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51. CMB 2016 (vol 59 pp. 403)

Zargar, Majid Rahro; Zakeri, Hossein
On Flat and Gorenstein Flat Dimensions of Local Cohomology Modules
Let $\mathfrak{a}$ be an ideal of a Noetherian local ring $R$ and let $C$ be a semidualizing $R$-module. For an $R$-module $X$, we denote any of the quantities $\mathfrak{d}_R X$, $\operatorname{\mathsf{Gfd}}_R X$ and $\operatorname{\mathsf{G_C-fd}}_RX$ by $\operatorname{\mathsf{T}}(X)$. Let $M$ be an $R$-module such that $\operatorname{H}_{\mathfrak{a}}^i(M)=0$ for all $i\neq n$. It is proved that if $\operatorname{\mathsf{T}}(X)\lt \infty$, then $\operatorname{\mathsf{T}}(\operatorname{H}_{\mathfrak{a}}^n(M))\leq\operatorname{\mathsf{T}}(M)+n$ and the equality holds whenever $M$ is finitely generated. With the aid of these results, among other things, we characterize Cohen-Macaulay modules, dualizing modules and Gorenstein rings.

Keywords:flat dimension, Gorenstein injective dimension, Gorenstein flat dimension, local cohomology, relative Cohen-Macaulay module, semidualizing module
Categories:13D05, 13D45, 18G20

52. CMB 2016 (vol 59 pp. 346)

Krantz, Steven
On a Theorem of Bers, with Applications to the Study of Automorphism Groups of Domains
We study and generalize a classical theorem of L. Bers that classifies domains up to biholomorphic equivalence in terms of the algebras of holomorphic functions on those domains. Then we develop applications of these results to the study of domains with noncompact automorphism group.

Keywords:Bers's theorem, algebras of holomorphic functions, noncompact automorphism group, biholomorphic equivalence
Categories:32A38, 30H50, 32A10, 32M99

53. CMB 2016 (vol 59 pp. 392)

Prajapati, S. K.; Sarma, R.
Total Character of a Group $G$ with $(G,Z(G))$ as a Generalized Camina Pair
We investigate whether the total character of a finite group $G$ is a polynomial in a suitable irreducible character of $G$. When $(G,Z(G))$ is a generalized Camina pair, we show that the total character is a polynomial in a faithful irreducible character of $G$ if and only if $Z(G)$ is cyclic.

Keywords:finite groups, group characters, total characters
Category:20C15

54. CMB 2016 (vol 59 pp. 311)

Ilten, Nathan; Teitler, Zach
Product Ranks of the $3\times 3$ Determinant and Permanent
We show that the product rank of the $3 \times 3$ determinant $\det_3$ is $5$, and the product rank of the $3 \times 3$ permanent $\operatorname{perm}_3$ is $4$. As a corollary, we obtain that the tensor rank of $\det_3$ is $5$ and the tensor rank of $\operatorname{perm}_3$ is $4$. We show moreover that the border product rank of $\operatorname{perm}_n$ is larger than $n$ for any $n\geq 3$.

Keywords:product rank, tensor rank, determinant, permanent, Fano schemes
Categories:15A21, 15A69, 14M12, 14N15

55. CMB 2016 (vol 59 pp. 363)

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

56. CMB 2015 (vol 59 pp. 73)

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

57. CMB 2015 (vol 59 pp. 170)

Martínez-Pedroza, Eduardo
A Note on Fine Graphs and Homological Isoperimetric Inequalities
In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected $2$-complex $X$ with a linear homological isoperimetric inequality, a bound on the length of attaching maps of $2$-cells and finitely many $2$-cells adjacent to any edge must have a fine $1$-skeleton. We provide a positive answer to this question. We revisit a homological characterization of relative hyperbolicity, and show that a group $G$ is hyperbolic relative to a collection of subgroups $\mathcal P$ if and only if $G$ acts cocompactly with finite edge stabilizers on an connected $2$-dimensional cell complex with a linear homological isoperimetric inequality and $\mathcal P$ is a collection of representatives of conjugacy classes of vertex stabilizers.

Keywords:isoperimetric functions, Dehn functions, hyperbolic groups
Categories:20F67, 05C10, 20J05, 57M60

58. CMB 2015 (vol 59 pp. 435)

Yao, Hongliang
On Extensions of Stably Finite C*-algebras (II)
For any $C^*$-algebra $A$ with an approximate unit of projections, there is a smallest ideal $I$ of $A$ such that the quotient $A/I$ is stably finite. In this paper, a sufficient and necessary condition is obtained for an ideal of a $C^*$-algebra with real rank zero is this smallest ideal by $K$-theory.

Keywords:extension, stably finite C*-algebra, index map
Categories:46L05, 46L80

59. CMB 2015 (vol 59 pp. 197)

Rajaee, Saeed
Quasi-copure Submodules
All rings are commutative with identity and all modules are unital. In this paper we introduce the concept of quasi-copure submodule of a multiplication $R$-module $M$ and will give some results of them. We give some properties of tensor product of finitely generated faithful multiplication modules.

Keywords:multiplication module, arithmetical ring, copure submodule, radical of submodules
Categories:13A15, 13C05, 13C13, , 13C99

60. CMB 2015 (vol 59 pp. 123)

Jensen, Gerd; Pommerenke, Christian
Discrete Space-time and Lorentz Transformations
Alfred Schild has established conditions that Lorentz transformations map world-vectors $(ct,x,y,z)$ with integer coordinates onto vectors of the same kind. The problem was dealt with in the context of tensor and spinor calculus. Due to Schild's number-theoretic arguments, the subject is also interesting when isolated from its physical background. The paper of Schild is not easy to understand. Therefore we first present a streamlined version of his proof which is based on the use of null vectors. Then we present a purely algebraic proof that is somewhat shorter. Both proofs rely on the properties of Gaussian integers.

Keywords:Lorentz transformation, integer lattice, Gaussian integers
Categories:22E43, 20H99, 83A05

61. CMB 2015 (vol 58 pp. 818)

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

62. CMB 2015 (vol 58 pp. 704)

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

63. CMB 2015 (vol 58 pp. 799)

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

64. CMB 2015 (vol 58 pp. 741)

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

65. CMB 2015 (vol 59 pp. 104)

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

66. CMB 2015 (vol 58 pp. 824)

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

67. CMB 2015 (vol 58 pp. 673)

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

68. CMB 2015 (vol 58 pp. 774)

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

69. CMB 2015 (vol 58 pp. 877)

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

70. CMB 2015 (vol 59 pp. 211)

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

71. CMB 2015 (vol 59 pp. 144)

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

72. CMB 2015 (vol 58 pp. 835)

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

73. CMB 2015 (vol 58 pp. 787)

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

74. CMB 2015 (vol 58 pp. 858)

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

75. CMB 2015 (vol 59 pp. 13)

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
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