A new construction of 3-separable matrices via an improved decoding of Macula s construction
|
|
- Richard McDaniel
- 6 years ago
- Views:
Transcription
1 Dscrete Optmzaton 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 s constructon Hung-Ln Fu, FK Hwang Department of Appled Mathematcs, Natonal Chao Tung Unversty, Hsn Chu,Tawan, ROC a r t c l e n f o a b s t r a c t Artcle hstory: Receved 30 July 004 Receved n revsed form 3 March 006 Accepted 16 Aprl 008 Avalable onlne 4 June 008 Keywords: 3-separable matrces Macula s constructon Macula proposed a novel constructon of poolng desgns whch can effectvely dentfy postve clones and also proposed a decodng method However, the probablty of an unresolved postve clone s hard to analyze In ths paper we propose an mproved decodng method and show that for d = 3 an exact probablty analyss s possble Further, we derve necessary and suffcent condtons for a postve clone to be unresolved and gave a modfed constructon whch avods ths necessary condton, thus resultng n a 3-separable matrx 008 Elsever BV All rghts reserved 1 Introducton A poolng desgn has many bologcal applcatons For convenence, we use the language of clone-lbrary screenng We have a set of n clones and a probe X whch s a short DNA sequence Let X denote the dual sequence of X, e, X s obtaned by frst reversng the order of the letters and then nterchangng A wth T and C wth G A clone s called postve f t contans X as a subsequence and negatve f not Typcally, there are a small number of postve clones, say, from 3 to 10, among the n clones The goal of a poolng desgn s to dentfy all postve clones through a small set of tests or pools performed parallelly A test can be appled to an arbtrary subset of clones wth two possble outcomes: a negatve outcome ndcates that the subset contans no postve clone, and a postve outcome ndcates otherwse Let M denote the ncdence matrx of a desgn wth clones labellng the columns and tests labellng the rows We wll treat a column as the subset of row labels where s n the column subset f and only f that column clone s n test M s called d-dsjunct f no column s contaned n the unon of any other d columns M s called d-separable f no two unons of dstnct sets of d columns are dentcal, and d-separable f d s changed to at most d It s well nown [1] that d-separable matrces can dentfy all postve clones f ther number s exactly d, whle d-separable or d-dsjunct matrces can dentfy all postve clones f ther number s at most d These matrces have become the major tools n constructng poolng desgns Many methods have been proposed to construct these matrces But ther exstence s stll rare for practcal need Recently, Macula [4] opened a new door by proposng the contanment constructon method More specfcally, let [m] = {1,, m} Then the columns of Mm,, d, d < are labelled by n random but dstnct -subsets, rows by all d-subsets, and M j = 1 f and only f the row label s contaned n the column label Macula proved that M s d-dsjunct Ths approach extends to many other contanment relatons as n partal orders [] and geometrcal structures [6] One problem wth ths constructon s that the number n of columns s bounded by m, hence m cannot be too small On the other hand, the number of tests s, so d must be small Macula [5] proposed usng Mm,, even though the actual m d Correspondng author E-mal address: hlfu@mathnctuedutw H-L Fu /$ see front matter 008 Elsever BV All rghts reserved do:101016/jdsopt
2 H-L Fu, FK Hwang / Dscrete Optmzaton number d of postve clones can be larger than In such an applcaton there may exst unresolved clones whose status of beng postve or negatve s unnown Let P + denote the probablty that a postve clone s unresolved The problem of computng P + under a gven decodng method turns out to be dffcult Macula [5] gave a smple decodng whle Hwang and Lu [3] mproved t, but wth a more complcate analyss where P + can be computed only for d = 3 In ths paper we further mprove the decodng method Although probablty analyss remans dffcult, we are able to accomplsh the followng for d = 3: 1 obtan