Cameron Liebler line classes in P G(n, 4)

Size: px
Start display at page:

Download "Cameron Liebler line classes in P G(n, 4)"

Transcription

1 Cameron Liebler line classes in P G(n, 4) Ivan Mogilnykh Sobolev Institute of Mathematics, Novosibirsk joint work with Alexander Gavrilyuk, Krasovski Institute of Mathematics and Mechanics, Ekaterinburg Finite Fields and their Applications, Magdeburg, July 23, 2013.

2 For a point P of P G(3, q) define: Star(P ) = {l : l is a line incident to P },

3 For a point P of P G(3, q) define: Star(P ) = {l : l is a line incident to P }, for a plane H in P G(3, q): line(h) = {l : l is a line in H}.

4 For a point P of P G(3, q) define: Star(P ) = {l : l is a line incident to P }, for a plane H in P G(3, q): line(h) = {l : l is a line in H}. for an incident point-hyperplane pair (P, H): pen(p, H) = Star(P ) line(h).

5 Cameron-Liebler classes in P G(3, q) Let χ L be the characteristic function of a set of lines L in P G(3, q). A set of lines L in P G(3, q) is a Cameron-Liebler line class with parameter x in P G(3, q) if one of the following holds: (i) For every line l, {m L \ {l} : m meets l} = (q + 1)x + (q 2 1)χ L (l).

6 Cameron-Liebler classes in P G(3, q) Let χ L be the characteristic function of a set of lines L in P G(3, q). A set of lines L in P G(3, q) is a Cameron-Liebler line class with parameter x in P G(3, q) if one of the following holds: (i) For every line l, {m L \ {l} : m meets l} = (q + 1)x + (q 2 1)χ L (l). (ii) For any incident point-plane pair (P, H) we have: Star(P ) L + line(h) L = x + (q + 1) pen(p, H) L.

7 Cameron-Liebler classes in P G(3, q) Let χ L be the characteristic function of a set of lines L in P G(3, q). A set of lines L in P G(3, q) is a Cameron-Liebler line class with parameter x in P G(3, q) if one of the following holds: (i) For every line l, {m L \ {l} : m meets l} = (q + 1)x + (q 2 1)χ L (l). (ii) For any incident point-plane pair (P, H) we have: Star(P ) L + line(h) L = x + (q + 1) pen(p, H) L. (iii) For a pair of skew lines l, l from P G(3, q) the number of lines meeting l and l from L is x + q(χ L (l) + χ L (l )).

8 Cameron-Liebler classes in P G(3, q) Let χ L be the characteristic function of a set of lines L in P G(3, q). A set of lines L in P G(3, q) is a Cameron-Liebler line class with parameter x in P G(3, q) if one of the following holds: (i) For every line l, {m L \ {l} : m meets l} = (q + 1)x + (q 2 1)χ L (l). (ii) For any incident point-plane pair (P, H) we have: Star(P ) L + line(h) L = x + (q + 1) pen(p, H) L. (iii) For a pair of skew lines l, l from P G(3, q) the number of lines meeting l and l from L is x + q(χ L (l) + χ L (l )). (iv) For any line-spread S we have S L = x.

9 Cameron-Liebler classes in P G(3, q) A set of lines L in P G(n, q) is a Cameron-Liebler line class with parameter x in PG(3,q) if one of the following holds: (i) For every line l, {m L \ {l} : m meets l} = (q + 1)x + (q 2 1)χ L (l). (ii) For any incident incident point-plane pair (P, H) we have: Star(P ) L + line(h) L = x + (q + 1) pen(p, H) L. (iii) For a pair of skew lines l, l from P G(3, q) the number of common neighbors from L is x + q(χ L (l) + χ L (l )) (iv) For any line-spread S we have S L = x. (v) L is a completely regular code in G q (4, 2) of valency (q + 1)x and strength 0 as a q-design.

10 Cameron-Liebler classes in P G(3, q): constructions Cameron and Liebler 82:, (x=0), line(h), Star(P ) (x=1), Star(P ) line(h) for nonincident pair (P,H) (x=2). Drudge 99: classification of Cameron-Liebler classes in P G(3, 3) (P G(n, 3)), a new example for x = 5. Bruen and Drudge 98: an example in P G(3, q), x = (q 2 + 1)/2 Govaerts and Pentilla 05: an example in P G(3, 4), x = 7 Rodgers 11: examples in P G(3, q) for some odd q 200, x = (q 2 1)/2.

