Introduction to Computational Manifolds and Applications
|
|
- Gladys Sharp
- 6 years ago
- Views:
Transcription
1 IMPA - Instituto de Matemática Pura e Aplicada, Rio de Janeiro, RJ, Brazil Introduction to Computational Manifolds and Applications Part 1 - Foundations Prof. Jean Gallier jean@cis.upenn.edu Department of Computer and Information Science University of Pennsylvania Philadelphia, PA, USA
2 Our definition of manifold is not constructive: it states what a manifold is by assuming that the space already exists. What if we are interested in constructing" a manifold? It turns out that a manifold can be built from what we call a set of gluing data. The idea is to glue open sets in E n in a controlled manner, and then embed them in E d. André Weil introduced this gluing process to define abstract algebraic varieties from irreducible affine sets in a book published in However, as far as we know, Cindy Grimm and John Hughes were the first to give a constructive definition of manifold. SIGGRAPH, 1995 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 2
3 The pioneering work of Grimm and Hughes allows us to create smooth 2-manifolds (i.e., smooth surfaces equipped with an atlas) in E 3 for the purposes of modeling and simulation. In this lecture we will introduce a formal definition of sets of gluing data, which fixes a problem in the definition given by Grimm and Hughes, and includes a Hausdorff condition. We also introduce the notion of parametric pseudo-manifolds. A parametric pseudo-manifold (PPM) is a topological space defined from a set of gluing data. Under certain conditions (which are often met in practice), PPM s are manifolds in E m. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 3
4 parametric pseudo-manifold θ i (Ω i ) E d M θ j (Ω j ) θ i θ j gluing data ϕ ij Ω i Ω ij Ω ji Ω j E n ϕ ji Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 4
5 Let I and K be (possibly infinite) countable sets such that I is nonempty. Definition 7.1. Let n be an integer, with n 1, and k be either an integer, with k 1, or k =. A set of gluing data is a triple, G = (Ω i ) i I, (Ω ij ) (i,j) I I, (ϕ ji ) (i,j) K, satisfying the following properties: Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 5
6 (1) For every i I, the set Ω i is a nonempty open subset of E n called parametrization domain, for short, p-domain, and any two distinct p-domains are pairwise disjoint, i.e., Ω i Ω j =, for all i = j. Ω 2 Ω 3 E n... Ω 1 Ω i Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 6
7 (2) For every pair (i, j) I I, the set Ω ij is an open subset of Ω i. Furthermore, Ω ii = Ω i and Ω ji = if and only if Ω ij =. Each nonempty subset Ω ij (with i = j) is called a gluing domain. Ω 21 Ω 2 Ω 3 E n Ω 12 Ω Ω 1 Ω i Ω 13 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 7
8 (3) If we let K = {(i, j) I I Ω ij = }, then ϕ ji : Ω ij Ω ji is a C k bijection for every (i, j) K called a transition (or gluing) map. Ω 21 Ω 2 Ω 3 E n ϕ 21 ϕ 12 Ω 12 ϕ 31 ϕ 13 Ω Ω 1 Ω i Ω 13 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 8
9 The transition functions must satisfy the following three conditions: (a) ϕ ii = id Ωi, for all i I, Ω i ϕ ii = id Ωi Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 9
10 (b) ϕ ij = ϕ 1 ji, for all (i, j) K, and Ω i ϕ ij Ω j p ϕ 1 ji Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 10
11 (c) For all i, j, k, if then Ω ji Ω jk =, ϕ ij (Ω ji Ω jk )=Ω ij Ω ik and ϕ ki (x) =ϕ kj ϕ ji (x), for all x Ω ij Ω ik. Ω ji Ω ik ϕ ij Ω j Ω ji Ω jk Ω i Ω ij Ω jk Ω ij Ω ik Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 11
12 ϕ ki (x) =(ϕ kj ϕ ji )(x), for all x (Ω ij Ω ik ). Ω ji Ω ij Ω j ϕ ji Ω i x Ω jk ϕ kj Ω k Ω ki ϕ ki = ϕ kj ϕ ji Ω ik Ω kj Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 12
13 The cocycle condition implies conditions (a) and (b): (a) ϕ ii = id Ωi, for all i I, and (b) ϕ ij = ϕ 1 ji, for all (i, j) K. Ω i ϕ ii = id Ωi Ω i ϕ ij Ω j p ϕ 1 ji Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 13
14 (4) For every pair (i, j) K, with i = j, for every x (Ω ij ) Ω i and y (Ω ji ) Ω j, there are open balls, V x and V y, centered at x and y, so that no point of V y Ω ji is the image of any point of V x Ω ij by ϕ ji. Ω ij ϕ ji Ω ji V x V y Ω i x ϕ ij y Ω j E n ϕ ji (V x Ω ij ) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 14
15 Given a set of gluing data, G, can we build a manifold from it? The answer is YES! Indeed, such a manifold is built by a quotient construction. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 15
16 The idea is to form the disjoint union, i I Ω i, of the Ω i and then identify Ω ij with Ω ji using ϕ ji. Formally, we define a binary relation,, on i I Ω i as follows: for all x, y i I Ω i, we have x y iff ( (i, j) K)(x Ω ij, y Ω ji, y = ϕ ji (x)). We can prove that is an equivalence relation, which enables us to define the space M G = i I Ω i /. We can also prove that M G is a Hausdorff and second-countable manifold. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 16