necessary and suffcent condtons for a postve clone to be unresolved; gve an exact probablty analyss; and 3 derve a smple necessary condton and show that by choosng the column ndces judcously, ths necessary condton can be avoded and hence the matrx obtaned s 3-separable The unque-graph decodng Consder Mm,, throughout ths secton We shall use u, v to denote an edge jonng u and v The outcome graph G has [m] as ts vertex-set and an edge u, v f the pool labelled by u, v s postve, e, t has a postve outcome Note that each postve clone nduces a -clque n G, but a -clque n G can correspond to a negatve clone or a -subset of [m] not chosen as a column label whle all ts edges actually come from some other postve clones The unque-graph decodng conssts of the followng rules: If a clone appears n a negatve pool, t s negatve If a -clque n G contans an edge not appearng n any other -clque n G, then t represents a postve clone Let p be the number of postve clones dentfed n Let G be obtaned from G by removng all p -clques dentfed n as postve clones and also removng solated vertces Let S denote the set of edges n these -clques wth both endponts n G Defne G = {G S : S S} If G contans a unque graph G S whch s the unon of a unque set of p -clques for some p wth p + p p, then each of these p -clques represents a postve clone v All clones not dentfed n,, are unresolved Note that rule dfferentates the unque-graph decodng from the orgnal Hwang Lu decodng The followng example llustrates the dfference Example 1 = 3, d = 3, 13, 14, 345 are postve Usng, only 345 s dentfed snce ether 3,5 or 4,5 s an edge not n any other trangle Usng, G conssts of two graphs whle only the frst one s the unon of two trangles Thus 13 and 14 are dentfed Even the unque-graph decodng can leave postve clones unresolved Example = 3, d = 3 Snce G s the unon of four dfferent sets of three trangles, the unque-graph decodng fals to dentfy any postve clone Defne A B = A \ B B \ A We have: Theorem 1 Under the unque-graph decodng, a postve clone A s unresolved f and only f there exsts a nonpostve -clque B such that all edges of A B are contaned n the other postve clones
3 70 H-L Fu, FK Hwang / Dscrete Optmzaton Fg 1 Intersecton of A, C, D Proof f We cannot tell whether A or B s postve snce they nduce the same outcome graph only f Suppose there does not exst a nonpostve -clque n G Then every -clque n G must be postve On the other hand, every postve -clque s clearly n G Therefore the number of -clques n G equals the true number of postve -clques; hence at most p Consequently, these postve -clques can be dentfed through rule of the unque-graph decodng The requrement that all edges n A \ B are contaned n the other postve clones follows from the fact that f a -clque K contans an edge whch s not n any other -clque, then K represents a postve clone Fnally, all edges n B\A are necessarly n the other postve clones snce otherwse B would not be n G Corollary If d = 3, then a necessary and suffcent condton for A to be unresolved s that each of the other two postve clones contans A B Proof Let C and D denote the other two postve clones For a A \ B, C D must contan all edges from a to A \ {a} But nether C nor D alone can contan all of them snce t requres C A or D A, an absurdty Therefore a C and a D Smlarly, we can prove that every b B \ A s n both C and D Corollary 3 For d = 3, a necessary condton for A to be unresolved s that G contans a -clque wth > Proof A B \ A = B A \ B s a -clque wth > 3 Exact probablty analyss for d = 3 The necessary and suffcent condton n Theorem 1 s not convenent for computng the probabltes of unresolved clones snce t nvolves an unspecfed negatve clone For d = 3, we transform the condton nto condtons nvolvng the three postve clones A, C, D only Fg 1 shows the ntersectons of A, B, C where the seven parts are labelled by A, C, D, AC, AD, CD and ACD A necessary condton for A to be unresolved s that A = φ, whch mples AC φ, AD φ otherwse C or D would contan A, an absurdty Furthermore CD cannot be empty snce otherwse the edges from AC