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 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 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 results for rooted minor problems

In this article, we consider the following problem. Given four distinct vertices v₁,v₂,v₃,v₄. How many edges guarantee the existence of seven connected disjoint subgraphs X_i for i = 1,…, 7 such that X_j contains v_j for j = 1, 2, 3, 4 and for j = 1, 2, 3, 4, X_j has a neighbor to each X_k with k = 5, 6, 7. This is the so called “rooted K_{3, 4}‐minor problem.” There are only few known results on rooted minor problems, for example, [15,6]. In this article, we prove that a 4‐connected graph with n vertices and at least 5n ? 14 edges has a rooted K_3,4‐minor. In the proof we use a lemma on graphs with 9 vertices, proved by computer search. We also consider the similar problems concerning rooted K_3,3‐minor problem, and rooted K_3,2‐minor problem. More precisely, the first theorem says that if G is 3‐connected and e(G) ≥ 4|G| ? 9 then G has a rooted K_3,3‐minor, and the second theorem says that if G is 2‐connected and e(G) ≥ 13/5|G| ? 17/5 then G has a rooted K_3,2‐minor. In the second case, the extremal function for the number of edges is best possible. These results are then used in the proof of our forthcoming articles 7 , 8 . © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 191–207, 2007     

Extremal problems for chromatic neighborhood sets

The chromatic neighborhood sequence of a graph G is the list of the chromatic numbers of the subgraphs induced by the neighborhoods of the vertices. We study the maximum multiplicity of this sequence, proving, amongst other things, that if a chromatic neighborhood sequence has t distinct values, the largest value being d_t, then there is a value with multiplicity at least . This bound is asymptotically tight. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 68–74, 2002     

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 on set systems

For a family $\boldmath{F}(k) = \{{\mathcal F}_1^{(k)}, {\mathcal F}_2^{(k)},\ldots,{\mathcal F}_t^{(k)}\}$ of k‐uniform hypergraphs let ex (n, F (k)) denote the maximum number of k‐tuples which a k‐uniform hypergraph on n vertices may have, while not containing any member of F (k). Let r_k(n) denote the maximum cardinality of a set of integers Z?[n], where Z contains no arithmetic progression of length k. For any k≥3 we introduce families $\boldmath{F}(k) = \{{\mathcal F}_1^{(k)},{\mathcal F}_2^{(k)}\}$ and prove that n^k?2r_k(n)≤ex (n_k², F (k))≤c_kn^k?1 holds. We conjecture that ex(n, F (k))=o(n^k?1) holds. If true, this would imply a celebrated result of Szemerédi stating that r_k(n)=o(n). By an earlier result o Ruzsa and Szemerédi, our conjecture is known to be true for k=3. The main objective of this article is to verify the conjecture for k=4. We also consider some related problems. © 2002 Wiley Periodicals, Inc. Random Struct. Alg. 20: 131–164, 2002.     

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,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 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.