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
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.
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.
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.
15784215
Support Vector Clustering for Brain Activation Detection
In this paper, we propose a new approach to detect activated time series in functional MRI using support vector clustering (SVC). We extract Fourier coefficients as the features of fMRI time series and cluster these features by SVC. In SVC, these features are mapped from their original feature space to a very high dimensional kernel space. By finding a compact sphere that encloses the mapped features in the kernel space, one achieves a set of cluster boundaries in the feature space. The SVC is an effective and robust fMRI activation detection method because of its advantages in (1) better discovery of real data structure since there is no cluster shape restriction, (2) high quality detection results without explicitly specifying the number of clusters, (3) the stronger robustness due to the mechanism in outlier elimination. Experimental results on simulated and real fMRI data demonstrate the effectiveness of SVC.
5552115
Synchrony and Desynchrony in Integrate-and-Fire Oscillators
Due to many experimental reports of synchronous neural activity in the brain, there is much interest in understanding synchronization in networks of neural oscillators and its potential for computing perceptual organization. Contrary to Hopfield and Herz (1995), we find that networks of locally coupled integrate-and-fire oscillators can quickly synchronize. Furthermore, we examine the time needed to synchronize such networks. We observe that these networks synchronize at times proportional to the logarithm of their size, and we give the parameters used to control the rate of synchronization. Inspired by locally excitatory globally inhibitory oscillator network (LEGION) dynamics with relaxation oscillators (Terman & Wang, 1995), we find that global inhibition can play a similar role of desynchronization in a network of integrate-and-fire oscillators. We illustrate that a LEGION architecture with integrate-and-fire oscillators can be similarly used to address image analysis.
15831894
Perceptual organization based on temporal dynamics
This paper presents a computational model for perceptual organization. A figure-ground segregation network is proposed based on a novel boundary pair representation. The system solves the figure-ground segregation problem through temporal evolution. Gestalt-like grouping rules are incorporated by modulating connections, which determines the temporal behavior and thus the perception of the system. The results are then fed to a surface completion module based on local diffusion. Different perceptual phenomena, such as modal and a modal completion, virtual contours, grouping and shape decomposition are explained by the model with a fixed set of parameters. Computationally, the system eliminates combinatorial optimization, which is common to many existing computational approaches. It also accounts for more examples that are consistent with psychological experiments. In addition, the boundary-pair representation is consistent with well-known on- and off-center cell responses and thus biologically more plausible.
44960189
GLIF3: a representation format for sharable computer-interpretable clinical practice guidelines
The Guideline Interchange Format (GLIF) is a model for representation of sharable computer-interpretable guidelines. The current version of GLIF (GLIF3) is a substantial update and enhancement of the model since the previous version (GLIF2). GLIF3 enables encoding of a guideline at three levels: a conceptual flowchart, a computable specification that can be verified for logical consistency and completeness, and an implementable specification that is intended to be incorporated into particular institutional information systems. The representation has been tested on a wide variety of guidelines that are typical of the range of guidelines in clinical use. It builds upon GLIF2 by adding several constructs that enable interpretation of encoded guidelines in computer-based decision-support systems. GLIF3 leverages standards being developed in Health Level 7 in order to allow integration of guidelines with clinical information systems. The GLIF3 specification consists of an extensible object-oriented model and a structured syntax based on the resource description framework (RDF). Empirical validation of the ability to generate appropriate recommendations using GLIF3 has been tested by executing encoded guidelines against actual patient data. GLIF3 is accordingly ready for broader experimentation and prototype use by organizations that wish to evaluate its ability to capture the logic of clinical guidelines, to implement them in clinical systems, and thereby to provide integrated decision support to assist clinicians.
35246774
Identifying reasoning strategies in medical decision making: A methodological guide
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.