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1. CMB Online first

Boynton, Jason Greene; Coykendall, Jim
On the Graph of Divisibility of an Integral Domain
It is well known that the factorization properties of a domain are reflected in the structure of its group of divisibility. The main theme of this paper is to introduce a topological/graph-theoretic point of view to the current understanding of factorization in integral domains. We also show that connectedness properties in the graph and topological space give rise to a generalization of atomicity.

Keywords:atomic, factorization, divisibility
Categories:13F15, 13A05

2. CMB Online first

Efrat, Ido; Matzri, Eliyahu
Vanishing of Massey products and Brauer groups
Let $p$ be a prime number and $F$ a field containing a root of unity of order $p$. We relate recent results on vanishing of triple Massey products in the mod-$p$ Galois cohomology of $F$, due to Hopkins, Wickelgren, Mináċ, and Tân, to classical results in the theory of central simple algebras. For global fields, we prove a stronger form of the vanishing property.

Keywords:Galois cohomology, Brauer groups, triple Massey products, global fields
Categories:16K50, 11R34, 12G05, 12E30

3. CMB Online first

Alfuraidan, Monther Rashed
The Contraction Principle for Multivalued Mappings on a Modular Metric Space with a Graph
We study the existence of fixed points for contraction multivalued mappings in modular metric spaces endowed with a graph. The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. This paper can be seen as a generalization of Nadler's and Edelstein's fixed point theorems to modular metric spaces endowed with a graph.

Keywords:fixed point theory, modular metric spaces, multivalued contraction mapping, connected digraph.
Categories:47H09, 46B20, 47H10, 47E10

4. CMB Online first

Lü, Guangshi; Sankaranarayanan, Ayyadurai
Higher Moments of Fourier Coefficients of Cusp Forms
Let $S_{k}(\Gamma)$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group $SL(2, \mathbb{Z})$. Let $\lambda_f(n)$, $\lambda_g(n)$, $\lambda_h(n)$ be the $n$th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms $f(z) \in S_{k_1}(\Gamma), g(z) \in S_{k_2}(\Gamma), h(z) \in S_{k_3}(\Gamma)$ respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as $\lambda_f(n)^4\lambda_g(n)^2$, $\lambda_g(n)^6$, $\lambda_g(n)^2\lambda_h(n)^4$, and $\lambda_g(n^3)^2$ twisted by the arithmetic function $\lambda_f(n)$.

Keywords:Fourier coefficients of automorphic forms, Dirichlet series, triple product $L$-function, Perron's formula
Categories:11F30, 11F66

5. CMB Online first

Tang, Xianhua
Ground state solutions of Nehari-Pankov type for a superlinear Hamiltonian elliptic system on RN
This paper is concerned with the following elliptic system of Hamiltonian type \[ \left\{ \begin{array}{ll} -\triangle u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N}, \\ -\triangle v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N}, \\ u, v\in H^{1}({\mathbb{R}}^{N}), \end{array} \right. \] where the potential $V$ is periodic and $0$ lies in a gap of the spectrum of $-\Delta+V$, $W(x, s, t)$ is periodic in $x$ and superlinear in $s$ and $t$ at infinity. We develop a direct approach to find ground state solutions of Nehari-Pankov type for the above system. Especially, our method is applicable for the case when \[ W(x, u, v)=\sum_{i=1}^{k}\int_{0}^{|\alpha_iu+\beta_iv|}g_i(x, t)t\mathrm{d}t +\sum_{j=1}^{l}\int_{0}^{\sqrt{u^2+2b_juv+a_jv^2}}h_j(x, t)t\mathrm{d}t, \] where $\alpha_i, \beta_i, a_j, b_j\in \mathbb{R}$ with $\alpha_i^2+\beta_i^2\ne 0$ and $a_j\gt b_j^2$, $g_i(x, t)$ and $h_j(x, t)$ are nondecreasing in $t\in \mathbb{R}^{+}$ for every $x\in \mathbb{R}^N$ and $g_i(x, 0)=h_j(x, 0)=0$.

