The Second Eigenvalue of Planar Graphs
|
|
- Reynard Perry
- 5 years ago
- Views:
Transcription
1 Spectral Graph Theory Lecture 20 The Second Egenvalue of Planar Graphs Danel A. Spelman November 11, 2015 Dsclamer These notes are not necessarly an accurate representaton of what happened n class. The notes wrtten before class say what I thnk I should say. I sometmes edt the notes after class to make them way what I wsh I had sad. There may be small mstakes, so I recommend that you check any mathematcally precse statement before usng t n your own work. These notes were last revsed on November 11, Overvew Spectral Graph theory frst came to the attenton of many because of the success of usng the second Laplacan egenvector to partton planar graphs and scentfc meshes [DH72, DH73, Bar82,?, Sm91]. In ths lecture, we wll attempt to explan ths success by provng, at least for planar graphs, that the second smallest Laplacan egenvalue s small. One can then use Cheeger s nequalty to prove that the correspondng egenvector provdes a good cut. Theorem Let G be a planar graph wth n vertces of maxmum degree d, and let λ 2 be the second-smallest egenvalue of ts Laplacan. Then, λ 2 8d n. The proof wll nvolve almost no calculaton, but wll use some specal propertes of planar graphs. However, ths proof has been generalzed to many planar-lke graphs, ncludng the graphs of wellshaped 3d meshes. I begn by recallng two defntons of planar graphs. Defnton A graph s planar f there exsts an embeddng of the vertces n IR 2, f : V IR 2 and a mappng of edges e E to smple curves n IR 2, f e : [0, 1] IR 2 such that the endponts of the curves are the vertces at the endponts of the edge, and no two curve ntersect n ther nterors. Defnton A graph s planar f there exsts an embeddng of the vertces n IR 2, f : V IR 2 such that for all pars of edges (a, b) and (c, d) n E, wth a, b, c, and d dstnct, the lne segment from f(a) to f(b) does not cross the lne segment from f(c) to f(d). 20-1
2 Lecture 20: November 11, These defntons are equvalent Geometrc Embeddngs We typcally upper bound λ 2 by evdencng a test vector. evdencng a test embeddng. The bound we apply s: Here, we wll upper bound λ 2 by Lemma For any d 1, λ 2 = mn v 1,...,v n IR d : v =0 (,j) E v v j 2 v 2. (20.1) Proof. Let v = (x, y,..., z ). We note that v v j 2 = (x x j ) 2 + (y y j ) (z z j ) 2. Smlarly, (,j) E (,j) E v 2 = x 2 + (,j) E y z 2. (,j) E It s now trval to show that λ 2 RHS: just let x = y = = z be gven by an egenvector of A+B+ +C λ 2. To show that λ 2 RHS, we apply my favorte nequalty: A +B + +C mn ( A A, B B,..., C ) C, and then recall that x = 0 mples (,j) E (x x j ) 2 x2 λ 2. For an example, consder the natural embeddng of the square wth corners (±1, ±1). The key to applyng ths embeddng lemma s to obtan the rght embeddng of a planar graph. Usually, the rght embeddng of a planar graph s gven by Koebe s embeddng theorem, whch I wll now explan. I begn by consderng one way of generatng planar graphs. Consder a set of crcles {C 1,..., C n } n the plane such that no par of crcles ntersects n ther nterors. Assocate a vertex wth each crcle, and create an edge between each par of crcles that meet at a boundary. The resultng graph s clearly planar. Koebe s embeddng theorem says that every planar graph results from such an embeddng. Theorem (Koebe). Let G = (V, E) be a planar graph. Then there exsts a set of crcles {C 1,..., C n } n IR 2 that are nteror-dsjont such that crcle C touches crcle C j f and only f (, j) E. Ths s an amazng theorem, whch I won t prove today. You can fnd a beautful proof n the book Combnatoral Geometry by Agarwal and Pach.
