Showing papers on "Algebraic number published in 1978"

Rational Quadratic Forms

Abstract: Publisher Summary This chapter presents local-global theorems and their limitations—that is, the information that can be obtained from p-adic considerations. Some of the results extend to algebraic number-fields or in other ways. There is a very close relation between the behavior of rational quadratic forms over Q and over the Qp. The chapter presents theorems on “weak Hasse principle” and “strong Hasse principle.” Strong Hasse principle theorem shows the advantage of the p-adic language both for enunciations and proofs. The chapter also introduces a new classification of lattices—namely, into spinor genera

Form rings and regular sequences

Abstract: Hironaka, in his paper [H 1 ] on desingularization of algebraic varieties over a field of characteristic 0, to deal with singular points develops the algebraic apparatus of the associated graded ring, introducing standard bases of ideals, numerical characters ν* and τ* etc. Such a point of view involves a deep investigation of the ideal b * generated by the initial forms of the elements of an ideal A of a local ring, with respect to a certain ideal a .

Connection between the maximal entropy and the scattering theoretic analyses of collision processes

Abstract: The information-theoretic maximal-entropy procedure for the analysis of collision processes is derived as a consequence of the dynamics, be they quantal or classical. The method centers attention on the minimal number of operators (the "dynamic constraints") whose expectation values are both necessary and sufficient to completely characterize the collision dynamics. For a given Hamiltonian and initial state, the constraints required to obtain an exact solution of the equations of motion are determined by a purely algebraic procedure. It is furthermore found possible to derive equations of motion for the conjugate Lagrange parameters. Immediate applications are noted, e.g., a family of similar reactions is shown to have a common set of dynamic constraints and simple illustrative applications are provided. The determination of the scattering matrix is discussed, with examples. The general formalism consists in solving the scattering problem in two stages. The first is purely algebraic. At the end of this stage one obtains the functional form of, say, the scattering matrix or of the density matrix after the collision expressed in terms of parameters whose number equals the number of dynamic constraints. The end result of this algebraic stage suffices to analyze the scattering pattern for any initial state. The second stage is the predictive procedure. Explicit coupled first-order nonlinear differential equations are obtained for the parameters.

Construction of Algebraic Cubature Rules Using Polynomial Ideal Theory

Abstract: Cubature formulae of fixed degree using the minimum number of nodes, the common zeros of a set of polynomials, are constructed by a number of techniques based on the theory of polynomial ideals. Examples demonstrate that known lower bounds to the number of nodes can be attained though usually these bounds are too severe. One example is shown to give rise to an interlacing family of rules, a two dimensional analogue of the Clenshaw–Curtis quadrature. An effective numerical procedure is also given for finding all the common zeros of a set of polynomials.

Combinatorial Structure of the Faces of the n-Cube

Abstract: This paper contains two results: 1) An algebraic characterization of the lattice of faces of the n-cube, based upon axioms independent of the dimension n. The resulting algebraic structure is suited for application to synthesis problems for Boolean functions. 2) Explicit construction of the partition of the lattice of faces of the cube into a minimal number of chains, based upon a new bracketing algorithm.

Stepwise Specification and Implementation of Abstract Data Types

Abstract: The algebraic approach to specification and implementation of abstract data type in the sense of Goguen, Thatcher and Wagner is extended to study problems of stepwise specification and implementation Two different concepts are introduced: 1 Stepwise specification by enrichment 2 Stepwise specification and implementation by functors

Some Results and Problems on Weak Systems of Arithmetic

Abstract: Publisher Summary This chapter discusses results and problems on weak systems of arithmetic. Shepherdson had investigated many systems of arithmetic, one of his aim being finding out the “algebraic effect” of various induction rules and axioms. One problem left unsolved there was to find an algebraic characterization of the open consequences of all the free variable induction axioms. The chapter presents a partial answer to this question for sentences of the form f (x→) ≠ 0 (where f(x→) is a polynomial with integer coefficients). The crucial lemma required in the case states that the full open induction axioms have precisely the same universal consequences as a much weaker system. Techniques are extended to models of P1—the induction scheme for universal formulae (without parameters)—and a connection is obtained between them and models of true arithmetic. The chapter shows that the set of all true universal sentences is the only universal theory implying P­1.

Algebraic integers whose conjugates lie near the unit circle

Abstract: — Let a be a non-zero algebraic integer of degree D (> 1), with conjugates a == 0.1, ..., do. The purpose of this note is to give a new proof of the following result due to BLANKSBY and MONTGOMERY. There exists a positive number C such that if n?=imax{l, |a.|}

A computer-aided root-locus method

Abstract: An efficient computer-aided root-locus method is described The approach is based on the concept of continuation methods in which the solution of a parameterized family of algebraic problems is converted into the solution of a differential equation. The root-locus plot is obtained in a systematic manner by numerical integration. Singularities are analyzed and classified according to the properties of higher order derivatives. Depending on their classification, singular points on the root loci are taken care of accordingly.

Tensor product of the octonionic Hilbert spaces and colour confinement

Abstract: The definition of the tensor product of the octonionic Hilbert spaces with complex geometry is proposed. This definition is based on the isomorphism (geometric and algebraic) of the octonionic Hilbert space with appropriate structure. It is found that the algebraic colour confinement holds only partially. In so called essentially octonionic theories the algebraic confinement of colour holds for all boson states.

Algebraic independence properties of the fredholm series

Abstract: We consider algebraic independence properties of series such as We show that the functions fr(z) are algebraically independent over the rational functions Further, if αrs (r = 2, 3, 4, hellip; s = 1, 2, 3, hellip) are algebraic numbers with 0 < |αrs|, we obtain an explicit necessary and sufficient condition for the algebraic independence of the numbers fr(αrs) over the rationals.

Iterative solution of constrained differential/algebraic systems

Abstract: This paper considers the large class of engineering systems modelled by coupled linear algebraic and differential/difference equations plus initial conditions and closed, convex state and control constraints. The control problem is formulated as the search for any point in the intersection of two closed, convex sets in some real Hilbert space and two iterative schemes are presented for numerical computation. The results are illustrated by application to an 8th-order nuclear reactor model with five algebraic constraints and input constraints.

Transcendence measures for exponentials and logarithms

Abstract: In the present paper, we derive transcendence measures for the numbers log α, e β , α β , (log α 1 )/(log α 2 ) from a previous lower bound of ours on linear forms in the logarithms of algebraic numbers.

Algebraic numbers, a constructive development.

Abstract: The theory of algebraic numbers is developed in the context of abstract fields with equality and inequality. Of classical interest is that any commutative local ring without nilpotent elements may be considered a field in this context. Procedures are given for deciding whether two complex algebraic numbers are equal or not, for factoring polynomials over algebraic number fields and for deciding whether a given algebraic number is in a given algebraic number field.