Keywords:Hamiltonian elliptic system, superlinear, ground state solutions of Nehari-Pankov type, strongly indefinite functionals
Categories:35J50, 35J55

6. CMB Online first

Li, Benling; Shen, Zhongmin
Ricci Curvature Tensor and Non-Riemannian Quantities
There are several notions of Ricci curvature tensor in Finsler geometry and spray geometry. One of them is defined by the Hessian of the well-known Ricci curvature. In this paper we will introduce a new notion of Ricci curvature tensor and discuss its relationship with the Ricci curvature and some non-Riemannian quantities. By this Ricci curvature tensor, we shall have a better understanding on these non-Riemannian quantities.

Keywords:Finsler metrics, sprays, Ricci curvature, non-Riemanian quantity
Categories:53B40, 53C60

7. CMB Online first

Matringe, Nadir
A specialisation of the Bump-Friedberg $L$-function
We study the restriction of the Bump-Friedberg integrals to affine lines $\{(s+\alpha,2s),s\in\mathbb{C}\}$. It has a simple theory, very close to that of the Asai $L$-function. It is an integral representation of the product $L(s+\alpha,\pi)L(2s,\Lambda^2,\pi)$ which we denote by $L^{lin}(s,\pi,\alpha)$ for this abstract, when $\pi$ is a cuspidal automorphic representation of $GL(k,\mathbb{A})$ for $\mathbb{A}$ the adeles of a number field. When $k$ is even, we show that for a cuspidal automorphic representation $\pi$, the partial $L$-function $L^{lin,S}(s,\pi,\alpha)$ has a pole at $1/2$, if and only if $\pi$ admits a (twisted) global period, this gives a more direct proof of a theorem of Jacquet and Friedberg, asserting that $\pi$ has a twisted global period if and only if $L(\alpha+1/2,\pi)\neq 0$ and $L(1,\Lambda^2,\pi)=\infty$. When $k$ is odd, the partial $L$-function is holmorphic in a neighbourhood of $Re(s)\geq 1/2$ when $Re(\alpha)$ is $\geq 0$.

Keywords:automorphic L functions
Categories:11F70, 11F66

8. CMB Online first

Han, Yanchang
Embedding theorem for inhomogeneous Besov and Triebel-Lizorkin spaces on RD-spaces
In this article we prove the embedding theorem for inhomogeneous Besov and Triebel-Lizorkin spaces on RD-spaces. The crucial idea is to use the geometric density condition on the measure.

Keywords:spaces of homogeneous type, test function space, distributions, Calderón reproducing formula, Besov and Triebel-Lizorkin spaces, embedding
Categories:42B25, 46F05, 46E35

9. CMB Online first

Laterveer, Robert
A brief note concerning hard Lefschetz for Chow groups
We formulate a conjectural hard Lefschetz property for Chow groups, and prove this in some special cases: roughly speaking, for varieties with finite-dimensional motive, and for varieties whose self-product has vanishing middle-dimensional Griffiths group. An appendix includes related statements that follow from results of Vial.

Keywords:algebraic cycles, Chow groups, finite-dimensional motives
Categories:14C15, 14C25, 14C30

10. CMB Online first

Hu, Pei-Chu; Li, Bao Qin
A simple proof and strengthening of a uniqueness theorem for L-functions
We give a simple proof and strengthening of a uniqueness theorem for functions in the extended Selberg class.

Keywords:meromorphic function, Dirichlet series, L-function, zero, order, uniqueness
Categories:30B50, 11M41

11. CMB Online first

Demirbas, Seckin
Almost Sure Global Well-posedness for Fractional Cubic Schrödinger Equation on Torus
In a previous paper, we proved that $1$-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed in $H^s$ for $s\gt \frac{1-\alpha}{2}$ and globally well-posed for $s\gt \frac{10\alpha-1}{12}$. In this paper we define an invariant probability measure $\mu$ on $H^s$ for $s\lt \alpha-\frac{1}{2}$, so that for any $\epsilon\gt 0$ there is a set $\Omega\subset H^s$ such that $\mu(\Omega^c)\lt \epsilon$ and the equation is globally well-posed for initial data in $\Omega$. We see that this fills the gap between the local well-posedness and the global well-posedness range in almost sure sense for $\frac{1-\alpha}{2}\lt \alpha-\frac{1}{2}$, i.e. $\alpha\gt \frac{2}{3}$ in almost sure sense.