to AD forces A to be postve Fnally, by Theorem 1 all edges from AC to AD must be n B, whch mples B AC AD So f B contans x vertces n CD, B must leave x vertces n ACD out to enforce A = B = Let P + n denote the probablty a gven postve clone s unresolved Theorem 31 Consder three postve clones A, C, D Then A s unresolved f and only f A s empty whle CD ACD Proof only f Shown n the precedng paragraph f Note that A = and CD 1 mples AC AD CD ACD + 1 B can be selected from AC AD CD ACD wth the provson that f B taes x vertces from CD, then t must leave x vertces n ACD untouched so that A \ B = B \ A We are now ready to gve the probablty formulas Defne m = mn{, j} Theorem 3 P + n = 1 =1 b 1 j=0 m j h= j + h b + j h / N 1 Proof b s the number of choces of C wth AC = The rest of the numerator gves the number of choces of D such that D taes all the vertces of A \ C plus j vertces of AC, h vertces of C \ A, and the rest outsde of A C Note that j + h The denomnator gves the unconstraned number of choces of C and D f we also consder the choce of the n 3 negatve clones, we merely add the term to both the numerator and denomnator N 3 n 3
4 H-L Fu, FK Hwang / Dscrete Optmzaton Note that P + n s ndependent of n Let P + n x denote the probablty that exactly x postve clones are undentfed Lemma 33 x = 3 f and only f G s a -clque wth > Proof Clearly, a trval necessary condton for x = 3 s that A = C = D = φ, or G = A C D s a -clque, wth > We now show that ths condton s also suffcent Suppose ACD = j Then, AC = AD = CD = j/ Therefore CD + ACD = j + j = +j = 3 forces j = 1, satsfyng the condton of Theorem 31 Hence A s unresolved Smlarly, we can prove that C and D are unresolved Corollary 34 P + n 3 = 1 j=0 Proof Gven A, there are j j b j j/ j/ / N 3 choces of j vertces n ACD, and choces for the remanng vertces of C Once A and C are chosen, D s fxed j choces of j/ vertces n AC and b j/ j/ Lemma 35 x = f and only f exactly two of A, C and D are empty, say, C and D, and CD Proof AC = ACD CD = AD 1, snce otherwse C = D, an absurdty Further, AC + ACD = AD + ACD = CD Lemma 35 follows from Theorem 31 mmedately Corollary 36 P + n = 3 1/ = 1 j= +1 j j / N 1 Proof Suppose A s the only dentfed postve clone Assume AC = and ACD = j Then AD = as shown n the proof of Lemma 35 Snce A s dentfed, Hence and = A > AC + AD + ACD = j j + 1, 1 Note that we set to guarantee CD = The multplcaton by 3 s because any of A, C, D can be the unque resolved one Let P + n be obtaned from P + n by changng the upper bound of h from m to m 1 n Theorem 3 to guarantee that nether C nor D s empty Then P + n s the probablty that A s the unque unresolved postve clone Hence: Lemma 37 P + n 1 = 3P+ n Theorem 38 P + n 0 = 1 P+ n 3 P+ n P+ n 1 Note that P + n x s ndependent of n for any x 4 A new constructon of 3-separable matrces Theorem 41 Suppose d 3 and G contans no + 1-clque Then the unque-graph decodng dentfes all d postve clones Proof Suppose d = 1 or Snce a sngle postve clone cannot cover the edges of another postve clone, rule of the unque-graph decodng always dentfes all postve clones Suppose d = 3 Then the proof follows from Corollary 3 We now show how to choose the labels of the columns such that G does not contan a + 1-clque Partton the m ndces of [m] evenly nto parts Then K s a legtmate label of columns f K conssts of one ndex from each part Now, a + 1-clque has + 1 ndces, hence two of whch, say, u and v, must come from the same part But no label contanng both u and v s chosen snce t s not legtmate Therefore G does not contan the edge u, v, and à fortor, does not contan the + 1-clque contanng u and v
