1. CMB 2015 (vol 59 pp. 50)
2. 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 

3. 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 

4. CMB 2015 (vol 58 pp. 250)
 Cartwright, Dustin; Jensen, David; Payne, Sam

Lifting Divisors on a Generic Chain of Loops
Let $C$ be a curve over a complete valued field with infinite
residue field whose skeleton is a chain of loops with generic
edge lengths. We prove that
any divisor on the chain of loops that is rational over the value
group lifts to a divisor of the same rank on $C$, confirming
a conjecture of Cools,
Draisma, Robeva, and the third author.
Keywords:tropical geometry, BrillNoether theory, special divisors on algebraic curves Categories:14T05, 14H51 

5. CMB 2015 (vol 58 pp. 381)
 Tang, Xiaomin; Liu, Taishun

The Schwarz Lemma at the Boundary of the Egg Domain $B_{p_1, p_2}$ in $\mathbb{C}^n$
Let $B_{p_1, p_2}=\{z\in\mathbb{C}^n:
z_1^{p_1}+z_2^{p_2}+\cdots+z_n^{p_2}\lt 1\}$
be an egg domain in $\mathbb{C}^n$. In this paper, we first
characterize the Kobayashi metric on $B_{p_1, p_2}\,(p_1\geq
1, p_2\geq 1)$,
and then establish a new type of the classical boundary Schwarz
lemma at $z_0\in\partial{B_{p_1, p_2}}$ for holomorphic selfmappings
of $B_{p_1, p_2}(p_1\geq 1, p_2\gt 1)$, where $z_0=(e^{i\theta},
0, \dots, 0)'$ and $\theta\in \mathbb{R}$.
Keywords:holomorphic mapping, Schwarz lemma, Kobayashi metric, egg domain Categories:32H02, 30C80, 32A30 

6. CMB 2014 (vol 58 pp. 188)
7. CMB 2014 (vol 57 pp. 708)
 Brannan, Michael

Strong Asymptotic Freeness for Free Orthogonal Quantum Groups
It is known that the normalized standard generators of the free
orthogonal quantum group $O_N^+$ converge in distribution to a free
semicircular system as $N \to \infty$. In this note, we
substantially improve this convergence result by proving that, in
addition to distributional convergence, the operator norm of any
noncommutative polynomial in the normalized standard generators of
$O_N^+$ converges as $N \to \infty$ to the operator norm of the
corresponding noncommutative polynomial in a standard free
semicircular system. Analogous strong convergence results are obtained
for the generators of free unitary quantum groups. As applications of
these results, we obtain a matrixcoefficient version of our strong
convergence theorem, and we recover a well known $L^2$$L^\infty$ norm
equivalence for noncommutative polynomials in free semicircular
systems.
Keywords:quantum groups, free probability, asymptotic free independence, strong convergence, property of rapid decay Categories:46L54, 20G42, 46L65 

8. CMB 2014 (vol 58 pp. 297)
 Khamsi, M. A.

Approximate Fixed Point Sequences of Nonlinear Semigroup in Metric Spaces
In this paper, we investigate the common
approximate fixed point sequences of nonexpansive semigroups of
nonlinear mappings $\{T_t\}_{t \geq 0}$, i.e., a family such that
$T_0(x)=x$, $T_{s+t}=T_s(T_t(x))$, where the domain is a metric space
$(M,d)$. In particular we prove that under suitable conditions, the
common approximate fixed point sequences set is the same as the common
approximate fixed point sequences set of two mappings from the family.
Then we use the Ishikawa iteration to construct a common approximate
fixed point sequence of nonexpansive semigroups of nonlinear
mappings.
Keywords:approximate fixed point, fixed point, hyperbolic metric space, Ishikawa iterations, nonexpansive mapping, semigroup of mappings, uniformly convex hyperbolic space Categories:47H09, 46B20, 47H10, 47E10 

