On colour-blind distinguishing colour pallets in regular graphs
|
|
- Austin Barnett
- 5 years ago
- Views:
Transcription
1 J Comb Optim ( : DOI /s x On colour-blin istinguishing colour pallets in regular graphs Jakub Przybyło Publishe online: 25 October 2012 The Author(s This article is publishe with open access at Springerlink.com Abstract Consier a graph G (V, E an a colouring of its eges with k colours. Then every vertex v V is associate with a pallet of incient colours together with their frequencies, which sum up to the egree of v. We say that two vertices have istinct pallets if they iffer in frequency of at least one colour. This is always the case if these vertices have istinct egrees. We consier an apparently the worse case, when G is regular. Suppose further that this coloure graph is being examine by a person who cannot name any given colour, but istinguishes one from another. Coul we colour the eges of G so that a person suffering from such colour-blinness is certain that colour pallets of every two ajacent vertices are istinct? Using the Lopsie Lovász Local Lemma, we prove that it is possible using colours for every -regular graph with 960. Keywors Neighbour-istinguishing colouring Lopsie Lovász Local Lemma Colour pallet 1 Distinguishing colour pallets by colour-blin Consier a simple graph G (V, E an an ege colouring c : E {1, 2,...,k}, not necessarily proper. Such colouring is calle neighbour istinguishing (or vertex colouring, see e.g., Aario-Berry et al if for every ege uv E, the multiset of colours incient with u is istinct from the multiset of colours incient with v. In other wors, if for every vertex v we set c(v (a 1,...,a k, where a i {w : wv E, c(wv i} for i 1,...,k, then the colouring c is neighbour istinguishing if c(u c(v for each ege uv of G. Clearly, one can fin such colouring if a graph J. Przybyło (B AGH University of Science an Technology, al. A. Mickiewicza 30, Krakow, Polan przybylo@wms.mat.agh.eu.pl
2 J Comb Optim ( : contains no isolate eges, e.g., by painting each ege ifferently. Karoński et al. (2004 first prove that in fact a finite number of 183 colours are always sufficient, or even 30 if the minimum egree δ of G is at least This was then greatly improve by Aario-Berry et al. (2005, who showe that four colours are sufficient, an these can be ecrease to three if δ 1, 000. Suppose now that such coloure graph is examine by a colour-blin person, i.e., someboy who cannot name colours but istinguishes one from another. Can such iniviual istinguish neighbours then? The answer is affirmative in many cases. It is ue to the fact that given a set of coloure eges, they are able to ivie it into monochromatic subsets an count their carinalities. Given any sequence c(v (a 1,...,a k, let us re-orer it non-ecreasingly. The obtaine sequence c (v ( 1,..., k we shall call a pallet of v. Note that there is a bijection between the set of all possible pallets one may obtain for a vertex v of egree an the set of all k-partitions of the integer, i.e., the set P(, k {( 1, 2,..., k N k : k an 0 i i+1 for i 1,...,k 1}. We say that a colour-blin person can istinguish neighbours in our colouring c : E {1,...,k} if c (u c (v for every ege uv E. The smallest integer k for which such colouring exists is calle the colourblin inex of G, an is enote by al(g. This notion refers to the English chemist John Dalton, who in 1798 wrote the first paper on colour-blinness. In fact, because of Dalton s work, the conition is often calle altonism. It has to be note that this parameter is unefine for some classes of graphs, in particular we must exclue graphs with isolate eges. However, thus far all known graphs with unefine colour-blin inex have minimum egree at most three, see Kalinowski et