Computer Algebra in Scientific Computing : 26th International Workshop, CASC 2024, Rennes, France, September 2–6, 2024, Proceedings / / edited by François Boulier, Chenqi Mou, Timur M. Sadykov, Evgenii V. Vorozhtsov

• Autore: Boulier François
• Pubblicazione: Cham : , : Springer Nature Switzerland : , : Imprint : Springer, , 2024
• Edizione: 1st ed. 2024.
• Descrizione fisica: 1 online resource (407 pages)
• Disciplina: 005.13
• Soggetto topico: Algorithms; Artificial intelligence; Computer science - Mathematics; Computer networks; Design and Analysis of Algorithms; Artificial Intelligence; Mathematics of Computing; Computer Communication Networks
• Altri autori: MouChenqi SadykovTimur M VorozhtsovEvgenii V
• Nota di contenuto: Intro -- Preface -- Organization -- Polynomial System Solving through Gröbner Bases in Scientific Computing (Invited Talk) -- Contents -- Recent Developments in Real Quantifier Elimination and Cylindrical Algebraic Decomposition (Extended Abstract of Invited Talk)*-12pt -- 1 Real Quantifier Elimination -- 2 Cylindrical Algebraic Decomposition -- 3 The Doubly Exponential Wall -- 4 CAD and Satisfiability Checking -- 4.1 SAT and SMT -- 4.2 CAD as Theory Solver -- 4.3 New CAD-Based Algorithms -- 4.4 SC2 -- 5 CAD and Machine Learning -- 5.1 CAD Variable Ordering Choice -- 5.2 Future Progress from Explainable AI? -- References -- Advances in Elimination Theory for Algebraic Differential and Difference Equations -- 1 Introduction -- 2 Basic Notions in Differential Algebra -- 3 Differential Chow Forms and Differential Chow Varieties -- 3.1 Differential Chow Forms -- 3.2 Differential Chow Varieties -- 3.3 Algorithms and Further Work -- 4 Differential Resultants, Sparse Differential Resultants and Sparse Difference Resultants -- 4.1 Differential Resultants -- 4.2 Sparse Differential Resultants -- 4.3 Sparse Difference Resultants -- 5 Effective Nullstellensatz and Elimination for Algebraic Differential-Difference Equations -- References -- A Modular Algorithm to Compute the Resultant of Multivariate Polynomials over Algebraic Number Fields Presented with Multiple Extensions -- 1 Introduction -- 1.1 Computing over Q(1,…,n) -- 1.2 Organization of the Paper -- 2 Preliminaries -- 2.1 Mapping Q(1,…,n) to a Single Extension Q() -- 2.2 Resultants -- 2.3 Computing Resultants of Univariate Polynomials -- 3 The Modular Resultant Algorithm -- 3.1 Algorithm PRES -- 3.2 Algorithm MRES -- 4 Implementation and Benchmarks -- 5 Complexity -- 6 Failure Probability -- 6.1 Lc-Bad Primes and Evaluation Points -- 6.2 Det-Bad Primes -- 7 Conclusion -- References. ; The Liouville Generator for Producing Integrable Expressions -- 1 Introduction -- 2 Existing Data Generation Methods -- 2.1 Deep Learning for Symbolic Mathematics -- 2.2 Generating Elementary Integrable Expressions -- 2.3 The Substitution Rule -- 3 Background Material for the New Method -- 3.1 Liouville's Theorem -- 3.2 Parallel Risch Algorithm -- 4 The LIOUVILLE Data Generation Method -- 4.1 The Main Idea -- 4.2 Design Choices -- 5 Discussion of the New Data Generation Method -- 5.1 Example -- 5.2 Benefits over FWD and BWD -- 5.3 Benefits over RISCH -- 5.4 The Effect of Normalisation -- 6 Conclusion and Future Work -- References -- Symbolic-Numeric Solving Boundary Value Problems: Collective Models of Atomic Nuclei -- 1 Introduction -- 2 Formulation of BVP and GTM and FEM Schemes -- 3 BVP for Five-Dimensional Quadrupole Hamiltonian -- 4 5D Harmonic Oscillator Model in Affine Coordinates -- 5 Benchmark Calculations of 154Gd in the RMF Model -- 6 Conclusions -- References -- On Formal Power Series Solutions of Regular Differential Chains -- 1 Introduction -- 2 Mathematical Background -- 3 The Prolongation Bounds -- 3.1 An Approximation Lemma -- 3.2 Examples -- 4 Basic Notions on Regular Chains -- 5 The Case of Series Depending on Parameters -- 6 Computation of the Nonnegative Integer Roots -- 7 Implementation -- References -- A Dataset for Suggesting Variable Orderings for Cylindrical Algebraic Decompositions -- 1 Introduction -- 2 Generation of the Dataset -- 2.1 The CAD Implementation to Use -- 2.2 The Initial Setting for Generating the Examples -- 2.3 Statistical Information of the Generated Datasets -- 2.4 Organization of the Datasets -- 3 Performance of the Heuristic Methods on the Datasets -- 4 Training and Testing on the Datasets -- 4.1 Feature Engineering -- 4.2 Training of the Models -- 4.3 Testing of the Models. ; 5 Conclusion and Future Work -- References -- Stability Analysis of a Differential Model for Quasi-Periodic Plasma Perturbations -- 1 Introduction -- 2 KCC Theory and Jacobi Stability of Dynamical Systems -- 3 Linear Stability Analysis of Model (1) -- 4 Jacobi Stability Analysis of Model (1) -- 4.1 The Nonlinear and Berwald Connections, and the KCC Invariants -- 4.2 Jacobi Stability of the Fixed Points -- 5 Dynamics of Deviation Vector in Model (1) -- 5.1 Dynamics of the Deviation Vector Near E0 -- 5.2 Dynamics of the Deviation Vector Near E+ and E- -- 5.3 The Curvature of the Deviation Vector -- 6 Conclusions -- References -- Counting the Integer Points of Parametric Polytopes: A Maple Implementation -- 1 Introduction -- 2 Generating Functions of Rational Cones and Polytopes -- 3 Quasi-polynomials -- 4 Generic Case Discussion -- 5 Counting the Integer Points -- 5.1 Counting the Integer Points of a Non-parametric Polytope -- 5.2 Counting the Integer Points of a Parametric Polytope -- 6 Experimentation -- References -- Algebraic Representations for Faster Predictions in Convolutional Neural Networks -- 1 Introduction -- 1.1 Machine Learning and Algebraic Geometry -- 1.2 Contributions and Structure -- 2 CNN Setup -- 3 Skip Connections in Linear Networks -- 3.1 Transformation Matrices for Resampling and Padding -- 3.2 Skip Connections -- 4 Removing Skip Connections with a Homotopy -- 5 Computational Experiments -- 5.1 Equipment, Software, and Data -- 5.2 Pre-computing CNNs -- 5.3 Practical Experiment for Removing Skip Connections -- 6 Future Directions -- 7 Conclusion -- References -- Computing a Basis of the Set of Isogenies Between Two Supersingular Elliptic Curves -- 1 Introduction -- 2 Mathematical Background -- 2.1 Elliptic Curves over Finite Fields -- 2.2 Properties of Supersingular Elliptic Curves. ; 3 Generators of Hom(E, E') for Supersingular Elliptic Curves E, E' -- 4 Computing a Basis of Hom(E, E') for Supersingular Elliptic Curves E, E' -- 4.1 Details and Implementation of Each Step -- 4.2 Numerical Examples -- 5 Conclusion -- References -- On the Radical of a Polynomial Ideal with Parameters -- 1 Introduction -- 2 Comprehensive Gröbner Systems -- 2.1 Preliminaries -- 2.2 Comprehensive Gröbner Systems -- 3 Tools for Parametric Ideals -- 3.1 Dimensions of a Parametric Ideal -- 3.2 Squarefree Part of a Univariate Polynomial with Parameters -- 3.3 Intersection of Parametric Ideals -- 3.4 Least Common Multiples of Parametric Polynomials -- 3.5 Saturation for a Parametric Ideal -- 4 Parametric Radical System -- 5 Zero Dimensional Case -- 6 Key Result -- 7 Non-zero Dimensional Case -- References -- Contribution to Integral Elimination -- 1 Introduction -- 2 Background -- 2.1 Integral Algebra -- 2.2 Ordering the Integral Monomials -- 2.3 Basic Rewriting Rules -- 3 Extended Rewriting Rules and Reduction -- 3.1 Reduced-Product Rule -- 3.2 