In this paper we prove that the only primitive solutions of the Thue inequality |x − 4cxy + (6c + 2)xy + 4cxy + y| ≤ 6c + 4, where c ≥ 4 is an integer, are (x, y) = (±1, 0), (0,±1), (1,±1), (−1,±1), (±1,∓2), (±2,±1).

Morse decompositions have been proposed to compute and represent the topological structure of steady vector fields. Compared to the conventional differential topology, Morse decomposition and the resulting Morse Connection Graph (MCG) is numerically stable. However, the granularity of the original Morse decomposition is constrained by the resolution of the underlying spatial discretization, whi...

In this paper we find all primitive solutions of Thue inequality |x4 + 2(1− n)xy + y4| ≤ 2n + 3, where n ≥ 3 is an integer.

In this paper, we solve a certain family of diophantine equations associated with a family of cyclic quartic number fields. In fact, we prove that for n ≤ 5× 106 and n ≥ N = 1.191× 1019, with n, n+ 2, n2 + 4 square-free, the Thue equation Φn(x, y) = x 4 − nxy − (n + 2n + 4n+ 2)xy − nxy + y = 1 has no integral solution except the trivial ones: (1, 0), (−1, 0), (0, 1), (0,−1).

An infinite dimensional Morse lemma is proved using the deformation lemma from singularity theory. It is shown that the versions of the Morse lemmas due to Palais and Tromba are special cases. An infinite dimensional splitting lemma is proved. The relationship of the work here to other approaches in the literature in discussed. Introduction. This paper shows that when the singularity theory pro...

Growing use of massive scan data in various engineering applications has necessitated research on point-set surfaces. A point-set surface is a continuous surface defined directly with a set of discrete points. This paper presents a new approach that extends our earlier work on slicing point-set surfaces into planar contours for rapid prototyping usage. This extended approach can decompose a poi...

We consider a kind of Yamabe problem whose scalar curvature vanishes in the unit ball B n and on boundary S − 1 mean is prescribed. By combining critical points at infinity approach with Morse theory we obtain new existence results higher dimensional case ≥ 5 , under suitable pinching conditions function.

The new 3and 4-manifold invariants recently constructed by Ozsváth and Szabó are based on a Floer theory associated with Heegaard diagrams. The following notes try to give an accessible introduction to their work. In the first part we begin by outlining traditional Morse theory, using the Heegaard diagram of a 3-manifold as an example. We then describe Witten’s approach to Morse theory and how ...

We consider the (minimal) distance function of a point in the plane to a set P of N points in the plane. The locus of non-di erentiability of this distance function consists (besides of the points of P) exactly of the Voronoi diagram of P. We show that the number of minima (m), maxima (M) and `saddle points' (s) of the distance function satisfy: m s +M = 1 This is similar to the Morse type of s...

a r t i c l e i n f o a b s t r a c t We present a topology-guided technique for improving performance of multifield volume rendering with peak finding and preintegration with 2D transfer functions. We apply Morse–Smale decomposition to segment the multidimensional transfer function domain. This segmentation helps to reduce the number of cases where sampling in transfer function space should be...