17 M G Sketching the proof: [x] For every i I, in i : Ω i i I Ω i is the natural injection. p in 1 p in 3 p in 2 [y] p in 1 Let p : i I Ω i M G be the quotient map, with p in n in 1 (Ω 1 ) in 2 (Ω 2 ) in 3 (Ω 3 ) in n (Ω n ) p(x) =[x]. For every i I, let τ i = p in i : Ω i M G. i I Ω i Let U i = τ i (Ω i ) and ϕ i = τ 1 i. It is immediately verified that (U i, ϕ i ) are charts and that this collection of charts forms a C k atlas for M G. in 1 in 2 in 3 in n x ϕ 21 (x) ϕ 31 (x) y Ω 1 Ω 2 Ω 3 ϕ n1 (y) Ω n Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 17
18 Sketching the proof: We now prove that the topology of M G is Hausdorff. Pick [x], [y] M G with [x] = [y], for some x Ω i and some y Ω j. Either τ i (Ω i ) τ j (Ω j )= or τ i (Ω i ) τ j (Ω j ) =. In the former case, as τ i and τ j are homeomorphisms, [x] and [y] belong to the two disjoint open sets τ i (Ω i ) and τ j (Ω j ). In the latter case, we must consider four subcases: Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 18
19 Sketching the proof: Ω ij Ω ji (1) x y (2) x y Ω i =Ω j Ω i Ω j Ω ij Ω ji Ω ij Ω ji (3) x y (4) x y Ω i Ω j Ω i Ω j Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 19
20 Sketching the proof: (1) If i = j then x and y can be separated by disjoint opens, V x and V y, and as τ i is a homeomorphism, [x] and [y] are separated by the disjoint open subsets τ i (V x ) and τ j (V y ). (1) x y Ω i =Ω j Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 20
21 Sketching the proof: (2) If i = j, x Ω i Ω ij and y Ω j Ω ji, then τ i (Ω i Ω ij ) and τ j (Ω j Ω ji ) are disjoint open subsets separating [x] and [y], where Ω ij and Ω ji are the closures of Ω ij and Ω ji, respectively. Ω ij Ω ji (2) x y Ω i Ω j Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 21
22 Sketching the proof: (3) If i = j, x Ω ij and y Ω ji, as [x] = [y] and y ϕ ij (y), then x = ϕ ij (y). We can separate x and ϕ ij (y) by disjoint open subsets, V x and V y, and [x] and [y] =[ϕ ij (y)] are separated by the disjoint open subsets τ i (V x ) and τ i (V y ). Ω ij Ω ji (3) x y Ω i Ω j Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 22
23 Sketching the proof: (4) If i = j, x (Ω ij ) Ω i and y (Ω ji ) Ω j, then we use condition 4 of Definition 7.1. This condition yields two disjoint open subsets, V x and V y, with x V x and y V y, such that no point of V x Ω ij is equivalent to any point of V y Ω ji, and so τ i (V x ) and τ j (V y ) are disjoint open subsets separating [x] and [y]. Ω ij Ω ji (4) x y Ω i Ω j Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 23
24 Sketching the proof: So, the topology of M G is Hausdorff and M G is indeed a manifold. M G is also second-countable (WHY?). Finally, it is trivial to verify that the transition maps of M G are the original gluing functions, ϕ ij, since ϕ i = τ 1 i and ϕ ji = ϕ j ϕ 1 i. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 24
25 Theorem 7.1. For every set of gluing data, G = (Ω i ) i I, (Ω ij ) (i,j) I I, (ϕ ji ) (i,j) K, there is an n-dimensional C k manifold, M G, whose transition maps are the ϕ ji s. Theorem 7.1 is nice, but... Our proof is not constructive; M G is an abstract entity, which may not be orientable, compact, etc. So, we know we can build a manifold from a set of gluing data, but that does not mean we know how to build a "concrete" manifold. For that, we need a formal notion of "concreteness". Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 25
26 Parametric Pseudo-Manifolds The notion of "concreteness" is realized as parametric pseudo-manifolds: Definition 7.2. Let n, d, and k be three integers with d > n 1 and k 1 or k =. A parametric C k pseudo-manifold of dimension n in E d (for short, parametric pseudo-manifold or PPM) is a pair, M =(G, (θ i ) i I ), such that G = (Ω i ) i I, (Ω ij ) (i,j) I I, (ϕ ji ) (i,j) K is a set of gluing data, for some finite set I, and each θ i : Ω i E d is C k and satisfies (C) For all (i, j) K, we have θ i = θ j ϕ ji. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 26
27 Manifolds Parametric Pseudo-Manifolds parametric pseudo-manifold θ i (Ω i ) E d M θ j (Ω j ) θ i θ j gluing data ϕ ij Ω i Ω ij Ω ji Ω j E n ϕ ji Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 27
28 Parametric Pseudo-Manifolds As usual, we call θ i a parametrization. The subset, M E d, given by M = i I θ i (Ω i ) is called the image of the parametric pseudo-manifold, M. Whenever n = 2 and d = 3, we say that M is a parametric pseudo-surface (or PPS, for short). We also say that M, the image of the PPS M, is a pseudo-surface. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 28
29 Parametric Pseudo-Manifolds Condition C of Definition 7.2, (C) For all (i, j) K, we have θ i = θ j ϕ ji, obviously implies that θ i (Ω ij )=θ j (Ω ji ), for all (i, j) K. Consequently, θ i and θ j are consistent parametrizations of the overlap θ i (Ω ij )=θ j (Ω ji ). Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 29
30 Parametric Pseudo-Manifolds θ i (Ω i ) E d M consistent! θ j (Ω j ) θ i θ j ϕ ij Ω i Ω ij Ω ji Ω j E n ϕ ji Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 30
31 Parametric Pseudo-Manifolds Thus, the set M, whatever it is, is covered by pieces, U i = θ i (Ω i ), not necessarily open. Each U i is parametrized by θ i, and each overlapping piece, U i U j, is parametrized consistently. The local structure of M is given by the θ i s and its global structure is given by the gluing data. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 31
