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  • A new algorithm for computing $μ$-bases of the univariate polynomial vector
    arXiv.cs.SC Pub Date : 2020-11-22
    Dingkang Wang; Hesong Wang; Fanghui Xiao

    In this paper, we characterized the relationship between Groebner bases and u-bases: any minimal Groebner basis of the syzygy module for n univariate polynomials with respect to the term-over-position monomial order is its u-basis. Moreover, based on the gcd computation, we construct a free basis of the syzygy module by the recursive way. According to this relationship and the constructed free basis

    更新日期:2020-11-25
  • Multi-experiment parameter identifiability of ODEs and model theory
    arXiv.cs.SC Pub Date : 2020-11-21
    Alexey Ovchinnikov; Anand Pillay; Gleb Pogudin; Thomas Scanlon

    Structural identifiability is a property of an ODE model with parameters that allows for the parameters to be determined from continuous noise-free data. This is natural prerequisite for practical identifiability. Conducting multiple independent experiments could make more parameters or functions of parameters identifiable, which is a desirable property to have. How many experiments are sufficient

    更新日期:2020-11-25
  • An effective method for computing Grothendieck point residue mappings
    arXiv.cs.SC Pub Date : 2020-11-18
    Shinichi Tajima; Katsusuke Nabeshima

    Grothendieck point residue is considered in the context of computational complex analysis. A new effective method is proposed for computing Grothendieck point residues mappings and residues. Basic ideas of our approach are the use of Grothendieck local duality and a transformation law for local cohomology classes. A new tool is devised for efficiency to solve the extended ideal membership problems

    更新日期:2020-11-19
  • Representation of hypergeometric products of higher nesting depths in difference rings
    arXiv.cs.SC Pub Date : 2020-11-17
    Evans Doe Ocansey; Carsten Schneider

    A non-trivial symbolic machinery is presented that can rephrase algorithmically a finite set of nested hypergeometric products in appropriately designed difference rings. As a consequence, one obtains an alternative representation in terms of one single product defined over a root of unity and nested hypergeometric products which are algebraically independent among each other. In particular, one can

    更新日期:2020-11-18
  • Sequence Positivity Through Numeric Analytic Continuation: Uniqueness of the Canham Model for Biomembranes
    arXiv.cs.SC Pub Date : 2020-11-16
    Stephen Melczer; Marc Mezzarobba

    We prove solution uniqueness for the genus one Canham variational problem arising in the shape prediction of biomembranes. The proof builds on a result of Yu and Chen that reduces the variational problem to proving non-negativity of a sequence defined by a linear recurrence relation with polynomial coefficients. We combine rigorous numeric analytic continuation of D-finite functions with classic bounds

    更新日期:2020-11-17
  • Symbolically Solving Partial Differential Equations using Deep Learning
    arXiv.cs.SC Pub Date : 2020-11-12
    Maysum Panju; Kourosh Parand; Ali Ghodsi

    We describe a neural-based method for generating exact or approximate solutions to differential equations in the form of mathematical expressions. Unlike other neural methods, our system returns symbolic expressions that can be interpreted directly. Our method uses a neural architecture for learning mathematical expressions to optimize a customizable objective, and is scalable, compact, and easily

    更新日期:2020-11-16
  • Invariants of Self-Intersected and Inversive N-Periodics in the Elliptic Billiard
    arXiv.cs.SC Pub Date : 2020-11-12
    Ronaldo Garcia; Dan Reznik

    We study more invariants in the elliptic billiard, including those manifested by self-intersected orbits and inversive polygons. We also derive expressions for some entries in "Eighty New Invariants of N-Periodics in the Elliptic Billiard" (2020), arXiv:2004.12497.

