Expand all Collapse all | Results 1 - 22 of 22 |
1. CMB Online first
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 process Categories:34A60, 49J99 |
2. CMB Online first
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 decay Categories:46L54, 20G42, 46L65 |
3. CMB 2012 (vol 56 pp. 621)
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 control Categories:49J15, 92D30 |
4. CMB 2012 (vol 56 pp. 534)
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, representations Categories:46H15, 46H25, 13N15 |
5. CMB 2012 (vol 56 pp. 584)
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 |
6. CMB 2011 (vol 56 pp. 378)
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 up Categories:35Q55, 35A05, 81Q99 |
7. CMB 2011 (vol 56 pp. 395)
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 transformations Categories:20F05, 20F99, 20J15 |
8. CMB 2011 (vol 56 pp. 258)
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 fields Categories:11A55, 11D45, 11D72, 11J61, 11J66 |
9. 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 maps Categories:54C05, 54C08, 54B10, 91A05 |
10. 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 problem Categories:31B05, 60G44, 60G40 |
11. CMB 2010 (vol 53 pp. 286)
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 ordinals Categories:54A25, 03E10, 03E75, 54A35 |
12. CMB 2010 (vol 53 pp. 223)
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, PI Categories:16D60, 16S50 |
13. CMB 2009 (vol 53 pp. 295)
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 compactness Categories:35Q53, 35B40, 35B41, 37L30 |
14. CMB 2008 (vol 51 pp. 372)
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 equations Categories:45G10, 47H99, 65J15 |
15. CMB 2008 (vol 51 pp. 283)
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 number Categories:14C15, 14C25 |
16. CMB 2008 (vol 51 pp. 146)
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 law Categories:60G57, 60J65 |
17. CMB 2007 (vol 50 pp. 191)
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 |
18. CMB 2006 (vol 49 pp. 237)
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-homotopic Categories:32E30, 32C18 |
19. CMB 2002 (vol 45 pp. 154)
On the Poisson Integral of Step Functions and Minimal Surfaces Applications of minimal surface methods are made to obtain information
about univalent harmonic mappings. In the case where the mapping arises
as the Poisson integral of a step function, lower bounds for the number
of zeros of the dilatation are obtained in terms of the geometry of the
image.
Keywords:harmonic mappings, dilatation, minimal surfaces Categories:30C62, 31A05, 31A20, 49Q05 |
20. CMB 2000 (vol 43 pp. 397)
Tournaments and Orders with the Pigeonhole Property A binary structure $S$ has the pigeonhole property ($\mathcal{P}$) if
every finite partition of $S$ induces a block isomorphic to $S$. We
classify all countable tournaments with ($\mathcal{P}$); the class of
orders with ($\mathcal{P}$) is completely classified.
Keywords:pigeonhole property, tournament, order Categories:05C20, 03C15 |
21. CMB 1999 (vol 42 pp. 335)
Cyclic Subgroup Separability of HNN-Extensions with Cyclic Associated Subgroups We derive a necessary and sufficient condition for HNN-extensions
of cyclic subgroup separable groups with cyclic associated
subgroups to be cyclic subgroup separable. Applying this, we
explicitly characterize the residual finiteness and the cyclic
subgroup separability of HNN-extensions of abelian groups with
cyclic associated subgroups. We also consider these residual
properties of HNN-extensions of nilpotent groups with cyclic
associated subgroups.
Keywords:HNN-extension, nilpotent groups, cyclic subgroup separable $(\pi_c)$, residually finite Categories:20E26, 20E06, 20F10 |
22. CMB 1998 (vol 41 pp. 41)
On the Clarke subdifferential of an integral functional on $L_p$, $1\leq p < \infty$ Given an integral functional defined on $L_p$, $1 \leq p <\infty$,
under a growth condition we give an upper bound of the Clarke
directional derivative and we obtain a nice inclusion between the
Clarke subdifferential of the integral functional and the set of
selections of the subdifferential of the integrand.
Keywords:Integral functional, integrand, epi-derivative Categories:28A25, 49J52, 46E30 |