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Convolutions of sets with bounded VC-dimension are uniformly continuousFeb 08 2018We introduce a notion of VC-dimension for subsets of groups, defining this for a set $A$ to be the VC-dimension of the family $\{ A \cap(xA) : x \in A\cdot A^{-1} \}$. We show that if a finite subset $A$ of an abelian group has bounded VC-dimension, then ... More

A derivative operator with respect to two conjugate kernels and applications to fractional calculusFeb 07 2018In this paper, we first introduce the class of k-integral operators, where k is a certain kernel-function, and we study some properties related to such operators. Next, we introduce the concept of conjugate kernels and define the new notion of (k,k')-derivative, ... More

On the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle IIFeb 07 2018Let either $R_k(t) := |P_k(e^{it})|^2$ or $R_k(t) := |Q_k(e^{it})|^2$, where $P_k$ and $Q_k$ are the usual Rudin-Shapiro polynomials of degree $n-1$ with $n=2^k$. In a recent paper we combined close to sharp upper bounds for the modulus of the autocorrelation ... More

Inequalities for integrals of the modified Struve function of the first kindFeb 06 2018Simple inequalities for some integrals involving the modified Struve function of the first kind $\mathbf{L}_{\nu}(x)$ are established. In most cases, these inequalities have best possible constant. We also deduce a tight double inequality, involving the ... More

A new method for proving some inequalities related to several special functionsFeb 06 2018In this paper we present a new approach to proving some exponential inequalities involving the sinc function. Power series expansions are used to generate new polynomial inequalities that are sufficient to prove the given exponential inequalities.

One-dimensional parameter-dependent boundary-value problems in Hölder spacesFeb 06 2018We study the most general class of linear boundary-value problems for systems of $r$-th order ordinary differential equations whose solutions range over the complex H\"older space $C^{n+r,\alpha}$, with $0\leq n\in\mathbb{Z}$ and $0<\alpha\leq1$. We prove ... More

Weak factorization of the Hardy space $H^p$ for small values of $p$, in the multilinear settingFeb 06 2018We give a weak factorization proof of the Hardy space $H^{p}(\mathbb{R}^{n})$ in the multilinear setting, for $\frac{n}{n+1} < p <1$. As a consequence, we obtain a characterization of the boundedness of the commutator $[b,T]$ from $L^{r_{1}}(\mathbb R^n) ... More

Weighted Morrey spaces related to Schrodinger operators with potentials satisfying a reverse Holder inequality and fractional integralsFeb 05 2018Let $\mathcal L=-\Delta+V$ be a Schr\"odinger operator on $\mathbb R^d$, $d\geq3$, where $\Delta$ is the Laplacian operator on $\mathbb R^d$ and the nonnegative potential $V$ belongs to the reverse H\"older class $RH_s$ for $s\geq d/2$. For given $0<\alpha<d$, ... More

Multi-parameter extensions of a theorem of PichoridesFeb 05 2018Extending work of Pichorides and Zygmund to the $d$-dimensional setting, we show that the supremum of $L^p$-norms of the Littlewood-Paley square function over the unit ball of the analytic Hardy spaces $H^p_A(\mathbb{T}^d)$ blows up like $(p-1)^{-d}$ ... More

On logarithmic bounds of maximal sparse operatorsFeb 03 2018Given sparse collections of measurable sets $\mathcal S_k$, $k=1,2,\ldots ,N$, in a general measure space $(X,\mathfrak M,\mu)$, let $ \Lambda_{\mathcal S_k}$ be the sparse operator, corresponding to $\mathcal S_k$. We show that the maximal sparse function ... More

Embeddings for spaces of Lorentz-Sobolev typeJan 31 2018The purpose of this paper is to characterize all embeddings for versions of Besov and Triebel-Lizorkin spaces where the underlying Lebesgue space metric is replaced by a Lorentz space metric. We include two appendices, one on the relation between classes ... More

Generalized reverse Young and Heinz inequalitiesJan 31 2018In this paper, we study the further improvements of the reverse Young and Heinz inequalities for the wider range of $v$, namely $v\in \mathbb{R}$. These modified inequalities are used to establish corresponding operator inequalities on a Hilbert space. ... More

A Class of Möbius Iterated Function SystemsJan 30 2018We give a procedure to produce M\"obius iterated function systems (MIFS) on the unit disc in the complex plane.

