We develop a distributional framework for the shearlet transform $mathcal{S}_{psi}colonmathcal{S}_0(mathbb{R}^2) omathcal{S}(mathbb{S})$ and the shearlet synthesis operator $mathcal{S}^t_{psi}colonmathcal{S}(mathbb{S}) omathcal{S}_0(mathbb{R}^2)$, where $mathcal{S}_0(mathbb{R}^2)$ is the Lizorkin test function space and $mathcal{S}(mathbb{S})$ is the space of highly localized test functions on the standard shearlet group $mathbb{S}$. These spaces and their duals $mathcal{S}_0^prime (mathbb R^2),, mathcal{S}^prime (mathbb{S})$ are called Lizorkin type spaces of test functions and distributions. We analyze the continuity properties of these transforms when the admissible vector $psi$ belongs to $mathcal{S}_0(mathbb{R}^2)$. Then, we define the shearlet transform and the shearlet synthesis operator of Lizorkin type distributions as transpose mappings of the shearlet synthesis operator and the shearlet transform, respectively. They yield continuous mappings from $mathcal{S}_0^prime (mathbb R^2)$ to $mathcal{S}^prime (mathbb{S})$ and from $mathcal{S}^prime (mathbb S)$ to $mathcal{S}_0^prime (mathbb{R}^2)$. Furthermore, we show the consistency of our definition with the shearlet transform defined by direct evaluation of a distribution on the shearlets. The same can be done for the shearlet synthesis operator. Finally, we give a reconstruction formula for Lizorkin type distributions, from which follows that the action of such generalized functions can be written as an absolutely convergent integral over the standard shearlet group.

We show that any toric K"ahler cone with smooth compact cross-section admits a family of Calabi-Yau cone metrics with conical singularities along its toric divisors. The family is parametrized by the Reeb cone and the angles are given explicitly in terms of the Reeb vector field. The result is optimal, in the sense that any toric Calabi-Yau cone metric with conical singularities along the toric divisor (and smooth elsewhere) belongs to this family. We also provide examples and interpret our results in terms of Sasaki-Einstein metrics.

A reaction-diffusion model was developed describing the spread of the COVID-19 virus considering the mean daily movement of susceptible, exposed and asymptomatic individuals. The model was calibrated using data on the confirmed infection and death from France as well as their initial spatial distribution. First, the system of partial differential equations is studied, then the basic reproduction number, R0 is derived. Second, numerical simulations, based on a combination of level-set and finite differences, shown the spatial spread of COVID-19 from March 16 to June 16. Finally, scenarios of unlockdown are compared according to variation of distancing, or partially spatial lockdown.

We consider the most general class of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of an arbitrary order whose solutions and right-hand sides belong to appropriate Sobolev spaces. For parameter-dependent problems from this class, we prove a constructive criterion for their solutions to be continuous in the Sobolev space with respect to the parameter. We also prove a two-sided estimate for the degree of convergence of these solutions to the solution of the nonperturbed problem.

In this article, we prove that if $R$ is a finitely generated ring over $mathbb{Z}$ of dimension $d, dgeq2, frac{1}{d!}in R$, then any unimodular row over $R[X]$ of length $d+1$ can be mapped to a factorial row by elementary transformations.

We offer an alternative proof of a result of Conlon, Fox, Sudakov and Zhao on solving translation-invariant linear equations in dense Sidon sets. Our proof generalises to equations in more than five variables and yields effective bounds.

In cite{S 2017}, Suh gave a non-existence theorem for Hopf real hypersurfaces in the complex quadric with parallel normal Jacobi operator. Motivated by this result, in this paper, we introduce some generalized conditions named $mathcal C$-parallel or Reeb parallel normal Jacobi operators. By using such weaker parallelisms of normal Jacobi operator, first we can assert a non-existence theorem of Hopf real hypersurfaces with $mathcal C$-parallel normal Jacobi operator in the complex quadric $Q^{m}$, $m geq 3$. Next, we prove that a Hopf real hypersurface has Reeb parallel normal Jacobi operator if and only if it has an $mathfrak A$-isotropic singular normal vector field.

