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In this paper, we use Aleksandrov's reflection principle to prove symmetry of solutions to F (∇ S n 2 u + u I) = f (u , u 2 + | ∇ S n u | 2 ) , where u is the support function of a convex body, and F is a function of principal radii. As a corollary, alongside [Ivaki, arXiv:2307.06252 ], we provide an alternative proof of the uniqueness of solution to the isotropic Gaussian-Minkowski problem. [ABSTRACT FROM AUTHOR]

In this paper we give a generalization of the problem that was posed by I. Raşa in [10] and was proved based on a probabilistic approach by Mrowiec, Rajba and Wąsowicz in [8] . We end this paper by proposing two open problems related to Raşa's proposed inequality. [ABSTRACT FROM AUTHOR]

This paper considers matrix convex sets invariant under several types of rotations. It is known that matrix convex sets that are free semialgebraic are solution sets of Linear Matrix Inequalities (LMIs); they are called free spectrahedra. We classify all free spectrahedra that are circular, that is, closed under multiplication by e i t : up to unitary equivalence, the coefficients of a minimal LMI defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal. A matrix convex set is called free circular if it is closed under left multiplication by unitary matrices. As a consequence of a Hahn–Banach separation theorem for free circular matrix convex sets, we show the coefficients of a minimal LMI defining a free circular free spectrahedron have, up to unitary equivalence, a block decomposition as above with only two blocks. This paper also gives a classification of those noncommutative polynomials invariant under conjugating each coordinate by a different unitary matrix. Up to unitary equivalence such a polynomial must be a direct sum of univariate polynomials. [ABSTRACT FROM AUTHOR]

This paper considers the problem of recovering the delays and amplitudes of a weighted superposition of pulses. This problem is motivated by a variety of applications, such as ultrasound and radar. We show that for univariate and bivariate stream of pulses, one can recover the delays and weights to any desired accuracy by solving a tractable convex optimization problem, provided that a pulse-dependent separation condition is satisfied. The main result of this paper states that the recovery is robust to additive noise or model mismatch. [ABSTRACT FROM AUTHOR]

In this paper, we establish an existence and uniqueness result for the system of quasilinear stochastic partial differential equations (SPDEs for short) with reflection in a convex domain in R K by analytical approach. The method is based on the approximation of the penalized systems of SPDEs. [ABSTRACT FROM AUTHOR]

Consider the semilinear elliptic equation − Δ u = λ f (u) in the unit ball B 1 ⊂ R N , with Dirichlet data u | ∂ B 1 = 0 , where λ ≥ 0 is a real parameter and f is a C 1 positive, nondecreasing and convex function in [ 0 , ∞) such that f (s) / s → ∞ as s → ∞. In this paper we study the behavior of f ′ (u ⁎) near the origin when u ⁎ , the extremal solution of the previous problem associated to λ = λ ⁎ , is singular. This answers to an open problems posed by Brezis and Vázquez [2, Open problem 5]. [ABSTRACT FROM AUTHOR]

In this paper we propose a primal-dual dynamical approach to the minimization of a structured convex function consisting of a smooth term, a nonsmooth term, and the composition of another nonsmooth term with a linear continuous operator. In this scope we introduce a dynamical system for which we prove that its trajectories asymptotically converge to a saddle point of the Lagrangian of the underlying convex minimization problem as time tends to infinity. In addition, we provide rates for both the violation of the feasibility condition by the ergodic trajectories and the convergence of the objective function along these ergodic trajectories to its minimal value. Explicit time discretization of the dynamical system results in a numerical algorithm which is a combination of the linearized proximal method of multipliers and the proximal ADMM algorithm. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to extract an explicit effective and uniform bound on the rate of asymptotic regularity of an iteration schema involving a finite family of nonexpansive mappings. The results presented in this paper contribute to the general project of proof mining as developed by the second author as well as generalize and improve various classical and corresponding quantitative results in the current literature. More precisely, we give a rate of asymptotic regularity of an iteration schema due to Kuhfittig for finitely many nonexpansive mappings in the context of uniformly convex hyperbolic spaces. The rate only depends on an upper bound on the distance between the starting point and some common fixed point, a lower bound , the error and a modulus of uniform convexity. [Copyright &y& Elsevier]