On the properties of the $(p,ν)$-extension of the Whittaker function $M_{κ,μ}(z)$Jan 30 2018In this paper, we obtain a $(p,\nu)$-extension of the Whittaker function $M_{\kappa,\mu}(z)$ by using the extended confluent hypergeometric function of the first kind $\Phi_{p,\nu}(b;c;z)$ introduced in Parmar et al. [J. Classical Anal. 11 (2017) 81--106]. ... More

Regularity and continuity of the multilinear strong maximal operatorsJan 30 2018Jan 31 2018Let $m\ge 1$, in this paper, our object of investigation is the regularity and and continuity properties of the following multilinear strong maximal operator $${\mathscr{M}}_{\mathcal{R}}(\vec{f})(x)=\sup_{\substack{R \ni x R\in\mathcal{R}}}\prod\limits_{i=1}^m\frac{1}{|R|}\int_{R}|f_i(y)|dy,$$ ... More

A note on expansion in prime fieldsJan 29 2018Let $\beta,\epsilon \in (0,1]$, and $k \geq \exp(122 \max\{1/\beta,1/\epsilon\})$. We prove that if $A,B$ are subsets of a prime field $\mathbb{Z}_{p}$, and $|B| \geq p^{\beta}$, then there exists a sum of the form $$S = a_{1}B \pm \ldots \pm a_{k}B, ... More

Operational calculus for Fourier transform on the group $GL(2,R)$Jan 29 2018Consider the Fourier transform on the group $GL(2,R)$ of real $2\times 2$-matrices. We show that Fourier-images of polynomial differential operators on $GL(2,R)$ are differential-difference operators with coefficients meromorphic in parameters of representations. ... More

The point equivalence problem for ordinary differential equations of the second orderJan 28 2018We use E. Cartan's method to solve the problem of equivalence of the second order ordinary differential equations with respect to the pseudogroup of point transformations.

Weighted Morrey spaces related to certain nonnegative potentials and Riesz transformsJan 28 2018Let $\mathcal L=-\Delta+V$ be a Schr\"odinger operator, where $\Delta$ is the Laplacian on $\mathbb R^d$ and the nonnegative potential $V$ belongs to the reverse H\"older class $RH_q$ for $q\geq d$. The Riesz transform associated with the operator $\mathcal ... More

Exact expansions of Hankel transforms and related integralsJan 28 2018The Hankel transform H_nu(q)=int_0^{infty}xf(x)J_nu(qx)dx is studied for a positive parameter q. Two particular cases nu=0 and nu=1 are investigated in much more detail. It is shown that the Hankel transforms are given by uniformly and absolutely convergent ... More

Set Theory and the AnalystJan 27 2018This survey is motivated by specific questions arising in the similarities and contrasts between (Baire) category and (Lebesgue) measure -- category-measure duality and non-duality, as it were. The bulk of the text is devoted to a summary, intended for ... More

Smoothness of Topological Equivalence on the Half Line for Nonautonomous SystemsJan 26 2018We study the differentiabilty of the topological equivalence between a uniformly asymptotically stable linear nonautonomous system and a perturbed system with suitable nonlinearities. For this purpose, we construct a uniform homeomorphism inspired in ... More

New bounds on the dimensions of planar distance setsJan 26 2018We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if $A\subset\mathbb{R}^2$ is a Borel set of Hausdorff dimension $s>1$, then its distance set has Hausdorff dimension at least ... More