Keywords:NLS, fractional Schrodinger equation, almost sure global wellposedness
Category:35Q55

12. CMB Online first

Casini, Emanuele; Miglierina, Enrico; Piasecki, Lukasz
Hyperplanes in the space of convergent sequences and preduals of $\ell_1$
The main aim of the present paper is to investigate various structural properties of hyperplanes of $c$, the Banach space of the convergent sequences. In particular, we give an explicit formula for the projection constants and we prove that an hyperplane of $c$ is isometric to the whole space if and only if it is $1$-complemented. Moreover, we obtain the classification of those hyperplanes for which their duals are isometric to $\ell_{1}$ and we give a complete description of the preduals of $\ell_{1}$ under the assumption that the standard basis of $\ell_{1}$ is weak$^{*}$-convergent.

Keywords:space of convergent sequences, projection, $\ell_1$-predual, hyperplane
Categories:46B45, 46B04

13. CMB Online first

Williams, Kenneth S.
Ternary quadratic forms and eta quotients
Let $\eta(z)$ $(z \in \mathbb{C},\;\operatorname{Im}(z)\gt 0)$ denote the Dedekind eta function. We use a recent product-to-sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly 10 eta quotients \[ f(z):=\eta^{a(m_1)}(m_1 z)\cdots \eta^{{a(m_r)}}(m_r z)=\sum_{n=1}^{\infty}c(n)e^{2\pi i nz},\quad z \in \mathbb{C},\;\operatorname{Im}(z)\gt 0, \] such that the Fourier coefficients $c(n)$ vanish for all positive integers $n$ in each of infinitely many non-overlapping arithmetic progressions. For example, it is shown that for $f(z)=\eta^4(z)\eta^{9}(4z)\eta^{-2}(8z)$ we have $c(n)=0$ for all $n$ in each of the arithmetic progressions $\{16k+14\}_{k \geq 0}$, $\{64k+56\}_{k \geq 0}$, $\{256k+224\}_{k \geq 0}$, $\{1024k+896\}_{k \geq 0}$, $\ldots$.

Keywords:Dedekind eta function, eta quotient, ternary quadratic forms, vanishing of Fourier coefficients, product-to-sum formula
Categories:11F20, 11E20, 11E25

14. CMB Online first

Totik, Vilmos
Universality under Szegő's condition
This paper presents a theorem on universality on orthogonal polynomials/random matrices under a weak local condition on the weight function $w$. With a new inequality for polynomials and with the use of fast decreasing polynomials, it is shown that an approach of D. S. Lubinsky is applicable. The proof works at all points which are Lebesgue-points both for the weight function $w$ and for $\log w$.

Keywords:universality, random matrices, Christoffel functions, asymptotics, potential theory
Categories:42C05, 60B20, 30C85, 31A15

15. CMB Online first

Vahidi, Alireza
Betti numbers and flat dimensions of local cohomology modules
Assume that $R$ is a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ is an ideal of $R$ and $X$ is an $R$--module. In this paper, we first study the finiteness of Betti numbers of local cohomology modules $\operatorname{H}_\mathfrak{a}^i(X)$. Then we give some inequalities between the Betti numbers of $X$ and those of its local cohomology modules. Finally, we present many upper bounds for the flat dimension of $X$ in terms of the flat dimensions of its local cohomology modules and an upper bound for the flat dimension of $\operatorname{H}_\mathfrak{a}^i(X)$ in terms of the flat dimensions of the modules $\operatorname{H}_\mathfrak{a}^j(X)$, $j\not= i$, and that of $X$.

