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An algorithm to determine regular singular Mahler systems arXiv.cs.SC Pub Date : 2021-02-22 Colin FaverjonICJ; Marina PouletICJ
This paper is devoted to the study of the analytic properties of Mahler systems at 0. We give an effective characterisation of Mahler systems that are regular singular at 0, that is, systems which are equivalent to constant ones. Similar characterisations already exist for differential and (q-)difference systems but they do not apply in the Mahler case. This work fill in the gap by giving an algorithm
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Root Radii and Subdivision for Polynomial Root-Finding arXiv.cs.SC Pub Date : 2021-02-22 Rémi Imbach; Victor Y. Pan
The recent subdivision algorithms for univariate polynomial Complex Root Clustering (CRC) and Real Root Isolation (RRI) approximate all roots in a fixed Region of Interest (RoI) and, like the algorithm of Pan (1995, 2002), achieve near optimal bit complexity for the so called benchmark problem. On the other hand, user's current choice for the approximation of all complex roots of a polynomial is the
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Metatheory.jl: Fast and Elegant Algebraic Computation in Julia with Extensible Equality Saturation arXiv.cs.SC Pub Date : 2021-02-15 Alessandro Cheli
We introduce Metatheory.jl: a lightweight and performant general purpose symbolics and metaprogramming framework meant to simplify the act of writing complex Julia metaprograms and to significantly enhance Julia with a native term rewriting system, based on state-of-the-art equality saturation techniques, and a dynamic first class Abstract Syntax Tree (AST) pattern matching system that is dynamically
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Lazy Hermite Reduction and Creative Telescoping for Algebraic Functions arXiv.cs.SC Pub Date : 2021-02-12 Shaoshi Chen; Lixin Du; Manuel Kauers
We present criteria on the existence of telescopers for trivariate rational functions in four mixed cases, in which discrete and continuous variables appear simultaneously. We reduce the existence problem in the trivariate case to the exactness testing problem, the separation problem and the existence problem in the bivariate case. The existence criteria we present help us determine the termination
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On FGLM Algorithms with Tate Algebras arXiv.cs.SC Pub Date : 2021-02-10 Xavier CarusoIMB, CNRS; Tristan VacconXLIM; Thibaut VerronJKU
Tate introduced in [Ta71] the notion of Tate algebras to serve, in the context of analytic geometry over the-adics, as a counterpart of polynomial algebras in classical algebraic geometry. In [CVV19, CVV20] the formalism of Gr{\"o}bner bases over Tate algebras has been introduced and advanced signature-based algorithms have been proposed. In the present article, we extend the FGLM algorithm of [FGLM93]
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On exact division and divisibility testing for sparse polynomials arXiv.cs.SC Pub Date : 2021-02-09 Pascal Giorgi; Bruno Grenet; Armelle Perret du Cray
Assessing that a sparse polynomial G divides another sparse polynomial F is not yet known to admit a polynomial time algorithm. While computing the quotient Q = F quo G can be done in polynomial time with respect to the sparsities of F, G and Q, it is not yet sufficient to get a polynomial time divisibility test in general. Indeed, the sparsity of the quotient Q can be exponentially larger than the
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Polynomial Linear System Solving with Random Errors: new bounds and early termination technique arXiv.cs.SC Pub Date : 2021-02-08 Guerrini Eleonora; Lebreton Romain; Zappatore Ilaria
This paper deals with the polynomial linear system solving with errors (PLSwE) problem. Specifically, we focus on the evaluation-interpolation technique for solving polynomial linear systems and we assume that errors can occur in the evaluation step. In this framework, the number of evaluations needed to recover the solution of the linear system is crucial since it affects the number of computations
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Fast real and complex root-finding methods for well-conditioned polynomials arXiv.cs.SC Pub Date : 2021-02-08 Guillaume MorozGAMBLE
Given a polynomial $p$ of degree $d$ and a bound $\kappa$ on a condition number of $p$, we present the first root-finding algorithms that return all its real and complex roots with a number of bit operations quasi-linear in $d \log^2(\kappa)$. More precisely, several condition numbers can be defined depending on the norm chosen on the coefficients of the polynomial. Let $p(x) = \sum\_{k=0}^d a\_k x^k
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Symbolic computation of hypergeometric type and non-holonomic power series arXiv.cs.SC Pub Date : 2021-02-08 Bertrand Teguia Tabuguia; Wolfram Koepf
