Lattice Points, Polyhedra, and Complexity
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ORTHOGONAL GROUP of CERTAIN INDEFINITE LATTICE Chang Heon Kim* 1. Introduction Given an Even Lattice M in a Real Quadratic Space
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 20, No. 1, March 2007 ORTHOGONAL GROUP OF CERTAIN INDEFINITE LATTICE Chang Heon Kim* Abstract. We compute the special orthogonal group of certain lattice of signature (2; 1). 1. Introduction Given an even lattice M in a real quadratic space of signature (2; n), Borcherds lifting [1] gives a multiplicative correspondence between vec- tor valued modular forms F of weight 1¡n=2 with values in C[M 0=M] (= the group ring of M 0=M) and meromorphic modular forms on complex 0 varieties (O(2) £ O(n))nO(2; n)=Aut(M; F ). Here NM denotes the dual lattice of M, O(2; n) is the orthogonal group of M R and Aut(M; F ) is the subgroup of Aut(M) leaving the form F stable under the natural action of Aut(M) on M 0=M. In particular, if the signature of M is (2; 1), then O(2; 1) ¼ H: O(2) £ O(1) and Borcherds' theory gives a lifting of vector valued modular form of weight 1=2 to usual one variable modular form on Aut(M; F ). In this sense in order to work out Borcherds lifting it is important to ¯nd appropriate lattice on which our wanted modular group acts. In this article we will show: Theorem 1.1. Let M be a 3-dimensional even lattice of all 2 £ 2 integral symmetric matrices, that is, ½µ ¶ ¾ AB M = j A; B; C 2 Z BC Received December 30, 2006. 2000 Mathematics Subject Classi¯cation: Primary 11F03, 11H56. -
ON the SHELLABILITY of the ORDER COMPLEX of the SUBGROUP LATTICE of a FINITE GROUP 1. Introduction We Will Show That the Order C
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 7, Pages 2689{2703 S 0002-9947(01)02730-1 Article electronically published on March 12, 2001 ON THE SHELLABILITY OF THE ORDER COMPLEX OF THE SUBGROUP LATTICE OF A FINITE GROUP JOHN SHARESHIAN Abstract. We show that the order complex of the subgroup lattice of a finite group G is nonpure shellable if and only if G is solvable. A by-product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups. 1. Introduction We will show that the order complex of the subgroup lattice of a finite group G is (nonpure) shellable if and only if G is solvable. The proof of nonshellability in the nonsolvable case involves the determination of the homotopy type of the order complexes of the subgroup lattices of many minimal simple groups. We begin with some history and basic definitions. It is assumed that the reader is familiar with some of the rudiments of algebraic topology and finite group theory. No distinction will be made between an abstract simplicial complex ∆ and an arbitrary geometric realization of ∆. Maximal faces of a simplicial complex ∆ will be called facets of ∆. Definition 1.1. A simplicial complex ∆ is shellable if the facets of ∆ can be ordered σ1;::: ,σn so that for all 1 ≤ i<k≤ n thereexistssome1≤ j<kand x 2 σk such that σi \ σk ⊆ σj \ σk = σk nfxg. The list σ1;::: ,σn is called a shelling of ∆. -
7 LATTICE POINTS and LATTICE POLYTOPES Alexander Barvinok
7 LATTICE POINTS AND LATTICE POLYTOPES Alexander Barvinok INTRODUCTION Lattice polytopes arise naturally in algebraic geometry, analysis, combinatorics, computer science, number theory, optimization, probability and representation the- ory. They possess a rich structure arising from the interaction of algebraic, convex, analytic, and combinatorial properties. In this chapter, we concentrate on the the- ory of lattice polytopes and only sketch their numerous applications. We briefly discuss their role in optimization and polyhedral combinatorics (Section 7.1). In Section 7.2 we discuss the decision problem, the problem of finding whether a given polytope contains a lattice point. In Section 7.3 we address the counting problem, the problem of counting all lattice points in a given polytope. The asymptotic problem (Section 7.4) explores the behavior of the number of lattice points in a varying polytope (for example, if a dilation is applied to the polytope). Finally, in Section 7.5 we discuss