Keywords:Betti numbers, flat dimensions, local cohomology modules
Categories:13D45, 13D05

16. CMB Online first

Brendle, Simon; Chodosh, Otis
On the maximum curvature of closed curves in negatively curved manifolds
Motivated by Almgren's work on the isoperimetric inequality, we prove a sharp inequality relating the length and maximum curvature of a closed curve in a complete, simply connected manifold of sectional curvature at most $-1$. Moreover, if equality holds, then the norm of the geodesic curvature is constant and the torsion vanishes. The proof involves an application of the maximum principle to a function defined on pairs of points.

Keywords:manifold, curvature
Category:53C20

17. CMB Online first

Duc, Dinh Thanh; Nhan, Nguyen Du Vi; Xuan, Nguyen Tong
Inequalities for partial derivatives and their applications
We present various weighted integral inequalities for partial derivatives acting on products and compositions of functions which are applied to establish some new Opial-type inequalities involving functions of several independent variables. We also demonstrate the usefulness of our results in the field of partial differential equations.

Keywords:inequality for integral, Opial-type inequality, Hölder's inequality, partial differential operator, partial differential equation
Categories:26D10, 35A23

18. CMB Online first

de Dios Pérez, Juan; Suh, Young Jin; Woo, Changhwa
Real Hypersurfaces in Complex Two-Plane Grassmannians with GTW Harmonic Curvature
We prove the non-existence of Hopf real hypersurfaces in complex two-plane Grassmannians with harmonic curvature with respect to the generalized Tanaka-Webster connection if they satisfy some further conditions.

Keywords:real hypersurfaces, complex two-plane Grassmannians, Hopf hypersurface, generalized Tanaka-Webster connection, harmonic curvature
Categories:53C40, 53C15

19. CMB Online first

Zaatra, Mohamed
Generating some symmetric semi-classical orthogonal polynomials
We show that if $v$ is a regular semi-classical form (linear functional), then the symmetric form $u$ defined by the relation $x^{2}\sigma u = -\lambda v$, where $(\sigma f)(x)=f(x^{2})$ and the odd moments of $u$ are $0$, is also regular and semi-classical form for every complex $\lambda $ except for a discrete set of numbers depending on $v$. We give explicitly the three-term recurrence relation and the structure relation coefficients of the orthogonal polynomials sequence associated with $u$ and the class of the form $u$ knowing that of $v$. We conclude with an illustrative example.

Keywords:orthogonal polynomials, quadratic decomposition, semi-classical forms, structure relation
Categories:33C45, 42C05

20. CMB Online first

He, Ziyi; Yang, Dachun; Yuan, Wen
Littlewood-Paley Characterizations of Second-Order Sobolev Spaces via Averages on Balls
In this paper, the authors characterize second-order Sobolev spaces $W^{2,p}({\mathbb R}^n)$, with $p\in [2,\infty)$ and $n\in\mathbb N$ or $p\in (1,2)$ and $n\in\{1,2,3\}$, via the Lusin area function and the Littlewood-Paley $g_\lambda^\ast$-function in terms of ball means.

Keywords:Sobolev space, ball means, Lusin-area function, $g_\lambda^*$-function
Categories:46E35, 42B25, 42B20, 42B35

21. CMB Online first

Hanson, Brandon
Character sums over Bohr sets
We prove character sum estimates for additive Bohr subsets modulo a prime. These estimates are analogous to classical character sum bounds of Pólya-Vinogradov and Burgess. These estimates are applied to obtain results on recurrence mod $p$ by special elements.

