Academic literature on the topic 'Symbolic computation continuation'

We introduce a symbolic representation of [Formula: see text]-fold harmonic sums at negative indices. This representation allows us to recover and extend some recent results by Duchamp et al., such as recurrence relations and generating functions for these sums. This approach is also applied to the study of the family of extended Bernoulli polynomials, which appear in the computation of harmonic sums at negative indices. It also allows us to reinterpret the Raabe analytic continuation of the multiple zeta function as both a constant term extension of Faulhaber’s formula, and as the result of a natural renormalization procedure for Faulhaber’s formula.

Let P(x) = 0 be a system of n polynomial equations in n unknowns. Denoting P = (p1,…, pn), we want to find all isolated solutions offor x = (x1,…,xn). This problem is very common in many fields of science and engineering, such as formula construction, geometric intersection problems, inverse kinematics, power flow problems with PQ-specified bases, computation of equilibrium states, etc. Elimination theory-based methods, most notably the Buchberger algorithm (Buchberger 1985) for constructing Gröbner bases, are the classical approach to solving (1.1), but their reliance on symbolic manipulation makes those methods seem somewhat unsuitable for all but small problems.

Abstract The aim of this paper is to grasp the relevant distinctions between various ways in which models and simulations in Artificial Intelligence (AI) relate to cognitive phenomena. In order to get a systematic picture, a taxonomy is developed that is based on the coordinates of formal versus material analogies and theory-guided versus pre-theoretic models in science. These distinctions have parallels in the computational versus mimetic aspects and in analytic versus exploratory types of computer simulation. The proposed taxonomy cuts across the traditional dichotomies between symbolic and embodied AI, general intelligence and symbol and intelligence and cognitive simulation and human/non-human-like AI. According to the taxonomy proposed here, one can distinguish between four distinct general approaches that figured prominently in early and classical AI, and that have partly developed into distinct research programs: first, phenomenal simulations (e.g., Turing’s “imitation game”); second, simulations that explore general-level formal isomorphisms in pursuit of a general theory of intelligence (e.g., logic-based AI); third, simulations as exploratory material models that serve to develop theoretical accounts of cognitive processes (e.g., Marr’s stages of visual processing and classical connectionism); and fourth, simulations as strictly formal models of a theory of computation that postulates cognitive processes to be isomorphic with computational processes (strong symbolic AI). In continuation of pragmatic views of the modes of modeling and simulating world affairs, this taxonomy of approaches to modeling in AI helps to elucidate how available computational concepts and simulational resources contribute to the modes of representation and theory development in AI research—and what made that research program uniquely dependent on them.

Nested binomial sums form a particular class of sums that arise in the context of particle physics computations at higher orders in perturbation theory within QCD and QED, but that are also mathematically relevant, e.g., in combinatorics. We present the package RICA (Rule Induced Convolutions for Asymptotics), which aims at calculating Mellin representations and asymptotic expansions at infinity of those objects. These representations are of particular interest to perform analytic continuations of such sums.

Abstract We calculate the helicity trace index B14 for $$\mathcal{N}$$ = 8 pure D-brane black holes using various techniques of computational algebraic geometry and find perfect agreement with the existing results in the literature. For these black holes, microstate counting is equivalent to finding the number of supersymmetric vacua of a multi-variable supersymmetric quantum mechanics which in turn is equivalent to solving a set of multi-variable polynomial equations modulo gauge symmetries. We explore four different techniques to solve a set of polynomial equations, namely Newton Polytopes, Homotopy continuation, Monodromy and Hilbert series. The first three methods rely on a mixture of symbolic and high precision numerics whereas the Hilbert series is symbolic and admit a gauge invariant analysis. Furthermore, exploiting various exchange symmetries, we show that quartic and higher order terms are absent in the potential, which if present would have spoiled the counting. Incorporating recent developments in algebraic geometry focusing on computational algorithms, we have extended the scope of one of the authors previous works [1, 2] and presented a new perspective for the black hole microstate counting problem. This further establishes the pure D-brane system as a consistent model, bringing us a step closer to $$\mathcal{N}$$ = 2 black hole microstate counting.

