CONTENTS -- Preface -- Multidegree for Bifiltered D-modules and Hypergeometric Systems R. Arcadias -- Introduction -- 1. Bifiltered free resolution of D-modules -- 2. Multidegree for bifiltered D-modules -- 3. Examples from the theory of hypergeometric systems -- 3.1. V -filtration along the origin -- 3.2. V -filtration along coordinate hyperplanes -- 3.3. Dependency of the multidegree on the parameters -- 3.4. Positivity -- 4. Proof of Theorem 2.1 -- Acknowledgements -- References -- Desingularization Algorithms: A Comparison from the Practical Point of View R. Blanco and A. Fruhbis-Kruger -- 1. Introduction -- 2. Algorithms re.ning Hironaka's approach in the general case -- 3. Combinatorial algorithms for the binomial case -- 4. Algorithmic resolution in low dimensions -- 4.1. Resolution of surfaces by Jung's approach -- 4.2. Beyond the geometric case: Lipman's construction for two dimensional schemes -- 5. Comparisons and timings -- Acknowledgments -- References -- Computing Localizations Iteratively F. J. Castro-Jimenez and A. Leykin -- Introduction -- 1. Preliminaries -- 1.1. Weyl algebra -- 1.2. Grobner bases -- 1.3. Holonomic D-modules -- 2. Iterative algorithm -- 2.1. Iterative approach -- 2.2. Stopping criterion -- 2.3. Annihilator order of a planar curve -- 3. Discussion and open problems -- 3.1. Isolated hypersurface singularities -- 3.2. Weyl closure -- 4. Acknowledgements -- References -- KNOPPIX/Math: A Live System for Mathematics T. Hamada and KNOPPIX/Math Committers -- 1. Introduction -- 2. History -- 3. The objectives of KNOPPIX/Math -- 4. How to boot KNOPPIX/Math -- References -- Running Markov Chain without Markov Basis H. Hara, S. Aoki and A. Takemura -- 1. Introduction -- 2. Markov basis and lattice basis -- 3. Sampling contingency tables with a lattice basis -- 3.1. Generating moves by using a lattice basis.

3.2. A lattice basis for higher Lawrence configuration -- 4. Numerical experiments -- 4.1. No-three-factor interaction model -- 4.2. Discrete logistic regression model -- References -- Degree Bounds for a Minimal Markov Basis for the Threestate Toric Homogeneous Markov Chain Model D. Haws, A. Martın del Campo and R. Yoshida -- 1. Introduction -- 2. Notation -- 2.1. Model (a) -- 2.2. Model (b) -- 2.3. Model (c) -- 2.4. Model (d) -- 2.5. Sufficient statistics, ideals, and Markov basis -- 2.6. State graph -- 3. Smith Normal Form -- 4. Semigroup -- 4.1. Model (a) -- 4.2. Model (b) -- 4.3. Model (c) -- 4.4. Model (d) -- 5. Polytope Structure -- 6. Computational Results -- 7. Conclusions and Open Problems -- Appendix A. Supporting Hyperplanes -- References -- First Steps toward the Geometry of Cophylogeny P. Huggins, M. Owen and R. Yoshida -- 1. Introduction -- 2. Spaces of cophylogenetic trees -- 3. Cophylogenetic reconstruction -- 3.1. Retraction onto spaces of cophylogenetic trees -- 3.2. Balanced minimum coevolution -- 4. Cophylogenetic invariants -- 5. Open problems -- 6. Proof of Theorem 2.2 -- Acknowledgements -- References -- Cones of Elementary Imsets and Supermodular Functions: A Review and Some New Results T. Kashimura, T. Sei, A. Takemura and K. Tanaka -- 1. Introduction -- 2. Basic facts on supermodular functions and imsets -- 3. Multiinformation and derivation of conditional independence using imsets -- 4. Results on extreme rays of the supermodular cone -- 5. Structure of faces of semi-elementary imsets -- 6. Small relations among imsets -- 7. Some computational results on Markov basis for imsets -- Appendix A -- References -- Non-vanishingness of Betti Numbers of Edge Ideals K. Kimura -- 1. Introduction -- 2. Strongly disjoint set of bouquets -- 3. Non-vanishingness of the graded Betti numbers -- 4. The case of chordal graphs.