    更新日期:2020-11-16
  • LDU factorization
    arXiv.cs.SC Pub Date : 2020-11-08
    Gennadi Malaschonok

    LU-factorization of matrices is one of the fundamental algorithms of linear algebra. The widespread use of supercomputers with distributed memory requires a review of traditional algorithms, which were based on the common memory of a computer. Matrix block recursive algorithms are a class of algorithms that provide coarse-grained parallelization. The block recursive LU factorization algorithm was obtained

    更新日期:2020-11-12
  • Quadratization of ODEs: Monomial vs. Non-Monomial
    arXiv.cs.SC Pub Date : 2020-11-08
    Foyez Alauddin

    Quadratization is a transform of a system of ODEs with polynomial right-hand side into a system of ODEs with at most quadratic right-hand side via the introduction of new variables. It has been recently used as a pre-processing step for new model order reduction methods, so it is important to keep the number of new variables small. Several algorithms have been designed to search for a quadratization

    更新日期:2020-11-12
  • A Neuro-Symbolic Method for Solving Differential and Functional Equations
    arXiv.cs.SC Pub Date : 2020-11-04
    Maysum Panju; Ali Ghodsi

    When neural networks are used to solve differential equations, they usually produce solutions in the form of black-box functions that are not directly mathematically interpretable. We introduce a method for generating symbolic expressions to solve differential equations while leveraging deep learning training methods. Unlike existing methods, our system does not require learning a language model over

    更新日期:2020-11-06
  • Connectivity in Semi-Algebraic Sets I
    arXiv.cs.SC Pub Date : 2020-11-04
    Hoon Hong; James Rohal; Mohab Safey El Din; Eric Schost

    A semi-algebraic set is a subset of the real space defined by polynomial equations and inequalities having real coefficients and is a union of finitely many maximally connected components. We consider the problem of deciding whether two given points in a semi-algebraic set are connected; that is, whether the two points lie in the same connected component. In particular, we consider the semi-algebraic

    更新日期:2020-11-06
  • Calcium: computing in exact real and complex fields
    arXiv.cs.SC Pub Date : 2020-11-03
    Fredrik JohanssonLFANT

    Calcium is a C library for real and complex numbers in a form suitable for exact algebraic and symbolic computation. Numbers are represented as elements of fields $\mathbb{Q}(a_1,\ldots,a_n)$ where the extensions numbers $a_k$ may be algebraic or transcendental. The system combines efficient field operations with automatic discovery and certification of algebraic relations, resulting in a practical

    更新日期:2020-11-04
  • Rounding Error Analysis of Linear Recurrences Using Generating Series
    arXiv.cs.SC Pub Date : 2020-11-02
    Marc MezzarobbaPEQUAN

    We develop a toolbox for the error analysis of linear recurrences with constant or polynomial coefficients, based on generating series, Cauchy's method of majorants, and simple results from analytic combinatorics. We illustrate the power of the approach by several nontrivial application examples. Among these examples are a new worst-case analysis of an algorithm for computing Bernoulli numbers, and

    更新日期:2020-11-03
  • The New Rewriting Engine of Dedukti
    arXiv.cs.SC Pub Date : 2020-10-30
    Gabriel HondetDEDUCTEAM, Inria, LSV, ENS Paris Saclay, CNRS; Frédéric BlanquiDEDUCTEAM, Inria, LSV, ENS Paris Saclay, CNRS

    Dedukti is a type-checker for the $\lambda$$\Pi$-calculus modulo rewriting, an extension of Edinburgh's logicalframework LF where functions and type symbols can be defined by rewrite rules. It thereforecontains an engine for rewriting LF terms and types according to the rewrite rules given by the user.A key component of this engine is the matching algorithm to find which rules can be fired. In thispaper

    更新日期:2020-11-02
  • Fast Minimal Presentations of Bi-graded Persistence Modules
    arXiv.cs.SC Pub Date : 2020-10-29
    Michael Kerber; Alexander Rolle

    Multi-parameter persistent homology is a recent branch of topological data analysis. In this area, data sets are investigated through the lens of homology with respect to two or more scale parameters. The high computational cost of many algorithms calls for a preprocessing step to reduce the input size. In general, a minimal presentation is the smallest possible representation of a persistence module

    更新日期:2020-10-30
  • On computation of the inverse of a polynomial map over finite fields using the reduced Koopman dual linear map
    arXiv.cs.SC Pub Date : 2020-10-26
    Ramachandran Anantharaman; Virendra Sule