We define local indices for projective umbilics and godrons (also called cusps of Gauss) on generic smooth surfaces in projective 3-space. By means of these indices, we provide formulas that relate the algebraic numbers of those characteristic points on a surface (and on domains of the surface) with the Euler characteristic of that surface (resp. of those domains). These relations determine the possible coexistences of projective umbilics and godrons on the surface. Our study is based on a "fundamental cubic form" for which we provide a closed simple expression.

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices. A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is $(k-1)$-colorable.

In this paper, we initiate a systematic study of the finiteness of $k$-vertex-critical graphs in subclasses of $P_5$-free graphs. Our main result is a complete classification of the finiteness of $k$-vertex-critical graphs in the class of $(P_5,H)$-free graphs for all graphs $H$ on 4 vertices. To obtain the complete dichotomy, we prove the finiteness for four new graphs $H$ using various techniques -- such as Ramsey-type arguments and the dual of Dilworth's Theorem -- that may be of independent interest.

We consider two special cases of the connection problem for the second Painlev'e equation (PII) using the method of uniform asymptotics proposed by Bassom et al.. We give a classification of the real solutions of PII on the negative (positive) real axis with respect to their initial data. By product, a rigorous proof of a property associate with the nonlinear eigenvalue problem of PII on the real axis, recently revealed by Bender and Komijani, is given by deriving the asymptotic behavior of the Stokes multipliers.

Given spherically symmetric characteristic initial data for the Einstein-scalar field system with a positive cosmological constant, we provide a criterion, in terms of the dimensionless size and dimensionless renormalized mass content of an annular region of the data, for the formation of a future trapped surface. This corresponds to an extension of Christodoulou's classical criterion by the inclusion of the cosmological term.

Classical studies on aspiration-based dynamics suggest that a dissatisfied individual changes strategy without taking into account the success of others. This promotes defection spreading. The imitation-based dynamics allow individuals to imitate successful strategies without taking into account their own-satisfactions. In this article, we propose to study a dynamic based on aspiration which takes into account imitation of successful strategies for dissatisfied individuals. This helps cooperative members to resist. Individuals compare their success to their desired satisfaction level before making a decision to update their strategies. This mechanism helps individuals with a minimum of self-satisfaction to maintain their strategies. If an individual is dissatisfied, it will learn from others by choosing successful strategies. We derive an exact expression of the fixation probability in well-mixed populations as in structured populations in networks. As a result, we show that selection may favor cooperation more than defection in well-mixed populations as in populations ranged over a regular graph. We show that the best scenario is a graph with small connectivity.

We study a 1-cocycle on the fatgraph complex of a punctured surface introduced by Penner. We present an explicit cobounding cochain for this cocycle, whose formula involves a summation over trivalent vertices of a trivalent fatgraph spine. In a similar fashion, we express the symplectic form of the underlying surface of a given fatgraph spine.

In this paper, we derive a formula for the sums of powers of the first $n$ positive integers that involves the hyperharmonic numbers and the Stirling numbers of the second kind. Then, using an explicit representation for the hyperharmonic numbers, we generalize this formula to the sums of powers of an arbitrary arithmetic progression. Moreover, as a by-product, we express the Bernoulli polynomials in terms of the hyperharmonic polynomials and the Stirling numbers of the second kind.

In this paper we study removable singularities for regular $(1,1/2)$-Lipschitz solutions of the heat equation in time varying domains. We introduce an associated Lipschitz caloric capacity and we study its metric and geometric properties and the connection with the $L^2$ boundedness of the singular integral whose kernel is given by the gradient of the fundamental solution of the heat equation.

The Baumslag Solitar group $BS(2,3)$, is a so-called non-Hopfian group, meaning that it has an epimorphism $phi$ onto itself, that is not injective. In particular this is equivalent to saying that $BS(2,3)$ has a quotient that is isomorphic to itself. As a consequence the Cayley graph of $BS(2,3)$ has a quotient that is isomorphic to itself up to change of generators. We describe this quotient on the graph-level and take a closer look at the most common epimorphism $phi$. We show its kernel is a free group of infinite rank with an explicit set of generators.

