Fundamental Parallelogram Research Articles

Doubly Periodic Functions and Floquet Theorem: DOUBLY PERIODIC FUNCTIONS AND FLOQUET THEOREM

In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions. Just as a periodic function of a real variable is defined by its values on an interval, an elliptic function is determined by its values on a fundamental parallelogram, which then repeat in a lattice. Such a doubly periodic function cannot be holomorphic, as it would then be a bounded entire function, and by Liouville's theorem every such function must be constant. Historically, elliptic functions were first discovered by Niels Henrik Abel as inverse functions of elliptic integrals, and their theory was improved by Carl Gustav Jacobi; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives. In this paper, we extend Floquet theorem and another theorem (which is mentioned in [1]) related to it, which are dependent on elliptic functions.

Two-dimensional Riemann problem for rigid representations on an elliptic curve

We consider a generalization of Riemann–Hilbert problem on elliptic curves. For a given elliptic curve and irreducible representation of free group with two generators we construct explicitly a semistable vector bundle of degree zero obeying a logarithmic connection such that its monodromy over fundamental parallelogram is equivalent to given free group representation, monodromy along a -cycle is trivial and monodromy along b -cycle belong to certain orbit.

Densest translational lattice packing of non-convex polygons

A translational lattice packing of k polygons P 1, P 2, P 3,…, P k is a (non-overlapping) packing of the k polygons which is replicated without overlap at each point of a lattice i 0 v 0+ i 1 v 1, where v 0 and v 1 are vectors generating the lattice and i 0 and i 1 range over all integers. A densest translational lattice packing is one which minimizes the area | v 0× v 1| of the fundamental parallelogram. An algorithm and implementation is given for densest translational lattice packing. This algorithm has useful applications in industry, particularly clothing manufacture.

On Totally Real Cubic Fields

The authors have constructed a table of the 26440 nonconjugate totally real cubic number fields of discriminant $D < 500000$ thereby extending the existing table of fields with $D < 100000$ by I. O. Angell [1]. Serious defects in Angell’s table are pointed out. For each field, running number, discriminant, coefficients of a generating polynomial, integral basis, class number, and a fundamental pair of units are listed. The article contains statistics about the following subjects: distribution of class numbers; fields in which every norm-positive unit is totally positive; nonconjugate fields with the same discriminant; fields with noncyclic class group. The fields are tabulated by means of a method due to Davenport and Heilbronn [7], [8] which leads to a unique normalized generating polynomial. The given units are chosen so that the fundamental parallelogram of the unit lattice determined by the corresponding vectors in the logarithmic space is reduced.

On totally real cubic fields

The authors have constructed a table of the 26440 nonconjugate totally real cubic number fields of discriminant D &gt; 500000 D &gt; 500000 thereby extending the existing table of fields with D &gt; 100000 D &gt; 100000 by I. O. Angell [1]. Serious defects in Angell’s table are pointed out. For each field, running number, discriminant, coefficients of a generating polynomial, integral basis, class number, and a fundamental pair of units are listed. The article contains statistics about the following subjects: distribution of class numbers; fields in which every norm-positive unit is totally positive; nonconjugate fields with the same discriminant; fields with noncyclic class group. The fields are tabulated by means of a method due to Davenport and Heilbronn [7], [8] which leads to a unique normalized generating polynomial. The given units are chosen so that the fundamental parallelogram of the unit lattice determined by the corresponding vectors in the logarithmic space is reduced.

The volume of a lattice polyhedron

Let L be the lattice of all points with integer coordinates in the real affine plane R2 (with respect to some fixed coordinate system). Let X be a finite rectilinear simplicial complex in R2 whose 0-simplexes are points of L. Suppose X is pure and the frontier Ẋ of X is a Jordan curve; then there is a well-known formula for the area of X in terms of the number of points of L which lie in X and Ẋ respectively, namelywhere L(X) (resp. L(Ẋ)) is the number of points of L which lie in X (resp. Ẋ), and μ(X) is the area of X, normalized so that a fundamental parallelogram of L has unit area.

A Tetrahedral Riemann Surface Model of a Closed Finite Locally-Euclidean Two-Space

The existence of a locally-Euclidean two-space whose fundamental region is a parallelogram, with boundary points identified as indicated in Figure 1, has been shown by Felix Klein.* The aim of the present paper is to show that this fundamental parallelogram may be folded in a Euclidean three-space, remaining metrically invariant, to form a closed finite two-sheeted Riemann surface shaped as a tetrahedron and having no metrically singular points whatsoever as a twospace.