Showing papers on "Algebraic number published in 1988"

Some algebraic and geometric computations in PSPACE

Abstract: We give a PSPACE algorithm for determining the signs of multivariate polynomials at the common zeros of a system of polynomial equations. One of the consequences of this result is that the “Generalized Movers' Problem” in robotics drops from EXPTIME into PSPACE, and is therefore PSPACE-complete by a previous hardness result [Rei]. We also show that the existential theory of the real numbers can be decided in PSPACE. Other geometric problems that also drop into PSPACE include the 3-d Euclidean Shortest Path Problem, and the “2-d Asteroid Avoidance Problem” described in [RS]. Our method combines the theorem of the primitive element from classical algebra with a symbolic polynomial evaluation lemma from [BKR]. A decision problem involving several algebraic numbers is reduced to a problem involving a single algebraic number or primitive element, which rationally generates all the given algebraic numbers.

Differential Algebraic Description of Beam Dynamics to Very High Orders

Abstract: The coordinates z contain positions and momenta of the particle. The vector b contains other parameters that influence the motion such as particle energy, mass, or charge or accelerator parameters such as certain multipole strengths. From this map \"u, quantities of interest for accelerators, such as tune shifts and chromaticities, can be extracted. This is described in detail in a companion paper. 1

A linear algebraic model of algorithm-based fault tolerance

Abstract: A linear algebraic interpretation is developed for previously proposed algorithm-based fault tolerance schemes. The concepts of distance, code space, and the definitions of detection and correction in the vector space R/sup n/ are explained. The number of errors that can be detected or corrected for a distance-(d+1) code is derived. It is shown why the correction scheme does not work for general weight vectors, and a novel fast-correction algorithm for a weighted distance-5 code is derived. >

Minimal rational threefolds. II

Abstract: The Enriques-Fano classification ([E.F], [F]) of the maximal connected algebraic subgroups of the three variable Cremona group, despite of its group theoretic feature, seems to be the most significant result on the rational threefolds so far known. In this paper as in [MU] we interpret the Enriques-Fano classification from a geometric view point, namely the geometry of minimal rational threefolds. We explained in [MU] the link between the two objects; the maximal algebraic subgroups and the minimal rational threefolds. Let (G, X) be a maximal algebraic subgroup of three variable Cremona group. We denote by l(G, X) the set of all the algebraic operations (G, Y) such that Y is non-singular and projective and such that (G, Y) is isomorphic to (G, X) as law chunks of algebraic operation: namely (G, Y) is birationally equivalent to (G, X). Then we define an order in l(G, X): for (G, Z), (G, W) ∊ l(G, X), (G, Z)>(G, W) if there exists an G-equivariant birational morphism of Z onto W.

Class Number Parity

Abstract: This book deals with classical questions of Algebraic Number Theory concerning the interplay between units, ideal class groups, and ramification for relative extensions of number fields. It includes a large collection of fundamental classical examples, dealing in particular with relative quadratic extensions as well as relative cyclic extensions of odd prime degree. The unified approach is exclusively algebraic in nature.

Reflexivity, Algebraic Reflexivity and Linear Interpolation

Abstract: Introduction. This paper was motivated by a discovery that "most" finite dimensional subalgebras of (B(H), for H an infinite dimensional Hilbert space, are reflexive, with the only obstructions to reflexivity being finite-rank considerations. Proofs (section 2) do not depend on multiplicative structure, nor on topology, so extend to linear subspaces of transformations in an abstract setting. Abstract reflexivity has been studied in [3, 4, 7], primarily for singly generated algebras. Reflexivity properties can be interpreted as linear interpolation properties, and we shall adopt this point of view. The results of section 2 extend, in section 3, to algebraic reflexivity counterparts for countably generated (algebraically) linear subspaces of bounded linear transformations acting on a Banach space. As consequences, in section 4 we obtain generalizations of two single operator results. One is a multivariate version of the result, due to Kaplansky [10, Theorem 15], that a bounded locally algebraic operator acting on a Banach space is algebraic. The other is a multivariate version of the result, due to Douglas and Foias [5] for Hilbert space and to Hadwin [7] for a general Banach space, extending work of Fillmore [6], that a bounded non-algebraic operator acting on a Banach space is (topologically) algebraically reflexive. In section 5 we give an application, kindly suggested by D. Hadwin, concerning joint strong similarity orbits of n-tupules of operators. We wish to thank E. Azoff, K. Davidson, J. Erdos, D. Hadwin, J. Kraus, C. Lance and A. Sourour for conversations, and some correspon-