Abstract: We study the best possible constants in the Brezis–Marcus inequalities for in balls . The quantity is known by our paper [F.G. Avkhadiev, K.-J. Wirths, Unified Poincaré and Hardy inequalities with sharp constants for convex domains, ZAMM Z. Angew. Math. Mech. 87 (8–9) 26 (2007) 632–642]. In the present paper we prove the estimate and the assertion which gives that the known lower estimates in [G. Barbatis, S. Filippas, and A. Tertikas in Comm. Cont. Math. 5 (2003), no. 6, 869–881] for , are asymptotically sharp as . Also, for the 3-dimensional ball we obtain a new Brezis–Marcus type inequality which contains two parameters , and has the following form where is the first zero of the Bessel function of order and the constants are sharp for all admissible values of parameters and . [Copyright &y& Elsevier]

Abstract In this paper we rigorously justify the propagation of chaos for the parabolic–elliptic Keller–Segel equation over bounded convex domains. The boundary condition under consideration is the no-flux condition. As intermediate steps, we establish the well-posedness of the associated stochastic equation as well as the well-posedness of the Keller–Segel equation for bounded weak solutions. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper we prove some representation theorems for t-Wright convex functions, as a consequence of a support theorem, which was proved by the author in earlier paper. [Copyright &y& Elsevier]

Abstract: A set is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set with a closed convex cone. The main result in this paper establishes that a closed convex set is Motzkin decomposable if and only if the set of extreme points of its intersection with the linear subspace orthogonal to its lineality is bounded. The paper characterizes the class of the extended functions whose epigraphs are Motzkin decomposable sets showing, in particular, that these functions attain their global minima when they are bounded from below. Calculus of Motzkin decomposable sets and functions is provided. [ABSTRACT FROM AUTHOR]

Abstract: In scalar optimization it is well known that a solution of a Minty variational inequality of differential type is a solution of the related optimization problem. This relation is known as “Minty variational principle.” In the vector case, the links between Minty variational inequalities and vector optimization problems were investigated in [F. Giannessi, On Minty variational principle, in: New Trends in Mathematical Programming, Kluwer Academic, Dordrecht, 1997, pp. 93–99] and subsequently in [X.M. Yang, X.Q. Yang, K.L. Teo, Some remarks on the Minty vector variational inequality, J. Optim. Theory Appl. 121 (2004) 193–201]. In these papers, in the particular case of a differentiable objective function f taking values in and a Pareto ordering cone, it has been shown that the vector Minty variational principle holds for pseudoconvex functions. In this paper we extend such results to the case of an arbitrary ordering cone and a nondifferentiable objective function, distinguishing two different kinds of solutions of a vector optimization problem, namely ideal (or absolute) efficient points and weakly efficient points. Further, we point out that in the vector case, the Minty variational principle cannot be extended to quasiconvex functions. [Copyright &y& Elsevier]

Abstract: In 1979, in an interesting paper, R.J. Morris introduced the notion of convex set function defined on an atomless finite measure space. After a short period this notion, as well as generalizations of it, began to be studied in several papers. The aim was to obtain results similar to those known for usual convex (or generalized convex) functions. Unfortunately several notions are ambiguous and the arguments used in the proofs of several results are not clear or not correct. In this way there were stated even false results. The aim of this paper is to point out that using some simple ideas it is possible, on one hand, to deduce the correct results by means of convex analysis and, on the other hand, to emphasize the reasons for which there are problems with other results. [Copyright &y& Elsevier]

