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Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We show that if there exists a continuous linear unbounded operator between l-K\"{o}the spaces, then there exists a continuous unbounded quasi-diagonal operator between them. Using this result, we study in terms of corresponding K\"{o}the matrices when every continuous linear operator between l-K\"{o}the spaces is bounded. As an application, we observe that the existence of an unbounded operator between l-K\"{o}the spaces, under a splitting condition, causes the existence of a common basic subspace.

Let Em and En be orthogonal Euclidean spaces of dimensions m and n respectively and with the origin of each as their only common point. In a previous paper (3) I gave what was intended to be a proof of the relationwhere the dimension of A, dim A, is the Besicovitch dimension, i.e. the number s such that the Hausdorff measure in any dimension greater than s is zero whilst that in any dimension less than s is infinite, where A and B are subsets of En and Em respectively and where A × B is the Cartesian product of A with B.

Within the class of offspring distributions with given meanm>1 and probability of no offspringp o, the probabilityq of ultimate extinction in a Galton-Watson branching process starting from one individual satisfiesp 0

M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.

In this short paper we study the d-Hermite condition about stably free modules for skew $$\textit{PBW}$$ extensions. For this purpose, we estimate the stable rank of these non-commutative rings. In addition, and closely related with these questions, we will prove Kronecker’s theorem about the radical of finitely generated ideals for some particular types of skew $$\textit{PBW}$$ extensions.

PROOF. By a theorem of Rudin a function gEH' in U whose boundary values are real a.e. on I can be analytically continued to D [3, p. 59]. The lemma follows on applying Rudin's result to gi= (1/2) (fl+f2) and g2=(i/2) (fi-f2). PROOF OF THEOREM 1. By a well-known decomposition theorem [2, p. 87], f(z)=B(z)F2(Z), where B(z) is a Blaschke product and F(z) EH1. Since the boundary values of B (z) have absolute value one a.e. on K, we have a.e. on I, f(ei0)= |f(eio)I, or B(ei0)F2(ei0) = F2(ei0) |, and hence

We shall present partial solutions to the conjecture such that $(q^{p}-1)/(q-1)$ does not divide $(p^{q}-1)/(p-1)$ for distinct primes $p < q$.

Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-27T11:25:28Z No. of bitstreams: 0Bitstream added on 2014-05-27T14:41:41Z : No. of bitstreams: 1 2-s2.0-79951846250.pdf: 494002 bytes, checksum: 56b6ee8beddda3e7dae971355d44a19f (MD5) Made available in DSpace on 2014-05-27T11:25:28Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-03-01 Item merged in doublecheck by Felipe Arakaki (arakaki@reitoria.unesp.br) on 2015-12-11T17:28:11Z Item was identical to item(s): 71803, 21370 at handle(s): http://hdl.handle.net/11449/72321, http://hdl.handle.net/11449/21804 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) The simplest necessary conditions for an entire function ψ(x) =∞ ∑ k=0 γk xk/k! to be in the Laguerre-Pólya class are the Turán inequalities γ2 k- γk+1γk-1 ≥ 0. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with ψ to be hyperbolic. The higher order Turán inequalities 4(γ2 n - γn-1γn+1)(γ2n +1 - γnγn+2) - (γnγn+1 - γn-1γn+2) 2 ≥ 0 are also necessary conditions for a function of the above form to belong to the Laguerre-Pólya class. In fact, these two sets of inequalities guarantee that the third degree generalized Jensen polynomials are hyperbolic. Pólya conjectured in 1927 and Csordas, Norfolk and Varga proved in 1986 that the Turán inequalities hold for the coefficients of the Riemann ψ-function. In this short paper, we prove that the higher order Turán inequalities also hold for the ψ-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three. © 2010 American Mathematical Society. Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP Departamento de matemática Aplicada IMECC UNICAMP, 13083-859 Campinas, SP Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP FAPESP: 03/01874-2 FAPESP: 06/60420-0 CNPq: 305622/2009-9 CAPES: DGU-160

In this paper we determine two asymptotic results for Jack measures on partitions, a model defined by two specializations of Jack polynomials proposed by Borodin-Olshanski in [European J. Combin. 26.6 (2005): 795-834]. Assuming these two specializations are the same, we derive limit shapes and Gaussian fluctuations for the anisotropic profiles of these random partitions in three asymptotic regimes associated to diverging, fixed, and vanishing values of the Jack parameter. To do so, we introduce a generalization of Motzkin paths we call "ribbon paths", show for general Jack measures that certain joint cumulants are weighted sums of connected ribbon paths on $n$ sites with $n-1+g$ pairings, and derive our two results from the contributions of $(n,g)=(1,0)$ and $(2,0)$, respectively. Our analysis makes use of Nazarov-Sklyanin's spectral theory for Jack polynomials. As a consequence, we give new proofs of several results for Schur measures, Plancherel measures, and Jack-Plancherel measures. In addition, we relate our weighted sums of ribbon paths to the weighted sums of ribbon graphs of maps on non-oriented real surfaces recently introduced by Chapuy-Dol\k{e}ga., Comment: Several results in this paper first appeared in the author's unpublished monograph arXiv:1508.03063. Version 2: revised and accepted for publication in International Mathematics Research Notices (IMRN)

The paper is devoted to the study of the metric properties of regular and semiregular polyhedra in Euclidean spaces. In the first part, we prove that every regular polytope of dimension greater or equal than 4, and different from 120-cell in $$\mathbb {E}^4 $$ is such that the set of its vertices is a Clifford–Wolf homogeneous finite metric space. The second part of the paper is devoted to the study of special properties of Archimedean solids. In particular, for each Archimedean solid, its description is given as the convex hull of the orbit of a suitable point of a regular tetrahedron, cube or dodecahedron under the action of the corresponding isometry group.

