Mathematics Commons^™

On The Noninterpolation Of Polyhedral Maps, Adrian Riskin, D.W. Barnette

Mathematics

In this paper we show that if attention is restricted to polyhedral embeddings of graphs, no theorem analogous to the Duke interpolation theorem for 2-cell embeddings is true. We also give two interesting classes of graphs: (i) a class in which the members have polyhedral embeddings in the torus and also in orientable manifolds of arbitrarily high genus, (ii) and another in which the members have polyhedral embeddings in the projective plane and also in orientable and nonorientable manifolds of arbitrarily low Euler characteristic.

The Splitting Theorem For Orbifolds, Joseph Borzellino, Shun-Hui Zhu

Mathematics

In this paper we wish to examine a generalization of the splitting theorem of Cheeger–Gromoll [CG] to Riemannian orbifolds. Roughly speaking, a Riemannian orbifold is a metric space locally modelled on quotients of Rie- mannian manifolds by finite groups of isometries. The term orbifold was coined by W. Thurston [T] sometime around the year 1976–77. The term is meant to suggest the orbit space of a group action on a manifold. A similar concept was introduced by I. Satake in 1956, where he used the term V–manifold (See [S1]). The “V” was meant to suggest a cone–like singularity. Since then, …

Isosymmetric Linear Transformations On Complex Hilbert Space, Mark Stankus

Mathematics

We explore the elementary operator theory of the equation
(0.1) ∑ c_m,nT^*nT^m=0
for c_m,n E C, c_m,n nonzero for only finitely many m, n and Ta bounded linear transformation on a complex Hilbert space in Chapters 1 and 2. We explore the equation

(0.2) T^*2T - T^*T² + T - T^* = 0
in greater depth in Chapters 4 and 5.

Chapter 1 explores the algebraic and C^*-algebraic aspects of the equation (0.1) and both the spectral picture of and growth conditions on …

Orbifolds Of Maximal Diameter, Joseph Borzellino

Mathematics

In this paper the Maximal Diameter Theorem of Riemannian geometry is proven for Riemannian orbifolds. In particular, it is shown that a complete Riemannian orbifold with Ricci curvature bounded below by (n−1) and diameter = π, must have constant sectional curvature 1, and must be a quotient of the sphere (Sn, can) of constant sectional curvature 1 by a subgroup of the orthogonal group O(n+1) acting discontinuously and isometrically on Sn. It is also shown that the singular locus of the orbifold forms a geometric barrier to the length minimization property of geodesics. We also extend the Bishop relative volume …

Nonstandard And Standard Compactifications Of Ordered Topological Spaces, Sergio Salbany, Todor D. Todorov

Mathematics

We construct the Nachbin ordered compactification and the ordered realcompactification, a notion defined in the paper, of a given ordered topological space as nonstandard ordered hulls. The maximal ideals in the algebras of the differences of monotone continuous functions are completely described. We give also a characterization of the class of completely regular ordered spaces which are closed subspaces of products of copies of the ordered real line, answering a question of T.H. Choe and Y.H. Hong. The methods used are topological (standard) and nonstandard.

Pointwise Kernels Of Schwartz Distributions, Todor D. Todorov

Mathematics

We show that Schwartz distributions have kernels in the class of the pointwise nonstandard functions.

Riemannian Geometry Of Orbifolds, Joseph Ernest Borzellino

Mathematics

We investigate generalizations of many theorems of Riemannian geometry to Riemannian orbifolds. Basic definitions and many examples are given. It is shown that Riemannian orbifolds inherit a natural stratified length space structure. A version of Toponogov's triangle comparison theorem for Riemannian orbifolds is proven. A structure theorem for minimizing curves shows that such curves cannot pass through the singular set. A generalization of the Bishop relative volume comparison theorem is presented. The maximal diameter theorem of Cheng is generalized. A finiteness result and convergence result is proven for good Riemannian orbifolds, and the existence of a closed geodesic is shown …

A P-Adic Cohomological Method For The Weierstrass Family And Its Zeta Invariants, Goro Kato

Mathematics

After a survey of the Weierstrass family and cohomology, we compute the lifted homology of the Weierstrass family with compact supports so that explicit formulae for the zeta function of each fibre of the Weierstrass family may be obtained. The (co-)homology theory that we use is found in [L₁], [L₂] and [L₃]. Therefore, this article can be regarded as an application of Lubkin's p-adic theory of cohomologies to an algebraic family called the Weierstrass scheme over the ring (Z/pZ)[g₂,g₃). The cohomological background for the computation will be rather carefully exploited.

A Binary Tree Decomposition Space Of Permutation Statistics, Don Rawlings

Mathematics

Based on the binary tree decomposition of a permutation, a natural vector space of permutation statistics is defined. Besides containing many well known permutation statistics, this space provides the general context for an archetypal recurrence relationship that contains most of the classic combinatorial sequences and some of their known generalizations.

Type 3 Diminimal Maps On The Torus, Adrian Riskin

Mathematics

A polyhedral map on the torus is diminimal if either shrinking or removing an edge yields a nonpolyhedral map. We show that all such maps on the torus fall into one of two classes, type 2 and type 3, and show that there are exactly two type 3 ones, which are given explicitly.

