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1. CMB Online first

Bahmanpour, Kamal; Naghipour, Reza
 Faltings' finiteness dimension of local cohomology modules over local Cohen-Macaulay rings Let $(R, \frak m)$ denote a local Cohen-Macaulay ring and $I$ a non-nilpotent ideal of $R$. The purpose of this article is to investigate Faltings' finiteness dimension $f_I(R)$ and equidimensionalness of certain homomorphic image of $R$. As a consequence we deduce that $f_I(R)=\operatorname{max}\{1, \operatorname{ht} I\}$ and if $\operatorname{mAss}_R(R/I)$ is contained in $\operatorname{Ass}_R(R)$, then the ring $R/ I+\cup_{n\geq 1}(0:_RI^n)$ is equidimensional of dimension $\dim R-1$. Moreover, we will obtain a lower bound for injective dimension of the local cohomology module $H^{\operatorname{ht} I}_I(R)$, in the case $(R, \frak m)$ is a complete equidimensional local ring. Keywords:Cohen Macaulay ring, equidimensional ring, finiteness dimension, local cohomologyCategories:13D45, 14B15

2. CMB Online first

Liu, Li; Weng, Peixuan
 Globally asymptotic stability of a delayed integro-differential equation with nonlocal diffusion We study a population model with nonlocal diffusion, which is a delayed integro-differential equation with double nonlinearity and two integrable kernels. By comparison method and analytical technique, we obtain globally asymptotic stability of the zero solution and the positive equilibrium. The results obtained reveal that the globally asymptotic stability only depends on the property of nonlinearity. As application, an example for a population model with age structure is discussed at the end of the article. Keywords:integro-differential equation, nonlocal diffusion, equilibrium, globally asymptotic stability, population model with age structureCategories:45J05, 35K57, 92D25

3. CMB Online first

Tang, Xianhua
 New super-quadratic conditions for asymptotically periodic SchrÃ¶dinger equation This paper is dedicated to studying the semilinear SchrÃ¶dinger equation $$\left\{ \begin{array}{ll} -\triangle u+V(x)u=f(x, u), \ \ \ \ x\in {\mathbf{R}}^{N}, \\ u\in H^{1}({\mathbf{R}}^{N}), \end{array} \right.$$ where $f$ is a superlinear, subcritical nonlinearity. It focuses on the case where $V(x)=V_0(x)+V_1(x)$, $V_0\in C(\mathbf{R}^N)$, $V_0(x)$ is 1-periodic in each of $x_1, x_2, \ldots, x_N$ and $\sup[\sigma(-\triangle +V_0)\cap (-\infty, 0)]\lt 0\lt \inf[\sigma(-\triangle +V_0)\cap (0, \infty)]$, $V_1\in C(\mathbf{R}^N)$ and $\lim_{|x|\to\infty}V_1(x)=0$. A new super-quadratic condition is obtained, which is weaker than some well known results. Keywords:SchrÃ¶dinger equation, superlinear, asymptotically periodic, ground state solutions of Nehari-Pankov typeCategories:35J20, 35J60

4. CMB Online first

Morimoto, Masaharu
 Cokernels of homomorphisms from Burnside rings to inverse limits Let $G$ be a finite group and let $A(G)$ denote the Burnside ring of $G$. Then an inverse limit $L(G)$ of the groups $A(H)$ for proper subgroups $H$ of $G$ and a homomorphism ${\operatorname{res}}$ from $A(G)$ to $L(G)$ are obtained in a natural way. Let $Q(G)$ denote the cokernel of ${\operatorname{res}}$. For a prime $p$, let $N(p)$ be the minimal normal subgroup of $G$ such that the order of $G/N(p)$ is a power of $p$, possibly $1$. In this paper we prove that $Q(G)$ is isomorphic to the cartesian product of the groups $Q(G/N(p))$, where $p$ ranges over the primes dividing the order of $G$. Keywords:Burnside ring, inverse limit, finite groupCategories:19A22, 57S17

