Showing papers on "Field (mathematics) published in 1985"

Elementary introduction to new generalized functions

Abstract: The author's previous book `New Generalized Functions and Multiplication of Distributions' (North-Holland, 1984) introduced `new generalized functions' in order to explain heuristic computations of Physics and to give a meaning to any finite product of distributions. The aim here is to present these functions in a more direct and elementary way. In Part I, the reader is assumed to be familiar only with the concepts of open and compact subsets of R , of C # functions of several real variables and with some rudiments of integration theory. Part II defines tempered generalized functions, i.e. generalized functions which are, in some sense, increasing at infinity no faster than a polynomial (as well as all their partial derivatives). Part III shows that, in this setting, the partial differential equations have new solutions. The results obtained show that this setting is perfectly adapted to the study of nonlinear partial differential equations, and indicate some new perspectives in this field.

Rational Quadratic Forms

Abstract: Historically the theory of quadratic forms has its origins in number-theoretic questions of the following type: Which integers can be written in the form x 2 + 2y 2 , which are sums of three squares, or more generally, which integers can be represented by an arbitrary quadratic form Σ a ij x i x j integral coefficients? This general question is exceptionally difficult and we are still quite far from a complete solution. It is natural and considerably simpler to first investigate these questions over the field of rational numbers, that is, to ask for rational instead of integral solutions to the equation Σ a ij x i x j = a. This leads to the problem of classification of quadratic forms over \( \mathbb{Q} \), which was first solved by Minkowski. His solution appears in this chapter basically unaltered, except for a few simplifications and the use of modern terminology. The Gaussian sums of Gauss and Dirich-let play a significant role in the more formal algebraic part of the theory.

About a New Method for Computing in Algebraic Number Fields

Abstract: • The other method cons is t s in fac tor iz ing PI(X) in ~[X] , then P2(X) in ~(C~l)[X] where ~(Ctl) = ~[X] / I I I (X) , for each i r reducible factor II 1 of P1 ' and so on, in o rde r to de te rmine al l the fields on which one has to compute. But a lgor i thms for factor izing a re heavy tools , and they requi re the knowledge of pr imi t ive e lements for Q(c~,cc 2) ..... ~(c~ 1 ..... a~_l).

Quantum theory of particle motion in a rapidly oscillating field

Abstract: We show that the motion of a particle of mass m in a high-frequency time-dependent potential V(x\ensuremath{\rightarrow})=v(x\ensuremath{\rightarrow})cos(\ensuremath{\Omega}t) is governed by a Schr\"odinger equation with time-independent effective potential ${V}_{\mathrm{eff}}$(x\ensuremath{\rightarrow})=\ensuremath{ abla}\ensuremath{\rightarrow}v(x\ensuremath{\rightarrow})\ensuremath{\cdot}\ensuremath{ abla}\ensuremath{\rightarrow}v(x\ensuremath{\rightarrow}) /4m${\ensuremath{\Omega}}^{2}$. The validity of this approximation and an exact formal solution based on the Wigner function are discussed for the case in which v(x\ensuremath{\rightarrow}) is quadratic.

Completely Analytical Gibbs Fields

Abstract: In this report we describe the class of interactions A for which the corresponding Gibbs field is unique and possess every possible virtue one can imagine. We mean by this the well-known regularity properties of Gibbs fields in the high-temperature region. This class of interactions is defined axiomatically by ten (!) very natural properties. But the reader should not be confused by their great amount because all of them turn out to be equivalent! This fact alone shows that the class considered is natural. We call the potentials of the class completely analitical. The boundary of A corresponds to phase transition surface. Furthermore, the region A can be defined constructively by a certain algorithm. Here we use the word “constructive” in the same sense, as we used it in report [1], which is conceptually close to this one. For the sake of simplicity, we restrict ourselves to random fields on Z V, with finite single-spin space S and finite-range, translation-invariant interactions with finite values. The ideas of the present report and those of [1] can be applied in more complex cases; in particular, we study perturbations of Gaussian fields in [2]. Throughout this report we shall use the notations of [1], §1.