problems with quantifiers. These problems are natural generalizations of the decision and counting problems. Whenever appropriate we address algorithmic issues. For general references in the area of computational complexity/algorithms see [AB09]. We summarize the computational complexity status of our problems in Table 7.0.1. TABLE 7.0.1 Computational complexity of basic problems. PROBLEM NAME BOUNDED DIMENSION UNBOUNDED DIMENSION Decision problem polynomial NP-hard Counting problem polynomial #P-hard Asymptotic problem polynomial #P-hard∗ Problems with quantifiers unknown; polynomial for ∀∃ ∗∗ NP-hard ∗ in bounded codimension, reduces polynomially to volume computation ∗∗ with no quantifier alternation, polynomial time 7.1 INTEGRAL POLYTOPES IN POLYHEDRAL COMBINATORICS We describe some combinatorial and computational properties of integral polytopes. -
INTEGER POINTS and THEIR ORTHOGONAL LATTICES 2 to Remove the Congruence Condition
INTEGER POINTS ON SPHERES AND THEIR ORTHOGONAL LATTICES MENNY AKA, MANFRED EINSIEDLER, AND URI SHAPIRA (WITH AN APPENDIX BY RUIXIANG ZHANG) Abstract. Linnik proved in the late 1950’s the equidistribution of in- teger points on large spheres under a congruence condition. The congru- ence condition was lifted in 1988 by Duke (building on a break-through by Iwaniec) using completely different techniques. We conjecture that this equidistribution result also extends to the pairs consisting of a vector on the sphere and the shape of the lattice in its orthogonal complement. We use a joining result for higher rank diagonalizable actions to obtain this conjecture under an additional congruence condition. 1. Introduction A theorem of Legendre, whose complete proof was given by Gauss in [Gau86], asserts that an integer D can be written as a sum of three squares if and only if D is not of the form 4m(8k + 7) for some m, k N. Let D = D N : D 0, 4, 7 mod8 and Z3 be the set of primitive∈ vectors { ∈ 6≡ } prim in Z3. Legendre’s Theorem also implies that the set 2 def 3 2 S (D) = v Zprim : v 2 = D n ∈ k k o is non-empty if and only if D D. This important result has been refined in many ways. We are interested∈ in the refinement known as Linnik’s problem. Let S2 def= x R3 : x = 1 . For a subset S of rational odd primes we ∈ k k2 set 2 D(S)= D D : for all p S, D mod p F× . -
A Parallelepiped Based Approach
3D Modelling Using Geometric Constraints: A Parallelepiped Based Approach Marta Wilczkowiak, Edmond Boyer, and Peter Sturm MOVI–GRAVIR–INRIA Rhˆone-Alpes, 38330 Montbonnot, France, [email protected], http://www.inrialpes.fr/movi/people/Surname Abstract. In this paper, efficient and generic tools for calibration and 3D reconstruction are presented. These tools exploit geometric con- straints frequently present in man-made environments and allow cam- era calibration as well as scene structure to be estimated with a small amount of user interactions and little a priori knowledge. The proposed approach is based on primitives that naturally characterize rigidity con- straints: parallelepipeds. It has been shown previously that the intrinsic metric characteristics of a parallelepiped are dual to the intrinsic charac- teristics of a perspective camera. Here, we generalize this idea by taking into account additional redundancies between multiple images of multiple parallelepipeds. We propose a method for the estimation of camera and scene parameters that bears strongsimilarities with some self-calibration approaches. Takinginto account prior knowledgeon scene primitives or cameras, leads to simpler equations than for standard self-calibration, and is expected to improve results, as well as to allow structure and mo- tion recovery in situations that are otherwise under-constrained. These principles are illustrated by experimental calibration results and several reconstructions from uncalibrated images. 1 Introduction This paper is about using partial information on camera parameters and scene structure, to simplify and enhance structure from motion and (self-) calibration. We are especially interested in reconstructing man-made environments for which constraints on the scene structure are usually easy to provide. -