5. CMB Online first

Le Fourn, Samuel
 Nonvanishing of central values of $L$-functions of newforms in $S_2 (\Gamma_0 (dp^2))$ twisted by quadratic characters We prove that for $d \in \{ 2,3,5,7,13 \}$ and $K$ a quadratic (or rational) field of discriminant $D$ and Dirichlet character $\chi$, if a prime $p$ is large enough compared to $D$, there is a newform $f \in S_2(\Gamma_0(dp^2))$ with sign $(+1)$ with respect to the Atkin-Lehner involution $w_{p^2}$ such that $L(f \otimes \chi,1) \neq 0$. This result is obtained through an estimate of a weighted sum of twists of $L$-functions which generalises a result of Ellenberg. It relies on the approximate functional equation for the $L$-functions $L(f \otimes \chi, \cdot)$ and a Petersson trace formula restricted to Atkin-Lehner eigenspaces. An application of this nonvanishing theorem will be given in terms of existence of rank zero quotients of some twisted jacobians, which generalises a result of Darmon and Merel. Keywords:nonvanishing of $L$-functions of modular forms, Petersson trace formula, rank zero quotients of jacobiansCategories:14J15, 11F67

6. CMB Online first

Jensen, Gerd; Pommerenke, Christian
 On the structure of the Schild group in Relativity Theory Alfred Schild has established conditions that Lorentz transformations map world-vectors $(ct,x,y,z)$ with integer coordinates onto vectors of the same kind. These transformations are called integral Lorentz transformations. The present paper contains supplements to our earlier work with a new focus on group theory. To relate the results to the familiar matrix group nomenclature we associate Lorentz transformations with matrices in $\mathrm{SL}(2,\mathbb{C})$. We consider the lattice of subgroups of the group originated in Schild's paper and obtain generating sets for the full group and its subgroups. Keywords:Lorentz transformation, integer lattice, Gaussian integers, Schild group, subgroupCategories:22E43, 20H99, 83A05

7. CMB Online first

Shravan Kumar, N.
 Invariant means on a class of von Neumann Algebras related to Ultraspherical Hypergroups II Let $K$ be an ultraspherical hypergroup associated to a locally compact group $G$ and a spherical projector $\pi$ and let $VN(K)$ denote the dual of the Fourier algebra $A(K)$ corresponding to $K.$ In this note, we show that the set of invariant means on $VN(K)$ is singleton if and only if $K$ is discrete. Here $K$ need not be second countable. We also study invariant means on the dual of the Fourier algebra $A_0(K),$ the closure of $A(K)$ in the $cb$-multiplier norm. Finally, we consider generalized translations and generalized invariant means. Keywords:ultraspherical hypergroup, Fourier algebra, Fourier-Stieltjes algebra, invariant mean, generalized translation, generalized invariant meanCategories:43A62, 46J10, 43A30, 20N20

8. CMB Online first

Azimi, Ali; Farrokhi Derakhshandeh Ghouchan, Mohammad
 Self $2$-distance graphs All finite simple self $2$-distance graphs with no square, diamond, or triangles with a common vertex as subgraph are determined. Utilizing these results, it is shown that there is no cubic self $2$-distance graph. Keywords:distance graph, regular graph, forbidden subgraphCategories:05C12, 05C60, 05C76

9. CMB Online first

Eroǧlu, Münevver Pınar; Argaç, Nurcan
 On Identities with Composition of Generalized Derivations Let $R$ be a prime ring with extended centroid $C$, $Q$ maximal right ring of quotients of $R$, $RC$ central closure of $R$ such that $dim_{C}(RC) \gt 4$, $f(X_{1},\dots,X_{n})$ a multilinear polynomial over $C$ which is not central-valued on $R$ and $f(R)$ the set of all evaluations of the multilinear polynomial $f\big(X_{1},\dots,X_{n}\big)$ in $R$. Suppose that $G$ is a nonzero generalized derivation of $R$ such that $G^2\big(u\big)u \in C$ for all $u\in f(R)$ then one of the following conditions holds: (I) there exists $a\in Q$ such that $a^2=0$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in R$; (II) there exists $a\in Q$ such that $0\neq a^2\in C$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in R$ and $f(X_{1},\dots,X_{n})^{2}$ is central-valued on $R$; (III) $char(R)=2$ and one of the following holds: (i) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all $x\in R$ and $a^{2}=b^{2}\in C$; (ii) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all $x\in R$, $a^{2}, b^{2}\in C$ and $f(X_{1},\ldots,X_{n})^{2}$ is central-valued on $R$; (iii) there exist $a \in Q$ and an $X$-outer derivation $d$ of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$ and $a^2+d(a)=0$; (iv) there exist $a \in Q$ and an $X$-outer derivation $d$ of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$, $a^2+d(a)\in C$ and $f(X_{1},\dots,X_{n})^{2}$ is central-valued on $R$. Moreover, we characterize the form of nonzero generalized derivations $G$ of $R$ satisfying $G^2(x)=\lambda x$ for all $x\in R$, where $\lambda \in C$. Keywords:prime ring, generalized derivation, composition, extended centroid, multilinear polynomial, maximal right ring of quotientsCategories:16N60, 16N25

