main highlight is Guionnet and Zeitouni’s large deviation theory proof [@GZ] of Matytsin’s heuristics [@Matytsin], which characterize the leading asymptotics of $I_N$ in terms of the flow of a compressible fluid. For the BGW integral, one has Johansson’s Toeplitz determinat proof [@Johansson:MRL] of Gross and Witten’s explicit formula [@GW] for the leading asymptotics of $J_N$ in the absence of external fields, a result which set the stage for the breakthrough work [@BDJ]. Another powerful technique is the use of Schwinger-Dyson “loop” equations [@Guionnet:book] to obtain both the leading [@CGM] and sub-leading [@GN] asymptotics of a large class of unitary matrix integrals containing the HCIZ and BGW integrals as prototypes. The common limitation of these prior works is that they are restricted to real asymptotics: they are obtained under the additional hypothesis that both the coupling constant and the eigenvalues of the external fields are real. This assumption is required in order to force the integrands of $I_N$ and $J_N$ to be positive functions on ${\mathrm{U}}(N),$ so that probabilistic methods can be applied. Indeed, all previous approaches to Conjecture \[conj:Main\] are, ultimately, elaborations of the classical Laplace method for the asymptotic evaluation of real integrals depending on a large real parameter. As soon as complex parameters are allowed, $I_N$ and $J_N$ become oscillatory integrals. The failure of previous works to treat the complex asymptotics of $I_N$ and $J_N$ is not just a technical limitation: many if not most situations in which one would like to invoke the conclusion of Conjecture \[conj:Main\] involve complex parameters in an essential way. For example, in order to analyze the spectral asymptotics of random matrices using characteristic functions, one needs the asymptotics of the orbital integral $I_N$ with complex coupling $z=i,$ which were previously inaccessible, except in certain degenerate scaling limits [@GM; @OlshVersh]. This is the sole reason that Fourier analysis has not been a viable technique in the asymptotic spectral analsysi of random matrices. For exactly the same reason, it has not been possible to make direct use of the Harish-Chandra/Kirillov formula [@HC; @K] in asymptotic representation theory. The results of this paper clear the way for a direct and unified approach to asymptotic random matrix theory and asymptotic representation theory based on Fourier analysis. Our results can moreover be applied to analyze certain asymptotic problems of physical problems interest which have been mired in confusion for some time [@McNov]. We have taken a conceptual as opposed to computational approach to the asymptotics of the HCIZ and BGW integrals by first constructing and understanding their $N=\infty$ stable limits and then using this insight to build $N \to \infty$ approximations. A first pass at this was made in [@GGN3], where Goulden, Guay-Paquet and the author succeeded in obtaining complete asymptotics for each fixed HCIZ string coefficient, but failed to understand the full string series at finite $N$ and its remarkable connection with longest increasing subsequences and Plancherel measure. Consequently, [@GGN3] failed to bridge the infinitely large gap between $N=\infty$ and $N \to \infty.$ Moreover, the fundamental fact that the relationship between the HCIZ and BGW integrals is precisely the relationship between double and single Hurwitz numbers was not perceived in [@GGN3], where the BGW integral was not considered. Indeed, prior to the present work, no matrix model for monotone single Hurwitz numbers was known, and it was an open question to find one [@ALS; @DDM]. Given that the known matrix model [@BEMS] for classical single Hurwitz numbers is somewhat contrived, it is remarkable that its counterpart for monotone single Hurwitz numbers is given by none other than the BGW integral, the basic special function of lattice gauge theory. Exact Formulas {#sec:Exact} ============== In this section, we prove Theorem \[thm:StringExpansions\], which is the starting point of our analysis. Symmetric polynomials --------------------- Given a Young diagram $\alpha$, the associated Newton power sum symmetric polynomial $p_\alpha$ in commuting variables $x_1,\dots,x_N$ is $$p_\alpha(x_1,\dots,x_N) = \prod_{i=1}^{\ell(\alpha)} \sum_{j=1}^N x_j^{\alpha_i}.$$ It is a classical result of Newton (see [@Macdonald; @Stanley:EC2]) that the polynomials $$p_\alpha(x_1,\dots,x_N), \quad \alpha \vdash d,$$ span the space $\Lambda_N^{(d)}$ of homogeneous degree $d$ symmetric polynomials in $x_1,\dots,x_N$. The Newton polynomials interface naturally with analysis: if $a_1,\dots,a_N$ is a point configuration in $\mathbb{C}$, then normalized power sums evaluated on these points are products of moments of the corresponding empirical probability measure $\mu.$ That is, we have $$\frac{p_\alpha(a_1,\dots,a_N)}{N^{\ell(\alpha)}} = \prod_{i=1}^{\ell(\alpha)} \int_{\mathbb{C}} \zeta^{\alpha_i} \mu(\mathrm{d}\zeta).$$ In particular, the normalized power sums which appear in Conjecture \[conj:Main\] are products of moments of the empirical eigenvalue distributions of the matrices $A,B$, and $C$. The power sums are the preferred basis for coupling expansions in lattice gauge theory, where they are referred to as “string states” [@BT]. There is another family of symmetric polynomials which play a role in what follows: the Schur polynomials. Given a Young diagram $\lambda$ with $d$ cells, let $(\mathsf{V}^\lambda,R^\lambda)$ denote the corresponding irreducible complex representation of the symmetric group ${\mathrm{S}}(d)$, and set $$\chi_\alpha(\lambda) = {\operatorname{Tr}}R^\lambda(\pi),$$ where $\pi \in {\mathrm{S}}(d)$ belongs to the conjugacy class $C_\alpha$ of permutations of cycle type $\alpha$. The Schur polynomials, $$s_\lambda(x_1,\dots,x_N) = \frac{1}{d!} \sum_{\alpha \vdash d} \|C_\alpha\| \chi_\alpha(\lambda) p_\alpha(a_1,\dots,a_N), \quad \lambda \vdash d, \ell(\lambda) \leq N.$$ form a basis of $\Lambda_N^{(d)}$, and the expansion of a given Newton polynomial in the Schur basis is $$p_\alpha(x_1,\dots,x_N) = \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} \chi_\alpha(\lambda) s_\lambda(x_1,\dots,x_N).$$ Evaluations of Schur polynomials at complex points also have representation-theoretic meaning: they are irreducible characters of the general linear group ${\mathrm{GL}}_N({\mathbb{C}}).$ More precisely, given a Young diagram $\lambda$ with at most $N$ rows, let $(\mathsf{W}^\lambda,S^\lambda)$ denote the corresponding irreducible polynomial representation of ${\mathrm{GL}}_N(\mathbb{C})$. Then $${\operatorname{Tr}}S^\lambda(A) = s_\lambda(a_1,\dots,a_N)$$ for any $A \in \mathrm{GL}_N(\mathbb{C})$ with eigenvalues $a_1,\dots,a_N$. Basic integrals --------------- Given a symmetric polynomial $f$ in $N$ variables and an $N \times N$ matrix matrix $A$, write $f(A)$ for the evaluation of $f$ on the spectrum of $A$. We shall need the following basic integration formulas, which are well-known manifestations of Schur orthogonality, see e.g. [@Macdonald]. We provide a proof for the sake of completeness. \[lem:BasicIntegrals\] For any Young diagrams $\lambda,\mu$ and matrices $A,B \in \mathrm{Mat}_N(\mathbb{C})$, we have $$\int_{{\mathrm{U}}(N)} s_\lambda(AUBU^{-1}) \mathrm{d}U = \frac{s_\lambda(A) s_\lambda(B)}{\dim \mathsf{W}^\lambda}$$ and $$\int_{{\mathrm{U}}(N)} s_\lambda(AU) s_\mu(BU^{-1}) \mathrm{d}U = \delta_{\lambda\mu}\frac{s_\lambda(AB)}{\dim \mathsf{W}^\lambda}.$$ Suppose first that $A,B \in {\mathrm{GL}}_N(\mathbb{C})$. Then $AUBU^{-1} \in {\mathrm{GL}}_N({\mathbb{C}})$, and we have $$\begin{aligned} s_\lambda(AUBU^{-1}) &= {\operatorname{Tr}}S^\lambda(AUBU^{-1}) \\ &= {\operatorname{Tr}}S^\lambda(A)S^\lambda(U)S^\lambda(B)S^\lambda(U^{-1}) \\ &= \sum_{i,j,k,l=1}^N S^\lambda(A)_{ij} S^\lambda(U)_{jk} S^\lambda(B)_{kl} S^\lambda(U^{-1})_{li}. \end{aligned}$$ Thus $$\int_{{\mathrm{U}}(N)} s_\lambda(AUBU^{-1}) \mathrm{d}U = \sum_{i,j,k,l=1}^N S^\lambda(A)_{ij} S^\lambda(B)_{kl} \int_{{\mathrm{U}}(N)} S^\lambda(U)_{jk}S^\lambda(U^{-1})_{li} \mathrm{d}U.$$ By Schur orthogonality for the matrix elements of an irreducible representation, we have $$\int_{{\mathrm{U}}(N)} S^\lambda(U)_{jk}S^\lambda(U^{-1})_{li} \mathrm{d}U = \frac{\delta_{ij}\delta_{kl}}{\dim \mathsf{W}^\lambda},$$ and hence $$\begin{aligned} \int_{{\mathrm{U}}(N)} s_\lambda(AUBU^{-1}) \mathrm{d}U &= \frac{1}{\dim \mathsf{W}^\lambda} \sum_{i=1}^N S^\lambda(A)_{ii} \sum_{k=1}^N S^\lambda(B)_{kk} \\ &= \frac{{\operatorname{Tr}}S^\lambda(A) {\operatorname{Tr}}S^\lambda(B)}{\dim \mathsf{W}^\lambda} \\ &= \frac{s_\lambda(A) s_\lambda(B)}{\dim \mathsf{W}^\lambda}. \end{aligned}$$ Similarly, if $A,B \in {\mathrm{GL}}_N({\mathbb{C}})$, then $AU,BU^{-1} \in {\mathrm{GL}}_N({\mathbb{C}}),$ and we have $$\begin{aligned} s_\lambda(AU) = {\operatorname{Tr}}S^\lambda(A)S^\lambda(U) &= \sum_{i,j=1}^N S^\lambda(A)_{ij} S^\lambda(U)_{ji} \\ s_\mu(BU^{-1}) = {\operatorname{Tr}}S^\mu(B)S^\mu(U^{-1}) &=\sum_{k,l=1}^N S^\mu(B)_{kl} S^\mu(U^{-1})_{lk}. \end{aligned}$$ Thus $$\int_{{\mathrm{U}}(N)} s_\lambda(AU) s_\mu(BU^{-1}) \mathrm{d}U = \sum_{i,j,k,l=1}^N S^\lambda(A)_{ij} S^\mu(B)_{kl} \int_{{\mathrm{U}}(N)} S^\lambda(U)_{ji} S^\mu(U^{-1})_{lk} \mathrm{d}U.$$ By Schur orthogonality for the matrix elements of different irreducible representations, $$\int_{{\mathrm{U}}(N)} S^\lambda(U)_{ji} S^\mu(U^{-1})_{lk} \mathrm{d}U = \frac{ \delta_{\lambda\mu} \delta_{il}\delta_{jk}}{\dim \mathsf{W}^\lambda},$$ and we conclude that $$\int_{{\mathrm{U}}(N)} s_\lambda(AU) s_\mu(BU^{-1}) \mathrm{d}U = \frac{\delta_{\lambda\mu}}{\dim \mathsf{W}^\lambda} \sum_{i,j=1}^N S^\lambda(A)_{ij} S^\mu(B)_{ji} = \delta_{\lambda\mu} \frac{s_\lambda(AB)}{\dim \mathsf{W}^\lambda}.$$ That these integral evaluations remain valid for arbitrary complex matrices $A$ and $B$ can be seen by taking limits. Let $(A_n)_{n=1}^\infty$ and $(B_n)_{n=1}^\infty$ be sequences in ${\mathrm{GL}}_N({\mathbb{C}})$ such that $$\lim_{n \to \infty} A_n = A \quad\text{ and }\quad \lim_{n \to \infty} B_n=B,$$ and apply the Dominated Convergence Theorem to obtain $$\begin{aligned} \int_{{\mathrm{U}}(N)} s_\lambda(AUBU^{-1}) \mathrm{d}U &= \int_{{\mathrm{U}}(N)} \lim_{n \to \infty} s_\lambda(A_nUB_nU^{-1}) \mathrm{d}U \\ &= \lim_{n \to \infty} \int_{{\mathrm{U}}(N)} s_\lambda(A_nUB_nU^{-1}) \mathrm{d}U \\ &= \lim_{n \to \infty} \frac{s_\lambda(A_n) s_\lambda(B_n)}{\dim \mathsf{W}^\lambda} \\ &= \frac{s_\lambda(A) s_\lambda(B)}{\dim \mathsf{W}^\lambda} \end{aligned}$$ and $$\begin{aligned} \int_{{\mathrm{U}}(N)} s_\lambda(AU) s_\mu(BU^{-1}) \mathrm{d}U &= \int_{{\mathrm{U}}(N)} \lim_{n \to \infty} s_\lambda(A_nU) s_\mu(B_nU^{-1}) \mathrm{d}U \\ &= \lim_{n \to \infty} \int_{{\mathrm{U}}(N)} s_\lambda(A_nU) s_\mu(B_nU^{-1}) \mathrm{d}U \\ &= \lim_{n \to \infty} \delta_{\lambda\mu} \frac{s_\lambda(A_nB_n)}{\dim \mathsf{W}^\lambda} \\ &= \delta_{\lambda\mu} \frac{s_\lambda(AB)}{\dim \mathsf{W}^\lambda}. \end{aligned}$$ Character expansions -------------------- Lemma leads to the following series representations of $I_N$ and $J_N$ in terms of Schur polynomials. Expansions of this sort appear in various forms in the physics literature, and were perhaps first utilized in work of James [@James] in multivariate statistics, where $I_N$ and $J_N$ are treated as hypergeometric functions with matrix arguments. \[thm:CharacterExpansion\] For any $z \in \mathbb{C}$, and any $A,B \in \mathrm{Mat}_N(\mathbb{C})$, we have $$\begin{aligned} \int_{{\mathrm{U}}(N)} e^{zN{\operatorname{Tr}}AUBU^{-1}} \mathrm{d}U &= 1 + \sum_{d=1}^\infty \frac{z^d}{d!} N^d \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} s_\lambda(a_1,\dots,a_N)s_\lambda(b_1,\dots,b_N) \frac{\dim \mathsf{V}^\lambda}{\dim \mathsf{W}^\lambda} \\ \int_{{\mathrm{U}}(N)} e^{zN{\operatorname{Tr}}(AU+BU^{-1})} \mathrm{d}U &= 1 + \sum_{d=1}^\infty \frac{z^{2d}}{d!d!} N^{2d} \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} s_\lambda(c_1,\dots,c_N) \frac{(\dim \mathsf{V}^\lambda)^2}{\dim \mathsf{W}^\lambda} , \end{aligned}$$ where $a_1,\dots,a_N$ are the eigenvalues of $A$, $b_1,\dots,b_N$ are the eigenvalues of $B$, and $c_1,\dots,c_N$ are the eigenvalues of $C=AB$. These series converge absolutely and uniformly on compact subsets of ${\mathbb{C}}^{2N+1}$ and ${\mathbb{C}}^{N+1},$ respectively. Consider first the HCIZ integral. Differentiating under the integral sign, the Maclaurin series of $I_N$ as an entire function of $z$ is $$\begin{aligned} I_N &= \int_{{\mathrm{U}}(N)} e^{zN p_1(AUBU^{-1})} \mathrm{d}U \\ &= 1+ \sum_{d=1}^\infty \frac{z^d}{d!} N^d \int_{{\mathrm{U}}(N)} p_{1^d}(AUBU^{-1}) \mathrm{d}U \\ &= 1+\sum_{d=1}^\infty \frac{z^d}{d!} N^d \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} (\dim \mathsf{V}^\lambda) \int_{{\mathrm{U}}(N)} s_\lambda(AUBU^{-1}) \mathrm{d}U \\ &= 1 + \sum_{d=1}^\infty \frac{z^d}{d!} N^d \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} s_\lambda(A) s_\lambda(B)\frac{\dim \mathsf{V}^\lambda}{\dim \mathsf{W}^\lambda}, \end{aligned}$$ by Lemma \[lem:BasicIntegrals\]. For the BGW integral, we have $$\begin{aligned} J_N &= \int_{{\mathrm{U}}(N)} e^{zNp_1(AU)} e^{zNp_1(BU^{-1})} \mathrm{d}U \\ &= 1 + \sum_{d=1}^\infty \frac{z^{2d}}{d!d!} N^{2d} \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} \sum_{\substack{\mu \vdash d \\ \ell(\mu) \leq N}} (\dim \mathsf{V}^\lambda)(\dim V^\mu) \int_{{\mathrm{U}}(N)} s_\lambda(AU) s_\mu(BU^{-1}) \mathrm{d}U \\ &= 1 + \sum_{d=1}^\infty \frac{z^{2d}}{d!d!} N^{2d} \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} s_\lambda(AB)\frac{(\dim \mathsf{V}^\lambda)^2}{\dim \mathsf{W}^\lambda}, \end{aligned}$$ by Lemma \[lem:BasicIntegrals\]. Let us perform a consistency check by examining these formulas in the absence of external fields, i.e. when both $A$ and $B$ are the identity matrix. For the HCIZ integral, we see directly from the definition that $I_N=e^{zN^2}$ when $A$ and $B$ are the identity. In this case the character expansion of $I_N$ becomes $$I_N = 1 + \sum_{d=1}^\infty \frac{z^d}{d!} N^d \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} (\dim \mathsf{V}^\lambda) (\dim \mathsf{W}^\lambda) = 1 + \sum_{d=1}^\infty \frac{z^d}{d!} N^{2d},$$ where we have used the isotypic decomposition of the space of $N$-dimensional tensors of rank $d$ as an ${\mathrm{S}}(d) \times {\mathrm{GL}}_N({\mathbb{C}})$ module, $$\left( \mathbb{C}^N \right)^{\otimes d} \simeq \bigoplus_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} \mathsf{V}^\lambda \otimes \mathsf{W}^\lambda.$$ For the BGW integral, in the case $AB=I$ the character expansion becomes $$J_N = 1 + \sum_{d=1}^\infty \frac{z^{2d}}{d!d!} N^{2d} \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} (\dim \mathsf{V}^\lambda)^2 = 1 + \sum_{d=1}^\infty \frac{z^{2d}}{d!} N^{2d} {\mathbb{P}}({\mathrm{LIS}}_d \leq N),$$ where in the final equality we used the Robinshon-Schensted correspondence (see below). This is exactly the Gessel-Rains identity. Generalizations of the Gessel-Rains identity to integrals over truncated unitary matrices were obtained in [@Novak:EJC; @Novak:IMRN], and analogues for the other classical groups may be found in [@BR; @Rains]. String expansions ----------------- In order to address Conjecture \[conj:Main\], we want expansions of $I_N$ and $J_N$ in terms of Newton polynomials rather than Schur polynomials — string expansions rather than character expansions. We will now obtain the string expansions of $I_N$ and $J_N$ from their character expansions. As is well-known [@Macdonald; @Stanley:EC2], the dimension of $\mathsf{V}^\lambda$ is equal to the number of standard Young tableaux of shape $\lambda$. Thus, by the Robinson-Schensted correspondence [@Stanley:EC2], we have $$\sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} (\dim \mathsf{V}^\lambda)^2 = \|{\mathrm{S}}_N(d)\|,$$ where ${\mathrm{S}}_N(d) \subseteq {\mathrm{S}}(d)$ is the set of permutations with no increasing subsequence of length $N+1$. It follows that $$\lambda \mapsto \frac{(\dim \mathsf{V}^\lambda)^2}{\|{\mathrm{S}}_N(d)\|}$$ is the mass function of a probability measure on the set of Young diagrams with $d$ cells an at most $N$ rows. This probability measure is known as the (row-restricted) Plancherel measure, see [@Kerov; @Romik]. We denote expectation with respect to Plancherel measure by angled brackets: $$\langle f \rangle = \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} f(\lambda) \frac{(\dim \mathsf{V}^\lambda)^2}{\|{\mathrm{S}}_N(d)\|}.$$ Note that if $N \geq d$ then the restriction on number of rows is vacuous, and the Plancherel measure is a probability measure on the full set of Young diagrams with $d$ cells whose normalization constant is $\|\mathrm{S}(d)\|=d!.$ For a Young diagram $\alpha \vdash d$, let us identify the conjugacy class $C_\alpha \subset {\mathrm{S}}(d)$ with the formal sum of its elements, so that $C_\alpha$ becomes a central element in the group algebra $\mathbb{C}{\mathrm{S}}(d)$. By Schur’s Lemma, $C_\alpha$ acts as a scalar operator in any irreducible representation $(\mathsf{V}^\lambda,R^\lambda)$ of $\mathbb{C}{\mathrm{S}}(d)$, i.e. $$R^\lambda(C_\alpha) = \omega_\alpha(\lambda) I_{\mathsf{V}^\lambda}$$ where $$\omega_\alpha(\lambda) = \frac{\|C_\alpha\| \chi_\alpha(\lambda)}{\dim \mathsf{V}^\lambda}$$ and $I_{\mathsf{V}^\lambda} \in \mathrm{End} \mathsf{V}^\lambda$ is the identity operator. Let us introduce the positive function $\Omega_N$ on Young diagrams with $d$ cells and at most $N$ rows defined by $$\Omega_N(\lambda) = \frac{d!}{N^d} \frac{\dim \mathsf{W}^\lambda}{\dim \mathsf{V}^\lambda} = \prod_{\Box \in \lambda} \left( 1 + \frac{c(\Box)}{N} \right).$$ Here we have used the dimension formulas [@Macdonald; @Stanley:EC2] $$\dim \mathsf{V}^\lambda = \frac{d!}{\prod_{\Box \in \lambda} h(\Box)} \quad\text{ and }\quad \dim \mathsf{W}^\lambda = \prod_{\Box \in \lambda} \frac{N + c(\Box)}{h(\Box)},$$ where $h(\Box)$ is the hook length of a given cell $\Box \in \lambda$ (number of cells to the right of $\Box$ plus number of cells below $\Box$ plus one) and $c(\Box)$ is its content (column index less row index), to render $\Omega_N(\lambda)$ as an explicit product. Note that $$\Omega_N^{-1}(\lambda) = \prod_{\Box \in \lambda} \frac{1}{1+\frac{c(\Box)}{N}}$$ is well-defined and positive since $\ell(\lambda) \leq N.$ The functions $\Omega_N^{\pm 1}$ seem to be closely related to the “$\Omega$-points” considered by physicists in the context of gauge/string dualities [@BT; @CMR; @GT1], but which seem not to have been fully understood in that context. In terms of $\Omega_N$, the Schur function expansions of $I_N$ and $J_N$ are $$\begin{aligned} I_N &= 1 + \sum_{d=1}^\infty z^d \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} s_\lambda(A) s_\lambda(B) \Omega_N^{-1}(\lambda) \\ J_N &= 1 + \sum_{d=1}^\infty \frac{z^{2d}}{d!} N^d \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} s_\lambda(C) \Omega_N^{-1}(\lambda) \dim \mathsf{V}^\lambda. \end{aligned}$$ We now prove Theorem \[thm:StringExpansions\], which we restate here using the notation just established. For any $z \in \mathbb{C}$ and any $A,B \in \mathrm{Mat}_N(\mathbb{C})$, we have $$\begin{aligned} \int_{{\mathrm{U}}(N)} e^{zN{\operatorname{Tr}}AUBU^{-1}} \mathrm{d}U &= 1 + \sum_{d=1}^\infty \frac{z^d}{d!} {\mathbb{P}}({\mathrm{LIS}}_d \leq N) \sum_{\alpha,\beta \vdash d} p_\alpha(A)p_\beta(B) \langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle,\\ \int_{{\mathrm{U}}(N)} e^{zN{\operatorname{Tr}}(AU+BU^{-1})} \mathrm{d}U &= 1 + \sum_{d=1}^\infty \frac{z^{2d}}{d!} N^d{\mathbb{P}}({\mathrm{LIS}}_d \leq N) \sum_{\beta \vdash d} p_\beta(C)\langle \Omega_N^{-1} \omega_\beta \rangle, \end{aligned}$$ where $C=AB$. For the HCIZ integral, we have $$\begin{aligned} I_N&= 1 + \sum_{d=1}^\infty z^d \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} s_\lambda(A) s_\lambda(B) \Omega_N^{-1}(\lambda) \\ &= 1+ \sum_{d=1}^\infty z^d \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} \left( \sum_{\alpha \vdash d} \frac{\|C_\alpha\| \chi_\alpha(\lambda)}{d!} p_\alpha(A) \right) \left( \sum_{\beta \vdash d} \frac{\|C_\beta\| \chi_\beta(\lambda)}{d!} p_\beta(B) \right) \Omega_N^{-1}(\lambda) \\ &= 1 + \sum_{d=1}^\infty \frac{z^d}{d!} \frac{\|S_N(d)\|}{d!} \sum_{\alpha,\beta \vdash d} p_\alpha(A) p_\beta(B) \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} \frac{\|C_\alpha\| \chi_\alpha(\lambda)}{\dim \mathsf{V}^\lambda}\Omega_N^{-1}(\lambda) \frac{\|C_\beta\| \chi_\beta(\lambda)}{\dim \mathsf{V}^\lambda} \frac{(\dim \mathsf{V}^\lambda)^2}{\|S_N(d)\|} \\ &= 1+ \sum_{d=1}^\infty \frac{z^d}{d!}{\mathbb{P}}({\mathrm{LIS}}_d \leq N)\sum_{\alpha,\beta \vdash d} p_\alpha(A) p_\beta(B) \left\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \right\rangle. \end{aligned}$$ For the BGW integral, we have $$\begin{aligned} J_N &= 1 + \sum_{d=1}^\infty \frac{z^{2d}}{d!} N^d \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} s_\lambda(AB) \Omega_N^{-1}(\lambda) \dim \mathsf{V}^\lambda \\ &= 1 + \sum_{d=1}^\infty \frac{z^{2d}}{d!} N^d \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} \left( \sum_{\beta \vdash d} \frac{\|C_\beta\|\chi_\beta(\lambda)}{d!} p_\beta(AB) \right) \Omega_N^{-1}(\lambda) \dim \mathsf{V}^\lambda \\ &= 1 + \sum_{d=1}^\infty \frac{z^{2d}}{d!} N^d \frac{\|S_N(d)\|}{d!} \sum_{\beta \vdash d} \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} \Omega_N^{-1}(\lambda)\frac{\|C_\beta\|\chi_\beta(\lambda)}{\dim \mathsf{V}^\lambda} \frac{(\dim \mathsf{V}^\lambda)^2}{\|S_N(d)\|} p_\beta(AB) \\ &= 1 + \sum_{d=1}^\infty \frac{z^{2d}}{d!} N^d {\mathbb{P}}({\mathrm{LIS}}_d \leq N) \sum_{\beta \vdash d} p_\beta(AB) \langle \Omega_N^{-1} \omega_\beta \rangle. \end{aligned}$$ Basic bounds ------------ Let us write the string expansions of $I_N$ and $J_N$ in the form $$I_N = 1 + \sum_{d=1}^\infty \frac{z^d}{d!}I_N(d) \quad\text{ and }\quad J_N = 1 + \sum_{d=1}^\infty \frac{z^{2d}}{d!}J_N(d).$$ Thus $I_N(d)$ and $J_N(d)$ are the symmetric polynomials $$\begin{aligned} I_N(d) &= N^d \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} s_\lambda(a_1,\dots,a_N) s_\lambda(b_1,\dots,b_N) \frac{\dim \mathsf{V}^\lambda}{\dim \mathsf{W}^\lambda} \\ &= {\mathbb{P}}({\mathrm{LIS}}_d \leq N)\sum_{\alpha,\beta \vdash d} p_\alpha(a_1,\dots,a_N)p_\beta(b_1,\dots,b_N) \langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle \end{aligned}$$ and $$\begin{aligned} J_N(d) &= \frac{N^{2d}}{d!} \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} s_\lambda(c_1,\dots,c_N) \frac{(\dim \mathsf{V}^\lambda)^2}{\dim \mathsf{W}^\lambda} \\ &= N^d{\mathbb{P}}({\mathrm{LIS}}_d \leq N)\sum_{\beta \vdash d} p_\beta(c_1,\dots,c_N) \langle \Omega_N^{-1}\omega_\beta \rangle. \end{aligned}$$ The following bounds — which say that $I_N(d)$ and $J_N(d)$ have maximum modulus in the case of trivial external fields — will be needed in Section \[sec:Functional\]. \[prop:BasicInequalities\] For any $d,N \in \mathbb{N}$ and any $a_1,\dots,a_N,b_1,\dots,b_N,c_1,\dots,c_N \in {\mathbb{C}}$ of modulus at most one, we have $$\|I_N(d)\| \leq N^{2d} \quad\text{ and }\quad \|J_N(d)\| \leq {\mathbb{P}}({\mathrm{LIS}}_d \leq N) N^{2d}.$$ This follows from the fact that the Schur polynomials are monomial positive. Stable Asymptotics {#sec:Stable} ================== In this section, we analyze the $N \to \infty$ asymptotics of the string coefficients of $I_N$ and $J_N.$ We obtain a convergent $N \to \infty$ asymptotic expansion for each fixed string coefficient, the coefficients of which count monotone walks on the symmetric groups with prescribed length and boundary conditions. These expansions are grouped together to form the stable HCIZ and BGW integrals $I$ and $J$, which are formal power series. The stable integrals $I$ and $J$ satisfy a formal power series version of Conjecture \[conj:Main\], the form of which points the way to an analytic solution. String coefficients ------------------- Our present goal is to determine the $N \to \infty$ asymptotics of the string coefficients of $I_N$ and $J_N,$ $${\mathbb{P}}({\mathrm{LIS}}_d \leq N)\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle,$$ in the regime where $\alpha,\beta \vdash d$ are fixed and $N \to \infty.$ In this regime we may assume $N \geq d,$ so that the string coefficients are pure Plancherel expectations: $$\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle = \sum_{\lambda \vdash d} \omega_\alpha(\lambda) \Omega_N^{-1}(\lambda) \omega_\beta(\lambda) \frac{(\dim \mathsf{V}^\lambda)^2}{d!}.$$ Since $$\lim_{N \to \infty} \Omega_N^{-1}(\lambda) = 1$$ for any fixed $\lambda,$ these Plancherel expectations are deformations of the usual inner product on the center of the group algebra ${\mathbb{C}}{\mathrm{S}}(d),$ with respect to which the functions $\omega_\alpha$ form an orthogonal basis: $$\langle \omega_\alpha \omega_\beta \rangle = \sum_{\lambda \vdash d} \omega_\alpha(\lambda)\omega_\beta(\lambda) \frac{(\dim \mathsf{V}^\lambda)^2}{d!} = \delta_{\alpha\beta} \|C_\alpha\|.$$ We thus have $$\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle = \delta_{\alpha\beta} \|C_\alpha\| + o(1)$$ as $N \to \infty,$ simply because of the orthogonality of irreducible characters. We will now quantify the error term in this approximation. Let $\hbar$ be a complex parameter, and consider the function on Young diagrams $\lambda$ defined by $$\Psi_\hbar(\lambda) = \prod_{\Box \in \lambda} (1-\hbar c(\Box)).$$ This is a polynomial function of $\hbar$ whose roots are the reciprocals of the contents of the off-diagonal cells of $\lambda.$ Explicitly, this polynomial is given by $$\Psi_\hbar(\lambda) = \sum_{r=0}^d (-\hbar)^r e_r(\lambda),$$ where $e_r(\lambda)$ is the degree $r$ elementary symmetric polynomial in $d$ variables, $$e_r = \sum_{\substack{ i \colon [r] \to [d] \\ i \text{ strictly increasing}}} x_{i(1)} \dots x_{i(r)},$$ evaluated on the contents of the diagram $\lambda$. Note that $e_r(\lambda)$ is a shifted symmetric function of $\lambda;$ see [@OP] for a discussion of shifted symmetric functions. The function $$\Psi_\hbar^{-1}(\lambda) = \frac{1}{\Psi_\hbar(\lambda)}$$ is a nonvanishing rational function of $\hbar$ whose poles are the roots of $\Psi_\hbar(\lambda).$ In particular, for any diagram $\lambda,$ the function $\Psi_\hbar^{-1}(\lambda)$ is analytic on the disc $$\|\hbar\| < \frac{1}{\max(\lambda_1-1,\ell(\lambda)-1)}$$ with Maclaurin series $$\Psi_\hbar^{-1}(\lambda) = \sum_{r=0}^\infty \hbar^r f_r(\lambda),$$ where $f_r(\lambda)$ is the degree $r$ complete symmetric polynomial in $d$ variables, $$f_r = \sum_{\substack{ i \colon [r] \to [d] \\ i \text{ weakly increasing}}} x_{i(1)} \dots x_{i(r)},$$ evaluated on the contents of $\lambda.$ The functions $\Omega_N^{\pm 1}$ are recovered from the functions $\Psi_\hbar^{\pm 1}$ by setting $$\hbar = - \frac{1}{N}.$$ In particular, the $N \to \infty$ asymptotics of $\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle$ may be obtained from the $\hbar \to 0$ asymptotics of $\langle \omega_\alpha \Psi_\hbar^{-1} \omega_\beta \rangle$, i.e. from its Taylor expansion around $\hbar = 0$ as derived above. We will now give a diagrammatic interpretation of this Maclaurin series. For a given pair of Young diagrams $\alpha,\beta \vdash d,$ we have the Taylor series $$\langle \omega_\alpha \Psi_\hbar^{-1} \omega_\beta \rangle = \sum_{r=0}^\infty \hbar^r \langle \omega_\alpha f_r \omega_\beta \rangle,$$ which is absolutely convergent for $\|\hbar\| < \frac{1}{d-1}.$ For any $\lambda \vdash d$, the observable $\omega_\alpha(\lambda) f_r(\lambda)\omega_\beta(\lambda)$ is the eigenvalue of the central element $C_\alpha f_r(X_1,\dots,X_d) C_\beta$ acting in the irreducible representation $\mathsf{V}^\lambda$ of the group algebra ${\mathbb{C}}{\mathrm{S}}(d)$ corresponding to $\lambda$, where $X_1,\dots,X_d \in {\mathbb{C}}{\mathrm{S}}(d)$ are the Jucys-Murphy elements [@DG; @OV]: $$X_j = \sum_{i < j} (i\ j), \quad 1 \leq j \leq d.$$ Thus, by the Fourier isomorphism, $${\mathbb{C}}{\mathrm{S}}(d) \simeq \bigoplus_{\lambda \vdash d} \mathrm{End} \mathsf{V}^\lambda,$$ the Plancherel expectation $\langle \omega_\alpha f_r \omega_\beta \rangle$ is the normalized character of the central element $C_\alpha f_r(X_1,\dots,X_d) C_\beta$ in the regular representation of ${\mathbb{C}}{\mathrm{S}}(d),$ i.e. the coefficient of the identity permutation in the sum $$\sum_{\rho \in C_\alpha, \sigma \in C_\beta} \sum_{\substack{ i,j \colon [r] \to [d] \\ i<j \text{ pointwise} \\ j \text{ weakly increasing}} } \rho \left( i(1)\ j(1) \right) \dots \left( i(r)\ j(r) \right) \sigma.$$ Adopting the convention that permutations are multiplied from left to right, this number may be visualized as follows. Identify the symmetric group ${\mathrm{S}}(d)$ with its right Cayley graph, as generated by the conjugacy class of transpositions. Introduce an edge labeling on this graph by marking each edge corresponding to the transposition $(i\ j)$ with $j$, the larger of the two elements interchanged. Thus, emanating from each vertex of ${\mathrm{S}}(d)$, one sees a single $2$-edge, two $3$-edges, three $4$-edges, etc. Figure \[fig:Cayley\] shows ${\mathrm{S}}(4)$ equipped with this edge labeling. A walk on ${\mathrm{S}}(d)$ is said to be monotone if the labels of the edges it traverses form a weakly increasing sequence. Given Young diagrams $\alpha,\beta \vdash d$ and a nonnegative integer $r$, let $\vec{W}^r(\alpha,\beta)$ denote the number of $r$-step monotone walks on ${\mathrm{S}}(d)$ which begin at a point of $C_\alpha$ and end at a point of $C_\beta$. Then from the calculation above we have the identity $$\langle \omega_\alpha f_r \omega_\beta \rangle = \vec{W}^r(\alpha,\beta).$$ Equivalently, $$\langle \omega_\alpha \Psi_\hbar^{-1} \omega_\beta \rangle = \sum_{r=0}^\infty \hbar^r \vec{W}^r(\alpha,\beta),$$ the generating function for monotone walks on ${\mathrm{S}}(d)$ with boundary conditions $\alpha,\beta,$ the sum being absolutely convergent for $\|\hbar\| < \frac{1}{d-1}.$ In the special case $\alpha=(1^d),$ we have $$\langle \Psi_\hbar^{-1} \omega_\beta \rangle = \sum_{r=0}^\infty \hbar^r \vec{W}^r(\beta),$$ where $\vec{W}^r(\beta) = \vec{W}^r(1^d,\beta)$ is the number of $r$-step monotone walks on ${\mathrm{S}}(d)$ which begin at the identity permutation and end at a permutation of cycle type $\beta.$ We may thus conclude the following. \[thm:FeynmanExpansions\] For any positive integers $1 \leq d \leq N$ and any Young diagrams $\alpha,\beta \vdash d$, we have $$\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle = \sum_{r=0}^\infty (-1)^r \frac{\vec{W}^r(\alpha,\beta)}{N^r} \quad\text{ and }\quad \langle \Omega_N^{-1} \omega_\beta \rangle = \sum_{r=0}^\infty (-1)^r \frac{\vec{W}^r(\beta)}{N^r}$$ and the series are absolutely convergent. Note that the $1/N$ expansions in Theorem \[thm:FeynmanExpansions\] are not actually alternating series: their nonzero terms are either all negative or all positive. Theorem \[thm:FeynmanExpansions\] gives a convergent $N \to \infty$ asymptotic expansion of the Plancherel expectation $\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle$ wherein monotone walks play the role of Feynman diagrams. As a consistency check, observe that $$\vec{W}^0(\alpha,\beta) = \delta_{\alpha\beta} \|C_\alpha\|,$$ corresponding to the fact that there exists a zero-step walk from $C_\alpha$ to $C_\beta$ if and only if these otherwise disjoint sets are equal, and in this case the number of such walks is just the cardinality of $C_\alpha.$ ![\[fig:Cayley\] Edge labeled Cayley graph of ${\mathrm{S}}(4).$ Figure by M. LaCroix.](Cayley.pdf) Stable integrals ---------------- For any fixed $N \in \mathbb{N},$ Theorem \[thm:FeynmanExpansions\] describes the first $N$ nonconstant terms in the string expansions of $I_N$ and $J_N$: we have $$I_N = 1 + \sum_{d=1}^N \frac{z^d}{d!} \sum_{\alpha,\beta \vdash d} p_\alpha(a_1,\dots,a_N) p_\beta(b_1,\dots,b_N) \sum_{r=0}^\infty (-1)^r \frac{\vec{W}^r(\alpha,\beta)}{N^r} + \text{ higher terms},$$ and $$J_N = 1 + \sum_{d=1}^N \frac{z^{2d}}{d!} N^d\sum_{\beta \vdash d} p_\beta(b_1,\dots,b_N) \sum_{r=0}^\infty (-1)^r \frac{\vec{W}^r(\beta)}{N^r} + \text{ higher terms}.$$ This description suggests that, as $N \to \infty,$ the integrals $I_N$ and $J_N$ approximate generating functions for monotone walks on all of the symmetric groups, of all possible lengths and boundary conditions. Unfortunately, for any finite $N,$ almost all terms of the string expansion are “higher terms.” To see past this analytic limitation, let us view $z$ as a formal variable, and replace the number $-\frac{1}{N}$ with a formal semiclassical parameter $\hbar.$ Furthermore, let us replace the eigenvalues of the matrices $A,B,$ and $C=AB$ with countably infinite alphabets of commuting indeterminates, these being formal stand-ins for the eigenvalues infinite-dimensional matrices. Let $\Lambda_A,\Lambda_B,\Lambda_C$ be the affiliated algebras of symmetric functions, i.e. the polynomial algebras $$\Lambda_A = {\mathbb{C}}[p_1(A),p_2(A),\dots],\quad \Lambda_B = {\mathbb{C}}[p_1(B),p_2(B),\dots],\quad \Lambda_C = {\mathbb{C}}[p_1(C),p_2(C),\dots],$$ where $$p_k(A) = \sum_{a \in A} a^k,\quad p_k(B) = \sum_{b \in B} b^k,\quad p_k(C) = \sum_{c \in C} c^k, \qquad k \in \mathbb{N},$$ are the pure power sums over these alphabets. Set $\Lambda_{A,B} = \Lambda_A \otimes \Lambda_B.$ We define the stable HCIZ integral to be the formal power series $$I = 1 + \sum_{d=1}^\infty \frac{z^d}{d!} \sum_{\alpha,\beta \vdash d} p_\alpha(A) p_\beta(B) \sum_{r=0}^\infty \hbar^r \vec{W}^r(\alpha,\beta),$$ which is an element of the ring $\Lambda_{A,B}[[z,\hbar]].$ Similarly, we define the stable BGW integral to be the formal power series $$J = 1 + \sum_{d=1}^\infty \frac{z^{2d}}{d!} (-1)^d\hbar^{-d} \sum_{\beta \vdash d}p_\beta(B) \sum_{r=0}^\infty \hbar^r \vec{W}^r(\beta),$$ which is an element of $\Lambda_C[[z,\hbar^{\pm 1}]].$ Thus $I$ and $J$ are “grand canonical” partition functions enumerating monotone walks of all possible lengths and boundary conditions, over all symmetric groups. \[thm:StableExponentials\] We have $$I = e^F \quad\text{ and }\quad J=e^G,$$ where $$F = \sum_{d=1}^\infty \frac{z^d}{d!} \sum_{\alpha,\beta \vdash d} p_\alpha(A) p_\beta(B) \sum_{r=0}^\infty \hbar^r {\vec{H}}^r(\alpha,\beta)$$ and ${\vec{H}}^r(\alpha,\beta)$ is the number of monotone $r$-step walks on ${\mathrm{S}}(d)$ which begin at a permutation of cycle type $\alpha,$ end at a permutation of cycle type $\beta,$ and have the property that their steps and endpoints together generate a transitive subgroup of ${\mathrm{S}}(d),$ and $$G = \sum_{d=1}^\infty \frac{z^d}{d!} (-1)^d \hbar^{-d}\sum_{\beta \vdash d} p_\beta(C) \sum_{r=0}^\infty \hbar^r {\vec{H}}^r(\beta)$$ with ${\vec{H}}^r(\beta) = {\vec{H}}^r(1^d,\beta).$ Theorem \[thm:StableExponentials\] follows from a fundamental result in enumerative combinatorics, the Exponential Formula [@Stanley:EC2 Chapter 5], according to which the exponential of a generating function for a class of “connected” combinatorial structures is a generating function for possibly “disconnected” structures of the same type. For walks on groups, the role of connectedness is played by transitivity. For a careful justification of the use of the Exponential Formula in the context of monotone walks on symmetric groups, see [@GGN1; @GGN2; @GGN4]. Topological expansion --------------------- The numbers ${\vec{H}}^r(\alpha,\beta)$ and ${\vec{H}}^r(\beta)$ appearing in Theorem \[thm:StableExponentials\] are known as the monotone double and single Hurwitz numbers, respectively. These enumerative quantities, introduced in [@GGN1; @GGN2; @GGN3] and studied in numerous articles since, are a combinatorial variant of the classical double and single Hurwitz numbers $H^r(\alpha,\beta)$ and $H^r(\beta)=H^r(1^d,\beta),$ which count transitive $r$-step walks $C_\alpha \to C_\beta$ without the monotonicity constraint. Clearly, ${\vec{H}}^r(\alpha,\beta) \leq H^r(\alpha,\beta),$ and in a sense monotone Hurwitz numbers are a “desymmetrized” version of classical Hurwitz numbers; see [@GGN1; @GGN2]. Reversing a classical construction due to Hurwitz [@Hurwitz] (see [@EEHS] for a modern treatment), the number $H^r(\alpha,\beta)$ may alternatively be interpreted as the number of isomorphism classes of degree $d$ branched covers of the Riemann sphere $\mathbf{P}^1({\mathbb{C}})$ by a compact, connected Riemann surface $\mathbf{S}$ which have profiles $\alpha,\beta \vdash d$ over $0,\infty \in \mathbf{P}^1({\mathbb{C}})$ and the simplest nontrivial branching over the $r$th roots of unity on the sphere. The monotone double Hurwitz number ${\vec{H}}^r(\alpha,\beta)$ is a signed enumeration of the same class of covers, see [@ACEH; @MN]. The genus $g$ of $\mathbf{S}$ is determined by the data $d,r,\alpha,\beta$ according to the Riemann-Hurwitz formula, $$g = \frac{r+2-\ell(\alpha)-\ell(\beta)}{2},$$ with the understanding that $H^r(\alpha,\beta)=0$ unless this formula returns a nonnegative integer. In particular, one may parameterize nonzero (classical and monotone) Hurwitz numbers by genus, setting $H_g(\alpha,\beta) := H^{2g-2+\ell(\alpha)+\ell(\beta)}(\alpha,\beta)$ and ${\vec{H}}_g(\alpha,\beta): = {\vec{H}}^{2g-2+\ell(\alpha)+\ell(\beta)}(\alpha,\beta).$ In the genus parameterization, Theorem \[thm:StableExponentials\] becomes the following topological expansion of the stable HCIZ and BGW integrals. \[thm:StableTopExpansions\] We have $$I = e^{\sum_{g=0}^\infty \hbar^{2g-2} F^{(g)}} \quad\text{ and }\quad J = e^{\sum_{g=0}^\infty \hbar^{2g-2} G^{(g)}},$$ where $$F^{(g)} = \sum_{d=1}^\infty \frac{z^d}{d!} \sum_{\alpha,\beta \vdash d} \hbar^{\ell(\alpha)+\ell(\beta)} p_\alpha(A) p_\beta(B) {\vec{H}}_g(\alpha,\beta).$$ and $$G^{(g)} = \sum_{d=1}^\infty \frac{z^{2d}}{d!} (-1)^d \sum_{\beta \vdash d} \hbar^{\ell(\beta)} p_\beta(C) {\vec{H}}_g(\beta).$$ Applying the Riemann-Hurwitz formula, we have $$\begin{aligned} F &= \sum_{d=1}^\infty \frac{z^d}{d!} \sum_{\alpha,\beta \vdash d} p_\alpha(A) p_\beta(B) \sum_{g=0}^\infty \hbar^{2g-2+\ell(\alpha)+\ell(\beta)}{\vec{H}}_g(\alpha,\beta) \\ &= \sum_{g=0}^\infty \hbar^{2g-2} \sum_{d=1}^\infty \frac{z^d}{d!} \sum_{\alpha,\beta \vdash d} \hbar^{\ell(\alpha)+\ell(\beta)}p_\alpha(A) p_\beta(B) {\vec{H}}_g(\alpha,\beta) \end{aligned}$$ and $$\begin{aligned} G &= \sum_{d=1}^\infty \frac{z^d}{d!} (-1)^d \hbar^{-d}\sum_{\beta \vdash d} p_\beta(B) \sum_{g=0}^\infty \hbar^{2g-2+d+\ell(\beta)} {\vec{H}}^r(\beta) \\ &= \sum_{g=0}^\infty \hbar^{2g-2} \sum_{d=1}^\infty \frac{z^d}{d!} (-1)^d \sum_{\beta \vdash d} \hbar^{\ell(\beta)} p_\beta(C) {\vec{H}}_g(\beta). \end{aligned}$$ Topological factorization ------------------------- For any nonnegative integer $k,$ the topological expansions of $I$ and $J$ given by Theorem \[thm:StableTopExpansions\] can be split into two corresponding factors, $$I = e^{\sum_{g=0}^k \hbar^{2g-2} F^{(g)}} e^{\sum_{g=k+1}^\infty \hbar^{2g-2} F^{(g)}} \quad\text{ and }\quad J = e^{\sum_{g=0}^k \hbar^{2g-2} G^{(g)}} e^{\sum_{g=k+1}^\infty \hbar^{2g-2} G^{(g)}}.$$ These factorizations have a clear enumerative meaning: the first factor is a generating fucntion enumerating possibly disconnected covers/walks in which each connected component has genus at most $k$, while the second factor is a generating function enumerating possibly disconnected covers/walks in which each connected component has genus at least $k+1.$ This may be equivalently stated as follows. Define the disconnected monotone double and single Hurwitz numbers by $${\vec{H}}_g^\bullet(\alpha,\beta) = \vec{W}^{r_g(\alpha,\beta)}(\alpha,\beta) \quad\text{ and }\quad {\vec{H}}_g^\bullet(\beta) = {\vec{H}}_g^\bullet(1^d,\beta),$$ where $g \in \mathbb{Z}$ and $r_g(\alpha,\beta) =2g-2+\ell(\alpha)+\ell(\beta).$ In particular, for disconnected Hurwitz numbers the genus $g$ may be negative (corresponding to the fact that the Euler characteristic is additive), but ${\vec{H}}_g^\bullet(\alpha,\beta)$ vanishes unless $r_g(\alpha,\beta) \geq 0.$ In terms of disconnected monotone Hurwitz numbers, the above factorization identities may be stated as follows. \[thm:StableTopFactorization\] For any $k \in \mathbb{N} \cup \{0\}$ $$\frac{I}{e^{\sum_{g=0}^k \hbar^{2g-2} F^{(g)}}} = 1 + \sum_{d=1}^\infty \frac{z^d}{d!} \sum_{\alpha,\beta \vdash d} \hbar^{\ell(\alpha)+\ell(\beta)} p_\alpha(A)p_\beta(B) \sum_{g=k+1}^\infty \hbar^{2g-2} {\vec{H}}_g^\bullet(\alpha,\beta)$$ and $$\frac{J}{e^{\sum_{g=0}^k \hbar^{2g-2} G^{(g)}}} = 1 + \sum_{d=1}^\infty \frac{z^{2d}}{d!} \sum_{\alpha,\beta \vdash d} (-1)^d \hbar^{\ell(\beta)}p_\beta(B) \sum_{g=k+1}^\infty \hbar^{2g-2} {\vec{H}}_g^\bullet(\beta).$$ As a corollary of Theorem \[thm:StableTopFactorization\], we obtain the following pair of “topological bounds,” which are significant since they indicate what sorts of bounds we may expect to be valid for the entire functions $I_N$ and $J_N$ at finite $N.$ Let us introduce the following formal order notation: given a formal power series $Z \in \Lambda_{A,B}[[z,\hbar]]$ and a nonnegative integer $n,$ we write $$Z = O(\hbar^n)$$ if $Z$ belongs to the principal ideal generated by $\hbar^n.$ We use the analogous order notation in $\Lambda_C[[z,\hbar^{\pm 1}]].$ \[cor:StableTopBound\] For each $k \in \mathbb{N} \cup \{0\},$ $$1 - \frac{I}{e^{\sum_{g=0}^k \hbar^{2g-2} F^{(g)}}} = O(\hbar^{2k}) \quad\text{ and }\quad 1 - \frac{J}{e^{\sum_{g=0}^k \hbar^{2g-2} G^{(g)}}} = O(\hbar^{2k}).$$ Functional Asymptotics {#sec:Functional} ====================== In this Section, we prove our main result, Theorem \[thm:Main\]. To achieve this, we must bridge the gap between the $N<\infty$ string expansions of the HCIZ and BGW integrals and their $N=\infty$ stable topological expansions. It is here that the mollifying effect of the ${\mathrm{LIS}}$ distribution plays a critical role: it controls the tail of the finite $N$ string expansions of $I_N$ and $J_N,$ effectively truncating them to polynomials of degree $O(N^2)$ for small $z$. The existence of this quadratic cutoff is a key feature of $I_N$ and $J_N$ that has not previously been recognized. Analytic candidates ------------------- Throughout this section, we will use the following notation for complex polydiscs. Given a real number $\rho$ and a positive integer $N$, we will ambiguously write $\overline{{\mathbb{D}}}_\rho^N$ to mean either of the closed polydiscs $$\overline{{\mathbb{D}}}_\rho \times \overline{{\mathbb{D}}}_1^N \times \overline{{\mathbb{D}}}_1^N \quad\text{ or }\quad \overline{{\mathbb{D}}}_\rho \times \overline{{\mathbb{D}}}_1^N,$$ where $\overline{{\mathbb{D}}}_\rho$ is the closed origin-centred disc of radius $\rho$ in the complex plane. Although the first of these domains lives in $\mathbb{C}^{2N+1}$ and the second lives in $\mathbb{C}^{N+1},$ which of the two domains $\overline{{\mathbb{D}}}_\rho^N$ is intended to represent will be clear from context. Let $\\|\cdot\\|_\rho$ denote the sup norm on $\overline{{\mathbb{D}}}_\rho^N$. Note that this is really a sequence of norms defined on a sequence of domains of growing dimension. Let $N \in \mathbb{N}$ be a positive integer, and let $a_1,\dots,a_N,b_1,\dots,b_N,c_1,\dots,c_N$ be any points sampled from the closed unit disc in ${\mathbb{C}}.$ Consider the corresponding specializations $$\Lambda_{A,B}[[z,\hbar]] \to {\mathbb{C}}[[z]] \quad\text{ and }\quad \Lambda_C[[z,\hbar]] \to {\mathbb{C}}[[z]]$$ defined by setting $\hbar=-1/N$ and $$A=\{a_1,\dots,a_N\}, B=\{b_1,\dots,b_N\}, C=\{c_1,\dots,c_N\},$$ and let $$\begin{aligned} F_N^{(g)} &= \sum_{d=1}^\infty \frac{z^d}{d!} \sum_{\alpha,\beta \vdash d} \frac{p_\alpha(a_1,\dots,a_N)}{N^{\ell(\alpha)}} \frac{p_\beta(b_1,\dots,b_N)}{N^{\ell(\beta)}} (-1)^{\ell(\alpha)+\ell(\beta)} {\vec{H}}_g(\alpha,\beta), \\ G_N^{(g)} &= \sum_{d=1}^\infty \frac{z^{2d}}{d!} \sum_{\beta \vdash d} \frac{p_\beta(c_1,\dots,c_N)}{N^{\ell(\beta)}} (-1)^{d+\ell(\beta)} {\vec{H}}_g(\beta). \end{aligned}$$ be the images of $F^{(g)}$ and $G^{(g)}$ under these specializations. A priori, $F_N^{(g)}$ and $G_N^{(g)}$ are only formal power series. However, they are in fact absolutely summable, and hence define analytic functions. This follows from an established result on the convergence of generating functions for monotone Hurwitz numbers [@GGN5]. \[thm:GenusSpecific\] For each $g \in \mathbb{N} \cup \{0\},$ the power series $$\begin{aligned} {\vec{H}}_g^{\text{simple}} &= \sum_{d=1}^\infty \frac{z^d}{d!} {\vec{H}}_g(1^d,1^d), \\ {\vec{H}}_g^{\text{single}} &= \sum_{d=1}^\infty \frac{z^d}{d!} \sum_{\beta \vdash d} {\vec{H}}_g(1^d,\beta), \\ {\vec{H}}_g^{\text{double}} &= \sum_{d=1}^\infty \frac{z^d}{d!} \sum_{\alpha,\beta \vdash d} {\vec{H}}_g(\alpha,\beta) \end{aligned}$$ have radii of convergence exactly $2/27,$ at least $1/27,$ and at least $1/54,$ respectively. The exact computation of the radius of convergence of the generating function for monotone simple Hurwitz numbers follows from a rational parameterization of this series in terms of the Gauss hypergeometric function, see [@GGN2; @GGN5]. A combinatorial argument based on sorting transpositions (a variant of the Hurwitz braid action) then shows that the radius of convergence drops by at most a factor of two for each new branch point added, see [@GGN5] for details. The author believes that the radius of convergence is in fact exactly $2/27$ in all three cases, but this has not been proved. It was pointed out to the author by Philippe Di Francesco that the number of isomorphism classes of finite groups of order $p^N,$ with $p$ prime, is known [@Pyber] to be asymptotically $p^{\frac{2}{27}N^3}$ as $N \to \infty.$ The author has no explanation for this numerical coincidence. For another interesting appearance of the number $2/27,$ see [@KT]. \[thm:AnalyticCandidates\] There exists $\delta > 0$ such that the series $F_N^{(g)}$ and $G_N^{(g)}$ converge absolutely and uniformly on $\overline{{\mathbb{D}}}_\delta^N$, for all $g \geq 0$ and $N \geq 1.$ For any Young diagrams $\alpha,\beta$, we have $$\left\| \frac{p_\alpha(a_1,\dots,a_N)}{N^{\ell(\alpha)}} \right\|, \left\| \frac{p_\beta(b_1,\dots,b_N)}{N^{\ell(\beta)}} \right\|, \left\| \frac{p_\beta(c_1,\dots,c_N)}{N^{\ell(\beta)}} \right\| \leq 1,$$ on $\overline{{\mathbb{D}}}_\delta^N$. We thus have $$\begin{aligned} \|F_N^{(g)}\| &\leq \sum_{d=1}^n \frac{\delta^d}{d!} \sum_{\alpha,\beta \vdash d} {\vec{H}}_g(\alpha,\beta) \\ \|G_N^{(g)}\| &\leq \sum_{d=1}^n \frac{\delta^{2d}}{d!} \sum_{\beta \vdash d} {\vec{H}}_g(\beta) \end{aligned}$$ uniformly on $\overline{{\mathbb{D}}}_\delta^N$ for any $n \in \mathbb{N},$ and the claim thus follows from Theorem \[thm:GenusSpecific\]. Let us fix $\delta >0$ so that $F_N^{(g)}$ and $G_N^{(g)}$ converge to define analytic functions on on $\overline{{\mathbb{D}}}_\delta^N,$ for all $N \in \mathbb{N}.$ Then, these functions are uniformly bounded in the following sense. \[cor:UniformFreeEnergies\] We have $$\sup_{N \in \mathbb{N}} \\|F_N^{(g)}\\|_\delta < \infty \quad\text{ and }\quad \sup_{N \in \mathbb{N}} \\|G_N^{(g)}\\|_\delta < \infty.$$ Polynomial approximation ------------------------ We now consider the behavior of the full string expansions of $I_N$ and $J_N$ given by Theorem \[thm:StringExpansions\] with $N$ large but finite. In this regime, the factor ${\mathbb{P}}({\mathrm{LIS}}_d \leq N)$ has a dramatic effect on the string expansions — it effectively truncates them to polynomials of degree $O(N^2)$. The mechanism behind this cutoff is the law of large numbers for longest increasing subsequences in random permutations, which is due to Vershik and Kerov [@Kerov]: we have $$\label{eqn:LISLLN} \lim_{d \to \infty} \frac{{\mathrm{LIS}}_d}{\sqrt{d}} = 2,$$ where the convergence is in probability. A detailed exposition of this LLN is given in [@Romik], which also presents the corresponding central limit theorem of Baik-Deift-Johansson [@BDJ], which asserts Tracy-Widom fluctuations of ${\mathrm{LIS}}_d$ around $2\sqrt{d}$ on the scale $d^{1/6}$. In particular, the distribution of the longest increasing subsequence in large uniformly random permutation is strongly concentrated around its mean. This implies that, for large $N$ we have the approximate step function behavior $${\mathbb{P}}({\mathrm{LIS}}_d \leq N) \approx \begin{cases} 1, \quad 1 \leq d \leq \frac{1}{4}N^2 \\ 0, \quad d > \frac{1}{4}N^2 \end{cases}.$$ Consequently, for $\|z\|$ small and $N$ large, $I_N$ and $J_N$ are well-approximated by their “string polynomials” $$\tilde{I}_N = 1 + \sum_{d=1}^{\lfloor \frac{1}{4}N^2 \rfloor} \frac{z^d}{d!} \sum_{\alpha,\beta \vdash d} p_\alpha(a_1,\dots,a_N) p_\beta(b_1,\dots,b_N) \langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle$$ and $$\tilde{J}_N = 1 + \sum_{d=1}^{\lfloor \frac{1}{4}N^2 \rfloor} \frac{z^{2d}}{d!} N^d\sum_{\beta \vdash d} p_\beta(c_1,\dots,c_N) \langle \Omega_N^{-1} \omega_\beta \rangle,$$ which are obtained from the string expansions of $I_N$ and $J_N$ as given by Theorem \[thm:StringExpansions\] by replacing the factor ${\mathbb{P}}({\mathrm{LIS}}_d \leq N)$ with the above step function. Feynman extension ----------------- The polynomial approximations $\tilde{I}_N$ and $\tilde{J}_N$ of $I_N$ and $J_N$ are only useful insofar as we are able to understand the Plancherel expectations $$\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle = \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} \omega_\alpha(\lambda) \Omega_N^{-1}(\lambda) \omega_\beta(\lambda) \frac{(\dim \mathsf{V}^\lambda)^2}{\|{\mathrm{S}}_N(d)\|}$$ in the range $1 \leq d \leq \frac{1}{4}N^2.$ This means that we must extend Theorem \[thm:FeynmanExpansions\], which gives the convergent $1/N$ expansion $$\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle = \sum_{r=0}^\infty \left( -\frac{1}{N} \right)^r \vec{W}^r(\alpha,\beta) = \frac{(-1)^{\ell(\alpha)+\ell(\beta)}}{N^{\ell(\alpha)+\ell(\beta)}} \sum_{\substack{g = -\infty \\ 2-2g \leq \ell(\alpha)+\ell(\beta)}} N^{2-2g} {\vec{H}}_g^\bullet(\alpha,\beta)$$ in the linear range $1 \leq d \leq N,$ to the range where $d$ may be as large as $\frac{1}{4}N^2.$ This may be done as follows. For any $d,N \in \mathbb{N}$ we may rewrite the expectation $\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle$ as a conditional expectation against the unrestricted Plancherel measure: we have $$\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle = \frac{1}{{\mathbb{P}}({\mathrm{LIS}}_d \leq N)} \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N}} \omega_\alpha(\lambda) \Omega_N^{-1}(\lambda) \omega_\beta(\lambda) \frac{(\dim \mathsf{V}^\lambda)^2}{d!}$$ Let us split this conditional expectation into two pieces: we write $$\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle = \langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle_1 + \langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle_2,$$ where $$\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle_1= \frac{1}{{\mathbb{P}}({\mathrm{LIS}}_d \leq N)} \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N \\ \lambda_1 \leq N}} \omega_\alpha(\lambda) \Omega_N^{-1}(\lambda) \omega_\beta(\lambda) \frac{(\dim \mathsf{V}^\lambda)^2}{d!}$$ and $$\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle_2 = \frac{1}{{\mathbb{P}}({\mathrm{LIS}}_d \leq N)} \sum_{\substack{\lambda \vdash d \\ \ell(\lambda) \leq N \\ \lambda_1 > N}} \omega_\alpha(\lambda) \Omega_N^{-1}(\lambda) \omega_\beta(\lambda) \frac{(\dim \mathsf{V}^\lambda)^2}{d!}.$$ In the range $1 \leq d \leq N,$ the second component of this decomposition vanishes. In the extended range $N < d \leq \frac{1}{4}N^2,$ when $N$ is large, the first component of this decomposition is virtually equal to $\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle,$ while the second is negligible. Indeed, it follows from the Vershik-Kerov limit shape theorem [@Kerov; @Romik] that for $N < d_N \leq \frac{1}{4}N^2,$ a Plancherel-random Young diagram with $d_N$ cells is contained in the $N \times N$ rectangular diagram $R(N,N)$ with overwhelming probability. Observe now that the massive component $\langle \omega_\alpha \Omega_N \omega_\beta \rangle_1$ of $\langle \omega_\alpha \Omega_N \omega_\beta \rangle$ admits an absolutely convergent $1/N$ expansion. Indeed, for any $\lambda \subseteq R(N,N),$ we have the absolutely convergent expansion $$\Omega_N^{-1}(\lambda) = \prod_{\Box \in \lambda} \frac{1}{1 + \frac{c(\Box)}{N}} = \sum_{r=0}^\infty \left( -\frac{1}{N} \right)^r f_r(\lambda),$$ so that $$\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle_1 = \sum_{r=0}^\infty \left( -\frac{1}{N} \right)^r \vec{W}^r_N(\alpha,\beta),$$ where $$\vec{W}^r_N(\alpha,\beta) = \sum_{\substack{\lambda \vdash d \\ \lambda \subseteq R(N,N)}} \omega_\alpha(\lambda) f_r(\lambda) \omega_\beta(\lambda) \frac{(\dim \mathsf{V}^\lambda)^2}{d!}.$$ agrees with $\vec{W}^r(\alpha,\beta)$ up to an exponentially small error. Thus, for any fixed but arbitrary $s \in \mathbb{N} \cup \{0\},$ we can replace the first $s$ coefficients of the massive component $\langle \omega_\alpha \Omega_N \omega_\beta \rangle_1$ with their $N$-independent counterparts up to an exponentially small error. Ignoring the negligible component $\langle \omega_\alpha \Omega_N \omega_\beta \rangle_2,$ this gives the $N \to \infty$ asymptotic approximation $$\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle = \sum_{r=0}^s \left( -\frac{1}{N} \right)^r \vec{W}^r(\alpha,\beta) + O\left(\frac{1}{N^{s+1}}\right),$$ which extends Theorem \[thm:FeynmanExpansions\] to the range $1 \leq d \leq \frac{1}{4}N^2.$ Note that this expansion implies the sharper estimate $$\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle = \sum_{r=0}^s \left( -\frac{1}{N} \right)^r \vec{W}^r(\alpha,\beta) + O\left(\frac{1}{N^{s+2}}\right),$$ since the numbers $\vec{W}^r(\alpha,\beta)$ which are nonzero correspond to either $r$ even, or $r$ odd. In particular, for any $k \in \mathbb{N} \cup \{0\},$ we have that $$\langle \omega_\alpha \Omega_N^{-1} \omega_\beta \rangle = \frac{(-1)^{\ell(\alpha)+\ell(\beta)}}{N^{\ell(\alpha)+\ell(\beta)}} \sum_{\substack{g = -\infty \\ 2-2g \leq \ell(\alpha) + \ell(\beta)}}^k N^{2-2g} {\vec{H}}_g^{\bullet}(\alpha,\beta) + O\left( N^{-2k}\right).$$ Topological bound ----------------- The following topological bound bridges the gap between formal asymptotics and functional asymptotics. Conceptually, this result is the unstable analytic shadow of the stable topological bounds appearing in Corollary \[cor:StableTopBound\]. \[thm:TopBound\] There exists $\gamma > 0$ such that, for each fixed $k \in \mathbb{N} \cup \{0\},$ we have $$\bigg{\\|} 1 - \frac{I_N}{e^{\sum_{g=0}^k N^{2-2g}F_N^{(g)}}} \bigg{\\|}_\gamma = O(N^{2-2k}) \quad\text{ and }\quad \bigg{\\|} 1 - \frac{J_N}{e^{\sum_{g=0}^k N^{2-2g}G_N^{(g)}}} \bigg{\\|} = O(N^{2-2k})$$ as $N \to \infty.$ We give the proof for the HCIZ integral; the argument for the BGW integral is essentially the same. With $\delta$ as in Theorem \[thm:AnalyticCandidates\], take $\gamma \leq \delta$ sufficiently small so that $I_N$ can be replaced with $\tilde{I}_N$ as $N \to \infty.$ Replacing the coefficients of $\tilde{I}_N$ with their asymptotic expansions to order $k+1$ and applying Theorem \[thm:StableTopFactorization\], we obtain $$\begin{aligned} 1-\frac{\tilde{I}_N}{e^{\sum_{g=0}^k N^{2-2g}F_N^{(g)}}} &= \sum_{d=1}^{\lfloor \frac{1}{4}N^2 \rfloor} \frac{z^d}{d!} \sum_{\alpha,\beta \vdash d} \frac{p_\alpha(a_1,\dots,a_N)}{N^{\ell(\alpha)}}\frac{p_\beta(b_1,\dots,b_N)}{N^{\ell(\beta)}} \left(N^{-2k}{\vec{H}}_{k+1}^\bullet(\alpha,\beta) + O(N^{-2k-2}) \right) \\ &+ O(z^{\lfloor \frac{1}{4}N^2\rfloor + 1}). \end{aligned}$$ On $\overline{{\mathbb{D}}}_\gamma^N,$ we have the estimate $$\begin{aligned} &\bigg{\|}\sum_{d=1}^{\lfloor \frac{1}{4}N^2 \rfloor} \frac{z^d}{d!} \sum_{\alpha,\beta \vdash d} \frac{p_\alpha(a_1,\dots,a_N)}{N^{\ell(\alpha)}}\frac{p_\beta(b_1,\dots,b_N)}{N^{\ell(\beta)}} \left(N^{-2k}{\vec{H}}_{k+1}^\bullet(\alpha,\beta) + O(N^{-2k-2}) \right) \bigg{\|} \\ &\leq N^{-2k} \sum_{d=1}^{\lfloor \frac{1}{4}N^2 \rfloor} \frac{\gamma^d}{d!} \sum_{\alpha,\beta \vdash d} \left({\vec{H}}_{k+1}^\bullet(\alpha,\beta) + O(N^{-2})\right) \\ &=O(N^{2-2k}). \end{aligned}$$ \[cor:NonVanishing\] For $N$ sufficiently large, the integrals $l_N$ and $J_N$ are non-vanishing on $\overline{{\mathbb{D}}}_{\gamma}^N.$ In particular, for $N$ sufficiently large, $\log I_N$ and $\log J_N$ are defined and analytic on $\overline{{\mathbb{D}}}_{\gamma}^N.$ This follows from the $k=2$ case of Theorem \[thm:TopBound\], which implies that $$\bigg{\|} 1 - \frac{I_N}{e^{N^2F_N^{(0)} + F_N^{(1)} + N^{-2}F_N^{(2)}}} \bigg{\|} <1 \quad\text{ and }\quad \bigg{\|} 1 - \frac{J_N}{e^{N^2G_N^{(0)} + G_N^{(1)} + N^{-2}G_N^{(2)}}} \bigg{\|} <1$$ on $\overline{{\mathbb{D}}}_{\gamma}^N$ for $N$ sufficiently large. These inequalities in turn imply the non-vanishing of $$\frac{I_N}{e^{N^2F_N^{(0)} + F_N^{(1)} + N^{-2}F_N^{(2)}}} \quad\text{ and }\quad \frac{J_N}{e^{N^2G_N^{(0)} + G_N^{(1)} + N^{-2}G_N^{(2)}}}$$ on $\overline{{\mathbb{D}}}_\gamma^N,$ from which we conclude the nonvanishing of $I_N$ and $J_N$ on this polydisc. Analytic error functions ------------------------ Set $\xi = \min(\gamma,\delta),$ where $\gamma$ is the positive constant in Theorem \[thm:TopBound\] and $\delta$ is the positive constant in Theorem \[thm:AnalyticCandidates\]. We may define an array of analytic functions on $\overline{{\mathbb{D}}}_\xi^N$ by $$\begin{aligned} \Delta_N^{(0)} &= N^{-2}\log I_N - F_N^{(0)} \\ \Delta_N^{(k)} &= N^2\Delta_N^{(k-1)} - F_N^{(k)}, \quad k \in \mathbb{N}. \end{aligned}$$ Explicitly, we have $$\Delta_N^{(k)} = N^{2k-2} \bigg{(} \log I_N - \sum_{g=0}^k N^{2-2g} F_N^{(g)} \bigg{)}, \quad k \in \mathbb{N} \cup \{0\}.$$ We could also have defined $\Delta_N^{(k)}$ using $J_N$ in place of $I_N$, and $G_N^{(g)}$ in place of $F_N^{(g)}$, and in what follows $\Delta_N^{(k)}$ may equally well be replaced with this function instead. Our main result, Theorem \[thm:Main\], is an immediate consequence of the following convergence theorem, the proof of which occupies the remainder of the paper. \[thm:MainReformulated\] For any $0 < \varepsilon < \xi$ we have $\lim_{N \to \infty} \\| \Delta_N^{(k)} \\|_\varepsilon = 0$ for each $k \in \mathbb{N}$. Reduction to uniform boundedness -------------------------------- By virtue of its definition, the function $\Delta_N^{(k)}$ admits the string expansion $$\Delta_N^{(k)} = \sum_{d=1}^\infty \frac{z^d}{d!} \sum_{\alpha,\beta \vdash d} \frac{p_\alpha(a_1,\dots,a_N)}{N^{\ell(\alpha)}} \frac{p_\beta(b_1,\dots,b_N)}{N^{\ell(\beta)}} \Delta_N^{(k)}(\alpha,\beta),$$ the coefficients of which are given by $$\Delta_N^{(k)}(\alpha,\beta) = N^{2k-2}\left( L_N(\alpha,\beta) - \sum_{g=0}^k \frac{{\vec{H}}_g(\alpha,\beta)}{N^{2g}} \right),$$ where $$\log I_N = \sum_{d=1}^\infty \frac{z^d}{d!} \sum_{\alpha,\beta \vdash d} \frac{p_\alpha(a_1,\dots,a_N)}{N^{\ell(\alpha)}} \frac{p_\beta(b_1,\dots,b_N)}{N^{\ell(\beta)}} L_N(\alpha,\beta),$$ and both series converge absolutely on $\overline{{\mathbb{D}}}_\gamma^N.$ From Section \[sec:Stable\], we know that, for each $k \in \mathbb{N} \cup \{0\},$ each fixed string coefficient of $\Delta_N^{(k)}$ converges to zero as $N \to \infty,$ $$\lim_{N \to \infty} \Delta_N^{(k)}(\alpha,\beta) = 0.$$ In fact, asymptotic vanishing of the string coefficients of $\Delta_N^{(k)}$ implies uniform asymptotic vanishing of string series provided we have uniform boundedness. Let $m \in \mathbb{N}$ be an arbitrary positive integer. By a “normalized string series” on $$\overline{{\mathbb{D}}}_\xi \times \overline{{\mathbb{D}}}_1^{mN},$$ we mean a power series of the form $$\Delta_N = \sum_{d=1}^\infty \frac{z^d}{d!} \sum_{\alpha^1,\dots,\alpha^m \vdash d} \prod_{i=1}^m \frac{p_{\alpha^i}(a_{i1},\dots,a_{iN})}{N^{\ell(\alpha^i)}}\Delta_N(\alpha^1,\dots,\alpha^m)$$ which converges absolutely on $\overline{{\mathbb{D}}}_\xi^N$. In order to prove Theorem \[thm:MainReformulated\], we will use the fact that, in the presence of uniform boundedness, uniform convergence of $\Delta_N$ on any closed proper subset of $\overline{{\mathbb{D}}}_\xi^N$ follows from the convergence of each of its string coefficients $\Delta_N(\alpha^1,\dots,\alpha^m)$. \[lem:ConvergenceLemma\] If $\sup_{N \in \mathbb{N}} \\|\Delta_N\\|_\xi < \infty$ and $$\lim_{N \to \infty} \Delta_N(\alpha^1,\dots,\alpha^m)=0$$ for any $d \in \mathbb{N}$ and $\alpha^1,\dots,\alpha^m \vdash d$, then $$\lim_{N \to \infty} \\|\Delta_N\\|_\varepsilon = 0$$ for any $0 < \varepsilon < \xi$. Fix $\varepsilon < \xi.$ Let $\kappa > 0$ be given. For any $n,N \in \mathbb{N}$, we have $$\begin{aligned} \left\\| \Delta_N \right\\|_\varepsilon \leq& \sum_{d=1}^n \frac{\varepsilon^d}{d!}\sum_{\alpha^1,\dots,\alpha^m \vdash d} \left\| \Delta_N(\alpha^1,\dots,\alpha^m) \right\| \\ +& \sum_{d=n+1}^\infty \frac{\varepsilon^d}{d!} \left\| \sum_{\alpha^1,\dots,\alpha^m \vdash d} \Delta_N(\alpha^1,\dots,\alpha^m) \prod_{i=1}^m \frac{p_{\alpha^i}(a_1,\dots,a_N)}{N^{\ell(\alpha^i)}} \right\|, \end{aligned}$$ by the triangle inequality. By Cauchy’s estimate, $$\frac{1}{d!} \left\| \sum_{\alpha^1,\dots,\alpha^m \vdash d} \Delta_N(\alpha^1,\dots,\alpha^m) \prod_{i=1}^m \frac{p_{\alpha^i}(a_1,\dots,a_N)}{N^{\ell(\alpha^i)}} \right\| \leq \frac{\\|\Delta_N\\|_\xi}{\xi^d}.$$ Thus $$\left\\| \Delta_N \right\\|_{\varepsilon} \\ \leq \ \sum_{d=1}^n \frac{\varepsilon^d}{d!}\sum_{\alpha^1,\dots,\alpha^m \vdash d} \left\| \Delta_N(\alpha^1,\dots,\alpha^m) \right\| + \left( \frac{\varepsilon}{\xi} \right)^{n+1} K,$$ where $$K = \frac{\sup_{N \in \mathbb{N}} \\|\Delta_N\\|_\xi}{1-\frac{\varepsilon}{\xi}}$$ is a constant. Since $$\lim_{n \to \infty} \left( \frac{\varepsilon}{\xi} \right)^{n+1} = 0,$$ we can choose $n_0$ sufficiently large so that $$\left( \frac{\varepsilon}{\xi} \right)^{n_0+1} K < \frac{\kappa}{2}.$$ Then, since $$\lim_{N \to \infty} \|\Delta_N(\alpha^1,\dots,\alpha^m)\| = 0$$ for each $d \in \mathbb{N}$ and all $\alpha^1,\dots,\alpha^m \vdash d$, we can choose $N_0$ sufficiently large so that $N \geq N_0$ implies $$\sum_{d=1}^{n_0} \frac{\varepsilon^d}{d!}\sum_{\alpha^1,\dots,\alpha^m \vdash d} \left\| \Delta_N(\alpha^1,\dots,\alpha^m) \right\| < \frac{\kappa}{2}.$$ We conclude that $N \geq N_0$ implies $$\left\\| \Delta_N \right\\|_{\varepsilon} < \kappa,$$ as required. Proof of uniform boundedness ---------------------------- In view of Lemma \[lem:ConvergenceLemma\], the following result completes the proof of Theorem \[thm:MainReformulated\] and hence also of Theorem \[thm:Main\], which proves Conjecture \[conj:Main\]. \[thm:UniformBoundedness\] For any $\varepsilon < \xi,$ we have $$\sup_{N \in \mathbb{N}} \\|\Delta_N^{(k)}\\|_\varepsilon < \infty$$ for each $k \in \mathbb{N} \cup \{0\}.$ Let $(\varepsilon_k)_{k=0}^\infty$ be a strictly decreasing sequence in the interval $(\varepsilon,\xi).$ Then, for each $N \in \mathbb{N}$, we have a corresponding sequence of nested closed polydiscs, $$\overline{{\mathbb{D}}}_\xi^N \supset \overline{{\mathbb{D}}}_{\varepsilon_0}^N \supset \overline{{\mathbb{D}}}_{\varepsilon_1}^N \supset \dots \supset \overline{{\mathbb{D}}}_{\varepsilon}^N.$$ Let $k \in \mathbb{N} \cup \{0\}$ be fixed. Observe that $$\frac{I_N}{e^{\sum_{g=0}^k N^{2-2g}F_N^{(g)}}} = e^{N^{2-2k}\Delta_N^{(k)}}.$$ Thus, by Theorem \[thm:TopBound\], we have $$\bigg{\|} 1 - e^{N^{2-2k}\Delta_N^{(k)}} \bigg{\|} \leq c_k N^{2-2k}$$ on $\overline{{\mathbb{D}}}_{\varepsilon_k}^N$ for $N$ sufficiently large, where $c_k$ is a positive constant depending only on $k.$ This in turn implies $$\left\| e^{N^{2-2k}\Delta_N^{(k)}} \right\| \leq 1 + c_kN^{2-2k}$$ on $\overline{{\mathbb{D}}}_{\varepsilon_k}^N$ for $N$ sufficiently large. Thus $$N^{2-2k} \operatorname{Re} \Delta_N^{(k)} \leq \log \left(1 + c_kN^{2-2k} \right).$$ on $\overline{{\mathbb{D}}}_{\varepsilon_k}^N$ for $N$ sufficiently large. If $k \neq 1,$ this yiels $$N^{2-2k} \operatorname{Re} \Delta_N^{(k)} \leq \log \left(1 + c_kN^{2-2k} \right) \leq c_kN^{2-2k}$$ on $\overline{{\mathbb{D}}}_{\varepsilon_k}^N$ for $N$ sufficiently large, so that $$\operatorname{Re} \Delta_N^{(k)} \leq c_k$$ on $\overline{{\mathbb{D}}}_{\varepsilon_k}^N$ for $N$ sufficiently large. If $k=1$ we obtain instead $$\operatorname{Re} \Delta_N^{(k)} \leq \log \left(1 + c_k \right)$$ on $\overline{{\mathbb{D}}}_{\varepsilon_k}^N$ for $N$ sufficiently large. Thus in all cases we have a uniform bound on the real part of $\Delta_N^{(k)}$, i.e. a bound of the form $$\operatorname{Re} \Delta_N^{(k)} \leq \tilde{c}_k$$ for some positive constant $\tilde{c}_k.$ In order to leverage this into a bound on the modulus, we apply the Borel-Carathéodory inequality (see e.g. [@Titchmarsh]), which bounds the sup norm of an analytic function on a closed disc in terms of the supremum of its real part on a larger closed disc. 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--- abstract: \| We prove that Ringel duality in the category strict polynomial functors can be interpreted as derived functors of non-additive functors (in the sense of Dold and Puppe). We give applications of this fact for both theories.\ AMS classification: 18G55, 20G05, 20G10. author: - Antoine Touzé title: 'Ringel duality and derivatives of non-additive functors' --- Introduction ============ The purpose of this paper is to give an explicit and simple relation between two fundamental theories: Ringel duality for the representations of Schur algebras, and the theory of derived functors of non-additive functors. Derived functors of non additive functors were introduced by Dold and Puppe in [@DP1; @DP2], and later generalized by Quillen [@Quillen]. These derived functors arose from the work on simplicial structures (the Dold-Kan correspondence), and came as a conceptual framework explaining many computations from algebraic topology. For example, the ‘quite bizarre’ functors $\Omega(\Pi)$ and $R(\Pi)$ discovered by Eilenberg and Mac Lane in their study [@EML2] of the low degrees of the integral homology of Eilenberg-Mac Lane spaces can be interpreted as the derived functors of the second symmetric power $S^2:\mathrm{Ab}\to \mathrm{Ab}$. More generally, the whole homology of Eilenberg-Mac Lane spaces can be described as derived functors of the symmetric algebra functor $S^$ or of the group ring functor. Derived functors are also related to the homology of symmetric products [@Do; @DP2], with group theory and the homotopy groups of Moore spaces (including the homotopy groups of spheres) [@Curtis; @MP; @BM]. Representations of Schur algebras arose in a quite different context, from the works of Schur on the general linear group [@Schur]. Modules over Schur algebras correspond to polynomial representations of $GL_n$. Representation theory of Schur algebras has known a great development in the eighties and the nineties, with the work of Akin, Buchsbaum and Weyman on characteristic free representation theory see e.g. [@ABW; @AB1; @AB], the development of the theory highest weight categories, see e.g. [@CPS; @Ringel], the work of Donkin, Green and many others (we refer the reader to the books [@Green; @Martin] for further references). A recent development of representations of Schur algebras is the introduction of the categories ${\mathcal{P}}_{d,{\Bbbk}}$ of strict polynomial functors by Friedlander and Suslin. Such functors can be thought of as functors $F:{\mathcal{V}}_{\Bbbk}\to {\mathcal{V}}_{\Bbbk}$ (where ${\mathcal{V}}_{\Bbbk}$ is the category of finitely generated projective ${\Bbbk}$-modules) defined by polynomial formulas (which are homogeneous of degree $d$). Typical examples are the symmetric powers $S^d$, the exterior powers $\Lambda^d$ or the divided powers $\Gamma^d$. Friedlander and Suslin proved [@FS Thm 3.2] an equivalence of categories $${\mathcal{P}}_{d,{\Bbbk}}\simeq S(n,d)\text{-mod}$$ between strict polynomial functors and modules over the Schur algebra $S(n,d)$, provided $n\ge d$. So, problems involving Schur algebras may be approached via strict polynomial functors. This point of view has been very fruitful for cohomological computations, see e.g. [@FS; @FFSS; @Chalupnik2; @TouzeEML], and it is the point of view which we adopt in the article. The category of strict polynomial functors is equipped with a duality operation $\Theta$ which produces for each functor $F$ a ‘signed version’ $\Theta F$ of $F$. For example, $\Theta S^d=\Lambda^d$. Actually, this duality has better properties at the level of derived categories, where it becomes an equivalence of categories. This equivalence of categories was first studied in the framework of highest weight categories and representations of Schur algebras in [@Ringel; @DonkinKos] and it is commonly called Ringel duality (although it has been called ‘Koszul duality’[^1] in the framework of strict polynomial functors in [@Chalupnik2]). Main result. {#main-result. .unnumbered} ------------ Let us now state our main result. Let ${\Bbbk}$ be a PID, and let $F\in{\mathcal{P}}_{d,{\Bbbk}}$ be a strict polynomial functor. If $V\in{\mathcal{V}}_{\Bbbk}$ is a free finitely generated ${\Bbbk}$-module, we build an isomorphism: $$L_{nd-i}F(V;n)\simeq H^i\left(\Theta^n F(V)\right) \quad ()$$ The objects on the left hand side of the isomorphism are the derived functors of $F$ (in the sense of Dold and Puppe) and the objects on the right hand side are the homology groups of $n$-th iterated Ringel dual of $F$. Actually, our main theorem \[thm-main\] is sharper: it asserts a version of isomorphism $()$ in the derived category, and it describes the slightly delicate compatibility of the isomorphism with tensor products. If $n=1$ or $n=2$, the homology groups of $\Theta^n F$ can be interpreted as extension groups, so the isomorphism $()$ takes a more concrete form. For example, for $V={\Bbbk}$, we have isomorphisms, natural in $F$: $$L_{d-i}F({\Bbbk};1)\simeq {{\mathrm{Ext}}}^i_{{\mathcal{P}}_{d,{\Bbbk}}}(\Lambda^d,F)\,,\quad L_{2d-i}F({\Bbbk};2)\simeq {{\mathrm{Ext}}}^i_{{\mathcal{P}}_{d,{\Bbbk}}}(S^d,F)$$ The existence of the isomorphism $()$ was hinted at in [@TouzeEML], where the author proved (without appealing to Ringel duality) that the extension groups ${{\mathrm{Ext}}}^_{{\mathcal{P}}_{{\Bbbk}}}(\Lambda^,\Gamma^)$ and ${{\mathrm{Ext}}}^_{{\mathcal{P}}_{{\Bbbk}}}(S^,\Gamma^)$ are related to the singular homology with ${\Bbbk}$ coefficients of the Eilenberg-Mac Lane spaces $K({\mathbb{Z}},3)$ and $K({\mathbb{Z}},4)$, which are respectively given by $L_\Gamma^({\Bbbk};1)$ and $L_\Gamma^({\Bbbk};2)$. Isomorphism $()$ yields other unexpected relations between representation theory of Schur algebras and algebraic topology. For example, let ${\mathcal{L}}^({\mathbb{Z}}^m)$ denote the free Lie algebra generated by ${\mathbb{Z}}^m$. Each degree of the free Lie algebra yields a strict polynomial functor ${\mathcal{L}}^d\in{\mathcal{P}}_{d,{\mathbb{Z}}}$, and their derived functors appear in the first page of the Curtis spectral sequence [@Curtis; @MP; @BM], which converges to the homotopy of Moore spaces. For instance, there are spectral sequences converging to the unstable homotopy groups of the spheres $\mathbb{S}^2$ and $\mathbb{S}^3$: $$\begin{aligned} &E^1_{i,j}={{\mathrm{Ext}}}^{j-i}_{{\mathcal{P}}_{j,{\mathbb{Z}}}}(\Lambda^j,{\mathcal{L}}^j)\Longrightarrow \pi_{j+1}(\mathbb{S}^2)\;,\\& E^1_{i,j}={{\mathrm{Ext}}}^{2j-i}_{{\mathcal{P}}_{j,{\mathbb{Z}}}}(S^j,{\mathcal{L}}^j)\Longrightarrow \pi_{j+1}(\mathbb{S}^3)\;.