The Toda and Painlevé systems associated with semiclassical matrix-valued orthogonal polynomials of Laguerre typeJan 26 2018Consider the Laguerre polynomials and deform them by the introduction in the measure of an exponential singularity at zero. In "Painlev\'e III and a singular linear statistics in Hermitian random matrix ensembles, I", the authors proved that this deformation ... More

A class of Meijer's G functions and further representations of the generalized hypergeometric functionsJan 26 2018In this paper we investigate the Meijer's $G$ function $G^{p,1}_{p+1,p+1}$ which for certain parameter values represents the Riemann-Liouville fractional integral of Meijer-N{\o}rlund function $G^{p,0}_{p,p}$. Our results for $G^{p,1}_{p+1,p+1}$ include: ... More

Biorthogonal Polynomial System Composed of X-Jacobi Polynomials from Different SequencesJan 26 2018The paper examines rational Darboux transformations (RDTs) of the Jacobi equation written in the canonical form, with emphasis on the Sturm-Liouville problems (SLPs) solved under the Dirichlet boundary conditions (DBCs) at the ends of the infinite interval ... More

Failure of $L^2$ boundedness of gradients of single layer potentials for measures with zero low densityJan 25 2018Consider a totally irregular measure $\mu$ in $\mathbb{R}^{n+1}$, that is, the upper density $\limsup_{r\to0}\frac{\mu(B(x,r))}{(2r)^n}$ is positive $\mu$-a.e.\ in $\mathbb{R}^{n+1}$, and the lower density $\liminf_{r\to0}\frac{\mu(B(x,r))}{(2r)^n}$ vanishes ... More

On a new $q$-analogue of Appell polynomialsJan 24 2018A new $q$-analogue of Appell polynomial sequences and their generalizations are introduced and their main characterizations are proved. As consequences new $q$-analogue of Bernoulli and Euler polynomials and numbers is introduced, their main representations ... More

A recurrence relation for Wronskian Hermite polynomialsJan 24 2018We consider polynomials that are defined as Wronskians of certain sets of Hermite polynomials. Our main result is a recurrence relation for these polynomials in terms of those of one or two degrees smaller, which generalizes the well-known three term ... More

General and Refined Montgomery LemmataJan 23 2018Montgomery's Lemma on the torus $\mathbb{T}^d$ states that a sum of $N$ Dirac masses cannot be orthogonal to many low-frequency trigonometric functions in a quantified way. We provide an extension to general manifolds that also allows for positive weights: ... More

On the quasi-arithmetic Gauss-type iterationJan 23 2018For a sequence of continuous, monotone functions $f_1,\dots,f_n \colon I \to \mathbb{R}$ ($I$ is an interval) we define the mapping $M \colon I^n \to I^n$ as a Cartesian product of quasi-arithmetic means generated by $f_j$-s. It is known that, for every ... More

Center of mass distribution of the Jacobi unitary ensembles: Painleve V, asymptotic expansionsJan 23 2018In this paper, we study the probability density function, $\mathbb{P}(c,\alpha,\beta, n)\,dc$, of the center of mass of the finite $n$ Jacobi unitary ensembles with parameters $\alpha\,>-1$ and $\beta >-1$; that is the probability that ${\rm tr}M_n\in(c, ... More

Uniform asymptotic stability of a fractional tuberculosis modelJan 22 2018We propose a Caputo type fractional-order mathematical model for the transmission dynamics of tuberculosis (TB). Uniform asymptotic stability of the unique endemic equilibrium of the fractional-order TB model is proved, for any $\alpha \in (0, 1)$. Numerical ... More

Spherical means on the Heisenberg group: Stability of a maximal function estimateJan 22 2018Consider the surface measure $\mu$ on a sphere in a nonvertical hyperplane on the Heisenberg group $\mathbb{H}^n$, $n\ge 2$, and the convolution $f*\mu$. Form the associated maximal function $Mf=\sup_{t>0}|f*\mu_t|$ generated by the automorphic dilations. ... More