    This paper proposes a symbolic representation of non-linear maps $F$ in $\ff^n$ in terms of linear combination of basis functions of a subspace of $(\ff^n)^0$, the dual space of $\ff^n$. Using this representation, it is shown that the inverse of $F$ whenever it exists can also be represented in a similar symbolic form using the same basis functions (using different coefficients). This form of representation

    更新日期:2020-10-30
  • Lexicographic Groebner bases of bivariate polynomials modulo a univariate one
    arXiv.cs.SC Pub Date : 2020-10-28
    Xavier Dahan

    Let T(x) in k[x] be a monic non-constant polynomial and write R=k[x]/(T) the quotient ring. Consider two bivariate polynomials a(x, y), b(x, y) in R[y]. In a first part, T = p^e is assumed to be the power of an irreducible polynomial p. A new algorithm that computes a minimal lexicographic Groebner basis of the ideal (a, b, p^e), is introduced. A second part extends this algorithm when T is general

    更新日期:2020-10-30
  • A Graph Theoretical Approach for Testing Binomiality of Reversible Chemical Reaction Networks
    arXiv.cs.SC Pub Date : 2020-10-23
    Hamid Rahkooy; Cristian Vargas Montero

    We study binomiality of the steady state ideals of chemical reaction networks. Considering rate constants as indeterminates, the concept of unconditional binomiality has been introduced and an algorithm based on linear algebra has been proposed in a recent work for reversible chemical reaction networks, which has a polynomial time complexity upper bound on the number of species and reactions. In this

    更新日期:2020-10-30
  • Optimized Multivariate Polynomial Determinant on GPU
    arXiv.cs.SC Pub Date : 2020-10-23
    Jianjun Wei; Liangyu Chen

    We present an optimized algorithm calculating determinant for multivariate polynomial matrix on GPU. The novel algorithm provides precise determinant for input multivariate polynomial matrix in controllable time. Our approach is based on modular methods and split into Fast Fourier Transformation, Condensation method and Chinese Remainder Theorem where each algorithm is paralleled on GPU. The experiment

    更新日期:2020-10-26
  • Logic Guided Genetic Algorithms
    arXiv.cs.SC Pub Date : 2020-10-21
    Dhananjay Ashok; Joseph Scott; Sebastian Wetzel; Maysum Panju; Vijay Ganesh

    We present a novel Auxiliary Truth enhanced Genetic Algorithm (GA) that uses logical or mathematical constraints as a means of data augmentation as well as to compute loss (in conjunction with the traditional MSE), with the aim of increasing both data efficiency and accuracy of symbolic regression (SR) algorithms. Our method, logic-guided genetic algorithm (LGGA), takes as input a set of labelled data

    更新日期:2020-10-26
  • Evaluation of Logic Programs with Built-Ins and Aggregation: A Calculus for Bag Relations
    arXiv.cs.SC Pub Date : 2020-10-20
    Matthew Francis-Landau; Tim Vieira; Jason Eisner

    We present a scheme for translating logic programs, which may use aggregation and arithmetic, into algebraic expressions that denote bag relations over ground terms of the Herbrand universe. To evaluate queries against these relations, we develop an operational semantics based on term rewriting of the algebraic expressions. This approach can exploit arithmetic identities and recovers a range of useful

    更新日期:2020-10-26
  • Algorithmic Reduction of Biological Networks With Multiple Time Scales
    arXiv.cs.SC Pub Date : 2020-10-20
    Niclas Kruff; Christoph Lüders; Ovidiu Radulescu; Thomas Sturm; Sebastian Walcher

    We present a symbolic algorithmic approach that allows to compute invariant manifolds and corresponding reduced systems for differential equations modeling biological networks which comprise chemical reaction networks for cellular biochemistry, and compartmental models for pharmacology, epidemiology and ecology. Multiple time scales of a given network are obtained by scaling, based on tropical geometry

    更新日期:2020-10-26
  • Optimal Descartes' Rule of Signs for Circuits
    arXiv.cs.SC Pub Date : 2020-10-19
    Frédéric Bihan; Alicia Dickenstein; Jens Forsgård

    We present an optimal version of Descartes' rule of signs to bound the number of positive real roots of a sparse system of polynomial equations in n variables with n+2 monomials. This sharp upper bound is given in terms of the sign variation of a sequence associated to the exponents and the coefficients of the system.

