Showing papers in "Experimental Mathematics in 1996"
TL;DR: In this paper, the problem of how to arrange n n-dimensional subspaces of m-dimensional Euclidean space so that they are as far apart as possible is addressed.
Abstract: We addressthe question: How should N n-dimensional subspaces of m-dimensional Euclidean space be arranged so that they are as far apart as possible? The resuIts of extensive computations for modest values of N, n, m are described, as well as a reformulation of the problem that was suggested by these computations The reformulation gives a way to describe n-dimensional subspaces of m-space as points on a sphere in dimension ½(m–l)(m+2), which provides a (usually) lowerdimensional representation than the Plucker embedding, and leads to a proof that many of the new packings are optimal The results have applications to the graphical display of multidimensional data via Asimov's grand tour method
700 citations
TL;DR: An implementation of the NFS is described, including the choice of two quadratic polynomials, both classical sieving and a special form of lattice sieving (line sieving), the block Lanczos method and a new square root algorithm.
Abstract: The Number Field Sieve (NFS) is the asymptotically fastest known factoring algorithm for large integers. This article describes an implementation of the NFS, including the choice of two quadratic polynomials, both classical sieving and a special form of lattice sieving (line sieving), the block Lanczos method and a new square root algorithm. Finally some data on factorizations obtained with this implementation are listed, including the record factorization of $12^{151}-1$.
52 citations
TL;DR: This work is mainly interested in three questions regarding the eigenvalues and eigenfunctions of the combinatorial Laplacian as q goes to infinity: How large is the second largest eigenvalue, in absolute value, compared with the graph's degree?
Abstract: We surveywhat is known about spectra of combinatorial Laplacians (or adjacency operators) of graphs on the simplest finite symmetric spaces. This work is joint with J. Angel, N. Celniker, A. Medrano, P. Myers, S. Poulos, H. Stark, C. Trimble, and E.Velasquez. For each finite field Fq with q odd, we consider graphs associated to finite Euclidean and non-Euclidean symmetric spaces over Fq. We are mainly interested in three questions regarding the eigenvalues and eigenfunctions of the combinatorial Laplacian as q goes to infinity: How large is the second largest eigenvalue, in absolute value, compared with the graph's degree? (The largesteigenvalue is the degree.) What can one say about the distribution of eigenvalues? What can one say about the “level curves” of the eigenfunctions?
31 citations
TL;DR: It is proved that for a geodesic automatic structure, the growth function for any fixed finite connected graph is a rational function and for a word-hyperbolic group, it is shown that one can choose the denominator of the rational function independently of the finite graph.
Abstract: In this paper we study growth functions of automatic and hyperbolic groups In addition to standard growth functions, we also want to count the number of finite graphs isomorphic to a given finite graph in the ball of radius n around the identity element in the Cayley graph This topic was introduced to us by K Saito [1991] We report on fast methods to compute the growth function once we know the automatic structure We prove that for a geodesic automatic structure, the growth function for any fixed finite connected graph is a rational function For a word-hyperbolic group, we show that one can choose the denominator of the rational function independently of the finite graph
30 citations
TL;DR: Theorems that predict the behavior of periodic orbits in the vicinity of an invariant surface on which the motion is conjugate to a Diophantine rotation for symplectic maps and quasiperiodic perturbations of symp eclectic maps are proved.
Abstract: The existence of an invariant surface in high-dimensional systems greatly influences the. behavior in a neighborhood of the invariant surface. We prove theorems that predict the behavior of periodic orbits in the vicinity of an invariant surface on which the motion is conjugate to a Diophantine rotation for symplectic maps and quasiperiodic perturbations of symplectic maps. Our results allow for efficient numerical algorithms that can serve as an indication for the breakdown of invariant surfaces.
27 citations
TL;DR: Numerical results for analyticity domains of invariant surfaces, behavior after breakdown, and a critical function describing breakdown of invariants surfaces as a function of their rotation vectors are presented.
