On avoidable and unavoidable claws
A claw of degree k is a directed tree consisting of k paths emerging from a common root. We prove that every claw of order n with degree less than 19 50 n appears in every n -vertex tournament. We also construct avoidable claws with degree approaching 11 23 n . Thus for large n , the maximum λ such that every claw with degree λn appears in every n -vertex tournament satisfies λ ⩽ 11 23 . This improves earlier bounds.

title: The strong Hall property and symmetric chain orders
abstract: Let G =( X , Y ; E ) be a bipartite graph with | X |⩾| Y |. For A ⊆ X , write φ ( A )=| A |−| N ( A )| and for a ⩽| X |, define φ(a)= max {φ(A) | A⊆X, |A|=a} . The graph G is said to have the strong Hall property if φ ( a )+ φ ( b )⩽| X |−| Y | for all nonnegative integers a and b with a + b ⩽| X |. We shall prove that any unimodal and self-dual poset with the strong Hall property is a symmetric chain order. This result will also be used to show that the inversion poset S 5 is a symmetric chain order.

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: Synchrony and Desynchrony in Integrate-and-Fire Oscillators
abstract: 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.

title: Geometry Machines: From AI to SMC
abstract: The existing techniques and software tools for automated geometry theorem proving (GTP) are examined and reviewed. The underlying ideas of various approaches are explained with a set of selected examples. Comments and analyses are provided to illustrate the encouraging success of GTP which interrelates AI and SMC. We also present some technological applications of GTP and discuss its challenges ahead.

title: Computer Algebra and Neurodynamics
abstract: We review the capabilities of computer algebra and the characteristics of neurodynamics. Possible applications of computer algebra to the study of neurodynamics are then discussed. We have investigated some of the applications, in particular, to the computer aided analysis and derivation of neurodynamics by using and developing tools in computer algebra systems. Part of our own work in this direction is summarized.

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.