32 Parametric Pseudo-Manifolds We can equip M with an atlas if we require the θ i s to be injective and to satisfy (C ) For all (i, j) K, (C ) For all (i, j) K, θ i (Ω i ) θ j (Ω j )=θ i (Ω ij )=θ j (Ω ji ). θ i (Ω i ) θ j (Ω j )=. Even if the θ i s are not injective, properties C and C are still desirable since they ensure that θ i (Ω i Ω ij ) and θ j (Ω j Ω ji ) are uniquely parametrized. Unfortunately, properties C and C may be difficult to enforce in practice (at least for surface constructions). Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 32
33 Parametric Pseudo-Manifolds Interestingly, regardless whether conditions C and C are satisfied, we can still show that M is the image in E d of the abstract manifold, M G, as stated by Proposition 7.2: Proposition 7.2. Let M =(G, (θ i ) i I ) be a parametric C k pseudo-manifold of dimension n in E d, where G = (Ω i ) i I, (Ω ij ) (i,j) I I, (ϕ ji ) (i,j) K is a set of gluing data, for some finite set I. Then, the parametrization maps, θ i, induce a surjective map, Θ : M G M, from the abstract manifold, M G, specified by G to the image, M E d, of the parametric pseudo-manifold, M, and the following property holds: θ i = Θ τ i, for every Ω i, where τ i : Ω i M G are the parametrization maps of the manifold M G. In particular, every manifold, M E d, such that M is induced by G is the image of M G by a map Θ : M G M. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 33
34 The Evil Cocycle Condition (c) For all i, j, k, if then Ω ji Ω jk =, ϕ ij (Ω ji Ω jk )=Ω ij Ω ik and ϕ ki (x) =ϕ kj ϕ ji (x), for all x Ω ij Ω ik. Ω ji Ω ik ϕ ij Ω j Ω ji Ω jk Ω i Ω ij Ω jk Ω ij Ω ik Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 34
35 The Evil Cocycle Condition ϕ ki (x) =(ϕ kj ϕ ji )(x), for all x (Ω ij Ω ik ). Ω ji Ω ij Ω j ϕ ji Ω i x Ω jk ϕ kj Ω k Ω ki ϕ ki = ϕ kj ϕ ji Ω ik Ω kj Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 35
36 The statement Parametric Pseudo-Manifolds The Evil Cocycle Condition if Ω ji Ω jk = then ϕ ij (Ω ji Ω jk )=Ω ij Ω ik is necessary for guaranteeing the transitivity of the equivalence relation. Ω ji Ω ik ϕ ij Ω j Ω ji Ω jk Ω i Ω ij Ω jk Ω ij Ω ik Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 36
37 The Evil Cocycle Condition Consider the p-domains (i.e., open line intervals) Ω 1 =]0, 3 [, Ω 2 =]4, 5 [, and Ω 3 =]6, 9 [. Ω 1 Ω 2 Ω E Consider the gluing domains Ω 12 =]0, 1 [ Ω 13 =]2, 3 [, Ω 21 = Ω 23 =]4, 5 [, Ω 32 =]8, 9 [ Ω 31 =]6, 7 [. Ω 12 Ω 13 Ω 21 = Ω 23 Ω 31 Ω E Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 37
38 The Evil Cocycle Condition Consider the transition maps: ϕ 21 (x) =x + 4, ϕ 32 (x) =x + 4 and ϕ 31 (x) =x + 4. Ω 12 Ω 13 Ω 21 = Ω 23 Ω 31 Ω E ϕ 21 ϕ 31 ϕ 32 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 38
39 The Evil Cocycle Condition Obviously, ϕ32 ϕ 21 (x) =x + 8, for all x Ω12. Ω 12 Ω 13 Ω 21 = Ω 23 Ω 31 Ω E ϕ 21 ϕ 32 ϕ 21 (0.5) =4.5 and ϕ 32 (4.5) =8.5 = and So, if were transitive, then we would have But... Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 39
40 The Evil Cocycle Condition it turns out that ϕ 31 is undefined at 0.5. Ω 12 Ω 13 Ω 21 = Ω 23 Ω 31 Ω E ϕ 31 So, The reason is that ϕ 31 and ϕ 32 ϕ 21 have disjoint domains. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 40
41 The Evil Cocycle Condition The reason they have disjoint domains is that condition "c" is not satisfied: if Ω 21 Ω 23 = then ϕ 12 (Ω 21 Ω 23 )=Ω 12 Ω 13. Ω 12 Ω 13 Ω 21 = Ω 23 Ω 31 Ω E Indeed but Ω 21 Ω 23 = Ω 2 =]4, 5 [ =, ϕ 12 (Ω 21 Ω 23 )=]0, 1 [ = = Ω 12 Ω 13. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 41
Outline. Sets of Gluing Data. Constructing Manifolds. Lecture 3 - February 3, PM
Constructing Manifolds Lecture 3 - February 3, 2009-1-2 PM Outline Sets of gluing data The cocycle condition Parametric pseudo-manifolds (PPM s) Conclusions 2 Let n and k be integers such that n 1 and
More informationIntroduction to Computational Manifolds and Applications
IMPA - Instituto de Matemática Pura e Aplicada, Rio de Janeiro, RJ, Brazil Introduction to Computational Manifolds and Applications Part - Constructions Prof. Marcelo Ferreira Siqueira mfsiqueira@dimap.ufrn.br
More informationDVA325 Formal Languages, Automata and Models of Computation (FABER)
DVA325 Formal Languages, Automata and Models of Computation (FABER) Lecture 1 - Introduction School of Innovation, Design and Engineering Mälardalen University 11 November 2014 Abu Naser Masud FABER November
More informationSOLUTIONS FOR PROBLEM SET 4
SOLUTIONS FOR PROBLEM SET 4 A. A certain integer a gives a remainder of 1 when divided by 2. What can you say about the remainder that a gives when divided by 8? SOLUTION. Let r be the remainder that a
More informationFinite and Infinite Sets
Finite and Infinite Sets MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007 Basic Definitions Definition The empty set has 0 elements. If n N, a set S is said to have
More informationCardinality revisited
Cardinality revisited A set is finite (has finite cardinality) if its cardinality is some (finite) integer n. Two sets A,B have the same cardinality iff there is a one-to-one correspondence from A to B
More informationTOPOLOGY, LIMITS OF COMPLEX NUMBERS. Contents 1. Topology and limits of complex numbers 1
