Computer Algebra 2006 : Latest Advances in Symbolic Algorithms - Proceedings of the Waterloo Workshop.

By: Kotsireas, IliasContributor(s): Zima, EugenePublisher: Singapore : World Scientific Publishing Co Pte Ltd, 2007Copyright date: ©2007Description: 1 online resource (220 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9789812778857Subject(s): Algebra -- Data processing -- Congresses.;Computer algorithms -- CongressesGenre/Form: Electronic books. Additional physical formats: Print version:: Computer Algebra 2006 : Latest Advances in Symbolic Algorithms - Proceedings of the Waterloo WorkshopDDC classification: 512.02 LOC classification: QA155.7.E4 -- C66 2006ebOnline resources: Click to View
Contents:
Intro -- CONTENTS -- Preface -- Hypergeometric Summation Revisited S. A. Abramov, M. Petkovsek -- 1. Introduction -- 2. Validity conditions of the discrete Newton-Leibniz formula -- 2.1. A criterion -- 2.2. Summation of proper hypergeometric sequences -- 2.3. When the interval I contains no leading integer singularity of L -- 3. The spaces VI(L) and WI(R(k), L) -- 3.1. The structure of WI(R(k), L) -- 3.2. When a rational solution of Gosper's equation is not unique -- 3.3. If Gosper's equation has a rational solution R(k) then WI(R, L) = 0 -- References -- Five Applications of Wilf-Zeilberger Theory to Enumeration and Probability M. Apagodu, D. Zeilberger -- Explicit Formulas vs. Algorithms -- The Holonomic Ansatz -- Why this Paper? -- The Maple packages AppsWZ and AppsWZmulti -- Asymptotics -- First Application: Rolling a Die -- Second Application: How many ways to have r people chip in to pay a bill of n cents -- Third Application: Hidden Markov Models -- Fourth Application: Lattice Paths Counting -- References -- Factoring Systems of Linear Functional Equations Using Eigenrings M. A. Barkatou -- 1. Introduction and notations -- 2. Preliminaries -- 3. Eigenrings and reduction of pseudo-linear equations -- Maximal Decompsition -- 4. Spaces of homomorphisms and factorization -- Appendix A. K[X -- φ, δ].modules and matrix pseudo-linear equations -- Appendix A.1. Pseudo-linear operators -- Appendix A.2. Similarity, reducibility, decomposability and complete reducibility -- Appendix A.3. The ring of endomorphisms of a pseudo-linear operator -- References -- Modular Computation for Matrices of Ore Polynomials H. Cheng, G. Labahn -- 1. Introduction -- 2. Preliminaries -- 2.1. Notation -- 2.2. Definitions -- 2.3. The FFreduce Elimination Algorithm -- 3. Linear Algebra Formulation -- 4. Reduction to Zp[t][Z] -- 4.1. Lucky Homomorphisms.
4.2. Termination -- 5. Reduction to Zp -- 5.1. Applying Evaluation Homomorphisms and Computation in Zp -- 5.2. Lucky Homomorphisms and Termination -- 6. Complexity Analysis -- 7. Implementation Considerations and Experimental Results -- 8. Concluding Remarks -- References -- Beta-Expansions of Pisot and Salem Numbers K. G. Hare -- 1. Introduction and History -- 2. Univoque Pisot Numbers -- 3. Algorithms and Implementation Issues -- 4. Conclusions and Open Questions -- References -- Logarithmic Functional and the Weil Reciprocity Law A. Khovanskii -- 1. Introduction -- 1.1. The Weil reciprocity law -- 1.2. Topological explanation of the reciprocity law over the field C -- 1.3. Multi-dimensional reciprocity laws -- 1.4. The logarithmic functional -- 1.5. Organization of material -- 2. Formulation of the Weil reciprocity law -- 3. LB-functional of the pair of complex valued functions of the segment on real variable -- 4. LB-functional of the pair of complex valued functions and one-dimensional cycle on real manifold -- 5. Topological proof of the Weil reciprocity law -- 6. Generalized LB-functional -- 7. Logarithmic function and logarithmic functional -- 7.1. Zero-dimensional logarithmic functional and logarithm -- 7.2. Properties of one-dimensional logarithmic functional -- 7.3. Prove of properties of logarithmic functional -- 8. Logarithmic functional and generalized LB-functional -- References -- On Solutions of Linear Functional Systems and Factorization of Laurent-Ore Modules M. Wu, Z. Li -- 1. Introduction -- 2. Preliminaries -- 3. Fully integrable systems -- 4. -finite systems -- 4.1. Generic solutions of linear algebraic equations -- 4.2. Laurent-Ore algebras -- 4.3. Modules of formal solutions -- 4.4. Fundamental matrices and Picard-Vessiot extensions -- 5. Computing linear dimension -- 6. Factorization of Laurent-Ore modules.