11 Cameron-Liebler classes in P G(3, q): nonexistence results Pentilla 91: x 3, 4 if q 5.

12 Cameron-Liebler classes in P G(3, q): nonexistence results Pentilla 91: x 3, 4 if q 5. Bruen, Drudge 98: x / {3,..., q}. Drudge 99: approach via blocking sets in P G(2, q), x / {3,..., e(q)}, where 1 + q + e(q) is the size of the smallest nontrivial blocking set in P G(2, q). De Beule, Hallez, Storme 08: x / {3,..., q/2}. Metsch 10: x / {3,..., q}.

13 Cameron-Liebler classes in P G(3, q): nonexistence results Govaerts, Pentilla 05: x 4, 5 in P G(3, 4).

14 Pattern Let L be a Cameron-Liebler line class in P G(3, q).

15 Pattern Let L be a Cameron-Liebler line class in P G(3, q). Consider line l P G(3, q) that belongs to Star(P 1 ),..., Star(P q+1 ), line(h 1 ),..., line(h q+1 ).

16 Pattern Let L be a Cameron-Liebler line class in P G(3, q). Consider line l P G(3, q) that belongs to Star(P 1 ),..., Star(P q+1 ), line(h 1 ),..., line(h q+1 ). The pattern w.r.t. l is the matrix T of order q + 1: T ij = (pen(p i, H j ) \ l) L for any i, j {1,..., q + 1}.

17 The properties of pattern Lemma Let L be a Cameron-Liebler line class in P G(3, q), T := (t ij ) be a pattern w.r.t. a line l. Then the following hold: 0 t ij q for all i, j {1,..., q + 1}; q+1 i,j=1 t ij = x(q + 1) + χ L (l)(q 2 1); q+1 q+1 t kj + t il = x + (q + 1)(t kl + χ L (l)), k, l; j=1 q+1 i,j=1 i=1 t 2 ij = (x χ L ) 2 + q(x χ L ) + χ L (l)q 2 (q + 1).

18 Cameron-Liebler line classes in P G(3, 4) The parameter x {0, 1, 2, 3, 4, 5, 6, 7, 8}

19 Cameron-Liebler line classes in P G(3, 4) The parameter x {0, 1, 2, 3, 4, 5, 6, 7, 8} Govaerts, Pentilla 05: x {0!, 1!, 2!, 3, 4, 5, 6?, 7!?, 8?}

20 Cameron-Liebler line classes in P G(3, 4) The parameter x {0, 1, 2, 3, 4, 5, 6, 7, 8} Govaerts, Pentilla 05: x {0!, 1!, 2!, 3, 4, 5, 6?, 7!?, 8?} The approach via patterns The cases x = 4, 8 are eliminated by the approach solely, the cases x = 5, 6, 7 are solved by the approach and additional considerations.

21 Cameron-Liebler line class with x = 7 in P G(3, 4) Govaerts, Pentilla 05: Let (P, H) be nonincident point-hyperplane pair in P G(3, 4), O be a hyperoval in H. Let C be the set of lines incident to the points of O and P. Then C, all 2-secants of C and all lines in H external to O form a Cameron Liebler line class with parameter x = 7.

22 Cameron-Liebler line class with x = 7 in P G(3, 4) Govaerts, Pentilla 05: Let (P, H) be nonincident point-hyperplane pair in P G(3, 4), O be a hyperoval in H. Let C be the set of lines incident to the points of O and P. Then C, all 2-secants of C and all lines in H external to O form a Cameron Liebler line class with parameter x = 7.

23 Cameron-Liebler line class with x = 7 in P G(3, 4) Let (P, H) be nonincident point-hyperplane pair in P G(3, 4), O be a hyperoval in H. Let C be the set of lines incident to the points of O and P. Then C, all 2-secants of C and all lines in H external to O form a Cameron Liebler line class with parameter x = 7. w.r.t. l L we have the following patterns: w.r.t. l / L: , ,,

24 Cameron-Liebler line class with x = 7 in P G(3, 4) Let (P, H) be nonincident point-hyperplane pair in P G(3, 4), O be a hyperoval in H. Let C be the set of lines incident to the points of O and P. Then C, all 2-secants of C and all lines in H external to O form a Cameron Liebler line class with parameter x = 7. w.r.t. l L we have the following patterns: w.r.t. l / L: , ,,

25 Cameron-Liebler line classes in P G(3, 4) Theorem There only Cameron-Liebler classes in P G(3, 4) are the classes with x = 0, 1, 2 and the class with x = 7 constructed by Govaerts and Pentilla unique up to collineation.