TOPOLOGY, LIMITS OF COMPLEX NUMBERS Contents 1. Topology and limits of complex numbers 1 1. Topology and limits of complex numbers Since we will be doing calculus on complex numbers, not only do we need
More informationClass 8 - Sets (Lecture Notes)
Class 8 - Sets (Lecture Notes) What is a Set? A set is a well-defined collection of distinct objects. Example: A = {1, 2, 3, 4, 5} What is an element of a Set? The objects in a set are called its elements.
More informationSets. Definition A set is an unordered collection of objects called elements or members of the set.
Sets Definition A set is an unordered collection of objects called elements or members of the set. Sets Definition A set is an unordered collection of objects called elements or members of the set. Examples:
More informationand problem sheet 7
1-18 and 15-151 problem sheet 7 Solutions to the following five exercises and optional bonus problem are to be submitted through gradescope by 11:30PM on Friday nd November 018. Problem 1 Let A N + and
More informationPermutation Groups. Every permutation can be written as a product of disjoint cycles. This factorization is unique up to the order of the factors.
Permutation Groups 5-9-2013 A permutation of a set X is a bijective function σ : X X The set of permutations S X of a set X forms a group under function composition The group of permutations of {1,2,,n}
More informationCSE 20 DISCRETE MATH. Fall
CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Define and compute the cardinality of a set. Use functions to compare the sizes of sets. Classify sets
More informationMath 127: Equivalence Relations
Math 127: Equivalence Relations Mary Radcliffe 1 Equivalence Relations Relations can take many forms in mathematics. In these notes, we focus especially on equivalence relations, but there are many other
More informationQUOTIENT AND PSEUDO-OPEN IMAGES OF SEPARABLE METRIC SPACES
proceedings of the american mathematical society Volume 33, Number 2, June 1972 QUOTIENT AND PSEUDO-OPEN IMAGES OF SEPARABLE METRIC SPACES PAUL L. STRONG Abstract. Ernest A. Michael has given a characterization
More informationOn the isomorphism problem of Coxeter groups and related topics
On the isomorphism problem of Coxeter groups and related topics Koji Nuida 1 Graduate School of Mathematical Sciences, University of Tokyo E-mail: nuida@ms.u-tokyo.ac.jp At the conference the author gives
More informationChapter 1. The alternating groups. 1.1 Introduction. 1.2 Permutations
Chapter 1 The alternating groups 1.1 Introduction The most familiar of the finite (non-abelian) simple groups are the alternating groups A n, which are subgroups of index 2 in the symmetric groups S n.
More informationLecture 3 Presentations and more Great Groups
Lecture Presentations and more Great Groups From last time: A subset of elements S G with the property that every element of G can be written as a finite product of elements of S and their inverses is
More informationOn uniquely k-determined permutations
On uniquely k-determined permutations Sergey Avgustinovich and Sergey Kitaev 16th March 2007 Abstract Motivated by a new point of view to study occurrences of consecutive patterns in permutations, we introduce
More informationOn first and second countable spaces and the axiom of choice
Topology and its Applications 143 (2004) 93 103 www.elsevier.com/locate/topol On first and second countable spaces and the axiom of choice Gonçalo Gutierres 1 Departamento de Matemática da Universidade
More informationCounting integral solutions
Thought exercise 2.2 25 Counting integral solutions Question: How many non-negative integer solutions are there of x 1 + x 2 + x 3 + x 4 =10? Give some examples of solutions. Characterize what solutions
More informationReading 14 : Counting
CS/Math 240: Introduction to Discrete Mathematics Fall 2015 Instructors: Beck Hasti, Gautam Prakriya Reading 14 : Counting In this reading we discuss counting. Often, we are interested in the cardinality
More informationNON-OVERLAPPING PERMUTATION PATTERNS. To Doron Zeilberger, for his Sixtieth Birthday
NON-OVERLAPPING PERMUTATION PATTERNS MIKLÓS BÓNA Abstract. We show a way to compute, to a high level of precision, the probability that a randomly selected permutation of length n is nonoverlapping. As
More informationSets. Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) August 6, Outline Sets Equality Subset Empty Set Cardinality Power Set
Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) August 6, 2012 Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) Gazihan Alankuş (Based on original slides by Brahim Hnich
More informationTHREE LECTURES ON SQUARE-TILED SURFACES (PRELIMINARY VERSION) Contents
THREE LECTURES ON SQUARE-TILED SURFACES (PRELIMINARY VERSION) CARLOS MATHEUS Abstract. This text corresponds to a minicourse delivered on June 11, 12 & 13, 2018 during the summer school Teichmüller dynamics,
More informationPermutation group and determinants. (Dated: September 19, 2018)
Permutation group and determinants (Dated: September 19, 2018) 1 I. SYMMETRIES OF MANY-PARTICLE FUNCTIONS Since electrons are fermions, the electronic wave functions have to be antisymmetric. This chapter
More informationLECTURE 3: CONGRUENCES. 1. Basic properties of congruences We begin by introducing some definitions and elementary properties.
