Showing papers on "Algebraic number published in 2005"

Algebraic Number Fields

Abstract: Subrings of fields Complete fields Decomposition groups and the Artin map Analytic methods and ray classes Class field theory Quadratic fields Appendix References Index.

A simple algebraic interface capturing scheme using hyperbolic tangent function

Abstract: This paper presents a simple and practical scheme for capturing moving interfaces or free boundaries in multi-fluid simulations. The scheme, which is called THINC (tangent of hyperbola for interface capturing), makes use of the hyperbolic tangent function to compute the numerical flux for the fluid fraction function, and gives a conservative, oscillation-less and smearing-less solution to the fluid fraction function even for the extremely distorted interfaces of arbitrary complexity. The numerical results from the THINC scheme possess adequate quality for practical applications, which make the extra geometric reconstruction, such as those in most of the volume of fluid (VOF) methods unnecessary. Thus the scheme is quite simple. The numerical tests show that the THINC scheme has competitive accuracy compared to most exiting methods. Copyright © 2005 John Wiley & Sons, Ltd.

An algebraic/numerical formalism for one-loop multi-leg amplitudes

Abstract: We present a formalism for the calculation of multi-particle one-loop amplitudes, valid for an arbitrary number N of external legs, and for massive as well as massless particles. A new method for the tensor reduction is suggested which naturally isolates infrared divergences by construction. We prove that for N>4, higher dimensional integrals can be avoided. We derive many useful relations which allow for algebraic simplifications of one-loop amplitudes. We introduce a form factor representation of tensor integrals which contains no inverse Gram determinants by choosing a convenient set of basis integrals. For the evaluation of these basis integrals we propose two methods: An evaluation based on the analytical representation, which is fast and accurate away from exceptional kinematical configurations, and a robust numerical one, based on multi-dimensional contour deformation. The formalism can be implemented straightforwardly into a computer program to calculate next-to-leading order corrections to multi-particle processes in a largely automated way.

On the Finite Element Solution of the Pure Neumann Problem

Abstract: This paper considers the finite element approximation and algebraic solution of the pure Neumann problem. Our goal is to present a concise variational framework for the finite element solution of the Neumann problem that focuses on the interplay between the algebraic and variational problems. While many of the results that stem from our analysis are known by some experts, they are seldom derived in a rigorous fashion and remain part of numerical folklore. As a result, this knowledge is not accessible (or appreciated) by many practitioners---both novices and experts---in one source. Our paper contributes a simple, yet insightful link between the continuous and algebraic variational forms that will prove useful.

Algebraic Flux Correction I. Scalar Conservation Laws

Abstract: An algebraic approach to the design of multidimensional high-resolution schemes is introduced and elucidated in the finite element context. A centered space discretization of unstable convective terms is rendered local extremum diminishing by a conservative elimination of negative off-diagonal coefficients from the discrete transport operator. This modification leads to an upwind-biased low-order scheme which is nonoscillatory but overly diffusive. In order to reduce the incurred error, a limited amount of compensating antidiffusion is added in regions where the solution is sufficiently smooth. Two closely related flux correction strategies are presented. The first one is based on a multidimensional generalization of total variation diminishing (TVD) schemes, whereas the second one represents an extension of the FEM-FCT paradigm to implicit time-stepping. Nonlinear algebraic systems are solved by an iterative defect correction scheme preconditioned by the low-order evolution operator which enjoys the M-matrix property. The dffusive and antidiffusive terms are represented as a sum of antisymmetric internodal fluxes which are constructed edge-by-edge and inserted into the global defect vector. The new methodology is applied to scalar transport equations discretized in space by the Galerkin method. Its performance is illustrated by numerical examples for 2D benchmark problems.

Insufficiency of linear coding in network information flow

Abstract: It is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite field alphabet. It is also known that this result does not generalize to arbitrary networks. There are several examples in the literature of solvable networks with no scalar linear solution over any finite field. However, each example has a linear solution for some vector dimension greater than one. It has been conjectured that every solvable network has a linear solution over some finite field alphabet and some vector dimension. We provide a counterexample to this conjecture. We also show that if a network has no linear solution over any finite field, then it has no linear solution over any finite commutative ring with identity. Our counterexample network has no linear solution even in the more general algebraic context of modules, which includes as special cases all finite rings and Abelian groups. Furthermore, we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field (exactly 10% greater), so the network is not even asymptotically linearly solvable. It follows that, even for more general versions of linearity such as convolutional coding, filter-bank coding, or linear time sharing, the network has no linear solution

