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26. CMB 2016 (vol 59 pp. 326)

Jiang, Chunlan; Shi, Rui
 On the Uniqueness of Jordan Canonical Form Decompositions of Operators by $K$-theoretical Data In this paper, we develop a generalized Jordan canonical form theorem for a certain class of operators in $\mathcal {L}(\mathcal {H})$. A complete criterion for similarity for this class of operators in terms of $K$-theory for Banach algebras is given. Keywords:strongly irreducible operator, similarity invariant, reduction theory of von Neumann algebras, $K$-theoryCategories:47A15, 47C15, 47A65

27. CMB Online first

De Nicola, Antonio; Yudin, Ivan
 Generalized Goldberg Formula In this paper we prove a useful formula for the graded commutator of the Hodge codifferential with the left wedge multiplication by a fixed $p$-form acting on the de Rham algebra of a Riemannian manifold. Our formula generalizes a formula stated by Samuel I. Goldberg for the case of 1-forms. As first examples of application we obtain new identities on locally conformally KÃ¤hler manifolds and quasi-Sasakian manifolds. Moreover, we prove that under suitable conditions a certain subalgebra of differential forms in a compact manifold is quasi-isomorphic as a CDGA to the full de Rham algebra. Keywords:graded commutator, Hodge codifferential, Hodge laplacian, de Rham cohomology, locally conformal Kaehler manifold, quasi-Sasakian manifoldCategories:53C25, 53D35

28. CMB Online first

Kachmar, Ayman
 A new formula for the energy of bulk superconductivity The energy of a type II superconductor submitted to an external magnetic field of intensity close to the second critical field is given by the celebrated Abrikosov energy. If the external magnetic field is comparable to and below the second critical field, the energy is given by a reference function obtained as a special (thermodynamic) limit of a non-linear energy. In this note, we give a new formula for this reference energy. In particular, we obtain it as a special limit of a linear energy defined over configurations normalized in the $L^4$-norm. Keywords:Ginzburg-Landau functionalCategories:35B40, 35P15, 35Q56

29. CMB 2016 (vol 59 pp. 279)

Dimca, Alexandru
 The PoincarÃ©-Deligne Polynomial of Milnor Fibers of Triple Point Line Arrangements is Combinatorially Determined Using a recent result by S. Papadima and A. Suciu, we show that the equivariant PoincarÃ©-Deligne polynomial of the Milnor fiber of a projective line arrangement having only double and triple points is combinatorially determined. Keywords:line arrangement, Milnor fiber, monodromy, mixed Hodge structuresCategories:32S22, 32S35, 32S25, 32S55

30. CMB Online first

Feng, Zhaosheng; Jiang, Yongxin; Wang, Wei
 Spatial Homogenization of Stochastic Wave Equation with Large Interaction A dynamical approximation of a stochastic wave equation with large interaction is derived. A random invariant manifold is discussed. By a key linear transformation, the random invariant manifold is shown to be close to the random invariant manifold of a second-order stochastic ordinary differential equation. Keywords:stochastic wave equation, homogeneous system, approximation, random invariant manifold, Neumann boundary conditionCategories:60F10, 60H15, 35Q55

31. CMB 2016 (vol 59 pp. 225)

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 inequalityCategories:26B25, 26A33, 39A12, 39A70, 26E70, 26D07, 26D10, 26D15

32. 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 moduleCategories:13D05, 13D45, 18G20

33. 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 equivalenceCategories:32A38, 30H50, 32A10, 32M99

34. 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 charactersCategory:20C15

35. 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 schemesCategories:15A21, 15A69, 14M12, 14N15

36. 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

37. 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 principleCategories:35J25, 35J92

38. 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 mapCategories:46L05, 46L80

39. 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 groupsCategories:20F67, 05C10, 20J05, 57M60

40. 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 submodulesCategories:13A15, 13C05, 13C13, , 13C99

41. 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 integersCategories:22E43, 20H99, 83A05

42. 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 cylinderCategories:34C35, 34D30

43. 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

44. 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 seriesCategories:11A55, 13J05

45. 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 ringCategories:16E30, 16E10, 16E60

46. 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, $g_\lambda^*$-functionCategories:46E35, 42B25, 42B20, 42B35

47. 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 modulesCategory:18G25

48. 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 matricesCategory:14K02

49. 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 PÃ³lya-Vinogradov and Burgess. These estimates are applied to obtain results on recurrence mod $p$ by special elements. Keywords:character sums, Bohr sets, finite fieldsCategories:11L40, 11T24, 11T23

50. 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 relationCategories:33C45, 42C05
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