The concepts of convergence and divergence, while not defined explicitly until early 19th century, have been studied since the third century BC. Throughout this time, mathematicians have largely focused upon convergent sequences and series, having seemingly little analytical reason to study series that diverged. However, as the area of mathematical analysis developed over the past few centuries...

We define united KK-theory for real C*-algebras A and B such that A is separable and B is σ-unital, extending united K-theory in the sense that KKCRT (R, B) = KCRT (B). United KK-theory contains real, complex, and self-conjugate KK-theory; but unlike unaugmented real KK-theory, it admits a universal coefficient theorem. For all separable A and B in which the complexification of B is in the boot...

In this paper we present two new types aggregation operators such as, induced Pythagorean fuzzy ordered weighted averaging aggregation operator and induced Pythagorean fuzzy hybrid averaging aggregation operator. We also discuss of important properties of these proposed operators and construct some examples to develop these operators.

Under certain conditions, a recent method of Barlet and Koziarz [2] constructs enough holomorphic functions to give a direct proof of the Stein condition for a cycle space. Here we verify those conditions for open G0–orbits on X, where G0 is the group of a bounded symmetric domain and X is its compact dual viewed as a flag quotient manifold of the complexification G of G0 . This Stein result wa...

We introduce real log canonical threshold and real jumping numbers for real algebraic functions. A real jumping number is a root of the b-function up to a sign if its difference with the minimal one is less than 1. The real log canonical threshold, which is the minimal real jumping number, coincides up to a sign with the maximal pole of the distribution defined by the complex power of the absol...

A Grauert tube, or adapted complex structure, gives a canonical complexification of a real analytic Riemannian manifold defined on a neighborhood of the zero section of the (co-)tangent bundle. Such a tube is called entire if it is defined on the whole tangent bundle of the real analytic Riemannian manifold M, g. Very few examples of such entire tubes are known, but a general statement has prov...

Digital fingerprinting is a technique to protect digital contents from illegal reproduction and redistribution by marking unique information for individual user. A powerful but simple attack to diminish fingerprint signals is averaging. While several fingerprinting schemes against collusion attacks were proposed, they often do not fully account for multimedia data. In this paper, we propose a z...

We review some aspects of the use of a technique known as group averaging, which provides a tool for the study of constrained systems. We focus our attention on the case where the gauge group is non–compact, and a ‘renormalized’ group averaging method must be introduced. We discuss the connection between superselection sectors and the rate of divergence of the group averaging integral.