Additional Theorems, Problems, and Proofs

Addition theorems have been indispensable tools for the reduction of quantum transition amplitudes. They are normally utilized at the start of the process to move the angular dependence within plane waves, Coulomb potentials, and the like, into a sum over spherical harmonics that allows the angular integration to be carried out. These have historically been “two-range” addition theorems, characterized by the two-fold notation r>=Max[r1,r2] and r<=Min[r1,r2] and comprising a single infinite series. More recently, “one-range” addition theorems have been created that have no such piecewise notation, but at the cost of the introduction of another infinite series. We use a very different approach to derive an infinite set of addition theorems for Slater orbitals, hydrogenic and Hylleraas wave functions, and so on, that retain the one-range variable dependence but have, at worst, a finite second series rather than an infinite one. In addition, unlike previous addition theorems, they are applicable to more than one coordinate system. One of these addition theorems may also be used for Yukawa-like functions that may appear late in the reduction of amplitude integrals, and we show its utility for an integral that has stubbornly defied reduction to analytic form for nearly sixty years. Finally, we craft indefinite integrals of 15 half-integer Macdonald functions multiplied by (inverse) powers and negative exponentials containing squares of the integration variable.

We study the regularity of solutions of functional equations of a generalized mean value type. In this paper we give sufficient conditions for the regularity by using hypoellipticity which is a concept of the theory of partial differential equations. We also give an affirmative answer to a conjecture of H. Światak. A part of the results was announced in the comprehensive paper [8] on our joint works. To prove the regularity of solutions of functional equations is one of the central problems in the theory of functional equations (see [1]).

Solutions /spl phi/(x) of the functional equation /spl phi/(/spl phi/(x)) = f (x) are called iterative roots of the given function f(x). They are of interest in dynamical systems, chaos and complexity theory and also in the modelling of certain industrial and financial processes. The problem of computing this square root in function (or operator) spaces remains a hard task and is, for the general case, still unsolved. While the theory of functional equations provides some insight for real and complex valued functions, iterative roots of mappings from R/sup n/ to R/sup n/ are not well understood by theory and there exists no published numerical algorithm for their computation. Here we prove existence of iterative roots of a certain class of monotonic mappings in R/sup n/ spaces and demonstrate how a method based on neural networks can find solutions to some examples that arise from simple physical systems.

Applications of measures of weak noncompactness and some classes of operators in the theory of functional equations in the lebesgue space

Solution and stability of mixed type functional equation in non-Archimedean random normed spaces

In this paper we define Cauchy and Pexider operators by applying the well-known Cauchy and Pexider differences and then consider some properties of such operators. Other type of operators (∆-operators, quadratic operators, Jensen operators) that are also very important in the theory of functional equations are investigated. Moreover, some new functional equations closely related to this problem, are discussed as well.

The greatest achievement of the seventeenth century was the calculus. From this source there stemmed major new branches of mathematics: differential equations, infinite series, differential geometry, the calculus of variations, functions of complex variables, and many others. Indeed, the beginnings of some of these subjects were already present in the works of Newton and Leibniz. The eighteenth century was devoted largely to the development of some of these branches of analysis. But before this could be accomplished, the calculus itself had to be extended. Newton and Leibniz had created basic methods, but much remained to be done. Many new functions of one variable and functions of two or more variables had either to be recognized explicitly or created; the techniques of differentiation and integration had to be extended to some of the existing functions and to others yet to be introduced; and the logical foundation of the calculus was still missing. The first goal was to expand the subject matter of the calculus and this is the subject of the present chapter and the next one.

Since the pioneering work of Landau and Nakanishi on general Feynman integrals, it has been known that the singularities of the S-matrix, causal Green's functions and related functions are described by the so called Landau equations. These Landau singularities were physically interpreted as the macroscopic causality by the theoretical physicists working in 5-matrix theory, and the notion of essential support was obtained (Chandler-Stapp [3], lagolnitzer-Stapp [5]). In the branch of mathematics, on the other side, the theory of microfunction has evolved and has been applied powerfully to the general theory of partial differential equations (Sato-Kawai-Kashiwara [11]). It contained the essential support theory as the singularity spectrum of a function (i.e. its support viewed as a microfunction), and was far-reaching because of its close connection with the theory of differential equations. Namely the method of microlocal analysis, based on the theory of holonomic systems, has provided a systematic way of handling functions with natural background and found most effective applications to various problems of mathematics, such as the theory of ^-functions and Fourier transformations (Sato [10], Kashiwara [6]). It was then recognized that the Landau equations give holonomic varieties, which led one to the holonomicity postulate of ^-matrix and related quantities (Sato [10]). In the present paper we shall study the holonomy structure of Landau singularities and Feynman integrals from this standpoint. First we review the notion of Landau varieties. In contrast with

By nowadays the general concepts of the Set theory are assumed to be a basis of every part of Mathematics as a science, and, simultaneously, in implicit or in explicit form - a basis of every training course of Mathematics at any level of education. It is quite clear that finite sets are the simplest ones from the point of view of their perception of a man anyway by the fact that in his practice, in his private life, a man does not deal with the other ones. Such sets are in the first place of attention in math courses at the level of general secondary education just therefore. But these courses cannot be limited by the consideration of the finite sets only already by the fact that there are infinite elements in the set of natural numbers, Euclidean line contains infinite number of points. Concept of an infinite set, of infinity, is a complex abstraction formed by humanity for thousand years. Thus, comprehension of the specific properties of infinite sets, mastering skills of operating with infinite sets in math courses of institutions of general secondary education cannot be a simple element of training. Countable sets are the simplest ones among the infinite sets, such of them that have the least possible potency. They are the first types of infinite sets that competitors of general secondary educations meet in the process of training. Determination the expedient ways of lightening the essence and main properties of countable sets in math courses of the indicated level represents non-trivial task of the corresponding methods of teaching. The article is devoted to the specification the role and the place of the content line «Countable sets» in the modern math courses of institutions of general secondary education during the all period of training. Thus, there are examined the math aspects of representation of countable sets in different parts of the theory of Number systems, in reasoning according the method of math induction, in the process of solving some problems of Geometry also, in the theory of sequences, in the theory of periodic functions, in trigonometry. The way the mathematical essence of the subject determines the expedient methods of training is demonstrated clearly.