10. CMB Online first

Louder, Larsen; Wilton, Henry
 Stackings and the $W$-cycles conjecture We prove Wise's $W$-cycles conjecture: Consider a compact graph $\Gamma'$ immersing into another graph $\Gamma$. For any immersed cycle $\Lambda:S^1\to \Gamma$, we consider the map $\Lambda'$ from the circular components $\mathbb{S}$ of the pullback to $\Gamma'$. Unless $\Lambda'$ is reducible, the degree of the covering map $\mathbb{S}\to S^1$ is bounded above by minus the Euler characteristic of $\Gamma'$. As a corollary, any finitely generated subgroup of a one-relator group has finitely generated Schur multiplier. Keywords:free groups, one-relator groups, right-orderabilityCategory:20F65

11. CMB Online first

Gurbuz, Ferit
 Some estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces In this paper, we establish $BMO$ estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively. Keywords:fractional integral operator, fractional maximal operator, rough kernel, generalized commutator, $A(p,q)$ weight, generalized weighted Morrey spaceCategories:42B20, 42B25

12. CMB Online first

Reichstein, Zinovy; Vistoli, Angelo
 On the dimension of the locus of determinantal hypersurfaces The characteristic polynomial $P_A(x_0, \dots, x_r)$ of an $r$-tuple $A := (A_1, \dots, A_r)$ of $n \times n$-matrices is defined as $P_A(x_0, \dots, x_r) := \det(x_0 I + x_1 A_1 + \dots + x_r A_r) \, .$ We show that if $r \geqslant 3$ and $A := (A_1, \dots, A_r)$ is an $r$-tuple of $n \times n$-matrices in general position, then up to conjugacy, there are only finitely many $r$-tuples $A' := (A_1', \dots, A_r')$ such that $p_A = p_{A'}$. Equivalently, the locus of determinantal hypersurfaces of degree $n$ in $\mathbf{P}^r$ is irreducible of dimension $(r-1)n^2 + 1$. Keywords:determinantal hypersurface, matrix invariant, $q$-binomial coefficientCategories:14M12, 15A22, 05A10

13. CMB Online first

Akbari, Saieed; Miraftab, Babak; Nikandish, Reza
 Co-Maximal Graphs of Subgroups of Groups Let $H$ be a group. The co-maximal graph of subgroups of $H$, denoted by $\Gamma(H)$, is a graph whose vertices are non-trivial and proper subgroups of $H$ and two distinct vertices $L$ and $K$ are adjacent in $\Gamma(H)$ if and only if $H=LK$. In this paper, we study the connectivity, diameter, clique number and vertex chromatic number of $\Gamma(H)$. For instance, we show that if $\Gamma(H)$ has no isolated vertex, then $\Gamma(H)$ is connected with diameter at most $3$. Also, we characterize all finite groups whose co-maximal graphs are connected. Among other results, we show that if $H$ is a finitely generated solvable group and $\Gamma(H)$ is connected and moreover the degree of a maximal subgroup is finite, then $H$ is finite. Furthermore, we show that the degree of each vertex in the co-maximal graph of a general linear group over an algebraically closed field is zero or infinite. Keywords:co-maximal graphs of subgroups of groups, diameter, nilpotent group, solvable groupCategories:05C25, 05E15, 20D10, 20D15