Curves of genus 2 with split Jacobian

Abstract: We say that an algebraic curve has split jacobian if its jacobian is isogenous to a product of elliptic curves. If X is a curve of genus 2, and f: X t E a map from X to an elliptic curve, then X has split jacobian. It is not true that a complement to E in the jacobian of X is uniquely determined, but, under certain conditions, there is a canonical choice of elliptic curve E' and algebraic f: X > E', and we give an algorithm for finding that curve. The construction works in any characteristic other than two. Applications of the algorithm are given to give explicit examples in characteristics 0 and 3. O. Introduction. We say that a curve has split jacobian if its jacobian is isogenous to a product of elliptic curves. In the later half of the nineteenth century a considerable body of work was done on the reduction of abelian integrals to elliptic. Krazer [1] gives a summary of the results obtained. Stated geometrically, the results are particular families of algebraic curves of genus 2 with maps of degree 2, 3 or 4 to elliptic curves. Such curves have split jacobian. The general question of split jacobian curves and particularly, those of genus 2, is of interest for several reasons. Split jacobian curves often have the maximal number of points over finite fields, e.g. the examples of Moret-Bailley [2] are one parameter families of curves of genus 2 over fields of order p2 whose jacobians are isomorphic to the square of the supersingular elliptic curare, and which have maximal numbers of points over fields of order p2n, n > 2. Split jacobian curves of genus 2 have also been used to exhibit nonisomorphic curves with the same jacobian; vide [3, 4]. The approach in these papers is through the algebraic geometry of abelian varieties, and the constructions are therefore far from explicit. Consider the following: Let X be a curve of genus 2, and f: X E a map from X to an elliptic curve. The jacobian of X is therefore isogenous to a product of E and another elliptic curve, E'. Problem: find E', e.g. what is its j-invariant? It is not clear, nor even true (vide T. Shioda [7]), that E' is uniquely determined. However, under certain conditions there is a canonical choice of complement, and we give an algorithm for finding that curve. Our aim is to provide explicit equations for the curves and the maps between them. We obtain a fairly complete combinatorial characterization of the splitting of the jacobians of curves of genus 2. The splitting is characterized by the degree of the map 5 above. Jacobi, generalizing an example of Legendre, gave the complete solution for degree 2. Given any involution Of pl, and three points not fixed by the involution, the curve of genus 2 which has its 6 Weierstrass points above the three points and their images under the involution, maps to the two elliptic curves represented as double covers of the quotient Of pl by the involution, ramified at Received by the editors July 20, 1987. 1980 Mathematicx Xubject (laxsiJication (1985 Revitsion). Primary 14H40, llG10. (ff)1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

Solving systems of nonlinear equations using the nonzero value of the topological degree

Abstract: Two algorithms are described here for the numerical solution of a system of nonlinear equations F(X) = Θ, Θ(0,0,…,0)∈ ℝ, and F is a given continuous mapping of a region 𝒟 in ℝn into ℝn. The first algorithm locates at least one root of the sy stem within n-dimensional polyhedron, using the non zero v alue of the topological degree of F at θ relative to the polyhedron; th e second algorithm applies a new generalized bisection method in order to compute an approximate solution to the system. Teh size of the original n-dimensional polyhedron is arbitrary, and the method is globally convergent in a residual sense. These algorithms, in the various function evaluations, only make use of the algebraic sign of F and do not require computations of the topological degree. Moreover, they can be applied to nondifferentiable continuous functions F and do not involve derivatives of F or approximations of such derivatives.

