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A Simple Proof of the Polar Decomposition Theorem

100%

Wójcik P.

tom 31

nr 1

165-171

In this expository paper, we present a new and easier proof of the Polar Decomposition Theorem. Unlike in classical proofs, we do not use the square root of a positive matrix. The presented proof is accessible to a broad audience.

On Jordan-Chevalley decomposition

75%

Bajorska B.

Zeszyty Naukowe. Matematyka Stosowana / Politechnika Śląska

2011

tom z. 1

7--27

Expressing a linear operator ƒ on a finite-dimensional vector space over any field K as a sum of two commuting operators – semisimple and nilpotent – is called the Jordan-Chevalley decomposition of ƒ. It is known that this decomposition exists for an arbitrary ƒ if only K is perfect. In this paper we give some methods for determining the decomposition.

Zapis operatora liniowego działającego na skończenie wymiarowej przestrzeni wektorowej nad dowolnym ciałem K w postaci sumy dwóch przemiennych operatorów – półprostego i nilpotentnego – nazywamy rozkładem Jordana-Chevalleya tego operatora.Wiadomo, że jeśli K jest ciałem doskonałym, to taki rozkład istnieje dla dowolnego operatora. Celem artykułu jest omówienie metod wyznaczenia postulowanego rozkładu.

Classes of functions defined by subordination

75%

Dziok J. , Stankiewicz J.

tom Vol. 30

53-59

In the paper, we define classes of analytic functions, in terms of subordination. We present some inclusion relations for defined classes.

Certain class of analytic functions associated with the wright generalized hypergeometric function

75%

Aouf M. K. , Dziok J.

tom Vol. 30

23-32

Using the Wrigh's generalized hypergeometric function, we introduce a new class W(q, s; A, B, γ) of analytic functions with nega-tive coefficients. In this paper we investigate coefficient estimates, distortion theorem and the radii of convexity and starlikeness.

On certain subclasses of meromorphically multivalent functions associated with a certain linear operator

63%

Aouf M. K. , Murugusundaramoorthy G.

tom Vol. 29

33-52

In this paper we introduce and study two subclasses (Rn,p(α, A, B)) and Sn,p(α, A, B)) of meromorphic p-valent functions of order α (0 ≤ α < p) defined by certain linear operator. We investigate the various important properties and characteristics of these subclasses. Some properties of neighborhoods of functions in these subclasses are investigated. Also we derive many interesting results for the Hadamard products of functions belonging to the class Sn,p(α, A, B).

On certain properties of neighborhoods of analytic functions of complex order

51%

Latha S. , Poornima N.

tom Vol. 30

83-90

Let A(n) denote the class of functions of the form [wzór] which are analytic in the open unit disk U = {z : |z| < 1}. In this note, the subclasses Sn (β, γ, a, c), Rn (β, γ, a, c; μ ), S(sup α) (sub n) (β, γ a, c) and R (sup α) (sub n) (β, γ, a, c; μ ) of A(n)(are defined and some properties of neighborhoods arę studied for functions of complex order in these classes.

Some parametric family of functions

51%

Dziok J.

tom Vol. 29

99-107

In the paper, we define classes of analytic functions, in terms of some linear operator. Coefficients estimates, distortion theorems, and the radii of convexity and starlikeness for this class of analytic functions are given here.