Showing papers on "Field (mathematics) published in 1970"

Quantum field theory on lightlike slabs

Abstract: Restricting the support of relativistic quantum fields to lightlike hyperplanes (e.g. x0+x3)=const) we find examples of such fields to exist as well-defined self-adjoint operators with properties however that differ vastly from those of fields on the usual spacelike surfaces. We show that on a lightlike hyperplane 1) the free-field algebra is irreducible (instead of Abelian, and in contrast to what one would expect of data on a characteristic surface) and 2) fields with different masses become unitarily equivalent (whereas they are inequivalent on spacelike planes). Furthermore the field algebra restricted to the space-time slab between two parallel lightlike planes is always irreducible (while there are counterexamples for spacelike slabs). We establish this directly for generalized free fields and rederive it for Wightman fields in gereral.

Forms of the affine line and its additive group

Abstract: Let k be a field, Xo an object (e.g., scheme, group scheme) defined over k. An object X of the same type and isomorphic to Xo over some field K z> k is called a form of Xo. If k is not perfect, both the affine line A1 and its additive group Gtt have nontrivial sets of forms, and these are investigated here. Equivalently, one is interested in ^-algebras R such that K ® k R = K[t] (the polynomial ring in one variable) for some field K => ky where, in the case of forms of G α, R has a group (or co-algebra) structure s\R—>R®kR such that (K®s)(t) = £ ® 1 + 1 ® ί. A complete classification of forms of Gα and their principal homogeneous spaces is given and the behaviour of the set of forms under base field extension is studied.

On the contents of polynomials

Abstract: In [13, p. 24], Prufer establishes the following result. (See also [2, Exercise 21, p. 97].) Let J be an integral domain with identity having quotient field K and letf, g, h:K [X] be such that h =fg. If A, B, and C denote the fractional ideals of J generated by the coefficients of f, g, and h, respectively, and if n is the degree of the polynomial g, then A C = A +B. This result is essentially what Krull in [7, p. 128] calls the Hilfssatz von Dedekind-Mertens, although the results which Dedekind in [4] and Mertens in [II ] prove are not so general as Prufer's theorem. In this note, we generalize the theorem cited above, first to the case where J is an arbitrary subring of the commutative ring K, and then to the case of polynomials in finitely many indeterminates. We conclude with some applications of the results obtained. We use consistently the following notation in this paper. R denotes a subring of a commutative ring S. If f is a polynominal over S, Yf denotes the set of coefficients of f, and Af denotes the R-submodule of S generated by Yf; we call Af the R-content of f. The following result is straightforward, but we list it here for the sake of reference.

On the Analogue of the Modular Group in Characteristic p

Abstract: It is well-known that the classical concept of modular forms may be introduced as follows Write G for one of the two algebraic groups GL(2), SL(2); take for k the field Q of rational numbers, R being then the completion k ∞ of k at its infinite place; let Γ be the subgroup G Z of G R (ie the group of the matrices in M2 (Z) with the determinant ±1 if G = GL(2), and + 1 if G = SL(2)) On G R , consider the complex-valued functions which are left-invariant under Γ (or at any rate under some congruence subgroup of Γ), behave in a prescribed manner under a translation belonging to the center of G R , and behave in a prescribed manner under the right translations belonging to the usual maximal compact subgroup of G R and under the Casimir operator for G R ; the two latter conditions ensure that this determines in the upper half-plane a modular form of prescribed degree which is an eigenfunction for the Beltrami operator (in particular, if the corresponding eigenvalue is O, it is holomorphic, or at any rate the sum of a holomorphic and an antiholomorphic function)

Degrees of sums in a separable field extension

Abstract: Let F be any field and suppose that E is a separable algebraic extension of F. For elements aGE, we let dga denote the degree of the minimal polynomial of a over F. Let a, O E, dga=m, dgo3=n and suppose (m, n) = 1. It is easy to see that [F(a, ,B): F] =nn, and by a standard theorem of field theory (for instance see Theorem 40 on p. 49 of [I]), there exists an element -y E such that F(a, r) -F(,y) and thus dgy = in. In fact, the usual proof of this theorem produces (for infinite F) an element of the form zy =-a +X3, with X E F. In this paper we show that in many cases the choice of X EF is completely arbitrary, as long as X#zO. In Theorem 63 on p. 71 of [1], it is shown that if n>m and n is a prime different from the characteristic of F, then dg(a + =rmn. The present result includes this.

