Abstract: In this paper, we study semi-slant submanifolds and their warped products in Kenmotsu manifolds. The existence of such warped products in Kenmotsu manifolds is shown by an example and a characterization. A sharp relation is obtained as a lower bound of the squared norm of second fundamental form in terms of the warping function and the slant angle. The equality case is also considered in this paper. Finally, we provide some applications of our derived results. PubDate: 2018-12-01

Abstract: Let \(g=(g_{ij})\) be a complete Riemmanian metric on \({\mathbb {R}}^2\) with finite total area and let \(I_g\) be the infimum of the quotient of the length of any closed simple curve \(\gamma \) in \({\mathbb {R}}^2\) and the sum of the reciprocal of the areas of the regions inside and outside \(\gamma \) respectively with respect to the metric g. Under some mild growth conditions on g we prove the existence of a minimizer for \(I_g\) . As a corollary we obtain a proof for the existence of a minimizer for \(I_{g(t)}\) for any \(0<t<T\) when the metric \(g(t)=g_{ij}(\cdot ,t)=u\delta _{ij}\) is the maximal solution of the Ricci flow equation \(\partial g_{ij}/\partial t=-2R_{ij}\) on \({\mathbb {R}}^2\times (0,T)\) (Daskalopoulos and Hamilton in Commun Anal Geom 12(1):143–164, 2004) where \(T>0\) is the extinction time of the solution. This existence of minimizer result is assumed and used without proof by Daskalopoulos and Hamilton (2004). PubDate: 2018-12-01

Abstract: At the turn of this century Durand, and Lagarias and Pleasants established that key features of minimal subshifts (and their higher-dimensional analogues) to be studied are linearly repetitive, repulsive and power free. Since then, generalisations and extensions of these features, namely \(\alpha \) -repetitive, \(\alpha \) -repulsive and \(\alpha \) -finite ( \(\alpha \ge 1\) ), have been introduced and studied. We establish the equivalence of \(\alpha \) -repulsive and \(\alpha \) -finite for general subshifts over finite alphabets. Further, we studied a family of aperiodic minimal subshifts stemming from Grigorchuk’s infinite 2-group G. In particular, we show that these subshifts provide examples that demonstrate \(\alpha \) -repulsive (and hence \(\alpha \) -finite) is not equivalent to \(\alpha \) -repetitive, for \(\alpha > 1\) . We also give necessary and sufficient conditions for these subshifts to be \(\alpha \) -repetitive, and \(\alpha \) -repulsive (and hence \(\alpha \) -finite). Moreover, we obtain an explicit formula for their complexity functions from which we deduce that they are uniquely ergodic. PubDate: 2018-12-01

Abstract: We present an operator calculus based on Krawtchouk polynomials, including Krawtchouk transforms and corresponding convolution structure which provides an inherently discrete alternative to Fourier analysis. This approach is well suited for applications such as digital image processing. This paper includes the theoretical aspects and some basic examples. PubDate: 2018-10-22

Abstract: Given a group-word w and a group G, the verbal subgroup w(G) is the one generated by all w-values in G. The word w is said to be concise if w(G) is finite whenever the set of w-values in G is finite. In the sixties P. Hall asked whether every word is concise but later Ivanov answered this question in the negative. On the other hand, Hall’s question remains wide open in the class of residually finite groups. In the present article we show that various generalizations of the Engel word are concise in residually finite groups. PubDate: 2018-09-25

Abstract: The paper is the first part of a program devoted to the study of the behavior of operator-valued multipliers in Morrey spaces. Embedding theorems and uniform separability properties involving E-valued Morrey spaces are proved. As a consequence, maximal regularity for solutions of infinite systems of anisitropic elliptic partial differential equations are established. PubDate: 2018-08-20

Abstract: In this paper Geronimus transformations for matrix orthogonal polynomials in the real line are studied. The orthogonality is understood in a broad sense, and is given in terms of a nondegenerate continuous sesquilinear form, which in turn is determined by a quasidefinite matrix of bivariate generalized functions with a well defined support. The discussion of the orthogonality for such a sesquilinear form includes, among others, matrix Hankel cases with linear functionals, general matrix Sobolev orthogonality and discrete orthogonal polynomials with an infinite support. The results are mainly concerned with the derivation of Christoffel type formulas, which allow to express the perturbed matrix biorthogonal polynomials and its norms in terms of the original ones. The basic tool is the Gauss–Borel factorization of the Gram matrix, and particular attention is paid to the non-associative character, in general, of the product of semi-infinite matrices. The Geronimus transformation, in where a right multiplication by the inverse of a matrix polynomial and an addition of adequate masses is performed, is considered. The resolvent matrix and connection formulas are given. Two different methods are developed. A spectral one, based on the spectral properties of the perturbing polynomial, and constructed in terms of the second kind functions. This approach requires the perturbing matrix polynomial to have a nonsingular leading term. Then, using spectral techniques and spectral jets, Christoffel–Geronimus formulas for the transformed polynomials and norms are presented. For this type of transformations, the paper also proposes an alternative method, which does not require of spectral techniques, that is valid also for singular leading coefficients. When the leading term is nonsingular a comparative of between both methods is presented. The nonspectral method is applied to unimodular Christoffel perturbations, and a simple example for a degree one massless Geronimus perturbation is given. PubDate: 2018-08-06