Let $S \subset \Bbb R^n$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\Bbb R^n$. If $X$ contains a ball of each radius, then the problem of determining the range of exponents $(p,q)$ for which the estimate $\| Ef \|_{L^q(X)} \leq C \| f \|_{L^p(S)}$ holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set $X$: there is a number $0 < \alpha \leq n$ such that $|X \cap B_R| \leq c \, R^\alpha$ for all balls $B_R$ in $\Bbb R^n$ of radius $R \geq 1$. On the left-hand side of this estimate, we are integrating the function $|Ef(x)|^q$ against the measure $\chi_X dx$. Our approach consists of replacing the characteristic function $\chi_X$ of $X$ by an appropriate weight function $H$, and studying the resulting estimate in three different regimes: small values of $\alpha$, intermediate values of $\alpha$, and large values of $\alpha$. In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted H\"{o}lder-type inequality that holds for general non-negative Lebesgue measurable functions on $\Bbb R^n$, and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du-Zhang theorem in the range $0 < \alpha < n/2$., Comment: 31 pages. Minor revision

A classical theorem of Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In the present paper we extend Koksma's theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every $x>1$ the pair correlations of the fractional parts of $(x^n)_{n=1}^{\infty}$ are asymptotically Poissonian. The proof is based on a martingale approximation method., Version 2: some minor changes. The paper will appear in the Israel Journal of Mathematics

Peg solitaire is a classical one-person game that has been played in various countries on different types of boards. Numerous studies have focused on the solvability of the games on these traditional boards and more recently on mathematical graphs. In this paper, we go beyond traditional peg solitaire and explore the solvability on graphs with pegs of more than one color and arrive at results that differ from previous works on the subject. This paper focuses on classifying the solvability of peg solitaire in three colors on several different types of common mathematical graphs, including the path, complete bipartite, and star. We also consider the solvability of peg solitaire on the Cartesian products of graphs.

This paper is devoted to a study of the concept of edge-group choosability of graphs. We say that G is edge- k -group choosable if its line graph is k -group choosable. In this paper, we study an edge-group choosability version of Vizing conjecture for planar graphs without 5-cycles and for planar graphs without noninduced 5-cycles (2010 Mathematics Subject Classification: 05C15, 05C20).

We revisit Rozansky’s construction of Khovanov homology for links in S 2 × S 1 , extending it to define the Khovanov homology Kh ( L ) for links L in M r = # r ( S 2 × S 1 ) for any r . The graded Euler characteristic of Kh ( L ) can be used to recover WRT invariants at certain roots of unity and also recovers the evaluation of L in the skein module S ( M r ) of Hoste and Przytycki when L is null-homologous in M r . The construction also allows for a clear path toward defining a Lee’s homology Kh ' ( L ) and associated s -invariant for such L , which we will explore in an upcoming paper. We also give an equivalent construction for the Khovanov homology of the knotification of a link in S 3 and show directly that this is invariant under handle-slides, in the hope of lifting this version to give a stable homotopy type for such knotifications in a future paper.

Michael Somos discovered several theta-function identities of various levels by computer and offered no proof for them. These identities highly resemble some of Ramanujan’s identities. The main focus of this paper is to prove some of these theta-function identities, in particular those of level 6 that have been discovered using computational searches. Some of the the Somos identities that we are discussing in this paper cannot be expressed in the form of P −−Q type. Furthermore, we establish certain colored partition identities for them.

For every monoid M of natural numbers defined a new class of periodic functions $$M_s^\alpha$$, which is a subclass of a known class of periodic functions Korobov $$E_s^\alpha$$. With respect to the norm $$\|f(\vec{x})\|_{E_s^\alpha}$$, the class $$M_s^\alpha$$ is an inseparable Banach subspace of class $$E_s^\alpha$$. It is established that the class $$M_s^\alpha$$ is closed with respect to the action of the Fredholm integral operator and the Fredholm integral equation of the second kind is solvable on this class. In this paper we obtain estimates of the image norm of the integral operator, which contain the kernel norm and the s-th degree of the Zeta function of the monoid M. Estimates are obtained for the parameter $$\lambda$$, in which the integral operator $$A_{\lambda,f}$$ is a compression. The theorem on the representation of the unique solution of Fredholm integral equation of the second kind in the form of Neumann series is proved. The paper deals with the problems of solving the partial differential equation with the differential operator $$Q\left(\frac{\partial }{\partial x_1},\ldots,\frac{\partial }{\partial x_s}\right)$$ in the space $$M^\alpha_{s}$$, which depends on the arithmetic properties of the spectrum of this operator. A paradoxical fact is found that for a monoid $$M_{q,1}$$ of numbers comparable to 1 modulo q, a quadrature formula with a parallelepiped grid for an admissible set of coefficients modulo q is exact on the class $$M_{q,1,s}^\alpha$$. Moreover, this statement remains true for the class $$M_{q,a,s}^\alpha$$ with 1 < a < q when q is a Prime number. Since the functions of class $$M_{q,a,s}^\alpha$$ with 1 < a < q do not have a zero Fourier coefficient $$C(\vec{0})$$, then for a simple q the sum of the function values at the nodes of the corresponding parallelepipedal grid will be zero.

