7488203
On the Parallelization of Characteristic-Set-Based Algorithms
This paper presents a parallelized version of algorithms for computing characteristic sets, characteristic series and irreducible characteristic series of sets of multivariate polynomials, decomposing algebraic varieties into irreducible components and proving theorems mechanically in elementary geometries. These algorithms have been implemented with up to 12 processors in MAPLE system by utilizing distributed workstations connected by a local network. The timing statistics on a set of test problems with remarks is given. The encountered problems of using parallelism for these algorithms are discussed.
20258295
Automated Generation of Diagrams with Maple and Java
This note shows how to draw diagrams automatically from the predicate specification of a given set of geometric relations among a set of points in the plane. It is done first in Maple by translating the geometric relations into polynomial equations, decomposing the obtained system of polynomials into irreducible representative triangular sets, and finding an adequate numerical solution from each triangular set. A Java class coding the solution and the polynomials in each triangular set is generated, compiled, and then executed with the main Java programs to draw a diagram. The whole process combining symbolic elimination in Maple with numerical computation, graphic drawing, and letter labeling in Java is fully automatic. The drawn diagrams may be animated and fine-tuned by mouse click and dragging and saved as PostScript files. We present the drawing method, discuss some techniques of implementation, and give several sample diagrams drawn by our program.
46198564
Reasoning about Surfaces Using Differential Zero and Ideal Decomposition
This paper presents methods for zero and ideal decomposition of partial differential polynomial systems and the application of these methods and their implementations to deal with problems from the local theory of surfaces. We show how to prove known geometric theorems and to derive unknown relations automatically. In particular, an algebraic relation between the first and the second fundamental coefficients in a very compact form has been derived, which is more general and has smaller degree than a relation discovered previously by Z. Li. Moreover, we provide symmetric expressions for Li's relation and clarify his statement. Some examples of theorem proving and computational difficulties encountered in our experiments are also discussed.
34533255
Clifford Term Rewriting for Geometric Reasoning in 3D
Clifford algebra formalism has been used recently in combination with standard term-rewriting techniques for proving many non-trivial geometric theorems. The key issue in this approach consists in verifying whether two Clifford expressions are equal. This paper is concerned with the generalization of the work to 3D geometric problems. A rewriting system is proposed and its theoretical properties are investigated. Some examples and potential applications are also presented.
35725775
A simple method for implicitizing rational curves and surfaces
Abstract This paper presents a simple method for converting rational parametric equations of curves and surfaces into implicit equations. The method proceeds via writing out the implicit polynomial F of estimated degree with indeterminate coefficients u i , substituting the rational expressions for the given parametric curve or surface into F to yield a rational expression g / h in the parameter s (or s and t ), equating the coefficients of g in terms of s (and t ) to 0 to generate a sparse, partially triangular system of linear equations in u i with constant coefficients, and finally solving the linear system for u i . If a nontrivial solution is found, then an implicit polynomial is obtained; otherwise, one repeats the same process, increasing the degree of F . Our experiments show that this simple method is efficient. It performs particularly well in the presence of base points and may detect the dependency of parameters incidentally.
39506333
GEOTHER: A Geometry Theorem Prover
40850385
Trading off between Misclassification, Recognition and Generalization in Data Mining with Continuous Features
This paper aims at developing a data mining approach for classification rule representation and automated acquisition from numerical data with continuous attributes. The classification rules are crisp and described by ellipsoidal regions with different attributes for each individual rule. A regularization model trading off misclassification rate, recognition rate and generalization ability is presented and applied to rule refinement. A regularizing data mining algorithm is given, which includes self-organizing map network based clustering techniques, feature selection using breakpoint technique, rule initialization and optimization, classifier structure and usage. An Illustrative example demonstrates the applicability and potential of the proposed techniques for domains with continuous attributes.
205025807
Depth-First Search as a Combinatorial Correspondence
Abstract A depth first search algorithm is used to establish the connection between labeled connected graphs and inversions of trees.
20244700
Pattern recognition: neural networks in perspective
Invariant pattern recognition will be a problem facing neural networks for some time, and the challenge is to overcome the limitation of Hamming distance generalization. Four representative architectures that are able to generalize are reviewed. The architectures are the backpropagation network, the ART architecture, the dynamic link architecture, and associate memories. Image representation, segmentation, and invariance are discussed.<<ETX>>
1461139
Lessons Learned from Managing a Petabyte
The amount of data collected and stored by the average business doubles each year. Many commercial databases are already approaching hundreds of terabytes, and at this rate, will soon be managing petabytes. More data enables new functionality and capability, but the larger scale reveals new problems and issues hidden in ''smaller'' terascale environments. This paper presents some of these new problems along with implemented solutions in the framework of a petabyte dataset for a large High Energy Physics experiment. Through experience with two persistence technologies, a commercial database and a file-based approach, we expose format-independent concepts and issues prevalent at this new scale of computing.
8658919
Using Combinatorial Optimization Methods for Quantification Scheduling
Model checking is the process of verifying whether a model of a concurrent system satisfies a specified temporal property. Symbolic algorithms based on Binary Decision Diagrams (BDDs) have significantly increased the size of the models that can be verified. The main problem in symbolic model checking is the image computation problem, i.e., efficiently computing the successors or predecessors of a set of states. This paper is an in-depth study of the image computation problem. We analyze and evaluate several newheuristics, metrics, and algorithms for this problem. The algorithms use combinatorial optimization techniques such as hill climbing, simulated annealing, and ordering by recursive partitioning to obtain better results than was previously the case. Theoretical analysis and systematic experimentation are used to evaluate the algorithms.