3 Lecture 20: November 11, Such an embeddng s often called a kssng dsk embeddng of the graph. From a kssng dsk embeddng, we obtan a natural choce of v : the center of dsk C. Let r denote the radus of ths dsk. We now have an easy upper bound on the numerator of (20.1): v v j 2 = (r + r j ) 2 2r 2 + 2r2 j. On the other hand, t s trcker to obtan a lower bound on v 2. In fact, there are graphs whose kssng dsk embeddngs result n (20.1) = Θ(1). These graphs come from trangles nsde trangles nsde trangles... Such a graph s depcted below: Graph Dscs We wll fx ths problem by lftng the planar embeddngs to the sphere by stereographc projecton. Gven a plane, IR 2, and a sphere S tangent to the plane, we can defne the stereographc projecton map, Π, from the plane to the sphere as follows: let s denote the pont where the sphere touches the plane, and let n denote the opposte pont on the sphere. For any pont x on the plane, consder the lne from x to n. It wll ntersect the sphere somewhere. We let ths pont of ntersecton be Π(x ). The fundamental fact that we wll explot about stereographc projecton s that t maps crcles to crcles! So, by applyng stereographc projecton to a kssng dsk embeddng of a graph n the plane, we obtan a kssng dsk embeddng of that graph on the sphere. Let D = Π(C ) denote the mage of crcle C on the sphere. We wll now let v denote the center of D, on the sphere. If we had v = 0, the rest of the computaton would be easy. For each, v = 1, so the denomnator of (20.1) s n. Let r denote the straght-lne dstance from v to the boundary of D. We then have v v j 2 (r + r j ) 2 2r 2 + 2r 2 j. So, the denomnator of (20.1) s at most 2d r2. On the other hand, the area of the cap encrcled by D s at least πr 2. As the caps are dsjont, we have πr 2 4π, whch mples that the denomnator of (20.1) s at most 2d r 2 8d.
4 Lecture 20: November 11, Puttng these nequaltes together, we see that mn v 1,...,v n IR d : v =0 (,j) E v v j 2 v 2. 8d n. Thus, we merely need to verfy that we can ensure that v = 0. (20.2) Note that there s enough freedom n our constructon to beleve that we could prove such a thng: we can put the sphere anywhere on the plane, and we could even scale the mage n the plane before placng the sphere. By carefully combnng these two operatons, t s clear that we can place the center of gravty of the v s close to any pont on the boundary of the sphere. It turns out that ths s suffcent to prove that we can place t at the orgn The center of gravty We need a nce famly of maps that transform our kssng dsk embeddng on the sphere. It s partcularly convenent to parameterze these by a pont ω nsde the sphere. For any pont α on the surface of the unt sphere, I wll let Π α denote the stereographc projecton from the plane tangent to the sphere at α. I wll also defne Π 1 α. To handle the pont α, I let Π 1 α ( α) =, and Π α ( ) = α. We also defne the map that dlates the plane tangent to the sphere at α by a factor a: Dα. a We then defne f ω (x ) def = Π ω/ ω ( D 1 ω ω/ ω ( )) Π 1 ω/ ω. For α S and ω = aα, ths map pushes everythng on the sphere to a pont close to α. As a approaches 1, the mass gets pushed closer and closer to α. Instead of provng that we can acheve (20.2), I wll prove a slghtly smpler theorem. The proof of the theorem we really want s smlar, but about just a few mnutes too long for class. We wll prove Theorem Let v 1,..., v n be ponts on the unt-sphere. Then, there exsts a crcle-preservng map from the unt-sphere to tself. The reason that ths theorem s dfferent from the one that we want to prove s that f we apply a crcle-preservng map from the sphere to tself, the center of the crcle mght not map to the center of the mage crcle. To show that we can acheve v = 0, we wll use the followng topologcal lemma, whch follows mmedately from Brouwer s fxed pont theorem. In the followng, we let B denote the ball of ponts of norm less than 1, and the the sphere of ponts of norm 1.