Let $G$ be a connected complex semi-simple Lie group, and let $Z_{\bf u}$ be an $n$-dimensional Bott-Samelson variety of $G$, where ${\bf u}$ is any sequence of simple reflections in the Weyl group of $G$. We study the Poisson structure $\pi_n$ on $Z_{\bf u}$ defined by a standard multiplicative Poisson structure $\pi_{\rm st}$ on $G$. We explicitly express $\pi_n$ on each of the $2^n$ affine coordinate charts, one for every subexpression of ${\bf u}$, in terms of the root strings and the structure constants of the Lie algebra of $G$. We show that the restriction of $\pi_n$ to each affine coordinate chart gives rise to a Poisson structure on the polynomial algebra ${\mathbb{C}}[z_1, \ldots, z_n]$ which is an {\it iterated Poisson Ore extension} of $\mathbb{C}$ compatible with a rational action by a maximal torus of $G$. For canonically chosen $\pi_{\rm st}$, we show that the induced Poisson structure on ${\mathbb{C}}[z_1, \ldots, z_n]$ for every affine coordinate chart is in fact defined over ${\mathbb Z}$, thus giving rise to an iterated Poisson Ore extension of any field ${\bf k}$ of arbitrary characteristic. The special case of $\pi_n$ on the affine chart corresponding to the full subexpression of ${\bf u}$ yields an explicit formula for the standard Poisson structures on {\it generalized Bruhat cells} in Bott-Samelson coordinates. The paper establishes the foundation on generalized Bruhat cells and sets up the stage for their applications, some of which are discussed in the Introduction of the paper.

Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Q n ( x , y ; N , p ) on the multinomial distribution which are sums of products of orthonormal polynomials in x and y of fixed total degree n = 0 , 1 , … , N . The Poisson kernel ∑ n = 0 N ρ n Q n ( x , y ; N , p ) arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new duplication formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The duplication formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The χ 2 cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.

In this paper, we introduce a general family of $q$-hypergeometric polynomials and investigate several $q$-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of $q$-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized $q$-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various $q$-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called $(p,q)$-variations of the $q$-results, which we have investigated here, because the additional parameter $p$ is obviously redundant., 14 pages

This paper studies the Sobolev-Lorentz capacity and its regularity in the Euclidean setting for $n \ge 1$ integer. We extend here our previous results on the Sobolev-Lorentz capacity obtained for $n \ge 2.$ Moreover, for $n \ge 2$ integer we obtain a few new results concerning the $n,1$ relative and global capacities. We obtain sharp estimates for the $n,1$ relative capacity of the concentric condensers $(\overline{B}(0,r), B(0,1))$ for all $r$ in $[0,1).$ As a consequence we obtain the exact value of the $n,1$ capacity of a point relative to all its bounded open neighborhoods from ${\mathbf{R}}^n$ when $n \ge 2.$ We also show that this aforementioned constant is the value of the $n,1$ global capacity of any point from ${\mathbf{R}}^n,$ where $n \ge 2$ is integer. This allows us to give a new proof of the embedding $H_{0}^{1,(n,1)}(\Omega) \hookrightarrow C(\overline{\Omega}) \cap L^{\infty}(\Omega),$ where $\Omega \subset {\mathbf{R}}^n$ is open and $n \ge 2$ is an integer. In the penultimate section of our paper we prove a new weak convergence result for bounded sequences in the non-reflexive spaces $H^{1,(p,1)}(\Omega)$ and $H_{0}^{1,(p,1)}(\Omega).$ The weak convergence result concerning the spaces $H^{1,(p,1)}(\Omega)$ is valid whenever $1

In 2005, using a famous lemma of Atkin and Swinnerton-Dyer (Some properties of partitions, Proc. Lond. Math. Soc. (3)4(1954), 84–106), Yesilyurt (Four identities related to third order mock theta functions in Ramanujan’s lost notebook, Adv. Math. 190(2005), 278–299) proved four identities for third order mock theta functions found on pages 2 and 17 in Ramanujan’s lost notebook. The primary purpose of this paper is to offer new proofs in the spirit of what Ramanujan might have given in the hope that a better understanding of the identities might be gained. Third order mock theta functions are intimately connected with ranks of partitions. We prove new dissections for two rank generating functions, which are keys to our proof of the fourth, and the most difficult, of Ramanujan’s identities. In the last section of this paper, we establish new relations for ranks arising from our dissections of rank generating functions.

Let rk(n) denote the number of lattice points on a k-dimensional sphere of radius $$ \sqrt{n} $$ . The generating function $$ {\zeta}_k(s)=\sum \limits_{n=1}^{\infty }{r}_k(n){n}^{-s},\kern0.5em k\ge 2, $$ is Epstein’s zeta function. The paper considers the Riesz mean of the type $$ {D}_{\rho}\left(x;{\zeta}_3\right)=\frac{1}{\Gamma \left(\rho +1\right)}\sum \limits_{n\le x}{\left(x-n\right)}^{\rho }{r}_3(n), $$ where ρ > 0; the error term Δρ(x; ζ3) is defined by $$ {D}_{\rho}\left(x;{\zeta}_3\right)=\frac{\uppi^{3/2}{x}^{\rho +3/2}}{\Gamma \left(\rho +5/2\right)}+\frac{x^{\rho }}{\Gamma \left(\rho +1\right)}{\zeta}_3(0)+{\Delta}_{\rho}\left(x;{\zeta}_3\right). $$ K. Chandrasekharan and R. Narasimhan (1962, MR25#3911) proved that $$ {\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{{\displaystyle \begin{array}{ll}O\Big({x}^{1/2+\rho /2\Big)}& \left(\rho >1\right),\\ {}{\Omega}_{\pm}\left({x}^{1/2+\rho /2}\right)& \left(\rho \ge 0\right).\end{array}} $$ In the present paper, it is proved that $$ {\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{{\displaystyle \begin{array}{ll}O\left(x\log x\right)& \left(\rho =1\right),\\ {}O\left({x}^{2/3+\rho /3+\varepsilon}\right)& \left(1/2

