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1. CMB Online first

Dorfmeister, Josef F.; Inoguchi, Jun-ichi; Kobayashi, Shimpei
 On the Bernstein problem in the three-dimensional Heisenberg group In this note we present a simple alternative proof for the Bernstein problem in the three-dimensional Heisenberg group $\operatorname{Nil}_3$ by using the loop group technique. We clarify the geometric meaning of the two-parameter ambiguity of entire minimal graphs with prescribed Abresch-Rosenberg differential. Keywords:Bernstein problem, minimal graphs, Heisenberg group, loop groups, spinorsCategories:53A10, 53C42

2. CMB Online first

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 Opial-type 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, Opial-type inequality, HÃ¶lder's inequality, partial differential operator, partial differential equationCategories: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, Brill-Noether theory, special divisors on algebraic curvesCategories: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 self-mappings 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 domainCategories:32H02, 30C80, 32A30

6. CMB 2014 (vol 58 pp. 188)

Wirths, Karl Joachim
 Telescoping Estimates for Smooth Series We derive telescoping majorants and minorants for some classes of series and give applications of these results. Keywords:telescoping series, Stietjes constant, Hardy's formula, Stirling's formulaCategories:26D15, 40A25, 97I30

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 non-commutative polynomial in the normalized standard generators of $O_N^+$ converges as $N \to \infty$ to the operator norm of the corresponding non-commutative 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 matrix-coefficient version of our strong convergence theorem, and we recover a well known $L^2$-$L^\infty$ norm equivalence for non-commutative polynomials in free semicircular systems. Keywords:quantum groups, free probability, asymptotic free independence, strong convergence, property of rapid decayCategories: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 spaceCategories:47H09, 46B20, 47H10, 47E10

9. CMB 2014 (vol 58 pp. 44)

Daniilidis, A.; Drusvyatskiy, D.; Lewis, A. S.
 Orbits of Geometric Descent We prove that quasiconvex functions always admit descent trajectories bypassing all non-minimizing critical points. Keywords:differential inclusion, quasiconvex function, self-contracted curve, sweeping processCategories:34A60, 49J99

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 Susceptible-Infectious-Susceptible (SIS) epidemic model with time-varying 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 controlCategories: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 Kaniuth-Lau-Pym and the second named author in the special case that $\pi \colon A \longrightarrow \mathbf C$ is a non-zero character on $A$. Keywords:Banach algebras, $\pi$-invariance, derivations, representationsCategories:46H15, 46H25, 13N15

12. CMB 2012 (vol 56 pp. 584)

Liau, Pao-Kuei; Liu, Cheng-Kai
 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 $[x,y]=xy-yx$ the commutator of $x$ and $y$. If $\sigma$ is a non-identity 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 Gross-Pitaevskii Equation with Trapped Dipolar Quantum Gases In this paper, we consider the Gross-Pitaevskii 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 so-called invariant manifold of the evolution flow. Keywords:Gross-Pitaevskii equation, sharp threshold, global existence, blow upCategories: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 2-generator groups whose abelianization epimorphism is not coessential. This family also provides counterexamples to the generalized Andrews--Curtis conjecture. Keywords:coessential epimorphism, Nielsen transformations, Andrew-Curtis transformationsCategories: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 $P(Y)=Y^{n}-\alpha XY^{n-1}-\alpha^n,$ 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 fieldsCategories:11A55, 11D45, 11D72, 11J61, 11J66

16. CMB 2011 (vol 56 pp. 55)

 Cliquishness and Quasicontinuity of Two-Variable 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 point-picking'' 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, point-picking game, quasicontinuity, separate continuity, two variable mapsCategories:54C05, 54C08, 54B10, 91A05

17. CMB 2011 (vol 55 pp. 597)

 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 problemCategories:31B05, 60G44, 60G40

18. CMB 2010 (vol 53 pp. 286)

Gorelic, Isaac
 Orders of π-Bases 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 π-base, point-countable π-base, free sequences, canonical form for ordinalsCategories:54A25, 03E10, 03E75, 54A35

19. CMB 2010 (vol 53 pp. 223)

Chuang, Chen-Lian; Lee, Tsiu-Kwen
 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 one-sided 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, PICategories: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 so-called asymptotic compactness property of the solution operator by energy equation approach. Keywords:global attractor, Fourier restriction norm, damping system, asymptotic compactnessCategories:35Q53, 35B40, 35B41, 37L30

21. CMB 2008 (vol 51 pp. 372)

Ezquerro, J. A.; Hernández, M. A.
 Picard's Iterations for Integral Equations of Mixed Hammerstein Type A new semilocal convergence result for the Picard method is presented, where the main required condition in the contraction mapping principle is relaxed. Keywords:nonlinear equations in Banach spaces, successive approximations, semilocal convergence theorem, Picard's iteration, Hammerstein integral equationsCategories:45G10, 47H99, 65J15

22. CMB 2008 (vol 51 pp. 283)

Ravindra, G. V.
 The Noether--Lefschetz Theorem Via Vanishing of Coherent Cohomology We prove that for a generic hypersurface in $\mathbb P^{2n+1}$ of degree at least $2+2/n$, the $n$-th Picard number is one. The proof is algebraic in nature and follows from certain coherent cohomology vanishing. Keywords:Noether--Lefschetz, algebraic cycles, Picard numberCategories:14C15, 14C25

23. CMB 2008 (vol 51 pp. 146)

Zhou, Xiaowen
 Stepping-Stone Model with Circular Brownian Migration In this paper we consider the stepping-stone 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 stepping-stone 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:stepping-stone model, circular coalescing Brownian motion, Arratia flow, duality, entrance lawCategories: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, $abc$-conjectureCategories:11R04, 11R06, 11R09, 11R33, 11D09

25. CMB 2006 (vol 49 pp. 237)

Gauthier, P. M.; Zeron, E. S.
 Approximation by Rational Mappings, via Homotopy Theory Continuous mappings defined from compact subsets $K$ of complex Euclidean space $\cc^n$ into complex projective space $\pp^m$ are approximated by rational mappings. The fundamental tool employed is homotopy theory. Keywords:Rational approximation, homotopy type, null-homotopicCategories:32E30, 32C18
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