A Many-Body Theory for Nearly-Free Electrons and Its Application to Annihilation of Positrons in a Periodic Field

Abstract: A many-body theory for nearly-free electrons has been constructed by introducing “ Bloch-pair operators,” α k and β k , which are related to free-particle operators through the transformation \begin{aligned} \begin{pmatrix} \alpha_{k}\\ \beta_{k} \end{pmatrix} = \begin{pmatrix} 1/\sqrt{2} & \theta_{h}/\sqrt{2}\\ -\theta_{h}/\sqrt{2} & 1/\sqrt{2} \end{pmatrix} \begin{pmatrix} a_{k}\\ a_{k+h} \end{pmatrix} , \end{aligned} where h is 2π times a reciprocal lattice vector, and θ h is the sign of the relevant Fourier component, v h , of the periodic field. This theory has been then applied to the annihilation of positrons in the periodic field. The interaction between a positron and a Bloch electron, has been, for the first time, taken into account through a few simplest diagrams. The result shows that a fairly sharp break of the angular-correlation curve occurs a the hexagonal-zone-face position in copper just as expected from the experiments previously made by the present author and his collaborator.

ON THE QUESTION OF THE STRUCTURE OF THE SUBFIELD OF INVARIANTS OF A CYCLIC GROUP OF AUTOMORPHISMS OF THE FIELD Q(x1,...,xn )

Abstract: The subfield L of the field K = Q(x1,...,xn) consisting of invariant functions relative to a cyclic permutation of the indeterminates is interpreted as the field of rational functions on a certain torus defined over Q. On this basis, a necessary condition is derived for pure transcendence of L over Q from which are obtained a number of counterexamples. A list is also given of fields L which are purely transcendental over Q.

Computable fields and arithmetically definable ordered fields

Abstract: Introduction. A computable field is one whose elements may be placed in one-one correspondence with the natural numbers in such a way that the number theoretic functions corresponding to the field operations are recursive. In the same vein a field is called arithmetically definable (AD for short) if its elements may be placed in one-one correspondence with the natural numbers in such a way that the number theoretic functions corresponding to the field operations are arithmetical. These notions clearly extend in an obvious way to ordered fields and indeed to algebraic structures in general. The term computable structure (group, ring, etc.) was probably introduced for the first time by M. 0. Rabin [4], however, a similar notion was discussed a few years earlier by Frohlich and Shepherdson [1]. Each of these references contains a number of interesting theorems on computable structures. Some results concerning AD structures appear in [2 ]. The main purpose of the present paper is to show that the fields of real algebraic numbers, constructible numbers, and solvable numbers, which were shown to be AD in [2], are in fact computable. This answers a question raised in footnote (2) of [2 ].

Computability Over Arbitrary Fields

Abstract: In most attempts to make precise the concept of a computable function, or decidable predicate, over a field F, it is considered necessary that the elements of F should be in some sense effectively describable, and hence that F itself should be countable. This is the attitude taken in the study of computable fields (see Rabin1). Our proposed definition of computability over arbitrary fields is based on the Shepherdson - Sturgis2 concept of an unlimited register machine.

Generalized Combining of Elements From Finite Fields

Abstract: In a recent paper Raktoe [1969] has presented a new approach and also a generalized technique to combining elements from distinct finite fields. The results however were related only to distinct prime fields i.e. each of the Galois fields in question consisted of residue classes modulo a prime. This paper solves the problem of combining elements of the most general case, i.e. the fields are not necessarily based on distinct primes and they can be prime powered.

On the mathematical description of quantized fields

Abstract: We start from Haag's proposal to describe quantum fields at a point, corresponding to the heuristic description by means of their matrix elements (A(x)Φ‖Ψ) between vectors of a dense linear manifoldD of the Hilbert space. We particularize this idea, so that the sesquilinear functional that describes the field at a point may be considered as an element of the sequential completion of a space of operators, endowed with a suitable “D-weak” topology.

