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 for trigonometric polynomials

Let be a trigonometric polynomial of degree The problem of finding the largest value for in the inequality is studied. We find exactly provided is the conjugate of an even integer and For general we get an interval estimate for where the interval length tends to as tends to

     

An extremal bandwidth problem for bipartite graphs

This paper discusses the problem of finding the maximum number of edges E(m, n, B) in a bipartite graph having partite set sizes m and n and bandwidth B. Exact values for E(m, n, B) are found for many cases. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 278–289, 2000     

An extremal problem for generalized Lelong numbers

We look for pointwise bounds on a plurisubharmonic function near its singularity point, given the value of its generalized Lelong number with respect to a plurisubharmonic weight. To this end, an extremal problem is considered. In certain cases, the problem is solved explicitly.     

An extremal problem for Graham-Rothschild parameter words

This paper exposes connections between the theory of Möbius functions and extremal problems, extending ideas of Frankl and Pach [8]. Extremal results concerning the trace of objects in geometric lattices and Graham—Rothschild parameter posets are proved, covering previous results due to Sauer [16] and Perles and Shelah [17].     

An extremal problem for subdivisions of K

It is proved that every graph G with ‖G‖ ≥ 2|G| − 5, |G| ≥ 6, and girth at least 5, except the Petersen graph, contains a subdivision of K, the complete graph on five vertices minus one edge. © 1999 John Wiley & Sons, Inc, J. Graph Theory 30: 261–276, 1999     

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.     

An extremal problem for real algebraic polynomials

Let G _n be the set of all real algebraic polynomials of degree at most n, positive on the interval (?1, 1) and without zeros inside the unit circle (|z| < 1). In this paper an inequality for the polynomials from the set G _n is obtained. In one special case this inequality is reduced to the inequality given by B. Sendov [5] and in another special case it is reduced to an inequality between uniform norm and norm in the L ₂ space for the Jacobi weight.     

An extremal problem in convolutions

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

An extremal problem for Fourier transforms of probabilities

We study an extremal problem concerning the supremum of the Fourier transforms (characteristic functions) of probability distributions under the constraint that the Fourier transforms vanish at a fixed point. This problem arises from the investigation of the survival amplitudes of quantum states driven by Schrödinger dynamics, and has general and curious implications for the evolution pictures of quantum systems. To cite this article: S. Luo, Z. Zhang, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

An extremal property for a class of positive linear operators

We generalize a recent result of de la Cal and Cárcamo concerning an extremal property of Bernstein operators.