Abstract: We study the behavior of invariant setsof a volume-preserving map that is a quasiperiodic perturbation of a symplectic map, using approximation by periodic orbits. We present numerical results for analyticity domains of invariant surfaces, behavior after breakdown, and a critical function describing breakdown of invariant surfacesas a function of their rotation vectors. We discuss implications of our results to the existence of a renormalization group operator describing breakdown of invariant surfaces.
23 citations
TL;DR: An elementary method for computing isolated values of M(x) = Σ n≤x μ(n), where μ is the Mobius function, and the complexity of the algorithm is O(x 2/3 (log log x)1/3) time and O( x 1/3(log log y) space.
Abstract: We describe an elementary method for computing isolated values of M(x) = Σ n≤x μ(n), where μ is the Mobius function. The complexity of the algorithm is O(x 2/3 (log log x)1/3) time and O(x 1/3(log log x)2/3) space. Certain values of M(x) for x up to 1016 are listed: for instance, M(1016) = −3195437.
22 citations
TL;DR: A method to construct irreducible rational matrix representations of finite groups, based on an efficient construction of fixed points of infinite groups acting on complex vector spaces is presented.
Abstract: We present a method to construct irreducible rational matrix representations of finite groups, basedon an efficient construction of fixed points of finite groups acting on complex vector spaces.
18 citations
TL;DR: It is shown that the norm of a composition operator on the classical Hardy space H 2 cannot be computed using only the set of H 2 reproducing kernels, answering a question raised by Cowen and MacCluer.
Abstract: We show that the norm of a composition operator on the classical Hardy space H 2 cannot be computed using only the set of H 2 reproducing kernels, answering a question raised by Cowen and MacCluer.
18 citations
TL;DR: The maximal finite irreducible subgroups of GL24 (Q), together with their natural lattices, are classified and new methods for finding the maximal finite supergroups of irredUCible cyclic groups are developed and applied.
Abstract: We classify maximal finite irreducible subgroups of GL24 (Q), together with their natural lattices. There are 65 conjugacy classes of such groups, 41 of which consist of primitive groups. New methods for finding the maximal finite supergroups of irreducible cyclic groups are developed and applied.
16 citations
TL;DR: Matzat has proved that the Mathieu group of degree 24 is a Galois group over the transcendent extension Q(T) by using a construction called nonrigid, proving the existence of a rat.
Abstract: Matzat a prouve que Ie groupe de Mathieu de degre 24 est groupe de Galois sur I'extension transcendante Q(T). Il utilise pour cela une construction dite non rigide et prouve I'existence d'un point rationnel dans un espacede Hurwitz adequate Nous donnonsici une telle extension explicitement. Nous en deduisons aussi I'existence d'une extension reguliere de K(T) de groupe de Galois M 23 pour tout K tel que I'equation x 2+y 2+z 2 = 0 ait une solution non triviale. Pour obtenir ces resultats, il a fallu remplacer les outils habituels du calcul formel par des constructions numeriques et retrouver ensuite les objets algebriques en parametrisant certaines courbes de genre 0. Cela nous permet d'illustrer la puissance des techniques de calcul de reveternents developpees dans [Couveignes 1994; Couveignes et Granboulan 1994]. Matzat has proved that the Mathieu group of degree 24 is a Galois group over the transcendent extension Q(T). He does this by using a construction called nonrigid, proving the existence of a rat...
TL;DR: Using a Cayley program, all firm, residually connected geometries whose rank-two residues satisfy the intersection are obtained.
Abstract: Using a Cayley program, we get all firm, residually connected geometries whose rank-two residues satisfy the intersection. property, on which M ll acts flag-transitively, and in which the stabilizer of each element is a maximal subgroup of M ll.
Journal Article•
TL;DR: Equivariant polynomial functions with the symmetries of the n-cube are completely determined in terms of permutations of exponents and strategies for random searchef linear combinations of these functions are described and used to generate interesting examples of attractors.
