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1. CMB Online first

Benamar, H.; Chandoul, A.; Mkaouar, M.
On the continued fraction expansion of fixed period in finite fields
The Chowla conjecture states that, if $t$ is any given positive integer, there are infinitely many prime positive integers $N$ such that $\operatorname{Per} (\sqrt{N})=t$, where $\operatorname{Per} (\sqrt{N})$ is the period length of the continued fraction expansion for $\sqrt{N}$. C. Friesen proved that, for any $k\in \mathbb{N}$, there are infinitely many square-free integers $N$, where the continued fraction expansion of $\sqrt{N}$ has a fixed period. In this paper, we describe all polynomials $Q\in \mathbb{F}_q[X] $ for which the continued fraction expansion of $\sqrt {Q}$ has a fixed period, also we give a lower bound of the number of monic, non-squares polynomials $Q$ such that $\deg Q= 2d$ and $ Per \sqrt {Q}=t$.

Keywords:continued fractions, polynomials, formal power series
Categories:11A55, 13J05

2. CMB Online first

Kong, Qingjun; Guo, Xiuyun
On $s$-semipermutable or $s$-quasinormally embedded subgroups of finite groups
Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. $H$ is said to be $s$-semipermutable in $G$ if $HG_{p}=G_{p}H$ for any Sylow $p$-subgroup $G_{p}$ of $G$ with $(p,|H|)=1$; $H$ is said to be $s$-quasinormally embedded in $G$ if for each prime $p$ dividing the order of $H$, a Sylow $p$-subgroup of $H$ is also a Sylow $p$-subgroup of some $s$-quasinormal subgroup of $G$. We fix in every non-cyclic Sylow subgroup $P$ of $G$ some subgroup $D$ satisfying $1\lt |D|\lt |P|$ and study the structure of $G$ under the assumption that every subgroup $H$ of $P$ with $|H|=|D|$ is either $s$-semipermutable or $s$-quasinormally embedded in $G$. Some recent results are generalized and unified.

Keywords:$s$-semipermutable subgroup, $s$-quasinormally embedded subgroup, saturated formation.
Categories:20D10, 20D20

3. CMB Online first

Gao, Zenghui
Homological Properties Relative to Injectively Resolving Subcategories
Let $\mathcal{E}$ be an injectively resolving subcategory of left $R$-modules. A left $R$-module $M$ (resp. right $R$-module $N$) is called $\mathcal{E}$-injective (resp. $\mathcal{E}$-flat) if $\operatorname{Ext}_R^1(G,M)=0$ (resp. $\operatorname{Tor}_1^R(N,G)=0$) for any $G\in\mathcal{E}$. Let $\mathcal{E}$ be a covering subcategory. We prove that a left $R$-module $M$ is $\mathcal{E}$-injective if and only if $M$ is a direct sum of an injective left $R$-module and a reduced $\mathcal{E}$-injective left $R$-module. Suppose $\mathcal{F}$ is a preenveloping subcategory of right $R$-modules such that $\mathcal{E}^+\subseteq\mathcal{F}$ and $\mathcal{F}^+\subseteq\mathcal{E}$. It is shown that a finitely presented right $R$-module $M$ is $\mathcal{E}$-flat if and only if $M$ is a cokernel of an $\mathcal{F}$-preenvelope of a right $R$-module. In addition, we introduce and investigate the $\mathcal{E}$-injective and $\mathcal{E}$-flat dimensions of modules and rings. We also introduce $\mathcal{E}$-(semi)hereditary rings and $\mathcal{E}$-von Neumann regular rings and characterize them in terms of $\mathcal{E}$-injective and $\mathcal{E}$-flat modules.