    更新日期:2020-10-20
  • Creative Telescoping on Multiple Sums
    arXiv.cs.SC Pub Date : 2020-10-18
    Christoph Koutschan; Elaine Wong

    We discuss the strategies and difficulties of determining a recurrence which a certain polynomial (in the form of a symbolic multiple sum) satisfies. The polynomial comes from an analysis of integral estimators derived via quasi-Monte Carlo methods.

    更新日期:2020-10-20
  • Projective isomorphisms between rational surfaces
    arXiv.cs.SC Pub Date : 2020-10-16
    Bert Jüttler; Niels Lubbes; Josef Schicho

    We present a method for computing projective isomorphisms between rational surfaces that are given in terms of their parametrizations. The main idea is to reduce the computation of such projective isomorphisms to five base cases by modifying the parametric maps such that the components of the resulting maps have lower degree. Our method can be used to compute affine, Euclidean and M\"obius isomorphisms

    更新日期:2020-10-19
  • New Remarks on the Factorization and Equivalence Problems for a Class of Multivariate Polynomial Matrices
    arXiv.cs.SC Pub Date : 2020-10-14
    Dong Lu; Dingkang Wang; Fanghui Xiao

    This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present some new criteria for the existence of matrix factorizations for a class of multivariate polynomial matrices, and obtain a necessary and sufficient condition for the equivalence of a square polynomial matrix and a diagonal matrix. Based on the constructive proof of the new criteria

    更新日期:2020-10-16
  • On Factor Left Prime Factorization Problems for Multivariate Polynomial Matrices
    arXiv.cs.SC Pub Date : 2020-10-14
    Dong Lu; Dingkang Wang; Fanghui Xiao

    This paper is concerned with factor left prime factorization problems for multivariate polynomial matrices without full row rank. We propose a necessary and sufficient condition for the existence of factor left prime factorizations of a class of multivariate polynomial matrices, and then design an algorithm to compute all factor left prime factorizations if they exist. We implement the algorithm on

    更新日期:2020-10-16
  • On Minor Left Prime Factorization Problem for Multivariate Polynomial Matrices
    arXiv.cs.SC Pub Date : 2020-10-14
    Dong Lu; Dingkang Wang; Fanghui Xiao

    A new necessary and sufficient condition for the existence of minor left prime factorizations of multivariate polynomial matrices without full row rank is presented. The key idea is to establish a relationship between a matrix and its full row rank submatrix. Based on the new result, we propose an algorithm for factorizing matrices and have implemented it on the computer algebra system Maple. Two examples

    更新日期:2020-10-16
  • A variational autoencoder for music generation controlled by tonal tension
    arXiv.cs.SC Pub Date : 2020-10-13
    Rui Guo; Ivor Simpson; Thor Magnusson; Chris Kiefer; Dorien Herremans

    Many of the music generation systems based on neural networks are fully autonomous and do not offer control over the generation process. In this research, we present a controllable music generation system in terms of tonal tension. We incorporate two tonal tension measures based on the Spiral Array Tension theory into a variational autoencoder model. This allows us to control the direction of the tonal

    更新日期:2020-10-14
  • On lattice point counting in $Δ$-modular polyhedra
    arXiv.cs.SC Pub Date : 2020-10-12
    D. V. Gribanov; N. Yu. Zolotykh

    Let a polyhedron $P$ be defined by one of the following ways: (i) $P = \{x \in R^n \colon A x \leq b\}$, where $A \in Z^{(n+k) \times n}$, $b \in Z^{(n+k)}$ and $rank\, A = n$; (ii) $P = \{x \in R_+^n \colon A x = b\}$, where $A \in Z^{k \times n}$, $b \in Z^{k}$ and $rank\, A = k$. And let all rank order minors of $A$ be bounded by $\Delta$ in absolute values. We show that the short rational generating