6.1. Constructions with modules over Laurent-Ore algebras -- 6.2. A module-theoretic approach to factorization -- 6.3. Eigenrings and decomposition of Laurent-Ore modules -- References -- The Vector Rational Function Reconstruction Problem Z. Olesh, A. Storjohann -- 1. Introduction -- 2. Reduced bases -- 3. Minimal approximant bases -- 3.1. An algorithm for simultaneous Pad´e approximation -- 4. Vector rational function reconstruction -- 5. Application to linear solving -- 6. Conclusion -- References -- Fast Algorithm for Computing Multipole Matrix Elements with Legendre Polynomials V. Yu. Papshev, S. Yu. Slavyanov -- Introduction -- 1. Hilbert transform for solutions of the equation for the product -- 2. Equation for the product of Legendre polynomials and its Hilbert transform -- 3. Calculation of particular cases of Clebsh-Gordon coefficients -- References -- Recurrence Relations for the Coe.cients in Hypergeometric Series Expansions L. Rebillard, H. Zakraj.sek -- 1. Introduction -- 2. Notations and basic properties -- 3. Associated families -- 4. Depression of the order -- 5. Normal form of a di.erential operator -- 5.1. σ has a double root ξ1 = ξ2 -- 5.2. σ is of degree one -- 6. Conclusion -- Acknowledgments -- References -- On Factorization and Solution of Multidimensional Linear Partial Di.erential Equations S. P. Tsarev -- 1. Introduction -- 2. Laplace and generalized Laplace transformations -- 3. Dini transformation: an example -- 4. Dini transformation: a general result for dim = 3, ord = 2 -- 5. Open problems -- Acknowledgments -- References -- Two Families of Algorithms for Symbolic Polynomials S. M. Watt -- 1. Introduction -- 2. Symbolic Polynomials -- 3. Multiplicative Properties -- 4. Extension Algorithms -- 5. Projection Methods -- 6. Finding Corresponding Terms -- 7. Generalizations -- 8. Conclusions -- References -- Author Index.
Summary: Key Features:Presents the most recent developments of fast and robust algorithms in many areas of symbolic computationsLargely devoted to accumulative reviews describing current state of the art in different areas of computer algebraContributors are world leading experts in computer algebra - S Abramov, D Zeilberger, B Salvy, M van Hoeji, A Hovanski, G Gonnet, G Egorychev, and others.
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Intro -- CONTENTS -- Preface -- Hypergeometric Summation Revisited S. A. Abramov, M. Petkovsek -- 1. Introduction -- 2. Validity conditions of the discrete Newton-Leibniz formula -- 2.1. A criterion -- 2.2. Summation of proper hypergeometric sequences -- 2.3. When the interval I contains no leading integer singularity of L -- 3. The spaces VI(L) and WI(R(k), L) -- 3.1. The structure of WI(R(k), L) -- 3.2. When a rational solution of Gosper's equation is not unique -- 3.3. If Gosper's equation has a rational solution R(k) then WI(R, L) = 0 -- References -- Five Applications of Wilf-Zeilberger Theory to Enumeration and Probability M. Apagodu, D. Zeilberger -- Explicit Formulas vs. Algorithms -- The Holonomic Ansatz -- Why this Paper? -- The Maple packages AppsWZ and AppsWZmulti -- Asymptotics -- First Application: Rolling a Die -- Second Application: How many ways to have r people chip in to pay a bill of n cents -- Third Application: Hidden Markov Models -- Fourth Application: Lattice Paths Counting -- References -- Factoring Systems of Linear Functional Equations Using Eigenrings M. A. Barkatou -- 1. Introduction and notations -- 2. Preliminaries -- 3. Eigenrings and reduction of pseudo-linear equations -- Maximal Decompsition -- 4. Spaces of homomorphisms and factorization -- Appendix A. K[X -- φ, δ].modules and matrix pseudo-linear equations -- Appendix A.1. Pseudo-linear operators -- Appendix A.2. Similarity, reducibility, decomposability and complete reducibility -- Appendix A.3. The ring of endomorphisms of a pseudo-linear operator -- References -- Modular Computation for Matrices of Ore Polynomials H. Cheng, G. Labahn -- 1. Introduction -- 2. Preliminaries -- 2.1. Notation -- 2.2. Definitions -- 2.3. The FFreduce Elimination Algorithm -- 3. Linear Algebra Formulation -- 4. Reduction to Zp[t][Z] -- 4.1. Lucky Homomorphisms.

