Decomposing Algebraic Varieties
An algebraic variety is a geometric figure defined by the zeros of a set of multivariate polynomials. This paper explains how to adapt two general zero decomposition methods for efficient decomposition of affine algebraic varieties into unmixed and irreducible components. Two devices based on Grobner bases are presented for computing the generators of the saturated ideals of triangular sets. We also discuss a few techniques and variants which, when properly used, may speed up the decomposition. Experiments for a set of examples are reported with comparison to show the performance and effectiveness of such techniques, variants and the whole decomposition methods. Several theoretical results are stated along with the description of algorithms. The paper ends with a brief mention of some applications of variety decomposition.

title: Automated Generation of Diagrams with Maple and Java
abstract: 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.

title: GEOTHER: A Geometry Theorem Prover

title: On Wu's Method for Proving Constructive Geometric Theorems
abstract: 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.

title: 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.

title: Reasoning about Surfaces Using Differential Zero and Ideal Decomposition
abstract: 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.

title: Using Combinatorial Optimization Methods for Quantification Scheduling
abstract: 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.

title: Nonlinear quantification scheduling in image computation
abstract: Computing the set of states reachable in one step from a given set of states, i.e. image computation, is a crucial step in several symbolic verification algorithms, including model checking and reachability analysis. So far, the best methods for quantification scheduling in image computation, with a conjunctively partitioned transition relation, have been restricted to a linear schedule. This results in a loss of flexibility during image computation. We view image computation as a problem of constructing an optimal parse tree for the image set. The optimality of a parse tree is defined by the largest BDD that is encountered during the computation of the tree. We present dynamic and static versions of a new algorithm, VarScore, which exploits the flexibility offered by the parse tree approach to the image computation. We show by extensive experimentation that our techniques outperform the best known techniques so far.

title: Classification of compound images based on transform coefficient likelihood
abstract: Applications like distance learning and teleconferencing often require compression of images that contain both text and graphics. Because text and graphics have different properties, a compression scheme can benefit by treating the textual and graphical portions of such compound images separately. In this paper, we propose new methods, called transform coefficient likelihood (TCL) schemes, for separating the textual and graphical portions of a compound image. TCL schemes examine the DCT coefficient values of an 8/spl times/8 block. For each coefficient, they refer to stored histograms that give the likelihood that a certain value occurs in a text block, or in a graphics block. They then examine the differences in these two likelihoods over all the coefficients in the block to decide whether it contains text or graphics. Experimental results show that the best TCL methods significantly outperform previously proposed techniques.

title: Identifying reasoning strategies in medical decision making: A methodological guide
abstract: Reasoning strategies are a key component in many medical tasks, including decision making, clinical problem solving, and understanding of medical texts. Identification of reasoning strategies used by clinicians may prove critical to the optimal design of decision support systems. This paper presents a formal method of cognitive-semantic analysis for the identification and characterization of reasoning strategies deployed in medical tasks and demonstrates its use through specific examples. Although semantic analysis was originally developed in the investigation of knowledge structures, it can also be applied to identify the reasoning and decision processes used by physicians and medical trainees in clinical tasks. Assumptions underlying the methods, as well as illustrations of their use in diagnostic explanation tasks, are presented. We discuss semantic analysis in the context of the current interests in developing medical ontologies and argue that a frame-based propositional analytic methodology can provide a systematic way of addressing the construction of such ontologies. Although the application of propositional analysis methods has some limitations, we show how such limitations are being addressed and present some examples of information tools that have been developed to ease, and make more systematic, the process of analysis.

title: On Temporal Generalization of Simple Recurrent Networks
abstract: Simple recurrent networks (Elman networks) have been widely used in temporal processing applications. In this study we investigate temporal generalization of simple recurrent networks, drawing comparisons between network capabilities and human performance. Elman networks are trained to generate temporal trajectories sampled at different rates. The networks are then tested with trajectories at the trained rates and other sampling rates, including trajectories representing mixtures of different sampling rates. It is found that for simple trajectories the networks show interval invariance, but not rate invariance. However, for complex trajectories which require greater contextural information, these networks do not seem to show any temporal generalization. Similar results are also obtained using measured speech data. These results suggest that this class of recurrent networks exhibits severe limitations in temporal generalization. Discussions are provided regarding rate invariance and possible ways to achieve it in neural networks.