Determining whether a finite group of outer automorphisms of a free group is geometric

In this article we give necessary and sufficient conditions for a given finite group of outer automorphisms to be induced by the action of a group of orientation-preserving homeomorphisms on the fundamental group of a punctured surface. When the group is abelian, necessary and sufficient conditions can also be given in the absence of orientability assumptions. These properties are formulated in terms of the finite automorphism groups which project into the given outer automorphism group: each non-trivial automorphism in any such group can fix at most a cyclic subgroup of the fundamental group.     

On the existence of solutions to the quadratic mixed-integer mean–variance portfolio selection problem

In the standard mean–variance portfolio selection approach, several operative features are not taken into account. Among these neglected aspects, one of particular interest is the finite divisibility of the (stock) assets, i.e. the obligation to buy/sell only integer quantities of asset lots whose number is pre-established. In order to consider such a feature, we deal with a suitably defined quadratic mixed-integer programming problem. In particular, we formulate this problem in terms of quantities of asset lots (instead of, as usual, in terms of capital per cent quotas). Secondly, we provide necessary and sufficient conditions for the existence of a non-empty mixed-integer feasible set of the considered programming problem. Thirdly, we present some rounding procedures for finding, in a finite number of steps, a feasible mixed-integer solution which is better than the one detected by the necessary and sufficient conditions in terms of the value assumed by the portfolio variance. Finally, we perform an extensive computational experiment by means of which we verify the goodness of our approach.     

Parametrized Gröbner–Shirshov bases

We consider the problem of describing Gröbner–Shirshov bases for free associative algebras in finite terms. To this end we consider parametrized elements of an algebra and give methods for working with them which under favorable conditions lead to a basis given by finitely many patterns. On the negative side we show that in general there can be no algorithm. We relate our study to the problem of verifying that a given set of words in certain groups yields Bokut’ normal forms (or groups with a standard basis).     

Monotonicity and finite element methods for hyperbolic initial value problems

A well known theórem about super- and subfunctions for the solution of hyperbolic initial value problems constructs differentiable functions as upper and lower bounds (see Walter [1], 21 XIII). The proof can be done by transforming the differential equation problem into a set of integral equations, using the monotonicity-properties of the arising integral operators. This proof needs an integral representation for twice differentiable functions. It is shown that this proceeding can be generalized to get upper and lower bounds in terms of finite element functions. To do this, we give an integral representation for continuous, piecewise differentiable functions, including the discontinuities of their derivatives. Then the generalization of the classical proof yields interface conditions for the finite element functions. Finally, it is demonstrated how to realize numerically these conditions.     

On the choice of parameters for the weighting method in vector optimization

We present a geometrical interpretation of the weighting method for constrained (finite dimensional) vector optimization. This approach is based on rigid movements which separate the image set from the negative of the ordering cone. We study conditions on the existence of such translations in terms of the boundedness of the scalar problems produced by the weighting method. Finally, using recession cones, we obtain the main result of our work: a sufficient condition under which weighting vectors yield solvable scalar problems. An erratum to this article can be found at     

Optimal control of stochastic differential equations with dynamical boundary conditions

In this paper we investigate the optimal control problem for a class of stochastic Cauchy evolution problems with nonstandard boundary dynamic and control. The model is composed by an infinite dimensional dynamical system coupled with a finite dimensional dynamics, which describes the boundary conditions of the internal system. In other terms, we are concerned with nonstandard boundary conditions, as the value at the boundary is governed by a different stochastic differential equation.     

超可解群的若干充分条件

本文研究了有限群的超可解性问题，利用子群的共轭可换性概念及极小反例法，获得了一个群为超可解的若干充分条件．举例说明了主要结果中的假设条件是不可少的．     

The path relation for directed planar graphs in rectangles, and its relation to the free diad

In this paper, we define the path relation of a directed graph to be the relation which relates two vertices if there is a path from the first to the second. We study the restriction of this relation to paths from sources to sinks, and consider the question of when two finite graphs embedded in a rectangle give the same relation. We find a set of local changes to these graphs which can be used to get between any two graphs for which this relation is the same. Furthermore, we classify the relations which can arise as this relation for a finite directed graph embedded in a rectangle as the triconvex relations between finite ordinals (defined in this paper).This work originated from some of the author’s work on category theory. It turns out that the category of finite ordinals and relations that can be the path relation of a directed graph embedded in a rectangle, is relevant to the study of diads—introduced by the author as a common generalisation of monads and comonads (note that the terms diad and dyad have been used to mean different things by other authors). More specifically, the referee of one of the author’s papers suggested that it would be useful to identify the category which plays the role for diads that the category of finite ordinals and order-preserving functions plays for monads. It turns out that the category of finite ordinals and relations that can be path relations of graphs embedded in a rectangle, is exactly the category that plays this role.     

On sum and stability of g-frames in Hilbert spaces

In this paper, we give some new results on sum and stability of g-frames in Hilbert spaces. Since the finite sum of g-frames may not be a g-frame for the Hilbert space, we give a necessary and sufficient condition and some sufficient conditions for the finite sum of g-frames to be a g-frame. We also show that every g-sequence in Hilbert space can be expanded to a tight g-frame by adding a linear bounded operator. Moreover, we obtain some sufficient conditions under which g-frames (and the finite sum of g-frames) are stable under small perturbations.     