In the paper [S1] I introduced a game, denoted by MG(J) (where J is a free ideal on some infinite set S) and called “the meager nowhere dense game for J”. The special case when J is the collection of finite subsets of the set S is called the countable-finite game on S. It proceeds as follows.First player ONE picks a countable set C1, then player TWO picks a finite set F1. Then in the second inning ONE picks a countable set C2 with C1 ⊂ C2 (unless explicitly indicated otherwise, “⊂” means “is a proper subset of”) and TWO responds with a finite set F2, and so on. The players construct a sequence (C1,F1,C2,F2,…,Ck,Fk,…) where for each positive integer k(i) Ck denotes ONE's countable set picked during the kth inning,(ii) Fk denotes TWO's finite set picked during the kth inning, and(iii) Ck ⊂ Ck + 1.Such a sequence is a play of the countable-finite game on S, and TWO wins this play if is contained in . The notion of a winning perfect information strategy is defined as usual (see, for example, [S1]). Zermelo-Fraenkel set theory together with the axiom of choice (denoted by ZFC; for a statement of the axioms see pp. xv–xvi of [K]) is a strong enough theory to build a winning perfect information strategy for player TWO in this game.Does TWO have a winning strategy requiring less than perfect information? Fix a positive integer k. A strategy of TWO which requires knowledge of only at the most the k most recent moves of ONE is said to be a k-tactic. For the countable-finite game on an infinite set S the following facts about the existence of winning k-tactics for TWO are proved in [S1]:1) TWO does not have a winning 1-tactic (Theorem 1 of [S1]).2) If the cardinality of S is less than ℵ2 then TWO has a winning 2-tactic (Corollary 4 of [S1]).3) If TWO has a winning k-tactic in the countable-finite game on an infinite set S, then TWO has a winning 3-tactic (Proposition 15 of [SI]).

Starting from one-range addition theorems for Slater-type functions, which are expansion in terms of complete and orthonormal functions based on the generalized Laguerre polynomials, Guseinov constructed addition theorems that are expansions in terms of Slater-type functions with a common scaling parameter and integral principal quantum numbers. This was accomplished by expressing the complete and orthonormal Laguerre-type functions as finite linear combinations of Slater-type functions and by rearranging the order of the nested summations. Essentially, this corresponds to the transformation of a Laguerre expansion, which in general only converges in the mean, to a power series, which converges pointwise. Such a transformation is not necessarily legitimate, and this contribution discusses in detail the difference between truncated expansions and the infinite series that result in the absence of truncation.

This chapter concerns mainly familiar old friends, the set ℕ of positive whole numbers variously known as the natural numbers or counting numbers, { 1, 2, 3, 4, …}. We think of counting as a very primitive notion firmly rooted in reality, yet already the innocent three dots in { 1, 2, 3, 4, …} may have taken us beyond reality into the realms of pure thought. The dots are usually interpreted as 'and so on for ever', which expresses our notion that ℕ is an infinite set. Cosmologists have still not made up their minds whether we live in a finite or an infinite universe, and in the former case there could be no such thing as an infinite set of real objects. However, mathematicians do not, in general, see this as a problem, and confidently assume that infinite sets (even if they are only sets of mental objects) can be handled with safety.

At present there can be no doubt to say that the quantum group theory has been one of the most important, modern and rapidly developing directions of mathematics and mathematical physics in the end of the twentieth century. Although initially (in the early 1980’s) the quantum group theory was formulated for solving problems in the theory of integrable systems and statistical physics, later the surprising connection of this theory with many branches of mathematics, and the theoretical and mathematical physics was discovered. Today the quantum group theory is connected with such mathematical fields as special functions (especially, with (q-orthogonal polynomials and basic hypergeometric series), the theory of difference and differential equations, combinatorial analysis and representation theory, matrix and operator algebras, noncommutative geometry, knot theory, topology, category theory and so on. From point of view of the mathematical physics there exists interconnection of the quantum group theory with the quantum inverse scattering method, conformai and quantum fields theory and so on. It is expected that the quantum groups will provide deeper understanding of concept of symmetry in physics.KeywordsQuantum GroupSimple RootBasic Hypergeometric SeriesReduction ChainGeneral Projection OperatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Toy Story is the first full length feature film produced entirely using the technology of computer animation. The main characters, Sheriff Woody and Space Ranger Buzz Lightyear; are toys that come to life when humans aren't around. Their story is one of rivalry, challenges, teamwork and redemption. Making this film required four years of effort, from writing the story and script, to illustrated storyboards, through modeling, animation, lighting, rendering, and filming. This paper examines various processes involved in producing the film.

In this paper, the basic theory of fractional differential equations in a Banach Space is discussed including flow invariance and theory of inequalities in cones.