LECTURE 3: CONGRUENCES 1. Basic properties of congruences We begin by introducing some definitions and elementary properties. Definition 1.1. Suppose that a, b Z and m N. We say that a is congruent to
More informationFermat s little theorem. RSA.
.. Computing large numbers modulo n (a) In modulo arithmetic, you can always reduce a large number to its remainder a a rem n (mod n). (b) Addition, subtraction, and multiplication preserve congruence:
More information18 Completeness and Compactness of First-Order Tableaux
CS 486: Applied Logic Lecture 18, March 27, 2003 18 Completeness and Compactness of First-Order Tableaux 18.1 Completeness Proving the completeness of a first-order calculus gives us Gödel s famous completeness
More information5 Symmetric and alternating groups
MTHM024/MTH714U Group Theory Notes 5 Autumn 2011 5 Symmetric and alternating groups In this section we examine the alternating groups A n (which are simple for n 5), prove that A 5 is the unique simple
More informationSimple permutations and pattern restricted permutations
Simple permutations and pattern restricted permutations M.H. Albert and M.D. Atkinson Department of Computer Science University of Otago, Dunedin, New Zealand. Abstract A simple permutation is one that
More information1. Functions and set sizes 2. Infinite set sizes. ! Let X,Y be finite sets, f:x!y a function. ! Theorem: If f is injective then X Y.
2 Today s Topics: CSE 20: Discrete Mathematics for Computer Science Prof. Miles Jones 1. Functions and set sizes 2. 3 4 1. Functions and set sizes! Theorem: If f is injective then Y.! Try and prove yourself
More informationMath236 Discrete Maths with Applications
Math236 Discrete Maths with Applications P. Ittmann UKZN, Pietermaritzburg Semester 1, 2012 Ittmann (UKZN PMB) Math236 2012 1 / 43 The Multiplication Principle Theorem Let S be a set of k-tuples (s 1,
More informationThe Hex game and its mathematical side
The Hex game and its mathematical side Antonín Procházka Laboratoire de Mathématiques de Besançon Université Franche-Comté Lycée Jules Haag, 19 mars 2013 Brief history : HEX was invented in 1942
More informationCardinality and Bijections
Countable and Cardinality and Bijections Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) August 13, 2012 Countable and Countable and Countable and How to count elements in a set? How
More informationFrom permutations to graphs
From permutations to graphs well-quasi-ordering and infinite antichains Robert Brignall Joint work with Atminas, Korpelainen, Lozin and Vatter 28th November 2014 Orderings on Structures Pick your favourite
More informationLecture 2: Sum rule, partition method, difference method, bijection method, product rules
Lecture 2: Sum rule, partition method, difference method, bijection method, product rules References: Relevant parts of chapter 15 of the Math for CS book. Discrete Structures II (Summer 2018) Rutgers
More informationMAT Modular arithmetic and number theory. Modular arithmetic
Modular arithmetic 1 Modular arithmetic may seem like a new and strange concept at first The aim of these notes is to describe it in several different ways, in the hope that you will find at least one
More informationMATH 433 Applied Algebra Lecture 12: Sign of a permutation (continued). Abstract groups.
MATH 433 Applied Algebra Lecture 12: Sign of a permutation (continued). Abstract groups. Permutations Let X be a finite set. A permutation of X is a bijection from X to itself. The set of all permutations
More informationCountability. Jason Filippou UMCP. Jason Filippou UMCP) Countability / 12
Countability Jason Filippou CMSC250 @ UMCP 06-23-2016 Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 1 / 12 Outline 1 Infinity 2 Countability of integers and rationals 3 Uncountability of R Jason
More informationMATHEMATICS 152, FALL 2004 METHODS OF DISCRETE MATHEMATICS Outline #10 (Sets and Probability)
MATHEMATICS 152, FALL 2004 METHODS OF DISCRETE MATHEMATICS Outline #10 (Sets and Probability) Last modified: November 10, 2004 This follows very closely Apostol, Chapter 13, the course pack. Attachments
More informationDiscrete Mathematics and Probability Theory Spring 2018 Ayazifar and Rao Midterm 2 Solutions
CS 70 Discrete Mathematics and Probability Theory Spring 2018 Ayazifar and Rao Midterm 2 Solutions PRINT Your Name: Oski Bear SIGN Your Name: OS K I PRINT Your Student ID: CIRCLE your exam room: Pimentel
More informationCardinality of Accumulation Points of Infinite Sets
International Mathematical Forum, Vol. 11, 2016, no. 11, 539-546 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2016.6224 Cardinality of Accumulation Points of Infinite Sets A. Kalapodi CTI
More informationarxiv: v1 [math.co] 11 Jul 2016
OCCURRENCE GRAPHS OF PATTERNS IN PERMUTATIONS arxiv:160703018v1 [mathco] 11 Jul 2016 BJARNI JENS KRISTINSSON AND HENNING ULFARSSON Abstract We define the occurrence graph G p (π) of a pattern p in a permutation
More informationWilson s Theorem and Fermat s Theorem
Wilson s Theorem and Fermat s Theorem 7-27-2006 Wilson s theorem says that p is prime if and only if (p 1)! = 1 (mod p). Fermat s theorem says that if p is prime and p a, then a p 1 = 1 (mod p). Wilson
More informationSome Fine Combinatorics
Some Fine Combinatorics David P. Little Department of Mathematics Penn State University University Park, PA 16802 Email: dlittle@math.psu.edu August 3, 2009 Dedicated to George Andrews on the occasion
More informationAn Erdős-Lovász-Spencer Theorem for permutations and its. testing
An Erdős-Lovász-Spencer Theorem for permutations and its consequences for parameter testing Carlos Hoppen (UFRGS, Porto Alegre, Brazil) This is joint work with Roman Glebov (ETH Zürich, Switzerland) Tereza
More informationCitation for published version (APA): Nutma, T. A. (2010). Kac-Moody Symmetries and Gauged Supergravity Groningen: s.n.
