51. 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 nonzero
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 

52. 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 nonself dual if every
proper subgroup of $G$
is self dual but $G$ is not self dual. In this paper, the structure
of minimal nonself dual groups is determined.
Keywords:minimal nonself dual group, finite group, metacyclic group, metabelian group Category:20D15 

53. 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 

54. CMB 2015 (vol 58 pp. 730)
 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, MinÃ¡Ä, and TÃ¢n, 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 

55. 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/graphtheoretic 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 

56. CMB 2015 (vol 58 pp. 651)
 Tang, Xianhua

Ground State Solutions of NehariPankov Type for a Superlinear Hamiltonian Elliptic System on ${\mathbb{R}}^{N}$
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 NehariPankov 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 NehariPankov type, strongly indefinite functionals Categories:35J50, 35J55 

57. CMB 2015 (vol 59 pp. 3)
 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 

58. CMB 2015 (vol 58 pp. 580)
 Matringe, Nadir

A Specialisation of the BumpFriedberg $L$function
We study the restriction of the BumpFriedberg 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 

59. CMB 2015 (vol 58 pp. 530)
 Li, Benling; Shen, Zhongmin

Ricci Curvature Tensor and NonRiemannian 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 wellknown 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
nonRiemannian quantities. By this Ricci curvature tensor, we shall
have a better understanding on these nonRiemannian quantities.
Keywords:Finsler metrics, sprays, Ricci curvature, nonRiemanian quantity Categories:53B40, 53C60 

60. CMB 2015 (vol 58 pp. 757)
61. CMB 2015 (vol 58 pp. 459)
 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 

62. CMB 2015 (vol 58 pp. 471)
 Demirbas, Seckin

Almost Sure Global Wellposedness for the Fractional Cubic SchrÃ¶dinger Equation on Torus
In a previous paper, we proved that $1$d periodic fractional
SchrÃ¶dinger equation with cubic nonlinearity is locally wellposed
in $H^s$ for $s\gt \frac{1\alpha}{2}$ and globally wellposed for
$s\gt \frac{10\alpha1}{12}$. 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
wellposed for initial data in $\Omega$. We see that this fills
the gap between the local wellposedness and the global wellposedness
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 

63. CMB 2015 (vol 58 pp. 713)
 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 

64. CMB 2015 (vol 58 pp. 486)
 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 Opialtype 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, Opialtype inequality, HÃ¶lder's inequality, partial differential operator, partial differential equation Categories:26D10, 35A23 

65. CMB 2015 (vol 58 pp. 620)
 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 nontrivial 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 $s=1n$.
Keywords:motivic cohomology, regulator, Artin Lfunctions Categories:11R42, 11R70, 14F42, 19F27 

66. CMB 2015 (vol 58 pp. 596)
 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 

67. CMB 2015 (vol 58 pp. 632)
 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 

68. CMB 2015 (vol 58 pp. 320)
 Llamas, Aurora; MartínezBernal, José

Cover Product and Betti Polynomial of Graphs
For disjoint graphs $G$ and $H$, with fixed
vertex covers
$C(G)$ and $C(H)$, their cover product is the graph $G
\circledast
H$ with vertex set
$V(G)\cup V(H)$ and edge set $E(G)\cup E(H)\cup\{\{i,j\}:i\in
C(G), j\in
C(H)\}$. We describe the graded Betti numbers of $G\circledast
H$ in terms of those of
$G$ and $H$. As applications we obtain: (i) For any positive
integer $k$ there
exists a connected bipartite graph $G$ such that $\operatorname{reg}
R/I(G)=\mu_S(G)+k$, where,
$I(G)$ denotes the edge ideal of $G$, $\operatorname{reg} R/I(G)$
is the CastelnuovoMumford
regularity of $R/I(G)$ and $\mu_S(G)$ is the induced or strong
matching number of
$G$; (ii) The graded Betti numbers of the complement of a tree
only depends upon
its number of vertices; (iii) The $h$vector of $R/I(G\circledast
H)$ is described in
terms of the $h$vectors of $R/I(G)$ and $R/I(H)$. Furthermore,
in a different
direction, we give a recursive formula for the graded Betti numbers
of chordal
bipartite graphs.
Keywords:CastelnuovoMumford regularity, chordal bipartite graph, edge ideal, graded Betti number, induced matching number, monomial ideal Categories:13D02, 05E45 

69. CMB 2015 (vol 58 pp. 402)
 Tikuisis, Aaron Peter; Toms, Andrew

On the Structure of Cuntz Semigroups in (Possibly) Nonunital C*algebras
We examine the ranks of operators in semifinite $\mathrm{C}^*$algebras
as measured by their densely defined lower semicontinuous traces.
We first prove that a unital simple $\mathrm{C}^*$algebra whose
extreme tracial boundary is nonempty and finite contains positive
operators of every possible rank, independent of the property
of strict comparison. We then turn to nonunital simple algebras
and establish criteria that imply that the Cuntz semigroup is
recovered functorially from the Murrayvon Neumann semigroup
and the space of densely defined lower semicontinuous traces.
Finally, we prove that these criteria are satisfied by notnecessarilyunital
approximately subhomogeneous algebras of slow dimension growth.
Combined with results of the firstnamed author, this shows that
slow dimension growth coincides with $\mathcal Z$stability,
for approximately subhomogeneous algebras.
Keywords:nuclear C*algebras, Cuntz semigroup, dimension functions, stably projectionless C*algebras, approximately subhomogeneous C*algebras, slow dimension growth Categories:46L35, 46L05, 46L80, 47L40, 46L85 