26 The pattern approach for small q Pairs (q, x) excluded by the approach. q x total 4 3,4,8 3 of 8 5 3,4,7,11 4 of ,4,5,6,7,11,12,14,15,19,20,22,23 13 of ,4,5,6,8,12,14,15,17,21,23,24,26,30,32 15 of ,4,5,7,8,9,11,13,14,15,18,19,23,24,27, 28,29,31,33,34,35, 38,39 23 of ,...,9,11,12,14,15,19,20,22,23,27 28,30,31,35,36,38,39,43,44,46,47,51,52,54,55,59,60 35 of 61

27 Cameron-Liebler line classes in P G(n, q) A set of lines L in P G(n, q) is a Cameron-Liebler line class with parameter x if one of the following holds: for every line l, {m L\{l} : m meets l} = (q+1)x+(q n q 2 1)χ L (l).

28 Cameron-Liebler line classes in P G(n, 4) Drudge s Lemma Let L be a Cameron Liebler line class in P G(n, q), n > 3, consider a 3-dimensional projective subspace P G(3, q) of P G(n, q). Then line(x) L is a Cameron Liebler line class in P G(3, q). Theorem The only Cameron-Liebler classes to exist in P G(n, 4) are: empty set (x = 0), line(h), Star(P ) (x=1), Star(P ) line(h) for nonincident pair (P, H) (x=2) and the class with x = 7 in P G(3, 4) constructed by Govaerts and Pentilla, which is unique up to a collineation.

29 Conclusion The patterns approach for studying Cameron-Liebler line classes. Classification of Cameron-Liebler line classes in P G(n, 4). Promising results for other values of q.

Cameron Liebler line classes

Cameron Liebler line classes Cameron Liebler line classes Alexander Gavrilyuk Krasovsky Institute of Mathematics and Mechanics, (Yekaterinburg, Russia) based on joint work with Ivan Mogilnykh, Institute of Mathematics (Novosibirsk,

More information

A modular equality for Cameron-Liebler line classes

A modular equality for Cameron-Liebler line classes A modular equality for Cameron-Liebler line classes Alexander L. Gavrilyuk Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579,

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

More information

COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS

COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS Alexander Burstein Department of Mathematics Howard University Washington, DC 259, USA aburstein@howard.edu Sergey Kitaev Mathematics

More information

SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

More information

1 Norms and Vector Spaces

1 Norms and Vector Spaces 008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)

More information

Systems of Linear Equations

Systems of Linear Equations Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

More information

3. INNER PRODUCT SPACES

3. INNER PRODUCT SPACES . INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

More information

Discernibility Thresholds and Approximate Dependency in Analysis of Decision Tables

Discernibility Thresholds and Approximate Dependency in Analysis of Decision Tables Discernibility Thresholds and Approximate Dependency in Analysis of Decision Tables Yu-Ru Syau¹, En-Bing Lin²*, Lixing Jia³ ¹Department of Information Management, National Formosa University, Yunlin, 63201,

More information

Lines in higgledy-piggledy position

Lines in higgledy-piggledy position Lines in higgledy-piggledy position arxiv:1402.4028v1 [math.co] 17 Feb 2014 Szabolcs L. Fancsali MTA-ELTE Geometric and Algebraic Combinatorics Research Group Péter Sziklai MTA-ELTE Geometric and Algebraic

More information

NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

More information

Let H and J be as in the above lemma. The result of the lemma shows that the integral

Let H and J be as in the above lemma. The result of the lemma shows that the integral Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;

More information

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

More information

1 Sets and Set Notation.

1 Sets and Set Notation. LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

More information

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers

More information

Math 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010

Math 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010 Math 550 Notes Chapter 7 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 7 Fall 2010 1 / 34 Outline 1 Self-Adjoint and Normal Operators

More information

ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath

ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath International Electronic Journal of Algebra Volume 7 (2010) 140-151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December

More information

Section 6.1 - Inner Products and Norms

Section 6.1 - Inner Products and Norms Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,