University of Groningen Kac-Moody Symmetries and Gauged Supergravity Nutma, Teake IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please
More informationThe Math Behind Futurama: The Prisoner of Benda
of Benda May 7, 2013 The problem (informally) Professor Farnsworth has created a mind-switching machine that switches two bodies, but the switching can t be reversed using just those two bodies. Using
More information1. The empty set is a proper subset of every set. Not true because the empty set is not a proper subset of itself! is the power set of A.
MAT 101 Solutions to Sample Questions for Exam 1 True or False Questions Answers: 1F, 2F, 3F, 4T, 5T, 6T, 7T 1. The empty set is a proper subset of every set. Not true because the empty set is not a proper
More informationAn elementary study of Goldbach Conjecture
An elementary study of Goldbach Conjecture Denise Chemla 26/5/2012 Goldbach Conjecture (7 th, june 1742) states that every even natural integer greater than 4 is the sum of two odd prime numbers. If we
More informationPermutations and codes:
Hamming distance Permutations and codes: Polynomials, bases, and covering radius Peter J. Cameron Queen Mary, University of London p.j.cameron@qmw.ac.uk International Conference on Graph Theory Bled, 22
More informationcode V(n,k) := words module
Basic Theory Distance Suppose that you knew that an English word was transmitted and you had received the word SHIP. If you suspected that some errors had occurred in transmission, it would be impossible
More informationIntroduction to Coding Theory
Coding Theory Massoud Malek Introduction to Coding Theory Introduction. Coding theory originated with the advent of computers. Early computers were huge mechanical monsters whose reliability was low compared
More informationSample Spaces, Events, Probability
Sample Spaces, Events, Probability CS 3130/ECE 3530: Probability and Statistics for Engineers August 28, 2014 Sets A set is a collection of unique objects. Sets A set is a collection of unique objects.
More informationCSE 312: Foundations of Computing II Quiz Section #2: Inclusion-Exclusion, Pigeonhole, Introduction to Probability (solutions)
CSE 31: Foundations of Computing II Quiz Section #: Inclusion-Exclusion, Pigeonhole, Introduction to Probability (solutions) Review: Main Theorems and Concepts Binomial Theorem: x, y R, n N: (x + y) n
More informationREU 2006 Discrete Math Lecture 3
REU 006 Discrete Math Lecture 3 Instructor: László Babai Scribe: Elizabeth Beazley Editors: Eliana Zoque and Elizabeth Beazley NOT PROOFREAD - CONTAINS ERRORS June 6, 006. Last updated June 7, 006 at :4
More information1.6 Congruence Modulo m
1.6 Congruence Modulo m 47 5. Let a, b 2 N and p be a prime. Prove for all natural numbers n 1, if p n (ab) and p - a, then p n b. 6. In the proof of Theorem 1.5.6 it was stated that if n is a prime number
More informationPermutation Groups. Definition and Notation
5 Permutation Groups Wigner s discovery about the electron permutation group was just the beginning. He and others found many similar applications and nowadays group theoretical methods especially those
More informationCSE 312: Foundations of Computing II Quiz Section #2: Inclusion-Exclusion, Pigeonhole, Introduction to Probability
CSE 312: Foundations of Computing II Quiz Section #2: Inclusion-Exclusion, Pigeonhole, Introduction to Probability Review: Main Theorems and Concepts Binomial Theorem: Principle of Inclusion-Exclusion
More informationFast Sorting and Pattern-Avoiding Permutations
Fast Sorting and Pattern-Avoiding Permutations David Arthur Stanford University darthur@cs.stanford.edu Abstract We say a permutation π avoids a pattern σ if no length σ subsequence of π is ordered in
More informationUNIT-III ASYNCHRONOUS SEQUENTIAL CIRCUITS TWO MARKS 1. What are secondary variables? -present state variables in asynchronous sequential circuits 2. What are excitation variables? -next state variables
More informationThe next several lectures will be concerned with probability theory. We will aim to make sense of statements such as the following:
CS 70 Discrete Mathematics for CS Fall 2004 Rao Lecture 14 Introduction to Probability The next several lectures will be concerned with probability theory. We will aim to make sense of statements such
More informationOn the isomorphism problem for Coxeter groups and related topics
On the isomorphism problem for Coxeter groups and related topics Koji Nuida (AIST, Japan) Groups and Geometries @Bangalore, Dec. 18 & 20, 2012 Koji Nuida (AIST, Japan) On the isomorphism problem for Coxeter
More informationA theorem on the cores of partitions
A theorem on the cores of partitions Jørn B. Olsson Department of Mathematical Sciences, University of Copenhagen Universitetsparken 5,DK-2100 Copenhagen Ø, Denmark August 9, 2008 Abstract: If s and t