5 Lecture 20: November 11, Lemma If φ : B B be a contnuous map that s the dentty on S. Then, there exsts an ω B such that φ(ω) = 0. We wll prove ths lemma usng Brouwer s fxed pont theorem: Theorem (Brouwer). If g : B B s contnuous, then there exsts an α B such that g(α) = α. Proof of Lemma Let b be the map that sends z B to z / z. The map b s contnuous at every pont other than 0. Now, assume by way of contradcton that 0 s not n the mage of φ, and let g(z ) = b(φ(z )). By our assumpton, g s contnuous and maps B to B. However, t s clear that g has no fxed pont, contradcton Brouwer s fxed pont theorem. Lemma , was our motvaton for defnng the maps f ω n terms of ω B. Now consder settng φ(ω) = 1 f ω (v ). n The only thng that stops us from applyng Lemma at ths pont s that φ s not defned on S, because f ω was not defned for ω S. To fx ths, we defne for α S { α f z α f α (z ) = α otherwse. We then encounter the problem that f α (z ) s not a contnuous functon of α. To fx ths, we set { 1 f dst(ω, z ) < 2 ɛ, and h ω (z ) = (2 dst(ω, z ))/ɛ otherwse. Now, the functon f α (z )h α (z ) s contnuous, because h α ( α) = 0. So, we may set φ(ω) def = f ω(v )h ω (v ) h ω/ ω (v ), whch s now contnuous and s the dentty map on S. So, for any ɛ > 0, we may now apply Lemma to fnd an ω for whch φ(ω) = 0. It s a smple exercse to verfy that for ɛ suffcently small, the ω we fnd wll have norm bounded away from 1, and so h ω (v ) = 1 for all, n whch case f ω (v ) = 0, as desred.
6 Lecture 20: November 11, Further progress Ths result has been mproved n the followng... References [Bar82] Earl R. Barnes. An algorthm for parttonng the nodes of a graph. SIAM Journal on Algebrac and Dscrete Methods, 3(4): , [DH72] W. E. Donath and A. J. Hoffman. Algorthms for parttonng graphs and computer logc based on egenvectors of connecton matrces. IBM Techncal Dsclosure Bulletn, 15(3): , [DH73] W. E. Donath and A. J. Hoffman. Lower bounds for the parttonng of graphs. IBM Journal of Research and Development, 17(5): , September [Sm91] Horst D. Smon. Parttonng of unstructured problems for parallel processng. Computng Systems n Engneerng, 2: , 1991.
Spectral Graph Theory and its Applications September 16, Lecture 5
Spectral Graph Theory and ts Applcatons September 16, 2004 Lecturer: Danel A. Spelman Lecture 5 5.1 Introducton In ths lecture, we wll prove the followng theorem: Theorem 5.1.1. Let G be a planar graph
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationEigenvalues of Random Graphs
Spectral Graph Theory Lecture 2 Egenvalues of Random Graphs Danel A. Spelman November 4, 202 2. Introducton In ths lecture, we consder a random graph on n vertces n whch each edge s chosen to be n the
More informationStanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011
Stanford Unversty CS359G: Graph Parttonng and Expanders Handout 4 Luca Trevsan January 3, 0 Lecture 4 In whch we prove the dffcult drecton of Cheeger s nequalty. As n the past lectures, consder an undrected
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationEdge Isoperimetric Inequalities
November 7, 2005 Ross M. Rchardson Edge Isopermetrc Inequaltes 1 Four Questons Recall that n the last lecture we looked at the problem of sopermetrc nequaltes n the hypercube, Q n. Our noton of boundary
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationLecture 3. Ax x i a i. i i
18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest
More informationLecture 10: May 6, 2013
TTIC/CMSC 31150 Mathematcal Toolkt Sprng 013 Madhur Tulsan Lecture 10: May 6, 013 Scrbe: Wenje Luo In today s lecture, we manly talked about random walk on graphs and ntroduce the concept of graph expander,
More informationprinceton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg
prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationErrors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
More information9 Characteristic classes
THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationU.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016
U.C. Berkeley CS94: Spectral Methods and Expanders Handout 8 Luca Trevsan February 7, 06 Lecture 8: Spectral Algorthms Wrap-up In whch we talk about even more generalzatons of Cheeger s nequaltes, and
More informationREAL ANALYSIS I HOMEWORK 1
REAL ANALYSIS I HOMEWORK CİHAN BAHRAN The questons are from Tao s text. Exercse 0.0.. If (x α ) α A s a collecton of numbers x α [0, + ] such that x α
More informationLinear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.
Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +
More informationThe path of ants Dragos Crisan, Andrei Petridean, 11 th grade. Colegiul National "Emil Racovita", Cluj-Napoca
Ths artcle s wrtten by students. It may nclude omssons and mperfectons, whch were dentfed and reported as mnutely as possble by our revewers n the edtoral notes. The path of ants 07-08 Students names and
More informationMAT 578 Functional Analysis
MAT 578 Functonal Analyss John Qugg Fall 2008 Locally convex spaces revsed September 6, 2008 Ths secton establshes the fundamental propertes of locally convex spaces. Acknowledgment: although I wrote these
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Handout 6 Luca Trevisan September 12, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 6 Luca Trevsan September, 07 Scrbed by Theo McKenze Lecture 6 In whch we study the spectrum of random graphs. Overvew When attemptng to fnd n polynomal
More informationDynamic Systems on Graphs
Prepared by F.L. Lews Updated: Saturday, February 06, 200 Dynamc Systems on Graphs Control Graphs and Consensus A network s a set of nodes that collaborates to acheve what each cannot acheve alone. A network,
More informationApproximate Smallest Enclosing Balls
Chapter 5 Approxmate Smallest Enclosng Balls 5. Boundng Volumes A boundng volume for a set S R d s a superset of S wth a smple shape, for example a box, a ball, or an ellpsod. Fgure 5.: Boundng boxes Q(P
More informationA 2D Bounded Linear Program (H,c) 2D Linear Programming
A 2D Bounded Lnear Program (H,c) h 3 v h 8 h 5 c h 4 h h 6 h 7 h 2 2D Lnear Programmng C s a polygonal regon, the ntersecton of n halfplanes. (H, c) s nfeasble, as C s empty. Feasble regon C s unbounded
More informationGraph Reconstruction by Permutations
Graph Reconstructon by Permutatons Perre Ille and Wllam Kocay* Insttut de Mathémathques de Lumny CNRS UMR 6206 163 avenue de Lumny, Case 907 13288 Marselle Cedex 9, France e-mal: lle@ml.unv-mrs.fr Computer
More information2 More examples with details
Physcs 129b Lecture 3 Caltech, 01/15/19 2 More examples wth detals 2.3 The permutaton group n = 4 S 4 contans 4! = 24 elements. One s the dentty e. Sx of them are exchange of two objects (, j) ( to j and
More informationINTERSECTION THEORY CLASS 13
INTERSECTION THEORY CLASS 13 RAVI VAKIL CONTENTS 1. Where we are: Segre classes of vector bundles, and Segre classes of cones 1 2. The normal cone, and the Segre class of a subvarety 3 3. Segre classes
More information10-801: Advanced Optimization and Randomized Methods Lecture 2: Convex functions (Jan 15, 2014)
0-80: Advanced Optmzaton and Randomzed Methods Lecture : Convex functons (Jan 5, 04) Lecturer: Suvrt Sra Addr: Carnege Mellon Unversty, Sprng 04 Scrbes: Avnava Dubey, Ahmed Hefny Dsclamer: These notes
More informationAppendix B. Criterion of Riemann-Stieltjes Integrability
Appendx B. Crteron of Remann-Steltes Integrablty Ths note s complementary to [R, Ch. 6] and [T, Sec. 3.5]. The man result of ths note s Theorem B.3, whch provdes the necessary and suffcent condtons for
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More information20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The first idea is connectedness.