al. for etails. It has been prove there that given a fixe R > 1, there always exists δ R such that al(g 6 for every graph with imum egree Rδ, provie that δ δ R. Unfortunately δ R tens to infinity along with R. It is thus not even known whether graphs with δ δ 0 have well efine colour-blin inex for any constant δ 0, though Kalinowski et al. conjecture that it is so (maybe even with δ 0 4. Situation with this mysterious parameter changes if we restrict ourselves to regular graphs exclusively. Using a Lovász Local Lemma, Kalinowski et al. prove that al(g 6 for every -regular graph G if its egree is greater than a huge constant, namely, if An application of the probabilistic metho in this context meets unusual obstacles. Unlike in many other similar problems, increasing the number of colours, helps only until a certain point. Then the probability of a ba event that vertices are inistinguishable for a colour-blin person (e.g., when the ege colouring is proper grows. In this paper we optimize this probabilistic approach in orer to significantly reuce the threshol for at the cost of a few more colours. We shall thus prove the following theorem. Theorem 1 For every -regular graph G of egree 960, al(g. The proof is base on the following variation of the Lovász Local Lemma, ue to Erős an Spencer (1991, sometimes referre to as the Lopsie Local Lemma. We
3 350 J Comb Optim ( : recall its symmetric versions from Alon an Spencer (2000 (see Corollary an the comments below. Theorem 2 (Lopsie Symmetric Local Lemma Let A be a family of (typically ba events in any probability space an let D (A, E be a irecte graph with imum out-egree +. Suppose that for each A A an every C A N + (A, Pr(A C C C p, (1 where ep( (2 Then Pr( A A A>0. 2 Proof of Theorem Ranom process an epenency igraph Suppose we are given a -regular graph G (V, E with 960. For each ege e E we inepenently an ranomly choose a colour from the available set {1, 2,...,}, each with equal probability, an enote it by c(e. In other wors, the eges of G are associate with a set of inepenent ranom variables (X e e E, each taking one of the values 1, 2,..., with probability 1/. Outcomes for these etermine an ege colouring of G, each occurring with probability 1/ E within the associate prouct probability space. By a ba event A e in our ranom process of generating c we shall mean obtaining c (u c (v for some ege e uv E. If no ba event occurs, the corresponing colouring shall meet our requirements. We thus nee to show that the probability of the event e E A e is positive in our probability space. We efine a igraph D (A, E, so calle epenency igraph, in the following manner. Let A {A e : e E}. Now for every ege e uv (i.e. e {u,v}ofg,we arbitrarily choose one of its en vertices, say v. Equivalently, we choose an orientation e (u,v of every ege e E, an the obtaine orientation of G we enote by G (V, E. Then for every ege e E with orientation e (u,v,werawan arc between A e an every event A e such that e is at istance at most 2 from v in a graph G e (where an ege incient with a vertex is at istance 1 from it, i.e., e is incient with some neighbour of v ifferent from u. The set of all such arcs we enote by E. Note that then + ( 1 2 1, (3 where + is the imum out-egree of D.