We construct semiglobal singular spacetimes for the Einstein equations coupled to a massless scalar field. Consistent with the heuristic analysis of Belinskii, Khalatnikov, Lifshitz or BKL for this system, there are no oscillations due to the scalar field. (This is much simpler than the oscillatory BKL heuristics for the Einstein vacuum equations.) Prior results are due to Andersson and Rendall in the real analytic case, and Rodnianski and Speck in the smooth near-spatially-flat-FLRW case. Similar to Andersson and Rendall we give asymptotic data at the singularity, which we refer to as final data, but our construction is not limited to real analytic solutions. This paper is a test application of tools (a graded Lie algebra formulation of the Einstein equations and a filtration) intended for the more subtle vacuum case. We use homological algebra tools to construct a formal series solution, then symmetric hyperbolic energy estimates to construct a true solution well-approximated by truncations of the formal one. We conjecture that the image of the map from final data to initial data is an open set of anisotropic initial data.

A natural problem in the context of the coupon collector's problem is the behavior of the maximum of independent geometrically distributed random variables (with distinct parameters). This question has been addressed by Brennan et al. (British J. of Math. & CS. 8 (2015), 330-336). Here we provide explicit asymptotic expressions for the moments of that maximum, as well as of the maximum of exponential random variables with corresponding parameters. We also deal with the probability of each of the variables being the maximal one.

The calculations lead to expressions involving Hurwitz's zeta function at certain special points. We find here explicitly the values of the function at these points. Also, the distribution function of the maximum we deal with is closely related to the generating function of the partition function. Thus, our results (and proofs) rely on classical results pertaining to the partition function.

We show that every 4-uniform hypergraph with $n$ vertices and minimum pair degree at least $(5/9+o(1))n^2/2$ contains a tight Hamiltonian cycle. This degree condition is asymptotically optimal.

This is the second of two papers in which we construct formal power series solutions in external parameters to the vacuum Einstein equations, implementing one bounce for the Belinskii-Khalatnikov-Lifshitz (BKL) proposal for spatially inhomogeneous spacetimes. Here we show that spatially inhomogeneous perturbations of spatially homogeneous elements are unobstructed. A spectral sequence for a filtered complex, and a homological contraction based on gauge-fixing, are used to do this.

An element in a ring $R$ is called clear if it is the sum of unit-regular element and unit. An associative ring is clear if every its element is clear. In this paper we defined clear rings and extended many results to wider class. Finally, we proved that a commutative B'ezout domain is an elementary divisor ring if and only if every full matrix order 2 over it is nontrivial clear.

We study a problem concerning parabolic induction in certain p-adic unitary groups. More precisely, for $E/F$ a quadratic extension of p-adic fields the associated unitary group $G=mathrm{U}(n,n)$ contains a parabolic subgroup $P$ with Levi component $L$ isomorphic to $mathrm{GL}_n(E)$. Let $pi$ be an irreducible supercuspidal representation of $L$ of depth zero. We use Hecke algebra methods to determine when the parabolically induced representation $iota_P^G pi$ is reducible.

We construct a smooth moduli stack of tuples consisting of genus two nodal curves, line bundles, and twisted fields. It leads to a desingularization of the moduli of genus two stable maps to projective spaces. The construction of this new moduli is based on systematical application of the theory of stacks with twisted fields (STF), which has its prototype appeared in arXiv:1906.10527 and arXiv:1201.2427 and is fully developed in this article. The results of this article are the second step of a series of works toward the resolutions of the moduli of stable maps of higher genera.

In this paper, we study the regularity criterion of weak solutions to the three-dimensional (3D) MHD equations. It is proved that the solution $(u,b)$ becomes regular provided that one velocity and one current density component of the solution satisfy% egin{equation} u_{3}in L^{frac{30alpha }{7alpha -45}}left( 0,T;L^{alpha ,infty }left( mathbb{R}^{3} ight) ight) ext{ with }frac{45}{7}% leq alpha leq infty , label{eq01} end{equation}% and egin{equation} j_{3}in L^{frac{2eta }{2eta -3}}left( 0,T;L^{eta ,infty }left( mathbb{R}^{3} ight) ight) ext{ with }frac{3}{2}leq eta leq infty , label{eq02} end{equation}% which generalize some known results.

