Generalization and extension of a problem from an American mathematics competition

ABSTRACT

The purpose of these notes is to generalize and extend a challenging geometry problem from a mathematics competition. The notes also contain solution sketches pertaining to the problems discussed.     

An interesting property of hexagons

These notes discuss several related problems in geometry that can be explored in a dynamic geometry environment. The problems involve an interesting property of hexagons.     

Duplicating the cube and other notes on constructions in the hyperbolic plane

The Euclidean classical problem ofDuplicating the Cube is investigated for the first time inhyperbolic geometry, and its partial solution presented. The other two associated problems ofTrisecting an Angle andSquaring the Circle in the hyperbolic plane were solved classically, but a few additions to these solutions are presented, along with some miscellaneous notes about straightedge and compass constructions in the hyperbolic plane.A preliminary version of this paper was presented at the MAA Undergraduate Student Paper Session, August 1987, Salt Lake City AMS/MAA Joint Meetings. Financial support was provided by a J.W.T. Youngs Undergraduate Award in Mathematics (UC Santa Cruz), and UCSD.     

Validation of Solutions of Construction Problems in Dynamic Geometry Environments

This paper discusses issues concerning the validation of solutions of construction problems in Dynamic Geometry Environments (DGEs) as compared to classic paper-and-pencil Euclidean geometry settings. We begin by comparing the validation criteria usually associated with solutions of construction problems in the two geometry worlds – the ‘drag test’ in DGEs and the use of only straightedge and compass in classic Euclidean geometry. We then demonstrate that the drag test criterion may permit constructions created using measurement tools to be considered valid; however, these constructions prove inconsistent with classical geometry. This inconsistency raises the question of whether dragging is an adequate test of validity, and the issue of measurement versus straightedge-and-compass. Without claiming that the inconsistency between what counts as valid solution of a construction problem in the two geometry worlds is necessarily problematic, we examine what would constitute the analogue of the straightedge-and-compass criterion in the domain of DGEs. Discovery of this analogue would enrich our understanding of DGEs with a mathematical idea that has been the distinguishing feature of Euclidean geometry since its genesis. To advance our goal, we introduce the compatibility criterion, a new but not necessarily superior criterion to the drag test criterion of validation of solutions of construction problems in DGEs. The discussion of the two criteria anatomizes the complexity characteristic of the relationship between DGEs and the paper-and-pencil Euclidean geometry environment, advances our understanding of the notion of geometrical constructions in DGEs, and raises the issue of validation practice maintaining the pace of ever-changing software.     

An unexpected property of quadrilaterals*

These notes discuss several related propositions in geometry that can be explored in a dynamic geometry environment. The propositions involve an unexpected property of quadrilaterals.     

Continuous dependence on geometry for the backward heat equation

In this paper we are concerned with the development of criteria for stabilizing inherently unstable initial-boundary value problems under small errors in the geometry of the underlying domain. We consider in particular the initial-boundary-value problem for the backward heat equation assuming that some error has been made in characterizing the geometry of the domain under consideration. It is shown that solutions which belong to an appropriately defined constraint set depend continuously in L₂ on errors in the geometry.     

Equidistant hypersurfaces of the bidisk

The following are notes on the geometry of the bidisk, H ² × H ². In particular, we examine the properties of equidistant surfaces in the bidisk.     

Using Root Functions for Eigenvalue Problems of Ordinary Differential Operators

We study eigenvalue problems for an ordinary differential operator L acting on L ₂(?)-spaces (Problem 1) and on L ₂(J)-spaces (Problem 2). Here J is a bounded but large interval. Assuming that in Problem 1 the spectral parameter s lies in the set of normal points of L, we show that the structure of eigenspaces for both problems is similar to the structure of finite complex-valued matrices. In the case of a finite matrix, the geometry of eigenspaces is described by the Jordan form. In the case of ordinary differential operators, the corresponding geometry is described by a sequence of root functions. Therefore, the main tool of our studies is root functions for complex-valued analytical matrix functions.     

Scanline algorithms on a grid

A number of important problems in computational geometry are solved efficiently on 2- or 3-dimensional grids by means of scanline techniques. In the time complexity of solutions to the maximal elements and closure problems, a factor logn is substituted by loglogn, wheren is the number of elements. Next, by using a data structure introduced in the paper, the interval trie, previous solutions to the rectangle intersection and connected component problems are improved upon. Finally, a fast intersection finding algorithm for arbitrarily oriented line segments is presented.     