Area, Volume and Surface Area
The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 11 AREA, VOLUME AND SURFACE AREA A guide for teachers - Years 8–10 June 2011 YEARS 810 Area, Volume and Surface Area (Measurement and Geometry: Module 11) For teachers of Primary and Secondary Mathematics 510 Cover design, Layout design and Typesetting by Claire Ho The Improving Mathematics Education in Schools (TIMES) Project 2009‑2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the international Centre of Excellence for Education in Mathematics (ICE‑EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution‑NonCommercial‑NoDerivs 3.0 Unported License. http://creativecommons.org/licenses/by‑nc‑nd/3.0/ The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 11 AREA, VOLUME AND SURFACE AREA A guide for teachers - Years 8–10 June 2011 Peter Brown Michael Evans David Hunt Janine McIntosh Bill Pender Jacqui Ramagge YEARS 810 {4} A guide for teachers AREA, VOLUME AND SURFACE AREA ASSUMED KNOWLEDGE • Knowledge of the areas of rectangles, triangles, circles and composite figures. • The definitions of a parallelogram and a rhombus. • Familiarity with the basic properties of parallel lines. • Familiarity with the volume of a rectangular prism. • Basic knowledge of congruence and similarity. • Since some formulas will be involved, the students will need some experience with substitution and also with the distributive law. -
Evaporationin Icparticles
The JapaneseAssociationJapanese Association for Crystal Growth (JACG){JACG) Small Metallic ParticlesProduced by Evaporation in lnert Gas at Low Pressure Size distributions, crystal morphologyand crystal structures KazuoKimoto physiosLaboratory, Department of General Educatien, Nago)ra University Various experimental results ef the studies on fine 1. Introduction particles produced by evaperation and subsequent conden$ation in inert gas at low pressure are reviewed. Small particles of metals and semi-metals can A brief historical survey is given and experimental be produced by evaporation and subsequent arrangements for the production of the particles are condensation in the free space of an inert gas at described. The structure of the stnoke, the qualitative low pressure, a very simple technique, recently particle size clistributions, small particle statistios and "gas often referred to as evaporation technique"i) the crystallographic aspccts ofthe particles are consid- ered in some detail. Emphasis is laid on the crystal (GET). When the pressure of an inert is in the inorphology and the related crystal structures ef the gas range from about one to several tens of Torr, the particles efsome 24 elements. size of the particles produced by GET is in thc range from several to several thousand nm, de- pending on the materials evaporated, the naturc of the inert gas and various other evaporation conditions. One of the most characteristic fea- tures of the particles thus produced is that the particles have, generally speaking, very well- defined crystal habits when the particle size is in the range from about ten to several hundred nm. The crystal morphology and the relevant crystal structures of these particles greatly interested sDme invcstigators in Japan, 4nd encouraged them to study these propenies by means of elec- tron microscopy and electron difliraction. -
Groups with Identical Subgroup Lattices in All Powers
GROUPS WITH IDENTICAL SUBGROUP LATTICES IN ALL POWERS KEITH A. KEARNES AND AGNES´ SZENDREI Abstract. Suppose that G and H are groups with cyclic Sylow subgroups. We show that if there is an isomorphism λ2 : Sub (G × G) ! Sub (H × H), then there k k are isomorphisms λk : Sub (G ) ! Sub (H ) for all k. But this is not enough to force G to be isomorphic to H, for we also show that for any positive integer N there are pairwise nonisomorphic groups G1; : : : ; GN defined on the same finite set, k k all with cyclic Sylow subgroups, such that Sub (Gi ) = Sub (Gj ) for all i; j; k. 1. Introduction To what extent is a finite group determined by the subgroup lattices of its finite direct powers? Reinhold