We construct a 2-equivalence $mathfrak{CohTheory}^ ext{op} simeq mathfrak{TypeSpaceFunc}$. Here $mathfrak{CohTheory}$ is the 2-category of positive theories and $mathfrak{TypeSpaceFunc}$ is the 2-category of type space functors. We give a precise definition of interpretations for positive logic, which will be the 1-cells in $mathfrak{CohTheory}$. The 2-cells are definable homomorphisms. The 2-equivalence restricts to a duality of categories, making precise the philosophy that a theory is `the same' as the collection of its type spaces (i.e. its type space functor).

In characterising those functors that arise as type space functors, we find that they are specific instances of (coherent) hyperdoctrines. This connects two different schools of thought on the logical structure of a theory.

The key ingredient, the Deligne completeness theorem, arises from topos theory, where positive theories have been studied under the name of coherent theories.

A perfect matroid design (PMD) is a matroid whose flats of the same rank all have the same size. In this paper we introduce the q-analogue of a PMD and its properties. In order to do that, we first establish new cryptomorphic definitions for q-matroids. We show that q-Steiner systems are examples of q-PMD's and we use this matroid structure to construct subspace designs from q-Steiner systems. We apply this construction to S(2, 13, 3; q) Steiner systems and hence establish the existence of subspace designs with previously unknown parameters.

The main goal of this paper is to achieve a parametrization of the solution set of the truncated matricial Hausdorff moment problem in the non-degenerate and degenerate situation. We treat the even and the odd cases simultaneously. Our approach is based on Schur analysis methods. More precisely, we use two interrelated versions of Schur-type algorithms, namely an algebraic one and a function-theoretic one. The algebraic version, worked out in our former paper arXiv:1908.05115, is an algorithm which is applied to finite or infinite sequences of complex matrices. The construction and discussion of the function-theoretic version is a central theme of this paper. This leads us to a complete description via Stieltjes transform of the solution set of the moment problem under consideration. Furthermore, we discuss special solutions in detail.

This paper develops a method for obtaining guaranteed outer approximations for global attractors of continuous and discrete time nonlinear dynamical systems. The method is based on a hierarchy of semidefinite programming problems of increasing size with guaranteed convergence to the global attractor. The approach taken follows an established line of reasoning, where we first characterize the global attractor via an infinite dimensional linear programming problem (LP) in the space of Borel measures. The dual to this LP is in the space of continuous functions and its feasible solutions provide guaranteed outer approximations to the global attractor. For systems with polynomial dynamics, a hierarchy of finite-dimensional sum-of-squares tightenings of the dual LP provides a sequence of outer approximations to the global attractor with guaranteed convergence in the sense of volume discrepancy tending to zero. The method is very simple to use and based purely on convex optimization. Numerical examples with the code available online demonstrate the method.

The characterization of phase dynamics in coupled oscillators offers insights into fundamental phenomena in complex systems. To describe the collective dynamics in the oscillatory system, order parameters are often used but are insufficient for identifying more specific behaviors. We therefore propose a topological approach that constructs quantitative features describing the phase evolution of oscillators. Here, the phase data are mapped into a high-dimensional space at each time point, and topological features describing the shape of the data are subsequently extracted from the mapped points. We extend these features to time-variant topological features by considering the evolution time, which serves as an additional dimension in the topological-feature space. The resulting time-variant features provide crucial insights into the time evolution of phase dynamics. We combine these features with the machine learning kernel method to characterize the multicluster synchronized dynamics at a very early stage of the evolution. Furthermore, we demonstrate the usefulness of our method for qualitatively explaining chimera states, which are states of stably coexisting coherent and incoherent groups in systems of identical phase oscillators. The experimental results show that our method is generally better than those using order parameters, especially if only data on the early-stage dynamics are available.

We introduce a Gaussian measure formally preserved by the 2-dimensional Primitive Equations driven by additive Gaussian noise. Under such measure the stochastic equations under consideration are singular: we propose a solution theory based on the techniques developed by Gubinelli and Jara in cite{GuJa13} for a hyperviscous version of the equations.

We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear elliptic PDEs on the form $$ F(x,u,Du,D^2u) = 0 $$ under suitable structure conditions on the equation allowing for non-Lipschitz growth in the gradient terms. In case of smooth boundaries, we also prove the Hopf lemma, the boundary Harnack inequality and that positive viscosity solutions vanishing on a portion of the boundary are comparable with the distance function near the boundary. Our results apply to weak solutions of an eigenvalue problem for the variable exponent $p$-Laplacian.

