Computer Mathematics: 8th Asian Symposium, ASCM 2007, Singapore, December 15-17, 2007, Revised and Invited Papers (Lecture Notes in Computer Science, 5081)

Title Page
Preface
Organization
Table of Contents
Computing the Minkowski Value of the Exponential Function over a Complex Disk
	Introduction
	Random–Coefficient Differential Equation
	Minkowski Exponential of a Real Interval
	Exponential Image of a Circular Disk
	Monomial Minkowski Exponential
	Monte Carlo Experiments
	Closure
	References
Unconstrained Parametric Minimization of a Polynomial: Approximate and Exact
	Introduction
	Algorithm for Lower Bound
	Examples
	The Minimum of a Quartic Polynomial
	Application to Integration
	References
The Nearest Real Polynomial with a Real Multiple Zero in a Given Real Interval
	Introduction
	Theoretical Background
		Existence of a Nearest Polynomial
		Main Theorem
	Computation Methods
		First Case
		Second Case
		Computational Complexity
		Examples
	Conclusion
	References
Practical and Theoretical Issues for the Computation of Generalized Critical Values of a Polynomial Mapping
	Introduction
	Preliminaries
	Main Results and Algorithms
		Geometric Results
		Algorithms and Complexity
	Practical Results
		Description of the Test-Suite
		Practical Results
	References
Which Symmetric Homogeneous Polynomials Can Be Proved Positive Semi-definite by Difference Substitution Method?
	Introduction
	Notation
	D-PSD Cone
	An Example for DevSubExp
	Relation of C-PSD and D-PSD
	Extension of D-PSD Cone
	Conclusions
	References
Basis-Independent Polynomial Division  Algorithm Applied to Division in Lagrange and Bernstein Basis
	Introduction
	Basis-Independent Framework for Polynomial Division
		Motivating Example
		Definition of Polynomial Division Algorithm
		Complexity of the Division Algorithm
		Correctness of the Division Algorithm
	Division in Lagrange Basis
		Definition of Basic Operators
		Complexity
		Correctness
	Division in Bernstein Basis
		Definition of Basic Operators
		Complexity
		Correctness
	Conclusion and Future Directions
	References
Computing the Greatest Common Divisor of  Polynomials Using the Comrade Matrix
	Introduction
	General Procedure and Related Work
	The GCD of Generalized Polynomials
	Computing Time Analysis
	Construction of the Comrade Matrix
	The Coefficient Matrix
	Concluding Remarks and Further Work
	References
Efficient Algorithms for Computing N$\\oe$ther Normalization
	Introduction
	$N\\oether$ Position Test
	Modular Noether Position Test
	Putting an Ideal in Noether Position
	Experiments and Remarks
	Conclusion
	References
Stability of GPBiCG AR Method Based on Minimization of Associate Residual
	Introduction
	GPBi-CG and GPBiCG AR Methods
		Preconditioned GPBi-CG and GPBiCG AR Methods
	NumericalExperiments
		Non Preconditioned GPBiCG AR and GPBi-CG Methods
		Preconditioned GPBiCG AR and GPBi-CG Methods
	Conclusions and Future Work
	References
Evaluation of a Java Computer Algebra System
	Introduction
		Related Work
		Outline
	Introduction to JAS
		Using the JAS Library
		JAS Class Overview
	Evaluation
		Interfaces as Types
		Generics and Inheritance
		Dependent Types
		Method Semantics
		Recursive Types
		Factory Pattern
		Code Reuse
		Performance
		Applications
		Parallelization
		Libraries
		Java Environment
	Conclusions
	References
A New Property of Hamming Graphs and Mesh of $d$-ary Trees
	Introduction
	Preliminaries
		Graph Definitions
		Group Definitions
	Introduction to $G$-Graphs
		Algorithmic Procedure
		Complexity and Example
	Hamming Graphs Are $G$-Graphs
	Meshof $d$-ary Trees
	References
An Interpolation Method That Minimizes an Energy Integral of Fractional Order
	Introduction
	The Problem and Its Solution
	The Procedure to Obtain the Solution
	What Happens When $0\\le\\alpha\\le\\frac{1}{2}$
	References
Solving Biomechanical Model Using Third-Order Runge-Kutta Methods
	Introduction
	The Biomechanics Model of a Cyclist
	Numerical Solution
	Conclusion
	References
An Efficient Fourth Order Implicit Runge-Kutta Algorithm for Second Order Systems
	Introduction
	The Implicit Runge-Kutta Method
	The Algorithm
	Numerical Experimentation
	Conclusions
	References
Laplace Equation Inside a Cylinder: Computational Analysis and Asymptotic Behavior of the Solution
	Introduction
	Solution to the Boundary Value Problem
	Simulation and Analysis Results
	Conclusion
	References
A Method and Its Implementation for Constructing B$\\\"{a}$cklund Transformations to Nonlinear Evolution Equations
	Introduction
	An Algorithm for B$\\\"{a}$cklund Transformation
		The First Case: Differential Form
		The Second Case: Differential-Free Form
	The Maple Package AutoBT
	The Application of the Package AutoBT
	Summary
	References