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that for all $x \geq R_n$ the interval $(x/2, x]$ contains at least $n$ primes. In this paper we undertake a study of the sequence $(\pi(R_n))_{n \geq 1}$, which tells us where the $n$th Ramanujan prime appears in the sequence of all primes. In the first part we establish new explicit upper and lower bounds for the number of primes up to the $n$th Ramanujan prime, which imply an asymptotic formula for $\pi(R_n)$ conjectured by Yang and Togbé. Then we use these explicit estimates to derive a result concerning an inequality involving $\pi(R_n)$ that was conjectured by Sondow, Nicholson, and Noe. In the second part of the paper, we apply the results derived in the first part to obtain some new results concerning the number of Ramanujan primes not exceeding $x$. Finally, we compute the limit of $((R_n - p_{2n})/n)_{n \geq 1}$.

In general, finding the sum of an infinite series is not always possible. However, there are infinite series whose sums can be computed. In our earlier paper, we derived a formula for computing the sum of a type of polylogarithm series involving multinomial coefficients. In this paper, we show that the formula leads to an elementary computation for the series $\sum_{k=1}^{\infty}\frac{k^n}{a^k}$ involving numbers obtained by a method similar to Pascal's triangle. We also show that our result is the number of ways of distributing $n$ distinct objects in $n$ or fewer distinct nonempty cells.

This paper sets out to extend the results in the paper Geodesic Continued Fractions to continued fractions with Gaussian integer coefficients. The Farey graph $\mathcal{F}$ , whose vertices are reduced Gaussian rationals in $\mathbb{Q}_{\infty }(i)$ and whose edges join Farey neighbors, is introduced. The graph is modeled by the concrete realization in $\mathbb{H}^{3}$ where Farey neighbors are joined by hyperbolic geodesics (Farey geodesics) as seen in the Farey tessellation of $\mathbb{H}^{3}$ by Farey octahedrons. A natural distance $\varrho $ on $\mathbb{Q}_{\infty }(i)$ is also recalled, where $\varrho (\infty ,w)=n$ is the least number of edges in $\mathcal{F}$ from $\infty $ to $w\in \mathbb{Q}(i)$ , where $n$ is called the generation of w and a relevant path in $\mathcal{F}$ is called a geodesic expansion for $w$ . The Farey neighborhood of a reduced Gaussian rational is introduced and partitioned into neighbors of generation $n-1$ , $n$ , and $n+1$ . Subsequently, it is seen that there can be at most four Farey neighbors of generation $n-1$ in the neighborhood. An ancestral path is introduced, and a bound on the number of geodesic paths to any $w$ is established. Central to the paper are conditions for a path to be a geodesic path. The paper also addresses conditions for the existence of an infinite geodesic Gaussian integer continued fraction and suggestions of extending the paper to continued fraction with integer quaternion entries.

We have studied the k-rainbow domination number of C n □ C m for k &ge, 4 (Gao et al. 2019), in which we present the 3-rainbow domination number of C n □ C m , which should be bounded above by the four-rainbow domination number of C n □ C m . Therefore, we give a rough bound on the 3-rainbow domination number of C n □ C m . In this paper, we focus on the 3-rainbow domination number of the Cartesian product of cycles, C n □ C m . A 3-rainbow dominating function (3RDF) f on a given graph G is a mapping from the vertex set to the power set of three colors { 1 , 2 , 3 } in such a way that every vertex that is assigned to the empty set has all three colors in its neighborhood. The weight of a 3RDF on G is the value &omega, ( f ) = &sum, v &isin, V ( G ) | f ( v ) | . The 3-rainbow domination number, &gamma, r 3 ( G ) , is the minimum weight among all weights of 3RDFs on G. In this paper, we determine exact values of the 3-rainbow domination number of C 3 □ C m and C 4 □ C m and present a tighter bound on the 3-rainbow domination number of C n □ C m for n &ge, 5 .

In 1900, at the international congress of mathematicians, Hilbert claimed that the Riemann zeta function $\zeta(s)$ is not the solution of any algebraic ordinary differential equations on its region of analyticity. In 2015, Van Gorder considered the question of whether $\zeta(s)$ satisfies a non-algebraic differential equation and showed that it formally satisfies an infinite order linear differential equation. Recently, Prado and Klinger-Logan extended Van Gorder's result to show that the Hurwitz zeta function $\zeta(s,a)$ is also formally satisfies a similar differential equation \begin{equation*}\label{HurDE} T\left[\zeta (s,a) - \frac{1}{a^s}\right] = \frac{1}{(s-1)a^{s-1}}. \end{equation*} But unfortunately in the same paper they proved that the operator $T$ applied to Hurwitz zeta function $\zeta(s,a)$ does not converge at any point in the complex plane $\mathbb{C}$. In this paper, by defining $T_{p}^{a}$, a $p$-adic analogue of Van Gorder's operator $T,$ we establish an analogue of Prado and Klinger-Logan's differential equation satisfied by $\zeta_{p,E}(s,a)$ which is the $p$-adic analogue of the Hurwitz-type Euler zeta functions \begin{equation*}\label{HEZ} \zeta_E(s,a)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+a)^s}. \end{equation*} In contrast with the complex case, due to the non-archimedean property, the operator $T_{p}^{a}$ applied to the $p$-adic Hurwitz-type Euler zeta function $\zeta_{p,E}(s,a)$ is convergent $p$-adically in the area of $s\in\mathbb{Z}_{p}$ with $s\neq 1$ and $a\in K$ with $|a|_{p}>1,$ where $K$ is any finite extension of $\mathbb{Q}_{p}$ with ramification index over $\mathbb{Q}_{p}$ less than $p-1.$, Comment: 18 pages. Final version. Dedicated to the memory of Prof. David Goss (1952-2017)