Identities in Jordan algebras

Abstract: This chapter describes the identities in Jordan algebras and presents the calculation of the dimensions of certain subspaces of Jordan algebras. The chapter discusses a fixed but arbitrary field of characteristic zero, the restriction on which can almost certainly be relaxed. The chapter presents an assumption in which w ∈ L ( n ), and where w is a sum of elements aR where R is a monomial in operators R ( x ) and each x is a monomial in some of the generators b, c, …. If x contains more than two generators, then in certain conditions, R ( x ) can be expanded as a sum of words R ( y ) where each y contains fewer generators than x.

The automorphisms of an algebraically closed field

Abstract: It is well known that the complex number field has infinitely many automorphisms. Moreover, it seems to be part of the folklore that the family of all automorphisms of the complex field has cardinality 2c, where c = 2ℵo. In this article the following generalization of this fact is proved: If k is any algebraically closed field then the family of all automorphisms of k has cardinality 2card k . The complex field has infinite transcendency degree over its prime subfield. For fields of this type the proof is accomplished by essentially permuting the elements in a transcendency basis and extending each permutation to an automorphism of the field.

Extending differential specializations

Abstract: The study of specializations in differential algebra has been marked by the appearance of several examples which behave very differently from their less complicated brethren in algebraic geometry. In fact, let a denote an ordinary differential field of characteristic zero, U a universal extension of a with field of constants K, with y and z differential indeterminates over U (these notational conventions will be observed from now on). Subscripts will denote derivatives. Then Kolchin has shown that if A=yz'+P(z), where P is a cubic polynomial having distinct roots with coefficients in anK, and if (ti, >) is a generic zero of the general component of A in a {y, z I then O is a differential specialization of ? over a but there is no element aEU such that (0, a) is a differential specialization of (i, t) or (77, D over 5. Another example, which goes back to Ritt [31, shows the existence of another prime differential ideal in a { y, z I whose generic zero (, q) has the property that 0 is a differential specialization of q? over , but not of t1 or of r over a. The situation arising out of the first example was dealt with in [II where it was shown that if R is a local differential domain, meaning that R is local and has a maximal ideal m which is differential, then if t he differential homomorphism 4: R-4R/m cannot be extended either to a or to j, there exist nonnegative integers i, j such that as13,fR. As a corollary it was shown that if a1/aeCR then 4 can be extended either to a differential homomorphism of R[{a or to one of Rtl/aI. In the reference cited a and # were restricted to the differential field of quotients of R, but it is easy to see that this assumption is not necessary. The present paper is concerned with the second of the above-mentioned phenomena and it will be shown that under a certain condition v or t must specialize to 0 over j. I would like to thank Professor Ellis Kolchin and Dr. Jerald Kovacic for their helpful thoughts on this subject. DEFINITION. Let R be a differential subring of U containing the rational numbers, Q. An element f3E U is said to be monic over.R if ,3 is a zero of a differential polynomial of the form y1+f(y) ERR{y where the total degree of f is less than n.

Desingularization of complex-analy tic varieties

Abstract: Several years ago a proof was given for the resolution of singularities of an arbitrary algebraic variety over a field of characteristi c zero ([1]). As was pointed out there, it was readily modified to give desingularizations of complex-analytic varieties on which global meromorphic functions give a local coordinate system at every point, such as complex Stein varieties. Later, Moishezon extended the desingularization theorem to compact complex-analytic varieties bimeromorphic to algebraic varieties, simply by an ingenuity of inductive reformulation of the problem ([2]). For the case of a general complex-analytic variety, however, we must go back to the essentials of the proof of desingulariza tion in the al­ gebraic case, and then develop a new technique to globalize the basic ideas found there. The essential difficulty in the complex-analytic case, as is compared with the algebraic case, is in fact the lack of global meromorphic functions and the inevidence of "sufficiently many" global subvarieties. The supporting philosophy toward the complex-analytic desingularization is that the more singular is the given variety, the more global subvarieties exist there. The problem is, of course, how to find them. In the resolution of singularities of complex-analytic varieties, we look for not only global subvarieties of a given variety but also global subvarieties of its transforms by successive blowing-ups having such subvarieties as their centers.