A maximal function approach to two-measure Poincaré inequalitiesJan 22 2018This paper extends the self-improvement result of Keith and Zhong in [16] to the two-measure case. Our main result shows that a two-measure $(p,p)$-Poincar\'e inequality for $1<p<\infty$ improves to a $(p,p-\varepsilon)$-Poincar\'e inequality for some ... More

Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifoldsJan 21 2018The sharp Wolff-type decoupling estimates of Bourgain--Demeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian manifolds, away from the ... More

The exact Power Law for Buffon's needle landing near some Random Cantor SetsJan 21 2018Jan 29 2018In this paper, we study the Favard length of some random Cantor sets of Hausdorff dimension 1. We start with a unit disk in the plane and replace the unit disk by $4$ disjoint subdisks (with equal distance to each other) of radius $1/4$ inside and tangent ... More

Interpolation of functional by integral continued C-fractionsJan 21 2018The functional interpolation problem on a continual set of nodes by an integral continued C-fraction is studied. The necessary and sufficient conditions for its solvability are found. As a particular case, the considered integral continued fraction contains ... More

Bilinear pseudo-differential operators with exotic symbols, IIJan 21 2018The boundedness from $H^p \times L^2$ to $L^r$, $1/p+1/2=1/r$, and from $H^p \times L^{\infty}$ to $L^p$ of bilinear pseudo-differential operators is proved under the assumption that their symbols are in the bilinear H\"ormander class $BS^m_{\rho,\rho}$, ... More

Bilinear pseudo-differential operators with exotic symbolsJan 21 2018The boundedness from $L^p \times L^q$ to $L^r$, $1<p,q \le \infty$, $0<1/p+1/q=1/r \le 1$, of bilinear pseudo-differential operators with symbols in the bilinear H\"ormander class $BS^m_{\rho,\rho}$, $0 \le \rho <1$, is proved for the critical order $m$. ... More

Ramanujan's Master Theorem and two formulas for zero-order Hankel transformJan 19 2018Using Ramanujan's Master Theorem, two formulas are derived which define the Hankel transforms of order zero with even functions by inverse Mellin transforms, provided these functions and their derivatives obey special conditions. Their validity is illustrated ... More

A Kotel'nikov Representation for WaveletsJan 17 2018This paper presents a wavelet representation using baseband signals, by exploiting Kotel'nikov results. Details of how to obtain the processes of envelope and phase at low frequency are shown. The archetypal interpretation of wavelets as an analysis with ... More

Orthogonality from linear combination of $R_I$ polynomialsJan 17 2018In this paper, a sequence of linear combination of $R_{I}$ type polynomials such that the terms in this sequence have a common zero is constructed. A biorthogonality relation arising from such a sequence is discussed. Besides a sequence of para-orthogonal ... More

Nonlocal Representation of the $sl(2,R)$ Algebra for the Chazy equationJan 15 2018A demonstration of how the point symmetries of the Chazy Equation become nonlocal symmetries for the reduced equation is discussed. Moreover we construct an equivalent third-order differential equation which is related to the Chazy Equation under a generalized ... More

On difference operators for symmetric Krall-Hahn polynomialsJan 15 2018It has been recently shown that the problem of finding measures whose orthogonal polynomials are also eigenfunctions of higher-order difference operators can be solved by multiplying the classical discrete measures by suitable polynomials. This solved ... More

Rational Solutions of the Painlevé-III EquationJan 13 2018All of the six Painlev\'e equations except the first have families of rational solutions, which are frequently important in applications. The third Painlev\'e equation in generic form depends on two parameters $m$ and $n$, and it has rational solutions ... More

Sequential derivativesJan 12 2018Consider a real valued function defined, but not differentiable at some point. We use sequences approaching the point of interest to define and study sequential concepts of secant and cord derivatives of the function at the point of interest. If the function ... More

Quantitative Fatou Theorems and Uniform RectifiabilityJan 04 2018We show that a suitable quantitative Fatou Theorem characterizes uniform rectifiability in the codimension 1 case.