More information

Finite dimensional C -algebras

Finite dimensional C -algebras Finite dimensional C -algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for self-adjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n

More information

Social Media Mining. Graph Essentials

Social Media Mining. Graph Essentials Graph Essentials Graph Basics Measures Graph and Essentials Metrics 2 2 Nodes and Edges A network is a graph nodes, actors, or vertices (plural of vertex) Connections, edges or ties Edge Node Measures

More information

Classification of Cartan matrices

Classification of Cartan matrices Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

More information

1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES 1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

More information

4. CLASSES OF RINGS 4.1. Classes of Rings class operator A-closed Example 1: product Example 2:

4. CLASSES OF RINGS 4.1. Classes of Rings class operator A-closed Example 1: product Example 2: 4. CLASSES OF RINGS 4.1. Classes of Rings Normally we associate, with any property, a set of objects that satisfy that property. But problems can arise when we allow sets to be elements of larger sets

More information

JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004

JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004 Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February

More information

The Assignment Problem and the Hungarian Method

The Assignment Problem and the Hungarian Method The Assignment Problem and the Hungarian Method 1 Example 1: You work as a sales manager for a toy manufacturer, and you currently have three salespeople on the road meeting buyers. Your salespeople are

More information

THE DESIGN OF AN EFFICIENT LOAD BALANCING ALGORITHM EMPLOYING BLOCK DESIGN. Ilyong Chung and Yongeun Bae. 1. Introduction

THE DESIGN OF AN EFFICIENT LOAD BALANCING ALGORITHM EMPLOYING BLOCK DESIGN. Ilyong Chung and Yongeun Bae. 1. Introduction J. Appl. Math. & Computing Vol. 14(2004), No. 1-2, pp. 343-351 THE DESIGN OF AN EFFICIENT LOAD BALANCING ALGORITHM EMPLOYING BLOCK DESIGN Ilyong Chung and Yongeun Bae Abstract. In order to maintain load

More information

ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES OF R n KEITH CONRAD ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x

More information

4: SINGLE-PERIOD MARKET MODELS

4: SINGLE-PERIOD MARKET MODELS 4: SINGLE-PERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market

More information

Hacking-proofness and Stability in a Model of Information Security Networks

Hacking-proofness and Stability in a Model of Information Security Networks Hacking-proofness and Stability in a Model of Information Security Networks Sunghoon Hong Preliminary draft, not for citation. March 1, 2008 Abstract We introduce a model of information security networks.

More information

Studia Scientiarum Mathematicarum Hungarica 41 (2), 243 266 (2004)

Studia Scientiarum Mathematicarum Hungarica 41 (2), 243 266 (2004) Studia Scientiarum Mathematicarum Hungarica 4 (), 43 66 (004) PLANAR POINT SETS WITH A SMALL NUMBER OF EMPTY CONVEX POLYGONS I. BÁRÁNY and P. VALTR Communicated by G. Fejes Tóth Abstract A subset A of

More information

Matrix Representations of Linear Transformations and Changes of Coordinates

Matrix Representations of Linear Transformations and Changes of Coordinates Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under

More information

Binary Codes and Caps

Binary Codes and Caps Binary Codes and Caps Aiden A. Bruen, 1 Lucien Haddad, 2 David L. Wehlau 3 1 Dept. of Mathematics, University of Western Ontario, London, Ontario, Canada N6A 3K7 2 Dept. of Mathematics and Computer Science,

More information

Scheduling Shop Scheduling. Tim Nieberg

Scheduling Shop Scheduling. Tim Nieberg Scheduling Shop Scheduling Tim Nieberg Shop models: General Introduction Remark: Consider non preemptive problems with regular objectives Notation Shop Problems: m machines, n jobs 1,..., n operations

More information

A note on the geometry of three circles

A note on the geometry of three circles A note on the geometry of three circles R. Pacheco, F. Pinheiro and R. Portugal Departamento de Matemática, Universidade da Beira Interior, Rua Marquês d Ávila e Bolama, 6201-001, Covilhã - Portugal. email:

More information

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column

More information

Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs

Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs Stavros Athanassopoulos, Ioannis Caragiannis, and Christos Kaklamanis Research Academic Computer Technology Institute

More information

CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction

CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction Acta Math. Univ. Comenianae Vol. LXXXI, 1 (2012), pp. 71 77 71 CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC E. BALLICO Abstract. Here we study (in positive characteristic) integral curves

More information

FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT MINING SYSTEM

FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT MINING SYSTEM International Journal of Innovative Computing, Information and Control ICIC International c 0 ISSN 34-48 Volume 8, Number 8, August 0 pp. 4 FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT

More information

Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and

Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study

More information

QUANTIZED PRINCIPAL COMPONENT ANALYSIS WITH APPLICATIONS TO LOW-BANDWIDTH IMAGE COMPRESSION AND COMMUNICATION. D. Wooden. M. Egerstedt. B.K.

