Through this study, the following has been proven, if  is an algebraically paranormal operator acting on separable Hilbert space, then  satisfies the ( ) property and  is also satisfies the ( ) property for all . These results are also achieved for  ( ) property.

   In addition, we prove that for a polaroid operator with finite ascent then after the property ( ) holds for  for all .

Through this study, the following has been proven, if  is an algebraically paranormal operator acting on separable Hilbert space, then  satisfies the ( ) property and  is also satisfies the ( ) property for all . These results are also achieved for  ( ) property.    In addition, we prove that for a polaroid operator with finite ascent then after the property ( ) holds for  for all.

     The objective of this paper is, first, study a new collection of sets such as field and we discuss the properties of this collection. Second, introduce a new concepts related to the field such as measure on field, outer measure on field and we obtain some important results deals with these concepts. Third, introduce the concept of null-additive on field as a generalization of the concept of measure on field. Furthermore, we establish new concept related to - field noted by weakly null-additive on field as a generalizations of the concepts of measure on and null-additive. Finally, we introduce the restriction of a set function  on field and many of its properties and characterizations are given.

In the present work, a program for calculating the coefficients of the Aplanatic Cassegrain Telescope (ACT) system, free from the effects of spherical and coma aberrations, were constructed. In addition, the two-mirrors of the optical system, as aspherical surfaces, were adopted. This means, that the two-equations of the mirrors are assumed to be polynomial function of five even terms only. The numerical method, least-squares curve fitting method to calculate the two-mirror coefficients system, was adopted. For choosing the values and ratios that give the best results, Rayleigh Criterion (Rayleigh Limit), for purpose of comparison and preference, was adopted.

The main goal of this paper is to study applications of the fractional calculus techniques for a certain subclass of multivalent analytic functions on Hilbert Space. Also, we obtain the coefficient estimates, extreme points, convex combination and hadamard product.

Although the concept of difference is as old as the foundational concept of similarity, the modern (and contemporary) understanding of difference as a working notion that not only differentiates, but also approximates conflicting elements in an all encompassing system owes a great deal to the German philosopher Georg Wilhelm Friedrich Hegel (1770-1831). An idealist to the backbone, Hegel bequeathed to modern philosophy the postulation that the identity of an individual rests not in itself but in the relationship that individual‟s identity entertains with other members of society. In his classic Phenomenology of Spirit, Hegel explains how humans come to consciousness (pivotal concept in Idealism) through a strenuous, albeit apparently i

       In this paper we offer two new subclasses of an open unit disk of r-fold symmetric bi-univalent functions. The Taylor-Maclaurin coefficients  have their coefficient bounds calculated. Furthermore, for functions in , we have solved Fekete-  functional issues. For the applicable classes, there are also a few particular special motivator results.