4.2. Termination -- 5. Reduction to Zp -- 5.1. Applying Evaluation Homomorphisms and Computation in Zp -- 5.2. Lucky Homomorphisms and Termination -- 6. Complexity Analysis -- 7. Implementation Considerations and Experimental Results -- 8. Concluding Remarks -- References -- Beta-Expansions of Pisot and Salem Numbers K. G. Hare -- 1. Introduction and History -- 2. Univoque Pisot Numbers -- 3. Algorithms and Implementation Issues -- 4. Conclusions and Open Questions -- References -- Logarithmic Functional and the Weil Reciprocity Law A. Khovanskii -- 1. Introduction -- 1.1. The Weil reciprocity law -- 1.2. Topological explanation of the reciprocity law over the field C -- 1.3. Multi-dimensional reciprocity laws -- 1.4. The logarithmic functional -- 1.5. Organization of material -- 2. Formulation of the Weil reciprocity law -- 3. LB-functional of the pair of complex valued functions of the segment on real variable -- 4. LB-functional of the pair of complex valued functions and one-dimensional cycle on real manifold -- 5. Topological proof of the Weil reciprocity law -- 6. Generalized LB-functional -- 7. Logarithmic function and logarithmic functional -- 7.1. Zero-dimensional logarithmic functional and logarithm -- 7.2. Properties of one-dimensional logarithmic functional -- 7.3. Prove of properties of logarithmic functional -- 8. Logarithmic functional and generalized LB-functional -- References -- On Solutions of Linear Functional Systems and Factorization of Laurent-Ore Modules M. Wu, Z. Li -- 1. Introduction -- 2. Preliminaries -- 3. Fully integrable systems -- 4. -finite systems -- 4.1. Generic solutions of linear algebraic equations -- 4.2. Laurent-Ore algebras -- 4.3. Modules of formal solutions -- 4.4. Fundamental matrices and Picard-Vessiot extensions -- 5. Computing linear dimension -- 6. Factorization of Laurent-Ore modules.

6.1. Constructions with modules over Laurent-Ore algebras -- 6.2. A module-theoretic approach to factorization -- 6.3. Eigenrings and decomposition of Laurent-Ore modules -- References -- The Vector Rational Function Reconstruction Problem Z. Olesh, A. Storjohann -- 1. Introduction -- 2. Reduced bases -- 3. Minimal approximant bases -- 3.1. An algorithm for simultaneous Pad´e approximation -- 4. Vector rational function reconstruction -- 5. Application to linear solving -- 6. Conclusion -- References -- Fast Algorithm for Computing Multipole Matrix Elements with Legendre Polynomials V. Yu. Papshev, S. Yu. Slavyanov -- Introduction -- 1. Hilbert transform for solutions of the equation for the product -- 2. Equation for the product of Legendre polynomials and its Hilbert transform -- 3. Calculation of particular cases of Clebsh-Gordon coefficients -- References -- Recurrence Relations for the Coe.cients in Hypergeometric Series Expansions L. Rebillard, H. Zakraj.sek -- 1. Introduction -- 2. Notations and basic properties -- 3. Associated families -- 4. Depression of the order -- 5. Normal form of a di.erential operator -- 5.1. σ has a double root ξ1 = ξ2 -- 5.2. σ is of degree one -- 6. Conclusion -- Acknowledgments -- References -- On Factorization and Solution of Multidimensional Linear Partial Di.erential Equations S. P. Tsarev -- 1. Introduction -- 2. Laplace and generalized Laplace transformations -- 3. Dini transformation: an example -- 4. Dini transformation: a general result for dim = 3, ord = 2 -- 5. Open problems -- Acknowledgments -- References -- Two Families of Algorithms for Symbolic Polynomials S. M. Watt -- 1. Introduction -- 2. Symbolic Polynomials -- 3. Multiplicative Properties -- 4. Extension Algorithms -- 5. Projection Methods -- 6. Finding Corresponding Terms -- 7. Generalizations -- 8. Conclusions -- References -- Author Index.

Key Features:Presents the most recent developments of fast and robust algorithms in many areas of symbolic computationsLargely devoted to accumulative reviews describing current state of the art in different areas of computer algebraContributors are world leading experts in computer algebra - S Abramov, D Zeilberger, B Salvy, M van Hoeji, A Hovanski, G Gonnet, G Egorychev, and others.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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