Neural networks, orientations of the hypercube, and algebraic threshold functions

Abstract: A class of possible generalizations of current neural networks models is described using local improvement algorithms and orientations of graphs. A notation of dynamical capacity is defined and, by computing bounds on the number of algebraic threshold functions, it is proven that for neural networks of size n and energy function of degree d, this capacity is O(n/sup d+1/). Stable states are studied, and it is shown that for the same networks the storage capacity is O(n/sup d+1/). In the case of random orientations, it is proven that the expected number of stable states is exponential. Applications to coding theory are indicated, and it is shown that usual codes can be embedded in neural networks but only at high cost. Cycles and their storage are also examined. >

Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers

Abstract: It is shown that the binary expansions of algebraic numbers do not form secure pseudorandom sequences, given sufficiently many initial bits of an algebraic number, its minimal polynomial can be reconstructed, and therefore the further bits of the algebraic number can be computed. This also enables the authors to devise a simple algorithm to factorise polynomials with rational coefficients. All algorithms work in polynomial time

Periods of Hecke Characters

Abstract: Algebraic hecke characters.- Motives for algebraic hecke characters.- The periods of algebraic hecke characters.- Elliptic integrals and the gamma function.- Abelian integrals with complex multiplication.- Motives of CM modular forms.

Hilbert’s tenth problem for a class of rings of algebraic integers

Abstract: We show that Z is diophantine over the ring of algebraic integers in any number field with exactly two nonreal embeddings into C of degree > 3 over Q. Introduction. Let R be a ring. A set S c Rm is called diophantine over R if it is of the form S = {x E Rm: 3y C Rn p(x, y) = 0}, where p is a polynomial in R[x, y]. A number field is a finite extension of the field Q of rational numbers. If K is a number field, we denote by OK the ring of elements of K which are integral over the ring Z of rational integers. N is the set {O, 1, 2,. ..} and No is the set {1, 2, 3, ..}. In this paper we prove THEOREM. Let K be a number field of degree n > 3 over Q with exactly two nonreal embeddings into the field C of complex numbers. Then Z is diophantirne over OK* An example of such a number field is Q(d) where d3 is a rational number which does not have a rational cube root. In order to prove the theorem, we use the methods of J. Denef in [3]. The terminology and enumeration of the lemmas is kept the same as in [3] so that the similarities and differences of the proofs are clear. The theorem implies COROLLARY. Let K be as in the theorem. Then Hilbert's Tenth Problem in OK is undecidable. The results of [3] and the present paper are the maximum that can be achieved using the present methods. Hence the general conjecture mnade in [4], namnely that Hilbert's Tenth Problem for the integers of any number field is undecidable, remains open. Let K be a number field of degree n > 3 over Q with exactly two nonreal embeddings into C. Let o,, i-1, 2,.. ,n, be all the embeddings of K into C, enumerated in such a way that Un-l and An are nonreal. Then the embedding u: K -k C such that v(x) = an(x) is distinct from An, and from all oa, i < n 2, Received by the editors July 16, 1987. 1980 Mathemattics Subject Cla8sificatton (1985 Revi'ston). Primary 03B25; Secondary 12B99. I would like to thank Professor Leonard Lipschitz for his encouragement and help during the preparation of this work. In the process of publication of this paper I was informed that Alexandra Shlapentokh obtained the same results as part of her thesis at Courant Institute of Mathematical Sciences. This paper has been supported in part by NSF Grant #DMS 8605-198. (?1988 American Mathematical Society 0002-9939/88 $1 00 + $ 25 per page

Defect correction method for the solution of algebraic Riccati equations

Abstract: The solution of discrete and continuous algebraic Riccati equations is considered. It is shown that if an approximate solution is obtained, then the defect for this solution again solves an algebraic Riccati equation of the same form, and that the system properties of detectability and stabilizability are inherited by this defect equation. On the basis of these results, a general defect-correction method is proposed and numerical examples are given for the use of this method in combination with A. Bunse-Gerstner and V. Mehrmann's (1986) SR method. >

Irreducible subgroups of exceptional algebraic groups

Abstract: Keywords: maximal closed connected subgroups ; simply connected simple algebraic group of exceptional type ; semisimple closed connected subgroups ; maximal tori ; root systems ; fundamental weights ; exceptional groups ; irreducible rational tensor indecomposable representation ; embeddings ; maximal parabolic subgroups Reference CTG-ARTICLE-1988-002 Record created on 2008-12-16, modified on 2017-05-12

Werte hypergeometrischer funktionen

Abstract: A partial result on a problem of Siegel is given: Hypergeometric functions with rational parameters have transcendental values in almost all algebraic points — up to some natural exceptions; these exceptions are the well-known algebraic functions and an (unexpected) second class of examples related to certain Shimura-curves.