9. CMB 2014 (vol 58 pp. 44)
10. CMB 2012 (vol 56 pp. 621)
 Shang, Yilun

Optimal Control Strategies for Virus Spreading in Inhomogeneous Epidemic Dynamics
In this paper, we study the spread of virus/worm in computer
networks with a view to addressing cyber security problems. Epidemic
models have been applied extensively to model the propagation of
computer viruses, which characterize the fact that infected machines
may spread malware to other hosts connected to the network. In our
framework, the dynamics of hosts evolves according to a modified
inhomogeneous SusceptibleInfectiousSusceptible (SIS) epidemic
model with timevarying transmission rate and recovery rate. The
infection of computers is subject to direct attack as well as
propagation among hosts. Based on optimal control theory, optimal
attack strategies are provided by minimizing the cost (equivalently
maximizing the profit) of the attacker. We present a threshold
function of the fraction of infectious hosts, which captures the
dynamically evolving strategies of the attacker and reflects the
persistence of virus spreading. Moreover, our results indicate that
if the infectivity of a computer worm is low and the computers are
installed with antivirus software with high reliability, the
intensity of attacks incurred will likely be low. This agrees with
our intuition.
Keywords:network securitypidemic dynamics, optimal control Categories:49J15, 92D30 

11. CMB 2012 (vol 56 pp. 534)
 Filali, M.; Monfared, M. Sangani

A Cohomological Property of $\pi$invariant Elements
Let $A$ be a Banach algebra and $\pi \colon A \longrightarrow \mathscr L(H)$
be a continuous representation of $A$ on a separable Hilbert space $H$
with $\dim H =\frak m$. Let $\pi_{ij}$ be the coordinate functions of
$\pi$ with respect to an orthonormal basis and suppose that for each
$1\le j \le \frak m$, $C_j=\sum_{i=1}^{\frak m}
\\pi_{ij}\_{A^*}\lt \infty$ and $\sup_j C_j\lt \infty$. Under these
conditions, we call an element $\overline\Phi \in l^\infty (\frak m , A^{**})$
left $\pi$invariant if $a\cdot \overline\Phi ={}^t\pi (a) \overline\Phi$ for all
$a\in A$. In this paper we prove a link between the existence
of left $\pi$invariant elements and the vanishing of certain
Hochschild cohomology groups of $A$. Our results extend an earlier
result by Lau on $F$algebras and recent results of KaniuthLauPym
and the second named author in the special case that $\pi \colon A
\longrightarrow \mathbf C$ is a nonzero character on $A$.
Keywords:Banach algebras, $\pi$invariance, derivations, representations Categories:46H15, 46H25, 13N15 

12. CMB 2012 (vol 56 pp. 584)
 Liau, PaoKuei; Liu, ChengKai

On Automorphisms and Commutativity in Semiprime Rings
Let $R$ be a semiprime ring with center
$Z(R)$. For $x,y\in R$, we denote by $[x,y]=xyyx$ the commutator of
$x$ and $y$. If $\sigma$ is a nonidentity automorphism of $R$ such
that
$$
\Big[\big[\dots\big[[\sigma(x^{n_0}),x^{n_1}],x^{n_2}\big],\dots\big],x^{n_k}\Big]=0
$$
for all $x \in R$, where $n_{0},n_{1},n_{2},\dots,n_{k}$ are fixed
positive integers, then there exists a map $\mu\colon R\rightarrow Z(R)$
such that $\sigma(x)=x+\mu(x)$ for all $x\in R$. In particular, when
$R$ is a prime ring, $R$ is commutative.
Keywords:automorphism, generalized polynomial identity (GPI) Categories:16N60, 16W20, 16R50 

13. CMB 2011 (vol 56 pp. 378)
 Ma, Li; Wang, Jing

Sharp Threshold of the GrossPitaevskii Equation with Trapped Dipolar Quantum Gases
In this paper, we consider the GrossPitaevskii equation for the
trapped dipolar quantum gases. We obtain the sharp criterion for the
global existence and finite time blow up in the unstable regime by
constructing a variational problem and the socalled invariant
manifold of the evolution flow.
Keywords:GrossPitaevskii equation, sharp threshold, global existence, blow up Categories:35Q55, 35A05, 81Q99 