\end{aligned}$$ Our result is also interesting from a computational point of view, because people working on each side of isomorphism $()$ use different techniques and understand different phenomena. Specialists of Schur algebras have developed combinatorics of partitions, highest weight categories, or use results coming from algebraic groups and algebraic geometry (e.g. Kempf theorem [@Jantzen Part II, Chap 4]). On the other side, homotopists use simplicial techniques, with an intuition coming from topology. Results which are well-known in the world of representations of Schur algebras translate into results which are unknown to homotopists and vice-versa. So we hope that our results can serve as a basis for fruitful new interactions between these two subjects. Organization of the paper. {#organization-of-the-paper. .unnumbered} -------------------------- The paper is more or less self-contained, and we have tried to give an elementary treatment of the subject. The first three sections are mainly introductory. Section \[sec-str\] is an introduction to Schur algebras, strict polynomial functors and their derived categories. Section \[section-Kos\] is a presentation of Ringel duality, based on Cha[ł]{}upnik’s definition from [@Chalupnik2]. Finally, section \[section-DP\] recalls the definition of derived functors of non-additive functors, and extends the classical definition to the derived category of strict polynomial functors. In section \[section-main\], we prove our main theorem, which gives the link between Ringel duality and derived functors of non additive functors. In section \[sec-applic\], we give some applications of our main theorem. All these applications follow the same principle: to obtain a theorem on one side of the isomorphism $()$, we prove a statement on the other side and we translate it using isomorphism $()$. In this way we obtain the following new results. 1. We prove a décalage formula for derived functors, generalizing the usual décalage formula due to Quillen [@Quillen2] and Bousfield [@Bousfield], and which also generalizes computations of Bott [@Bott]. 2. We find a formula to compute Ringel duals of plethysms (i.e. composites of functors). This formula yields many ${{\mathrm{Ext}}}$-computations in ${\mathcal{P}}_{\Bbbk}$. In particular it provides obstructions to the existence of certain ‘universal filtrations’ (the non-existence of these filtrations was conjectured in [@Boffi]). 3. We show how to use block theory of Schur algebras to prove vanishing results for derived functors. 4. We supplement these applications by giving an example of a computation of derived functors using Ringel duality (thus retrieving a result of [@BM]). For the sake of completeness, we have included: a conversion table to help the reader read the literature on Schur functors in section \[subsubsec-schur\], a short discussion about the target category of our strict polynomial functors in section \[sec-arb\], and an appendix providing an elementary proof that the category of strict polynomial functors is equivalent to the category of modules over the Schur algebra. Strict polynomial functors and their derived categories {#sec-str} ======================================================= This section is an introduction to the theory of strict polynomial functors originally developed in [@FS Sections 2 and 3] (and in [@SFB Section 2] over an arbitrary commutative ring). We first give the basics of the theory over an arbitrary commutative base ring ${\Bbbk}$. When ${\Bbbk}$ is a Dedekind ring (e.g. a PID), the category has an internal ${{\mathrm{Hom}}}$ and we recall its properties. Finally, we give a brief account of the derived category of strict polynomial functors over a PID. Recollections of Strict Polynomial Functors {#subsec-SPF} ------------------------------------------- Strict polynomial functors were originally defined over a field ${\Bbbk}$ in [@FS] but as remarked in [@SFB], the generalization over a commutative ring ${\Bbbk}$ is straightforward. In this more general setting, we let ${\mathcal{V}}_{\Bbbk}$ be the category of finitely generated projective ${\Bbbk}$-modules and ${\Bbbk}$-linear maps. (This notation and the letters $V$, $W$, etc. denoting the objects of ${\mathcal{V}}_{\Bbbk}$ come from the field case where ${\mathcal{V}}_{\Bbbk}$ is the category of finite dimensional vector spaces). ### Schur algebras, categories $\Gamma^d{\mathcal{V}}_{\Bbbk}$, and their representations {#subsubsec-SGR} Let ${\Bbbk}$ be a commutative ring and let $n$ and $d$ be positive integers. The tensor product ${{\Bbbk}^n}^{\,\otimes d}$ is acted on by the symmetric group ${\mathfrak{S}}_d$ which permutes the factors of the tensor product. The Schur algebra $S(n,d)$ is the algebra ${{\mathrm{End}}}_{{\mathfrak{S}}_d}({{\Bbbk}^n}^{\,\otimes d})$ of ${\mathfrak{S}}_d$-equivariant ${\Bbbk}$-linear homomorphisms [@Green 2.6c]. The categories $\Gamma^d{\mathcal{V}}_{\Bbbk}$ generalize Schur algebras. The objects of $\Gamma^d{\mathcal{V}}_{\Bbbk}$ are the finitely generated projective ${\Bbbk}$-modules. The homomorphism modules ${{\mathrm{Hom}}}_{\Gamma^d{\mathcal{V}}_{\Bbbk}}(V,W)$ are the ${\mathfrak{S}}_d$-equivariant maps from $V^{\otimes d}$ to $W^{\otimes d}$ and the composition is just the composition of ${\mathfrak{S}}_d$-equivariant maps. In particular, $S(n,d)$ is nothing but the algebra of endomorphisms of the object ${\Bbbk}^n\in\Gamma^d{\mathcal{V}}_{\Bbbk}$. By abuse, we identify $S(n,d)$ with the full subcategory of $\Gamma^d{\mathcal{V}}_{\Bbbk}$ with ${\Bbbk}^n$ as unique object. The category of ${\Bbbk}$-linear representations of $\Gamma^d{\mathcal{V}}_{\Bbbk}$ in ${\mathcal{V}}_{\Bbbk}$ (i.e. the category of functors $F:\Gamma^d{\mathcal{V}}_{\Bbbk}\to{\mathcal{V}}_{\Bbbk}$ whose action on morphisms $f\mapsto F(f)$ is ${\Bbbk}$-linear) is denoted by ${\mathcal{P}}_{d,{\Bbbk}}$ and is commonly called the category of degree $d$ homogeneous strict polynomial functors[^2]. Restriction of $\Gamma^d{\mathcal{V}}_{\Bbbk}$-representations to the full subcategory with unique object ${\Bbbk}^n$ yields a functor $${\mathcal{P}}_{d,{\Bbbk}}\to S(n,d)\text{-mod}\;.$$ (Here $S(n,d)\text{-mod}$ stands for the category of modules over the Schur algebra, which are finitely generated and projective as ${\Bbbk}$-modules.) Friedlander and Suslin proved [@FS; @SFB] that it is an equivalence of categories if $n\ge d$ (we give a direct proof of this in appendix \[app\]). The strict polynomial functor $\otimes^d$ sends an object $V$ of $\Gamma^d{\mathcal{V}}_{\Bbbk}$ to $V^{\otimes d}$ and sends a morphism $f\in {{\mathrm{Hom}}}_{\Gamma^d{\mathcal{V}}_{\Bbbk}}(V,W)$ to the same $f$, but viewed as an element of ${{\mathrm{Hom}}}_{\Bbbk}(V^{\otimes d},W^{\otimes d})$. Sums, products, kernels, cokernels, etc. in the category ${\mathcal{P}}_{d,{\Bbbk}}$ are computed objectwise in the target category ${\mathcal{V}}_{\Bbbk}$, so that the structure of ${\mathcal{P}}_{d,{\Bbbk}}$ inherits many properties from ${\mathcal{V}}_{\Bbbk}$. In particular, if ${\Bbbk}$ is a field, then ${\mathcal{P}}_{d,{\Bbbk}}$ is an abelian category, and more generally over an arbitrary commutative ring ${\Bbbk}$, ${\mathcal{P}}_{d,{\Bbbk}}$ is an exact category in the sense of Quillen [@Buehler; @Keller], the admissible short exact sequences $F\xrightarrow[]{\iota} G\xrightarrow[]{p} H $ (i.e. the conflations $(\iota,p)$ according to the terminology of [@Keller]) being the ones which become short exact sequences after evaluation on any $V\in{\mathcal{V}}_{\Bbbk}$. We denote by ${\mathcal{P}}_{0,{\Bbbk}}$ the category of constant functors from ${\mathcal{V}}_{\Bbbk}$ to ${\mathcal{V}}_{\Bbbk}$. The category ${\mathcal{P}}_{{\Bbbk}}$ of strict polynomial functors of finite degree is defined by $${\mathcal{P}}_{\Bbbk}=\textstyle\bigoplus_{d\ge 0}{\mathcal{P}}_{d,{\Bbbk}}\;,$$ where the right hand side term denotes the subcategory of $\Pi_{d\ge 0} {\mathcal{P}}_{d,{\Bbbk}}$ whose objects are the finite products (only a finite number of terms are non zero). If $F=\oplus_{i\le d} F_i$, with $F_i\in{\mathcal{P}}_{i,{\Bbbk}}$ and $F_d\ne 0$, then $F$ is said to have strict polynomial degree $d$. ### Strict polynomial functors vs ordinary functors There is an exact forgetful functor from the category ${\mathcal{P}}_{{\Bbbk}}$ to the category ${\mathcal{F}}_{\Bbbk}$ of ordinary functors from ${\mathcal{V}}_{\Bbbk}$ to ${\Bbbk}$-modules: $${\mathcal{U}}:{\mathcal{P}}_{\Bbbk}\to {\mathcal{F}}_{\Bbbk}={\mathrm{Fct}}({\mathcal{V}}_{\Bbbk},{\Bbbk}\text{-mod})\;,$$ defined in the following way. If $F\in{\mathcal{P}}_{0,{\Bbbk}}$ then $F$ is already a (constant) functor. If $d\ge 1$, the functor ${\mathcal{U}}$ sends an element $F\in{\mathcal{P}}_{d,{\Bbbk}}$ to the precomposition $F\circ \gamma^d$, where $\gamma^d:{\mathcal{V}}_{\Bbbk}\to \Gamma^d{\mathcal{V}}_{\Bbbk}$ is the functor which is the identity on objects and sends a map $f$ to $f^{\otimes d}$. ${\mathcal{U}}(\otimes^d)$ is the functor $V\mapsto V^{\otimes d}$ in the usual sense. One can prove that all the functors in the image of ${\mathcal{U}}$ are polynomial in the sense of Eilenberg and Mac-Lane [@EML2], and the Eilenberg Mac-Lane degree of ${\mathcal{U}}F$ is less or equal to the strict polynomial degree of $F$. The inequality may very well be strict, e.g. if ${\Bbbk}=\mathbb{F}_p$, the Frobenius twist functor $I^{(1)}$ [@FS (v) p. 224] has strict polynomial degree $p$, but ${\mathcal{U}}I^{(1)}$ is the identity functor, whose Eilenberg Mac Lane degree equals one. ### Further structure of ${\mathcal{P}}_{\Bbbk}$ over a commutative ring ${\Bbbk}$ We now briefly summarize further structure borne by the category of strict polynomial functors over a commutative ring ${\Bbbk}$. Tensor products. : One can take tensor products of strict polynomial functors in the target category $(F\otimes G)(V) :=F(V)\otimes G(V)$. This yields bi-exact functors: $$\otimes:{\mathcal{P}}_{d,{\Bbbk}}\times{\mathcal{P}}_{e,{\Bbbk}}\to {\mathcal{P}}_{d+e,{\Bbbk}}\quad\text{ and }\quad\otimes:{\mathcal{P}}_{\Bbbk}\times{\mathcal{P}}_{\Bbbk}\to {\mathcal{P}}_{\Bbbk}\;.$$ Composition. : One can also compose strict polynomial functors: $(F\circ G)(V)=F(G(V))$. In this way we obtain exact functors: $$\circ:{\mathcal{P}}_{d,{\Bbbk}}\times{\mathcal{P}}_{e,{\Bbbk}}\to {\mathcal{P}}_{de,{\Bbbk}}\quad\text{ and }\quad\circ:{\mathcal{P}}_{\Bbbk}\times{\mathcal{P}}_{\Bbbk}\to {\mathcal{P}}_{\Bbbk}\;.$$ Duality. : We let $V^\vee$ be the ${\Bbbk}$-module ${{\mathrm{Hom}}}_{\Bbbk}(V,{\Bbbk})$. The formula $F^\sharp(V):=F(V^\vee)^\vee$ defines equivalences of categories (which are self inverse): $$^\sharp: ({\mathcal{P}}_{d,{\Bbbk}})^{\mathrm{op}}\xrightarrow[]{\simeq}{\mathcal{P}}_{d,{\Bbbk}}\quad\text{ and }\quad \,^\sharp:({\mathcal{P}}_{\Bbbk})^{\mathrm{op}}\xrightarrow[]{\simeq}{\mathcal{P}}_{\Bbbk}\;.$$ Projectives. : We denote by $\Gamma^{d,V}$ the strict polynomial functor defined by (the first equality is the definition, and the last two equalities are the canonical identifications): $$\Gamma^{d,V}(W)={{\mathrm{Hom}}}_{\Gamma^d{\mathcal{V}}_{\Bbbk}}(V,W)=({{\mathrm{Hom}}}_{\Bbbk}(V,W)^{\otimes d})^{{\mathfrak{S}}_d}=\Gamma^{d}({{\mathrm{Hom}}}_{\Bbbk}(V,W))\;.$$ The Yoneda lemma yields an isomorphism (natural in $V$ and $F$): $${{\mathrm{Hom}}}_{{\mathcal{P}}_{d,{\Bbbk}}}(\Gamma^{d,V},F)\simeq F(V)\;,$$ so the $\Gamma^{d,V}$ are actually projectives of ${\mathcal{P}}_{d,{\Bbbk}}$. In fact, the functors $\Gamma^{d,V}$ for $V\in{\mathcal{V}}_{\Bbbk}$, form a projective generator of ${\mathcal{P}}_{d,{\Bbbk}}$. To be more specific, the canonical map $\Gamma^{d,V}(W)\otimes G(V)\to G(W)$ is an epi if $V$ contains ${\Bbbk}^d$ as a direct summand, see [@FS Thm 2.10] or appendix \[app\]. We simply denote by $\Gamma^d$ the functor $\Gamma^{d,{\Bbbk}}$ (if $d=0$, it is the constant functor with value ${\Bbbk}$). For all $n$-tuples $\mu=(\mu_1,\dots,\mu_n)$, we denote by $\Gamma^\mu$ the tensor product $\bigotimes_{i=1}^n \Gamma^{\mu_i}$. The functor $\Gamma^{d,{\Bbbk}^n}$ decomposes as the direct sum $\bigoplus_\mu \Gamma^\mu$, the sum being taken over all $n$-tuples $\mu$ of nonnegative integers of weight $\sum\mu_i=d$. This has two consequences. First, the $\Gamma^\mu$ form a projective generator. Second, the tensor product of projectives is once again projective. Injectives. : By duality, the functors $S^d_V = (\Gamma^{d,V})^\sharp:W\mapsto S^d(W\otimes V)$ form an injective cogenerator of ${\mathcal{P}}_{d,{\Bbbk}}$. We denote by $S^d$ the functor $S^{d}_{\Bbbk}$ and for all $n$-tuples $\mu$ of nonnegative integers we denote by $S^\mu$ the tensor product $\otimes_{i=1}^n S^{\mu_i}$. All these functors are injectives and the family $(S^\mu)_\mu$ indexed by tuples of weight $d$ forms an injective cogenerator of ${\mathcal{P}}_{d,{\Bbbk}}$. Extensions. : Since the ${\mathcal{P}}_{d,{\Bbbk}}$ (hence ${\mathcal{P}}_{\Bbbk}$) are exact categories with enough projectives and injectives, there is no problem in defining extension groups [@Buehler; @Keller]. Since ${\mathcal{P}}_{\Bbbk}=\textstyle\bigoplus_{d\ge 0}{\mathcal{P}}_{d,{\Bbbk}}$, if $F$ and $G$ are homogeneous strict polynomial functors then ${{\mathrm{Ext}}}^_{{\mathcal{P}}_{\Bbbk}}(F,G)$ equals ${{\mathrm{Ext}}}^_{{\mathcal{P}}_{d,{\Bbbk}}}(F,G)$ if $F$ and $G$ have the same degree, and zero otherwise. \[rk-PT\] The category ${\mathcal{P}}_{d,{\Bbbk}}$ is a full exact subcategory of the abelian category $\widetilde{{\mathcal{P}}}_{d,{\Bbbk}}$ of functors from $\Gamma^d{\mathcal{V}}_{\Bbbk}$ to arbitrary ${\Bbbk}$-modules. A projective resolution ${\mathcal{P}}_{d,{\Bbbk}}$ yields a projective resolution in $\widetilde{{\mathcal{P}}}_{d,{\Bbbk}}$, so the inclusion functor ${\mathcal{P}}_{d,{\Bbbk}}\hookrightarrow \widetilde{{\mathcal{P}}}_{d,{\Bbbk}}$ induces an isomorphism (see also section \[sec-arb\]): $${{\mathrm{Ext}}}^_{{\mathcal{P}}_{{\Bbbk},d}}(F,G)\simeq {{\mathrm{Ext}}}^_{\widetilde{{\mathcal{P}}}_{{\Bbbk},d}}(F,G)\;.$$ In this article, we prefer to work in ${\mathcal{P}}_{d,{\Bbbk}}$ because one loses duality and exactness of tensor products for functors with arbitrary values. The internal ${{\mathrm{Hom}}}$ over Dedekind rings --------------------------------------------------- In this section, we restrict our attention to the category ${\mathcal{P}}_{d,{\Bbbk}}$ of homogeneous strict polynomial functors of degree $d$. We assume that ${\Bbbk}$ is a Dedekind ring (for example a PID). It ensures that ${{\mathrm{Hom}}}$-groups in ${\mathcal{P}}_{d,{\Bbbk}}$ are finitely generated and projective ${\Bbbk}$-modules. Thus, we can introduce a parameter to obtain internal ${{\mathrm{Hom}}}$s. This idea is already used e.g. in [@Chalupnik2; @TouzeEML]. We recall here their definition and main properties. We begin with a few notations about parameterized functors. ### Parameterized functors {#subsubsec-param} If $F$ is a strict polynomial functor and $V\in {\mathcal{V}}_{\Bbbk}$, we form a strict polynomial functor $F^V$ with parameter $V$, by letting $F^V(W)=F({{\mathrm{Hom}}}_{\Bbbk}(V,W))$. It is straightforward to check that this actually yields a functor: $$(\Gamma^d{\mathcal{V}}_{\Bbbk})^{{\mathrm{op}}}\times\Gamma^d{\mathcal{V}}_{\Bbbk}\to {\mathcal{V}}_{\Bbbk},\quad (V,W)\mapsto F^V(W).$$ The parameter ‘$V$’ is written as an exponent to indicate the contravariance in $V$. The notation for parameterized functors is coherent with the notation for projectives: $(\Gamma^d)^V=\Gamma^{d,V}$, and more generally, $(\Gamma^{d,V_1})^{V_2}=\Gamma^{d,V_1\otimes V_2}$. Similarly, one can introduce a covariant parameter $V$ (hence written as an index), by letting $F_V(W)=F(V\otimes W)$. Observe that $(F_V)^{\sharp}\simeq (F^\sharp)^V$, and that this notation agrees with the notation for injectives: $(S^d_{V_1})_{V_2}\simeq S^d_{V_1\otimes V_2}$. We shall use heavily the following fact in the sequel. We can replace the parameter $V$ by a simplicial free ${\Bbbk}$-module $X$. In this case $F_X$ becomes a simplicial object in ${\mathcal{P}}_{d,{\Bbbk}}$, with $(F_X)_n=F_{X_n}$ with face operators $F(d_i):F_{X_n}\to F_{X_{n-1}}$ and degeneracy operators $F(s_i):F_{X_n}\to F_{X_{n+1}}$. Similarly $F^X$ becomes a cosimplicial object in ${\mathcal{P}}_{d,{\Bbbk}}$. This generalizes to multisimplicial objects: if $X$ is a bisimplicial free ${\Bbbk}$-module then $F_X$ is a bisimplicial strict polynomial functor, and so on. ### The internal ${{\mathrm{Hom}}}$ in ${\mathcal{P}}_{d,{\Bbbk}}$ {#subsubsec-internal} We recall the construction and properties of the internal ${{\mathrm{Hom}}}$ in ${\mathcal{P}}_{{\Bbbk},d}$. For the proofs (which are elementary) we refer the reader to [@TouzeEML section 4]. Let ${\Bbbk}$ be a Dedekind ring. Then for all $F,G\in{\mathcal{P}}_{d,{\Bbbk}}$, the ${\Bbbk}$-module ${{\mathrm{Hom}}}_{{\mathcal{P}}_{\Bbbk}}(F,G)$ is finitely generated and projective. In particular, the formula $V\mapsto {{\mathrm{Hom}}}_{{\mathcal{P}}_{\Bbbk}}(F^V,G)$ defines an element of ${\mathcal{P}}_{d,{\Bbbk}}$, which we denote by ${\mathbb{H}}(F,G)$, and we have a bifunctor: $$\begin{array}{cccc} {\mathbb{H}}:&{\mathcal{P}}_{d,{\Bbbk}}^{\mathrm{op}}\times{\mathcal{P}}_{d,{\Bbbk}}&\to&{\mathcal{P}}_{d,{\Bbbk}}\\ & (F,G)&\mapsto & {\mathbb{H}}(F,G) \end{array}\;.$$ The bifunctor ${\mathbb{H}}$ enjoys the following properties. First, the Yoneda lemma and duality respectively yield isomorphisms (natural in $F,G,U$): $${\mathbb{H}}(\Gamma^{d,U},G)\simeq G_U\;,\qquad {\mathbb{H}}(F,G)\simeq {\mathbb{H}}(G^\sharp,F^\sharp)\;.$$ The bifunctor ${\mathbb{H}}$ is also compatible with tensor products. For $i=1,2$, let $F_i,G_i$ be homogeneous strict polynomial functors of degree $d_i$. Tensor products induce a morphism of strict polynomial functors (natural in $F_i, G_i$): $${\mathbb{H}}(F_1,G_1)\otimes{\mathbb{H}}(F_2,G_2)\xrightarrow[]{\otimes} {\mathbb{H}}(F_1\otimes F_2,G_1\otimes G_2)\;.$$ Finally, if $X^$ denotes $S^,\Lambda^$ or $\Gamma^$, we may postcompose this tensor product by the map induced by the comultiplication $X^{d_1+d_2}\to X^{d_1}\otimes X^{d_2}$ to get an isomorphism [@TouzeEML Lemma 5.5]: $$\begin{aligned} {\mathbb{H}}(X^{d_1},G_1)\otimes{\mathbb{H}}(X^{d_2},G_2)\xrightarrow[]{\simeq} {\mathbb{H}}(X^{d_1+d_2},G_1\otimes G_2)\;.\end{aligned}$$ The bifunctor ${\mathbb{H}}$ is an internal ${{\mathrm{Hom}}}$ in the usual sense. In particular, it is adjoint to the symmetric monoidal product $\bullet$ in ${\mathcal{P}}_{{\Bbbk},d}$ defined by $F\bullet G={\mathbb{H}}(F,G^\sharp)^\sharp$. However, the facts recalled above will be sufficient for our purposes, and we refer the reader to [@Krause] for a nice exposition of this symmetric monoidal product. We have used the notation ‘${\mathbb{H}}$’ instead of the more standard notation $\underline{{{\mathrm{Hom}}}}_{{\mathcal{P}}_{d,{\Bbbk}}}$ for typographical reasons (to keep formulas compact). Derived categories ------------------ Now we assume that ${\Bbbk}$ is a PID. This ensures [@DonkinHDim; @AB] that the category of modules over the Schur algebra, hence the category ${\mathcal{P}}_{d,{\Bbbk}}$, has finite homological dimension. So we can work without trouble in the bounded derived category. The assumption on ${\Bbbk}$ also allows us to use internal ${{\mathrm{Hom}}}$s. ### Chain complexes, etc. {#subsec-conventions} We recall the conventions and notations for complexes which we use in the article. In what follows, ${\mathcal{A}}$ is an additive category, enriched over a commutative ring ${\Bbbk}$ (e.g. the category of ${\Bbbk}$-modules, ${\mathcal{P}}_{d,{\Bbbk}}$, modules over a ${\Bbbk}$-algebra, etc.) Complexes. : We let ${\mathrm{Ch}}({\mathcal{A}})$ be the category of complexes in ${\mathcal{A}}$. Gradings are indifferently denoted using the homological convention or the cohomological one. As usual, the conversion between the two conventions is realized by the formula $C^i=C_{-i}$. We denote by ${\mathrm{Ch}}^b({\mathcal{A}})$, ${\mathrm{Ch}}^+({\mathcal{A}})$, ${\mathrm{Ch}}^-({\mathcal{A}})$ the full subcategories of bounded, bounded below, and bounded above cochain complexes. We also denote by ${\mathrm{Ch}}_{\ge 0}$ the full subcategory of ${\mathrm{Ch}}^-({\mathcal{A}})$ consisting of nonnegatively graded chain complexes. We let ${\mathbf{K}}({\mathcal{A}})$, ${\mathbf{K}}^b({\mathcal{A}})$, ${\mathbf{K}}^+({\mathcal{A}})$, ${\mathbf{K}}^-({\mathcal{A}})$, ${\mathbf{K}}_{\ge 0}$ the corresponding homotopy categories. Suspension. : If $C$ is a complex, its $i$-th suspension $C[i]$ is defined by $C[i]^n=C^{n-i}$ (or $C[i]_n=C_{i+n}$) and $d_{C[i]}= (-1)^{i}d_C$. The suspension of a morphism of complexes $f:C\to D$ is given by $f[d]_i=f_{d+i}$ Tensor products. : If ${\mathcal{A}}$ is equipped with a symmetric, associative and unital tensor product (in other words: ${\mathcal{A}}$ is a strict symmetric monoidal category), then so are the categories ${\mathrm{Ch}}^b({\mathcal{A}})$, ${\mathrm{Ch}}^+({\mathcal{A}})$, ${\mathrm{Ch}}^-({\mathcal{A}})$ and ${\mathrm{Ch}}_{\ge 0}({\mathcal{A}})$. Differentials in tensor products of complexes are defined as usual by the Koszul convention: $$d(x\otimes y)=dx\otimes y + (-1)^{\deg x}x\otimes dy\;,$$ and we denote by $\tau$ the symmetry isomorphism $C\otimes D\simeq D\otimes C$, which maps $x\otimes y$ to $(-1)^{\deg x\deg y}y\otimes x$. Bicomplexes. : A bicomplex $B$ is a bigraded object in ${\mathcal{A}}$, equipped with two differentials $d_1:B^{i,j}\to B^{i+1,j}$ and $d_2:B^{i,j}\to B^{i,j+1}$ which commute. The total complex ${\mathrm{Tot}\,}B$ associated to $B$ is defined as usual by using the Koszul sign convention. Thus $({\mathrm{Tot}\,}B)^k=\bigoplus_{i+j=k}B^{i,j}$ and for $x\in B^{i,j}$, $dx=d_1x+(-1)^id_2x$. ### Quasi-isomorphisms and derived categories {#subsec-derived} Now we assume that ${\Bbbk}$ is a PID. A chain map $f:C\to D$ is a quasi-isomorphism if it satisfies one of the conditions of the following lemma. \[lm-caracqis\] Let ${\Bbbk}$ be a PID. Let $C,D\in{\mathrm{Ch}}({\mathcal{P}}_{d,{\Bbbk}})$ and let $f:C\to D$ be a chain map. The following conditions are equivalent. - For all $V\in{\mathcal{V}}_{\Bbbk}$, the morphism of complexes of ${\Bbbk}$-modules $f_V:C(V)\to D(V)$ induces an isomorphism in homology. - For all $V\in{\mathcal{V}}_{\Bbbk}$, the mapping cone $M(V)$ of $f_V$ is a complex of ${\Bbbk}$-modules with trivial homology. - The mapping cone $M$ of $f$ is a complex of strict polynomial functor, which decomposes as the Yoneda splice of admissible short exact sequences $Z^n\hookrightarrow M^n \twoheadrightarrow Z^{n+1}$. The equivalence between (i) and (ii) is standard [@Weibel Cor 1.5.4]. It is trivial that (iii) implies (ii). The converse uses that ${\Bbbk}$ is a PID. Since ${\Bbbk}$ is a PID, the cycles of a complex are functors from $\Gamma^d{\mathcal{V}}_{\Bbbk}$ to ${\mathcal{V}}_{\Bbbk}$, i.e. they are genuine strict polynomial functors. The $Z^n\hookrightarrow M^n \twoheadrightarrow Z^{n+1}$ are admissible short exact sequences since they are exact after evaluation on $V\in{\mathcal{V}}_{\Bbbk}$. Condition (i) mimics the definition quasi-isomorphism in abelian categories, and it is handy since it allows traditional spectral sequence argument to check that a map is a quasi-isomorphism. Condition (iii) is the standard definition (see [@Buehler Def. 10.16] or [@Keller Section 11]) of quasi-isomorphisms in exact categories (like ${\mathcal{P}}_{d,{\Bbbk}}$). Let $$ denote the symbol $+$, $-$ or $b$ or the empty symbol. Since ${\mathcal{P}}_{d,{\Bbbk}}$, resp. ${\mathcal{P}}_{\Bbbk}$ are exact categories, there are associated derived categories ${\mathbf{D}}^({\mathcal{P}}_{d,{\Bbbk}})$, resp. ${\mathbf{D}}^({\mathcal{P}}_{{\Bbbk}})$ [@Keller Section 11], or [@Buehler Section 10.4]. The derived categories are localizations of ${\mathbf{K}}^({\mathcal{P}}_{d,{\Bbbk}})$, resp. ${\mathbf{K}}^({\mathcal{P}}_{{\Bbbk}})$, with respect to quasi-isomorphisms, exactly as in the case of an abelian category [@Weibel Chap 10]. So, the objects of the derived categories ${\mathbf{D}}^({\mathcal{P}}_{d,{\Bbbk}})$, resp. ${\mathbf{D}}^({\mathcal{P}}_{{\Bbbk}})$ are the same as the ones of ${\mathrm{Ch}}^({\mathcal{P}}_{d,{\Bbbk}})$, resp. ${\mathrm{Ch}}^({\mathcal{P}}_{{\Bbbk}})$. Morphisms in the derived categories, with source $C$ and target $D$ are represented by diagrams $C\rightarrow D' \leftarrow D$, where the first map is a chain map and the second map is a quasi-isomorphism. Two diagrams $C\to D'\leftarrow D$ and $C\to D''\leftarrow D$ represent the same morphism if and only if they fit into a commutative diagram (where the vertical arrows are quasi-isomorphisms): $$\xymatrix@R=0.3cm{ & D'\ar[d] &\\ C\ar[ru]\ar[r]\ar[rd] &D'''& D\ar[lu]\ar[l]\ar[ld]\\ & D''\ar[u] & }.$$ As in the case of abelian categories, the derived categories ${\mathbf{D}}^({\mathcal{P}}_{d,{\Bbbk}})$ and ${\mathbf{D}}^({\mathcal{P}}_{{\Bbbk}})$ are triangulated categories. The exact triangles are (rotates of) the ones isomorphic to the standard triangle $C\to D\to M\to C[-1] $ where $M$ denotes the mapping cone of the morphism $C\to D$. Let the symbol $\ast$ stand for $+$, $-$ or $b$. By bi-exactness, tensor products of complexes induce tensor products at the level of the derived categories $$\begin{aligned} &{\mathbf{D}}^({\mathcal{P}}_{d,{\Bbbk}})\times {\mathbf{D}}^({\mathcal{P}}_{e,{\Bbbk}})\xrightarrow[]{\otimes} {\mathbf{D}}^({{\mathcal{P}}_{d+e,{\Bbbk}}})\,,\\ &{\mathbf{D}}^({\mathcal{P}}_{{\Bbbk}})\times {\mathbf{D}}^({\mathcal{P}}_{{\Bbbk}})\xrightarrow[]{\otimes} {\mathbf{D}}^({{\mathcal{P}}_{{\Bbbk}}})\;.\end{aligned}$$ These tensor products are symmetric, associative and unital. In particular, ${\mathbf{D}}^({\mathcal{P}}_{{\Bbbk}})$ is a strict symmetric monoidal category. Actually, we shall work mainly in bounded derived categories. We list below some extra properties of these categories. Decomposition. : The derived category ${\mathbf{D}}^b({\mathcal{P}}_{{\Bbbk}})$ decomposes as the direct sum of its full subtriangulated categories ${\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}})$: $${\mathbf{D}}^b({\mathcal{P}}_{\Bbbk})=\bigoplus_{d\ge 0}{\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}})\;.$$ Duality. : By exactness, duality induce functors: $$\begin{aligned} &{\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}})\xrightarrow[]{^\sharp}{\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}}^{\mathrm{op}})\simeq ({\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}}))^{\mathrm{op}}\;,\\ &{\mathbf{D}}^b({\mathcal{P}}_{{\Bbbk}})\xrightarrow[]{^\sharp}{\mathbf{D}}^b({\mathcal{P}}_{{\Bbbk}}^{\mathrm{op}})\simeq ({\mathbf{D}}^b({\mathcal{P}}_{{\Bbbk}}))^{\mathrm{op}}\;.\end{aligned}$$ Injectives and projectives. : Since ${\Bbbk}$ is a PID, ${\mathcal{P}}_{d,{\Bbbk}}$ has finite homological dimension [@DonkinHDim; @AB]. So each complex in ${\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}})$ or ${\mathbf{D}}^b({\mathcal{P}}_{{\Bbbk}})$ is quasi-isomorphic to a finite complex of injectives and to a bounded complex of projectives. Thus ${\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}})$ and ${\mathbf{D}}^b({\mathcal{P}}_{{\Bbbk}})$ are equivalent to their full subcategory of complexes of projective functors, and also to their full subcategory of injective objects. In particular we have equivalences of categories: $${\mathbf{D}}^b({\mathcal{P}}_{\Bbbk})\simeq {\mathbf{K}}^b({\mathrm{Inj}}({\mathcal{P}}_{{\Bbbk}}))\;,\quad {\mathbf{D}}^b({\mathcal{P}}_{\Bbbk})\simeq {\mathbf{K}}^b({\mathrm{Proj}}({\mathcal{P}}_{{\Bbbk}}))\;.$$ Since tensor products of injectives (resp. projectives) remain injective (resp. projective), the equivalences above may be realized by monoidal functors. Internal Hom. : Let $C\in{\mathcal{P}}_{d,{\Bbbk}}$. The internal ${{\mathrm{Hom}}}$ functor induces a derived functor: $${\mathbf{R}}{\mathbb{H}}(C,-): {\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}})\to {\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}})\;.$$ Tensor products induce morphisms natural with respect to the complexes $C,C',D,D'$, associative, commutative and unital: $${\mathbf{R}}{\mathbb{H}}(C,D)\otimes {\mathbf{R}}{\mathbb{H}}(C',D')\to {\mathbf{R}}{\mathbb{H}}(C\otimes C',D\otimes D')\;.$$ If $D$ is a complex of ${\mathbb{H}}(C,-)$-acyclic objects, there is an isomorphism ${\mathbf{R}}{\mathbb{H}}(C,D)\simeq {\mathbb{H}}(C,D)$. Ringel duality {#section-Kos} ============== In this short section, we present Ringel duality. We adopt the point of view of Cha[ł]{}upnik [@Chalupnik2], which we generalize over a PID ${\Bbbk}$. For an explanation of the combinatorial ideas encoded in Ringel duality, we refer the reader to [@Chalupnik2 Section 2]. Definition of Ringel duality ---------------------------- Let ${\Bbbk}$ be a PID. The Ringel duality functor $\Theta$ is the triangle functor: $$\begin{array}{cccc} \Theta: &{\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}})&\to &{\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}})\\ & C &\mapsto & {\mathbf{R}}{\mathbb{H}}(\Lambda^d,C) \end{array}.$$ Functors of degree zero are constant functors, and $\Lambda^0$ is the constant functor with value ${\Bbbk}$. In particular, ${\mathbb{H}}(\Lambda^0,-)$ is the identity functor of ${\mathcal{P}}_{0,{\Bbbk}}$. So $\Theta$ is the identity map if $d=0$. Now we describe the compatibility of $\Theta$ with tensor products. For $C\in{\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}})$ and $D\in{\mathbf{D}}^b({\mathcal{P}}_{e,{\Bbbk}})$, the following composite, where the last map is induced by the comultiplication $\Lambda^{d+e}\to\Lambda^d\otimes\Lambda^e$, is an isomorphism: $${\mathbf{R}}{\mathbb{H}}(\Lambda^d,C)\otimes{\mathbf{R}}{\mathbb{H}}(\Lambda^e,D)\xrightarrow[]{\otimes}{\mathbf{R}}{\mathbb{H}}(\Lambda^d\otimes\Lambda^e,C\otimes D)\to {\mathbf{R}}{\mathbb{H}}(\Lambda^{d+e},C\otimes D)\;,$$ Indeed, it is true for complexes of injectives since in this case it reduces to the isomorphism ${\mathbb{H}}(\Lambda^d,C)\otimes{\mathbb{H}}(\Lambda^e,D)\xrightarrow[]{\simeq}{\mathbb{H}}(\Lambda^{d+e},C\otimes D)$ from section \[subsubsec-internal\]. We denote this composite by ${\square}$. $${\square}:(\Theta C)\otimes (\Theta D)\xrightarrow[]{\simeq} \Theta(C\otimes D)\;.$$ The morphism ${\square}$ is associative and unital since ${\mathbb{H}}(\Lambda^d,-)$ is. So, gathering all possible degrees $d$, we obtain the following result. \[prop-square\] Ringel duality yields a monoidal functor: $$(\Theta,\square,{{\mathrm{Id}}})\;:\; ({\mathbf{D}}^b({\mathcal{P}}_{{\Bbbk}}),\otimes,{\Bbbk})\to ({\mathbf{D}}^b({\mathcal{P}}_{{\Bbbk}}),\otimes,{\Bbbk})$$ \[rk-sgn-Ringel\] Ringel duality is not a symmetric monoidal functor. Indeed, the comultiplication $\Lambda^{d+e}\to \Lambda^d\otimes\Lambda^e$ composed with the permutation $\Lambda^d\otimes\Lambda^e\simeq \Lambda^e\otimes\Lambda^d$ equals $(-1)^{de}$ times the comultiplication. So the following diagram commutes up to a $(-1)^{de}$ sign. $$\xymatrix{ \Theta(C)\otimes\Theta(D)\ar[r]_-\simeq^-{{\square}}\ar[d]_-\simeq^-{\tau}& \Theta(C\otimes D)\ar[d]_-\simeq^-{\Theta(\tau)}\\ \Theta(D)\otimes\Theta(C)\ar[r]_-\simeq^-{{\square}}& \Theta(D\otimes C) }.$$ How to compute Ringel duals --------------------------- Let $C\in{\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}})$. Then $\Theta(C)$ is quasi-isomorphic to the complex ${\mathbb{H}}(\Lambda^d,D)$, where $D$ is a complex of ${\mathbb{H}}(\Lambda^d,-)$-acyclic objects quasi-isomorphic to $C$. The following lemma shows that many concrete functors are ${\mathbb{H}}(\Lambda^d,-)$-acyclic, so there is often a huge choice of complexes $D$ available for explicit computations. \[lm-acyclic\] The class of ${\mathbb{H}}(\Lambda^d,-)$-acyclic objects is stable under tensor products. It contains symmetric powers, exterior powers and more generally Schur functors. Stability of the class of ${\mathbb{H}}(\Lambda^d,-)$-acyclic objects by tensor product follows from the fact that ${\square}$ is an isomorphism. Symmetric powers are ${\mathbb{H}}(\Lambda^d,-)$-acyclic because they are injective. For exterior powers, this well known fact is proved e.g. in [@TouzeEML Remark 7.5]. Finally, it is proved in [@AB] that Schur functors admit finite resolutions by tensor products of exterior powers. Hence they are ${\mathbb{H}}(\Lambda^d,-)$-acyclic[^3]. If $X^$ denotes the symmetric exterior or divided power algebra and if $\lambda=(\lambda_1,\dots,\lambda_n)$ is a tuple of nonnegative integers of weight $\sum\lambda_i=d$, we denote by $X^\lambda$ the tensor product $\bigotimes_{i=1}^n X^{\lambda_i}$. For the reader’s convenience, we describe the (well-known) action of the functor ${\mathbb{H}}(\Lambda^d,-)$ on the ${\mathbb{H}}(\Lambda^d,-)$-acyclic objects $S^\lambda$ and $\Lambda^\lambda$ in the following lemma. \[lm-comput\] Let $d$ be a positive integer. The following computations hold in ${\mathcal{P}}_{d,{\Bbbk}}$. 1. ${\mathbb{H}}(\Lambda^d,\otimes^d)\simeq \otimes^d$. If we denote by $\sigma:\otimes^d\to \otimes^d$ the morphism induced by a permutation $\sigma\in{\mathfrak{S}}_d$, then ${\mathbb{H}}(\Lambda^d,\sigma)= \epsilon(\sigma)\sigma$. 2. If $\lambda$ is a partition of weight $d$, ${\mathbb{H}}(\Lambda^d,S^\lambda)\simeq \Lambda^\lambda$. Moreover, the multiplication $\otimes^d\twoheadrightarrow S^\lambda$ is sent by ${\mathbb{H}}(\Lambda^d,-)$ to the multiplication $\otimes^d\twoheadrightarrow \Lambda^\lambda$. 3. If $\lambda$ is a partition of weight $d$, ${\mathbb{H}}(\Lambda^d,\Lambda^\lambda)\simeq \Gamma^\lambda$. Moreover, the comultiplication $\Lambda^\lambda\hookrightarrow\otimes^d$ is sent by ${\mathbb{H}}(\Lambda^d,-)$ to the comultiplication $\ \Gamma^\lambda\hookrightarrow\otimes^d$. Let us prove (i). For $d=1$ it reduces to the Yoneda isomorphism ${\mathbb{H}}(\Lambda^1,S^1)\simeq \Lambda^1$. For $d\ge 2$, we use the fact that ${\square}$ is an isomorphism. \(ii) is a little trickier. Using the isomorphism ${\square}$, one reduces the proof of (ii) to the case of $S^\lambda=S^d$. By duality ${\mathbb{H}}(\Lambda^d,S^d)$ is isomorphic to ${\mathbb{H}}(\Gamma^d,\Lambda^d)$ which is isomorphic to $\Lambda^d$ by the Yoneda isomorphism. It remains to compute the image of the multiplication $\otimes^d\twoheadrightarrow S^d$ by ${\mathbb{H}}(\Lambda^d,-)$. By duality, this amounts to computing the map ${\mathbb{H}}(\otimes^d,\Lambda^d)\to {\mathbb{H}}(\Gamma^d,\Lambda^d)$ induced by the comultiplication $\Gamma^d\hookrightarrow\otimes^d$. First, $\Gamma^{d,{\Bbbk}^d}\simeq \bigoplus \Gamma^\lambda$, the sum being taken over all $d$-tuples $\lambda=(\lambda_1,\dots,\lambda_d)$ of nonnegative integers of weight $d$. Thus $\otimes^d$ identifies with a direct summand of $\Gamma^{d,{\Bbbk}^d}$, and one checks that the composite $\Gamma^d\hookrightarrow \otimes^d\hookrightarrow \Gamma^{d,{\Bbbk}^d}$ equals the map $\Gamma^d=\Gamma^{d,{\Bbbk}}\hookrightarrow \Gamma^{d,{\Bbbk}^d}$ induced by the map $\Sigma:{\Bbbk}^d\to {\Bbbk}$, $(x_1,\dots,x_d)\mapsto \sum x_i$. The map ${\mathbb{H}}(\Sigma,\Lambda^d)$ identifies through the Yoneda isomorphism with $\Lambda^d_{\Sigma}$. Now $\Lambda^_\Sigma:\Lambda^_{{\Bbbk}^d}\simeq \Lambda^{\otimes d}\to \Lambda^$ is the $d$-fold multiplication, so restricting our attention to the summand $\otimes^d$ of $\Lambda^d_{{\Bbbk}^d}$ we get the result. Finally, to prove (iii), we first use the isomorphism ${\square}$ to reduce the proof to the case of $\Lambda^\lambda=\Lambda^d$. Then $\Lambda^d$ fits into an exact sequence $$\Lambda^d\hookrightarrow\textstyle\otimes^d\to \bigoplus_{i=1}^{d-1}(\otimes^{i-1})\otimes S^2\otimes (\otimes^{d-1-i})\;,$$ where the first map is the comultiplication and the components of the second map are obtained by tensoring the multiplication $\otimes^2\to S^2$ by identities. By (i), (ii) and by left exactness of ${\mathbb{H}}(\Lambda^d,-)$, ${\mathbb{H}}(\Lambda^d,\Lambda^d)$ is the kernel of the same map but with $S^2$ replaced by $\Lambda^2$. Hence it equals $\Gamma^d$, and the map ${\mathbb{H}}(\Lambda^d,\Lambda^d)\hookrightarrow {\mathbb{H}}(\Lambda^d,\otimes^d)$ identifies with the comultiplication $\Gamma^d\to \otimes^d$. As a consequence of the elementary computations of lemma \[lm-comput\] we have: \[lm-iso\] The functor ${\mathbb{H}}(\Lambda^d,-)$ induces isomorphisms: $${{\mathrm{Hom}}}_{{\mathcal{P}}_{d,{\Bbbk}}}(S^\mu,S^\lambda)\simeq {{\mathrm{Hom}}}_{{\mathcal{P}}_{d,{\Bbbk}}}(\Lambda^\mu,\Lambda^\lambda)\simeq {{\mathrm{Hom}}}_{{\mathcal{P}}_{d,{\Bbbk}}}(\Gamma^\mu,\Gamma^\lambda)\;.$$ These isomorphisms fit into a diagram: $$\xymatrix{ {{\mathrm{Hom}}}_{{\mathcal{P}}_{d,{\Bbbk}}}(S^\mu,S^\lambda)\ar[d]^{\sharp}\ar[r]^-{\simeq}&{{\mathrm{Hom}}}_{{\mathcal{P}}_{d,{\Bbbk}}}(\Lambda^\mu,\Lambda^\lambda)\ar[d]^{\sharp}\\ {{\mathrm{Hom}}}_{{\mathcal{P}}_{d,{\Bbbk}}}(\Gamma^\lambda,\Gamma^\mu)&{{\mathrm{Hom}}}_{{\mathcal{P}}_{d,{\Bbbk}}}(\Lambda^\lambda,\Lambda^\mu)\ar[l]_{\simeq} }.$$ We first treat the particular case $S^\lambda=S^\mu=\otimes^d$. In that case, the morphisms $\sigma:\otimes^d\to\otimes^d$ induced by the permutations $\sigma\in{\mathfrak{S}}_d$ (acting on $\otimes^d$ by permuting the factors of $\otimes^d$) form a basis of ${{\mathrm{Hom}}}_{{\mathcal{P}}_{d,{\Bbbk}}}(\otimes^d,\otimes^d)$. So lemma \[lm-comput\](i) completely describes ${\mathbb{H}}(\Lambda^d,-)$ on tensor products, and the result follows. Now we prove the general case. First, we check the commutativity of the diagram of lemma \[lm-iso\]. Let $f\in{{\mathrm{Hom}}}_{{\mathcal{P}}_{d,{\Bbbk}}}(S^\lambda,S^\mu)$. By projectivity of $\otimes^d$, the morphism $f$ fits into a commutative square (where the vertical arrows are induced by the multiplication) $$\xymatrix{ \otimes^d\ar@{->>}[d]\ar[r]^-{\overline{f}}&\otimes^d\ar@{->>}[d]\\ S^\lambda\ar[r]^-{f} &S^\mu }.$$ If we apply to this square on the one hand duality, and on the other hand first ${\mathbb{H}}(\Lambda^d,-)$, then duality and once again ${\mathbb{H}}(\Lambda^d,-)$, we obtain commutative squares (where the vertical arrows are the comultipications) $$\xymatrix{ \otimes^d&\otimes^d\ar[l]_-{\overline{f}^\sharp}\\ \Gamma^\lambda\ar@{^{(}->}[u] &\Gamma^\mu\ar@{^{(}->}[u]\ar[l]_-{f^\sharp} }, \quad\; \xymatrix{ \otimes^d&&&\otimes^d\ar[lll]_-{{\mathbb{H}}(\Lambda^d,{\mathbb{H}}(\Lambda^d,\overline{f})^\sharp)}\\ \Gamma^\lambda\ar@{^{(}->}[u] &&&\Gamma^\mu\ar@{^{(}->}[u]\ar[lll]_-{{\mathbb{H}}(\Lambda^d,{\mathbb{H}}(\Lambda^d,f)^\sharp)} }.$$ By the particular case $S^\lambda=S^\mu=\otimes^d$, we get that ${\mathbb{H}}(\Lambda^d,{\mathbb{H}}(\Lambda^d,\overline{f})^\sharp)$ equals $\overline{f}^\sharp$. Thus, ${\mathbb{H}}(\Lambda^d,{\mathbb{H}}(\Lambda^d,f)^\sharp)$ equals $f^\sharp$. So the diagram commutes. To conclude the proof, we observe that the vertical arrows of the diagram are isomorphisms. So the horizontal maps must be isomorphisms. Now, using lemma \[lm-iso\], it is easy to prove the following result. [[@Chalupnik2]]{} The functor $\Theta$ is an equivalence of categories, with inverse the functor $C\mapsto (\Theta(C^\sharp))^\sharp$. To check $\Theta(\Theta(C^\sharp)^\sharp)\simeq C$ we can restrict to complexes of projectives. Then the result follows by lemma \[lm-iso\]. To check that $\Theta(\Theta(C)^\sharp)^\sharp\simeq C$, we can restrict to complexes of injectives. Once again the result follows by lemma \[lm-iso\]. To give the reader an intuition of the behaviour of $\Theta$, we gather a few explicit computations $\Theta F$ following from lemmas \[lm-acyclic\] and \[lm-comput\]. \[exemple\] We have the following isomorphisms in ${\mathbf{D}}^b({\mathcal{P}}_{{\Bbbk}})$: $$\Theta(S^\lambda)\simeq\Lambda^\lambda\;,\quad \Theta(\Lambda^\lambda)\simeq\Gamma^\lambda \;,$$ and $\Theta(\Gamma^2)$ is isomorphic to the cochain complex with $\otimes^2$ in degree $0$, $\Gamma^2$ in degree $1$ and differential $\otimes^2\to \Gamma^2$ given by the multplication (sending $v_1\otimes v_2$ onto $v_1\otimes v_2+v_2\otimes v_1\in \Gamma^2(V)=(V^{\otimes 2})^{{\mathfrak{S}}_2}$). Observe that the degree zero homology of $\Theta(\Gamma^2)$ equals $\Lambda^2$ if $2\ne 0$ in the ground ring ${\Bbbk}$, and $\Gamma^2$ otherwise. Moreover, $\Theta(\Gamma^2)$ has trivial homology in degree $1$ if and only if $2$ is invertible in the ground ring ${\Bbbk}$. Derived functors à la Dold-Puppe {#section-DP} ================================ The derived functors $L_F(V;n)$ of a non-additive functor $F$ were introduced by Dold and Puppe in [@DP1; @DP2]. In section \[subsec-rappel-DP\], we recall the classical definition of derived functors. In section \[subsec-DP-strict\], we adapt derivation of functors to the framework of strict polynomial functors. Derivation of ordinary functors {#subsec-rappel-DP} ------------------------------- ### The Dold-Kan correspondence Let ${\mathcal{A}}$ be an abelian category and let $s{\mathcal{A}}$ denote the category of simplicial objects in ${\mathcal{A}}$. The Dold-Kan correspondence asserts that the normalized chain functor ${\mathcal{N}}$ yields an equivalence of categories (with inverse denoted by ${\mathcal{K}}$): $${\mathcal{N}}:s{\mathcal{A}}\leftrightarrows {\mathrm{Ch}}_{\ge 0}({\mathcal{A}}):{\mathcal{K}}\;,$$ and moreover that ${\mathcal{N}}$ and ${\mathcal{K}}$ preserve homotopy relations between maps [@DP2 Section 3], [@Do], [@Weibel Thm 8.4.1]. For the homotopy types of complexes, it is equivalent to use the normalized chain functor ${\mathcal{N}}$ and the unnormalized chain functor ${\mathcal{C}}$. Indeed, the canonical projection ${\mathcal{C}}X\to {\mathcal{N}}X$ is a homotopy equivalence [@DP2 Satz 3.22], [@ML VIII Thm 6.1]. ### Derived functors Let $F:{\mathcal{A}}\to {\mathcal{B}}$ be a (non-necessarily additive) functor between two abelian categories, and assume that ${\mathcal{A}}$ has enough projectives. Let $V\in {\mathcal{A}}$ and let $n\ge 0$. Dold and Puppe defined the derived functors $L_qF(V;n)$ (called the $q$-th derived functor of $F$ with height $n$) by the following formula: $$L_qF(V;n):= H_q\big({\mathcal{C}}F{\mathcal{K}}(P[-n])\big)\;,$$ where $P$ is a projective resolution of $V$ in ${\mathcal{A}}$, and $[-n]$ refers to the suspension in ${\mathrm{Ch}}_{\ge 0}({\mathcal{A}})$ (thus $P[-n]_i= P_{i-n}$). If $F$ is additive, one proves [@DP2 4.7] that $L_q F(V;n)$ equals the usual left derived functor $L_{q-n}F(V)$, as defined e.g. in [@Weibel Chap. 2]. Using the language of homotopical algebra, the definition of Dold and Puppe may be rephrased as follows. Consider the standard model structure on $s{\mathcal{A}}$ [@Quillen II.4]. Then $F$ induces a derived functor (in the sense of Quillen) $LF:\mathrm{ho}(s{\mathcal{A}})\to \mathrm{ho}(s{\mathcal{B}})$. Now $L_q F(V;n)$ is just the $q$-th homotopy group of the value of $L F$ on ${\mathcal{K}}(V;n)$, where the latter denotes a simplicial object with trivial homotopy groups except for $\pi_n({\mathcal{K}}(V;n))=V$. ### Products and shuffle maps Let ${\mathcal{A}}$ be an additive category, equipped with a symmetric monoidal product $\otimes:{\mathcal{A}}\times{\mathcal{A}}\to {\mathcal{A}}$, which is additive with respect to both variables. Then both ${\mathrm{Ch}}_{\ge 0}({\mathcal{A}})$ and $s{\mathcal{A}}$ inherit a symmetric monoidal product. So we can associate two chain complexes to a pair $(X,Y)$ of simplicial objects in ${\mathcal{A}}$, namely $({\mathcal{C}}X) \otimes ({\mathcal{C}}Y)$ and ${\mathcal{C}}(X\otimes Y)$. The Eilenberg-Zilber theorem asserts that the shuffle map: $$\nabla:({\mathcal{C}}X) \otimes ({\mathcal{C}}Y)\to {\mathcal{C}}(X\otimes Y)$$ is a homotopy equivalence. It is natural with respect to $X,Y$ and associative, symmetric and unital (in other words: the chain functor ${\mathcal{C}}$ is a lax symmetric monoidal functor), as we can see it from the explicit expression of $\nabla$ [@DP2 Satz 2.15], [@ML VIII Thm 8.8], [@Weibel Section 8.5.3]. In particular, if our functors take values in the category of ${\Bbbk}$-modules, the shuffle map induces a morphism (natural in $F,G,V$): $$L_pF(V;n)\otimes L_qG(V;n) \to L_{p+q}(F\otimes G)(V;n).$$ Indeed, the left hand side injects (via the Künneth morphism) into the homology of the complex ${\mathcal{C}}F{\mathcal{K}}(P[-n])\otimes {\mathcal{C}}G{\mathcal{K}}(P[-n])$, while the right hand side is the homology of the complex ${\mathcal{C}}\left( F{\mathcal{K}}(P[-n]) \otimes G{\mathcal{K}}(P[-n])\right)$. Derivation of strict polynomial functors {#subsec-DP-strict} ---------------------------------------- We shall denote for short by $K(n)$ the simplicial free ${\Bbbk}$-module ${\mathcal{K}}({\Bbbk}[-n])$. If $V$ is a ${\Bbbk}$-module, the definition of the Dold-Kan functor ${\mathcal{K}}$ yields an isomorphism, natural with respect to $V$: $$K(n)\otimes V\simeq {\mathcal{K}}(V[-n]).\qquad()$$ We observe that the definition of derived functors from the preceding section makes sense in the framework of strict polynomial functors. To be more specific, for $F\in{\mathcal{P}}_{d,{\Bbbk}}$, $q\ge 0$ and $n\ge 0$, we define the $q$-th derived functor of $F$ with height $n$ $$L_qF(-;n):\Gamma^d{\mathcal{V}}_{\Bbbk}\to {\Bbbk}\text{-mod}$$ as the $q$-th homology group of the complex ${\mathcal{C}}F_{K(n)}$. \[rq-1\] For strict polynomial functors of degree $d=0$, the above formula still makes sense. A functor of degree zero is just a constant functor with value $F$, so the simplicial ${\Bbbk}$-module $F_{K(n)}$ equals $F$ in each degree, and the face and degeneracy operators are identity maps. So $L_q F(V;n)$ equals zero in positive degrees and $F$ in degree zero. Derivation of strict polynomial functors extends the derivation of ordinary functors. Indeed, thanks to isomorphism $()$, there is a commutative diagram where the horizontal morphisms are induced by taking the $q$-th derived functors with height $n$ (and $\widetilde{{\mathcal{P}}}_{d,{\Bbbk}}$ is the category of strict polynomial functors with values in arbitrary ${\Bbbk}$-modules, cf \[rk-PT\]). $$\xymatrix{ {\mathcal{P}}_{d,{\Bbbk}}\ar[r]\ar[d]^-{{\mathcal{U}}}&\widetilde{{\mathcal{P}}}_{d,{\Bbbk}}\ar[d]^-{{\mathcal{U}}}\\ {\mathcal{F}}_{\Bbbk}\ar[r] &{\mathcal{F}}_{\Bbbk}}.$$ Finally, we can take the direct sum of the categories ${\mathcal{P}}_{d,{\Bbbk}}$ and interpret the $q$-th derived functor with height $n$ as a functor ${\mathcal{P}}_{{\Bbbk}}\to \widetilde{{\mathcal{P}}}_{{\Bbbk}}$. Derivation in ${\mathbf{D}}^-({\mathcal{P}}_{{\Bbbk}})$ ------------------------------------------------------- In this section, ${\Bbbk}$ is a PID. We lift the definition of derivation to the level of the derived category ${\mathbf{D}}^-({\mathcal{P}}_{{\Bbbk}})$. Let us first introduce the chain functor ${\overline{{\mathcal{C}}}}$ for mixed bicomplexes. A mixed bicomplex in an additive category ${\mathcal{A}}$ is just an object of ${\mathrm{Ch}}^-(s{\mathcal{A}})$. If $X$ is a mixed bicomplex in ${\mathcal{A}}$, we may apply the chain functor ${\mathcal{C}}$ to each object $X_i$ of $X$. In this way we obtain a bicomplex $(X_{i})_{j}$, whose first differential $d_1^X:(X_i)_j\to (X_{i-1})_j$ is induced by the differential of $X$ and whose second differential $d_2^X:(X_i)_j\to (X_i)_{j-1}$ is induced by the simplicial structure of $X_i$. We denote by ${\overline{{\mathcal{C}}}}X$ the total complex associated to this bicomplex. Let $n$ be a positive integer and let $C\in{\mathrm{Ch}}^-({\mathcal{P}}_{{\Bbbk}})$. The derived complex of $C$ is the complex $L(C;n)\in {\mathrm{Ch}}^-({\mathcal{P}}_{{\Bbbk}})$ defined by: $$L(C;n)={\overline{{\mathcal{C}}}}(C_{K(n)})\;.$$ Derivation of complexes yields an additive functor: $$\begin{array}{cccc} L(-;n): & {\mathrm{Ch}}^-({\mathcal{P}}_{{\Bbbk}}) & \to & {\mathrm{Ch}}^-({\mathcal{P}}_{{\Bbbk}})\\ & C & \mapsto & L(C;n) \end{array}.$$ \[prop-DDPd\] The functor $L(-;n)$ induces a triangle functor at the level of the derived categories, still denoted by $L(-;n)$: $$L(-;n):{\mathbf{D}}^-({\mathcal{P}}_{{\Bbbk}})\to {\mathbf{D}}^-({\mathcal{P}}_{{\Bbbk}})\;.$$ This is a standard verification. First, derivation preserves homotopies: if $f:C\to D$ is homotopic to zero via a homotopy $h$, then $L(f;n)$ is homotopic to zero via the homotopy $L(h;n)$. In particular, derivation induces a functor at the level of the homotopy categories ${\mathbf{K}}^-({\mathcal{P}}_{{\Bbbk}})$. Derivation commutes with suspension, actually we have an equality of complexes $L(C[1];n)= L(C;n)[1]$. Derivation also preserves triangles of ${\mathbf{K}}^-({\mathcal{P}}_{{\Bbbk}})$. Indeed, if $f$ is a chain map, the chain complex $L(\mathrm{Cone}(f);n)$ is equal to $\mathrm{Cone}(L(f;n))$. So $L(-;n)$ induces a triangle endofunctor of ${\mathbf{K}}^-({\mathcal{P}}_{{\Bbbk}})$. By a standard spectral sequence argument on the bicomplexes $(C_i)_j$, this triangle functor preserves quasi-isomorphisms, whence the result. Now we give details about the compatibility of derivation with tensor products. If $X$ and $Y$ are mixed bicomplexes in ${\mathcal{P}}_{\Bbbk}$, their tensor product $X{\widehat{\otimes}}Y$ is defined as follows. As a bigraded object we have: $$(X{\widehat{\otimes}}Y)_{k,\ell}=\bigoplus_{i+j=k} X_{i,\ell}\otimes Y_{j,\ell}\;.$$ The differential of an element $x\otimes y\in X_{i,\ell}\otimes Y_{j,\ell}$ is given by $$\partial(x\otimes y)=\partial_X(x)\otimes y+(-1)^i x\otimes \partial_Y(y)\;,$$ and the simplicial structure is the diagonal one: $d_t=d_t^X\otimes d^Y_t$, $s_t=s_t^X\otimes s_t^Y$. The tensor product of mixed bicomplexes is a symmetric monoidal product on ${\mathrm{Ch}}^-(s{\mathcal{P}}_{\Bbbk})$. The unit for ${\widehat{\otimes}}$ is the simplicial object ${\mathcal{K}}({\Bbbk})$ (obtained by applying the Dold-Kan functor ${\mathcal{K}}$ the constant functor with value ${\Bbbk}$) considered as a complex concentrated in degree zero, and the symmetry isomorphism $\tau:X{\widehat{\otimes}}Y\to Y{\widehat{\otimes}}X$ sends $x\otimes y\in X_{i,\ell}\otimes Y_{j,\ell}$ to $(-1)^{ij}y\otimes x$. There are shuffle maps for mixed bicomplexes in ${\mathcal{P}}_{\Bbbk}$. To be more specific, we define a morphism of complexes (natural in $X,Y$): $${\overline{\nabla}}: {\overline{{\mathcal{C}}}}X \otimes {\overline{{\mathcal{C}}}}Y\to {\overline{{\mathcal{C}}}}\,(X{\widehat{\otimes}}Y)$$ by sending an element $x\otimes y\in (X_i)_k\otimes (Y_j)_\ell$ to $(-1)^{jk}\nabla(x\otimes y)$, where $\nabla$ denotes the usual shuffle map (the sign is needed to get a morphism of complexes). If $C$ and $D$ are bounded above cochain complexes of strict polynomial functors, the mixed bicomplex $(C\otimes D)_{K(n)}$ equals $C_{K(n)}{\widehat{\otimes}}D_{K(n)}$, so the complex $L(C\otimes D;n)$ equals ${\overline{{\mathcal{C}}}}\,(C_{K(n)}{\widehat{\otimes}}D_{K(n)})$. Hence, the shuffle map ${\overline{\nabla}}$ yields a morphism of complexes, natural with respect to $C$ and $D$: $$L(C;n)\otimes L(D;n)\xrightarrow[]{{\overline{\nabla}}} L(C\otimes D;n).$$ As already observed in remark \[rq-1\], if ${\Bbbk}$ denotes the constant functor with value ${\Bbbk}$, the complex $L({\Bbbk};n)$ equals ${\Bbbk}$ in each degree. So if we consider ${\Bbbk}$ as a complex concentrated in degree zero, there is a (unique) quasi-isomorphism $$\phi: {\Bbbk}\to L({\Bbbk};n)$$ which equals the identity map in degree zero. \[prop-sym\] Derivation with height $n$ induces a lax symmetric monoidal functor at the level of complexes $$(L(-;n),{\overline{\nabla}},\phi)\;:\;({\mathrm{Ch}}^-({\mathcal{P}}_{\Bbbk}),\otimes,{\Bbbk})\to ({\mathrm{Ch}}^-({\mathcal{P}}_{\Bbbk}),\otimes,{\Bbbk}),$$ and a symmetric monoidal functor at the level of the derived category: $$(L(-;n),{\overline{\nabla}},\phi)\;:\;({\mathbf{D}}^-({\mathcal{P}}_{{\Bbbk}}),\otimes,{\Bbbk}) \to ({\mathbf{D}}^-({\mathcal{P}}_{{\Bbbk}}),\otimes,{\Bbbk}).$$ We shall prove slightly more general facts. First, let us denote by $\phi:{\mathcal{C}}{\mathcal{K}}({\Bbbk})\to {\Bbbk}$ the morphism of chain complexes which is the identity map in degree zero. Then we have a lax monoidal functor: $$({\overline{{\mathcal{C}}}},{\overline{\nabla}},\phi):({\mathrm{Ch}}^-(s{\mathcal{P}}_{\Bbbk}),{\widehat{\otimes}},{\mathcal{K}}({\Bbbk}))\to ({\mathrm{Ch}}^-({\mathcal{P}}_{\Bbbk}),\otimes,{\Bbbk})\;.$$ Indeed, unity, associativity and commutativity for ${\overline{\nabla}}$ follow from the fact that the shuffle map $\nabla$ is unital, associative and commutative (not up to homotopy!), as we can see it from the explicit formula of [@ML VIII, Thm 8.8]. In particular, when we restrict to mixed bicomplexes of the form $C_{K(n)}$, we obtain the first part of proposition \[prop-sym\]. The second part of proposition \[prop-sym\] follows from the slightly more general fact that for arbitrary mixed bicomplexes $X$ and $Y$, the shuffle morphism ${\overline{\nabla}}: {\overline{{\mathcal{C}}}}X \otimes {\overline{{\mathcal{C}}}}Y\to {\overline{{\mathcal{C}}}}\,(X{\widehat{\otimes}}Y)$ is a quasi-isomorphism. To prove this, we write ${\overline{\nabla}}$ as the composition of two quasi-isomorphisms $\mu$ and $\nu$ defined as follows. The morphism of quadrigraded objects $({\mathcal{C}}X_{i})_k\otimes ({\mathcal{C}}Y_{j})_\ell\to ({\mathcal{C}}X_{i})_k\otimes ({\mathcal{C}}Y_{j})_\ell$ mapping $x\otimes y$ to $(-1)^{jk}x\otimes y$ induces an isomorphism $\nu$ between the chain complex ${\overline{{\mathcal{C}}}}(X)\otimes{\overline{{\mathcal{C}}}}(Y)$ and the totalization of the bicomplex $$\begin{aligned} &({\mathcal{C}}X\otimes{\mathcal{C}}Y)_{m,n}:=\bigoplus_{i+j=m,k+\ell=n} ({\mathcal{C}}X_{i})_k\otimes ({\mathcal{C}}Y_{j})_\ell\;.\end{aligned}$$ (The differentials of this bicomplex are defined as follows. The restriction of the first differential of $({\mathcal{C}}X\otimes{\mathcal{C}}Y)_{m,n}$ to $({\mathcal{C}}X_{i})_k\otimes ({\mathcal{C}}Y_{j})_\ell$ equals $d_1^X\otimes{{\mathrm{Id}}}+(-1)^{i} {{\mathrm{Id}}}\otimes d_1^Y$, and the restriction of the second differential equals $d_2^X\otimes{{\mathrm{Id}}}+(-1)^k {{\mathrm{Id}}}\otimes d_2^Y$.) There is a morphism of bicomplexes from $({\mathcal{C}}X\otimes{\mathcal{C}}Y)_{m,n}$ to the bicomplex $$\begin{aligned} &{\mathcal{C}}(X{\widehat{\otimes}}Y)_{m,n}:= \bigoplus_{i+j=m}({\mathcal{C}}(X{\widehat{\otimes}}Y)_{i+j})_{n}\;.\end{aligned}$$ whose restriction to the $n$-th rows equals $\nabla: ({\mathcal{C}}X\otimes{\mathcal{C}}Y)_{,n}\to {\mathcal{C}}(X{\widehat{\otimes}}Y)_{,n}$. Its totalization $\mu$ is a quasi-isomorphism. Indeed, $\nabla$ is a quasi-isomorphism (it is even a homotopy equivalence) so we prove this by comparing the spectral sequences associated to the bicomplexes. Now ${\overline{{\mathcal{C}}}}(X{\widehat{\otimes}}Y)$ is the total complex associated to ${\mathcal{C}}(X{\widehat{\otimes}}Y)_{m,n}$ and ${\overline{\nabla}}=\mu\circ\nu$, so ${\overline{\nabla}}$ is a quasi-isomorphism. Iterated derivations -------------------- Let $m$ and $n$ be positive integers. Then we may compose derivation with height $m$ and derivation with height $n$ to get a functor $$L(-;n)\circ L(-;m): {\mathbf{D}}^-({\mathcal{P}}_{\Bbbk})\to {\mathbf{D}}^-({\mathcal{P}}_{\Bbbk})\;.$$ Since unnormalized derivations are triangle functors and symmetric monoidal functors, so is the composite $L(-;n)\circ L(-;m)$. The main result about iterated derivations is the following. \[thm-iter\] There is an isomorphism of endofunctors of ${\mathbf{D}}^-({\mathcal{P}}_{{\Bbbk}})$: $$L(-;n)\circ L(-;m)\simeq L(-;n+m)\;.$$ This isomorphism is an isomorphism of triangle functors as well as an isomorphism of the symmetric monoidal functors. \[cor-iter\] There is an isomorphism of endofunctors of ${\mathbf{D}}^-({\mathcal{P}}_{{\Bbbk}})$: $$L(-;n)\simeq \underbrace{L(-;1)\circ\dots \circ L(-;1)}_{\text{$n$ times}}\;.$$ This isomorphism is an isomorphism of triangle functors as well as an isomorphism of the symmetric monoidal functor. We will actually prove theorem \[thm-iter\] at the level of chain complexes. To be more specific, let us consider $L(-;m)$ and $L(-;n)$ as endofunctors of the category ${\mathrm{Ch}}^-({\mathcal{P}}_{{\Bbbk}})$. Then theorem \[thm-iter\] is a consequence of the following stronger statement. \[thm-iter-strong\] Let $C\in {\mathrm{Ch}}^-({\mathcal{P}}_{d,{\Bbbk}})$ and let $n$ be a positive integer. There is a morphism of complexes, natural with respect to $C$, $$L(L(C;n);m)\xrightarrow[]{\zeta} L(C;n+m)\;,$$ satisfying the following properties: 1. $\zeta$ is a homotopy equivalence, 2. $\zeta$ commutes with suspension, i.e. there is a commutative diagram: $$\xymatrix{ L(L(C[1];n);m)\ar[r]^-{\zeta}\ar[d]^-{=}& L(C[1];n+m)\ar[d]^-{=}\\ L(L(C;n);m)[1]\ar[r]^-{\zeta[1]}& L(C;n+m)[1] }.$$ 3. If $C\in{\mathrm{Ch}}^-({\mathcal{P}}_{d,{\Bbbk}})$ and $D\in{\mathrm{Ch}}^-({\mathcal{P}}_{e,{\Bbbk}})$, the following diagram commutes up to homotopy: $$\xymatrix{ L(L(C;n);m)\otimes L(L(D;n);m)\ar[r]^-{\zeta^{\otimes 2}}\ar[d]^-{{\overline{\nabla}}}& L(C;n+m)\otimes L(D;n+m)\ar[dd]^-{{\overline{\nabla}}}\\ L(L(C;n)\otimes L(D;n);m)\ar[d]^-{L({\overline{\nabla}};m)}\\ L(L(C\otimes D;n);m)\ar[r]^-{\zeta}& L(C\otimes D;n+m) }.$$ The remainder of the section is devoted to the proof of theorem \[thm-iter-strong\]. The proof will essentially result from the generalized Eilenberg-Zilber Theorem. ### The generalized Eilenberg-Zilber theorem In this paragraph, ${\mathcal{A}}$ is an additive full subcategory of the category of modules over a ring $R$, containing $R$ as an object. For example, take ${\mathcal{A}}={\mathcal{P}}_{d,{\Bbbk}}$, which is an additive subcategory of the category of modules over the Schur algebra (cf. section \[subsubsec-SGR\] and appendix \[app\]). An $n$-simplicial object in ${\mathcal{A}}$ is an $n$-graded object $X_{i_1,\dots,i_n}$ equipped with $n$ simplicial structures (each simplicial structure being relative to one of the indices of $X$), which commute with one another. We denote by ${\mathcal{C}}X$ the total chain complex associated to an $n$-simplicial object. It is the total complex of the multicomplex with $n$ differentials obtained by applying the chain functor to each one of the simplicial structures of $X$. If $\alpha:\{1,\dots,n\}\twoheadrightarrow \{1,\dots,d\}$ is a surjective map, we denote by ${\mathrm{diag}\,}^\alpha X$ the associated partial diagonal. That is, ${\mathrm{diag}\,}^\alpha X$ is the $d$-simplicial object with $$({\mathrm{diag}\,}^\alpha X)_{k_1,\dots,k_d}= X_{k_{\alpha(1)}, k_{\alpha(2)},\dots, k_{\alpha(n)}}$$ and the corresponding diagonal simplicial structure. We shall often denote surjective maps $\alpha:\{1,\dots,n\}\twoheadrightarrow \{1,\dots,d\}$ as $d$-tuples where the $i$-th element of the list consists of the elements of the set $\alpha^{-1}\{i\}$, separated by symbols ‘$+$’. For example the map $\alpha:\{1,2,3\}\to \{1,2\}$ with $\alpha(1)=\alpha(3)=2$ and $\alpha(2)=1$ will be denoted by $(2,1+3)$. With this more suggestive notation, we have: $$({\mathrm{diag}\,}^{(2,1+3)}X)_{i,j}=X_{j,i,j}\;,$$ and the first simplicial structure of ${\mathrm{diag}\,}^{(2,1+3)}X$ is given by the second simplicial structure of $X$, while the second simplicial structure of ${\mathrm{diag}\,}^{(2,1+3)}X$ is given by taking the diagonal simplicial structure of the first and the third simplicial structure of $X$. The formula of the shuffle map $\nabla$ involved in the classical Eilenberg-Zilber theorem [@ML VIII Thm 8.8] actually yields a chain map, which is natural with respect to the bisimplicial object $X$, and which equals the identity in degree zero: $$\nabla:{\mathcal{C}}X={\mathcal{C}}{\mathrm{diag}\,}^{(1,2)}X\to {\mathcal{C}}{\mathrm{diag}\,}^{(1+2)}X\;.$$ This example justifies the following terminology. Let $\alpha$ and $\beta$ be surjective maps. An Eilenberg-Zilber map is a morphism of complexes, natural with respect to $n$-simplicial objects $X$, and which equals the identity map in degree zero: $${\mathcal{C}}{\mathrm{diag}\,}^\alpha(X)\to {\mathcal{C}}{\mathrm{diag}\,}^\beta(X).$$ 1. The shuffle map $\nabla:{\mathcal{C}}{\mathrm{diag}\,}^{(1,2)}X\to {\mathcal{C}}{\mathrm{diag}\,}^{(1+2)}X$ is an Eilenberg-Zilber map. 2. The composite of Eilenberg-Zilber maps is an Eilenberg-Zilber map. 3. If $\sigma\in{\mathfrak{S}}_n$ is a permutation of $\{1,\dots,n\}$, there is an Eilenberg-Zilber map: $$\begin{array}{cccc} \sigma:& {\mathcal{C}}X &\to & {\mathcal{C}}{\mathrm{diag}\,}^\sigma X\\ & x &\mapsto & \epsilon(\sigma,x) x \end{array}$$ where $\epsilon(\sigma,x)$ is a sign defined as follows. If $x$ has $n$-degree $(i_1,\dots,i_n)$, we consider a free graded ring $R$ over generators $e_1,\dots,e_n$ such that each $e_k$ has degree $i_k$. Then $\epsilon(\sigma,x)\in\{\pm 1\}$ is the sign such that $$e_1\cdot\dots\cdot e_n = \epsilon(\sigma,x)\,e_{\sigma(1)}\cdot\dots\cdot e_{\sigma(n)}\;.$$ 4. If $\alpha:\{1,\dots,n\}\twoheadrightarrow \{1,\dots,d\}$ and $\beta:\{1,\dots,m\}\twoheadrightarrow \{1,\dots,e\}$ are surjective maps we denote by $\alpha\|\beta$ their concatenation: $$\begin{array}{cccl} \alpha\|\beta:& \{1,\dots,m+n\} &\twoheadrightarrow & \{1,\dots,d+e\}\\ & i &\mapsto & \left\{\begin{array}{l}\alpha(i)\text{ if $i\le n$,}\\\beta(i-n)+d\text{ if $i> n$.}\end{array}\right. \end{array}$$ Given two Eilenberg-Zilber maps $f_i:{\mathcal{C}}{\mathrm{diag}\,}^{\alpha_i}(X)\to {\mathcal{C}}{\mathrm{diag}\,}^{\beta_i}(X)$ for $i=1,2$, we may concatenate them to get another Eilenberg-Zilber map: $$f_1\|f_2: {\mathcal{C}}{\mathrm{diag}\,}^{\alpha_1\|\alpha_2}(X)\to {\mathcal{C}}{\mathrm{diag}\,}^{\beta_1\|\beta_2}(X)\;.$$ 5. If $\gamma:\{1,\dots,m\}\twoheadrightarrow \{1,\dots,n\}$ is surjective and if $f:{\mathcal{C}}{\mathrm{diag}\,}^{\alpha}(X)\to {\mathcal{C}}{\mathrm{diag}\,}^{\beta}(X)$ is an Eilenberg-Zilber map natural with respect to $n$-simplicial objects, we can restrict it to $n$-simplicial objects of the form ${\mathrm{diag}\,}^\gamma Y$ to get an Eilenberg-Zilber map $$f:{\mathcal{C}}{\mathrm{diag}\,}^{\alpha\circ \gamma}(Y)\to {\mathcal{C}}{\mathrm{diag}\,}^{\beta\circ\gamma}(Y)\;.$$ The following result is well-known [@DP2 Bemerkung 2.16], and it is proved by the methods of acyclic models. \[prop-general-EZ\] Let $X$ be an $n$-simplicial object in the additive category ${\mathcal{A}}$, and let $\alpha:\{1,\dots,n\}\twoheadrightarrow \{1,\dots,d\}$ and $\beta:\{1,\dots,n\}\twoheadrightarrow \{1,\dots,e\}$ be surjective maps. There exists an Eilenberg-Zilber map: $${\mathcal{C}}{\mathrm{diag}\,}^\alpha(X)\to {\mathcal{C}}{\mathrm{diag}\,}^\beta(X),$$ and two such Eilenberg-Zilber maps are chain homotopic, via a homotopy which is natural with respect to $X$. First, an $n$-simplicial object in ${\mathcal{A}}$ is a functor $(\Delta^{\mathrm{op}})^{\times n}\to {\mathcal{A}}$. Since ${\mathcal{A}}$ is a subcategory of $R\text{-mod}$ containing the free $R$-modules of finite rank, $n$-simplicial objects in ${\mathcal{A}}$ include the acyclic models $M_I:=R(\Delta[i_1]\times\dots\times\Delta[i_n])$ for all $n$-tuples $I=(i_1,\dots,i_n)$ of nonnegative integers (i.e. $\Delta[i_1]\times\dots\times\Delta[i_n]$ is the $n$-simplicial set $\prod{{\mathrm{Hom}}}_{\Delta}(-,[i_k])$, and $M_I$ is the $n$-simplicial $R$-module obtained by taking the free $R$-module over it). Moreover, by the Yoneda lemma, homomorphisms of $n$-simplicial objects from such an acyclic model $M_I$ to an $n$-simplicial object $X$ is determined by an isomorphism: ${{\mathrm{Hom}}}(M_I,X)\simeq X_I$. Hence, for all $x\in X_I$ there exists a unique map $\phi:M_I\to X$ such that $x=\phi(\iota_I)$ where $\iota_I$ is the characteristic simplex of $\Delta[i_1]\times\dots\times\Delta[i_n]$, i.e. the identity map of $[i_1]\times\dots \times [i_n]\in (\Delta^{\mathrm{op}})^{\times n}$. So, if $t$ is an Eilenberg-Zilber map (or a homotopy between Eilenberg-Zilber maps), the commutative diagram: $$\xymatrix{ {\mathcal{C}}{\mathrm{diag}\,}^\alpha(M_I)\ar[r]^-{t}\ar[d]^-{C{\mathrm{diag}\,}^\alpha(\phi)}& {\mathcal{C}}{\mathrm{diag}\,}^\beta(M_I)\ar[d]^-{C{\mathrm{diag}\,}^\beta(\phi)}\\ {\mathcal{C}}{\mathrm{diag}\,}^\alpha(X)\ar[r]^-{t}& {\mathcal{C}}{\mathrm{diag}\,}^\beta(X) }$$ shows that the value of $t$ on $x\in X_I$ is completely determined by the restriction of $t$ to the acyclic model $M_I$. Hence, $t$ is completely determined by its restriction to the full subcategory of ${\mathcal{A}}$ whose objects are the acyclic models $M_I$. Conversely, if $t$ is defined on the full subcategory of ${\mathcal{A}}$ whose objects are the acyclic models $M_I$, the commutative diagram (and the uniqueness of $\phi$) show that $t$ may be extended uniquely to ${\mathcal{A}}$. So it suffices to prove the proposition when $X$ is an acyclic model. In that case, the result is standard, and proved exactly as in the proof of [@ML VIII, lemmas 8.2 and 8.3]. Eilenberg-Zilber maps are homotopy equivalences. ### Proof of theorem \[thm-iter-strong\] The proof consists of several steps. [Step 1. A replacement for $L(C;n+m)$.]{} We denote by $K(n)\boxtimes K(m)$ the bisimplicial ${\Bbbk}$-module obtained as tensor product of $K(n)$ and $K(m)$, and by $K(n,m)$ its diagonal. We consider the lax monoidal functor $$({\mathcal{L}}(-;n+m),{\overline{\nabla}},\phi):({\mathrm{Ch}}^-({\mathcal{P}}_{\Bbbk}),\otimes,{\Bbbk})\to ({\mathrm{Ch}}^-({\mathcal{P}}_{\Bbbk}),\otimes,{\Bbbk})$$ where ${\mathcal{L}}(C;n+m)={\overline{{\mathcal{C}}}}\left( C_{K(n,m)}\right)$, and ${\overline{\nabla}}$ is the shuffle map for mixed complexes. By the Eilenberg-Zilber theorem and the Künneth theorem, the homology of ${\mathcal{C}}K(n,m)$ equals ${\Bbbk}$ concentrated in homological degree $n+m$. Hence, the Dold-Kan correspondence yields a homotopy equivalence of simplicial ${\Bbbk}$-modules $K(n,m)\simeq K(n+m)$. This homotopy equivalence induces a homotopy equivalence of chain complexes: $${\mathcal{L}}(C;n+m)={\overline{{\mathcal{C}}}}\left( C_{K(n,m)}\right)\to {\overline{{\mathcal{C}}}}\left( C_{K(n+m)}\right)=L(C;n+m)\;,$$ natural with respect to $C$, commuting with suspension and with ${\overline{\nabla}}$. Hence we can replace $L(C;n+m)$ by ${\mathcal{L}}(C;n+m)$ in the proof. [Step 2. Construction of $\zeta$.]