An algebraic/numerical formalism for one-loop multi-leg amplitudes

Abstract: We present a formalism for the calculation of multi-particle one-loop amplitudes, valid for an arbitrary number N of external legs, and for massive as well as massless particles. A new method for the tensor reduction is suggested which naturally isolates infrared divergences by construction. We prove that for N>4, higher dimensional integrals can be avoided. We derive many useful relations which allow for algebraic simplifications of one-loop amplitudes. We introduce a form factor representation of tensor integrals which contains no inverse Gram determinants by choosing a convenient set of basis integrals. For the evaluation of these basis integrals we propose two methods: An evaluation based on the analytical representation, which is fast and accurate away from exceptional kinematical configurations, and a robust numerical one, based on multi-dimensional contour deformation. The formalism can be implemented straightforwardly into a computer program to calculate next-to-leading order corrections to multi-particle processes in a largely automated way.

Distributions in algebraic dynamics

Abstract: The complex dynamic system is a subject to study iterations on P1 or PN with respect to complex topology. It originated from the study of Newton method and the three body problem in the end of 19th century and was highly developed in 20th century. It is a unique visualized subject in pure math because of the beautiful and intricate pictures of the Julia sets generated by computer. The subject of this paper, algebraic dynamics, is a subject to study iterations under Zariski topology and is still in its infancy. If the iteration is defined over a number field, then we are in the situation of arithmetical dynamics where the Galois group and heights will be involved. Here we know very little besides very symmetric objects like abelian varieties and multiplicative groups. The development of arithmetical dynamics was initiated by Northcott in his study of heights on projective space [47], 1950. He showed that the set of rational preperiodic points of any endomorphism of PN of degree ≥ 2 is always finite. The modern theory of canonical heights was developed by Call and Silverman in [11]. Their theory generalized earlier notions of Weil heights on projective spaces and Néron-Tate heights on abelian varieties. Thus many classical questions about abelian varieties and multiplicative groups can be asked again for the dynamical system, such as the size of

On the Algebraic Immunity of Symmetric Boolean Functions.

Abstract: In this paper, we analyze the algebraic immunity of symmetric Boolean functions. The algebraic immunity is a property which measures the resistance against the algebraic attacks on symmetric ciphers. We identify a set of lowest degree annihilators for symmetric functions and propose an efficient algorithm for computing the algebraic immunity of a symmetric function. The existence of several symmetric functions with maximum algebraic immunity is proven. In this way, we have found a new class of functions which have good implementation properties and maximum algebraic immunity.

Morse theory from an algebraic viewpoint

Abstract: Forman's discrete Morse theory is studied from an algebraic viewpoint, and we show how this theory can be extended to chain complexes of modules over arbitrary rings. As applications we compute the homologies of a certain family of nilpotent Lie algebras, and show how the algebraic Morse theory can be used to derive the classical Anick resolution as well as a new two-sided Anick resolution.

Walks in the quarter plane: Kreweras’ algebraic model

Abstract: We consider planar lattice walks that start from (0,0), remain in the first quadrant i,j≥0, and are made of three types of steps: North-East, West and South. These walks are known to have remarkable enumerative and probabilistic properties: • they are counted by nice numbers [Kreweras, Cahiers du B.U.R.O 6 (1965) 5–105], • the generating function of these numbers is algebraic [Gessel, J. Statist. Plann. Inference 14 (1986) 49–58], • the stationary distribution of the corresponding Markov chain in the quadrant has an algebraic probability generating function [Flatto and Hahn, SIAM J. Appl. Math. 44 (1984) 1041–1053]. These results are not well understood, and have been established via complicated proofs. Here we give a uniform derivation of all of them, which is more elementary that those previously published. We then go further by computing the full law of the Markov chain. This helps to delimit the border of algebraicity: the associated probability generating function is no longer algebraic, unless a diagonal symmetry holds. Our proofs are based on the solution of certain functional equations, which are very simple to establish. Finding purely combinatorial proofs remains an open problem.

