Bernstein's theorem for compact groups

In this paper we generalize the classical Bernstein theorem concerning the absolute convergence of the Fourier series of Lipschitz functions. More precisely, we consider a group G which is finite dimensional, compact, and separable and has an infinite, closed, totally disconnected, normal subgroup D, such that $\text{G}\text{D}$ is a Lie group. Using this structure, we define in a natural way the notion of Lipschitz condition, and then prove that a function which satisfies a Lipschitz condition of order greater than $\text{(}\text{dim}\text{G + 1)}\text{2}$ belongs to the Fourier algebra of G.     

Optimal packings of K4's into a Kn

In this paper we determine the maximum cardinality of a packing of K₄'s into K_n, that is, construct optimal constant weight codes with weight 4 and minimum distance 6.     

On finding the K best cuts in a network

We show that the O(K · n⁴) algorithm of Hamacher (1982) for finding the K best cut-sets fails because it may produce cuts rather than cut-sets. With the convention that two cuts $\text{(X,}\text{X}\text{)}$ and $\text{(Y,}\text{Y}\text{)}$ are different whenever X ≠ Y the K best cut problem can be solved in O(K · n⁴).     

Newton’s method and high-order algorithms for the nth root computation

Two modifications of Newton’s method to accelerate the convergence of the n$n$th root computation of a strictly positive real number are revisited. Both modifications lead to methods with prefixed order of convergence p∈N,p≥2$p\in \mathbb{N},p\ge 2$. We consider affine combinations of the two modified p$p$th-order methods which lead to a family of methods of order p$p$ with arbitrarily small asymptotic constants. Moreover the methods are of order p+1$p+1$ for some specific values of a parameter. Then we consider affine combinations of the three methods of order p+1$p+1$ to get methods of order p+1$p+1$ again with arbitrarily small asymptotic constants. The methods can be of order p+2$p+2$ with arbitrarily small asymptotic constants, and also of order p+3$p+3$ for some specific values of the parameters of the affine combination. It is shown that infinitely many p$p$th-order methods exist for the n$n$th root computation of a strictly positive real number for any p≥3$p\ge 3$.     

Inequalities for the derivative of a polynomial

Let P(z) be a polynomial of degreen which does not vanish in ¦z¦ <k, wherek > 0. Fork ≤ 1, it is known that

$$\mathop {\max }\limits_{|z| = 1} |P'(z)| \leqslant \frac{n}{{1 + k^n }}\mathop {\max }\limits_{|z| = 1} |P(z)|$$

, provided ¦P’(z)¦ and ¦Q’(z)¦ become maximum at the same point on ¦z¦ = 1, where\(Q(z) = z^n \overline {P(1/\bar z)} \). In this paper we obtain certain refinements of this result. We also present a refinement of a generalization of the theorem of Tu?an.

     

An asymptotic estimate related to Selberg's sieve

Suppose 1 ≤ z₁ ≤ z₂ ≤ N, and let $\text{Λ}{}_{\mathrm{i}}\text{(d) = μ(d)}\text{max(log}\text{(}\text{z}{}_{\mathrm{i}}\text{d}\text{), 0)}$ for i = 1, 2. We show that

$\text{∑}\text{N?n}\text{(}\text{∑}\text{d|n}\text{Λ}{}_1\text{(d))(}\text{∑}\text{e|n}\text{Λ}{}_1\text{(e)) = N}\text{log z}{}_1\text{+ O(N).}$

We then use this to improve a result of Barban-Vehov which has applications to zero-density theorems.     

On an upper bound of Sanov's inequality for the probability of a large deviation

In this paper a sharp upper bound of Sanov's inequality for the large probability of large deviations under the multinomial distribution is obtained. Examples of computing the upper bounds for the probabilities of misclassification in discriminant analysis are also given.     

Ramsey's theorem—A new lower bound

This paper gives improved asymptotic lower bounds to the Ramsey function R(k, t). Section 1 considers the symmetric case k = t while the more general case is considered in Section 2.     

A lower bound for the asymptotic characteristics of classes of functions with dominating mixed derivative

We obtain a new representation for classes of functions with dominating mixed derivative; using this representation we can obtain new lower bounds for the -entropy and for the errors in integration and interpolation for such classes.Translated from Matematicheskie Zametki, Vol. 12, No. 6, pp. 655–664, December, 1972.     

On the derivative of a polynomial

For an entire function f(z), let . If p(z) is a polynomial of degree n, then, in general, it is difficult to obtain a lower bound for M(p′, 1). But if the zeros of the polynomial are close to the origin, then various lower bounds for M(p′, 1) have been obtained in the past. In this paper, we have considered polynomials having all their zeros in ⋎z⋎≤k(k≥1), with a possible zero of order m(m≥0) at the origin and have obtained a lower bound for M(p′, 1), which is better than most of the known lower bounds. Our bound is sharp for m=0.     

Distributions de type K-positif sur l'espace tangent

Let K be a compact subgroup of the isometry group of $\text{R}$ⁿ. A distribution T is said to be of K-positif type if it is K-invariant and if $\text{〈T, ? 1}\text{}\text{?}\text{gj〉 = ∝∝ ?(x + y)}\text{?(Y) dT(x) ? 0}$ for every K-invariant $\text{b}$^∞ function ? with compact support. We look for an integral representation of these distributions (i.e., an analog of the Bochner-Schwartz theorem). In this paper we obtain such a representation for distributions with growth of exponential type in the following case: K is the maximal compact subgroup of a semi-simple connected Lie group G with finite center, acting by the adjoint action on the tangent space of $\text{G}\text{K}$. The main step is to prove that it suffices to work with distributions of W-positif type (where W is the Weyl group associated with $\text{G}\text{K}$). This is achieved following ideas of a paper of S. Helgason [Advan. in Math.36 (1980) 297]. The end of the proof follows from the case where K is finite [N. Bopp, in “Analyse harmonique sur les groupes de Lie,” Lecture Notes in Mathematics No. 739, p. 15, Springer-Verlag, Berlin/New York 1979].     

Some inequalities for the polar derivative of a polynomial

LetP(z) be a polynomial of degreen which does not vanish in |z|<1. In this paper, we estimate the maximum and minimum moduli of thekth polar derivative ofP(z) on |z|=1 and thereby obtain compact generalizations of some known results, which among other results, yields interesting refinements of Erdos-Lax theorem and a theorem of Ankeny and Rivlin.     

Inequalities for a polynomial and its derivative

Let , 1?μ?n, be a polynomial of degree n such that p(z)≠0 in |z|<k, k>0, then for 0<r?R?k, Dewan, Yadav and Pukhta [K.K. Dewan, R.S. Yadav, M.S. Pukhta, Inequalities for a polynomial and its derivative, Math. Inequal. Appl. 2 (2) (1999) 203-205] proved