The first named author is supported in part by NSF grant DMS-1855464, ISF grant 281/17, and the Simons Foundation. The second named author is supported in part by an OTKA NK 104183 grant. The third named author is supported in part by a NSERC and an OTKA NK 104183 grant. In their seminal paper Erdös and Szemerédi formulated conjectures on the size of sumset and product set of integers. The strongest form of their conjecture is about sums and products along the edges of a graph. In this paper we show that this strong form of the Erdös-Szemerédi conjecture does not hold. We give upper and lower bounds on the cardinalities of sumsets, product sets, and ratio sets along the edges of graphs.

This paper studies zeta functions of the form $\sum_{n=1}^{\infty} \chi(n) n^{-s}$, with $\chi$ a completely multiplicative function taking only unimodular values. We denote by $\sigma(\chi)$ the infimum of those $\alpha$ such that the Dirichlet series $\sum_{n=1}^{\infty} \chi(n) n^{-s}$ can be continued meromorphically to the half-plane $\operatorname{Re} s>\alpha$, and denote by $\zeta_{\chi}(s)$ the corresponding meromorphic function in $\operatorname{Re} s>\sigma(\chi)$. We construct $\zeta_{\chi}(s)$ that have $\sigma(\chi)\le 1/2$ and are universal for zero-free analytic functions on the half-critical strip $1/21$. Finally, we show that there exists $\zeta_{\chi}(s)$ with $\sigma(\chi) \le 1/2$ and zeros at any discrete multiset in the strip $1/21/2$; on the Riemann hypothesis, this strip may be replaced by the half-critical strip $1/2 < \operatorname{Re} s < 1$., Comment: This is the final version of the paper which has been accepted for publication in Journal d'Analyse Math\'{e}matique

In this paper, we are interested in the multiple solutions of the following fractional critical problem \[ \begin{cases} (-\Delta)^s u = |u|^{2_s^*-2} u + \lambda u &\textrm{in $\Omega$}, \\ u = 0 &\textrm{on $\mathbb{R}^N \setminus \Omega$}, \end{cases} \] where $s \in (0,1)$, $N \gt 4s$, $2^*_s = 2N/(N-2s)$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ and $(-\Delta)^s$ is the fractional Laplace operator. Because the nonlocal property of fractional Laplacian makes the variational functional of the fractional critical problem different from the one of local operator $-\Delta$. To the best of our knowledge, it is still unknown whether multiple solutions of the fractional critical problem exist for all $\lambda \gt 0$. In this paper, we give a partial answer. Precisely, by introducing some new ideas and careful estimates, we prove that for any $s \in (0,1)$, the fractional critical problem has at least $[(N+1)/2]$ pairs of nontrivial solutions if $0 \lt \lambda \neq \lambda_n$, and has $[(N+1-l)/2]$ pairs if $\lambda = \lambda_n$ with multiplicity number $0 \lt l \lt \min \{n,N+2\}$, via constraint method and Krasnoselskii genus. Here $\lambda_n$ denotes the $n$-th eigenvalue of $(-\Delta)^s$ with zero Dirichlet boundary data in $\Omega$ and $[a]$ denotes the least positive integer $k$ such that $k \geq a$.

In this paper, we improve two known invariant subspace theorems. More specifically, we show that a closed linear subspace M in the Hardy space H p ( D ) ( 1 &le, p &lt, &infin, ) is invariant under the shift operator M z on H p ( D ) if and only if it is hyperinvariant under M z , and that a closed linear subspace M in the Lebesgue space L 2 ( &part, D ) is reducing under the shift operator M e i &theta, on L 2 ( &part, D ) if and only if it is hyperinvariant under M e i &theta, At the same time, we show that there are two large classes of invariant subspaces for M e i &theta, that are not hyperinvariant subspaces for M e i &theta, and are also not reducing subspaces for M e i &theta, Moreover,we still show that there is a large class of hyperinvariant subspaces for M z that are not reducing subspaces for M z . Furthermore, we gave two new versions of the formula of the reproducing function in the Hardy space H 2 ( D ) , which are the analogue of the formula of the reproducing function in the Bergman space A 2 ( D ) . In addition, the conclusions in this paper are interesting now, or later if they are written into the literature of invariant subspaces and function spaces.

This paper studies Menon-type identities involving both multiplicative characters and additive characters. In the paper, we shall give the explicit formula of the following sum \[ \sum_{\substack{a \in \mathbb{Z}_n^{\ast} \\ b_1, \ldots, b_k \in \mathbb{Z}_n}} \gcd(a-1, b_1, \ldots, b_k, n) \chi(a) \lambda_1(b_1) \cdots \lambda_k(b_k), \] where for a positive integer $n$, $\mathbb{Z}_n^{\ast}$ is the group of units of the ring $\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z}$, $\gcd$ represents the greatest common divisor, $\chi$ is a Dirichlet character modulo $n$, and for a nonnegative integer $k$, $\lambda_1, \ldots, \lambda_k$ are additive characters of $\mathbb{Z}_n$. Our formula further extends the previous results by Sury [13], Zhao-Cao [17] and Li-Hu-Kim [4].