QUANTIZED PRINCIPAL COMPONENT ANALYSIS WITH APPLICATIONS TO LOW-BANDWIDTH IMAGE COMPRESSION AND COMMUNICATION. D. Wooden. M. Egerstedt. B.K. International Journal of Innovative Computing, Information and Control ICIC International c 5 ISSN 1349-4198 Volume x, Number x, x 5 pp. QUANTIZED PRINCIPAL COMPONENT ANALYSIS WITH APPLICATIONS TO LOW-BANDWIDTH

More information

Network (Tree) Topology Inference Based on Prüfer Sequence

Network (Tree) Topology Inference Based on Prüfer Sequence Network (Tree) Topology Inference Based on Prüfer Sequence C. Vanniarajan and Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology Madras Chennai 600036 vanniarajanc@hcl.in,

More information

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

More information

Tree sums and maximal connected I-spaces

Tree sums and maximal connected I-spaces Tree sums and maximal connected I-spaces Adam Bartoš drekin@gmail.com Faculty of Mathematics and Physics Charles University in Prague Twelfth Symposium on General Topology Prague, July 2016 Maximal and

More information

MATH1231 Algebra, 2015 Chapter 7: Linear maps

MATH1231 Algebra, 2015 Chapter 7: Linear maps MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter

More information

Linear Threshold Units

Linear Threshold Units Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear

More information

Method of Stationary phase. Reference: Hormander vol I. Steve Zelditch Department of Mathematics Northwestern University

Method of Stationary phase. Reference: Hormander vol I. Steve Zelditch Department of Mathematics Northwestern University Method of Stationary phase Reference: Hormander vol I Steve Zelditch Department of Mathematics Northwestern University 1 Method of Stationary Phase We now describe the method of stationary phase, which

More information

3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices.

3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices. Exercise 1 1. Let A be an n n orthogonal matrix. Then prove that (a) the rows of A form an orthonormal basis of R n. (b) the columns of A form an orthonormal basis of R n. (c) for any two vectors x,y R

More information

Factorization Algorithms for Polynomials over Finite Fields

Factorization Algorithms for Polynomials over Finite Fields Degree Project Factorization Algorithms for Polynomials over Finite Fields Sajid Hanif, Muhammad Imran 2011-05-03 Subject: Mathematics Level: Master Course code: 4MA11E Abstract Integer factorization is

More information

Chapter 1 A Pri Characterization of T m e Pairs w in Proceedings NCUR V. (1991), Vol. I, pp. 362{366. Jerey F. Gold Department of Mathematics, Department of Physics University of Utah DonH.Tucker Department

More information

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

More information

Computer Science Department. Technion - IIT, Haifa, Israel. Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there

Computer Science Department. Technion - IIT, Haifa, Israel. Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there - 1 - THREE TREE-PATHS Avram Zehavi Alon Itai Computer Science Department Technion - IIT, Haifa, Israel Abstract Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there

More information

Inner Product Spaces

Inner Product Spaces Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

More information

2.3 Convex Constrained Optimization Problems

2.3 Convex Constrained Optimization Problems 42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

More information

Removing Even Crossings

Removing Even Crossings EuroComb 2005 DMTCS proc. AE, 2005, 105 110 Removing Even Crossings Michael J. Pelsmajer 1, Marcus Schaefer 2 and Daniel Štefankovič 2 1 Department of Applied Mathematics, Illinois Institute of Technology,

More information

State of Stress at Point

State of Stress at Point State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

More information

Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

More information

INTRODUCTORY SET THEORY

INTRODUCTORY SET THEORY M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

More information

Extremal Wiener Index of Trees with All Degrees Odd

Extremal Wiener Index of Trees with All Degrees Odd MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (2013) 287-292 ISSN 0340-6253 Extremal Wiener Index of Trees with All Degrees Odd Hong Lin School of

More information

Introduction to Modern Algebra

Introduction to Modern Algebra Introduction to Modern Algebra David Joyce Clark University Version 0.0.6, 3 Oct 2008 1 1 Copyright (C) 2008. ii I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write

More information

Die ganzen zahlen hat Gott gemacht

Die ganzen zahlen hat Gott gemacht Die ganzen zahlen hat Gott gemacht Polynomials with integer values B.Sury A quote attributed to the famous mathematician L.Kronecker is Die Ganzen Zahlen hat Gott gemacht, alles andere ist Menschenwerk.