A term $a_n$ is $m$-fold hypergeometric, for a given positive integer $m$, if the ratio $a_{n+m}/a_n$ is a rational function over a field $K$ of characteristic zero. We establish the structure of holonomic recurrence equation, i.e. linear and homogeneous recurrence equations having polynomial coefficients, that have $m$-fold hypergeometric term solutions over $K$, for any positive integer $m$. Consequently
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Separability Problems in Creative Telescoping arXiv.cs.SC Pub Date : 2021-02-07 Shaoshi Chen; Ruyong Feng; Pingchuan Ma; Michael F. Singer
For given multivariate functions specified by algebraic, differential or difference equations, the separability problem is to decide whether they satisfy linear differential or difference equations in one variable. In this paper, we will explain how separability problems arise naturally in creative telescoping and present some criteria for testing the separability for several classes of special functions
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Algorithms for Linearly Recurrent Sequences of Truncated Polynomials arXiv.cs.SC Pub Date : 2021-02-06 Seung Gyu Hyun; Vincent Neiger; Éric Schost
Linear recurrent sequences are those whose elements are defined as linear combinations of preceding elements, and finding relations of recurrences is a fundamental problem in computer algebra. In this paper, we focus on sequences whose elements are vectors over the ring $\mathbb{A} = \mathbb{K}[x]/(x^d)$ of truncated polynomials. We present three methods for finding the ideal of canceling polynomials:
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On Two Signature Variants Of Buchberger's Algorithm Over Principal Ideal Domains arXiv.cs.SC Pub Date : 2021-02-05 Maria Francis; Thibaut Verron
Signature-based algorithms have brought large improvements in the performances of Gr\"obner bases algorithms for polynomial systems over fields. Furthermore, they yield additional data which can be used, for example, to compute the module of syzygies of an ideal or to compute coefficients in terms of the input generators. In this paper, we examine two variants of Buchberger's algorithm to compute Gr\"obner
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Solving linear difference equations with coefficients in rings with idempotent representations arXiv.cs.SC Pub Date : 2021-02-05 Jakob Ablinger; Carsten Schneider
We introduce a general reduction strategy that enables one to search for solutions of parameterized linear difference equations in difference rings. Here we assume that the ring itself can be decomposed by a direct sum of integral domains (using idempotent elements) that enjoys certain technical features and that the coefficients of the difference equation are not degenerated. Using this mechanism
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Combinatorial Differential Algebra of $x^p$ arXiv.cs.SC Pub Date : 2021-02-05 Rida Ait El Manssour; Anna-Laura Sattelberger
We link $n$-jets of the affine monomial scheme defined by $x^p$ to the stable set polytope of some perfect graph. We prove that, as $p$ varies, the dimension of the coordinate ring of the scheme of $n$-jets as a $\mathbb{C}$-vector space is a polynomial of degree $n+1,$ namely the Erhart polynomial of the stable set polytope of that graph. One main ingredient for our proof is a result of Zobnin who
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Proceedings 11th International Workshop on Computing with Terms and Graphs arXiv.cs.SC Pub Date : 2021-02-02 Patrick BahrIT University of Copenhagen
Graphs, and graph transformation systems, are used in many areas within Computer Science: to represent data structures and algorithms, to define computation models, as a general modelling tool to study complex systems, etc. Research in term and graph rewriting ranges from theoretical questions to practical implementation issues. Relevant research areas include: the modelling of first- and higher-order
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Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation arXiv.cs.SC Pub Date : 2021-02-02 Carsten Schneider
A general overview of the existing difference ring theory for symbolic summation is given. Special emphasis is put on the user interface: the translation and back translation of the corresponding representations within the term algebra and the formal difference ring setting. In particular, canonical (unique) representations and their refinements in the introduced term algebra are explored by utilizing
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Computing Limits of Quotients of Multivariate Real Analytic Functions arXiv.cs.SC Pub Date : 2021-02-02 Adam Strzebonski
We present an algorithm for computing limits of quotients of real analytic functions. The algorithm is based on computation of a bound on the Lojasiewicz exponent and requires the denominator to have an isolated zero at the limit point.