Baer proved results in 1939 implying that an abelian group G is determined up to isomorphism by Sub (G3) (cf. [1]). Michio Suzuki proved in 1951 that a finite simple group G is determined up to isomorphism by Sub (G2) (cf. [10]). Roland Schmidt proved in 1981 that if G is a finite, perfect, centerless group, then it is determined up to isomorphism by Sub (G2) (cf. [6]). Later, Schmidt proved in [7] that if G has an elementary abelian Hall normal subgroup that equals its own centralizer, then G is determined up to isomorphism by Sub (G3). It has long been open whether every finite group G is determined up to isomorphism by Sub (G3). (For more information on this problem, see the books [8, 11].) One may ask more generally to what extent a finite algebraic structure (or algebra) is determined by the subalgebra lattices of its finite direct powers. -
Crystallography Ll Lattice N-Dimensional, Infinite, Periodic Array of Points, Each of Which Has Identical Surroundings
crystallography ll Lattice n-dimensional, infinite, periodic array of points, each of which has identical surroundings. use this as test for lattice points A2 ("bcc") structure lattice points Lattice n-dimensional, infinite, periodic array of points, each of which has identical surroundings. use this as test for lattice points CsCl structure lattice points Choosing unit cells in a lattice Want very small unit cell - least complicated, fewer atoms Prefer cell with 90° or 120°angles - visualization & geometrical calculations easier Choose cell which reflects symmetry of lattice & crystal structure Choosing unit cells in a lattice Sometimes, a good unit cell has more than one lattice point 2-D example: Primitive cell (one lattice pt./cell) has End-centered cell (two strange angle lattice pts./cell) has 90° angle Choosing unit cells in a lattice Sometimes, a good unit cell has more than one lattice point 3-D example: body-centered cubic (bcc, or I cubic) (two lattice pts./cell) The primitive unit cell is not a cube 14 Bravais lattices Allowed centering types: P I F C primitive body-centered face-centered C end-centered Primitive R - rhombohedral rhombohedral cell (trigonal) centering of trigonal cell 14 Bravais lattices Combine P, I, F, C (A, B), R centering with 7 crystal systems Some combinations don't work, some don't give new lattices - C tetragonal C-centering destroys cubic = P tetragonal symmetry 14 Bravais lattices Only 14 possible (Bravais, 1848) System Allowed centering Triclinic P (primitive) Monoclinic P, I (innerzentiert) Orthorhombic P, I, F (flächenzentiert), A (end centered) Tetragonal P, I Cubic P, I, F Hexagonal P Trigonal P, R (rhombohedral centered) Choosing unit cells in a lattice Unit cell shape must be: 2-D - parallelogram (4 sides) 3-D - parallelepiped (6 faces) Not a unit cell: Choosing unit cells in a lattice Unit cell shape must be: 2-D - parallelogram (4 sides) 3-D - parallelepiped (6 faces) Not a unit cell: correct cell Stereographic projections Show or represent 3-D object in 2-D Procedure: 1. -
Polyhedral Combinatorics, Complexity & Algorithms
POLYHEDRAL COMBINATORICS, COMPLEXITY & ALGORITHMS FOR k-CLUBS IN GRAPHS By FOAD MAHDAVI PAJOUH Bachelor of Science in Industrial Engineering Sharif University of Technology Tehran, Iran 2004 Master of Science in Industrial Engineering Tarbiat Modares University Tehran, Iran 2006 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY July, 2012 COPYRIGHT c By FOAD MAHDAVI PAJOUH July, 2012 POLYHEDRAL COMBINATORICS, COMPLEXITY & ALGORITHMS FOR k-CLUBS IN GRAPHS Dissertation Approved: Dr. Balabhaskar Balasundaram Dissertation Advisor Dr. Ricki Ingalls Dr. Manjunath Kamath Dr. Ramesh Sharda Dr. Sheryl Tucker Dean of the Graduate College iii Dedicated to my parents iv ACKNOWLEDGMENTS I would like to wholeheartedly express my sincere gratitude to my advisor, Dr. Balabhaskar Balasundaram, for his continuous support of my Ph.D. study, for his patience, enthusiasm, motivation, and valuable knowledge. It has been a great honor for me to be his first Ph.D. student and I could not have imagined having a better advisor for my Ph.D. research and role model for my future academic career. He has not only been a great mentor during the course of my Ph.D. and professional development but also a real friend and colleague. I hope that I can in turn pass on the research values and the dreams that he has given to me, to my future students. I wish to express my warm and sincere thanks to my wonderful committee mem- bers, Dr. Ricki Ingalls, Dr. Manjunath Kamath and Dr. Ramesh Sharda for their support, guidance and helpful suggestions throughout my Ph.D. -
