A class of meromorphic multivalent functions with negative coefficients defined by a differential operator

For 0≤μ≤λ, 0<α≤1, 1/(p+1)<τ≤1, m∈N₀=N∪{0} and p∈N, a new class of meromorphic p-valent functions defined by means of the differential operator is introduced and systematically studied. Membership in this class is characterized through coefficient estimates. Distortion, radii of starlikeness and convexity of order , Hadamard convolution product, an integral operator, convex combinations and neighborhoods are also investigated.     

On sandwich theorems for multivalent functions involving a generalized differential operator

The purpose of this paper is to derive some subordination and superordination results for multivalent functions involving certain differential operator.     

A unified presentation of some classes of meromorphically multivalent functions

The authors introduce and investigate various properties of a general class

U_k[p,a,β,A,B]

p, k ε N {1, 2, 3,…,}; 0 α < p; β 0

;

−1 A < B 1; 0 < B 1)

,which unifies and extends several (known or new) subclasses of meromorphically multivalent functions. The properties and characteristics of this general class, which are presented here, include growth and distortion theorems; they also involve Hadamard products (or convolution) of functions belonging to the class U_k[p,α,β,A,B].     

Compactly supported functions generated by a polyharmonic operator

Atomic functions of two and three variables are proposed that are infinitely differentiable compactly supported solutions of functional-differential equations of a special type. Existence theorems and computational procedures are presented for these functions. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 141–156, September–October 2006.     

Classe of functions defined by transducers

Translated from Kibernetika, No. 6, pp. 1–7, 34, November–December, 1990.     

Neighborhoods of a new class of harmonic multivalent functions

We introduce and investigate a new subclass of harmonic multivalent functions defined by using a differential operator. We obtain coefficient conditions, distortion bounds, extreme points, convex combination for the above class of harmonic multivalent functions. We also, derive inclusion relationships involving the neighborhoods of harmonic multivalent functions.     

Certain classes of meromorphic multivalent functions

In this paper, we introduce and investigate two new classes of meromorphic multivalent functions. Such results as subordination properties, coefficient inequalities, convolution properties and integral representations are proved. Several sufficient conditions for meromorphic multivalent starlikeness and convexity are also derived.     

Approximation of functions by means of a new generalized Bernstein operator

In this paper we first use a probabilistic method to construct a linear positive polynomial operatorL _{m, r} ^α,β Bernstein type, depending on a non-negative integer parameterr and on two real parameters α and β, such that 0≤α≤β. Then we investigate the approximation properties of this operator mapping into itself the Banach spaceC[0,1] of real-valued continuous functions on [0,1]. A special attention is accorded to the case of the operatorL _m,r=L _m,r ^0,0 . We prove that the remainder of the approximation formula of a functionfεC[0,1] byL _m,r f can be represented either by means of divided differences, or in an integral form, obtained by using a classical theorem of Peano. We give also an asymptotic estimate for this remainder. The operatorL _m,r enjoys the variation diminishing property—in the sense of I. J. Schoenberg [15]. By extending the known inequalities of T. Popoviciu [12] and G. G. Lorentz [7], we evaluate the orders of approximation in terms of the modulus of continuity of the functionf or of its derivative. In the last section of this paper we determine the point spectrum of the operatorL _m,r and , finally, we present a quadrature formula which can be constructed by means of this operator. Dedicated to Professor Aldo Ghizzetti on his 75th birthday     

Distance functions defined by variable neighborhood sequences

In the field of pattern recognition, many researchers adopt the definition that distance between two points x and y on digitized space is the length of the shortest path from x to y determined by a specific sequence of neighborhood forms. The diamond distance and various octagonal distances are typical examples of these kinds of distances. However, not necessarily every sequence of neighborhood forms does define a distance function. Thus, in this paper, we present a necessary and sufficient condition for a sequence of neighborhood forms to define a distance function. Two applications of this condition are also presented.     

On curves defined by generalised cubic blending functions

As the relationship between the control polygon and the curve specified by generalised cubic blending functions has been defined rather vaguely previously, the present paper proves that the curve is tangent to the middle side of its control polygon only when the blending functions reduce to those of Timmer which are a special case of the former. Otherwise, the curve looses the convex hull property of the Bézier curve formulation and may be discontinuous.     

Some properties of certain classes of meromorphically p-valent functions involving extended multiplier transformations

In this paper, the authors investigate interesting properties of certain subclasses of meromorphically multivalent functions which are defined here by means of extended multiplier transformations.     

Analysis of families of hash functions defined by automata over a finite ring

Some general properties of families of hash functions defined by strongly connected automata without output function over a finite ring are analyzed. The probabilities of randomly choosing a sequence for which a hash function assumes a given value and also two different sequences of the same length for which the values of a hash function coincide. The computational security of the investigated hash-functions is characterized.     

Superpositions of continuous functions defined on a Baire space

We consider uniformly continuous functions on a Baire space and introduce the notion of a continuity modulus of a function. We formulate a condition on the growth of the continuity modulus φ guaranteeing that superpositions of n-ary functions with continuity modulus φ do not exhaust all (n + 1)-ary functions with continuity modulus φ for any n. Moreover, negating this property leads to the inverse effect.     

On the normalization of numbers and functions defined by radicals

Based on a criterion due to Kneser, we present new results for the degree of field extensions generated by finitely many radicals. We also improve an algorithm given by Zippel for computing a basis for such an extension field. This algorithm may be effectively used by computer algebra systems to normalize expressions involving radicals.     

Construction of convex continuations for functions defined on a hypersphere

Construction of convex continuations for functions defined on the vertices of some combinatorial polyhedra, in particular the permutation polyhedron and the arrangement polyhedron, has been studied in [1, 2]. Subsequently this result has been generalized to functions defined at the extreme points of an arbitrary polyhedron [3]. For purposes of combinatorial optimization [4-6] it is relevant to consider the existence and construction of convex continuations from continua, in particular, when the function is defined on a hypersphere in thek-dimensional space. Unfortunately, passage to the limit from discrete sets to continua does not produce positive results in this case. We are thus forced to develop special approaches to investigating the existence of convex continuations of functions defined on continua. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 27–36, March–April, 1998.     

Control of damping oscillations by fractional differential operator with time-dependent order

The aim of the present paper is application of a differential operator of variable order in constitutive relations for viscoelastic materials. This operator is used for studying the vibrations of a one-degree-of-freedom oscillator, in which the viscoelastic deformation is governed by a servo order function. A method for numerical determination of the eigenfrequency of the fractional differential equation is proposed.