Computer Algebra 2006: Latest Advances in Symbolic Algorithms

CONTENTS......Page 8
Preface......Page 6
1. Introduction......Page 10
2.1. A criterion......Page 13
2.2. Summation of proper hypergeometric sequences......Page 14
2.3. When the interval I contains no leading integer singularity of L......Page 16
3.1. The structure of WI(R(k), L)......Page 17
3.2. When a rational solution of Gosper’s equation is not unique......Page 18
References......Page 19
Explicit Formulas vs. Algorithms......Page 21
Why this Paper?......Page 22
First Application: Rolling a Die......Page 23
Second Application: How many ways to have r people chip in to pay a bill of n cents......Page 24
Third Application: Hidden Markov Models......Page 25
Fourth Application: Lattice Paths Counting......Page 27
References......Page 30
1. Introduction and notations......Page 31
2. Preliminaries......Page 34
3. Eigenrings and reduction of pseudo-linear equations......Page 36
Maximal Decompsition......Page 41
4. Spaces of homomorphisms and factorization......Page 44
Appendix A.1. Pseudo-linear operators......Page 47
Appendix A.2. Similarity, reducibility, decomposability and complete reducibility......Page 48
References......Page 50
1. Introduction......Page 52
2.2. Definitions......Page 55
2.3. The FFreduce Elimination Algorithm......Page 57
3. Linear Algebra Formulation......Page 59
4. Reduction to Zp[t][Z]......Page 62
4.1. Lucky Homomorphisms......Page 63
4.2. Termination......Page 64
5. Reduction to Zp......Page 66
5.1. Applying Evaluation Homomorphisms and Computation in Zp......Page 67
5.2. Lucky Homomorphisms and Termination......Page 69
6. Complexity Analysis......Page 70
7. Implementation Considerations and Experimental Results......Page 72
8. Concluding Remarks......Page 74
References......Page 75
1. Introduction and History......Page 76
2. Univoque Pisot Numbers......Page 85
3. Algorithms and Implementation Issues......Page 87
4. Conclusions and Open Questions......Page 90
References......Page 91
1.1. The Weil reciprocity law......Page 94
1.2. Topological explanation of the reciprocity law over the field C......Page 95
1.3. Multi-dimensional reciprocity laws......Page 96
1.4. The logarithmic functional......Page 97
2. Formulation of the Weil reciprocity law......Page 98
3. LB-functional of the pair of complex valued functions of the segment on real variable......Page 99
4. LB-functional of the pair of complex valued functions and one-dimensional cycle on real manifold......Page 103
5. Topological proof of the Weil reciprocity law......Page 105
6. Generalized LB-functional......Page 107
7. Logarithmic function and logarithmic functional......Page 108
7.1. Zero-dimensional logarithmic functional and logarithm......Page 109
7.2. Properties of one-dimensional logarithmic functional......Page 110
7.3. Prove of properties of logarithmic functional......Page 111
8. Logarithmic functional and generalized LB-functional......Page 116
References......Page 117
1. Introduction......Page 118
2. Preliminaries......Page 120
3. Fully integrable systems......Page 121
4.1. Generic solutions of linear algebraic equations......Page 124
4.2. Laurent–Ore algebras......Page 125
4.3. Modules of formal solutions......Page 126
4.4. Fundamental matrices and Picard–Vessiot extensions......Page 128
5. Computing linear dimension......Page 130
6. Factorization of Laurent–Ore modules......Page 133
6.1. Constructions with modules over Laurent–Ore algebras......Page 134
6.2. A module-theoretic approach to factorization......Page 135
6.3. Eigenrings and decomposition of Laurent–Ore modules......Page 140
References......Page 144
1. Introduction......Page 146
2. Reduced bases......Page 149
3. Minimal approximant bases......Page 150
3.1. An algorithm for simultaneous Pad´e approximation......Page 152
4. Vector rational function reconstruction......Page 153
5. Application to linear solving......Page 156
References......Page 157
Introduction......Page 159
1. Hilbert transform for solutions of the equation for the product......Page 160
2. Equation for the product of Legendre polynomials and its Hilbert transform......Page 162
3. Calculation of particular cases of Clebsh-Gordon coefficients......Page 164
References......Page 166
1. Introduction......Page 167
2. Notations and basic properties......Page 170
3. Associated families......Page 175
4. Depression of the order......Page 181
5. Normal form of a di.erential operator......Page 184
5.1. σ has a double root ξ1 = ξ2......Page 186
6. Conclusion......Page 187
References......Page 188
1. Introduction......Page 190
2. Laplace and generalized Laplace transformations......Page 191
3. Dini transformation: an example......Page 195
4. Dini transformation: a general result for dim = 3, ord = 2......Page 196
5. Open problems......Page 198
References......Page 200
1. Introduction......Page 202
2. Symbolic Polynomials......Page 204
3. Multiplicative Properties......Page 205
4. Extension Algorithms......Page 206
5. Projection Methods......Page 209
6. Finding Corresponding Terms......Page 213
7. Generalizations......Page 217
8. Conclusions......Page 218
References......Page 219
Author Index......Page 220