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Choosing the Variable Ordering for Cylindrical Algebraic Decomposition via Exploiting Chordal Structure arXiv.cs.SC Pub Date : 2021-02-01 Haokun Li; Bican Xia; Huiying Zhang; Tao Zheng
Cylindrical algebraic decomposition (CAD) plays an important role in the field of real algebraic geometry and many other areas. As is well-known, the choice of variable ordering while computing CAD has a great effect on the time and memory use of the computation as well as the number of sample points computed. In this paper, we indicate that typical CAD algorithms, if executed with respect to a special
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Certified evaluations of Hölder continuous functions at roots of polynomials arXiv.cs.SC Pub Date : 2021-01-30 Parker B. Edwards; Jonathan D. Hauenstein; Clifford D. Smyth
Various methods can obtain certified estimates for roots of polynomials. Many applications in science and engineering additionally utilize the value of functions evaluated at roots. For example, critical values are obtained by evaluating an objective function at critical points. For analytic evaluation functions, Newton's method naturally applies to yield certified estimates. These estimates no longer
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A Companion Curve Tracing Method for Rank-deficient Polynomial Systems arXiv.cs.SC Pub Date : 2021-01-29 Wenyuan Wu; Changbo Chen
We propose a method for tracing implicit real algebraic curves defined by polynomials with rank-deficient Jacobians. For a given curve $f^{-1}(0)$, it first utilizes a regularization technique to compute at least one witness point per connected component of the curve. We improve this step by establishing a sufficient condition for testing the emptiness of $f^{-1}(0)$. We also analyze the convergence
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Extensions of the AZ-algorithm and the Package MultiIntegrate arXiv.cs.SC Pub Date : 2021-01-27 Jakob Ablinger
We extend the (continuous) multivariate Almkvist-Zeilberger algorithm in order to apply it for instance to special Feynman integrals emerging in renormalizable Quantum field Theories. We will consider multidimensional integrals over hyperexponential integrands and try to find closed form representations in terms of nested sums and products or iterated integrals. In addition, if we fail to compute a
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Accelerated Polynomial Evaluation and Differentiation at Power Series in Multiple Double Precision arXiv.cs.SC Pub Date : 2021-01-22 Jan Verschelde
The problem is to evaluate a polynomial in several variables and its gradient at a power series truncated to some finite degree with multiple double precision arithmetic. To compensate for the cost overhead of multiple double precision and power series arithmetic, data parallel algorithms for general purpose graphics processing units are presented. The reverse mode of algorithmic differentiation is
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There are EXACTLY 1493804444499093354916284290188948031229880469556 Ways to Derange a Standard Deck of Cards (ignoring suits) [and many other such useful facts] arXiv.cs.SC Pub Date : 2021-01-25 Shalosh B. Ekhad; Christoph Koutschan; Doron Zeilberger
In this memorial tribute to Joe Gillis, who taught us that Special Functions count, we show how the seminal Even-Gillis integral formula for the number of derangements of a multiset, in terms of Laguerre polynomials, can be used to efficiently compute not only the number of the title, but much harder ones, when it is interfaced with Wilf-Zeilberger algorithmic proof theory.
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MultivariateApart: Generalized Partial Fractions arXiv.cs.SC Pub Date : 2021-01-20 Matthias Heller; Andreas von Manteuffel
We present a package to perform partial fraction decompositions of multivariate rational functions. The algorithm allows to systematically avoid spurious denominator factors and is capable of producing unique results also when being applied to terms of a sum separately. The package is designed to work in Mathematica, but also provides interfaces to the Form and Singular computer algebra systems.