Keywords:character sums, Bohr sets, finite fields
Categories:11L40, 11T24, 11T23

22. CMB Online first

Chen, Caisheng; Song, Hongxue; Yan, Qinglun
Existence of multiple solutions for a $p$-Laplacian system in $\textbf{R}^{N}$ with sign-changing weight functions
In this paper, we consider the quasi-linear elliptic problem \[ \left\{ \begin{aligned} & -M \left(\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla u|^{p}dx \right){\rm div} \left(|x|^{-ap}|\nabla u|^{p-2}\nabla u \right) \\ & \qquad=\frac{\alpha}{\alpha+\beta}H(x)|u|^{\alpha-2}u|v|^{\beta}+\lambda h_{1}(x)|u|^{q-2}u, \\ & -M \left(\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla v|^{p}dx \right){\rm div} \left(|x|^{-ap}|\nabla v|^{p-2}\nabla v \right) \\ & \qquad=\frac{\beta}{\alpha+\beta}H(x)|v|^{\beta-2}v|u|^{\alpha}+\mu h_{2}(x)|v|^{q-2}v, \\ &u(x)\gt 0,\quad v(x)\gt 0, \quad x\in \mathbb{R}^{N} \end{aligned} \right. \] where $\lambda, \mu\gt 0$, $1\lt p\lt N$, $1\lt q\lt p\lt p(\tau+1)\lt \alpha+\beta\lt p^{*}=\frac{Np}{N-p}$, $0\leq a\lt \frac{N-p}{p}$, $a\leq b\lt a+1$, $d=a+1-b\gt 0$, $M(s)=k+l s^{\tau}$, $k\gt 0$, $l, \tau\geq0$ and the weight $H(x), h_{1}(x), h_{2}(x)$ are continuous functions which change sign in $\mathbb{R}^{N}$. We will prove that the problem has at least two positive solutions by using the Nehari manifold and the fibering maps associated with the Euler functional for this problem.

Keywords:Nehari manifold, quasilinear elliptic system, $p$-Laplacian operator, concave and convex nonlinearities
Category:35J66

23. CMB Online first

Sands, Jonathan W.
$L$-functions for Quadratic Characters and Annihilation of Motivic Cohomology Groups
Let $n$ be a positive even integer, and let $F$ be a totally real number field and $L$ be an abelian Galois extension which is totally real or CM. Fix a finite set $S$ of primes of $F$ containing the infinite primes and all those which ramify in $L$, and let $S_L$ denote the primes of $L$ lying above those in $S$. Then $\mathcal{O}_L^S$ denotes the ring of $S_L$-integers of $L$. Suppose that $\psi$ is a quadratic character of the Galois group of $L$ over $F$. Under the assumption of the motivic Lichtenbaum conjecture, we obtain a non-trivial annihilator of the motivic cohomology group $H_\mathcal{M}^2(\mathcal{O}_L^S,\mathbb{Z}(n))$ from the lead term of the Taylor series for the $S$-modified Artin $L$-function $L_{L/F}^S(s,\psi)$ at $s=1-n$.

Keywords:motivic cohomology, regulator, Artin L-functions
Categories:11R42, 11R70, 14F42, 19F27

24. CMB Online first

Silberman, Lior
Quantum unique ergodicity on locally symmetric spaces: the degenerate lift
Given a measure $\bar\mu_\infty$ on a locally symmetric space $Y=\Gamma\backslash G/K$, obtained as a weak-{*} limit of probability measures associated to eigenfunctions of the ring of invariant differential operators, we construct a measure $\bar\mu_\infty$ on the homogeneous space $X=\Gamma\backslash G$ which lifts $\bar\mu_\infty$ and which is invariant by a connected subgroup $A_{1}\subset A$ of positive dimension, where $G=NAK$ is an Iwasawa decomposition. If the functions are, in addition, eigenfunctions of the Hecke operators, then $\bar\mu_\infty$ is also the limit of measures associated to Hecke eigenfunctions on $X$. This generalizes results of the author with A. Venkatesh in the case where the spectral parameters stay away from the walls of the Weyl chamber.

Keywords:quantum unique ergodicity, microlocal lift, spherical dual
Categories:22E50, 43A85

25. CMB Online first

Ongaro, Jared; Shapiro, Boris
A note on planarity stratification of Hurwitz spaces
One can easily show that any meromorphic function on a complex closed Riemann surface can be represented as a composition of a birational map of this surface to $\mathbb{CP}^2$ and a projection of the image curve from an appropriate point $p\in \mathbb{CP}^2$ to the pencil of lines through $p$. We introduce a natural stratification of Hurwitz spaces according to the minimal degree of a plane curve such that a given meromorphic function can be represented in the above way and calculate the dimensions of these strata. We observe that they are closely related to a family of Severi varieties studied earlier by J. Harris, Z. Ran and I. Tyomkin.

Keywords:Hurwitz spaces, meromorphic functions, Severi varieties
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