Extremal problems for convex polygons

Consider a convex polygon V _n with n sides, perimeter P _n , diameter D _n , area A _n , sum of distances between vertices S _n and width W _n . Minimizing or maximizing any of these quantities while fixing another defines 10 pairs of extremal polygon problems (one of which usually has a trivial solution or no solution at all). We survey research on these problems, which uses geometrical reasoning increasingly complemented by global optimization methods. Numerous open problems are mentioned, as well as series of test problems for global optimization and non-linear programming codes.     

Extremal problems for triple systems

In this article Turán-type problems for several triple systems arising from (k, k ? 2)-configurations [i.e. (k ? 2) triples on k vertices] are considered. It will be shown that every Steiner triple system contains a (k, k ? 2)-configuration for some k < c log n/ log log n. Moreover, the Turán numbers of (k, k ? 2)-trees are determined asymptotically to be ((k ? 3)/3).(ⁿ₂) (1?o(1)). Finally, anti-Pasch hypergraphs avoiding (5, 3) -and (6, 4)-Configurations are considered. © 1993 John Wiley & Sons, Inc.     

Extremal problems for Fredholm eigenvalues

The study of extremal problems for Fredholm eigenvalues was initiated by Schiffer in the context of the existence of conformal maps onto canonical domains. We present a different approach to solving rather general extremal problems for Fredholm eigenvalues related to appropriate univalent functions with quasiconformal extensions. It involves the complex geometry of the universal Teichmüller space.     

Extremal problems for numerical series

In this paper, we consider extremal problems for numerical positive series. The terms of these series are pairwise products of the elements of two sequences, one of which is fixed and the other varies within a given set of sequences. We obtain exact solutions for a number of such problems. As one of the possible applications of the results obtained, we find solutions of some extremal problems related to best n-term approximations of periodic functions.     

Extremal signatures for bivariate Chebyshev approximation problems

The problem of finding a Chebyshev solution of the real matrix equationAX+YB=C, whereC is anm×n matrix, is considered. This equation is equivalent to a linear system [I _n A,B ^T I _m ]z=d. The characterization and the computation of best linear Chebyshev approximations are connected with the notion of extremal signature. The purpose of this paper is to analyze the extremal signatures of this problem.     

Extremal problems for linear functionals on the Tchebycheff spaces

The study of Tchebycheff spaces (generalizing the space of algebraic polynomials) and extremal problems related to them began one and a half centuries ago. Recently, many facts of approximation theory have been understood and reinterpreted from the point of view of general principles of the theory of extremum and convex duality. This approach not only allowed one to prove the previously known results for algebraic polynomials and generalized polynomials in a unified way, but also enabled one to obtain new results. In this paper, we work out this direction with special attention to the optimal recovery problems. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 87–100, 2005.     

Extremal problems in graph theory

The aim of this note is to give an account of some recent results and state a number of conjectures concerning extremal properties of graphs.     

Extremal problems involving neighborhood unions

We examine several extremal problems for graphs satisfying the property |N(x) ∪ N(y)| ? s for every pair of nonadjacent vertices x, y ? V(G). In particular, values for s are found that ensure that the graph contains an s-matching, a 1-factor, a path of a specific length, or a cycle of a specific length.     

Extremal problems for hypotheses testing with set-valued decisions

We consider a class of extremal problems for multiple hypothesis testing with set-valued decisions and given total variation distances between hypotheses. The quality of a test is measured by an arbitrary piecewise linear continuous function of the error probabilities. We show that the extremal value of the test quality may be found as a solution of some linear programming problem, so the original infinite-dimensional problem is reduced to a certain finite-dimensional one.     

Extremal eigenvalue problems for composite membranes,I

Given an open bounded connected set R ^N and a prescribed amount of two homogeneous materials of different density, for smallk we characterize the distribution of the two materials in that extremizes thekth eigenvalue of the resulting clamped membrane. We show that these extremizers vary continuously with the proportion of the two constituents. The characterization of the extremizers in terms of level sets of associated eigenfunctions provides geometric information on their respective interfaces. Each of these results generalizes toN dimensions the now classical one-dimensional work of M. G. Krein.The work of the first author was supported in part by NSF Grant DMS-8201719 (A. Manitius, P. I.), an IBM fellowship, a GE teaching incentive, and DARPA Contract F49620-87-C-0065. That of the second author was supported in part by ONR Grant N00014-84-5-516, AFOSR Grant AFOSR-86-0180, and NSF Grant DMS-8713722.     

Extremal eigenvalue problems for composite membranes,II

Given an open bounded connected set R ^N and a prescribed amount of two homogeneous materials of different density, for smallk we characterize the distribution of the two materials in that extremizes thekth eigenvalue of the resulting clamped membrane. We show that these extremizers vary continuously with the proportion of the two constituents. The characterization of the extremizers in terms of level sets of associated eigenfunctions provides geometric information on their respective interfaces. Each of these results generalizes toN dimensions the now classical one-dimensional work of M. G. Krein.The work of the first author was supported in part by NSF Grant DMS-8201719 (A. Manitius, P. I.), an IBM fellowship, a GE teaching incentive, and DARPA Contract F49620-87-C-0065. That of the second author was supported in part by ONR Grant N00014-84-5-516, AFOSR Grant AFOSR-86-0180, and NSF Grant DMS-8713722.     

Extremal problems concerning Kneser graphs

Let and be two intersecting families of k-subsets of an n-element set. It is proven that | | ≤ (_k−1ⁿ⁻¹) + (_k−1ⁿ⁻¹) holds for , and equality holds only if there exist two points a, b such that {a, b} ∩ F ≠ for all F . For an example showing that in this case max | | = (1−o(1))(_kⁿ) is given. This disproves an old conjecture of Erdös [7]. In the second part we deal with several generalizations of Kneser's conjecture.     

Extremal problems with unstabilized integrands

Institute for Applied Problems of Mechanics and Mathematics, Academy of Sciences of the Ukrainian SSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 22, No. 1, pp. 81–82, January–March, 1988.     

Extremal problems in finite-dimensional spaces

A theory of the first variation in the finite-dimensional case is presented using the concepts of essential covering.