The lighting situation in which a picture was taken has an impact on its color. Illuminant estimation is crucial in computer vision because the colors of objects vary as illumination changes. For this reason, numerous methods for estimating the illuminant have been suggested. In this paper, we suggest a novel statistic-based method for estimating single and multiple illuminants using convex functions. In this respect, convex functions are used in the two subsequent steps of normalization and weight creation. After using weighted K-means to segment the picture, each segment's associated illuminations are determined. The illumination map for the input image is estimated as a final stage. In this study, we also analyze the effect of convexity on color constancy algorithms and present proofs for the convexity of some statistic-based algorithms. Four different single and multi-illuminant datasets have been used to evaluate the proposed algorithm in terms of two evaluation metrics; recovery and reproduction angular error. We believe that the proposed method could be considered one of the statistical state-of-the-art algorithms. In addition, it has competitive results when compared to most learning-based and deep-learning methods. Further advantages of the proposed algorithm include its simplicity of implementation and low execution time. [Display omitted] • A new statistic-based method with easy implementation and no learning required. • A general statistic-based method for estimating single and multiple illuminations. • Presenting an efficient method that is one of the statistical state-of-the-art methods. • Estimating the local and global illuminations using convex functions. • Algorithm stability against changing the number of light sources and databases. [ABSTRACT FROM AUTHOR]

In this paper, we construct new barriers for the Pucci extremal operators with unbounded RHS. The geometry of these barriers is given by a Harnack inequality up to the boundary type estimate. Under the possession of these barriers, we prove a new quantitative version of the Hopf–Oleĭnik Lemma for quasilinear elliptic equations with g -Laplace type growth. Finally, we prove (sharp) regularity for ω -semiconvex supersolutions for some nonlinear PDEs. These results are new even for second order linear elliptic equations in nondivergence form. Moreover, these estimates extend and improve a classical a priori estimate proven by L. Caffarelli, J.J. Kohn, J. Spruck and L. Nirenberg in [13] in 1985 as well as a more recent result on the C 1 , 1 regularity for convex supersolutions obtained by C. Imbert in [33] in 2006. [ABSTRACT FROM AUTHOR]

A well known result of Forstnerić [15] states that most real-analytic strictly pseudoconvex hypersurfaces in complex space are not holomorphically embeddable into spheres of higher dimension. A more recent result by Forstnerić [16] states even more: most real-analytic hypersurfaces do not admit a holomorphic embedding even into a merely algebraic hypersurface of higher dimension, in particular, a hyperquadric. We emphasize that both cited theorems are proved by showing that the set of embeddable hypersurfaces is a set of first Baire category. At the same time, the classical theorem of Webster [30] asserts that every real-algebraic Levi-nondegenerate hypersurface admits a transverse holomorphic embedding into a nondegenerate real hyperquadric in complex space. In this paper, we provide effective results on the non-embeddability of real-analytic hypersurfaces into a hyperquadric. We show that, under the codimension restriction N ≤ 2 n , the defining functions φ ( z , z ¯ , u ) of all real-analytic hypersurfaces M = { v = φ ( z , z ¯ , u ) } ⊂ C n + 1 containing Levi-nondegenerate points and locally transversally holomorphically embeddable into some hyperquadric Q ⊂ C N + 1 satisfy an universal algebraic partial differential equation D ( φ ) = 0 , where the algebraic-differential operator D = D ( n , N ) depends on n ≥ 1 , n < N ≤ 2 n only. To the best of our knowledge, this is the first effective result characterizing real-analytic hypersurfaces embeddable into a hyperquadric of higher dimension. As an application, we show that for every n , N as above there exists μ = μ ( n , N ) such that a Zariski generic real-analytic hypersurface M ⊂ C n + 1 of degree ≥ μ is not transversally holomorphically embeddable into any hyperquadric Q ⊂ C N + 1 . We also provide an explicit upper bound for μ in terms of n , N . To the best of our knowledge, this gives the first effective lower bound for the CR-complexity of a Zariski generic real-algebraic hypersurface in complex space of a fixed degree. [ABSTRACT FROM AUTHOR]

