Search results for "Symbolic"
showing 10 items of 449 documents
RationalizeRoots: Software Package for the Rationalization of Square Roots
2019
The computation of Feynman integrals often involves square roots. One way to obtain a solution in terms of multiple polylogarithms is to rationalize these square roots by a suitable variable change. We present a program that can be used to find such transformations. After an introduction to the theoretical background, we explain in detail how to use the program in practice.
A novel approach to integration by parts reduction
2015
Integration by parts reduction is a standard component of most modern multi-loop calculations in quantum field theory. We present a novel strategy constructed to overcome the limitations of currently available reduction programs based on Laporta's algorithm. The key idea is to construct algebraic identities from numerical samples obtained from reductions over finite fields. We expect the method to be highly amenable to parallelization, show a low memory footprint during the reduction step, and allow for significantly better run-times.
Conceptual Spaces for Cognitive Architectures: A lingua franca for different levels of representation
2017
During the last decades, many cognitive architectures (CAs) have been realized adopting different assumptions about the organization and the representation of their knowledge level. Some of them (e.g. SOAR [Laird (2012)]) adopt a classical symbolic approach, some (e.g. LEABRA [O'Reilly and Munakata (2000)]) are based on a purely connectionist model, while others (e.g. CLARION [Sun (2006)] adopt a hybrid approach combining connectionist and symbolic representational levels. Additionally, some attempts (e.g. biSOAR) trying to extend the representational capacities of CAs by integrating diagrammatical representations and reasoning are also available [Kurup and Chandrasekaran (2007)]. In this p…
REDUCTION OF CONSTRAINT SYSTEMS
1993
Geometric modeling by constraints leads to large systems of algebraic equations. This paper studies bipartite graphs underlaid by systems of equations. It shows how these graphs make possible to polynomially decompose these systems into well constrained, over-, and underconstrained subsystems. This paper also gives an efficient method to decompose well constrained systems into irreducible ones. These decompositions greatly speed up the resolution in case of reducible systems. They also allow debugging systems of constraints.
Symbolic integration of hyperexponential 1-forms
2019
Let $H$ be a hyperexponential function in $n$ variables $x=(x_1,\dots,x_n)$ with coefficients in a field $\mathbb{K}$, $[\mathbb{K}:\mathbb{Q}] <\infty$, and $\omega$ a rational differential $1$-form. Assume that $H\omega$ is closed and $H$ transcendental. We prove using Schanuel conjecture that there exist a univariate function $f$ and multivariate rational functions $F,R$ such that $\int H\omega= f(F(x))+H(x)R(x)$. We present an algorithm to compute this decomposition. This allows us to present an algorithm to construct a basis of the cohomology of differential $1$-forms with coefficients in $H\mathbb{K}[x,1/(SD)]$ for a given $H$, $D$ being the denominator of $dH/H$ and $S\in\mathbb{K}[x…
Constructing Antidictionaries in Output-Sensitive Space
2021
A word $x$ that is absent from a word $y$ is called minimal if all its proper factors occur in $y$. Given a collection of $k$ words $y_1,y_2,\ldots,y_k$ over an alphabet $\Sigma$, we are asked to compute the set $\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{k}}$ of minimal absent words of length at most $\ell$ of word $y=y_1\#y_2\#\ldots\#y_k$, $\#\notin\Sigma$. In data compression, this corresponds to computing the antidictionary of $k$ documents. In bioinformatics, it corresponds to computing words that are absent from a genome of $k$ chromosomes. This computation generally requires $\Omega(n)$ space for $n=|y|$ using any of the plenty available $\mathcal{O}(n)$-time algorithms. This is because a…
Determinantal sets, singularities and application to optimal control in medical imagery
2016
International audience; Control theory has recently been involved in the field of nuclear magnetic resonance imagery. The goal is to control the magnetic field optimally in order to improve the contrast between two biological matters on the pictures. Geometric optimal control leads us here to analyze mero-morphic vector fields depending upon physical parameters , and having their singularities defined by a deter-minantal variety. The involved matrix has polynomial entries with respect to both the state variables and the parameters. Taking into account the physical constraints of the problem, one needs to classify, with respect to the parameters, the number of real singularities lying in som…
The infinite dihedral group
2022
We describe the infinite dihedral group as automaton group. We collect basic results and give full proofs in details for all statements.
Group-Analytic Family Psychotherapy: A Transcultural Perspective
1997
Group-analytic family psychotherapy is a methodology based on a development of Group-analytic theory. The family is defined as a mental field formed by the symbolic plot of `us' in a double relationship: with the cultural history of the family group on one side, and with external groups on the other. The symbolic plot thus has a tribal characteristic which connects the genealogical trees to the ancestral foundation of the group. In cases of psychotic and borderline patients, Group-analytic family psychotherapy has indicated two types of family: those that are embedded in the past, or families that are cut off from the past. After outlining the circumstances of Italian families, this articl…
A uniform quantificational logic for algebraic notions ofcontext
2002
A quantificational framework of formal reasoning is proposed, which emphasises the pattern of entering and exiting context. Contexts are modelled by an algebraic structure which reflects the order and manner in which context is entered into and exited from. The equations of the algebra partitions context terms into equivalence classes. A formal semantics is defined, containing models that map equivalence classes of certain context terms to sets of first order structures. The corresponding Hilbert system incorporates the algebraic equations as axioms asserted in context. In this way a uniform logic for arbitrary algebras of context is obtained. Soundness and completeness are proved. In semig…