Showing papers on "Algebraic number published in 1970"
01 Jan 1970
TL;DR: In this article, the authors describe knots and links and some of their algebraic properties, and show that any knot diagram can be obtained by substituting algebraic tangles for the vertices of some basic polyhedron P in fact P, and the manner of substitution is essentially unique.
Abstract: Publisher Summary This chapter describes knots and links, and some of their algebraic properties. An edge-connected 4-valent planar map is called a polyhedron, and a polyhedron is basic if no region has just 2 vertices. The term region includes the infinite region, which is regarded in the same light as the others. Knot diagrams can be obtained from polyhedra by substituting tangles for their vertices, for instance, tangles 1 or −1 could always be substituted. A knot diagram K can be obtained by substituting algebraic tangles for the vertices of some nonbasic polyhedron P. There is a polyhedron Q with fewer vertices than P obtained by shrinking some 2-vertex region of P, and K can simply be obtained by substituting algebraic tangles for the vertices of Q. Any knot diagram can be obtained by substituting algebraic tangles for the vertices of some basic polyhedron P in fact P, and the manner of substitution is essentially unique.
876 citations
TL;DR: In this paper, the authors present a deterministic procedure for factoring polynomials over finite fields, which reduces the problem of factoring an arbitrary polynomial over the Galois field GF(p m) to finding the roots in GF(m) of certain other polynomorphisms over GF (m).
Abstract: This paper reviews some of the known algorithms for factoring polynomials over finite fields and presents a new deterministic procedure for reducing the problem of factoring an arbitrary polynomial over the Galois field GF(p m) to the problem of finding the roots in GF(p) of certain other polynomials over GF(p). The amount of computation and the storage space required by these algorithms are algebraic in both the degree of the polynomial to be factored and the logarithm of the order of the finite field. Certain observations on the application of these methods to the factorization of polynomials over the rational integers are also included.
506 citations
295 citations
TL;DR: In this paper, the geometric situation giving rise to variation of Hodge structure is discussed and a number of results about holomorphic and locally constant cohomology classes are given. But these results are restricted to local monodromy of homology.
Abstract: 0. Introduction 229 Par t I. Summary of main results 231 1. The geometric situation giving rise to variation of Hodge structure. . . . 231 2. Data given by the variation of Hodge structure 232 3. Theorems about monodromy of homology 235 4. Theorems about Picard-Fuchs equations (Gauss-Manin connex ion) . . . . 237 5. Global theorems about holomorphic and locally constant cohomology classes 242 6. Global results on variation of Hodge structure 246 Par t I I . Problems and conjectures 247 7. Problems on Torelli-type theorems 247 8. Problems on local monodromy and variation of Hodge structure 248 9. Questions on compactification and the behavior of periods at infinity. . 251
244 citations
TL;DR: In this paper, the conservation laws derived in an earlier paper for the KortewegdeVries equation are proved to be the only ones of polynomial form, and an algebraic operator formalism is developed to obtain explicit formulas for them.
Abstract: The conservation laws derived in an earlier paper for the Korteweg‐deVries equation are proved to be the only ones of polynomial form. An algebraic operator formalism is developed to obtain explicit formulas for them.
201 citations
148 citations
148 citations
131 citations
Book•
01 Jan 1970
TL;DR: An algebraic prelude Continuity of automorphisms and derivations $C^*$-algebra axiomatics and basic results Derivations of $C*$ -algebras Homogeneous $C *$-cCR-algebases $W *$ and $AW *$ -cCRalgebs as discussed by the authors.
Abstract: An algebraic prelude Continuity of automorphisms and derivations $C^*$-algebra axiomatics and basic results Derivations of $C^*$-algebras Homogeneous $C^*$-algebras CCR-algebras $W^*$ and $AW^*$-algebras Miscellany Mappings preserving invertible elements Nonassociativity Bibliography.
130 citations
TL;DR: The floating-point number system is contrasted with the real numbers as discussed by the authors, and the author then illustrates the variety of computational pitfalls a person can fall into who merely translates information gained from pure mathematics courses into computer programs, including summing a Taylor series, solving a quadratic equation, solving linear algebraic systems, solving ordinary and partial differential equations, and finding polynomial zeros.
Abstract: The floating-point number system is contrasted with the real numbers. The author then illustrates the variety of computational pitfalls a person can fall into who merely translates information gained from pure mathematics courses into computer programs. Examples include summing a Taylor series, solving a quadratic equation, solving linear algebraic systems, solving ordinary and partial differential equations, and finding polynomial zeros. It is concluded that mathematics courses should be taught with a greater awareness of automatic computation.
