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At least 109 records · Page 6Linked to original sources

arXiv2009

ASF+ --- eine ASF-aehnliche Spezifikationssprache

Maintaining the main aspects of the algebraic specification language ASF as presented in [Bergstra&al.89] we have extend ASF with the following concepts: While once exported names in ASF must stay visible up to the top the module hierarchy, ASF+ permits a more sophisticated hiding of signature names. The erroneous merging of distinct structures that occurs when importing different actualizations of the same parameterized module in ASF is avoided in ASF+ by a more adequate form of parameter binding. The new ``Namensraum''-concept of ASF+ permits the specifier on the one hand directly to identify the origin of hidden names and on the other to decide whether an imported module is only to be accessed or whether an important property of it is to be modified. In the first case he can access one single globally provided version; in the second he has to import a copy of the module. Finally ASF+ permits semantic conditions on parameters and the specification of tasks for a theorem prover.

cs.AI↗

arXiv2009

A bound on the minimum of a real positive polynomial over the standard simplex

We consider the problem of bounding away from 0 the minimum value m taken by a polynomial P of Z[X_1,...,X_k] over the standard simplex, assuming that m>0. Recent algorithmic developments in real algebraic geometry enable us to obtain a positive lower bound on m in terms of the dimension k, the degree d and the bitsize of the coefficients of P. The bound is explicit, and obtained without any extra assumption on P, in contrast with previous results reported in the literature.

cs.SC↗

arXiv2009

From Moments to Functions in Quantum Chromodynamics

Single-scale quantities, like the QCD anomalous dimensions and Wilson coefficients, obey difference equations. Therefore their analytic form can be determined from a finite number of moments. We demonstrate this in an explicit calculation by establishing and solving large scale recursions by means of computer algebra for the anomalous dimensions and Wilson coefficients in unpolarized deeply inelastic scattering from their Mellin moments to 3-loop order.

hep-ph↗

arXiv2009

A Graph Analysis of the Linked Data Cloud

The Linked Data community is focused on integrating Resource Description Framework (RDF) data sets into a single unified representation known as the Web of Data. The Web of Data can be traversed by both man and machine and shows promise as the \textit{de facto} standard for integrating data world wide much like the World Wide Web is the \textit{de facto} standard for integrating documents. On February 27$^\text{th}$ of 2009, an updated Linked Data cloud visualization was made publicly available. This visualization represents the various RDF data sets currently in the Linked Data cloud and their interlinking relationships. For the purposes of this article, this visual representation was manually transformed into a directed graph and analyzed.

cs.CY↗

arXiv2009

Heuristic Reasoning on Graph and Game Complexity of Sudoku

The Sudoku puzzle has achieved worldwide popularity recently, and attracted great attention of the computational intelligence community. Sudoku is always considered as Satisfiability Problem or Constraint Satisfaction Problem. In this paper, we propose to focus on the essential graph structure underlying the Sudoku puzzle. First, we formalize Sudoku as a graph. Then a solving algorithm based on heuristic reasoning on the graph is proposed. The related r-Reduction theorem, inference theorem and their properties are proved, providing the formal basis for developments of Sudoku solving systems. In order to evaluate the difficulty levels of puzzles, a quantitative measurement of the complexity level of Sudoku puzzles based on the graph structure and information theory is proposed. Experimental results show that all the puzzles can be solved fast using the proposed heuristic reasoning, and that the proposed game complexity metrics can discriminate difficulty levels of puzzles perfectly.

cs.AI↗

arXiv2010

Combinatorial Deformations of Algebras: Twisting and Perturbations

The framework used to prove the multiplicative law deformation of the algebra of Feynman-Bender diagrams is a \textit{twisted shifted dual law} (in fact, twice). We give here a clear interpretation of its two parameters. The crossing parameter is a deformation of the tensor structure whereas the superposition parameters is a perturbation of the shuffle coproduct of Hoffman type which, in turn, can be interpreted as the diagonal restriction of a superproduct. Here, we systematically detail these constructions.