4 J Comb Optim ( : Conitional probability of a ba event Consier any event A e with e (u,van some family of events C A (N + (A e {A e }, where N + (A e is the set of out-neighbours of A e in D.WeassumethatA e / C, since inequality (1 is obvious otherwise (for every p 0. Note that every event C A f C (hence also C is etermine by the values of the ranom variables X e with e at istance at most 1 from f (i.e., sharing a vertex with f. By our construction of D, neither of such e is incient with v, except possibly when f e.lete 1, e 2,...,e enote the eges incient with v, where e e, an let e +1,...,e m enote the remaining eges of G. Then the event C C C is etermine by the outcomes for (part of the ranom variables X ei with i. Denote [] {1, 2,...,} an let Z be a set of all (partial colourings of the eges e,...,e m for which C C C hols, i.e., the set of vectors c (c,...,c m [] m +1 such that (X e,...,x em c guarantees C C C. Then (if C C C, hence Pr( C C C>0: Pr(A e C Pr(A e C C C Pr( C C C C C c Z Pr(A e (X e,...,x em cpr((x e,...,x em c Pr( C C C Pr(A e (X e,...,x em c c Z Pr((X e,...,x em c c Z Pr( C C C Pr(A e (X e,...,x em c c Z Pr(A e (X e,...,x em c. c [] m +1 Since A e is etermine by the outcomes for ranom variables associate with eges at istance at most 1 from e, an the pallet of v by the outcomes for X e1,...,x e,we thus obtain: Pr(A e C C C Pr(c (v c X e c, (4 where the imum is taken over all partitions c ( 1,..., of an all c []. Note however that since a colour blin person cannot name a single colour, in particular the colour c, then the boun from (4 is equivalent to the following one: Pr(A e C C C c P(, Pr(c (v c. (5 (Note that the same upper boun as in (5 hols also for Pr(A e. Consier any fixe c ( 1,..., P(,, an enote the lengths of its consecutive imal subsequences of ientical integers by l 1, l 2,...,l q,where
5 352 J Comb Optim ( : l i 1fori 1,...,q an l 1 + +l q. In other wors, c is of the form ( 1,..., }{{} 1, l1 +1,..., l1 +1,..., lq +1,..., lq +1. Then }{{}}{{} l 1 l 2 Pr(c (v c! l 1!...l q!, (6 where is just the number of istinct partitions of elements (eges 1... into ( (enumerate subsets S 1,...,S of carinalities 1,...,, resp., hence! ! 2!!, the factor! appears ue to the colour-blinness of a person trying to istinguish neighbours, for which every (bijective assignment of colours 1,..., to the sets S 1,...,S yiels the same pallet, while l 1!...l q! counts how many times a given colouring has been taken into account in our calculations. Let us enote by r( 1,..., (or r(c the number of repetitions in c, where we call i a repetition if i j for some j < i (hence r( 1,..., q, an note that by (6, Pr(c (v c! 1... l q 2 r(c 1. (7 In fact such estimation has (almost no influence on the result we are able to prove, but significantly simplifies calculations. By (1, (2, (3, (5 an (7, in orer to prove Theorem 1, it is sufficient to show that e 2! r(c 1 1 (8 for every c ( 1,..., P(,. We shall prove this inequality consecutively for the elements of an ascening family P 0 (, P 1 (,...P 14 (, P(, of subsets of P(,, where P r (, {c P(, : r(c r} is just the set of all -partitions of with at most r repetitions, r 0,..., Partitions without repetitions We first consier c P 0 (,, for which all i are istinct. We shall prove that a : e 2 ( 1,..., P 0 (, 1! ( for every 960. Given a not necessarily monotone sequence of non-negative integers k 1,...,k summing up to,bytightening its two elements k i, k j satisfying k j k i + 2 we shall mean substituting these with the elements k i + 1 an k j 1. Note that such operation
6 J Comb Optim ( : always increase the value of, since if without lost of generality, i k 1...k ( 1 an j, i.e, k k > k 1 + 1, then k 1 + 1k 2...k 14 k 1 k k 1 +1 > ( k 1...k k 1...k. Moreover, the minimum an imum of this sequence shall be calle its leftan right borers, resp., an every integer which oes not appear in the sequence, but is between its borers shall be calle a gap. Let c,0 ( 1,..., be an element of P 0 (, for which the value of is imal. Then this sequence has at most one gap, since otherwise we 1... coul tighten the element preceing the smallest gap an the element succeeing the largest gap creating no repetitions in the obtaine one. For every (sufficiently large there is only one such sequence, ue to the fact that its elements must sum up to. Namely, for s (mo, s {0,...,14}, wehave: a e 2! ( s + 8 s! ( s 7! ( s 6!... ( s + 8!! 