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Telescopers for differential forms with one parameter arXiv.cs.SC Pub Date : 2021-01-17 Shaoshi Chen; Ruyong Feng; Ziming Li; Michael F. Singer; Stephen Watt
Telescopers for a function are linear differential (resp. difference) operators annihilated by the definite integral (resp. definite sum) of this function. They play a key role in Wilf-Zeilberger theory and algorithms for computing them have been extensively studied in the past thirty years. In this paper, we introduce the notion of telescopers for differential forms with $D$-finite function coefficients
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Covering rational surfaces with rational parametrization images arXiv.cs.SC Pub Date : 2021-01-18 Jorge Caravantes; J. Rafael Sendra; David Sevilla; Carlos Villarino
Let $S$ be a rational projective surface given by means of a projective rational parametrization whose base locus satisfies a mild assumption. In this paper we present an algorithm that provides three rational maps $f,g,h:\mathbb{A}^2 --\to S\subset \mathbb{P}^n$ such that the union of the three images covers $S$. As a consequence, we present a second algorithm that generates two rational maps $f,
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Weakly nonlocal Poisson brackets: tools, examples, computations arXiv.cs.SC Pub Date : 2021-01-16 Matteo Casati; Paolo Lorenzoni; Daniele Valeri; Raffaele Vitolo
We implement an algorithm for the computation of Schouten bracket of weakly nonlocal Hamiltonian operators in three different computer algebra systems: Maple, Reduce and Mathematica. This class of Hamiltonian operators encompass almost all the examples coming from the theory of (1+1)-integrable evolutionary PDEs
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Multi-Source Anomaly Detection in Distributed IT Systems arXiv.cs.SC Pub Date : 2021-01-13 Jasmin Bogatinovski; Sasho Nedelkoski
The multi-source data generated by distributed systems, provide a holistic description of the system. Harnessing the joint distribution of the different modalities by a learning model can be beneficial for critical applications for maintenance of the distributed systems. One such important task is the task of anomaly detection where we are interested in detecting the deviation of the current behaviour
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Proceedings 6th International Workshop on Symbolic-Numeric methods for Reasoning about CPS and IoT arXiv.cs.SC Pub Date : 2021-01-01 Thao DangVerimag/CNRS, France; Stefan RatschanInstitute of Computer Science, Czech Academy of Sciences
The proceedings of the 6th International Workshop on Symbolic-Numeric Methods for Reasoning about CPS and IoT (SNR 2020) contains papers underlying talks presented at the workshop. SNR focuses on the combination of symbolic and numeric methods for reasoning about Cyber-Physical Systems and the Internet of Things to facilitate model identification, specification, verification, and control synthesis
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Object-Level Reasoning with Logics Encoded in HOL Light arXiv.cs.SC Pub Date : 2021-01-11 Petros PapapanagiotouUniversity of Edinburgh; Jacques FleuriotUniversity of Edinburgh
We present a generic framework that facilitates object level reasoning with logics that are encoded within the Higher Order Logic theorem proving environment of HOL Light. This involves proving statements in any logic using intuitive forward and backward chaining in a sequent calculus style. It is made possible by automated machinery that take care of the necessary structural reasoning and term matching
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A New Type of Bases for Zero-dimensional Ideals arXiv.cs.SC Pub Date : 2021-01-10 Sheng-Ming Ma
We formulate a substantial improvement on Buchberger's algorithm for Gr\"obner bases of zero-dimensional ideals. The improvement scales down the phenomenon of intermediate expression swell as well as the complexity of Buchberger's algorithm to a significant degree. The idea is to compute a new type of bases over principal ideal rings instead of over fields like Gr\"obner bases. The generalizations
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Theorem Proving and Algebra arXiv.cs.SC Pub Date : 2021-01-07 Joseph A. Goguen
This book can be seen either as a text on theorem proving that uses techniques from general algebra, or else as a text on general algebra illustrated and made concrete by practical exercises in theorem proving. The book considers several different logical systems, including first-order logic, Horn clause logic, equational logic, and first-order logic with equality. Similarly, several different proof