Class number in constant extensions of elliptic function fields

Abstract: For F/K a function field of genus one having the finite field K as field of constants and E the constant extension of degree n we give explicitly the class number of the field E as a poly- nomial expression in terms of the class number of F and the order of the field K. Applications are made to determine the degree of a con- stant extension E necessary to have a predetermined prime p occur as a divisor of the class number of the field E. Let F/K be a function field in one variable with exact field of con- stants K, a finite field having q elements. The order of the finite group of divisor classes of degree zero is the class number hF. Let E denote the constant extension of degree n and hE the class number of E. It is known that hE= khF for some integer k. In this note we give an ex- plicit determination of k in the particular case that F has genus one and give several applications of it. Precisely, we prove the THEOREM. If F/K is a function field with genus one and EIF is the constant extension of degree n then

The image of marketing as a field of study among business students

Abstract: Review of the Study Solutions to three basic problems were sought in this study, 1. Does the image of marketing held by business students differ significantly from these students’ images of accounting, economics, finance and management as fields of study? 2. Does the image of marketing as a field of study differ significantly among various classes from within the population of business students? 3. What are some specific characteristics which contribute to the formulation of favorable and unfavor­ able images of marketing as a field of study? Images were operationally defined as mental repre­ sentations of anything not actually present to the senses; mental pictures formed as a result of stimuli. Students selected for this study were presented stimuli in the form of attitude statements, descriptive adjectives and value statements related to fields of study in business. Favorable images were revealed by responses indicating agreement with statements and adjectives

On a result of Cassels

Abstract: Let α be an irrational algebraic number of degree k over the rationals. Let K denote the field generated by α over the rationals and let a denote the ideal denominator of α. Then Cassels [3] has shown that for sufficiently large integral N > 0 distinctly more than half the integers n , are such that ( n +α)a is divisible by a prime ideal p n which does not divide ( m +α)a for any integer m ≠ n satisfying . The purpose of this note is to point out that minor modification of Cassel's proof enables the extension of the interval for n from to , and to derive results on the proportion of values n , for which the values f(n) of a given integral polynomial in n are divisible by a prime p > N .

Visual search through random walk number fields

Abstract: Serial search through ordered number fields (resembling two-dimensional random walk patterns) was studied. A method employing generating discs allowed for constructing the number fields in such a way as to vary systematically the serial constraint (step size and variability of step size) between consecutive numbers in the field. In Experiment I, performances on constrained number fields were compared with those on a randomly generated number field. The number of points serially connected in 30-sec trials increased with increasing constraint in the construction of the number field. In Experiment 2, step size and variability of step size were combined orthogonally in the construction of the number fields. Both variables had significant effects on number of points connected. In addition, learning effects were shown to occur over the 10 trials of practice. Two explanations of serial search behavior are mentioned, one based on local density in the number field and the other on area searched through in seeking a target.

The Siegel-Brauer theorem concerning parameters of algebraic number fields

Abstract: An upper bound is obtained for the product of the number of classes of ideals and the regulator of algebraic number fields

A direct numerical method for the solution of field problems

Abstract: A numerical method for the analysis of field problems is described. The algorithm is based upon the generalized Betti-Maxwell theorem. Using a set of known solutions to problems with similar boundary conditions produces a set of ‘integral’ equations for the required solution. Using any convenient numerical integration formula reduces the problem to the solution of a set of simultaneous algebraic equations. The accuracy of the solution depends upon the accuracy of the integration formula as applied to the problem under consideration and is independent of the known auxiliary solutions. The method is described in detail as applied to harmonic problems.

Fields with two linear orderings

Abstract: We characterize fields which are maximal with respect to the property of having two different linear orderings. The Galois group of the algebraic closure of a maximal field is described. An example of non-uniqueness of the maximal extension is mentioned.