    更新日期:2020-10-13
  • An Algorithm for the Factorization of Split Quaternion Polynomials
    arXiv.cs.SC Pub Date : 2020-10-12
    Daniel F. Scharler; Hans-Peter Schröcker

    We present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we present also geometric interpretations in terms of rulings on the quadric of non-invertible split quaternions. However, suitable real polynomial multiples of split

    更新日期:2020-10-13
  • Exploiting Knowledge Graphs for Facilitating Product/Service Discovery
    arXiv.cs.SC Pub Date : 2020-10-11
    Sarika Jain

    Most of the existing techniques to product discovery rely on syntactic approaches, thus ignoring valuable and specific semantic information of the underlying standards during the process. The product data comes from different heterogeneous sources and formats giving rise to the problem of interoperability. Above all, due to the continuously increasing influx of data, the manual labeling is getting

    更新日期:2020-10-13
  • A Categorical Programming Language
    arXiv.cs.SC Pub Date : 2020-10-11
    Tatsuya Hagino

    A theory of data types based on category theory is presented. We organize data types under a new categorical notion of F,G-dialgebras which is an extension of the notion of adjunctions as well as that of T-algebras. T-algebras are also used in domain theory, but while domain theory needs some primitive data types, like products, to start with, we do not need any. Products, coproducts and exponentiations

    更新日期:2020-10-13
  • Deterministic computation of the characteristic polynomial in the time of matrix multiplication
    arXiv.cs.SC Pub Date : 2020-10-09
    Vincent Neiger; Clément Pernet

    This paper describes an algorithm which computes the characteristic polynomial of a matrix over a field within the same asymptotic complexity, up to constant factors, as the multiplication of two square matrices. Previously, to our knowledge, this was only achieved by resorting to genericity assumptions or randomization techniques, while the best known complexity bound with a general deterministic

    更新日期:2020-10-12
  • Exact Symbolic Inference in Probabilistic Programs via Sum-Product Representations
    arXiv.cs.SC Pub Date : 2020-10-07
    Feras A. Saad; Martin C. Rinard; Vikash K. Mansinghka

    We present the Sum-Product Probabilistic Language (SPPL), a new system that automatically delivers exact solutions to a broad range of probabilistic inference queries. SPPL symbolically represents the full distribution on execution traces specified by a probabilistic program using a generalization of sum-product networks. SPPL handles continuous and discrete distributions, many-to-one numerical transformations

    更新日期:2020-10-08
  • A Simple and Efficient Tensor Calculus for Machine Learning
    arXiv.cs.SC Pub Date : 2020-10-07
    Sören Laue; Matthias Mitterreiter; Joachim Giesen

    Computing derivatives of tensor expressions, also known as tensor calculus, is a fundamental task in machine learning. A key concern is the efficiency of evaluating the expressions and their derivatives that hinges on the representation of these expressions. Recently, an algorithm for computing higher order derivatives of tensor expressions like Jacobians or Hessians has been introduced that is a few

    更新日期:2020-10-08
  • Factorization of Dual Quaternion Polynomials Without Study's Condition
    arXiv.cs.SC Pub Date : 2020-10-05
    Johannes Siegele; Martin Pfurner; Hans-Peter Schröcker

    In this paper we investigate factorizations of polynomials over the ring of dual quaternions into linear factors. While earlier results assume that the norm polynomial is real ("motion polynomials"), we only require the absence of real polynomial factors in the primal part and factorizability of the norm polynomial over the dual numbers into monic quadratic factors. This obviously necessary condition

    更新日期:2020-10-06
  • On Differentially Algebraic Generating Series for Walks in the Quarter Plane
    arXiv.cs.SC Pub Date : 2020-10-02
    Charlotte Hardouin; Michael F Singer

    We refine necessary and sufficient conditions for the generating series of a weighted model of a quarter plane walk to be differentially algebraic. In addition, we give algorithms based on the theory of Mordell-Weil lattices, that, for each weighted model, yield polynomial conditions on the weights determining this property of the associated generating series.