More information

Duality of linear conic problems

Duality of linear conic problems Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

More information

Chapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors

Chapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors Chapter 9. General Matrices An n m matrix is an array a a a m a a a m... = [a ij]. a n a n a nm The matrix A has n row vectors and m column vectors row i (A) = [a i, a i,..., a im ] R m a j a j a nj col

More information

TAKE-AWAY GAMES. ALLEN J. SCHWENK California Institute of Technology, Pasadena, California INTRODUCTION

TAKE-AWAY GAMES. ALLEN J. SCHWENK California Institute of Technology, Pasadena, California INTRODUCTION TAKE-AWAY GAMES ALLEN J. SCHWENK California Institute of Technology, Pasadena, California L INTRODUCTION Several games of Tf take-away?f have become popular. The purpose of this paper is to determine the

More information

1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0

1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 Solutions: Assignment 4.. Find the redundant column vectors of the given matrix A by inspection. Then find a basis of the image of A and a basis of the kernel of A. 5 A The second and third columns are

More information

Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

More information

On the crossing number of K m,n

On the crossing number of K m,n On the crossing number of K m,n Nagi H. Nahas nnahas@acm.org Submitted: Mar 15, 001; Accepted: Aug 10, 00; Published: Aug 1, 00 MR Subject Classifications: 05C10, 05C5 Abstract The best lower bound known

More information

Prime Numbers and Irreducible Polynomials

Prime Numbers and Irreducible Polynomials Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.

More information

Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I. Ronald van Luijk, 2012 Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

More information

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function 17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):

More information

Name: Section Registered In:

Name: Section Registered In: Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are

More information

Finding and counting given length cycles

Finding and counting given length cycles Finding and counting given length cycles Noga Alon Raphael Yuster Uri Zwick Abstract We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected

More information

The Ideal Class Group

The Ideal Class Group Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

More information

(Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties

(Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties Lecture 1 Convex Sets (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties 1.1.1 A convex set In the school geometry

More information

BINARY SELF-DUAL CODES OF LENGTH 24

BINARY SELF-DUAL CODES OF LENGTH 24 BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 8, Number 6, November 1974 BINARY SELF-DUAL CODES OF LENGTH BY VERA PLESS 1 AND N. J. A. SLOANE Communicated by Olga Taussky Todd, February 28, 1974

More information

Network Design and Protection Using Network Coding

Network Design and Protection Using Network Coding Network Design and Protection Using Network Coding Salah A. Aly Electrical & Computer Eng. Dept. New Jersey Institute of Technology salah@njit.edu Ahmed E. Kamal Electrical & Computer Eng. Dept. Iowa State

More information

Dragoš Cvetković, Tatjana Davidović MULTIPROCESSOR INTERCONNECTION NETWORKS

Dragoš Cvetković, Tatjana Davidović MULTIPROCESSOR INTERCONNECTION NETWORKS Dragoš Cvetković, Tatjana Davidović MULTIPROCESSOR INTERCONNECTION NETWORKS Abstract. Homogeneous multiprocessor systems are usually modelled by undirected graphs. Vertices of these graphs represent the

More information

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

More information

THE AVERAGE DEGREE OF AN IRREDUCIBLE CHARACTER OF A FINITE GROUP

THE AVERAGE DEGREE OF AN IRREDUCIBLE CHARACTER OF A FINITE GROUP THE AVERAGE DEGREE OF AN IRREDUCIBLE CHARACTER OF A FINITE GROUP by I. M. Isaacs Mathematics Department University of Wisconsin 480 Lincoln Dr. Madison, WI 53706 USA E-Mail: isaacs@math.wisc.edu Maria

More information

THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

More information

SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

More information

Copyright. Rohit Ghosh

Copyright. Rohit Ghosh Copyright by Rohit Ghosh 2007 The Dissertation Committee for Rohit Ghosh certifies that this is the approved version of the following dissertation: Incompleteness of the Giulietti-Ughi arc for Large Primes