1. We shall first show that the sequence (a 960 consists of ecreasing subsequences (a n+ j n 64, j 0, 1,...,14. For this purpose consier the following proportion (for s (mo : a a 2 ( 2 2 ( 2 [ 14 ] ( i0 ( i s + 8 s 7( s 6... ( 1 s + 8 ( s 14 i0 ( i ( 7 ( 6... ( The last inequality is obvious for s 0, while for the remaining s, the fact that s s s s s + 9 s or, equivalently, log 0.5 ( 7 + log 0.5 ( 6 + +log 0.5 ( + 7 s log log 0.5 ( s + 9 s + +log 0.5 ( s + 7 s + +log 0.5 ( s + 8 (10
7 354 J Comb Optim ( : is a simple consequence of the following Theorem 3 on so calle majorization inequality, applie for the function f (x log 0.5 x. Theorem 3 (Karamata s inequality, Kaelburg et al Let I be an interval of the real line an let f enote a real-value convex function efine on I. If x 1 x 2 x n an y 1 y 2 y n are numbers in I such that (x 1,...,x n majorizes (y 1,...,y n, i.e., x 1 + +x n y y n an x 1 + +x i y y i for i 1,...,n 1, then By (10 we thus have: f (x f (x n f (y f (y n. a a 2 14 i0 ( i ( 2 ( 7 ( 6...( + 7 [ 5 ( ] ( 2i + 1( 2i ( (i + 2( + (i + 2 i1 ( ( < 1, ( 11( 12 ( ( ( + ( 13( 14 ( ( 30( + 30 (11 because ( 11( 12 ( ( ( < 0 for 32, ( 13( 14 ( ( 30( < 0for 76, an (since 960 > 4 2, 2 (4i 1+2i(2i 1 < 2 (4i (16i (i (i ( 2i+1( 2i ( (i+2(+(i+2 < 1fori 1,...,5. 2 (i Computing a < , a , a , a , a , a , a , a , a , a , a , a , a , a , a , by inequality (11 we thus obtain a < for 960, i.e., (9 hols. Now we shall exten this result, an show that e 2 ( 1,..., P i (, P i 1 (,! i 1 (12 for every 960 an i 0, 1,...,14 (where P 1 (, :. This will imply (8 an finalize the proof. It is then the more sufficient to show that a,i : e 2 ( 1,..., P i (,! i 1 (13 for 960 an i 0, 1,...,14. The proof of this fact shall be inuctive with respect to i. The case of i 0 being alreay consiere, let us fix i 1, i 14, an assume that (13 hols for a,i with i < i. Letc,i ( 1,..., be an element
8 J Comb Optim ( : of P i (, for which the value of is imal. If c 1...,i P i 1 (,, then (13 hols by inuction hypothesis. Assume then that our c,i contains exactly i repetitions (i 1. Observe then that c,i cannot contain any gaps, since otherwise we coul tighten two of its element creating no aitional repetitions (contraicting imality of c,i. Inee, if any repetition of c,i was larger than some gap, then we coul tighten this repetition an the element of c,i preceing its smallest (left-most gap, an analogously in the opposite case (i.e., when a repetition was smaller than some gap. Finally note that to prove (13 for our fixe i, by inuction hypothesis, it is sufficient to prove for every 960 that at least one of the following two conitions hols: ( 1,..., P i (, ( 1,..., P i (, ( 1,..., ( 1,..., P i 1(, P i 2(, ( (the later for i 2. In fact this is exactly what we shall o for almost every i. We will have to by slightly more careful with the case of i 1 though. (14 ( 2.4 Partitions with one repetition Assume that i 1, hence c,i c,1 ( 1, 1 + 1,..., 1 + t, 1 + t, 1 + t + 1,..., for some t {0,...,13}. Consequently, t , an hence Note that our repetition must be closer to one of the borers of c,1, an let us enote the smaller of these two istances by b, i.e., b {0,...,6} is an integer such that {t, 13 t} {b, 13 b}. Then we may make of c,1 a partition of without repetitions by substituting its elements 1 + b an b with 1 1 an 1 +14, respectively (an orering non-ecreasingly. For the obtaine partition ( 1,..., we then have: b j b + j 1 + j j j. 1 + j Since j j0 1 + j is a ecreasing function of 1 (for 1 1 an (for 960, we thus obtain: j j ( j , but since ( 1,..., P 0(,, by(9 wehave: e 2! 1.