The proof of conjecture of Brutman and PassowJan 04 2018We consider the conjecture of Brutman and Pasow on a totality divided differences and prove the conjecture for continuous functions.

Symmetry breaking for representations of rank one orthogonal groups IIDec 30 2017For a pair $(G,G')=(O(n+1,1), O(n,1))$ of reductive groups, we investigate intertwining operators (symmetry breaking operators) between principal series representations $I_\delta(V,\lambda)$ of $G$, and $J_\varepsilon(W,\nu)$ of the subgroup $G'$. The ... More

Economic interpretation of fractional derivativesDec 27 2017An economic interpretation of the Caputo derivatives of non-integer orders is proposed. The suggested economic interpretation of the fractional derivatives is based on a generalization of average and marginal values of economic indicators. We formulate ... More

On the sharp constant in "magnetic" 1D embedding theoremDec 23 2017We study the sharp constant in the embedding theorem $H^1(0,2\pi)\to L_q(0,2\pi)$ in the case where the norm in $H^1(0,2\pi)$ is defined via the energy of a 1D magnetic Schr\"odinger operator.

From Symmetry to MonotonicityDec 22 2017We offer an alternative and shorter proof to a result by Jan J.Ub\o{}e about monotonicity properties of a one-dimensional function that appeared in the Mathematical Intelligencer in 2015. Our proof is based on reducing the problem to symmetry properties ... More

An Application of the Schur Complement to Truncated Matricial Power Moment ProblemsDec 19 2017The main goal of this paper is to reconsider a phenomenon which was treated in earlier work of the authors' on several truncated matricial moment problems. Using a special kind of Schur complement we obtain a more transparent insight into the nature of ... More

Strong convergence of two--dimensional Vilenkin-Fourier seriesDec 15 2017We prove that certain means of the quadratical partial sums of the two-dimensional Vilenkin-Fourier series are uniformly bounded operators from the Hardy space $H_{p}$ to the space $L_{p}$ for $0<p\leq 1.$ We also prove that the sequence in the denominator ... More

Asymptotics of Chebyshev Polynomials, III. Sets Saturating SzegŐ, Schiefermayr, and Totik--Widom BoundsDec 10 2017We determine which sets saturate the Szeg}o and Schiefermayr lower bounds on the norms of Chebyshev Polynomials. We also discuss sets that saturate the Totik--Widom upper bound.

Dimension structuresNov 02 2017Dec 14 2017We are going to introduce a new algebraic, analytic structure that is a kind of generalization of the Hausdorff dimension and measure. We give many examples and study the basic properties and relations of such systems.

Matricial Canonical Moments and Parametrization of Matricial Hausdorff Moment SequencesNov 02 2017In this paper we study moment sequences of matrix-valued measures on compact intervals. A complete parametrization of such sequences is obtained via a symmetric version of matricial canonical moments. Furthermore, distinguished extensions of finite moment ... More

Asymptotics of Chebyshev Polynomials, II. DCT Subsets of $\mathbb{R}$Sep 20 2017We prove Szeg\H{o}-Widom asymptotics for the Chebyshev polynomials of a compact subset of $\mathbb{R}$ which is regular for potential theory and obeys the Parreau-Widom and DCT conditions.

Uniqueness of Limit Cycles for Quadratic Vector FieldsSep 04 2017This article deals with the study of the number of limit cycles surrounding a critical point of a quadratic planar vector field, which, in normal form, can be written as $x'= a_1 x-y-a_3x^2+(2 a_2+a_5)xy + a_6 y^2$, $y'= x+a_1 y + a_2x^2+(2 a_3+a_4)xy ... More

Sharpening Hölder's inequalityAug 29 2017Nov 20 2017We strengthen H\"older's inequality. The new family of sharp inequalities we obtain might be thought of as an analog of Pythagorean theorem for the $L^p$ spaces. Our reasonings rely upon Bellman functions of four variables.