    更新日期:2020-10-05
  • Iteration in ACL2
    arXiv.cs.SC Pub Date : 2020-09-29
    Matt KaufmannUniv. of Texas at Austin; J Strother MooreUniv. of Texas at Austin

    Iterative algorithms are traditionally expressed in ACL2 using recursion. On the other hand, Common Lisp provides a construct, loop, which -- like most programming languages -- provides direct support for iteration. We describe an ACL2 analogue loop$ of loop that supports efficient ACL2 programming and reasoning with iteration.

    更新日期:2020-09-30
  • Formal Verification of Arithmetic RTL: Translating Verilog to C++ to ACL2
    arXiv.cs.SC Pub Date : 2020-09-29
    David M. RussinoffArm

    We present a methodology for formal verification of arithmetic RTL designs that combines sequential logic equivalence checking with interactive theorem proving. An intermediate model of a Verilog module is hand-coded in Restricted Algorithmic C (RAC), a primitive subset of C augmented by the integer and fixed-point register class templates of Algorithmic C. The model is designed to be as abstract and

    更新日期:2020-09-30
  • KG-BART: Knowledge Graph-Augmented BART for Generative Commonsense Reasoning
    arXiv.cs.SC Pub Date : 2020-09-26
    Ye Liu; Yao Wan; Lifang He; Hao Peng; Philip S. Yu

    Generative commonsense reasoning which aims to empower machines to generate sentences with the capacity of reasoning over a set of concepts is a critical bottleneck for text generation. Even the state-of-the-art pre-trained language generation models struggle at this task and often produce implausible and anomalous sentences. One reason is that they rarely consider incorporating the knowledge graph

    更新日期:2020-09-29
  • A Low-Level Index for Distributed Logic Programming
    arXiv.cs.SC Pub Date : 2020-09-22
    Thomas ProkoschInstitute for Informatics, Ludwig-Maximilian University of Munich, Germany

    A distributed logic programming language with support for meta-programming and stream processing offers a variety of interesting research problems, such as: How can a versatile and stable data structure for the indexing of a large number of expressions be implemented with simple low-level data structures? Can low-level programming help to reduce the number of occur checks in Robinson's unification

    更新日期:2020-09-23
  • Enhancing Linear Algebraic Computation of Logic Programs Using Sparse Representation
    arXiv.cs.SC Pub Date : 2020-09-22
    Tuan Nguyen QuocNational Institute of Informatics; Katsumi InoueNational Institute of Informatics; Chiaki SakamaWakayama University

    Algebraic characterization of logic programs has received increasing attention in recent years. Researchers attempt to exploit connections between linear algebraic computation and symbolic computation in order to perform logical inference in large scale knowledge bases. This paper proposes further improvement by using sparse matrices to embed logic programs in vector spaces. We show its great power

    更新日期:2020-09-23
  • Deriving Theorems in Implicational Linear Logic, Declaratively
    arXiv.cs.SC Pub Date : 2020-09-22
    Paul TarauUniversity of North Texas; Valeria de PaivaTopos Institute

    The problem we want to solve is how to generate all theorems of a given size in the implicational fragment of propositional intuitionistic linear logic. We start by filtering for linearity the proof terms associated by our Prolog-based theorem prover for Implicational Intuitionistic Logic. This works, but using for each formula a PSPACE-complete algorithm limits it to very small formulas. We take a

    更新日期:2020-09-23
  • Modeling Hierarchical System with Operads
    arXiv.cs.SC Pub Date : 2020-09-15
    Spencer BreinerNIST; Blake PollardNIST; Eswaran SubrahmanianCMU; Olivier Marie-RosePrometheus Computing

    This paper applies operads and functorial semantics to address the problem of failure diagnosis in complex systems. We start with a concrete example, developing a hierarchical interaction model for the Length Scale Interferometer, a high-precision measurement system operated by the US National Institute of Standards and Technology. The model is expressed in terms of combinatorial/diagrammatic structures

    更新日期:2020-09-22
  • Characterizing Positively Invariant Sets: Inductive and Topological Methods
    arXiv.cs.SC Pub Date : 2020-09-08
    Khalil Ghorbal; Andrew Sogokon