Abstract: Equivariant polynomial functions with the symmetries of the n-cube are completely determined in terms of permutations of exponents. Strategiesfor random searchesof linear combinations of these functions are described and used to generate interestingexamples of attractors. These attractors have symmetries that are an admissible subgroup of the symmetries of the square, cube and 4-cube. A central projection ofthe 4-cube with partial inversion is used for the illustrations of attractors in four dimensions.
TL;DR: This is an experimental study of the rank of elliptic curves over Q obtained by specializing to integer values the parameter t of curves over Z(t) having rank 0 to 4.
Abstract: This is an experimental study of the rank of elliptic curves over $\Q$ obtained by specializing to integer values the parameter $t$ of curves over $\Q(t)$ having rank 0 to 4 On etudie experimentalement le rang des courbes elliptiques sur $\Q$ obtenues par specialisation entiere du parametre $t$ de courbes de rang allant de 0 a 4 sur $\Q(t)$
TL;DR: In this article, the authors present the results of many factorization runs with the PMPQS and PPMPQSon SGI workstations and on a Cray C90 vector computer.
Abstract: This article is concerned with the large-prime variations of the multipolynomial quadratic sieve factorization method: the PMPQS (one large prime) and the PPMPQS (two). We present the results of many factorization runs with the PMPQS and PPMPQSon SGI workstations and on a Cray C90 vector computer. Experimentsshow that for our Cray C90 implementations PPMPQS beats PMPQS for numbers of more than 80 digits, and that this crossover point goes down with the amount of available central memory. For PMPQS we give a formula to predict the total running time based on a short test run. The accuracy of the prediction is within 10% of the actual running time. For PPMPQS we do not have such a formula. Yet in order to provide measurements to help determining a good choice of the parameters in PPMPQS, we factored many numbers. In addition we give an experimental prediction formula for PPMPQS suitable if one wishes to factor many large numbers of about the same size.
TL;DR: This paper contains a report on a numerical study of dspec Γ for various noncocompact groups Γ, and particularly interesting are the results for some nonarithmetic groupsΓ.
Abstract: Let H3 be three-dimensional hyperbolic space and Γ a group of isometries of H3 that acts discontinuously on H3 and that has a fundamental domain of finite hyperbolic volume. The laplace operator –δ of H3 gives rise to a positive, essentiallv selfadjoint operator on L 2 (Γ\H3). The nature of its discrete spectrum dspec Γ is still not well understood if Γ is not cocompact. This paper contains a report on a numerical study of dspec Γ for various noncocompact groups Γ. Particularly interesting are the results for some nonarithmetic groups Γ.
TL;DR: TheclassicalCaratheodoryConjecturestatesthateverysmo othconvexemb edding of a2-sphereinR3musthaveatleasttwoumbilics, and the convexityhyp othesis is notrelevantforthis argument.
Abstract: We show that, under mild nonflatness conditions, for any $r\ge 3$ and any $C^r$-immersion of a surface into $\R^3$ with an isolated umbilic point there exist an analytic surface with an isolated umbilic of the same index. The connection of this with Caratheodory's Conjecture on umbilics is discussed.
TL;DR: The smallest value of m is obtained such that n is (m, k)-perfect, for 1 ≤ n ≤ 1000, and questions concerning the limiting behaviour of σ m+1(n)/σ m(n) and (σ m (n))l/m are addressed, as m → ∞.
Abstract: Let $\sigma^0(n) = n$ and $\sigma^m(n) = \sigma(\sigma^{m-1}(n))$, where $m\ge1$ and $\sigma$ is the sum-of-divisors function. We say that $n$ is $(m,k)$-perfect if $\sigma^m(n) = kn$. We have tabulated all $(2,k)$-perfect numbers up to $10^9$ and all $(3,k)$- and $(4,k)$-perfect numbers up to $2\cdot10^8$. These tables have suggested several conjectures, some of which we prove here. We ask in particular: For any fixed $m\ge1$, are there infinitely many $(m,k)$-perfect numbers? Is every positive integer $(m,k)$-perfect, for sufficiently large $m\ge1$? In this connection, we have obtained the smallest value of $m$ such that $n$ is $(m,k)$-perfect, for $1\le n\le1000$. We also address questions concerning the limiting behaviour of $\sigma^{m+1}(n)/\sigma^m(n)$ and $(\sigma^m(n))^{1/m}$, as $m\to\infty$.