{} Let $C$ be a chain complex. We first observe that we have equalities of complexes: $$L(L(C;n);m)= {\mathrm{Tot}\,}\left({\mathrm{Tot}\,}^{(1+2,3)}C''\right) = {\mathrm{Tot}\,}C'' = {\mathrm{Tot}\,}\left({\mathrm{Tot}\,}^{(1,2+3)} C''\right)\;,$$ In these equalities, $C''$ denotes the tricomplex $(C_i)_{K(n)_j\otimes K(m)_k}$, with first differential induced by the differential of $C$, and with second (resp. third) differential induced by the simplicial structure of $K(n)$ (resp. $K(m)$), and ${\mathrm{Tot}\,}^{(1,2+3)}C''$ is the bicomplex obtained by totalizing the second and third differential of $C''$. For each $i$, the simplicial object $(C_i)_{K(n,m)}$ is the diagonal of the bisimplicial object $(C_i)_{K(n)\otimes K(m)}$. So, proposition \[prop-general-EZ\] yields an Eilenberg-Zilber map: $$f: {\mathcal{C}}\left((C_i)_{K(n)\otimes K(m)}\right)\to {\mathcal{C}}\left( (C_i)_{K(n,m)}\right)\;.$$ natural with respect to $C_i$, hence a morphism of bicomplexes, natural with respect to $C$: $$f:{\mathrm{Tot}\,}^{(1,2+3)} C''\to C'\;.$$ We define $\zeta$ as the morphism of chain complexes (natural with respect to $C$) induced at the level of total complexes: $$\zeta: L(L(C;n);m)= {\mathrm{Tot}\,}{\mathrm{Tot}\,}^{(1,2+3)} C''\to {\mathrm{Tot}\,}C'={\mathcal{L}}(C;m+n)\;.$$ We readily check from the definition of $\zeta$ that condition (2) is satisfied. So it remains to check conditions (1) and (3). [Step 3. Reduction to nonnegative complexes.]{} Since condition (2) is satisfied, we see that $\zeta$ is a homotopy equivalence for $C[1]$ if and only if it is a homotopy equivalence for $C$. Hence, it suffices to prove condition (1) with $C\in{\mathrm{Ch}}_{\ge 0}({\mathcal{P}}_{d,{\Bbbk}})$. Similarly the diagram involved in condition (3) commutes up to homotopy for given $C$ and $D$ if and only if it commutes up to homotopy for their suspensions $C[2i]$ and $D[2j]$ (taking even suspensions enables us to avoid checking signs). Hence it suffices to prove condition (3) with $C\in{\mathrm{Ch}}_{\ge 0}({\mathcal{P}}_{d,{\Bbbk}})$ and $D\in {\mathrm{Ch}}_{\ge 0}({\mathcal{P}}_{e,{\Bbbk}})$. [Step 4. Reduction to the simplicial case.]{} For $C\in{\mathrm{Ch}}_{\ge 0}({\mathcal{P}}_{d,{\Bbbk}})$, the Dold-Kan correspondence $${\mathcal{N}}:s(\widetilde{{\mathcal{P}}}_{d,{\Bbbk}})\leftrightarrows {\mathrm{Ch}}^-(\widetilde{{\mathcal{P}}}_{d,{\Bbbk}}):{\mathcal{K}}$$ ensures that ${\mathcal{C}}{\mathcal{K}}(C)$ is homotopy equivalent to $C$. Moreover the explicit formula for ${\mathcal{K}}$ [@Weibel Section 8.4] ensures that ${\mathcal{C}}{\mathcal{K}}(C)$ takes finitely generated projective values, i.e. is an element of ${\mathrm{Ch}}_{\ge 0}({\mathcal{P}}_{d,{\Bbbk}})$. Since derivation of functors preserves homotopy equivalences, conditions (1) and (3) hold for $C$ (and $D$) if and only if they hold for the complexes ${\mathcal{C}}{\mathcal{K}}(C)$ (and ${\mathcal{C}}{\mathcal{K}}(D)$). Therefore, to prove conditions (1) and (3), it suffices to treat the case where there are $X\in s({\mathcal{P}}_{d,{\Bbbk}})$ and $Y\in s({\mathcal{P}}_{e,{\Bbbk}})$ such that $C={\mathcal{C}}X$ and $D={\mathcal{C}}Y$. [Step 5. Proof of condition (1): simplicial case.]{} If $C={\mathcal{C}}X$ with $X\in s({\mathcal{P}}_{d,{\Bbbk}})$, then $\zeta:L(L(C;n,m))\to {\mathcal{L}}(C,n+m)$ is the evaluation of the Eilenberg-Zilber map ${{\mathrm{Id}}}\|f: {\mathcal{C}}Z \to {\mathcal{C}}{\mathrm{diag}\,}^{(1,2+3)}Z$ on the $3$-simplicial object $(X_i)_{K(n)_j\otimes K(m)_k}$. Hence it is a homotopy equivalence. [Step 6. Proof of condition (3): simplicial case.]{} We assume that $C={\mathcal{C}}X$ and $D={\mathcal{C}}Y$ with $X\in s({\mathcal{P}}_{d,{\Bbbk}})$ and $Y\in s({\mathcal{P}}_{e,{\Bbbk}})$. In that case, we may rewrite the various morphisms in the diagram of condition (3), hence their compositions, as Eilenberg-Zilber maps, so commutation up to homotopy follows from proposition \[prop-general-EZ\]. To be more specific, let us denote by $Z$ the $6$-simplicial object $$Z_{i,j,k,\ell,r,s}=(X_i)_{K(n)_j\otimes K(m)_k}\otimes (Y_\ell)_{K(n)_r\otimes K(m)_s}\;.$$ Then $\zeta\otimes\zeta$ is the Eilenberg-Zilber map $${\mathcal{C}}Z\xrightarrow[]{{{\mathrm{Id}}}\|f\|{{\mathrm{Id}}}\|f}{\mathcal{C}}{\mathrm{diag}\,}^{(1,2+3,4,5+6)}Z\;,$$ the map ${\overline{\nabla}}$ on the right hand side is the Eilenberg-Zilber map given as the composite (where $\sigma\in{\mathfrak{S}}_4$ exchanges $2$ and $3$), $${\mathcal{C}}{\mathrm{diag}\,}^{(1,2+3,4,5+6)}Z\xrightarrow[]{\sigma} {\mathcal{C}}{\mathrm{diag}\,}^{(1,4,2+3,5+6)}Z\xrightarrow[]{{{\mathrm{Id}}}\|{{\mathrm{Id}}}\|\nabla}{\mathcal{C}}{\mathrm{diag}\,}^{(1,4,2+3+5+6)}Z\;,$$ the map ${\overline{\nabla}}$ on the left hand side is the composite (where $\sigma\in{\mathfrak{S}}_6$ satisfies $\sigma(3)=4$, $\sigma(4)=5$ and $\sigma(5)=3$): $${\mathcal{C}}Z\xrightarrow[]{\sigma} {\mathcal{C}}{\mathrm{diag}\,}^{(1,2,4,5,3,6)}Z\xrightarrow[]{{{\mathrm{Id}}}\|{{\mathrm{Id}}}\|{{\mathrm{Id}}}\|{{\mathrm{Id}}}\|\nabla}{\mathcal{C}}{\mathrm{diag}\,}^{(1,2,4,5,3+6)}Z\;,$$ the map $L({\overline{\nabla}};m)$ is the composite (where $\sigma\in {\mathfrak{S}}_5$ exchanges $2$ and $3$) $${\mathcal{C}}{\mathrm{diag}\,}^{(1,2,4,5,3+6)}Z \xrightarrow[]{\sigma} {\mathcal{C}}{\mathrm{diag}\,}^{(1,4,2,5,3+6)}Z\xrightarrow[]{{{\mathrm{Id}}}\|{{\mathrm{Id}}}\|\nabla\|{{\mathrm{Id}}}}{\mathcal{C}}{\mathrm{diag}\,}^{(1,4,2+5,3+6)}Z\;,$$ and the map $\zeta$ is the Eilenberg-Zilber map $${\mathcal{C}}{\mathrm{diag}\,}^{(1,4,2+5,3+6)}Z\xrightarrow[]{{{\mathrm{Id}}}\|{{\mathrm{Id}}}\|f}{\mathcal{C}}{\mathrm{diag}\,}^{(1,4,2+5+3+6)}Z\;.$$ This finishes the proof of theorem \[thm-iter-strong\]. Main theorem {#section-main} ============ The functor $\Sigma$ -------------------- For all $d\ge 0$ and all $C\in {\mathrm{Ch}}^b({\mathcal{P}}_{d,{\Bbbk}})$, we denote by $\Sigma C$ the shifted complex $C[-d]$ (recall the convention $C[-d]^i=C^{i+d}$). Gathering all $d$ together, we get a functor: $$\Sigma:{\mathrm{Ch}}^b({\mathcal{P}}_{{\Bbbk}})\to {\mathrm{Ch}}^b({\mathcal{P}}_{{\Bbbk}})$$ For $C\in {\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}})$ and $D\in {\mathbf{D}}^b({\mathcal{P}}_{e,{\Bbbk}})$, we denote by $\xi_{d,e}$ the isomorphism of complexes (the sign ensures that $\xi_{d,e}$ commutes with the differentials) $$\begin{array}{cccc} \xi_{d,e}:& (C[-d])_i\otimes (D[-e])_j & \to & (C\otimes D)[-d-e]_{i+j}\\ & x\otimes y &\mapsto & (-1)^{e(i+d)} x\otimes y \end{array}$$ Gathering all indices $d,e$ together, we get a natural isomorphism: $$\xi: \Sigma(C)\otimes \Sigma(D)\xrightarrow[]{\,\simeq\,}\Sigma(C\otimes D)\;.$$ The following statement is straightforward from the definitions. The triple $(\Sigma,\xi,{{\mathrm{Id}}})$ is a monoidal endofunctor of the symmetric monoidal category $({\mathrm{Ch}}^b({\mathcal{P}}_{{\Bbbk}}),\otimes,{\Bbbk})$. It induces a triangle monoidal endofunctor, still denoted by $(\Sigma,\xi,{{\mathrm{Id}}})$, of the symmetric monoidal category $({\mathbf{D}}^b({\mathcal{P}}_{{\Bbbk}}),\otimes,{\Bbbk})$ \[rk-sgn-Sigma\] The monoidal functor $(\Sigma,\xi,{{\mathrm{Id}}})$ is not symmetric. Indeed, if $C\in {\mathrm{Ch}}^b({\mathcal{P}}_{d,{\Bbbk}})$ and $D\in {\mathrm{Ch}}^b({\mathcal{P}}_{e,{\Bbbk}})$, the following diagram commutes up to a $(-1)^{de}$ sign: $$\xymatrix{ (\Sigma C)\otimes (\Sigma D)\ar[r]^-{\xi}\ar[d]^-{\tau} & \Sigma(C\otimes D)\ar[d]^-{\Sigma(\tau)}\\ (\Sigma D)\otimes (\Sigma C)\ar[r]^-{\xi}& \Sigma(D\otimes C)\;. }$$ The functor $(\Sigma,\xi,{{\mathrm{Id}}})$ is actually a monoidal automorphism of the category of the symmetric monoidal category $({\mathrm{Ch}}^b({\mathcal{P}}_{{\Bbbk}}),\otimes,{\Bbbk})$. That is, there is a monoidal functor $(\Sigma^{-1},\xi^{-1},{{\mathrm{Id}}})$ such that the composite of these two monoidal functors equals the monoidal functor $({{\mathrm{Id}}},{{\mathrm{Id}}},{{\mathrm{Id}}})$. To be more specific, the monoidal inverse sends $C\in{\mathrm{Ch}}^b({\mathcal{P}}_{d,{\Bbbk}})$ to $\Sigma^{-1}(C)= C[d]$, and the restriction of the map $\xi^{-1}$ to complexes of homogeneous functors of degrees $C$ and $D$ is the map $$\begin{array}{cccc} (\xi^{-1})_{d,e}: &C[d]_i\otimes D[e]_j&\to& (C\otimes D)[d+e]_{i+j}\;. \\ & x\otimes y & \mapsto &(-1)^{ei} x\otimes y \end{array}$$ Observe that the sign appearing in the definition of $(\xi^{-1})_{d,e}$ is different from the sign appearing in the definition of $\xi_{d,e}$. Other signs would be possible in the definition of $\xi_{d,e}$ (hence in the definition of $\xi^{-1}_{d,e}$). Our choice of signs is the good one to make theorem \[thm-main\] work. Since $\Sigma$ is defined from the suspension functors, and since triangle functors commute with suspension, the following result holds. \[lm-commut\] Let $F:{\mathbf{D}}^b({\mathcal{P}}_{\Bbbk})\to {\mathbf{D}}^b({\mathcal{P}}_{\Bbbk})$ be a triangle functor. The natural transformation $F(C [-1])\simeq F(C)[-1]$ induces an isomorphism of triangle functors: $$\Sigma\circ F\simeq F\circ \Sigma\;.$$ Moreover, if $F=\Theta$, the isomorphism above is also an isomorphism of monoidal functors. Finally, it follows from remarks \[rk-sgn-Sigma\] and \[rk-sgn-Ringel\] that the functor $\Sigma\circ \Theta$ is symmetric monoidal (although neither $\Sigma$ nor $\Theta$ is). Main theorem and consequences ----------------------------- \[thm-main\] Let ${\Bbbk}$ be a PID, and let $n$ be a positive integer. The derivation functor $L(-;n):{\mathbf{D}}^-({\mathcal{P}}_{\Bbbk})\to {\mathbf{D}}^-({\mathcal{P}}_{\Bbbk})$ induces a (triangle, symmetric monoidal) functor $$L(-;n):{\mathbf{D}}^b({\mathcal{P}}_{\Bbbk})\to {\mathbf{D}}^b({\mathcal{P}}_{\Bbbk})\;,$$ and there is an isomorphism of functors $$L(-;n)\simeq \Sigma^n\circ\Theta^n\;.$$ Moreover, this isomorphism is an isomorphism of triangle functors as well as an isomorphism of monoidal functors. Let us make clear what the first part of the statement means. The canonical monoidal functor (induced by the inclusion) $${\mathbf{D}}^b({\mathcal{P}}_{\Bbbk})\to {\mathbf{D}}^-({\mathcal{P}}_{\Bbbk})$$ is fully faithful, and induces an equivalence of monoidal categories between ${\mathbf{D}}^b({\mathcal{P}}_{\Bbbk})$ and the full subcategory $({\mathbf{D}}^b)'$ of ${\mathbf{D}}^-({\mathcal{P}}_{\Bbbk})$ whose objects are complexes satisfying $H^n(C)=0$ for $n\ll 0$ (see e.g. [@Keller Lemma 11.7] for the equivalence of categories, the fact that all the functors involved are monoidal is a straightforward verification). The first part of theorem \[thm-main\] means that there exist a triangle monoidal functor (the dashed arrow) making the following diagram commute: $$\xymatrix{ {\mathbf{D}}^-({\mathcal{P}}_{\Bbbk})\ar[r]^-{L(-;n)}&{\mathbf{D}}^-({\mathcal{P}}_{\Bbbk})\\ ({\mathbf{D}}^b)'\ar@{^{(}->}[u]\ar@{-->}[r]& ({\mathbf{D}}^b)'\ar@{^{(}->}[u] }$$ and we still denote by $L(-;n)$ the composite $${\mathbf{D}}^b({\mathcal{P}}_{\Bbbk})\simeq ({\mathbf{D}}^b)'\to ({\mathbf{D}}^b)'\simeq {\mathbf{D}}^b({\mathcal{P}}_{\Bbbk})\;.$$ The first part of theorem \[thm-main\] is actually equivalent to saying that strict polynomial functors have bounded derivatives. So it may be seen as a strict polynomial version of [@DP2 Satz 4.22]. Now let us spell out some consequences of theorem \[thm-main\]. Restricting to functors and taking homology, we obtain the statement alluded to in the introduction. \[cor-1\] Let $F\in{\mathcal{P}}_{d,{\Bbbk}}$ and let $V\in{\mathcal{V}}_{\Bbbk}$ be a free ${\Bbbk}$-module of finite rank. There is an isomorphism, natural with respect to $V$ and $F$: $$H^{i}(\Theta^n F)(V)\simeq L_{nd-i}F(V;n)\;.$$ Moreover, if $F\in{\mathcal{P}}_{d,{\Bbbk}}$ and $G\in{\mathcal{P}}_{e,{\Bbbk}}$, these isomorphisms fit into a diagram which commutes up to a $(-1)^{nei}$ sign (and where $\kappa$ denotes the usual Künneth morphism [@ML V (10.1)]): $$\xymatrix{ H^{i}(\Theta^n F)(V)\otimes H^j(\Theta^n G)(V)\ar[d]^-{\kappa}\ar[rr]^-{\simeq} && L_{nd-i}F(V;n)\otimes L_{ne-j}G(V;n)\ar[dd]^-{\nabla}\\ H^{i+j}(\Theta^n F\otimes \Theta^n G)(V)\ar[d]^-{H^{i+j}({\square}^n)}&&\\ H^{i+j}(\Theta^n(F\otimes G))(V)\ar[rr]^-{\simeq}&& L_{nd+ne-i-j}(F\otimes G)(V;n) }.$$ As it has been said in the introduction, corollary \[cor-1\] has concrete interpretations in terms of extension groups. If $F,G\in{\mathcal{P}}_{d,{\Bbbk}}$ and if $V$ is a finitely generated projective ${\Bbbk}$-module $V$, we denote by ${\mathbb{E}}(F,G)(V)$ the parameterized extension groups: $${\mathbb{E}}^(F,G)(V):= {{\mathrm{Ext}}}^_{{\mathcal{P}}_{d,{\Bbbk}}}(F^V,G)\;.$$ Thus, ${\mathbb{E}}(F,G)$ is a functor from $\Gamma^d{\mathcal{V}}_{\Bbbk}$ to graded ${\Bbbk}$-modules and ${\mathbb{E}}^(F,G)({\Bbbk})$ equals ${{\mathrm{Ext}}}^_{{\mathcal{P}}_{d,{\Bbbk}}}(F,G)$. By definition, the homology of $\Theta F$ equals ${\mathbb{E}}^(\Lambda^d,F)$. Moreover, after taking homology, the morphism ${\square}$ is nothing but the usual convolution product of extensions (used e.g. in [@FFSS; @Chalupnik2; @TouzeClassical; @TouzeEML]): $${\mathbb{E}}^i(\Lambda^{d},F)\otimes{\mathbb{E}}^j(\Lambda^{e},G)\to {\mathbb{E}}^{i+j}(\Lambda^{d+e},F\otimes G)\;.$$ Thus, corollary \[cor-1\] may be reinterpreted in the following way. Let ${\Bbbk}$ be a PID, let $F\in{\mathcal{P}}_{d,{\Bbbk}}$ and let $V\in{\mathcal{V}}_{\Bbbk}$. There are isomorphisms (natural in $F,V$): $${\mathbb{E}}^i(\Lambda^d,F)(V)\simeq L_{d-i}F(V;1)\;.$$ Moreover, for $F\in{\mathcal{P}}_{d,{\Bbbk}}$ and $G\in{\mathcal{P}}_{e,{\Bbbk}}$, the pairing $$L_{d-i}F(V;1)\otimes L_{e-j}G(V;1)\to L_{d+e-i-j}(F\otimes G)(V;1)$$ identifies through this isomorphism, up to a $(-1)^{ie}$ sign, with the pairing $${\mathbb{E}}^i(\Lambda^{d},F)(V)\otimes{\mathbb{E}}^j(\Lambda^{e},G)(V)\to {\mathbb{E}}^{i+j}(\Lambda^{d+e},F\otimes G)(V)\;.$$ Similarly, the $2$-fold iteration of Ringel duality has an interpretation in terms of extension groups. Indeed, $\Theta^2 F$ equals ${\mathbf{R}}{\mathbb{H}}(\Lambda^d,\Theta F)$, which is isomorphic to ${\mathbf{R}}{\mathbb{H}}(\Theta^{-1}\Lambda^d,F)$ since $\Theta$ is an equivalence of categories. Now, $\Theta^{-1}\Lambda^d=S^d$, so corollary \[cor-1\] yields an isomorphism: $$L_{2d-i}F(V;2)\simeq {\mathbb{E}}^i(S^d,F)(V)\;.$$ Proof of theorem \[thm-main\] {#subsec-proof-main} ----------------------------- Lemma \[lm-commut\] yields an isomorphism of triangle monoidal functors $\Sigma^n\circ \Theta^n\simeq (\Sigma\circ \Theta)^n$ and similarly, corollary \[cor-iter\] yields an isomorphism of triangle monoidal functors $L(-;n)\simeq L(-;1)^n$. So it suffices to prove theorem \[thm-main\] for $n=1$. In the latter case, theorem \[thm-main\] is a consequence of the following statement. \[prop-main\] The following two composites are isomorphic as triangle and monoidal functors. $$\begin{aligned} &{\mathbf{D}}^b({\mathcal{P}}_{{\Bbbk}})\xrightarrow[]{\Theta} {\mathbf{D}}^b({\mathcal{P}}_{\Bbbk})\xrightarrow[]{\Sigma} {\mathbf{D}}^b({\mathcal{P}}_{\Bbbk})\to {\mathbf{D}}^-({\mathcal{P}}_{\Bbbk})\\ &{\mathbf{D}}^b({\mathcal{P}}_{\Bbbk})\to {\mathbf{D}}^-({\mathcal{P}}_{\Bbbk})\xrightarrow[]{L(-;1)} {\mathbf{D}}^-({\mathcal{P}}_{\Bbbk}).\end{aligned}$$ The remainder of section \[subsec-proof-main\] is devoted to the proof of proposition \[prop-main\]. ### Conventions for ${\mathbb{H}}(C,D)$ {#subsubsec-convention} For the proof, we need to consider ${\mathbb{H}}(C,D)$ when both $C$ and $D$ are complexes. In this paragraph, we give sign conventions for ${\mathbb{H}}(C,D)$ and their consequences. If $C\in {\mathrm{Ch}}^+({\mathcal{P}}_{\Bbbk})$ and $D\in {\mathrm{Ch}}^-({\mathcal{P}}_{\Bbbk})$, we define ${\mathbb{H}}(C,D)\in {\mathrm{Ch}}^-({\mathcal{P}}_{\Bbbk})$ as the total complex of the bicomplex $({\mathbb{H}}(C_i,D^j),{\mathbb{H}}(\partial_C,D^j),{\mathbb{H}}(C_i,\partial_D))$. Thus, $$\begin{aligned}&{\mathbb{H}}(C,D)^n=\bigoplus_{i+j=n}{\mathbb{H}}(C_i,D^j)\;,\\ &\partial(f)=f\circ \partial_C + (-1)^{i}\partial_D\circ f\quad \text{ if $f\in {\mathbb{H}}(C_i,D^j)$.} \end{aligned}$$ With this convention, we have the following compatibility results with the tensor products, suspension and duality. Tensor products. : Tensor products yield a morphism of complexes $$\begin{array}{cccc} {\overline{\otimes}}: & {\mathbb{H}}(C_i,D^j)\otimes {\mathbb{H}}(E_k,F^\ell) &\to & {\mathbb{H}}(C_i\otimes E_k, D^j,F^\ell)\;.\\ & f\otimes g &\mapsto & (-1)^{jk}f\otimes g \end{array}$$ Suspension. : There is an isomorphism of complexes $$\psi_d: {\mathbb{H}}(C,D[d])\xrightarrow[]{\simeq} {\mathbb{H}}(C,D)[d]\;,$$ which sends $f\in {\mathbb{H}}(C_i,D[d]^j)$ to $\psi(f)=(-1)^{di} f$. Moreover, the following diagram commutes $$\xymatrix{ {\mathbb{H}}(C,D[s])\otimes {\mathbb{H}}(E,F[t]) \ar[rr]^-{\psi_s\otimes\psi_t}\ar[d]^-{{\overline{\otimes}}}&& {\mathbb{H}}(C,D)[s]\otimes {\mathbb{H}}(E,F)[t]\ar[d]^-{\xi_{s,t}}\\ {\mathbb{H}}(C\otimes E,D[s]\otimes F[t])\ar[d]^-{{\mathbb{H}}(C\otimes E,\xi_{s,t})}&& \left({\mathbb{H}}(C,D)\otimes {\mathbb{H}}(E,F)\right)[s+t]\ar[d]^-{{\overline{\otimes}}[s+t]}\\ {\mathbb{H}}(C\otimes E,(D\otimes F)[s+t])\ar[rr]^-{\psi_{s+t}}&&{\mathbb{H}}(C\otimes E,D\otimes F)[s+t]\;. }$$ Duality. : The dual of a complex $C$ is the complex $C^\sharp$ with $(C^\sharp)_i=(C^i)^\sharp$ and $\partial_{C^\sharp}= (\partial_C)^\sharp$ (no sign on the differential of the dual). Duality commutes with tensor products: $C^\sharp\otimes D^\sharp=(C\otimes D)^\sharp$. There is an isomorphism of complexes: $$\begin{array}{cccc} ^\sharp:&{\mathbb{H}}((C^{\sharp})_i, D^j) & \xrightarrow[]{\simeq} & {\mathbb{H}}((D^{\sharp})_j, C^i)\;.\\ &f & \mapsto & (-1)^{ij} f^\sharp \end{array}$$ Moreover, the following diagram commutes: $$\xymatrix{ {\mathbb{H}}(C^\sharp, D)\otimes {\mathbb{H}}(E^\sharp, F) \ar[d]^-{{\overline{\otimes}}}\ar[r]^-{^\sharp\otimes^\sharp}& {\mathbb{H}}(D^\sharp, C)\otimes {\mathbb{H}}(F^\sharp, E)\ar[d]^-{{\overline{\otimes}}}\\ {\mathbb{H}}(C^\sharp\otimes E^\sharp, D\otimes F)\ar[r]^-{^\sharp}&{\mathbb{H}}(D^\sharp\otimes F^\sharp, C\otimes E)\;. }$$ ### Plan of the proof of proposition \[prop-main\]. We first give an equivalent definition of $\Theta$. By duality $\Theta:{\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}})\to {\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}})$ is isomorphic to the right derived functor of $F\mapsto {\mathbb{H}}(F^\sharp,\Lambda^d)$, and the morphism ${\square}$ may be described as the composite (where the last map is induced by the multiplication $\Lambda^d\otimes\Lambda^e\to \Lambda^{d+e}$). $${\mathbf{R}}{\mathbb{H}}(C^\sharp,\Lambda^d)\otimes {\mathbf{R}}{\mathbb{H}}(D^\sharp,\Lambda^e)\xrightarrow[]{\otimes}{\mathbf{R}}{\mathbb{H}}(C^\sharp\otimes D^\sharp,\Lambda^d\otimes\Lambda^e)\to {\mathbf{R}}{\mathbb{H}}((C\otimes D)^\sharp, \Lambda^{d+e})\;.$$ Now assume that there is a family of complexes of injectives $J(d)\in {\mathrm{Ch}}^-({\mathcal{P}}_{d,{\Bbbk}})$, together with quasi-isomorphisms $\phi_d:\Lambda^d\to J(d)$ and with morphisms $f_{d,e}: J(d)\otimes J(e)\to J(d+e)$ such that the following diagram commutes in ${\mathrm{Ch}}^-({\mathcal{P}}_{\Bbbk})$: $$\xymatrix{\Lambda^d\otimes \Lambda^e\ar[d]^-{\phi_d\otimes\phi_e}\ar[r]^-{\mathrm{mult}}&\Lambda^{d+e}\ar[d]^-{\phi_{d+e}}\\ J(d)\otimes J(e)\ar[r]^-{f_{d,e}}& J(d+e)\;.}$$ Then $\Theta$ is isomorphic to the localization of the functor $$\begin{array}{cccc} {\widetilde{\Theta}}: &{\mathrm{Ch}}^b({\mathcal{P}}_{d,{\Bbbk}})&\to &{\mathrm{Ch}}^-({\mathcal{P}}_{d,{\Bbbk}})\;.\\ & C &\mapsto &{\mathbb{H}}(C^\sharp,J(d)) \end{array}$$ Moreover, for $C\in {\mathrm{Ch}}^-({\mathcal{P}}_{d,{\Bbbk}})$ and $D\in {\mathrm{Ch}}^-({\mathcal{P}}_{e,{\Bbbk}})$, the morphism ${\square}$ corresponds to the natural transformation ${\widetilde{{\square}}}$ defined as the composite (where the last map is induced by $f_{d,e}$): $${\mathbb{H}}(C^\sharp,J(d))\otimes{\mathbb{H}}(D^\sharp,J(e))\xrightarrow[]{{\overline{\otimes}}} {\mathbb{H}}(C^\sharp\otimes D^\sharp, J(d)\otimes J(e))\xrightarrow[]{} {\mathbb{H}}((C\otimes D)^\sharp, J(d+e))\;,$$ and the unit $\phi: {\Bbbk}\to \Theta({\Bbbk})$ is induced by the morphism $\widetilde{\phi}$ $${\Bbbk}={\mathbb{H}}({\Bbbk},{\Bbbk})={\mathbb{H}}({\Bbbk},\Lambda^0)\to {\mathbb{H}}({\Bbbk},J(0))\;.$$ The proof of proposition \[prop-main\] is organized as follows. - In a first step, we make explicit choices of coresolutions $J(d)$ and maps $f_{d,e}$. Thus we get an explicit monoidal functor $({\widetilde{\Theta}},{\widetilde{{\square}}},\widetilde{\phi})$. - In a second step, we prove that with our choices, the functors $\Sigma\circ {\widetilde{\Theta}}$ and $L(-;1)$ are isomorphic as functors from ${\mathrm{Ch}}^b({\mathcal{P}}_{d,{\Bbbk}})$ to ${\mathrm{Ch}}^-({\mathcal{P}}_{d,{\Bbbk}})$, and that this isomorphism commutes with suspension and monoidal structures. ### Construction of $({\widetilde{\Theta}},{\widetilde{{\square}}},\widetilde{\phi})$. To define $({\widetilde{\Theta}},{\widetilde{{\square}}},\widetilde{\phi})$, we take the complexes $J(d):= ({\mathcal{C}}S^d_{K(1)})[d]$ and the maps $f_{d,e}$ provided by the following lemma. For all $d\ge 0$, there is a quasi-isomorphism of complexes: $$\Lambda^d\xrightarrow[]{\phi_d} ({\mathcal{C}}S^d_{K(1)})[d]\;.$$ Moreover, if $\nabla$ denotes the shuffle map, let $f_{d,e}$ denote the composite: $$\begin{aligned} ({\mathcal{C}}S^d_{K(1)})[d]&\otimes ({\mathcal{C}}S^e_{K(1)})[e]\xrightarrow[]{(\xi^{-1})_{d,e}}({\mathcal{C}}S^d_{K(1)})\otimes ({\mathcal{C}}S^e_{K(1)})[d+e]\\&\xrightarrow[]{\nabla[d+e]} {\mathcal{C}}(S^d\otimes S^e)_{K(1)}[d+e]\xrightarrow[]{\mathrm{mult}[d+e]} {\mathcal{C}}(S^{d+e})_{K(1)}[d+e]\;,\end{aligned}$$ then the following diagram is commutative: $$\xymatrix{\Lambda^d\otimes \Lambda^e\ar[d]^-{\phi_d\otimes\phi_e}\ar[r]^-{\mathrm{mult}}&\Lambda^{d+e}\ar[d]^-{\phi_{d+e}}\\ ({\mathcal{C}}S^d_{K(1)})[d]\otimes ({\mathcal{C}}S^e_{K(1)})[e]\ar[r]^-{f_{d,e}}& ({\mathcal{C}}S^{d+e}_{K(1)})[d+e]\;.}$$ Since the shuffle map $\nabla$ is strictly associative and strictly commutative, the composite $${\mathcal{C}}S^d_{K(1)}\otimes {\mathcal{C}}S^e_{K(1)}\xrightarrow[]{\nabla}{\mathcal{C}}(S^d\otimes S^e)_{K(1)}\xrightarrow[]{\mathrm{mult}} {\mathcal{C}}S^{d+e}_{K(1)}$$ makes ${\mathcal{C}}S_{K(1)}=\bigoplus_{d\ge 0} {\mathcal{C}}S^d_{K(1)}$ a commutative monoid in ${\mathrm{Ch}}^-({\mathcal{P}}_{\Bbbk})$. By definition ${\mathcal{C}}S^1_{K(1)}= ({\mathcal{C}}K(1))\otimes S^1$ has homology equal to the functor $S^1=\Lambda^1$, placed in degree $1$. Choosing a cycle in the complex of ${\Bbbk}$-modules ${\mathcal{C}}K(1)$ and tensoring by $\Lambda^1=S^1$, we obtain a quasi-isomorphism of complexes of strict polynomial functors (put the trivial differential on the left hand side): $$\Lambda^1[-1]\hookrightarrow {\mathcal{C}}S^1_{K(1)}\;.$$ Since ${\mathcal{C}}S_{K(1)}$ is graded commutative, the universal property of exterior algebras yield a morphism of commutative monoids in ${\mathrm{Ch}}^-({\mathcal{P}}_{\Bbbk})$: $$A=\bigoplus_{d\ge 0}\Lambda^d[-d]\hookrightarrow \bigoplus_{d\ge 0} {\mathcal{C}}S^d_{K(1)}={\mathcal{C}}S_{K(1)}\;. \qquad(\dag)$$ It is well known that $(\dag)$ is a quasi-isomorphism. Indeed, for all free finitely generated ${\Bbbk}$-modules $U$, $V$, there is a commutative diagram (where the horizontal quasi-isomorphisms are defined via multiplications): $$\xymatrix{ A(U)\otimes A(V)\ar@{^{(}->}[d]\ar[rr]^-{\simeq}&& A(U\oplus V)\ar@{^{(}->}[d]\\ {\mathcal{C}}S_{K(1)}(U)\otimes {\mathcal{C}}S_{K(1)}(V)\ar[rr]^-{\mathrm{mult}\,\circ\nabla}&&{\mathcal{C}}S_{K(1)}(U\oplus V) }$$ so it suffices to prove that ${\Bbbk}[-1]=A({\Bbbk})\hookrightarrow {\mathcal{C}}S_{K(1)}({\Bbbk})$ is a quasi-isomorphism. By construction the map ${\Bbbk}[-1]\hookrightarrow S^1_{K(1)}({\Bbbk})=K(1)$ is a quasi isomorphism, so we only have to check that the homology of $\bigoplus_{d\ne 1}{\mathcal{C}}S^d_{K(1)}({\Bbbk})$ vanishes. We readily check that ${\mathcal{N}}S(K(1))=\overline{B} (S({\Bbbk}))$ (the bar complex of the symmetric algebra on one generator), so the vanishing follows from the equalities (the last equality follows e.g. from [@ML VII Thm 2.2]) $$H({\mathcal{C}}S_{K(1)}({\Bbbk}))\simeq H({\mathcal{N}}S_{K(1)}({\Bbbk}))= H(\overline{B}(S({\Bbbk})))=\mathrm{Tor}^{S(k)}({\Bbbk},{\Bbbk})={\Bbbk}[-1]\;.$$ Restricting the quasi-isomorphism $(\dag)$ to the homogeneous part of degree $d$ and shifting by $d$ we get the required isomorphism: $$\phi_d:\Lambda^d=(\Lambda^d[-d])[d]\hookrightarrow ({\mathcal{C}}S^d_{K(1)})[d]\;.$$ Moreover, since $(\dag)$ is a morphism of monoids, we have commutative diagrams: $$\xymatrix{(\Lambda^d[-d])[d]\otimes (\Lambda^e[-e])[e]\ar[d]^-{\phi_d\otimes\phi_e}\ar[rrr]^-{\mathrm{mult}[d+e]\circ(\xi^{-1})_{d,e}} &&&\Lambda^{d+e}[-d-e][d+e]\ar[d]^-{\phi_{d+e}}\\ ({\mathcal{C}}S^d_{K(1)})[d]\otimes ({\mathcal{C}}S^e_{K(1)})[e]\ar[rrr]^-{f_{d,e}}&&& ({\mathcal{C}}S^{d+e}_{K(1)})[d+e]}.$$ To finish the proof, we observe that the sign induced by $(\xi^{-1})_{d,e}$ is equal to $1$. Hence the upper horizontal arrow identifies with the multiplication $\Lambda^d\otimes\Lambda^e\to \Lambda^{d+e}$. ### Proof of the isomorphism $\Sigma\circ {\widetilde{\Theta}}\simeq L(-;1)$. To prove the isomorphism between $\Sigma\circ {\widetilde{\Theta}}$ and $L(-;1)$ commuting with suspension and monoidal structures, we first introduce yet another monoidal functor $$L':{\mathrm{Ch}}^b({\mathcal{P}}_{\Bbbk})\to {\mathrm{Ch}}^-({\mathcal{P}}_{\Bbbk})\;.$$ Namely, if $C\in{\mathrm{Ch}}^b({\mathcal{P}}_{d,{\Bbbk}})$, we let $$L'(C)={\mathbb{H}}(C^\sharp, {\mathcal{C}}S^d_{K(1)})\;.$$ We observe that $L'(C[1])$ equals $L'(C)[1]$, so that $L'$ commutes with suspension. The monoidal structure of $L'$ is defined for $C\in{\mathrm{Ch}}^-({\mathcal{P}}_{d,{\Bbbk}})$ and $D\in{\mathrm{Ch}}^-({\mathcal{P}}_{e,{\Bbbk}})$ as the composite: $$\begin{aligned} {\mathbb{H}}(C^\sharp, {\mathcal{C}}S^d_{K(1)})\otimes {\mathbb{H}}(D^\sharp, {\mathcal{C}}S^e_{K(1)})& \xrightarrow[]{{\overline{\otimes}}} {\mathbb{H}}(C^\sharp\otimes D^\sharp, {\mathcal{C}}S^d_{K(1)}\otimes {\mathcal{C}}S^e_{K(1)})\\& \to{\mathbb{H}}(C^\sharp\otimes D^\sharp, {\mathcal{C}}(S^d\otimes S^e)_{K(1)}) \\ &\to{\mathbb{H}}(C^\sharp\otimes D^\sharp, {\mathcal{C}}(S^{d+e})_{K(1)})\;, \end{aligned}$$ where the second map is induced by the shuffle map $\nabla: {\mathcal{C}}S^d_{K(1)}\otimes {\mathcal{C}}S^e_{K(1)}\to {\mathcal{C}}(S^d\otimes S^e)_{K(1)}$, and the last one is induced by the multiplication $S^d\otimes S^e\to S^{d+e}$. The unit morphism is the identity map in degree zero: $${\Bbbk}={\mathbb{H}}({\Bbbk},{\Bbbk})\to {\mathbb{H}}({\Bbbk},{\Bbbk}_{K(1)})={\Bbbk}_{K(1)}\;.$$ To construct the isomorphism $\Sigma\circ {\widetilde{\Theta}}\simeq L(-;1)$ we compose the isomorphisms $L'\simeq L(-;1)$ and $\Sigma\circ {\widetilde{\Theta}}\simeq L'$ given by the two following two lemmas. This will finish the proof of proposition \[prop-main\]. There is an isomorphism $L'\simeq L(-;1)$. This isomorphism commutes with suspension and with the monoidal structures. Recall from section \[subsubsec-internal\] the isomorphism: $${\mathbb{H}}(F^\sharp,S^d_U)\simeq {\mathbb{H}}(\Gamma^{d,U},F)\simeq F_U\;.\qquad(\star)$$ Since this isomorphism is natural with respect to $F$ and $U$, we can replace $F$ by a complex $C\in {\mathrm{Ch}}^b({\mathcal{P}}_{d,{\Bbbk}})$ and $U$ by the simplicial ${\Bbbk}$-module $K(1)$ to get an isomorphism of mixed complexes ${\mathbb{H}}(C^\sharp,S^d_{K(1)})\simeq C_{K(1)}$, hence an isomorphism of complexes $$L'(C)={\mathbb{H}}(C^\sharp, {\mathcal{C}}S^d_{K(1)})={\overline{{\mathcal{C}}}}{\mathbb{H}}(C^\sharp,S^d_{K(1)})\simeq {\overline{{\mathcal{C}}}}C_{K(1)}=L(C;1)\;.$$ It is obvious from the definition that this isomorphism commutes with suspension. If $d=0$ and $C={\Bbbk}$, this isomorphism is the identity in degree zero, so it preserves the units of the monoidal functors $L'$ and $L(-;1)$. So it remains to check that the following diagram is commutative. $$\xymatrix{ L'(C)\otimes L'(D)\ar[d]\ar[r]^-{\simeq}& L(C;1)\otimes L(D;1)\ar[d]^-{{\overline{\nabla}}}\\ L'(C\otimes D)\ar[r]^-{\simeq} & L(C\otimes D;1)\;. }\qquad \text{(D1)}$$ Since the functors are additive, we may restrict to the case $C\in {\mathrm{Ch}}^b({\mathcal{P}}_{d,{\Bbbk}})$ and $D\in {\mathrm{Ch}}^b({\mathcal{P}}_{e,{\Bbbk}})$. We proceed in two steps. [Step 1. A commutative diagram.]{} If $f\in {{\mathrm{Hom}}}_{{\mathcal{V}}_{\Bbbk}}(U,W)$ and $g\in {{\mathrm{Hom}}}_{{\mathcal{V}}_{\Bbbk}}(V,W)$ we denote by $\mathrm{mult}(f,g)$ the following composite, where the first map is induced by $f$ and $g$ and the second map is induced by the multiplication $S^d\otimes S^e\to S^{d+e}$: $$\mathrm{mult}(f,g): S^d_{U}\otimes S^e_V\to S^d_{W}\otimes S^e_W \to S^{d+e}_W\;.$$ Then one readily checks that for all $F\in{\mathcal{P}}_{d,{\Bbbk}}$ and $G\in {\mathcal{P}}_{e,{\Bbbk}}$, the isomorphisms $(\star)$ fit into a commutative diagram: $$\xymatrix{ {\mathbb{H}}(F^\sharp,S^d_U)\otimes {\mathbb{H}}(G^\sharp,S^d_V)\ar[d]^-{\otimes}\ar[rr]^-{\simeq} && F_U\otimes G_V\ar[dd]^-{F_f\otimes F_g}\\ {\mathbb{H}}(F^\sharp\otimes G^\sharp,S^d_U\otimes S^d_V)\ar[d]^-{{\mathbb{H}}(F^\sharp,G^\sharp,\mathrm{mult}(f,g))} &&\\ {\mathbb{H}}(F^\sharp\otimes G^\sharp, S^{d+e}_W)\ar[rr]^-{\simeq}&& F_W\otimes G_W \;. }\qquad\text{(D2)}$$ [Step 2. Proof of the commutativity of diagram (D1).]{} Restriction of diagram (D1) to the indices $i,j,p,q$ yields the following diagram. $$\xymatrix{ {\mathbb{H}}((C^i)^\sharp,S^d_{K(1)_p})\otimes {\mathbb{H}}((D^j)^\sharp,S^d_{K(1)_q})\ar[d]\ar[r]^-{\simeq}_-{(\star)\otimes(\star)}& C^i_{K(1)_{p}}\otimes D^j_{K(1)_{q}}\ar[d]\\ {\mathbb{H}}((C^i)^\sharp\otimes (D^j)^\sharp,S^{d+e}_{K(1)_{p+q}})\ar[r]^-{\simeq}_-{(\star)}& C^i_{K(1)_{p+q}}\otimes D^j_{K(1)_{p+q}}\;.} \quad \text{(D3)}$$ So, to prove the commutativity of diagram (D1), it suffices to prove that (D3) commutes for all indices $i,j,p,q$. Let us describe explicitly the vertical arrows in diagram (D3). By [@ML VIII Thm 8.8], if $X$ is a simplicial object, the shuffle map $\nabla:X_p\otimes X_q\to X_{p+q}\otimes X_{p+q}$ is the sum over all $(p,q)$-shuffles $\mu$ $$\nabla=\sum \epsilon(\mu) f_\mu\otimes g_\mu$$ where $f_\mu:X_p\to X_{p+q}$ and $g_{\mu}:X_q\to X_{p+q}$ are the composites (with the $\sigma_i$ denoting the degeneracy operators of $X$): $$f_\mu=\sigma_{\mu(p+q)}\circ \sigma_{\mu(p+q-1)}\circ\cdots \circ \sigma_{\mu(p+1)}\,,\quad g_\mu=\sigma_{\mu(p)}\circ \cdots \circ \sigma_{\mu(1)}\;.$$ As a consequence, the right vertical arrow of (D3) is equal, up to a $(-1)^{jp}$ sign, to the sum over all $(p,q)$-shuffles $\mu$ of the maps $\epsilon(\mu)F_{f_\mu}\otimes G_{g_\mu}$, while the left vertical arrow of (D3) is equal up to a $(-1)^{jp}$ sign, to the sum over all $(p,q)$-shuffles $\mu$ of the postcomposition by $\epsilon(\mu)\mathrm{mult}(f_\mu,g_\mu)$: $$x\otimes y\mapsto \epsilon(\mu)\mathrm{mult}(f_\mu,g_\mu)\circ (x\otimes y)\;.$$ Since diagram (D2) commutes, by taking $U=K(1)_p$, $V=K(1)_q$, $W=K(1)_{p+q}$, $F=C^i$, $G=D^j$, $f=f_\mu$ and $g=g_\mu$, we get that postcomposition by $\epsilon(\mu)\mathrm{mult}(f_\mu,g_\mu)$ identifies through isomorphism $(\star)$ with $\epsilon(\mu)F_{f_\mu}\otimes G_{g_\mu}$. By summing over all $(p,q)$-shuffles, we obtain that diagram (D3) commutes. There is an isomorphism $\Sigma\circ {\widetilde{\Theta}}\simeq L'$. This isomorphism commutes with suspension and with the monoidal structures. We are actually going to prove an isomorphism of functors ${\widetilde{\Theta}}\simeq \Sigma^{-1}\circ L'$, compatible with suspensions and monoidal structures. If $C\in{\mathrm{Ch}}^b({\mathcal{P}}_{d,{\Bbbk}})$ compatibility of ${\mathbb{H}}$ is with suspension (cf. section \[subsubsec-convention\]) yields an isomorphism $$\psi_d:{\widetilde{\Theta}}(C)={\mathbb{H}}(C^\sharp,{\mathcal{C}}S^d_{K(1)}[d])\simeq {\mathbb{H}}(C^\sharp,{\mathcal{C}}S^d_{K(1)})[d]= L'(C)[d]=(\Sigma^{-1}\circ L')(C)\;.$$ Gathering all these isomorphisms, we obtain a morphism $\psi:{\widetilde{\Theta}}\simeq \Sigma^{-1}\circ L'$. This isomorphism commutes with suspension. Moreover, $\psi_0$ is the identity map, so this isomorphism preserves the units of the monoidal structures. So it remains to check that the following diagram commutes $$\xymatrix{ {\widetilde{\Theta}}(C)\otimes {\widetilde{\Theta}}(D)\ar[d]^-{{\widetilde{{\square}}}}\ar[r]^-{\psi\otimes\psi}& (\Sigma^{-1}\circ L')(C)\otimes (\Sigma^{-1}\circ L')(D)\ar[d]\\ {\widetilde{\Theta}}(C\otimes D)\ar[r]^-{\psi} & (\Sigma^{-1}\circ L')(C\otimes D)\;. }\qquad \text{(D1)}$$ By additivity of the functors appearing in the diagram, we may restrict to the case where $C\in{\mathrm{Ch}}^b({\mathcal{P}}_{d,{\Bbbk}})$ and $D\in {\mathrm{Ch}}^b({\mathcal{P}}_{e,{\Bbbk}})$. Since the map $f_{d,e}$ equals the composite $(\mathrm{mult}\,\circ \nabla)[d+e]\circ (\xi^{-1})_{d,e}$, diagram (D1) can be rewritten as follows. $$\xymatrix{ {\mathbb{H}}(C^\sharp,{\mathcal{C}}S^d_{K(1)}[d])\otimes {\mathbb{H}}(D^\sharp,{\mathcal{C}}S^e_{K(1)}[e])\ar[d]^-{{\overline{\otimes}}}\ar[r]^-{\psi_d\otimes\psi_e} & {\mathbb{H}}(C^\sharp,{\mathcal{C}}S^d_{K(1)})[d]\otimes {\mathbb{H}}(D^\sharp,{\mathcal{C}}S^e_{K(1)})[e]\ar[d]^{(\xi^{-1})_{d,e}}\\ {\mathbb{H}}(C^\sharp\otimes D^\sharp,{\mathcal{C}}S^d_{K(1)}[d]\otimes {\mathcal{C}}S^e_{K(1)}[e])\ar[d]^-{{\mathbb{H}}(C^\sharp\otimes D^\sharp,(\xi^{-1})_{d,e})}& {\mathbb{H}}(C^\sharp,{\mathcal{C}}S^d_{K(1)})\otimes {\mathbb{H}}(D^\sharp,{\mathcal{C}}S^e_{K(1)})[d+e]\ar[d]^{{\overline{\otimes}}[d+e]}\\ {\mathbb{H}}(C^\sharp\otimes D^\sharp,{\mathcal{C}}S^{d}_{K(1)}\otimes {\mathcal{C}}S^e_{K(1)}[d+e]) \ar[d]^-{{\mathbb{H}}(C^\sharp\otimes D^\sharp,(\mathrm{mult}\,\circ \nabla)[d+e])}\ar[r]^-{\psi_{d+e}}& {\mathbb{H}}(C^\sharp\otimes D^\sharp,{\mathcal{C}}S^{d}_{K(1)}\otimes {\mathcal{C}}S^e_{K(1)})[d+e]\ar[d]^-{{\mathbb{H}}(C^\sharp\otimes D^\sharp,\mathrm{mult}\,\circ \nabla)[d+e]}\\ {\mathbb{H}}(C^\sharp\otimes D^\sharp,{\mathcal{C}}S^{d+e}_{K(1)}[d+e])\ar[r]^-{\psi_{d+e}}\ar[r]^-{\psi_{d+e}}& {\mathbb{H}}(C^\sharp\otimes D^\sharp,{\mathcal{C}}S^{d+e}_{K(1)})[d+e] }$$ The upper square commutes by the compatibility properties of tensor products suspensions and ${\mathbb{H}}$, cf section \[subsubsec-convention\], and the lower square commutes since $\psi_{d+e}$ is a natural transformation. Thus diagram (D1) commutes. Applications {#sec-applic} ============ Décalages --------- ### Recollections of Schur functors and Weyl functors {#subsubsec-schur} If $\lambda=(\lambda_1,\dots,\lambda_k)$ is a partition of weight $\sum\lambda_i=d$, we denote by $\lambda'$ the conjugate partition. The Schur functor $S_\lambda$ associated to the partition $\lambda$ is defined as the image of the composite: $$d_\lambda:\Lambda^{\lambda'}\hookrightarrow \otimes^d\xrightarrow[]{\sigma_\lambda}\otimes^d\twoheadrightarrow S^\lambda\;,$$ where the first map is the canonical inclusion of $\Lambda^{\lambda'}=\bigotimes_j \Lambda^{\lambda'_j}$ into $\otimes^d$, the last map is the canonical projection onto $S^\lambda=\bigotimes_i S^{\lambda_i}$ and the middle map is the isomorphism induced by sending $v_1\otimes\dots v_d$ to $v_{\sigma_\lambda(1)}\otimes\dots\otimes v_{\sigma_\lambda(d)}$ where $\sigma_\lambda\in{\mathfrak{S}}_d$ is the permutation defined as follows. Let $t_\lambda$ be the Young tableau with standard filling: $1,\dots,\lambda_1$ in the first row, $\lambda_1+1,\dots,\lambda_2$ in the second row, etc. Then $\sigma_\lambda$, in one-line notation, is the row-reading of the conjugate tableau $t_{\lambda'}$. As particular cases of Schur functors, we recover symmetric and exterior powers: $$S_{(1^d)}=\Lambda^d\,,\quad S_{(d)}=S^d\,.$$ These Schur functors were first defined (in arbitrary characteristic) in [@ABW]. They are denoted there by a letter ‘$L$’, but we prefer to denote them by the letter ‘$S$’ and to keep the letter ‘$L$’ for simple objects, as it is done in [@Jantzen; @Martin]. Also, conjugate partitions are used in [@ABW] to index Schur functors, but we prefer the other convention, which agrees with [@Green; @Jantzen; @Martin; @MacDonald; @FultonHarris]. Schur functors have various notations and names, depending on the context. For the reader interested in reading the sources we have quoted, the following table provides the translation. Reference This article [@FultonHarris] [@MacDonald] [@ABW] [@Green] [@Martin] [@Jantzen] ----------- -------------- ---------------------- -------------- ---------------- ----------------------- -------------- ---------------- Notation $S_\lambda$ $\mathbb{S}_\lambda$ $F(\lambda)$ $L_{\lambda'}$ $D_{\lambda,{\Bbbk}}$ $M(\lambda)$ $H^0(\lambda)$ The notations in this table all occur in slightly different contexts. In [@FultonHarris Lecture 6] and in [@MacDonald I, App.A], $\mathbb{S}_\lambda$ and $F(\lambda)$ refer to Schur functors defined over complex numbers. In [@Green; @Martin], the notations refer to modules over the Schur algebra, the table means that $S_\lambda({\Bbbk}^n)$ coincides with $D_{\lambda,{\Bbbk}}$ and $M(\lambda)$ as a module over the Schur algebra $S(n,d)$ ($M(\lambda)$ is called a Schur module in [@Martin], and $D_{\lambda,{\Bbbk}}$ is called a dual Weyl module in [@Green]). Finally, in [@Jantzen], $H^0(\lambda)$ refers to a $GL_n$-module, so the table means that $S_\lambda({\Bbbk}^n)$ coincides with $H^0(\lambda)$ as a $GL_n$-module. It is obvious that the objects $S_\lambda$, $\mathbb{S}_\lambda$, $L_{\lambda'}$ coincide. To see that $S_\lambda({\Bbbk}^n)$ coincides with $M(\lambda)$, use the embedding embedding of $M(\lambda)\subset S^\lambda({\Bbbk}^n)$ [@Martin Example (1) p.73], and [@ABW Thm II.2.16]. Finally, $M(\lambda)$ and $H^0(\lambda)$ coincide by a theorem of James, cf [@Martin Thm 3.2.6] (see also [@Martin Thm 3.4.1]). The Weyl functor $W_\lambda$ is the dual of the Schur functor: $W_\lambda=S_{\lambda}^\sharp$. It may be defined as the image of the composite $$\Gamma^\lambda\hookrightarrow \otimes^d\xrightarrow[]{\sigma_{\lambda'}} \otimes^d\twoheadrightarrow \Lambda^{\lambda'}\;.$$ Just as in the case of Schur functors, these functors (or the associated representations) have various notations (and names) depending on the context. For example, $W_\lambda$ is called a coSchur functor in [@ABW], and denoted there by $K_\lambda$ (see [@ABW Prop II.4.1] for the description as the dual of $S_\lambda$). No notation is used for Weyl functors in [@FultonHarris; @MacDonald] because over a field of characteristic zero there is an isomorphism $W_\lambda\simeq S_\lambda$ [@FultonHarris Exercise 6.14]. Here is the conversion table. Reference This article [@ABW] [@Green] [@Martin] [@Jantzen] ----------- -------------- ------------- ----------------------- -------------- -------------- Notation $W_\lambda$ $K_\lambda$ $V_{\lambda,{\Bbbk}}$ $V(\lambda)$ $V(\lambda)$ Finally, in the context of highest weight categories, Schur functors $S_\lambda$ and Weyl functors $W_\lambda$ are respectively called costandard modules and standard modules, and they are often respectively denoted by $\nabla(\lambda)$ and $\Delta(\lambda)$ (although they are respectively denoted by $A(\lambda)$ and $V(\lambda)$ in [@CPS]). ### Formality and décalages A complex $C\in {\mathbf{D}}^b({\mathcal{P}}_{\Bbbk})$ is formal if there is an isomorphism $C\simeq H_(C)$ in ${\mathbf{D}}^b({\mathcal{P}}_{\Bbbk})$. The following lemma gives a sufficient condition for the formality of $\Theta^n F$. \[lm-formal\] Let $F\in{\mathcal{P}}_{{\Bbbk}}$. Assume that $H^i(\Theta^n(F))=0$ for $i\ne 0$. Then there is an isomorphism in the derived category $$H^0(\Theta^n F)\simeq \Theta^n(F)\;.$$ Observe that by construction, $\Theta^n(F)$ is isomorphic to a complex $C$ with $C^i=0$ if $i<0$. Hence we may produce the requested isomorphism as the composite $H^0(F)\simeq H^0(C)\simeq C\simeq \Theta^n(F)$. In particular, if $F$ is a Schur functor, then $H^i(\Theta F)=0$ for $i>0$ by lemma \[lm-acyclic\], hence $\Theta(F)$ is formal. Moreover it is easy to compute $H^0(\Theta S_\lambda)$. Let $\lambda$ be a partition of weight $d$, and let ${\Bbbk}$ be a PID. There is an isomorphism $H^0(\Theta S_\lambda)=W_{\lambda'}$. Let us fix a partition $\lambda=(\lambda_1,\dots,\lambda_m)$. It is proved in [@ABW Thm II.2.16] that the Schur functor $S_{\lambda'}$ associated to the conjugate of $\lambda$ is the cokernel of the map $[]_{\lambda}:\bigoplus_{\mu}\Lambda^\mu\to\Lambda^{\lambda} $, where the sum is taken over all tuples of positive integers of the form $(\lambda_1,\dots,\lambda_i+k,\lambda_{i+1}-k,\dots, \lambda_n)$ for all $1\le i\le n-1$ and all $1\le k\le \lambda_{i+1}$ and the restriction of $[]_{\lambda}$ to the summand indexed by the tuple $(\lambda_1,\dots,\lambda_i+k,\lambda_{i+1}-k,\dots, \lambda_n)$ is built by tensoring identities with the composite $$\Lambda^{\lambda_{i+1}-k}\otimes \Lambda^{\lambda_{i+1}-k}\xrightarrow[]{\mathrm{comult}\otimes 1}\Lambda^{\lambda_{i+1}}\otimes \Lambda^k\otimes \Lambda^{\lambda_{i+1}-k} \xrightarrow[]{1\otimes \mathrm{mult}}\Lambda^{\lambda_{i+1}}\otimes \Lambda^{\lambda_{i+1}}\;.$$ Hence $W_{\lambda'}$ is the kernel of the map $[]_{\lambda}^\sharp:\Lambda^{\lambda}\to \bigoplus_{\mu}\Lambda^\mu$. Similarly, it is proved in [@ABW Thm II.3.16] that the Weyl functor $W_\lambda$ is the cokernel of a similar map $[]'_\lambda:\bigoplus_\mu \Gamma^\mu\to\Gamma^\lambda$, hence $S_\lambda$ is the kernel of the map $([]'_\lambda)^\sharp$. Now tensors of exterior powers are ${\mathbb{H}}(\Lambda^d,-)$-acyclic, so $$H^0(\Theta S_\lambda)\simeq\ker({\mathbb{H}}(\Lambda^d,([]_{\lambda'})^\sharp)=\ker (([]_{\lambda'})^\sharp)=W_{\lambda'}.$$ So we have an isomorphism $W_{\lambda'}\simeq \Theta S_\lambda$ in ${\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}})$. Thus, theorem \[thm-main\] yields the following result. \[prop-decalage\] Let ${\Bbbk}$ be a PID and let $\lambda$ be a partition of weight $d$ and $\lambda'$ be the dual partition. There are décalage isomorphisms (for all integers $i$ and $n$): $$L_iW_{\lambda'}(V;n)\simeq L_{i+d}S_\lambda(V;n+1)\;.$$ This proposition generalizes the décalage isomorphisms of Quillen [@Quillen2] and Bousfield [@Bousfield]: $$L_i\Lambda^d(V;n)\simeq L_{i+d}S^d(V;n+1)\;,\quad L_i\Gamma^d(V;n)\simeq L_{i+d}\Lambda^d(V;n+1)$$ (These isomorphisms correspond to the cases $\lambda=(d)$ and $\lambda=(1,\dots,1)$). It also generalizes a result of Bott [@Bott], who computed derived functors of Schur functors in characteristic zero. Indeed, in characteristic zero $W_\lambda\simeq S_{\lambda}$ so proposition \[prop-decalage\] yields the following result. If ${\Bbbk}$ is a field of characteristic zero, the derived functors of a Schur functor indexed by a partition $\lambda$ of weight $d$ are given by: $$L_(S_\lambda;n)\simeq S_{\lambda'}[-nd]\text{ if $n$ is odd, and }L_(S_\lambda;n)\simeq S_{\lambda}[-nd]\text{ if $n$ is even.}$$ Plethysms --------- The study of plethysms, i.e. representations given by composites of functors, is a hard problem of representation theory (even over a field of characteristic zero, see [@MacDonald Chap I]). For example, the composition series of the functor $F\circ G$ is usually unknown, even when the functors $F$ and $G$ are well understood. In this section, we our simplicial model for iterated Ringel duality, namely: $$\Theta^n C [-dn]\simeq {\overline{{\mathcal{C}}}}C_{K(n)}$$ to give explicit formulas for Ringel duals of plethysms. ### Composition by Frobenius twists. Let ${\Bbbk}$ be a field of positive characteristic $p$. The $r$-th Frobenius twist functor $I^{(r)}$ is the subfunctor of $S^{p^r}$ generated by $p^r$-th powers: $$I^{(r)}(V):=V^{(r)}=\langle v^{p^r}\;\|\; v\in V\rangle \subset S^{p^r}(V)\;.$$ Plethysms of the form $F\circ I^{(r)}$ and $I^{(r)}\circ F$ are of central importance for the representation theory over finite fields, see e.g. [@CPSVdK; @FS; @FFSS; @TVdK]. The following proposition was first proved in [@Chalupnik2], relying on the computations of [@FFSS] and [@Chalupnik1]. With our simplicial model, we can give an elementary proof. \[prop-chal\] Let ${\Bbbk}$ be a field of positive characteristic $p$. There are isomorphisms, natural with respect to $C\in{\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}})$: $$\begin{aligned} \text{(i)}&& \Theta^n(C\circ I^{(r)}) \simeq \left( \Theta^n(C)\circ I^{(r)} \right) [dn(p^r-1)]\;, \\ \text{(ii)}&& \Theta^n(I^{(r)}\circ C) \simeq \left( I^{(r)}\circ\Theta^n(C) \right) [dn(p^r-1)]\;.\end{aligned}$$ We shall prove (i), the proof of the second statement is similar. Since $I^{(r)}(V\otimes W)\simeq I^{(r)}(V)\otimes I^{(r)}(W)$ we have isomorphisms of mixed complexes: $$(C\circ I^{(r)})_{K(n)}\simeq (C_{K(n)^{(r)}})\circ I^{(r)}\;.$$ Now $I^{(r)}({\Bbbk})\simeq {\Bbbk}$ and $I^{(r)}$ is additive, hence exact as a functor from ${\Bbbk}$-vector spaces to ${\Bbbk}$-vector spaces, so the complex of ${\Bbbk}$-vector spaces ${\mathcal{C}}K(n)^{(r)}$ has homology concentrated in homological degree $n$ and equal to ${\Bbbk}$ in this degree. So we have a quasi-isomorphism of complexes of ${\Bbbk}$-vector spaces ${\mathcal{N}}K(n)^{(r)}\simeq {\Bbbk}[-n]$, hence a homotopy equivalence between these complexes of ${\Bbbk}$-vector spaces. Now the Dold-Kan correspondence ensures that we have a homotopy equivalence $K(n)^{(r)}\simeq K(n)$ of simplicial ${\Bbbk}$-vector spaces. Thus the complexes of strict polynomial functors ${\overline{{\mathcal{C}}}}(C_{K(n)^{(r)}})$ and ${\overline{{\mathcal{C}}}}(C_{K(n)})$ are homotopy equivalent. Hence: $$\Theta^n(C\circ I^{(r)})[-np^rd]\simeq \left({\overline{{\mathcal{C}}}}(C_{K(n)})\right)\circ I^{(r)})\simeq \left( \Theta^n(C)\circ I^{(r)} \right)[-nd]\;.$$ This finishes the proof of proposition \[prop-chal\]. ### Plethysm under a vanishing condition Now we go back to the case of an arbitrary PID ${\Bbbk}$. The following theorem provides an efficient way to compute Ringel duals of many plethysms. \[thm-pleth\]Let ${\Bbbk}$ be a PID. Let $G\in {\mathcal{P}}_{d,{\Bbbk}}$ and let $n$ be a positive integer such that $G$ satisfies the following vanishing condition $$H^i(\Theta^n G)=0 \quad\text{for all $i>0$}\;.$$ Then for all $F\in {\mathcal{P}}_{{\Bbbk}}$, there is an isomorphism (in the derived category), natural with respect to $F$ and $G$: $$\Theta^{n}(F\circ G)\simeq \Theta^{nd}(F)\circ H^0(\Theta^n G) \;.$$ Before we prove theorem \[thm-pleth\], we examine the statement and give a few consequences. A lot of functors satisfy the vanishing condition appearing in theorem \[thm-pleth\]. Let us give a list of examples. 1. If $n=1$, the functors satisfying the vanishing condition are the ${\mathbb{H}}(\Lambda^d,-)$-acyclic functors. Hence by lemma \[lm-acyclic\], they include symmetric powers, exterior powers, and more generally Schur functors. Sums, tensor products, or direct summands of functors satisfying the vanishing condition also satisfy it, as well as filtered functors whose graded pieces satisfy the vanishing condition[^4]. For example, plethysms of the form $S^k\circ S^2$ and $S^k\circ \Lambda^2$ have a filtration whose graded pieces are Schur functors by a result of Boffi [@Boffi], hence they satisfy the vanishing condition. 2. If $n=2$, the functors satisfying the vanishing condition are the ${\mathbb{H}}(S^d,-)$-acyclic functors. Hence they include tensor products of symmetric powers and more generally injective functors. 3. Tensor powers satisfy the vanishing condition for all $n$ (indeed, $\Theta^n (\otimes^d)\simeq\otimes^d$ by example \[exemple\]). If the ground ring ${\Bbbk}$ is a field of positive characteristic, functors of degree strictly less than the characteristic also satisfy the vanishing condition since they are direct summands of tensor powers. If $n$ equals $1$ or $2$, the homology of $\Theta^n(F\circ G)$ may be interpreted as extension groups. So we deduce from theorem \[thm-pleth\] the following ${{\mathrm{Ext}}}$-computations involving plethysms. As usual, we denote by ${\mathbb{E}}^i(E,F\circ G)$ the functor assigning to $V$ the extension groups ${{\mathrm{Ext}}}^i_{{\mathcal{P}}_{\Bbbk}}(E^V, F\circ G)$. In particular, ${\mathbb{E}}^i(E,F\circ G)({\Bbbk})$ equals ${{\mathrm{Ext}}}^i_{{\mathcal{P}}_{\Bbbk}}(E, F\circ G)$. \[cor-Exthigher\] Let ${\Bbbk}$ be a PID, and let $F\in {\mathcal{P}}_{e,{\Bbbk}}$. The following isomorphisms hold. $$\begin{aligned} &{\mathbb{E}}^(S^{de}, F\circ \otimes^d)\simeq H^(\Theta^{2d} F)\circ \otimes^d \\ &{\mathbb{E}}^(S^{de}, F\circ S^d)\simeq H^(\Theta^{2d} F)\circ \Gamma^d\\ &{\mathbb{E}}^(\Lambda^{de}, F\circ \otimes^d)\simeq H^(\Theta^d F)\circ \otimes^d \\ &{\mathbb{E}}^(\Lambda^{de}, F\circ S_\lambda)\simeq H^(\Theta^d F)\circ W_{\lambda'}\label{eqn}\end{aligned}$$ In particular, if $\lambda$ is not the partition $(1,\dots,1)$, that is if $S_\lambda$ is not an exterior power, then $W_{\lambda'}({\Bbbk})=0$ so that $${{\mathrm{Ext}}}^_{{\mathcal{P}}_{\Bbbk}}(\Lambda^{de}, F\circ S_\lambda)=0\;.$$ And if $S_\lambda=\Lambda^e$ is an exterior power, then $W_{\lambda'}({\Bbbk})=\Gamma^e({\Bbbk})={\Bbbk}$ so that $${{\mathrm{Ext}}}^_{{\mathcal{P}}_{\Bbbk}}(\Lambda^{de}, F\circ \Lambda^d)=H^(\Theta^d F)({\Bbbk})\;.$$ Corollary \[cor-Exthigher\] shows that iterated Ringel duals $\Theta^d(F)$ for arbitrary large $d$ also have an interpretation in terms of extension groups in the category of strict polynomial functors. We finish by an application of the formulas of corollary \[cor-Exthigher\]. In [@Boffi], Boffi proves that the plethysm $S^k\circ S^2$ and $S^k\circ \Lambda^2$ have a universal filtration, that is a filtration defined over the ground ring ${\mathbb{Z}}$, whose graded pieces are Schur functors. Then he notices that ‘there is little hope’ that plethysms of the form $S^k\circ S^d$ or $S^k\circ \Lambda^d$, for $d>2$ have similar universal filtrations and proves that this indeed fails for $d=3$. With the formulas of corollary \[cor-Exthigher\], we are able to prove the general non-existence result. Let ${\Bbbk}={\mathbb{Z}}$ be the ground ring. For $d>2$ and for $k\ge 2$, the plethysms $S^k\circ S^d$ and $S^k\circ \Lambda^d$ have no filtration whose graded pieces are Schur functors. We prove the case of $S^k\circ S^d$, the other case is similar. If $S^k\circ S^d$ had a filtration whose graded pieces are Schur functors, then Schur functors are ${\mathbb{H}}(\Lambda^{kd},-)$ acyclic, so we would have for all positive $i$: $${\mathbb{E}}^i_{{\mathcal{P}}_{\mathbb{Z}}}(\Lambda^{kd}, S^k\circ S^d)=0\;.$$ Moreover, we know that $${\mathbb{E}}^0_{{\mathcal{P}}_{\mathbb{Z}}}(\Lambda^{kd}, S^k\circ S^d)={\mathbb{H}}(\Lambda^{kd}, S^k\circ S^d)$$ has ${\mathbb{Z}}$-free values. In particular, ${\mathbb{E}}^_{{\mathcal{P}}_{\mathbb{Z}}}(\Lambda^{kd}, S^k\circ S^d)$ has values in free graded ${\mathbb{Z}}$-modules (concentrated in degree zero). By corollary \[cor-Exthigher\](\[eqn\]) this would mean that $H^(\Theta^{d} S^k)\circ \Lambda^d$ has values in ${\mathbb{Z}}$-free graded modules (concentrated in degree zero). In particular $L_{}S^k({\mathbb{Z}};d)=L_S^k(\Lambda^d({\mathbb{Z}}^d);d)$ would be a graded free ${\mathbb{Z}}$-module (concentrated in degree $dk$). But the values of $L_{}S^k({\mathbb{Z}};d)$ are well-known [@TouzeEML], they are a well-identified direct summand (the direct summand of weight $k$) of the homology of the Eilenberg-Mac Lane space $K({\mathbb{Z}},d)$. In particular, if $d>2$ they have non trivial $p$-torsion when $p$ is a prime dividing $k$. This contradiction proves the result. ### Proof of theorem \[thm-pleth\] Let $X,Y\in s({\mathcal{P}}_{\Bbbk})$ be simplicial strict polynomial functors. We say that a morphism $f:X\to Y$ is a weak equivalence if ${\mathcal{N}}f:{\mathcal{N}}X\to {\mathcal{N}}Y$ is a quasi-isomorphism of chain complexes (or equivalently if ${\mathcal{C}}F:{\mathcal{C}}X\to {\mathcal{C}}Y$ is a quasi-isomorphism). The proof of theorem \[thm-pleth\] uses the following property of weak equivalences. \[lm-we\] Let $f\in {{\mathrm{Hom}}}_{s({\mathcal{P}}_{\Bbbk})}(X,Y)$ be a weak equivalence. Then for all $F\in {\mathcal{P}}_{\Bbbk}$, the morphism $F(f)\in {{\mathrm{Hom}}}_{s({\mathcal{P}}_{\Bbbk})}(F\circ X,F\circ Y)$ is also a weak equivalence. As proved in lemma \[lm-caracqis\](i), $f$ is a weak equivalence if and only if for all $V\in{\mathcal{V}}_{\Bbbk}$, the morphism of complexes of ${\Bbbk}$-modules ${\mathcal{N}}f_V:{\mathcal{N}}X(V)\to {\mathcal{N}}Y(V)$ is a quasi-isomorphism. By definition, functors in ${\mathcal{P}}_{\Bbbk}$ have ${\Bbbk}$-projective values, hence ${\mathcal{N}}X(V)$ and ${\mathcal{N}}Y(V)$ are complexes of projective ${\Bbbk}$-modules. Hence ${\mathcal{N}}f_V$ is a homotopy equivalence in ${\mathrm{Ch}}_{\ge 0}({\Bbbk}\text{-mod})$ (although ${\mathcal{N}}f$ is not a homotopy equivalence in ${\mathrm{Ch}}_{\ge 0}({\mathcal{P}}_{\Bbbk})$ in general). Using the Dold-Kan correspondence $s({\Bbbk}\text{-mod})\leftrightarrows {\mathrm{Ch}}_{\ge 0}({\Bbbk}\text{-mod})$, we conclude that $f_V:X_V\to Y_V$ is a homotopy equivalence of simplicial ${\Bbbk}$-modules (we use here the simple observation that the normalized chain functors ${\mathcal{N}}$, and their inverses ${\mathcal{N}}^{-1}={\mathcal{K}}$, commute with evaluation functors). Hence $F(f_V)=F(f)_V$ is a homotopy equivalence of ${\Bbbk}$-modules. In particular, ${\mathcal{N}}F(f)_V$ is a homotopy equivalence in ${\mathrm{Ch}}_{\ge 0}({\Bbbk}\text{-mod})$, so that ${\mathcal{N}}F(f)$ is a quasi-isomorphism. First, by lemma \[lm-formal\] and theorem \[thm-main\], the vanishing hypothesis yields an isomorphism in ${\mathbf{D}}^b({\mathcal{P}}_{d,{\Bbbk}})$: $$H^0(\Theta^n G) [-nd]\simeq {\mathcal{C}}G_{K(n)}\simeq {\mathcal{N}}G_{K(n)}\;.$$ Such an isomorphism may be represented by a zig-zag in ${\mathrm{Ch}}_{\ge 0}({\mathcal{P}}_{d,{\Bbbk}})$: $$H^0(\Theta^n G) [-nd]\leftarrow P \rightarrow {\mathcal{N}}G_{K(n)} \;,$$ in which the two arrows are quasi-isomorphisms, and $P$ is a complex of projective objects in ${\mathcal{P}}_{d,{\Bbbk}}$. We can take the complex $P$ in the middle as a complex concentrated in positive homological degrees because if $Q$ is a bounded below complex of projectives, with $H_i(Q)=0$ with $i<0$, then we can replace it by its truncation $Q'$ (with $Q'_i=0$ if $i<0$ and $Q'_0=\ker (Q_0\to Q_{-1})$), which is also a complex of projectives. Applying the Dold-Kan functor ${\mathcal{K}}={\mathcal{N}}^{-1}$ we get a zigzag of weak equivalences: $${\mathcal{K}}(H^0(\Theta^n G) [-nd])\leftarrow {\mathcal{K}}(P) \rightarrow {\mathcal{K}}{\mathcal{N}}G_{K(n)}\simeq G_{K(n)}\;.$$ Moreover, the explicit formula for ${\mathcal{K}}$ [@Weibel section 8.4] shows that $${\mathcal{K}}(H^0(\Theta^n G) [-nd])\simeq {\mathcal{K}}({\Bbbk}[-nd])\otimes H^0(\Theta^n G)=K(nd)\otimes H^0(\Theta^n G)\;.$$ Hence, by lemma \[lm-we\], applying $F$ to our zigzag of weak equivalences yields a zigzag of weak equivalences: $$F_{K(nd)}\circ H^0(\Theta^n G)\leftarrow F\circ {\mathcal{K}}(P)\rightarrow F\circ (G_{K(n)})= (F\circ G)_{K(n)}\;.$$ Applying the chain functor, we thus get an isomorphism in ${\mathbf{D}}^b({\mathcal{P}}_{de,{\Bbbk}})$: $$(\Theta^{nd}F)\circ H^0(\Theta^n G) [-nd]\simeq \Theta^n(F\circ G) [-nd]\;.$$ This finishes the proof of theorem \[thm-pleth\]. Block theory and vanishing -------------------------- ### Brief recollections of block theory In this section we recall basics of block theory for ${\mathcal{P}}_{d,{\Bbbk}}$. We refer the reader to [@Martin Chap 5] for further details. Let ${\Bbbk}$ be a field of positive characteristic $p$. The simple objects of ${\mathcal{P}}_{d,{\Bbbk}}$ are classified by partitions of weight $d$. To be more specific, the Schur functors $S_{\lambda}$ have a simple socle, denoted by $L_\lambda$. The $L_\lambda$, for $\lambda$ a partition of weight $d$, are exactly the simple objects of ${\mathcal{P}}_{d,{\Bbbk}}$ [@Martin Thm 3.4.2]. The blocks of ${\mathcal{P}}_{d,{\Bbbk}}$ are the equivalent classes of the set of simple objects of ${\mathcal{P}}_{d,{\Bbbk}}$ under the following equivalence relation. To simple objects $L_\lambda,L_\mu$ are equivalent if there is a sequence of simple functors $$L_\lambda=L_1,L_2,\dots,L_r=L_\mu$$ such that for all $i$, ${{\mathrm{Ext}}}^1_{{\mathcal{P}}_{d,{\Bbbk}}}(L_i,L_{i+1})\ne 0$. We let ${\mathcal{B}}_{d,{\Bbbk}}$ be the blocks of ${\mathcal{P}}_{d,{\Bbbk}}$. For all functors $F$, and for all blocks $b\in {\mathcal{B}}_{d,{\Bbbk}}$, we denote by $F_b$ the sum of all subfunctors of $F$ whose composition factors belong to $b$. It is not hard to prove that $F=\oplus_{b\in{\mathcal{B}}_{d,{\Bbbk}}} F_b$. Moreover, if $F,G\in{\mathcal{P}}_{d,{\Bbbk}}$ and $b\ne b'$ are two blocks, then ${{\mathrm{Hom}}}_{{\mathcal{P}}_{d,{\Bbbk}}}(F_b,G_{b'})=0$. So if we denote by $({\mathcal{P}}_{d,{\Bbbk}})_b$ the full subcategory of ${\mathcal{P}}_{d,{\Bbbk}}$ whose objects are the $F_b$, for $F\in{\mathcal{P}}_{d,{\Bbbk}}$, the abelian category ${\mathcal{P}}_{d,{\Bbbk}}$ splits as a direct sum: $$\textstyle{\mathcal{P}}_{d,{\Bbbk}} =\bigoplus_{b\in{\mathcal{B}}_{d,{\Bbbk}}}({\mathcal{P}}_{d,{\Bbbk}})_b\;.$$ In particular, if no composition factors of $F$ and $G$ belong to the same block, then ${{\mathrm{Ext}}}^_{{\mathcal{P}}_{d,{\Bbbk}}}(F,G)=0$. The blocks of ${\mathcal{P}}_{d,{\Bbbk}}$ were determined by Donkin in [@DonkinHDim]. Two simple functors $L_\lambda,L_\mu$ lie in the same block if and only if the partitions $\lambda$ and $\mu$ have the same $p$-core (this is known as ‘Nakayama rule’). See [@JamesKerber 6.2] for the definition of a $p$-core and [@Martin Thm 5.3.1] for another combinatorial formulation of the statement. ### Base change Let ${\Bbbk}$ be a field. By [@SFB], there is an exact base change functor for strict polynomial functors $${\mathcal{P}}_{d,{\mathbb{Z}}}\to {\mathcal{P}}_{d,{\Bbbk}}\;,\quad F\mapsto F_{{\Bbbk}}\;.$$ The base change functor formalizes the fact that the integral equations defining $F$ may be interpreted in ${\Bbbk}$ (through the ring morphism ${\mathbb{Z}}\to{\Bbbk}$) to define an element of ${\mathcal{P}}_{d,{\Bbbk}}$. For example, the base change functor sends symmetric (resp. exterior, resp. divided, resp. tensor) powers of ${\mathcal{P}}_{d,{\mathbb{Z}}}$ to the symmetric (resp. exterior, resp. divided, resp. tensor) powers, viewed as elements of ${\mathcal{P}}_{d,{\Bbbk}}$. In general, the functor $F_{{\Bbbk}}$ is characterized by the following condition. For all free and finitely generated ${\mathbb{Z}}$-modules $V$, $F(V)\otimes {\Bbbk}$ is naturally isomorphic to $F_{{\Bbbk}}(V\otimes {\Bbbk})$. In particular we have isomorphisms of complexes of ${\Bbbk}$-modules $$LF(V;n)\otimes {\Bbbk}\simeq LF_{\Bbbk}(V\otimes{\Bbbk};n)\;.$$ ### Vanishing results Now we indicate how to use block theory to get vanishing results for derived functors. We work here over the base ring ${\Bbbk}={\mathbb{Z}}$. Let $F\in{\mathcal{P}}_{d,{\mathbb{Z}}}$. By base change and universal coefficient theorem, we have injective morphisms for all $i$: $$L_iF({\mathbb{Z}}^m;n)\otimes \mathbb{F}_p\hookrightarrow L_iF_{\mathbb{F}_p}(\mathbb{F}_p^m;n)\;.$$ So vanishing of $L_iF({\mathbb{Z}}^m;n)\otimes \mathbb{F}_p$ follows from vanishing of $L_iF_{\mathbb{F}_p}(\mathbb{F}_p^m;n)$. Now our main theorem gives isomorphisms: $$L_iF_{\mathbb{F}_p}(\mathbb{F}_p^m;1)\simeq {{\mathrm{Ext}}}^{d-i}_{{\mathcal{P}}_{d,\mathbb{F}_p}}(\Lambda^{d,\mathbb{F}_p^m}, F_{\mathbb{F}_p})\;, \; L_iF_{\mathbb{F}_p}(\mathbb{F}_p^m;2)\simeq {{\mathrm{Ext}}}^{d-i}_{{\mathcal{P}}_{d,\mathbb{F}_p}}(S^{d,\mathbb{F}_p^m}, F_{\mathbb{F}_p})\;.$$ Thus, if no composition factor of $F_{\mathbb{F}_p}$ and $\Lambda^{d,\mathbb{F}_p^m}$ (or $S^{d,\mathbb{F}_p^m}$) belongs to the same block, then the ${{\mathrm{Ext}}}$-groups on the right hand side, hence the corresponding derived functors, must vanish. As an example of this approach, we prove vanishing results for some derived functors of the Schur functor $S_{(d-1,1)}$ (which is more concretely described as the kernel of the multiplication $S^{d-1}\otimes S^1\to S^d$). To put our result in context, we first recall some easily obtained information on torsion of the derived functors of $S_{(d-1,1)}$. \[lm-tors\] Let $d\ge 3$ and let $n$ be a positive integer. For all $i$ and all $m$, the ${\mathbb{Z}}$-module $L_iS_{(d-1,1)}({\mathbb{Z}}^m;n)$ may have $r$-torsion elements only for $r$ dividing $d!(d-1)!$. We know the derived functors of tensor products: $L_i\otimes^d({\mathbb{Z}}^m;n)$ equals zero if $i\ne nd$ and $({\mathbb{Z}}^m)^{\otimes d}$ if $i=nd$ (by a direct computation, or use example \[exemple\] and theorem \[thm-main\]). Multiplication by $d!$, as a mormphism from $S^d$ to $S^d$ factors as the composite of the comultiplication $S^d\to \otimes^d$ and of the multiplication $\otimes^d\to S^d$. So, multiplication by $d!$ anihilates the torsion part of $L_iS^d({\mathbb{Z}}^m;n)$ for all $i$, $n$, $d$, $m$. Similarly, multiplication by $(d-1)!$ anihilates the torsion part of $L_i(S^{d-1}\otimes S^1)({\mathbb{Z}}^m;n)$. To prove lemma \[lm-tors\], it suffices to prove that multiplication by $(d-1)!d!$ anihilates the torsion part of $L_iS_{(d-1,1)}({\mathbb{Z}}^m;n)$. This results from the information on the torsion of $S^d$ and $S^{d-1}\otimes S^1$ and the long exact sequence of derived functors $$\dots\to L_{i+1}S^d({\mathbb{Z}}^m;n)\xrightarrow[]{\partial} L_iS_{(d-1,1)}({\mathbb{Z}}^m;n)\to L_i(S^{d-1}\otimes S^1)({\mathbb{Z}}^m;n)\to \dots$$ arising from the exact sequence $0\to S_{(d-1,1)}\to S^{d-1}\otimes S^1\to S^d\to 0$. We also recall that we can compute $L_iS_{(d-1,1)}({\mathbb{Z}}^m;1)$ by décalage. It equals zero for $i\ne d$, and $W_{(2,1^{d-2})}({\mathbb{Z}}^m)$ for $i=d$. So we are only interested in computing $L_iS_{(d-1,1)}({\mathbb{Z}}^m;n)$ for $n\ge 2$. We now give our vanishing result. \[prop-vanish\] Let $d\ge 3$ and let $p$ be a prime. If $d\ne 0\text{ mod }p$, then $$L_S_{(d-1,1)}({\mathbb{Z}};2)\otimes\mathbb{F}_p =0 .$$ If $d\ne 0\text{ mod }p$ and if $d\ne 2\text{ mod }p$, then $$L_S_{(d-1,1)}({\mathbb{Z}};3)\otimes\mathbb{F}_p =0 .$$ First, using décalage isomorphisms, we know that $L_S_{(d-1,1)}({\mathbb{Z}};i+1)$ is isomorphic to $L_{-d}W_{(2,1^{d-2})}({\mathbb{Z}},i)$. Thus we have to prove the vanishing of $L_W_{(2,1^{d-2})}({\mathbb{Z}},i)\otimes\mathbb{F}_p$, for $i=1$ or $2$. To do this, it suffices to prove the vanishing of $L_W_{(2,1^{d-2}), \mathbb{F}_p}(\mathbb{F}_p,i)$. So, in the remainder of the proof, we work over the ground field $\mathbb{F}_p$. Since the socle of a Schur functor is simple, all its composition factors belong to the same block. Taking duality, we obtain that all the composition factors of a Weyl functor belong to the same block. Moreover, simple functors are self-dual [@Martin Thm 3.4.9], so the head (= cosocle = largest semi-simple quotient) of $W_{(2,1^{d-2})}$ is $L_{(2,1^{d-2})}$. So, the composition factors of $W_{(2,1^{d-2})}$ all belong to the same block as the simple functor $L_{(2,1^{d-2})}$. Thus, to prove proposition \[prop-vanish\], it suffices to prove that the partitions $(2,1^{d-2})$ and $(1^d)$ do not correspond to the same block if $d\ne 0\text{ mod }p$, and that the partitions $(2,1^{d-2})$ and $(d)$ do not correspond to the same block if $d\ne 0\text{ mod }p$ and $d\ne 2\text{ mod }p$. Now the result follows from an easy application of Nakayama rule. Indeed, if $d= qp+r$ with $0\le r<p$, the $p$-core of $(1^d)$ is $(1^r)$ and the $p$-core of $(d)$ is $(r)$, while the $p$-core of $(2,1^{d-2})$ equals $(2,1^{r-2})$ if $r\ge 2$, $(2,1^{p-1})$ if $r=1$, and $(0)$ if $r=0$. If we want to generalize proposition \[prop-vanish\] to get vanishing results for $L_{}F({\mathbb{Z}}^m,i)$ for $m>1$, we need to determine the blocks which appear through the composition factors of $\Lambda^{d,\mathbb{F}_p}$ and $S^{d,\mathbb{F}_p}$. This can be easily done using Pieri rules [@AB1 Thm (3) p.168]. It is interesting to compare proposition \[prop-vanish\] with the discussion at the end of section 8.4 of [@BM], where some computations of the derived functors of $S_{(d-1,1)}$ are discussed (the Schur functor $S_{(d-1,1)}$ is denoted by $J^d$ in this reference). In particular, it seems that the vanishing of $L_S_{(d-1,1)}({\mathbb{Z}};3)\otimes\mathbb{F}_p$ we have obtained is optimal: there is $3$ and $5$ torsion in $L_S_{(4,1)}({\mathbb{Z}};3)$, and there is $11$ and $13$ torsion in $L_S_{(12,1)}({\mathbb{Z}};3)$. An example of computation ------------------------- We finish by an example of elementary computation of derived functors involving Ringel duality, namely we compute $L_ S^p(V;n)$ over a field of positive characteristic $p$, and we retrieve from it a computation of [@BM]. \[prop-calcSp\] Let ${\Bbbk}$ be a field of odd characteristic $p$ and let $n$ be a positive integer. The derived functors of $S^p$ satisfy - $L_{np} S^p(V;n)$ equals $\Lambda^p$ if $n$ is odd, and $\Gamma^p$ if $n$ is even. If $n=1$ or $n=2$ the derived functors are trivial in the other degrees. If $n\ge 3$, the only nontrivial summands of $L_S^p(V;n)$ are: - if $n$ is odd, and $1\le i\le (n-1)/2$, the summands $$L_{np-i(2p-2)+p}S^p(V;n)=L_{np-i(2p-2)+p-1}S^p(V;n)=V^{(1)}\;,$$ - if $n$ is even and $1\le i\le n/2-1$, the summands $$L_{np-i(2p-2)+1}S^p(V;n)=L_{np-i(2p-2)}S^p(V;n)=V^{(1)}\;.$$ We already know that $\Theta S^p=\Lambda^p$ and $\Theta \Lambda^p =\Gamma^p$. So we have to compute the iterated Ringel duals of $\Gamma^p$. Recall the exact Koszul complex (see e.g. [@FS Section 4]): $$\Lambda^p\to \Lambda^{p-1}\otimes S^1\to \Lambda^{p-2}\otimes S^2\to \cdots\to \Lambda^1\otimes S^{p-1}\to S^p\;.\quad ()$$ Its differentials are defined by using first the comultiplication $\Lambda^i\to \Lambda^{i-1}\otimes \Lambda^1$ (tensored by the identity of $S^{p-i}$) and then the multiplication $\Lambda^1\otimes S^{p-i}\to S^{p-i+1}$ (tensored by the identity of $\Lambda^{i-1}$). Taking duality, we obtain another exact complex, the dual Koszul complex $$\Gamma^p\to \Gamma^{p-1}\otimes \Lambda^1 \to \cdots\to \Gamma^1\otimes \Lambda^{p-1}\to \Lambda^p.$$ Splicing these two complexes together, we get a coresolution of $\Gamma^p$: $$\Gamma^{p-1}\otimes \Lambda^1 \to \cdots\to \Gamma^1\otimes \Lambda^{p-1}\xrightarrow[]{\partial} \Lambda^{p-1}\otimes S^1\to \cdots\to S^p\;.$$ This is actually an injective coresolution of $\Gamma^p$ ($S^p$ is injective, and all the other objects are direct summands of $\otimes^p$). Let us denote by $T$ the truncation of the Koszul complex $()$ obtained by removing $\Lambda^p$ and $S^p$. Then the injective coresolution of $\Gamma^p$ can be informally written as $ T^\sharp \xrightarrow[]{\partial} T \to S^p$. The Ringel dual of $\Gamma^p$ is the complex $ {\mathbb{H}}(\Lambda^p,T^\sharp \xrightarrow[]{\partial} T \to S^p)$. We can compute it explicitly. Indeed, ${\mathbb{H}}(\Lambda^p,\Lambda^i\otimes S^{p-i})=\Gamma^i\otimes \Lambda^{p-i}$ and ${\mathbb{H}}(\Lambda^p,-)$ sends the comultiplication $\Lambda^i\to \Lambda^{i-1}\otimes \Lambda^1$ to $\Gamma^i\to \Gamma^{i-1}\otimes \Gamma^1$ and the multiplication $\Lambda^1\otimes S^{p-i}\to S^{p-i+1}$ to $\Gamma^1\otimes \Lambda^{p-i}\to \Lambda^{p-i+1}$. Thus we obtain: $${\mathbb{H}}(\Lambda^p,T)=T^\sharp.$$ For $i<p$, $S^i$ is canonically isomorphic to $\Gamma^i$. The comultiplication $S^i\to S^{i-1}\otimes S^1$ identifies through this isomorphism with $\Gamma^i\to \Gamma^{i-1}\otimes \Gamma^1$, and the multiplication identifies $S^{i-1}\otimes S^1\to S^i$ identifies with $\Gamma^{i-1}\otimes \Gamma^1\to \Gamma^i$. So we also have: $${\mathbb{H}}(\Lambda^p,T^\sharp)=T.$$ Finally, ${\mathbb{H}}(\Lambda^p,-)$ sends the map $\partial$ onto the composite $$d:\Lambda^1\otimes S^{p-1}\to S^p\to \Gamma^p \to \Gamma^{p-1}\otimes \Lambda^1.$$ (to see this embed $\Gamma^1\otimes \Lambda^{p-1}$ and $\Lambda^{p-1}\otimes S^1$ in $\otimes^p$, lift the map $\partial$, apply ${\mathbb{H}}(\Lambda^p,-)$ to the lift and use left exactness of ${\mathbb{H}}(\Lambda^p,-)$). So we conclude that the Ringel dual of $\Gamma^p$ is given by $$\Theta\Gamma^p\simeq \left(T\xrightarrow[]{d} T^{\sharp}\to \Lambda^p\right)\simeq \left(T\xrightarrow[]{d} T^{\sharp}\xrightarrow[]{\partial} T\to S^p\right)\;.$$ To compute $\Theta^2\Gamma^p)$ we apply ${\mathbb{H}}(\Lambda^p,-)$ to the complex of injectives from the right hand side. The functor ${\mathbb{H}}(\Lambda^p,-)$ sends $d$ to $\partial$, so that $$\Theta^2 \Gamma^p\simeq\left(T^\sharp\xrightarrow[]{\partial} T\xrightarrow[]{d} T^{\sharp}\to \Lambda^p\right)\simeq\left(T^\sharp\xrightarrow[]{\partial} T\xrightarrow[]{d} T^{\sharp}\to T\to S^p\right)\;.$$ By induction we prove that: $$\Theta^n\Gamma^p\simeq \left\{\begin{array}{l} T^\sharp\xrightarrow[]{\partial} (T\xrightarrow[]{d} T^{\sharp})^{n/2}\to \Lambda^p\quad \text{if $n$ is even,}\\ (T\xrightarrow[]{d} T^{\sharp})^{(n+1)/2}\to \Lambda^p \quad\text{if $n$ is odd.} \end{array}\right.