70. CMB 2015 (vol 58 pp. 306)
 Khoshkhah, Kaveh; Zaker, Manouchehr

On the Largest Dynamic Monopolies of Graphs with a Given Average Threshold
Let $G$ be a graph and $\tau$ be an assignment of nonnegative
integer thresholds to the vertices of $G$. A subset of vertices,
$D$ is said to be a $\tau$dynamic monopoly, if $V(G)$ can be
partitioned into subsets $D_0, D_1, \ldots, D_k$ such that $D_0=D$
and for any $i\in \{0, \ldots, k1\}$, each vertex $v$ in $D_{i+1}$
has at least $\tau(v)$ neighbors in $D_0\cup \ldots \cup D_i$.
Denote the size of smallest $\tau$dynamic monopoly by $dyn_{\tau}(G)$
and the average of thresholds in $\tau$ by $\overline{\tau}$.
We show that the values of $dyn_{\tau}(G)$ over all assignments
$\tau$ with the same average threshold is a continuous set of
integers. For any positive number $t$, denote the maximum $dyn_{\tau}(G)$
taken over all threshold assignments $\tau$ with $\overline{\tau}\leq
t$, by $Ldyn_t(G)$. In fact, $Ldyn_t(G)$ shows the worstcase
value of a dynamic monopoly when the average threshold is a given
number $t$. We investigate under what conditions on $t$, there
exists an upper bound for $Ldyn_{t}(G)$ of the form $cG$, where
$c\lt 1$. Next, we show that $Ldyn_t(G)$ is coNPhard for planar
graphs but has polynomialtime solution for forests.
Keywords:spread of influence in graphs, irreversible dynamic monopolies, target set selection Categories:05C69, 05C85 

71. CMB 2015 (vol 58 pp. 271)
72. CMB 2015 (vol 58 pp. 285)
 Karpukhin, Mikhail

Spectral Properties of a Family of Minimal Tori of Revolution in Fivedimensional Sphere
The normalized eigenvalues $\Lambda_i(M,g)$ of the LaplaceBeltrami
operator can be considered as functionals on the space of all
Riemannian metrics $g$ on a fixed surface $M$. In recent papers
several explicit examples of extremal metrics were provided.
These metrics are induced by minimal immersions of surfaces in
$\mathbb{S}^3$ or $\mathbb{S}^4$. In the present paper a family
of extremal metrics induced by minimal immersions in $\mathbb{S}^5$
is investigated.
Keywords:extremal metric, minimal surface Category:58J50 

73. CMB 2015 (vol 58 pp. 415)
 Willson, Benjamin

A Fixed Point Theorem and the Existence of a Haar Measure for Hypergroups Satisfying Conditions Related to Amenability
In this paper we present a fixed point property for amenable
hypergroups which is analogous to Rickert's fixed point theorem
for semigroups. It equates the existence of a left invariant
mean on the space of weakly right uniformly continuous functions
to the existence of a fixed point for any action of the hypergroup.
Using this fixed point property, a certain class of hypergroups
are shown to have a left Haar measure.
Keywords:invariant measure, Haar measure, hypergroup, amenability, function translations Categories:43A62, 43A05, 43A07 

74. CMB 2015 (vol 58 pp. 281)
 Kalus, Matthias

On the Relation of Real and Complex Lie Supergroups
A complex Lie supergroup can be described as a real Lie supergroup
with integrable almost complex structure. The necessary and
sufficient conditions on an almost complex structure on a real
Lie supergroup for defining a complex Lie supergroup are deduced.
The classification of real Lie supergroups with such almost
complex
structures yields a new approach to the known classification
of complex Lie supergroups by complex HarishChandra superpairs.
A universal complexification of a real Lie supergroup is
constructed.
Keywords:Lie supergroup, almost complex structure, HarishChandra pair, universal complexification Categories:32C11, 58A50 

75. CMB 2015 (vol 58 pp. 241)
 Botelho, Fernanda

Isometries and Hermitian Operators on Zygmund Spaces
In this paper we characterize the isometries of subspaces of the little Zygmund space. We show that the isometries of these spaces are surjective and represented as integral operators. We also show that all hermitian operators on these settings are bounded.
Keywords:Zygmund spaces, the little Zygmund space, Hermitian operators, surjective linear isometries, generators of oneparameter groups of surjective isometries Categories:46E15, 47B15, 47B38 