This paper contains a number of results related to volumes of projective perturbations of convex bodies and the Laplace transform on convex cones. First, it is shown that a sharp version of Bourgain's slicing conjecture implies the Mahler conjecture for convex bodies that are not necessarily centrally-symmetric. Second, we find that by slightly translating the polar of a centered convex body, we may obtain another body with a bounded isotropic constant. Third, we provide a counter-example to a conjecture by Kuperberg on the distribution of volume in a body and in its polar. [ABSTRACT FROM AUTHOR]

We consider the L ∞ weak solutions to a type of compressible Euler equation with dissipation effects. Several studies [6,14,30] have obtained the L ∞ weak solutions to this type of system by using numerical schemes and the compensated compactness method. Therefore, the uniform boundedness of approximate solutions and the H l o c − 1 compactness of the corresponding entropy dissipation measures must be considered. It should be noted that the obtained L ∞ bounds typically increase over time. However, getting a time-independent uniform bound is important to consider the large time behavior of weak solutions. In this paper, by using invariant region theory, we prove that the L ∞ weak solutions derived by the Lax–Friedrichs scheme are uniformly bounded in time. [ABSTRACT FROM AUTHOR]

Based on convex analysis, a novel consensus algorithm of dynamical points in a CAT(0) space is developed in this paper, in which the associated communication graph uniformly contains a directed spanning tree. The proposed algorithm provides an efficient method of solving consensus problems in a general CAT(0) space, while having certain robustness against weak communication. The application of the new algorithm to the distributed fusion of phylogenetic trees is shown with demonstrative-case simulations, together with a study on the algorithm's robustness and efficiency. [ABSTRACT FROM AUTHOR]

Let X be a separable Banach space endowed with a non-degenerate centered Gaussian measure μ . The associated Cameron–Martin space is denoted by H . Consider two sufficiently regular convex functions U : X → R and G : X → R . We let ν = e − U μ and Ω = G − 1 ( − ∞ , 0 ] . In this paper we are interested in the W 2 , 2 regularity of the weak solutions of elliptic equations of the type (0.1) λ u − L ν , Ω u = f , where λ > 0 , f ∈ L 2 ( Ω , ν ) and L ν , Ω is the self-adjoint operator associated with the quadratic form ( ψ , φ ) ↦ ∫ Ω 〈 ∇ H ψ , ∇ H φ 〉 H d ν ψ , φ ∈ W 1 , 2 ( Ω , ν ) . In addition we will show that if u is a weak solution of problem (0.1) then it satisfies a Neumann type condition at the boundary, namely for ρ -a.e. x ∈ G − 1 ( 0 ) 〈 Tr ( ∇ H u ) ( x ) , Tr ( ∇ H G ) ( x ) 〉 H = 0 , where ρ is the Feyel–de La Pradelle Hausdorff–Gauss surface measure and Tr is the trace operator. [ABSTRACT FROM AUTHOR]

Perspective functions arise explicitly or implicitly in various forms in applied mathematics and in statistical data analysis. To date, no systematic strategy is available to solve the associated, typically nonsmooth, optimization problems. In this paper, we fill this gap by showing that proximal methods provide an efficient framework to model and solve problems involving perspective functions. We study the construction of the proximity operator of a perspective function under general assumptions and present important instances in which the proximity operator can be computed explicitly or via straightforward numerical operations. These results constitute central building blocks in the design of proximal optimization algorithms. We showcase the versatility of the framework by designing novel proximal algorithms for state-of-the-art regression and variable selection schemes in high-dimensional statistics. [ABSTRACT FROM AUTHOR]