Keywords:injectively resolving subcategory, \mathcal{E}-injective module (dimension), \mathcal{E}-flat module (dimension), cover, preenvelope, \mathcal{E}-(semi)hereditary ring
Categories:16E30, 16E10, 16E60

4. 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

5. CMB Online first

Atıcı, Ferhan M.; Yaldız, Hatice
Convex Functions on Discrete Time Domains
In this paper, we introduce the definition of a convex real valued function $f$ defined on the set of integers, ${\mathbb{Z}}$. We prove that $f$ is convex on ${\mathbb{Z}}$ if and only if $\Delta^{2}f \geq 0$ on ${\mathbb{Z}}$. As a first application of this new concept, we state and prove discrete Hermite-Hadamard inequality using the basics of discrete calculus (i.e. the calculus on ${\mathbb{Z}}$). Second, we state and prove the discrete fractional Hermite-Hadamard inequality using the basics of discrete fractional calculus. We close the paper by defining the convexity of a real valued function on any time scale.

Keywords:discrete calculus, discrete fractional calculus, convex functions, discrete Hermite-Hadamard inequality
Categories:26B25, 26A33, 39A12, 39A70, 26E70, 26D07, 26D10, 26D15

6. CMB Online first

Luo, Xiu-Hua
Exact Morphism category and Gorenstein-projective representations
Let $Q$ be a finite acyclic quiver, $J$ be an ideal of $kQ$ generated by all arrows in $Q$, $A$ be a finite-dimensional $k$-algebra. The category of all finite-dimensional representations of $(Q, J^2)$ over $A$ is denoted by $\operatorname{rep}(Q, J^2, A)$. In this paper, we introduce the category $\operatorname{exa}(Q,J^2,A)$, which is a subcategory of $\operatorname{rep}{}(Q,J^2,A)$ of all exact representations. The main result of this paper explicitly describes the Gorenstein-projective representations in $\operatorname{rep}{}(Q,J^2,A)$, via the exact representations plus an extra condition. As a corollary, $A$ is a self-injective algebra, if and only if the Gorenstein-projective representations are exactly the exact representations of $(Q, J^2)$ over $A$.

Keywords:representations of a quiver over an algebra, exact representations, Gorenstein-projective modules
Category:18G25

7. CMB Online first

Gasiński, Leszek; Papageorgiou, Nikolaos S.
Positive solutions for the generalized nonlinear logistic equations
We consider a nonlinear parametric elliptic equation driven by a nonhomogeneous differential operator with a logistic reaction of the superdiffusive type. Using variational methods coupled with suitable truncation and comparison techniques, we prove a bifurcation type result describing the set of positive solutions as the parameter varies.

Keywords:positive solution, bifurcation type result, strong comparison principle, nonlinear regularity, nonlinear maximum principle
Categories:35J25, 35J92

8. CMB Online first

Li, Dan; Ma, Wanbiao
Dynamical Analysis of a Stage-Structured Model for Lyme Disease with two delays
In this paper, a nonlinear stage-structured model for Lyme disease is considered. The model is a system of differential equations with two time delays. The basic reproductive rate, $R_0(\tau_1,\tau_2)$, is derived. If $R_0(\tau_1,\tau_2)\lt 1$, then the boundary equilibrium is globally asymptotically stable. If $R_0(\tau_1,\tau_2)\gt 1$, then there exists a unique positive equilibrium whose local asymptotical stability and the existence of Hopf bifurcations are established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions is derived by using the normal form and the center manifold theory. Some numerical simulations are performed to confirm the correctness of theoretical analysis. At last, some conclusions are given.

Keywords:Lyme disease, stage-structure, time delay, Lyapunov functional stability Hopf bifurcation.
Category:34D20

9. 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

10. CMB Online first

Achter, Jeffrey; Williams, Cassandra
Local Heuristics and an Exact Formula for Abelian Surfaces Over Finite Fields
Consider a quartic $q$-Weil polynomial $f$. Motivated by equidistribution considerations, we define, for each prime $\ell$, a local factor that measures the relative frequency with which $f\bmod \ell$ occurs as the characteristic polynomial of a symplectic similitude over $\mathbb{F}_\ell$. For a certain class of polynomials, we show that the resulting infinite product calculates the number of principally polarized abelian surfaces over $\mathbb{F}_q$ with Weil polynomial $f$.