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Polynomial modular product verification and its implications arXiv.cs.SC Pub Date : 2021-01-06 Pascal Giorgi; Bruno Grenet; Armelle Perret du Cray
Polynomial multiplication is known to have quasi-linear complexity in both the dense and the sparse cases. Yet no truly linear algorithm has been given in any case for the problem, and it is not clear whether it is even possible. This leaves room for a better algorithm for the simpler problem of verifying a polynomial product. While finding deterministic methods seems out of reach, there exist probabilistic
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PTOPO: Computing the Geometry and the Topology of Parametric Curves arXiv.cs.SC Pub Date : 2021-01-06 Christina Katsamaki; Fabrice Rouillier; Elias Tsigaridas
We consider the problem of computing the topology and describing the geometry of a parametric curve in $\mathbb{R}^n$. We present an algorithm, PTOPO, that constructs an abstract graph that is isotopic to the curve in the embedding space. Our method exploits the benefits of the parametric representation and does not resort to implicitization. Most importantly, we perform all computations in the parameter
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Methods for computing $b$-functions associated with $μ$-constant deformations -- Case of inner modality 2 -- arXiv.cs.SC Pub Date : 2021-01-05 Katsusuke Nabeshima; Shinichi Tajima
New methods for computing parametric local $b$-functions are introduced for $\mu$-constant deformations of semi-weighted homogeneous singularities. The keys of the methods are comprehensive Gr\"obner systems in Poincar\'e-Birkhoff-Witt algebra and holonomic ${\mathcal D}$-modules. It is shown that the use of semi-weighted homogeneity reduces the computational complexity of $b$-functions associated
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Some fast algorithms multiplying a matrix by its adjoint arXiv.cs.SC Pub Date : 2021-01-04 Jean-Guillaume DumasCASC; Clément PernetCASC; Alexandre SedoglavicCRIStAL
We present a non-commutative algorithm for the multiplication of a 2 x 2 block-matrix by its adjoint, defined by a matrix ring anti-homomorphism. This algorithm uses 5 block products (3 recursive calls and 2 general products)over C or in positive characteristic. The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its adjoint to general matrix product, improving
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Border Basis Computation with Gradient-Weighted Norm arXiv.cs.SC Pub Date : 2021-01-02 Hiroshi Kera
Normalization of polynomials plays an essential role in the approximate basis computation of vanishing ideals. In computer algebra, coefficient normalization, which normalizes a polynomial by its coefficient norm, is the most common method. In this study, we propose gradient-weighted normalization for the approximate border basis computation of vanishing ideals, inspired by the recent results in machine
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Monomial-agnostic computation of vanishing ideal arXiv.cs.SC Pub Date : 2021-01-01 Hiroshi Kera; Yoshihiko Hasegawa
The approximate basis computation of vanishing ideals has recently been extensively studied and applied both in computer algebra and data-driven applications such as machine learning. However, the symbolic computation and the dependency on the monomial ordering remain an essential gap between the two fields. In the present paper, we propose the first efficient monomial-agnostic approximate basis computation
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Notes on Computational Graph and Jacobian Accumulation arXiv.cs.SC Pub Date : 2020-12-30 Yichong Zhou
The optimal calculation order of a computational graph can be represented by a set of algebraic expressions. Computational graph and algebraic expression both have close relations and significant differences, this paper looks into these relations and differences, making plain their interconvertibility. By revealing different types of multiplication relations in algebraic expressions and their elimination
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Using tropical differential equations arXiv.cs.SC Pub Date : 2020-12-28 Ethan Coterill; Cristhian Garay; Johana Luviano
In this note, we develop the theory of tropical differential algebraic geometry from scratch and show how it may be used to extract combinatorial information about the set of power series solutions to a given system of differential equations.