We study properties of sets having the minimum length (one-dimensional Hausdorff measure) in the class of closed connected sets Σ ⊂ ℝ2 satisfying the inequality max yϵM dist (y, Σ) ≤ r for a given compact set M ⊂ ℝ2 and given r > 0. Such sets play the role of the shortest possible pipelines arriving at a distance at most r to every point of M where M is the set of customers of the pipeline. In this paper, it is announced that every maximum distance minimizer is a union of finitely many curves having one-sided tangent lines at every point. This shows that a maximum distance minimizer is isotopic to a finite Steiner tree even for a “bad” compact set M, which distinguishes it from a solution of the Steiner problem (an example of a Steiner tree with infinitely many branching points can be found in a paper by Paolini, Stepanov, and Teplitskaya). Moreover, the angle between these lines at each point of a maximum distance minimizer is at least 2π/3. Also, we classify the behavior of a minimizer Σ in a neighborhood of any point of Σ. In fact, all the results are proved for a more general class of local minimizers.

Completely multiplicative functions whose sum is zero ($CMO$). The paper deals with $CMO$, meaning completely multiplicative ($CM$) functions $f$ such that $f(1)=1$ and $\sum\limits_1^\infty f(n)=0$. $CM$ means $f(ab)=f(a)f(b)$ for all $(a,b)\in \N^{*2}$, therefore $f$ is well defined by the $f(p)$, $p$ prime. Assuming that $f$ is $CM$, give conditions on the $f(p)$, either necessary or sufficient, both is possible, for $f$ being $CMO$ : that is the general purpose of the authors. The $CMO$ character of $f$ is invariant under slight modifications of the sequence $(f(p))$ (theorem~3). The same idea applies also in a more general context (theorem~4). After general statements of that sort, including examples of $CMO$ (theorem~5), the paper is devoted to ``small'' functions, that is, functions of the form $\frac{f(n)}{n}$, where the $f(n)$ are bounded. Here is a typical result : if $|f(p)|\le 1$ and $Re\, f(p)\le0$ for all $p$, a necessary and sufficient condition for $\big(\frac{f(n)}{n}\big)$ to be $CMO$ is $\sum \, Re\, f(p)/p=-\infty$ (theorem~8). Another necessary and sufficient condition is given under the assumption that $|1+f(p)|\le 1$ and $f(2)\not=-2$ (theorem~7). A third result gives only a sufficient condition (theorem~9). The three results apply to the particular case $f(p)=-1$, the historical example of Euler. Theorems 7 and 8 need auxiliary results, coming either from the existing literature (Hal\'asz, Montgomery--Vaughan), or from improved versions of classical results (Ingham, Ska\l ba) about $f(n)$ under assumptions on the $f*1(n)$, * denoting the multiplicative convolution (theorems~10~and~11).

The aim of the paper is to study spaces of multipliers acting from the Bessel potential space H (ℝ n ) to the other Bessel potential space H (ℝ n ). We obtain conditions ensuring the equivalence of uniform and standard multiplier norms on the space of multipliers $$M\left[ {H_p^s({\mathbb{R}^n}) \to H_q^t({\mathbb{R}^n})} \right]fors,t \in \mathbb{R},p,q > 1.$$ In the case $$p,q > 1,p \leqslant q,s > \frac{n}{p},t \geqslant 0,s - \frac{n}{p} \geqslant t - \frac{n}{q}$$ , the space M[H (ℝ n ) → H (ℝ n ) can be described explicitly. Namely, we prove in this paper that the latter space coincides with the space H , unif (ℝ n ) of uniformly localized Bessel potentials introduced by Strichartz. It is also proved that if both smoothness indices s and t are nonnegative, then such a description is possible only for the given values of the indices.

In this paper we show that any Bohr-zero non-periodic set B B of traceless integer-valued matrices, denoted by Λ \Lambda , intersects non-trivially the conjugacy class of any matrix from Λ \Lambda . As a corollary, we obtain that the family of characteristic polynomials of B B contains all characteristic polynomials of matrices from Λ \Lambda . The main ingredient used in this paper is an equidistribution result for an S L d ( Z ) SL_d(\mathbb {Z}) random walk on a finite-dimensional torus deduced from Bourgain-Furman-Lindenstrauss-Mozes work [J. Amer. Math. Soc. 24 (2011), 231–280].

In 1874, Franz Mertens proved the so-called Mertens’ theorem, and in 1974, Kenneth S. Williams showed Mertens’ theorem associated with a character. In a previous paper, we presented a graph-theoretic analogue to Williams’ theorem. In this paper, we generalize our previous work in the sense that a character is extended to a representation. To our knowledge, a number-theoretic analogue to our result is not yet known. So, we expect that, by using our methods, it can be proven in the future.

The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a "minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitely as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$., Comment: 22 pages; Minor corrections

The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin’s axiom for σ \sigma -centered posets. In his 1973 paper he showed under this assumption that both ω 1 {\omega }_{1} and the reals can be embedded. Analogous results were obtained later for the coarser notion of Tukey reducibility. We prove in this paper that Martin’s axiom for σ \sigma -centered posets implies that the Boolean algebra P ( ω ) / FIN \mathcal {P}(\omega ) / \operatorname {FIN} equipped with its natural partial order can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility. Consequently, the continuum hypothesis implies that every partial order of size at most continuum embeds into the P-points under both notions of reducibility.

