42132999
Polynomial Systems from Certain Differential Equations
In this paper, combined elimination techniques are applied to establish relations among center conditions for certain cubic differential systems initially investigated by Kukles in 1944. The obtained relations clarify recent rediscoveries of some known conditions of Cherkas. The computational difficulties of establishing the complete center conditions for Kukles? system, a problem that is still open, are illustrated by interactive elimination. Some generated polynomial systems that need be solved are made available electronically for other developers to test elimination algorithms and their implementations.
10779017
On Wu's Method for Proving Constructive Geometric Theorems
In 1977 Wu Wen-tsiin discovered an efficient method for mechanical theorem proving. This method has been applied to prove and discover theorems in various kinds of geometries. Based on an extension of one of Wu's algorithmic procedures, the author implemented a prover CPS for proving so-called constructive geometric theorems, as well as a general theorem prover PS. Using this prover we proved more than one hundred non-trivial theorems and discovered also a number of interesting ones. This paper is a summarization of our early work, as a abbreviation of the chapter 2 from my Chinese doctoral dissertation.
39506333
GEOTHER: A Geometry Theorem Prover
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.
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.
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.
13039133
On Connectedness: A Solution Based on Oscillatory Correlation
A long-standing problem in neural computation has been the problem of connectedness, first identified by Minsky and Papert (1969). This problem served as the cornerstone for them to establish analytically that perceptrons are fundamentally limited in computing geometrical (topological) properties. A solution to this problem is offered by a different class of neural networks: oscillator networks. To solve the problem, the representation of oscillatory correlation is employed, whereby one pattern is represented as a synchronized block of oscillators and different patterns are represented by distinct blocks that desynchronize from each other. Oscillatory correlation emerges from LEGION (locally excitatory globally inhibitory oscillator network), whose architecture consists of local excitation and global inhibition among neural oscillators. It is further shown that these oscillator networks exhibit sensitivity to topological structure, which may lay a neurocomputational foundation for explaining the psychophysical phenomenon of topological perception.
15453012
Perceiving without Learning: From Spirals to Inside/Outside Relations
As a benchmark task, the spiral problem is well known in neural networks. Unlike previous work that emphasizes learning, we approach the problem from a generic perspective that does not involve learning. We point out that the spiral problem is intrinsically connected to the inside/ outside problem. A generic solution to both problems is proposed based on oscillatory correlation using a time delay network. Our simulation results are qualitatively consistent with human performance, and we interpret human limitations in terms of synchrony and time delays, both biologically plausible. As a special case, our network without time delays can always distinguish these figures regardless of shape, position, size, and orientation.
8554279
High level verification of control intensive systems using predicate abstraction
Predicate abstraction has been widely used for model checking hardware/software systems. However, for control intensive systems, existing predicate abstraction techniques can potentially result in a blowup of the size of the abstract model. We deal with this problem by retaining important control variables in the abstract model. By this method we avoid having to introduce an unreasonable number of predicates to simulate the behavior of the control variables. We also show how to improve predicate abstraction by extracting useful information from a high level representation of hardware/software systems. This technique works by first extracting relevant branch conditions. These branch conditions are used to invalidate spurious abstract counterexamples through a new counterexample-based lazy refinement algorithm. Experimental results are included to demonstrate the effectiveness of our methods.
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>>
12267265
On the Verge of One Petabyte - the Story Behind the BaBar Database System
The BaBar database has pioneered the use of a commercial ODBMS within the HEP community. The unique object-oriented architecture of Objectivity/DB has made it possible to manage over 700 terabytes of production data generated since May'99, making the BaBar database the world's largest known database. The ongoing development includes new features, addressing the ever-increasing luminosity of the detector as well as other changing physics requirements. Significant efforts are focused on reducing space requirements and operational costs. The paper discusses our experience with developing a large scale database system, emphasizing universal aspects which may be applied to any large scale system, independently of underlying technology used.