583 research outputs found
Foundations of abstract probability theory
Using the ideas of abstract algebra, we introduce the basic concepts of
abstract probability theory that generalize the Kolmogorov's probability
theory, possibility theory and other theories that deal with uncertainty. Based
on abstract addition and multiplication, we define an abstract measure and
abstract Lebesgue integral. System of Kolmogorov's axioms is criticized, after
which we introduce an abstract probability measure and abstract conditional
probability, show that they have recognizable probability properties. In
addition, we define an abstract expected value operator as the abstract
Lebesgue integral and prove its properties.Comment: 12 page
Representation of maxitive measures: an overview
Idempotent integration is an analogue of Lebesgue integration where
-maxitive measures replace -additive measures. In addition to
reviewing and unifying several Radon--Nikodym like theorems proven in the
literature for the idempotent integral, we also prove new results of the same
kind.Comment: 40 page
Approximation Theory and Related Applications
In recent years, we have seen a growing interest in various aspects of approximation theory. This happened due to the increasing complexity of mathematical models that require computer calculations and the development of the theoretical foundations of the approximation theory. Approximation theory has broad and important applications in many areas of mathematics, including functional analysis, differential equations, dynamical systems theory, mathematical physics, control theory, probability theory and mathematical statistics, and others. Approximation theory is also of great practical importance, as approximate methods and estimation of approximation errors are used in physics, economics, chemistry, signal theory, neural networks and many other areas. This book presents the works published in the Special Issue "Approximation Theory and Related Applications". The research of the world’s leading scientists presented in this book reflect new trends in approximation theory and related topics
Complex Fuzzy Set-Valued Complex Fuzzy Measures and Their Properties
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 from above, autocontionuous from below, and autocontinuity of the defined fuzzy complex number-valued measures. Properties of fuzzy complex number-valued measures are studied in detail
Fuzzy-valued integrals based on a constructive methodology
summary:The procedures for constructing a fuzzy number and a fuzzy-valued function from a family of closed intervals and two families of real-valued functions, respectively, are proposed in this paper. The constructive methodology follows from the form of the well-known “Resolution Identity” (decomposition theorem) in fuzzy sets theory. The fuzzy-valued measure is also proposed by introducing the notion of convergence for a sequence of fuzzy numbers. Under this setting, we develop the fuzzy-valued integral of fuzzy-valued function with respect to fuzzy-valued measure. Finally, we provide a Dominated Convergence Theorem for fuzzy-valued integrals
Aggregation functions: Means
The two-parts state-of-art overview of aggregation theory summarizes the essential information concerning aggregation issues. Overview of aggregation properties is given, including the basic classification of aggregation functions. In this first part, the stress is put on means, i.e., averaging aggregation functions, both with fixed arity (n-ary means) and with open arity (extended means).
Quantum measures and integrals
We show that quantum measures and integrals appear naturally in any
-Hilbert space . We begin by defining a decoherence operator
and it's associated -measure operator on . We show that
these operators have certain positivity, additivity and continuity properties.
If is a state on , then D_\rho (A,B)=\rmtr\sqbrac{\rho D(A,B)} and
have the usual properties of a decoherence
functional and -measure, respectively. The quantization of a random variable
is defined to be a certain self-adjoint operator \fhat on . Continuity
and additivity properties of the map f\mapsto\fhat are discussed. It is shown
that if is nonnegative, then \fhat is a positive operator. A quantum
integral is defined by \int fd\mu_\rho =\rmtr (\rho\fhat\,). A tail-sum
formula is proved for the quantum integral. The paper closes with an example
that illustrates some of the theory.Comment: 16 page
Integral Transformation, Operational Calculus and Their Applications
The importance and usefulness of subjects and topics involving integral transformations and operational calculus are becoming widely recognized, not only in the mathematical sciences but also in the physical, biological, engineering and statistical sciences. This book contains invited reviews and expository and original research articles dealing with and presenting state-of-the-art accounts of the recent advances in these important and potentially useful subjects
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