9 356 J Comb Optim ( : Consequently, thus (13 hols for i ! 1 a,1 e e 2! , 2.5 Partitions with at least two repetitions Assume now that i 2. Since 1 an i are the borers of c,i, then analogously as above, 1 + ( ( i + i( i 1 + ( i(14 i 2 + i(14 i 1 + (+i(14 i (for i 2, hence Consier first the case when at least one of the repetitions of c,i,say 1 + t,is at istance at most 3 from one of the borers, i.e., there exists an integer b {0,...,3} such that {t, 14 i t} {b, 14 i b}. Then we may make of c,i a partition of with at most i 1 repetitions by substituting its elements 1 + b an i b with 1 1 an 1 + i, respectively. For the obtaine partition ( 1,..., P i 1 (, we then have: b j0 b 1... j j j ( 1,..., P i 1(, 1 + i b + j 1 + j b + j 1 + j j 1 + j j j 1,... hence (13 hols by (14. Assume then that every repetition of c,i is between an i (hence 2 i 6. Choose any two repetitions 1 +t an 1 +t of c,i with 4 t t 10 i (possibly t t, an let a {t, 14 i t }, hence t t +a 8, an thus a 8. Then we may make of c,i a partition of with at most i 2 repetitions by substituting its elements 1 +t an 1 +t with 1 +t a 1 < 1 an 1 +t +a+1 > i, respectively. For the obtaine partition ( 1,..., P i 2(, we then have:
10 J Comb Optim ( : a 1 + t j j0 1 + t a + j a 1 + (8 + t a j 1... j0 1 + t a + j t j 1... j0 1 + t 8 + j j j0 4 + j ( 1,..., P i 2(, 1,... hence (13 hols by (. The proof of Theorem 1 is thus complete. Acknowlegments This research was partly supporte by the National Science Centre Grant No. DEC- 2011/01/D/ST1/044 an by the Polish Ministry of Science an Higher Eucation. I enclose special thanks to Anrzej Żak for fruitful iscussions in 306A. Open Access This article is istribute uner the terms of the Creative Commons Attribution License which permits any use, istribution, an reprouction in any meium, provie the original author(s an the source are creite. References Aario-Berry L, Alre REL, Dalal K, Ree BA (2005 Vertex colouring ege partitions. J Combin Theory Ser B 94(2: Alon N, Spencer JH (2000 The probabilistic metho, 2n en. Wiley, New York Erős P, Spencer J (1991 Lopsie Lovász Local Lemma an Latin transversals. Discret Appl Math 30:1 4 Kaelburg Z, Duki D, Luki M, Mati I (2005 Inequalities of Karamata, Schur an Muirhea, an some applications. Teach Math 8(1:31 45 Kalinowski R, Pilśniak M, Przybyło J, Woźniak M, Can colour-blin istinguish colour pallets? (submitte Karoński M, Łuczak T, Thomason A (2004 Ege weights an vertex colours. J Combin Theory Ser B 91:1 7
On decomposing graphs of large minimum degree into locally irregular subgraphs
On decomposing graphs of large minimum degree into locally irregular subgraphs Jakub Przyby lo AGH University of Science and Technology al. A. Mickiewicza 0 0-059 Krakow, Poland jakubprz@agh.edu.pl Submitted:
More informationRamsey numbers of some bipartite graphs versus complete graphs
Ramsey numbers of some bipartite graphs versus complete graphs Tao Jiang, Michael Salerno Miami University, Oxfor, OH 45056, USA Abstract. The Ramsey number r(h, K n ) is the smallest positive integer
More informationLecture 5. Symmetric Shearer s Lemma
Stanfor University Spring 208 Math 233: Non-constructive methos in combinatorics Instructor: Jan Vonrák Lecture ate: January 23, 208 Original scribe: Erik Bates Lecture 5 Symmetric Shearer s Lemma Here
More informationAcute sets in Euclidean spaces