More informationHarmonic numbers, Catalan s triangle and mesh patterns
Harmonic numbers, Catalan s triangle and mesh patterns arxiv:1209.6423v1 [math.co] 28 Sep 2012 Sergey Kitaev Department of Computer and Information Sciences University of Strathclyde Glasgow G1 1XH, United
More informationEquilateral k-isotoxal Tiles
Equilateral k-isotoxal Tiles R. Chick and C. Mann October 26, 2012 Abstract In this article we introduce the notion of equilateral k-isotoxal tiles and give of examples of equilateral k-isotoxal tiles
More informationTheory of Probability - Brett Bernstein
Theory of Probability - Brett Bernstein Lecture 3 Finishing Basic Probability Review Exercises 1. Model flipping two fair coins using a sample space and a probability measure. Compute the probability of
More informationarxiv:math/ v1 [math.cv] 12 Dec 2005
arxiv:math/0512241v1 [math.cv] 12 Dec 2005 The pluri-fine topology is locally connected Said El Marzguioui and Jan Wiegerinck November 1, 2018 Abstract We prove that the pluri-fine topology on any open
More informationNon-overlapping permutation patterns
PU. M. A. Vol. 22 (2011), No.2, pp. 99 105 Non-overlapping permutation patterns Miklós Bóna Department of Mathematics University of Florida 358 Little Hall, PO Box 118105 Gainesville, FL 326118105 (USA)
More informationRestricted Permutations Related to Fibonacci Numbers and k-generalized Fibonacci Numbers
Restricted Permutations Related to Fibonacci Numbers and k-generalized Fibonacci Numbers arxiv:math/0109219v1 [math.co] 27 Sep 2001 Eric S. Egge Department of Mathematics Gettysburg College 300 North Washington
More informationThe topic for the third and final major portion of the course is Probability. We will aim to make sense of statements such as the following:
CS 70 Discrete Mathematics for CS Spring 2006 Vazirani Lecture 17 Introduction to Probability The topic for the third and final major portion of the course is Probability. We will aim to make sense of
More informationPermutation groups, derangements and prime order elements
Permutation groups, derangements and prime order elements Tim Burness University of Southampton Isaac Newton Institute, Cambridge April 21, 2009 Overview 1. Introduction 2. Counting derangements: Jordan
More informationPRIMES 2017 final paper. NEW RESULTS ON PATTERN-REPLACEMENT EQUIVALENCES: GENERALIZING A CLASSICAL THEOREM AND REVISING A RECENT CONJECTURE Michael Ma
PRIMES 2017 final paper NEW RESULTS ON PATTERN-REPLACEMENT EQUIVALENCES: GENERALIZING A CLASSICAL THEOREM AND REVISING A RECENT CONJECTURE Michael Ma ABSTRACT. In this paper we study pattern-replacement
More informationOlympiad Combinatorics. Pranav A. Sriram
Olympiad Combinatorics Pranav A. Sriram August 2014 Chapter 2: Algorithms - Part II 1 Copyright notices All USAMO and USA Team Selection Test problems in this chapter are copyrighted by the Mathematical
More informationFormal Description of the Chord Protocol using ASM
Formal Description of the Chord Protocol using ASM Bojan Marinković 1, Paola Glavan 2, Zoran Ognjanović 1 Mathematical Institute of the Serbian Academy of Sciences and Arts 1 Belgrade, Serbia [bojanm,
More informationEnumeration of Two Particular Sets of Minimal Permutations
3 47 6 3 Journal of Integer Sequences, Vol. 8 (05), Article 5.0. Enumeration of Two Particular Sets of Minimal Permutations Stefano Bilotta, Elisabetta Grazzini, and Elisa Pergola Dipartimento di Matematica
More informationLocalization for Group of Robots using Matrix Contractors
for Group of Robots using Matrix Contractors Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, Manchester Nisha Rani Mahato, Luc Jaulin, Snehashish Chakraverty Swim-Smart 2017, for
More informationCompound Probability. Set Theory. Basic Definitions
Compound Probability Set Theory A probability measure P is a function that maps subsets of the state space Ω to numbers in the interval [0, 1]. In order to study these functions, we need to know some basic
More informationTwo-person symmetric whist
Two-person symmetric whist Johan Wästlund Linköping studies in Mathematics, No. 4, February 21, 2005 Series editor: Bengt Ove Turesson The publishers will keep this document on-line on the Internet (or
More informationarxiv: v3 [math.co] 4 Dec 2018 MICHAEL CORY
CYCLIC PERMUTATIONS AVOIDING PAIRS OF PATTERNS OF LENGTH THREE arxiv:1805.05196v3 [math.co] 4 Dec 2018 MIKLÓS BÓNA MICHAEL CORY Abstract. We enumerate cyclic permutations avoiding two patterns of length
More informationFinite homomorphism-homogeneous permutations via edge colourings of chains
Finite homomorphism-homogeneous permutations via edge colourings of chains Igor Dolinka dockie@dmi.uns.ac.rs Department of Mathematics and Informatics, University of Novi Sad First of all there is Blue.