On functional equations characterizing derivations: methods and examplesAug 25 2017Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and automorphisms are additive ... More

Reifenberg Flatness and Oscillation of the Unit Normal VectorAug 17 2017We show (under mild topological assumptions) that small oscillation of the unit normal vector implies Reifenberg flatness. We then apply this observation to the study of chord-arc domains and to a quantitative version of a two-phase free boundary problem ... More

Measuring sets by meansAug 16 2017We are going to classify sets by a given mean in two ways. Firstly we study small and big sets regarding a given mean. Secondly we study sets that have the same weight according to a mean. We also generalize the notion of roundness and get another way ... More

The magnitude of odd balls via Hankel determinants of reverse Bessel polynomialsAug 10 2017Magnitude is an invariant of metric spaces with origins in enriched category theory. Using potential theoretic methods, Barcel\'o and Carbery gave an algorithm for calculating the magnitude of any odd dimensional ball in Euclidean space, and they proved ... More

Cyclicity in $\ell^p$ spaces and zero sets of the Fourier transformsJul 31 2017Jan 10 2018We study the cyclicity of vectors $u$ in $\ell^p(\mathbb{Z})$. It is known that a vector $u$ is cyclic in $\ell^2(\mathbb{Z})$ if and only if the zero set, $\mathcal{Z}(\widehat{u})$, of its Fourier transform, $\widehat{u}$, has Lebesgue measure zero ... More

A variation on the theme of NicomachusJul 29 2017We prove some conjectures of K. Stolarsky concerning the first and third moment of the Beatty sequences with the golden section and its square.

Non-local self-improving properties: A functional analytic approachJul 19 2017Jul 21 2017A functional analytic approach to obtaining self-improving properties of solutions to linear non-local elliptic equations is presented. It yields conceptually simple and very short proofs of some previous results due to Kuusi-Mingione-Sire and Bass-Ren. ... More

On the truncated matricial Stieltjes moment problem $\mathsf{M}[[α,\infty);(s_j)_{j=0}^m,\leq]$Jul 19 2017This paper gives via Stieltjes transform a complete description of the solution set of a matricial truncated Stieltjes-type power moment problem in the non-degenerate and degenerate cases. The approach is based on the Schur type algorithm which was worked ... More

Non-local Gehring lemmasJul 07 2017We prove a self-improving property for reverse H{\"o}lder inequalities with non-local right hand side. We attempt to cover all the most important situations that one encounters when studying elliptic and parabolic partial differential equations as well ... More

Interior of sums of planar sets and curvesJul 05 2017Recently, considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the interior $\left(A+\Gamma\right)^{\circ}$, when $\Gamma$ is a piecewise $\mathcal{C}^2$ curve and $A\subset \mathbb{R}^2.$ ... More

Dimension and measure of sums of planar sets and curvesJul 05 2017Considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the measure and dimension of $A+\Gamma:=\left\{a+g:a\in A, g\in B\right\}$ when $\Gamma$ is a piecewise $\mathcal{C}^2$ curve and $A\subset ... More

Lagrangian Transport Through Surfaces in Compressible FlowsJul 03 2017Sep 28 2017A material-based, i.e., Lagrangian, methodology for exact integration of flux by volume-preserving flows through a surface has been developed recently in [Karrasch, SIAM J. Appl. Math., 76 (2016), pp. 1178-1190]. In the present paper, we first generalize ... More

On regularity of weak solutions to parabolic systemsJun 26 2017We establish a new regularity property for weak solutions of parabolic systems with coefficients depending measurably on time as well as on all spatial variables. Namely, weak solutions are locally H\"older continuous Lp valued functions for some p > ... More

Measures by means, means by measuresJun 12 2017Aug 03 2017We construct measures which determines ordinary means in a very natural way. Using that measure we can extend the mean to infinite sets as well. E.g. we can calculate the geometric mean of any set with positive Lebesgue measure. We also study the properties ... More