Reduced-Power Rule -- 3.3 Reduction -- 4 Critical Pairs -- 5 Exponentials -- 5.1 Algorithms find_A_A0_G_Fand update_exp -- 6 Integral Elimination Prototype -- 7 Examples -- 7.1 Intra-Host Model of Malaria (taken from ch13malaria) -- 7.2 SIWR Model Cholera (taken from ch13dong2022differential) -- References -- Sparse Tensors and Subdivision Methods for Finding the Zero Set of Polynomial Equations -- 1 Introduction -- 1.1 Amortized Evaluation on a Grid of Boxes -- 1.2 Amortized Evaluation on a Sparse Subset of a Grid -- 1.3 Notations -- 1.4 Main Result -- 2 Evaluating Polynomials with Compressed Sparse Fibers -- 2.1 Sparse Tensor Data Structure -- 2.2 Evaluation Algorithm -- 3 Applications -- 3.1 The Fast Fourier Transform Revisited -- 3.2 Subdivision Algorithm -- 4 Experiments -- 4.1 Random Polynomials. ; 4.2 Polynomials Coming from Applications -- References -- Merging Adjacent Cells During Single Cell Construction -- 1 Introduction -- 2 Preliminaries -- 2.1 Cylindrical Algebraic Decomposition, McCallum's Projection, and Levelwise Single Cell Construction -- 3 Detecting ``Irrelevant'' Roots -- 4 Local Delineability and Modified Projection -- 5 Weak Orderings and (Half-)closed Intervals -- 6 Experimental Results -- 7 Conclusion -- References -- Computing Clipped Products -- 1 Introduction -- 2 Preliminaries -- 3 Straightforward Methods -- 4 Clipped Polynomial Products -- 5 Clipped Integer Multiplication -- 6 Combining Methods -- 7 Further Thoughts and Conclusions -- References -- Fast Integral Bases Computation -- 1 Introduction -- 2 OM Algorithm and Local Integral Basis -- 2.1 Local Integral Basis and OM Factorisation -- 2.2 Precision of the OM Algorithm -- 3 Triangular p-Integral Bases -- 3.1 Reduced Triangular p-integral Bases -- 3.2 The MaxMin Algorithm -- 3.3 Global Integral Bases -- 4 Bases of Fractional Ideals -- 4.1 Fractional Ideals -- 4.2 p-Bases of Fractional Ideals -- 4.3 Improvements via S-basis -- 4.4 Global Triangular Bases of Fractional Ideals -- 5 Complexity and Proofs of the Main Results -- 6 An Illustrative Example -- References -- On Rational Recursion for Holonomic Sequences -- 1 Introduction -- 2 Problem Statement in Difference Algebra -- 3 Lower-Degree Holonomic Difference Polynomials -- 4 Proof and Algorithms -- 5 Concluding Remarks -- References -- GPU Accelerated Newton for Taylor Series Solutions of Polynomial Homotopies in Multiple Double Precision -- 1 Introduction -- 1.1 Problem Statement -- 1.2 Multiprecision Arithmetic -- 1.3 Numerical Condition of Taylor Series -- 2 Linearized Series and Newton's Method -- 3 Columns of Monomials -- 4 Staggered Computations -- 5 Accelerating Newton's Method. ; 5.1 Arithmetic Intensity of Convolutions.
• Sommario/riassunto: This book constitutes the refereed proceedings of the 26th International Workshop on Computer Algebra in Scientific Computing, CASC 2024, which took place in Rennes, France, during September 2 - September 6, 2024. The 19 full papers included in this book were carefully reviewed and selected from 23 submissions. The annual International Workshop CASC 2024 aims to bring together researchers in theoretical computer algebra (CA), engineers, scholars, as well as other allied professionals applying CA tools for solving problems in industry and in various branches of scientific computing to explore and discuss advancements, challenges, and innovations related to CA. .
• Titolo autorizzato: Computer Algebra in Scientific Computing
• ISBN: 9783031690709; 9783031690693
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910882888103321
• Lo trovi qui: Univ. Federico II
• Serie: Lecture Notes in Computer Science, . 1611-3349 ; ; 14938