In this paper we consider a nonlinear viscoelastic equation with minimal conditions on the L 1 ( 0 , ∞ ) relaxation function g namely g ′ ( t ) ≤ − ξ ( t ) H ( g ( t ) ) , where H is an increasing and convex function near the origin and ξ is a nonincreasing function. With only these very general assumptions on the behavior of g at infinity, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when H ( s ) = s p and p covers the full admissible range [ 1 , 2 ) . We get the best decay rates expected under this level of generality and our new results substantially improve several earlier related results in the literature. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study shape preserving properties of univariate Lototsky–Bernstein operators L n ( f ) based on Lototsky–Bernstein basis functions. The Lototsky–Bernstein basis functions b n , k ( x ) ( 0 ≤ k ≤ n ) of order n are constructed by replacing x in the i th factor of the generating function for the classical Bernstein basis functions of degree n by a continuous nondecreasing function p i ( x ) , where p i ( 0 ) = 0 and p i ( 1 ) = 1 for 1 ≤ i ≤ n . These operators L n ( f ) are positive linear operators that preserve constant functions, and a non-constant function γ n p ( x ) . If all the p i ( x ) are strictly increasing and strictly convex, then γ n p ( x ) is strictly increasing and strictly convex as well. Iterates L n M ( f ) of L n ( f ) are also considered. It is shown that L n M ( f ) converges to f ( 0 ) + ( f ( 1 ) − f ( 0 ) ) γ n p ( x ) as M → ∞ . Like classical Bernstein operators, these Lototsky–Bernstein operators enjoy many traditional shape preserving properties. For every ( 1 , γ n p ( x ) ) -convex function f ∈ C [ 0 , 1 ] , we have L n ( f ; x ) ≥ f ( x ) ; and by invoking the total positivity of the system { b n , k ( x ) } 0 ≤ k ≤ n , we show that if f is ( 1 , γ n p ( x ) ) -convex, then L n ( f ; x ) is also ( 1 , γ n p ( x ) ) -convex. Finally we show that if all the p i ( x ) are monomial functions, then for every ( 1 , γ n + 1 p ( x ) ) -convex function f , L n ( f ; x ) ≥ L n + 1 ( f ; x ) if and only if p 1 ( x ) = ⋯ = p n ( x ) = x . [ABSTRACT FROM AUTHOR]

A well known result of H. Attouch states that the Mosco convergence of a sequence of proper convex lower semicontinuous functions defined on a Hilbert space is equivalent to the pointwise convergence of the associated Moreau envelopes. In the present paper we generalize this result to Hadamard spaces. More precisely, while it has already been known that the Mosco convergence of a sequence of convex lower semicontinuous functions on a Hadamard space implies the pointwise convergence of the corresponding Moreau envelopes, the converse implication was an open question. We now fill this gap. Our result has several consequences. It implies, for instance, the equivalence of the Mosco and Frolík–Wijsman convergences of convex sets. As another application, we show that there exists a complete metric on the cone of proper convex lower semicontinuous functions on a separable Hadamard space such that a sequence of functions converges in this metric if and only if it converges in the sense of Mosco. [ABSTRACT FROM AUTHOR]

In this paper we consider support points for the family S g 0 ( B 2 ) of mappings with g -parametric representation on the Euclidean unit ball B 2 in C 2 , where g is a univalent function on the unit disc U in C , which satisfies certain natural assumptions. We shall use the shearing process recently introduced by Bracci, to prove the existence of bounded support points for the family S g 0 ( B 2 ) . This result is in contrast to the one dimensional case, where all support points of the family S are unbounded. We also study the case of time- log ⁡ M reachable families R ˜ log ⁡ M ( id B 2 , M g ) generated by the Carathéodory family M g , and obtain certain results and applications, which show a basic difference between the theory in the case of one complex variable and that in higher dimensions. Of particular interest is the case where g is a convex (univalent) function on U . Finally, various consequences and certain conjectures are also considered. [ABSTRACT FROM AUTHOR]

In this paper we study the structure, weak continuity and approximability properties for the integer multiplicity rectifiable currents carried by the graphs of maximal semi-monotone set-valued maps on an n -dimensional convex domain. Especially, we give an enhanced version of approximation theorem for the subgradients of semi-convex functions. [ABSTRACT FROM AUTHOR]