The present paper studies several types of compatible adjacency relations for digital products such as a $C$-compatible adjacency (or the $L_C$-property in \cite{H13}), an $S$-compatible adjacency in \cite{H19} (or the $L_S$-property in \cite{H13}), which can contribute to the study of product properties of digital spaces (or digital images). Furthermore, to study an automorphism group of a Cartesian product of two digital coverings which do not satisfy a radius $2$ local isomorphism, which remains open, the paper uses some properties of an ultra regular covering in \cite{H16}. By using this approach, we can substantially classify digital products. In particular, using a $C$-compatible adjacency (or the $L_C$-property), we address a product problem of a digital isomorphism (see Theorems 3.6 and 4.1).

In this paper we consider the existence of Hamilton cycles in the random graph $G=G_{n,m}^{��\geq 3}$. This a random graph chosen uniformly from the set of graphs with vertex set $[n]$, $m$ edges and minimum degree at least 3. Our ultimate goal is to prove that if $m=cn$ and $c>3/2$ is constant then $G$ is Hamiltonian w.h.p. In an earlier paper the second author showed that $c\geq 10$ is sufficient for this and in this paper we reduce the lower bound to $c>2.662...$. This new lower bound is the same lower bound found in Frieze and Pittel \cite{FP} for the expansion of so-called P��sa sets., arXiv admin note: text overlap with arXiv:1107.4947

This paper deals with a complex third order linear measure differential equation \begin{equation*} i\mathrm{d}\left( y^{\prime }\right) ^{\bullet }+2iq\left( x\right) y^{\prime }\mathrm{d}x+y\left( i\mathrm{d}q\left( x\right) +\mathrm{d}p\left( x\right) \right) = \lambda y\mathrm{d}x \end{equation*} on a bounded interval with boundary conditions presenting a mixed aspect of the Dirichlet and the periodic problems. The dependence of eigenvalues on the coefficients $p$, $q$ is investigated. We prove that the $n$-th eigenvalue is continuous in $p$, $q$ when the norm topology of total variation and the weak$^*$ topology are considered. Moreover, the Fr\'{e}chet differentiability of the $n$-th eigenvalue in $p$, $q$ with the norm topology of total variation is also considered. To deduce these conclusions, we investigate the dependence of solutions of the above equation on the coefficients $p$, $q$ with different topologies and establish the counting lemma of eigenvalues according to the estimates of solutions., Comment: This paper has been accepted for publication in SCIENCE CHINA Mathematics

We prove that there are arbitrarily large values of $t$ such that $|\zeta(1+it)| \geq e^{\gamma} (\log_2 t + \log_3 t) + \mathcal{O}(1)$. This essentially matches the prediction for the optimal lower bound in a conjecture of Granville and Soundararajan. Our proof uses a new variant of the "long resonator" method. While earlier implementations of this method crucially relied on a "sparsification" technique to control the mean-square of the resonator function, in the present paper we exploit certain self-similarity properties of a specially designed resonator function., Comment: 7 pages. Versions 2: several minor changes. The paper will appear in IMRN

The paper deals with a new object of study --- hyperbolic Hurwitz zeta function, which is given in the right \(\alpha\)-semiplane \( \alpha = \sigma + it \), \( \sigma> 1 \) by the equality $$ \zeta_H(\alpha; d, b) = \sum_{m \in \mathbb Z} \left(\, \overline{dm + b} \, \right)^{-\alpha}, $$ where \( d \neq0 \) and \( b \) --- any real number. Hyperbolic Hurwitz zeta function \( \zeta_H (\alpha; d, b) \), when \( \left\| \frac {b} {d} \right\|> 0 \) coincides with the hyperbolic zeta function of shifted one-dimensional lattice \( \zeta_H (\Lambda (d, b) | \alpha) \). The importance of this class of one-dimensional lattices is due to the fact that each Cartesian lattice is represented as a union of a finite number of Cartesian products of one-dimensional shifted lattices of the form \( \Lambda (d, b) = d \mathbb{Z} + b \). Cartesian products of one-dimensional shifted lattices are in substance shifted diagonal lattices, for which in this paper the simplest form of a functional equation for the hyperbolic zeta function of such lattices is given. The connection of the hyperbolic Hurwitz zeta function with the Hurwitz zeta function \( \zeta^* (\alpha; b)\) periodized by parameter \(b\) and with the ordinary Hurwitz zeta function \( \zeta (\alpha; b) \) is studied. New integral representations for these zeta functions and an analytic continuation to the left of the line \( \alpha = 1 + it \) are obtained. All considered hyperbolic zeta functions of lattices form an important class of Dirichlet series directly related to the development of the number-theoretical method in approximate analysis. For the study of such series the use of Abel's theorem is efficient, which gives an integral representation through improper integrals. Integration by parts of these improper integrals leads to improper integrals with Bernoulli polynomials, which are also studied in this paper.

