Global solutions of partitioning problems ∗

In this paper, a special kind of optimization problems is considered. The total cost function for a homogeneous group of participants– each with the same convex-concave cost function – is to be minimized. The particular structure of this problem makes it possible to find global minimizers in dependence on the linking parameter. The symmeric partition is a stationary solution and the optimal one for small parameter values. Beyond a critical parameter value one obtains certain non-symmetric solutions. Two geometrical applications are discussed. They are related to perimeter and volume partitions     

Adaptive methods for solvings minimax problems ∗

We consider finite-dimensional minimax problems for two traditional models: firstly,with box constraints at variables and,secondly,taking into account a finite number of tinear inequalities. We present finite exact primal and dual methods. These methods are adapted to a great extent to the specific structure of the cost function which is formed by a finite number of linear functions. During the iterations of the primal method we make use of the information from the dual problem, thereby increasing effectiveness. To improve the dual method we use the “long dual step” rule (the principle of ullrelaxation).The results are illustrated by numerical experiments.     

Sample solutions of stochastic boundary value problems ∗

We prove existence theorems for nonlinear stochastic Sturm-Liouville problems which improve results from [4]. In the simplest case this is done by means of a known result about measurable selections of multivalued maps and a new fixed point theorem for stochastic nonlinear operators which is more realistic than existing ones     

An alogrithm for monotonic global optimization problems ∗

We propose an algorithm to locate a global maximum of an increasing function subject to an increasing constraint on the cone of vectors with nonnegative coordinates. The algorithm is based on the outer approximation of the feasible set. We eastablish the con vergence of the algorithm and provide a number of numerical experiments. We also discuss the types of constraints and objective functions for which the algorithm is best suited     

Controllability completion problems of partial upper triangular matrices ∗

The main result of this paper states sufficient conditions for the existence of a completion A_c of an n × n partial upper triangular matrix A, such that the pair (A_c B) has prescribed controllability indices, being B an n×m matrix. If A is a partial Hessenberg matrix some conditions may be dropped. An algorithm that obtains a completion A_c of A such that pair (A_c e_k ) is completely controllable, where e_k is a unit vector, is used to proof the results.     

On stochastic programming ii: dynamic problems under risk ∗

This paper surveys recent work on dynamic stochastic programming problems and their applications. New results are included on the measurability and interpretation-in terms of the expected value of perfect information (EVPI)-of the dual multiplier processes corresponding to these problems. A final section reports preliminary computational experiments with algorithms for 2-stage problems     

State constrained optimal control problems governed by semilinear equations sup∗

This work deals with necessary conditions of optimality for optimal control problems governed by semilinear differential equations in Hilbert spaces.     

A posteriori finite element error estimators for indefinite elliptic boundary value problems ∗

A general construction technique is presented for a posteriori error estimators of finite element solutions of elliptic boundary value problems that satisfy a Gång inequality. The estimators are obtained by an element–by–element solution of ‘weak residual’ with or without considering element boundary residuals. There is no order restriction on the finite element spaces used for the approximate solution or the error estimation; that is, the design of the estimators is applicable in connection with either one of the h–p–, or hp– formulations of the finite element method. Under suitable assumptions it is shown that the estimators are bounded by constant multiples of the true error in a suitable norm. Some numerical results are given to demonstrate the effectiveness and efficiency of the approach.     

On the continuity of the minima for a family of constrained optimization problems ∗

We consider a family of problems Py dealing with the minimization of a given function on a constraint set, both depending on a parameter y. We study continuity properties, with respect to a parameter, of the value and of the solution set of the problems. Working with convex functions and convex constraint sets, we show how the well-posedness of the problem allows to avoid compactness hypotheses usually requested to get the same stability results.     

The coupling of boundary integral and finite element methods for nonmonotone nonlinear problems ∗

In this paper, we apply the coupling of the boundary integral and finite element methods to study the weak solvability of certain nonmonotone nonlinear exterior boundary value problems. In order to convert the original exterior problem into an equivalent nonlocal boundary value problem on a finite region, we employ two different approaches based on the use of one and two integral equations on the coupling boundary. Existence of a solution for the associated weak formulation, and convergence properties of the corresponding Galerkin approximations are deduced from fundamental results in nonlinear functional analysis. Indeed, the main arguments of our proofs are based on a surjectivity theorem for mappings of type (S) and on the Fredholm alternative for nonlinear A-proper mappings.     

On ∗-minimal ∗-ideals and ∗-biideals in involution rings

For semiprime involution rings, we determine some ∗-minimal ∗-ideals using idempotent elements. Nevertheless, ∗-minimal ∗-biideals are characterized by idempotent elements. Moreover, the involutive version of a theorem due to Steinfeld, which investigates a semiprime involution ring A if A=SocA, is given. Finally, semiprime involution rings having no proper nonzero ∗-biideals are characterized.     

Symmetric-line problems

In analytic geometry, senior students usually run into some“Symmetric-line problems”. Here, I will introduce you some ingenious(巧妙的) sol- ving processes about them. Question 1: line L_1 and L_2 are symmetri- cal about line:y=3, and the equation of L_1 is known as:x y-6=0, so the equation of L_2     

Taximetry∗

In this article the authors present some interesting geometrical ideas and give their experiences of teaching these to young children. Possible extensions of these ideas to the concept of functions are also indicated.

     

Characterizing stable complete Erdős space

Every Polish group is not free whereas some F _σ group may be free. Also, every automorphism group of a structure of cardinality, e.g. ℶ _ω , is not free.     

∗-Polynomial Identities of a Nonsymmetric ∗-Minimal Algebra

Let F be an infinite field of characteristic ≠?2. We study the ?-polynomial identities of the ?-minimal algebra R?=?UT _?(F?⊕?F, F). We describe the generators of T _?(R) and a linear basis of the relatively free algebra of R. When char.F?=?0, these results allow us to provide a complete list of polynomials generating irreducible GL × GL-modules decomposing the proper part of the relatively free algebra of R. Finally, the ?-codimension sequence of R is explicitly computed.     

Characterizing C6＋P2-graphic Sequences

For a given graph H, a graphic sequence π = （d1, d2, ···, dn） is said to be potentially H-graphic if π has a realization containing H as a subgraph. In this paper, we characterize the potentially C6 ＋P2-graphic sequences where C6 ＋P2 denotes the graph obtained from C6 by adding two adjacent edges to the three pairwise nonadjacent vertices of C6. Moreover, we use the characterization to determine the value ofσ（C6＋P2, n）.     

Lucas' theorem refined ∗

We prove a refined version of the classical Lucas' theorem: if p is a polynomial with zeros a ₁,…,a _n+1 all having modulus one and φis the Blaschke product whose zeros are those of the derivative p ¹, then the compression of the shift S(φ) has its numerical range circumscribed about by the (n+ l)-gon a ₁…a _n+1 with tangent points the midpoints of then+ l sides of the polygon. This is proved via a special matrix representation of S(φ) and is a generalization of the known case for n= 2.     

E ∗-dense E-semigroups

We obtain a covering theorem for E ^?-dense E-semigroups showing that such a semigroup has an E ^?-dense, strongly E ^?-unitary E-semigroup as a cover and describe the structure of the latter semigroups.