Means of infinite sets IIMay 17 2017Aug 05 2017We continue the study of how one can define means of infinite sets. We introduce many new properties, investigate their relations to each other and how they can typify a mean. We collect the properties in property groups e.g. for monotonicity and continuity ... More

Density solutions to a class of integro-differential equationsApr 26 2017We consider the integro-differential equation ${\rm I}^{\alpha}_{0+}f= x^m f$ on the half-line. We show that there exists a density solution, which is then unique and can be expressed in terms of the Beta distribution, if and only if $m> \alpha.$ These ... More

Means of infinite sets IApr 24 2017May 17 2017We begin the study of how one can define means on infinite sets. We investigate many definitions, their properties and their relations to each other. One method is based on sequences of ideals, other deal with accumulation/isolated points, other with ... More

On bounding exact models of epidemic spread on networksApr 06 2017In this paper we use comparison theorems from classical ODE theory in order to rigorously show that the N-Intertwined Mean-Field Approximation (NIMFA) model provides an upper estimate on the exact stochastic process. The proof of the results relies on ... More

Extending means to several variablesApr 05 2017We begin the study of how to extend few variable means to several variable ones and how to shrink means of several variables to less variables. With the help of one of the techniques we show that it is enough to check an inequality between two quasi-arithmetic ... More

Cyclicity in weighted $\ell^p$ spacesMar 08 2017We study the cyclicity in weighted $\ell^p(\mathbb{Z})$ spaces. For $p \geq 1$ and $\beta \geq 0$, let $\ell^p\_\beta(\mathbb{Z})$ be the space of sequences $u=(u\_n)\_{n\in \mathbb{Z}}$ such that $(u\_n |n|^{\beta})\in \ell^p(\mathbb{Z}) $. We obtain ... More

On spectral gaps of Markov mapsJan 31 2017Feb 16 2017It is shown that if a Markov map $T$ on a noncommutative probability space $\mathcal{M}$ has a spectral gap on $L_2(\mathcal{M})$, then it also has one on $L_p(\mathcal{M})$ for $1<p<\infty$. For fixed $p$, the converse also holds if $T$ is factorizable. ... More

Digit frequencies and self-affine sets with non-empty interiorJan 24 2017In this paper we study digit frequencies in the setting of expansions in non-integer bases, and self-affine sets with non-empty interior. Within expansions in non-integer bases we show that if $\beta\in(1,1.787\ldots)$ then every $x\in(0,\frac{1}{\beta-1})$ ... More

On the structure of Hausdorff moment sequences of complex matricesJan 16 2017Jan 25 2017The paper treats several aspects of the truncated matricial $[\alpha,\beta]$-Hausdorff type moment problems. It is shown that each $[\alpha,\beta]$-Hausdorff moment sequence has a particular intrinsic structure. More precisely, each element of this sequence ... More

Revisiting Lie integrability by quadratures from a geometric perspectiveJan 14 2017After a short review of the classical Lie theorem, a finite dimensional Lie algebra of vector fields is considered and the most general conditions under which the integral curves of one of the fields can be obtained by quadratures in a prescribed way ... More

Counting the number of master integrals for sunrise-type Feynman diagrams via Mellin-Barnes representationDec 20 2016Jun 30 2017A number of irreducible master integrals for L-loop sunrise-type and bubble Feynman diagrams with generic values of masses and external momenta are explicitly evaluated via Mellin-Barnes representation.