More informationA NEW COMPUTATION OF THE CODIMENSION SEQUENCE OF THE GRASSMANN ALGEBRA
A NEW COMPUTATION OF THE CODIMENSION SEQUENCE OF THE GRASSMANN ALGEBRA JOEL LOUWSMA, ADILSON EDUARDO PRESOTO, AND ALAN TARR Abstract. Krakowski and Regev found a basis of polynomial identities satisfied
More informationSolutions for the Practice Questions
Solutions for the Practice Questions Question 1. Find all solutions to the congruence 13x 12 (mod 35). Also, answer the following questions about the solutions to the above congruence. Are there solutions
More informationAlgebra. Recap: Elements of Set Theory.
January 14, 2018 Arrangements and Derangements. Algebra. Recap: Elements of Set Theory. Arrangements of a subset of distinct objects chosen from a set of distinct objects are permutations [order matters]
More informationProblem 2A Consider 101 natural numbers not exceeding 200. Prove that at least one of them is divisible by another one.
1. Problems from 2007 contest Problem 1A Do there exist 10 natural numbers such that none one of them is divisible by another one, and the square of any one of them is divisible by any other of the original
More informationLecture 18 - Counting
Lecture 18 - Counting 6.0 - April, 003 One of the most common mathematical problems in computer science is counting the number of elements in a set. This is often the core difficulty in determining a program
More informationCombinatorics: The Fine Art of Counting
Combinatorics: The Fine Art of Counting Week 6 Lecture Notes Discrete Probability Note Binomial coefficients are written horizontally. The symbol ~ is used to mean approximately equal. Introduction and
More informationEnhanced Turing Machines
Enhanced Turing Machines Lecture 28 Sections 10.1-10.2 Robb T. Koether Hampden-Sydney College Wed, Nov 2, 2016 Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, 2016 1 / 21
More informationTopology and its Applications
Topology and its Applications 157 (2010) 1541 1547 Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol On a core concept of Arhangel skiĭ Franklin D. Tall
More informationEXPLAINING THE SHAPE OF RSK
EXPLAINING THE SHAPE OF RSK SIMON RUBINSTEIN-SALZEDO 1. Introduction There is an algorithm, due to Robinson, Schensted, and Knuth (henceforth RSK), that gives a bijection between permutations σ S n and
More informationPermutations. = f 1 f = I A
Permutations. 1. Definition (Permutation). A permutation of a set A is a bijective function f : A A. The set of all permutations of A is denoted by Perm(A). 2. If A has cardinality n, then Perm(A) has
More information29. Army Housing (a) (b) (c) (d) (e) (f ) Totals Totals (a) (b) (c) (d) (e) (f) Basketball Positions 32. Guard Forward Center
Infinite Sets and Their Cardinalities As mentioned at the beginning of this chapter, most of the early work in set theory was done by Georg Cantor He devoted much of his life to a study of the cardinal
More informationA State Equivalence and Confluence Checker for CHR
A State Equivalence and Confluence Checker for CHR Johannes Langbein, Frank Raiser, and Thom Frühwirth Faculty of Engineering and Computer Science, Ulm University, Germany firstname.lastname@uni-ulm.de
More informationConvexity Invariants of the Hoop Closure on Permutations
Convexity Invariants of the Hoop Closure on Permutations Robert E. Jamison Retired from Discrete Mathematics Clemson University now in Asheville, NC Permutation Patterns 12 7 11 July, 2014 Eliakim Hastings
More informationConnected Identifying Codes
Connected Identifying Codes Niloofar Fazlollahi, David Starobinski and Ari Trachtenberg Dept. of Electrical and Computer Engineering Boston University, Boston, MA 02215 Email: {nfazl,staro,trachten}@bu.edu
More informationPD-SETS FOR CODES RELATED TO FLAG-TRANSITIVE SYMMETRIC DESIGNS. Communicated by Behruz Tayfeh Rezaie. 1. Introduction
Transactions on Combinatorics ISSN (print): 2251-8657, ISSN (on-line): 2251-8665 Vol. 7 No. 1 (2018), pp. 37-50. c 2018 University of Isfahan www.combinatorics.ir www.ui.ac.ir PD-SETS FOR CODES RELATED
More informationMinimal generating sets of Weierstrass semigroups of certain m-tuples on the norm-trace function field
Minimal generating sets of Weierstrass semigroups of certain m-tuples on the norm-trace function field Gretchen L. Matthews and Justin D. Peachey Abstract. The norm-trace function field is a generalization
More informationON SPLITTING UP PILES OF STONES
ON SPLITTING UP PILES OF STONES GREGORY IGUSA Abstract. In this paper, I describe the rules of a game, and give a complete description of when the game can be won, and when it cannot be won. The first
More informationLECTURE 7: POLYNOMIAL CONGRUENCES TO PRIME POWER MODULI
LECTURE 7: POLYNOMIAL CONGRUENCES TO PRIME POWER MODULI 1. Hensel Lemma for nonsingular solutions Although there is no analogue of Lagrange s Theorem for prime power moduli, there is an algorithm for determining
More informationSYMMETRIES OF FIBONACCI POINTS, MOD m
PATRICK FLANAGAN, MARC S. RENAULT, AND JOSH UPDIKE Abstract. Given a modulus m, we examine the set of all points (F i,f i+) Z m where F is the usual Fibonacci sequence. We graph the set in the fundamental
More information