Polynomials and spatial Pick-type theorems

Pick's theorem about the area of a simple lattice planar polygon has many extensions and generalizations even in the planar case. The theorem has also higher-dimensional generalizations, which are not as commonly known as the 2-dimensional case. The aim of the paper is, on one hand, to give a few new higher-dimensional generalizations of Pick's theorem and, on the other hand, collect known ones. We also study some relationships between lattice points in a lattice polyhedron which lead to some new Pick-type formulae. Another purpose of this paper is to pose several problems related to the subject of higher-dimensional Pick-type theorems. We hope that the paper may popularize the idea of determining the volume of a lattice polyhedron P by reading information contained in a lattice and the tiling of the space generated by the lattice.     

Local convexity and starshaped sets

Previously [7] we proved among other results that a closed connected set inE _n which has a unique point of local nonconvexity is starshaped. Here we characterize a fairly large class of plane sets whose points of local nonconvexity are so arranged that starshapedness follows. This theory determines as a special case the simple closed polygonal regions which are starshaped. In order to proceed simply we utilize the following notations and definitions. The preparation of this paper was supported in part by NSF Grant GP-1988.     

The boundary characteristic and Pick's theorem in the Archimedean planar tilings

The tiling of the plane by unit squares is only one of the 11 Archimedean tilings which use regular polygons and have only one type of vertex. In this paper, the boundary characteristic of a lattice polygon is defined for every Archimedean tiling, and related enumeration formulae are found. Pick's theorem (on the area of a lattice polygon in the tilling by squares) is then generalized for lattice polygons in each of the Archimedean tilings, by enumerating the number of tiles of each type in the polygon.     

A Note on Collars of Simple Closed Geodesics

For compact hyperbolic Riemann surfaces, the collar theorem gives a lower bound on the distance between a simple closed geodesic and all other simple closed geodesics that do not intersect the initial geodesic. Here it is shown that there are two possible configurations, and in each configuration there is a natural collar width associated to a simple closed geodesic. If one extends the natural collar of a simple closed geodesic α by ε >0, then the extended collar contains an infinity of simple closed geodesics that do not intersect α.Mathematics Subject Classiffications (2000). primary: 30F45; secondary: 32G07     

A THEOREM OF BÄCKLUND REVISITED

Abstract

Bäcklund's theorem states that the most general contact transformation is an extended point transformation whenever both the number of independent variables and the number of dependent variables exceed one. A partial circumvention of Bäcklund's theorem is obtained by assigning each dependent variable its own distinct manifold of independent variables. This gives rise to extended symplectic product structures. sequences of extended Hamiltonians, and Lie groups of regular maps that satisfy systems of extended Hamilton-Jacobi equations provided the initial data is determined by a regular map. These ideas are applied to the study of systems of nonlinear second order partial differential equations. Lie groups of solutions are shown to be obtained by solving systems of extended Hamilton-Jacobi equations provided the initial data defines a solution.     

On a conjecture of S. Reich

Simeon Reich (1974) proved that the fixed point theorem for single-valued mappings proved by Boyd and Wong can be generalized to multivalued mappings which map points into compact sets. He then asked (1983) whether his theorem can be extended to multivalued mappings whose range consists of bounded closed sets. In this note, we provide an affirmative answer for a certain subclass of Boyd-Wong contractive mappings.

     

McShane’s identity,using elliptic elements

We introduce a new method to establish McShane’s Identity. Elliptic elements of order two in the Fuchsian group uniformizing the quotient of a fixed once-punctured hyperbolic torus act so as to exclude points as being highest points of geodesics. The highest points of simple closed geodesics are already given as the appropriate complement of the regions excluded by those elements of order two that factor hyperbolic elements whose axis projects to be simple. The widths of the intersection with an appropriate horocycle of the excluded regions sum to give McShane’s value of 1/2. The remaining points on the horocycle are highest points of simple open geodesics, we show that this set has zero Hausdorff dimension.      

Minkowski’s theorem for arbitrary convex sets

V. Klee extended a well-known theorem of Minkowski to non-compact convex sets. We generalize Minkowski’s theorem to convex sets which are not necessarily closed.     

Kostant's convexity theorem and the compact classical groups

The relationship between the classical Schur-Horn's theorem on the diagonal elements of a Hermitian matrix with prescribed eigenvalues and Kostant's convexity theorem in the context of Lie groups. By using Kostant's convexity theorem, we work out the statements on the special orthogonal group and the symplectic group explicitly. Schur-Horn's result can be stated in terms of a set of inequalities. The counterpart in the Lie-theoretic context is related to a partial ordering, introduced by Atiyah and Bott, defined on the closed fundamental Weyl chamber. Some results of Thompson on the diagonal elements of a matrix with prescribed singular values are recovered. Thompson-Poon's theorem on the convex hull of Hermitian matrices with prescribed eigenvalues is also generalized. Then a result of Atiyah-Bott is recovered.     

Class numbers of quadratic function fields and continued fractions

A function field version of a theorem of F. Hirzebruch relating continued fractions to class numbers of quadratic number fields is established. Our approach is based on Artin's thesis and Zagier's proof of Hirzebruch's theorem. Some of our results seem to be of independent interest, e.g. explicit formulas for Zeta functions of real quadratic function fields.     

On a class of matrices generated by certain generalized permutation matrices and applications

ABSTRACT

In this paper, we study a particular class of matrices generated by generalized permutation matrices corresponding to a subgroup of some permutation group. As applications, we first present a technique from which we can get closed formulas for the roots of many families of polynomial equations with degree between 5 and 10, inclusive. Then, we describe a tool that shows how to find solutions to Fermat's last theorem and Beal's conjecture over the square integer matrices of any dimension. Finally, simple generalizations of some of the concepts in number theory to integer square matrices are presented.     

Irreducible values of polynomials

Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove a quantitative arithmetic analog of this conjecture for polynomial rings over pseudo algebraically closed fields. This implies results over large finite fields via model theory. A main tool in the proof is an irreducibility theorem à la Hilbert.     

Raney and Catalan

Raney’s lemma is often used in a counting argument to prove the formula for (generalized) Catalan numbers. It ensures the existence of “good” cyclic shifts of certain sequences, i.e. cyclic shifts for which all partial sums are positive.We introduce a simple algorithm that finds these cyclic shifts and also those with a slightly weaker property. Moreover it provides simple proofs of lemma’s of Raney type.A similar clustering procedure is also used in a simple proof of a theorem on probabilities of which many well-known results (e.g. on lattice paths and on generalized Catalan numbers) can be derived as corollaries. The theorem generalizes generalized Catalan numbers. In the end it turns out to be equivalent to a formula of Raney.     

Some topological properties of support points of convex sets

It is shown, as a corollary to Trojanski’s renorming theorem, that the set of support points (of certain closed convex sets) is connected. Some connectedness properties are established for support functionals, and it is also shown that the support points form an F_σ.     

On the number of singularities,zero curvature points and vertices of a simple convex space curve

We prove a generalization of the 4 vertex theorem forC ³ closed simple convex space curves including singular and zero curvature points.Work partially supported by CNPq. The second author is also grateful to the Universidade Federal de Viçosa (Brasil) for hospitality during the production of this work.     

Acute Triangulations of Polygons

This paper proposes a combinatorial approach to planning non-colliding trajectories for a polygonal bar-and-joint framework with n vertices. It is based on a new class of simple motions induced by expansive one-degree-of-freedom mechanisms, which guarantee noncollisions by moving all points away from each other. Their combinatorial structure is captured by pointed pseudo-triangulations, a class of embedded planar graphs for which we give several equivalent characterizations and exhibit rich rigidity theoretic properties. The main application is an efficient algorithm for the Carpenter’s Rule Problem: convexify a simple bar-and-joint planar polygonal linkage using only non-self-intersecting planar motions. A step of the algorithm consists in moving a pseudo-triangulation-based mechanism along its unique trajectory in configuration space until two adjacent edges align. At the alignment event, a local alteration restores the pseudo-triangulation. The motion continues for O(n3) steps until all the points are in convex position. An erratum to this article is available at .     

Stability regions of linear canonical systems with periodic coefficients

A new definition of the index of the stability region of a canonical system of linear differential equations with periodic coefficients is proposed. A simple proof of the Gel'fand-Lidskii theorem [1] on the structure of stability regions is given and a theorem on the directional convexity of such regions is proved. It follows from this theorem, in particular, that stability regions of parametric oscillations in a system with a sign-definite Hamiltonian are convex with respect to the frequency of parametric perturbation.     

A Shortcoming in the Geometrically Non‐Linear Shakedown Theorem

The original static shakedown theorem of Melan [1], valid for geometrically linear theory, was extended for geometrically non‐linear theory e.g. by Polizzotto and Borino [2], who presented a proof for large rotations with small strains. However, a counterexample to this extended Melan's theorem has been found. The reason of the failure is investigated and is corrected by an additional condition in the theorem. The outline of the proof is given.     

Stability of solutions for Ky Fan's section theorem with some applications

In this paper, we prove that “most of” problems in Ky Fan's section theorem (in the sense of Baire category) are essential and that for any problem in Ky Fan's section theorem, there exists at least one essential component of its solution set. As applications, we deduce both the existence of essential components of the set of Ky Fan's points based on Ky Fan's minimax inequality theorem and the existence of essential components of the set of Nash equilibrium points for general n-person non-cooperative games with non-concave payoffs.     

Polygonal chains cannot lock in 4D

We prove that, in all dimensions d4, every simple open polygonal chain and every tree may be straightened, and every simple closed polygonal chain may be convexified. These reconfigurations can be achieved by algorithms that use polynomial time in the number of vertices, and result in a polynomial number of “moves”. These results contrast to those known for d=2, where trees can “lock”, and for d=3, where open and closed chains can lock.