A high order projection method for nonlinear two point boundary value problems

Summary Approximations to the solutions of a general class of 2m-th order nonlinear boundary value problems are developed in spaces of polynomial splines of degree 2m+1 by requiring the residual to be orthogonal to a class of polynomial splines of degree 2m–1 over the same mesh. Conditions are given for existence and uniqueness of approximations along with theoretical error rates. In some cases these rates are shown to be of the same order as the best approximation to the solution over the approximating spline spaces. Some computational notes and the results of numerical experiments are also given.     

Discrete real time flows with quasi-discrete spectra and algebras generated by expq(t)

These notes (essentially unedited) were sent to W. Parry in 1964. The first two parts are complete and in a letter to Parry at that time Hahn indicated his intention to publish them. Evidently he did not manage to do this. The remainder of these notes represents an attempt to establish a theory of quasidiscrete spectra for discrete one-parameter flows. Hahn indicates the gaps and in a following note Parry clarifies his theory. The first part of these notes presents a characteristic example of a discrete one-parameter flow with quasidiscrete spectrum. Ergodicity, minimality and distality are established. The second part examines the Banach algebra of functions onR generated by {expq(t): q a real polynomial of degree <n+1} and shows that the shift isometries arise from a discrete one-parameter flow on its maximal ideal space Λ _n and that ifn is finite this flow is isomorphic to the example examined in the first part. Deceased     

The geometry of multiple integrals depending on parameters

We consider the geometry of an integral in the situation where the multiple integral depends on a finite number of parameters; the problems of exhibiting properties invariant with respect to certain classes of transformations (involving the underlying manifold and the parameter manifold) and the corresponding classification of integrals are solved by the methods of modern differential geometry.Translated fromItogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 22, 1990, pp. 37–58.     

Geometric Applications of a Randomized Optimization Technique

We propose a simple, general, randomized technique to reduce certain geometric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra logarithmic factors in previous, often more complicated, deterministic approaches (such as parametric searching). Faster algorithms are thus obtained for a variety of problems in computational geometry: finding minimal k -point subsets, matching point sets under translation, computing rectilinear p -centers and discrete 1-centers, and solving linear programs with k violations. Received May 23, 1998, and in revised form March 29, 1999.     

Notes on Gromov’s systolic estimate

These notes cover some of the main results of Gromov’s paper Filling Riemannian manifolds. The goal of these notes is to make the results and proofs accessible to more people. The main result is that if (M,g) is a Riemannian manifold of dimension n, then there is a non-contractible curve in (M,g) of length at most C _n Vol(M,g)^1/n .     

Problems on Bieberbach Groups and Flat Manifolds

We present about twenty conjectures, problems and questions about flat manifolds. Many of them build the bridges between the flat world and representation theory of the finite groups, hyperbolic geometry and dynamical systems. In memory of Charles B. Thomas     

基于担保的中小企业融资的集合票据的定价研究

在简约化模型框架下,考虑担保机构的违约对集合发债融资的中小企业有违约传染的影响,通过引进一个几何双曲线衰减函数,得到了集合票据的定价公式,在此基础上对担保集合票据所隐含的信用风险进行分析.结果表明:担保机构的存在能显著降低集合票据的信用利差,提高其市场发行价格;且有担保下,担保机构的违约传染风险因子越大,相应的集合票据价格就越低,违约概率越大,信用利差越高,担保价值越低.     

Lectures on Wonderful Varieties

These notes are an introduction to wonderful varieties. We discuss some general results on their geometry, their role in the theory of spherical varieties, several aspects of the combinatorics arising from these varieties, and some examples.     

Continuous dependence on the geometry and on the initial time for a class of parabolic problems I

In this paper, we investigate the continuous dependence on the geometry and the initial time for solutions u( x , t) of a class of nonlinear parabolic initial‐boundary value problems. Copyright © 2007 John Wiley & Sons, Ltd.     

On the Intrinsic Complexity of Elimination Theory

We consider the intrinsic complexity of selected algorithmic problems of classical elimination theory in algebraic geometry. The inputs and outputs of these problems are given by finite sets of polynomials which we represent alternatively in dense form or by straight line programs. We begin with an overview on the known upper bounds for the sequential and parallel time complexity of these problems and show then that in the most important cases these bounds are tight. Our lower bound results include both the relative and the absolute viewpoint of complexity theory. On one side we give reductions of fundamental questions of elimination theory to NP- and P^#-complete problems and on the other side we show that some of these questions may have exponential size outputs. In this way we confirm the intrinsically exponential character of algorithmic problems in elimination theory whatever the type of data structure may be.     

An application of radon inversion to diffraction tomography

Basic equations of linear computer tomography are considered; a connection of the mathermatical formulation of problems of computer tomography with problems of integral geometry is indicated; the properties of the Radon transform for even- and odd-dimensional spaces are analyzed, and applications of Radon inversion to problems of diffraction tomography are presented. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 264–277. Translated by V. N. Troyan.