5. Another characterization of the projective dimension -- Acknowledgments -- References -- Abstract Tubes Associated with Perturbed Polyhedra with Applications to Multidimensional Normal Probability Computations S. Kuriki, T. Miwa and A. J. Hayter -- 1. Introduction -- 2. Abstract tubes for a perturbed inequality system -- 3. Construction of the abstract tube -- 4. Tukey's studentized range -- References -- An Algorithm of Computing Inhomogeneous Difference Equations for a Definite Sum H. Nakayama -- 1. Introduction -- 2. Algorithm of Computing Inhomogeneous Difference Equations for a Definite Sum -- References -- Incomplete A-Hypergeometric Systems K. Nishiyama and N. Takayama -- 1. Introduction -- 2. General definition -- 3. Algorithms deriving contiguity relations -- 4. Incomplete 1 × 1-hypergeometric functions -- 4.1. Homogeneous system -- 4.2. Contiguity relations -- 4.4. Connection formulas -- 4.5. Monodromy formula -- References -- Implementation of a Primary Decomposition Package M. Noro -- 1. Introduction -- 2. Computation of minimal associated primes -- 2.1. SL algorithm -- 2.2. Minimal associated primes of zero-dimensional ideals -- 2.3. Computation of minimal polynomials -- 2.4. Minimal associated primes of I : Q -- 3. Computation of saturations -- 4. Computation of intersection of ideals -- 5. Parallelization -- 6. Experiments -- References -- On Computation of the Characteristic Polynomials of the Discriminantal Arrangements and the Arrangements Generated by Generic Points Y. Numata and A. Takemura -- 1. Introduction -- 2. Intersection lattice and order ideal -- 3. Computation of the Mobius function and the characteristic polynomial -- 3.1. Mobius function and characteristic polynomial -- 3.2. Number of elements of the intersection lattice -- 4. Examples of computations -- 4.1. The case when l = 0 -- 4.2. The case when l = 1.

4.3. The case when l = 2 -- 4.4. The case when l 3 -- 5. Combinatorics of discriminantal arrangements -- 5.1. The case when k = 0 -- 5.2. The case when k = 1 -- 5.3. The case when k = 2 -- 6. Computation of Ad+3,d -- References -- A Dictionary of Grobner Bases of Toric Ideals H. Ohsugi -- Introduction -- 2. How to construct new configurations and Grober bases -- 2.1. Sorting orders and Segre-Veronese configurations -- 2.2. Nested configurations -- 3. Toric ideals arising from graphs -- 4. Toric ideals arising from contingency tables -- 4.1. No n-way interaction models -- 4.2. Graphical models -- 4.3. Binary graph models -- 5. Important examples -- 5.1. Toric ideals arising from root systems -- 5.1.1. The root system An-1 -- 5.1.2. The root system Bn -- 5.1.3. The root system Cn -- 5.1.4. The root system Dn -- 5.2. Toric ideals arising from matroids -- Acknowledgments -- References -- Log-linear Model Estimation for Stratified Educational Data T. Otsu -- 1. Background of the data -- 2. Subject selection data of the NCT -- 3. Log-linear model estimation based on conditional probability -- 4. Data enumeration by constraint logic programming -- References -- Toric Statistical Models: Ising and Markov G. Pistone and M. P. Rogantin -- 1. Introduction -- 2. Lattice exponential families -- Notes -- 3. A-model -- Notes -- 4. Toric ideals and the closure of the A-model -- 4.1. Example: the binomial -- 4.2. Example: 3 binary identical RVs, no 3-way interaction -- Notes -- 5. Differentiation of the normalizing constant -- 5.1. Example: 3 binary variables, no 23- and 123-interactions -- Notes -- 6. Markov chain, toric Markov chain -- Notes -- 7. Reversible Markov chains -- Notes -- Discussion -- References -- Algebraic Reliability Based on Monomial Ideals: A Review E. Saenz-de-Cabezon and H. P. Wynn -- 1. Introduction.

2. Algebraic analysis of system reliability -- 2.1. The coherence property and the ideal of the system -- 2.2. Reliability computation and Hilbert series -- 2.3. Bounds, improvements and Betti numbers -- 2.4. Computations -- 3. Some relevant examples -- 3.1. k-out-of-n systems -- 3.2. Consecutive k-out-of-n systems -- 3.3. Series-parallel systems -- 4. Further topics -- 4.1. Asymptotics -- 4.2. Distributional assumptions -- 4.3. Networks and other applications -- Acknowledgments -- References -- On Irreducibility of Algebroid Curves over the Complex Number Field T. Shibuta -- 1. Introduction -- 2. Local tropical varieties -- 2.1. definition of local tropical varieties -- 2.2. Basic properties of local tropical varieties -- 3. Irreducibility criterion for algebroid curves -- 4. The structure of the initial ideal inω(p)(p) -- Acknowledgments -- References -- Polyhedral Approach to Statistical Learning Graphical Models M. Studeny, D. Haws, R. Hemmecke and S. Lindner -- 1. Introduction -- 2. Basic concepts -- 2.1. Graphical concepts -- 2.2. Learning Bayesian network structure -- 2.3. Algebraic approach to learning -- 3. Summary of recent results -- 3.1. Towards the outer description of the polytope -- 3.2. Geometric neighborhood -- 3.3. Lattice points in the polytope and affine transformation -- 3.4. Integer programming approach -- 4. Characteristic imsets -- 5. Complexity of learning (special) chordal graphs -- Our assumptions -- 5.1. Learning undirected forests and spanning trees -- 5.2. Learning forests and trees with a degree bound -- 5.3. Learning chordal graphs -- 5.4. Learning chordal graphs with bounded size of cliques -- 6. Geometric neighborhood -- 7. Computational experiments -- 8. Conclusions -- Acknowledgments -- References -- Author Index.