In this paper, for any symplectic matrix P , the existence of subharmonic P l -solutions of the first order non-autonomous superquadratic Hamiltonian systems is considered. Under the convex condition, the existence of infinitely many geometrically distinct P l -solutions is proved. [ABSTRACT FROM AUTHOR]

Recently Nikodem, Rajba and Wąsowicz compared the classes of n -Wright-convex functions and n -Jensen-convex functions by showing that the first one is a proper subclass of the latter one, whenever n is an odd natural number. Till now the case of even n was an open problem. In this paper the complete solution is given: it is shown that the inclusion is proper for any natural n . The classes of strongly n -Wright-convex and strongly n -Jensen-convex functions are also compared (with the same assertion). [ABSTRACT FROM AUTHOR]

In this paper we study the automorphism group of smoothly bounded convex domains. We show that such a domain is biholomorphic to a “polynomial ellipsoid” (that is, a domain defined by a weighted homogeneous balanced polynomial) if and only if the limit set of the automorphism group intersects at least two closed complex faces of the set. The proof relies on a detailed study of the geometry of the Kobayashi metric and ideas from the theory of non-positively curved metric spaces. We also obtain a number of other results including the Greene–Krantz conjecture in the case of uniform non-tangential convergence, new results about continuous extensions (of biholomorphisms and complex geodesics), and a new Wolff–Denjoy theorem. [ABSTRACT FROM AUTHOR]

In this paper we establish new rules for the calculus of the subdifferential mapping of the sum of two convex functions. Our results are established under conditions which are at an intermediate level of generality among those leading to the Hiriart-Urruty and Phelps formula (Hiriart-Urruty and Phelps, 1993 [15] ), involving the approximate subdifferential, and the stronger assumption used in the well-known Moreau–Rockafellar formula (Rockafellar 1970, [23] ; Moreau 1966, [20] ), which only uses the exact subdifferential. We give an application to derive asymptotic optimality conditions for convex optimization. [ABSTRACT FROM AUTHOR]

In this paper we consider Bernstein’s Lethargy Theorem (BLT) in the context of Fréchet spaces. Let X be an infinite-dimensional Fréchet space and let V = { V n } be a nested sequence of subspaces of X such that V n ¯ ⊆ V n + 1 for any n ∈ N . Let e n be a decreasing sequence of positive numbers tending to 0. Under one additional but necessary condition on sup { dist ( x , V n ) } , we prove that there exist x ∈ X and n o ∈ N such that e n 3 ≤ dist ( x , V n ) ≤ 3 e n for any n ≥ n o . By using the above theorem, as a corollary we obtain classical Shapiro’s (1964) and Tyuriemskih’s (1967) theorems for Banach spaces. Also we prove versions of both Shapiro’s (1964) and Tyuriemskih’s (1967) theorems for Fréchet spaces. Considering rapidly decreasing sequences, other versions of the BLT theorem in Fréchet spaces will be discussed. We also give a theorem improving Konyagin’s (2014) result for Banach spaces. Finally, we present some applications of the above mentioned result concerning particular classes of Fréchet spaces, such as Orlicz spaces generated by s -convex functions and locally bounded Fréchet spaces. [ABSTRACT FROM AUTHOR]