5 704 H-L Fu, FK Hwang / Dscrete Optmzaton Theorem 4 The number of legtmate labels s approxmately m Therefore, by Theorem 41, we have a constructon of a 3-separable matrx wth m tests and m clones We now explan why ths constructon does not lead to a 3-dsjunct matrx Note that rule of the unque-graph decodng dentfes postve clones even when there exst unresolved nonpostve -clques see Example 1, where trangles 14 and 134 are nonpostve and undentfed On the other hand, a d-dsjunct matrx has the property that all clones whch are not dentfed as negatve are dentfed as postve, a clear conflct wth rule The followng comparson shows the advantage and dsadvantage respectvely of usng ths new constructon Gven n clones, we want to choose m, such that m n wth the mnmum m so the number of tests s mnmzed Approxmate 1 by equalty Then 1 m = n 1 To mnmze the rght-hand sde sze of wth respect to, we obtan 0 = ln n Consequently, m 0 = e ln n So the number of tests s e ln n To compare wth Mm,,, we frst choose m, such that m n and m s mnmum Snce = m maxmzes m 3 There the number of tests s mn, we replace by m n 3 Denote by m n the m mnmzng the modfed Example Let n = 1000 Then m 0 = 0 and 0 = 6 snce = 0, and > 1000 Thus our 3-separable matrx requres 0 = 190 tests On the other hand, m 1000 = 13 = 6 snce 13 6 > 1000 > 1 6 Hence M13, 6, requres 78 tests However, to dentfy three postve clones, we need to use M13, 6, 3 whch requres 86 tests Acnowledgments The authors would le to than the referees for ther helpful comments and suggestons 3 References [1] DZ Du, FK Hwang, Combnatoral Group Testng and ts Applcatons, nd ed, World Scentfc, Sngapore, 000 [] TY Huang, CW Weng, Poolng spaces and nonadaptve poolng desgns, Dscrete Math [3] FK Hwang, YC Lu, Random poolng desgns under varous structures, J Comb Optm [4] AJ Macula, A smple constructon of d-dsjunct matrces wth certan constant weghts, Dscrete Appl Math [5] AJ Macula, Probablstc nonadaptve and two-stage group testng wth relatvely small pools and DNA lbrary screenng, J Comb Optm [6] H Ngo, DZ Du, New constructons of nonadaptve and error-tolerance poolng desgns, Dscrete Math
Every 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 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 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 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 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 informationThe lower and upper bounds on Perron root of nonnegative irreducible matrices
Journal of Computatonal Appled Mathematcs 217 (2008) 259 267 wwwelsevercom/locate/cam The lower upper bounds on Perron root of nonnegatve rreducble matrces Guang-Xn Huang a,, Feng Yn b,keguo a a College
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 informationDiscrete Mathematics
Dscrete Mathematcs 30 (00) 48 488 Contents lsts avalable at ScenceDrect Dscrete Mathematcs journal homepage: www.elsever.com/locate/dsc The number of C 3 -free vertces on 3-partte tournaments Ana Paulna
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 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 informationDiscrete Mathematics. Laplacian spectral characterization of some graphs obtained by product operation
Dscrete Mathematcs 31 (01) 1591 1595 Contents lsts avalable at ScVerse ScenceDrect Dscrete Mathematcs journal homepage: www.elsever.com/locate/dsc Laplacan spectral characterzaton of some graphs obtaned
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 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 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 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 informationA Simple Research of Divisor Graphs
The 29th Workshop on Combnatoral Mathematcs and Computaton Theory A Smple Research o Dvsor Graphs Yu-png Tsao General Educaton Center Chna Unversty o Technology Tape Tawan yp-tsao@cuteedutw Tape Tawan
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 informationLinear Algebra and its Applications
Lnear Algebra and ts Applcatons 4 (00) 5 56 Contents lsts avalable at ScenceDrect Lnear Algebra and ts Applcatons journal homepage: wwwelsevercom/locate/laa Notes on Hlbert and Cauchy matrces Mroslav Fedler
More informationTHE CHVÁTAL-ERDŐS CONDITION AND 2-FACTORS WITH A SPECIFIED NUMBER OF COMPONENTS
Dscussones Mathematcae Graph Theory 27 (2007) 401 407 THE CHVÁTAL-ERDŐS CONDITION AND 2-FACTORS WITH A SPECIFIED NUMBER OF COMPONENTS Guantao Chen Department of Mathematcs and Statstcs Georga State Unversty,