ON SCOTT MODULES AND p-PERMUTATION MODULES: AN APPROACH THROUGH THE BRAUER MORPHISM

Abstract: Following Lluis Puig we give a presentation of the theory of p- permutation modules (also called "trivial source modules") by a systematic use of the generalized Brauer morphism. The aim of this paper is to present a somewhat new and self-contained treatment of p-permutation modules and Scott modules by a systematic use of the Brauer morphism, as suggested by L. Puig (private communication). By "self-contained" we mean that only knowledge of basic facts of representation theory is needed: elementary theory of vertices and sources as presented in (5) (see also (4)), as well as classical results about lifting idempotents. Burry's result about the module-theoretic interpretation of the coefficients of lower defect groups is obtained as a by-product of that presentation. Let G be a finite group, and let 0 be a commutative ring, complete for a discrete valuation, with maximal ideal p and residual field F = 0/p. We assume F has characteristic p > 0. Note that 0 may be equal to F. (0.1) DEFINITION. Let M be an 0-free OG-module. We say that M is a p- permutation module if, whenever P is a p-subgroup of G, there is an 0-basis of M which is stabilized by P.

A Lower Bound for the Formula Size of Rational Functions

Abstract: We establish a lower bound for the formula size of quolynomials over arbitrary fields. Our basic formula operations are addition, subtraction, multiplication and division. The proof is based on Neciporuk’s [Soviet Math. Doklady, 7 (1966), pp. 999–1000] lower bound for Boolean functions and uses formal power series. This result immediately yields a lower bound for the formula size of rational functions over infinite fields. We also show how to adapt Neciporuk’s method to rational functions over finite fields. These results are then used to show that, over any field, the $n \times n$ determinant function has formula size at least $\Omega (n^3 )$. We thus have an algebraic analogue to the $\Omega (n^3 )$ lower bound for the Boolean determinant due to Kloss [Soviet Math. Doklady, 7 (1966), pp. 1537–1540].

Fast Algorithms for Signal Processing and Error Control

Abstract: Algorithms for computation are a central part of digital signal processing and of decoders for error-control codes. When restricted to the study of their computational algorithms, there is not much to distinguish those two subjects. Only the arithmetic field is different; in one case the real or complex field, and a Galois field in the other. Even this distinction is hard to defend; signal processing problems may use Galois fields, and error-control codes in the real or complex field are now under study.

The geometry of conservative systems of hydrodynamic type. The method of averaging for field-theoretical systems

Abstract: CONTENTSIntroduction § 1. Definitions § 2. Geometry and algebra § 3. The method of averaging. Conservativity § 4. Total integrability § 5. The most important examplesReferences

Stably Free Modules

Abstract: In [Su. Prob. 3], Suslin had asked the following question: Let A be any affine algebra of dimension n over an algebraically closed field. What is the smallest integer m such that all stably free projective modules of rank bigger than m are free? All the examples in the literature of stably free non-free modules have rank less than or equal to (n 1)/2. The aim of this note is to construct examples of such modules of large rank. We construct rank p stably free non-free modules over (p + 2)-dimensional affine algebras over algebraically closed fields, wherep is any prime. These varieties are actually smooth and rational. Over C, these are trivial as holomorphic vector bundles. [Forp > 2, this is classical. Forp = 2, see [MS]]. So these are strictly algebraic examples. I had described this construction in [MK 1] and proved the result for p = 2. We will reproduce the construction with necessary modifications in this note. Let p be any prime number and k any field. Letf (x) be any polynomial of degree p over k. Letf (0) = a E k* and Fi (xo, xl) = F(xo,x1) = xP *f (x0 /x1 ). Also let

Methods of constructing hyperfields

Abstract: In this paper we introduce a class of hyperfields which contains non quotient hyperfields. Thus we give a negative answer to the question of whether every hyperfield is isomorphic to a quotient KG of a field K by some subgroup G of its multiplicative group.

