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Articles 181 - 188 of 188
Full-Text Articles in Mathematics
The R-Major Index, Don Rawlings
The R-Major Index, Don Rawlings
Mathematics
The r-major index is a new permutation statistic that is suggested by the work of Carlitz and Gausner on Foulkes' skew hook rule for computing the r-Eulerian numbers. The new statistic (1) generalizes both the major index and the inversion number of a permutation and (2) leads to a q-analog of the r-Eulerian numbers.
Quasi-Extended Asymptotic Functions, Todor D. Todorov
Quasi-Extended Asymptotic Functions, Todor D. Todorov
Mathematics
The class F of "quasi-extended asymptotic functions" introduced in the present paper contains all extended asymptotic functions [8, (3.1)] (in particular, all examples constructed in [9, Sec. 1 ]). But F contains also some new asymptotic functions very similar to tht Schwartz distributions. On the other hand, every two quasi-extended asymptotic functions can be multiplied as opposed to the Schwartz distributions; in particular, the square &# 948;2 of an asymptotic function &# 948; similar to Dirac's delta-function is constructed as an example. The connection with the asymptotic functions introduced in [2] and [4] is established.
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 …
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 …
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 …