14. CMB Online first

Liu, Zhongyun; Qin, Xiaorong; Wu, Nianci; Zhang, Yulin
 The shifted classical circulant and skew circulant splitting iterative methods for Toeplitz matrices It is known that every Toeplitz matrix $T$ enjoys a circulant and skew circulant splitting (denoted by CSCS) i.e., $T=C-S$ with $C$ a circulant matrix and $S$ a skew circulant matrix. Based on the variant of such a splitting (also referred to as CSCS), we first develop classical CSCS iterative methods and then introduce shifted CSCS iterative methods for solving hermitian positive definite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical CSCS iterative methods work slightly better than the Gauss-Seidel (GS) iterative methods if the CSCS is convergent, and that there is always a constant $\alpha$ such that the shifted CSCS iteration converges much faster than the Gauss-Seidel iteration, no matter whether the CSCS itself is convergent or not. Keywords:Hermitian positive definite, CSCS splitting, Gauss-Seidel splitting, iterative method, Toeplitz matrixCategories:15A23, 65F10, 65F15

15. CMB Online first

Bouchemakh, Isma; Fatma, Kaci
 On the dual KÃ¶nig property of the order-interval hypergraph of two classes of N-free posets Let $P$ be a finite N-free poset. We consider the hypergraph $\mathcal{H}(P)$ whose vertices are the elements of $P$ and whose edges are the maximal intervals of $P$. We study the dual KÃ¶nig property of $\mathcal{H}(P)$ in two subclasses of N-free class. Keywords:poset, interval, N-free, hypergraph, KÃ¶nig property, dual KÃ¶nig propertyCategory:05C65

16. CMB Online first

Chung, Jaeyoung; Ju, Yumin; Rassias, John
 Cubic functional equations on restricted domains of Lebesgue measure zero Let $X$ be a real normed space, $Y$ a Bancch space and $f:X \to Y$. We prove the Ulam-Hyers stability theorem for the cubic functional equation \begin{align*} f(2x+y)+f(2x-y)-2f(x+y)-2f(x-y)-12f(x)=0 \end{align*} in restricted domains. As an application we consider a measure zero stability problem of the inequality \begin{align*} \|f(2x+y)+f(2x-y)-2f(x+y)-2f(x-y)-12f(x)\|\le \epsilon \end{align*} for all $(x, y)$ in $\Gamma\subset\mathbb R^2$ of Lebesgue measure 0. Keywords:Baire category theorem, cubic functional equation, first category, Lebesgue measure, Ulam-Hyers stabilityCategory:39B82

17. CMB Online first

Sickel, Winfried; Yang, Dachun; Yuan, Wen; Zhuo, Ciqiang
 Characterizations of Besov-Type and Triebel-Lizorkin-Type Spaces via Averages on Balls Let $\ell\in\mathbb N$ and $\alpha\in (0,2\ell)$. In this article, the authors establish equivalent characterizations of Besov-type spaces, Triebel-Lizorkin-type spaces and Besov-Morrey spaces via the sequence $\{f-B_{\ell,2^{-k}}f\}_{k}$ consisting of the difference between $f$ and the ball average $B_{\ell,2^{-k}}f$. These results give a way to introduce Besov-type spaces, Triebel-Lizorkin-type spaces and Besov-Morrey spaces with any smoothness order on metric measure spaces. As special cases, the authors obtain a new characterization of Morrey-Sobolev spaces and $Q_\alpha$ spaces with $\alpha\in(0,1)$, which are of independent interest. Keywords:Besov space, Triebel-Lizorkin space, ball average, CalderÃ³n reproducing formulaCategories:42B25, 46E35, 42B35

18. CMB Online first

Stoyanov, Luchezar
 On Gibbs measures and spectra of Ruelle transfer operators We prove a comprehensive version of the Ruelle-Perron-Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the HÃ¶lder constant of the function generating the operator appears only polynomially, not exponentially as in previous known estimates. Keywords:subshift of finite type, Ruelle transfer operator, Gibbs measureCategories:37A05, 37B10

19. CMB Online first

Diestel, Geoff
 An extension of Nikishin's factorization theorem A Nikishin-Maurey characterization is given for bounded subsets of weak-type Lebesgue spaces. New factorizations for linear and multilinear operators are shown to follow. Keywords:factorization, type, cotype, Banach spacesCategories:46E30, 28A25