The Kirillov-Reshetikhin conjecture and solutions of T-systems

Abstract: We prove the Kirillov-Reshetikhin conjecture for all untwisted quantum affine algebras : we prove that the character of Kirillov-Reshetikhin modules solve the Q-system and we give an explicit formula for the character of their tensor products. In the proof we show that the Kirillov-Reshetikhin modules are special in the sense of monomials and that their q-characters solve the T-system (functional relations appearing in the study of solvable lattice models). Moreover we prove that the T-system can be written in the form of an exact sequence. For simply-laced cases, these results were proved by Nakajima with geometric arguments which are not available in general. The proof we use is different and purely algebraic, and so can be extended uniformly to non simply-laced cases.

Zero-Range Process with Open Boundaries

Abstract: We calculate the exact stationary distribution of the one-dimensional zero-range process with open boundaries for arbitrary bulk and boundary hopping rates. When such a distribution exists, the steady state has no correlations between sites and is uniquely characterized by a space-dependent fugacity which is a function of the boundary rates and the hopping asymmetry. For strong boundary drive the system has no stationary distribution. In systems which on a ring geometry allow for a condensation transition, a condensate develops at one or both boundary sites. On all other sites the particle distribution approaches a product measure with the finite critical density ρ c . In systems which do not support condensation on a ring, strong boundary drive leads to a condensate at the boundary. However, in this case the local particle density in the interior exhibits a complex algebraic growth in time. We calculate the bulk and boundary growth exponents as a function of the system parameters.

On two‐grid convergence estimates

Abstract: We derive a new representation for the exact convergence factor of classical two-level and two-grid preconditioners. Based on this result, we establish necessary and sufficient conditions for constructing the components of efficient algebraic multigrid (AMG) methods. The relation of the sharp estimate to the classical two-level hierarchical basis methods is discussed as well. Lastly, as an application, we give an optimal two-grid convergence proof of a purely algebraic ‘window’-AMG method. Published in 2005 by John Wiley & Sons, Ltd.

An algebraic method to compute the critical points of the distance function between two Keplerian orbits

Abstract: We describe an efficient algorithm to compute all the critical points of the distance function between two Keplerian orbits (either bounded or unbounded) with a common focus The critical values of this function are important for different purposes, for example to evaluate the risk of collisions of asteroids or comets with the Solar system planets Our algorithm is based on the algebraic elimination theory: through the computation of the resultant of two bivariate polynomials, we find a 16th degree univariate polynomial whose real roots give us one component of the critical points We discuss also some degenerate cases and show several examples, involving the orbits of the known asteroids and comets

Minkowski’s Conjecture, Well-Rounded Lattices and Topological Dimension

Abstract: Let A C SLn(R) be the diagonal subgroup, and identify SLn(R)/SLn(Z) with the space of unimodular lattices in Rn. In this paper we show that the closure of any bounded orbit A.LcSLn(R)/SLn(Z) meets the set of well-rounded lattices. This assertion implies Minkowski's conjecture for n = 6 and yields bounds for the density of algebraic integers in totally real sextic fields. The proof is based on the theory of topological dimension, as reflected in the combinatorics of open covers of Rn and Tn.

Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation

Abstract: The generalized Korteweg–de Vries equation has the property that solutions with initial data that are analytic in a strip in the complex plane continue to be analytic in a strip as time progresses. Established here are algebraic lower bounds on the possible rate of decrease in time of the uniform radius of spatial analyticity for these equations. Previously known results featured exponentially decreasing bounds.

Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants

Abstract: Let X be a real algebraic convex 3-manifold whose real part is equipped with a Pin i structure. We show that every irreducible real rational curve with non-empty real part has a canonical spinor state belonging to f§1g. The main result is then that the algebraic count of the number of real irreducible rational curves in a given numerical equivalence class passing through the appropriate number of points does not depend on the choice of the real conflguration of points, provided that these curves are counted with respect to their spinor states. These invariants provide lower bounds for the total number of such real rational curves independantly of the choice of the real conflguration of points.

Spectral analysis of percolation Hamiltonians

Abstract: We study the family of Hamiltonians which corresponds to the adjacency operators on a percolation graph. We characterise the set of energies which are almost surely eigenvalues with finitely supported eigenfunctions. This set of energies is a dense subset of the algebraic integers. The integrated density of states has discontinuities precisely at this set of energies. We show that the convergence of the integrated densities of states of finite box Hamiltonians to the one on the whole space holds even at the points of discontinuity. For this we use an equicontinuity-from-the-right argument. The same statements hold for the restriction of the Hamiltonian to the infinite cluster. In this case we prove that the integrated density of states can be constructed using local data only. Finally we study some mixed Anderson-Quantum percolation models and establish results in the spirit of Wegner, and Delyon and Souillard.