$$ Now we compute the homology of these complexes. If $n$ is odd, $H^0(\Theta^n\Gamma^p)= \Lambda^p$ since it is the kernel of the map $\Lambda^{p-1}\otimes S^1\to \Lambda^{p-2}\otimes S^2$ appearing in the Koszul complex $()$. If $n$ is even $H^0(\Theta^n\Gamma^p)= \Gamma^p$ since it is the kernel of the map $\Gamma^{p-1}\otimes \Lambda^1\to \Gamma^{p-2}\otimes \Lambda^2$ appearing in the dual Koszul complex. This proves assertion (1). To prove assertion (2) and (3) we observe that the homology of the complex of the form: $$\dots\xrightarrow[]{\partial}T\xrightarrow[]{d}T^{\sharp}\xrightarrow[]{\partial} T \xrightarrow[]{d}T^{\sharp}\xrightarrow[]{\partial}\dots$$ is zero everywhere (as splices of exact complexes) except for the homology groups located at the source and the target of the maps $d$. Indeed the map $S^p\to \Gamma^p$ has kernel $I^{(1)}$ and cokernel $I^{(1)}$ so we obtain that the homology groups at the source or the target of these maps equal $I^{(1)}$. Now let us work over the integers. For all free ${\mathbb{Z}}$-modules $V$, the composite $S^d(V)\hookrightarrow V^{\otimes d}\twoheadrightarrow S^d(V)$ equals multiplication by the scalar $d!$. But $L\otimes^d(V;n)$ is quasi isomorphic to $V^{\otimes d}[-nd]$, so by functoriality $d!$ annihilates the torsion part of $LS^d(V;n)$. Taking $d=p$, we see that the $p$-primary part of $L_iS^p(V;n)$ consists only of $p$-torsion. Hence, using base change and the universal coefficient theorem, we can retrieve from proposition \[prop-calcSp\] the $p$-primary part of $L_iS^p(V;n)$. We obtain the following result. \[cor-calc-LSp\] Let $p$ be an odd prime, and let $V$ be a finitely generated free ${\mathbb{Z}}$-module. For $n\ge 3$, the $p$-primary part of $L_{} S^p(V;n)$ is zero, with the following exceptions. - If $n$ is odd, and $1\le i\le (n-1)/2$, the $p$-primary part of $L_{np-i(2p-2)+p-1}S^p(V;n)$ equals $(V\otimes\mathbb{F}_p)^{(1)}$. - If $n$ is even and $1\le i\le n/2-1$, the $p$-primary part of $L_{np-i(2p-2)}S^p(V;n)$ equals $(V\otimes\mathbb{F}_p)^{(1)}$. In [@BM Thm 8.2], Breen and Mikhailov compute the derived functors of the third Lie functor ${\mathcal{L}}^3$ which is the kernel of the map $S^2\otimes S^1\to S^3$ (so ${\mathcal{L}}^3$ is nothing but the Schur functor $S_{(2,1)}$). Their result give the values of $L{\mathcal{L}}^3(V;n)$ over an arbitrary abelian group $V$. As an illustration of our techniques, we use corollary \[cor-calc-LSp\] to recover their result when $V$ is free and finitely generated. Let $V$ be a free finitely generated ${\mathbb{Z}}$-module. The derived functors $L_i{\mathcal{L}}^3(V;n)$ are trivial in all degrees except the following degrees. - If $n$ is odd, $L_{3n}{\mathcal{L}}^3(V;n)$ is the kernel of the multiplication $\Lambda^2\otimes \Lambda^1\to \Lambda^3$ and for $1\le i\le (n-1)/2$, $L_{3n-4i+1}{\mathcal{L}}^3(V;n)$ equals $(V\otimes\mathbb{F}_3)^{(1)}$. - If $n$ is even, $L_{3n}{\mathcal{L}}^3(V;n)={\mathcal{L}}^3(V)$ and for $1\le i\le n/2$, $L_{3n-4i-1}{\mathcal{L}}^3(V;n)$ equals $(V\otimes\mathbb{F}_3)^{(1)}$. ${\mathcal{L}}^3(V)$ is formed by the elements of $S^2(V)\otimes S^1(V)$ of the form $ab\otimes c - bc\otimes a$. Thus, the composite (where $\tau_2$ is the transposition $(2,3)$) $${\mathcal{L}}^3\hookrightarrow S^2\otimes S^1\to \otimes^3 \xrightarrow[]{1-\tau_{2}}\otimes^3\to S^2\otimes S^1$$ (where $\tau_2$ is the transposition $(2,3)$) surjects on ${\mathcal{L}}^3$, and actually equals multiplication by the scalar $3$. So derived functors of ${\mathcal{L}}^3$ have only $3$-torsion. The short exact sequence ${\mathcal{L}}^3\hookrightarrow S^2\otimes S^1\twoheadrightarrow S^3$ induces a long exact sequence $$\cdots\to L_i(S^2\otimes S^1)(V;n)\to L_i S^3(V;n)\to L_{i-1}{\mathcal{L}}^3(V;n)\to \cdots\;.$$ Since $S^1\otimes S^2$ has no $3$-torsion (the composite $S^2\otimes S^1\to \otimes^3\to S^2\otimes S^1$ is multiplication by $2$), this long exact sequence splits into exact sequences (we use theorem \[thm-main\] to identify derived functors in degree $3n$): $$0\to L_{3n}{\mathcal{L}}^3(V;n)\to H^0\Theta^n (S^2\otimes S^1) \to H^0\Theta^n S^3\to L_{3n-1}{\mathcal{L}}^3(V;n)\to 0\;.$$ $$0\to L_j S^3(V;n)\otimes\mathbb{F}_3\to L_{j-1}{\mathcal{L}}^3(V;n)\to 0\;,\; \text{ for $j<3n$.}$$ We identify the map $H^0\Theta^n (S^2\otimes S^1) \to H^0\Theta^n S^3$ by successive applications of the functor ${\mathbb{H}}(\Lambda^3,-)$ to the multiplication $S^2\otimes S^1\to S^3$. Applying it once, we obtain the multiplication $\Lambda^2\otimes\Lambda^1\to \Lambda^3$. Applying it once again we obtain the multiplication $\Gamma^2\otimes\Gamma^1\to \Gamma^3$, applying it once again we obtain again the multiplication $\Lambda^2\otimes\Lambda^1 \to\Lambda^3$. Thus, if $n$ is odd, $$L_{3n}{\mathcal{L}}^3(V;n)= \ker\{\Lambda^2\otimes\Lambda^1\to \Lambda^3\}\;,\;\text{ and } L_{3n-1}{\mathcal{L}}^3(V;n)=0\;.$$ Now, if $n$ is even, the injective morphism of algebras $S^\to \Gamma^$ yields a commutative diagram: $$\xymatrix{ \Gamma^2\otimes\Gamma^1\ar[rr]^-{\mathrm{mult}}&&\Gamma^3\\ S^2\otimes S^1\ar@{^{(}->}[u]^-{(1)}\ar[rr]^-{\mathrm{mult}}&&S^3\ar@{^{(}->}[u] }.$$ An elementary computation shows the map $(1)$ induces an isomorphism from ${\mathcal{L}}^3$ onto the kernel of $\Gamma^2\otimes\Gamma^1\to \Gamma^3$. Thus, for $n$ even: $$L_{3n}{\mathcal{L}}^3(V;n)= {\mathcal{L}}^3(V)\;,\;\text{ and } L_{3n-1}{\mathcal{L}}^3(V;n)= (V\otimes\mathbb{F}_3)^{(1)}\;.$$ The other short exact sequences give $L_i{\mathcal{L}}^3(V;n)$ in lower degrees. Strict polynomial functors with non free values {#sec-arb} =============================================== In the previous sections, we have dealt with the category ${\mathcal{P}}_{\Bbbk}$ of strict polynomial functors having values in finitely generated free modules over the PID ${\Bbbk}$. This framework is sufficient for the applications of section \[sec-applic\], and it slightly simplifies the proofs since we don’t have to derive tensor products. However, the case of functors with non-free values may be interesting, so in this section we briefly explain how to deal with this apparently more general case. Let ${\Bbbk}$ be a PID. Let us denote by $\widetilde{{\mathcal{P}}}_{d,{\Bbbk}}$ the category of ${\Bbbk}$-linear functors from $\Gamma^d{\mathcal{V}}_{\Bbbk}$ to the abelian category of finitely generated ${\Bbbk}$-modules, and by $\widetilde{{\mathcal{P}}}_{\Bbbk}$ the direct sum of the categories $\widetilde{{\mathcal{P}}}_{d,{\Bbbk}}$. Thus the category ${\mathcal{P}}_{\Bbbk}$ is a full exact subcategory of $\widetilde{{\mathcal{P}}}_{\Bbbk}$. The category $\widetilde{{\mathcal{P}}}_{\Bbbk}$ has enough projectives. A projective generator is provided by the functors $\Gamma^{d,V}$, $d\ge 0, V\in{\mathcal{V}}_{\Bbbk}$. We observe that it is also the projective generator of ${\mathcal{P}}_{\Bbbk}$. The following lemma is a formal consequence of this observation (together with the existence of finite projective resolutions [@DonkinHDim; @AB] in the bounded case). \[lm-equiv\] Let the symbol $$ stand for $-$ or $b$. The functor ${\mathcal{P}}_{\Bbbk}\to \widetilde{{\mathcal{P}}}_{\Bbbk}$ induces equivalences of monoidal triangulated categories: $${\mathbf{D}}^({\mathcal{P}}_{\Bbbk})\simeq {\mathbf{D}}^(\widetilde{{\mathcal{P}}}_{{\Bbbk}})\;.$$ We have a commutative diagram of monoidal triangulated functors, where the two vertical arrows are equivalences of categories, with triangulated monoidal inverses: $$\xymatrix{ {\mathbf{K}}^(\mathrm{Proj}({\mathcal{P}}_{\Bbbk}))\ar[d]^-{\simeq}\ar[r]^-{=}& {\mathbf{K}}^(\mathrm{Proj}(\widetilde{{\mathcal{P}}}_{\Bbbk}))\ar[d]^-{\simeq}\\ {\mathbf{D}}^({\mathcal{P}}_{\Bbbk})\ar[r]&{\mathbf{D}}^(\widetilde{{\mathcal{P}}}_{\Bbbk}) }\;.$$ The definition of the functors $L(-,n)$, $\Sigma$ and $\Theta$ can be generalized without change when working in $\widetilde{{\mathcal{P}}}_{\Bbbk}$. Now lemma \[lm-equiv\] implies that our main theorem \[thm-main\] stays valid with ${\mathcal{P}}_{\Bbbk}$ replaced by $\widetilde{{\mathcal{P}}}_{\Bbbk}$. Appendix: representations of ${\Bbbk}$-linear categories {#app} ======================================================== We fix a commutative ring ${\Bbbk}$, and we denote by ${\mathcal{V}}_{\Bbbk}$ the category of finitely generated projective ${\Bbbk}$-modules and ${\Bbbk}$-linear maps. We consider a category ${\mathcal{C}}$ enriched over ${\mathcal{V}}_{\Bbbk}$ (That is, ${{\mathrm{Hom}}}$s are finitely generated projective ${\Bbbk}$-modules, and composition is bilinear). In this appendix, we describe the relations between the following two categories. - The category ${\mathcal{C}}\text{-mod}$ of ${\Bbbk}$-linear representations of ${\mathcal{C}}$, that is the category of ${\Bbbk}$-linear functors ${\mathcal{C}}\to{\mathcal{V}}_{\Bbbk}$. - If $P\in{\mathcal{C}}$, then ${{\mathrm{End}}}_{\mathcal{C}}(P)$ is an algebra (for the composition), and we denote by ${{\mathrm{End}}}_{\mathcal{C}}(P)\text{-mod}$ the category of ${{\mathrm{End}}}_{\mathcal{C}}(P)$-modules, which are finitely generated and projective as ${\Bbbk}$-modules. Direct sums, products, kernels and cokernels in ${\mathcal{C}}\text{-mod}$ are computed objectwise in the target category. So ${\mathcal{C}}\text{-mod}$ inherits the structure of ${\mathcal{V}}_{\Bbbk}$. So it is an exact category, that is an additive category equipped with a collection of admissible short exact sequences [@Keller; @Buehler]. To be more specific, the admissible short exact sequences are the pairs of morphisms $F\hookrightarrow G\twoheadrightarrow H$ which become short exact sequences of ${\Bbbk}$-modules after evaluation on the objects of ${\mathcal{C}}$. (If ${\Bbbk}$ is a field, ${\mathcal{C}}\text{-mod}$ is even an abelian category). Similarly, the category ${{\mathrm{End}}}_{\mathcal{C}}(P)\text{-mod}$ is an exact category, the admissible short exact sequences are the pairs of morphisms $F\hookrightarrow G\twoheadrightarrow H$ which become short exact sequences of ${\Bbbk}$-modules after forgetting the action of ${{\mathrm{End}}}_{\mathcal{C}}(P)$. The two categories are related via evaluation on $P$. If $F\in{\mathcal{C}}\text{-mod}$, the functoriality of $F$ makes the ${\Bbbk}$-module $F(P)$ into a ${{\mathrm{End}}}_{\mathcal{C}}(P)$-module. Thus we have an evaluation functor: $$\begin{array}{ccc} {\mathcal{C}}\text{-mod}&\to &{{\mathrm{End}}}_{\mathcal{C}}(P)\text{-mod}\\ F&\mapsto & F(P) \end{array}.$$ This functor is additive and exact but in general it does not behave well with projectives. For example, ${{\mathrm{End}}}_{\mathcal{C}}(P)$ is a projective generator of ${{\mathrm{End}}}_{\mathcal{C}}(P)\text{-mod}$, whereas the projective functor $ {\mathcal{C}}^P:X\mapsto{{\mathrm{Hom}}}_{\mathcal{C}}(P;X)$ need not be a projective generator of ${\mathcal{C}}\text{-mod}$. The following theorem gives a condition on ${\mathcal{C}}$ so that the evaluation functor is an equivalence of categories (compare [@Kuhn lemma 3.4], where a similar statement is given in the case when ${\Bbbk}$ is a field and ${\mathcal{C}}$ is a category constructed from a functor with products, which by [@Kuhn example 3.5] encompasses the case of Schur algebras). \[prop-app\] Let ${\mathcal{C}}$ be a category enriched over ${\mathcal{V}}_{\Bbbk}$. Assume that there exists an object $P\in{\mathcal{C}}$ such that for all $X,Y\in{\mathcal{C}}$, the composition induces a surjective map $${{\mathrm{Hom}}}_{\mathcal{C}}(X,P)\otimes {{\mathrm{Hom}}}_{\mathcal{C}}(P,Y)\twoheadrightarrow {{\mathrm{Hom}}}_{\mathcal{C}}(X,Y)\;.$$ Then the following holds. 1. For all $F\in{\mathcal{C}}\text{-mod}$ and all $Y\in{\mathcal{C}}$ the canonical map $F(P)\otimes {{\mathrm{Hom}}}_{\mathcal{C}}(P,Y)\to F(Y)$ is surjective. 2. The functor ${\mathcal{C}}^P:X\mapsto {{\mathrm{Hom}}}_{\mathcal{C}}(P;X)$ is a projective generator of ${\mathcal{C}}\text{-mod}$. 3. Evaluation on $P$ induces an equivalence of categories ${\mathcal{C}}\text{-mod}\simeq {{\mathrm{End}}}_{\mathcal{C}}(P)\text{-mod}$. Let us prove (i). The canonical map is the map $x\otimes f\mapsto F(f)(x)$. Since ${{\mathrm{Hom}}}_{\mathcal{C}}(Y,P)\otimes {{\mathrm{Hom}}}_{\mathcal{C}}(P,Y)\twoheadrightarrow {{\mathrm{Hom}}}_{\mathcal{C}}(Y,Y)$ is surjective, there exists a finite family of maps $\alpha_i\in {{\mathrm{Hom}}}_{\mathcal{C}}(Y,P)$ and $\beta_i\in {{\mathrm{Hom}}}_{\mathcal{C}}(P,Y)$ such that $\sum_i \beta_i\circ \alpha_i={{\mathrm{Id}}}_Y$. For all $y\in F(Y)$ the element $\sum_i F(\alpha_i)(x)\otimes \beta_i\in F(P)\otimes {{\mathrm{Hom}}}_{\mathcal{C}}(P,Y)$ is sent onto $y$ by the canonical map. Let us prove (ii). The Yoneda isomorphism ${{\mathrm{Hom}}}_{{\mathcal{C}}\text{-mod}}({\mathcal{C}}^P,F)\simeq F(P)$ ensures that ${\mathcal{C}}^P$ is projective. Moreover (i) yields an epimorphism $F(P)\otimes {\mathcal{C}}^P\twoheadrightarrow F$. This epimorphism is admissible. Indeed, for all $X\in{\mathcal{C}}$, the kernel $K(X)$ of the map $F(P)\otimes {\mathcal{C}}^P(X)\twoheadrightarrow F(X)$ has values in ${\mathcal{V}}_{\Bbbk}$ (because the sequence of ${\Bbbk}$-modules $K(X)\hookrightarrow F(P)\otimes {\mathcal{C}}^P(X)\twoheadrightarrow F(X)$ is exact and the two other objects of the sequence are finitely generated and projective ${\Bbbk}$-modules). So the canonical map has a kernel in ${\mathcal{C}}\text{-mod}$. To prove (iii), we have to show that the evaluation map is fully faithful and essentially surjective. The evaluation map fits into a commutative triangle (where the horizontal arrow is the Yoneda isomorphism and the diagonal arrow is the evaluation on the unit of ${{\mathrm{End}}}_{\mathcal{C}}(P)$): $$\xymatrix{ {{\mathrm{Hom}}}_{{\mathcal{C}}\text{-mod}}({\mathcal{C}}^P,F)\ar[d]\ar[r]^-{\simeq}& F(P)\\ {{\mathrm{Hom}}}_{{{\mathrm{End}}}_{\mathcal{C}}(P)\text{-mod}}({{\mathrm{End}}}_{\mathcal{C}}(P),F(P))\ar[ru]_-{\simeq} }.$$ Thus it induces an isomorphism. Now, by additivity of ${{\mathrm{Hom}}}$s, we can extend this result to the functors of the form $V\otimes{\mathcal{C}}^P$, with $V\in{\mathcal{V}}_{\Bbbk}$. Finally, by (ii), any $G\in{\mathcal{C}}\text{-mod}$ has a presentation by functors of the form $V\otimes{\mathcal{C}}^P$. So by left exactness of ${{\mathrm{Hom}}}_{{\mathcal{C}}\text{-mod}}(-,F)$ and ${{\mathrm{Hom}}}_{{{\mathrm{End}}}_{\mathcal{C}}(P)\text{-mod}}(-,F(P))$, we obtain that evaluation on $P$ is fully faithful. It remains to prove that evaluation on $P$ is essentially surjective. Let $M$ be an ${{\mathrm{End}}}_{\mathcal{C}}(P)$, we may find a presentation of $M$ of the form: $V_2\otimes {{\mathrm{End}}}_{\mathcal{C}}(P)\xrightarrow[]{\psi} V_1\otimes {{\mathrm{End}}}_{\mathcal{C}}(P)\twoheadrightarrow M$ with $V_1,V_2\in{\mathcal{V}}_{\Bbbk}$. Since evaluation on $P$ is fully faithful, there exists a unique morphism $\phi:V_2\otimes{\mathcal{C}}^P\to V_1\otimes{\mathcal{C}}^P$ which coincides with $\psi$ after evaluation on $P$. We define a functor $F_M:{\mathcal{C}}\to{\Bbbk}\text{-mod}$ by $F_M(X)=\mathrm{coker}\phi_X$ (the cokernel is taken in the category of ${\Bbbk}$-modules). Then $F_M(P)\simeq M$ is finitely generated and projective. Moreover for all $Y\in{\mathcal{C}}$, $F_M(Y)$ is a direct summand of $F_M(P)\otimes{{\mathrm{Hom}}}_C(P,Y)$ (indeed, the canonical map $F_M(P)\otimes{{\mathrm{Hom}}}_C(P,Y)\to F_M(X)$ is a retract of the map $x\mapsto \sum_iF_M(\alpha_i)(x)\otimes\beta_i$). So $F_M$ is actually a functor with finitely generated projective values. To sum up, we have found $F_M\in{\mathcal{C}}\text{-mod}$ whose evaluation on $P$ is isomorphic to $M$. That is, evaluation on $P$ is essentially surjective. The category $\Gamma^d{\mathcal{V}}_{\Bbbk}$ satisfies the hypotheses of proposition \[prop-app\] with $P={\Bbbk}^n$, $n\ge d$, see e.g. [@TouzeClassical lemma 2.3]. Hence we get a direct proof of the equivalence of categories $\Gamma^d{\mathcal{V}}_{\Bbbk}\text{-mod}\simeq S(n,d)\text{-mod}$, without appealing to the fact that $\Gamma^{d}{\mathcal{V}}_{\Bbbk}\text{-mod}$ is isomorphic to the category ${\mathcal{P}}_{d,{\Bbbk}}$ as defined by Friedlander and Suslin [@FS]. 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Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 1994. xiv+450 pp. ISBN: 0-521-43500-5; 0-521-55987-1 [^1]: A first version of this article was entitled ‘Koszul duality and derivatives of non-additive functors’ because the author was not aware of the name ‘Ringel duality’. Since $\Theta$ is not Koszul duality in the sense of [@Priddy; @BGS], the title has been modified to avoid confusion. [^2]: This is not the original definition of Friedlander and Suslin, but it is not hard to show it is equivalent, see e.g. [@FPan p. 41]. [^3]: Alternative proofs of the ${\mathbb{H}}(\Lambda^d,-)$-acyclicity of Schur functors can be provided by the highest weight category structure of ${\mathcal{P}}_{d,{\Bbbk}}$ [@CPS Thm 3.11], or by Kempf vanishing [@Jantzen II, Prop 4.13] combined with [@FS Cor 3.13]. For these alternative proofs, one must identify Schur functors in a slightly different context, see section \[subsubsec-schur\]. [^4]: By the equivalence of categories ${\mathcal{P}}_{d,{\Bbbk}}\simeq S(n,d)\text{-mod}$, a functor $F$ is equivalent to a $S(n,d)$-module which is free of finite rank as a ${\Bbbk}$-module. Hence a filtration of $F$ must be finite. To prove the vanishing condition for $F$ from the vanishing condition for $\mathrm{Gr}(F)$, use the long exact sequences induced by short exact sequences at the level of ${{\mathrm{Ext}}}$s.
--- abstract: 'Fractional spectral collocation (FSC) method based on fractional Lagrange interpolation has recently been proposed to solve fractional differential equations. Numerical experiments show that the linear systems in FSC become extremely ill-conditioned as the number of collocation points increases. By introducing suitable fractional Birkhoff interpolation problems, we present fractional integration preconditioning matrices for the ill-conditioned linear systems in FSC. The condition numbers of the resulting linear systems are independent of the number of collocation points. Numerical examples are given.' author: - 'Kui Du[^1]' title: Preconditioning fractional spectral collocation --- Fractional Lagrange interpolation, fractional Birkhoff interpolation, fractional spectral collocation, preconditioning 65L60, 41A05, 41A10 Introduction ============ Fractional spectral collocation (FSC) methods [@zayernouri2014fract; @zayernouri2015fract; @fatone2015optim] based on fractional Lagrange interpolation have recently been proposed to solve fractional differential equations. By a spectral theory developed in [@zayernouri2013fract] for fractional Sturm-Liouville eigenproblems, the corresponding fractional differential matrices can be obtained with ease. However, numerical experiments show that the involved linear systems become extremely ill-conditioned as the number of collocation points increases. Typically, the condition number behaves like $\mcalo(N^{2\nu})$, where $N$ is the number of collocation points and $\nu$ is the order of the leading fractional term. Efficient preconditioners are highly required when solving the linear systems by an iterative method. Recently, Wang, Samson, and Zhao [@wang2014well] proposed a well-conditioned collocation method to solve linear differential equations with various types of boundary conditions. By introducing a suitable Birkhoff interpolation problem, they constructed a pseudospectral integration preconditioning matrix, which is the exact inverse of the pseudospectral discretization matrix of the $n$th-order derivative operator together with $n$ boundary conditions. Essentially, the linear system in the well-conditioned collocation method [@wang2014well] is the one obtained by right preconditioning the original linear system; see [@du2015prersc]. By introducing suitable fractional Birkhoff interpolation problems and employing the same techniques in [@wang2014well], Jiao, Wang, and Huang [@jiao2015well] proposed fractional integration preconditioning matrices for linear systems in fractional collocation methods base on Lagrange interpolation. In the Riemann-Liouville case, it is necessary to modify the fractional derivative operator in order to absorb singular fractional factors (see [@jiao2015well §3]). In this paper, we extend the Birkhoff interpolation preconditioning techniques in [@wang2014well; @jiao2015well] to the fractional spectral collocation methods [@zayernouri2014fract; @zayernouri2015fract; @fatone2015optim] based on fractional Lagrange interpolation. Unlike that in [@jiao2015well], there are no singular fractional factors in the Riemann-Liouville case. Numerical experiments show that the condition number of the resulting linear system is independent of the number of collocation points. The rest of the paper is organized as follows. In §2, we review several topics required in the following sections. In §3, we introduce fractional Birkhoff interpolation problems and the corresponding fractional integration matrices. In §4, we present the preconditioning fractional spectral collocation method. Numerical examples are also reported. We present brief concluding remarks in §5. Preliminaries ============= Fractional derivatives ---------------------- The definitions of fractional derivatives of order $\nu\in(n-1,n), n\in \mbbn$, on the interval $[-1,1]$ are as follows [@kilbas2006theor]: - Left-sided Riemann-Liouville fractional derivative: $$\rlfdl u(x)=\frac{1}{\Gamma(n-\nu)}\frac{\rmd^n}{\rmd x^n}\int^x_{-1}\frac{u(\xi)}{(x-\xi)^{\nu-n+1}}\rmd\xi,$$ - Right-sided Riemann-Liouville fractional derivative: $$\rlfdr u(x)=\frac{(-1)^n}{\Gamma(n-\nu)}\frac{\rmd^n}{\rmd x^n}\int^1_x\frac{u(\xi)}{(\xi-x)^{\nu-n+1}}\rmd\xi,$$ - Left-sided Caputo fractional derivative: $$\cfdl u(x)=\frac{1}{\Gamma(n-\nu)}\int^x_{-1}\frac{u^{(n)}(\xi)}{(x-\xi)^{\nu-n+1}}\rmd\xi,$$ - Right-sided Caputo fractional derivative: $$\cfdr u(x)=\frac{(-1)^n}{\Gamma(n-\nu)}\int^1_x\frac{u^{(n)}(\xi)}{(\xi-x)^{\nu-n+1}}\rmd\xi.$$ By the definitions of fractional derivatives, we have \[rlcl\] u(x)=\_[i=0]{}\^[n-1]{}(x+1)\^[i-]{}+ u(x),and \[rlcr\] u(x)=\_[i=0]{}\^[n-1]{}(1-x)\^[i-]{}+ u(x). Therefore, u(x)=u(x),u\^[(i)]{}(-1)=0,i=0,1,…,n-1,and u(x)=u(x),u\^[(i)]{}(1)=0,i=0,1,…,n-1. In this paper, we mainly deal with the left-sided Riemann-Liouville fractional problems with homogeneous boundary/initial conditions. By a simple change of variables, (\[rlcl\]) and (\[rlcr\]), the extension to other fractional problems is easy. Fractional Lagrange interpolation --------------------------------- Throughout the paper, let $\{x_j\}_{j=1}^N$ be a set of distinct points satisfying \[iptsx\] -1<x\_1<<x\_[N-1]{}<x\_N1. Given $\mu\in(0,1)$, the [$\mu$-fractional Lagrange interpolation basis]{} associated with the points $\{x_j\}_{j=1}^N$ is defined as \[mufl\] \_[j]{}\^(x)=ł()\^\_[n=1,nj]{}\^N,j=1,…,N. For a function $u(x)$ with $u(-1)=0$, the $\mu$-fractional Lagrange interpolant $u_N(x)$ of $u(x)$ takes the form $$u_N(x)=\sum_{j=1}^Nu(x_j)\ell_{j}^\mu(x).$$ Computations of ${^{RL}_{-1}\mcald^{\mu}_x} \ell_{j}^{\mu}(x)$ and ${^{RL}_{-1}\mcald^{1+\mu}_x} \ell_{j}^{\mu}(x)$ with $\mu\in(0,1)$ -------------------------------------------------------------------------------------------------------------------------------------- Note that $\ell_{j}^\mu(x)$, $j=1,\ldots,N,$ can be represented exactly as \[hexpan\] \_[j]{}\^(x)=ł()\^\_[n=1,nj]{}\^N=(x+1)\^\_[n=1]{}\^N\_[nj]{} P\^[(-,)]{}\_[n-1]{}(x),where $P^{(\alpha,\beta)}_n(x)$ denote the standard Jacobi polynomials. The coefficients $\alpha_{nj}$ can be obtained by solving the linear system $$\sum_{n=1}^N (x_i+1)^{\mu}P^{(-\mu,\mu)}_{n-1}(x_i)\alpha_{nj}=\delta_{ij},\quad i=1,\ldots,N.$$ \[rem1\] Let $\{x_j\}_{j=1}^N$ and $\{\omega_j\}_{j=1}^N$ be the Gauss-Jacobi quadrature nodes and weights with the Jacobi polynomial $P_N^{(-\mu,\mu)}(x)$. Then, \_[nj]{}=\_jP\_[n-1]{}\^[(-,)]{}(x\_j). We now compute ${^{RL}_{-1}\mcald^{\mu}_x} \ell_{j}^{\mu}(x)$ and ${^{RL}_{-1}\mcald^{1+\mu}_x} \ell_{j}^{\mu}(x)$. Let $P_n(x)$ denote the Legendre polynomial of order $n$. By (see [@zayernouri2013fract]) ł((x+1)\^P\_[n-1]{}\^[(-,)]{}(x))=P\_[n-1]{}(x)and $${^{RL}_{-1}\mcald^{1+\mu}_x} \ell_{j}^{\mu}(x)=\frac{\rmd}{\rmd x}\l({^{RL}_{-1}\mcald^\mu_x}\ell_{j}^{\mu}(x)\r),$$ we have $${^{RL}_{-1}\mcald^\mu_x} \ell_{j}^{\mu}(x)=\sum_{n=1}^N\alpha_{nj} \frac{\G(n+\mu)}{\G(n)}P_{n-1}(x)$$ and \_[j]{}\^(x)&=&\_[n=2]{}\^N\_[nj]{} P\_[n-1]{}’(x)\ &=&\_[n=2]{}\^N\_[nj]{} P\^[(1,1)]{}\_[n-2]{}(x). Riemann-Liouville fractional Birkhoff interpolation =================================================== Let $\mbbp_n$ be the set of all algebraic polynomials of degree at most $n$. Define the space $$\mbbs^\mu_N=(x+1)^\mu\mbbp_{N-1}.$$ In the following, we consider two special cases. The case $\nu=\mu\in(0,1)$ -------------------------- For a function $u(x)$ with $u(-1)=0$, given $N$ distinct points $\{y_j\}_{j=1}^N$ satisfying $$-1<y_1<\cdots <y_{N-1}<y_N\leq 1,$$ consider the Riemann-Liouville $\nu$-fractional Birkhoff interpolation problem: \[birkhoff1\] [Find]{} p(x)\_N\^ [such that]{} p(y\_j)=u(y\_j),j=1,…,N. \[tcase1\] The interpolant $u_N^\nu(x)$ for the Riemann-Liouville $\nu$-fractional Birkhoff problem [(\[birkhoff1\])]{} of a function $u(x)$ with $u(-1)=0$ takes the form $$u_N^\nu(x)=\sum_{j=1}^{N}\rlfdl u(y_j)B_j^\nu(x),$$ where B\_j\^(x)=(x+1)\^\_[n=1]{}\^N\_[nj]{}P\_[n-1]{}\^[(-,)]{}(x),j=1,…,N,with $\wt\alpha_{nj}$ satisfying $$\sum_{n=1}^N\wt\alpha_{nj}\frac{\G(n+\nu)}{\G(n)}P_{n-1}(y_i)=\delta_{ij},\quad i=1,\ldots,N.$$ By $\rlfdl B_j^\nu(y_i)=\delta_{ij}$, the proof of Theorem \[tcase1\] is straightforward. \[rem2\] Let $\{y_j\}_{j=1}^{N}$ and $\{\omega_j\}_{j=1}^{N}$ be the Gauss-Legendre quadrature nodes and weights with the Legendre polynomial $P_N(x)$. Then, $$\wt\alpha_{nj}=\frac{2n-1}{2}\frac{\G(n)}{\G(n+\nu)}\omega_jP_{n-1}(y_j).$$ Define the matrices $${\bf D}^{(\nu)}_{{\bf x}\mapsto {\bf y}}=\l[\rlfdl \ell_{j}^{\mu}(y_i)\r]_{i,j=1}^{N},\qquad {\bf B}_{{\bf y}\mapsto {\bf x}}^{(-\nu)}=\l[B_j^\nu(x_i)\r]_{i,j=1}^N.$$ It is easy to show that \[case1\][D]{}\^[()]{}\_[[x]{}]{}[B]{}\_[[y]{}]{}\^[(-)]{}=[I]{}\_N. The case $\nu=1+\mu\in(1,2)$ ---------------------------- For a function $u(x)$ with $u(\pm1)=0$, given $N-1$ distinct points $\{y_j\}_{j=1}^{N-1}$ satisfying $$-1<y_1<\cdots <y_{N-1}<1,$$ consider the Riemann-Liouville $\nu$-fractional Birkhoff interpolation problem: \[birkhoff2\] [Find]{} p(x)\_N\^ [such that]{} ł{ [ll]{}p(1)=0, &\ p(y\_j)=u(y\_j),& j=1,…,N-1. . \[tcase2\] The interpolant $u_N^\nu(x)$ for the Riemann-Liouville $\nu$-fractional Birkhoff problem [(\[birkhoff2\])]{} of a function $u(x)$ with $u(\pm 1)=0$ takes the form $$u_N^\nu(x)=\sum_{j=1}^{N-1}\rlfdl u(y_j)B_j^\nu(x),$$ where $$B_j^\nu(x)=(x+1)^\mu\sum_{n=1}^{N-1}\wt\beta_{nj}\l(P_{n}^{(-\mu,\mu)}(x)-P_{n}^{(-\mu,\mu)}(1)\r),\quad j=1,\ldots,N-1,$$ with $\mu=\nu-1$ and $\wt\beta_{nj}$ satisfying \_[n=1]{}\^[N-1]{}\_[nj]{}P\_[n-1]{}\^[(1,1)]{}(y\_i)=\_[ij]{},i=1,…,N-1. By $\rlfdl B_j^\nu(y_i)=\delta_{ij}$, the proof of Theorem \[tcase2\] is straightforward. \[rem3\] Let $\{y_j\}_{j=1}^{N-1}$ and $\{\omega_j\}_{j=1}^{N}$ be the Gauss-Jacobi quadrature nodes and weights with the Jacobi polynomial $P_{N-1}^{(1,1)}(x)$. Then, \_[nj]{}=\_jP\_[n-1]{}\^[(1,1)]{}(y\_j). In this subsection, let $x_N=1$. Define the matrices $$\wt{\bf D}^{(\nu)}_{{\bf x}\mapsto {\bf y}}=\l[\rlfdl \ell_{j}^{\nu}(y_i)\r]_{i,j=1}^{N-1}, \qquad \wt{\bf B}_{{\bf y}\mapsto {\bf x}}^{(-\nu)}=\l[B_j^\nu(x_i)\r]_{i,j=1}^{N-1}.$$ It is easy to show that \[case2\]\^[()]{}\_[[x]{}]{}\_[[y]{}]{}\^[(-)]{}=[I]{}\_[N-1]{}. Preconditioning fractional spectral collocation (PFSC) ====================================================== In this section, we use two examples to introduce the preconditioning scheme. An initial value problem ------------------------ Consider the fractional differential equation of the form \[fde1\]u(x)+a(x)u(x)=f(x), (0,1);u(-1)=0.The fractional spectral collocation scheme leads to the following linear system \[fsc1\] ł([D]{}\_[[x]{}]{}\^[()]{}+{[a]{}}[D]{}\_[[x]{}]{}\^[(0)]{})[u]{}=[f]{}, where $${\bf D}_{{\bf x}\mapsto{\bf y}}^{(0)}=\l[\ell_{j}^\nu(y_i)\r]_{i,j=1}^N,$$ and &&[a]{}=ł\[ [cccc]{}a(y\_1)& a(y\_2)& &a(y\_[N]{}) \]\^,\ &&[f]{}=ł\[ [cccc]{}f(y\_1)& f(y\_2)& &f(y\_[N]{}) \]\^.The unknown vector ${\bf u}$ is an approximation of the vector of the exact solution $u(x)$ at the points $\{x_j\}_{j=1}^{N}$, i.e., $$\l[\begin{array}{cccc}u(x_1)& u(x_2)& \cdots &u(x_{N})\end{array}\r ]^\rmt.$$ Consider the matrix ${\bf B}_{{\bf y}\mapsto {\bf x}}^{(-\nu)}$ as a right preconditioner for the linear system (\[fsc1\]). By (\[case1\]), we have the right preconditioned linear system \[pfsc1\]ł([I]{}\_N+{[a]{}}[D]{}\_[[x]{}]{}\^[(0)]{}[B]{}\_[[y]{}]{}\^[(-)]{})[v]{}=[f]{}. It is easy to show that $${\bf D}_{{\bf x}\mapsto {\bf y}}^{(0)}{\bf B}_{{\bf y}\mapsto {\bf x}}^{(-\nu)}={\bf B}_{{\bf y}\mapsto {\bf y}}^{(0-\nu)},$$ where $${\bf B}_{{\bf y}\mapsto {\bf y}}^{(0-\nu)}=\l[B_j^{\nu}(y_i)\r]_{i,j=1}^N.$$ Then, the equation (\[pfsc1\]) reduces to \[pfsc1t\]ł([I]{}\_N+{[a]{}}[B]{}\_[[y]{}]{}\^[(0-)]{})[v]{}=[f]{}.After solving (\[pfsc1t\]), we obtain ${\bf u}$ by ${\bf u}={\bf B}_{{\bf y}\mapsto {\bf x}}^{(-\nu)}{\bf v}.$ ### Example 1 {#example-1 .unnumbered} We consider the fractional differential equation (\[fde1\]) with $$\nu=0.8,\qquad a(x)=2+\sin(25x).$$ The function $f(x)$ is chosen such that the exact solution of (\[fde1\]) is $$u(x)=\rme^{x+1}-1+(x+1)^{46/7}.$$ Let $\{x_j\}_{j=1}^N$ be the Gauss-Jacobi points as in Remark \[rem1\] and $\{y_j\}_{j=1}^N$ be the Gauss-Legendre points as in Remark \[rem2\]. We compare condition numbers, number of iterations (using BiCGSTAB in Matlab with TOL$=10^{-9}$) and maximum point-wise errors of FSC and PFSC (see Figure \[e1f1\]). Observe from Figure \[e1f1\] (left) that the condition number of FSC behaves like $\mcalo(N^{1.6})$, while that of PFSC scheme remains a constant even for $N$ up to $1024$. As a result, PFSC scheme only requires about 7 iterations to converge (see Figure \[e1f1\] (middle)), while the usual FSC scheme requires much more iterations with a degradation of accuracy as depicted in Figure \[e1f1\] (right). A boundary value problem ------------------------ Consider the fractional differential equation of the form \[fde2\]u(x)+a(x)u’(x)+b(x)u(x)=f(x), =1+(1,2);u(1)=0.The fractional spectral collocation method leads to the following linear system \[fsc2\] ł(\_[[x]{}]{}\^[()]{}+{[a]{}}\_[[x]{}]{}\^[(1)]{}+{[b]{}}\_[[x]{}]{}\^[(0)]{})[u]{}=[f]{}, where $$\wt{\bf D}_{{\bf x}\mapsto{\bf y}}^{(1)}=\l[\frac{\rmd}{\rmd x}\l(\ell^\mu_j(x)\r)\Big\|_{x=y_i}\r]_{i,j=1}^{N-1},\qquad \wt{\bf D}_{{\bf x}\mapsto{\bf y}}^{(0)}=\l[\ell^\mu_j(y_i)\r]_{i,j=1}^{N-1},$$ and &&[a]{}=ł\[ [cccc]{}a(y\_1)& a(y\_2)& &a(y\_[N-1]{}) \]\^,\ &&[b]{}=ł\[ [cccc]{}b(y\_1)& b(y\_2)& &b(y\_[N-1]{}) \]\^,\ &&[f]{}=ł\[ [cccc]{}f(y\_1)& f(y\_2)& &f(y\_[N-1]{}) \]\^.The unknown vector ${\bf u}$ is an approximation of the vector of the exact solution $u(x)$ at the points $\{x_j\}_{j=1}^{N-1}$, i.e., $$\l[\begin{array}{cccc}u(x_1)& u(x_2)& \cdots &u(x_{N-1})\end{array}\r ]^\rmt.$$ Consider the matrix $\wt{\bf B}_{{\bf y}\mapsto {\bf x}}^{(-\nu)}$ as a right preconditioner for the linear system (\[fsc2\]). By (\[case2\]), we have the right preconditioned linear system \[pfsc2\]ł([I]{}\_[N-1]{}+{[a]{}}\_[[x]{}]{}\^[(1)]{}\_[[y]{}]{}\^[(-)]{}+{[b]{}}\_[[x]{}]{}\^[(0)]{}\_[[y]{}]{}\^[(-)]{})[v]{}=[f]{}.It is easy to show that $$\wt {\bf D}_{{\bf x}\mapsto {\bf y}}^{(1)}\wt{\bf B}_{{\bf y}\mapsto {\bf x}}^{(-\nu)}=\wt{\bf B}_{{\bf y}\mapsto {\bf y}}^{(1-\nu)},\qquad \wt {\bf D}_{{\bf x}\mapsto {\bf y}}^{(0)}\wt{\bf B}_{{\bf y}\mapsto {\bf x}}^{(-\nu)}=\wt{\bf B}_{{\bf y}\mapsto {\bf y}}^{(0-\nu)},$$ where $$\wt{\bf B}_{{\bf y}\mapsto {\bf y}}^{(1-\nu)}=\l[\frac{\rmd}{\rmd x}\l(B_j^{\nu}(x)\r)\Big\|_{x=y_i}\r]_{i,j=1}^{N-1},\qquad \wt{\bf B}_{{\bf y}\mapsto {\bf y}}^{(0-\nu)}=\l[B_j^{\nu}(y_i)\r]_{i,j=1}^{N-1}.$$ Then, the equation (\[pfsc2\]) reduces to \[pfsc2t\]ł([I]{}\_[N-1]{}+{[a]{}}\_[[y]{}]{}\^[(1-)]{}+{[b]{}}\_[[y]{}]{}\^[(0-)]{})[v]{}=[f]{}.After solving (\[pfsc2t\]), we obtain ${\bf u}$ by ${\bf u}=\wt{\bf B}_{{\bf y}\mapsto {\bf x}}^{(-\nu)}{\bf v}.$ ### Example 2 {#example-2 .unnumbered} We consider the fractional differential equation (\[fde2\]) with $$\nu=1.9,\qquad a(x)=2+\sin(4\pi x),\qquad b(x)=2+\cos x.$$ The function $f(x)$ is chosen such that the exact solution of (\[fde2\]) is $$u(x)=\rme^{x+1}-x-2-\frac{\rme^2-3}{4}(x+1)^2+(x+1)^{46/7}-2(x+1)^{39/7}.$$ Let $\{x_j\}_{j=0}^N$ be the Chebyshev points of the second kind (also known as Gauss-Chebyshev-Lobatto points) defined as $$x_j=-\cos\frac{j\pi}{N},\qquad j=0,1,\ldots, N,$$ and $\{y_j\}_{j=1}^{N-1}$ be the Gauss-Jacobi points as in Remark \[rem3\]. We compare condition numbers, number of iterations (using BiCGSTAB in Matlab with TOL$=10^{-11}$) and maximum point-wise errors of FSC and PFSC (see Figure \[e2f1\]). Observe from Figure \[e2f1\] (left) that the condition number of FSC behaves like $\mcalo(N^{3.8})$, while that of PFSC scheme remains a constant even for $N$ up to $1024$. As a result, PFSC scheme only requires about 13 iterations to converge (see Figure \[e2f1\] (middle)), while the FSC scheme fails to converge (when $N\geq 16$) within $N$ iterations as depicted in Figure \[e2f1\] (right). Concluding remarks ================== We numerically show that the Birkhoff interpolation preconditioning techniques in [@wang2014well; @jiao2015well] are still effective for fractional spectral collocation schemes [@zayernouri2014fract; @zayernouri2015fract; @fatone2015optim] based on fractional Lagrange interpolation. The preconditioned coefficient matrix is a perturbation of the identity matrix. The condition number is independent of the number of collocation points. The preconditioned linear system can be solved by an iterative solver within a few iterations. The application of the preconditioning FSC scheme to multi-term fractional differential equations is straightforward. [1]{} , [Preconditioning rectangular spectral collocation]{}, arXiv preprint arXiv:1510.00195, (2015). , [Optimal collocation nodes for fractional derivative operators]{}, SIAM J. Sci. Comput., 37 (2015), pp. A1504–A1524. , [Well-conditioned fractional collocation methods using fractional birkhoff interpolation basis]{}, arXiv preprint arXiv:1503.07632, (2015). , [Theory and applications of fractional differential equations]{}, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 2006. , [A well-conditioned collocation method using a pseudospectral integration matrix]{}, SIAM J. Sci. Comput., 36 (2014), pp. A907–A929. , [Fractional [S]{}turm-[L]{}iouville eigen-problems: theory and numerical approximation]{}, J. Comput. Phys., 252 (2013), pp. 495–517. height 2pt depth -1.6pt width 23pt, [Fractional spectral collocation method]{}, SIAM J. Sci. Comput., 36 (2014), pp. A40–A62. height 2pt depth -1.6pt width 23pt, [Fractional spectral collocation methods for linear and nonlinear variable order [FPDE]{}s]{}, J. Comput. Phys., 293 (2015), pp. 312–338. [^1]: School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computation, Xiamen University, Xiamen 361005, China ([kuidu@xmu.edu.cn]{}). The research of this author was supported by the National Natural Science Foundation of China (No.11201392 and No.91430213), the Doctoral Fund of Ministry of Education of China (No.20120121120020), and the Fundamental Research Funds for the Central Universities (No.2013121003).
--- abstract: 'I review the subjects of non-solar cosmic rays (CRs) and long-duration gamma-ray bursts (GRBs). Of the various interpretations of these phenomena, the one best supported by the data is the following. Accreting compact objects, such as black holes, are seen to emit relativistic puffs of plasma: ‘cannonballs’ (CBs). The inner domain of a rotating star whose core has collapsed resembles such an accreting system. This suggests that core-collapse supernovae (SNe) emit CBs, as SN1987A did. The fate of a CB as it exits a SN and travels in space can be studied as a function of the CB’s mass and energy, and of ‘ambient’ properties: the encountered matter- and light- distributions, the composition of the former, and the location of intelligent observers. The latter may conclude that the interactions of CBs with ambient matter and light generate CRs and GRBs, all of whose properties can be described by this ‘CB model’ with few parameters and simple physics. GRB data are still being taken in unscrutinized domains of energy and timing. They agree accurately with the model’s predictions. CR data are centenary. Their precision will improve, but new striking predictions are unlikely. Yet, a one-free-parameter description of all CR data works very well. This is a bit as if one discovered QED today and only needed to fit $\alpha$.' author: - \| Alvaro De Rújula\ [CERN & Boston University]{}\ title: \| An introduction to Cosmic Rays and Gamma-Ray Bursts,\ and to their simple understanding --- =11.6pt Introduction ============ This is a version of an introductory talk to high-energy physicists. Cosmic rays (CRs) were the first item in their field, and will remain the energy record-breakers for the foreseeable future. I shall argue that nothing ‘besides the standard model’ is required to understand CRs of any energy, subtracting from their interest. ‘Long’ gamma-ray bursts (GRBs) are flashes of mainly sub-MeV photons, originating in supernova (SN) explosions. The $\gamma$-rays are highly collimated. Hence, GRBs are not the publicized ‘highest-energy explosions after the big bang’, but more modest torches occasionally pointing to the observer. GRBs are of interest because their understanding is intimately related to that of CRs. It might have been more precise to say ‘my understanding’ of GRBs and CRs, for the work of my coauthors and I is viewed as unorthodox. What a start! I have already admitted that our stance is not trendy and that the subject is of no post-standard interest. But our claims are based on clear hypothesis, which may be proven wrong, and very basic physics, which is precise enough, very pretty, understandable to undergraduates, and successful. The information about GRBs and CRs is overwhelming. GRBs are known since the late 60’s and CRs since 1912. Surprisingly, no theories have arisen that are both accepted (‘standard’) and acceptable (transparent, predictive and successful). I cannot refer to a representative subset of the $\sim\!70\!+\!70$ kilo-papers on CRs and GRBs. For reviews of the standard views on CRs, see e.g. Hillas[@Hillas] or Hoerandel[@Hoer2]. For the accepted ‘fireball’ model of GRBs see e.g. Meszaros[@Mes] or Piran[@Pir]. Fewer self-citations and many more references, particularly to data, appear in DDD02[@AGoptical], DDD03[@AGradio], DD04[@DDGRB]and DD06[@DDCR]. Most of what you may want to recall about Cosmic Rays ===================================================== In CR physics ‘all-particle’ refers to nuclei: all charged CRs but electrons. The CR spectra being fairly featureless, it is customary to weigh them with powers of energy, to over-emphasize their features. The $E^3\,dF/dE$ all-particle spectrum is shown in fig. \[AllPart\]a, not updated for recent data at the high-energy tail. At less than TeV energies the CR flux is larger than $\rm 1\,m^{-2}\,s^{-1}\,sr^{-1}$ and it is possible to measure the charge $Z$ and mass number $A$ of individual particles with, e.g., a magnetic spectrometer in a balloon, or in orbit. Some low-energy results for H and He are shown in fig. \[AllPart\]b. They vary with solar activity. The CR fluxes of the lightest 30 elements at $E\!=\!1$ TeV (of a nucleus, not per-nucleon) are shown in fig. \[abundances\]a, and compared with the relative abundances in the interstellar medium (ISM) of the solar neighbourhood. Elements such as Li, Be and B are relatively enhanced in the CRs, they result from collisional fragmentation of heavier and abundant ‘primaries’ such as C, N and O. Otherwise, the solar-ISM and CR $Z$-distributions are akin, but for H and He. In fig. \[abundances\]b the abundances relative to H of the CR primary elements up to Ni ($Z\!=\!28$) are plotted as blue squares. The stars are solar ISM abundances. CR positrons and antiprotons attract attention as putative dark matter products, but it is nearly impossible to prove that their fluxes are not entirely secondary. The Galaxy has a complex magnetic-field structure with $B_{_{\rm G}}\!=\!{\cal{O}}(1\,\rm \mu G)$ and coherent domains ranging in size up to $\sim\!1$ kpc, $\sim\!1/8$ of our distance to the Galactic center. In such a field, a nucleus of $E\!\simeq\!p\!>\!Z\,(3\!\times\!10^9)$ GeV would hardly be deflected. For $Z\!=\!1$, this energy happens to be the ‘ankle’ energy, at which the flux of fig. \[AllPart\] bends up. CRs originating within the Galaxy and having $E\!>\!E_{\rm ankle}$ would escape practically unhindered. The CR flux does not bend [down]{} at that energy, thus the generally agreed conclusion that CRs above the ankle are mainly extragalactic. CRs of Galactic origin and $E\!<\!E_{\rm ankle}$ are ‘confined’, implying that their observed and source fluxes obey: $$F_{\rm o}\propto\tau_{\rm conf}\,F_{\rm s},\;\;\; \tau_{\rm conf}\propto(Z/p)^{\beta_{\rm conf}},\;\;\; \beta_{\rm conf}\sim\!0.6\pm\!0.1, \label{confinement}$$ with $\tau_{\rm conf}$ a ‘confinement time’, deduced from the study of stable and unstable CRs and their fragments. At $E\!=\!10^6$-$10^8$ GeV the all-particle spectrum of fig. \[AllPart\]a bends in one or two ‘knees’. The knee flux is too small to measure directly its energy and composition, which are inferred from the properties of the CR shower of hadrons, $\gamma$’s, $e$’s and $\mu$’s, initiated by the CR in the upper atmosphere. The results for H, He and Fe are shown in fig. \[KASKADE\]. Note that even the same data leads to incompatible results, depending on the Monte-Carlo program used to analize the showers. But the spectra of the various elements seem to have ‘knees’ which scale roughly with $A$ or $Z$, the data not been good enough to distinguish. The high-$E$ end of the $E^3$-weighed CR spectrum is shown in fig. \[highE\]a. These data and the more recent ones of HIRES and Auger, clearly show a cutoff, predicted by Greisen, Zatsepin and Kuzmin (GZK) as the result of the inevitable interactions of extragalactic CR protons with the microwave background radiation. The reactions $p\!+\!\gamma\!\to\!n\!+\!\pi^+;\,p\!+\!\pi^0$ cut off the flux at $E\!>\!E_{_{\rm GZK}}\!\sim\!A\times 10^{11}$ GeV, from distances larger than tens of Mpc. Similarly, extragalactic nuclei of $E\!>\!10^9$ GeV are efficiently photo-dissociated in the cosmic infrared radiation, the corresponding CR flux should not contain many. At very high energies, rough measures of the CR $A$-distribution are extracted from the ‘depth of shower maximum’, $X$, the number of grams/cm$^2$ of atmosphere travelled by a CR shower before its $e^\pm$/$\gamma$ constituency reaches a maximum. At a fixed energy, $X$ decreases with $A$, since a nucleus is an easily broken bag of nucleons of energy $\sim\!E/A$. As in fig. \[highE\]b, the data are often presented as $\langle ln[A](E) \rangle$, which approximately satisfies $X(A)\!\sim\!X(1)-x\, ln[A]$, with $x\!\sim\!37$ grams/cm$^2$ the radiation length in air. If CRs are chiefly Galactic in origin, their accelerators must compensate for the escape of CRs from the Galaxy, to sustain the observed CR flux: it is known from meteorite records that the flux has been steady for the past few Giga-years. The Milky Way’s luminosity in CRs must therefore satisfy: $${L_{_{\rm CR}} ={4\pi \over c} \int {1\over \tau_{\rm conf}} \,E\,{dF_{\rm o}\over dE}\,dE\,dV} \sim 1.5\times 10^{41}{\rm ~erg~s^{-1}}\, , \label{CRlum}$$ where $V$ is a CR-confinement volume. The quoted standard estimate of $L_{_{\rm CR}}$ is very model-dependent[@DDLum]. More than you ever wanted to know about Gamma-Ray Bursts ======================================================== Two $\gamma$-ray count rates of GRBs, peaking at $dN/dt\!=\!{\cal{O}}(10^4)\,\rm s^{-1}$, are shown in fig. \[try\]. The typical energy of the $\gamma$-ray of GBBs is $\sim\!250$ keV. The total [‘isotropic equivalent’]{} energy of a source of such photons at a typical redshift, $z\!=\!{\cal{O}}(1)$, is $E_\gamma^{\rm iso}\!\sim\!10^{53}$ erg, similar to the available energy in a core-collapse SN explosion, i.e. half of the binding energy of a solar-mass neutron star, maybe a bit more for a black-hole remnant. It is hard to imagine a process with $>\!1$% efficiency for $\gamma$-ray production. Since GRBs are observed to be made by SNe, either the parent stars are amazingly special, or the $\gamma$-rays are narrowly beamed. The total-duration distribution of the $\gamma$-rays of GRBs has two peaks, with a trough at $\sim\!2$s dividing (by definition) two distinct types. ‘Long’ GRBs are more common and better measured than short ones; one is more confident discussing mainly the former, as I shall. The long GRB light curves of fig. \[try\] are not atypical. The ‘pulses’ of a given GRB vary in intensity, but have similar widths, a fairly universal exponential rise, and a power decay $\propto\!t^{-a}$, $a\!\sim\!2$. The number of ‘clear pulses’ averages to $\sim\!5$, it may reach $\sim\!12$. The pulse-to-pulse delays are random, extending from ${\cal{O}}$(1s) to ${\cal{O}}(10^2$s). Put all the above in a random-generator and, concerning long GRBs, ‘you have seen them all’. -.5truecm GRBs are not often seen more than once a day, they are baptized with their observation date. GRBs 980425 and 030329, shown in fig. \[try\], originated at $z\!=\!0.0085$ (the record smallest) and $0.168$, respectively. How are the redshifts known? GRBs have “afterglows" (AGs): they are observable in radio to X-rays for months after their $\gamma$-ray signal peters out. The AG of GRB 030329 in the ‘R-band’ (a red-light interval) and radio is shown in fig. \[try2\]a-c. Once the object is seen in optical or radio, its direction can be determined with much greater precision than via $\gamma$ rays. Very often the source is localized within a galaxy, whose lines can be measured to determine $z$ (in some cases a lower limit on $z$ is deduced from absorption lines in intervening material). .3cm GRB 980425 was [‘associated’]{} with a supernova called SN1998bw: within directional errors and within a timing uncertainty of $\sim\!1$ day, they coincided. The luminosity of a 1998bw-like SN peaks at $\sim 15\,(1+z)$ days. The SN light competes at that time and frequency with the AG of its GRB, and it is not always easily detectable. Iff one has a predictive theory of AGs, one may test whether GRBs are associated with ‘standard torch’ SNe, akin to SN1998bw, ‘transported’ to the GRBs’ redshifts. The test was already conclusive (to us) in 2001[@AGoptical]. One could even foretell [the date]{} in which a GRB’s SN would be discovered. For example, GRB 030329 was so ‘very near’ at $z\!=\!0.168$, that one could not resist posting such a daring prediction[@SN030329] during the first few days of AG observations. The prediction and the subsequent SN signal are shown in fig. \[try2\]a,b. The spectrum of this SN was very well measured and seen to coincide snugly with that of SN1998bw, and this is why the SN/GRB association ceased to be doubted: [long GRBs are made by core-collapse SNe.]{} Astrophysicists classify SNe in Types, mainly depending on the composition of their ejecta. Within very limited statistics the SNe associated with GRBs are of Type Ib/c. These constitute some 15% of core-collapse SNe, the fascinating ones which beget neutrinos, neutron stars and presumably black holes. Type Ia SNe are probably mere explosions of accreting white dwarfs, but they are very luminous, and of cosmological standard-candle fame. GRBs have many ‘typical’ properties. Their spherical-equivalent number of $\gamma$-rays is $\sim\!10^{59}$. Their spectrum at fixed $t$ is very well approximated by: $${dN\over dE}\biggm\|_t \propto \left[{T(t)\over E}\right]^\alpha\; e^{-E/T(t)}+b\; \left[1-e^{-E/T(t)}\right]\; {\left[T(t)\over E\right]}^\beta \label{totdist}$$ with $b\!\sim\!1$, $\alpha\!\sim\!1$, $\beta\!\sim\!2.1$. Early in the evolution of a pulse, the ‘peak energy’ (characterizing the photons carrying most of the GRB’s energy) is $E_p\!\sim\!T[0]\!\sim\!250$ keV, evolving later to $T(t)\!\sim\!t^{-2}$. A pulse’s shape at fixed $E$ is well fit by: $${dN\over dt}\biggm\|_E\approx \Theta[t]\,e^{-[{\Delta t(E)/t}]^2} \left\{ 1 - e^{-[{\Delta t(E)/t}]^2} \right\};\;{\Delta t(E_1)\over \Delta t(E_2)} \approx \left[{E_2\over E_1} \right]^{1\over 2}, \label{shape}$$ with $\Delta t\!\sim\!{\cal{O}}(1$s) at $E\!\sim\!E_p$. Eq. (\[shape\]) reflects an approximate spectro-temporal correlation whereby $E\,dN/(dE\,dt)\!\approx\!F[E\,t^2]$, which we call the $E\,t^2$ ‘law’. The values of $E_p$; of the isotropic-equivalent energy and luminosity, $E_\gamma^{\rm iso}$ and $L_p^{\rm iso}$; of a pulse’s rise-time $t_{\rm rise}$; or of its ‘lag-time’ $t_{\rm lag}$ (a measure of how a pulse peaks at a later time in a lower energy interval) vary from GRB to GRB over orders of magnitude. But they are strongly correlated, as shown in Figs. (\[f1\]a-d). It is patently obvious that such an organized set of results is carrying a strong and simple message, which we shall decipher. X-ray flashes (XRFs) are lower-energy kinsfolk of GRBs. They are defined by having $E_p\!<\!50$ keV. Their pulses are wider than the ones of GRBs and their overlap is more pronounced, since the total durations of (multi-pulse) XRFs and GRBs are not significantly different. In fig. \[f1\]e I show the time at which the single pulse of XRF 060218 peaked (measured from the start of the count-rate rise) as measured in different energy intervals. This is an impressive validation of the $E\,t^2$ law (the red line), also screaming for a simple explanation. Analytical expressions summarizing the behaviour of GRB and XRF afterglows in time (from seconds to months) and frequency (from radio to X-rays) do exist (DDD02/03), but they are somewhat more complex than Eqs. (\[totdist\],\[shape\]). The typical AG behaviour is shown in fig. \[f1\]f, as a function of frequency, at 1, 10... 300 days after burst (the value of $p$ is $\sim\!2.2\pm0.2$). This simple figure reflects a rich behaviour in time and frequency. ‘Chromatic bends’ (called ‘breaks’ in the literature) are an example. At a fixed time, the spectra steepen from $\sim\!\nu^{-0.5}$ to $\sim\!\nu^{-1.1}$ at the dots in the figure. Around a given frequency, such as the optical one marked by a dotted line, the optical spectrum makes this same transition as a function of time (at $t\!\sim\!3$d, for the parameters of this example), while the spectral shape at X-ray frequencies stays put. The Swift era ------------- Physicists, unlike ordinary year-counting mortals, live in ‘eras’. Many are waiting for the LHC era or the Plank era, GRB astronomers are in their ‘Swift era’. Various satellites are currently contributing to a wealth of new data on GRBs and XRFs. Swift is one of them. Within 15 seconds after detection, its 15-150 keV Burst Alert Telescope sends to ground a 1 to 4 arcmin position estimate, for use by robotic optical ground telescopes. In 20 to 75 s, Swift slews to bring the burst location into the field of view of its 0.3-10 keV X-ray Telescope and its 170-650 nm UV/Optical Telescope. With nominal celerity, Swift has filled a gap in GRB data: the very ‘prompt’ X-ray and optical radiations. Swift has established a [canonical behaviour]{} of the X-ray and optical AGs of a large fraction of GRBs. The X-ray fluence decreases very fast from a ‘prompt’ maximum. It subsequently turns into a ‘plateau’. After a time of ${\cal{O}}(1$d), the fluence bends (has a ‘break’, in the usual parlance) and steepens to a power-decline. In fig. \[fpreSwift\]a, this is shown for a Swift GRB. This bend is achromatic: the UV and optical light curves vary in proportion to it. Although all this is considered a surprise, it is not. In fig. \[fpreSwift\]b I show a pre-Swift AG and its interpretation in two models. In fig. \[fpreSwift\]c one can see that the bend of this GRB was achromatic. Even the good old GRB 980425, the first to be clearly associated with a SN, sketched a canonical X-ray light curve, see fig. \[fpreSwift\]d. The $\gamma$ rays of a GRB occur in a series of [pulses]{}, 1 and 2 in the examples of fig. \[try\]. Swift has clearly established that somewhat wider X-ray [flares]{} coincide with the $\gamma$ pulses, having, within errors, the same start-up time. On occasion, even wider optical [humps]{} are seen, as in fig. \[postSwift1\]a. The X-ray counterpart of the second hump in this figure is clearly seen in fig. \[postSwift1\]b. In an XRF the X-ray flares can be very wide, as in the one-flare example of fig. \[postSwift2\]a. In such a case, the accompanying optical ‘humps’ peak very late, at $t\!=\!{\cal{O}}(1$d), as in fig. \[postSwift2\]b. All these interconnected $\gamma$-pulses, X-ray flares and optical humps are described by Eqs. (\[totdist\],\[shape\]). They are obviously manifestations of a common underlying phenomenon, which we shall dig out. Finally, Swift has discovered that not all X-ray light curves are smooth after the onset of their fast decay, as the one in fig. \[postSwift2\]a is. Well after $\gamma$ pulses are no longer seen, relatively weak X-ray flares may still be observable, as is the case in figs. \[postSwift1\]c,\[postSwift2\]d. Breath-taking entities: the astrophysical jets of cannonballs ============================================================= A look at the web –or at the sky, if you have the means– results in the realization that jets are emitted by many astrophysical systems. One impressive case is the quasar Pictor A, shown in figs. (\[Pictor\]a,b). [Somehow]{}, its active galactic nucleus is discontinuously spitting [something that does not appear to expand sideways before it stops and blows up]{}, having by then travelled almost $10^6$ light years. Many such systems have been observed. They are very relativistic: the Lorentz factors (LFs) $\gamma\!\equiv\! E/(mc^2)$ of their ejecta are typically of ${\cal{O}}(10)$. The mechanism responsible for these mighty ejections —suspected to be due to episodes of violent accretion into a very massive black hole— is not understood. In our galaxy there are ‘micro-quasars’, whose central black hole’s mass is a few $M_\odot$. The best studied is GRS 1915+105. In a non-periodic manner, about once a month, this object emits two oppositely directed [cannonballs]{}, travelling at $v\sim 0.92\, c$. When this happens, the continuous X-ray emissions —attributed to an unstable accretion disk— temporarily decrease. Another example is the $\mu$-quasar XTE J1550-564, shown in fig. \[Pictor\]c. The process reminds one of the blobs thrown up as the water closes into the ‘hole’ made by a stone dropped onto its surface, but it is not understood; for quasars and $\mu$-quasars, the ‘cannon’s’ relativistic, general-relativistic, catastrophic, magneto-hydro-dynamic details remain to be filled in! Atomic lines of many elements have been seen in the CBs of $\mu$-quasar SS 433. Thus, at least in this case, the ejecta are made of ordinary matter, and not of a fancier substance, such as $e^+e^-$ pairs. The Cannonball Model: summary ============================= The ‘cannon’ of the CB model is analogous to the ones of quasars and microquasars. In an [ordinary core-collapse]{} SN event, due to the parent star’s rotation, an accretion disk is produced around the newly-born compact object, either by stellar material originally close to the surface of the imploding core, or by more distant stellar matter falling back after the shock’s passage [@GRB1; @ADR]. A CB made of [ordinary-matter plasma]{} is emitted, as in microquasars, when part of the accretion disk falls abruptly onto the compact object. [Long-duration]{} GRBs and [non-solar]{} CRs are produced by these jetted CBs. To agree with observations, CBs must be launched with LFs, $\gamma_0\!\sim\!10^3$, and baryon numbers $N_{_{\rm B}}\!=\!{\cal{O}}(10^{50})$, corresponding to $\sim\!1/2$ of the mass of Mercury, a miserable $\sim\!10^{-7}$th of a solar mass. Two jets, each with $n_{_{\rm CB}}\!=\!\langle n_{_{\rm CB}}\rangle\!\sim\!5$ CBs, carry $$E_{\rm jets}=2\,n_{_{\rm CB}}\,\gamma_0\,N_{_{\rm B}}\,m_p\,c^2 \sim 1.5\times 10^{51}\;\rm erg, \label{Ejets}$$ comparable to the energy of the SN’s non-relativistic shell, that is ${\cal{O}}(1\%)$ of the explosion’s energy, $\sim\!98$% of which is carried away by thermal neutrinos. We have seen that long GRBs are indeed made by SNe, as advocated in the CB model well before the pair GRB030329/SN2003dh convinced the majority. But do SNe emit cannonballs? Until 2003[@SLum030329], there was only one case with data good enough to tell: SN1987A, the core-collapse SN in the LMC, whose neutrino emission was seen. Speckle interferometry data taken 30 and 38 days after the explosion[@Costas] did show two back-to-back relativistic CBs, see fig. \[try2\]e,f. The approaching one was [superluminal]{}: seemingly moving at $v\!>\!c$. A summary of the CB model is given in Fig. \[figCB\]. The [‘inverse’ Compton scattering]{} (ICS) of light by electrons within a CB produces a forward beam of higher-energy photons: a pulse of a GRB or an XRF. The target light is in a temporary reservoir: the [glory]{}, illuminated by the early SN light and illustrated by analogy in fig. \[Pictor\]d. A second mechanism, [synchrotron radiation]{} (SR), takes over later and generally dominates the AG. The $\gamma$ rays ionize the ISM on the CB’s path. The CBs collide with the ISM electrons and nuclei, boosting them to cosmic ray status. The ISM penetrating a CB’s plasma creates turbulent magnetic fields within it. The ISM electrons moving in this field emit the mentioned SR. This paradigm accounts for all properties of GRBs and CRs. -1.cm 2truecm The observed properties of a CB’s radiation depend crucially on the angle $\theta$ of its motion relative to the line of sight to the observer, via the Doppler factor $$\delta = 1/[\gamma(1 - \beta\cos\theta)] \approx 2\gamma/(1 + \theta^2\gamma^2) \label{delta}$$ by which a photon’s energy is boosted from the CB’s rest system to that of a (cosmologically nearby) observer. For an isotropic emission in the CB’s system, the observed photon number, energy flux and luminosity are $\propto \delta,\,\delta^3,\,\delta^4$, respectively, just as in a $\nu$ beam from $\pi$ decay. That makes GRBs observable only extremely close to one of their bipolar CB axes, $\theta\!=\!{\cal{O}}(1/\gamma)\!\sim\!1$ mrad \[typically $\gamma(t=0)\!=\!\gamma_0\!\sim\!\delta_0\!\sim\!10^3$; and AGs are observed till $\gamma(t)\!\sim\!\gamma_0/2$\]. The relation between CB travel-time in the host galaxy, $dt_\star\!=\!dx/(\beta\,c)$, and observer’s time, $t$, is $dt_\star/dt\!=\!\gamma\,\delta/(1\!+\!z)$. Stop in awe at this gigantic factor: a CB whose AG is observed for 1 day may have travelled for ${\cal{O}}(10^6)$ light days, what a fast-motion video! A CB with $\theta\!=\!1/\gamma\!=10^{-3}$ moves in the sky at an apparent transverse velocity of $2000\, c$, yet another large Doppler aberration. GRB afterglows in the CB model =============================== Historically, two GRB phases were distinguished: a prompt one, and the [after]{}-glow. Swift data have filled the gap, there is no longer a very clear distinction. Nor is there a profound difference between the CB-model’s radiation mechanisms, since synchrotron radiation is but Compton scattering on virtual photons and, in a universe whose age is finite, all observed photons were virtual. In the understanding of GRBs in the CB model, SR-dominated AGs came first. The CB-model AG analysis is strictly a ‘model’: it contains many simplifications. But the comparison with data determines the distributions of the relevant parameters. Given these, the predictions for CRs and for the ICS-dominated phase of GRBs (such as all properties of the $\gamma$-ray pulses) involve only independent observations, basic physics and no ‘modeling’. For the reader who might want to move to the more decisive sections, I anticipate the contents of this one. The distribution of $\gamma_0$ and $\gamma_0\,\delta_0$ values of pre-Swift GRBs are shown in fig. \[Distributions\]a,b. The radius of a CB evolves as in fig. \[Distributions\]d. A CB does not expand inertially; for most of its trajectory it has a slowly changing radius, as a common [cannonball]{} does. The baryon number of observed CBs is of ${\cal{O}}(10^{50})$. -1.5cm To determine the fate of a CB, we make the following assumptions. CBs initially expand at $\beta_s\,c\!=\!{\cal{O}}(c/\sqrt{3})$, the relativistic speed of sound, swiftly becoming spherical in their rest system and losing memory of their initial size. For the CB’s baryon number returned by the analysis, this means that CBs become ‘collisionless’ fast: their nuclei and electrons do not often collide with the ISM ones they encounter. Hadron and Thompson cross sections being similar, CBs also become transparent, except to long radio waves, losing their radiation pressure. In agreement with first-principle calculations of the relativistic merger of two plasmas[@Frede], a chaotic magnetic field is generated within a CB by the ISM particles it sweeps in. In accordance with observations of similar plasmas (such as the ISM itself and the CRs it contains) the CB’s magnetic field is in energy equipartition with the impinging ISM, resulting in: $$B_{_{\rm CB}}(\gamma)=3\;{\rm G}\;{(\gamma/ 10^3})\; \left[{n/ (10^{-2}\;{\rm cm}^{-3}})\right]^{1/2}\; , \label{B}$$ where $n$ is the ISM baryonic number density, normalized to a typical value in the ‘superbubbles’ in which most SNe and GRBs are born. In a CB’s rest system the motion of its constituents is an inertial memory of the initial radial expansion, whose kinetic energy is larger than the one of the CB’s magnetic field. An ISM proton entering a CB will meander it its magnetic field and be isotropically reemitted (in the CB’s rest system). The rate of radial momentum loss per unit surface is a surface pressure countering the expansion. We assume that the dominant effect of this pressure is to counteract the expansion. We use Newton’s law to compute the ensuing radial deceleration and the CB’s radius $R(\gamma)$. The results are shown in fig. \[Distributions\]d. A CB initially expands quasi-inertially. It subsequently settles into a slowly evolving radius till it blows up as its motion becomes non-relativistic (DD06), obeying: $$R(\gamma)\approx R_c\,(\gamma_0/\beta\gamma)^{2/3},\;\;{\rm with\,} R_c={\cal{O}}(10^{14}\rm cm),\;for\,typical\, parameters. \label{Radius}$$ This is a complex problem, and ours is a big simplification, once assessed by a cunning referee as “almost baron Munchhausen”. Yet, the result describes the surprising ‘jet self-focusing’ observed, e.g., in Pictor A, see fig. \[Pictor\]a. The collisions with the ISM continuously decelerate a CB. For a given $R(t)$ and ISM baryon number per unit volume $n$, energy-momentum conservation dictates the explicit form of the CB’s diminishing Lorentz factor $\gamma(t)$. Typically $\gamma(t)$ is roughly constant for a day or so of observer’s time, steepening to $\propto\,t^{-1/4}$ thereafter. During the short $\gamma$-ray emission time, $\gamma(t)\!\approx\!\gamma_0$. We assume that practically all of the energy of the ISM electrons (of number density $n_e\!\simeq\!n$) entering a CB is reemitted fast in the form of SR, so that the corresponding observed frequency-integrated AG power per unit area is $dF/(dt\,d\Omega)\!=\!\pi\,R^2\,n_e\,m_e\,c^3\,\gamma^2\,\delta^4/(4\,\pi\,D_L^2)$, with $D_L$ the luminosity distance. The CB deceleration law, dictated by energy-momentum conservation, is equally simple: $M_0\,\gamma_0\,d\gamma\!=\!-\pi\,R^2\,n\,m_p\,\gamma^3\,dx$, for an H-dominated ISM and in the extreme in which the re-emission time of protons is long on the scale of the CB’s deceleration time. For constant $n$, the distance travelled by a CB is: $$x(\gamma)\approx L\,\left[\left({\gamma_0/ \gamma}\right)^{2/3}-1\right],\;\;\; L={3\,N_{_{\rm B}}/( 2\pi\,R_c^2\,n\,\gamma_0^3)}=180\,{\rm pc,}\, \label{distance}$$ where the number is for $N_{_{\rm B}}\!=\!10^{50}$, $n\!=\!10^{-2}\,{\rm cm}^{-3}$, $R_c\!=\!10^{14}\,{\rm cm}$, $\gamma_0\!=\!10^3$. Given all this, it appears easy to extract from an AG’s normalization and shape the values of $\theta$, $\gamma_0$ and $N_{_{\rm B}}$, if one trusts the estimate of $R(\gamma)$ and uses a typical $n$. Limited observational information makes life a bit harder. The spectrum of fig. \[f1\]f is actually the one predicted by the CB model, illustrated for a typical choice of parameters. The chromatic ‘bends’ shown as dots in this figure, for instance, are ‘injection bends’: the typical SR energy, in the CB’s magnetic field, of the electrons entering it, at time $t$, with a (relative) LF $\gamma=\gamma(t)$. The small portion of the spectrum above the bend is emitted by a tiny fraction of electrons ‘Fermi-accelerated’ in the CB’s turbulent magnetic fields to a pre-synchrotron-cooling spectrum $E_e^{-p}$, with $p\!\sim\!2.2$. The prediction fits with no exception the AGs of the first score of well measured GRBs (DDD02/03) of known $z$. But only on rare occasions can one clearly see in an AG the contributions of the various CBs seen in the $\gamma$-ray count rate (a counterexample is the GRB of figs. \[try\]b, \[try2\]a,b). Thus, generally, the parameters extracted from AG fits refer to a dominant CB or to an average over CBs. After an observer’s day or so, the optical and X-ray AG are typically SR-dominated, are above the injection bend, and are of the approximate form: $$F_\nu\propto n_{_{\rm CB}}\,n^{1.05}\,R_c^2\,\gamma^{2.3}\,\delta^{4.1}\,\nu^{-1.1}, \label{SRFlux}$$ where the unwritten proportionality factors, such as $D_L^{-2}$, are known. From a fit to the shape of $F_\nu(t)$ one obtains $\theta\,\gamma_0$ and the combination $L$ of Eq. (\[distance\]). At late times $F_\nu\!\propto\!\gamma(t)^{6.4}\!\propto\!\gamma_0^{6.4}$ with a coefficient determined by the other fit parameters. The value of $\gamma_0$ is extracted from the 6.4 root of the inverse of Eq. (\[SRFlux\]), so that, for a result within a factor of 2, one can tolerate large errors in the chosen $n$ or in the estimate of $R_c^2$. Trusting these, one can extract $N_{_{\rm B}}$ from $L$, perhaps with an uncertainty of one order of magnitude[^1]. Eq. (\[SRFlux\]) has been used to fit, after the early fast fall-off, the X-ray and optical data of figs. \[fpreSwift\],\[postSwift1\],\[postSwift2\]. The required form of $\gamma(t)$ is Eq. (\[fpreSwift\]), supplemented by the relation between CB’s mileage and observer’s time, see the end of Sec. 5. A GRB’s $\gamma$ rays in the CB model (DD06) ============================================ A pulse of a GRB is made by a CB crossing the parent star’s [glory]{}. The glory is a reservoir of non-radially directed light, fed by the parent star’s luminosity, as in the artist’s view of fig. \[Pictor\]d. For the best studied GRB-associated SN, 1998bw, and for ${\cal{O}}(1\rm d)$ after the explosion, the luminosity was $L_{_{\rm SN}}\sim\!5\!\times\!10^{52}$ erg/s, in photons of typical energy $E_i\!\sim\! 1$ eV. We adopt these values as ‘priors’ (parameters to be used in calculations, but independent of the CB model). Massive stars destined to ‘go supernova’ eject solar-mass amounts of matter in successive explosions during their last few thousand pre-SN years. At the ‘close’ distances of ${\cal{O}}(10^{16}$ cm) relevant here, these stellar coughs generate a thick layer of ‘wind-fed’ material with an approximate density profile $\rho\!\propto\!r^{-2}$ and normalization $\rho\,r^2\!\sim \! 10^{16}$ g cm$^{-1}$, the last prior we need. The very early UV flash of the SN suffices to ionize the wind-fed matter. The Thompson cross section $\sigma_{_{\rm T}}$ is such that this matter is semitransparent: $\sigma_{_{\rm T}}\,\rho\,r^2/m_p\!=\!4\!\times\!10^{15}$ cm. This means that the number of times a SN photon reinteracts on its way out –becoming ‘non-radial’– is of ${\cal{O}}(1)$, and that the number density of such photons is $n_\gamma(r)\!\sim\!L_{_{\rm SN}}/(4\pi\,r^2\,c\,E_i)$. From emission-time to the time it is still one $\gamma$-ray interaction length inside the ‘wind’, a CB has travelled for $$t_{\rm tr}^{\rm w}=(0.3 \,{\rm s})\,{\rho\,r^2\over 10^{16}\,{\rm g\,cm}^{-1}} \,{1+z\over 2}\,{10^6\over \gamma_0\,\delta_0} \label{ttrans}$$ of observer’s time. That is a typical $\gamma$-ray pulse rise-time in a GRB, and the reason why, closing the loop, distances of ${\cal{O}}(10^{16}$ cm) were relevant. In the collapse of a rotating star, material from ‘polar’ directions should fall more efficiently than from equatorial directions. The CBs would then be emitted into relatively empty space. We assume that the wind-material is also under-dense in the polar directions. This is not the case for the glory’s photons, which have been scattered by the wind’s matter, and partially isotropised. During the production of $\gamma$-rays by ICS, $\gamma\!\simeq\!