    Set positive invariance is an important concept in the theory of dynamical systems and one which also has practical applications in areas of computer science, such as formal verification, as well as in control theory. Great progress has been made in understanding positively invariant sets in continuous dynamical systems and powerful computational tools have been developed for reasoning about them;

    更新日期:2020-09-22
  • Guessing Gr{ö}bner Bases of Structured Ideals of Relations of Sequences
    arXiv.cs.SC Pub Date : 2020-09-11
    Jérémy BerthomieuPolSys; Mohab Safey El DinPolSys

    Assuming sufficiently many terms of a n-dimensional table defined over a field are given, we aim at guessing the linear recurrence relations with either constant or polynomial coefficients they satisfy. In many applications, the table terms come along with a structure: for instance, they may be zero outside of a cone, they may be built from a Gr{\"o}bner basis of an ideal invariant under the action

    更新日期:2020-09-14
  • PolyAdd: Polynomial Formal Verification of Adder Circuits
    arXiv.cs.SC Pub Date : 2020-09-07
    Rolf Drechsler

    Only by formal verification approaches functional correctness can be ensured. While for many circuits fast verification is possible, in other cases the approaches fail. In general no efficient algorithms can be given, since the underlying verification problem is NP-complete. In this paper we prove that for different types of adder circuits polynomial verification can be ensured based on BDDs. While

    更新日期:2020-09-08
  • On FGLM Algorithms with Tropical Gröbner bases
    arXiv.cs.SC Pub Date : 2020-09-04
    Yuki IshiharaXLIM; Tristan VacconXLIM; Kazuhiro Yokoyama

    Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Gr{\"o}bner bases has proved to provide settings for computations over polynomial rings over a p-adic field that are more stable than that of classical Gr{\"o}bner bases. In this article

    更新日期:2020-09-08
  • Strong Consistency and Thomas Decomposition of Finite Difference Approximations to Systems of Partial Differential Equations
    arXiv.cs.SC Pub Date : 2020-09-03
    Vladimir P. Gerdt; Daniel Robertz; Yuri A. Blinkov

    For a wide class of polynomially nonlinear systems of partial differential equations we suggest an algorithmic approach that combines differential and difference algebra to analyze s(trong)-consistency of finite difference approximations. Our approach is applicable to regular solution grids. For the grids of this type we give a new definition of s-consistency for finite difference approximations which

    更新日期:2020-09-05
  • On a non-archimedean broyden method
    arXiv.cs.SC Pub Date : 2020-09-03
    Xavier DahanXLIM; Tristan VacconXLIM

    Newton's method is an ubiquitous tool to solve equations, both in the archimedean and non-archimedean settings -- for which it does not really differ. Broyden was the instigator of what is called "quasi-Newton methods". These methods use an iteration step where one does not need to compute a complete Jacobian matrix nor its inverse. We provide an adaptation of Broyden's method in a general non-archimedean

    更新日期:2020-09-05
  • Computing critical points for invariant algebraic systems
    arXiv.cs.SC Pub Date : 2020-09-02
    Jean-Charles FaugèrePolSys; George LabahnSCG; Mohab Safey El DinPolSys; Éric SchostSCG; Thi Xuan VuPolSys, SCG

    Let $\mathbf{K}$ be a field and $\phi$, $\mathbf{f} = (f_1, \ldots, f_s)$ in $\mathbf{K}[x_1, \dots, x_n]$ be multivariate polynomials (with $s < n$) invariant under the action of $\mathcal{S}_n$, the group of permutations of $\{1, \dots, n\}$. We consider the problem of computing the points at which $\mathbf{f}$ vanish and the Jacobian matrix associated to $\mathbf{f}, \phi$ is rank deficient provided

    更新日期:2020-09-03
  • Homotopy techniques for solving sparse column support determinantal polynomial systems
    arXiv.cs.SC Pub Date : 2020-09-02
    George LabahnSCG; Mohab Safey El DinPolSys; Éric SchostSCG; Thi Xuan VuPolSys, SCG