20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The frst dea s connectedness. Essentally, we want to say that a space cannot be decomposed
More information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationMin Cut, Fast Cut, Polynomial Identities
Randomzed Algorthms, Summer 016 Mn Cut, Fast Cut, Polynomal Identtes Instructor: Thomas Kesselhem and Kurt Mehlhorn 1 Mn Cuts n Graphs Lecture (5 pages) Throughout ths secton, G = (V, E) s a mult-graph.
More informationAn Explicit Construction of an Expander Family (Margulis-Gaber-Galil)
An Explct Constructon of an Expander Famly Marguls-Gaber-Gall) Orr Paradse July 4th 08 Prepared for a Theorst's Toolkt, a course taught by Irt Dnur at the Wezmann Insttute of Scence. The purpose of these
More informationWeek 2. This week, we covered operations on sets and cardinality.
Week 2 Ths week, we covered operatons on sets and cardnalty. Defnton 0.1 (Correspondence). A correspondence between two sets A and B s a set S contaned n A B = {(a, b) a A, b B}. A correspondence from
More information2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification
E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 12 10/21/2013. Martingale Concentration Inequalities and Applications
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.65/15.070J Fall 013 Lecture 1 10/1/013 Martngale Concentraton Inequaltes and Applcatons Content. 1. Exponental concentraton for martngales wth bounded ncrements.
More informationCurvature and isoperimetric inequality
urvature and sopermetrc nequalty Julà ufí, Agustí Reventós, arlos J Rodríguez Abstract We prove an nequalty nvolvng the length of a plane curve and the ntegral of ts radus of curvature, that has as a consequence
More informationp 1 c 2 + p 2 c 2 + p 3 c p m c 2
Where to put a faclty? Gven locatons p 1,..., p m n R n of m houses, want to choose a locaton c n R n for the fre staton. Want c to be as close as possble to all the house. We know how to measure dstance
More informationMATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS
MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples
More informationRandomness and Computation
Randomness and Computaton or, Randomzed Algorthms Mary Cryan School of Informatcs Unversty of Ednburgh RC 208/9) Lecture 0 slde Balls n Bns m balls, n bns, and balls thrown unformly at random nto bns usually
More informationSome basic inequalities. Definition. Let V be a vector space over the complex numbers. An inner product is given by a function, V V C
Some basc nequaltes Defnton. Let V be a vector space over the complex numbers. An nner product s gven by a functon, V V C (x, y) x, y satsfyng the followng propertes (for all x V, y V and c C) (1) x +
More informationAssortment Optimization under MNL
Assortment Optmzaton under MNL Haotan Song Aprl 30, 2017 1 Introducton The assortment optmzaton problem ams to fnd the revenue-maxmzng assortment of products to offer when the prces of products are fxed.
More informationA new construction of 3-separable matrices via an improved decoding of Macula s construction
Dscrete Optmzaton 5 008 700 704 Contents lsts avalable at ScenceDrect Dscrete Optmzaton journal homepage: wwwelsevercom/locate/dsopt A new constructon of 3-separable matrces va an mproved decodng of Macula
More informationModeling curves. Graphs: y = ax+b, y = sin(x) Implicit ax + by + c = 0, x 2 +y 2 =r 2 Parametric:
Modelng curves Types of Curves Graphs: y = ax+b, y = sn(x) Implct ax + by + c = 0, x 2 +y 2 =r 2 Parametrc: x = ax + bxt x = cos t y = ay + byt y = snt Parametrc are the most common mplct are also used,
More informationOne Dimension Again. Chapter Fourteen
hapter Fourteen One Dmenson Agan 4 Scalar Lne Integrals Now we agan consder the dea of the ntegral n one dmenson When we were ntroduced to the ntegral back n elementary school, we consdered only functons
More informationLearning Theory: Lecture Notes
Learnng Theory: Lecture Notes Lecturer: Kamalka Chaudhur Scrbe: Qush Wang October 27, 2012 1 The Agnostc PAC Model Recall that one of the constrants of the PAC model s that the data dstrbuton has to be
More informationLecture 20: Lift and Project, SDP Duality. Today we will study the Lift and Project method. Then we will prove the SDP duality theorem.
prnceton u. sp 02 cos 598B: algorthms and complexty Lecture 20: Lft and Project, SDP Dualty Lecturer: Sanjeev Arora Scrbe:Yury Makarychev Today we wll study the Lft and Project method. Then we wll prove
More informationProblem Do any of the following determine homomorphisms from GL n (C) to GL n (C)?