Formal, mathematically rigorous programming language semantics are the essential prerequisite for the design of logics and calculi that permit automated reasoning about concurrent programs. We propose a novel modular semantics designed to align smoothly with program logics used in deductive verification and formal specification of concurrent programs. Our semantics separates local evaluation of expressions and statements performed in an abstract, symbolic environment from their composition into global computations, at which point they are concretised. This makes incremental addition of new language concepts possible, without the need to revise the framework. The basis is a generalisation of the notion of a program trace as a sequence of evolving states that we enrich with event descriptors and trailing continuation markers. This allows to postpone scheduling constraints from the level of local evaluation to the global composition stage, where well-formedness predicates over the event structure declaratively characterise a wide range of concurrency models. We also illustrate how a sound program logic and calculus can be defined for this semantics.

Les systèmes polynomiaux multivariés apparaissant dans de nombreuses applications ont des structures spéciales et les systèmes invariants apparaissent dans un large éventail d'applications telles que dans l’optimisation polynomiale et des questions connexes en géométrie algébrique réelle. Le but de cette thèse est de fournir des algorithmes efficaces pour résoudre de tels systèmes structurés. Afin de résoudre le premier type de systèmes, nous concevons des algorithmes efficaces en utilisant les techniques d’homotopie symbolique. Alors que les méthodes d'homotopie, à la fois numériques et symboliques, sont bien comprises et largement utilisées dans la résolution de systèmes polynomiaux pour les systèmes carrés, l'utilisation de ces méthodes pour résoudre des systèmes surdéterminés n'est pas si claire. Hors, les systèmes déterminants sont surdéterminés avec plus d'équations que d'inconnues. Nous fournissons des algorithmes d'homotopie probabilistes qui tirent parti de la structure déterminantielle pour calculer des points isolés dans les ensembles des zéros de tels systèmes. Les temps d'exécution de nos algorithmes sont polynomiaux dans la somme des multiplicités des points isolés et du degré de la courbe d'homotopie. Nous donnons également des bornes sur le nombre de points isolés que nous devons calculer dans trois contextes: toutes les termes de l'entrée sont dans des anneaux polynomiaux classiques, tous ces polynômes sont creux, et ce sont des polynômes à degrés pondérés. Dans la seconde moitié de la thèse, nous abordons le problème de la recherche de points critiques d'une application polynomiale symétrique sur un ensemble algébrique invariant. Nous exploitons les propriétés d'invariance de l'entrée pour diviser l'espace de solution en fonction des orbites du groupe symétrique. Cela nous permet de concevoir un algorithme qui donne une description triangulaire de l'espace des solutions et qui s'exécute en temps polynomial dans le nombre de points que nous devons calculer. Nos résultats sont illustrés par des applications à l'étude d'ensembles algébriques réels définis par des systèmes polynomiaux invariants au moyen de la méthode des points critiques
Multivariate polynomial systems arising in numerous applications have special structures. In particular, determinantal structures and invariant systems appear in a wide range of applications such as in polynomial optimization and related questions in real algebraic geometry. The goal of this thesis is to provide efficient algorithms to solve such structured systems. In order to solve the first kind of systems, we design efficient algorithms by using the symbolic homotopy continuation techniques. While the homotopy methods, in both numeric and symbolic, are well-understood and widely used in polynomial system solving for square systems, the use of these methods to solve over-detemined systems is not so clear. Meanwhile, determinantal systems are over-determined with more equations than unknowns. We provide probabilistic homotopy algorithms which take advantage of the determinantal structure to compute isolated points in the zero-sets of determinantal systems. The runtimes of our algorithms are polynomial in the sum of the multiplicities of isolated points and the degree of the homotopy curve. We also give the bounds on the number of isolated points that we have to compute in three contexts: all entries of the input are in classical polynomial rings, all these polynomials are sparse, and they are weighted polynomials. In the second half of the thesis, we deal with the problem of finding critical points of a symmetric polynomial map on an invariant algebraic set. We exploit the invariance properties of the input to split the solution space according to the orbits of the symmetric group. This allows us to design an algorithm which gives a triangular description of the solution space and which runs in time polynomial in the number of points that we have to compute. Our results are illustrated by applications in studying real algebraic sets defined by invariant polynomial systems by the means of the critical point method