Shape characterization of some molecular model surfaces

Abstract: An algebraic method is proposed to represent and to characterize in a concise way the shape of an arbitrarily asymmetrical surface composed from spherical pieces. These surfaces include, among others, the well-known van der Waals surfaces. The procedure is based on the computation of a hierarchy of homology groups (“shape groups”) of algebraic topology, for a family of objects defined by the original surface. The technique uses the same input information as that necessary to produce a graphical display of the molecular surface. However, the actual figure is not necessary for the computation of the shape groups. Only a classification of the points on the surface, according to their position with respect to the intersection of two or more spheres, is needed. The result is a purely algebraic characterization that can be obtained and stored by a computer, and that may prove to be useful when comparing shapes of different molecules. Illustrative examples are provided for different molecules, as well as for different conformations of the same molecule.

Exactly solvable models and new link polynomials. III: Two-variable topological invariants

Abstract: New link polynomials, reported in I and II of the series, are extended into those with two variables. A concept of composite string is introduced. It is shown that the composite string representation and the generalized Ocneanu's trace lead to a sequence of two-variable link polynomials. In addition, algebraic aspects of the composite string representation are studied in some detail.

An algebraic theory for multigrid methods for variational problems

Abstract: A convergence theory is developed for multigrid methods for symmetric, positive definite problems in a variational setting. The theory is based on a single algebraic approximation assumption, which is satisfied for finite element discretizations of elliptic boundary value problems, although the theory can be applied to problems without any continuous background as well. In contrast to previous results, we prove fast convergence with any positive number of smoothing steps for V- and W-cycles under discrete analogues of the $H^2 $ and $H^{1 + \alpha } $ regularity assumptions, respectively. We analyze a wide class of smoothers, including arbitrary symmetric and nonsymmetric preconditioned iterations, arbitrarily ordered Gauss–Seidel, steepest descent, Chebyshev iteration and conjugate gradients. Our estimates exhibit the usual asymptotic behavior for a large number of smoothing steps.

On real algebraic morphisms into even-dimensional spheres

Abstract: Given affine real algebraic varieties X and Y let us denote by R( X, Y) the set of regular mappings (real algebraic morphisms) from X into Y (for definitions and notions of real algebraic geometry see [2], where the theory of real algebraic varieties is treated systematically). Our aim in this paper is to study the set M(X,S') of regular mappings from affine nonsingular real algebraic varieties X into S ', the unit sphere EZn+?1x2 = 1. Earlier we obtained several results in this direction [3], [4]; cf. also [2], Chap. 13. In particular, it was shown in [4] that given a compact connected nonsingular orientable real algebraic subset X of RP, of odd dimension k, either each smooth mapping from X to Sk is homotopic to a regular mapping, or precisely those mappings of even topological degree have this property. Now we shall study the case of regular mappings into even-dimensional spheres. Strangely enough the situation then is radically different from that mentioned above for odd-dimensional spheres. For a very large class of compact smooth manifolds of even dimension 2k, "most" algebraic models X of these manifolds have the property that every regular mapping from X into S2k is null homotopic (the meaning of "most" will be made precise later; cf. Remark 1.6, Theorem 2.1, Example 2.3). This class of manifolds contains all compact Co hypersurfaces of R2k?1*

A note on the transcendency of Painlevé’s first transcendent

Abstract: Here we shall prove that Painleve’s first transcendent, a solution of the equation y″ = 6y 2 + x , can not be described as any combination of solutions of first order algebraic differential equations and those of linear differential equations. This result gives an answer to the question whether the function is truely new or not.

Algebraic probability theory

Abstract: Decomposition in semigroups decomposition of probability measures homomorphisms of the convolution semigroup arithmetically important classes of distributions other operations.