Homework 8 solutons. Problem 16.1. Whch of the followng defne homomomorphsms from C\{0} to C\{0}? Answer. a) f 1 : z z Yes, f 1 s a homomorphsm. We have that z s the complex conjugate of z. If z 1,z 2
More informationNP-Completeness : Proofs
NP-Completeness : Proofs Proof Methods A method to show a decson problem Π NP-complete s as follows. (1) Show Π NP. (2) Choose an NP-complete problem Π. (3) Show Π Π. A method to show an optmzaton problem
More informationMaximizing the number of nonnegative subsets
Maxmzng the number of nonnegatve subsets Noga Alon Hao Huang December 1, 213 Abstract Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what s the maxmum
More informationVolume 18 Figure 1. Notation 1. Notation 2. Observation 1. Remark 1. Remark 2. Remark 3. Remark 4. Remark 5. Remark 6. Theorem A [2]. Theorem B [2].
Bulletn of Mathematcal Scences and Applcatons Submtted: 016-04-07 ISSN: 78-9634, Vol. 18, pp 1-10 Revsed: 016-09-08 do:10.1805/www.scpress.com/bmsa.18.1 Accepted: 016-10-13 017 ScPress Ltd., Swtzerland
More informationfind (x): given element x, return the canonical element of the set containing x;
COS 43 Sprng, 009 Dsjont Set Unon Problem: Mantan a collecton of dsjont sets. Two operatons: fnd the set contanng a gven element; unte two sets nto one (destructvely). Approach: Canoncal element method:
More informationStatistical Mechanics and Combinatorics : Lecture III
Statstcal Mechancs and Combnatorcs : Lecture III Dmer Model Dmer defntons Defnton A dmer coverng (perfect matchng) of a fnte graph s a set of edges whch covers every vertex exactly once, e every vertex
More informationRandom Walks on Digraphs
Random Walks on Dgraphs J. J. P. Veerman October 23, 27 Introducton Let V = {, n} be a vertex set and S a non-negatve row-stochastc matrx (.e. rows sum to ). V and S defne a dgraph G = G(V, S) and a drected
More informationEvery planar graph is 4-colourable a proof without computer
Peter Dörre Department of Informatcs and Natural Scences Fachhochschule Südwestfalen (Unversty of Appled Scences) Frauenstuhlweg 31, D-58644 Iserlohn, Germany Emal: doerre(at)fh-swf.de Mathematcs Subject
More informationDECOUPLING THEORY HW2
8.8 DECOUPLIG THEORY HW2 DOGHAO WAG DATE:OCT. 3 207 Problem We shall start by reformulatng the problem. Denote by δ S n the delta functon that s evenly dstrbuted at the n ) dmensonal unt sphere. As a temporal
More informationSTEINHAUS PROPERTY IN BANACH LATTICES
DEPARTMENT OF MATHEMATICS TECHNICAL REPORT STEINHAUS PROPERTY IN BANACH LATTICES DAMIAN KUBIAK AND DAVID TIDWELL SPRING 2015 No. 2015-1 TENNESSEE TECHNOLOGICAL UNIVERSITY Cookevlle, TN 38505 STEINHAUS
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More information12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product
12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton
More informationBOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS
BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve sem-defnte for all
More informationPerron Vectors of an Irreducible Nonnegative Interval Matrix
Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of
More informationFinding Dense Subgraphs in G(n, 1/2)
Fndng Dense Subgraphs n Gn, 1/ Atsh Das Sarma 1, Amt Deshpande, and Rav Kannan 1 Georga Insttute of Technology,atsh@cc.gatech.edu Mcrosoft Research-Bangalore,amtdesh,annan@mcrosoft.com Abstract. Fndng
More informationCaps and Colouring Steiner Triple Systems
Desgns, Codes and Cryptography, 13, 51 55 (1998) c 1998 Kluwer Academc Publshers, Boston. Manufactured n The Netherlands. Caps and Colourng Stener Trple Systems AIDEN BRUEN* Department of Mathematcs, Unversty
More informationEuropean Journal of Combinatorics
European Journal of Combnatorcs 0 (009) 480 489 Contents lsts avalable at ScenceDrect European Journal of Combnatorcs journal homepage: www.elsever.com/locate/ejc Tlngs n Lee metrc P. Horak 1 Unversty