\gamma_0$. Consider an electron, comoving with a CB at $\gamma\!=\!\gamma_0$, and a photon of energy $E_i$ moving at an angle $\theta_i$ relative to $\vec r$. They Compton-scatter. The outgoing photon is viewed at an angle $\theta$. Its energy is totally determined: $$E_p \!=\! {\gamma\,\delta\over 1\!+\!z}\, (1\!+\!\cos\theta_i)\, E_i \!=\! (250\;{\rm keV})\; \sigma\;{1\!+\!\cos\theta_i\over 1/2}\, {E_i\over 1\;\rm eV},\; \sigma\! \equiv\! {\gamma\,\delta\over 10^6}\,{2\over 1\!+\!z}\; , \label{boosting}$$ where I set $\beta\!\approx\! 1$ and, for a [semi]{} transparent wind, $\langle\cos\theta_i\rangle\!\sim\!-1/2$. For pre-Swift GRBs $\langle z \rangle\!\approx\! 1$ and, for the typical $\gamma$ and $\delta$, $E\!=\!250$ keV, the average peak or ‘break’ energy in Eq. (\[totdist\]). From the fits to the AGs of the subset of known $z$, we could determine the distribution of $\sigma$ values, see fig. \[Distributions\]b. Its fitted result is used in Eq. (\[boosting\]) to predict the overall $E_p$ distribution, see fig. \[Distributions\]c. The rest of the properties of a GRB’s pulse can be derived on similarly trivial grounds and with hardly more toil. During the GRB phase a CB is still expanding inertially at a speed $\beta_s\,c$. It becomes transparent when its radius is $R_{\rm tr}\!\sim\![3\,\sigma_{_{\rm T}}\,N_{_{\rm B}}/(4\pi)]^{1/2}$, at an observer’s time very close to that of Eq. (\[ttrans\]), for typical parameters. One can simply count the number of ICS interactions of a CB’s electrons with the glory, multiply by their energy, Eq. (\[boosting\]), and figure out the isotropic-equivalent energy deduced by an observer at an angle $\theta$: $$E_\gamma^{\rm iso} \simeq {\delta^3\, L_{_{\rm SN}}\,N_{_{\rm CB}}\,\beta_s\over 6\, c}\, \sqrt{\sigma_{_{\rm T}}\, N_{_{\rm B}}\over 4\, \pi}\sim 3.2\! \times\! 10^{53}\,{\rm erg}, \label{eiso}$$ where the number is for our typical parameters, and agrees with observation. ?‘ Is it Inverse Compton Scattering ... --------------------------------------- The $\gamma$ and $\delta$ dependance in Eqs. (\[boosting\],\[eiso\]) is purely ‘kinematical’, but specific to ICS: it would be different for self-Compton or synchrotron radiation. To verify that the $\gamma$ rays of a GRB are made by ICS, as proposed[@SD] by Shaviv and Dar, we may look at the correlations[@corr1; @corr2] between GRB observables. In the CB model, the $(\gamma,\delta,z)$ dependences of the peak isotropic luminosity of a GRB, $L_p^{\rm iso}$; its pulse rise-time, $t_{\rm rise}$; and the lag-time between the peaks of a pulse at different energies, $t_{\rm lag}$; are also simply derived[@corr2] to be: $$%E_\gamma^{\rm iso}\propto\delta^3,\;\;\; (1+z)^2\,L_p^{\rm iso}\propto \delta^4,\;\;\; %(1+z)\,E_p\propto \gamma\,\delta,\;\;\; t_{\rm rise}\propto (1+z)/(\gamma\,\delta),\;\;\; t_{\rm lag}\propto (1+z)^2/(\delta^2\,\gamma^2). %V\propto \gamma\,\delta/(1+z). \label{brief}$$ I have not specified the numerical coefficients in Eqs. (\[brief\]), which are explicit, as in Eqs. (\[boosting\],\[eiso\]). Of all the parameters and priors in these expressions, the one explicitly varying by orders of magnitude by simply changing the observer’s angle is $\delta(\gamma,\theta)$, making it the prime putative cause of case-by-case variability. For such a cause, Eqs. (\[eiso\]) and the first of (\[brief\]) imply that $E_\gamma^{\rm iso}\!\propto\! [(1+z)^2\,L_p^{\rm iso}]^{3/4}$. This is tested in fig. \[f1\]a. A most celebrated correlation is the $[E_p\,,E_\gamma^{\rm iso}]$ one, see Fig. \[f1\]b. It evolves from $E_p\!\propto\![E_\gamma^{\rm iso}]^{1/3}$ for small $E_p$, to $E_p\!\propto\![E_\gamma^{\rm iso}]^{2/3}$ for large $E_p$. This is because the angle subtended by a moving CB from its place of origin is $\beta_s/\gamma$, comparable to the beaming aperture, $1/\gamma$, of the radiation from a point on its surface. Integration over this surface implies that, for $\theta\!\ll\! 1/\gamma $, $\delta\!\propto\!\gamma$, while in the opposite case $\delta$ varies independently. The straight lines in fig. \[f1\]b are the central expectations of Eqs. (\[boosting\],\[eiso\]), the data are fit to the predicted evolving power law. The predicted $[t_{\rm lag},L_p^{\rm iso}]$ and $[t_{\rm rise},L_p^{\rm iso}]$ correlations are tested in figs. \[f1\]c,d. The seal of authenticity of inverse Compton scattering —by a quasi-point-like electron beam— is unmistakable in all of this, QED. ... on a Glory’s light ? ------------------------ The ‘target’ photons subject to ICS by the CB’s electrons have very specific properties. Their number-density, $n_\gamma(r)\!\propto\!L_{_{\rm SN}}/r^2$, translates into the $\sim\!t^{-2}$ late-time dependence of the number of photons in a pulse since, once a CB is transparent to radiation, ICS by its electrons simply ‘reads’ the target-photon distribution. As a CB exits the wind-fed domain, the photons it scatters are becoming more radial, so that $1\!+\!\cos\theta_i\!\to\!r^{-2}\!\propto\! t^{-2}$ in Eq. (\[boosting\]). For a semi-transparent wind material, which we have studied in analytical approximations and via Montecarlo, this asymptotic behaviour is reached fast and is approximately correct at all $t$. This means that the energies of the scattered photons evolve with observer’s time as $t^{-2}$: the ‘$E\,t^2$ law’ of Eq. (\[shape\]) and fig. \[f1\]e. The pulse shape and the spectrum -------------------------------- The spectrum of a GRB, Eq. (\[totdist\]), and the time-dependence of its pulses, Eq. (\[shape\]), describe the data well, and are actually analytical approximations to the results of ICS of an average CB on a typical glory. The spectrum of a semitransparent glory has a ‘thermal bremsstrahlung’ shape, $dn_\gamma/dE_i\!\propto\!(T_i/E_i)^\alpha\,{\rm Exp}[-E_i/T_i]$, with $\alpha\!\sim\!1$ and $T_i\!\sim\! 1$ eV. The first term in Eq. (\[totdist\]) is this same spectrum, boosted by ICS as in Eq. (\[boosting\]), by electrons comoving with the CB, $E_e\!=\!\gamma\,m_e\,c^2$. The second term is due to ICS by ‘knock-on’ electrons (generated while the CB is not yet collisionless) and electrons ‘Fermi-accelerated’ by the CB’s turbulent magnetic fields. They both have a spectrum $dn_e/dE_e\!\propto\!E_e^{-\beta}$, with $\beta\!\sim\!2$ to 2.2. They are a small fraction of the CB’s electrons, reflected in the parameter $b$, which we cannot predict. The temporal shape of a pulse has an exponential rise due to the CB and the windy material becoming transparent at a time $\sim\!t_{\rm tr}^{\rm w}$, see Eq. (\[ttrans\]), the width of pulse (in $\gamma$ rays) is a few $t_{\rm tr}^{\rm w}$, the subsequent decay is $\propto\!t^{-2}$. The time-energy correlations obey the ‘$E\,t^2$ law’. All as observed. Polarization ------------ A tell-tale signature of ICS is the high degree of polarization. For a pointlike CB the prediction[@SD] is $\Pi\!\approx\!2\,\theta^2\,\gamma^2/(1+\theta^4\,\gamma^4)$, peaking at 100% at $\theta=1/\gamma$, the most probable $\theta$, corresponding to $90^{\rm o}$ in the CB’s system. For an expanding CB, $\Pi$ is a little smaller. For SR, which dominates the AGs at sufficiently late times, the expectation is $\Pi\!\approx\!0$. The $\gamma$-ray polarization has been measured, with considerable toil, in 4 GRBs. It is always compatible, within very large errors, with 100%. The situation is unresolved[@Pol]. I shall not discuss it. Detailed Swift light curves and hardness ratios =============================================== Swift has abundantly filled its goal to provide X-ray, UV and optical data starting briefly after the detection of a GRB: compare the Swift result of fig. 8a to the pre-Swift data in fig. 8b. In the CB-model description of the data in figs. 8,9,10, the abruptly falling signal is the tail of one or several $\gamma$-ray pulses or X-ray flares, produced by ICS and jointly described by Eqs. (\[totdist\],\[shape\]). The following ‘afterglow’, its less pronounced decay and subsequent achromatic ‘bend’ are due to the CBs’ synchrotron radiation, described by Eq. (\[SRFlux\]). Thanks to the quality of SWIFT data one can proceed to test these CB-model predictions in detail. The two prompt optical ‘humps’ of GRB 060206 in fig. \[postSwift1\]a are the ICS low-energy counterparts of its two late X-ray flares of fig. \[postSwift1\]b, simultaneoulsly fit by Eqs. (\[totdist\],\[shape\]). Swift provides a rough measure of a GRB’s spectrum: the [hardness ratio]{} of count rates in the \[1.5-10\] keV and \[0.3-1.5\] keV intervals. Given the case-by-case parameters of a CB-model fit to the \[0.3-10\] keV light curve, one can estimate the corresponding hardness ratio[@DDDDecline]. This is done in figs. \[postSwift1\]c,d and \[postSwift2\]a,c for GRB 060904 and XRF 060218, respectively. This last XFR is observed at a ‘large’ angle, $\theta\!\sim\!5$ mrad and a correspondingly small $\delta_0$, its single X-ray pulse is, in accordance with Eq. (\[ttrans\]), relatively wide. The optical and UV counterparts of the X-ray pulse are clearly visible as the ‘humps’ in the optical data of fig. \[postSwift2\]b. Given the ‘$E\,t^2$ law’ of Section 7.3, the pulse peak times at different frequencies are simply related: $t_{\rm peak}\!\propto\!E^{-1/2}$. The prediction, an example of the ubiquitous $1/r^2$ law of 3-D physics, is tested in fig. \[f1\]e. The peak fluxes at all frequencies are also related as dictated[@XSwift] by Eq. (\[totdist\]). The adequacy of the CB model over many decades in flux and time is exemplified by the X-ray light curve of GRB 061121 in fig. \[postSwift2\]d. The predictions for the peak $\gamma$-ray energy of Eq. (\[boosting\]), its distribution as in fig. \[Distributions\]c, the GRB spectrum of Eq. (\[totdist\]), and the correlations of figs. \[f1\]a-d are clear proof that ICS is the prompt GRB mechanism. The test of the $E\,t^2$ law in fig. \[f1\]e corroborates that the ‘target light’ becomes increasingly radially directed with distance: [Inverse Compton Scattering on a ‘glory’s light’ by the electrons in CBs is responsible for the $\gamma$-ray pulses of a GRB and their sister X-ray flares and optical humps.]{} The properties of the subsequent [synchrotron-dominated afterglows]{} are also in accordance with the CB model. The GRB/SN association in the CB model ====================================== We have gathered very considerable evidence that the LFs and viewing angles of [observed]{} GRBs are $\gamma_0\!=\!{\cal{O}}(10^3)$ and $\theta\!=\!{\cal{O}}(1)$ mrad. The fraction of GRBs beamed towards us is $\sim\!\theta^2\!=\!{\cal{O}}(10^{-6})$. The number of such observed GRBs (with a hypothetical $4\,\pi$ coverage) is a few a day. The same coverage would result in the observation of a few million core-collapse SN per day, in the visible Universe. These numbers are compatible with the extreme conclusion that $all$ these SNe emit GRBs, but the estimates and errors are sufficient to accommodate a one order of magnitude smaller fraction, which would be compatible with most Type Ib/c emitting (long) GRBs. Short Hard $\gamma$-ray Bursts (SHBs) ===================================== SHBs share with (long) GRBs the properties not reflected in their name. A good fraction of SHBs have ‘canonical’ X-ray light curves. The origin of SHBs is not well established, in contrast to that of GRBs and XRFs. Clues to the origin and production mechanism of SHBs are provided by their similarity to long GRBs. The X-ray light curves of some well-sampled SHBs are ‘canonical’. The similarities suggest common mechanisms generating the GRB and SHB radiations. This is expected in the CB model, wherein both burst types are produced by highly relativistic, narrowly collimated, bipolar jets of CBs, ejected in stellar processes[@SD]. The mechanisms for their prompt and AG emissions (ICS and synchrotron) coincide with the ones of GRBs. The ‘engine’ is different; it is a core-collapse supernova for GRBs and XRFs, in SHBs it may be a merger (of two neutron stars or a neutron star and a black hole), the result of mass accretion episodes on compact objects in close binaries (e.g. microquasars), or phase transitions in increasingly compactified stars (neutron stars, hyper-stars or quark stars), induced by accretion, cooling, or angular-momentum loss. In the CB model, the ‘master formulae’ describing prompt and afterglow emissions in long GRBs are directly applicable to SHBs, provided the parameters of the CBs, of the glory, and of the circumburst environment, are replaced by those adequate for SHBs. This results in a good description of the data[@DDDSHB]. Cosmic Rays in the CB model =========================== In the CB model, CRs are as simple to understand as GRBs. If relativistic CBs are indeed ejected by a good fraction of core-collapse SNe, it is inevitable to ask what they do as they travel in the ISM. The answer is that they make CRs with the observed properties, simply by interacting with the constituents of the ISM, previously ionized by the $\gamma$ rays of the accompanying GRB. Early in their voyage, CBs act as [Compton relativistic rackets]{}, in boosting a glory’s photon to $\gamma$-ray status. Analogously, all along their trajectories, CBs act as [Lorentz relativistic rackets,]{} in boosting an ISM nucleus or electron to CR status. Once again, the necessary input is two-fold. On the one hand, there are the properties of CBs: the average number of significant GRB pulses (or CBs) per jet (5), the $\gamma_0$ distribution of fig. \[Distributions\]a, and the $N_{_{\rm B}}\!\sim\! 10^{50}$ estimate. On the other hand, there are a few ‘priors’, items of information independent of the CR properties: the rate of core-collapse SNe, the relative abundances, $n_{_{A}}$ (of the elements of atomic number $A$) in the ISM, and the properties of Galactic magnetic fields. We shall see that the CB-model predictions for the normalization of CR spectra are correct to within a factor of ${\cal{O}}(3)$, while the ratios between elements are correct within errors. In figs. 1, 3 and 4a, the predictions have been made to adjust the data, not reflecting the common overall normalization uncertainty. Relativistic rackets: The knees ------------------------------- Our simplest result concerns the ‘knees’ of the all-particle spectrum in fig. \[AllPart\]a and of the main individual elements in fig. \[KASKADE\]. The essence of their understanding is kinematical and trivial. In an [elastic]{} interaction of a CB at rest with ISM electrons or ions of LF $\gamma$, the light recoiling particles (of mass $m\!\approx\!A\,m_p$) retain their incoming energy. Viewed in the ISM rest system, they have, for large $\gamma$, a flat spectrum extending up to $E\simeq 2\,\gamma^2\,m\,c^2$ \[this is recognizable as the forward, massive-particle, $z\!=\!0$, analog of Eq. (\[boosting\])\]. Thus, a moving CB is a gorgeous [Lorentz-boost accelerator:]{} the particles it elastically scatters reach up to, for $\gamma=\gamma_0\!=\!(1\,{\rm to}\,1.5)\!\times\!10^3$, an $A$-dependent [knee]{} energy $E_{{\rm knee}}(A)\!\sim\! (2\,{\rm to}\,4)\!\times\! 10^{15}\,A\,{\rm eV.}$ If this trivial process is the main accelerator of CRs, there must be a feature in the CR spectra: endpoints at $E_{{\rm knee}}(A)$. The arrows in fig. \[KASKADE\] show that the H and He data are compatible with this prediction. So does the second knee of fig. \[AllPart\]a, the predicted Fe knee. The CR flux above the H knee, to which we shall return, is $\sim\!10^{-15}$ of the total. The spectra below the knee -------------------------- The ‘elastic’ scattering we just described is dominant below the knees. To compute the resulting spectrum, we assume that the ISM particles a CB intercepts, trapped in its magnetic mesh, reexit it by diffusion, isotropically in the CB’s system, and with the same ‘confinement’ law, Eq. (\[confinement\]), as in the Galaxy (the opposite assumption, that they are immediately elastically scattered, yields a slightly different spectral index). The CB deceleration law is Eq. (\[distance\]), its radius evolves as in Eq. (\[Radius\]). A modest amount of algebra gives a simple result (DD06), which, for $\gamma\!>\! 2$ and to a good approximation, reads[^2]: $${dF_{\rm elast}\over d\gamma_{_{\rm CR}}} \propto n_{_A} \left({A\over Z}\right)^{\beta_{\rm conf}} \int_1^{\gamma_0}{d\gamma\over \gamma^{7/3}} \, \int_{\rm max[\gamma,\gamma_{_{\rm CR}}/(2\,\gamma)]} ^{\rm min[\gamma_0,2\,\gamma\,\gamma_{_{\rm CR}}]} {d\gamma_{\rm co}\over \gamma_{\rm co}^{4}}\; , \label{NRFlux}$$ where $\gamma_{_{\rm CR}}$ is the CR’s LF, and $\beta_{\rm conf}$ is the same confinement index as in Eq. (\[confinement\]). The flux $dF_{\rm elast}/d\gamma_{_{\rm CR}}$ depends on the priors $n_{_A}$, ${\beta_{\rm conf}}$, and $\gamma_0$, but not on any parameter specific to the mechanism of CR acceleration. But for the normalization, this flux is $A$-independent. In the large range in which it is roughly a power law, $dF_{\rm elast}/d\gamma_{_{\rm CR}}\!\propto\![\gamma_{_{\rm CR}}]^{-\beta_{_{\rm CR}}}$, with $\beta_{_{\rm CR}}\!=\!13/6\!\approx\!2.17$. The H, He and Fe fluxes of fig. \[KASKADE\] are given by Eq. (\[NRFlux\]), modified by the Galactic confinement $\tau$-dependence of Eq. (\[confinement\]), with $\beta_{\rm conf}\!=\!0.6$. The fastest-dropping curve in fig. \[KASKADE\]a corresponds to a fixed $\gamma_0$. The other two curves are for the $\gamma_0$ distribution of fig. \[Distributions\]a, and one twice as wide. The low-energy data of fig. \[AllPart\]b are also described by Eq. (\[NRFlux\]), whose shape in this region (the ‘hip’, also visible in fig. \[KASKADE\]c for Fe) is insensitive to $\gamma_0$ and, thus, parameter-independent. The relative abundances ----------------------- It is customary to discuss the composition of CRs at a fixed energy $E_{_A}=1$ TeV. This energy is relativistic, below the corresponding knees for all $A$, and in the domain wherein the fluxes are dominantly elastic and well approximated by a power law of index $\beta_{\rm th}\!=\!\beta_{\rm elast}\!+\!\beta_{\rm conf}\!\simeq\!2.77$. Expressed in terms of energy ($E_{_A}\!\propto\! A\,\gamma$), and modified by confinement as in Eq. (\[confinement\]), Eq. (\[NRFlux\]) becomes: $${dF_{\rm obs}/ dE_{_A}}\propto \bar{n}_{_A}\,A^{\beta_{\rm th}-1} \,E_{_A}^{-\beta_{\rm th}},\;\;\; X_{_{\rm CR}}(A)=(\bar{n}_{_A}/ \bar{n}_p)\,A^{1.77}, \label{compo}$$ with $\bar{n}_{_A}$ an average ISM abundance and $X_{_{\rm CR}}(A)$ the CR abundances relative to H, at fixed $E$. The results, for input $\bar{n}_{_A}$’s in the ‘superbubbles’ wherein most SNe occur, are shown in Fig. \[abundances\]b. In these regions, the abundances are a factor $\sim\!3$ more ‘metallic’ than solar (a ‘metal’ is anything with $Z\!>\!2$). Eq. (\[compo\]) snugly reproduces the large enhancements of the heavy-CR relative abundances, in comparison with solar or superbubble abundances (e.g. $A^{1.77}\!=\! 1242$ for Fe). The essence of this result is deceptively simple: in the kinematics of the collision of a heavy object (a CB) and a light one (the ISM nucleus), their mass ratio ($N_{_{\rm B}}/A\sim\!\infty$!) is irrelevant. Above the knees --------------- We discussed around Eq. (\[B\]) the generation of turbulently-moving magnetic fields (MFs) in the merger of two plasmas. Charged particles interacting with these fields tend to gain energy: a relativistic-injection, ‘Fermi’ acceleration process, for which numerical analyses[@Frede] result in a spectrum $dN/dE\!\propto\!E^{-2.2}$, $p\!\sim\!2.2$. For the ISM/CB merger, we (DD04) approximate the spectrum of particles accelerated within a CB, in its rest system, as: $${dN/d\gamma_{_A}}\propto\gamma_{_A}^{-2.2}\, \Theta(\gamma_{_A}\!-\!\gamma)\,\Theta[\gamma_{\rm max}(\gamma)\!-\!\gamma_{_A}], \, \gamma_{\rm max} \simeq 10^5\;\gamma_0^{2/3}\;(Z/A)\; \gamma^{1/3}, \label{gammaA}$$ The first $\Theta$ function reflects the fact that it is much more likely for the light particles to gain than to lose energy in their elastic collisions with the heavy ‘particles’ (the CB’s turbulent MF domains). The second $\Theta$ is the Larmor cutoff implied by the finite radius and MF of a CB, with a numerical value given for the typical adopted parameters. But for the small dependence of $\gamma_{\rm max}$ on the nuclear identity (the factor $Z/A$), the spectrum of Eq. (\[gammaA\]) is universal. Boosted by the CB’s motion, an accelerated and re-emitted particle may reach a Larmor-limited $\gamma_{_{\rm CR}}{[\rm max]}\!=\!2\,\gamma\,\gamma_{\rm max}$, a bit larger, for $\gamma\!=\!\gamma_0\!\sim\!1.5\!\times\!10^3$, than the corresponding GZK cutoffs. Our model has a [single source]{}, CBs, for the acceleration of CRs from the lowest to the largest observed energies. The calculation of the ‘elastic’ spectrum of Eq. (\[NRFlux\]) was done for the bulk of the ISM particles entering a CB, assuming that they were not significantly Fermi-accelerated, but kept their incoming energy, i.e. $dN/d\gamma_{_A}\!\propto\!\delta(\gamma_{_A}-\gamma)$. The ‘inelastic’ spectrum, with $dN/d\gamma_{_A}$ as in Eq. (\[NRFlux\]), yields an equally simple result. The two $E^2$-weighed spectra are shown (for H) in fig. \[DDelinel\]. The inelastic contribution is a tiny fraction of the total, and is negligible below the knee, a point at which we may compare the ratio of fluxes, $f$, the only parameter freely fit to the CR data. The boost of ISM particles by a CB and their acceleration within it are mass-independent, so that the ratio $f$ is universal. \[DDelinel\] The $E^3$-weighed [source]{} spectra for the main elements are shown in fig. \[Groups\]a. They are very different from the [observed]{} spectra of fig. \[Groups\]b, for many reasons. Below the ankle(s) the slopes differ due to Galactic confinement, see Eq. (\[confinement\]). Above the ankles the flux from Galactic sources is strongly suppressed: we would see their straight-moving CRs only for CB jets pointing to us. The CRs above the ankle are mainly extragalactic in origin, and they also cross the Galaxy just once. Extragalactic CRs of $A\!>\!1$ are efficiently photo-dissociated by the cosmic infrared light. Extragalactic CRs are GZK-cutoff. All this can be modeled with patience and fair confidence. Below the ankle extragalactic CRs may have to fight the CR ‘wind’ of our Galaxy, analogous to that of the Sun. We have covered our lack of information on this subject by choosing two extreme possibilities (DD06), resulting in the two curves of figs. \[AllPart\]a and \[highE\]a,b. \[Groups\] In fig. \[highE\]b I have converted the results of fig. \[Groups\]b into a prediction for $\langle{\rm Ln}\,A(E)\rangle$. The flux at the second knee is dominated by Galactic Fe at its knee. Thereafter this flux decreases abruptly to let extragalactic H dominate all the way from the ankle to the nominal position of the proton’s GZK cutoff. Above that point the high-energy tail of Galactic Fe may dominate again. The CR luminosity, and the overall normalization of the CR flux --------------------------------------------------------------- The rate of core-collapse SNe in our Galaxy is $R_{_{\rm SN}}\!\sim\!2$ per century. In the CB model, we contend that $\sim\!50$% of the energy of CRs is transfered to the magnetic fields [they]{} generate[@DDMag]. If all core-collapse SNe emit CBs, the Galactic CR luminosity should be $L_{_{\rm CR}}\!\sim\!R_{_{\rm SN}}\,E_{\rm jets}/2\!\sim\!4.7\!\times\!10^{41}$ erg s$^{-1}$, with $E_{\rm jets}$ as in Eq. (\[Ejets\]). This is 3 times larger than the rhs of Eq. (\[CRlum\]). The ‘discrepancy’ is not worrisome. A smaller fraction of SNe may generate high-$\gamma_0$ CBs. The rhs of Eq. (\[CRlum\]) is for ‘standard’ CRs, but the confinement volume and time of the CB model are non-standard by factors of $\sim\!10$. All inputs are fairly uncertain. The calculation of the flux above the ankle is lengthy but straightforward. But for the GZK effect, its shape is that of the source H flux, since protons at that energy should escape other galaxies directly, and enter ours unhindered. Its normalization per SN is fixed. The SN rate per unit volume is measured in the local Universe. The overall flux is the result of the integration over redshift of the flux from past SNe. The integrand must be properly red-shifted and weighed with the star-formation rate as a function of $z$ (SN progenitors have short lives on Hubble-time scales). The integration in $z$ is an integration over look-back time, as opposed to distance, since CRs do not travel straight. The error in the result is hard to estimate, its central value is within a factor of 2 of the observations (DD06). This explains the coincidence that the ankle is the escape energy of protons from the Galaxy [and]{} the place where the extragalactic flux –not enhanced by confinement and thus less steep– begins to dominate. CR diffusion, CR electrons, and the $\gamma$ background radiation ----------------------------------------------------------------- In the standard paradigm CRs are accelerated by the nonrelativistic ejecta of SNe. SNe occur mainly in the central realms of the Galaxy, so that CRs must diffuse to arrive to our location. A directional asymmetry is predicted, and not observed[@Plaga]. For CR electrons the problem is even more severe: their cooling time in the Galaxy’s light, and magnetic fields, is so short that they should have lost all their energy on their way here. Our source distribution is totally non-standard, CBs generate CRs all along their many-kpc-long trajectories, see Eq. (\[distance\]) and fig. \[figCB\]c. Depending on the ISM density profile they encounter, CBs may travel for up to tens of kpc before they become nonrelativistic. It takes some $6\!\times\! 10^4$ years to travel 20 kpc at $v\!\sim\!c$. If a Galactic SN occurs every 50 years, and emits an average of 10 CBs, there are currently several thousand CBs in the Galaxy and its halo. This is a very diffuse CR source, satisfactory in view of the previous paragraph. We have not yet studied the CR source-distribution and diffusion in detail. Below their knee at $2\,\gamma_0^2\,m_e\,c^2\!\sim\!2.3$ TeV, the source spectrum of CB-accelerated electrons has the same index as that of nuclei: $dN/dE_e\!\propto\!E^{-\beta_{_{\rm CR}}}$, $\beta_{_{\rm CR}}\!\approx\!2.17$. The predicted spectrum[@GBR], steepened by radiative energy loses, has an index $\beta_e\!=\!\beta_{_{\rm CR}}+1\!\approx\!3.17$. Its observed slope[@Slope] is $3.2 \pm 0.1$ above $E_e\!\sim\!10$ GeV, an energy below which other losses should dominate (DD06). The Gamma Background Radiation (GBR), measured by EGRET from a few MeV to $\sim\!10^5$ MeV, was argued to be dominantly of cosmological origin, in directions above the disk of the Galaxy and away from its bulge[@EGRET]. A more careful analysis reveals a significant correlation of its intensity with our position relative to the Galactic centre[@GBR]. The CB-model reproduces this correlation, provided a good fraction of the GBR is generated by CR electrons at high galactic latitudes, as they cool radiatively by the very same process that steepens their spectrum. The predicted index of the radiated GBR photons is $\beta_\gamma\!=\!(\beta_e\!-\!1)/2\!=\!2.08$. The observed one[@EGRET] is $2.10\!\pm\!0.03$. If CBs are so pervasive, why are they not readily observed? =========================================================== The answer is simple. The CBs of SNe are tiny astrophysical objects: their typical mass is half of the mass of Mercury. Their energy flux at all frequencies is $\propto\!\delta^3$, large only when their Lorentz factors are large. But then, the radiation is also extraordinarily collimated, it can only be seen nearly on-axis. Typically, observed SNe are too far to [photograph]{} their CBs with sufficient resolution. Only in two SN explosions that took place close enough, the CBs were in practice observable. One case was SN1987A, located in the LMC, whose approaching and receding CBs were photographed, see fig. \[try2\]e,f. The other case was SN2003dh, associated with GRB030329, at $z=0.1685$. In the CB model interpretation, its two approaching CBs were first ‘seen’, and fit, as the two-peak $\gamma$-ray light curve of fig. \[try\]b and the two-shoulder AG of fig. \[try2\]a,b. This allowed us to estimate the time-varying angle of their superluminal motion in the sky[@SLum030329]. Two sources or ‘components’ were indeed clearly seen in radio observations at a certain date, coincident with an optical AG rebrightening. We claim that the data agree with our expectations[^3], including the predicted inter-CB separation[@SLum030329] of fig. \[try2\]d. The observers claimed the contrary, though the evidence for the weaker ‘second component’ is $>20\sigma$. They report[@Taylor] that this component is ‘not expected in the standard model’. The unpublished and no-doubt spectacular-discovery picture of the two superluminally moving sources would have been worth a thousand words... in support of the CB model. Other CR sources ================ We have defended the simplistic view that CBs from SNe is all one needs to generate CRs at all energies. The recent data of Auger[@Auger] show a significant correlation between the arrival directions of ultra-high energy CRs (UHECRs) and Active Galactic Nuclei (AGNs) located within a distance of 75 Mpc ($z\!\sim\!0.02$), comparable to the GZK ‘horizon’ at the observed energies, $E\!>\! 56$ EeV. As the authors discuss, this does not mean that AGNs are the actual sources[@Rus], for AGNs are themselves correlated with matter and with active regions of enhanced stellar birth and death. More work on correlations is no doubt in progress. A search for correlations with GRBs is less hopeful, for the fraction of them observed from within $z\!\sim\!0.02$ is negligible, and the observed ‘correlated’ CRs may have been bent by magnetic fields up to a few degrees, implying a possible delay of millenia between the arrival times of $\gamma$’s and CRs. Observations of AGNs are an ingredient of the ‘inspiration’ of the CB model, as we have discussed in connection with Pictor A, see fig. \[Pictor\]a,b. Naturally, we have estimated their contribution to the UHECR proton flux, concluding that they may constitute at most 1 to 10% of the flux generated by extragalactic SNe (DD04). The estimate is for the energy-integrated flux; in applying it to UHECRs we assumed the same energy dependence for the flux generated –by the same mechanisms– by the CBs of SNe and AGNs (a small difference of spectral index implies an enormous uncertainty). In a subsequent study[@DDGBR2] of the CR electron flux, assumed to be in a fixed proportion to the proton flux, we used more recent inputs, and simplified and modified our upper limit to $40$%. But we forgot[^4] to extract the putative consequences for the AGN contribution to UHECR protons! Discussion and conclusion ========================= We do not have a solid understanding of accretion onto black holes or neutron stars. But such processes are observed to result in the ejection of relativistic and highly collimated jets. We assumed that a similar process takes place as a stellar core collapses, leading to a supernova event. We posited that the SN’s relativistic ejecta –two jets of cannonballs– are the sources of GRBs and CRs. The association between SNe and (long) GRBs is now established. We argued that the electrons in a CB, by inverse Compton scattering on the illuminated surroundings of the exploding star, generate the $\gamma$ rays, X-rays, UV and optical light of the ‘prompt’ phase of a GRB. The ensuing results are the simplest and most predictive, they are a firm ‘theory’. In this paper I have, however, followed the historical development, in which the CB parameters were first extracted from the observations of the afterglow of GRBs. This involves a ‘model’, a set of arguable but simple hypothesis leading to the prediction of the properties of the AG –dominated by synchrotron radiation by the ISM electrons that a CB intercepts– as a function of frequency and time. In the historical order the ‘prompt’ results for GRBs are predictions of the theory. Some results for Cosmic Rays –the ISM particles that CBs scatter in their journey– are also ‘theory’, others can be viewed as further tests of the ‘model’. The results for GRB afterglows may be based on a simplified model, but they work with no exception all the way from radio to X-rays (DDD02,03). In particular, they describe correctly GRB 980425, located at a redshift two orders of magnitude closer than average. Its associated SN is the one we ‘transported’ to conclude –thanks to the reliability of our AG model– that core-collapse SNe generate long GRBs (DDD02). The X-ray light curve of GRB 980425 and a few others, with extremely scarce data, was described with the ‘canonical’ properties later observed in detail in many SWIFT-era GRBs. It is not recognized that the two CBs of GRB 030329 were seen, or that their separation in the sky was the predicted ‘hyper-luminal’ one. In view of the overall success of the CB model, this is a durable hurdle: GRBs so close and luminous are very rare. The accuracy of the predictions for the prompt phase of GRBs amazes even the CB-model’s proponents. The typical values and the correlations between the $\gamma$-ray prompt observables leave little doubt that the production mechanism is inverse Compton scattering on ‘ambient’ light of $\sim\!1$ eV energy. The approximate scaling law $E\,dN_\gamma/dE dt\!\propto\!F(E\,t^2)$ –spectacularly confirmed in the case of XRF 060218– demonstrates that the light is that of a ‘glory’: the early SN light scattered by the ‘windy’ pre-SN ejecta. A GRB spectrum that works even better[@HessLady] than the phenomenological ‘Band’ expression is also predicted. The flux and its spectral evolution during the prompt and rapid-decline phases are the expected ones, as we tested in minute detail with SWIFT data. In the internal-external fireball model of GRBs, highly relativistic thin conical shells of $e^+e^-$ pairs, sprinkled with a finely tuned baryon ‘load’, collide with each other generating a shock that accelerates their constituents and creates magnetic fields. Each collision of two shells produces a GRB pulse by synchrotron radiation. The ensemble of shells collides with the ISM to produce the AG by the same mechanism. The energy available to produce the GRB pulse –as two shells moving in the [same]{} direction collide– is more than one order of magnitude smaller than that of the merged shells as they collide with the ISM at rest. The ratio of observed GRB and AG energies is more than one order of magnitude, but in the opposite direction. This ‘energy crisis’ in the comparison of bolometric prompt and AG fluences[@Pir] is not resolved. Moreover, the GRB spectrum cannot be accommodated on grounds of synchrotron radiation[@GCL], the ‘standard’ prompt mechanism. The SWIFT-era observations also pose decisive problems to the standard model, whose microphysics[@Petal], reliance on shocks[@Ketal] and correlations based on the jet-opening angle[@Setal] have to be abandoned, according to the cited authors. In spite of the above, the defenders of the fireball model are not discouraged. Their attitude towards the CB model, whose observational support is so remarkable, is not equally supportive. This may be due to cultural differences. Particle physicists believe that complex phenomena may have particularly simple explanations. They thrive on challenging their standard views. Doubting or abandoning a previous consensus in astrophysics is less easy. The CB-model description of Cosmic Rays is also simplistic: there is only one source of (non-solar) CRs at all energies, and only one parameter to be fit. The model has a certain inevitability: if CBs with the properties deduced from GRB physics are a reality, what do they do as they scatter the particles of the interstellar medium? We have argued that they transmogrify them into CRs with all of their observed properties. The mechanism is entirely analogous to the ICS responsible for a GRB’s prompt radiation. Suffice it to substitute the CB’s electron, plus the ambient photon, by a moving-CB’s inner magnetic field, plus an ambient nucleus or electron. After a century of CR measurements, the CB-model results lack the glamour of predictions. Yet, the expectations for the knee energies, and for the relative abundances of CRs, are ‘kinematical’, simple, and verified. They constitute evidence, in my opinion, that the underlying model is basically correct. The prediction for the shape of the spectra: the low energy hips, the large energy stretch very well described by a power-law of (source) index $\beta_s\!=\!13/6$, and the steepening at the knees, are also verified. The index $\beta_s$ is measured well enough for the adequacy of the prediction to be sensitive to the details of the underlying model, such as the form of the function $R(\gamma)$ in Eq. (\[Radius\]). I cannot claim that the fact that the prediction is right on the mark is much more than a consistency test, for the physics underlying this aspect of the problem may be terrifyingly complex. The same CR source –cannonballs from supernovae, this time extragalactic– satisfactorily describes the CR data above the ankle. Finally, the properties of CR electrons, and of the high-latitude Gamma ‘Background’ Radiation, are also correctly reproduced. Most CR scholars agree with the ‘standard’ paradigm that the flux well below the knee is produced by the acceleration of the ISM in the frontal shocks of the nonrelativistic ejecta of SNe. In spite of recent observations of large magnetic fields[@Yasu] in collisions of SN shells and molecular clouds, nobody has been able to argue convincingly that this process can accelerate particles up to the (modest) energy of the knee, [and]{} to show that the number and efficiency of the putative sources suffices to generate the observed CR luminosity (to my satisfaction, I add, to make these statements indisputable). From this point on, there is no ‘standard’ consensus on the origin of CRs, e.g., of the highest-energy ones. In this sense, the CB model is regarded as yet another model, which it is. After all, we are only saying that CRs are accelerated by the jetted relativistic ejecta of SNe, as opposed to the quasi-spherical, non-relativistic ones. Yet, the CB model is also rejected by the CR experts, sometimes even in print[@Hillas], though it survives the critique[@HResponse]. But, concerning CRs, the model does not trigger the same indignant wrath as in the GRB realm. I have shown that the problem of GRBs is convincingly –i.e. predictively– solved and that, on the same simple basis, all properties of CRs can be easily derived. Only an overwhelmed minority recognizes these facts, in contradiction with Popper’s and Ockham’s teachings. I would conclude with a dictum attributed to Lev Landau: [‘In astrophysics, theories never die, only people do.’]{} .4cm [Acknowledgement]{} I thank Shlomo Dado, Arnon Dar and Rainer Plaga for our collaboration. [99]{} A.M. Hillas, arXiv:astro-ph/0607109 J.R. Hoerandel, arXiv:astro-ph/0702370v2 P. Meszaros, arXiv:astro-ph/0605208 T. Piran, Rev. Mod. Phys. [76]{}, 1143 (2004); Phys. Rep. [333]{}, 529 (2000). S. Dado, A. Dar & A. De Rújula, Astron. & Astrophys. [388]{}, 1079 (2002). S. Dado, A. Dar & A. De Rújula, Astron. & Astrophys. [401]{}, 243 (2003). A. Dar & A. De Rújula, Phys. Reps. [405]{}, 203, (2004). A. Dar & A. De Rújula, arXiv:astro-ph/0606199 S. Dado, A. Dar & A. De Rújula, arXiv:0706.0080 A. Dar & A. De Rújula, Astroph. J. [547]{}, L33, (2001). S. Dado, A. Dar & A. De Rújula, Astroph. J. [594]{}, L89, (2003). P. Nisenson & C. Papaliolios, Astrophys. J. [518]{}, L29, (1999). E. Maiorano [et al]{}. Astron. & Astroph. [438]{}, 821 (2005). S. Campana, et al. Nature, [442]{}, 1008 (2006). A. Dar & A. De Rújula, arXiv:astro-ph/0008474 A. De Rújula, Phys. Lett. [B193]{}, 514 (1987). S. Dado, A. Dar & A. De Rújula, arXiv:astro-ph/0402374, 0406325 J.K. Frederiksen [et al]{}., Astroph. J. [608]{}, L13 (2004); astro-ph/0303360 N. J. Shaviv & A. Dar, Astroph. J. [447]{}, 863 (1995). A. Dar & A. De Rújula, arXiv:astro-ph/0012227 S. Dado, A. Dar & A. De Rújula, Astroph. J. [663]{}, 400, (2007). S. Dado, A. Dar & A. De Rújula, arXiv:astro-ph/0701294 S. Dado, A. Dar & A. De Rújula, arXiv:0709.4307 S. Dado, A. Dar & A. De Rújula, to be published. A. Dar & A. De Rújula, Phys. Rev. [D72]{}, 123002, (2005). R. Plaga, arXiv:astro-ph/0111555 A. Dar & A. De Rújula, Mon. Not. Roy, Astron. Soc. [323]{}, 391, (2001). M. Aguilar [et al.]{}, Phys. Rep. [366]{}, 331 (2002). P. Sreekumar [et al.,]{} Astrophys. J. [494]{}, 523 (1998). G.B. Taylor [et al]{}. Astrophys. J. [609]{}, L1, (2004). The Pierre Auger collaboration, Science [318]{}, 938 (2007). V.L. Ginzburg & S.I. Syrovatskii, [The origin of Cosmic Rays]{} (Pergamon, Oxford, 1964). S. Dado, A. Dar & A. De Rujula Nuc, Phys. Proc. Suppl. [B165]{}, 103 (2007) C. Wigger [et al]{}. arXiv:0710.2858 G. Ghisellini, A. Celotti, & D. Lazzati, MNRAS [316]{}, L5 (2000). A. Panaitescu [et al]{}. MNRAS [369]{}, 2059 (2006). P. Kumar [et al]{}. MNRAS [376]{}, L57, (2007). G. Sato [et al]{}. Astroph. J. [657]{}, 359, (2007);\ D. N. Burrows & J. Racusin, arXiv:astro-ph/0702633 Y. Uchiyama [et al.]{} Nature [449]{}, 576, (2007). S. Dado, A. Dar & A. De Rújula, arXiv:astro-ph/0611369 [^1]: This is what we did in DDD02/03 but not quite what we wrote. I am indebted to J. Steinberger for noticing this error. [^2]: I am keeping factors of $A/Z$ for kicks. Numerically, they are irrelevant: the theory and data are not so precise, and $(A/Z)^{0.6}$ is 1 for H, 1.6 for Fe. [^3]: The size of a CB is small enough to expect its radio image to scintillate, arguably more than observed[@Taylor]. Admittedly, we only realized a posteriori that the ISM electrons a CB scatters, synchrotron-radiating in the ambient magnetic field, would significantly contribute at radio frequencies, somewhat blurring the CBs’ radio image[@SLum030329]. [^4]: In these days of large experimental collaborations percolated by theorists, rumours herald publications. It might have been possible to turn this comment into a renewed[@Rus] timely ‘prediction’, prior to the Auger announcement.
--- abstract: \| Recent analyses show evidence for a thermal emission component that accompanies the non-thermal emission during the prompt phase of GRBs. First, we show the evidence for the existence of this component; Second, we show that this component is naturally explained by considering emission from the photosphere, taking into account high latitude emission from optically thick relativistically expanding plasma. We show that the thermal flux is expected to decay at late times as $F_{\rm BB} \sim t^{-2}$, and the observed temperature as $T \sim t^{-\alpha}$, with $\alpha \approx 1/2 - 2/3$. These theoretical predictions are in very good agreement with the observations. Finally, we discuss three implications of this interpretation: (a) The relation between thermal emission and high energy, non-thermal spectra observed by [Fermi]{}. (b) We show how thermal emission can be used to directly measure the Lorentz factor of the flow and the initial radius of the jet. (c) We show how the lack of detection of the thermal component can be used to constrain the composition of GRB jets. author: - 'A. Pe’er$^1$' - 'F. Ryde$^2$' title: 'Observations, theory and implications of thermal emission from gamma-ray bursts' --- Introduction {#sec:intro} ============ Despite many efforts, a clear understanding of the physical origin of the photons observed during the prompt emission phase in GRBs is still lacking. The prompt emission spectra are often fitted with a broken power law model (known as the “Band” function, [@Band93]). However, recent [Fermi]{}- LAT results show that this fit is inadequate in some cases in which high energy photons are observed (e.g., [@Abdo+09]). Even more importantly, the “Band” function fit does not, by itself, carry any explanation about the physical origin of the observed photons. A common interpretation is that the peak observed at sub-Mev range is due to synchrotron emission [@Frontera00]. However, while the synchrotron interpretation is consistent with many afterglow observations, this interpretation is inconsistent with the majority of the prompt spectra, due to steep low energy spectral slopes observed [@Preece98]. This inconsistency motivated us to search for an alternative explanation. Arguably, the most natural ingredient is a thermal component, that should exist in the outflow. In principle, thermal photons can originate either by the initial explosion, or by any dissipation of the kinetic energy that occurs deep enough in the flow, in region where the optical depth is $\gg 1$, so that the emitted photons thermalize before escaping the plasma. In a series of papers [@Ryde04; @Ryde05; @PMR05; @PMR06; @PRWMR07; @Peer08; @RP09; @ZP09; @Ryde+09] we have extensively studied the contribution of a thermal emission component to the overall (non-thermal) GRB prompt spectra. Our research is focused on both the observational properties [@Ryde04; @Ryde05; @RP09; @Ryde+09], theoretical modeling [@PMR05; @PMR06; @Peer08] and implications of the existence of this component [@PRWMR07; @ZP09]. We give here a brief summary of our key results. Observational clues {#sec:obs} =================== Two main difficulties exist in interpreting the observed spectrum. The first is that clearly, in addition to the thermal component there is a strong non-thermal part. The second is that the properties of the thermal component, (temperature and flux) may be time-dependent, hence its signal is smeared in a time-integrated analysis, as is frequently done. In order to overcome the second problem, one needs to carry a [time-dependent]{} analysis. Such an analysis was indeed carried by [Ryde]{} ([@Ryde04; @Ryde05]). In these works, a hybrid model (thermal + a single broken power law) was found to adequately describe the prompt spectra of 9 burst over the limited BATSE energy band. Moreover, the temperature of the thermal component showed a repetitive behaviour: a broken power law in time, with power law index $T(t) \propto t^{-2/3}$ after the break time at few seconds. Repeating a similar analysis on a larger sample of 56 bursts, we found ([@RP09]) that the same repetitive behaviour is ubiquitous. Moreover, in this work we also considered the evolution of the thermal flux, and found that it too shows a similar behavior: a broken power law, with index $F(t) \propto t^{-2}$ at late times. The break time, not surprisingly, is the same within the errors to the break time found in the temperature behaviour. Histograms of the late time power law indices are shown in figure \[fig:1\]. ![(Left, center): Histograms of the late time decay indices of the temperature (left) and thermal flux (center) in a sample of 56 bursts (taken from [@RP09]). Right: theoretical results of the flux decay at late times, showing $F(t) \propto t^{-2}$ (taken from [@Peer08])[]{data-label="fig:1"}](histogram.ps "fig:"){width="8.0cm"} ![(Left, center): Histograms of the late time decay indices of the temperature (left) and thermal flux (center) in a sample of 56 bursts (taken from [@RP09]). Right: theoretical results of the flux decay at late times, showing $F(t) \propto t^{-2}$ (taken from [@Peer08])[]{data-label="fig:1"}](f4.eps "fig:"){width="4.4cm"} Theoretical interpretation {#sec:theory} ========================== As thermal photons originate from below the photosphere, one needs to study the properties of the photosphere in a relativistically expanding plasma. In such a plasma, Lorentz aberration plays a significant role: for example, the photospheric radius strongly depends on the angle to the line of sight, $\theta$, via $r_{ph}(\theta) \propto (\theta^2/3+\Gamma^{-2})$, where $\Gamma$ is the factor Lorentz of the outflow [@Peer08]. This strong dependence implies that photons emitted from the photosphere at high angles are significantly delayed with respect to photons emitted on the line of sight: $\Delta t^{ob} \approx 30 L_{52} \Gamma_2^{-1} \theta_{-1}^4$ s, where $L$ is the GRB luminosity and $Q_x = Q/10^x$ in cgs units is used. Calculations of the expected decay of the flux and temperature at late times are carried under the assumption that the source (the inner engine) terminates abruptly at a given time. Photons emitted off axis are delayed (“high latitude” emission in [optically thick]{} expanding plasma), resulting in flux decay at late times. Moreover, due to both weaker Doppler boosting and energy losses to the expanding plasma at large radii, these photons’ observed temperature is also lower. By integrating over equal arrival time surface in the entire space, it was shown ([@Peer08]) that the flux decays at late times as $F(t) \propto t^{-2}$ and the temperature as $T(t) \propto t^{-\alpha}$, with $\alpha \approx 1/2 - 2/3$. The results of the theoretical calculations of the flux decay (both numerical and analytic) are shown in figure \[fig:1\] (right). Clearly, the theoretical predictions are in excellent agreement with the observations. While this by itself does not prove that indeed the photons that we see are thermal, we find the agreement between theory and observations, as well as the repetitive behaviour seen, two very strong, independent indications that we indeed are able to properly identify the thermal emission component and discriminate it from the non-thermal part. Implications {#sec:implications} ============ The existence of a thermal emission component, which, as described, is a natural outcome of the fireball model, can bring a breakthrough in our understanding of the physics of GRB prompt emission. We discuss here three important implications of it. [The relation between thermal and non-thermal emission]{}. As GRB prompt spectra is non-thermal, clearly, in addition to any thermal component there is a non-thermal part. Thus, according to our picture, the observed spectrum is composed of (at least) two separated ingredients, thermal component originating from the photosphere, and non-thermal component originating from the kinetic energy dissipation above the photosphere. The true properties of this non-thermal part can only be deduced after subtracting the contribution from the thermal part. This, however, is not an easy task: since thermal photons serve as seed photons to Compton scattering by energetic electrons (produced by the dissipation mechanism above the photosphere), they contribute to the cooling of these electrons. Hence, the resulting non-thermal spectrum (from, e.g., synchrotron emission or Compton scattering) depends not only on the properties of the acceleration mechanism (e.g., the power law) but also on the relative contribution of the thermal photons. This can lead to a variety of very complex spectra (see [@PMR05; @PMR06]), which depend on the various parameters - the photospheric radius, the dissipation radius, magnetic field strength, etc. Interestingly, we were able to show that for a relatively large parameter space region, a single power law may be sufficient to model the non-thermal part over a limited energy range. This result is indeed consistent with the single-power law fitting of the non-thermal part of the spectrum seen in several recent [Fermi]{} bursts (e.g., GRB090902B, [@Abdo+09]) [Measuring the parameters of the outflow]{}. One major advantage of identifying the thermal emission, is that its radius of origin is known - it is the photosphere. The ratio of the thermal flux and temperature, $\mathcal{R} \equiv (F/ \sigma T^4)^{1/2}$ ($\sigma$ is Stefan’s constant) must therefore be proportional to the photospheric radius (for photons emitted on the line of sight). In fact, due to Lorentz aberration, we showed ([@PRWMR07]) that $\mathcal{R} \propto r_{ph}(\theta=0)/\Gamma$. In the classical “fireball” model, the photospheric radius depends only on two parameters: the (kinetic) luminosity, and the Lorentz factor. Hence, for bursts with known redshift, one can use the three measurable quantities of emission at the photosphere (thermal flux, temperature and GRB distance) to deduce the values of the three unknowns - the luminosity[^1], the Lorentz factor and the photospheric radius itself. Moreover, in the classical fireball model, the dynamics below the photosphere is fully determined by the conservation of energy and entropy. One can therefore use the values of the luminosity and Lorentz factor at the photospheric radius to deduce the radius at which the initial acceleration began. This radius is denoted here as $r_0$. Implementing these ideas, we were able to show that for GRB970828, at redshift z=0.96, the terminal Lorentz factor is $\Gamma=305 \pm 28$, and $r_0 = (2.9\pm 1.8)\times 10^8$ cm. The statistical errors in the estimate of the Lorentz factor, $\pm 10\%$ are by far the smallest from all the methods known today. [Deducing the outflow composition]{}. GRBs show a wide variety of properties from burst to burst. While in some bursts, thermal emission component is very pronounced (e.g., in GRB090902B the low energy spectrum is so steep and the peak is so narrow that it is very difficult to find an alternative explanation to the peak; see [@Ryde+09]), in other bursts it is much less pronounced. As one example, in the bright burst GRB080916C, the “Band” model provides a good fit to the spectrum, up to the highest observed photons energies, at 13.2 GeV [@Abdo09]. Opacity arguments can easily show that these energetic photons cannot originate from the photosphere, but from some (significantly) larger radius [^2], $R_\gamma \gsim 10^{15}$ cm. The lack of the detection of a thermal component as predicted by the baryonic models strongly suggests that a significant fraction of the outflow energy is initially not in the “fireball” form. Thus, we found [@ZP09] the most plausible alternative to be Poynting flux entrained with the baryonic matter. The ratio between the Poynting and the baryonic flux in this burst is at least $\sim(15 −- 20)$. Summary {#sec:summary} ======= The existence of a thermal emission component, which, as described, is a natural outcome of the fireball model, has a potential to bring about a breakthrough in our understanding of the physics of GRB prompt emission. It is therefore the subject of an extensive on-going research, from all aspects. We stress, that the different physical environment prevents using the tools developed for studying the afterglow in the study of the prompt emission.\ [Acknowledgments]{}\ We would like to thank Ralph Wijers, Peter Mészáros, Martin Rees and Bing Zhang for significant contributions to these works. [8]{} Band, D. , , 413 (1993) 281. Abdo, A.A. (the Fermi collaboration), , 706 (2009) L138. Frontera, F. , , 127 (2000) 59. Preece, R.D. , , 506 (1998), L23. Ryde, F., , 614 (2004) 827. Ryde, F., , 625 (2005) L95. Pe’er, A., Mészáros, P., & Rees, M.J., , 635 (2005) 476. Pe’er, A., Mészáros, P., & Rees, M.J., , 642 (2006) 995. Pe’er, A., Ryde, F., Wijers, R.A.M.J., Mészáros, P., & Rees, M.J., , 664 (2007) L1 . Pe’er, A., , 682 (2008) 463. Ryde, F., & Pe’er, A., , 702 (2009) 1211. Zhang, B., & Pe’er, A., , 700 (2009) L65. Ryde, F. , , 709 (2010) L172. Abdo, A.A., (the Fermi collaboration), Science, 323, (2009) 1688 [^1]: Note that there is an uncertainty in estimating the kinetic luminosity from the flux. This uncertainty can be removed once afterglow measurements are available [^2]: Note that opacity argument by itself does not give the emission radius; in order to obtain that, one needs to specify the relation between the radius and the Lorentz factor. The results often used in the literature, $r = \Gamma^2 c \delta t$ rely on assumed knowledge of the variability time $\delta t$, which is highly uncertain.
--- abstract: 'We present early results from the ongoing Hydrogen Accretion in LOcal GAlaxieS (HALOGAS) Survey, which is being performed with the Westerbork Synthesis Radio Telescope (WSRT). The HALOGAS Survey aims to detect and characterize the cold gas accretion process in nearby spirals, through sensitive observations of neutral hydrogen ([[Hi]{}]{}) 21-cm line emission. In this contribution, we present an overview of ongoing analyses of several HALOGAS targets.' author: - 'George Heald$^1$, John Allan$^2$, Laura Zschaechner$^3$, Peter Kamphuis$^4$, Rich Rand$^3$, Gyula Józsa$^1$,' - Gianfranco Gentile$^5$ title: \| The Westerbork HALOGAS Survey:\ Status and Early Results --- Survey description and status {#section:intro} ============================= The scientific motivation for the HALOGAS Survey, and a detailed description of the observational setup, are provided by [@heald_etal_2011]. Briefly, we have selected a sample of 24 edge-on and moderately inclined nearby galaxies[^1], using neutral criteria, for deep (120 hr) [[Hi]{}]{} line observations at WSRT. The observations are sensitive to faint, diffuse gas (typical column density sensitivity $\sim10^{19}\,\mathrm{cm^{-2}}$) and to small unresolved [[Hi]{}]{} clouds (typical mass sensitivity of order $10^5\,M_{\odot}$). Within the HALOGAS project, each galaxy will be carefully studied to ascertain whether signs of gas accretion are present. Results from the individual targets will be combined to give the first indication of the ubiquity and general characteristics of the cold gas accretion process in nearby spirals. At the time of writing, the [[Hi]{}]{} observations are nearly completed; we anticipate that the guaranteed WSRT observations (20 of 22 targets) will be finished before summer 2011. A number of ancillary programs are also in progress and will supplement the neutral gas observations: deep UV (GALEX) and optical (INT) images are being obtained, and a novel multi-slit technique will obtain 3D optical spectroscopy of the ionized gas in the edge-on survey targets ([@wu_etal_2011]). Early results ============= First results, based on the first semester of survey observations (the HALOGAS Pilot Survey), are presented by [@heald_etal_2011]. The survey targets presented in that work are UGC 2082, NGC 672, NGC 925, and NGC 4565. In this contribution we present preliminary analysis for three additional survey targets. A more complete analysis for each of these is in preparation. NGC 1003 {#subsection:NGC1003} -------- NGC 1003 has a star formation rate of $0.34\,M_\odot\,\mathrm{yr}^{-1}$ ([@heald_etal_2011]). The width of the [[Hi]{}]{} disk is $21^\prime$, compared to the roughly $5^\prime$ optical disk (see Figure \[Figure:NGC1003\]). Assuming $D\,=\,11\,\mathrm{Mpc}$, we obtain a total [[Hi]{}]{} mass of $5.34\times10^9\,M_\odot$. Our tilted-ring modeling (described below) indicates a strongly warped, flaring thick disk. Several distinct [[Hi]{}]{} features are detected, as shown in Figure \[Figure:NGC1003\]. None of the clouds, nor the accretion complex, are visible in the optical DSS plates. Cloud 1 has an [[Hi]{}]{} mass of $2\times10^5\,M_\odot$. Cloud 2 has $3.3\times10^5\,M_\odot$ of [[Hi]{}]{}. Cloud 3 is an unresolved object within the accretion complex with an [[Hi]{}]{} mass of $3\times10^5\,M_\odot$. The accretion complex, excluding Cloud 3, has an [[Hi]{}]{} mass of $2.9\times10^6\,M_\odot$. The Milky Way and M31 are both known to have intermediate- and high-velocity clouds similar to these. The masses we obtain here, as well as the heights above the disk, are consistent with those derived by [@wakker_etal_2008] for Milky Way clouds and complexes. Over a dynamical time, they can provide fresh gas compensating for only $\sim2\%$ of the SFR in NGC 1003. ![Neutral hydrogen in NGC 1003. Left: Total [[Hi]{}]{} (moment-0) map overlaid on DSS image. Center: Total [[Hi]{}]{} (moment-0) map constructed from HVC analog [[Hi]{}]{} emission only. Right: Velocity field (moment-1 map), with cloud positions indicated.[]{data-label="Figure:NGC1003"}](n1003mom0_all.pdf "fig:"){width="25.00000%"}![Neutral hydrogen in NGC 1003. Left: Total [[Hi]{}]{} (moment-0) map overlaid on DSS image. Center: Total [[Hi]{}]{} (moment-0) map constructed from HVC analog [[Hi]{}]{} emission only. Right: Velocity field (moment-1 map), with cloud positions indicated.[]{data-label="Figure:NGC1003"}](n1003mom0_hvcs.pdf "fig:"){width="25.00000%"}![Neutral hydrogen in NGC 1003. Left: Total [[Hi]{}]{} (moment-0) map overlaid on DSS image. Center: Total [[Hi]{}]{} (moment-0) map constructed from HVC analog [[Hi]{}]{} emission only. Right: Velocity field (moment-1 map), with cloud positions indicated.[]{data-label="Figure:NGC1003"}](n1003mom1.png "fig:"){width="50.00000%"} The GIPSY program [GALMOD]{} was used to hand-craft a tilted-ring model of NGC 1003. Literature values for scale height, rotational velocity, and other parameters were used, and inclination and position angle were estimated. A radial profile was made and used for surface brightness values. This model was refined until further improvements could no longer be made, and the best parameters were fed into the Tilted Ring Fitting Code (TiRiFiC) developed by [@jozsa_etal_2007]. The program fits an arbitrary number of parameters at a time for each of the galaxy’s rings. We varied a limited number of parameters at once (typically two to three) and then fixed them at their optimum values. Three final models were produced: a thin-disk model, in which scale height was fixed at around $200\,\mathrm{pc}$; a thick-disk model, in which scale height was varied as a unit for all rings (and resulted in a scale height of $\sim1\,\mathrm{kpc}$), and a flare model where the scale height was varied individually for all rings. The east and west halves of the galaxy were modeled separately because of an asymmetrical “blob” feature in the eastern side of the disk. NGC 4244 -------- NGC 4244, having a star formation rate of $0.058\,M_{\odot}\,\mathrm{yr}^{-1}$ ([@heald_etal_2011]), is on the low star-forming end of the HALOGAS sample. Its high inclination ($88\,^{\circ}$) allows for readily determining the vertical [[Hi]{}]{} structure, as well as an accurate assessment of the kinematics of the neutral gas. For these reasons it will provide substantial insight concerning any connection between star formation and [[Hi]{}]{} halo properties. The total [[Hi]{}]{} map is shown in Figure \[Figure:NGC4244\_NGC5023\]. The disk is clearly warped, and does not show evidence for a bright diffuse halo component. Through careful tilted-ring modeling (similar to that described in §\[subsection:NGC1003\] but without the use of TiRiFiC), the disk of NGC 4244 is found to be warped both parallel to, as well as perpendicular to, the line of sight. The modeling confirms that we do not detect a substantially extended [[Hi]{}]{} halo. We do, however, detect vertical gradients (lags) in the rotational speed, with magnitudes $-12\,\pm\,2\,\mathrm{km\,s^{-1}\,kpc^{-1}}$ and $-9\,\pm\,2\,\mathrm{km\,s^{-1}\,kpc^{-1}}$ in the approaching and receding halves respectively. These lags decrease in magnitude to $-7\,\pm\,2\,\mathrm{km\,s^{-1}\,kpc^{-1}}$ near a radius of $9\,\mathrm{kpc}$. The detection of the radial variation in the vertical velocity structure will be valuable information for comparison with physical models of the disk-halo interface in spiral galaxies. Additionally, a few prominent localized features are detected in NGC 4244. Among them is a shell in the approaching half, located directly above a region of star formation. A second feature in the receding half displays an elongated, curved path in the [[Hi]{}]{} contours, extending away from the major axis to approximately $1.5^\prime$ above the midplane. This feature also appears to correspond with star formation, exhibiting a possible hole or vent in the [[Hi]{}]{}, which is partially filled by H$\alpha$ emission ([@hoopes_etal_1999]), indicating a possible connection between the two. Finally, a pronged, faint streak is seen over a velocity range of $\approx\,50\,\mathrm{km\,s^{-1}}$ (cf. the rotation speed, $95-100\,\mathrm{km\,s}^{-1}$) in the receding half. This feature shows no apparent connection to star formation and exhibits traits similar to ram pressure stripping. ![Left: Integrated [[Hi]{}]{} map of NGC 4244, corrected for primary beam attenuation. Contours begin at $N_{\mathrm{HI}}\,=\,1.4\times10^{20}\,\mathrm{cm}^{-2}$ and increase by factors of two. Right: Integrated [[Hi]{}]{} map of NGC 5023 overlaid on a false-color optical picture constructed from DSS images. The contours begin at $N_{\mathrm{HI}}\,=\,1.9\times10^{19}\,\mathrm{cm}^{-2}$ and increase by factors of two. The arrows indicate individual extraplanar features of interest.[]{data-label="Figure:NGC4244_NGC5023"}](NGC4244_v3.pdf "fig:"){width="50.00000%"}![Left: Integrated [[Hi]{}]{} map of NGC 4244, corrected for primary beam attenuation. Contours begin at $N_{\mathrm{HI}}\,=\,1.4\times10^{20}\,\mathrm{cm}^{-2}$ and increase by factors of two. Right: Integrated [[Hi]{}]{} map of NGC 5023 overlaid on a false-color optical picture constructed from DSS images. The contours begin at $N_{\mathrm{HI}}\,=\,1.9\times10^{19}\,\mathrm{cm}^{-2}$ and increase by factors of two. The arrows indicate individual extraplanar features of interest.[]{data-label="Figure:NGC4244_NGC5023"}](NGC5023optHI.png "fig:"){width="50.00000%"} NGC 5023 -------- The edge-on galaxy NGC 5023 is a small ($v_{\mathrm{rot}}\,\sim\,80\,\mathrm{km\,s}^{-1}$), slightly warped galaxy at a distance of $6.6\,\mathrm{Mpc}$ ([@heald_etal_2011]). Figure \[Figure:NGC4244\_NGC5023\] shows an integrated [[Hi]{}]{} (moment-0) map. The 21-cm line emission in this galaxy extends on average up to a projected distance of $\sim\,80^{\prime\prime}$ from the plane, with individual features extending to $\sim\,100^{\prime\prime}$ (indicated in the figure with arrows). At the adopted distance, these vertical distances translate to 2.6 kpc and 3.2 kpc, respectively. The most vertically extended features also clearly show up in the individual channel maps of the data. Even though the bulk of the high-latitude emission could be projected above the major axis due to a line-of-sight warp, it is more difficult to explain the individual features in this manner. The extraplanar gas also has lower projected rotational velocities than the gas in the mid-plane. Detailed modeling is in progress to show whether the kinematics is due to a line-of sight warp, a vertical gradient in the rotation curve (as seen in e.g. NGC 4244), or some other effect. The star formation rate in NGC 5023 is extremely low ($0.032\,M_\odot\,\mathrm{yr}^{-1}$; [@heald_etal_2011]). Attributing the extended extraplanar [[Hi]{}]{} features to galactic fountain activity would therefore seem unlikely. However, deep H$\alpha$ imaging performed by [@rand_1996] shows faint diffuse [ionized]{} emission in the extraplanar regions – at the same radial locations as the [[Hi]{}]{} features indicated in Figure \[Figure:NGC4244\_NGC5023\] (they are however not detected to the same vertical distances as the [[Hi]{}]{} features). The correlation between extraplanar diffuse ionized gas (EDIG) features and star formation has been well established (e.g. [@rand_1996]). This may indicate that the extraplanar [[Hi]{}]{} features in NGC 5023 are indeed the result of star formation activity. Our 3D optical spectroscopic observations (mentioned in §\[section:intro\]) of this target will be very interesting for comparison with the [[Hi]{}]{} data. Intermediate conclusions and ongoing work ========================================= As the HALOGAS Survey progresses, the number of galaxies with well-studied [[Hi]{}]{} morphology and kinematics continues to grow. The extraplanar [[Hi]{}]{} features detected in HALOGAS targets have so far been more subtle than, e.g., the massive multiphase gaseous halo in NGC 891. However, it is not yet clear which characteristics of the underlying galaxy dictate the gaseous content of spiral galaxy halos. It does not seem to be the case that halos are tied in a straightforward way to the SFR of the host galaxy. NGC 4244, for example, has a low SFR and no halo. As shown here, NGC 5023, with a similarly low SFR, does have extended extraplanar [[Hi]{}]{} features. UGC 7321, with an even lower SFR, also has an [[Hi]{}]{} halo as shown by [@matthews_wood_2003]. Which other galactic properties are relevant to the characteristics of gas in halos? What role does cold gas accretion play? The full HALOGAS Survey will provide access to these questions, by providing uniformly sensitive data over a large sample of galaxies, with a broad range in properties such as mass, SFR, and environment. 2011, A&A, 526, 118 1999, ApJ, 522, 669 2007, A&A, 468, 731 2003, ApJ, 593, 721 1996, ApJ, 462, 712 2008, ApJ, 672, 298 2011, BAAS, 43, 246.21 [^1]: Two of the 24 HALOGAS sample galaxies, NGC 891 and NGC 2403, have already been the subject of deep [[Hi]{}]{} observations and are not reobserved in our program.