TL;DR: In this paper, necessary and sufficient conditions for a sum-free set to be ultimately periodic were presented, and shown how these conditions can be used to test specific sets, and these tests produce the first evidence of a positive nature that certain sets are, in fact, not ultimately periodic.
Abstract: Cameron has introduced a natural one-to-one correspondence between infinite binary sequences and setsof positive integers with the property that no two elements add up to a third. He observed that, if a sum-free set is ultimately periodic, so is the corresponding binary sequence, and asked if the converse also holds. We present here necessary and sufficient conditions for a sum-free set to be ultimately periodic, and show how these conditions can be used to test specific sets. These tests produce the first evidence of a positive nature that certain sets are, in fact, not ultimately periodic.
TL;DR: Can two latin squares L, M have the same determinant, up to a renaming of the variables, apart from the obvious cases when L is obtained from M by a sequence of row interchanges, column interchanges; it is shown that it is yes for n = 8.
Abstract: A latin square is an n × n array of n symbols in which each symbol appears exactly once in each row and column. Regarding each symbol as a variable and taking the determinant, we get a degree-n polynomial in n variables. Can two latin squares L, M have the same determinant, up to a renaming of the variables, apart from the obvious cases when L is obtained from M by a sequence of row interchanges, column interchanges, renaming of variables, and transposition? The answer was known to beno if n ≤ 7; we show that it is yes for n = 8. The latin squaresfor which this situation occurs have interesting special characteristics.
TL;DR: Knuth's conjecture is proved by applying Zeilberger's algorithm as well as classical hypergeometric machinery to imply an identity of type Ln = |Rn |, involving the floor function.
Abstract: From numerical experiments, D. E. Knuth conjectured that 0 < D n+4 < D n for a combinatorial sequence (Dn ) defined as the difference Dn = Rn – Ln of two definite hypergeometric sums. The conjecture implies.an identity of type Ln = |Rn |, involving the floor function. We prove Knuth's conjecture by applying Zeilberger's algorithm as well as classical hypergeometric machinery.
TL;DR: This work describes a computational, heuristic approach to the problem of deciding whether or not a given finitely presented group has a free quotient of rank two or more, and describes successful computations with sections of the Picard group SL2(Z[i]).
Abstract: We describe a computational, heuristic approach to the problem of deciding whether or not a given finitely presented group has a free quotient of rank two or more. Our strategy is to construct a finite nilpotent quotient of the given group, to search for quotients that are free within a variety containing that quotient, and then lift to the original group. We give theoretical justification to our strategy, and describe successful computations with sections of the Picard group SL2(Z[i]).
TL;DR: In this paper, a methode for the construction of formes primitives based on the result of Langlands and Weil is proposed, and the coefficients of their Fourier expansions at infinity can then be computed so as to provide tables.
Abstract: Un resultat de langlands et Weil permet d'associer a toute representation galoisienne de type octaedral et de determinant impair du groupe de Galois absolu de Q une forme primitive de poids 1. En nous appuyant sur les travaux de Bayer et Frey, nous proposons une methode de construction de formes primitives basee sur ce resultat, le calcul des coefficients de leur developpement de Fourier a I'infini, que nous avons implemente sur machine, permet la construction de tables. Le cas des formes de niveau pair est etudie avec precision. It is known by a result of Langlands and Weil that one can associate to each representation of the absolute Galois group of Q with odd determinant and octahedral type a newform of weight one. Using the work of Bayer and Frey, we provide a method for constructing such newforms. The calculation of the coefficients of their Fourier expansions at infinity can then be computed so as to provide tables. The case of forms of even level is studied in detail.