Knot Tightening By Constrained Gradient Descent

We present new computations of approximately length-minimizing polygons with fixed thickness. These curves model the centerlines of "tight" knotted tubes with minimal length and fixed circular cross-section. Our curves approximately minimize the ropelength (or quotient of length and thickness) for polygons in their knot types. While previous authors have minimized ropelength for polygons using simulated annealing, the new idea in our code is to minimize length over the set of polygons of thickness at least one using a version of constrained gradient descent. We rewrite the problem in terms of minimizing the length of the polygon subject to an infinite family of differentiable constraint functions. We prove that the polyhedral cone of variations of a polygon of thickness one which do not decrease thickness to first order is finitely generated, and give an explicit set of generators. Using this cone we give a first-order minimization procedure and a Karush-Kuhn-Tucker criterion for polygonal ropelength criticality. Our main numerical contribution is a set of 379 almost-critical prime knots and links, covering all prime knots with no more than 10 crossings and all prime links with no more than 9 crossings. For links, these are the first published ropelength figures, and for knots they improve on existing figures. We give new maps of the self-contacts of these knots and links, and discover some highly symmetric tight knots with particularly simple looking self-contact maps.Comment: 45 pages, 16 figures, includes table of data with upper bounds on ropelength for all prime knots with no more than 10 crossings and all prime links with no more than 9 crossing

PMR BERBASIS WML PADA LUAS DAN VOLUME KERUCUT (WEBSITE-MOBILE-BASED LEARNING OF REALISTIC MATHEMATICS EDUCATION (RME) ON THE MEASUREMENT OF THE AREAN AND VOLUME OF CONE)

Despite of its misuse, such as ethical disobidience by the students in the classroom during the teaching and learning process, handphone (HP) is an urgent device in education that facilitates learning.The problem of the research is “How is the implementation of Website Mobile Learning by Using HP to develop Realistics Mathematics Learning on the measurement of the area and the volume of cone?’. The long-term objective of the research is to develop HP as the media of realistics mathematics learning on the measurement of the area and the volume of cone. While the short-term objective of this research is (1) to create website program on the measurement of the area and the volume of cone that can be accessed through HP. (2) The occurence of the development of Realistics Mathematics Education on the measurement of the area and the volume of cone.In general, this research is an attempt to develop a software application on the measurement of the area and the volume of cone for Website Mobile Learning. It used Developmental research in nature The steps of this research include pre-liminary study covering observation and literary study, system analysis, system design, system development, system testing, system verification and validation, sytem revising and review, system tral, and output analysis.Keyword: Realistic Mathematics Education, Wapstie Mobile Learning, Arean And Volume Of Con

Computational Approaches to Lattice Packing and Covering Problems

We describe algorithms which address two classical problems in lattice geometry: the lattice covering and the simultaneous lattice packing-covering problem. Theoretically our algorithms solve the two problems in any fixed dimension d in the sense that they approximate optimal covering lattices and optimal packing-covering lattices within any desired accuracy. Both algorithms involve semidefinite programming and are based on Voronoi's reduction theory for positive definite quadratic forms, which describes all possible Delone triangulations of Z^d. In practice, our implementations reproduce known results in dimensions d <= 5 and in particular solve the two problems in these dimensions. For d = 6 our computations produce new best known covering as well as packing-covering lattices, which are closely related to the lattice (E6)*. For d = 7, 8 our approach leads to new best known covering lattices. Although we use numerical methods, we made some effort to transform numerical evidences into rigorous proofs. We provide rigorous error bounds and prove that some of the new lattices are locally optimal.Comment: (v3) 40 pages, 5 figures, 6 tables, some corrections, accepted in Discrete and Computational Geometry, see also http://fma2.math.uni-magdeburg.de/~latgeo

Inhomogeneous extreme forms

G.F. Voronoi (1868-1908) wrote two memoirs in which he describes two reduction theories for lattices, well-suited for sphere packing and covering problems. In his first memoir a characterization of locally most economic packings is given, but a corresponding result for coverings has been missing. In this paper we bridge the two classical memoirs. By looking at the covering problem from a different perspective, we discover the missing analogue. Instead of trying to find lattices giving economical coverings we consider lattices giving, at least locally, very uneconomical ones. We classify local covering maxima up to dimension 6 and prove their existence in all dimensions beyond. New phenomena arise: Many highly symmetric lattices turn out to give uneconomical coverings; the covering density function is not a topological Morse function. Both phenomena are in sharp contrast to the packing problem.Comment: 22 pages, revision based on suggestions by referee, accepted in Annales de l'Institut Fourie