--- abstract: 'Bipolar disorder, an illness characterized by manic and depressive episodes, affects more than 60 million people worldwide. We present a preliminary study on bipolar disorder prediction from user-generated text on Reddit, which relies on users’ self-reported labels. Our benchmark classifiers for bipolar disorder prediction outperform the baselines and reach accuracy and F1-scores of above 86%. Feature analysis shows interesting differences in language use between users with bipolar disorders and the control group, including differences in the use of emotion-expressive words.' author: - \| Ivan Sekuli[ć]{} Matej Gjurkovi[ć]{} Jan [Š]{}najder\ Text Analysis and Knowledge Engineering Lab\ Faculty of Electrical Engineering and Computing, University of Zagreb\ Unska 3, 10000 Zagreb, Croatia\ {ivan.sekulic,matej.gjurkovic,jan.snajder}@fer.hr bibliography: - 'emnlp2018.bib' title: 'Not Just Depressed: Bipolar Disorder Prediction on Reddit' ---
--- abstract: 'The optical Fundamental Plane of black hole activity relates radio continuum luminosity of Active Galactic Nuclei to \[O[<span style="font-variant:small-caps;">iii</span>]{}\] luminosity and black hole mass. We examine the environments of low redshift ($z<0.2$) radio-selected AGN, quantified through galaxy clustering, and find that halo mass provides similar mass scalings to black hole mass in the Fundamental Plane relations. AGN properties are strongly environment-dependent: massive haloes are more likely to host radiatively inefficient (low-excitation) radio AGN, as well as a higher fraction of radio luminous, extended sources. These AGN populations have different radio – optical luminosity scaling relations, and the observed mass scalings in the parent AGN sample are built up by combining populations preferentially residing in different environments. Accounting for environment-driven selection effects, the optical Fundamental Plane of supermassive black holes is likely to be mass-independent, as predicted by models.' author: - \| Stanislav S. Shabala$^{1}$[^1]\ $^{1}$ School of Natural Sciences, Private Bag 37, University of Tasmania, Hobart, TAS 7001, Australia bibliography: - 'FPletter\_bibliography.bib' date: 'Accepted XXX. Received YYY; in original form ZZZ' title: The role of environment in the observed Fundamental Plane of radio Active Galactic Nuclei --- \[firstpage\] black hole physics – galaxies: active – galaxies: jets Introduction ============ The Fundamental Plane (FP) of black hole activity defines a low-redshift correlation between radio continuum luminosity, X-ray luminosity and black hole mass of the form $L_{\rm radio} \propto L_X^{a} M^{b}$. It holds over 10 orders of magnitude in radio luminosity, and connects Active Galactic Nuclei (AGN) hosted by supermassive black holes with (much more radio-quiet) X-ray black hole binaries (XRBs). Work by various authors [@MerloniEA03; @KoerdingEA06; @GultekinEA09; @BonchiEA13; @NisbetBest16] has confirmed the existence of the FP, with exponents in the range $a=0.4-0.8$, $b=0.6-1.0$, depending on the sample used. @HeinzSunyaev03 pointed out that the expected scaling between X-ray and radio luminosity depends on whether the X-ray emission is dominated by the hot disk corona, or the jet itself through optically thin synchrotron or synchrotron self-Compton emission. @FalckeEA04 showed that, for jet-dominated X-ray emission, the mass dependence of the FP naturally comes about due to the changing break frequency between the optically thin and thick parts of the jet. Typical spectral indices[^2] of $\alpha=+0.1$ and $-0.6$ for the optically thick and thin parts of the SED, respectively, yield $a=0.7$ and $b=0.6$, consistent with observed scalings in the low/hard state of black hole X-ray binaries [@KoerdingEA06; @CorbelEA13]; similar scalings are obtained if the X-rays are produced by a radiatively inefficient accretion flow, rather than the jet [@MerloniEA03]. For a self-absorbed synchrotron jet with X-ray emission coming from the accretion disk, theory predicts a mass-[independent]{} scaling $L_{\rm radio} \propto L_X^{1.4}$ [@FalckeEA04], consistent with observations of at least some radio-faint XRBs [e.g. H1743-322; @CoriatEA11 but see @RushtonEA16] and Atoll-type neutron star X-ray binaries [@MigliariFender06; @TetarenkoEA16]. @SaikiaEA15 recently reported the existence of the so-called optical FP, which uses the \[O[<span style="font-variant:small-caps;">iii</span>]{}\] narrow-line luminosity instead of X-ray luminosity. Line emission from the narrow-line region scales with the strength of the ionising radiation field, and has been shown to trace AGN bolometric luminosity [@HeckmanEA04]; hence it is a good proxy for the black hole accretion rate. By converting the measured \[O[<span style="font-variant:small-caps;">iii</span>]{}\] luminosities to hard (2–10 keV) X-ray luminosities, @SaikiaEA15 compared their extrapolated FP for 39 AGN (12 Seyferts, 20 LINERs, 7 transition objects) to observations of XRBs, finding that the observed XRB population has higher radio luminosities than expected from the AGN-only FP. This result, which brings into question the FP unification of supermassive and stellar mass black holes, has since been confirmed using a sample of lower-luminosity AGN (LINERs) by @NisbetBest16. In this paper, we suggest that the observed mass dependence in the optical FP relations for supermassive black holes is due to environmental effects. High and low-mass radio AGN preferentially inhabit different environments. As a consequence, these AGN are found in different accretion modes, and have different scalings between jet kinetic power and radio continuum luminosity. Sample {#sec:sample} ====== The starting point for our sample of radio AGN comes from @BestHeckman12. These authors cross-matched the seventh data release of the SDSS spectroscopic sample [@AbazajianEA09] with the NVSS [@CondonEA98] and FIRST [@BeckerEA95] 1.4 GHz radio continuum surveys. @BestHeckman12 used a combination of indicators to identify radio AGN, including the 4000Å break, ratio of radio luminosity to stellar mass, and optical emission line diagnostics. The @BestHeckman12 sample consists of 7302 radio AGN with redshifts $0.01 \leq z \leq 0.3$ and host galaxy [r]{}-band magnitude brighter than 17.77. These authors further separated radio AGN into High (HERG) and Low (LERG) Excitation Radio Galaxies, based on optical line ratios of six species, as well as the \[OIII\] line equivalent width. Although only around one third of the objects in the @BestHeckman12 sample had sufficiently strong emission lines to be classified spectroscopically, the majority of unclassified sources were found towards the back of the sample volume. In the analysis below, we restrict the AGN sample to $z < 0.2$. Early theoretical [@MerloniEA03] and observational [@KoerdingEA06] work on the Fundamental Plane explicitly excluded radiatively efficient AGN, however much subsequent work [e.g. @GultekinEA09; @BonchiEA13; @SaikiaEA15] has employed samples containing both high and low-excitation radio galaxies. As shown in Section \[sec:results\], this introduces a systematic bias in the derived relations. To study the correlation between radio and narrow line luminosities, we need to construct a complete subsample. NVSS is 99 percent complete at 3.4mJy [@CondonEA98]. Above a narrow-line luminosity integrated flux of $S_{\rm [O\,{\textsc{iii}}]} > 20 \times 10^{-17}$ ergs/s/cm$^2$, we found 95% of the AGN have a signal-to-noise ratio of at least two. Applying these two cuts to the @BestHeckman12 sample yields 1356 LERGs and 138 HERGs at $z<0.2$. To quantify environments, we use the SDSS group catalogues of @YangEA08. These authors used an adaptive halo-based group finder to associate nearby galaxies with a common dark matter halo, both in rich and poor systems. Tests with mock galaxy catalogues suggest a typical accuracy for the halo mass estimates of 0.3 dex. Cross-matching with the @YangEA08 catalogue yields 836 AGN with halo masses. Of these, 726 are classified by @BestHeckman12 as LERGs, 68 as HERGs, and the remaining 42 objects do not have an optical classification; we do not consider objects without a spectroscopic classification in our analysis. We further restrict our sample to sources which we can reliably classify as compact or extended (Section \[sec:OIII\_radio\_relation\]). Our final sample consists of 714 LERGs and 67 HERGs with halo masses. Theoretical considerations {#sec:theory} ========================== Extended radio sources ---------------------- Analytical models of powerful double (Fanaroff-Riley type II; FR-II [@FR74]) radio sources [e.g. @Scheuer74; @KA97; @TS15] describe supersonic expansion of radio lobes inflated by backflow of shocked plasma from the termination point of an initially relativistic jet. Conservation of energy within the cocoon and the shocked gas shell, together with the ram pressure condition at the hotspot, are sufficient for describing cocoon dynamics. Similar models apply to lobed lower-power FR-I sources. In jetted FR-Is, the initially conical jets are not collimated sufficiently by the external medium before stalling [@Alexander06; @KrauseEA12]; beyond the stalling point the jet suffers significant entrainment, yielding diffuse plumes of radio emission. The synchrotron emission is usually calculated under the assumption of a constant fraction of the total lobe energy being in the magnetic field [@KDA97]. It can be shown [e.g. @SG13 see their Eqn 4] that, in the absence of loss processes[^3], the relationship between monochromatic lobe luminosity $L_{\nu}$ and jet kinetic power $Q_{\rm jet}$ is $$\label{eqn:Lradio_Qjet_scaling} L_{\rm \nu} = A_1 Q_{\rm jet}^{\frac{5+p}{6}} \left( \frac{\nu}{\nu_0} \right)^{\frac{1-p}{2}}$$ Here, the initial electron Lorentz factor distribution at the site of acceleration is given by $n(\gamma) \propto \gamma^{-p}$, and $A_1$ is a constant. The optically thin synchrotron spectral index is related to $p$ via $\alpha=(1-p)/2$. Let jet power generation efficiency be $\eta$, i.e. $Q_{\rm jet}=\eta \dot{M}_{\rm BH} c^2$; and bolometric luminosity be $L_{\rm bol}=\epsilon_{\rm rad} \dot{M}_{\rm BH} c^2$. Then, $Q_{\rm jet} = \frac{\eta}{\epsilon_{\rm rad}} L_{\rm bol}$ and Equation \[eqn:Lradio\_Qjet\_scaling\] gives the relationship between radio and bolometric luminosities, $$\label{eqn:extended_radioBol_scaling} \log ( L_{\rm \nu} ) = \left[ \log A_2 + \left( {\frac{5+p}{6}} \right) \log \left( \frac{\eta}{\epsilon_{\rm rad}} \right) \right] + \left( {\frac{5+p}{6}} \right) \log L_{\rm bol}$$ where the frequency dependence has been absorbed into the constant $A_2$. For $p=2.6$ (corresponding to spectral index $\alpha=-0.8$) the slope of the $\log L_{\rm \nu}$ – $ \log L_{\rm bol}$ relation is 1.27. Compact sources {#sec:compact_theory} --------------- @FalckeBiermann95 and @HeinzSunyaev03 derived a general expression for the integrated flux density of a self-similar jet. The scaling between jet kinetic power and flux density (or radio luminosity) depends on the details of the jet model. For jet models in which the energy density in the magnetic field and relativistic particles both scale with total pressure, $L_{\nu} \propto Q_{\rm jet}^{17/12 + \alpha/3}$. For steep-spectrum jets ($\alpha=-0.8$) this gives an exponent of 1.15, or 1.42 for flat-spectrum jets with $\alpha \sim 0$. Synchrotron self-absorption, which naturally produces flat spectra [@BlandfordKonigl79], takes place on parsec scales, and for typical jet powers the synchrotron emission should be optically thin on kiloparsec scales which are of interest in this work. On scales comparable to galactic disk thickness, the observed spectra of (intrinsically) optically thin jets can still show relatively flat or even peaked spectra due to free-free absorption by the interstellar medium [@BicknellEA97]. The assumption of proportionality between magnetic, particle and total pressure is also made in standard radio lobe models, and hence it is not surprising that for both extended lobes and compact jets we expect similar scalings between jet power (or a proxy for this quantity, such as bolometric luminosity) and synchrotron radio luminosity. For gas-dominated disks the scaling is slightly different [@HeinzSunyaev03], $L_{\nu} \propto Q_{\rm jet}^{1.65 + 0.45 \alpha}$; for $\alpha=-0.8$ this yields a slope of $1.29$, similar to the above values for both compact and extended sources. Comparing compact and extended emission {#sec:theory_summary} --------------------------------------- As shown above, both compact and extended steep-spectrum sources are expected to follow $L_{\rm radio} \propto L_{\rm bol}^x$ for $x=1.15-1.42$. The main difference is in the normalization: in a large, low-redshift ($z<0.3$) sample of powerful radio galaxies, @HardcastleEA98 find the ratio of core to total luminosity at 178 MHz to be between $0.001$ and $0.4$, with the median value of $0.02$. We therefore expect compact sources to have substantially lower radio luminosities at a given bolometric luminosity. There are two further noteworthy points. First, the above discussion implicitly assumes that core and extended radio flux relate to the same jet kinetic power. However, these two measurements probe different timescales: extended radio luminosity is a proxy for time-averaged jet power on timescales of tens to hundreds of Myrs, while core luminosity traces the (quasi-)instantaneous jet power. For variable accretion rate these two measurements will in general be different, although they should converge to similar values for a continuously active jet, i.e. the time-averaged jet power should be similar the average instantaneous power. In the case of intermittent jet activity, the jet power inferred from extended radio luminosity will be lower by a factor corresponding to the jet duty cycle. The majority of radio sources in the local Universe are hosted by massive galaxies with high duty cycles [@BestEA05], hence we do not expect this effect to be important to the present work. Second, recognising the need to probe nuclear emission, a number of authors [e.g. @MerloniEA03; @KoerdingEA06; @GultekinEA09] have used compact (arcsec-scale) 5 GHz or even 15 GHz [@SaikiaEA15] radio flux densities. However, for all but the nearest AGN these observations still probe scales of several kpc; the problem is exacerbated at higher redshifts [e.g. @BonchiEA13]. Below, we show that the contribution of extended flux is environment-dependent. ![1.4 GHz radio luminosity - \[O[<span style="font-variant:small-caps;">iii</span>]{}\] line luminosity correlation. Circles are the complete low-luminosity @BestHeckman12 sample with halo masses. Pentagons are powerful 3C radio sources from [@ButtiglioneEA10]; filled symbols denote an \[O[<span style="font-variant:small-caps;">iii</span>]{}\] detection, open symbols are upper limits on \[O[<span style="font-variant:small-caps;">iii</span>]{}\] luminosity. High-excitation sources (blue) are offset from the low-excitation (red) population. Powerful extended radio galaxies clearly trace out the upper (high radio luminosity) envelope of the underlying NVSS population.[]{data-label="fig:Lnvss_vs_Loiii_complete"}](./L1p4_vs_Loiii_BH12complete_Buttiglione.png){width="1.0\columnwidth"} ![image](./LnvssExcess_vs_Mhalo_compactExtended_LERGs_hist.png){width="1.0\columnwidth"} ![image](./LnvssExcess_vs_Mhalo_compactExtended_HERGs_hist.png){width="1.0\columnwidth"} The optical Fundamental Plane {#sec:results} ============================= Optical – radio luminosity relation {#sec:OIII_radio_relation} ----------------------------------- The left panel of Figure \[fig:Lnvss\_vs\_Loiii\_complete\] shows the relationship between \[O[<span style="font-variant:small-caps;">iii</span>]{}\][^4] and radio continuum luminosity in our sample. High and low-excitation radio AGN both show a correlation, as previously reported by other authors [@WillottEA99; @ButtiglioneEA10]. The different normalizations for HERG and LERG populations are expected due to their different jet production efficiencies $\eta / \epsilon_{\rm rad}$ (Equation \[eqn:extended\_radioBol\_scaling\]). The low radio luminosity end is dominated by LERGs, while number counts of LERGs and HERGs become comparable at the highest ($L_{\rm 1.4}>10^{25}$ W/Hz) luminosities (Figure \[fig:Lnvss\_vs\_Loiii\_complete\]; see also Figure 4 of @BestHeckman12). For comparison, we overplot the sample of extended 3C radio galaxies of @ButtiglioneEA10, who also used spectroscopic diagnostics to classify AGN as High or Low-Excitation. Extended radio galaxies follow similar trends to the lower-luminosity sample, but clearly occupy the upper envelope of the distribution; in other words, they have high radio luminosities relative to their \[O[<span style="font-variant:small-caps;">iii</span>]{}\] luminosity. It is instructive to consider the residuals to the best-fit relation between these variables. Figure \[fig:Lnvss\_vs\_Loiii\_resids\] shows the excess in radio luminosity (traced by NVSS) as a function of halo mass, plotted separately for LERGs and HERGs. Here, the radio excess has been calculated by subtracting from each source the best-fit to the appropriate $L_{\rm 1.4} - L_{\rm [O\,{\textsc{iii}}]}$ relation. Each population has further been split into compact and extended sources, using the classifications of @BestHeckman12. These authors classified sources into four categories: (1) those sources which consist of single components in both FIRST and NVSS; (2) sources with a single NVSS component but multiple components in FIRST; (3) NVSS sources without a FIRST counterpart; and (4) NVSS sources with multiple components. We classify sources in categories (2) and (4) as extended. We classify category (1) sources as extended if their NVSS to FIRST flux density ratio exceeds 5, i.e. they exhibit significant low surface brightness emission which is resolved out by FIRST. Category (1) sources in which this flux density ratio is below the threshold are classified as compact. This approach follows @KimballIvezic08 [@SinghEA15], albeit with a higher value of the flux density ratio (those authors adopt a threshold value of 1.4); our results are qualitatively similar for lower values of the flux density ratio threshold. Finally, we exclude category (3) objects from our analysis, noting that only 1 HERG and 12 LERGs fall into this category. There are 138 extended and 576 compact LERGs; and 19 extended, 48 compact HERGs. There is no statistically significant (as given by a Kolmogorov-Smirnov test at the 10 percent level) difference in the redshift distributions of compact and extended sources. As a caveat, we note that compact sources defined using 1.4 GHz flux densities may encompass more non-jetted emission than higher frequency samples [e.g. @KoerdingEA06; @GultekinEA09; @SaikiaEA15]. Three features are immediately apparent in Figure \[fig:Lnvss\_vs\_Loiii\_resids\]. First, High-Excitation Radio Galaxies preferentially reside in poorer environments, with a median halo mass of $6 \times 10^{12} M_{\odot}/h$ compared to $1.6 \times 10^{13} M_{\odot}/h$ for LERGs; this result echoes the findings of @SabaterEA13. Second, both HERGs and LERGs show a clear dichotomy between compact and extended sources, with extended sources consistently exhibiting radio emission in excess of the median value for the combined (compact plus extended) population. Third, compact and extended source counts depend on environment: despite compact sources greatly outnumbering extended sources in both HERGs and LERGs, the numbers are comparable at the highest masses (above $10^{13} M_{\odot}/h$ for HERGs, and $10^{14} M_{\odot}/h$ for LERGs). Hence, at a fixed \[O[<span style="font-variant:small-caps;">iii</span>]{}\] luminosity, massive haloes are more likely to host extended, high-luminosity radio sources. The most radio bright sources will, therefore, be preferentially found in rich environments. Because halo mass correlates with AGN host galaxy mass and therefore black hole mass, any mass-dependent scaling will push the extended, high-luminosity sources to the right in $L_{1.4} - L_{[O\,{\textsc{iii}}]}$ plane, thereby strengthening any existing correlation. The separation between HERGs and LERGs (Figure \[fig:Lnvss\_vs\_Loiii\_complete\]) will also be reduced with a mass scaling, since HERGs preferentially inhabit poor environments; however a separate relation, corresponding to a different $\eta / \epsilon_{\rm rad}$ value from the LERG population, would be a more physically-motivated alternative. Thickness of the Fundamental Plane {#sec:scatter} ---------------------------------- A number of authors [e.g. @MerloniEA03; @Daly16] have suggested that parameters such as black hole spin add scatter to the Fundamental Plane. The above analysis suggests that environmental effects are also likely to contribute. Table \[tab:fit\_params\] shows the results of fitting a plane in radio luminosity, \[OIII\] luminosity and mass (either halo or black hole, estimated from velocity dispersion using the relation of @TremaineEA02[^5]) to our data, using the [`Hyperfit`]{}[^6] package [@RobothamObreschkow15]. `Hyperfit` uses traditional likelihood methods to estimate a best-fitting model to multi-dimensional data, in the presence of parameter covariances, intrinsic scatter and heteroscedastic errors on individual data points. It assumes that both the intrinsic scatter and uncertainties on individual measurements are Gaussian, and allows for error covariance between orthogonal directions. Parameter estimates in Table \[tab:fit\_params\] have been obtained using the Conjugate Gradients method; similar results are found using a quasi-Newtonian method. The scatter about the best-fit relation of $\sim 0.5$ dex is comparable with results reported by other authors [@MerloniEA03; @BonchiEA13; @SaikiaEA15; @NisbetBest16]. Compact sources show a marginally steeper dependence of radio luminosity on \[OIII\] luminosity than extended sources, and substantially less scatter about this relation. This is expected from synchrotron ageing and jet-environment interaction in extended sources, discussed in Section \[sec:theory\], and has also been reported by @MiraghaeiBest17. Compact and extended sources have different normalisations of the radio – optical luminosity relation, and preferentially reside in different environments (Figure \[fig:Lnvss\_vs\_Loiii\_complete\]). This likely accounts for the reduced scatter when mass is introduced as an additional variable: the mass scalings (either halo or black hole) of the combined population are steeper than for compact and extended sources individually; a similar point applies to combining the HERG and LERG samples. Radio - optical luminosity correlations depend only weakly on mass in individual subsamples, with pronounced mass scalings built up by combining different populations. A key result of this work is that the scatter in the Fundamental Plane which uses halo mass is the same as that using black hole mass. This is consistent with our interpretation that black hole mass is a proxy for environment. Our radio – \[O[<span style="font-variant:small-caps;">iii</span>]{}\] luminosity slope is consistent with the results of @SaikiaEA15. Both our relation and that of @SaikiaEA15 are shallower than the slope predicted by Equation \[eqn:extended\_radioBol\_scaling\], and also the observed scaling relations in powerful radio sources [@WillottEA99]. Different jet production efficiencies $\eta / \epsilon_{\rm rad}$ across the AGN populations will contribute to flattening this slope. Recently, @SbarratoEA14 have argued that at the lowest accretion rates the ionising luminosity (traced by \[O[<span style="font-variant:small-caps;">iii</span>]{}\]) can drop significantly below a linear scaling. Such a decrease would lead to higher $L_{1.4} / L_{\rm [O\,{\textsc{iii}}]}$ ratios at the lowest \[O[<span style="font-variant:small-caps;">iii</span>]{}\] luminosities, and again flatten the observed relation. =0.08cm [rccccccccc]{} & & No mass & & $M_{\rm halo}$ & & & $M_{\rm bh}$ & &\ & $N_{\rm AGN}$ & $a_{\rm OIII}$ & $\sigma$ & $a_{\rm OIII}$ & $a_{\rm mass}$ & $\sigma$ & $a_{\rm OIII}$ & $a_{\rm mass}$ & $\sigma$\ \ [All sources]{} & 781 &$ 0.85 \pm 0.05 $ & $ 0.45 $ & $ 0.87 \pm 0.05 $ & $ 0.28 \pm 0.04 $ & $ 0.42 $ & $ 0.91 \pm 0.05 $ & $ 0.25 \pm 0.05 $ & $ 0.42 $\ Compact & 576 &$ 0.90 \pm 0.04 $ & $ 0.33 $ & $ 0.91 \pm 0.04 $ & $ 0.11 \pm 0.03 $ & $ 0.32 $ & $ 0.93 \pm 0.04 $ & $ 0.18 \pm 0.03 $ & $ 0.31 $\ Extended & 138 &$ 0.67 \pm 0.10 $ & $ 0.47 $ & $ 0.73 \pm 0.10 $ & $ 0.18 \pm 0.10 $ & $ 0.45 $ & $ 0.74 \pm 0.10 $ & $ 0.20 \pm 0.11 $ & $ 0.45 $\ [LERGs]{}: All & 714 &$ 0.94 \pm 0.06 $ & $ 0.43 $ & $ 0.93 \pm 0.06 $ & $ 0.25 \pm 0.04 $ & $ 0.40 $ & $ 0.97 \pm 0.06 $ & $ 0.16 \pm 0.06 $ & $ 0.41 $\ Compact & 576 &$ 1.02 \pm 0.05 $ & $ 0.30 $ & $ 1.03 \pm 0.05 $ & $ 0.05 \pm 0.03 $ & $ 0.29 $ & $ 1.03 \pm 0.05 $ & $ 0.12 \pm 0.04 $ & $ 0.29 $\ Extended & 138 &$ 0.88 \pm 0.15 $ & $ 0.43 $ & $ 0.93 \pm 0.14 $ & $ 0.17 \pm 0.10 $ & $ 0.40 $ & $ 0.92 \pm 0.14 $ & $ 0.02 \pm 0.13 $ & $ 0.42 $\ [HERGs]{}: All & 67 &$ 0.80 \pm 0.16 $ & $ 0.52 $ & $ 0.60 \pm 0.97 $ & $ 0.62 \pm 0.95 $ & $ 0.45 $ & $ 0.76 \pm 0.16 $ & $ 0.55 \pm 0.14 $ & $ 0.43 $\ Compact & 48 &$ 0.62 \pm 0.18 $ & $ 0.45 $ & $ 0.34 \pm 0.27 $ & $ 0.73 \pm 0.22 $ & $ 0.37 $ & $ 0.61 \pm 0.16 $ & $ 0.32 \pm 0.11 $ & $ 0.39 $\ Discussion {#sec:discussion} ========== Dimensionless Fundamental Plane {#sec:dimless_FP} ------------------------------- @Daly16 presented a novel approach to non-dimensionalising the Fundamental Plane analysis, plotting the ratio $Q_{\rm jet} / L_{\rm bol}$ as a function of $L_{\rm bol} / L_{\rm Edd}$ for a sample of powerful classical double radio sources. Using a sample of high and low-excitation powerful radio galaxies, @Daly16 argued that there appears to be a trend to lower jet production efficiencies (i.e. lower $Q_{\rm jet} / L_{\rm bol}$ values) with increasing accretion efficiency. @XieYuan17 performed a similar analysis using radio and X-ray luminosities, and argued for a change in radio-loudness at the lowest accretion rates. In Figure \[fig:dimless\_jetEfficiency\], we plot the ratio of radio to bolometric luminosity as a function of dimensionless accretion rate[^7]. We recover the result of @Daly16 and @XieYuan17 that lower accretion rate systems are more radio-loud. Our work also shows that environment is a hidden variable in these relations: massive haloes host a higher fraction of low accretion rate (in Eddington units) systems. Moreover, as Figures \[fig:dimless\_jetEfficiency\] and \[fig:Lnvss\_vs\_Loiii\_complete\] show, massive haloes are also more likely to host extended, luminous radio sources. ![image](./LradioEfficiency_vs_Lbol_edd_logMhalo14p21.png){width="0.68\columnwidth"} ![image](./LradioEfficiency_vs_Lbol_edd_logMhalo13p24.png){width="0.68\columnwidth"} ![image](./LradioEfficiency_vs_Lbol_edd_logMhalo12p24.png){width="0.68\columnwidth"} Implications of the environmental dependence of radio emission {#sec:invisibleLobes} -------------------------------------------------------------- The majority of sources in our sample appear compact, consistent with the earlier results of @BestEA05 [@SAAR08; @BaldiEA15]. Using the definition of Section \[sec:results\], only 18 of 594 compact LERGs (our largest sub-sample) have significant excess of diffuse emission, $S_{\rm NVSS} / S_{\rm FIRST} > 5$. Relaxing the compactness criterion to $S_{\rm NVSS} / S_{\rm FIRST} > 1.38$ [@SinghEA15] increases the number of sources with extended emission to 170. This fraction is mass-dependent, with $24 \pm 3$% of unresolved AGN in low-mass ($M_{\rm halo}<10^{13} M_\odot / h$) haloes having extended emission, compared with $33 \pm 4$% in high-mass haloes ($M_{\rm halo}>3 \times 10^{13} M_\odot / h$). A possible interpretation of this compact population is that the compact jets are not young or frustrated, but instead have low-surface brightness radio lobes which are too faint for detection, similar to the subsample of compact, VLBI-detected AGN in poor environments studied by @ShabalaEA17. Extended low surface brightness radio emission is naturally expected in steep environments where the radio jets cannot be collimated to form classical Fanaroff-Riley extended structures. In this scenario, only emission on scales comparable to the galactic disk – where the jet has a sufficient working surface – will be seen. Free-free absorption by the multi-phase interstellar medium will produce characteristic GHz-peaked spectra on these kpc-scales, potentially explaining the overabundance of observed low-luminosity, compact, flat-spectrum sources [@WhittamEA17]. @YuanWang12 found a broad relation between core and lobe radio luminosities in both radio galaxy and quasar samples, with a (logarithmic) slope of less than unity; in other words, faint radio sources have a higher extended-to-core luminosity ratio. These low-luminosity jets tend to be LERGs in dense environments [@BestHeckman12; @SabaterEA13], and hence the shallow slope of the @YuanWang12 relation is consistent with our results above. The possibility of lobes contributing to detected radio emission has two potentially important implications. As discussed in Section \[sec:theory\], compact and extended sources have different jet power – radio luminosity scalings; misinterpreting lobe-dominated emission will therefore tend to overestimate jet kinetic powers, with implications for galaxy formation models [@RaoufEA17]. Moreover, the conversion from total to core luminosity [e.g. using the relations of @YuanWang12] must take environment into account. The second effect will be on using the Fundamental Plane relations to estimate black hole masses, as has been recently done for intermediate [@GultekinEA14; @KoliopanosEA17] and ultra-massive [@HlavacekLarrondoEA12] black hole populations. Here, any lobe radio emission will lead to overestimates in black hole mass. Observations sensitive to low-surface brightness emission, such as the MWA GLEAM survey [@HurleyWalkerEA17], should soon quantify such selection effects and answer the question of whether the majority of radio AGN sources are genuinely compact, or their full extent is simply invisible to conventional interferometers at gigahertz frequencies. Conclusions {#sec:conclusions} =========== This short paper considers the environmental dependence on the optical Fundamental Plane of black hole activity. SDSS group catalogues were used to quantify environments around a sample of low-redshift ($z<0.2$) AGN. \[O[<span style="font-variant:small-caps;">iii</span>]{}\] line luminosities are used the estimate the AGN bolometric luminosities, and VLA FIRST and NVSS surveys to quantify the 1.4 GHz radio continuum emission associated with the AGN on two spatial scales. The main result of this work is that mass of the AGN host halo is a hidden variable in the Fundamental Plane relations. Specifically: - High and low-excitation radio galaxies occupy different mass haloes. These AGN follow similar jet power – radio luminosity relations, but with different normalizations due to their different jet production efficiencies. When these populations are considered together, the preference of low-excitation radio AGN towards massive haloes naturally introduces a mass-dependent “tilt” in the AGN Fundamental Plane. - Even at fixed accretion mode, the fraction of compact and extended radio AGN depends on halo mass. Massive haloes are more likely to host AGN with substantial lobe, rather than jet, emission. The relationship between lobe radio luminosity and jet power is similar to that involving jet luminosity, but again with a different normalization. Lobe contribution to AGN radio luminosity also contributes to the mass-dependent tilt in the FP. - Scatter in the FP relations is reduced significantly when AGNs in different accretion modes, and with different jet/lobe contributions to radio emission, are treated separately. For compact low-excitation AGN, our largest sub-sample, scatter in the FP relations reduces by over 40 percent, and the halo mass dependence becomes statistically insignificant at the $2\sigma$ level. Similar results are found when black hole mass (estimated using stellar velocity dispersions) is used instead of halo mass. This is expected due to the well-known correlation between these two parameters, and suggests something rather important: our results above are consistent with a scenario in which the role of black hole mass in the observed optical Fundamental Plane of black hole activity is simply as a proxy for the real variable, namely the environment in which the AGN resides. Acknowledgements {#acknowledgements .unnumbered} ================ I am grateful to Ivy Wong, Simon Ellingsen, Leith Godfrey and Martin Krause for many interesting conversations. I thank the anonymous referee for a constructive report which greatly improved the paper, particularly aspects related to stellar mass black holes. This work was partly funded by an Australian Research Council Early Career Fellowship (DE130101399). \[lastpage\] [^1]: E-mail: stanislav.shabala@utas.edu.au [^2]: We use the convention $F_{\nu} \propto \nu^{\alpha}$ throughout this paper. [^3]: Equation \[eqn:Lradio\_Qjet\_scaling\] is evaluated under the assumption of no synchrotron or Inverse Compton losses in the emitting population. In reality, these losses become progressively more important as the source ages, and manifest themselves via a systematic shift in the radio luminosity – jet power relation with increasing source size (see Figure 2 of @SG13). [^4]: Our \[O[<span style="font-variant:small-caps;">iii</span>]{}\] luminosities are not extinction-corrected, consistent with the luminosity-independent bolometric correction reported by @HeckmanEA04. @SaikiaEA15 showed that using dust-corrected \[O[<span style="font-variant:small-caps;">iii</span>]{}\] luminosity results in lower $a_{\rm OIII}$ and higher $a_{\rm mass}$ values (see Table \[tab:fit\_params\]) by 0.3-0.4 dex, due to the luminosity dependence of the bolometric correction [@LamastraEA09]. [^5]: As shown by @NisbetBest16, using the steeper relation of @McConnellMa13 will change $a_{\rm mass}$ by 0.2-0.3 dex. [^6]: hyperfit.icrar.org [^7]: Using jet generation efficiency instead of radio luminosity, $\log \left( Q_{\rm jet} / L_{\rm bol} \right) = \frac{6}{5+p} \log L_{\rm radio} - \log L_{\rm bol} + {\rm const}$ (Equation \[eqn:extended\_radioBol\_scaling\]), yields similar results.
--- address: 'Institut f. Theoret. Physik, Univ. Würzburg D–97074 Würzburg, [Federal Republic of Germany]{}' author: - 'H. Feldmann and R. Oppermann' bibliography: - 'oppgroup.bib' title: 'Random Magnetic Interactions and Spin Glass Order Competing with Superconductivity: Interference of the Quantum Parisi Phase' ---
--- abstract: 'Here we consider a variant of the 5 dimensional Kaluza-Klein theory within the framework of Einstein-Cartan formalism that includes torsion. By imposing a set of constraints on torsion and Ricci rotation coefficients, we show that the torsion components are completely expressed in terms of the metric. Moreover, the Ricci tensor in 5D corresponds exactly to what one would obtain from torsion-free general relativity on a 4D hypersurface. The contributions of the scalar and vector fields of the standard K-K theory to the Ricci tensor and the affine connections are completely nullified by the contributions from torsion. As a consequence, geodesic motions do not distinguish the torsion free 4D space-time from a hypersurface of 5D space-time with torsion satisfying the constraints. Since torsion is not an independent dynamical variable in this formalism, the modified Einstein equations are different from those in the general Einstein-Cartan theory. This leads to important cosmological consequences such as the emergence of cosmic acceleration.' author: - 'Karthik H. Shankar and Kameshwar C. Wali' title: 'Kaluza-Klein Theory with Torsion confined to the Extra-dimension' --- In any attempt to link fundamental matter fields with intrinsic spin to gravity, it becomes necessary to extend the Riemannian space-time to include torsion, defined to be the antisymmetric part of affine connection. The resulting theory of gravity, known as the Einstein-Cartan theory[@Hehl] treats the metric and torsion as two independent geometrical characteristics of space-time [@Shapiro]. Historically, beginning with the Kaluza-Klein (KK) theory, there has been a great interest in introducing extra dimensions of space-time to unify gravity with elementary particle interactions. In the KK theory, the scalar and vector fields, which are the extra dimensional components of the metric tensor contribute to the affine connection and the Ricci tensor and hence modify their values from the corresponding values in 4D space-time[@Wesson]. The contribution of these fields to the Einstein tensor are normally interpreted as gravity induced matter. In this work, we incorporate torsion into 5D KK theory[@Kalinowski]. The inclusion of torsion introduces free parameters in the affine connection and the Ricci tensor in addition to the contributions from extra dimensional metric components that occur in the torsion free KK theory. In this paper we impose a minimal set of conditions so as to restrict torsion to purely extra dimensional components and determine all its components in terms of the metric. Interestingly, the imposed conditions lead to a complete cancellation between the modifications induced by the extra dimensional metric components and the contributions from the torsion. Thus the Ricci tensor in 5D space-time in the resulting formalism is exactly the Ricci tensor in a torsion free 4D space-time. In the second part of the paper, we apply the action principle to derive the equations of motion of this formalism. The modified Einstein equations thus derived are finally applied to Robertson-Walker cosmology that leads to a novel expansion history for the universe (see figure \[FRW\]). To describe the constraints to be imposed, we start, for the sake of completeness, with a brief overview on the relationship between tetrads, torsion and affine connection. $\\$ Setting up the framework: Since the constraints we propose to impose can be more simply stated with reference to a locally flat inertial system (allowed by the equivalence principle), we proceed to define and collect together from reference [@Nakaharabook], the relevant standard relations we need in both the coordinate and the inertial frames. Let ($i, j,k,...$) and (${\textrm{{\scriptsize A}}},{\textrm{{\scriptsize B}}},...$) denote coordinate and inertial frame indices respectively and ${\ensuremath{\mathbf{\hat{e}}_{i}}}=\partial_{i}$ and ${\ensuremath{\mathbf{\hat{\theta}}^{i}}}=dx^{i}$ be the basis of the tangent and dual spaces at each point in space-time. We define the corresponding inertial basis to be ${\ensuremath{\mathbf{\hat{e}}_{{\textrm{{\scriptsize A}}}}}}={\ensuremath{\mathrm{e}^{i}_{\cdot {\textrm{{\scriptsize A}}}}}} {\ensuremath{\mathbf{\hat{e}}_{i}}}$ and ${\ensuremath{\mathbf{\hat{\theta}}^{{\textrm{{\scriptsize A}}}}}}={\ensuremath{\mathrm{e}^{\cdot {\textrm{{\scriptsize A}}}}_{i}}}{\ensuremath{\mathbf{\hat{\theta}}^{i}}}$, where the vielbeins ${\ensuremath{\mathrm{e}^{i}_{\cdot {\textrm{{\scriptsize A}}}}}}$ and ${\ensuremath{\mathrm{e}^{\cdot {\textrm{{\scriptsize B}}}}_{j}}}$ satisfy the orthonormality conditions, $${\ensuremath{\mathrm{e}^{\cdot {\textrm{{\scriptsize A}}}}_{i}}} {\ensuremath{\mathrm{e}^{j}_{\cdot {\textrm{{\scriptsize A}}}}}}={\delta}^{j}_{i}; \qquad {\ensuremath{\mathrm{e}^{\cdot {\textrm{{\scriptsize A}}}}_{i}}}{\ensuremath{\mathrm{e}^{i}_{\cdot {\textrm{{\scriptsize B}}}}}}={\delta}^{{\textrm{{\scriptsize A}}}}_{{\textrm{{\scriptsize B}}}}. \label{eq:orthonormality}$$ By definition, the metric in the inertial frame is Minkowskian $\eta_{AB}$, and the metric tensor in the coordinate system is $\mathbf{g}_{i j}={\ensuremath{\mathrm{e}^{\cdot {\textrm{{\scriptsize A}}}}_{i}}} {\ensuremath{\mathrm{e}^{\cdot {\textrm{{\scriptsize B}}}}_{j}}} \eta_{AB}$ and $\mathbf{g}^{i j}={\ensuremath{\mathrm{e}^{i}_{\cdot {\textrm{{\scriptsize A}}}}}} {\ensuremath{\mathrm{e}^{j}_{\cdot {\textrm{{\scriptsize B}}}}}} \eta^{AB}$. The covariant derivative operator ($\tilde{\nabla}$) can be defined in terms of the coordinate basis, or equivalently in terms of the inertial frame basis, $${\tilde{\nabla}}_{{\ensuremath{\mathbf{\hat{e}}_{i}}}}{\ensuremath{\mathbf{\hat{e}}_{j}}}={\ensuremath{ \tilde{\Gamma}^{k}_{\cdot \, i j}}}{\ensuremath{\mathbf{\hat{e}}_{k}}}, \qquad {\tilde{\nabla}}_{{\ensuremath{\mathbf{\hat{e}}_{{\textrm{{\tiny A}}}}}}}{\ensuremath{\mathbf{\hat{e}}_{{\textrm{{\scriptsize B}}}}}}={\ensuremath{ \mathrm{\omega}^{{\textrm{{\scriptsize C}}}}_{\cdot \, {\textrm{{\scriptsize A}}}{\textrm{{\scriptsize B}}}}}}{\ensuremath{\mathbf{\hat{e}}_{{\textrm{{\scriptsize C}}}}}},$$ where ${\ensuremath{ \tilde{\Gamma}^{i}_{\cdot \, j k}}}$ and ${\ensuremath{ \mathrm{\omega}^{{\textrm{{\scriptsize A}}}}_{\cdot \, {\textrm{{\scriptsize B}}}{\textrm{{\scriptsize C}}}}}}$ are the affine and the Ricci rotation coefficients respectively. The relationship between these two quantities follows from the transformation laws between the coordinate frame([$\mathbf{\hat{e}}_{i}$]{}) and inertial frame([$\mathbf{\hat{e}}_{{\textrm{{\scriptsize A}}}}$]{}), $${\ensuremath{ \mathrm{\omega}^{{\textrm{{\scriptsize A}}}}_{\cdot \, {\textrm{{\scriptsize B}}}{\textrm{{\scriptsize C}}}}}}={\ensuremath{\mathrm{e}^{i}_{\cdot {\textrm{{\scriptsize B}}}}}} ( {\tilde{\nabla}}_{{\ensuremath{\mathbf{\hat{e}}_{i}}}} {\ensuremath{\mathrm{e}^{j}_{\cdot {\textrm{{\scriptsize C}}}}}} ) {\ensuremath{\mathrm{e}^{\cdot {\textrm{{\scriptsize A}}}}_{j}}}. \label{def:omega}$$ The affine connection by itself is not a tensor, but its antisymmetric part, the torsion, is a tensor. [^1] $${\ensuremath{ \mathrm{T}^{i}_{\cdot \, j k}}}={\ensuremath{ \tilde{\Gamma}^{i}_{\cdot \, j k}}}-{\ensuremath{ \tilde{\Gamma}^{i}_{\cdot \, k j}}} = 2 {\ensuremath{ \tilde{\Gamma}^{i}_{\cdot \, [j k]}}}$$ Again, using the transformation laws between the coordinate and the inertial frames, we have $${\ensuremath{ \mathrm{T}^{i}_{\cdot \, j k}}} = {\ensuremath{\mathrm{e}^{\cdot {\textrm{{\scriptsize B}}}}_{j}}}{\ensuremath{\mathrm{e}^{\cdot {\textrm{{\scriptsize C}}}}_{k}}} {\ensuremath{\mathrm{e}^{i}_{\cdot {\textrm{{\scriptsize A}}}}}}{\ensuremath{ \mathrm{T}^{{\textrm{{\scriptsize A}}}}_{\cdot \, {\textrm{{\scriptsize B}}}{\textrm{{\scriptsize C}}}}}}. \label{torsion_transform}$$ Furthermore, with the standard assumption of metric compatibility, namely $\nabla_{{\ensuremath{\mathbf{\hat{e}}_{i}}}} g_{jk}=0$, we obtain $${\ensuremath{ \tilde{\Gamma}^{i}_{\cdot \, j k}}} = {\ensuremath{ \hat{\Gamma}^{i}_{\cdot \, j k}}} + {\ensuremath{ \mathrm{K}^{i}_{\cdot \, j k}}}, \label{chris1}$$ where ${\ensuremath{ \hat{\Gamma}^{i}_{\cdot \, j k}}}$ is the Christoffel connection, $${\ensuremath{ \hat{\Gamma}^{i}_{\cdot \, j k}}}= \Big\{ {}_{j} {}^{i} {}_{k} \Big\} =\frac{1}{2}\mathbf{g}^{im}[{\partial_{j}\mathbf{g}_{km}+\partial_{k}\mathbf{g}_{jm}-\partial_{m}\mathbf{g}_{jk}}] , \label{def:Chris}$$ and ${\ensuremath{ \mathrm{K}^{i}_{\cdot \, j k}}}$ is the contorsion tensor. $${\ensuremath{ \mathrm{K}^{i}_{\cdot \, j k}}}= \frac{1}{2} \left[ {\ensuremath{ \mathrm{T}^{i}_{\cdot \, j k}}} + {\ensuremath{\mathrm{T}^{\cdot \, i}_{j\, \cdot k}}} + {\ensuremath{\mathrm{T}^{\cdot \, i}_{k\, \cdot j}}} \right] . \label{def:contorsion}$$ Constructing the 5D space-time: We shall now focus on applying the above formalism to a five dimensional space-time. Consider a foliation of the 5D space-time in terms of a family of 4D hypersurfaces, which are parametrized by the coordinate system $\{x^{\mu} \}$, where $(\mu, \nu,...)$ denote the coordinate indices on these hypersurfaces. Let $x^{5}$ denote the parametrization of the family, and $5$ denote the corresponding coordinate index. The hypersurface coordinates $\{x^{\mu}\}$ together with $x^{5}$ will then span the entire 5D space-time. Let the metric and its inverse on each of the hypersurface be $\mathrm{g}_{\mu \nu}$ and $\mathrm{g}^{\mu \nu}$ respectively, which can in principle depend on the $x^{5}$ coordinate. Let $\{{\ensuremath{\mathrm{e}^{\mu}_{\cdot a}}},{\ensuremath{\mathrm{e}^{\cdot a}_{\mu}}}\}$ denote the tetrad system on these hypersurfaces satisfying the orthonormality relations ${\ensuremath{\mathrm{e}^{\mu}_{\cdot a}}} {\ensuremath{\mathrm{e}^{\cdot b}_{\mu}}}=\mathbf{\delta}^{a}_{b}$ and ${\ensuremath{\mathrm{e}^{\mu}_{\cdot a}}}{\ensuremath{\mathrm{e}^{a}_{\cdot \nu}}}=\mathbf{\delta}^{\mu}_{\nu}$. Here $(a,b,...)$ denote the tetrad indices on these 4D hypersurfaces. The metric on the hypersurface is then given by $\mathrm{g}_{\mu \nu}={\ensuremath{\mathrm{e}^{\cdot a}_{\mu}}} {\ensuremath{\mathrm{e}^{\cdot b}_{\nu}}} \eta_{ab}$ and its inverse $\mathrm{g}^{\mu \nu}={\ensuremath{\mathrm{e}^{\mu}_{\cdot a}}} {\ensuremath{\mathrm{e}^{\nu}_{\cdot b}}} \eta^{ab}$. We will now construct the vielbiens in the 5D space-time by extending the tetrad system on the 4D hypersurfaces. We take the components of the 5D vielbeins to be ${\ensuremath{\mathrm{e}^{i}_{\cdot {\textrm{{\scriptsize A}}}}}}=({\ensuremath{\mathrm{e}^{\mu}_{\cdot a}}}, {\ensuremath{\mathrm{e}^{\mu}_{\cdot {\ensuremath{\dot{5}}}}}}, {\ensuremath{\mathrm{e}^{5}_{\cdot a}}}, {\ensuremath{\mathrm{e}^{5}_{\cdot {\ensuremath{\dot{5}}}}}})$ and ${\ensuremath{\mathrm{e}^{\cdot {\textrm{{\scriptsize A}}}}_{i}}}= ({\ensuremath{\mathrm{e}^{\cdot a}_{\mu}}}, {\ensuremath{\mathrm{e}^{\cdot a}_{5}}}, {\ensuremath{\mathrm{e}^{\cdot {\ensuremath{\dot{5}}}}_{\mu}}},{\ensuremath{\mathrm{e}^{\cdot {\ensuremath{\dot{5}}}}_{5}}} )$. The index ${\ensuremath{\dot{5}}}$ corresponds to the fifth dimension of the inertial frame. The orthonormality relations in 5D (eq. \[eq:orthonormality\]) immediately leads to [^2] $$\begin{aligned} {\ensuremath{\mathrm{e}^{\mu}_{\cdot {\ensuremath{\dot{5}}}}}}=0, \,\, {\ensuremath{\mathrm{e}^{5}_{\cdot a}}}=-{\ensuremath{\mathrm{e}^{\mu}_{\cdot a}}}\mathrm{{\ensuremath{\mathrm{A}}}}_{\mu}, \, \, {\ensuremath{\mathrm{e}^{5}_{\cdot {\ensuremath{\dot{5}}}}}} = \Phi^{-1}, \nonumber \\ {\ensuremath{\mathrm{e}^{\cdot a}_{5}}}=0, \,\, {\ensuremath{\mathrm{e}^{\cdot {\ensuremath{\dot{5}}}}_{\mu}}}={\ensuremath{\mathrm{A}}}_{\mu} \Phi, \,\, {\ensuremath{\mathrm{e}^{\cdot {\ensuremath{\dot{5}}}}_{5}}}= \Phi. \label{eq:vielbeins}\end{aligned}$$ The metric in the 5D is then given by $\mathbf{g}_{i j}={\ensuremath{\mathrm{e}^{\cdot {\textrm{{\scriptsize A}}}}_{i}}} {\ensuremath{\mathrm{e}^{\cdot {\textrm{{\scriptsize B}}}}_{j}}} \eta_{AB}$ and $\mathbf{g}^{i j}={\ensuremath{\mathrm{e}^{i}_{\cdot {\textrm{{\scriptsize A}}}}}} {\ensuremath{\mathrm{e}^{j}_{\cdot {\textrm{{\scriptsize B}}}}}} \eta^{AB}$. $$\begin{aligned} \mathbf{g}_{\mu \nu} &=& \mathrm{g}_{\mu \nu}+\epsilon {\ensuremath{\mathrm{A}}}_{\mu}{\ensuremath{\mathrm{A}}}_{\nu}\Phi^{2},\, \mathbf{g}_{\mu 5} = \epsilon {\ensuremath{\mathrm{A}}}_{\mu}\Phi^{2}, \, \mathbf{g}_{5 5}=\epsilon \Phi^{2}, \nonumber \\ \mathbf{g}^{\mu \nu} &=& \mathrm{g}^{\mu \nu}, \, \mathbf{g}^{\mu 5}=-{\ensuremath{\mathrm{A}}}^{\mu}, \, \mathbf{g}^{5 5} = {\ensuremath{\mathrm{A}}}_{\lambda} {\ensuremath{\mathrm{A}}}^{\lambda} + \epsilon \Phi^{-2} . \label{eq:5Dmetric}\end{aligned}$$ The raising and lowering of indices on the vector field is done w.r.t the 4D metric $\mathrm{g}_{\mu \nu}$. The parameter $\epsilon= \pm 1$ denotes whether the extra dimension is space-like or time-like. Note that the induced metric on the hypersurfaces (induced by the 5D geometry), $\mathrm{g}_{\mu \nu}+\epsilon {\ensuremath{\mathrm{A}}}_{\mu}{\ensuremath{\mathrm{A}}}_{\nu}\Phi^{2}$ is different from the 4D metric $\mathrm{g}_{\mu \nu}$ on them, but quite evidently related by a gauge transformation. We will now impose a set of constraints on torsion and the Ricci rotation coefficients consistent with Cartan’s structure equation [@Nakaharabook] that relates torsion and connection coefficients. With a minimal modification of the standard general relativity in mind, we chose the set of constraints [@Viet] so that the torsion components tangential to the 4D hypersurface vanish (see also [@Bohmer]), while leaving non-vanishing torsion components completely determined in terms of the metric components of the 5D space-time. In the absence of torsion, the metric compatibility condition serves to evaluate the connection coefficients (Christoffel symbols, eq. \[def:Chris\]) in terms of the metric components. In the presence of torsion, the imposed constraints can be viewed as an extension to the metric compatibility condition in the spirit that they together serve to determine the connection coefficients in terms of the metric components. Furthermore, the imposed constraints are covariant conditions on the 4D hypersurface. That is, the conditions are form invariant with respect to within-hypersurface coordinate transformations (that do not mix $\mu,\nu$ with $5$). $$\textbf{Condition 1} : {\ensuremath{ \mathrm{T}^{a}_{\cdot \, {\textrm{{\scriptsize B}}}{\textrm{{\scriptsize C}}}}}}=0$$ Using eq. \[torsion\_transform\] and noting that ${\ensuremath{\mathrm{e}^{\mu}_{\cdot {\ensuremath{\dot{5}}}}}}=0$, we find ${\ensuremath{ \mathrm{T}^{\mu}_{\cdot \, i k}}}=0$. This implies that the only nonzero components of torsion are [$ \mathrm{T}^{5}_{\cdot \, i k}$]{}. To determine these, we impose the following condition on the Ricci rotation coefficients, $$\textbf{Condition 2} : {\ensuremath{ \mathrm{\omega}^{{\ensuremath{\dot{5}}}}_{\cdot \, {\textrm{{\scriptsize B}}}{\textrm{{\scriptsize C}}}}}}=0$$ This condition along with metric compatibility implies ${\ensuremath{ \mathrm{\omega}^{{\textrm{{\scriptsize A}}}}_{\cdot \, {\textrm{{\scriptsize B}}}{\ensuremath{\dot{5}}}}}}=0$. We can now use eq. \[def:omega\] to write, $${\ensuremath{ \mathrm{\omega}^{{\textrm{{\scriptsize A}}}}_{\cdot \, {\textrm{{\scriptsize B}}}{\ensuremath{\dot{5}}}}}}= {\ensuremath{\mathrm{e}^{i}_{\cdot {\textrm{{\scriptsize B}}}}}} ( {\tilde{\nabla}}_{{\ensuremath{\mathbf{\hat{e}}_{i}}}} {\ensuremath{\mathrm{e}^{j}_{\cdot {\ensuremath{\dot{5}}}}}}) {\ensuremath{\mathrm{e}^{\cdot {\textrm{{\scriptsize A}}}}_{j}}}= {\ensuremath{\mathrm{e}^{i}_{\cdot {\textrm{{\scriptsize B}}}}}} ( \partial_{i} {\ensuremath{\mathrm{e}^{j}_{\cdot {\ensuremath{\dot{5}}}}}} + {\ensuremath{ \tilde{\Gamma}^{j}_{\cdot \, i k}}} {\ensuremath{\mathrm{e}^{k}_{\cdot {\ensuremath{\dot{5}}}}}} ) {\ensuremath{\mathrm{e}^{\cdot {\textrm{{\scriptsize A}}}}_{j}}} =0$$ Since ${\ensuremath{\mathrm{e}^{\mu}_{\cdot {\ensuremath{\dot{5}}}}}}= {\ensuremath{\mathrm{e}^{a}_{\cdot 5}}}=0$ , the above equation implies, $${\ensuremath{ \tilde{\Gamma}^{\mu}_{\cdot \, i 5}}}=0, \qquad {\ensuremath{ \tilde{\Gamma}^{5}_{\cdot \, i 5}}} = -{\ensuremath{\mathrm{e}^{\cdot {\ensuremath{\dot{5}}}}_{5}}} \partial_{i} {\ensuremath{\mathrm{e}^{5}_{\cdot {\ensuremath{\dot{5}}}}}}$$ Using the above equations along with eq.\[chris1\], we can express the contorsion in terms of the Christoffel symbols [$ \hat{\Gamma}^{}_{\cdot \, }$]{}and the vielbeins, $$\begin{aligned} {\ensuremath{ \mathrm{K}^{\mu}_{\cdot \, i 5}}} &=& -{\ensuremath{ \hat{\Gamma}^{\mu}_{\cdot \, i 5}}}, \nonumber \\ {\ensuremath{ \mathrm{K}^{5}_{\cdot \, i 5}}}&=&-\left( {\ensuremath{ \hat{\Gamma}^{5}_{\cdot \, i 5}}} + {\ensuremath{\mathrm{e}^{\cdot {\ensuremath{\dot{5}}}}_{5}}} \partial_{i} {\ensuremath{\mathrm{e}^{5}_{\cdot {\ensuremath{\dot{5}}}}}} \right) =- {\ensuremath{ \hat{\Gamma}^{5}_{\cdot \, i 5}}} + \mathrm{J}_{i}, \label{c2}\end{aligned}$$ where $\mathrm{J}_{i} \equiv \Phi^{-1} \partial_{i}\Phi$. Note that the above equations give only a subset of the components [$ \mathrm{K}^{i}_{\cdot \, j k}$]{}, but along with eq. \[def:contorsion\], they are sufficient to determine all the components of the torsion. The torsion thus obtained can in turn be substituted in eq. \[def:contorsion\] to determine the remaining components of contorsion. The components of torsion are $$\begin{aligned} {\ensuremath{ \mathrm{T}^{\mu}_{\cdot \, i j}}}&=&{\ensuremath{ \mathrm{T}^{5}_{\cdot \, 5 5}}}= 0, \nonumber \\ {\ensuremath{ \mathrm{T}^{5}_{\cdot \, \mu \nu}}} &=& 2 \partial_{[\mu} {\ensuremath{\mathrm{A}}}_{\nu]} +2 {\ensuremath{\mathrm{J}}}_{[\mu} {\ensuremath{\mathrm{A}}}_{\nu]}, \nonumber \\ {\ensuremath{ \mathrm{T}^{5}_{\cdot \, \mu 5}}} &=& {\ensuremath{\mathrm{J}}}_{\mu} - \partial_{5} {\ensuremath{\mathrm{A}}}_{\mu} -{\ensuremath{\mathrm{A}}}_{\mu} {\ensuremath{\mathrm{J}}}_{5}, \label{torsions}\end{aligned}$$ In addition to yielding the non-vanishing components of torsion in terms of the metric components, the solution to eq. \[c2\] also yields the following condition on the metric on the 4D hypersurfaces. $$\partial_{5} \mathrm{g}_{\mu \nu}=0.$$ This implies all the hypersurfaces in the foliating family have the same 4D metric. To place things in perspective, we observe that in the standard Kaluza-Klein theory, the assumption of cylindrical condition makes all the quantities, namely $\mathrm{g}_{\mu \nu}$, ${\ensuremath{\mathrm{A}}}_{\mu}$ and $\Phi$ independent of $x^{5}$. Whereas in our formulation, the constraints automatically imply $\mathrm{g}_{\mu \nu}$ is independent of $x^{5}$, while ${\ensuremath{\mathrm{A}}}_{\mu}$ and $\Phi$ could still depend on $x^{5}$. With all the [$ \mathrm{K}^{i}_{\cdot \, j k}$]{} determined from eqns. \[torsions\] and \[def:contorsion\], we now use eq. \[chris1\] to calculate all the affine connection coefficients. $$\begin{aligned} {\ensuremath{ \tilde{\Gamma}^{\lambda}_{\cdot \, 5 5}}}&=&{\ensuremath{ \tilde{\Gamma}^{\lambda}_{\cdot \, \nu 5}}}={\ensuremath{ \tilde{\Gamma}^{\lambda}_{\cdot \, 5 \nu}}}=0, \nonumber \\ {\ensuremath{ \tilde{\Gamma}^{5}_{\cdot \, \mu \nu}}}&=&\nabla_{\mu}{\ensuremath{\mathrm{A}}}_{\nu}+ {\ensuremath{\mathrm{J}}}_{\mu} {\ensuremath{\mathrm{A}}}_{\nu}, \nonumber \\ {\ensuremath{ \tilde{\Gamma}^{5}_{\cdot \, 5 \mu}}}&=& \partial_{5} {\ensuremath{\mathrm{A}}}_{\mu} + {\ensuremath{\mathrm{J}}}_{5} {\ensuremath{\mathrm{A}}}_{\mu}, \nonumber \\ {\ensuremath{ \tilde{\Gamma}^{5}_{\cdot \, \mu 5}}} &=& {\ensuremath{\mathrm{J}}}_{\mu}, \,\,\, {\ensuremath{ \tilde{\Gamma}^{5}_{\cdot \, 5 5}}} = {\ensuremath{\mathrm{J}}}_{5}, \,\,\, {\ensuremath{ \tilde{\Gamma}^{\lambda}_{\cdot \, \mu \nu}}}={\ensuremath{\Gamma^{\lambda}_{\cdot \, \mu \nu}}}. \label{connections}\end{aligned}$$ Here [$\Gamma^{\lambda}_{\cdot \, \mu \nu}$]{} corresponds to the Christoffel symbols (analogous to eq. \[def:Chris\]) obtained from torsion free 4D space-time with metric $\mathrm{g}_{\mu \nu}$. Note that the components of 5D Christoffel symbols along the hypersurface coordinates is not equal to the Christoffel symbols calculated on 4D spacetime, that is, ${\ensuremath{ \hat{\Gamma}^{\lambda}_{\cdot \, \mu \nu}}} \neq {\ensuremath{\Gamma^{\lambda}_{\cdot \, \mu \nu}}} $. The symbol $\nabla_{\mu}$ corresponds to the covariant derivative operator on the torsion-free 4D space-time, where the Christoffel symbols are exactly the affine connection coefficients. Substituting the above connection coefficients in the Ricci tensor defined by $$\tilde{R}_{ik}=\partial_{k}{\ensuremath{ \tilde{\Gamma}^{j}_{\cdot \, j i}}} - \partial_{j} {\ensuremath{ \tilde{\Gamma}^{j}_{\cdot \, k i}}} +{\ensuremath{ \tilde{\Gamma}^{j}_{\cdot \, k m}}}{\ensuremath{ \tilde{\Gamma}^{m}_{\cdot \, j i}}}- {\ensuremath{ \tilde{\Gamma}^{j}_{\cdot \, j m}}}{\ensuremath{ \tilde{\Gamma}^{m}_{\cdot \, k i}}},$$ we find $$\tilde{R}_{\mu \nu}= R_{\mu \nu}, \,\, \tilde{R}_{\mu 5} = \tilde{R}_{5 \mu} = \tilde{R}_{5 5}=0.$$ Here $R_{\mu \nu}$ represents the Ricci tensor on the torsion-free 4D space-time. It also follows that the 5D Ricci scalar is exactly the same as the Ricci scalar in the torsion free 4D space time, that is $\tilde{R} =R$. We also note, in general, the presence of torsion makes the Ricci tensor non-symmetric, but the constraints we have imposed on the torsion leaves the Ricci tensor symmetric. It is straightforward to see that the formalism and the results obtained thus far are not specific to 4 and 5 dimensions, they can be generalized to any arbitrary D and D+1 dimensions. We shall now consider some implications of the formalism with respect to geodesic equations and solutions to Einstein equations. $\\$ Geodesic Equations: The 5D geodesic equations split into $$\begin{aligned} &\overset{..}{x}^{5}& + {\ensuremath{ \tilde{\Gamma}^{5}_{\cdot \, \mu \nu}}} \dot{x}^{\mu} \dot{x}^{\nu} + \left( {\ensuremath{ \tilde{\Gamma}^{5}_{\cdot \, \mu 5}}} + {\ensuremath{ \tilde{\Gamma}^{5}_{\cdot \, 5 \mu}}} \right) \dot{x}^{\mu} \dot{x}^{5} + {\ensuremath{ \tilde{\Gamma}^{5}_{\cdot \, 5 5}}} \left( \dot{x}^{5} \right)^{2}=0, \nonumber \\ &\overset{..}{x}^{\lambda}& +{\ensuremath{\Gamma^{\lambda}_{\cdot \, \mu \nu}}} \dot{x}^{\mu} \dot{x}^{\nu} =0 \label{eq:geodesic}\end{aligned}$$ We note that the components of the geodesic equations along the hypersurface are exactly the same as the geodesic equations in the torsion free 4D space-time. This is in contrast with the conventional Kaluza Klein theory where the 4D geodesic equations are modified by the presence of the fields ${\ensuremath{\mathrm{A}}}_{\mu}$ and $\Phi$. In this formalism, the presence of torsion completely nullifies the effect of these fields in the 4D geodesic equations. Furthermore, it is worth noting that the geodesic of a particle can be confined to a 4D hypersurface by requiring $\dot{x}^{5}=0$. From eq. \[eq:geodesic\], we see that this requires the additional condition, $${\ensuremath{ \tilde{\Gamma}^{5}_{\cdot \, \mu \nu}}}=\nabla_{\mu}{\ensuremath{\mathrm{A}}}_{\nu}+ {\ensuremath{\mathrm{J}}}_{\mu} {\ensuremath{\mathrm{A}}}_{\nu}=0.$$ If this condition can be satisfied, then there will be no observable difference, as far as a test particle is concerned, whether we live in a torsion free 4D space-time or on a hypersurface within the 5D space-time with torsion. However, it is apparent that this condition is a strong constraint requiring the vector and scalar fields satisfy the above equation for a given 4D metric. It is conceivable that for some 4D metrics, no choice of vector and scalar fields would satisfy the above constraint. In such cases, the particle would be free to move in the $x^{5}$ direction unless constrained by an external force or if the fifth co-ordinate is compact and small as usually assumed in most adaptations of the Kaluza-Klein theory. $\\$ Einstein’s equations: To obtain the equations of motion, we need to vary the action with respect to the independent dynamical fields of the theory. In general, if one includes torsion in the theory, torsion and metric are two independent dynamical fields and consequently one needs to vary the action with respect to both these variables \[1, 4\]. In our approach, with torsion determined in terms of the metric, metric components are the only dynamical variables. Taking the action to be $S= \int \tilde{R} \sqrt{-\mathbf{g}} \, d^{5}x $, its variation yields [^3] $$\int \left[ \tilde{R} \, \delta \sqrt{-\mathbf{g}} + \tilde{R}_{i k} \sqrt{-\mathbf{g}} \, \delta \mathbf{g}^{ik} + \mathbf{g}^{ik} \sqrt{-\mathbf{g}} \, \delta \tilde{R}_{i k} \right] d^{5}x$$ One can set this variation to zero to obtain the modified Einstein equations. The first two terms give rise to the usual symmetric Einstein tensor. In the absence of torsion, the third term becomes a boundary integral that contributes nothing to the equation. While in the presence of torsion, the third term can be shown to contribute $$\int \left[ {\ensuremath{ \mathrm{T}^{m}_{\cdot \, j m}}} \mathbf{g}^{ik} \delta {\ensuremath{ \tilde{\Gamma}^{j}_{\cdot \, k i}}} - {\ensuremath{ \mathrm{T}^{m}_{\cdot \, k m}}} \mathbf{g}^{ik} \delta {\ensuremath{ \tilde{\Gamma}^{j}_{\cdot \, j i}}} + {\ensuremath{ \mathrm{T}^{m}_{\cdot \, k j}}} \mathbf{g}^{ik} \delta {\ensuremath{ \tilde{\Gamma}^{j}_{\cdot \, m i}}} \right] \sqrt{-\mathbf{g}} \, d^{5}x \nonumber$$ Treating the variation in the torsion and metric components independent, one can obtain the Einstein-Cartan equations derived by Hehl et.al [@Hehl]. However, by expressing $\delta {\ensuremath{ \tilde{\Gamma}^{}_{\cdot \, }}}$ purely in terms of variation of the 5D metric components (from eq. \[connections\]), it turns out that the above expression can be simplified to $$\int {\ensuremath{\mathrm{H}}}_{\mu \nu} \delta g^{\mu \nu} \, \sqrt{-\mathbf{g}} \, d^{5}x \label{extraterm}$$ where $$\begin{aligned} {\ensuremath{\mathrm{H}}}_{\mu \nu} &\equiv& \nabla_{( \mu} {\ensuremath{\mathrm{B}}}_{\nu) } - (\nabla \cdot {\ensuremath{\mathrm{B}}}) g_{\mu \nu} + {\ensuremath{\mathrm{J}}}_{( \mu} {\ensuremath{\mathrm{B}}}_{\nu )} - ({\ensuremath{\mathrm{J}}}\cdot {\ensuremath{\mathrm{B}}}) g_{\mu \nu}, \nonumber \\ {\ensuremath{\mathrm{B}}}_{\mu} & \equiv&{\ensuremath{ \mathrm{T}^{5}_{\cdot \, \mu 5}}} = {\ensuremath{\mathrm{J}}}_{\mu} - \partial_{5} {\ensuremath{\mathrm{A}}}_{\mu} -{\ensuremath{\mathrm{A}}}_{\mu} {\ensuremath{\mathrm{J}}}_{5},\end{aligned}$$ are interpreted as tensors in torsion free 4D space. Since eq. \[extraterm\] is the only additional term that contributes to the variation of action, ${\ensuremath{\mathrm{H}}}_{\mu \nu}$ is essentially the modification to the standard torsion free Einstein tensor. The modified Einstein equations then turns out to be $$\begin{aligned} R_{\mu}^{\,\, \nu} -\frac{1}{2} R \delta^{\,\, \nu}_{\mu} + {\ensuremath{\mathrm{H}}}^{\,\, \nu}_{\mu} &=& \Sigma_{\mu}^{\,\, \nu} \label{EE1} \\ -{\ensuremath{\mathrm{A}}}^{\alpha} R_{\mu \alpha} - {\ensuremath{\mathrm{A}}}^{\alpha} {\ensuremath{\mathrm{H}}}_{\mu \alpha} &=& \Sigma_{\mu}^{\,\, 5} \label{EE2} \\ -\frac{1}{2} R &=& \Sigma_{5}^{\,\, 5} \label{EE3}\end{aligned}$$ Here $\Sigma$ is the stress tensor that one would obtain when matter fields are included in the Lagrangian prior to variation of action. Its 4D components will be the observed stress energy tensor and can be identified with the stress energy tensor of the usual 4D torsion free general relativity. Its conservation immediately yields $$\nabla_{\nu} \Sigma^{\,\, \nu}_{\mu} =0 \Longrightarrow \nabla_{\nu} {\ensuremath{\mathrm{H}}}^{\,\, \nu}_{\mu} =0. \label{conservation}$$ In general, the components $\Sigma_{\mu}^{\,\, 5}$ and $\Sigma_{5}^{\,\, 5}$ cannot be obtained from observations on the 4D hypersurface. Hence, without an explicit 5D matter Lagrangian specifying these components, it is not possible to solve equations \[EE2\] and \[EE3\]. Thus, we are left with just equations \[EE1\] and \[conservation\] to solve for the metric components $g_{\mu \nu}$, ${\ensuremath{\mathrm{A}}}_{\mu}$ and $\Phi$. ![The dashed-dot curve denotes $a(t)$, the dotted curve denotes $\dot{a}(t)$, and the solid curve denotes $\overset{..}{a}(t)$. The time axis is in units of ${\ensuremath{\mathrm{H}}}_{o}^{-1}$. []{data-label="FRW"}](grfig1.pdf) $\\$ Robertson-Walker cosmology: To illustrate an application of the formalism, let us consider spatially flat homogenous and isotropic universe with metric $$\mathrm{ds}^2 = -dt^2 +a^{2}(t) \left( dx^{2} + dy^{2} + dz^{2} \right).$$ The assumption of homogeneity and isotropy of the 4D geometry requires that ${\ensuremath{\mathrm{A}}}_{\mu} =({\ensuremath{\mathrm{A}}}_{t}, 0,0,0)$ and ${\ensuremath{\mathrm{A}}}_{t}$ and $\Phi$ are functions of $t$ and not the spatial coordinates. To further simplify, we shall also assume that the fields do not depend on $x^{5}$. From eq. \[torsions\], these constraints imply that ${\ensuremath{\mathrm{B}}}_{\mu}={\ensuremath{\mathrm{J}}}_{\mu}$ and the only non vanishing component of ${\ensuremath{\mathrm{J}}}_{\mu}$ is ${\ensuremath{\mathrm{J}}}_{t}$, and of ${\ensuremath{\mathrm{A}}}_{\mu}$ is ${\ensuremath{\mathrm{A}}}_{t}$. Applying the conservation equation (eq. \[conservation\]) yields $$\begin{aligned} (i) \, \dot{ \Phi}= 0, {\ensuremath{\mathrm{J}}}_{t}=0, \,\, \mathrm{or} \qquad (ii) \, \Phi= \dot{a}(t), {\ensuremath{\mathrm{J}}}_{t}=\overset{..}{a}/\dot{a}. \\ \nonumber\end{aligned}$$ Case $(i)$ yields ${\ensuremath{\mathrm{H}}}_{\mu \nu}=0$, and eq. \[EE1\] yields the usual Friedman equation along with matter conservation. $$\left( \dot{a}/a \right)^{2} = \frac{8 \pi}{3} \rho.$$ Case $(ii)$ yields non vanishing ${\ensuremath{\mathrm{H}}}_{\mu \nu}$, which when applied to eq. \[EE1\] gives the modified Friedman equation $$\left( \dot{a}/a \right)^{2} + \left( \overset{..}{a}/a \right) = \frac{8 \pi}{3} \rho, \label{FE}$$ along with matter conservation which implies $\rho a^3= \rho_{o}$ is a constant in a matter dominated universe. To solve the equation, we specify initial conditions at the current instant of time, $a=1$, $\dot{a}={\ensuremath{\mathrm{H}}}_{o}$, the Hubble constant and $\overset{..}{a}= - q_{o}{\ensuremath{\mathrm{H}}}_{o}^2 $, where $q_{o}$ is the current deceleration parameter. These conditions can be used to calculate $\rho_{o}$, the current matter density (including dark matter). Taking $q_{o}=-0.5$, which is consistent with the current observations [@acc_history], the solution to eq. \[FE\] is plotted in figure \[FRW\]. From the dashed-dot curve (scale factor), note that the universe started expanding at $t= -0.518 \,{\ensuremath{\mathrm{H}}}_{o}^{-1}$, from a size of $a=2/3$, prior to which it was in a contracting phase. This is in contrast to the solution of the usual Friedmann equations which yields a big bang ($a=0$) at $t= -0.667 \,{\ensuremath{\mathrm{H}}}_{o}^{-1}$. From the solid curve, note that the acceleration is currently positive but decreasing and would become negative beyond $t=0.319 \,{\ensuremath{\mathrm{H}}}_{o}^{-1}$. This is qualitatively consistent with the analysis of observed data in [@Shafieloo], but is in sharp contrast with the standard $\Lambda$CDM model which predicts that the acceleration would continue to increase for ever. Clearly, more detailed studies are needed to understand the full implications of this formalism on cosmology. In conclusion, the formalism in this paper provides a general mathematical result pertaining to affine connection and Ricci tensor in Einstein-Cartan theory in higher dimensions. It provides an alternative way to confine gravity from (D+1) to D dimensions, with modifications that can be significant to cosmology. Acknowledgements: This work was supported in part (K. C. Wali) by the U. S. Department of Energy (DOE) under contract no. DE-FG02-85ER40237. [10]{} F. W. Hehl, P. v.d. Hyde, G. D. Kerlick and J. M. Nester, Rev. Mod. Phys. [48]{}, 393 (1976). For a historical review of concept of torsion and its role in constructing a non-Riemanninan structure of space-time. I. L. Shapiro, Phys. Rept. [357]{}, 113 (2002). The report reviews the physical aspects of torsion including both classical and quantum aspects and the necessity of torsion to couple the spin of matter fields to space-time. J. M. Overduin and P. S. Wesson, Phys. Rept. [283]{}, 303 (1997). For various aspects and recent developments of torsion free Kaluza-Klein theory. M. W. Kalinowski, Int. J. Theo. Phys. [20]{}, 563 (1981). For an analysis of Kaluza-Klein theory in the presence of torsion. M. Nakahara, sections 7.8.1 and 7.8.4, [Geometry, Topology and Physics]{}, Second edition (IOP Publishing Ltd., London, 2003). N. A. Viet and K. C. Wali, Phys. Rev. [D67]{}, 124029 (2003). These minimal conditions on torsion were utilized in Viet, Wali to obtain a fully covariant action involving all the metric fields in a discretized Kaluza-Klein theory. See also N. A. Viet, arxiv: hep-th/0303128. C. G. Bohmer and L. Fabbri, Mod. Phys. Lett. [A22]{}, 2727 (2007). A. Shafieloo, V. Sahni and A. A. Starobinsky, arXiv: astro-ph/0903.5141. [^1]: Throughout this paper, square brackets enclosing indices denote the conventional anti-symmetrization, while the regular brackets enclosing the indices denotes symmetrization. [^2]: There exists another class of vielbeins that satisfy the orthonormality relations. But they are essentially related to eq. \[eq:vielbeins\] by gauge. Here, we choose to work with the vielbeins given by eq. \[eq:vielbeins\] to make the formalism readily comparable to the existing Kaluza-Klein literature. [^3]: We are using geometric units where $G=c=1$. We also note that the action in general will include a matter Lagrangian term which would also contribute to the variation.
