This paper proposes a heuristic algorithm which, given a basis of a subspace of the space of cuspforms of weight 2 for Γ0(N) which is invariant for the action of the Hecke operators, tests whether the subspace corresponds to a quotient A of the jacobian of the modular curve X0(N) such that A is the jacobian of a curve C. Moreover, equations for such a curve C are computed which make the quotien...

We are given a two-dimensional square grid of size N N, where N := 2 n and n 0. A space lling curve (SFC) is a numbering of the cells of this grid with numbers from c + 1 to c + N 2 , for some c 0. We call a SFC recursive (RSFC) if it can be recursively divided into four square RSFCs of equal size. Examples of well-known RSFCs include the Hilbert curve, the z-curve, and the Gray code. We prove ...

Let Dn denote the n-punctured disk in the complex plane, where the punctures are on the real axis. An n-braid α is said to be reducible if there exists an essential curve system C in Dn, called a reduction system of α, such that α ∗ C = C where α ∗ C denotes the action of the braid α on the curve system C. A curve system C in Dn is said to be standard if each of its components is isotopic to a ...

Each orientable surface with nonempty boundary can be associated with a planar model, whose edges can then be labeled with letters that read out a surface word. Then, the curve word of a free homotopy class of closed curves on a surface is the minimal sequence of edges of the planar model through which a curve in the class passes. The length of a class of curves is defined to be the number of l...

It is shown that a simple closed curve in $${\mathbb {C}}^n$$ uniform limit of rectifiable curves each which has nontrivial polynomial hull itself hull. In case the rectifiable, to be hulls. also every , $$n\ge 2$$ can approximated total variation norm by polynomially convex, coincides with original except on an arbitrarily small segment. As corollary, it arc contained curve.

We describe each multiple curve on an orientable surface of genus-1, with n punctures and one boundary component by using this curve?s geometric intersection numbers the embedded curves in surface.

Let {pn(z)}n=0 be a sequence of complex polynomials (pn of degree n) that are orthonormal with respect to the area measure over the interior domain of an analytic Jordan curve. We prove that each pn of sufficiently large degree has a primitive that can be expanded in a series of functions recursively generated by a couple of integral transforms whose kernels are defined in terms of the degree n...

Elliptic curve cryptographic protocols often make use of the inherent hardness of the discrete logarithm problem, which is to solve kG = P for k. There is an abundance of evidence suggesting that elliptic curve cryptography is more secure than the classical case. One reason for this is the best known general-purpose algorithm to solve the elliptic curve discrete logarithm problem is Pollard’s R...