    Let $\mathbf{K}$ be a field of characteristic zero with $\overline{\mathbf{K}}$ its algebraic closure. Given a sequence of polynomials $\mathbf{g} = (g_1, \ldots, g_s) \in \mathbf{K}[x_1, \ldots , x_n]^s$ and a polynomial matrix $\mathbf{F} = [f_{i,j}] \in \mathbf{K}[x_1, \ldots, x_n]^{p \times q}$, with $p \leq q$, we are interested in determining the isolated points of $V_p(\mathbf{F},\mathbf{g})$

    更新日期:2020-09-03
  • Robots, computer algebra and eight connected components
    arXiv.cs.SC Pub Date : 2020-08-31
    Jose CapcoJKU; Mohab Safey El DinPolSys; Josef SchichoRISC

    Answering connectivity queries in semi-algebraic sets is a long-standing and challenging computational issue with applications in robotics, in particular for the analysis of kinematic singularities. One task there is to compute the number of connected components of the complementary of the singularities of the kinematic map. Another task is to design a continuous path joining two given points lying

    更新日期:2020-09-01
  • Towards a noncommutative Picard-Vessiot theory
    arXiv.cs.SC Pub Date : 2020-08-25
    G. DuchampLIPN; Viincel Hoang Ngoc Minh; Vu Nguyen Dinh

    A Chen generating series, along a path and with respect to $m$ differential forms, is a noncommutative series on $m$ letters and with coefficients which are holomorphic functions over a simply connected manifold in other words a series with variable (holomorphic) coefficients. Such a series satisfies a first order noncommutative differential equation which is considered, by some authors, as the universal

    更新日期:2020-08-26
  • Exact $p$-adic computation in Magma
    arXiv.cs.SC Pub Date : 2020-08-24
    Christopher Doris

    We describe a new arithmetic system for the Magma computer algebra system for working with $p$-adic numbers exactly, in the sense that numbers are represented lazily to infinite $p$-adic precision. This is the first highly featured such implementation. This has the benefits of increasing user-friendliness and speeding up some computations, as well as forcibly producing provable results. We give theoretical

    更新日期:2020-08-26
  • Computing singular elements modulo squares
    arXiv.cs.SC Pub Date : 2020-08-24
    Przemysław Koprowski

    The group of singular elements was first introduced by Helmut Hasse and later it has been studied by numerous authors including such well known mathematicians as: Cassels, Furtw\"{a}ngler, Hecke, Knebusch, Takagi and of course Hasse himself; to name just a few. The aim of the present paper is to present algorithms that explicitly construct groups of singular and $S$-singular elements (modulo squares)

    更新日期:2020-08-25
  • Computing the Real Isolated Points of an Algebraic Hypersurface
    arXiv.cs.SC Pub Date : 2020-08-24
    Huu Phuoc Le; Mohab Safey El Din; Timo de Wolff

    Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role for studying rigidity properties of mechanism in material designs. In this paper, we design an algorithm which solves this problem. It is based on the computations

    更新日期:2020-08-25
  • A Simple and Fast Algorithm for Computing the $N$-th Term of a Linearly Recurrent Sequence
    arXiv.cs.SC Pub Date : 2020-08-20
    Alin Bostan; Ryuhei Mori

    We present a simple and fast algorithm for computing the $N$-th term of a given linearly recurrent sequence. Our new algorithm uses $O(\mathsf{M}(d) \log N)$ arithmetic operations, where $d$ is the order of the recurrence, and $\mathsf{M}(d)$ denotes the number of arithmetic operations for computing the product of two polynomials of degree $d$. The state-of-the-art algorithm, due to Charles Fiduccia

    更新日期:2020-08-21
  • Competition Report: CHC-COMP-20
    arXiv.cs.SC Pub Date : 2020-08-07
    Philipp RümmerUppsala University, Sweden

    CHC-COMP-20 is the third competition of solvers for Constrained Horn Clauses. In this year, 9 solvers participated at the competition, and were evaluated in four separate tracks on problems in linear integer arithmetic, linear real arithmetic, and arrays. The competition was run in the first week of May 2020 using the StarExec computing cluster. This report gives an overview of the competition design

    更新日期:2020-08-10
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