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Articles 211 - 218 of 218
Full-Text Articles in Entire DC Network
Asymptotic Functions And The Problem Of Multiplication Of Distributions, Todor D. Todorov
Asymptotic Functions And The Problem Of Multiplication Of Distributions, Todor D. Todorov
Mathematics
The asymptotic functions are a new type of generalized functions. But they are not functionals on some space of test-functions as the Schwartz distributions. They are mappings of the set of the asymptotic numbers (1, 3, 5, 6) into itself. On its part, the set of the asymptotic numbers is a totally-ordered set of generalized numbers including the systems of real and complex numbers, as well as infinitesimals and infinitely large numbers. Every two asymptotic functions can be multiplied. On the other hand, the Schwartz distributions have realizations, in a certain sense, as asymptotic functions. The motivations of this work …
Asymptotic Numbers: Ii. Order Relation, Infinitesimals And Interval Topology, Todor D. Todorov
Asymptotic Numbers: Ii. Order Relation, Infinitesimals And Interval Topology, Todor D. Todorov
Mathematics
It has been shown in [8] that the set of asympototic numbers A is a system of generalized numbers including isomorphically the set of real numbers R, as well as the field of formal power (asymptotic) series. In the present paper, which is a continuation of [8], an order relation in A is introduced due to A turning out to be a totally-ordered set. The consistency between the order relation and the algebraic operations in A is investigated and in particular, it is shown that the inequalities in A can be added and multiplied as in the set of …
Asymptotic Numbers: I. Algebraic Properties, Todor D. Todorov
Asymptotic Numbers: I. Algebraic Properties, Todor D. Todorov
Mathematics
The set of asymptotic numbers A introduced in Refs. [1] and [3] is a system of generalized numbers including the system of real numbers R, as well as infinitely small (infinitesimals) and infinitely large numbers. The purpose of this paper is to study in detail the algebraic properties of A which are a little unusual, in a cenain sense, as compared with the known algebraic structures (rings. fields, etc.) This is necessary for the investigation of the class of asymptotic functions [2.4], which are on their part, generalized functions similar to the distributions of Schwartz but allowing the operation of …
On The Existence Of Good Markov Strategies, Theodore P. Hill
On The Existence Of Good Markov Strategies, Theodore P. Hill
Research Scholars in Residence
In contrast to the known fact that there are gambling problems based on a finite state space for which no stationary family of strategies is at all good, in every such problem there always exist ε-optimal Markov families (in which the strategy depends only on the current state and time) and also ε-optimal tracking families (in which the strategy depends only on the current state and the number of times that state has been previously visited). More generally, this result holds for all finite state gambling problems with a payoff which is shift and permutation invariant.
Second Leray Spectral Sequence Of Relative Hypercohomology, Saul Lubkin, Goro C. Kato
Second Leray Spectral Sequence Of Relative Hypercohomology, Saul Lubkin, Goro C. Kato
Mathematics
A second Leray spectral sequence of relative hypercohomology is constructed. (This is skew in generality to an earlier one constructed by S. Lubkin [(1968) Ann. Math. 87, 105-255].) The Mayer-Vietoris sequence of relative hypercohomology [Lubkin, S. (1968) Ann. Math. 87, 105-255] is also generalized.
Lipschitz Spaces Of Distributions On The Surface Of Unit Sphere In Euclidean N-Space, Harvey Greenwald
Lipschitz Spaces Of Distributions On The Surface Of Unit Sphere In Euclidean N-Space, Harvey Greenwald
Mathematics
In this paper Lipschitz spaces of distributions are defined and various inclusion relations are shown. Certain properties such as completeness, separability, and the density of the testing space for appropriate Lipschitz spaces are proved. The Littlewood-Paley function is defined and used to prove inclusion relationships between Lipschitz and Lebesgue spaces.
Asymptotic Numbers: Algebraic Operations With Them, Christo Ya. Christov, Todor D. Todorov
Asymptotic Numbers: Algebraic Operations With Them, Christo Ya. Christov, Todor D. Todorov
Mathematics
The main subject of the present paper is to define the four algebraic operations - additions, subtraction, multiplication and division in the set of the asymptotic numbers A [7] and to deduce the corresponding formulas for the components of the asymptotic number, representing the result as functions of the components of the arguments. The definitions of the operations, in fact, are introduced as a special case of the more general notion of a quasiclassical function - one special class of functions defined on A. The discussion of the algebraic and some other properties of the asymptotic numbers is put …
Lipschitz Spaces On The Surface Of Unit Sphere In Euclidean N-Space, Harvey Greenwald
Lipschitz Spaces On The Surface Of Unit Sphere In Euclidean N-Space, Harvey Greenwald
Mathematics
This paper is concerned with defining Lipschitz spaces on Σn-1 the surface of the unit sphere in Rn. The importance of this example is that Σn-1 is not a group but a symmetric space. One begins with functions in Lp(Σn-1),1≤p≤∞. Σn-1 is a symmetric space and is related in a natural way to the rotation group SO(n). One can then use the group SO(n) to define first and second differences for functions in Lp(Σn-1). Such a function is the boundary value of its Poisson integral. This enables one to work with functions which are harmonic. Differences can then be replaced …