More informationComplete subgraphs in multipartite graphs
Complete subgraphs n multpartte graphs FLORIAN PFENDER Unverstät Rostock, Insttut für Mathematk D-18057 Rostock, Germany Floran.Pfender@un-rostock.de Abstract Turán s Theorem states that every graph G
More informationarxiv: v1 [math.co] 1 Mar 2014
Unon-ntersectng set systems Gyula O.H. Katona and Dánel T. Nagy March 4, 014 arxv:1403.0088v1 [math.co] 1 Mar 014 Abstract Three ntersecton theorems are proved. Frst, we determne the sze of the largest
More informationOn the Operation A in Analysis Situs. by Kazimierz Kuratowski
v1.3 10/17 On the Operaton A n Analyss Stus by Kazmerz Kuratowsk Author s note. Ths paper s the frst part slghtly modfed of my thess presented May 12, 1920 at the Unversty of Warsaw for the degree of Doctor
More informationMatrix Approximation via Sampling, Subspace Embedding. 1 Solving Linear Systems Using SVD
Matrx Approxmaton va Samplng, Subspace Embeddng Lecturer: Anup Rao Scrbe: Rashth Sharma, Peng Zhang 0/01/016 1 Solvng Lnear Systems Usng SVD Two applcatons of SVD have been covered so far. Today we loo
More informationFirst day August 1, Problems and Solutions
FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve
More informationExercise Solutions to Real Analysis
xercse Solutons to Real Analyss Note: References refer to H. L. Royden, Real Analyss xersze 1. Gven any set A any ɛ > 0, there s an open set O such that A O m O m A + ɛ. Soluton 1. If m A =, then there
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationChapter Twelve. Integration. We now turn our attention to the idea of an integral in dimensions higher than one. Consider a real-valued function f : D
Chapter Twelve Integraton 12.1 Introducton We now turn our attenton to the dea of an ntegral n dmensons hgher than one. Consder a real-valued functon f : R, where the doman s a nce closed subset of Eucldean
More information42. Mon, Dec. 8 Last time, we were discussing CW complexes, and we considered two di erent CW structures on S n. We continue with more examples.
42. Mon, Dec. 8 Last tme, we were dscussng CW complexes, and we consdered two d erent CW structures on S n. We contnue wth more examples. (2) RP n. Let s start wth RP 2. Recall that one model for ths space
More informationLecture 17: Lee-Sidford Barrier
CSE 599: Interplay between Convex Optmzaton and Geometry Wnter 2018 Lecturer: Yn Tat Lee Lecture 17: Lee-Sdford Barrer Dsclamer: Please tell me any mstake you notced. In ths lecture, we talk about the
More informationRandić Energy and Randić Estrada Index of a Graph
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL
More informationSpectral graph theory: Applications of Courant-Fischer
Spectral graph theory: Applcatons of Courant-Fscher Steve Butler September 2006 Abstract In ths second talk we wll ntroduce the Raylegh quotent and the Courant- Fscher Theorem and gve some applcatons for
More informationSpectral Clustering. Shannon Quinn
Spectral Clusterng Shannon Qunn (wth thanks to Wllam Cohen of Carnege Mellon Unverst, and J. Leskovec, A. Raaraman, and J. Ullman of Stanford Unverst) Graph Parttonng Undrected graph B- parttonng task:
More informationStatistical Inference. 2.3 Summary Statistics Measures of Center and Spread. parameters ( population characteristics )
Ismor Fscher, 8//008 Stat 54 / -8.3 Summary Statstcs Measures of Center and Spread Dstrbuton of dscrete contnuous POPULATION Random Varable, numercal True center =??? True spread =???? parameters ( populaton
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationModule 2. Random Processes. Version 2 ECE IIT, Kharagpur
Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be
More informationCase A. P k = Ni ( 2L i k 1 ) + (# big cells) 10d 2 P k.