This is the first of two papers concerning saddle-shaped solutions to the semilinear equation L K u = f (u) in R 2 m , where L K is a linear elliptic integro-differential operator and f is of Allen-Cahn type. Saddle-shaped solutions are doubly radial, odd with respect to the Simons cone { (x ′ , x ″) ∈ R m × R m : | x ′ | = | x ″ | } , and vanish only on this set. By the odd symmetry, L K coincides with a new operator L K O which acts on functions defined only on one side of the Simons cone, { | x ′ | > | x ″ | } , and that vanish on it. This operator L K O , which corresponds to reflect a function oddly and then apply L K , has a kernel on { | x ′ | > | x ″ | } which is different from K. In this first paper, we characterize the kernels K for which the new kernel is positive and therefore one can develop a theory on the saddle-shaped solution. The necessary and sufficient condition for this turns out to be that K is radially symmetric and τ ↦ K (τ) is a strictly convex function. Assuming this, we prove an energy estimate for doubly radial odd minimizers and the existence of saddle-shaped solution. In a subsequent article, part II, further qualitative properties of saddle-shaped solutions will be established, such as their asymptotic behavior, a maximum principle for the linearized operator, and their uniqueness. [ABSTRACT FROM AUTHOR]

A recent paper by Cordero-Erausquin and Klartag provides a characterization of the measures μ on R d which can be expressed as the moment measures of suitable convex functions u , i.e. are of the form ( ∇ u ) # e − u for u : R d → R ∪ { + ∞ } and finds the corresponding u by a variational method in the class of convex functions. Here we propose a purely optimal-transport-based method to retrieve the same result. The variational problem becomes the minimization of an entropy and a transport cost among densities ρ and the optimizer ρ turns out to be e − u . This requires to develop some estimates and some semicontinuity results for the corresponding functionals which are natural in optimal transport. The notion of displacement convexity plays a crucial role in the characterization and uniqueness of the minimizers. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of logarithmic Schrödinger equations with a potential which may change sign. When the potential is coercive, we obtain infinitely many solutions by adapting some arguments of the Fountain theorem, and in the case of bounded potential we obtain a ground state solution, i.e. a nontrivial solution with least possible energy. The functional corresponding to the problem is the sum of a smooth and a convex lower semicontinuous term. [ABSTRACT FROM AUTHOR]

This paper studies traveling fronts to cooperation–diffusion systems in R N for N ≥ 3 . We consider ( N − 2 ) -dimensional smooth surfaces as boundaries of strictly convex compact sets in R N − 1 , and define an equivalence relation between them. We prove that there exists a traveling front associated with a given surface and show its stability. The associated traveling fronts coincide up to phase transition if and only if the given surfaces satisfy the equivalence relation. [ABSTRACT FROM AUTHOR]

In this paper, we study locally strictly convex affine hypersurfaces for which the center map is centroaffine congruent with the original hypersurface. By the equiaffine support function ρ , we show that the hypersurface is locally isometric to a warped product R × | ρ | N , where the gradient direction of ρ is along R . As a main result, we complete the classification when grad ρ is the eigenvector of affine shape operator, which shows how to explicitly construct such hypersurfaces starting from one (or two) low dimensional affine hypersphere(s). [ABSTRACT FROM AUTHOR]

In this paper, we study closed convex hulls of unitary orbits in various C ⁎ -algebras. For unital C ⁎ -algebras with real rank zero and a faithful tracial state determining equivalence of projections, a notion of majorization describes the closed convex hulls of unitary orbits for self-adjoint operators. Other notions of majorization are examined in these C ⁎ -algebras. Combining these ideas with the Dixmier property, we demonstrate unital, infinite dimensional C ⁎ -algebras of real rank zero and strict comparison of projections with respect to a faithful tracial state must be simple and have a unique tracial state. Also, closed convex hulls of unitary orbits of self-adjoint operators are fully described in unital, simple, purely infinite C ⁎ -algebras. [ABSTRACT FROM AUTHOR]

In this paper, author proves that if X 1 and X 2 are Gâteaux differentiable space, then X 1 and X 2 have the ball-covering property if and only if ( X 1 × X 2 , ‖ ⋅ ‖ p ) and ( X 1 × X 2 , ‖ ⋅ ‖ ∞ ) have the ball-covering property, where ‖ ( x , y ) ‖ p = ( ‖ x ‖ 1 p + ‖ y ‖ 2 p ) 1 p , p ∈ [ 1 , + ∞ ) and ‖ ( x , y ) ‖ ∞ = max ⁡ { ‖ x ‖ , ‖ y ‖ } . [ABSTRACT FROM AUTHOR]