cs.SC↗

arXiv2009

Supernodal Analysis Revisited

In this paper we show how to extend the known algorithm of nodal analysis in such a way that, in the case of circuits without nullors and controlled sources (but allowing for both, independent current and voltage sources), the system of nodal equations describing the circuit is partitioned into one part, where the nodal variables are explicitly given as linear combinations of the voltage sources and the voltages of certain reference nodes, and another, which contains the node variables of these reference nodes only and which moreover can be read off directly from the given circuit. Neither do we need preparational graph transformations, nor do we need to introduce additional current variables (as in MNA). Thus this algorithm is more accessible to students, and consequently more suitable for classroom presentations.

cs.SC↗

arXiv2009

Computations modulo regular chains

The computation of triangular decompositions are based on two fundamental operations: polynomial GCDs modulo regular chains and regularity test modulo saturated ideals. We propose new algorithms for these core operations relying on modular methods and fast polynomial arithmetic. Our strategies take also advantage of the context in which these operations are performed. We report on extensive experimentation, comparing our code to pre-existing Maple implementations, as well as more optimized Magma functions. In most cases, our new code outperforms the other packages by several orders of magnitude.

cs.SC↗

arXiv2009

Computing Cylindrical Algebraic Decomposition via Triangular Decomposition

Cylindrical algebraic decomposition is one of the most important tools for computing with semi-algebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set $F \subset {\R}[y_1, ..., y_n]$ we apply comprehensive triangular decomposition in order to obtain an $F$-invariant cylindrical decomposition of the $n$-dimensional complex space, from which we extract an $F$-invariant cylindrical algebraic decomposition of the $n$-dimensional real space. We report on an implementation of this new approach for constructing cylindrical algebraic decompositions.

cs.SC↗

arXiv2011

Dependency Pairs and Polynomial Path Orders

We show how polynomial path orders can be employed efficiently in conjunction with weak innermost dependency pairs to automatically certify polynomial runtime complexity of term rewrite systems and the polytime computability of the functions computed. The established techniques have been implemented and we provide ample experimental data to assess the new method.

cs.LO↗

arXiv2009

On the Cooperation of the Constraint Domains H, R and FD in CFLP

This paper presents a computational model for the cooperation of constraint domains and an implementation for a particular case of practical importance. The computational model supports declarative programming with lazy and possibly higher-order functions, predicates, and the cooperation of different constraint domains equipped with their respective solvers, relying on a so-called Constraint Functional Logic Programming (CFLP) scheme. The implementation has been developed on top of the CFLP system TOY, supporting the cooperation of the three domains H, R and FD, which supply equality and disequality constraints over symbolic terms, arithmetic constraints over the real numbers, and finite domain constraints over the integers, respectively. The computational model has been proved sound and complete w.r.t. the declarative semantics provided by the $CFLP$ scheme, while the implemented system has been tested with a set of benchmarks and shown to behave quite efficiently in comparison to the closest related approach we are aware of. To appear in Theory and Practice of Logic Programming (TPLP)

cs.PL↗

arXiv2009

A Symbolic Summation Approach to Find Optimal Nested Sum Representations

We consider the following problem: Given a nested sum expression, find a sum representation such that the nested depth is minimal. We obtain a symbolic summation framework that solves this problem for sums defined, e.g., over hypergeometric, $q$-hypergeometric or mixed hypergeometric expressions. Recently, our methods have found applications in quantum field theory.

cs.SC↗

arXiv2009

La Résolvante de Lagrange et ses Applications

In this paper, the changes of representations of a group are used in order to describe its action as algebraic Galois group of an univariate polynomial on the roots of factors of any Lagrange resolvent. By this way, the Galois group of resolvent factors are pre-determinated. In follows, different applications are exposed; in particular, some classical results of algebraic Galois theory.

cs.SC↗

arXiv2010

Successive Difference Substitution Based on Column Stochastic Matrix and Mechanical Decision for Positive Semi-definite Forms

The theory part of this paper is sketched as follows. Based on column stochastic average matrix $T_n$ selected as a basic substitution matrix, the method of advanced successive difference substitution is established. Then, a set of necessary and sufficient conditions for deciding positive semi-definite form on $\R^n_+$ is derived from this method. And furthermore, it is proved that the sequence of SDS sets of a positive definite form is positively terminating. Worked out according to these results, the Maple program TSDS3 not only automatically proves the polynomial inequalities, but also outputs counter examples for the false. Sometimes TSDS3 does not halt, but it is very useful by experimenting on so many examples.

