In this work, the notion of a slant helix is extended to the space E. First, we introduce the type-2 harmonic curvatures of a regular curve. Thereafter, by using this, we present some necessary and sufficient conditions for a curve to be a slant helix in Euclidean n-space. We also express some integral characterizations of such curves in terms of the curvature functions. Finally, we give some c...

We study the integral expression of a knot invariant obtained as the second coeecient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves and it is related with invariants of generic plane curves recently deened by Arnold, with deep mot...

CONTENTS Introduction 1. Elliptic Curves Defined by Simplest Cubic Fields 2. Linear Forms in Elliptic Logarithms 3. Computation of Integral Points 4. Tables of Results 5. General Results about Integral Points on the Elliptic Curves y2 = x3 + mx2 (m+3)x + 1 References Let f(X) be a cubic polynomial defining a simplest cubic field in the sense of Shanks. We study integral points on elliptic curve...

Gorsky et al. presented an explicit construction of Whitham deformations of the Seiberg-Witten curve for the SU(N+1) N = 2 SUSY Yang-Mills theory. We extend their result to all classical gauge groups and some other cases such as the spectral curve of the A (2) 2N affine Toda Toda system. Our construction, too, uses fractional powers of the superpotential W (x) that characterizes the curve. We a...

The discrete Nahm equations, a system of matrix valued difference equations, arose in the work of Braam and Austin on half-integral mass hyperbolic monopoles. We show that the discrete Nahm equations are completely integrable in a natural sense: to any solution we can associate a spectral curve and a holomorphic linebundle over the spectral curve, such that the discrete-time DN evolution corres...

An automatic algorithm for fairing B-spline curves of general order is presented. This work was motivated by a method of Farin and Sapidis about fairing planar cubic B-spline curves by subsequently removing and reinserting knots. Instead our new algorithm is based on the idea of subsequently changing one control point of a given B-spline curve so that the new curve minimizes the integral of the...

The linking integral is an invariant of the link-type of two manifolds immersed in a Euclidean space. It is shown that the ordinary Gauss integral in three dimensions may be simplified to a winding number integral in two dimensions. This result is then generalized to show that in certain circumstances the linking integral between arbitrary manifolds may be similarly reduced to a lower dimension...

one of the main steps within the geoid computation methodology without applying the stokes formula is downward continuation of the harmonic residual observables from the surface of the earth down to the surface of the reference ellipsoid. this downward continuation is done via the abel-poisson integral and its derivatives. this integral in which the unknowns, i.e. harmonic residual potential va...

A pseudo-line of a real plane curve C is a global real branch of C(R) that is not homologically trivial in P(R). A geometrically integral real plane curve C of degree d has at most d− 2 pseudo-lines, provided that C is not a real projective line. Let C be a real plane curve of degree d having exactly d − 2 pseudo-lines. Suppose that the genus of the normalization of C is equal to d− 2. We show ...

Thus Tf(x) can be viewed as averaging the function f over the curve t → γ(x, t) with respect to the fractional integral kernel k(t). The condition γ(x, 0) = x is a way of saying the curve at x is ”centered at x”, and the condition that ∂γ ∂t (x, t) 6= 0 ensures that the averaging is smooth. In fact, the arguments of this paper will go through with k(t) replaced by k(x, t) satisfying appropriate...