14. CMB 2011 (vol 56 pp. 395)
 Oancea, D.

Coessential Abelianization Morphisms in the Category of Groups
An epimorphism $\phi\colon G\to H$ of groups, where $G$ has rank $n$, is called
coessential if every (ordered) generating $n$tuple of $H$ can be
lifted along $\phi$ to a generating $n$tuple for $G$. We discuss this
property in the context of the category of groups, and establish a criterion
for such a group $G$ to have the property that its abelianization
epimorphism $G\to G/[G,G]$, where $[G,G]$ is the commutator subgroup, is
coessential. We give an example of a family of 2generator groups whose
abelianization epimorphism is not coessential.
This family also provides counterexamples to the generalized AndrewsCurtis conjecture.
Keywords:coessential epimorphism, Nielsen transformations, AndrewCurtis transformations Categories:20F05, 20F99, 20J15 

15. CMB 2011 (vol 56 pp. 258)
 Chandoul, A.; Jellali, M.; Mkaouar, M.

The Smallest Pisot Element in the Field of Formal Power Series Over a Finite Field
Dufresnoy and Pisot characterized the smallest
Pisot number of degree $n \geq 3$ by giving explicitly its minimal
polynomial. In this paper, we translate Dufresnoy and Pisot's
result to the Laurent series case.
The
aim of this paper is to prove that the minimal polynomial
of the smallest Pisot element (SPE) of degree $n$ in the field of
formal power series over a finite field
is given by $P(Y)=Y^{n}\alpha XY^{n1}\alpha^n,$ where $\alpha$
is the least element of the finite field $\mathbb{F}_{q}\backslash\{0\}$
(as a finite total ordered set). We prove that the sequence of
SPEs of degree $n$ is decreasing and converges to $\alpha X.$
Finally, we show how to obtain explicit continued fraction
expansion of the smallest Pisot element over a finite field.
Keywords:Pisot element, continued fraction, Laurent series, finite fields Categories:11A55, 11D45, 11D72, 11J61, 11J66 

16. CMB 2011 (vol 56 pp. 55)
 Bouziad, A.

Cliquishness and Quasicontinuity of TwoVariable Maps
We study the existence of continuity points for mappings
$f\colon X\times Y\to Z$ whose $x$sections $Y\ni y\to f(x,y)\in Z$ are
fragmentable and $y$sections $X\ni x\to f(x,y)\in Z$ are
quasicontinuous, where $X$ is a Baire space and $Z$
is a metric space. For the factor $Y$, we consider two
infinite ``pointpicking'' games $G_1(y)$ and $G_2(y)$ defined respectively
for each $y\in Y$ as follows: in the $n$th
inning, Player I gives a dense set $D_n\subset Y$, respectively, a dense open set $D_n\subset Y$. Then
Player II picks a point $y_n\in D_n$;
II wins if $y$ is in the closure of ${\{y_n:n\in\mathbb N\}}$, otherwise
I wins. It is shown that
(i) $f$ is
cliquish
if II has a winning strategy in $G_1(y)$ for every $y\in Y$, and (ii) $
f$ is quasicontinuous if
the $x$sections of $f$ are continuous and the set of $y\in Y$
such that II has a winning strategy in $G_2(y)$ is dense in $Y$. Item (i) extends substantially
a result of Debs and item (ii) indicates that
the problem of Talagrand on separately continuous maps has a positive answer for a wide
class of ``small'' compact spaces.
Keywords:cliquishness, fragmentability, joint continuity, pointpicking game, quasicontinuity, separate continuity, two variable maps Categories:54C05, 54C08, 54B10, 91A05 

17. CMB 2011 (vol 55 pp. 597)
 Osękowski, Adam

Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales
We determine the best constants $C_{p,\infty}$ and $C_{1,p}$,
$1 < p < \infty$, for which the following holds. If $u$, $v$ are
orthogonal harmonic functions on a Euclidean domain such that $v$ is
differentially subordinate to $u$, then
$$ \v\_p \leq C_{p,\infty}
\u\_\infty,\quad
\v\_1 \leq C_{1,p} \u\_p.
$$
In particular, the inequalities are still sharp for the conjugate
harmonic functions on the unit disc of $\mathbb R^2$.
Sharp probabilistic versions of these estimates are also studied.
As an application, we establish a sharp version of the classical logarithmic inequality of Zygmund.
Keywords: harmonic function, conjugate harmonic functions, orthogonal harmonic functions, martingale, orthogonal martingales, norm inequality, optimal stopping problem Categories:31B05, 60G44, 60G40 