A subset of vertices in a graph $G$ is called a maximum dissociation set if it induces a subgraph with vertex degree at most 1 and the subset has maximum cardinality. The dissociation number of $G$, denoted by $psi(G)$, is the cardinality of a maximum dissociation set. A subcubic tree is a tree of maximum degree at most 3. In this paper, we give the lower and upper bounds on the dissociation number in a subcubic tree of order $n$ and show that the number of maximum dissociation sets of a subcubic tree of order $n$ and dissociation number $psi$ is at most $1.466^{4n-5psi+2}$.

We observe that the DeTurck Laplacian flow of G2-structures introduced by Bryant and Xu as a gauge fixing of the Laplacian flow can be regarded as a flow of G2-structures (not necessarily closed) which fits in the general framework introduced by Hamilton in [4].

We apply the paracontrolled calculus to study the asymptotic behavior of a certain quasilinear PDE with smeared mild noise, which originally appears as the space-time scaling limit of a particle system in random environment on one dimensional discrete lattice. We establish the convergence result and show a local in time well-posedness of the limit stochastic PDE with spatial white noise. It turns out that our limit stochastic PDE does not require any renormalization. We also show a comparison theorem for the limit equation.

The main aim of this paper is to detect dynamical properties of the Gy"orgyi-Field model of the Belousov-Zhabotinsky chemical reaction. The corresponding three-variable model given as a set of nonlinear ordinary differential equations depends on one parameter, the flow rate. As certain values of this parameter can give rise to chaos, the analysis was performed in order to identify different dynamics regimes. Dynamical properties were qualified and quantified using classical and also new techniques. Namely, phase portraits, bifurcation diagrams, the Fourier spectra analysis, the 0-1 test for chaos, and approximate entropy. The correlation between approximate entropy and the 0-1 test for chaos was observed and described in detail. Moreover, the three-stage system of nested subintervals of flow rates, for which in every level the 0-1 test for chaos and approximate entropy was computed, is showing the same pattern. The study leads to an open problem whether the set of flow rate parameters has Cantor like structure.

In this work, the generalized $N$-component Fokas-Lenells(FL) equations, which have been studied by Guo and Ling (2012 J. Math. Phys. 53 (7) 073506) for $N=2$, are first investigated via Riemann-Hilbert(RH) approach. The main purpose of this is to study the soliton solutions of the coupled Fokas-Lenells(FL) equations for any positive integer $N$, which have more complex linear relationship than the analogues reported before. We first analyze the spectral analysis of the Lax pair associated with a $(N+1) imes (N+1)$ matrix spectral problem for the $N$-component FL equations. Then, a kind of RH problem is successfully formulated. By introducing the special conditions of irregularity and reflectionless case, the $N$-soliton solution formula of the equations are derived through solving the corresponding RH problem. Furthermore, take $N=2,3$ and $4$ for examples, the localized structures and dynamic propagation behavior of their soliton solutions and their interactions are discussed by some graphical analysis.

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. We consider the system $mathcal L (H)$ of all sets of lengths of $H$ and study when $mathcal L (H)$ contains or is contained in a system $mathcal L (H')$ of a Krull monoid $H'$ with finite class group $G'$, prime divisors in all classes and Davenport constant $mathsf D (G')=mathsf D (G)$. Among others, we show that if $G$ is either cyclic of order $m ge 7$ or an elementary $2$-group of rank $m-1 ge 6$, and $G'$ is any group which is non-isomorphic to $G$ but with Davenport constant $mathsf D (G')=mathsf D (G)$, then the systems $mathcal L (H)$ and $mathcal L (H')$ are incomparable.

Let $Gamma$ be a discrete group acting properly discontinuously and isometrically on the three-dimensional anti-de Sitter space $mathrm{AdS}^{3}$, and $square$ the Laplacian which is a second-order hyperbolic differential operator. We study linear independence of a family of generalized Poincar'{e} series introduced by Kassel-Kobayashi [Adv. Math. 2016], which are defined by the $Gamma$-average of certain eigenfunctions on $mathrm{AdS}^{3}$. We prove that the multiplicities of $L^{2}$-eigenvalues of the hyperbolic Laplacian $square$ on $Gammaackslashmathrm{AdS}^{3}$ are unbounded when $Gamma$ is finitely generated. Moreover, we prove that the multiplicities of extit{stable $L^{2}$-eigenvalues} for compact anti-de Sitter $3$-manifolds are unbounded.