Generalized Sobolev orthogonal polynomials, matrix moment problems and integrable systemsDec 15 2016We introduce a large class of Sobolev bi-orthogonal polynomial sequences arising from a $LU$-factorizable moment matrix and associated with a suitable measure matrix that characterizes the Sobolev bilinear form. A theory of deformations of Sobolev bilinear ... More

Inequalities for means of two mean variablesDec 08 2016Motivated by the work of Anderson, Vamanamurthy and Vuorinen \cite{avv}, in this paper authors study the log-convexity and log-concavity of Power mean, Identric mean, weighted Power mean, Lehmer mean, Modified Alzer mean, and establish the relation of ... More

Uniform rectifiability, elliptic measure, square functions, and $\varepsilon$-approximabilityDec 08 2016Let $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 2$, be an open set with Ahlfors-David regular boundary satisfying the corkscrew condition. We consider a uniformly elliptic operator $L$ in divergence form associated with a matrix with real and merely bounded ... More

Discrete Reifenberg-type theoremDec 07 2016The paper proves that a bound on the averaged Jones' square function of a measure implies an upper bound on the measure. Various types of assumptions on the measure are considered. The theorem is a generalization of a result due to A. Naber and D. Valtorta ... More

Non-differentiable solutions for local fractional nonlinear Riccati differential equationsDec 07 2016We investigate local fractional nonlinear Riccati differential equations (LFNRDE) by transforming them into local fractional linear ordinary differential equations. The case of LFNRDE with constant coefficients is considered and non-differentiable solutions ... More

On the Laplace transform of absolutely monotonic functionsDec 07 2016We obtain necessary and sufficient conditions on a function in order that it be the Laplace transform of an absolutely monotonic function. Several closely related results are also given.

Total positivity of sums, Hadamard products and Hadamard powers: Results and counterexamplesDec 07 2016We show that, for Hankel matrices, total nonnegativity (resp. total positivity) of order r is preserved by sum, Hadamard product, and Hadamard power with real exponent t \ge r-2. We give examples to show that our results are sharp relative to matrix size ... More

Extended relativistic Toda lattice and L-orthogonal polynomials on the real line and on the unit circleDec 06 2016The coefficients of the recurrence relation of orthogonal polynomials, when the measure varies parametrically in a certain way, satisfy the so-called Toda lattice. In this paper we show that the coefficients of the recurrence relation of L-orthogonal ... More

Asymptotically sharp reverse Hölder inequalities for flat Muckenhoupt weightsDec 06 2016We present reverse H\"older inequalities for Muckenhoupt weights in $\mathbb{R}^n$ with an asymptotically sharp behavior for flat weights, namely $A_\infty$ weights with Fujii-Wilson constant $(w)_{A_\infty}\to 1^+$. That is, the local integrability exponent ... More

An Ikehara type Tauberian theorem with $O(x^γ)$ remainderDec 06 2016Motivated by analytic number theory, we prove a remainder version of Ikehara's Tauberian theorem, yielding a result $f(x)=Ax+O(x^\gamma)$ with $\gamma<1$. The result is weaker than what one might naively hope for, but we prove that it is essentially optimal. ... More

A Maximal Extension of the Best-Known Bounds for the Furstenberg-Sárközy TheoremDec 06 2016We show that if $h\in \mathbb{Z}[x]$ is a polynomial of degree $k \geq 2$ such that $h(\mathbb{N})$ contains a multiple of $q$ for every $q\in \mathbb{N}$, known as an $\textit{intersective polynomial}$, then any subset of $\{1,2,\dots,N\}$ with no nonzero ... More

A linear system of differential equations related to vector-valued Jack polynomials on the torusDec 05 2016For each irreducible module of the symmetric group $\mathcal{S}_{N}$ there is a set of parametrized nonsymmetric Jack polynomials in $N$ variables taking values in the module. These polynomials are simultaneous eigenfunctions of a commutative set of operators, ... More

Asymptotics for moments of certain cotangent sums for arbitrary exponentsDec 05 2016In this paper we extend a result on the asymptotics of moments of certain cotangent sums associated to the Estermann and Riemann zeta functions established in a previous paper for integer exponents to arbitrary positive real exponents.

Real solutions of the first Painlevé equation with large initial dataDec 05 2016We consider three special cases of the initial value problem of the first Painlev\'e equation (PI). Our approach is based on the method of uniform asymptotics introduced by Bassom, Clarkson, Law and McLeod. A rigorous proof of a property of the PI solutions ... More