20. CMB Online first

Deng, Hanyuan; Tang, Zikai
 Degree Kirchhoff index of bicyclic graphs Let $G$ be a connected graph with vertex set $V(G)$. The degree Kirchhoff index of $G$ is defined as $S'(G) =\sum_{\{u,v\}\subseteq V(G)}d(u)d(v)R(u,v)$, where $d(u)$ is the degree of vertex $u$, and $R(u, v)$ denotes the resistance distance between vertices $u$ and $v$. In this paper, we characterize the graphs having maximum and minimum degree Kirchhoff index among all $n$-vertex bicyclic graphs with exactly two cycles. Keywords:degree Kirchhoff index, resistance distance, bicyclic graph, extremal graphCategories:05C12, 05C35

21. CMB Online first

Pathak, Siddhi
 On a conjecture of Livingston In an attempt to resolve a folklore conjecture of ErdÃ¶s regarding the non-vanishing at $s=1$ of the $L$-series attached to a periodic arithmetical function with period $q$ and values in $\{ -1, 1\}$, Livingston conjectured the $\bar{\mathbb{Q}}$ - linear independence of logarithms of certain algebraic numbers. In this paper, we disprove Livingston's conjecture for composite $q \geq 4$, highlighting that a new approach is required to settle ErdÃ¶s's conjecture. We also prove that the conjecture is true for prime $q \geq 3$, and indicate that more ingredients will be needed to settle ErdÃ¶s's conjecture for prime $q$. Keywords:non-vanishing of L-series, linear independence of logarithms of algebraic numbersCategories:11J86, 11J72

22. CMB Online first

Xu, Xu; Zhu, Laiyi
 Rational function operators from Poisson integrals In this paper, we construct two classes of rational function operators by using the Poisson integrals of the function on the whole real axis. The convergence rates of the uniform and mean approximation of such rational function operators on the whole real axis are studied. Keywords:rational function operators, Poisson integrals, convergence rate, uniform approximation, mean approximationCategories:41A20, 41A25, 41A35

23. CMB Online first

Oubbi, Lahbib
 On Ulam stability of a functional equation in Banach modules Let $X$ and $Y$ be Banach spaces and $f : X \to Y$ an odd mapping. For any rational number $r \ne 2$, C. Baak, D. H. Boo, and Th. M. Rassias have proved the Hyers-Ulam stability of the following functional equation: \begin{align*} r f \left(\frac{\sum_{j=1}^d x_j}{r} \right) & + \sum_{\substack{i(j) \in \{0,1\} \\ \sum_{j=1}^d i(j)=\ell}} r f \left( \frac{\sum_{j=1}^d (-1)^{i(j)}x_j}{r} \right) = (C^\ell_{d-1} - C^{\ell -1}_{d-1} + 1) \sum_{j=1}^d f(x_j) \end{align*} where $d$ and $\ell$ are positive integers so that $1 \lt \ell \lt \frac{d}{2}$, and $C^p_q := \frac{q!}{(q-p)!p!}$, $p, q \in \mathbb{N}$ with $p \le q$. In this note we solve this equation for arbitrary nonzero scalar $r$ and show that it is actually Hyers-Ulam stable. We thus extend and generalize Baak et al.'s result. Different questions concerning the *-homomorphisms and the multipliers between C*-algebras are also considered. Keywords:linear functional equation, Hyers-Ulam stability, Banach modules, C*-algebra homomorphisms.Categories:39A30, 39B10, 39A06, 46Hxx

24. CMB 2016 (vol 59 pp. 806)

Izumiya, Shyuichi
 Geometric Interpretation of Lagrangian Equivalence As an application of the theory of graph-like Legendrian unfoldings, relations of the hidden structures of caustics and wave front propagations are revealed. Keywords:wave front propagations, big wave fronts, graph-like Legendrian unfoldings, causticsCategories:58K05, 57R45, 58K60

25. CMB 2016 (vol 59 pp. 776)

Gauthier, Paul M; Sharifi, Fatemeh
 The CarathÃ©odory Reflection Principle and Osgood-CarathÃ©odory Theorem on Riemann Surfaces The Osgood-CarathÃ©odory theorem asserts that conformal mappings between Jordan domains extend to homeomorphisms between their closures. For multiply-connected domains on Riemann surfaces, similar results can be reduced to the simply-connected case, but we find it simpler to deduce such results using a direct analogue of the CarathÃ©odory reflection principle. Keywords:bordered Riemann surface, reflection principle, Osgood-CarathÃ©odoryCategories:30C25, 30F99
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