An extremal problem concerning matrices of 0's and 1 's

We determine the smallest number f(n,k) such that every (0,1)-matrix of order n what zero main diagonal which has at least f(n,k) 1's contains an irreducible, principal submatrix of order K. We characterize those matrices with f(n,k)?1 l's having no irreducible, principal submatrix of order k     

An extremal problem for positive defintie matrices

A problem studied by Flanders (1975) is minimize the function f(R)=tr(SR+TR^-1) over the set of positive definite matrices R, where S and T are positive semi-definite matrices. Alternative proofs that may have some intrinsic interest are provided. The proofs explicitly yield the infimum of f(R). One proof is based on a convexity argument and the other on a sequence of reductions to a univariate problem.     

Summability matrices and sequences of 0's and 1's

Summary Conditions are given on a nonnegative regular summability matrix A to ensure that for a given number α, 0 ≤ α ≤ 1, there exists a sequence x consisting of 0's and 1's such that Ax converges to α.     

An extremal problem for operators

Let A be a Banach algebra, F a compact set in the complex plane, and h a function holomorphic in some neighborhood of the set F. Thus h(a) is meaningful for each element a ε A whose spectrum σ(a) is contained in F, and it is possible to evaluate the norm |h(a)|. Problem: Compute the supremum of the norms |h(a) as a ranges over all elements of A with spectrum contained in F and whose norm does not exceed one; that is, compute sup{|h(a)|; a ε A, σ(a) ⊂ F, |a| ⩽ 1}. This problem was first formulated and treated by the author in the particular case where A is the algebra of all linear operators on a finite-dimensional Hilbert space and F is the disc {z; |z| ⩽ r} for a given positive number r<1. The paper discusses motivation, connections with complex function theory, convergence of iterative processes, critical exponents, and the infinite companion matrix.     

An extremal problem for polynomials

We give a solution to an extremal problem for polynomials, which asks for complex numbers α₀,…,α_n${\alpha }_0,\dots ,{\alpha }_n$ of unit magnitude that minimise the largest supremum norm on the unit circle for all polynomials of degree n whose k  -th coefficient is either α_k${\alpha }_k$ or −α_k$-{\alpha }_k$.     

An extremal problem in convolutions

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 3, pp. 412–414, March, 1989.     

Linear bound on extremal functions of some forbidden patterns in 0-1 matrices

In this note by saying that a 0-1 matrix A avoids a pattern P given as a 0-1 matrix we mean that no submatrix of A either equals P or can be transformed into P by replacing some 1 entries with 0 entries. We present a new method for estimating the maximal number of the 1 entries in a matrix that avoids a certain pattern. Applying this method we give a linear bound on the maximal number of the 1 entries in an n by n matrix avoiding pattern L₁ and thereby we answer the question that was asked by Gábor Tardos. Furthermore, we use our approach on patterns related to L₁.     

An extremal problem of quasiconformal mappings

In this paper, the following problem is studied. Let and be two domains in the complex plane with . Suppose that are two quasiconformal mappings satisfying . Let be the mapping in defined by (). If both and are uniquely extremal, is always uniquely extremal? It is shown in this paper that the answer to this problem is no.

     

An extremal problem in geodetic graphs

An upper bound for the number of lines in a geodetic block of diameter d on p points is obtained, using some new general properties of geodetic blocks which are also of independent interest.     

An extremal problem related to probability

Summary We consider a walk from a stateA ₁ to a stateA _n+1 in which the probability of remaining atA _i isp _i , and the probability of progressing fromA _i toA _i+1 is 1 –p _i . The probabilityW _nk of reachingA _n+1 fromA ₁ in exactlyn + k steps can then be expressed as a polynomial of degreen + k in then variablesp ₁,,p _n . We determine the maximum value ofW _nk and the (unique) choice (p ₁,,p _n ) for which this extremum occurs.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday