Short cycles of Poncelet’s conics

Main results of this paper are the following:1. A closed N-gon interscribed between two conics exists if and only if a specially constructed polygon with a smaller number of sides (n) is closed. To verify the closure of this n-gon, we need to find a periodic solution of a dynamical system of order n. The proof is based on the connection of Poncelet’s curves and matrices that admit unitary bordering [4,9,10,16]. Application of this criterion makes sense when n?N, in particular when n≈log₂N (see Table 4 where n=m₁). So for example we may say that a polygon with 2049 sides interscribed between two circles is closed if and only if some specially constructed 11-gon is closed.2. A closed N-gon interscribed between two confocal ellipses (the billiard case) exists if and only if an N-gon interscribed between two special nested circles is closed.     

Metric properties of poncelet polygonal lines

A generalization of the Radi?-Kaliman theorem on metric relations in Poncelet polygonal lines is obtained in the paper. A general principle of determination of a closure condition Poncelet polygonal line for two circles is derived from this result. In particular, explicit formulas for inscribed-circumscribed n-gons for n = 5, 6, 8 are obtained and an algorithm for determination of such formulas for all n is found.     

Cyclic polygons with rational sides and area

We generalise the notion of Heron triangles to rational-sided, cyclic n-gons with rational area using Brahmagupta's formula for the area of a cyclic quadrilateral and Robbins' formulæ for the area of cyclic pentagons and hexagons. We use approximate techniques to explore rational area n-gons for n greater than six. Finally, we produce a method of generating non-Eulerian rational area cyclic n-gons for even n and conjecturally classify all rational area cyclic n-gons.     

Distance-j-Ovoids in Regular Near 2d-Gons

In [8] valuations were introduced and it was shown that these were important objects for classifying near 2n-gons. Several classes were given including one arising from so-called distance-2j-ovoids. Here we introduce pseudo-valutions and explain why these objects can be important for classifying near (2n+1)-gons. Every valuation of a near polygon gives rise to pseudo-valuations and almost all known examples of pseudo-valuations arise in this way. We show that every distance-(2j+1)-ovoid gives rise to a pseudo-valuation which does not come from a valuation. Subsequently, we study distance-j-ovoids in regular near polygons. We are able to calculate the number of elements of a distance-j-ovoid in two ways, yielding a relation between the parameters of the regular near polygon. We will discuss some cases where this relation can be solved. Postdoctoral Fellow of the Research Foundation - Flanders     

An extension of Asgeirsson's mean value theorem for solutions of the ultra-hyperbolic equation in dimension four

In 1937 Asgeirsson established a mean value property for solutions of the general ultra-hyperbolic equation in 2n variables. In the case of four variables, it states that the integrals of a solution over certain pairs of conjugate circles are the same. In this paper we extend this result to non-degenerate conjugate conics, which include the original case of conjugate circles and adds the new case of conjugate hyperbolae.The broader context of this result is the geometrization of Fritz John's 1938 analysis of the ultra-hyperbolic equation. Solutions of the equation arise as the condition for functions on line space to come from line integrals of functions in Euclidean 3-space, and hence it appears as a compatibility condition for tomographic data.The introduction of the canonical neutral Kaehler metric on the space of oriented lines clarifies the relationship and broadens the paradigm to allow new insights. In particular, it is proven that a solution of the ultra-hyperbolic equation has the mean value property over any pair of curves that arise as the image of John's conjugate circles under a conformal map. These pairs of curves are then shown to be conjugate conics, which include circles and hyperbolae.John identified conjugate circles with the two rulings of a hyperboloid of 1-sheet. Conjugate hyperbolae are identified with the two rulings of either a piece of a hyperboloid of 1-sheet or a hyperbolic paraboloid.     

On the determination of the potential function from given orbits

The paper deals with the problem of finding the field of force that generates a given (N ? 1)-parametric family of orbits for a mechanical system with N degrees of freedom. This problem is usually referred to as the inverse problem of dynamics. We study this problem in relation to the problems of celestial mechanics. We state and solve a generalization of the Dainelli and Joukovski problem and propose a new approach to solve the inverse Suslov’s problem. We apply the obtained results to generalize the theorem enunciated by Joukovski in 1890, solve the inverse Stäckel problem and solve the problem of constructing the potential-energy function U that is capable of generating a bi-parametric family of orbits for a particle in space. We determine the equations for the sought-for function U and show that on the basis of these equations we can define a system of two linear partial differential equations with respect to U which contains as a particular case the Szebehely equation. We solve completely a special case of the inverse dynamics problem of constructing U that generates a given family of conics known as Bertrand’s problem. At the end we establish the relation between Bertrand’s problem and the solutions to the Heun differential equation. We illustrate our results by several examples.     

Dividing a polygon into two similar polygons

If n≥7, then there is no n-gon P that can be divided into two n-gons each being similar to P.     

Limits of chromatic zeros of some families of maps

Two families of maps are considered, one consisting of maps with two pentagons separated by n 5-rings, the other of maps with two n-gons separated by two n-rings. For each family, a homogeneous linear recursion is derived for the corresponding family of chromatic polynomials. It is shown that B₅, B₇, and B₁₀ are limits of sequences of zeros from one or another of the families, where $\text{B}{}_{\mathrm{n}}\text{= 2(1+}\text{cos}\text{2π}\text{n}\text{)}$.     

Graphs Satisfying a Richness Condition and One Concerning Forbidden Subgraphs

We characterize several classes of graphs which satify a richness condition (R) and a condition (F) which concerns forbidden subgraphs. In particular we treat the case where (R) is: Each two vertices have a common neighbor. And this in combination with (F): G has no 5-circle C ₅ resp. no 6-circle C ₆ as subgraph. Another result deals with the case where (R) and (F) concern existence or non-existence of circles C _n .     

Extended Formulations for Polygons

The extension complexity of a polytope?P is the smallest integer?k such that?P is the projection of a polytope?Q with?k facets. We study the extension complexity of n-gons in the plane. First, we give a new proof that the extension complexity of regular n-gons is O(logn), a result originating from work by Ben-Tal and Nemirovski (Math. Oper. Res. 26(2), 193?C205, 2001). Our proof easily generalizes to other permutahedra and simplifies proofs of recent results by Goemans (2009), and Kaibel and Pashkovich (2011). Second, we prove a lower bound of $\sqrt{2n}$ on the extension complexity of generic n-gons. Finally, we prove that there exist n-gons whose vertices lie on an O(n)×O(n ²) integer grid with extension complexity $\varOmega (\sqrt{n}/\sqrt{\log n})$ .     

Minimizing a Quadratic Cost Function of Waiting Times in Single-Machine Scheduling

This paper deals with scheduling n jobs on a single machine in order to minimize the weighted sum of squared waiting times of the jobs. We present a powerful decomposition mechanism, based on a precedence relation concept, that easily handles problems of the size n = 50 and 100 where the processing times and penalties are independently drawn from a uniform distribution. This mechanism is incorporated along with new branching rules in a branch-and-bound scheme that efficiently handles tough problems of the size 20 and 50.     

Non-Hamiltonian simple 3-polytopes having just two types of faces

It is shown that for every value of an integer k, k?11, there exist 3-valent 3-connected planar graphs having just two types of faces, pentagons and k-gons, and which are non- Hamiltonian. Moreover, there exist ?=?(k) > 0, for these values of k, and sequences (G_n)^∞_n=1 of the said graphs for which V(G_n)→∞ and the size of a largest circuit of G_n is at most (1??)V(G_n); similar result for the size of a largest path in such graphs is established for all k, k?11, except possibly for k = 14, 17, 22 and k = 5m+ 5 for all m?2.     

3-Webs generated by confocal conics and circles

We consider families of confocal conics and two pencils of Apollonian circles having the same foci. We will show that these families of curves generate trivial 3-webs and find the exact formulas describing them.     

Rational Functions and Association Scheme Parameters

The parameters of metric, cometric, symmetric association schemes with q ± 1 (the same as the parameters of the underlying orthogonal polynomials) can be given in general by evaluating a single rational function of degree (4, 4) in the complex variable q ^j. But in all known examples, save the simple n-gons, these reduce to polynomials of degree at most 2 in q ^j with q an integer. One reason this occurs is that the rational function can have singularities at points which would determine some of the parameters. This paper deals with the case in which not all of the singularities are removable, thus giving some reason why the n-gons might naturally be the only exceptions to schemes with parameters being polynomials of degree at most 2 in q ^j , except possibly for schemes of very small diameter.     

The tricriterion shortest path problem with at least two bottleneck objective functions

The focus of this paper is on the tricriterion shortest path problem where two objective functions are of the bottleneck type, for example MinMax or MaxMin. The third objective function may be of the same kind or we may consider, for example, MinSum or MaxProd. Let p(n) be the complexity of a classical single objective algorithm responsible for this third function, where n is the number of nodes and m be the number of arcs of the graph. An O(m²p(n)) algorithm is presented that can generate the minimal complete set of Pareto-optimal solutions. Finding the maximal complete set is also possible. Optimality proofs are given and extensions for several special cases are presented. Computational experience for a set of randomly generated problems is reported.     

The classification of quasi-regular polyhedra of genus 2

The method of chamber systems is used to provide a complete list of all possible tessellations of the closed, orientable surface of genus 2 by (topological)n-gons andm-gons (n, m>2) satisfying a certain local symmetry condition. Using a computer program it is shown that (up to homeomorphism) there are precisely 379 such quasi-regular polyhedra. S. Bilinski constructed the first one for each of the 17 possible combinations ofm- andn-gons using geometrical methods. It is the intention of the authors to demonstrate the usefulness and suitability of chamber systems in dealing with problems of the above type.     

Generalized n-gons and Chebychev polynomials

Two sequences of orthogonal polynomials are given whose weight functions consist of an absolutely continuous part and two point masses. Combinatorial proofs of the orthogonality relations are given. The polynomials include natural q-analogs of the Chebychev polynomials. The technique uses association schemes of generalized n-gons to find approximating discrete orthogonality relations. The Feit-Higman Theorem is a corollary of these orthogonality relations for the polynomials.     

Planar curve interpolation by piecewise conics of arbitrary type

Five points in general position inR ² always lie on a unique conic, and three points plus two tangents also have a unique interpolating conic, the type of which depends on the data. These well-known facts from projective geometry are generalized: an odd number 2n+1≥5 of points inR ², if they can be interpolated at all by a smooth curve with nonvanishing curvature, will have a uniqueGC ² interpolant consisting of pieces of conics of varying type. This interpolation process reproduces conics of arbitrary type and preserves strict convexity. Under weak additional assumptions its approximation order is ?(h ⁵), whereh is the maximal distance of adjacent data pointsf(t _i ) sampled from a smooth and regular planar curvef with nonvanishing curvature. Two algorithms for the construction of the interpolant are suggested, and some examples are presented.