in this paper. (1) we determine the complex-valued solutions of the following variant of van vleck's functional equation $$int_{s}f(sigma(y)xt)dmu(t)-int_{s}f(xyt)dmu(t) = 2f(x)f(y), ;x,yin s,$$ where $s$ is a semigroup, $sigma$ is an involutive morphism of $s$, and $mu$ is a complex measure that is linear combinations of dirac measures $(delta_{z_{i}})_{iin i}$, such that for all $iin i$,...

We propose a simple real-valued generalization of the well known integer-valued Erdös number as a topological, non-metric measure of the ‘closeness’ felt between two nodes in an undirected, weighted graph. These real-valued Erdös numbers are asymmetric and are able to distinguish between network topologies that standard distance metrics view as identical. We use this measure to study some simpl...

We explicitly identify the possible probability entrance laws for a class of measure-valued processes that are constructed by taking a particular measure-valued Markov branching process and conditioning it to stay away from the zero measure trap. The set of extreme points of the entrance space is larger than the state space of the conditioned process, and contains elements which correspond to "...

Interval-valued intuitionistic fuzzy set (IVIFS) has developed to cope with the uncertainty of imprecise human thinking. In the present communication, new entropy and similarity measures for IVIFSs based on exponential function are presented and compared with the existing measures. Numerical results reveal that the proposed information measures attain the higher association with the existing me...

Let F*(K) be the set of all fuzzy complex numbers. In this paper some classical and measure-theoretical notions are extended to the case of complex fuzzy sets. They are fuzzy complex number-valued distance on F*(K), fuzzy complex number-valued measure on F*(K), and some related notions, such as null-additivity, pseudo-null-additivity, null-subtraction, pseudo-null-subtraction, autocontionuous f...

Representations of measure-valued processes in terms of countable systems of particles are constructed for models with spatially varying birth and death rates. In previous constructions for models with birth and death rates not depending on location or type, the particles were assigned integer-valued \levels", the joint distributions of the particle types were exchangeable, and the measure-valu...

This paper shows the best operators for sumof-products expressions. We first describe conditions of functions for product and sum operations. We examine all two-variable functions and select those that meet the conditions and then evaluate the number of product terms needed in the minimum sum-of-products expressions when each combination of selected product and sum functions is used. As a resul...

In this note we define Riemann integrabillty for real valued functions defined on a compact metric space accompanied by a finite Borel measure. If the measure of each open ball equals the measure of its corresponding closed ball, then a bounded function is Riemann integrable if and only if its set of points of discontinuity has measure zero. Let denote the algebra of sets generated by the open ...

The single-valued neutrosophic set is a subclass of neutrosophic set, and has been proposed in recent years. An important application for single-valued neutrosophic sets is to solve multicriteria decision-making problems. The key to using neutrosophic sets in decision-making applications is to make a similarity measure between single-valued neutrosophic sets. In this paper, a new method to meas...

Probability theory is an important tool of modeling randomness in a practical problem. But besides randomness, in the real world, there exists other kind of uncertainties such as impreciseness or vagueness. Set-valued functions are employed to model the impreciseness in applied field such as in Economics, control theory (see for example [1]). Integrals of set-valued functions have been received...