This paper introduces an algorithmic approach to the analysis of Jacobi stability of systems of second order ordinary differential equations (ODEs) via the Kosambi–Cartan–Chern (KCC) theory. We develop an efficient symbolic program using Maple for computing the second KCC invariant for systems of second order ODEs in arbitrary dimension. The program allows us to systematically analyze Jacobi stability of a system of second order ODEs by means of real solving and solution classification using symbolic computation. The effectiveness of the proposed approach is illustrated by a model of wound strings, a two-dimensional airfoil model with cubic nonlinearity in supersonic flow and a 3-DOF tractor seat-operator model. The computational results on Jacobi stability of these models are further verified by numerical simulations. Moreover, our algorithmic approach allows us to detect hand-guided computation errors in published papers.

The problem of collision detection plays an important role in many fields of science and engineering. This article presents a collision detection method for general convex objects bounded by pieces of implicit surfaces. There are two key ideas that underlie our method: one is the introduction of a new kind of pseudodistance, called the$\delta$-distance, for implicitly represented convex objects which has the desired properties of convexity and square differentiability; the other is the use of$\delta$-distance functions to construct a virtual potential field in the real space, so that the problem of collision detection can be reduced to a problem of unconstrained convex optimization. The method is extended and applied to detect whether two objects collide when they are moving continuously along linearly translational trajectories, which is a special case of one of the continuous collision detection subproblems. We have implemented collision detection algorithms in C++ and conducted a large number of experiments, with test examples involving objects modeled by planar, quadric, superquadric, superellipsoidal, and hyperquadric surfaces, as well as pieces of them, in both stationary and linearly translational moving states. The experimental results show that our method has good performance and it is computationally efficient and widely applicable.

The classical theory of Kosambi–Cartan–Chern (KCC) developed in differential geometry provides a powerful method for analyzing the behaviors of dynamical systems. In the KCC theory, the properties of a dynamical system are described in terms of five geometrical invariants, of which the second corresponds to the so-called Jacobi stability of the system. Different from that of the Lyapunov stability that has been studied extensively in the literature, the analysis of the Jacobi stability has been investigated more recently using geometrical concepts and tools. It turns out that the existing work on the Jacobi stability analysis remains theoretical and the problem of algorithmic and symbolic treatment of Jacobi stability analysis has yet to be addressed. In this paper, we initiate our study on the problem for a class of ODE systems of arbitrary dimension and propose two algorithmic schemes using symbolic computation to check whether a nonlinear dynamical system may exhibit Jacobi stability. The first scheme, based on the construction of the complex root structure of a characteristic polynomial and on the method of quantifier elimination, is capable of detecting the existence of the Jacobi stability of the given dynamical system. The second algorithmic scheme exploits the method of semi-algebraic system solving and allows one to determine conditions on the parameters for a given dynamical system to have a prescribed number of Jacobi stable fixed points. Several examples are presented to demonstrate the effectiveness of the proposed algorithmic schemes.

This paper studies the representation of the set of zeros with multiplicities for an ideal generated by a given set of multivariate polynomials in terms of triangular regular sequences, whose dimensions and degrees can be read out directly. A new algebro-geometric approach is proposed that enables one to decompose any polynomial ideal into finitely many triangular regular sequences of polynomials such that certain implicit relations between the Hilbert polynomials and explicit relations between the sets of zeros of the ideals generated by the regular sequences are preserved. The decomposition algorithms make use of the properties and computations of W-characteristic sets of polynomial ideals and perform simultaneous sum-and-quotient operation, a key technique that is used implicitly in the recursive process of computing Hilbert polynomials. The present work elaborates and reveals inherent connections between some commonly used concepts in the algorithmic theories of triangular sets, Gröbner bases, and Hilbert polynomials. Examples are provided to illustrate the computational aspects and differences of our approach from that of pseudo-division-based triangular decomposition.

This paper presents an algorithm that decomposes an arbitrary set F of multivariate polynomials involving parameters into finitely many sets Γi of (lexicographical) Gröbner bases G ij such that associated with each Γi there is a system Ai of parametric constraints, all the Ai's partition the parameter space, all the Gij's in each Γi remain Gröbner bases under specialization of the parameters satisfying the constraints in Ai, and for each i the Gröbner bases Gij together with their corresponding W-characteristic sets form a normal characteristic decomposition of F. The sets of Gröbner bases computed by the algorithm provide a comprehensive characteristic decomposition of F that is structure-invariant under the associated constraints and possesses many algebraic and geometric properties, including most of the notable properties on comprehensive Gröbner systems and comprehensive triangular decomposition. Some of these properties are highlighted in the paper and advantages of the proposed algorithm are discussed briefly from the aspects of methodological simplicity and computational performance using illustrative examples and preliminary experiments.

This paper presents a method for computing uniform Gröbner bases for certain ideals generated by polynomials with parametric exponents. The method proceeds by replacing monomials involving parametric exponents in the generators of an ideal with new variables, computing the reduced Gröbner basis for the resulting ideal with respect to a special monomial order, and then verifying whether the leading monomial ideal of the Gröbner basis satisfies some consistency conditions according to two criteria (of which one is derived from Buchberger graphs). When the consistency conditions are verified, a uniform Gröbner basis for the original ideal is obtained by substituting the new variables back to original monomials. The effectiveness and practical value of the method are demonstrated by its application to a family of ideals coming from the modeling of biological systems.

This paper presents a new and general approach for analyzing the stability of a large class of biological networks, modeled as autonomous systems of differential equations, using real solving and solution classification. The proposed approach, based on the classical technique of linearization from the qualitative theory of ordinary differential equations yet with exact symbolic computation, is applied to analyzing the local stability of the Cdc2-cyclin B/Wee1 system and the Mos/MEK/p42 MAPK cascade, two well-known models for cell and protein signaling that have been studied extensively in the literature. We provide rigorous proofs and generalizations for some of the previous results established experimentally and report our new findings.

A general approach we have proposed for automatically proving geometric theorems requires both Clifford algebraic reduction and term-rewriting. This paper shows how efficient techniques and software tools developed in the areas of algebraic computati

The theoretical analysis and derivation of artificial neural systems, which consists essentially of manipulating symbolic mathematical objects according to certain mathematical and biological knowledge, can be done more efficiently with computer assistance by using and extending methods and systems of symbolic computation. After presenting the mathematical characteristics of neural systems and a brief review on Lyapunov stability theory, the authors present some features and capabilities of existing systems and the extension for manipulating objects occurring in the analysis of neural systems. Some strategies and a toolkit developed in MACSYMA for computer-aided analysis and derivation are described. A concrete example is given to demonstrate the derivation of a hybrid neural system, i.e. a system which in its learning rule combines elements of supervised and unsupervised learning. Future work and research directions are indicated.>

In this paper we present the design and implementation of a toolkit in MACSYMA for the manipulation of indefinite summations which is of interest and commonly used in mathematical derivations. The differentiation and integration of indefinite summations with respect to indexed variables are considered in particular. The development of this toolkit was motivated by and has its major purpose for our investigation in the computer aided analysis and derivation of artificial neural networks. The application of the toolkit to this subject is briefly discussed.