Generating polynomial invariants for hybrid systems

Abstract: We present a powerful computational method for automatically generating polynomial invariants of hybrid systems with linear continuous dynamics. When restricted to linear continuous dynamical systems, our method generates a set of polynomial equations (algebraic set) that is the best such over-approximation of the reach set. This shows that the set of algebraic invariants of a linear system is computable. The extension to hybrid systems is achieved using the abstract interpretation framework over the lattice defined by algebraic sets. Algebraic sets are represented using canonical Grobner bases and the lattice operations are effectively computed via appropriate Grobner basis manipulations.

Lattice Image Processing: A Unification of Morphological and Fuzzy Algebraic Systems

Abstract: This paper explores some aspects of the algebraic theory of mathematical morphology from the viewpoints of minimax algebra and translation-invariant systems and extends them to a more general algebraic structure that includes generalized Minkowski operators and lattice fuzzy image operators. This algebraic structure is based on signal spaces that combine the sup-inf lattice structure with a scalar semi-ring arithmetic that possesses generalized `additions' and ?-`multiplications'. A unified analysis is developed for: (i) representations of translation-invariant operators compatible with these generalized algebraic structures as nonlinear sup-? convolutions, and (ii) kernel representations of increasing translation-invariant operators as suprema of erosion-like nonlinear convolutions by kernel elements. The theoretical results of this paper develop foundations for unifying large classes of nonlinear translation-invariant image and signal processing systems of the max or min type. The envisioned applications lie in the broad intersection of mathematical morphology, minimax signal algebra and fuzzy logic.

On the complexity of algebraic numbers II. Continued fractions

Abstract: The continued fraction expansion of an irrational number $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real numbers of degree at least three. Because of some numerical evidence and a belief that these numbers behave like most numbers in this respect, it is often conjectured that their partial quotients form an unbounded sequence. More modestly, we may expect that if the sequence of partial quotients of an irrational number $\alpha$ is, in some sense, "simple", then $\alpha$ is either quadratic or transcendental. The term "simple" can of course lead to many interpretations. It may denote real numbers whose continued fraction expansion has some regularity, or can be produced by a simple algorithm (by a simple Turing machine, for example), or arises from a simple dynamical system... The aim of this paper is to present in a unified way several new results on these different approaches of the notion of simplicity/complexity for the continued fraction expansion of algebraic real numbers of degree at least three.

Solving Algebraic Computaitonal problems in Geodesy and Geoinformatics : The Answer to Modern Challenges

Abstract: Basics of Ring Theory.- Basics of Polynominal Theory.- Groebner Basis.- Polynomial Resultants.- Gauss-Jacobi Combinatorial.- Procrustes Algorithm.- Local (LPS) and Global Positioning Systems.- Positioning by Ranging.- Gauss Coordinates in Geometry and Gravity Space.- Positioning by Resection Method.- Positioning by Intersection Method.- Algebraic Approach to GPS Meteorology.- Algebraic Diagnosis of Outliers.- 7-Parameter Datum Conformal C7 Transformation.- Computer Algebra Software.

A Lower Bound for the Height of a Rational Function at S-unit Points

Abstract: Let a,b be given, multiplicatively independent positive integers and let e>0. In a recent paper jointly with Y. Bugeaud we proved the upper bound exp(en) for g.c.d.(an−1, bn−1); shortly afterwards we generalized this to the estimate g.c.d.(u−1,v−1)

Polynomials over the reals in proofs of termination : from theory to practice

Abstract: This paper provides a framework to address termination problems in term rewriting by using orderings induced by algebras over the reals. The generation of such orderings is parameterized by concrete monotonicity requirements which are connected with different classes of termination problems: termination of rewriting, termination of rewriting by using dependency pairs, termination of innermost rewriting, top-termination of infinitary rewriting, termination of context-sensitive rewriting, etc. We show how to define term orderings based on algebraic interpretations over the real numbers which can be used for these purposes. Prom a practical point of view, we show how to automatically generate polynomial algebras over the reals by using constraint-solving systems to obtain the coefficients of a polynomial in the domain of the real or rational numbers. Moreover, as a consequence of our work, we argue that software systems which are able to generate constraints for obtaining polynomial interpretations over the naturals which prove termination of rewriting (e.g., AProVE, CiME, and TTT), are potentially able to obtain suitable interpretations over the reals by just solving the constraints in the domain of the real or rational numbers.