In this paper we solve a problem raised by Gutiérrez and Montanari about comparison principles for H -convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous H -convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples. [ABSTRACT FROM AUTHOR]

We investigate the asymptotic behavior, as t goes to infinity, for a semilinear hyperbolic equation with asymptotically small dissipation and convex potential. We prove that if the damping term behaves like K t α as t → + ∞ , for some K > 0 and α ∈ ] 0 , 1 [ , then every global solution converges weakly to an equilibrium point. This result is a positive answer to a question left open in the paper of Cabot and Frankel (2012) [6] . [ABSTRACT FROM AUTHOR]

In this paper, the periodic solutions for nonlinear differential inclusion governed by convex subdifferential and different perturbations are studied. It is firstly proved that the differential inclusion has unique periodic solution, if the perturbation function is a single-valued function. Then, by Schauder's fixed point theorem and Kakutani's fixed point theorem, we prove that the differential inclusion has at least a periodic solution, when the perturbation function is an upper semicontinuous (or lower semicontinuous) multifunction. Moreover, the existence of the extremal solution for the differential inclusion is also studied. Finally, based on one-sided Lipschitz (OSL) assumption, we prove the related relaxation theorem. [ABSTRACT FROM AUTHOR]

It turns out that there exist general covariance matrices associated not only to a random vector itself but also to its general moments. In this paper we introduce and characterize general covariance matrices of a random vector that are associated to some important general moments, which are determined by a specific class of convex functions. As special cases, the original covariance matrices of a random vector, as well as the p th covariance matrices characterized recently, are included. The covariance matrices associated to the p -power function distribution and the logistic distribution are characterized as by-products. [ABSTRACT FROM AUTHOR]

A remarkable result by S. Artstein-Avidan and V. Milman states that, up to pre-composition with affine operators, addition of affine functionals, and multiplication by positive scalars, the only fully order preserving mapping acting on the class of lower semicontinuous proper convex functions defined on R n is the identity operator, and the only fully order reversing one acting on the same set is the Fenchel conjugation. Here fully order preserving (reversing) mappings are understood to be those which preserve (reverse) the pointwise order among convex functions, are invertible, and such that their inverses also preserve (reverse) such order. In this paper we establish a suitable extension of these results to order preserving and order reversing operators acting on the class of lower semicontinuous proper convex functions defined on arbitrary infinite dimensional Banach spaces. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdré). Then we give a quantitative form of the isoperimetric inequality, leading to a somehow anomalous behavior. Indeed, it could be that a set is very close to be optimal, in the sense that the isoperimetric inequality is almost an equality, but at the same time is very far (in the sense of the symmetric difference between sets) from any extremal sets! From the results on sets we derive quantitative functional inequalities of weak Cheeger type. [ABSTRACT FROM AUTHOR]

The paper concerns the limit shape (under some probability measure) of convex polygonal lines with vertices on Z + 2 , starting at the origin and with the right endpoint n = ( n 1 , n 2 ) → ∞ . In the case of the uniform measure, an explicit limit shape γ ⁎ : = { ( x 1 , x 2 ) ∈ R + 2 : 1 − x 1 + x 2 = 1 } was found independently by Vershik (1994) [19] , Bárány (1995) [3] , and Sinaĭ (1994) [16] . Recently, Bogachev and Zarbaliev (1999) [5] proved that the limit shape γ ⁎ is universal for a certain parametric family of multiplicative probability measures generalizing the uniform distribution. In the present work, the universality result is extended to a much wider class of multiplicative measures, including (but not limited to) analogs of the three meta-types of decomposable combinatorial structures — multisets, selections, and assemblies. This result is in sharp contrast with the one-dimensional case where the limit shape of Young diagrams associated with integer partitions heavily depends on the distributional type. [ABSTRACT FROM AUTHOR]