Cross Product Review
12.4 Cross Product Review: The dot product of uuuu123, , and v vvv 123, , is u v uvuvuv 112233 uv u u u u v u v cos or cos uv u and v are orthogonal if and only if u v 0 u uv uv compvu projvuv v v vv projvu cross product u v uv23 uv 32 i uv 13 uv 31 j uv 12 uv 21 k u v is orthogonal to both u and v. u v u v sin Geometric description of the cross product of the vectors u and v The cross product of two vectors is a vector! • u x v is perpendicular to u and v • The length of u x v is u v u v sin • The direction is given by the right hand side rule Right hand rule Place your 4 fingers in the direction of the first vector, curl them in the direction of the second vector, Your thumb will point in the direction of the cross product Algebraic description of the cross product of the vectors u and v The cross product of uu1, u 2 , u 3 and v v 1, v 2 , v 3 is uv uv23 uvuv 3231,, uvuv 1312 uv 21 check (u v ) u 0 and ( u v ) v 0 (u v ) u uv23 uvuv 3231 , uvuv 1312 , uv 21 uuu 123 , , uvu231 uvu 321 uvu 312 uvu 132 uvu 123 uvu 213 0 similary: (u v ) v 0 length u v u v sin is a little messier : 2 2 2 2 2 2 2 2uv 2 2 2 uvuv sin22 uv 1 cos uv 1 uvuv 22 uv now need to show that u v2 u 2 v 2 u v2 (try it..) An easier way to remember the formula for the cross products is in terms of determinants: ab 12 2x2 determinant: ad bc 4 6 2 cd 34 3x3 determinants: An example Copy 1st 2 columns 1 6 2 sum of sum of 1 6 2 1 6 forward backward 3 1 3 3 1 3 3 1 diagonal diagonal 4 5 2 4 5 2 4 5 products products determinant = 2 72 30 8 15 36 40 59 19 recall: uv uv23 uvuv 3231, uvuv 1312 , uv 21 i j k i j k i j u1 u 2 u 3 u 1 u 2 now we claim that uvu1 u 2 u 3 v1 v 2 v 3 v 1 v 2 v1 v 2 v 3 iuv23 j uv 31 k uv 12 k uv 21 i uv 32 j uv 13 u v uv23 uv 32 i uv 13 uv 31 j uv 12 uv 21 k uv uv23 uvuv 3231,, uvuv 1312 uv 21 Example: Let u1, 2,1 and v 3,1, 2 Find u v. -
Groups with Almost Modular Subgroup Lattice Provided by Elsevier - Publisher Connector
Journal of Algebra 243, 738᎐764Ž. 2001 doi:10.1006rjabr.2001.8886, available online at http:rrwww.idealibrary.com on View metadata, citation and similar papers at core.ac.uk brought to you by CORE Groups with Almost Modular Subgroup Lattice provided by Elsevier - Publisher Connector Francesco de Giovanni and Carmela Musella Dipartimento di Matematica e Applicazioni, Uni¨ersita` di Napoli ‘‘Federico II’’, Complesso Uni¨ersitario Monte S. Angelo, Via Cintia, I 80126, Naples, Italy and Yaroslav P. Sysak1 Institute of Mathematics, Ukrainian National Academy of Sciences, ¨ul. Tereshchenki¨ska 3, 01601 Kie¨, Ukraine Communicated by Gernot Stroth Received November 14, 2000 DEDICATED TO BERNHARD AMBERG ON THE OCCASION OF HIS 60TH BIRTHDAY 1. INTRODUCTION A subgroup of a group G is called modular if it is a modular element of the lattice ᑦŽ.G of all subgroups of G. It is clear that everynormal subgroup of a group is modular, but arbitrarymodular subgroups need not be normal; thus modularitymaybe considered as a lattice generalization of normality. Lattices with modular elements are also called modular. Abelian groups and the so-called Tarski groupsŽ i.e., infinite groups all of whose proper nontrivial subgroups have prime order. are obvious examples of groups whose subgroup lattices are modular. The structure of groups with modular subgroup lattice has been described completelybyIwasawa wx4, 5 and Schmidt wx 13 . For a detailed account of results concerning modular subgroups of groups, we refer the reader towx 14 . 1 This work was done while the third author was visiting the Department of Mathematics of the Universityof Napoli ‘‘Federico II.’’ He thanks the G.N.S.A.G.A.