More information

MCS 563 Spring 2014 Analytic Symbolic Computation Wednesday 9 April. Hilbert Polynomials

MCS 563 Spring 2014 Analytic Symbolic Computation Wednesday 9 April. Hilbert Polynomials Hilbert Polynomials For a monomial ideal, we derive the dimension counting the monomials in the complement, arriving at the notion of the Hilbert polynomial. The first half of the note is derived from

More information

FIXED POINT SETS OF FIBER-PRESERVING MAPS

FIXED POINT SETS OF FIBER-PRESERVING MAPS FIXED POINT SETS OF FIBER-PRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 e-mail: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics

More information

TECHNIQUES OF. C.T. Pan 1. C.T. Pan

TECHNIQUES OF. C.T. Pan 1. C.T. Pan TECHNIQUES OF CIRCUIT ANALYSIS C.T. Pan 1 4.1 Introduction 4.2 The Node-Voltage Method ( Nodal Analysis ) 4.3 The Mesh-Current Method ( Mesh Analysis ) 4.4 Fundamental Loop Analysis 4.5 Fundamental Cutset

More information

Graphical degree sequences and realizations

Graphical degree sequences and realizations swap Graphical and realizations Péter L. Erdös Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences MAPCON 12 MPIPKS - Dresden, May 15, 2012 swap Graphical and realizations Péter L. Erdös

More information

A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE

A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE DANIEL A. RAMRAS In these notes we present a construction of the universal cover of a path connected, locally path connected, and semi-locally simply

More information

MA106 Linear Algebra lecture notes

MA106 Linear Algebra lecture notes MA106 Linear Algebra lecture notes Lecturers: Martin Bright and Daan Krammer Warwick, January 2011 Contents 1 Number systems and fields 3 1.1 Axioms for number systems......................... 3 2 Vector

More information

CSG 399 Lecture. The Inductive Approach to Protocol Analysis p.1

CSG 399 Lecture. The Inductive Approach to Protocol Analysis p.1 The Inductive Approach to Protocol Analysis CSG 399 Lecture The Inductive Approach to Protocol Analysis p.1 Last Time CSP approach: Model system as a CSP process A specification is a property of traces

More information

11 Ideals. 11.1 Revisiting Z

11 Ideals. 11.1 Revisiting Z 11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

More information

Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

More information

AP CALCULUS AB 2009 SCORING GUIDELINES

AP CALCULUS AB 2009 SCORING GUIDELINES AP CALCULUS AB 2009 SCORING GUIDELINES Question 5 x 2 5 8 f ( x ) 1 4 2 6 Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for selected points in

More information

Vector Spaces 4.4 Spanning and Independence

Vector Spaces 4.4 Spanning and Independence Vector Spaces 4.4 and Independence October 18 Goals Discuss two important basic concepts: Define linear combination of vectors. Define Span(S) of a set S of vectors. Define linear Independence of a set

More information

it is easy to see that α = a

it is easy to see that α = a 21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

More information

Notes on Factoring. MA 206 Kurt Bryan

Notes on Factoring. MA 206 Kurt Bryan The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

More information

Notes on the representational possibilities of projective quadrics in four dimensions

Notes on the representational possibilities of projective quadrics in four dimensions bacso 2006/6/22 18:13 page 167 #1 4/1 (2006), 167 177 tmcs@inf.unideb.hu http://tmcs.math.klte.hu Notes on the representational possibilities of projective quadrics in four dimensions Sándor Bácsó and

More information

. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9

. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9 Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a

More information

Communities and Emerging Semantics in Semantic Link Network: Discovery and Learning

Communities and Emerging Semantics in Semantic Link Network: Discovery and Learning IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, MANUSCRIPT ID: TKDE-2007-07-0321.R1 1 Communities and Emerging Semantics in Semantic Link Network: Discovery and Learning Hai Zhuge, Senior Member,

More information

Irreducible Representations of Wreath Products of Association Schemes

Irreducible Representations of Wreath Products of Association Schemes Journal of Algebraic Combinatorics, 18, 47 52, 2003 c 2003 Kluwer Academic Publishers. Manufactured in The Netherlands. Irreducible Representations of Wreath Products of Association Schemes AKIHIDE HANAKI

More information

MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =

MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A = MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the

More information