On the Invariant Properties of Hyperbolic Bivariate Third-Order Linear Partial Differential Operators
	Introduction
	Definitions and Notations
	Factorization Via Invariants
	Factorization Type $(p^X + q^Y )(XY )$
	Factorization Type (X)(Y S)
	Factorization Types $(p^X + q^Y )(X)(Y )$ and $(X)(p^X + q^Y )(Y )$
	Formal Adjoint
	SymbolofConstantCoefficients
	Conclusion
	References
Symbolic Solution to Magnetohydrodynamic Hiemenz Flow in Porous Media
	Introduction
	Boundary Layer Equations of MHD Hiemenz Flow
	Solution Procedure
	Results and Discussion
	Conclusions
	References
Local Similarity Solutions for Laminar BoundaryLayer Flow a long a Moving Cylinder in a Parallel Stream
	Introduction
	Problem Formulation
	Solution Procedure
		Local Similarity Assumption
		Finite-Difference Method
		Newton’s Method
		Block-Elimination Method
	Results and Discussion
	References
An Algorithm for Transforming Regular Chain into Normal Chain
	Introduction
	Preliminaries
	An Algorithm to Compute the Inverse of a Polynomial Modulo an Ideal
	Transforming Regular Chain into Normal Chain
	Examples
	Conclusions
	References
A Modified Van der Waerden Algorithm to Decompose Algebraic Varieties and Zero-Dimensional Radical Ideals
	Introduction
	Preliminaries
	Outline of the VDW Decomposition Algorithm
	Irredundant Decomposition
	Decomposing Zero-Dimensional Radical Ideals
		Decomposing Zero-Dimensional Radical Ideals Using VDW Algorithm
		Decomposing Zero-Dimensional Radical Ideals Using Macaulay Resultant
	Conclusions
	References
Regular Decompositions
	Introduction
	Regular Set
	Applications
	Algorithm
		Main Idea of the Algorithm
		Recursion Step
		Complete Algorithm
		Examples
	Optimizations
		Pruning the Tree $(I)$
		Degree ordering
		Fraction Field
		Zero-Dimensional Case
		Pruning the Tree $(II)$
	PracticalBehaviour
	References
Floating-Point Gr¨obner Basis Computation with Ill-conditionedness Estimation
	Introduction
	Instability Due to Self-reduction
		Clones and Self-reduction Caused by Small Leading Terms
		Self-reduction in Three Other Cases
	Analysis of Self-reductions Given in Sect. 2
		Analysis of Self-reduction by Double Clones
		Estimation of Amount of Main-Term Cancellation
	New Method of Stabilization
		Supporting Theorem
		Effective Floating-Point Numbers
		Description of New Method
	Implementation Details
		Representation of Clones
		Reducer Subtraction
		Estimating the Amount of Intrinsic Cancellation
	Concluding Remarks
	References
The Maximality of the Dixon Matrix on Corner-Cut Monomial Supports
	Introduction
	The Structure of Cornet-Cut Monomial Supports
		Rectangular and Arbitrary Lattice Sets
		Bidegree Hulls
		Exterior Points and Exposed Points
		Presentation Convention
		An Example
	TheDixonConstruction
		The Dixon Polynomial P(S) for a Support S
		The Dixon Matrix D(S) for a Support S
		The Row Support R(S) and Column Support C(S) for the Dixon Matrix D(S)
		An Example
	Exterior Points Simplify the Dixon Matrix
		Translating Points of S to Points of R(S) and C(S)
		Exterior Points Eliminate Rows and Columns
		Exterior Points Eliminate Brackets
		Examples
	Excess Degrees and Quotient Dixon Resultants
	Supports with Three or Fewer Exposed Points Preserve Maximality
		All Possible Exposed Points
		Columns Indexed by Exposed Points in the Column Support
		Corresponding Row Indices
		Proving the Maximality of D(S)
		Provision for Degeneracies
	Conclusions
	References
Properties of Ascending Chains for Partial Difference Polynomial Systems
	Introduction
	Preliminaries
		Difference Polynomials and Difference Chains
		Invertibility of Algebraic Polynomials
	Coherent Chains
		Prolongation of Chains
		Coherent Chains
	Regular Chains
	Characteristic Set of Reflexive Prime Difference Ideals
	Algorithms of Zero Decomposition
	Conclusion
	References
Some Mathematical Problems in Cryptanalysis
A Reduction Attack on Algebraic Surface Public-Key Cryptosystems
	Introduction
	Algebraic Surface Public-Key Cryptosystem [1, 3, 4]
	Attack on Algebraic Surface Public-Key Cryptosystem under the Assumption [5]
	Attack on Algebraic Surface Public-Key Cryptosystem
		Attack by Straightforward Generalization in $IF_p^(t) [x, y]$
		Attack by Utilizing Gr$\\\"{o}$bner Base Techniques in $IF_p^[x, y, t,A]$
	Conclusion
	References
The Four Colour Theorem: Engineering of a Formal Proof
On the Computation of Elimination Ideals of Boolean Polynomial Rings
	Introduction
	Boolean Polynomial Ring
	Boolean Gr$\\\"{o}$bner Bases
	Comprehensive Boolean Gr$\\\"{o}$bner Bases
	New Algorithm for Elimination Ideals
	Conclusions and Remarks
	References
Computer Search for Large Sets of Idempotent Quasigroups
	Introduction
	Preliminaries
		Definitions
		The Problems
	Search for LIQs of Small Orders
	SearchforLSPQ(11)
	Discussion
	Conclusion
	References
Author Index