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationSingular Value Decomposition: Theory and Applications
Sngular Value Decomposton: Theory and Applcatons Danel Khashab Sprng 2015 Last Update: March 2, 2015 1 Introducton A = UDV where columns of U and V are orthonormal and matrx D s dagonal wth postve real
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 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 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 informationCOMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
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 informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationSociété de Calcul Mathématique SA
Socété de Calcul Mathématque SA Outls d'ade à la décson Tools for decson help Probablstc Studes: Normalzng the Hstograms Bernard Beauzamy December, 202 I. General constructon of the hstogram Any probablstc
More informationThe L(2, 1)-Labeling on -Product of Graphs
Annals of Pure and Appled Mathematcs Vol 0, No, 05, 9-39 ISSN: 79-087X (P, 79-0888(onlne Publshed on 7 Aprl 05 wwwresearchmathscorg Annals of The L(, -Labelng on -Product of Graphs P Pradhan and Kamesh
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 informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationLecture 5 Decoding Binary BCH Codes
Lecture 5 Decodng Bnary BCH Codes In ths class, we wll ntroduce dfferent methods for decodng BCH codes 51 Decodng the [15, 7, 5] 2 -BCH Code Consder the [15, 7, 5] 2 -code C we ntroduced n the last lecture
More information= z 20 z n. (k 20) + 4 z k = 4
Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5
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 informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
More informationEXPANSIVE MAPPINGS. by W. R. Utz
Volume 3, 978 Pages 6 http://topology.auburn.edu/tp/ EXPANSIVE MAPPINGS by W. R. Utz Topology Proceedngs Web: http://topology.auburn.edu/tp/ Mal: Topology Proceedngs Department of Mathematcs & Statstcs
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationSelf-complementing permutations of k-uniform hypergraphs
Dscrete Mathematcs Theoretcal Computer Scence DMTCS vol. 11:1, 2009, 117 124 Self-complementng permutatons of k-unform hypergraphs Artur Szymańsk A. Paweł Wojda Faculty of Appled Mathematcs, AGH Unversty
More informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
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 informationMATH 5707 HOMEWORK 4 SOLUTIONS 2. 2 i 2p i E(X i ) + E(Xi 2 ) ä i=1. i=1
MATH 5707 HOMEWORK 4 SOLUTIONS CİHAN BAHRAN 1. Let v 1,..., v n R m, all lengths v are not larger than 1. Let p 1,..., p n [0, 1] be arbtrary and set w = p 1 v 1 + + p n v n. Then there exst ε 1,..., ε
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 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 informationChapter 8 Indicator Variables
Chapter 8 Indcator Varables In general, e explanatory varables n any regresson analyss are assumed to be quanttatve n nature. For example, e varables lke temperature, dstance, age etc. are quanttatve n
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 informationThe Minimum Universal Cost Flow in an Infeasible Flow Network
Journal of Scences, Islamc Republc of Iran 17(2): 175-180 (2006) Unversty of Tehran, ISSN 1016-1104 http://jscencesutacr The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran
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 informationHMMT February 2016 February 20, 2016
HMMT February 016 February 0, 016 Combnatorcs 1. For postve ntegers n, let S n be the set of ntegers x such that n dstnct lnes, no three concurrent, can dvde a plane nto x regons (for example, S = {3,
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 informationOn the correction of the h-index for career length
1 On the correcton of the h-ndex for career length by L. Egghe Unverstet Hasselt (UHasselt), Campus Depenbeek, Agoralaan, B-3590 Depenbeek, Belgum 1 and Unverstet Antwerpen (UA), IBW, Stadscampus, Venusstraat
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