Staircase tableaux are combinatorial objects that were first introduced due to a connection with the asymmetric simple exclusion process (ASEP) and Askey-Wilson polynomials. Since their introduction, staircase tableaux have been the object of study in many recent papers. Relevant to this paper, Hitczenko and Janson proved that distribution of parameters on the first diagonal is asymptotically normal. In addition, they conjectured that other diagonals would be asymptotically Poisson. Since then, only the second and the third diagonal were proven to follow the conjecture. This paper builds upon those results to prove the conjecture for the kth diagonal where k is fixed. In particular, we prove that the distribution of the number of α's (β's) on the kth diagonal, k > 1, is asymptotically Poisson with parameter 1/2. In addition, we prove that symbols on the kth diagonal are asymptotically independent and thus, collectively follow the Poisson distribution with parameter 1. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 795–818, 2016

In this note we point out an error in [2]. We show how to repair the proof in dimension 5. The results are true in general as can easily be seen from recent work of Borman, Eliashberg and Murphy [1]. The proof of Lemma 3.4 in [2] is incorrect. Below we will describe the problem with the proof and then show how it can easily be repaired in dimension 5. We then observe that Lemma 3.4, and thus the main results of the paper, is true in all dimensions based on recent work of Borman, Eliashberg and Murphy [1]. However this approach does not give an explicit construction and hence goes against the sprit of the original paper and in addition all the results of [2] follow directly from [1]. Acknowledgement: We thank Yasha Eliashberg for pointing out the error in the proof of Lemma 3.4 in [2]. The first author was partially supported by a grant from the Simons Foundation (#342144) and NSF grant DMS-1309073. 1. Exact Lagrangians, Liouville flows, and the error in the proof of Lemma 3.4 We begin by recalling the statement of Lemma 3.4 from [2]. To state the lemma we first establish some notation (that is slightly different that what was used in [2]). Consider T 2 × [0, 1] with coordinates (θ, φ, r) and the contact structure ξi = kerαi, i = 1, 2, where αi = ki(r) dθ + li(r) dφ. Here we have k1(r) = cos π 2 r and l(r) = sin π 2 r, and for i = 2 we have k2 and l2 agreeing with k1 and l1 near r = 0 and 1, and the curve (k2(r), l2(r)) in R has 5π/2 winding about the origin. In particular notice that ξ2 is obtained from ξ1 by adding Giroux torsion. Lemma 3.4 from [2] now reads as follows. Lemma 1. Let W be a manifold with contact form λ, there is a contact structure on W × [0, 1] × ([0, 1]× T ) such that the following properties are satisfied: (1) near W×{0}× [0, 1]×T 2 and W× [0, 1]×{0, 1}×T 2 the contact structure is contactomorphic to λ+ eα1, and (2) near W × {1} × [0, 1]× T 2 the contact structure is contactomorphic to λ+ eα2. Here t is the coordinate on the first [0, 1] factor. See [2] for details on how the main constructions and theorems of the paper follow from this lemma. The strategy of the proof in [2] was: (1) To construct a contact structure on W×[0, 1]×T 3 that near W×{0}×T 3 is given by λ+eβ0 and near W ×{1}× T 3 is given by λ+ e × β1, where βi is the contact structure on T 3 with Giroux torsion i and we are thinking of T 3 as S×T 2 with the S-factors Legendrian curves. (2) Then cut W × [0, 1]×T 3 along W × [0, 1]× ({θ0, θ1}×T ) so that one of the resulting pieces is as described in the lemma. To try to arrange this let β = p1 dθ1+p1 dθ2 be the Liouville form on T ∗T 2 = R×T 2 with coordinates (p1, p2, θ1, θ2). Notice that α = λ + β is a contact form on W × T ∗T . We will see below that we can arrange the two items above that are needed for our proof if there is a radial vector field v in R centered at a point p whose flow expands dβ (that is, Lvdβ = dβ) and a Lagrangian torus T 2 in a small neighborhood of {q}×T 2 ⊂ T ∗T 2 that is exact with respect to ιvdβ that is isotopic to {q}×T 2 by an isotopy disjoint from {p} × T . One may easily arrange all of this except for either the last 2010 Mathematics Subject Classification. 57R17 (primary), and 53D35(secondary).

The Farey sequence of order n consists of all reduced fractions a / b between 0 and 1 with positive denominator b less or equal to n. The sums of the inverse denominators 1 / b of the Farey fractions in prescribed intervals with rational bounds have a simple main term, but the deviations are determined by an interesting sequence of polygonal functions \(f_n\). In a former paper we obtained a limit function for \(n \rightarrow \infty \), which describes a scaling behaviour of the functions \(f_n\) in the vicinity of any fixed rational number a / b and which is independent of a / b. In this paper we prove that \(f_n(a/b)\) tends to zero for \(n \rightarrow \infty \) by using elementary representation formulas for \(f_n(a/b)\) as well as a variant of the prime number theorem. An application of this result immediately gives a global version of the scaling behaviour of the functions \(f_n\) around the rational numbers.

In the present paper, we introduce a class of $\alpha$-Carleson measures $\mathcal{C}_{\alpha,v}(\mathbb{R}^{n+1}_+)$, which is called by the vanishing $\alpha$-Carleson measures. We prove that $\mathcal{C}_{1/p,v}(\mathbb{R}^{n+1}_+)$ is just a predual of the tent space $\widetilde{T}_{\infty}^p$ ($0 \lt p \lt 1$). Furthermore, we construct the $\alpha$-Carleson measures and the vanishing $\alpha$-Carleson measures by the Campanato functions and its a subclass, respectively. Moreover, a characterization of the vanishing $\alpha$-Carleson measure by the compactness of Poisson integral is given in this paper. Finally, as some applications, we give the $(L^{2/\alpha},L^2)$ boundedness and compactness for some paraproduct operators.