--- abstract: 'Understanding the nature of galactic populations of double compact binaries (where both stars are a neutron star or black hole) has been a topic of interest for many years, particularly the coalescence rate of these binaries. The only observed systems thus far are double neutron star systems containing one or more radio pulsars. However, theorists have postulated that short duration gamma-ray bursts may be evidence of coalescing double neutron star or neutron star-black hole binaries, while long duration gamma-ray bursts are possibly formed by tidally enhanced rapidly rotating massive stars that collapse to form black holes (collapsars). The work presented here examines populations of double compact binary systems and tidally enhanced collapsars. We make use of <span style="font-variant:small-caps;">binpop</span> and <span style="font-variant:small-caps;">binkin</span>, two components of a recently developed population synthesis package. Results focus on correlations of both binary and spatial evolutionary population characteristics. Pulsar and long duration gamma-ray burst observations are used in concert with our models to draw the conclusions that: double neutron star binaries can merge rapidly on timescales of a few million years (much less than that found for the observed double neutron star population), common envelope evolution within these models is a very important phase in double neutron star formation, and observations of long gamma-ray burst projected distances are more centrally concentrated than our simulated coalescing double neutron star and collapsar Galactic populations. Better agreement is found with dwarf galaxy models although the outcome is strongly linked to the assumed birth radial distribution. The birth rate of the double neutron star population in our models range from $4-160~$Myr$^{-1}$ and the merger rate ranges from $3-150~$Myr$^{-1}$. The upper and lower limits of the rates results from including electron capture supernova kicks to neutron stars and decreasing the common envelope efficiency respectively. Our double black hole merger rates suggest that black holes should receive an asymmetric kick at birth.' author: - \| Paul D. Kiel$^{1}$[^1], Jarrod R. Hurley$^{1}$ and Matthew Bailes$^{1}$\ $^{1}$Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, Victoria, 3122, Australia title: 'The long and the short of it: modelling double neutron star and collapsar Galactic dynamics' --- \[firstpage\] binaries: close – stars: evolution – stars: pulsar – stars: neutron – Galaxy: stellar content – Gamma-rays: bursts Introduction {#s:intro} ============ Pulsars, magnetic oblate spheriods of nuclear densities $20-30$ kilometers in diameter, have been found rotating at speeds of up to almost one thousand times a second (Hessels et al. 2006; see also Galloway 2008) in some of the most exotic settings in the known Universe. For example, some pulsars are found within X-ray binaries (Liu, van Paradijs & van den Heuvel 2007) and others with compact object companions in close binaries (Hulse & Taylor 1975) thought to be emitting gravitational radiation (see Landau & Lifshitz 1951; Paczyński 1967; Clark & Eardley 1977). Recently, the number of known binary pulsars has been rapidly increasing (Lorimer et al. 2006a; Galloway et al. 2008). There are now in excess of $100$ Galactic disk binary pulsars. This includes rotation powered pulsars (radio pulsars: ATNF Pulsar Catalogue Manchester, Hobbs, Teoh & Hobbs 2005[^2]) and accretion powered pulsars (X-ray binary pulsars: Liu, van Paradijs & van den Heuvel 2007; Galloway 2008; Galloway et al. 2008). More than $20$ of these are accreting from a range of stellar masses and companion types (Galloway 2008), $9$ are thought to orbit another neutron star (van den Heuvel 2007; Stairs 2008), while others ($> 70$) dwell in detached systems with white dwarfs companions (ATNF Pulsar Catalogue, Manchester et al. 2005) and possibly main sequence (MS) companions (Champion et al. 2008). From observations of pulsars in such systems it is possible to constrain computational modelling of binary evolutionary phases that are general to many non-pulsar related systems including, but not limited to, tidal evolution, Roche-lobe overflow and common envelope (CE) evolution. It is the most rapidly rotating pulsars that best constrain uncertainties in the theory related to these processes. Such work has been attempted in Kiel et al. (2008) and Kiel & Hurley (2009: KH09), where models of the complete Galactic pulsar population have been made. However, we note that these models are yet to include selection effects which are critical when interpreting the results of pulsar surveys (Taylor & Manchester 1977; Oslowski et al. 2009). Short gamma-ray bursts provide an alternative method to study the physics of compact stars. Here compact stars are considered to be the most compact of remnants: neutron stars (NSs) and black holes (BHs). We define a double compact binary (DCB) to be any combination of these compact objects within a binary system (without any limit on the orbital period) and close DCBs are those systems with orbital periods of less than a few days. The DCBs that merge within the assumed $10~$Gyr age of the Galaxy (i.e. during our simulations) are defined as coalescing DCBs. It is postulated that gamma-ray emission is produced during the coalescence of these systems and that radiation of gravitational waves occurs during the preceding in-spiral phase (Clark & Eardley 1977). In particular it is thought that short gamma-ray bursts are produced by coalescing double neutron star systems (Paczynski 1986). Gamma-ray burst observations are very interesting in themselves, however, DCB systems offer other observational features of importance. Not only can many tests of general relativity be performed but they offer insights into a host of observable phenomena (e.g. Taylor, Fowler & McCulloch 1979; Lyne et al. 2004). The formation of close DCBs requires two stars of sufficient mass to interact gravitationally, triggering mechanisms to decrease the separation between them during their stellar lifetimes. What seems to be the most important binary evolutionary mechanism in forming close DCBs is the common-envelope phase (Paczyński 1976) where the evolution following the first supernova (SN) generally requires at least one such event. The modelling of the CE phase is associated with much uncertainty (see, for example, Dewi, Podsiadlowski & Sena 2006; Belczynski et al. 2007a) and will be discussed further in the following section. Interestingly, Voss & Tauris (2003) found that $1/3$ of BH-NS DCB systems can be formed via the direct-SN mechanism (Kalogera 1998) and therefore a CE phase is not required in all cases. Adding further uncertainty to the mix Brown (1995) suggested that unless the initial mass ratio is very close to unity NSs spiraling-in within a CE should always accrete enough matter to collapse and form BHs. This work examines population characteristics of DCBs, including predictions of double neutron star (NS-NS) distributions and correlations between orbital properties and location. The results are extended to detail both long and short gamma-ray bursts (GRBs) and their progenitors, modelling tidally enhanced collapsars and coalescing DCBs. In particular, projected distances from the host galaxy of model GRBs are compared to observations. No detailed conclusions from direct comparison to observations are attempted, as in Belczynski, Bulik & Rudak (2002) who account for redshift and different galaxy masses. Nor do we consider the evolution of systems in globular clusters (see e.g. Ivanova et al. 2008; Sadowski et al. 2008). Section \[s:bpbk\] briefly outlines the population synthesis tool used in this body of work. The results are spread over Sections \[s:DCBNS-NS\], \[s:coal\] and \[s:grb\], which examine the bound DCB and NS-NS populations (even if they go on to eventually coalesce), coalescing DCB populations and GRB populations, respectively. Population synthesis models {#s:bpbk} =========================== The following work utilizes theoretical studies into NS binary and stellar evolution (Kiel et al. 2008) and Galactic kinematics (KH09). The modelling of stellar and binary evolution is performed by <span style="font-variant:small-caps;">binpop</span>, a Monte-Carlo population synthesis scheme that incorporates the detailed binary stellar evolution (<span style="font-variant:small-caps;">bse</span>) code, developed by Hurley, Tout & Pols (2002) and updated in Kiel et al. (2008). <span style="font-variant:small-caps;">bse</span> facilitates modelling of the latest theoretical stellar and binary evolution prescriptions such as: tides, stellar composition, magnetic braking, gravitational radiation, mass accretion over a range of timescales and realistic angular mometum evolution of pulsar systems. All of which are important evolutionary features when modelling populations of DCBs. An aspect of binary evolution crucial to the formation and characteristics of close DCBs is common-envelope evolution (Paczyński 1976; Webbink 1984; Iben & Livio 1993). The treatment of CE evolution within <span style="font-variant:small-caps;">binpop</span>/<span style="font-variant:small-caps;">bse</span> has previously been described in detail (Hurley, Tout & Pols 2002; Kiel & Hurley 2006). However, considering the potential influence of this phase on the outcomes of close binary population synthesis we also give an overview here so as to assist the reader in the interpretation of our results, noting that a full investigation of how uncertainties involved in modeling this phase affect the properties of the DCB populations will be left to future work. A CE phase is assumed to be initiated if a giant star (with a sizeable convective envelope) fills its Roche-lobe and the ensuing mass-transfer is calculated to occur on a dynamical timescale. The envelope of the giant then rapidly expands and envelops both the companion star and the degenerate core of the giant, thus forming a CE. As suggested by Paczyński (1976), the companion star and giant core will then spiral towards each other while transferring orbital energy to the envelope via dynamical friction. The eventual outcome of this process is essentially determined by a race between the spiral-in and the removal of the envelope owing to the energy transfer: if the companion and the giant core come into contact first then a merger results, otherwise the result is a close binary composed of the former core (now a remnant star) and the companion. Although some detailed hydrodynamic models of this process have been attempted (Bodenheimer & Taam 1984; Taam & Sandquist 2000; Ricker & Taam 2008) the treatment within population synthesis codes remains simplistic as a detailed theory is lacking. In <span style="font-variant:small-caps;">binpop</span>/<span style="font-variant:small-caps;">bse</span> modelling of the CE phase is summed up by the equation: $$\frac{M \left( M - M_{\rm c} \right)}{\lambda R} = \frac{\alpha_{\rm CE} M_{\rm c} m}{2} \left( \frac{1}{a_{\rm f}} - \frac{1}{a_{\rm i}} \right) \,$$ which describes the energy balance between the binding energy of the envelope and change in orbital energy. These are related by an efficiency parameter, $\alpha_{\rm CE}$, which encapsulates the uncertainty of the model within what is known as the $\alpha$-formalism (Nelemans & Tout 2005). Also included in the calculation are the mass of the giant star, $M$, the mass of the giant core $M_{\rm c}$, the mass of the companion, $m$, the gravitational constant, $G$, the structure constant of the giant, $\lambda$, the radius of the giant, $R$, and the initial and final orbital separation, $a_{\rm i}$ and $a_{\rm f}$. As documented in Hurley, Tout & Pols (2002) and Kiel & Hurley (2006), this is roughly the equivalent of using $\alpha^{'}_{\rm CE} = 1$ in the alternative formulation given by Iben & Livio (1993: where we have taken the liberty of using $\alpha^{'}_{\rm CE}$ to distinguish between the two methods) and thus corresponds to efficient transfer of energy within the CE. Kiel & Hurley (2006) found that using $\alpha_{\rm{CE}} = 3$ aided comparison of the relative NS-low-mass X-ray binary and BH-low-mass X-ray binary formation rates to observations. Previous use of $\alpha_{\rm CE} = 1$ produced a dearth of BH-low-mass X-ray binaries compared to NS-low-mass X-ray binaries. Within the $\alpha$-formalism the greater the value of $\alpha_{\rm CE}$ the more efficient the spiral in process is at driving away the envelope. Values greater than unity may suggest an energy source other than gravitational potential energy is also important in driving off the envelope. The effect of the assumed value of $\alpha_{\rm CE}$ (or $\alpha^{'}_{\rm CE}$) on the results of binary population synthesis have been documented previously (e.g. Tutukov & Yungleson 1996; Belczynski, Bulik & Rudak 2002; Hurley, Tout & Pols 2002; Voss & Tauris 2003; Kiel & Hurley 2006). A large $\alpha_{\rm CE}$ prescription is also equivalent to a large value of $\lambda$ – a diffuse stellar envelope. In the past there have been three methods of accounting for the structure of the giant donor star with $\lambda$ (see Voss & Tauris 2003 for a similar discussion on this point). One method is to include it with the CE efficiency and to simply vary a combined $\alpha_{\rm CE} \lambda$ parameter (e.g. Belczynski et al. 2002a,b; Nelemans & Tout 2005; Pfahl, Podsiadlowski & Rappaport 2005; Belczynski et al. 2007a). Another method is to assume a constant value of $\lambda$ (with $0.5$ typically used) separate to the assumed $\alpha_{\rm CE}$ value (numerically consistent with the previous method: Portegies Zwart & Yungleson 1998; Hurley, Tout & Pols 2002; Dewi, Podsiadlowski & Sena 2006; Kiel & Hurley 2006). The final method is to include an algorithm which provides varying values of $\lambda$ depending upon (see Tauris & Dewi 2001) the mass (core and envelope) and stellar evolutionary phase of the donor star (e.g. Voss & Tauris 2003; Podsiadlowski, Rappaport, Han 2003; Kiel & Hurley 2006). Tauris & Dewi (2001) show that $\lambda$ values may range between $0.02-0.7$ (the average for massive stars being $\lambda = 0.1$: Dewi & Tauris 2001; Dewi, Podsiadlowski & Sena 2006). In this body of work we use a variable $\lambda$ that ranges from $0.01-0.5$ depending upon the donor type and we assume $\alpha_{\rm CE}$ to be either $1$ or $3$. Another feature of the CE phase which has been shown to clearly affect the resultant populations of double compact binaries is the possibility that CE initiated by stars on the Hertzsprung Gap (HG) always leads to coalescence (Belczynski et al. 2007a). Following this assumption Belczynski et al. (2007a) found that the merger rate of BH-BH binaries decreased by $\sim 2$ orders of magnitude compared to models without such a restriction. This concept is based on the study of Ivanova & Taam (2003) who, in agreement with the suggestion of Podsiadlowski, Rappaport & Han (2003) found that all their detailed models in which a HG star initiates CE resulted in coalescence of the two cores. At this stage we do not impose such an outcome explicitly. However, the possibility of coalescence is increased by the use of a varying $\lambda$ during CE. Here the value of $\lambda$ may drop to as low as $\sim 0.01$ during the HG phase (e.g. Podsiadlowski, Rappaport & Han 2003) accounting for the lack of a convective envelope especially for those stars near the Hayashi track. During the rapid and explosive formation of compact objects impulsive asymmetric mass-loss can lead to the transmission of momentum to the central compact object. To model this transfer of momentum we assume an instantaneous SN velocity kick for NSs in the form of a Maxwellian distribution with a dispersion of $190~{\rm km s}^{-1}$ (see Kiel et al. 2008). For BH systems there are difficulties in statistically determining a realistic description for the SN kick magnitudes of the population. The magnitude of stellar BH velocity kicks is believed to vary between $0$ and $\sim 300~{\rm km s}^{-1}$ (e.g. Jonker & Nelemans 2004; Willems et al. 2005). To account for this uncertainty we produce models where BHs receive no kicks or where BH kicks are drawn from a Maxwellian distribution with a dispersion of $190~{\rm km s}^{-1}$. We also can include electron capture SN (EC SN; Miyaji et al. 1980) events in our models. The capture of electrons onto primarily magnesium atoms within an oxygen-neon-magnesium core depletes the core of electron pressure resulting in the collapse and supernova explosion. The details of our EC SN model are found in Kiel et al. (2008) and here we provide models with and without this assumption. During EC SNe the SN kick is taken from a Maxwellian distribution with a dispersion assumed to be $20~{\rm km s}^{-1}$ (typical values range anywhere from $0-50~{\rm km s}^{-1}$). The lower dispersion value arises because of the lower energy yield of EC SNe as compared to typical SNe. Modelling of stellar and binary kinematic evolution within a galaxy is completed using <span style="font-variant:small-caps;">binkin</span>. This code, developed in KH09, solves four coupled equations of motion to evolve forward in time the galactic positions of systems of interest. <span style="font-variant:small-caps;">binkin</span> uses a recipe for the galactic gravitational potential structure, initial birth positions, initial system velocities and output from <span style="font-variant:small-caps;">binpop</span> to calculate the kinematic details for a population of stellar systems within a galaxy. The particular output from <span style="font-variant:small-caps;">binpop</span> of interest for <span style="font-variant:small-caps;">binkin</span> is information on SN recoil velocities (time, direction and magnitude) and the birth time of each system within the Galaxy (randomly selected from a flat distribution between $0-10~$Gyr). Previously, both <span style="font-variant:small-caps;">binpop</span> (Kiel et al. 2008) and <span style="font-variant:small-caps;">binkin</span> (KH09) were used to examine pulsar systems exclusively. DCB and GRB systems, on the other hand, have not been considered directly or in such detail by us until now. We outline the modelling of GRBs systems below and consider their population characteristics in Section \[s:grb\]. Gamma-ray bursts are found to occur on two timescales. There are long GRBs (LGRBs), where the observed bursting event occurs over timescales of typically $20$s, and short GRBs (SGRBs), with a median burst duration of $0.3$s (Woosley & Bloom 2006). The majority of population synthesis works have examined the evolution of what is believed to be SGRB progenitors – coalescing DCB systems (Portegies Zwart & Yungelson 1998; Bloom, Sigurdsson & Pols 1999; Hurley, Tout & Pols 2002; Belczynski, Bulik & Rudak 2002; Voss & Tauris 2003; Belczynski et al. 2008; Sadowski et al. 2008) and similarly Paczyński (1990) modelled GRBs taking their locations as those of an old NS population. At present it is believed that LGRBs are formed during core-collapse type Ibc supernovae (SNe) of massive stars whose cores at the time of explosion are rapidly rotating (Woosley 1993; Woosley & Bloom 2006; termed as the ‘collapsar’ model by MacFadyen & Woosley 1999). Models suggest that the fall back accretion disks of SNe are more energetic and have longer timescales than models of double compact coalescence (Woosley 1993; Woosley & Bloom 2006). This leads to a longer GRB event for the former. Some alternative models of progenitor SGRBs are outlined in the review of Nakar (2007; and references within). These models include accretion induced collapse of NSs into BHs, type Ia SNe, magnetar giant flares and phase conversion from NSs to quark or hyper-stars. Recently, realistic population synthesis models of GRBs have examined the formation and evolution of LGRBs produced from Population II progenitors (see Bogomazov, Lipunov & Tutukov 2008 and Detmers, Langer, Podsiadlowski & Izzard 2008) and Population III progenitors (Belczynski et al. 2007b). It has been shown that magnetic fields extract angular momentum from Wolf-Rayet (LGRB progenitor) cores, slowing the spin of the core below the rate required by GRB theory (Petrovic, Langer, Yoon & Heger 2005; Woosley 1993). Detailed modelling can produce rapidly rotating Wolf-Rayet cores at collapse if the initial stellar metallicity is reduced below solar quantities (Yoon, Langer & Norman 2006). However, in terms of population synthesis both Bogomazov, Lipunov & Tutukov (2008) and Detmers et al. (2008) assumed that the formation of LGRBs must have occurred within a binary system. Here a companion star can exert a tidal influence on the Wolf-Rayet core, keeping the system tidally locked until the SN and formation of LGRB and black hole. Bogomazov, Lipunov & Tutukov (2008) examined in some detail the effect modifying the metallicity and wind mass loss had on the formation of LGRBs, in line with the detailed models of Yoon, Langer & Norman (2006). Detmers et al. (2008) investigated, via detailed stellar models, whether the effect of tides in close binary systems at solar metallicities can spin up a Wolf-Rayet star sufficiently enough to form a collapsar-LGRB. Although Detmers et al. (2008) produced collapsar systems they found it could only arise from those systems that had some mass transfer and/or merger event during their lifetimes. Detmers et al. (2008) then perform a population synthesis study to compare their model birthrate to observations. They found that the tidally spun-up collapsar model could only produce a small fraction of the LGRB formation rate, however, carbon-oxygen and/or helium star mergers with BHs occur more readily and could possibly form some fraction of GRB systems. For our LGRB models we follow the method of Detmers et al. (2008) which requires a system that contains a carbon-oxygen star to be tidally influenced by a BH. The rapidly rotating (tidally locked) carbon-oxygen star then goes on to form a BH and in the process a LGRB. It is not relevant if the resultant BH-BH system becomes disassociated. We consider a binary system to be tidally locked when the ratio of carbon-oxygen star radius to the orbital separation is greater than $0.2$ (Portegies Zwart & Verbunt 1996). The focus of this paper is on DCB and GRB systems produced in population synthesis models. To compare with our previous pulsar population synthesis work we provide analysis of Model C$^{'''}$ from KH09. This model was introduced briefly in KH09 but is used in detail here. It involves the evolution of $10^{9}$ binary systems which is two orders of magnitude greater than the main suite of models presented in KH09. Model C$^{'''}$ incorporates the favoured binary evolutionary model of Kiel et al. (2008), Model Fd, and the favoured kinematics of KH09. Details for the model can be found in Table 1 of Kiel et al. (2008) and Table 1 of KH09. For clarity we show the full set of possible parameters in <span style="font-variant:small-caps;">binpop</span> and <span style="font-variant:small-caps;">binkin</span> within Table \[t:binpopbinkin\] along with their values for Model C$^{'''}$. To address some possible deficiencies of Model C$^{'''}$ we also examine the resultant DCB and GRB populations of an additional set of models that include different evolutionary assumptions. The model changes include assuming: (i) $\alpha_{\rm CE} = 1$; (ii) BHs receive velocity kicks at birth (from the same SN kick distribution as NSs); and, (iii) BHs receive kicks and NSs may form via EC SN. Including these effects allow us to determine the extent by which such assumptions affect the final model DCB and LGRB populations. The initial number of binary systems in these additional models is $10^8$, an order of magnitude less than in Model C$^{'''}$. As shown in KH09 this decrease in model initial binary number still provides statistical significant results for binary evolution and Galactic kinematics, while being preferable for computational efficiency. In our models we define the Galactic region of interest to include all stars with $R \leq 30~$kpc and $\|z\| \leq 10~$kpc. We note that when discussing DCB systems the order of compact star formation is depicted by the placement within the system name. For example the NS is created first in a NS-BH binary (i.e. before any SN the initially more massive star transfers much of its mass to the initially less massive star facilitating NS formation prior to BH formation). <span style="font-variant:small-caps;">binpop</span> ------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------ ------------------------------------- -------------------- Value/ Varied in choice Kiel et al. (2008) $Z$ Zero-age main sequence metallicity $0.02$ $\times$ $M_{\rm 1i,range}$ Primary star birth mass range ($M_{\rm 1i,min} - M_{\rm 1i,max}$) $5 - 80~M_\odot$ $\times$ $\xi(M_1)$ Birth distribution of primary masses KTG93 $\times$ $M_{\rm 2i,range}$ Secondary star birth mass range ($M_{\rm 2i,min} - M_{\rm 2i,max}$) $0.1 - 80~M_\odot$ $\times$ $\phi(M_2)$ Birth distribution of secondary masses $n(q) = 1$ $\times$ $P_{\rm orbi,range}$ Orbital period birth range ($P_{\rm orbi,min} - P_{\rm orbi,max}$) $1 - 3000~$days $\times$ $ \Omega(\log \rm{P}_{\rm orb})$ Birth distribution of orbital period Flat $\times$ $e_{\rm i}$ Initial eccentricity distribution $0$ $\times$ $\alpha_{\rm CE}$ Common envelope efficiency parameter $3$ $\times$ $\lambda$ Binding energy factor for common envelope evolution Variable $\left(0.01 - 0.5 \right)$ $\times$ $M_{\rm NS,max}$ Maximum NS mass $3~M_\odot$ $\times$ $\eta$ Reimers mass-loss coefficient $0.5$ $\times$ $\mu_{\rm nHe}$ Helium star wind mass loss efficiency parameter $0.5$ $\times$ $B_{\rm eml}$ Binary (only) enhanced mass loss parameter $0.0$ $\times$ $\beta_{\rm wind}$ Wind velocity factor $1/8$ $\times$ $\alpha_{\rm wind}$ Bondi-Hoyle wind accretion factor $3/2$ $\times$ $\Xi_{\rm wind}$ Wind accretion efficiency factor $1.0$ $\surd$ $\epsilon_{\rm nova}$ Fraction of accreted matter retained in nova eruption $0.0001$ $\times$ $\rm{E}_{\rm FAC}$ Eddington limit factor for mass transfer $1.0$ $\times$ $\rm{T}_{\rm flag}$ Activates tidal circularisation ‘on’ $\times$ $\rm{BH}_{\rm flag}$ Allows velocity kick at BH formation ‘off’ $\times$ $\rm{NS}_{\rm flag}$ Takes NS/BH mass from BKB02 ‘on’ $\times$ $\rm{Be}_{\rm flag}$ Allows Be star evolution and sets size of Be circumstellar disk ‘off’ $\times$ $\rm{Be}_{\rm method}$ Sets method used in Be/X-ray evolution: wind ($< 0$) or RLOF ($> 0$) N/A $\times$ $\rm{Be}_{\rm \dot{M}}$ Mass loss rate from Be star $(10^{-9} - 10^{-12}~M_\odot/\rm{yr})$ N/A $\times$ $\tau_{\rm B}$ Pulsar magnetic field decay timescale $2~$Gyr $\surd$ $k$ Pulsar magnetic field decay parameter during accretion $3000$ $\surd$ $\rm{P}_{\rm flag}$ Allows propeller evolution ‘off’ $\surd$ $\rm{EC}_{\rm flag}$ Triggers electron capture supernova evolution ‘off’ $\surd$ $f(\rm{P})$ Beaming fraction of pulsar according to TM98 ‘on’ $\surd$ $\rm{DL}_{\rm flag}$ Allows pulsar death line ‘on’ $\surd$ $\rm{SN}_{\rm link}$ Initial pulsar parameters linked to supernova ‘on’ $\surd$ $P_{\rm 0min}$[^3] Initial minimum pulsar spin period $0.02~$s $\surd$ $P_{\rm 0max}$ Initial maximum pulsar spin period $0.16~$s $\surd$ $B_{\rm 0min}$ Initial minimum pulsar magnetic field $5\times10^{11}~$G $\surd$ $B_{\rm 0max}$[^4] Initial maximum pulsar magnetic field $4\times10^{12}~$G $\surd$ <span style="font-variant:small-caps;">binkin</span> Value/ Varied in choice KH09 $N$ Number of systems evolved (also used in <span style="font-variant:small-caps;">binpop</span>) $10^9$ $\surd$ $V_\sigma$ Maxwellian dispersion for the supernova kick speed (also used in <span style="font-variant:small-caps;">binpop</span>) $190~$km s$^{-1}$ $\surd$ $t_{\rm max}$ Age of the galaxy (also used in <span style="font-variant:small-caps;">binpop</span>) $10~$Gyr $\surd$ $\Phi$ Gravitational potential Pac90 $\surd$ $\alpha_{\rm g}$ Galaxy size and mass scaling parameter normalised to Milky Way $1$ $\times$ $R_{\rm init}$ Galactic radial stellar birth distribution YK04 $\surd$ $\|z_{\rm imax}\|$ Maximum possible birth height off the galactic plane $0.075~$kpc $\surd$ $\|z_{\rm max}\|$ Maximum height from the galactic plane used to calculate statistics $10~$kpc $\surd$ Bound double neutron star and black hole population characteristics {#s:DCBNS-NS} =================================================================== In this section, the population characteristics of DCBs are investigated. Model C$^{'''}$ was primarily used by KH09 to check the accuracy of scale heights drawn from models with smaller binary populations. The scale height is twice the e-folding distance of the population in $\|z\|$ and is calculated by counting all systems within $10~$kpc of the plane. Exploring the DCB scale heights of Model C$^{'''}$, Table \[t:table2\] shows the clear difference in NS-NS scale heights compared to those systems with at least one BH. This is primarily due to our assumption that BHs do not receive any velocity kick during their formation. It is interesting that BH-NS systems have a greater scale height than NS-BH systems. This results from the order in which the SN kick occurs, further discussion on this point follows below. For interest we note that very large distances from the plane can occur for DCB systems – up to $5.7~$Mpc for the NS-NS systems – indicating that ejection of DCBs into the intergalactic medium can occur. Such systems are relatively rare, and are those which receive velocity kicks greater than $\sim 500~$km s$^{-1}$ (and integrated for close to $10~$Gyr). Very few ($< 1\%$) BH-BH systems receive relatively large recoil velocities during SN solely from instantaneous mass loss (see Section \[s:recoilvels\]). Model C$^{'''}$ produces a high relative number of BH-BH systems which remain bound after passing through two SNe as opposed to systems which receive one or two kicks in the process (see also the formation rates calculated in Section \[s:fratesNS-NS\]), once again a result of the assumption in Model C$^{'''}$ that BHs do not receive kicks. When BHs are allowed to receive kicks the models show an expected increase in the scale height of BH-BH DCBs and all systems that include BHs. The relative number of BH-BH systems has also decreased, although they still dominate the DCB numbers. Such a large increase in scale heights of NS-BH systems, especially compared to BH-NS systems, is because BH kicks in our model are taken from a Maxwellian distribution with no regard to the amount of fall-back onto the BH. BHs with large mass progenitors ($M>40~M_\odot$) have complete fall-back; therefore, binary systems are not unbound owing to instantaneous SN mass-loss from binary (as there is none) which allows the possibility of greater SN recoil velocities to be imparted onto the system than otherwise. Decreasing the efficiency of removing the envelope during the CE phase significantly reduces the NS-NS scale height. A lower $\alpha_{\rm CE}$ results in more merger events during CE and tighter orbits of those systems that do survive. Although tighter pre-second supernova binary system would allow at first glance greater recoil velocities to occur (owing to larger allowed SN kicks) we find that the magnitude of the recoil velocity after the second SN in NS-NS systems is typically smaller ($\sim 200~{\rm km s}^{-1}$) than that in Model C$^{'''}$ ($\sim 500~{\rm km s}^{-1}$, see Figure \[f:fig15NS-NS\]), although the same range is covered. NS-NS systems in this model, therefore, typically merge on a smaller time scale than otherwise (see Section \[s:coal\]). As expected including EC SN kicks reduces the NS-NS scale height by half, while slightly decreasing the scale heights of other systems that include a NS. The relative number of NS-NS systems increases by two orders of magnitude in this model. ----------------------- -------- -------------- ---------- -- -- -- -- -- Model System Scale height Relative type \[kpc\] number C$^{'''}$ NS-NS $1.53$ $0.003$ NS-BH $0.10$ $0.008$ BH-NS $0.24$ $0.006$ BH-BH $0.03$ $0.982$ $\alpha_{\rm CE} = 1$ NS-NS $0.90$ $0.002$ NS-BH $0.04$ $0.003$ BH-NS $0.15$ $0.008$ BH-BH $0.03$ $0.987$ BH kicks NS-NS $1.55$ $0.111$ NS-BH $1.18$ $0.063$ BH-NS $0.46$ $0.036$ BH-BH $0.78$ $0.790$ BH kicks & NS-NS $0.73$ $0.290$ EC SNe NS-BH $1.21$ $0.048$ BH-NS $0.37$ $0.048$ BH-BH $0.78$ $0.614$ ----------------------- -------- -------------- ---------- -- -- -- -- -- : For each model the scale heights (kpc) and relative numbers are provided for all double compact binaries within $10~$kpc of the Galactic plane at a Galactic age of $10~$Gyr. \[t:table2\] Recoil velocities {#s:recoilvels} ----------------- Figure \[f:fig12NS-NS\] depicts the center of mass recoil velocities directly after the initial SN kick for Model C$^{'''}$. These are shown as a function of final height from the Galactic plane for each system at the end of the simulation (or the point of coalescence). We compare recoil velocities of the four double compact binary types. We limit the figures to solely those of Model C$^{'''}$, although our discussion of the resultant populations cover the other models. The initial SN velocity kick of those systems that form DCBs is typically less than the average velocity kick a NS in our models is given. This was also found by Voss & Tauris (2003, see Figure 11 of their paper) and arises because typical SN kick velocity magnitudes tend to disrupt binary systems, thus preventing DCB formation. NS-NS progenitors can survive the first SN with a greater recoil velocity than their NS-BH counterparts (who also receive a kick on the first SN). The cause of this is partly due to the greater average total system mass of NS-BH progenitors compared to that of NS-NS progenitors. However, for the formation of NS-BH binaries, the necessity of: mass transfer prior to any SN event, mass and orbital angular momentum loss via winds and the instantaneous mass loss during SN events (with the possibility of fall back onto the stellar remnants), greatly complicates matters. Other than a weak trend for NS-BH systems there does not seem to be any significant trend with final system height off the plane and the recoil velocity after the first SN. ![ Distribution of the centre-of-mass recoil velocities after the 1st SN kick and the final system height above the plane in $\|z\|$ at $10~$Gyr for all double compact systems that formed within Model C$^{'''}$. This includes both coalesced and non-coalesced systems. Provided are contours ranging from $10\%$ through to the outer contours of $90\%$. Clockwise from the top left is NS-NSs, BH-BHs, BH-NS (BH formed first) and NS-BH (NS formed first). Because of the large number of BH-BH systems (that may or may not coalesce) produced in this model we have had to limit the number of points we plot to every $10$th (the contours are calculated from the complete sample). \[f:fig12NS-NS\]](figure1DNS.eps){width="84mm"} ![ As for Figure \[f:fig12NS-NS\] but now showing the centre-of-mass recoil velocity induced by the 2nd SN only. Contours are provided as in Figure \[f:fig12NS-NS\]. \[f:fig15NS-NS\]](figure2DNS.eps){width="84mm"} Figure \[f:fig15NS-NS\] illustrates the system centre-of-mass recoil velocity induced solely by the second SN. The range of resultant velocities is much greater than that provided by the first SN kick. Again, like Voss & Tauris (2003), we find that the double compact systems can survive, on average, greater second SN kicks than that which may occur for the first SN. Due to the massive nature of DCB progenitor stars it is typical that such systems have passed through at least one CE phase (see Sections \[s:eccporb\] and \[s:coal\]; although see also Dewi, Podsiadlowski & Sena 2006). A large fraction of these systems have engaged in CE evolution in the intervening time between SN events because, when they survive the CE, the likelihood of binary disassociation owing to SNe decreases. Therefore, DCBs are typically tighter prior to the second SN compared to the first SN. Thus the SN kick must overcome greater orbital binding energy to disassociate the system which means a greater SN kick velocity is allowed that may in turn induce a greater recoil velocity into the system (ignoring mass loss). For some systems the only way that they may survive is from a sufficiently large and well directed kick, as discussed in Portegies Zwart & Yungelson (1998), and similar to the low-mass X-ray binary direct formation mechanism first recognised in Kalogera (1996). The mass of the double compact progenitors, at the second SN, is less than the mass at the first SN and as such the systems can end with much greater recoil velocities after the second SN. We note that Figure \[f:fig15NS-NS\] contains all systems that remain bound after the second SN regardless of whether they subsequently coalesce or not. Focusing on the NS-NSs (top left panel of Figure \[f:fig15NS-NS\]) we find that of the NS-NS systems in Model C$^{'''}$ that survived the two supernovae that $90\%$ coalesce within $10~$Gyr. Because of this the contours mainly trace the coalesced systems, which slope downwards in $\|z\|$ with increasing recoil velocity. This inverse trend arises because we simply plot the merger site – the remnant evolution is ignored here – and systems that receive a greater recoil velocity (on average greater kick velocity and thus higher eccentricity) are more likely to merge faster than those that receive medium to low recoil velocities. However, the height off the plane for systems that remain bound and do not coalesce within our assumed age of the Galaxy is proportional to the recoil velocity of the second SN (and, of course, to the age of the system). This is shown by the population of points in the upper right of the panel (noting that the trend continues to low recoil velocity and $\|z\|$). The top right panel depicts the BH-BHs. The majority of systems are contained at small recoil velocity values. However, there is a population of systems with recoil velocities greater than $\sim 200~$km s$^{-1}$. These systems are ones that have coalesced within $10~$Gyr of their formation. The NS-BH systems are given in the lower left panel of Figure \[f:fig15NS-NS\]. Both coalesced and bound NS-BH systems are tightly constrained to low recoil velocities and both tend to have increased $\|z\|$ values with higher recoil velocities, in this low recoil velocity range. However, beyond a recoil velocity of $\sim 200~$km s$^{-1}$ there is a population of coalescing NS-BH systems which appear constrained to $\|z\| < 1~$kpc from the plane of the Galaxy. There is also a small population of NS-BH systems, which do not coalesce, that reside within a narrow recoil velocity band between $500$ and $600~$km s$^{-1}$. Such systems extend from $\|z\| \sim 0.01~$kpc up to $\|z\| \sim 1000~$kpc. These systems do not receive SN velocity kicks for the second SN, so their formation and large $\|z\|$ does not rely on a sympathetic kick direction but instead on being tightly bound prior to BH formation and that for these systems there has been enough time in their evolution to reach such great distances. The BH-NS systems within the bottom right panel contain two quite different distributions. The bound BH-NS systems extend from very low recoil velocities and small $\|z\|$ values to quite large recoil velocities and high $\|z\|$ values. In contrast the distribution of BH-NSs that coalesce begins at recoil velocities of $\sim 200~$km s$^{-1}$ and $\|z\| \sim 1~$kpc and stretches to high recoil velocities and a height above the plane range of $0.0001 < \|z\| < 1~$kpc. When BHs receive SN kicks their recoil velocity-$\|z\|$ distribution takes on the characteristics of the NS counterparts (although with generally smaller magnitudes). Providing NSs with the option to receive lower kick velocities during EC SNe assembles the resultant NS-NS distribution as an amalgamation between all the Model C$^{'''}$ distributions. When assuming a less efficient CE phase the NS-NS systems have a bimodal recoil velocity distribution with a group clustered between velocities of $0~{\rm km s}^{-1}$ to $\sim 30~{\rm km s}^{-1}$ and another centered around $\sim 100~{\rm km s}^{-1}$ and extending either side by roughly $40~{\rm km s}^{-1}$. Eccentricities and orbital periods {#s:eccporb} ---------------------------------- ![ Eccentricity and orbital period parameter space snapshot of NS-NS systems at a Galactic age of $10~$Gyr and within $4.5 < R < 12.5~$kpc. Included over our model points are $8$ observed systems suspected to be NS-NSs according to van den Heuvel (2007) and Stairs (2008). The observed systems are, J0737-3039 (Lyne et al. 2004), J1518+4904 (Nice et al. 1996), B1534+12, J1811-1736, B1913+16 (Stairs 2004), J1756-2251 (Faulkner et al. 2005), J1829+2456 (Champion et al. 2004), J1906+0746 (Lorimer et al. 2006b) and B2127+11C (Anderson et al. 1990; detected within M15 with an eccentricity of 0.68 and orbital period of 8.05 hours). \[f:fig222NS-NS\]](figure3DNS.eps){width="84mm"} To examine NS-NS systems in greater detail Figure \[f:fig222NS-NS\] depicts the eccentricity-orbital period population distribution that occurs at the simulation end. With greater numbers of NS-NSs now known (9 systems, 8 within the Galactic disk) and more realistic mass estimates of their stellar components available, recently it has become clear that this diagram sheds light on a possible inconsistency between observations and stellar binary theory (van den Heuvel 2007). Similar distributions to those shown in Figure \[f:fig222NS-NS\] have been discussed in varying detail previously in population synthesis works (Portegies Zwart & Yungelson 1998; Bloom, Sigurdsson & Pols 1999; Voss & Tauris 2003; Ihm, Kalogera & Belczynski 2006). Unlike these previous works the systems considered here are restricted to reside within the solar neighbourhood ($4.5 \geq R \geq 12.5~$kpc). There is also no assumption on the ages of the observed NS-NS systems used for comparison – that is, the complete randomly born NS-NS model population is shown. Like Portegies Zwart & Yungelson (1998) this model finds that the addition of SN kicks within NS formation smears the eccentricity and orbital period distributions. Portegies Zwart & Yungelson (1998) also detail the importance of CE evolution in shaping the eccentricity-orbital period distribution. The typical formation pathway of NS-NSs systems that reside in Figure \[f:fig222NS-NS\] pass through stable mass-transfer prior to any SN event. Owing to this mass-transfer either the mass ratio $q$ inverts or, if $q \sim 1$, the secondary stars (initially less massive accreting stars) grow significantly more massive driving $q$ to small values. After the first SN event the systems have a large range of eccentricities and orbital periods, however, with the onset of secondary star evolution these systems eventually pass through a CE evolutionary phase, circularising the orbit. The binary systems must survive CE and the secondary stars evolve to explode, forming NSs. It is the orbital parameters at the time of this SN and the event (kick) itself which is important in regulating the resultant eccentricity-orbital period NS-NS population distribution. Such an evolutionary pathway compares well with the models of Portegies Zwart & Yungelson (1998), though, for young NS-NSs Portegies Zwart & Yungelson find many systems that pass through multiple CE phases between the two SN events. Multiple CE systems are also found in this model, however, these systems coalesce rapidly (few Myr), as such a discussion of these systems is left until Section \[s:coal\]. The distribution of NS-NS systems within Figure \[f:fig222NS-NS\] covers well the nine observed systems believed to be NS-NSs (van den Heuvel 2007; Stairs 2008). However, there is a greater relative number of observed NS-NSs with low eccentricities than that produced in this model. The majority of modelled systems within Figure \[f:fig222NS-NS\] have eccentricities greater than $0.7$ and only $17\%$ have $e < 0.5$. Of the observed Galactic disk NS-NS systems $75\%$ have $e < 0.5$. This discrepancy may be due to low number statistics owing to pulsar observational selection effects or incomplete theoretical modelling. One possible evolutionary phase that may assist in lowering the average eccentricity of model NS-NSs, as suggested by van den Heuvel (2007), is electron capture SNe (Miyaji et al. 1980; and as discussed in Section \[s:bpbk\]). EC SNe would induce smaller SN kicks into these systems potentially rendering them less eccentric. We are now in a position to examine this possibility with our EC SN model. We find that with EC SN included the percentage of systems with $e < 0.5$ increases only slightly to $20\%$. The overabundance of highly eccentric model NS-NS systems still remains a problem. We note here that a full parameter space search that includes EC SNe kicks can produce models that match the observed eccentricity distribution (Andrews, Kalogera & Belczynski, in preparation). However, we show here that this is not the case for standard model assumptions, of typical CE efficiencies and core collapse SN velocity kicks. Eccentricities and height above the plane ----------------------------------------- ![ The eccentricity-$\|z\|$ distributions for each double compact system type. Top left is NS-NSs, top right BH-BHs, bottom left NS-BH where the NS formed first and bottom right BH-NS where the NS formed second. The $90\%$ contour is shown for guidance. We also provide a representation of the median eccentricity of the distribution in steps of $\|z\| = 0.5~$kpc (plus symbols). When the statistical significance of each median eccentricity value is poor (less than $10$ systems in the $\|z\|$ bin) we do not plot the point. \[f:fig16NS-NS\]](figure4DNS.eps){width="84mm"} It is also possible to examine the Galactic dynamics of double compact binaries to look for correlations between orbital properties and location. For such an analysis we make use of Figure \[f:fig16NS-NS\] which depicts the $\|z\|$-eccentricity parameter space for the four double compact binary types. The kinematics of NS-NS systems is as expected: the large relative number of eccentric systems typically extend in greater numbers further out in $\|z\|$ than less eccentric systems. The contour shows a strong trend within $1~$kpc of increasing eccentricity with increasing $\|z\|$. However, there is effectively no variation above $1~$kpc, the median eccentricity does not vary significantly with increasing height above the plane (see plus symbols within Figure \[f:fig16NS-NS\]). This is not the case for BH-BH systems of which the majority reside close to the Galactic plane with little to no eccentricity. Here the median eccentricity increases with rising $\|z\|$ until low number statistics dominate at $\|z\| \sim 2~$kpc. This suggests that the recoil velocity after the SN is greater with greater induced eccentricity. Because BH-BH systems receive no SN velocity kick, the resultant eccentricity and centre-of-mass recoil velocity relies upon the initial orbital stellar velocity components and the instantaneous mass loss from the system. According to the SN fall back prescription used within Kiel et al. (2008; from Belczynski, Kalogera & Bulik 2002) the greater the SN progenitor the less mass is lost from the system during SN and for many massive BHs complete fall back occurs. Therefore, in some instances there is little to no recoil velocity induced onto the system. Similar to BH-BH systems, although shifted to higher eccentricities, is the $\|z\|$-eccentricity trend of NS-BH systems. There are more eccentric binaries in this population than in the BH-BH population. Owing to the kick velocity imparted during the NS formation, many systems are still eccentric when the BH is formed. The eccentricity ratio trend of the NS-BH systems, in $\|z\|$, is similar to that of the BH-BH population because the final SN produces a BH – the higher recoil velocity is directly proportional to the induced eccentricity. This is why there is no such trend in the BH-NS population: the eccentricity-recoil velocity link is weakened by the SN kick (the eccentricity depends mostly upon the kick strength, the recoil velocity upon the kick strength, direction and orbital parameters). Because distributions such as in Figure \[f:fig16NS-NS\] rely on SN kicks and binary evolution further observations of pulsar populations measuring parameters such as eccentricity, orbital period and height from the Galactic plane may help to constrain these evolutionary features. Of the 8 observed Galactic disk NS-NS systems all reside within $\|z\| < 1~$kpc and have a median eccentricity of $0.2$. This is significantly less than the medium eccentricity produced in our model. Of the observed systems four reside within $0.05~$kpc of the plane and while we do find that the median eccentricity of the model NS-NSs decreases with $\|z\|$ (for $\|z\| < 1~$kpc) we do not get below a value of $0.5$. As shown in our analysis of Figure \[f:fig222NS-NS\] this discrepancy is not resolved when accounting for core collapse electron capture. Even limiting the region of the Galaxy considered to $\|z\| < 1~{\rm kpc}$ and $4.5 < R < 2.5~{\rm kpc}$ does not rectify the situation, only increasing the percentage of systems with $e < 0.5$ to $\sim 25\%$. Of course observational selection effects occur and unless these are also accounted for any direct observational comparisons are incomplete.[^5] Orbital period and Galactic kinematics -------------------------------------- ![image](figure5DNS.eps){width="168mm"} It is informative to examine the distribution of NS-NS systems in Galactic coordinates, in particular, to investigate the optimal regions within the Galaxy to survey when searching for such systems. Figure \[f:fig210NS-NS\] depicts this for those NS-NSs with orbital periods less than $1~$day in an Aitoff projection (as designed by Hammer 1892: see Steers 1970). We provide two populations: those with eccentricities less than $0.5~$ (grey circles) and those with eccentricities greater than $0.5~$ (dark circles). NS-NS systems, according to our model, will most likely be observed towards the Galactic centre. In Galactic Cartesian coordinates the distribution in $R$ peaks at $R \sim 5~$kpc and the systems preferentially reside close to the Galactic plane. Unfortunately for observational predictions the populations with low and high eccentricities trace each other quite well. We provide the 9 detected NS-NS systems as a guide. Coalescing double compact binaries {#s:coal} ================================== We now examine the population of double compact systems that coalesce within the simulations (i.e. within the assumed age of the Galaxy). The scale heights of these merger events are given in Table \[t:table3\]. The most interesting aspect of Table \[t:table3\] is the scale height of coalescing NS-NSs compared to coalescing BH-NSs. It is surprising that the systems which typically receive greater combined recoil velocities (from both SN events) have a lower scale height. This scale height difference does not arise from low number statistics but rather from the shorter merger time scales of NS-NS systems compared to BH-NSs. Once a close NS-NS is formed, if it is to coalesce within a Hubble time, it will generally coalesce after a few Myr (see also Chaurasia & Bailes 2005). Whereas after BH-NS formation coalescence typically occurs after a few Gyr. Of course, this story changes somewhat when we include EC SN into the models. Now the scale height for merging double NSs increases owing to an increase in the typical merger time scale. Including kicks to BHs obviously helps to increase the scale height and decrease the merger time scale. Assuming $\alpha_{\rm CE} = 1$ causes NS-NS systems to be closer after the final supernova explosion, which allows these systems to merge faster and decreases their merger scale height. ----------------------- -------- -------------- ---------- -- -- -- -- -- Model System scale height Relative type \[kpc\] number C$^{'''}$ NS-NS $0.51$ $0.239$ NS-BH $0.11$ $0.029$ BH-NS $0.69$ $0.041$ BH-BH $0.05$ $0.691$ $\alpha_{\rm CE} = 1$ NS-NS $0.40$ $0.051$ NS-BH $0.06$ $0.001$ BH-NS $0.05$ $0.055$ BH-BH $0.05$ $0.893$ BH kicks NS-NS $0.53$ $0.536$ NS-BH $1.06$ $0.049$ BH-NS $0.69$ $0.029$ BH-BH $0.66$ $0.386$ BH kicks & NS-NS $0.82$ $0.827$ EC SNe NS-BH $1.07$ $0.018$ BH-NS $0.67$ $0.011$ BH-BH $0.65$ $0.144$ ----------------------- -------- -------------- ---------- -- -- -- -- -- : Model scale heights (kpc) and relative numbers for all DCBs that have coalesced for the four models. \[t:table3\] The coalescence times for the four system types in Model C$^{'''}$ are given in Figure \[f:fig182NS-NS\] with each distribution normalised to unity. The coalescence times shown in Figures \[f:fig182NS-NS\] to \[f:BH-BHmergertime\] are for those systems that merged during the simulation and represent the time each system took to coalesce (the length of time the DCBs lived). We note that there are in fact many more BH-BH and NS-NS systems than NS-BH and BH-NSs (an order of magnitude) and also we do not include the projected coalescence times for these systems that do not merge within our model. A model selection effect is then introduced at large times, where a turn over in the curves occur. Much can be gleaned from the incidence of the merger timescales peaks between each system type. The peaks of NS-NS and NS-BH systems correlate well, as do the peaks for BH-BH and BH-NSs. From this alone we see the importance of the first SN on the final system merger timescale. The systems in which a NS forms from the first SN – imparting an asymmetric kick into the system – typically merge faster than those in which a BH is formed from the first SN. Also those systems that pass through multiple CEs merge fastest. In DCB formation mass transfer prior to any SN is an important factor in determining the number of CE events beyond the first SN event. If unstable mass transfer occurs prior to any SN and a CE occurs it is unlikely that two CE events will occur following the first SN event. However, if stable mass transfer occurs prior to the first SN it is possible that if the orbit is wide enough after the first SN then the companion star will initiate unstable mass transfer, while, say, core helium burning takes place, leaving a tight enough orbit for the resultant helium star to evolve and overflow its Roche-lobe and cause unstable mass transfer. Tightening the orbit once more. This is also more likely to occur – in a relative sense – for NS-NSs and NS-BHs than for BH-BHs and BH-NSs, because the NS SN kick can cause a tight binary system to expand more readily than for a BH SN event. These systems (NS-NSs and NS-BHs) also have a high eccentricity which allows unstable mass transfer to occur even though the separation of the two stars would otherwise be greater. The BH-BH and BH-NS systems that are wide (tens of thousands of solar radii) after the first SN must be wide to begin with. Unfortunately for these systems the relative stellar velocities are small which means that little to no eccentricities are imparted to the orbit. If after the first SN the secondary initiates mass transfer while core helium burning is taking place then, with such little eccentricities and large separations, these systems are likely to have stable mass transfer and expand their orbits. Even if a CE phase does occur, after spiral in, the systems are still hundreds of solar radii apart. The NS-NS and NS-BH systems that merge quickly all survive the first CE phase with a separation of $1-2~R_\odot$. The helium giant secondary star may then evolve to initiate another CE phase, in which the second NS is formed. Again the system survives with a separation of $1-2~R_\odot$. Such short orbital periods allows rapid (few Myr) coalescence. BH-BH and BH-NS systems that merge within $10~$Gyr generally survive the CE phase with separations greater than $5~R_\odot$. The secondary star evolves and forms either a BH or NS but the separation at formation is typically of order $10~R_\odot$, for which a system will coalesce on a Gyr timescale. Therefore, these systems are susceptible to changes in the assumptions of CE evolution. However, usually such changes only result in shifts in the progenitor masses and orbital periods that produce such systems – the evolution pathway forming each system type remains viable (Kiel & Hurley 2006) but the relative numbers between system types change. Population birth/death rates (an important tool in model comparisons to observations) on the other hand are sensitive to changes in progenitor properties (Belczynski, Kalogera & Bulik 2002; O’Shaughnessy et al. 2005a). In particular rates are sensitive to changes in the initial stellar mass ranges and distributions and orbital period distributions (Kiel & Hurley 2006). The asymmetric kicks in NS-NS and NS-BH systems also help in forming both very close and disrupted systems. This is indirectly shown by the greater width of the NS-NS merger time peak in Figure \[f:fig182NS-NS\] to that of the NS-BH peak. The final NS-NS system separation depends upon the second SN kick, which is a random distribution, and causes a large array of separation values, which in turn forces variation into the coalescence times – smearing the merger time peak. The BH-BH merger time scale peak is not quite aligned to the BH-NS peak. Basically, this arises owing to the nature of gravitationally induced coalescence: heavier systems of the same separation after the second SN will naturally merger faster – even if the orbits are more circular. Of course, there is some cross over with system types and the mass transfer phases that occur which is why the four merger time curves have multiple peaks. However, the above analysis examines the main formation pathway for each system type. We note here that this picture changes somewhat when the random birth age of each system is accounted for. In this case the number of merging systems steadily increases over time, reaching a maximum at the end of our simulation. Take NS-NS systems for example where the typical system age at the time of coalescence in Model C$^{'''}$ is $20-100~$Myr. Therefore, with random birth ages included the coalescence times for the model Galaxy begin at $\sim 10^7~$year. From that point on the continuous birth of systems between $0-10~$Gyr means that the number of coalescing systems increases until the end of star formation. Of the NS-NS systems observed in the Galaxy approximately half are expected to merge within $10~$Gyr – based on calculating the merger time scale owing to gravitational radiation from their orbital parameters. The double pulsar J0737-3039 has the shortest measured merger time scale of $85~$Myr (Burgay et al. 2003). Therefore, the merger times of the observed NS-NSs all fall within the tail of our model distribution. This means that the model predicts that there is a large population of unobserved NS-NSs with merger time scales shorter than those already observed. Future detection of such a system has the potential to significantly increase the empirical Galactic NS-NS merger rate (Kalogera et al. 2007). When accounting for other evolutionary scenarios and assumptions the merger times of systems can change. Figures \[f:NS-NSmergertime\] and \[f:BH-BHmergertime\] depict the changes in merger times of NS-NS and BH-BH systems respectively, for a variety of models. When including EC SN evolution into our model the typical time NS-NS systems take to merge increases, as denoted by comparison of the peaks between the black line (Model C$^{'''}$) and the grey line (model including EC SN) within Figure \[f:NS-NSmergertime\]. Such an outcome was expected because the small merger timescales of Model C$^{'''}$ are driven by strong well directed kicks that force the orbital separation to contract. Decreasing the common envelope efficiency from $3$ to unity decreases the number of DCBs produced, which explains the erratic nature of the dashed curve within Figure \[f:NS-NSmergertime\]. The $\alpha_{\rm CE} = 1$ curve peaks at a slightly lower coalescence time than the other two models depicted. The less efficient the CE phase the greater the number of systems that merge or survive with smaller orbital separations. This last point means that systems which go on to form DCBs typically merge faster. There are, however, systems that normally would not have survived the second SN which now do (owing to the stronger gravitational well the exploding star resides in). There are a greater relative number of these large separation NS-NS systems in lower CE efficiency models, which can also be seen at long coalescence times in Figure \[f:NS-NSmergertime\]. Note that the curves in Figures \[f:fig182NS-NS\], \[f:NS-NSmergertime\] and \[f:BH-BHmergertime\] end prior to the age of the Galaxy because the number of systems drop to extremely low values. There are DCB populations with longer lifetimes than these end points suggest, however, the lifetimes are much to long to allow coalescence during our simulations (the DCB population characteristics are similar to the MSP-BH population described in detail within Section 4.4.2 of KH09). Figure \[f:BH-BHmergertime\] depicts BH-BH coalescence times for Model C$^{'''}$ (black line), BHs receiving kicks (grey line) and the $\alpha_{\rm CE} = 1$ (dashed line) models. When SN kicks are introduced into BH evolution BH-BH systems merge slightly faster than otherwise. When $\alpha_{\rm CE} = 1$ the coalescence time peaks at a similar value to that of Model C$^{'''}$ but there is a greater relative number of BH-BH systems that merge very quickly ($< 0.1~{\rm Myr}$). ![ The coalescence times after DCB formation for Model C$^{'''}$ measured as the time elapsed between DCB formation and merger of the two stars. Full black line gives NS-NSs, dashed line is BH-BHs, full grey line is NS-BHs while the dotted line is BH-NSs. The distribution for each type of system is normalised to unity. This figure only counts those DCBs that coalesced within our simulation, which introduces a selection effect at large coalescence times. \[f:fig182NS-NS\]](figure6DNS.eps){width="84mm"} ![ The coalescence times after NS-NS formation for variants on Model C$^{'''}$. The full black line depicts Model C$^{'''}$ (as in Fig. \[f:fig182NS-NS\]). The model including electron capture SNe is given by the grey line and the dashed line depicts the model with $\alpha_{\rm CE} = 1$. Each distribution is normalised to unity. \[f:NS-NSmergertime\]](figure7DNS.eps){width="84mm"} ![ The coalescence times after BH-BH formation for variants on Model C$^{'''}$. The full black line depicts Model C$^{'''}$ (corresponding to the dashed line in Fig. \[f:fig182NS-NS\]). The model including SN velocity kicks for BHs is denoted by the grey line and the dashed line depicts the model with $\alpha_{\rm CE} = 1$. Each distribution is normalised to unity. \[f:BH-BHmergertime\]](figure8DNS.eps){width="84mm"} Formation and merger rates {#s:fratesNS-NS} -------------------------- ------- ------- ------------------ ----------------------- ------------------ ------------------ Model C$^{'''}$ $\alpha_{\rm CE} = 1$ BH kicks BH kicks $+$ EC SNe Type ${\rm Myr}^{-1}$ ${\rm Myr}^{-1}$ ${\rm Myr}^{-1}$ ${\rm Myr}^{-1}$ NS-NS $38$ $4$ $37$ $162$ NS-BH $10$ $2$ $4$ $5$ BH-NS $10$ $7$ $3$ $3$ BH-BH $820$ $750$ $42$ $45$ ------- ------- ------------------ ----------------------- ------------------ ------------------ : Formation rates for double compact binaries in a range of models over a span of $10\,$Gyr. Models considered are Model C$^{'''}$ from KH09 and three variations on this model which are, in turn, a reduction of $\alpha_{\rm CE}$, the inclusion of BH velocity kicks, and the inclusion of BH kicks as well as EC SNe. \[t:tableFormRates\] ------- ------- ------------------ ----------------------- ------------------ ------------------ Model C$^{'''}$ $\alpha_{\rm CE} = 1$ BH kicks BH kicks $+$ EC SNe Type ${\rm Myr}^{-1}$ ${\rm Myr}^{-1}$ ${\rm Myr}^{-1}$ ${\rm Myr}^{-1}$ NS-NS $36$ $3$ $35$ $154$ NS-BH $4$ $0.04$ $3$ $3$ BH-NS $6$ $3$ $2$ $2$ BH-BH $107$ $56$ $25$ $27$ ------- ------- ------------------ ----------------------- ------------------ ------------------ : As for Table \[t:tableFormRates\] but now showing merger rates for DCBs. \[t:tableMergeRates\] In Tables \[t:tableFormRates\] and \[t:tableMergeRates\] we show the formation and merger rates respectively for the four DCB system types in our models. To calculate the rates we first count the fraction of stars in the model that led to a Type II supernova, combine this with the assumption that the fraction of binaries in the Galaxy is 0.5, and normalize this to the empirical Galactic type II SNe rate ($\sim 0.01 \, {\rm yr}^{-1}$: Cappellaro, Evans & Turatto 1999). The relative numbers of the DCB systems and mergers are then calibrated against this to produce the rates. This method of calculation is commonly employed in binary population synthesis (Belczynski, Kalogera & Bulik 2002; Belczynski, Bulik & Rudak 2002; Voss & Tauris 2003; Pfahl et al. 2005, for example) so facilitates easy comparison with previous work. An alternative method uses the Galactic star formation rate (e.g. Kiel & Hurley 2006). With either method the uncertainties involved in the calibration make it more appropriate to discuss relative rather than absolute rates. This is also true of the uncertainties in rates owing to variations in the parameters of binary evolution models. These uncertainties have been well documented in the past. For example, Belczynski, Bulik & Rudak (2002) find NS-NS merger rates in the range $3$–$300\,$Myr$^{-1}$ as a result of a comprehensive exploration of the available parameter space. While O’Shaughnessy et al. (2005b) favor models with NS-NS merger rates within the range $2.5$–$25\,$Myr$^{-1}$, although the total model range of merger rates extends far beyond these limits. Conversely, we find that rates quoted for a particular model (with a set choice of parameter values) vary by only a few percent at most with repeated realisations using distinct random number distributions (as was also found by Belczynski, Bulik & Rudak 2002). We note that in calculating our rates we have assumed an age of $10\,$Gyr for the Galaxy. If we instead take an age of $15\,$Gyr the quoted rates increase by less than 10%. We quote rates for our Model C$^{'''}$ – with parameter values and distribution functions as listed in Table \[t:binpopbinkin\] – and a set of comparison models in which we alter the common-envelope efficiency parameter, the treatment of BH velocity kicks, and the possibility of NS formation via electron-capture SNe. A number of trends are immediately evident from Tables \[t:tableFormRates\] and \[t:tableMergeRates\]: (i) decreasing $\alpha_{\rm CE}$ can reduce the rates by as much as an order of magnitude; (ii) without kicks for BHs the BH-BH rates are very high; and, (iii) the inclusion of EC SNe leads to a sharp increase in the NS-NS rates. Looking first at the NS-NS systems the merger rates of all models are in agreement with the empirical estimates of $3$–$190\,$Myr$^{-1}$ made recently by Kim, Kalogera & Lorimer (2006). They are also in agreement with the ranges found in the binary population synthesis works of Belczynski, Bulik & Rudak (2002) and O’Shaughnessy et al. (2005b), as is the noted behaviour of rates with variations in $\alpha_{\rm CE}$. The NS-NS merger rate found by Belczynski, Bulik & Rudak (2002) in their standard model was $53\,$Myr$^{-1}$ which is higher than in our Model C$^{'''}$ which has a rate of $36\,$Myr$^{-1}$. Belczynski, Bulik & Rudak (2002) used $\alpha^{'}_{\rm CE} \lambda = 1$ in their standard model which, assuming $\lambda = 0.5$, means $\alpha^{'}_{\rm CE} = 2$ and consequently a more efficient CE process than in Model C$^{'''}$ (which effectively uses $\alpha^{'}_{\rm CE} \lambda \simeq 1$: see Section 2). Thus the difference is consistent with the expected trend in CE efficiency but there are numerous more subtle differences between the models which could also play a role. A noticeable outcome of our models is the high formation and merger rates for BH-BH binaries predicted by Model C$^{'''}$, which are above previous predictions in the literature (e.g. Lipunov et al 1997; Portegies Zwart & Yungleson 1998; Voss & Tauris 2003; Belczynski et al. 2007a), in some cases by more than an order of magnitude. Following the methodology of Belczynski et al. (2007a) the BH-BH merger rate of Model C$^{'''}$, given in Table \[t:tableMergeRates\], sets a detection rate for the current LIGO gravitational wave detector at approximately $1~{\rm yr}^{-1}$. LIGO has been running in detection mode for longer than this without a detection and therefore the Model C$^{'''}$ rate is inconsistent and thus an over-estimate. Reducing the efficiency of the CE spiral-in process halves the predicted BH-BH merger rate which is a step in the right direction. A further reduction result from the introduction of BH velocity kicks. In our model where BHs are given velocity kicks from the same distribution as for NSs (Maxwellian distribution with dispersion of $190~$km s$^{-1}$) the BH-BH formation and merger rates decrease to $42~{\rm Myr}^{-1}~{\rm and}~25~$Myr$^{-1}$, respectively. We now find that the merger rate of NS-NS systems is larger than for BH-BH systems in agreement with the majority of previous works (other than Voss & Tauris 2003). This is further accentuated with the inclusion of EC SNe which increases the NS-NS merger rate to $154$ Myr$^{-1}$ while leaving the BH-BH rate untouched. The models with BH velocity kicks lead to more plausible merger rate predictions as they are safely below the limit requiring a LIGO detection. They also agree well with the rates from the models of Belczynski, Kalogera & Bulik (2002) with similar input parameters and assumptions (see Belczynski et al. 2007a for a discussion of these rates). As mentioned in Section 2, Belczynski et al. (2007a) describe a radically different evolutionary pathway to that presented in our models for Hertzsprung Gap stars which initiate CE. That is, these systems never survive CE. Belczynski et al. (2007a) show that adopting this pathway reduces the merger rates of BH-BH systems by a factor of 500 compared to a reduction of only a factor of 5 for NS-NS mergers. This swing would account for the stark difference in the BH-BH/NS-NS merger ratios found in our models compared to Belczynski et al. (2007a) and also makes a BH-BH gravitational wave detection less likely. The detection of gravitational waves with Advanced LIGO (Lazzarini 2007) will go a long way towards constraining the NS-NS and BH-BH merger rates and importantly reduce the uncertainty in the BH-BH merger rates predicted by models. This has been highlighted previously by Belczynski et al. (2007a). The detection of a pulsar orbiting a black hole would also help to constrain many features of compact binary and pulsar evolution. This has been explored in Pfahl et al. (2005) and more recently in KH09 in terms of millisecond pulsar-black hole systems. Fortunately the models, owing to a greater number of possible observational comparisons, are more robust concerning predictions for coalescing NS-NS systems (as described above) and from this point on NS-NS systems are the focus of this work. Long and short gamma-ray burst Galactic kinematics {#s:grb} ================================================== ![ The projected distance from host galaxy for our different double compact systems that coalesce (SGRB progenitors). The projected distance is $\pi/4$ times the Model C$^{'''}$ Galactic radial distance (as in Voss & Tauris 2003). Full grey line gives NS-NSs, dashed line is BH-BHs and the dotted line is the two NS and BH combinations. We note that to model SGRBs via double compact coalescence requires that the progenitor system must contain at least one NS, for the formation of an accretion disk (see Voss & Tauris 2003 and references therein). The LGRB observations of Bloom, Kulkarni & Djorgovski (2002) are depicted by the jagged histogram. \[f:fig19NS-NS\]](figure9DNS.eps){width="84mm"} ![ The projected distance from host galaxy for our different GRB progenitors. The projected distance is $\pi/4$ times the Model C$^{'''}$ GRB radial distances (as in Voss & Tauris 2003). Full black line gives NS-NS-SGRBs, full grey line is collapsar-LGRBs assuming the birth distribution of Yuslifov & Kucuk (2004) and dashed-dotted grey line is the collapsar-LGRB distribution assuming a flat Galactic radial distribution between $0.01 < R < 20~$kpc. The observations of Bloom, Kulkarni & Djorgovski (2002) are depicted by the jagged histogram. \[f:fig21NS-NS\]](figure10DNS.eps){width="84mm"} We now examine the Galactic spatial distributions for our model GRBs, which require the following assumptions. We assume the coalesced NS-NSs which result in a BH are short gamma-ray bursts while our collapsar model (see Section \[s:bpbk\]) is assumed to form the long duration gamma-ray bursts. Figure \[f:fig19NS-NS\] shows projected distances from the host galaxy center for our Model C$^{'''}$ coalescing DCB populations. The curves for all coalescing DCB system types are provided, noting that BH-BH mergers can not form gamma-ray bursts. We see that the different SGRB progenitor populations all follow very similar curves. The only slight difference is the BH-BH systems which receive reduced recoil velocities on average (see Figures \[f:fig12NS-NS\] and \[f:fig15NS-NS\]) causing them to coalesce slightly closer to their original radial birth location compared to the other system types. Importantly, even the NS-NS cumulative curve follows the stellar radial birth cumulative curve (not shown) closely. At first glance such a result is surprising: previously it has been argued that because NSs receive large recoil velocities (Hobbs et al. 2005) SGRBs should not typically appear in star forming regions (Bloom, Kulkarni & Djorgovski 2002; Woosley & Bloom 2006). This is in contrast to LGRBs whose progenitors are thought to be massive stars – young stellar systems – which should correlate well with star forming regions (Bloom, Kulkarni & Djorgovski 2002). Yet owing to a population of close double compact systems, and thus a large population of systems that coalesce rather promptly (see Figure \[f:fig182NS-NS\] which is in line with those produced in Belczynski, Bulik & Rudak 2002 and Voss & Tauris 2003), many of the model SGRBs merge at similar radial positions to their birth positions. As noted in Section \[s:coal\] this feature of coalescing NS-NSs is also responsible for the smaller scale height of coalescing NS-NSs compared to that of BH-NS coalescence (see Table \[t:table3\]). As shown in KH09 the form of the host galaxy potential plays an important role in determining the resultant stellar kinematics. The Galactic potential of our model, taken from Paczyński (1990), is more dominant than that used in Voss & Tauris (2003) but we do not see a significant difference in the SGRB distributions. We find that our model projected distances compare best with model f of Bloom, Sigurdsson & Pols (1998) which assumes a similar circular velocity ($225\,$km s$^{-1}$ compared to our $220\,$km s$^{-1}$) and an identical SN velocity kick dispersion of $190~$km s$^{-1}$. In Figure \[f:fig19NS-NS\] we compare our SGRB models to observations of projected LGRB distances from their host galaxy centers according to Bloom, Kulkarni & Djorgovski (2002). We see that all Model C$^{'''}$ coalescing DCB populations fail to reproduce the LGRB observations. This is not exactly a surprise and agrees with previous studies (e.g. Bloom, Sigurdsson & Pols 1999; Belczynski, Bulik & Rudak 2002; Voss & Tauris 2003). What we emphasise here is that the difference does not arise from SNe recoil velocities but instead could be telling us something about the assumed initial birth distribution of the stars and binaries. In our model the birth distribution is based on observations, in particular the distributions of OB stars and HII in the Galaxy as determined by Yusifov & Kucuk (2004). Of course, at this stage, we are reliant upon making comparisons of two different populations, observed long gamma-ray bursts and model short gamma-ray bursts, and we note that statistics of observed SGRBs do not yet allow for meaningful comparisons (Savaglio, Glazebrook & Le Borgne 2009). Another factor is that our models so far have been for the Milky Way whereas the LGRB observations are predominantly from dwarf galaxies (see Figure \[f:fig22NS-NS\] and associated text). To understand the GRB distributions further we next provide results of LGRB models (rather than our previous SGRB models) to compare with the observations of long-GRBs (Bloom, Kulkarni & Djorgovski 2002). The method of determining LGRB progenitors is outlined in Section 2 and the important point to note is that the average age of each system (at the time of the LGRB) is young, less than $\sim 10\,$Myr, with little dependence of the formation mechanism on uncertain features of binary evolution. So it is safe to assume that the LGRBs will trace the birth positions of their progenitors very closely. This is distinct from the SGRB models where the lifetimes of coalescing systems depend upon a number of evolutionary features and assumptions (common-envelope evolution and SNe velocity kicks, for example) as discussed in detail in Sections \[s:DCBNS-NS\] and \[s:coal\] and Belczynski, Bulik & Rudak (2002). ![ The projected distance from host galaxy for SGRBs of different models. The projected distance is $\pi/4$ times the Model C$^{'''}$ GRB radial distances (as in Voss & Tauris 2003). The grey line is the SGRB distribution assuming $\alpha_{\rm CE} = 1$ and the full black line is Model C$^{'''}$ SGRBs. The observations of Bloom, Kulkarni & Djorgovski (2002) are depicted by the jagged histogram. \[f:alphaComparison\]](figure11DNS.eps){width="84mm"} The model LGRB projected distance distribution is compared in Figure \[f:fig21NS-NS\] to the LGRB observations and our previous NS-NS-SGRB distribution. This clearly shows that both our model GRBs – long and short – cannot reproduce the observations of GRB distances from their host galaxies. This is the case even when we allow an unrealistic radial birth description that is flat in Galactic radius (dashed-dotted line of Figure \[f:fig21NS-NS\]). Assuming a flat radial birth distribution does increase the number of LGRBs that occur in the inner Galactic regions compared to our standard LGRB and SGRB models (though not enough compared to observations), however, it also produces too many GRBs beyond $\sim 2-4~$kpc compared to the other models. Hence, the collapsar model requires birth placement which is incompatible with Population I stars in Milky Way like galaxies. ![ The projected distance from host dwarf galaxy for our different GRB progenitor systems. The projected distance is $\pi/4$ times the model radial distances (as in Voss & Tauris 2003). The dashed-dotted grey line is the collapsar-LGRB distribution assuming a scaled down model of our Galaxy ($\alpha_{\rm g} = 0.01$). Full black line and the full grey line gives the NS-NS-SGRBs and collapsar-LGRBs within a scaled spherical potential ($\alpha_{\rm g} = 0.1419$). The birth distribution for both dwarf galaxy models is that of Yuslifov & Kucuk (2004) scaled by $\alpha_{\rm g} = 0.01$. The dashed line shows the NS-NS-SGRB distribution from a population of low metallicity stars ($Z = 0.0001$). The observations of Bloom, Kulkarni & Djorgovski (2002) are depicted by the jagged histogram. \[f:fig22NS-NS\]](figure12DNS.eps){width="84mm"} The majority of NS-NS systems merge rapidly when we assume $\alpha_{\rm CE} = 1$, as shown in Figure \[f:NS-NSmergertime\]. We compare this model SGRB projected distance distribution with LGRB observations and Model C$^{'''}$ in Figure \[f:alphaComparison\]. The EC SN model is not shown as it does not differ greatly from Model C$^{'''}$ and has typically longer merger times – which will not help here. The only noticeable difference between Model C$^{'''}$ and our $\alpha_{\rm CE} = 1$ model is the distributions beyond about $10~{\rm kpc}$. Here the greater relative number of systems with large merger times cause the distribution to slope upward to unity slightly slower. Unfortunately it is the other end of the distribution, at small distances, where the models depart from observations. This leads us to examine models of what is believed to be a more typical GRB host galaxy – a dwarf galaxy. Dwarf galaxies contain less mass than the Galaxy, with masses of order $10^9~M_\odot$, and are typically modelled by simple potentials such as in Bloom, Sigurdsson & Pols (1999). In fact, Belczynski, Bulik & Rudak found that the best match to GRB observations arose from a galaxy mass of $0.01\times M_{\rm MW}$ – the typical mass of a LGRB host galaxy. Bloom, Sigurdsson & Pols (1999) and Voss & Tauris (2003) also accounted for different galaxy masses, however, their initial stellar birth positions were not based on observations of OB stars but instead followed the exponential disk potentials – something which can be estimated from stellar kinematics (e.g. Kuijken & Gilmore 1989) and light profiles (e.g. de Vaucouleurs 1948). However, such observations measure stellar systems that differ from GRB progenitors. To examine the effect such a small mass galaxy and gravitational potential has on our LGRB and SGRB populations we have produced two new galaxy models. The first model is a scaled down version of our Galaxy where, in line with Bloom, Sigurdsson & Pols (1999), Belczynski, Bulik & Rudak (2002) and Voss & Tauris (2003), we scale our previous model Galaxy mass, as described in KH09, by the scale parameter $\alpha_{\rm g}$, giving us $M_{\rm galaxy} = \alpha_{\rm g} M_{\rm MW}$. Assuming a constant density distribution between models, $$\alpha_{\rm g} M_{\rm MW} \propto R_{MW}^3,$$ we then rescale the model scale lengths, scale heights and initial birth positions, also described in KH09, by $\alpha_{\rm G}^{1/3}$. This first dwarf galaxy model assumes $\alpha_{\rm g} = 0.01$ (giving us a similar galaxy mass and size of the favoured galaxy model in the work of Belczynski, Bulik & Rudak 2002), from which the mass of the dwarf galaxy is $M_{\rm galaxy} = 1.419\times 10^9~M_\odot$. The second model takes a spherical Hernquist (1990) potential as used in KH09 but scaled so the galaxy mass is the same as in our first dwarf galaxy model, thus $\alpha_{\rm g} = 0.1419$. However, we keep $\alpha_{\rm g} = 0.01$ when scaling the birth distribution in galactic $z$ and $R$. We then evolve our SGRB and LGRB progenitor systems in each of the dwarf galaxy models. The resultant projected distance curves are compared with the observations of Bloom, Kulkarni & Djorgovski (2002) in Figure \[f:fig22NS-NS\] (note we do not plot the SGRB model for the first scenario). All populations now more closely fit the observations than those in Figure \[f:fig21NS-NS\], however, our LGRB models seem to over-estimate the number of GRB systems displaced between $1-10~$kpc from the host galaxy central region and under-estimate the number of GRB systems displaced less than $1~$kpc from the host galaxy. It is interesting that the best fit to the observations arises from our SGRB model, although this also under-estimates the number of GRB systems that occur in the inner galactic region. The observed GRB host galaxies span a great range in red shift ($0 < z < 6.3$: Savaglio, Glazebrook & Le Borgne 2009) and as such many GRB host galaxies are low metallicity galaxies (Fruchter et al. 2006) – the average host galaxy metallicity being roughly one quarter solar (Savaglio, Glazebrook & Le Borgne 2009). The initial stellar metallicity has an important effect on governing the progenitor mass range for the formation of compact stars, in particular we note that NSs may emerge from lower mass stars in low metallicity environments than could possibly occur for higher metallicity populations. Therefore, compared to solar metallicity galaxies the resulting stellar and binary evolution in low metallicity galaxies could possibly extend the lifetimes of pre- and post-NS-NS formation, allowing such systems more time to evolve kinematically. This motivated us to examine what effect a low metallicity environment would have on the resultant SGRB population. To complete this we produced a SGRB model in our spherical Hernquist potential and assumed a metallicity of $Z = 0.0001$, which is depicted within Figure \[f:fig22NS-NS\]. Even with such a large difference in metallicity of the two SGRB models shown there does not appear to be a significant difference between their projected distance distributions. The number of model systems found close to the host galaxy (projected distance $< 1~$kpc) is still underestimated in this model although the slope of the distribution does become less steep. Although this suggests that metallicity does not play a major role in kinematics of SGRBs (and it does not within the assumptions used here), if mass loss rates are found to vary strongly with Z (Vink, de Koter & Lamens, 2001; Belczynski et al. 2010) and the kick strength prescription is allowed to vary with mass of the exploding star (e.g. from fallback as assumed in Belczynski et al. 2008) then there could indeed be a greater level of dependence on metallicity for SGRB kinematics. Our models appear to confirm current observations that LGRBs are born in star forming regions (Kelly, Kirshner & Pahre, 2008; Dado & Dar 2009). We find that the final model GRB projected distance distributions are close to their respective birth distributions, while, when compared to observations, we under-estimate the number of GRBs that occur in the inner galactic region. Recently Fruchter et al. (2006) have suggested that there is a difference in LGRB and core collapse SN galactic environments. This could indicate that there is a difference in the birth distributions of the progenitors of these systems or that they arise from the same star formation distribution but from distinct progenitor mass ranges. Within our models we use the same birth distribution for both our LGRB and SGRB progenitors, namely the distribution according to Yusifov & Kucuk (2004) which should be realistic for SN progenitors and indeed was shown in KH09 to work well for pulsar models. However, as observations of the Galaxy suggest (Muno et al. 2006a; Muno et al. 2006b) there appears to be a relatively large population of massive stars within the inner $\sim 300~$pc. This may be the case for many galaxies and suggests that the Yusifov & Kucuk (2004) radial birth distribution is inappropriate for modelling the central galactic region. Thus it is quite possible that massive stellar systems may be born with a different galactic birth distribution than standard stars. In particular, the massive stellar birth distribution may be a combination of the the typical stellar distribution and a more centralised distribution. This scenario would suit both our models and observations that the material surrounding LGRBs is being highly irradiated by approximately $10-100$ times more intensive UV radiation than that of the solar neighbourhood (Tumlinson et al. 2007) and that LGRBs occur in relatively low stellar density regions (Hammer et al. 2006; Tumlinson et al. 2007). It is also possible that this difference in the radial birth distribution may arise from open clusters with more massive stellar components falling quickly into the central galactic regions and becoming tidally disassociated (as in Muno et al. 2006b) or perhaps these more massive stars are formed from a metal poor distribution, allowing the birth of more massive stars that can evolve in isolation and still cause a LGRB (Yoon, Langer & Norman 2006). The issue of whether or not LGRB and SGRB progenitors follow the same radial birth distributions has important consequences for the role that population synthesis studies can play in understanding GRB observations. If they do follow the same birth distribution then population synthesis models can not reliably distinguish between LGRB and SGRB progenitors via the resultant GRB-galaxy off-sets. This is because both GRB populations trace star forming regions (although given time SGRBs can also occur in old stellar systems). However, if the two GRB populations have different galactic birth distributions, as raised in the above discussions, then further modelling of GRB populations can figure prominently in shedding light on the continuing GRB progenitor problem. Summary ======= This work investigated the stellar, binary and Galactic kinematical evolutionary features of double compact binary systems and possible short gamma-ray burst (coalescing NS-NSs) and long gamma-ray burst (tidally influenced collapsars) objects. The main conclusions from this investigation are summarised here: - NS-NS systems are typically more kinematically active and have the greatest scale height of all DCBs, eclipsing the BH-BH population especially when BHs are assumed not to receive velocity kicks at birth. Including electron capture supernovae into models alters this somewhat. When assuming electron capture supernovae occur and that BHs receive kicks the BH-BH population has a comparable scale height to NS-NSs. - We find the double neutron-star formation rate ranges between $4~$Myr$^{-1}$ and $162~{\rm Myr}^{-1}$. The lower value is set by assuming $\alpha_{\rm CE} = 1$, the upper value by assuming EC SNe occurs. Our model without EC SNe and with $\alpha_{\rm CE} = 3$ gives a double neutron-star formation rate of $38~{\rm Myr}^{-1}$. - Our model double black-hole formation rate ranges between $42~{\rm Myr}^{-1}$ and $820~{\rm Myr}^{-1}$. The lower value arises when we assume BHs receive kicks, the upper value arises when we do not assume BHs receive kicks. - If NS-NSs merge it is likely they do so within a time scale of a few million years and with a merger rate that ranges between $3~$Myr$^{-1}$ and $154~{\rm Myr}^{-1}$ for our models. The double neutron-star merger rates closely reflect their birth rates. - BH-BH systems typically merge more slowly than NS-NS systems and this difference is enhanced when BHs do not receive kicks. The range of merger rates for double black holes is $25~{\rm Myr}^{-1}$ to $107~{\rm Myr}^{-1}$. The most limiting factor is the addition of SN kicks to BH formation, while assuming $\alpha_{\rm CE} = 1$ results in a merger rate of $56~{\rm Myr}^{-1}$. The maximum double BH merger rate arises when we assume they do not feel kicks at birth. Such a high merger rate suggest that at least one merger event should have been detected with LIGO. The null detection of gravitational waves by LIGO gives credence to the latest findings that BHs should receive kicks at birth. - Common envelope evolution is reconfirmed to be a very important process for DCB formation. In particular, merger time scales are sensitive to the number of CE phases that occur within the intervening time between SN events. Decreasing the efficiency of the CE process decreases the typical merger time scale and final number of NS-NS systems. The CE phase is also important in shaping the final NS-NS eccentricity-orbital period distribution. - We find good agreement of the shape of eccentricity-orbital period distribution when compared to observations but poor agreement with observations on the relative number of high eccentricy to low eccentricy systems. Including electron capture SN evolution into population synthesis models does not rectify the situation, as has been suggested previously. - We find poor agreement between the projected distance of long gamma-ray bursts from their host galaxy and the projected distances of NS-NS systems within a Milky Way model. - We find much better agreement between observations and models when the model galaxy mass and size is scaled down by a factor $\alpha_{\rm G} = 0.01$. Owing to the short life times of our model systems (both short gamma-ray burst and long gamma-ray burst) the final projected distances depend heavily on the assumed radial birth distribution. Acknowledgements {#acknowledgements .unnumbered} ================ PDK thanks Swinburne University of Technology for a PhD scholarship. The authors thank the referee for help with this paper. 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--- abstract: 'Training models on highly unbalanced data is admitted to be a challenging task for machine learning algorithms. Current studies on deep learning mainly focus on data sets with balanced class labels or unbalanced data, but with massive amount of samples available, like in speech recognition. However, the capacities of deep learning on imbalanced data with little samples is not deeply investigated in literature, while it is a very common application context in numerous industries. To contribute to fill this gap, this paper compares the performances of several popular machine learning algorithms previously applied with success to unbalanced data set with deep learning algorithms. We conduct those experiments on a highly unbalanced data set, used for credit scoring. We evaluate various configuration including neural network optimisation techniques and try to determine their capacities when they operate with imbalanced corpora.' author: - \| Louis Marceau\ National Bank Of Canada\ 600 de la Gauchetière\ Montréal, Québec\ `louis.marceau@bnc.ca`\ Lingling Qiu\ National Bank Of Canada\ 600 de la Gauchetière\ Montréal, Québec\ `lingling.qiu@bnc.ca`\ Nick Vandewiele\ National Bank Of Canada\ 600 de la Gauchetière\ Montréal, Québec\ `nick.vandewiele@bnc.ca`\ Eric Charton\ National Bank Of Canada\ 600 de la Gauchetière\ Montréal, Québec\ `eric.charton@bnc.ca`\ bibliography: - 'references.bib' title: A Comparison of Deep Learning Performances with Other Machine Learning Algorithms on Credit Scoring Unbalanced Data --- Introduction ============ Forecasting a score of credit is a key aspect of risk management in financial institutions. The aim of credit scoring is essentially to classify credit product (like credit card or loans) applicants into two classes: good payers (i.e., those who are likely to keep up with their repayments) and bad payers (i.e., those who are likely to default on their credit card loan). The particularity of this scoring task is that classes are highly unbalanced as bad behavior represent usually a small percentage of a customer base in a banking context. Training models on highly unbalanced data is admitted to be a challenging task for machine learning (ML) algorithms. A wide range of classification techniques have already been proposed in the credit scoring literature to tackle the specific aspect of unbalanced credit risk data, including statistical techniques, such as linear discriminant analysis, logistic regression, and non-parametric models, such nearest neighbour and decision trees. Deep learning (DL) ML techniques have become increasingly popular in both academic and industrial areas in the past years, as it is common to show that modern implementations of DL algorithms outperforms non-DL algorithms in numerous existing and well documented tasks. That being said, it is currently unclear from the literature if DL techniques performs well on unbalanced data set, and specifically on credit scoring data. Various domains including pattern recognition, computer vision, and natural language processing have witnessed the performances of deep neural networks. However, current studies on DL mainly focus on data sets with balanced class labels, or unbalanced data sets with massive amount of samples available. Its performance on more classical imbalanced data sets is not widely examined. In this paper, we conduct a compared experiment of traditional ML and DL algorithms on a highly unbalanced data set representing credit behavior. We use a data set coming from a 24 months credit card user data. We use various ML algorithms, including DL ones, to forecast the user behavior as good or bad in 12 months. This paper is organised as follows: in section \[sec:related\] we investigate related work in the specific domain of classification tasks applied to unbalanced data. In section \[sec:context\], we explain the context of this study, risk management in finance, and more specifically the domain of credit scoring. We explain how we built our experimental corpus, and what features we extracted from it. In section \[sec:experiments\] we conduct our experiments in two phases: first we establish what is the best ML classifier on each of our experimental corpus, then we apply DL algorithms. We used those two set of experiments to compare the performances of the two ML families of algorithms. After a discussion on the results, we finally conclude in section \[sec:conclusions\]. Related work {#sec:related} ============ In their review, [@haixiang2017learning] of methods and applications for learning on unbalanced data, authors cite as domains extensively explored the topic Taxonomy, Chemical engineering, financial management, information technology, energy management, security management. Most of the related experimentation tend to show that since years, decision trees are the best performing ML algorithms on unbalanced data sets. In [@cieslak01], authors outline the performance of several popular decision tree splitting criteria – information gain, Gini measure, and DKM – can be used to form decision trees, and improve performances of tree construction method applied to unbalanced data. Authors of [@almeida2014machine] discuss the use of 108 different classification models to determine the most fitting model, capable of dealing with the imbalanced data issue coming from a biomedical literature corpus and achieve the most satisfying results with a LMT decision tree classifier previously used with success in a unbalanced multi-label classification task related to taxonomy by [@charton2014using]. In the domain of credit applications, authors of [@brown2012experimental] compare several techniques that can be used in the analysis of imbalanced credit scoring data sets. The results from this empirical study indicate that the random forest and gradient boosting classifiers perform very well in a credit scoring context and are able to cope comparatively well with pronounced class imbalances in these data sets. Various domains including pattern recognition, computer vision, and natural language processing have witnessed the great power of deep neural networks. However, current studies on DL mainly focus on data sets with balanced class labels, while its performance on imbalanced data is not well examined. Context of this study {#sec:context} ===================== Risk management in finance and banking industry includes determining in diverse forms how risk can evolve for a given use case. Forecasting the evolution of various risk related indicators is a key activity, previously handled through rule based or regression models, and more recently with ML generated classification models. As well as using traditional classification techniques such as logistic regression, neural networks and decision trees, gradient boosting, support vector machines and random forests have been tested in the past on various aspect of client default prediction. Our objective in this experiment was to forecast, for a given credit card account, a probability of default after 12 months. We used for this a set of credit card account holder characteristics along with their historical credit behavior to train a ML system and see how this model could forecast good and bad labels. In a credit scoring context, the imbalance issue can interfere directly on the classifier performance, which is biased by the majority class. Because the majority class (the good labeled samples) is more heavily represented in the data set than the minority class (the bad labeled samples), it tends to have more influence under uncertainty cases, since the class distribution can influence learning criteria. In addition, according to [@weiss2001effect], a classifier presents a lower error rate when classifying an instance belonging to the majority class, since it will have learned more information from the examples of the majority class, compared to the information learned in fewer examples from the minority class. ![Corpus construction.[]{data-label="fig:fig1"}](prescorpus.png){width="100.00000%"} Data set -------- We used as a data set of two years of historical data randomly extracted from our organisation accounts data between 2015 and 2018. The extracted set is made of 1.4M accounts. This data set includes samples built from credit status, customer information and transactional data. Each sample is labelled with the value Good or Bad. We defined as a Bad label for this experiment, an account with 60 days past due, 12 months after the time the data are collected (see figure \[fig:fig1\]). On this data set, we kept 288,558 accounts representation (20%) used as test corpus. We used the remaining data to build the train corpora. Downsizing and re-balancing can have various impact on ML applied to unbalanced data. For instance, a strong re-balancing, as it reduce substantially the amount of samples available to train the ML algorithm, can have a negative effect on the precision of the model trained. As there is no clear theoretical proposition to estimate by calculation the most adapted re-balancing amplitude we built two train corpora. We built two down-sampling configurations for train (DS1 and DS2) as shown in table \[tab:tablecorpus\]. - DS1 : a corpus with a simple downsizing (the smaller class is 7.2% of the total). - DS2 : a corpus with a strong re-balancing (the smaller class is 35% of the total). The test corpus reflect the exact proportion as Bad as the original corpus as show in last column of table \[tab:tablecorpus\]. [llll]{}\ Corpus & Good Samples & Bad Samples & smaller class\ Total & 1442787 & 67896 & 4.38%\ Train DS1 & 700 000 & 54317 & 7.2%\ Train DS2 & 100 000 & 54317 & 35.2%\ Test & 288 558 & 13579 & 4.38 %\ \[tab:tablecorpus\] ### Features There is 66 features used by the model. Features includes account information (like credit card limit, current balance), credit status (like the delinquency cycle) and 6 categories of transactional data aggregated on the 3 months before the date of the account information collection. Experiments {#sec:experiments} =========== The main purpose of our experiment is to compare ML approaches commonly applied on unbalanced data (like decision trees or SVM) with DL model. We also want to evaluate the capacities of optimization algorithms used to maximize the performance of a DL model, like AutoML, in this experimental context. Our experimental plan will consist in 2 main steps : 1. First, training common ML algorithms used for unbalanced data on our data set, and select the best configuration. 2. Then optimising a DL algorithm on our data set from the best configuration selected. We describe those two sequences of experiment in the two following subsections. Machine learning algorithms selection ------------------------------------- Our first set of experiments consist in applying 4 algorithms, Logistic Regression, SVM, Random Forest, XGBoost to the 2 corpus configurations, DS1 (re-balanced) and DS2 (downsampled). ### Results Results of the first set of experiments are in table \[tab:tableDS2\] and table \[tab:tableDS1\]. On both experiments, with downsized corpus (DS1) and rebalanced corpus (DS2), XGBoost classifiers overperform logistic regression, SVM and Random Forest as measured by F-Score. However, the XGBoost model provides its best performances with a F-Score of 0.549 as shown in table \[tab:tableDS1\]. This set of experiments shows that in our experimental context, downsizing is the best solution to counteract the effects of unbalanced data on the training process. (r)[1-2]{} Model Precision Recall F-Score AUC Run time (s) --------------------- ----------- -------- ------------ -------- -------------- Logistic Regression 0.5104 0.5366 0.5232 0.7562 22.8 SVM 0.5364 0.5144 0.5251 0.7467 149.9 Random Forest 0.6324 0.4037 0.4928 0.6963 55.4 XGBoost 0.5138 0.5907 0.5496 0.7822 299.8 : Basic classifiers applied on 12 month forecast, 60 days past due, DS2 train corpus - Re-balanced corpus \[tab:tableDS1\] (r)[1-2]{} Model Precision Recall F-Score AUC Run time (s) --------------------- ----------- -------- --------- ------- -------------- Logistic Regression 0.298 0.744 0.426 0.830 22.8 SVM 0.314 0.730 0.439 0.827 12.0 Random Forest 0.308 0.77 0.44 0.844 55.4 XGBoost 0.304 0.789 0.439 0.852 64.3 : Basic classifiers applied on 12 month forecast, 60 days past due, DS2 train corpus - Re-balanced corpus \[tab:tableDS2\] Deep learning experiments ------------------------- We now apply DL algorithms and AutoML techniques on the selected corpus from the first sequence of ML experiments, the downsampled corpus DS1, and we will compare the performances of both DL and AutoML with the best classifier on DS1, XGBoost. ### AutoML and Deep Learning When training a neural network on a data set a DL practitioner is trying to optimize and balance a neural network architecture that lends itself to the nature of the data set. After the proper architecture defined, a second step is intended to tune a set of hyper-parameters over many experiments. Typical hyper-parameters that need to be tuned include the optimizer algorithm (SGD, Adam, etc), learning rate, learning rate scheduling and regularization to name a few. Depending on the data set and problem, it can take numerous experiments to find a balance between the best neural network architecture and hyper-parameters. The complexity of having to find both an architecture and a combination of hyper-parameters is a very specific aspect of the difficulty to use DL. A difficulty that is not encountered at the same scale with ML algorithms like SVM or Decision trees. To answer to this problem AutoML techniques were recently proposed. Their purpose is to automatically determine a performing set of hyperparameters and a neural network architecture to train an optimized neural networks. AutoML is defined by [@feurer2015efficient] as the problem of automatically (without human input) producing test set predictions for a new data set within a fixed computational budget. Numerous open frameworks are made available now to conduct AutoML experiments. We mention here Auto-Keras[^1][@jin2018efficient], Auto-Weka [^2][@kotthoff2017auto] and Auto-Sklearn[^3][@feurer2015efficient]. Both Google’s AutoML and Auto-Keras are based on the Neural Architecture Search (NAS) technique [@zoph2016neural][@zoph2018learning][^4]. We observe that none of the proposed AutoML frameworks were tested in their respective descriptive papers on unbalanced data sets like ours ([@zoph2016neural] is tested on an image recognition framework and experiment in [@jin2018efficient] for Auto-Keras are based on MNIST, CIFAR and FASHION, 3 images recognition data sets). Considering the specific nature and difficulty of comparing traditional ML algorithmic approaches with DL approaches, we decided to include in our experimental plan an AutoML training, using Auto-Keras. We considered important to do so, as, according to the plasticity of any DL model configuration, one could legitimately claim that we did not found the proper neural network configuration or architecture in our experiments, making it difficult to compare with other algorithms used. We considered AutoML output as an acceptable experiment to potentially refute this legitimate claim. As of today, we did not found in literature previous application of NAS technique to optimize a DL model and compare it to non DL models on unbalanced data set like ours. ### Results The results of the comparison between the best selected classifier, XGBoost, with DL and AutoML are presented in in table \[tab:tableDL\]. They show that XGBoost over-perform a deep neural network tuned by hand (DNN) F-Score by nearly one point. Comparison between DNN and its optimisation with AutoKeras NAS technique shows an improvement in precision but a loss of recall.The final F-Score of DNN not tuned with NAS over-perform the AutoKeras NAS configuration found with a 4-hour search time. Once the hyperparameters are set, the computational cost of DNN and AutoKeras is in both cases more expensive than the cost of XGBoost calculations. (r)[1-2]{} Model Precision Recall F-Score AUC Run time (s) --------------------------- ----------- -------- --------- -------- -------------- XGBoost 0.5138 0.5907 0.5496 0.7822 299.8 DNN 0.5001 0.5865 0.5399 0.7794 450.1 Auto-Keras Neural Network 0.5146 0.5659 0.5390 0.7704 611 : Gradient Boosted Trees compared to DNN and Auto-Keras applied on 12 month forecast, 60 days past due, DS1 Train corpus \[tab:tableDL\] Discussion ---------- In this experiment, the AutoML algorithm with a 4-hour search time did not find a neural layer organisation that would outperform the decision tree algorithms. There is still need in our specific application context, of carefully engineered features used to train decision tree family of algorithms and obtain the best classification performances. Even in the case of hybrid solution involving fusion or ensemble methods to merge outputs from decision trees and DL models, there still is a need for an engineering step to construct the final classification system. Indeed, the assumption frequently claimed to promote NAS technique that, using AutoML algorithms, an operator with minimal machine learning expertise can obtain state-of-the-art performance with very little effort proved to be refuted in our experimental context. We also note that a new field of research emerge to investigate the energy cost of computation generated by AutoML techniques. In their paper, authors of [@strubell2019energy] estimate the cost of AutoML and NAS algorithms for state of the art recent NLP applications. They show that computational costs - and associated carbon impact implied by energy consumption - of modelling using NAS technique on applications like BERT or some transformers, increased along the last 5 years. The authors suggest that it would be beneficial to directly compare different models to perform a cost-benefit analysis. The comparison conducted in this paper could be easily adapted to estimate the cost / value ratio of various algorithms in an application like ours. Conclusions and perspectives {#sec:conclusions} ============================ The results from this empirical study indicate that the random forest and gradient boosting classifiers obtain the best performances in a user credit card behavior forecasting experiment and are able to cope comparatively well with pronounced class imbalances. Those results are consistent with previously published work on similar topic. We have shown that deep learning models directly applied, using the same features as those selected for decision tree models do not outperform decision tree approaches on our highly unbalanced data set. We also found that the AutoML techniques like NAS, used to automatically configure the neural architecture and hyperparameters, instead of performing well like it have been shown for example with image recognition data sets, slightly underperforms when applied on highly unbalanced data set and small sized corpus like the one used in this study. Perspectives ------------ However, since the AutoML method has been shown to increase substantially the precision of the classifier while reducing its recall, we would like to investigate, as potential future work, how the fusion in an hybrid system of the decision tree and DL classifiers optimized with NAS could improve the overall results of classification. [^1]: <https://autokeras.com/> [^2]: <https://www.cs.ubc.ca/labs/beta/Projects/autoweka/> [^3]: <https://github.com/automl/auto-sklearn> [^4]: There is an implementation difference between Google’s AutoML implementation according to [@zoph2018learning] is based on TensorFlow framework while Auto-Keras is based on Pytorch framework. There is also some marginal algorithmic differences as Auto-Keras authors introduces various heuristics to improve the performances of their system.
