An extremal property of the student distribution

The extrema of the distribution function of Student's ratio for observations of unequal accuracy are found.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 153, pp. 16–26, 1986.     

An extremal property of Hermite polynomials

Let H_n be the nth Hermite polynomial, i.e., the nth orthogonal on polynomial with respect to the weight w(x)=exp(−x²). We prove the following: If f is an arbitrary polynomial of degree at most n, such that |f||H_n| at the zeros of H_n+1, then for k=1,…,n we have f^(k)H_n^(k), where · is the norm. This result can be viewed as an inequality of the Duffin and Schaeffer type. As corollaries, we obtain a Markov-type inequality in the norm, and estimates for the expansion coefficients in the basis of Hermite polynomials.     

An extremal property of Fekete polynomials

The Fekete polynomials are defined as

where is the Legendre symbol. These polynomials arise in a number of contexts in analysis and number theory. For example, after cyclic permutation they provide sequences with smallest known norm out of the polynomials with coefficients.

The main purpose of this paper is to prove the following extremal property that characterizes the Fekete polynomials by their size at roots of unity.

Theorem 0.1. Let with odd and . If

then must be an odd prime and is . Here

This result also gives a partial answer to a problem of Harvey Cohn on character sums.

     

An extremal property of rectangular distributions

It is proved that the correlation coefficient between the elements of the order statistics is maximal for a rectangularly (uniformly) distributed population.     

An extremal property of convex hexagons

The following conjecture is discussed: if K is a plane convex figure and T is a triangle of maximal area contained in K, then K is contained in ?5 \sqrt {5} T. It is shown that it suffices to check the conjecture in the case where K is a convex hexagon, but the conjecture is proved only in the case where K is a pentagon. Bibliography: 2 titles.     

An extremal property of outer functions

Our main result is the following: iff (z) is in the space H², and F(z) is its outer part, then F⁽ⁿ⁾_H2F⁽ⁿ⁾_H2(n=1,2,...), the left side being finite if the right side is finite. Under certain essential restrictions, this. inequality was proved by B. I. Korenblyum and V. S. Korolevich [1].Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 53–56, July, 1971.     

An extremal property of the Möbius function

Archiv der Mathematik -     

An extremal sparsity property of the Jordan canonical form

We prove that among all the matrices that are similar to a given square complex matrix, the Jordan canonical form has the largest number of off-diagonal zero entries. We also characterize those matrices that attain this largest number.     

An identity between the determinant and the permanent of Hessenberg-type matrices

In this short note we provide an extension of the notion of Hessenberg matrix and observe an identity between the determinant and the permanent of such matrices. The celebrated identity due to Gibson involving Hessenberg matrices is consequently generalized.     

An extremal property of positive operators and its spectral implications

The Perron-Frobenius theory of a positive operator T (defined on an ordered space E) is developed from an extremal characterization of its largest eigenvalue. This characterization has manifest intrinsic interest. Additionally, it is used to give a particularly revealing derivation of the basic results concerning the existence, multiplicity, and distribution of the eigenvalues of T of maximum modulus. A significant feature of this derivation is that the customary assumptions that the space E be complete and/or that its positive cone have a nonvoid interior are often unnecessary or can be replaced by weaker hypotheses more amenable to practical applications (see 1, 3). The extremal characterization proof of the distribution properties of the eigenvalues of maximum modulus is new in the infinite dimensional case. Also, several new results on the extent of applicability of the extremal characterization are given     

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.     

A geometric property of extremal surfaces

Let the surface R³ be defined by the equation z = f(x, y), where f(x, y) is a function 3 times continuously differentiable in R². It is proved that if the total (Gaussian) curvature of the surface is nonzero almost everywhere on in the sense of Lebesgue measure in R²), then is extremal, i.e., for almost all (x,y) R² the inequality max (||qx||, ||qy, qf (x, y)) > q^–1/s–. holds for all integral q qo (f), where x is the distance from the real number x to the nearest integer and > 0 is arbitrarily small.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 177–181, February, 1978.In conclusion, the author thanks V. G. Sprindzhuk for suggesting the problem.     

Economic equilibrium: Existence and extremal property

The existence and an extremal property of the Arrow—Debreu economic equilibrium are investigated. The optimality of the equilibrium is also considered.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol. 19, pp. 23–58, 1982.     

An extremal property of Jacobi polynomials in two-sided Chernoff-type inequalities for higher order derivatives

For a weight function generating the classical Jacobi polynomials, the sharp double estimate of the distance from the subspace of all polynomials of an arbitrary fixed order is established.