Keywords:abelian surfaces, finite fields, random matrices
Category:14K02

11. CMB Online first

Llibre, Jaume; Zhang, Xiang
On the limit cycles of linear differential systems with homogeneous nonlinearities
We consider the class of polynomial differential systems of the form $\dot x= \lambda x-y+P_n(x,y)$, $\dot y=x+\lambda y+ Q_n(x,y),$ where $P_n$ and $Q_n$ are homogeneous polynomials of degree $n$. For this class of differential systems we summarize the known results for the existence of limit cycles, and we provide new results for their nonexistence and existence.

Keywords:polynomial differential system, limit cycles, differential equations on the cylinder
Categories:34C35, 34D30

12. 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

13. 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

14. 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

15. 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

16. CMB Online first

Aulaskari, Rauno; Chen, Huaihui
On classes $Q_p^\#$ for hyperbolic Riemann surfaces
The $Q_p$ spaces of holomorphic functions on the disk, hyperbolic Riemann surfaces or complex unit ball have been studied deeply. Meanwhile, there are a lot of papers devoted to the $Q^\#_p$ classes of meromorphic functions on the disk or hyperbolic Riemann surfaces. In this paper, we prove the nesting property (inclusion relations) of $Q^\#_p$ classes on hyperbolic Riemann surfaces. The same property for $Q_p$ spaces was also established systematically and precisely in earlier work by the authors of this paper.

Keywords:$Q_p^\#$ class, hyperbolic Riemann surface, spherical Dirichlet function,
Categories:30D50, 30F35

17. CMB Online first

Kitabeppu, Yu; Lakzian, Sajjad
Non-branching RCD$(0,N)$ Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups
In this paper, we generalize the finite generation result of Sormani to non-branching $RCD(0,N)$ geodesic spaces (and in particular, Alexandrov spaces) with full support measures. This is a special case of the Milnor's Conjecture for complete non-compact $RCD(0,N)$ spaces. One of the key tools we use is the Abresch-Gromoll type excess estimates for non-smooth spaces obtained by Gigli-Mosconi.

Keywords:Milnor conjecture, non negative Ricci curvature, curvature dimension condition, finitely generated, fundamental group, infinitesimally Hilbertian
Categories:53C23, 30L99

18. 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

19. 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

20. CMB Online first

Donovan, Diane M.; Griggs, Terry S.; McCourt, Thomas A.; Opršal, Jakub; Stanovský, David
Distributive and anti-distributive Mendelsohn triple systems
We prove that the existence spectrum of Mendelsohn triple systems whose associated quasigroups satisfy distributivity corresponds to the Loeschian numbers, and provide some enumeration results. We do this by considering a description of the quasigroups in terms of commutative Moufang loops. In addition we provide constructions of Mendelsohn quasigroups that fail distributivity for as many combinations of elements as possible. These systems are analogues of Hall triple systems and anti-mitre Steiner triple systems respectively.

Keywords:Mendelsohn triple system, quasigroup, distributive, Moufang loop, Loeschian numbers
Categories:20N05, 05B07

21. CMB 2015 (vol 58 pp. 664)

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

22. CMB 2015 (vol 58 pp. 538)

Li, Lili; Chen, Guiyun
Minimal Non Self Dual Groups
A group $G$ is self dual if every subgroup of $G$ is isomorphic to a quotient of $G$ and every quotient of $G$ is isomorphic to a subgroup of $G$. It is minimal non-self dual if every proper subgroup of $G$ is self dual but $G$ is not self dual. In this paper, the structure of minimal non-self dual groups is determined.

Keywords:minimal non-self dual group, finite group, metacyclic group, metabelian group
Category:20D15

23. CMB 2015 (vol 58 pp. 548)

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

24. CMB 2015 (vol 58 pp. 449)

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

25. 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
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