Abstract: In this article we analyze the notions of amenability and paradoxical decomposition from an algebraic perspective. We consider this dichotomy for locally finite extended metric spaces and for general algebras over fields. In the context of algebras we also study the relation of amenability with proper infiniteness. We apply our general analysis to two important classes of algebras: the unital Leavitt path algebras and the translation algebras on locally finite extended metric spaces. In particular, we show that the amenability of a metric space is equivalent to the algebraic amenability of the corresponding translation algebra. PubDate: 2018-08-01

Abstract: Weighted Sobolev spaces play a main role in the study of Sobolev orthogonal polynomials. The aim of this paper is to prove several important properties of weighted Sobolev spaces: separability, reflexivity, uniform convexity, duality and Markov-type inequalities. PubDate: 2018-08-01

Abstract: We define a distance function on the bordered punctured disk \(0< z \le 1/e\) in the complex plane, which is comparable with the hyperbolic distance of the punctured unit disk \(0< z <1.\) As an application, we will construct a distance function on an n-times punctured sphere which is comparable with the hyperbolic distance. We also propose a comparable quantity which is not necessarily a distance function on the punctured sphere but easier to compute. PubDate: 2018-08-01

Abstract: We consider a nonlinear parametric Dirichlet problem driven by the p-Laplace differential operator and a reaction which has the competing effects of a parametric singular term and of a Carathéodory perturbation which is ( \(p-1\) )-linear near \(+\infty \) . The problem is uniformly nonresonant with respect to the principal eigenvalue of \((-\Delta _p,W^{1,p}_0(\Omega ))\) . We look for positive solutions and prove a bifurcation-type theorem describing in an exact way the dependence of the set of positive solutions on the parameter \(\lambda >0\) . PubDate: 2018-07-20

Abstract: Consider a function F(X, Y) of pairs of positive matrices with values in the positive matrices such that whenever X and Y commute \(F(X,Y)= X^pY^q.\) Our first main result gives conditions on F such that \(\mathrm{Tr}[ X \log (F(Z,Y))] \le \mathrm{Tr}[X(p\log X + q \log Y)]\) for all X, Y, Z such that \(\mathrm{Tr}Z =\mathrm{Tr}X\) . (Note that Z is absent from the right side of the inequality.) We give several examples of functions F to which the theorem applies. Our theorem allows us to give simple proofs of the well known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables X, Y, Z instead of just X, Y alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy \(D(X Y) = \mathrm{Tr}[X(\log X-\log Y])\) , and two others, the Donald relative entropy \(D_D(X Y)\) , and the Belavkin–Stasewski relative entropy \(D_{BS}(X Y)\) . They are known to satisfy \(D_D(X Y) \le D(X Y)\le D_{BS}(X Y)\) . We prove that the Donald relative entropy provides the sharp upper bound, independent of Z on \(\mathrm{Tr}[ X \log (F(Z,Y))]\) in a number of cases in which F(Z, Y) is homogeneous of degree 1 in Z and −1 in Y. We also investigate the Legendre transforms in X of \(D_D(X Y)\) and \(D_{BS}(X Y)\) , and show how our results about these Legendre transforms lead to new refinements of the Golden–Thompson inequality. PubDate: 2018-05-29

Abstract: In this paper, we study global regularity for oblique boundary value problems of augmented Hessian equations for a class of general operators. By assuming a natural convexity condition of the domain together with appropriate convexity conditions on the matrix function in the augmented Hessian, we develop a global theory for classical elliptic solutions by establishing global a priori derivative estimates up to second order. Besides the known applications for Monge–Ampère type operators in optimal transportation and geometric optics, the general theory here embraces Neumann problems arising from prescribed mean curvature problems in conformal geometry as well as general oblique boundary value problems for augmented k-Hessian, Hessian quotient equations and certain degenerate equations. PubDate: 2018-05-21

Abstract: In this paper we prove the existence and uniqueness of solutions of an inverse problem of the simultaneous recovery of the evolution of two coefficients in the Korteweg-de Vries equation. PubDate: 2018-05-18

Abstract: As a class of Lévy type Markov generators, nonlocal Waldenfels operators appear naturally in the context of investigating stochastic dynamics under Lévy fluctuations and constructing Markov processes with boundary conditions (in particular the construction with jumps). This work is devoted to prove the weak and strong maximum principles for ‘parabolic’ equations with nonlocal Waldenfels operators. Applications in stochastic differential equations with \(\alpha \) -stable Lévy processes are presented to illustrate the maximum principles. PubDate: 2018-05-16

Abstract: We derive trace formulas of the Buslaev–Faddeev type for quantum star graphs. One of the new ingredients is high energy asymptotics of the perturbation determinant. PubDate: 2018-04-01

Abstract: We consider a class of Schrödinger operators with complex decaying potentials on the lattice. Using some classical results from complex analysis we obtain some trace formulae and use them to estimate zeros of the Fredholm determinant in terms of the potential. PubDate: 2018-03-13

Abstract: We give an overview of existing enhancement techniques for derived and trianguated categories based on the notion of a stable model category, and show how it can be applied to the problem of gluing triangulated categories. The article is mostly expository, but we do prove some new results concerning existence of model structures. PubDate: 2018-02-07