18. CMB 2010 (vol 53 pp. 286)
 Gorelic, Isaac

Orders of πBases
We extend the scope of B. Shapirovskii's results on the order of $\pi$bases in compact spaces and answer some questions of V. Tkachuk.
Keywords:Shapirovskii πbase, pointcountable πbase, free sequences, canonical form for ordinals Categories:54A25, 03E10, 03E75, 54A35 

19. CMB 2010 (vol 53 pp. 223)
 Chuang, ChenLian; Lee, TsiuKwen

Density of Polynomial Maps
Let $R$ be a dense subring of $\operatorname{End}(_DV)$, where $V$ is a left vector space over a division ring $D$. If $\dim{_DV}=\infty$, then the range of any nonzero polynomial $f(X_1,\dots,X_m)$ on $R$ is dense in $\operatorname{End}(_DV)$. As an application, let $R$ be a prime ring without nonzero nil onesided ideals and $0\ne a\in R$. If $af(x_1,\dots,x_m)^{n(x_i)}=0$ for all $x_1,\dots,x_m\in R$, where $n(x_i)$ is a positive integer depending on $x_1,\dots,x_m$, then $f(X_1,\dots,X_m)$ is a polynomial identity of $R$ unless $R$ is a finite matrix ring over a finite field.
Keywords:density, polynomial, endomorphism ring, PI Categories:16D60, 16S50 

20. CMB 2009 (vol 53 pp. 295)
 Guo, Boling; Huo, Zhaohui

The Global Attractor of a Damped, Forced Hirota Equation in $H^1$
The existence of the global attractor of a damped
forced Hirota equation in the phase space $H^1(\mathbb R)$ is proved. The
main idea is to establish the socalled asymptotic compactness
property of the solution operator by energy equation approach.
Keywords:global attractor, Fourier restriction norm, damping system, asymptotic compactness Categories:35Q53, 35B40, 35B41, 37L30 

21. CMB 2008 (vol 51 pp. 372)
22. CMB 2008 (vol 51 pp. 283)
23. CMB 2008 (vol 51 pp. 146)
 Zhou, Xiaowen

SteppingStone Model with Circular Brownian Migration
In this paper we consider the steppingstone model on a circle with
circular Brownian migration. We first point out a connection between
Arratia flow on the circle and the marginal distribution of this
model. We then give a new representation for the steppingstone
model using Arratia flow and circular coalescing Brownian motion.
Such a representation enables us to carry out some explicit
computations. In particular, we find the distribution for the first
time when there is only one type
left across the circle.
Keywords:steppingstone model, circular coalescing Brownian motion, Arratia flow, duality, entrance law Categories:60G57, 60J65 

24. CMB 2007 (vol 50 pp. 191)
 Drungilas, Paulius; Dubickas, Artūras

Every Real Algebraic Integer Is a Difference of Two Mahler Measures
We prove that every real
algebraic integer $\alpha$ is expressible by a
difference of two Mahler measures of integer polynomials.
Moreover, these polynomials can be chosen in such a way that they
both have the same degree as that of $\alpha$, say
$d$, one of these two polynomials is irreducible and
another has an irreducible factor of degree $d$, so
that $\alpha=M(P)bM(Q)$ with irreducible polynomials
$P, Q\in \mathbb Z[X]$ of degree $d$ and a
positive integer $b$. Finally, if $d \leqslant 3$, then one can take $b=1$.
Keywords:Mahler measures, Pisot numbers, Pell equation, $abc$conjecture Categories:11R04, 11R06, 11R09, 11R33, 11D09 

25. CMB 2006 (vol 49 pp. 237)