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 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 informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More informationCalculation of time complexity (3%)
Problem 1. (30%) Calculaton of tme complexty (3%) Gven n ctes, usng exhaust search to see every result takes O(n!). Calculaton of tme needed to solve the problem (2%) 40 ctes:40! dfferent tours 40 add
More informationISSN: ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 1, July 2013
ISSN: 2277-375 Constructon of Trend Free Run Orders for Orthogonal rrays Usng Codes bstract: Sometmes when the expermental runs are carred out n a tme order sequence, the response can depend on the run
More information5 The Rational Canonical Form
5 The Ratonal Canoncal Form Here p s a monc rreducble factor of the mnmum polynomal m T and s not necessarly of degree one Let F p denote the feld constructed earler n the course, consstng of all matrces
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationx = , so that calculated
Stat 4, secton Sngle Factor ANOVA notes by Tm Plachowsk n chapter 8 we conducted hypothess tests n whch we compared a sngle sample s mean or proporton to some hypotheszed value Chapter 9 expanded ths to
More informationStructure and Drive Paul A. Jensen Copyright July 20, 2003
Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationYong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )
Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often
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 informationn ). This is tight for all admissible values of t, k and n. k t + + n t
MAXIMIZING THE NUMBER OF NONNEGATIVE SUBSETS NOGA ALON, HAROUT AYDINIAN, AND HAO HUANG Abstract. Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what
More informationAnti-van der Waerden numbers of 3-term arithmetic progressions.
Ant-van der Waerden numbers of 3-term arthmetc progressons. Zhanar Berkkyzy, Alex Schulte, and Mchael Young Aprl 24, 2016 Abstract The ant-van der Waerden number, denoted by aw([n], k), s the smallest
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationarxiv:cs.cv/ Jun 2000
Correlaton over Decomposed Sgnals: A Non-Lnear Approach to Fast and Effectve Sequences Comparson Lucano da Fontoura Costa arxv:cs.cv/0006040 28 Jun 2000 Cybernetc Vson Research Group IFSC Unversty of São
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 informationExercises of Chapter 2
Exercses of Chapter Chuang-Cheh Ln Department of Computer Scence and Informaton Engneerng, Natonal Chung Cheng Unversty, Mng-Hsung, Chay 61, Tawan. Exercse.6. Suppose that we ndependently roll two standard
More informationSpectral 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 informationThe Second Eigenvalue of Planar Graphs
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
More informationExpected Value and Variance
MATH 38 Expected Value and Varance Dr. Neal, WKU We now shall dscuss how to fnd the average and standard devaton of a random varable X. Expected Value Defnton. The expected value (or average value, or
More informationCOS 521: Advanced Algorithms Game Theory and Linear Programming
COS 521: Advanced Algorthms Game Theory and Lnear Programmng Moses Charkar February 27, 2013 In these notes, we ntroduce some basc concepts n game theory and lnear programmng (LP). We show a connecton
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More informationGames of Threats. Elon Kohlberg Abraham Neyman. Working Paper
Games of Threats Elon Kohlberg Abraham Neyman Workng Paper 18-023 Games of Threats Elon Kohlberg Harvard Busness School Abraham Neyman The Hebrew Unversty of Jerusalem Workng Paper 18-023 Copyrght 2017
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 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 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 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 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 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 informationComparison of the Population Variance Estimators. of 2-Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method