TL;DR: In this paper, it was shown that the theory of surgery does not fit into the already existing concepts of stable algebra, and that the systematization of this material is most naturally carried through from the point of view of an algebraic version of the hamiltonian formalism over rings with involution.
Abstract: The complicated and intricate algebraic material in smooth topology (the theory of surgery) does not fit into the already existing concepts of stable algebra. It turns out that the systematization of this material is most naturally carried through from the point of view of an algebraic version of the hamiltonian formalism over rings with involution. The present article is devoted to this task. The first part contains a development of the algebraic concepts.
01 May 1970
TL;DR: In this paper, it was shown that Siegel's argument cannot provide an algorithm for determining all the integer points on the curve of genus ≥ 1, and that such an algorithm cannot be found in the case of curves of genus 1.
Abstract: 1. Introduction. A well-known theorem of Siegel(5) states that there exist only a finite number of integer points on any curve of genus ≥ 1. Siegel's proof, published in 1929, depended, inter alia, on his earlier work concerning rational approximations to algebraic numbers and on Weil's recently established generalization of Mordell's finite basis theorem. Both of these possess a certain non-effective character and thus it is clear that Siegel's argument cannot provide an algorithm for determining all the integer points on the curve. The purpose of the present paper is to establish such an algorithm in the case of curves of genus 1.
TL;DR: In this paper, a self-contained algebraic formulation of the theory of screws is presented, which is both general and well-adapted to kinematic analysis and numerical methods.
01 Jan 1970
TL;DR: In this article, the authors deal mainly with problems connected with topological and homotopic invariance of Pontrjagin classes and some closely related problems of algebraic topology and stable algebra.
Abstract: This paper deals mainly with problems connected with topological and homotopic invariance of Pontrjagin classes and some closely related problems of algebraic topology and stable algebra. These questions have arisen from the authors’ papers on topological and homotopic invariance of rational Pontrjagin classes, based on the discovery of deep connections between characteristic classes and the fundamental group. There are a number of new stable algebraic problems connected with the diffeomorphism problem and Pontrjagin classes of nonsimply connected manifold (especially when π1= Z × • • • × Z).
Journal Article•
TL;DR: In this paper, the authors present an agreement with the Foundation Compositio Mathematica which implique l'accord avec les conditions generales d'utilisation (http://www.compositio.org/legal.php).
Abstract: © Foundation Compositio Mathematica, 1970, tous droits reserves. L’acces aux archives de la revue « Compositio Mathematica » (http:// http://www.compositio.nl/) implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systematique est constitutive d’une infraction penale. Toute copie ou impression de ce fichier doit contenir la presente mention de copyright.
01 Oct 1970
TL;DR: In this paper, absolute stability, optimality and passivity algebraic criterion in terms of real even polynomial coefficients for dynamic systems are defined. But they do not consider real-time systems.
Abstract: Dynamic systems absolute stability, optimality and passivity algebraic criterion in terms of real even polynomial coefficients
TL;DR: In this article, a sufficient condition for controllability of linear time-varying delay-differential systems is established, which reduces to the conventional one for ordinary differential systems when the delay term is absent.
Abstract: A new algebraic sufficient condition for controllability of linear time-varying delay-differential systems is established, which reduces to the conventional one for ordinary differential systems when the delay term is absent.
TL;DR: In this article, an inequality by Bombieri and Davenport is generalized to arbitrary algebraic number fields of finite degree over the rationals and applied to obtain upper estimates for prime numbers in parallelepipeds.
TL;DR: In this article, the authors present a systematic study of the processes by which the original Euclidean invariance of a quantum statistical theory can be broken to produce pure phases with a lower symmetry.
Abstract: We present a systematic study of the processes by which the original Euclidean invariance of a quantum statistical theory can be broken to produce pure phases with a lower symmetry. Our results provide a rigorous basis for Landau's argument on the nonexistence of critical point in the liquid‐solid phase transition. A classification of the possible residual symmetries is obtained, and its connection with spectral and cluster properties is established. Our tools are those of the algebraic approach to statistical mechanics; in particular, we make an extensive use of the KMS condition. None of our proofs involves the separability of the algebra of quasilocal observables.
TL;DR: In this article, several additional observations which serve to revise and further unify the proof of the approximation theorem are presented, and they serve as a basis for the proof in this paper.
Abstract: This paper contains several additional observations which serve to revise and further unify the proof of the approximation theorem.
01 Apr 1970
TL;DR: 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 arithmetically definable in this paper, are in fact computable.
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 ].