The Algebraic Complexity of Shortest Paths in Polyhedral Spaces

Abstract: In this paper we show that the problem of finding the shortest path between two points in Euclidean 3-space, bounded by a finite collection of polyhedral obstacles, is in general not solvable by radicals over the field of rationals. The problem is shown to be not solvable even for the case when only two obstacle edges are encountered in the shof[est path in 3-space. One direct consequence of the non-solvability by radicals is that for the shortest path problem there cannot exist an exac:t algorhhm under models of computation where the root of an algebraic equation is obtained using arithmetic operations and the extraction of err roots. This leaves only numerical or :;ymbolic approximations to the solutions, where the complexity of the approximations is primarily a function of the algebraic degree of the optimum solu~ tion. For special relative orientations of the polyhedral obstacles however the shortest path is ShOWll to be straight-edge and compass constructible. Simple polynomial time exact algorithms are known for such cases. The Algebraic Complexity of Sbortest Paths in P.:JLy:J.edral Spnt:c~

On computing logarithms over finite fields

Abstract: The problem of computing logarithms over finite fields has proved to be of interest in different fields [4]. Subexponential time algorithms for computing logarithms over the special cases GF(p), GF(p2) and GF(pm) for a fixed p and m ? ? have been obtained. In this paper, we present some results for obtaining a sub exponential time algorithms for the remaining cases GF(pm) for p ? ? and fixed m ? 1, 2. The algorithm depends on mapping the fieLd GF(pm) into a suitable cyclotomic extension of the integers (or rationals). Once an isomorphism between GF(pm) and a subset of the cyclotomic field Q(?q) is obtained, the algorithms becomes similar to the previous algorithms for m = 1, 2.A rigorous proof for subexponential time is not yet available, but using some heuristic arguments we can show how it could be proved. If a proof would be obtained, it would use results on the distribution of certain classes of integers and results on the distribution of some ideal classes in cyclotomic fields.

Algebraic fields, signal processing, and error control

Abstract: This survey paper is intended to integrate the subjects of digital signal processing and error control codes by studying their common dependence on the properties of the discrete Fourier transform. The two subjects are traditionally studied in different algebraic fields. Usually, the computations of digital signal processing are done using the complex number system, while the computations of error control codes are done using the arithmetic of Galois fields. We will argue that this dichotomy may be partly a historical accident. By viewing the two problems in the opposite number system, we shall find that there are parallels and that many techniques can be shared by the two subjects. The new material included within the paper is introduced in order to extend known techniques used in one algebraic field into another algebraic field where those techniques are not yet used.

Perfect matchings in hexagonal systems

Abstract: This paper deals with perfect matchings in hexagonal systems. Counterexamples are given to Sachs's conjecture in this field. A necessary and sufficient condition for a hexagonal system to have a perfect matching is obtained.

On Totally Real Cubic Fields

Abstract: The authors have constructed a table of the 26440 nonconjugate totally real cubic number fields of discriminant D < 500000 thereby extending the existing table of fields with D < 100000 by I. 0. Angell (1). Serious defects in Angell's table are pointed out. For each field, running number, discriminant, coefficients of a generating polynomial, integral basis, class number, and a fundamental pair of units are listed. The article contains statistics about the following subjects: distribution of class numbers; fields in which every norm-positive unit is totally positive; nonconjugate fields with the same discriminant; fields with noncyclic class group. The fields are tabulated by means of a method due to Davenport and Heilbronn (7), (8) which leads to a unique normalized generating polynomial. The given units are chosen so that the fundamental parallelogram of the unit lattice determined by the corresponding vectors in the logarithmic space is reduced.