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Augmenting Policy Learning with Routines Discovered from a Demonstration arXiv.cs.SC Pub Date : 2020-12-23 Zelin Zhao; Chuang Gan; Jiajun Wu; Xiaoxiao Guo; Joshua Tenenbaum
Humans can abstract prior knowledge from very little data and use it to boost skill learning. In this paper, we propose routine-augmented policy learning (RAPL), which discovers routines composed of primitive actions from a single demonstration and uses discovered routines to augment policy learning. To discover routines from the demonstration, we first abstract routine candidates by identifying grammar
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Method for estimating hidden structures determined by unidentifiable state-space models and time-series data based on the Groebner basis arXiv.cs.SC Pub Date : 2020-12-22 Mizuka Komatsu; Takaharu Yaguchi
In this study, we propose a method for extracting the hidden algebraic structures of model parameters that are uniquely determined by observed time-series data and unidentifiable state-space models, explicitly and exhaustively. State-space models are often constructed based on the domain, for example, physical or biological. Such models include parameters that are assigned specific meanings in relation
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A variant of van Hoeij's algorithm to compute hypergeometric term solutions of holonomic recurrence equations arXiv.cs.SC Pub Date : 2020-12-21 Bertrand Teguia Tabuguia
Linear homogeneous recurrence equations with polynomial coefficients are said to be holonomic. Such equations have been introduced in the last century for proving and discovering combinatorial and hypergeometric identities. Given a field K of characteristic zero, a term a(n) is called hypergeometric with respect to K, if the ratio a(n+1)/a(n) is a rational function over K. The solutions space of holonomic
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SymFields: An Open Source Symbolic Fields Analysis Tool for General Curvilinear Coordinates in Python arXiv.cs.SC Pub Date : 2020-12-19 Nan Chu
An open source symbolic tool for vector fields analysis 'SymFields' is developed in Python. The SymFields module is constructed upon Python symbolic module sympy, which could only conduct scaler field analysis. With SymFields module, you can conduct vector analysis for general curvilinear coordinates regardless whether it is orthogonal or not. In SymFields, the differential operators based on metric
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Repairing dynamic models: a method to obtain identifiable and observable reparameterizations with mechanistic insights arXiv.cs.SC Pub Date : 2020-12-17 Gemma Massonis; Julio R. Banga; Alejandro F. Villaverde
Mechanistic dynamic models allow for a quantitative and systematic interpretation of data and the generation of testable hypotheses. However, these models are often over-parameterized, leading to non-identifiability and non-observability, i.e. the impossibility of inferring their parameters and state variables. The lack of structural identifiability and observability (SIO) compromises a model's ability
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Investigating ADR mechanisms with knowledge graph mining and explainable AI arXiv.cs.SC Pub Date : 2020-12-16 Emmanuel Bresso; Pierre Monnin; Cédric Bousquet; François-Elie Calvier; Ndeye-Coumba Ndiaye; Nadine Petitpain; Malika Smaïl-Tabbone; Adrien Coulet
Adverse Drug Reactions (ADRs) are characterized within randomized clinical trials and postmarketing pharmacovigilance, but their molecular mechanism remains unknown in most cases. Aside from clinical trials, many elements of knowledge about drug ingredients are available in open-access knowledge graphs. In addition, drug classifications that label drugs as either causative or not for several ADRs,
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Fast Computation of the $N$-th Term of a $q$-Holonomic Sequence and Applications arXiv.cs.SC Pub Date : 2020-12-15 Alin Bostan; Sergey Yurkevich
In 1977, Strassen invented a famous baby-step/giant-step algorithm that computes the factorial $N!$ in arithmetic complexity quasi-linear in $\sqrt{N}$. In 1988, the Chudnovsky brothers generalized Strassen's algorithm to the computation of the $N$-th term of any holonomic sequence in essentially the same arithmetic complexity. We design $q$-analogues of these algorithms. We first extend Strassen's
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SONC Optimization and Exact Nonnegativity Certificates via Second-Order Cone Programming arXiv.cs.SC Pub Date : 2020-12-14 Victor Magron; Jie Wang
The second-order cone (SOC) is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a representation using SOCs, given that they have a strong expressive ability. In this paper, we prove constructively that the cone of sums of nonnegative circuits
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Parallel Software to Offset the Cost of Higher Precision arXiv.cs.SC Pub Date : 2020-12-11 Jan Verschelde
Hardware double precision is often insufficient to solve large scientific problems accurately. Computing in higher precision defined by software causes significant computational overhead. The application of parallel algorithms compensates for this overhead. Newton's method to develop power series expansions of algebraic space curves is the use case for this application.
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Hexapods with a small linear span arXiv.cs.SC Pub Date : 2020-12-09 Hans-Christian Graf von Bothmer; Matteo Gallet; Josef Schicho
The understanding of mobile hexapods, i.e., parallel manipulators with six legs, is one of the driving questions in theoretical kinematics. We aim at contributing to this understanding by employing techniques from algebraic geometry. The set of configurations of a mobile hexapod with one degree of freedom has the structure of a projective curve, which hence has a degree and an embedding dimension.
