(a) The curves z(s) and z(s, t) were not required to be simple curves. This made the task of approximating them with continuously differentiable curves very easy. (b) The Cauchy integral theorem also holds if the function z(s) describing the curve C is merely piecewise continuously differentiable. For in this case we carry out the integration by parts in (6) over each subinterval on which zs(s)...

A geometric proof of the Matsuki orbit duality for flag manifolds is established in [2] by analyzing the gradient flow of the normsquared of a moment map. In the present paper, we investigate explicit formulas for integral curves associated with this flow, leading to a correspondence between certain integral curves and Cayley transforms. In addition, an exhaustive collection of curves is presen...

Let C ⊂ An be an irreducible affine curve of (geometric) genus 0 defined by a finite family of polynomials having integer coefficients. In this note we give a necessary and sufficient condition for C to possess infinitely many integer points, correcting a statement of J. H. Silverman (Theoret. Comput. Sci., 2000). Let C be an irreducible affine curve of (geometric) genus 0 in the affine space A...

FENG, SHUO. 3D Integral Invariant Signatures And Their Application on Face Recognition. (Under the direction of Professor Hamid Krim). Curves are important features in computer vision and pattern recognition, and their classification under a variety of transformations, such as Euclidean, affine or projective, poses a great challenge. Invariant features of these curves turn out to be crutial to ...

1. Introduction. By a famous theorem of Siegel [S], the number of integral points on an elliptic curve E over an algebraic number field K is finite. A conjecture of Lang and Demyanenko (see [L3], p. 140) states that, for a quasiminimal model of E over K, this number is bounded by a constant depending only on the rank of E over K and on K (see also [HSi], [Zi4]). This conjecture was proved by Si...

the aim of this paper is to show how some measures of noncompactness in the banach space of continuous functions defined on two variables can be applied to the solvability of a general system of functional integral equations . the results obtained generalize and extend several equations . an illustrative example is also presented .

We provide a precise description of the integer points on elliptic curves of the shape y2 = x3 − N2x, where N = 2apb for prime p. By way of example, if p ≡ ±3 (mod 8) and p > 29, we show that all such points necessarily have y = 0. Our proofs rely upon lower bounds for linear forms in logarithms, a variety of old and new results on quartic and other Diophantine equations, and a large amount of ...

We prove results generalizing the classical Riemann Singularity Theorem to the case of integral, singular curves. The main result is a computation of the multiplicity of the theta divisor of an integral, nodal curve at an arbitrary point. We also suggest a general formula for the multiplicity of the theta divisor of a singular, integral curve at a point and present some evidence that this formu...

By a famous theorem of Siegel [S], the number of integral points on an elliptic curve E over an algebraic number field K is finite. A conjecture of Lang and Demjanenko [L3] states that, for a quasiminimal model of E over K, this number is bounded by a constant depending only on the rank of E over K and on K (see also [HSi], [Zi4]). This conjecture was proved by Silverman [Si1] for elliptic curv...