A new calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F , called F α-integral, where α is the dimension of F . A derivative along the fractal curve called Fα-derivative, is also defined. The mass function, a measurelike algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algori...

In this paper, a new basis, to be called C-Bézier basis, is constructed for the space Γn = span{1, t, t2, . . . , tn−2, sin t, cos t} by an integral approach. Based on this basis, we define C-Bézier curves. We then show that such basis and curves share the same properties as the Bernstein basis and the Bézier curves in polynomial spaces respectively.  2003 Elsevier Science B.V. All rights rese...

Eichler integrals play an integral part in the modular parametrizations of elliptic curves. In her master’s thesis, Kodgis conjectures several dozen zeros of Eichler integrals for elliptic curves with conductor ≤ 179. In this paper we prove a general theorem which confirms many of these conjectured zeros. We also provide two ways to generate infinite families of elliptic curves with certain zer...

The paper studies boundary integral operators of the bi{Laplacian on piecewise smooth curves with corners and describes their mapping properties in the trace spaces of variational solutions of the biharmonic equation. We formulate a direct integral equation method for solving mixed boundary value problems for the biharmonic equation on a nonsmooth plane domain, analyse the solvability of the co...

It is frequently important to approximate a rational Bézier curve by an integral, i.e., polynomial one. This need will arise when a rational Bézier curve is produced in one CAD system and is to be imported into another system, which can only handle polynomial curves. The objective of this paper is to present an algorithm to approximate rational Bézier curves with polynomial curves of higher deg...

Let X be a normal projective variety admitting an action of a semisimple group with a unique closed orbit. We construct finitely many rational curves in X , all having a common point, such that every effective one–cycle on X is rationally equivalent to a unique linear combination of these curves with non–negative rational coefficients. When X is nonsingular, these curves are projective lines, a...

Max Noether’s Theorem asserts that if ω is the dualizing sheaf of a nonsingular nonhyperelliptic projective curve then the natural morphisms SymH(ω) → H(ω) are surjective for all n ≥ 1. This is true for Gorenstein nonhyperelliptic curves as well. We prove this remains true for nearly Gorenstein curves and for all integral nonhyperelliptic curves whose non-Gorenstein points are unibranch. The re...

§0. Introduction §1. The Étale Integral Structure on the Universal Extension §2. The Étale Integral Structure for an Ordinary Elliptic Curve §2.1. Some p-adic Function Theory §2.2. The Verschiebung Morphism §3. Compactified Hodge Torsors §4. The Étale Integral Structure on the Hodge Torsors §4.1. Notation and Set-Up §4.2. Degenerating Elliptic Curves §4.3. Ordinary Elliptic Curves §4.4. The Gen...

There exist five families of Lipschitz curves on the unit square such that any continuous function is uniquely defined by the values of its integral (properly defined) along these curves. We present this uniqueness result as a consequence of the Kolmogorov’s superposition theorem.

We prove that, when elliptic curves E/Q are ordered by height, the average number of integral points #|E(Z)| is bounded, and in fact is less than 66 (and at most 8 9 on the minimalist conjecture). By “E(Z)” we mean the integral points on the corresponding quasiminimal Weierstrass model EA,B : y2 = x3 + Ax + B with which one computes the naı̈ve height. The methods combine ideas from work of Silve...