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Counting Positive Roots in Polynomial-Time for Systems Supported on Circuits arXiv.cs.SC Pub Date : 2020-12-09 J. Maurice Rojas
Suppose $A=\{a_1,\ldots,a_{n+2}\}\subset\mathbb{Z}^n$ has cardinality $n+2$, with all the coordinates of the $a_j$ having absolute value at most $d$, and the $a_j$ do not all lie in the same affine hyperplane. Suppose $F=(f_1,\ldots,f_n)$ is an $n\times n$ polynomial system with generic integer coefficients at most $H$ in absolute value, and $A$ the union of the sets of exponent vectors of the $f_i$
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A SAT-based Resolution of Lam's Problem arXiv.cs.SC Pub Date : 2020-12-08 Curtis Bright; Kevin K. H. Cheung; Brett Stevens; Ilias Kotsireas; Vijay Ganesh
In 1989, computer searches by Lam, Thiel, and Swiercz experimentally resolved Lam's problem from projective geometry$\unicode{x2014}$the long-standing problem of determining if a projective plane of order ten exists. Both the original search and an independent verification in 2011 discovered no such projective plane. However, these searches were each performed using highly specialized custom-written
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A Generic and Executable Formalization of Signature-Based Gröbner Basis Algorithms arXiv.cs.SC Pub Date : 2020-12-03 Alexander Maletzky
We present a generic and executable formalization of signature-based algorithms (such as Faug\`ere's $F_5$) for computing Gr\"obner bases, as well as their mathematical background, in the Isabelle/HOL proof assistant. Said algorithms are currently the best known algorithms for computing Gr\"obner bases in terms of computational efficiency. The formal development attempts to be as generic as possible
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An Algebraic Graph Transformation Approach for RDF and SPARQL arXiv.cs.SC Pub Date : 2020-12-03 Dominique DuvalCNRS and Univ. Grenoble Alpes, France; Rachid EchahedCNRS and Univ. Grenoble Alpes, France; Frédéric ProstCNRS and Univ. Grenoble Alpes, France
We consider the recommendations of the World Wide Web Consortium (W3C) about RDF framework and its associated query language SPARQL. We propose a new formal framework based on category theory which provides clear and concise formal definitions of the main basic features of RDF and SPARQL. We define RDF graphs as well as SPARQL basic graph patterns as objects of some nested categories. This allows one
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Proceedings of the Eleventh International Workshop on Graph Computation Models arXiv.cs.SC Pub Date : 2020-12-02 Berthold HoffmannUniversität Bremen; Mark MinasUniversität der Bundeswehr München
Graphs are common mathematical structures that are visual and intuitive. They constitute a natural and seamless way for system modelling in science, engineering and beyond, including computer science, biology, business process modelling, etc. Graph computation models constitute a class of very high-level models where graphs are first-class citizens. The aim of the International GCM Workshop series
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Solving parametric systems of polynomial equations over the reals through Hermite matrices arXiv.cs.SC Pub Date : 2020-11-28 Huu Phuoc Le; Mohab Safey El Din
We design a new algorithm for solving parametric systems having finitely many complex solutions for generic values of the parameters. More precisely, let $f = (f_1, \ldots, f_m)\subset \mathbb{Q}[y][x]$ with $y = (y_1, \ldots, y_t)$ and $x = (x_1, \ldots, x_n)$, $V\subset \mathbb{C}^{t+n}$ be the algebraic set defined by $f$ and $\pi$ be the projection $(y, x) \to y$. Under the assumptions that $f$
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Why Charles Can Pen-test: an Evolutionary Approach to Vulnerability Testing arXiv.cs.SC Pub Date : 2020-11-26 Gabriele Costa; Andrea Valenza
Discovering vulnerabilities in applications of real-world complexity is a daunting task: a vulnerability may affect a single line of code, and yet it compromises the security of the entire application. Even worse, vulnerabilities may manifest only in exceptional circumstances that do not occur in the normal operation of the application. It is widely recognized that state-of-the-art penetration testing
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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
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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
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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
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