Splitting‐characterizations of the Papangelou process

For point processes we establish a link between integration‐by‐parts‐ and splitting‐formulas which can also be considered as integration‐by‐parts‐formulas of a new type. First we characterize finite Papangelou processes in terms of their splitting kernels. The main part then consists in extending these results to the case of infinitely extended Papangelou and, in particular, Pólya and Gibbs processes.     

On Polynomial Optimization Over Non-compact Semi-algebraic Sets

The optimal value of a polynomial optimization over a compact semi-algebraic set can be approximated as closely as desired by solving a hierarchy of semidefinite programs and the convergence is finite generically under the mild assumption that a quadratic module generated by the constraints is Archimedean. We consider a class of polynomial optimization problems with non-compact semi-algebraic feasible sets, for which an associated quadratic module, that is generated in terms of both the objective function and the constraints, is Archimedean. For such problems, we show that the corresponding hierarchy converges and the convergence is finite generically. Moreover, we prove that the Archimedean condition (as well as a sufficient coercivity condition) can be checked numerically by solving a similar hierarchy of semidefinite programs. In other words, under reasonable assumptions, the now standard hierarchy of semidefinite programming relaxations extends to the non-compact case via a suitable modification.     

MEAN-SQUARE STABILITY OF NONLINEAR SYSTEMS WITH TIME-VARYING,RANDOM DELAY

A class of nonlinear systems with a time-varying delay is considered.The delay is modeled by a continuous-time Markov process with a finite number of states. Systems of this type may arise in real-time control applications. Employing a “delay-averaging” approach we demonstrate how certain mean-square stochastic stability conditions can be derived in terms of transition functions of the Markov process and stability properties of a system with a constant delay.     

On the spatial blow-up and decay for some nonlinear parabolic equations with nonlinear boundary conditions

In this note we investigate the spatial behavior of several nonlinear parabolic equations with nonlinear boundary conditions. Under suitable conditions on the nonlinear terms we prove that the solutions either cease to exist for a finite value of the spatial variable or else they decay algebraically. The main tool used is the weighted energy method. Our results can be applied to several situations concerning heat conduction. Received: April 4, 2004; revised: September 20, 2004     

Zeta functions of trinomial curves and maximal curves

We determine the zeta functions of trinomial curves in terms of Jacobi sums, and obtain an explicit formula of the genus of a trinomial curve over a finite field, and we study the conditions for this curve to be a maximal curve over a finite field.     

Finite rank kernels for multi-task learning

Motivated by the importance of kernel-based methods for multi-task learning, we provide here a complete characterization of multi-task finite rank kernels in terms of the positivity of what we call its associated characteristic operator. Consequently, we are led to establishing that every continuous multi-task kernel, defined on a cube in an Euclidean space, not only can be uniformly approximated by multi-task polynomial kernels, but also can be extended as a multi-task kernel to all of the Euclidean space. Finally, we discuss the interpolation of multi-task kernels by multi-task finite rank kernels.     

Simpler Maltsev conditions for (weak) difference terms in locally finite varieties

This paper is motivated by a practical question: given a finite algebra A in a finite language, how can we best program a computer to decide whether the variety generated by A has a difference term, and how hard is it to find the difference term? To help address this question, we produce a simple Maltsev condition which characterizes difference terms in the class of locally finite varieties. We do the same for weak difference terms.     

降维梯度法

§1.引言 本文研究降维梯度法,它具有共轭梯度法的一切性质.对于正定二次函数,用不着精确的一维搜索,只要在每步加入两个校正项,即可将高阶问题转化为低阶问题,保证了二     

Structure of extreme correlated equilibria: a zero-sum example and its implications

We exhibit the rich structure of the set of correlated equilibria by analyzing the simplest of polynomial games: the mixed extension of matching pennies. We show that while the correlated equilibrium set is convex and compact, the structure of its extreme points can be quite complicated. In finite games the ratio of extreme correlated to extreme Nash equilibria can be greater than exponential in the size of the strategy spaces. In polynomial games there can exist extreme correlated equilibria which are not finitely supported; we construct a large family of examples using techniques from ergodic theory. We show that in general the set of correlated equilibrium distributions of a polynomial game cannot be described by conditions on finitely many moments (means, covariances, etc.), in marked contrast to the set of Nash equilibria which is always expressible in terms of finitely many moments.     

A semilinear parabolic system with generalized Wentzell boundary condition

In this paper, we consider a weak coupled semilinear parabolic system with general Wentzell boundary condition. We prove the well-posedness of the problem and derive different conditions in terms of the powers of the nonlinear terms under which the global solution exists and finite time blow-up occurs.     

On the summation of Schlömilch's series

ABSTRACT

Schlömilch's series is named after the German mathematician Oscar Xavier Schlömilch, who derived it in 1857 as a Fourier series type expansion in terms of the Bessel function of the first kind. However, except for Bessel functions, here we consider an expansion in terms of Struve functions or Bessel and Struve integrals as well. The method for obtaining a sum of Schlömilch's series in terms of the Bessel or Struve functions is based on the summation of trigonometric series, which can be represented in terms of the Riemann zeta and related functions of reciprocal powers and in certain cases can be brought in the closed form, meaning that the infinite series are represented by finite sums. By using Krylov's method we obtain the convergence acceleration of the trigonometric series.