10384078
Mechanical Manipulation for a Class of Differential Systems
In this paper we describe a mechanical procedure for computing the Liapunov functions and Liapunov constants for a class of differential systems. These functions and constants are used for establishing the stability criteria, the conditions for the existence of a center and for the investigation of limit cycles. Some problems for handling the computed constants, which are usually large polynomials in terms of the coefficients of the differential system, and an approach towards their solution by using computer algebraic methods are proposed. This approach has been successfully applied to check some known results mechanically. The author has implemented a system DEMS on an HP1000 and in Scratchpad II on an IBM4341 for computing and manipulating the Liapunov functions and Liapunov constants. As examples, two particular cubic systems are discussed in detail. The explicit algebraic relations between the computed Liapunov constants and the conditions given by Saharnikov are established, which leads to a rediscovery of the incompleteness of his conditions. A class of cubic systems with 6-tuple focus is presented to demonstrate the feasibility of our approach for finding systems with higher multiple focus.
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.
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.
39506333
GEOTHER: A Geometry Theorem Prover
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.
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.
14446930
Novel Self-Organizing Takagi Sugeno Kang Fuzzy Neural Networks Based on ART-like Clustering
The existing Self-Organizing Takagi Sugeno Kang Fuzzy Neural Networks (S-TSKfnn) structure uses virus infection clustering (VIC) method to generate fuzzy rules. In this paper, we propose a novel architecture called Modified S-TSKfnn (MS-TSKfnn) that uses ART-like clustering called discrete incremental clustering (DIC). By doing so, MS-TSKfnn is able to handle online data input, and its performance is also enhanced. Most importantly, the accurate clustering in the fuzzy set derivation has significantly reduced the number of fuzzy TSK rules necessary to describe a problem. Extensive simulations are conducted using MS-TSKfnn and its performance is encouraging when benchmarked against other established neuro-fuzzy systems. The empirical work also firmly demonstrated the importance of clustering within a fuzzy neural reasoning system in ensuring a compact and expressive fuzzy rate base.
1058945
A Classification-based Cocktail-party Processor
At a cocktail party, a listener can selectively attend to a single voice and filter out other acoustical interferences. How to simulate this perceptual ability remains a great challenge. This paper describes a novel supervised learning approach to speech segregation, in which a target speech signal is separated from interfering sounds using spatial location cues: interaural time differences (ITD) and interaural intensity differences (IID). Motivated by the auditory masking effect, we employ the notion of an ideal time-frequency binary mask, which selects the target if it is stronger than the interference in a local time-frequency unit. Within a narrow frequency band, modifications to the relative strength of the target source with respect to the interference trigger systematic changes for estimated ITD and IID. For a given spatial configuration, this interaction produces characteristic clustering in the binaural feature space. Consequently, we perform pattern classification in order to estimate ideal binary masks. A systematic evaluation in terms of signal-to-noise ratio as well as automatic speech recognition performance shows that the resulting system produces masks very close to ideal binary ones. A quantitative comparison shows that our model yields significant improvement in performance over an existing approach. Furthermore, under certain conditions the model produces large speech intelligibility improvements with normal listeners.
33401390
Image segmentation using local spectral histograms
We propose a new algorithm for image segmentation. We use the spectral histogram, which is a vector consisting of marginal distributions of responses from chosen filters as a generic feature for texture as well as intensity images. Motivated by a new segmentation energy functional, we derive an iterative and deterministic approximation algorithm for segmentation. Based on the relationships between different scales and neighboring windows, we also develop an algorithm which can automatically detect homogeneous regions in an input image, which may consist of texture regions. To reduce the boundary uncertainty due to the large spatial window used for spectral histograms, we propose a novel local feature by building precise probability models based on current segmentation results. We have applied our algorithm to intensity, texture, and natural images and obtained good results with accurate texture boundaries.
45620473
Theoretical Foundation for Nonlinear Edge-Preserving Regularized Learning Image Restoration
Image restoration is an important issue in image processing, which helps recovering of degraded images caused by various factors in different circumstances. Neural networks models have been successfully applied in handling image restoration problems and some progress and promising results have been reported in the past. A popular neural networks model for image restoration is the Hopfield network due to its ability on dealing with optimization problems. A degraded image may have multiple corresponding solutions, i.e., the restored images, and the obtained solution can be sub-optimal which is related to a local minimum point in the weight space of the Hopfield network. This paper gives an algebraic characterization on images and shows that images with lower complexity can be restored uniquely from their degraded images and edge information. The obtained result in this paper establishes a mathematical basis for employing the mapping neural networks to realize a learning based image restoration scheme. The linear image restoration model is firstly generalized to a nonlinear one to broaden the scope of application. Secondly, we view the image restoration task as a set of approximation problems in a high dimensional space, and a mapping relationship between the degraded images with edge information and the source images is then built using feed-forward neural networks and the well trained neural network can be used to restore the degraded images in realtime. Computer simulations demonstrate the effectiveness of the learning image restoration techniques proposed in this paper.
2942056
Automated Abstraction Refinement for Model Checking Large State Spaces Using SAT Based Conflict Analysis
We introduce a SAT based automatic abstraction refinement framework for model checking systems with several thousand state variables in the cone of influence of the specification. The abstract model is constructed by designating a large number of state variables as invisible. In contrast to previous work where invisible variables were treated as free inputs we describe a computationally more advantageous approach in which the abstract transition relation is approximated by pre-quantifying invisible variables during image computation. The abstract counterexamples obtained from model-checking the abstract model are symbolically simulated on the concrete system using a state-of-the-art SAT checker. If no concrete counterexample is found, a subset of the invisible variables is reintroduced into the system and the process is repeated. The main contribution of this paper are two new algorithms for identifying the relevant variables to be reintroduced. These algorithms monitor the SAT checking phase in order to analyze the impact of individual variables. Our method is complete for safety properties (AGp) in the sense that - performance permitting - a property is either verified or disproved by a concrete counterexample. Experimental results are given to demonstrate the power of our method on real-world designs.