cs.SC↗

arXiv2010

Multihomogeneous Resultant Formulae for Systems with Scaled Support

Constructive methods for matrices of multihomogeneous (or multigraded) resultants for unmixed systems have been studied by Weyman, Zelevinsky, Sturmfels, Dickenstein and Emiris. We generalize these constructions to mixed systems, whose Newton polytopes are scaled copies of one polytope, thus taking a step towards systems with arbitrary supports. First, we specify matrices whose determinant equals the resultant and characterize the systems that admit such formulae. Bezout-type determinantal formulae do not exist, but we describe all possible Sylvester-type and hybrid formulae. We establish tight bounds for all corresponding degree vectors, and specify domains that will surely contain such vectors; the latter are new even for the unmixed case. Second, we make use of multiplication tables and strong duality theory to specify resultant matrices explicitly, for a general scaled system, thus including unmixed systems. The encountered matrices are classified; these include a new type of Sylvester-type matrix as well as Bezout-type matrices, known as partial Bezoutians. Our public-domain Maple implementation includes efficient storage of complexes in memory, and construction of resultant matrices.

cs.SC↗

arXiv2011

Castelnuovo-Mumford Regularity and Computing the de Rham Cohomology of Smooth Projective Varieties

We describe a parallel polynomial time algorithm for computing the topological Betti numbers of a smooth complex projective variety $X$. It is the first single exponential time algorithm for computing the Betti numbers of a significant class of complex varieties of arbitrary dimension. Our main theoretical result is that the Castelnuovo-Mumford regularity of the sheaf of differential $p$-forms on $X$ is bounded by $p(em+1)D$, where $e$, $m$, and $D$ are the maximal codimension, dimension, and degree, respectively, of all irreducible components of $X$. It follows that, for a union $V$ of generic hyperplane sections in $X$, the algebraic de Rham cohomology of $X\setminus V$ is described by differential forms with poles along $V$ of single exponential order. This yields a similar description of the de Rham cohomology of $X$, which allows its efficient computation. Furthermore, we give a parallel polynomial time algorithm for testing whether a projective variety is smooth.

math.AG↗

arXiv2009

Programming Realization of Symbolic Computations for Non-linear Commutator Superalgebras over the Heisenberg--Weyl Superalgebra: Data Structures and Processing Methods

We suggest a programming realization of an algorithm for verifying a given set of algebraic relations in the form of a supercommutator multiplication table for the Verma module, which is constructed according to a generalized Cartan procedure for a quadratic superalgebra and whose elements are realized as a formal power series with respect to non-commuting elements. To this end, we propose an algebraic procedure of Verma module construction and its realization in terms of non-commuting creation and annihilation operators of a given Heisenberg--Weyl superalgebra. In doing so, we set up a problem which naturally arises within a Lagrangian description of higher-spin fields in anti-de-Sitter (AdS) spaces: to verify the fact that the resulting Verma module elements obey the given commutator multiplication for the original non-linear superalgebra. The problem setting is based on a restricted principle of mathematical induction, in powers of inverse squared radius of the AdS-space. For a construction of an algorithm resolving this problem, we use a two-level data model within the object-oriented approach, which is realized on a basis of the programming language C#. The program allows one to consider objects (of a less general nature than non-linear commutator superalgebras) that fall under the class of so-called $GR$-algebras, for whose treatment one widely uses the module \emph{Plural} of the system \emph{Singular} of symbolic computations for polynomials.

hep-th↗

arXiv2009

Spherical Distribution of 5 Points with Maximal Distance Sum

In this paper, we mainly consider the problem of spherical distribution of 5 points, that is, how to configure 5 points on a sphere such that the mutual distance sum attains the maximum. It is conjectured that the sum of distances is maximal if 5 points form a bipyramid configuration in which case two points are positioned at two poles of the sphere and the other three are positioned uniformly on the equator. We study this problem using interval methods and related technics, and give a proof for the conjecture through computers in finite time.

cs.DM↗