Relative Lubin-Tate groups

Abstract: We construct a class of formal groups that generalizes LubinTate groups. We formulate the major properties of these groups and indicate their relation to local class field theory. The aim of this note is to introduce a certain family of formal groups generalizing Lubin-Tate groups. Although the construction, basic properties and relation with local class field theory are all similar to Lubin-Tate theory, the author is unaware of previous references to these groups. We remark, however, that they are complementary in some sense to the formal groups studied by Honda in [2]. Since we want to keep this note short, all the proofs are omitted. The reader who is acquainted with Lubin-Tate theory as in [4 or 5] will be able to supply them without any difficulties. I would like to acknowledge my debt to K. Iwasawa. His beautiful exposition of local class field theory [3] motivated this note. 1. Let k be a finite extension of Qp, v: kx -Z the normalized valuation (normalized in the sense that v(kX) = Z), 0 and p its ring of integers and maximal ideal, and k = 0/ the residue field, a finite field of characteristics p and q elements. kalg denotes an algebraic closure of k and kUr the maximal unramified extension of k in it. We also fix a completion of kalg, Q, and let K be the closure of k"r in it. We write (p for the Frobenius automorphism of kur/k, characterized by p(x) = mod pur, for all x E Our. It extends by continuity to an automorphism of K/k, still denoted by p. If k' is another finite extension of Qp, the corresponding objects will be denoted by ', e.g. (p', q', etc. If A is any ring, A[[X,... ., Xn]] will denote the power series ring in Xi. If f and g are elements of it, f -g mod deg m means that the power series f g involves only monomials of degree at least m. 2. Fix the field k. For each integer d let Ed be the set of all ( E k, v(s) = d. Fix also d > 0 and let k' be the unique unramified extension of k of degree d. Let E Ed and consider = {f E 0'[[X]]If _ 7r'Xmoddeg2, Nk,/k(1r') = ( and f-Xq mod O'}. THEOREM 1. For each f E 7e there is a unique one-dimensional commutative formal group law Ff E 0'[[X, Y]] satisfying FJ' o f = f o Ff. In others words, f is a homomorphism of Ff to Fr. Received by the editors March 2, 1984. 1980 Mathematics Subject Claification. Primary 12B25.

On matrix spaces with zero determinant

Abstract: Let g be a linear space of n×n matrices of determinant zero over an infinite (or suitably large finite) field It is proved that if the dimension of L exceeds n 2−2n+2, then either L or its transpose has a common null vector This extends a result due to Dieudonne and solves a recent research problem posed by S Pierce in this journal We also consider the problem of classifying all maximal matrix spaces with zero determinant, and offer some examples and observations

Yang-Mills equations on the two-dimensional sphere

Abstract: We construct a 1:1 correspondence between the equivalence classes of Yang-Mills fields over S2 and the conjugacy classes of closed geodesies of the structure group. Furthermore, we give an explicit isolation theorem for any Yang-Mills field over S2.

Time-varying linear systems and invariants of system equivalence

Abstract: In the paper we consider time-varying systems with coefficients depending mero-morphically on time. In differential operator representations these systems are described by matrices over a skew polynomial ring with coefficients in the field of real meromorphic functions. Different kinds of indices (controllability, minimal, geometric and dynamical) are introduced and it is proved that they essentially coincide. The input module and the formal transfer matrix are defined and used for an algebraic description of time-varying systems. A characterization of system equivalence is given in these terms and also a complete list of invariants of similarity for time-varying state-space systems.

Sum-ordered partial semirings

Abstract: If we endow the set of partial functions from a data set to itself with an addition (disjoint-domain sums) and a multiplication (functional composition), then any iterative algorithm may be described formally as the solution to a matrix equation, where the matrix entries are partial functions which describe the parts of the algorithm. This suggests that algorithms may be transformed by manipulation of matrices of partial functions. Hence, it becomes necessary to understand how such matrices behave. The partial functions under disjoint-domain sums and functional composition do not form a field, and thus conventional linear algebra is not applicable. However, they can be regarded as a sum-ordered partial semiring or "so-ring", an algebraic structure possessing a natural partial ordering, an infinitary partial addition, and a binary multiplication, subject to a set of axioms. The majority of this dissertation is devoted to a detailed study of the properties and interesting substructures of so-rings themselves; preliminary results illustrating the behavior of matrices over so-rings are also presented. We hope that this study in part provides a basis for a matrix theory of algorithm transformation.

On the Sprindžuk-Weissauer approach to universal Hilbert subsets

Abstract: The works of both Sprindžuk (1979–80) and Weissauer (1980) consider the relation between Hilbert subsets of Q and sets consisting of powers of primes. A comparison of their results leads to generalizations and new proofs devoid of eitherp-adic diophantine approximation or of nonstandard arithmetic (§3 and §4). Results of Weissauer, giving new Hibertian infinite extensions of every Hilbertian field, receive short direct standard proofs, and a negative answer is given to a question of Roquette on the relation between Hilbert sets and value sets.