Appled Mathematcal Scences, Vol. 7, 0, no. 47, 07-0 HIARI Ltd, www.m-hkar.com Comparson of the Populaton Varance Estmators of -Parameter Exponental Dstrbuton Based on Multple Crtera Decson Makng Method
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 informationMath 426: Probability MWF 1pm, Gasson 310 Homework 4 Selected Solutions
Exercses from Ross, 3, : Math 26: Probablty MWF pm, Gasson 30 Homework Selected Solutons 3, p. 05 Problems 76, 86 3, p. 06 Theoretcal exercses 3, 6, p. 63 Problems 5, 0, 20, p. 69 Theoretcal exercses 2,
More informationarxiv: v3 [cs.dm] 7 Jul 2012
Perfect matchng n -unform hypergraphs wth large vertex degree arxv:1101.580v [cs.dm] 7 Jul 01 Imdadullah Khan Department of Computer Scence College of Computng and Informaton Systems Umm Al-Qura Unversty
More informationNeryškioji dichotominių testo klausimų ir socialinių rodiklių diferencijavimo savybių klasifikacija
Neryškoj dchotomnų testo klausmų r socalnų rodklų dferencjavmo savybų klasfkacja Aleksandras KRYLOVAS, Natalja KOSAREVA, Julja KARALIŪNAITĖ Technologcal and Economc Development of Economy Receved 9 May
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
More informationGeometric drawings of K n with few crossings
Geometrc drawngs of K n wth few crossngs Bernardo M. Ábrego, Slva Fernández-Merchant Calforna State Unversty Northrdge {bernardo.abrego,slva.fernandez}@csun.edu ver 9 Abstract We gve a new upper bound
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More information1 The Mistake Bound Model
5-850: Advanced Algorthms CMU, Sprng 07 Lecture #: Onlne Learnng and Multplcatve Weghts February 7, 07 Lecturer: Anupam Gupta Scrbe: Bryan Lee,Albert Gu, Eugene Cho he Mstake Bound Model Suppose there
More informationTAIL BOUNDS FOR SUMS OF GEOMETRIC AND EXPONENTIAL VARIABLES
TAIL BOUNDS FOR SUMS OF GEOMETRIC AND EXPONENTIAL VARIABLES SVANTE JANSON Abstract. We gve explct bounds for the tal probabltes for sums of ndependent geometrc or exponental varables, possbly wth dfferent
More informationDynamic Programming. Preview. Dynamic Programming. Dynamic Programming. Dynamic Programming (Example: Fibonacci Sequence)
/24/27 Prevew Fbonacc Sequence Longest Common Subsequence Dynamc programmng s a method for solvng complex problems by breakng them down nto smpler sub-problems. It s applcable to problems exhbtng the propertes
More information763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1.
7636S ADVANCED QUANTUM MECHANICS Soluton Set 1 Sprng 013 1 Warm-up Show that the egenvalues of a Hermtan operator  are real and that the egenkets correspondng to dfferent egenvalues are orthogonal (b)
More informationLecture 4: November 17, Part 1 Single Buffer Management
Lecturer: Ad Rosén Algorthms for the anagement of Networs Fall 2003-2004 Lecture 4: November 7, 2003 Scrbe: Guy Grebla Part Sngle Buffer anagement In the prevous lecture we taled about the Combned Input
More informationHomework Notes Week 7
Homework Notes Week 7 Math 4 Sprng 4 #4 (a Complete the proof n example 5 that s an nner product (the Frobenus nner product on M n n (F In the example propertes (a and (d have already been verfed so we
More informationIntroduction to Algorithms
Introducton to Algorthms 6.046J/8.40J Lecture 7 Prof. Potr Indyk Data Structures Role of data structures: Encapsulate data Support certan operatons (e.g., INSERT, DELETE, SEARCH) Our focus: effcency of
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 information4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA
4 Analyss of Varance (ANOVA) 5 ANOVA 51 Introducton ANOVA ANOVA s a way to estmate and test the means of multple populatons We wll start wth one-way ANOVA If the populatons ncluded n the study are selected
More informationarxiv: v1 [math.co] 12 Sep 2014
arxv:1409.3707v1 [math.co] 12 Sep 2014 On the bnomal sums of Horadam sequence Nazmye Ylmaz and Necat Taskara Department of Mathematcs, Scence Faculty, Selcuk Unversty, 42075, Campus, Konya, Turkey March
More information