THE CELLULAR METHOD In ths lecture, we ntroduce the cellular method as an approach to ncdence geometry theorems lke the Szemeréd-Trotter theorem. The method was ntroduced n the paper Combnatoral complexty
More informationNon-negative Matrices and Distributed Control
Non-negatve Matrces an Dstrbute Control Yln Mo July 2, 2015 We moel a network compose of m agents as a graph G = {V, E}. V = {1, 2,..., m} s the set of vertces representng the agents. E V V s the set of
More informationMA 323 Geometric Modelling Course Notes: Day 13 Bezier Curves & Bernstein Polynomials
MA 323 Geometrc Modellng Course Notes: Day 13 Bezer Curves & Bernsten Polynomals Davd L. Fnn Over the past few days, we have looked at de Casteljau s algorthm for generatng a polynomal curve, and we have
More informationCSCE 790S Background Results
CSCE 790S Background Results Stephen A. Fenner September 8, 011 Abstract These results are background to the course CSCE 790S/CSCE 790B, Quantum Computaton and Informaton (Sprng 007 and Fall 011). Each
More informationThe Second Anti-Mathima on Game Theory
The Second Ant-Mathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2-player 2-acton zero-sum games 2. 2-player
More informationMath1110 (Spring 2009) Prelim 3 - Solutions
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.
More informationBezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0
Bezer curves Mchael S. Floater August 25, 211 These notes provde an ntroducton to Bezer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the
More informationLecture 10 Support Vector Machines. Oct
Lecture 10 Support Vector Machnes Oct - 20-2008 Lnear Separators Whch of the lnear separators s optmal? Concept of Margn Recall that n Perceptron, we learned that the convergence rate of the Perceptron
More informationNotes on Frequency Estimation in Data Streams
Notes on Frequency Estmaton n Data Streams In (one of) the data streamng model(s), the data s a sequence of arrvals a 1, a 2,..., a m of the form a j = (, v) where s the dentty of the tem and belongs to
More informationThe Number of Ways to Write n as a Sum of ` Regular Figurate Numbers
Syracuse Unversty SURFACE Syracuse Unversty Honors Program Capstone Projects Syracuse Unversty Honors Program Capstone Projects Sprng 5-1-01 The Number of Ways to Wrte n as a Sum of ` Regular Fgurate Numbers
More informationVapnik-Chervonenkis theory
Vapnk-Chervonenks theory Rs Kondor June 13, 2008 For the purposes of ths lecture, we restrct ourselves to the bnary supervsed batch learnng settng. We assume that we have an nput space X, and an unknown
More informationComplex Variables. Chapter 18 Integration in the Complex Plane. March 12, 2013 Lecturer: Shih-Yuan Chen
omplex Varables hapter 8 Integraton n the omplex Plane March, Lecturer: Shh-Yuan hen Except where otherwse noted, content s lcensed under a BY-N-SA. TW Lcense. ontents ontour ntegrals auchy-goursat theorem
More informationCollege of Computer & Information Science Fall 2009 Northeastern University 20 October 2009
College of Computer & Informaton Scence Fall 2009 Northeastern Unversty 20 October 2009 CS7880: Algorthmc Power Tools Scrbe: Jan Wen and Laura Poplawsk Lecture Outlne: Prmal-dual schema Network Desgn:
More information