A Nonstandard Delta Function, Todor D. Todorov

Mathematics

We prove that the Dirac delta distribution has a kernel in the class of the pointwise nonstandard functions.

Lifted P-Adic Homology With Compact Supports Of The Weierstrass Family And Its Zeta Endomorphism, Goro Kato

Mathematics

The relations among the generators for the lifted p-adic homology with compact supports of the various subfamilies of the Weierstrass family in characteristic p > 0 (p ≠ 2, 3) are explicitly given in Section 2. Then, the universal coefficient spectral sequence and the zeta endomorphism in Section 3 enable one to compute explicitly the lifted p-adic homology with compact supports of all fibres, including all the elliptic curves and all their singular degenerations in the family.

Multicolored Simon Newcomb Problems, Don Rawlings

Mathematics

Recent progress made by Desarmenien and Foata in the area of permutation statistics indirectly points out a certain gap in the theory of sequence enumeration. The remedy of the situation lies in the consideration of some colorful extensions of the Simon Newcomb problem.

Alexander's Subbase Lemma, Todor D. Todorov, S. Salbany

Mathematics

No abstract provided.

Operations With Distribution Vectors, Brian Fisher, Todor D. Todorov

Mathematics

No abstract provided.

Asymptotic Functions As Kernels Of The Schwartz Distributions, Todor D. Todorov

Mathematics

Using a version of the sequential method we introduce a class of generalized functions called here "asymptotic functions''. This class contains kernels of all Schwartz distributions and is equipped with a correctly defined multiplication operation. So, in a sense, one solves the problem of "multiplication of Schwartz distributions" although the solution refers to the class of the asymptotic functions and not to the Schwartz distributions themselves. The paper is a continuation of a series of works [1-10] but here only part of the results of [5], [6] and [8] will be needed.

On The Notion Of Precohomology, Goro Kato

Mathematics

ABSTRACT. For a cochain complex one can have the cohomology functor. In this paper we introduce the notion of precohomology for a cochain that is not a complex, i. e., d^q+1 o d^q may not be zero. Such a cochain, with objects and morphisms of an abelian category A, is called a cochain precomplex whose category is denoted by Pco (A). If a cochain precomplex is actually a cochain complex, then the notion of precohomology coincides with that of cohomology, i. e., precohomology is a gene¬ralization of cohomology. For a left exact functor F from an abelian category …

The Products Δn2(X), Δ(X). X-N, ϴ(X). X-N, Etc. In The Class Of The Asymptotic Functions, Todor D. Todorov

Mathematics

Several products like δⁿ(x), δ(x)ϴ(x), δ^(m)(x). X^-n, ϴ(x). X^-n, etc., where δ(x), ϴ(x), X^-n, etc., are kernels of the corresponding Schwartz distributions, are studied in the framework of the class of the asymptotic functions F₀ introduced in a previous paper [11]. In some particular cases many formulae are derived and several examples are presented. The work is of mathematical type but its motivations lie in some problems in quantum theory. It is closely …

Enumeration Of Permutations By Descents, Idescents, Imajor Index, And Basic Components, Don Rawlings

Mathematics

Multivariable extensions of classic permutation cycle structure results are obtained by counting permutations by descents, idescents, imajor index, and basic components.

The Combinatorics Of Certain Products, Don Rawlings

Mathematics

A combinatorial interpretation for the coefficients in the expansion of Π(1 + ux^jy^k)(1 - ux^jy^k)^-1 is given.

On The Generators Of The First Homology With Compact Supports Of The Weierstrass Family In Characteristic Zero, Goro Kato

Mathematics

No abstract provided.

On The Theory Of Homogeneous Lipschitz Spaces And Campanato Spaces, Harvey Greenwald

Mathematics

In this paper the equivalence between the Campanato spaces and homogeneous Lipschitz spaces is shown through the use of elementary and constructive means. These Lipschitz spaces can be defined in terms of derivatives as well as differences.

Zeta Matrices Of Elliptic Curves, Goro Kato, Saul Lubkin

Mathematics

Let O=lim_nZ/pⁿZ, , let A=O[g₂,g₃] Δ, where g₂ and g₃ are coefficients of the elliptic curve: Y² = 4X³ − g₂X − g₃ over a finite field and Δ = g₂³ − 27g₃² and let B=A[X,Y]/(Y²-4X³+g₂X+g₃). Then the p-adic cohomology theory will be applied to compute explicitly the zeta matrices of the elliptic curves, induced by the pth power map on the free AzQ -module H1(X, AzQ). Main results are; Theorem 1.1: X …

The (Q, R)-Simon Newcomb Problem, Don Rawlings

Mathematics

A new statistic, the r-major index, is defined for sequences. A linear recurrence is then derived that enumerates sequences by r-descent number and r-major index.

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 …

Generalized Worpitzky Identities With Applications To Permutation Enumeration, Don Rawlings

Mathematics

The enumeration of permutations by inversions often leads to a q -analog of the usual generating function. In this paper, two generalizations of the Worpitzky identity for the Eulerian numbers are obtained and used to enumerate permutations by the descent number and the major index of their inverses. The resulting (t, q)-generating series do in fact generalize the q-series obtained when counting by inversions.

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

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

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

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 …