Let $(K, u)$ be a valued field, the notions of emph{augmented valuation}, of emph{limit augmented valuation} and of emph{admissible family} of valuations enable to give a description of any valuation $mu$ of $K [x]$ extending $ u$. In the case where the field $K$ is algebraically closed, this description is particularly simple and we can reduce it to the notions of emph{minimal pair} and emph{pseudo-convergent family}. Let $(K, u )$ be a henselian valued field and $ar u$ the unique extension of $ u$ to the algebraic closure $ar K$ of $K$ and let $mu$ be a valuation of $ K [x]$ extending $ u$, we study the extensions $armu$ from $mu$ to $ar K [x]$ and we give a description of the valuations $armu_i$ of $ar K [x]$ which are the extensions of the valuations $mu_i$ belonging to the admissible family associated with $mu$.

In the present paper by the Fourier transform we show that every linear differential equations of $n$-th order has a solution in $L^1(Bbb{R})$ which is infinitely differentiable in $Bbb{R} setminus {0}$. Moreover the Hyers-Ulam stability of such equations on $L^1(Bbb{R})$ is investigated.

In this paper, we give an alternate definition of the well-known Parry measure on an aperiodic subshift of finite type using correlation between the forbidden words. We use the concept of the local escape rate to obtain this definition. We also compute Perron eigenvectors corresponding to the Perron root of the associated adjacency matrix.

Our goal in this article is to study the global Lorentz estimates for gradient of weak solutions to $p$-Laplace double obstacle problems involving the Schr"odinger term: $-Delta_p u + mathbb{V}|u|^{p-2}u$ with bound constraints $psi_1 le u le psi_2$ in non-smooth domains. This problem has its own interest in mathematics, engineering, physics and other branches of science. Our approach makes a novel connection between the study of Calder'on-Zygmund theory for nonlinear Schr"odinger type equations and variational inequalities for double obstacle problems.

Let $E$ be the self-similar set generated by the {it iterated function system} {[ f_0(x)=frac{x}{eta},quad f_1(x)=frac{x+1}{eta}, quad f_{eta+1}=frac{x+eta+1}{eta} ]}with $etage 3$. {Then} $E$ is a self-similar set with complete {overlaps}, i.e., $f_{0}circ f_{eta+1}=f_{1}circ f_1$, but $E$ is not totally self-similar.

We investigate all its generating iterated function systems, give the spectrum of $E$, and determine the Hausdorff dimension and Hausdorff measure of $E$ and of the sets which contain all points in $E$ having finite or infinite different triadic codings.

In this paper we construct an equivariant embedding of the affine space $mathbb{A}^n$ with the translation group action into a complete non-projective algebraic variety $X$ for all $n geq 3$. The theory of toric varieties is used as the main tool for this construction. In the case of $n = 3$ we describe the orbit structure on the variety $X$.

In this short note we give a proof of the famous conjecture of Erd"{o}s-Straus for the case $nequiv13 extrm{ mod } 24.$ The Erd"{o}s--Straus conjecture states that the equation $frac{4}{n}=frac{1}{x}+frac{1}{y}+frac{1}{z}$ has positive integer solutions $x,y,z$ for every $ngeq 2$. It is open for $nequiv 1 extrm{ mod } 12$. Indeed, in all of the other cases the solutions are always easy to find. We prove that the conjecture is true for every $nequiv 13 extrm{ mod } 24$. Therefore, to solve it completely, it remains to find solutions for every $nequiv 1 extrm{ mod } 24$.

In information theory, the so-called log-sum inequality is fundamental and a kind of generalization of the non-nagativity for the relative entropy. In this paper, we show the generalized log-sum inequality for two functions defined for scalars. We also give a new result for commutative matrices. In addition, we demonstrate further results for general non-commutative positive semi-definite matrices.

Let $Nge 2$ and $ hoin(0,1/N^2]$. The homogenous Cantor set $E$ is the self-similar set generated by the iterated function system

[

left{f_i(x)= ho x+frac{i(1- ho)}{N-1}: i=0,1,ldots, N-1 ight}.