--- abstract: 'The problem of finding a point in the intersection of closed sets can be solved by the method of alternating projections and its variants. It was shown in earlier papers that for convex sets, the strategy of using quadratic programming (QP) to project onto the intersection of supporting halfspaces generated earlier by the projection process can lead to an algorithm that converges multiple-term superlinearly. The main contributions of this paper are to show that this strategy can be effective for super-regular sets, which are structured nonconvex sets introduced by Lewis, Luke and Malick. Manifolds should be approximated by hyperplanes rather than halfspaces. We prove the linear convergence of this strategy, followed by proving that superlinear and quadratic convergence can be obtained when the problem is similar to the setting of the Newton method. We also show an algorithm that converges at an arbitrarily fast linear rate if halfspaces from older iterations are used to construct the QP.' author: - 'C.H. Jeffrey Pang' bibliography: - '../refs.bib' title: \| Nonconvex set intersection problems:\ From projection methods to the Newton method\ for super-regular sets --- Introduction ============ For finitely many closed sets $K_{1},\dots,K_{m}$ in $\mathbb{R}^{n}$, the Set Intersection Problem (SIP) is stated as: $$\mbox{(SIP):}\quad\mbox{Find }x\in K:=\bigcap_{i=1}^{m}K_{i}\mbox{, where }K\neq\emptyset.\label{eq:SIP}$$ One assumption on the sets $K_{i}$ is that projecting a point in $\mathbb{R}^{n}$ onto each $K_{i}$ is a relatively easy problem. A popular method of solving the SIP is the Method of Alternating Projections (MAP), where one iteratively projects a point through the sets $K_{i}$ to find a point in $K$. For more on the background and recent developments of the MAP and its variants, we refer the reader to [@BB96_survey; @BR09; @EsRa11], as well as [@Deustch01 Chapter 9] and [@BZ05 Subsubsection 4.5.4]. We refer to the references mentioned earlier for a commentary on the applications of the SIP for the convex case (i.e., when all the sets $K_{i}$ in are convex) The convex SIP -------------- One problem of the MAP is slow convergence. As discussed in the previously mentioned references, in the presence of a regular intersection property, one can at best expect linear convergence of the MAP. A few acceleration methods were explored. The papers [@GPR67; @GK89; @BDHP03] explored the acceleration of the MAP using a line search in the case where $K_{i}$ are linear subspaces. See also the papers [@HernandezRamosEscalanteRaydan2011; @Pang_subBAP] for newer research for this particular setting. In [@cut_Pang12], we looked at a different method for the convex SIP (i.e., the SIP when the sets $K_{i}$ are all convex). Each projection generates a halfspace containing the intersection of the sets $K$, and one can project onto the intersection of a number of these halfspaces using standard methods in quadratic programming (for example an active set method [@Goldfarb_Idnani] or an interior point method). We call this the SHQP (supporting halfspace and quadratic programming) strategy. This strategy is illustrated in Figure \[fig:alt-proj-compare\]. We refer to [@cut_Pang12] for more on the history on the SHQP strategy, and we point out a few earlier papers that had some ideas of the SHQP strategy [@Pierra84; @G-P98; @G-P01; @BausCombKruk06; @Polak_Mayne79; @Mayne_Polak_Heunis81]. ![\[fig:alt-proj-compare\]Refer to the diagram on the left. The method of alternating projections on two convex sets $K_{1}$ and $K_{2}$ in $\mathbb{R}^{2}$ with starting iterate $x_{0}$ arrives at $x_{3}$ in three iterations. The point $x_{4}$ is the projection of $x_{1}$ onto the intersection of halfspaces generated by projecting onto $K_{1}$ and $K_{2}$ earlier. One can see that $d(x_{4},K_{1}\cap K_{2})<d(x_{3},K_{1}\cap K_{2})$, illustrating the potential of the SHQP (supporting halfspace and quadratic programming) strategy elaborated in [@cut_Pang12]. The diagram on the right shows that such a heuristic need not be effective for nonconvex sets.](heur "fig:") ![\[fig:alt-proj-compare\]Refer to the diagram on the left. The method of alternating projections on two convex sets $K_{1}$ and $K_{2}$ in $\mathbb{R}^{2}$ with starting iterate $x_{0}$ arrives at $x_{3}$ in three iterations. The point $x_{4}$ is the projection of $x_{1}$ onto the intersection of halfspaces generated by projecting onto $K_{1}$ and $K_{2}$ earlier. One can see that $d(x_{4},K_{1}\cap K_{2})<d(x_{3},K_{1}\cap K_{2})$, illustrating the potential of the SHQP (supporting halfspace and quadratic programming) strategy elaborated in [@cut_Pang12]. The diagram on the right shows that such a heuristic need not be effective for nonconvex sets.](noncon "fig:") -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The main result in [@cut_Pang12] is to show the following: For a convex SIP satisfying the linearly regular intersection property (Definition \[def:lin-reg-int\]), we have an algorithm that achieves multiple-term superlinear convergence if enough halfspaces generated from earlier projections are stored to form the quadratic programs to be solved in later iterations. While the proof of this result suggests keeping an impractically huge number of halfspaces to guarantee the fast convergence, simple examples like the one in Figure \[fig:alt-proj-compare\] suggests that the number of halfspaces that need to be used to obtain the fast convergence can actually be quite small. The nonconvex SIP ----------------- We quote from [@LLM09_lin_conv_alt_proj] on the applications and background of the SIP in the nonconvex case (i.e., when the sets $K_{i}$ in are not known to be convex): An example of a nonconvex set that is easy to project onto is the set of matrices with some fixed rank. The method of alternating projections for nonconvex problems appear in areas such as inverse eigenvalue problems [@ChenChu96SINUM; @Chu95SIMAX], pole placement [@Orsi06SIMAX; @YangOrsi06], information theory [@TroppDhillonHeathStrohmer05], low-order control design [@GrigoriadisBeran2000SIAM; @GrigoriadisSkelton96; @OrsiHelmkeMoore06], and image processing [@BauschkeCombettesLuke02; @Marchesini_Tu_Wu_2014; @WeberAllebach86]. Previous convergence results on nonconvex alternating projection algorithms have been uncommon, and have either focused on a very special case (see, for example [@ChenChu96SINUM; @LewisMalick08]), or have been much weaker than for the convex case [@CombettesTrussell90; @TroppDhillonHeathStrohmer05]. For more discussion, see [@LewisMalick08]. More recent works on the nonconvex SIP include [@BauschkeLukePhanWang13a; @BauschkeLukePhanWang13b; @HesseLuke12]. See also [@AttouchBolteRedontSoubeyran2010]. For the nonconvex problem, the projection onto a nonconvex set need not generate a supporting halfspace. It is easy to construct examples such that the halfspace generated by the projection process will not contain any point in the intersection. (See for example the diagram on the right in Figure \[fig:alt-proj-compare\].) The notion of super-regularity (See Definition \[def:super-regular\]) was first defined in [@LLM09_lin_conv_alt_proj]. They also showed how super-regularity is connected to various other well-known properties in variational analysis. In the presence of super-regularity, they established the linear convergence of the MAP. \[sub:Contrib\]Contributions of this paper ------------------------------------------ The main contribution of this paper is to make two observations about super-regular sets. The first observation is that once a point is close enough to a super-regular set, the projection onto this set produces a halfspace that locally separates a point from the set. (This observation is used to prove Claim (a) in Theorem \[thm:loc-lin-conv\].) With this observation, the SHQP strategy can be carried over to super-regular sets. The second observation is that if one of the sets is a manifold, then we can use a hyperplane to approximate the manifold instead of using a halfspace in the QP subproblem and still obtain convergence of our algorithms. See . In Section \[sec:local-strat\], we show that under typical conditions in the study of alternating projections, an algorithm (Algorithm \[alg:basic-alg\]) that has a sequence of projection steps and SHQP steps that visits all the sets will converge linearly to a point in the intersection. In Section \[sec:Newton-method\], we show that the SHQP strategy applied to find a point in the intersection of manifolds and super-regular sets with only one unit normal on its boundary points will converge superlinearly. The convergence is quadratic under added conditions. This makes a connection to the Newton method. Lastly, in Section \[sec:fast-alg\], we show that arbitrary fast linear convergence is possible when enough halfspaces from previous iterations are kept to form the quadratic programs to accelerate later iterations. Notation -------- The notation we use are fairly standard. We let $\mathbb{B}(x,r)$ be the closed ball with center $x$ and radius $r$, and we denote the projection onto a set $C$ by $P_{C}(\cdot)$. Preliminaries ============= In this section, we recall some definitions in nonsmooth analysis and some basic background material on the theory of alternating projections that will be useful for the rest of the paper. \[def:normal-cones\](Normal cones and Clarke regularity) For a closed set $C\subset\mathbb{R}^{n}$, the regular normal cone at $\bar{x}$ is defined as $$\hat{N}_{C}(\bar{x}):=\{y\mid\langle y,x-\bar{x}\rangle\leq o(\\|x-\bar{x}\\|)\mbox{ for all }x\in C\}.\label{eq:def-normal-1}$$ The limiting normal cone at $\bar{x}$ is defined as $$N_{C}(\bar{x}):=\{y\mid\mbox{there exists }x_{i}\xrightarrow[C]{}\bar{x},\, y_{i}\in\hat{N}_{C}(x_{i})\mbox{ such that }y_{i}\to y\}.\label{eq:def-normal-2}$$ When $\hat{N}_{C}(\bar{x})=N_{C}(\bar{x})$, then $C$ is Clarke regular at $\bar{x}$. If $C$ is Clarke regular at all points, then we simply say that it is Clarke regular. An important tool for our analysis for the rest of the paper is the following notion of regularity of nonconvex sets. \[def:super-regular\][@LLM09_lin_conv_alt_proj Proposition 4.4](Super-regularity) A closed set $C\subset\mathbb{R}^{n}$ is super-regular at a point $\bar{x}\in C$ if, for all $\delta>0$ we can find a neighborhood $V$ of $\bar{x}$ such that $$\langle z-y,v\rangle\leq\delta\\|z-y\\|\\|v\\|\mbox{ for all }z,y\in C\cap V\mbox{ and }v\in N_{C}(y).$$ We say that $C$ is super-regular if it is super-regular at all points. The discussion in [@LLM09_lin_conv_alt_proj] also shows that 1. \[enu:Super-reg-Clarke\]Super-regularity at a point implies Clarke regularity there [@LLM09_lin_conv_alt_proj Corollary 4.5]. (The converse is not true [@LLM09_lin_conv_alt_proj Example 4.6].) 2. Either amenability at a point or prox-regularity at a point implies super-regularity there [@LLM09_lin_conv_alt_proj Propositions 4.8 and 4.9]. We assume that all the sets involved in this paper are super-regular. In view of property , we will not need to distinguish between $\hat{N}_{C}(\bar{x})$ and $N_{C}(\bar{x})$ for the rest of the paper. (On manifolds) It is clear that if $M$ is a smooth manifold in the usual sense, then $M$ is super-regular. Moreover, $$\mbox{For all }x\in M\mbox{, }v\in N_{M}(x)\mbox{ implies }-v\in N_{M}(x).\label{eq:manifold-ppty}$$ For the rest of our discussions, we shall let a manifold be a super-regular set satisfying . The following property relates $d(x,\cap_{l=1}^{m}K_{l})$ to $\max_{1\leq l\leq m}d(x,K_{l})$. \[def:Loc-lin-reg\](Local metric inequality) We say that a collection of closed sets $K_{l}\subset\mathbb{R}^{n}$, $l=1,\dots,m$ satisfies the local metric inequality at $\bar{x}$ if there is a $\beta>0$ and a neighborhood $V$ of $\bar{x}$ such that $$d(x,\cap_{l=1}^{m}K_{l})\leq\beta\max_{1\leq l\leq m}d(x,K_{l})\mbox{ for all }x\in V.\label{eq:loc-metric-ineq}$$ A concise summary of further studies on the local metric inequality appears in [@Kruger_06], who in turn referred to [@BBL99; @Iof00; @Ngai_Thera01; @NgWang04] on the topic of local metric inequality and their connection to metric regularity. Definition \[def:Loc-lin-reg\] is sufficient for our purposes. The local metric inequality is useful for proving the linear convergence of alternating projection algorithms [@BB93_Alt_proj; @LLM09_lin_conv_alt_proj]. See [@BB96_survey] for a survey. \[def:lin-reg-int\](Linearly regular intersection) For closed sets $K_{l}\subset\mathbb{R}^{n}$, we say that $\{K_{l}\}_{l}$ has linearly regular intersection at $x\in K:=\cap_{l=1}^{m}K_{l}$ if the following condition holds: $$\mbox{If }\sum_{l=1}^{m}v_{l}=0\mbox{ for some }v_{l}\in N_{K_{l}}(x)\mbox{, then }v_{l}=0\mbox{ for all }l\in\{1,\dots,r\}.\label{eq:CQ-1}$$ The linearly regular intersection property appears in [@RW98 Theorem 6.42] as a condition for proving that $N_{\cap_{l=1}^{m}K_{l}}(x)=\sum_{l=1}^{m}N_{K_{l}}(x)$. As discussed in [@Kruger_06] and related papers, linearly regular intersection is related to the sensitivity analysis of the SIP . Linearly regular intersection implies the linear convergence of the method of alternating projections. Furthermore, linearly regular intersection implies local metric inequality, but the converse is not true. The following easy and well known principle is used to prove the Fejér monotonicity of iterates in Theorems \[lem:conv-alg\] and \[thm:arb-lin-conv\]. \[prop:fejer-principle\](Fejér monotonicity) Suppose $C$ is a closed convex set in $\mathbb{R}^{n}$, with $x\notin C$ and $y\in C$. Then for any $\lambda\in[0,1]$, $$\\|y-[P_{C}(x)+\lambda(P_{C}(x)-x)]\\|\leq\\|y-x\\|,$$ and the inequality is strict if $\lambda\in[0,1)$. \[sec:local-strat\]Basic local convergence for super-regular SIP ================================================================ In the absence of additional information on the global structure of a nonconvex SIP, the analysis of convergence must necessarily be local. In this section, we discuss how super-regularity can give a halfspace that locally separates a point from the intersection of the sets. This leads to the local linear convergence of an alternating projection algorithm that incorporates QP steps whenever possible. We begin with the algorithm that we study for this section. \[alg:basic-alg\](Basic algorithm) Let $K_{l}$ be (not necessarily convex) closed sets in $\mathbb{R}^{n}$ for $l\in\{1,\dots,m\}$. From a starting point $x_{0}\in\mathbb{R}^{n}$, this algorithm finds a point in the intersection $K:=\cap_{l=1}^{m}K_{l}$.\ $\,$\ 01 For iteration $i=0,1,\dots$\ 02 $\quad$Set $x_{i}^{0}=x_{i}$.\ 03 $\quad$Find sets $S_{1}$, $\dots$, $S_{m}\subset\{1,\dots,m\}$ such that $\cup_{i=1}^{m}S_{i}=\{1,\dots,m\}$.\ 04 $\quad$For $j=1,\dots,m$\ 05 $\quad\quad$Find $x_{i,j,l}\in P_{K_{l}}(x_{i}^{j-1})$ for all $l\in S_{j}$\ 06 $\quad\quad$For $l\in S_{j}$, define halfspace/ hyperplane $H_{i,j,l}$ by $$H_{i,j,l}:=\begin{cases} \{x:\langle x_{i}^{j-1}-x_{i,j,l},x-x_{i,j,l}\rangle=0\} & \mbox{ if }K_{l}\mbox{ is a manifold}\\ \{x:\langle x_{i}^{j-1}-x_{i,j,l},x-x_{i,j,l}\rangle\leq0\} & \mbox{ otherwise}. \end{cases}$$ 07 $\quad\quad$Define the polyhedron $F_{i}^{j}$ by $F_{i}^{j}=\cap_{(k,l)\in\tilde{S}_{i}^{j}}H_{i,k,l}$, where\ 08 $\quad\quad$$\tilde{S}_{i}^{j}\subset\{1,\dots,m\}\times\{1,\dots,m\}$ is such that $\{j\}\times S_{j}\subset\tilde{S}_{i}^{j}$ and \[eq:def-S-i-j\] $$\begin{aligned} \tilde{S}_{i}^{j}: & = & \big\{(k,l):l\in S_{k},k\in\{1,\dots,j\},\mbox{ and }\label{eq:def-S-i-j-1}\\ & & \phantom{\big\{(k,l):}(k_{1},l),(k_{2},l)\in\tilde{S}_{i}^{j}\mbox{ implies }k_{1}=k_{2}\big\}.\label{eq:def-S-i-j-2}\end{aligned}$$ 09 $\quad$$\quad$Set $x_{i}^{j}=P_{F_{i}^{j}}(x_{i}^{j-1})$.\ 10 $\quad$end for\ 11 $\quad$Set $x_{i+1}=x_{i}^{m}$.\ 12 end We allow some of the $S_{j}$’s to be empty as long as the condition $\cup_{i=1}^{m}S_{i}=\{1,\dots,m\}$ is satisfied. When $S_{j}=\{j\}$ and $\tilde{S}_{i}^{j}=\{(j,j)\}$ for all $i,j$, Algorithm \[alg:basic-alg\] reduces to the alternating projection algorithm. Algorithm \[alg:basic-alg\] has the given design because we believe that by performing QP steps with polyhedra that bound the sets $K_{l}$ better, the convergence to a point in $K$ can be accelerated. Yet, we still retain the flexibility of the size of the QPs so that each step can be performed with a reasonable amount of effort. \[rem:mass-proj\](Mass projection) Another particular case of Algorithm \[alg:basic-alg\] we will study in Section \[sec:Newton-method\] is when $S_{1}=\{1,\dots,m\}$, $S_{j}=\emptyset$ for all $j\in\{2,\dots,m\}$, and $\tilde{S}_{i}^{j}=\{j\}\times S_{j}$ for all $i,j\in\{1,\dots,m\}$. In such a case, Algorithm \[alg:basic-alg\] is simplified to $$\begin{aligned} x_{i,1,l} & \in & P_{K_{l}}(x_{i})\\ H_{i,1,l} & = & \begin{cases} \{x:\langle x_{i}-x_{i,1,l},x-x_{i,1,l}\rangle=0\} & \mbox{ if }K_{l}\mbox{ is a manifold}\\ \{x:\langle x_{i}-x_{i,1,l},x-x_{i,1,l}\rangle\leq0\} & \mbox{ otherwise} \end{cases}\\ x_{i+1} & = & P_{\cap_{l=1}^{m}H_{i,1,l}}(x_{i}).\end{aligned}$$ \[rem:sometimes-empty-intersect\](On the polyhedron $F_{i}^{j}$) The polyhedron $F_{i}^{j}$ is defined by intersecting some of the halfspaces/ hyperplanes $H_{i,k,l}$. The line in defining $\tilde{S}_{i}^{j}$ ensures that no two of the halfspaces/ hyperplanes $H_{i,k,l}$ that are intersected to form $F_{i}^{j}$ come from projecting onto the same set. To see why we need , observe that we can draw two tangent lines to a manifold in $\mathbb{R}^{2}$ that do not intersect, which would lead to $F_{i}^{j}=\emptyset$. (Treatment of manifolds) Another feature of this algorithm is that when $K_{l}$ is a manifold, the set $H_{i,j,l}$ is a hyperplane instead. Manifolds are super-regular sets. We take advantage of property of manifolds to create a more logical algorithm. The hyperplane is a better approximate of a manifold than a halfspace, and we may expect faster convergence to a point in $K$ when we use hyperplanes instead. Another advantage of using hyperplanes is that quadratic programming algorithms resolve equality constraints (which are always tight) better than they resolve inequality constraints (where determining whether each constraint is tight at the optimal solution requires some effort). The lemma below will be useful in studying the convergence of the algorithms throughout this paper. \[lem:lin-conv-backbone\](Linear convergence conditions) Let $K$ be a set in $\mathbb{R}^{n}$. Suppose an algorithm generates iterates $\{x_{i}\}$ such that 1. There exists some $\rho\in(0,1)$ such that $d(x_{i+1},K)\leq\rho d(x_{i},K)$, and 2. there exists a constant $c>0$ such that $\\|x_{i+1}-x_{i}\\|\leq cd(x_{i},K)$. Then the sequence $\{x_{i}\}$ converges to a point $\bar{x}\in K$, and we have, for all $i\geq0$, - $\\|x_{i}-\bar{x}\\|\leq\frac{c}{1-\rho}d(x_{i},K)\leq\frac{c\rho^{i}}{1-\rho}d(x_{0},K)$, and - $\mathbb{B}(x_{i+1},\frac{c}{1-\rho}d(x_{i+1},K))\subset\mathbb{B}(x_{i},\frac{c}{1-\rho}d(x_{i},K))$. For any $j\geq0$, we have $$\\|x_{i+j+1}-x_{i+j}\\|\leq cd(x_{i+j},K)\leq c\rho^{j}d(x_{i},K).$$ Standard arguments in analysis shows that $\{x_{i}\}$ is a Cauchy sequence which converges to a point $\bar{x}\in K$. Both parts (a) and (b) are straightforward. The next result shows how such derived halfspaces relate to the original halfspaces. \[lem:derived-halfspaces\](Derived supporting halfspaces) Let $\bar{x}\in\mathbb{R}^{n}$, and suppose $H_{1}$, $H_{2}$, $\dots$, $H_{k}$ are $k$ halfspaces containing $\bar{x}$ such that $d(\bar{x},\partial H_{i})$, the distance from $\bar{x}$ to the boundary of each halfspace $H_{i}$, is at most $\alpha$. Suppose the normal vectors of each halfspace $H_{i}$ is $v_{i}$, where $\\|v_{i}\\|=1$, and the constant $\eta$ defined by $$\eta:=\min\left\{ \left\Vert \sum_{i=1}^{k}\lambda_{i}v_{i}\right\Vert :\sum_{i=1}^{k}\lambda_{i}=1,\,\lambda_{i}\geq0\mbox{ for all }i\in\{1,\dots,k\}\right\} \label{eq:eta-defn}$$ is positive. (i.e., $\eta\neq0$.) Let $F$ be the intersection of these halfspaces. Let $\tilde{H}$ be the halfspace containing $F$ produced by projecting from a point $x^{\prime}\notin F$ onto $F$. In other words, the halfspace $\tilde{H}$ is defined by $$\{x:\langle x^{\prime}-P_{F}(x^{\prime}),x-P_{F}(x^{\prime})\rangle\leq0\}.$$ Then the distance of $\bar{x}$ from the boundary of $\tilde{H}$ is at most $\frac{1}{\eta}\alpha$. As a consequence, suppose $H_{i}$ are defined by $H_{i}=\{x:\langle v_{i},x\rangle\leq\alpha\}$. Let $v=\frac{\sum_{i=1}^{k}\lambda_{i}v_{i}}{\\|\sum_{i=1}^{k}\lambda_{i}v_{i}\\|}$ for some nonzero vector $\lambda\in\mathbb{R}^{k}$ that has nonnegative components, and $H$ be $H=\{x:\langle v,x\rangle\leq\frac{\alpha}{\eta}\}$. Then we have $\cap_{i=1}^{k}H_{i}\subset\tilde{H}\subset H$. We remark that $\eta$ is the distance of the origin to the convex hull of $\{v_{i}\}_{i=1}^{k}$. We can eliminate halfspaces if necessary and assume that $k\geq1$, and that $P_{F}(x^{\prime})$ lies on the boundaries of all the halfspaces. The KKT condition tells us that $x^{\prime}-P_{F}(x^{\prime})$ lies in the conical hull of $\{v_{i}\}_{i=1}^{k}$. By Caratheodory’s theorem, we can assume that $k$ is not more than the dimension $n$. We can also eliminate halfspaces if necessary so that the vectors $\{v_{i}\}_{i=1}^{k}$ are linearly independent. Suppose each halfspace $H_{i}$ is defined by $\{x:\langle v_{i},x\rangle\leq b_{i}\}$, where $b_{i}\in\mathbb{R}$. Since $P_{F}(x^{\prime})$ lies on the boundaries of the halfspaces $H_{i}$, we have $$\langle v_{i},P_{F}(x^{\prime})\rangle=b_{i}\mbox{ for all }i.\label{eq:inn-pdt-eq-b-i}$$ Define the hyperslab $S_{i}$ by $$S_{i}:=\{x:\langle v_{i},x\rangle\in[b_{i}-\alpha,b_{i}]\}.\label{eq:S-i-hyperslab}$$ Since the distance from $\bar{x}$ to the boundaries of each halfspace $H_{i}$ were assumed to be at most $\alpha$, the point $\bar{x}$ is inside all the hyperslabs $S_{i}$. Let $\bar{v}$ be the vector $\frac{x^{\prime}-P_{F}(x^{\prime})}{\\|x^{\prime}-P_{F}(x^{\prime})\\|}$. We now study the problem $$\begin{aligned} & \min_{x} & \left\langle \bar{v},x\right\rangle \label{eq:min-slab-pblm}\\ & \mbox{s.t.} & x\in S_{i}\mbox{ for all }i\in\{1,\dots,k\}.\nonumber \end{aligned}$$ If the above problem were a maximization problem instead, then an optimizer is $P_{F}(x^{\prime})$. Consider the point $P_{F}(x^{\prime})-\alpha d$, where $d$ is the direction defined through $$\langle v_{i},d\rangle=1\mbox{ for all }i\mbox{, and }d\in\mbox{span}(\{v_{i}\}_{i=1}^{k}).\label{eq:defn-dirn-d}$$ Since the vectors $\{v_{i}\}_{i=1}^{k}$ are linearly independent, such a $d$ exists, and can be calculated by $d=QR^{-T}\mathbf{1}$, where $\mathbf{1}$ is the vector of all ones, $QR$ is the QR factorization of $V$, and $V$ is the matrix formed by concatenating the vectors $\{v_{i}\}_{i=1}^{k}$. We can use and to calculate that $$\langle v_{i},P_{F}(x^{\prime})-\alpha d\rangle=b_{i}-\alpha\mbox{ for all }i,$$ so $P_{F}(x^{\prime})-\alpha d$ is on the other boundary of all the hyperslabs $S_{i}$. Furthermore, since $N_{\cap_{i=1}^{k}S_{i}}(P_{F}(x^{\prime})-\alpha d)=-N_{\cap_{i=1}^{k}S_{i}}(P_{F}(x^{\prime}))$, we have $-\bar{v}\in N_{\cap_{i=1}^{k}S_{i}}(P_{F}(x^{\prime})-\alpha d)$. Hence $P_{F}(x^{\prime})-\alpha d$ is a minimizer of . We proceed to find the optimal value of . Since $\bar{v}$ lies in the conical hull of $\{v_{i}\}_{i=1}^{k}$, $\bar{v}$ can be written as $\frac{V\lambda}{\\|V\lambda\\|}$, where $\lambda\in\mathbb{R}_{+}^{k}$ is a vector with nonnegative elements such that its elements sum to one. We can calculate $$\begin{aligned} \left(\frac{V\lambda}{\\|V\lambda\\|}\right)^{T}d & = & \frac{1}{\\|V\lambda\\|}\lambda^{T}V^{T}QR^{-T}\mathbf{1}\\ & = & \frac{1}{\\|V\lambda\\|}\lambda^{T}R^{T}Q^{T}QR^{-T}\mathbf{1}=\frac{1}{\\|V\lambda\\|}\lambda^{T}\mathbf{1}=\frac{1}{\\|V\lambda\\|}.\end{aligned}$$ By the definition of $\eta$, we have $\frac{1}{\\|V\lambda\\|}\geq\frac{1}{\eta}$. This means that the minimum value of is at least $\left\langle \bar{v},P_{F}(x^{\prime})-\alpha d\right\rangle =\left\langle \bar{v},P_{F}(x^{\prime})\right\rangle -\frac{1}{\eta}\alpha$. Since $\bar{x}\in S_{i}$ for all $i\in\{1,\dots,k\}$, we can deduce that $\bar{x}$ lies in the hyperslab $$\{x:\langle\bar{v},x\rangle\in[\langle\bar{v},P_{F}(x^{\prime})\rangle-\nicefrac{\alpha}{\eta},\langle\bar{v},P_{F}(x^{\prime})\rangle]\}.$$ In other words, $\bar{x}$ lies in the halfspace $\{x:\langle\bar{v},x\rangle\leq\langle\bar{v},P_{F}(x^{\prime})\rangle\}$, and the distance from $\bar{x}$ to the boundary of this halfspace is at most $\frac{1}{\eta}\alpha$, which is the conclusion we seek. The final paragraph is easily deduced from the main result. (The formula $\eta$) We remark that the use of the notation $\eta$ in Lemma \[lem:derived-halfspaces\] is consistent with the notation of [@Kruger_06] and related papers, where the relationship of the constants related to the sensitivity analysis of the SIP and linearly regular intersection are studied. We now prove our result on the convergence of Algorithm \[alg:basic-alg\]. \[thm:loc-lin-conv\](Local linear convergence of general Algorithm) Suppose $K_{l}$, where $l\in\{1,\dots,m\}$, are super-regular at $x^{}\in K=\cap_{l=1}^{m}K_{l}$. Suppose that $\eta$ defined by $$\eta:=\min\left\{ \left\Vert \sum_{i=1}^{m}v_{i}\right\Vert :v_{i}\in N_{K_{i}}(x^{}),\, x^{}\in K_{i},\,\sum_{i=1}^{m}\\|v_{i}\\|=1\right\}$$ is positive. (i.e., $\eta\neq0$.) This is equivalent to $\{K_{l}\}_{l=1}^{m}$ having linear regular intersection at $x^{}$, which in turn implies that the local metric inequality holds at $x^{}$. If $x_{0}$ is sufficiently close to $x^{}$, then Algorithm \[alg:basic-alg\] converges to a point in $K$ Q-linearly (i.e., at a rate bounded above by a geometric sequence). Since the local metric inequality holds at $x^{}$, let $\beta\geq1$ and $V$ be a neighborhood of $x^{}$ such that $$d(x,K)\leq\beta\max_{l}d(x,K_{l})\mbox{ for all }x\in V.$$ Let $$\begin{aligned} \rho & = & \sqrt{1+\frac{1}{\beta^{2}m^{3}}+\frac{1}{4\beta^{4}m^{6}}-\frac{1}{\beta^{2}m^{2}}+\frac{1}{2\beta^{3}m^{5}}-\frac{1}{16\beta^{4}m^{8}}+\frac{1}{16\beta^{4}m^{6}}},\label{eq:linear-rho}\\ \mbox{ and }c & = & \sqrt{m}\sqrt{\left[1+\frac{1}{4m^{3}\beta^{2}}\right]^{2}+\frac{1}{16m^{6}\beta^{4}}}\label{eq:linear-c}\end{aligned}$$ It is clear to see that if $m\geq2$, then $\rho<1$. Choose $\delta>0$ such that $\delta\leq\frac{(1-\rho)\eta}{16m^{4}\beta^{2}c}$. Since $x^{}$ is super-regular at all sets $K_{l}$, where $l\in\{1,\dots,m\}$, we can shrink the neighborhood $V$ if necessary so that for all $l\in\{1,\dots,m\}$, we have $$\langle v,z-y\rangle\leq\delta\\|v\\|\\|z-y\\|\mbox{ for all }z,y\in K_{l}\cap V\mbox{ and }v\in N_{K_{l}}(y).$$ By the outer semicontinuity of the normal cones, we can shrink $V$ if necessary so that for all $x\in V$, we have $$\min\left\{ \left\Vert \sum_{i=1}^{m}v_{i}\right\Vert :v_{i}\in N_{K_{i}}(x),\, x\in K_{i},\,\sum_{i=1}^{m}\\|v_{i}\\|=1\right\} \geq\frac{\eta}{2}.$$ Suppose $x_{0}$ is close enough to $x^{}$ such that $\mathbb{B}(x_{0},\frac{c}{1-\rho}d(x_{0},K))\subset V$. Provided that we prove conditions (1) and (2) in Lemma \[lem:lin-conv-backbone\], we have the convergence of the iterates $\{x_{i}\}$ to some point $\bar{x}\in K$. The convergence of $\{x_{i}\}$ to $\bar{x}$ would be at the rate suggested in Lemma \[lem:lin-conv-backbone\](a). If $x\in K\cap\mathbb{B}(x_{i},\frac{c}{1-\rho}d(x_{i},K))$ and $x_{i}^{j-1},x_{i,j,l}\in\mathbb{B}(x_{i},\frac{c}{1-\rho}d(x_{i},K))$, then $$\begin{aligned} \left\langle \frac{x_{i}^{j-1}-x_{i,j,l}}{\\|x_{i}^{j-1}-x_{i,j,l}\\|},x-x_{i,j,l}\right\rangle & \leq & \delta\\|x-x_{i,j,l}\\|\label{eq:halfspace-QP-alg}\\ & \leq & \delta\frac{2c}{1-\rho}d(x_{i},K)\nonumber \\ & \leq & \frac{\eta}{8m^{4}\beta^{2}}d(x_{i},K).\nonumber \end{aligned}$$ Define the halfspace $H_{i,j,l}^{+}$ by $$H_{i,j,l}^{+}:=\left\{ x:\left\langle \frac{x_{i}^{j-1}-x_{i,j,l}}{\\|x_{i}^{j-1}-x_{i,j,l}\\|},x-x_{i,j,l}\right\rangle \leq\frac{\eta}{8m^{4}\beta^{2}}d(x_{i},K)\right\} .$$ (Note that the halfspace $H_{i,j,l}$ defined in Algorithm \[alg:basic-alg\] is similar to $H_{i,j,l}^{+}$ with the exception that the right hand side of the inequality is zero.) We have $K_{l}\cap\mathbb{B}(x_{i},\frac{c}{1-\rho}d(x_{i},K))\subset H_{i,j,l}^{+}$. Note that $x_{i}^{j}$ is the projection of $x_{i}^{j-1}$ onto $F_{i}^{J}$. Define the halfspace $H_{i,j}^{+}$ by $$H_{i,j}^{+}:=\left\{ x:\left\langle \frac{x_{i}^{j-1}-x_{i}^{j}}{\\|x_{i}^{j-1}-x_{i}^{j}\\|},x-x_{i}^{j}\right\rangle \leq\frac{1}{4m^{4}\beta^{2}}d(x_{i},K)\right\} .\label{eq:agg-halfspace-defn}$$ By Lemma \[lem:derived-halfspaces\], we have $$K\cap\mathbb{B}(x_{i},\frac{c}{1-\rho}d(x_{i},K))\subset\cap_{(k,l)\in\tilde{S}_{i}^{j}}H_{i,k,l}^{+}\subset H_{i,j}^{+}.\label{eq:agg-halfspace-works}$$ Note that almost exactly the same arguments works if the set $K_{l}$ is a manifold, but we may have to take $-\frac{x_{i}^{j-1}-x_{i}^{j}}{\\|x_{i}^{j-1}-x_{i}^{j}\\|}$ as the normal vector of $H_{i,j,l}^{+}$ instead and define $H_{i,j}^{+}$ differently, depending on the multipliers in the KKT condition. Claim: \(a) If $\\|x_{i}^{j}-x_{i}^{j-1}\\|\geq\frac{1}{2m^{4}\beta^{2}}d(x_{i},K)$, then $\qquad\qquad d(x_{i}^{j},K)^{2}\leq d(x_{i}^{j-1},K)^{2}-\left[\\|x_{i}^{j}-x_{i}^{j-1}\\|-\frac{1}{4m^{4}\beta^{2}}d(x_{i},K)\right]^{2}$ $\qquad\qquad\phantom{d(x_{i}^{j},K)^{2}\leq}+\left[\frac{1}{4m^{4}\beta^{2}}d(x_{i},K)\right]^{2}.$ \(b) If $\\|x_{i}^{j}-x_{i}^{j-1}\\|\leq\frac{1}{2m^{4}\beta^{2}}d(x_{i},K)$, then $d(x_{i}^{j},K)\leq d(x_{i}^{j-1},K)+\frac{1}{2m^{4}\beta^{2}}d(x_{i},K)$. Part (b) is obvious. We now prove part (a). Let $y$ be any point in $P_{K}(x_{i}^{j-1})$, and let $z=P_{H_{i,j}^{+}}(x_{i}^{j-1})$. See Figure \[fig:apply-cosine-rule\], where $d_{2}=\frac{1}{4m^{4}\beta^{2}}d(x_{i},K)$ in view of . Noting that $\angle zyx_{i}^{j}\geq\pi/2$, we apply cosine rule to get $$\begin{aligned} & & d(x_{i}^{j},K)^{2}\\ & \leq & \\|y-x_{i}^{j}\\|^{2}\\ & \leq & \\|y-z\\|^{2}+\\|z-x_{i}^{j}\\|^{2}\\ & \leq & \\|y-x_{i}^{j-1}\\|^{2}-\\|x_{i}^{j}-z\\|^{2}+\\|z-x_{i}^{j}\\|^{2}\\ & = & d(x_{i}^{j-1},K)^{2}-\left[\\|x_{i}^{j}-x_{i}^{j-1}\\|-\frac{1}{4m^{4}\beta^{2}}d(x_{i},K)\right]^{2}+\left[\frac{1}{4m^{4}\beta^{2}}d(x_{i},K)\right]^{2}.\end{aligned}$$ This completes the proof of the claim. ![\[fig:apply-cosine-rule\]This figure illustrates the proof in the claim of Theorem \[thm:loc-lin-conv\]. Note that $d_{1}=\\|x_{i}^{j}-x_{i}^{j-1}\\|-\frac{1}{4m^{4}\beta^{2}}d(x_{i},K)$ and $d_{2}=\frac{1}{4m^{4}\beta^{2}}d(x_{i},K)$.](fejer) It now remains the prove conditions (1) and (2) of Lemma \[lem:lin-conv-backbone\]. By local metric inequality, there is some $j\in\{1,\dots,m\}$ such that $d(x_{i},K_{j})\geq\frac{1}{\beta}d(x_{i},K)$. Hence there is a distance $\\|x_{i}^{j}-x_{i}^{j-1}\\|$ that will be at least $\frac{1}{m\beta}d(x_{i},K)$. Making use of the claim earlier, we have the following estimate of $d(x_{i+1},K)$. $$\begin{aligned} & & d(x_{i+1},K)^{2}\label{eq:first-ineq-block}\\ & \leq & \left[d(x_{i},K)+\frac{m}{2m^{4}\beta^{2}}d(x_{i},K)\right]^{2}-\left[\frac{1}{m\beta}d(x_{i},K)-\frac{1}{4m^{4}\beta^{2}}d(x_{i},K)\right]^{2}\nonumber \\ & & +\left[\frac{m}{4m^{4}\beta^{2}}d(x_{i},K)\right]^{2}\nonumber \\ & = & \left[1+\frac{1}{\beta^{2}m^{3}}+\frac{1}{4\beta^{4}m^{6}}-\frac{1}{\beta^{2}m^{2}}+\frac{1}{2\beta^{3}m^{5}}-\frac{1}{16\beta^{4}m^{8}}+\frac{1}{16\beta^{4}m^{6}}\right]d(x_{i},K)^{2}\nonumber \\ & = & \rho^{2}d(x_{i},K)^{2}.\nonumber \end{aligned}$$ This proves that $d(x_{i+1},K)\leq\rho d(x_{i},K)$. Next, $$\begin{aligned} & & \\|x_{i+1}-x_{i}\\|\label{eq:2nd-ineq-block}\\ & \leq & \sum_{j=1}^{m}\\|x_{i}^{j}-x_{i}^{j-1}\\|\nonumber \\ & \leq & \frac{1}{4m^{3}\beta^{2}}d(x_{i},K)+\sum_{j=1}^{m}\max\left\{ \\|x_{i}^{j}-x_{i}^{j-1}\\|-\frac{1}{4m^{4}\beta^{2}}d(x_{i},K),0\right\} \nonumber \\ & \leq & \frac{1}{4m^{3}\beta^{2}}d(x_{i},K)+\underbrace{\sqrt{m\sum_{j=1}^{m}\max\left\{ \\|x_{i}^{j}-x_{i}^{j-1}\\|-\frac{1}{4m^{4}\beta^{2}}d(x_{i},K),0\right\} ^{2}}}_{()}.\nonumber \end{aligned}$$ By the analysis in , the fact that $d(x_{i+1},K)^{2}\geq0$ gives $$\begin{aligned} 0 & \leq & d(x_{i+1},K)^{2}\\ & \leq & \left[d(x_{i},K)+m\frac{1}{2m^{4}\beta^{2}}d(x_{i},K)\right]^{2}\\ & & +\left[\frac{m}{4m^{4}\beta^{2}}d(x_{i},K)\right]^{2}-\sum_{j=1}^{m}\max\left\{ \\|x_{i}^{j}-x_{i}^{j-1}\\|-\frac{1}{4m^{4}\beta^{2}}d(x_{i},K),0\right\} ^{2}.\end{aligned}$$ We thus deduce that the term marked $()$ in is at most $$\sqrt{m}\sqrt{\left[1+\frac{1}{2m^{3}\beta^{2}}\right]^{2}+\frac{1}{16m^{6}\beta^{4}}}d(x_{i},K).$$ Thus the constant $c$ in Lemma \[lem:lin-conv-backbone\] can be taken to be what was given in . (On the condition $\eta>0$ in Theorem \[thm:loc-lin-conv\]) The condition $\eta>0$ is required in the proof of Theorem \[thm:loc-lin-conv\] only when $\|S_{l}\|>1$, when halfspaces are aggregated. So in the case of alternating projections, the weaker condition of local metric inequality is sufficient. \[sec:Newton-method\]Connections with the Newton method ======================================================= To find a point in $\{x\in\mathbb{R}^{n}:F(x)=0\}$ for some smooth $F:\mathbb{R}^{n}\to\mathbb{R}^{m}$, the method of choice is to use the Newton method provided that the linear system in the Newton method can be solved quickly enough. Note that the set $\{x:F(x)=0\}$ can be written as the intersection of the manifolds $M_{j}:=\{x:F_{j}(x)=0\}$ for $j\in\{1,\dots,m\}$, where $F_{j}:\mathbb{R}^{n}\to\mathbb{R}$ is the $j$th component of $F(\cdot)$. Note that the manifolds $M_{j}$ are of codimension 1. This section gives conditions for which the SHQP strategy can converge superlinearly or quadratically when the sets involved satisfy the conditions for fast convergence in the Newton method. The following result was proved in [@cut_Pang12] for convex sets, but is readily generalized to Clarke regular sets, which we do so now. \[thm:radiality\](Supporting hyperplane near a point) Suppose $C\subset\mathbb{R}^{n}$ is Clarke regular, and let $\bar{x}\in C$. Then for any $\epsilon>0$, there is a $\delta>0$ such that for any point $x\in[\mathbb{B}_{\delta}(\bar{x})\cap C]\backslash\{\bar{x}\}$ and supporting hyperplane $A$ of $C$ with unit normal $v\in N_{C}(x)$ at the point $x$, we have $$\left\langle v,x-\bar{x}\right\rangle \leq\epsilon\\|\bar{x}-x\\|.\label{eq:little-SOSH}$$ Let $\delta$ be small enough so that for any $x\in[\mathbb{B}_{\delta}(\bar{x})\cap C]\backslash\{\bar{x}\}$ and unit normal $v\in N_{C}(x)$, we can find $\bar{v}\in N_{C}(\bar{x})$ such that $\\|v-\bar{v}\\|<\frac{\epsilon}{2}$ and that $\langle\bar{v},x-\bar{x}\rangle\leq\frac{\epsilon}{2}\\|x-\bar{x}\\|$. Then we have $$\begin{aligned} \langle v,x-\bar{x}\rangle & = & \langle v-\bar{v},x-\bar{x}\rangle+\langle\bar{v},x-\bar{x}\rangle\\ & \leq & \\|v-\bar{v}\\|\\|x-\bar{x}\\|+\frac{\epsilon}{2}\\|x-\bar{x}\\|\\ & \leq & \epsilon\\|x-\bar{x}\\|.\end{aligned}$$ Thus we are done. We identify a property that will give multiple-term quadratic convergence. Compare this property to that in Theorem \[thm:radiality\]. \[def:SOSH\](Second order supporting hyperplane property) Suppose $C\subset\mathbb{R}^{n}$ is a closed convex set, and let $\bar{x}\in C$. We say that $C$ has the second order supporting hyperplane (SOSH) property at $\bar{x}$ (or more simply, $C$ is SOSH at $\bar{x}$) if there are $\delta>0$ and $M>0$ such that for any point $x\in[\mathbb{B}_{\delta}(\bar{x})\cap C]\backslash\{\bar{x}\}$ and $v\in N_{C}(x)$ such that $\\|v\\|=1$, we have $$\left\langle v,x-\bar{x}\right\rangle \leq M\\|\bar{x}-x\\|^{2}.\label{eq:SOSH-1}$$ It is clear how compares with . The next two results show that SOSH is prevalent in applications. \[prop:C2-SOSH\](Smoothness implies SOSH) Suppose function $f:\mathbb{R}^{n}\to\mathbb{R}$ is $\mathcal{C}^{2}$ at $\bar{x}$. Then the set $C=\{x\mid f(x)\leq0\}$ is SOSH at $\bar{x}$. Consider $\bar{x},x\in C$. In order for the problem to be meaningful, we shall only consider the case where $f(\bar{x})=0$. We also assume that $f(x)=0$ so that $C$ has a tangent hyperplane at $x$. An easy calculation gives $N_{C}(\bar{x})=\mathbb{R}_{+}\{\nabla f(\bar{x})\}$ and $N_{C}(x)=\mathbb{R}_{+}\{\nabla f(x)\}$. Without loss of generality, let $\bar{x}=0$. We have $$\begin{aligned} & & f(x)=f(0)+\nabla f(0)x+\frac{1}{2}x^{T}\nabla^{2}f(0)x+o(\\|x\\|^{2}).\\ & \Rightarrow & f(0)x+\frac{1}{2}x^{T}\nabla^{2}f(0)x=o(\\|x\\|^{2}).\end{aligned}$$ Since $f(x)=f(0)=0$ and $[\nabla f(0)-\nabla f(x)]x=x^{T}\nabla^{2}f(0)x+o(\\|x\\|^{2})$, we have $$-\nabla f(x)(x)=[\nabla f(0)-\nabla f(x)]x+\frac{1}{2}x^{T}\nabla^{2}f(0)x+o(\\|x\\|^{2})=O(\\|x\\|^{2}).$$ Therefore, we are done. \[prop:SOSH-intersect\](SOSH under intersection) Suppose $K_{l}\subset\mathbb{R}^{n}$ are closed sets that are SOSH at $\bar{x}$ for $l\in\{1,\dots,m\}$. Let $K:=\cap_{l=1}^{m}K_{l}$, and suppose that $\{K_{l}\}_{l=1}^{m}$ satisfy the linear regular intersection property at $\bar{x}$. Then $K$ is SOSH at $\bar{x}$. Since each $K_{l}$ is SOSH at $\bar{x}$, we can find $\delta>0$ and $M>0$ such that for all $l\in\{1,\dots,m\}$ and $x\in K_{l}\cap\mathbb{B}_{\delta}(\bar{x})$ and $v\in N_{K_{l}}(x)$, we have $$\left\langle v,x-\bar{x}\right\rangle \leq M\\|v\\|\\|\bar{x}-x\\|^{2}.$$ ** $$\begin{aligned} & & \sum_{l=1}^{m}v_{l}=0,\, v_{l}\in N_{K_{l}}(x)\mbox{ and }x\in K\cap\mathbb{B}_{\delta}(\bar{x})\label{eq:CQ-all-x}\\ & & \qquad\qquad\mbox{ implies }v_{l}=0\mbox{ for all }l\in\{1,\dots,m\}.\nonumber \end{aligned}$$ Suppose otherwise. Then we can find $\{x_{i}\}_{i=1}^{\infty}\in K$ such that $\lim x_{i}=\bar{x}$ and for all $i>0$, there exists $v_{l,i}\in N_{K_{l}}(x_{i})$ such that $\sum_{l=1}^{m}v_{l,i}=0$ but not all $v_{l,i}=0$. We can normalize so that $\\|v_{l,i}\\|\leq1$, and for each $i$, $\max_{l}\\|v_{l,i}\\|=1$. By taking a subsequence if necessary, we can assume that $\lim v_{l,i}$, say $\bar{v}_{l}$, exists for all $l$. Not all $\bar{v}_{l}$ can be zero, but $\sum_{l=1}^{m}\bar{v}_{l}=0$. The outer semicontinuity of the normal cone mapping implies that $\bar{v}_{l}\in N_{K_{l}}(\bar{x})$. This contradicts the linear regular intersection property, which ends the proof of Claim 1. ** Suppose otherwise. Then for each $i$, there exists $x_{i}\in\mathbb{B}_{\delta}(\bar{x})\cap K$ and $\tilde{v}_{l,i}\in N_{K_{l}}(x_{i})$ such that $\tilde{v}_{i}=\sum_{l=1}^{m}\tilde{v}_{l,i}$, $\\|\tilde{v}_{i}\\|\leq\frac{1}{i}$, and $\max_{l}\\|\tilde{v}_{l,i}\\|=1$ for all $i$. As we take limits to infinity, this would imply that is violated, a contradiction. This ends the proof of Claim 2. Since is satisfied, this means that $N_{K}(x)=\sum_{l=1}^{m}N_{K_{l}}(x)$ for all $x\in\mathbb{B}_{\delta}(\bar{x})\cap K$ by the intersection rule for normal cones in [@RW98 Theorem 6.42]. Then each $v\in N_{K}(x)$ can be written as a sum of elements in $N_{K_{l}}(x)$, say $v=\sum_{l=1}^{m}v_{l}$, where $v_{l}\in N_{K_{l}}(x)$, and $\max\\|v_{l}\\|\leq M^{\prime}\\|v\\|$. Then $$\begin{aligned} \langle v,x-\bar{x}\rangle & = & \sum_{l=1}^{m}\langle v_{l},x-\bar{x}\rangle\\ & \leq & M\\|\bar{x}-x\\|^{2}\sum_{l=1}^{m}\\|v_{l}\\|\quad\leq\quad M\\|\bar{x}-x\\|^{2}mM^{\prime}\\|v\\|.\end{aligned}$$ Thus we are done. We now make a connection to the Newton method. Consider the mass projection algorithm. (Connection to Newton method) Consider Algorithm \[alg:basic-alg\] for the case when $S_{1}=\{1,\dots,m\}$ and $S_{j}=\emptyset$ for all $j\in\{2,\dots,m\}$ at all iterations $i$, and $\tilde{\ensuremath{S}}_{i}^{j}=\{j\}\times S_{j}$ for all $j\in\{1,\dots,m\}$. See Remark \[rem:mass-proj\]. Let $x^{}\in K:=\cap_{l=1}^{m}K_{l}$. Suppose the following hold 1. Each set $K_{l}$ is super-regular. 2. For each $l\in\{1,\dots,m\}$, $K_{l}$ is either a manifold, or $N_{K_{l}}(x)$ contains at most one point of norm 1 for all $x\in K_{l}$ near $x^{}$. 3. The sets $\{K_{l}\}_{l=1}^{m}$ has linearly regular intersection at $x^{}$. Then provided $x_{0}$ is close enough to $x^{}$, the convergence of the iterates $\{x_{i}\}$ to some $\bar{x}\in K$ is superlinear. Furthermore, the convergence is quadratic if all the sets $K_{i}$ satisfy the SOSH property. By Theorem \[thm:loc-lin-conv\], the convergence of the iterates $\{x_{i}\}$ to $\bar{x}$ is assured. What remains is to prove that the convergence is actually superlinear, or quadratic under the additional assumption. Without loss of generality, let $\bar{x}=0$. We first prove the superlinear convergence. The proof in Theorem \[thm:loc-lin-conv\] assures that there is some $\beta\geq1$ such that $d(x_{i},K)\leq\beta\max_{l}d(x_{i},K_{l})$ for all iterates $x_{i}$. Let $x_{i}$ be an iterate. Recall that $x_{i,1,j}=P_{K_{j}}(x_{i})$. The projection of $x_{i}$ onto the polyhedron gives $x_{i+1}$. Let $v_{j}^{+}$ be the unit normal in $N_{K_{j}}(x_{i+1,1,j})$ in the direction of $x_{i+1}-x_{i+1,1,j}$, and let $v_{j}^{\circ}$ be the unit normal in $N_{K_{j}}(x_{i,1,j})$ that is close to $v_{j}^{+}$. The proof of Theorem \[thm:loc-lin-conv\] uses Lemma \[lem:lin-conv-backbone\]. Hence there are constants $c$ and $\rho\in(0,1)$ such that $\\|x_{i}\\|\leq\frac{c}{1-\rho}d(x_{i},K)$ for all $i$. By local metric inequality, let the index $j$ be such that $d(x_{i+1},K)\leq\beta d(x_{i+1},K_{j})$. We let $\kappa=\frac{c\beta}{(1-\rho)}$. Then $$\begin{aligned} \langle v_{j}^{+},x_{i+1}-x_{i+1,1,j}\rangle & = & \\|x_{i+1}-x_{i+1,1,j}\\|\label{eq:first-chain}\\ & = & d(x_{i+1},K_{j})\geq\frac{1}{\beta}d(x_{i+1},K)\geq\frac{1}{\kappa}\\|x_{i+1}\\|.\nonumber \end{aligned}$$ Consider the neighborhood $U$ such that if $x\in U$ and $v\in N_{K_{j}}(x)\backslash\{0\}$, then ${\\|\frac{v}{\\|v\\|}-\frac{\bar{v}}{\\|\bar{v}\\|}\\|}\leq\frac{1}{4\kappa}$ for some $\bar{v}\in N_{K_{j}}(\bar{x})\backslash\{0\}$. If $i$ is large enough, then $x_{i}\in U$ and $x_{i,1,j}\in U$ for all $j\in\{1,\dots,m\}$, which leads to $$\\|v_{j}^{\circ}-v_{j}^{+}\\|\leq\\|v_{j}^{\circ}-\bar{v}_{j}\\|+\\|v_{j}^{+}-\bar{v}_{j}\\|\leq\frac{1}{2\kappa},\label{eq:second-chain}$$ where $\bar{v}_{j}$ is the appropriate unit vector in $N_{K_{j}}(\bar{x})$. For any $\delta>0$, we can reduce the neighborhood $U$ if necessary so that by super-regularity, $$\langle v_{j}^{+},0-x_{i+1,1,j}\rangle\leq\delta\\|x_{i+1,1,j}\\|.\label{eq:super-reg-conseq}$$ Claim: $\\|x_{i+1,1,j}\\|\leq\frac{1}{\sqrt{1-\delta^{2}}}\\|x_{i+1}\\|$. We know that $x_{i+1}=x_{i+1,1,j}+tv_{j}^{+}$, where $t=\\|x_{i+1}-x_{i+1,1,j}\\|>0$. By super-regularity, we have $\cos^{-1}\delta\leq\angle x_{i+1}x_{i+1,1,j}0$. Note that $\sqrt{1-\delta^{2}}=\sin\cos^{-1}\delta$. Some simple trigonometry ends the proof of the claim. Choose $\delta$ small enough so that $\frac{\delta}{\sqrt{1-\delta^{2}}}\leq\frac{1}{4\kappa}$. From , we have $$\langle v_{j}^{+},0-x_{i+1,1,j}\rangle\leq\delta\\|x_{i+1,1,j}\\|\leq\frac{\delta}{\sqrt{1-\delta^{2}}}\\|x_{i+1}\\|\leq\frac{1}{4\kappa}\\|x_{i+1}\\|.\label{eq:third-chain}$$ Then combining , and , we get $$\begin{aligned} \langle v_{j}^{\circ},x_{i+1}\rangle & = & \langle v_{j}^{+},x_{i+1}-x_{i+1,1,j}\rangle+\langle v_{j}^{+},x_{i+1,1,j}\rangle+\langle v_{j}^{\circ}-v_{j}^{+},x_{i+1}\rangle\label{eq:suplin-set-1}\\ & \geq & \frac{1}{\kappa}\\|x_{i+1}\\|-\frac{1}{4\kappa}\\|x_{i+1}\\|-\frac{1}{2\kappa}\\|x_{i+1}\\|=\frac{1}{4\kappa}\\|x_{i+1}\\|.\nonumber \end{aligned}$$ Choose any $\epsilon>0$. Theorem \[thm:radiality\] implies that $\langle v_{j}^{\circ},x_{i,1,j}\rangle\leq\epsilon\\|x_{i,1,j}\\|$ for all $i$ large enough. We have the following set of inequalities. $$\langle v_{j}^{\circ},x_{i+1}\rangle\leq\langle v_{j}^{\circ},x_{i,1,j}\rangle\leq\epsilon\\|x_{i,1,j}\\|\leq\frac{\epsilon}{\sqrt{1-\delta^{2}}}\\|x_{i}\\|.\label{eq:suplin-set-2}$$ (The first inequality comes from the fact that $x_{i+1}$ has to lie in the halfspaces constructed by the previous projection. If $K_{j}$ is a manifold, then the first inequality is in fact an equation. The last inequality is from the highlighted claim above.) Combining and gives $\\|x_{i+1}\\|\leq\frac{4\kappa\epsilon}{\sqrt{1-\delta^{2}}}\\|x_{i}\\|$, which is what we need. In the case where $K_{j}$ has the SOSH property near $\bar{x}$, can be changed to give $\langle v_{j}^{\circ},x_{i+1}\rangle\leq\frac{M}{1-\delta^{2}}\\|x_{i}\\|^{2}$ for some constant $M$, which gives $\\|x_{i+1}\\|\leq\frac{4\kappa M}{1-\delta^{2}}\\|x_{i}\\|^{2}$. This completes the proof. \[sec:fast-alg\]An algorithm with arbitrary fast linear convergence ==================================================================== In this section, we show the arbitrary fast linear convergence of Algorithm \[alg:Mass-proj-alg\] for the nonconvex SIP when the sets are super-regular. Motivated by the fast convergent algorithm in [@cut_Pang12], Algorithm \[alg:Mass-proj-alg\] collects old halfspaces from previous projections to try to accelerate the convergence in later iterations. We now present an algorithm that can achieve arbitrarily fast linear convergence. \[alg:Mass-proj-alg\] (Local super-regular SHQP) Let $K_{l}$ be (not necessarily convex) closed sets in $\mathbb{R}^{n}$ for $l\in\{1,\dots,m\}$. From a starting point $x_{0}\in\mathbb{R}^{n}$, this algorithm finds a point in the intersection $K:=\cap_{l=1}^{m}K_{l}$. Step 0: Set $i=1$, and let $\bar{p}$ be some positive integer. Step 1: Choose $\bar{j}_{i}\in\arg\max_{j}\{d(x_{i-1},K_{j})\}$. (i.e., we take only an index which give the largest distance.) Step 2: Choose some $\tau_{i}\in[0,1)$. Define $x_{i}^{(\bar{j}_{i})}\in\mathbb{R}^{n}$, $a_{i}^{(\bar{j}_{i})}\in\mathbb{R}^{n}$ and $b_{i}^{(\bar{j}_{i})}\in\mathbb{R}$ by \[eq:alg-x-a-b\] $$\begin{aligned} x_{i}^{(\bar{j}_{i})} & \in & P_{K_{j}}(x_{i-1}),\label{eq:alg-x}\\ a_{i}^{(\bar{j}_{i})} & = & x_{i-1}-x_{i}^{(\bar{j}_{i})},\label{eq:alg-a}\\ \mbox{and }b_{i}^{(\bar{j}_{i})} & = & \langle a_{i}^{(\bar{j}_{i})},x_{i}^{(\bar{j}_{i})}\rangle+\tau_{i}\langle a_{i}^{(\bar{j}_{i})},x_{i-1}-x_{i}^{(\bar{j}_{i})}\rangle\label{eq:alg-b}\\ & = & \langle a_{i}^{(\bar{j}_{i})},(1-\tau_{i})x_{i}^{(\bar{j}_{i})}+\tau_{i}x_{i-1}\rangle.\nonumber \end{aligned}$$ Let $x_{i}=P_{\tilde{F}_{i}}(x_{i-1})$, where the set $\tilde{F}_{i}\subset\mathbb{R}^{n}$ is defined by $$\tilde{F}_{i}:=\big\{ x\mid\langle a_{l}^{(\bar{j}_{l})},x\rangle\leq b_{l}^{(\bar{j}_{l})}\mbox{ for }\max(1,i-\bar{p})\leq l\leq i\big\}.\label{eq:def-F}$$ Step 3: Set $i\leftarrow i+1$, and go back to step 1. There are some differences between Algorithm \[alg:Mass-proj-alg\] and that of [@cut_Pang12 Algorithm 5.1]. Firstly, in step 1, we take only one index $j$ in $\{1,\dots,m\}$ that gives the largest distance $d(x_{i-1},K_{j})$. Secondly, the term $\tau_{i}\langle a_{i}^{(\bar{j}_{i})},x_{i-1}-x_{i}^{(\bar{j}_{i})}\rangle$ is added in to account for the nonconvexity of the set $K_{\bar{j}_{i}}$. The parameter $\tau_{i}$ in Algorithm \[alg:Mass-proj-alg\] requires tuning to achieve fast convergence. This tuning may not be easy to perform. \[lem:conv-alg\](Convergence of Algorithm \[alg:Mass-proj-alg\]) Suppose that in Algorithm \[alg:Mass-proj-alg\], the sets $K_{l}$ are all super-regular at a point $x^{}\in K=\cap_{l=1}^{m}K_{l}$ for all $l\in\{1,\dots,m\}$, and the local metric inequality holds, i.e., there is a $\beta>0$ and a neighborhood $V_{1}$ of $x^{}$ such that $$d(x,\cap_{l=1}^{m}K_{l})\leq\beta\max_{1\leq l\leq m}d(x,K_{l})\mbox{ for all }x\in V_{1}.\label{eq:due-to-beta}$$ Then for any $\tau\in(0,1)$, we can find a neighborhood $U$ of $x^{}$ such that - For any $x_{0}\in U$, Algorithm \[alg:Mass-proj-alg\] with $\tau_{i}=\tau$ for all $i$ generates a sequence $\{x_{i}\}$ that converges to some $\bar{x}\in V_{1}$ so that $$\begin{aligned} & & \\|x_{i+1}-\bar{x}\\|\leq\\|x_{i}-\bar{x}\\|\mbox{ for all }i\geq0,\label{eq:fej-mon}\\ & \mbox{and } & \\|x_{i}-\bar{x}\\|\leq L\max_{l\in\{1,\dots,m\}}d(x_{i},K_{l}),\label{eq:L-bdd}\end{aligned}$$ where $$\rho:=\frac{\sqrt{\beta^{2}-(1-\tau)^{2}}}{\beta}\mbox{ and }L:=\frac{\beta}{1-\rho}.\label{eq:r-and-L}$$ By the super-regularity of the sets $K_{l}$, for any $\delta>0$, there exists a neighborhood $V_{2}$ of $x^{}$ such that for any $l\in\{1,\dots,m\}$, we have $$\langle z-y,v\rangle\leq\delta\\|z-y\\|\\|v\\|\mbox{ for all }z,y\in K_{l}\cap V_{2},v\in N_{K_{l}}(y).\label{eq:angle-small}$$ We choose $\delta\geq0$ to be small enough so that $\delta\leq\frac{\tau(1-\rho)}{2\beta^{2}}$. Claim: If $x_{i-1}$ are such that $\mathbb{B}(x_{i-1},\frac{1}{1-\rho}d(x_{i-1},K))\subset V_{1}\cap V_{2}$, then $K\cap\mathbb{B}(x_{i-1},\frac{1}{1-\rho}d(x_{i-1},K))\subset H_{i}$, where the halfspace $H_{i}:=\{x:\langle a_{i}^{(\bar{j}_{i})},x\rangle\leq b_{i}^{(\bar{j}_{i})}\}$ is defined by . Proof of Claim: Suppose $x^{\prime}\in K\cap\mathbb{B}(x_{i-1},\frac{1}{1-\rho}d(x_{i-1},K))$. Since $K\cap V_{2}$, we have $$\langle x_{i-1}-x_{i}^{(\bar{j}_{i})},x^{\prime}-x_{i}^{(\bar{j}_{i})}\rangle\leq\delta\\|x_{i-1}-x_{i}^{(\bar{j}_{i})}\\|\\|x^{\prime}-x_{i}^{(\bar{j}_{i})}\\|,$$ where $x_{i}^{(\bar{j}_{i})}$ is the point in $P_{K_{j_{i}}}(x_{i-1})\subset K_{j_{i}}$ in . Also, $x^{\prime}$ was assumed to lie in $\mathbb{B}(x_{i-1},\frac{1}{1-\rho}d(x_{i-1},K))$. Note that $\\|x_{i-1}-x_{i}^{(\bar{j}_{i})}\\|\leq d(x_{i-1},K)$. So we have $$\\|x^{\prime}-x_{i}^{(\bar{j}_{i})}\\|\leq\\|x^{\prime}-x_{i-1}\\|+\\|x_{i-1}-x_{i}^{(\bar{j}_{i})}\\|\leq\left(\frac{1}{1-\rho}+1\right)d(x_{i-1},K).$$ Note that $\frac{1}{1-\rho}+1\leq\frac{2}{1-\rho}$. From the above inequality, we have $$\langle x_{i-1}-x_{i}^{(\bar{j}_{i})},x^{\prime}-x_{i}^{(\bar{j}_{i})}\rangle\leq\delta\\|x_{i-1}-x_{i}^{(\bar{j}_{i})}\\|\\|x^{\prime}-x_{i}^{(\bar{j}_{i})}\\|\leq\frac{2\delta}{1-\rho}d(x_{i-1},K)^{2}.$$ Recall that $\delta\leq\frac{\tau(1-\rho)}{2\beta^{2}}$. Local metric inequality gives $\\|x_{i-1}-x_{i}^{(\bar{j}_{i})}\\|\geq\frac{1}{\beta}d(x_{i},K)$, so $$\langle x_{i-1}-x_{i}^{(\bar{j}_{i})},x^{\prime}-x_{i}^{(\bar{j}_{i})}\rangle\leq\frac{\tau}{\beta^{2}}d(x_{i-1},K)^{2}\leq\tau\\|x_{i-1}-x_{i}^{(\bar{j}_{i})}\\|^{2}.$$ The above inequality is precisely $\langle a_{i}^{(\bar{j}_{i})},x^{\prime}\rangle\leq b_{i}^{(\bar{j}_{i})}$, so $x^{\prime}\in H_{i}$. This ends the proof of the claim. Suppose $\mathbb{B}(x_{0},\frac{1}{1-\rho}d(x_{0},K))\subset V_{1}\cap V_{2}$. If the conditions of Lemma \[lem:lin-conv-backbone\] are satisfied, then we have convergence to some $\bar{x}$. We try to prove that $d(x_{i+1},K)\leq\rho d(x_{i},K)$. Recall that $x_{i+1}=P_{\tilde{F}_{i+1}}(x_{i})$. By making use of the claim above, the previous halfspaces generated all contain $K\cap\mathbb{B}(x_{i},\frac{1}{1-\rho}d(x_{i},K))$, so $\tilde{F}_{i+1}$ is a polyhedron that contains $K\cap\mathbb{B}(x_{i},\frac{1}{1-\rho}d(x_{i},K))$. It is clear that $K\cap\mathbb{B}(x_{i},\frac{1}{1-\rho}d(x_{i},K))\neq\emptyset$, so $\tilde{F}_{i+1}$ is nonempty. It is obvious that $d(x_{i},\tilde{F}_{i+1})\leq d(x_{i},K)$, so $\\|x_{i}-x_{i+1}\\|\leq(1-\tau)d(x_{i},K)$. The distance $d(x_{i},H_{i+1})$ is at least $\frac{1}{\beta}d(x_{i},K)$, so $\\|x_{i}-x_{i+1}\\|\geq\frac{1}{\beta}d(x_{i},K)$. We then have $$\begin{aligned} d(x_{i+1},K)^{2} & \leq & d(x_{i},K)^{2}-\\|x_{i}-x_{i+1}\\|^{2}\\ & \leq & d(x_{i},K)^{2}-\frac{(1-\tau)^{2}}{\beta^{2}}d(x_{i},K)^{2}\\ & = & \frac{\sqrt{\beta^{2}-(1-\tau)^{2}}}{\beta}d(x_{i},K)^{2}.\end{aligned}$$ We can now apply Lemma \[lem:lin-conv-backbone\]. The conclusion comes from the fact that $\{x_{i}\}$, by construction, is obtained by projection onto convex sets that contain $\bar{x}$ and the theory of Fejér monotonicity. The conclusion is straightforward from Lemma \[lem:lin-conv-backbone\](a) and local metric inequality. We now prove the theorem on the arbitrary fast multiple-term linear convergence of Algorithm \[alg:Mass-proj-alg\]. \[thm:arb-lin-conv\](Arbitrary fast linear convergence) Consider the setting of Theorem \[lem:conv-alg\]. If $\bar{p}$ in Algorithm \[alg:Mass-proj-alg\] is finite and sufficiently large, then for all $\tau\in(0,0.5)$ (independent of $\bar{p}$) we can find a neighborhood $U$ of $x^{}$ such that if $x_{0}\in U$, then the iterates of Algorithm \[alg:Mass-proj-alg\] with $\tau_{i}\equiv\tau$ converge to some $\bar{x}\in K$. Moreover, $$\limsup_{i\to\infty}\frac{\\|x_{i+\bar{p}}-\bar{x}\\|}{\\|x_{i}-\bar{x}\\|}\leq8\bar{L}\tau,\label{eq:arb-lin-concl}$$ where $\bar{L}=\frac{\beta}{1-\bar{\rho}}$ and $\bar{\rho}=\frac{\sqrt{\beta^{2}-1/4}}{\beta}$. The basic strategy is to prove the inequalities and like in [@cut_Pang12 Theorem 5.12], with a bit more attention put into handling the nonconvexity. By Lemma \[lem:conv-alg\], the convergence of the iterates $\{x_{i}\}$ to some $\bar{x}\in K$ is assured. Without loss of generality, suppose that $\bar{x}=0$. Let $v_{i}^{}:=\frac{x_{i}-x_{i+1}^{(\bar{j}_{i+1})}}{\\|x_{i}-x_{i+1}^{(\bar{j}_{i+1})}\\|}$, where $x_{i+1}^{(\bar{j}_{i+1})}$ is defined through . The sphere $S^{n-1}:=\{w\in\mathbb{R}^{n}:\\|w\\|=1\}$ is compact. Suppose $\bar{p}$ is such that we can cover $S^{n-1}$ with $\bar{p}$ balls of radius $\frac{1}{4\bar{L}}$. By the pigeonhole principle, we can find $j$ and $k$ such that $i\leq j<k\leq i+\bar{p}$ and $v_{j}^{}$ and $v_{k}^{}$ belong to the same ball of radius $\frac{1}{4\bar{L}}$ covering $S^{n-1}$. We thus have $\\|v_{j}^{}-v_{k}^{}\\|\leq\frac{1}{2\bar{L}}$. (The key in choosing $\bar{p}$ is to obtain the last inequality.) We shall prove that if $i$ is large enough, we have the two inequalities $$\begin{aligned} & & \langle v_{j}^{},x_{k}\rangle\leq2\tau\\|x_{j}\\|\label{eq:money-1}\\ & \mbox{ and } & \frac{1}{4\bar{L}}\\|x_{k}\\|\leq\langle v_{j}^{},x_{k}\rangle.\label{eq:money-2}\end{aligned}$$ In view of the Fejér monotonicity condition , these two inequalities give $\\|x_{i+\bar{p}}\\|\leq\\|x_{k}\\|\leq8\bar{L}\tau\\|x_{j}\\|\leq8\bar{L}\tau\\|x_{i}\\|$, which gives the conclusion we seek. We first prove . Since $x_{k}$ lies in $\tilde{F}_{k}$, it lies in the halfspace with normal $v_{j}^{}$ passing through $(1-\tau)x_{j+1}^{(\bar{j}_{j+1})}+\tau x_{j}$. (Recall that $x_{j+1}^{(\bar{j}_{j+1})}$ was defined in , and lies in $P_{K_{\bar{j}_{j+1}}}(x_{j})$.) This gives us $$\begin{aligned} \langle v_{j}^{},x_{k}\rangle & \leq & \langle v_{j}^{},(1-\tau)x_{j+1}^{(\bar{j}_{j+1})}+\tau x_{j}\rangle\label{eq:money-1-a}\\ & = & (1-\tau)\langle v_{j}^{}-\bar{v},x_{j+1}^{(\bar{j}_{j+1})}\rangle+(1-\tau)\langle\bar{v},x_{j+1}^{(\bar{j}_{j+1})}\rangle+\tau\langle v_{j}^{},x_{j}\rangle,\nonumber \end{aligned}$$ where $\bar{v}$ is some vector with norm 1 in $N_{K_{\bar{j}_{i+1}}}(\bar{x})$. Since $\lim_{i\to\infty}x_{i}=\bar{x}$, we can assume that $\{x_{i}\}$ is sufficiently close to $\bar{x}$ so that: 1. the vector $\bar{v}$, by the outer semicontinuity of the normal cone mapping $x\mapsto N_{K_{\bar{j}_{j+1}}}(x)$, can be chosen to be such that $\\|v_{j}^{}-\bar{v}\\|\leq\frac{\tau}{3}$, and 2. by the super-regularity of $K_{\bar{j}_{j+1}}$ at $\bar{x}$, we have $\langle\bar{v},x_{j+1}^{(\bar{j}_{i+1})}\rangle\leq\frac{\tau}{3}\\|x_{j+1}^{(\bar{j}_{i+1})}\\|$. Note that $(1-\tau)x_{j+1}^{(\bar{j}_{i+1})}+\tau x_{j}$ is the projection of $x_{j}$ onto one of the halfspaces defining $\tilde{F}_{j+1}$ and that $\tau<\frac{1}{2}$. From the principle in Proposition \[prop:fejer-principle\], we have $\\|x_{j+1}^{(\bar{j}_{i+1})}\\|\leq\\|x_{j}\\|$. Since $\\|v_{j}^{}\\|=1$, we have $\langle v_{j}^{},x_{j}\rangle\leq\\|x_{j}\\|$. Continuing the arithmetic in , we have $$\begin{aligned} \langle v_{j}^{},x_{k}\rangle & \leq & (1-\tau)\langle v_{j}^{}-\bar{v},x_{j+1}^{(\bar{j}_{j+1})}\rangle+(1-\tau)\langle\bar{v},x_{j+1}^{(\bar{j}_{j+1})}\rangle+\tau\langle v_{j}^{},x_{j}\rangle\\ & \leq & (1-\tau)\Big(\frac{\tau}{3}+\frac{\tau}{3}\Big)\\|x_{j+1}^{(\bar{j}_{j+1})}\\|+\tau\\|x_{j}\\|\\ & \leq & \frac{2\tau}{3}\\|x_{j}\\|+\tau\\|x_{j}\\|<2\tau\\|x_{j}\\|.\end{aligned}$$ This ends the proof of . Next, we prove . Recall that $x_{k+1}^{(\bar{j}_{k+1})}\in P_{K_{\bar{j}_{k+1}}}(x_{k})$ was defined in . Note that provided $\tau<\frac{1}{2}$, the $\rho=\frac{\sqrt{\beta^{2}-(1-\tau)^{2}}}{\beta}$ in is less than $\bar{\rho}=\frac{\sqrt{\beta^{2}-1/4}}{\beta}$. Hence the $L=\frac{\beta}{1-\rho}$ in is less than $\bar{L}=\frac{\beta}{1-\bar{\rho}}$. By using the definition of $v_{k}^{}=\frac{x_{k}-x_{k+1}^{(\bar{j}_{k+1})}}{\\|x_{k}-x_{k+1}^{(\bar{j}_{k+1})}\\|}$ and , we have $$\langle v_{k}^{},x_{k}-x_{k+1}^{(\bar{j}_{k+1})}\rangle=\\|x_{k}-x_{k+1}^{(\bar{j}_{k+1})}\\|=d(x_{k},K_{\bar{j}_{k+1}})\geq\frac{1}{L}\\|x_{k}\\|\geq\frac{1}{\bar{L}}\\|x_{k}\\|.\label{eq:money-2a}$$ By the super-regularity of $K_{\bar{j}_{k+1}}$ at $\bar{x}$ and the fact that $\bar{x}=\lim_{i\to\infty}x_{i}$, we can assume that $x_{k}$ is close enough to $\bar{x}$ so that $$\langle v_{k}^{},0-x_{k+1}^{(\bar{j}_{k+1})}\rangle\leq\frac{1}{4\bar{L}}\\|x_{k+1}^{(\bar{j}_{k+1})}\\|\leq\frac{1}{4\bar{L}}\\|x_{k}\\|.\label{eq:money-2b}$$ (Note that the inequality on the right follows from the same proof of $\\|x_{j+1}^{(\bar{j}_{j+1})}\\|\leq\\|x_{j}\\|$.) Combining and as well as $\\|v_{j}^{}-v_{k}^{}\\|\leq\frac{1}{2\bar{L}}$ gives us $$\begin{aligned} \langle v_{j}^{},x_{k}\rangle & = & \langle v_{k}^{},x_{k}-x_{k+1}^{(\bar{j}_{k+1})}\rangle+\langle v_{k}^{},x_{k+1}^{(\bar{j}_{k+1})}\rangle+\langle v_{j}^{}-v_{k}^{},x_{k}\rangle\\ & \geq & \frac{1}{\bar{L}}\\|x_{k}\\|-\frac{1}{4\bar{L}}\\|x_{k}\\|-\frac{1}{2\bar{L}}\\|x_{k}\\|=\frac{1}{4\bar{L}}\\|x_{k}\\|.\end{aligned}$$ This ends the proof of , which concludes the proof of our result. The large parameter $\bar{p}$ is an upper bound on when we can find $v_{j}^{}$ and $v_{k}^{}$ such that $\\|v_{j}^{}-v_{k}^{}\\|\leq\frac{1}{2\bar{L}}$. We hope that the upper bound needed in a practical implementation would be much smaller than $\bar{p}$. (Towards superlinear convergence) The coefficient of $8$ in can be reduced, but this does not detract us from the point that as $\tau\searrow0$, the right hand side of goes to zero. So there is a choice of parameters $\{\tau_{i}\}_{i=1}^{\infty}$ that can be chosen at each iteration of Algorithm \[alg:Mass-proj-alg\] so that superlinear convergence is achieved, even though there doesn’t seem to be a good way of choosing how the parameters $\tau$ go to zero. If the parameter $\tau$ goes to zero too fast, the Fejér monotonicity of the iterates may not be maintained, which may mean that Lemma \[lem:conv-alg\] may not hold, i.e., the iterates $\{x_{i}\}_{i=1}^{\infty}$ may not converge. Contrast this to the convex SIP in [@cut_Pang12], where setting $\tau\equiv0$ gives multiple-term superlinear convergence* $$\lim_{i\to\infty}\frac{\\|x_{i+\bar{p}}-\bar{x}\\|}{\\|x_{i}-\bar{x}\\|}=0$$ instead of multiple-term arbitrary linear convergence . In view of nonconvexity, the observation in Remark \[rem:sometimes-empty-intersect\] has to be overcome, so we believe that this arbitrary fast convergence is difficult to improve on in general. (Simplification in ) The inequality $d(x_{k},K_{\bar{j}_{k+1}})\geq\frac{1}{L}\\|x_{k}\\|$ in follows easily from . But in [@cut_Pang12], some effort was spent to prove the inequality $\limsup_{k\to\infty}\frac{1}{\\|x_{k}\\|}d(x_{k},K_{\bar{j}_{k+1}})\geq\frac{1}{\beta}$. The proof of the multiple-term superlinear convergent algorithm for convex problems in [@cut_Pang12] can thus be shortened considerably. If some of the sets $K_{l}$ are known to be convex sets or affine subspaces, then this information can be taken into account by setting the appropriate $\tau_{i}$ to zero when creating the halfspaces defined by . Two step SHQP ============= The algorithms in this paper need not guarantee that $\{d(x_{i},K)\}$ is nonincreasing. In this section, we give an example of additional conditions needed for the SHQP to have this property. Consider the following algorithm. \[alg:2-SHQP\](2-SHQP) Let $K_{1}$, $K_{2}$ be two closed sets in $\mathbb{R}^{n}$, and $K=K_{1}\cap K_{2}$. This algorithm tries to find a point $x\in K$ using a starting iterate $x_{0}$. 01 Set $i=0$ 02 Loop 03 $\quad$Set $x_{i+1}$ to be an element in $P_{K_{1}}(x_{i})$ and $i\leftarrow i+1$. 04 $\quad$Set $x_{i+1}$ to be an element in $P_{K_{2}}(x_{i})$ and $i\leftarrow i+1$. 05 $\quad$If $\angle x_{i-2}x_{i-1}x_{i}<\pi/2$, then 06 $\quad\quad$set $x_{i+1}=P_{\{x:\langle x-x_{i-1},x_{i-2}-x_{i-1}\rangle\leq0,\langle x-x_{i},x_{i-1}-x_{i}\rangle\leq0\}}(x_{i})$. 07 $\quad$else 08 $\quad$$\quad$set $x_{i+1}=x_{i}$ 09 $\quad$end if 10 $\quad$$i\leftarrow i+1$ 11 end loop In line 6, $x_{i+1}$ is the projection of $x_{i}$ onto the polyhedron formed by intersecting the last two halfspaces generated by the projection process. See Figure \[fig:no-t-fig\] for an illustration of the first few iterates $x_{1}$, $x_{2}$ and $x_{3}$ formed by a single iteration of the loop. If the “if” block in lines 5 to 9 is removed, then the algorithm reduces to an alternating projection algorithm. We now analyze the effectiveness of this “if” block. \[prop:2-SHQP\](2-SHQP) Consider Algorithm \[alg:2-SHQP\]. Let $\delta\in(0,1)$. Let $x^{}\in K$ and let a neighborhood $V$ of $x^{}$ be such that 1. $\langle v,y-z\rangle\leq\delta\\|v\\|\\|y-z\\|\mbox{ for all }y,z\in K_{l}\cap V\mbox{, }l\in\{1,2\}\mbox{ and }v\in N_{K_{l}}(z),$and 2. $d(x,K)\leq\beta\max_{l\in\{1,2\}}d(x,K_{l})$ for all $x\in V$. Let $x_{1}$, $x_{2}$ be successive iterates of Algorithm \[alg:2-SHQP\]. Suppose $\mathbb{B}(x_{1},(\beta+1)\\|x_{1}-x_{2}\\|)\subset V$. Let $\theta:=\angle x_{2}x_{1}x_{0}<\pi/2$. If $$\delta[\beta\cos\theta+(\beta+1)]<\frac{1}{2}\cos\theta,\label{eq:no-t-concl}$$ then $d(x_{3},K)<d(x_{2},K)$. Conditions (1) and (2) are consequences of the super-regularity condition and local metric inequality condition respectively, so they will be satisfied when close to $K$. Since property (2) and the fact that $x_{1}\in K_{1}$ implies that $$d(x_{2},K)\leq\beta\max_{l\in\{1,2\}}d(x_{2},K_{l})=\beta d(x_{2},K_{1})\leq\beta\\|x_{1}-x_{2}\\|,$$ the set $\mathbb{B}(x_{2},\beta\\|x_{2}-x_{1}\\|)\cap K$ is not empty. Hence $$\emptyset\neq\mathbb{B}(x_{2},\beta\\|x_{2}-x_{1}\\|)\subset\mathbb{B}(x_{1},(\beta+1)\\|x_{2}-x_{1}\\|)\subset V.\label{eq:ball-chain}$$ Let $y$ be any point in $\mathbb{B}(x_{2},\beta\\|x_{2}-x_{1}\\|)\cap K$. By property (1), we have $$\langle y-x_{2},x_{1}-x_{2}\rangle\leq\delta\\|y-x_{2}\\|\\|x_{1}-x_{2}\\|\leq\beta\delta\\|x_{1}-x_{2}\\|^{2}.\label{eq:first-ball-nonempty}$$ In other words, $\mathbb{B}(x_{2},\beta\\|x_{2}-x_{1}\\|)\cap K\subset H_{2}$, where $H_{2}$ is the halfspace defined by $$H_{2}:=\{x:\langle x-x_{2},x_{1}-x_{2}\rangle\leq\beta\delta\\|x_{1}-x_{2}\\|^{2}\}.$$ Next, from , we can make use of the argument similar to to prove that $$\mathbb{B}(x_{1},(\beta+1)\\|x_{2}-x_{1}\\|)\cap K\subset H_{1},$$ where $H_{1}$ is the halfspace defined by $$H_{1}:=\{x:\langle x-x_{1},x_{0}-x_{1}\rangle\leq(\beta+1)\delta\\|x_{0}-x_{1}\\|^{2}\}.$$ This implies that $$\emptyset\neq\mathbb{B}(x_{2},\beta\\|x_{2}-x_{1}\\|)\cap K\subset H_{1}\cap H_{2}.\label{eq:poly-contain}$$ ![\[fig:no-t-fig\]This figure illustrates the proof of Proposition \[prop:2-SHQP\]. The dotted lines show the boundaries of $H_{1}$ and $H_{2}$. ](no_t) We refer to Figure \[fig:no-t-fig\], which shows the two dimensional cross section containing $x_{0}$, $x_{1}$ and $x_{2}$. The point $x_{3}$ is also shown in the figure, and is the projection of $x_{2}$ onto $H_{1}\cap H_{2}$. We now calculate the minimal value of $\langle\frac{x_{3}-x_{2}}{\\|x_{3}-x_{2}\\|},x\rangle$, where $x$ ranges over $H_{1}\cap H_{2}$. This minimal value can be seen to be $d_{1}-d_{2}-d_{3}$, where $d_{1}$, $d_{2}$ and $d_{3}$ are the distances as indicated in Figure \[fig:no-t-fig\]. These distances can be calculated to be $$\begin{aligned} d_{1} & = & \\|x_{2}-x_{3}\\|=\\|x_{1}-x_{2}\\|\cot\theta,\\ d_{2} & = & \beta\delta\\|x_{1}-x_{2}\\|\cot\theta,\\ \mbox{ and }d_{3} & = & (\beta+1)\delta\\|x_{1}-x_{2}\\|/\sin\theta.\end{aligned}$$ We can check that is equivalent to $d_{2}+d_{3}<\frac{1}{2}d_{1}$. As long as holds, the region $H_{1}\cap H_{2}$ lies on the same side as $x_{3}$ of the perpendicular bisector of the points $x_{2}$ and $x_{3}$. Hence all the points in $H_{1}\cap H_{2}$ are closer to $x_{3}$ than to $x_{2}$. Since $H_{1}\cap H_{2}$ contains all the points in $P_{K}(x_{2})$ by , we thus have $d(x_{3},K)<d(x_{2},K)$ as needed. Note that if $\theta<\pi/2$ is too close to $\pi/2$, then the condition can fail. In fact, if $\theta>\cos^{-1}\delta$, one can check that condition (1) in Proposition \[prop:2-SHQP\] does not rule out $x_{2}$ being inside $K_{1}$, so there would be no point calculating $x_{3}$. The supporting halfspaces as calculated by the projection process can be too aggressive for super-regular sets. For example, one can draw a manifold in $\mathbb{R}^{2}$ such that the intersection the manifold and a halfspace generated by the projection process consists of only one point. Two halfspaces of this kind would give an empty intersection with the manifold. Therefore, one has to relax the halfspaces. We remark that the procedure in shows how to construct halfspaces under the super-regularity condition, and can be augmented into Algorithm \[alg:Mass-proj-alg\] as long as we have a good estimate for $\delta$. \[sec:global-strat\]Global strategies ===================================== In this section, we discuss methods for when local methods of the nonconvex SIP are not appropriate. In Example \[exa:backtrack\], we show that while the theory for the convex SIP suggests that one should not backtrack, backtracking is however suggested for the nonconvex problem, which can lead to the Maratos effect and slows down convergence. The problem of finding a point in the intersection of a finite number of closed sets $K_{l}\subset\mathbb{R}^{n}$, where $l=1,\dots,m$, can be equivalently cast as the problem of finding a point that minimizes $f(x)$, where $f(x)$ can be chosen as \[eq:all-d-forms\] $$\begin{aligned} & & d(x,\cap_{l=1}^{m}K_{l}),\label{eq:d-int-form}\\ & & \sum_{l=1}^{m}d(x,K_{l})^{2},\label{eq:d-2-norm-form}\\ & & \max_{l\in\{1,\dots,m\}}\{d(x,K_{l})\},\label{eq:d-max-form}\end{aligned}$$ or some other function similar to those presented above. In the event that the intersection $\cap_{l=1}^{m}K_{l}$ is nonempty, then any point in $K:=\cap_{l=1}^{m}K_{l}$ would be a global minimizer of $f(\cdot)$. The function in is the function of choice, but $\cap_{l=1}^{m}K_{l}$ can be only be estimated well locally with the techniques in Section \[sec:local-strat\]. Instead of trying to minimize $f(\cdot)$, the problem that really needs to be solved is the one of finding an $x$ in $\{\tilde{x}:f(\tilde{x})\leq0\}$. This is a simpler problem which can be solved by a subgradient projection method that is somewhat simpler than the minimization problem. A bundle method [@HiriartUrrutyLamerechal93a; @BonnansGilbertLemarechalSagastizabal06] adapted for a nonconvex objective function can be used to solve the nonconvex SIP. (See also [@BauschkeWangWangXu14; @Pang_nonconvex_ineq] for the principles of a finitely convergent algorithm for this setting. This idea of finite convergence goes back to [@Polak_Mayne79; @Mayne_Polak_Heunis81; @Fukushima82; @DePierroIusem88] for the convex case and the smooth case.) A standard procedure in optimization algorithms is the line search procedure. A search direction is calculated, and the next solution is obtained by a line search along this search direction. For the nonconvex SIP, the search direction can be calculated by projecting onto a polyhedron formed by intersecting a number of previously generated halfspaces. There are two ways we can backtrack to obtain decrease in some objective function (in or otherwise). Firstly, one can remove halfspaces that describe the polyhedron. It is sensible to remove the older halfspaces since they become less reliable. This has the effect of reducing the distance from the current iterate to the polyhedron, so the search direction is more likely to give decrease. The problem of projecting onto the polyhedron with one halfspace removed can be solved effectively from the old solution using a warmstart quadratic programming algorithm (for example, the active set method of [@Goldfarb_primal]). Secondly, one can use the usual backtracking line search. We note however that in the pursuit of obtaining decrease in the objective function, we may encounter the Maratos effect (see [@NW06 Section 15.5], who in turn cited [@Maratos]) which slows convergence. \[exa:backtrack\](Backtracking slows convergence) In this example, we show how the SHQP strategy for a convex SIP converges quickly for a problem, but would be slowed down by backtracking when treated as a nonconvex SIP. Consider the sets $K_{1},K_{2}\subset\mathbb{R}^{3}$ where $K_{1}=H_{1}$ and $K_{2}=H_{2}\cap H_{3}$, where the halfspaces $H_{1}$, $H_{2}$ and $H_{3}$ are defined by $$\begin{aligned} H_{1} & := & \{x\in\mathbb{R}^{3}:(0,1,0)x\leq0\},\\ H_{2} & := & \{x\in\mathbb{R}^{3}:(\nicefrac{1}{3},-1,0)x\leq-2\},\\ \mbox{and }H_{3} & := & \{x\in\mathbb{R}^{3}:(-1,-1,1)x\leq0\}.\end{aligned}$$ Let the point $x_{0}$ be $(0,1,0)$. The projection of $x_{0}$ onto $K_{1}$ and $K_{2}$ generates the halfspaces $H_{1}$ and $H_{2}$ respectively. The projection of $x_{0}$ onto $H_{1}\cap H_{2}$ is $x_{1}:=(-6,0,0)$. We can calculate that $$d(x_{0},K_{1})=1,\mbox{ }d(x_{0},K_{2})=\nicefrac{3}{\sqrt{10}},\mbox{ }d(x_{1},K_{1})=0\mbox{ and }d(x_{1},K_{2})=2\sqrt{3}.\label{eq:exa-vals}$$ The projection of $x_{1}$ onto $K_{2}$ generates $H_{3}$, and once we project $x_{1}$ onto $H_{1}\cap H_{2}\cap H_{3}$, we found a point in $K_{1}\cap K_{2}$. If this SIP were solved as a nonconvex SIP, the values in fitted into the objective function or suggests that one has to backtrack in some manner, and this actually slows down the convergence. (See Figure \[fig:exa-backtr\] for an illustration.) ![\[fig:exa-backtr\]This figure illustrates the two dimensional cross section in $\{x\in\mathbb{R}^{3}:x_{3}=0\}$ in the example in Example \[exa:backtrack\]. Note that the projection of $x_{1}$ onto $K_{2}$ lies outside this cross section.](backtr) We recall the method of averaged projections for finding a point in $\cap_{l=1}^{m}K_{l}$, where $K_{l}\subset\mathbb{R}^{n}$ for all $l\in\{1,\dots,m\}$, is defined by $$x_{i+1}=\frac{1}{m}\sum_{l=1}^{m}P_{K_{l}}(x_{i}).\label{eq:avg-proj}$$ It was noticed that this formula corresponds to the method of alternating projections between the two sets in $\mathbb{R}^{nm}$ defined by $$\begin{aligned} \mathbf{D} & := & \{(x,x,\dots,x):x\in\mathbb{R}^{n}\}\\ \mbox{and }\mathbf{K} & := & K_{1}\times K_{2}\times\cdots\times K_{m}.\end{aligned}$$ It is easy to see that $f(x_{i+1})\leq f(x_{i})$ if $x_{i+1}$ is defined by and $f(\cdot)$ is defined by since $\sqrt{f(x)}$ is the distance of $(x,\cdots,x)\in\mathbf{D}$ to $\mathbf{K}$. Moreover, if $f(x_{i+1})=f(x_{i})$, then $x_{i}$ is the minimizer. In the SHQP strategy for nonconvex problems, we can use backtracking to find the next iterate $x_{i}$ of the form $tP_{\tilde{F}_{i}}(x_{i-1})+(1-t)x_{i-1}$, where $t\in(0,1]$ and $\tilde{F}_{i}$ is the polyhedron defined by intersecting previously generated halfspaces like in Algorithm \[alg:Mass-proj-alg\]. We can instead find an iterate of the form $$tP_{\tilde{F}_{i}}(x_{i-1})+(1-t)\frac{1}{m}\sum_{l=1}^{m}P_{K_{l}}(x_{i-1}).$$ Other heuristics for the nonconvex problem are also possible. For example, if one is certain that the intersection is nonempty, then one can try to avoid points in the balls $\mathbb{B}(x_{i},d(x_{i},K_{l}))$ for all $i\geq0$ and $l\in\{1,\dots,m\}$. If some of the sets are spectral sets (i.e., the set of symmetric matrices solely described by their eigenvalues), then the results in [@LewisMalick08] can also be applied. Conclusion ========== We hope our results make the case that in solving feasibility problems involving super-regular sets, one should use the SHQP procedure as much as possible to accelerate convergence once close enough to the intersection. The size of the QPs to be solved can be kept to be of a manageable size if we combine with projection methods like in Algorithm \[alg:basic-alg\].
--- author: - \| Seth Gilbert\ National University of Singapore\ [seth.gilbert@comp.nus.edu.sg]{} - \| James Maguire\ Georgetown University\ [jrm346@georgetown.edu]{} - \| Calvin Newport\ Georgetown University\ [cnewport@cs.georgetown.edu]{} bibliography: - 'bio.bib' - 'bib.bib' title: \| On Bioelectric Algorithms:\ A Novel Application of Theoretical Computer Science to\ Core Problems in Developmental Biology --- Acknowledgments =============== This work was support in part by NSF award \#1649484.

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