]

Let $s=dim_H E$ be the Hausdorff dimension of $E$, and let $mu=mathcal H^s|_E$ be the $s$-dimensional Hausdorff measure restricted to $E$. In this paper we describe, for each $xin E$, the pointwise lower $s$-density $Theta_*^s(mu,x)$ and upper $s$-density $Theta^{*s}(mu, x)$ of $mu$ at $x$. This extends some early results of Feng et al. (2000). Furthermore, we determine two critical values $a_c$ and $b_c$ for the sets

[

E_*(a)=left{xin E: Theta_*^s(mu, x)ge a ight}quad extrm{and}quad E^*(b)=left{xin E: Theta^{*s}(mu, x)le b ight}

] respectively, such that $dim_H E_*(a)>0$ if and only if $a<a_c$, and that $dim_H E^*(b)>0$ if and only if $b>b_c$. We emphasize that both values $a_c$ and $b_c$ are related to the Thue-Morse type sequences, and our strategy to find them relies on ideas from open dynamics and techniques from combinatorics on words.

We investigate the quantum twistor bundle constructed as a $U(1)$-quotient of the quantum instanton bundle of Bonechi, Ciccoli and Tarlini. It is an example of a locally trivial noncommutative bundle fulfilling conditions of the framework recently proposed by Brzezi'nski and Szyma'nski. In particular, we give a detailed description of the corresponding $C^*$-algebra of 'continuous functions' on its noncommutative total space. Furthermore, we analyse a different construction of a quantum instanton bundle due to Landi, Pagani and Reina, find a basis of its polynomial algebra and discover an intriguing and unexpected feature of its enveloping $C^*$-algebra.

The problem of minimizing an entropy functional subject to linear constraints is a useful example of partially finite convex programming. In the 1990s, Borwein and Lewis provided broad and easy-to-verify conditions that guarantee strong duality for such problems. Their approach is to construct a function in the quasi-relative interior of the relevant infinite-dimensional set, which assures the existence of a point in the core of the relevant finite-dimensional set. We revisit this problem, and provide an alternative proof by directly appealing to the definition of the core, rather than by relying on any properties of the quasi-relative interior. Our approach admits a minor relaxation of the linear independence requirements in Borwein and Lewis' framework, which allows us to work with certain piecewise-defined moment functions precluded by their conditions. We provide such a computed example that illustrates how this relaxation may be used to tame observed Gibbs phenomenon when the underlying data is discontinuous. The relaxation illustrates the understanding we may gain by tackling partially-finite problems from both the finite-dimensional and infinite-dimensional sides. The comparison of these two approaches is informative, as both proofs are constructive.

The congruence subgroup problem for a finitely generated group $Gamma$ and $Gleq Aut(Gamma)$ asks whether the map $hat{G} o Aut(hat{Gamma})$ is injective, or more generally, what is its kernel $Cleft(G,Gamma ight)$? Here $hat{X}$ denotes the profinite completion of $X$. In the case $G=Aut(Gamma)$ we denote $Cleft(Gamma ight)=Cleft(Aut(Gamma),Gamma ight)$.

Let $Gamma$ be a finitely generated group, $ar{Gamma}=Gamma/[Gamma,Gamma]$, and $Gamma^{*}=ar{Gamma}/tor(ar{Gamma})congmathbb{Z}^{(d)}$. Denote $Aut^{*}(Gamma)= extrm{Im}(Aut(Gamma) o Aut(Gamma^{*}))leq GL_{d}(mathbb{Z})$. In this paper we show that when $Gamma$ is nilpotent, there is a canonical isomorphism $Cleft(Gamma ight)simeq C(Aut^{*}(Gamma),Gamma^{*})$. In other words, $Cleft(Gamma ight)$ is completely determined by the solution to the classical congruence subgroup problem for the arithmetic group $Aut^{*}(Gamma)$.

In particular, in the case where $Gamma=Psi_{n,c}$ is a finitely generated free nilpotent group of class $c$ on $n$ elements, we get that $C(Psi_{n,c})=C(mathbb{Z}^{(n)})={e}$ whenever $ngeq3$, and $C(Psi_{2,c})=C(mathbb{Z}^{(2)})=hat{F}_{omega}$ = the free profinite group on countable number of generators.