Acute sets in Eucliean spaces Viktor Harangi April, 011 Abstract A finite set H in R is calle an acute set if any angle etermine by three points of H is acute. We examine the maximal carinality α() of
More informationIterated Point-Line Configurations Grow Doubly-Exponentially
Iterate Point-Line Configurations Grow Doubly-Exponentially Joshua Cooper an Mark Walters July 9, 008 Abstract Begin with a set of four points in the real plane in general position. A to this collection
More informationThe chromatic number of graph powers
Combinatorics, Probability an Computing (19XX) 00, 000 000. c 19XX Cambrige University Press Printe in the Unite Kingom The chromatic number of graph powers N O G A A L O N 1 an B O J A N M O H A R 1 Department
More informationPermanent vs. Determinant
Permanent vs. Determinant Frank Ban Introuction A major problem in theoretical computer science is the Permanent vs. Determinant problem. It asks: given an n by n matrix of ineterminates A = (a i,j ) an
More informationarxiv: v1 [math.co] 15 Sep 2015
Circular coloring of signe graphs Yingli Kang, Eckhar Steffen arxiv:1509.04488v1 [math.co] 15 Sep 015 Abstract Let k, ( k) be two positive integers. We generalize the well stuie notions of (k, )-colorings
More informationLecture 22. Lecturer: Michel X. Goemans Scribe: Alantha Newman (2004), Ankur Moitra (2009)
8.438 Avance Combinatorial Optimization Lecture Lecturer: Michel X. Goemans Scribe: Alantha Newman (004), Ankur Moitra (009) MultiFlows an Disjoint Paths Here we will survey a number of variants of isjoint
More informationSurvey Sampling. 1 Design-based Inference. Kosuke Imai Department of Politics, Princeton University. February 19, 2013
Survey Sampling Kosuke Imai Department of Politics, Princeton University February 19, 2013 Survey sampling is one of the most commonly use ata collection methos for social scientists. We begin by escribing
More informationarxiv: v1 [math.co] 13 Dec 2017
The List Linear Arboricity of Graphs arxiv:7.05006v [math.co] 3 Dec 07 Ringi Kim Department of Mathematical Sciences KAIST Daejeon South Korea 344 an Luke Postle Department of Combinatorics an Optimization
More informationA Weak First Digit Law for a Class of Sequences
International Mathematical Forum, Vol. 11, 2016, no. 15, 67-702 HIKARI Lt, www.m-hikari.com http://x.oi.org/10.1288/imf.2016.6562 A Weak First Digit Law for a Class of Sequences M. A. Nyblom School of
More informationLarge Triangles in the d-dimensional Unit-Cube (Extended Abstract)
Large Triangles in the -Dimensional Unit-Cube Extene Abstract) Hanno Lefmann Fakultät für Informatik, TU Chemnitz, D-0907 Chemnitz, Germany lefmann@informatik.tu-chemnitz.e Abstract. We consier a variant
More informationDEBRUIJN-LIKE SEQUENCES AND THE IRREGULAR CHROMATIC NUMBER OF PATHS AND CYCLES
DEBRUIJN-LIKE SEQUENCES AND THE IRREGULAR CHROMATIC NUMBER OF PATHS AND CYCLES MICHAEL FERRARA, CHRISTINE LEE, PHIL WALLIS DEPARTMENT OF MATHEMATICAL AND STATISTICAL SCIENCES UNIVERSITY OF COLORADO DENVER
More informationDiophantine Approximations: Examining the Farey Process and its Method on Producing Best Approximations
Diophantine Approximations: Examining the Farey Process an its Metho on Proucing Best Approximations Kelly Bowen Introuction When a person hears the phrase irrational number, one oes not think of anything
More informationDiscrete Mathematics
Discrete Mathematics 309 (009) 86 869 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: wwwelseviercom/locate/isc Profile vectors in the lattice of subspaces Dániel Gerbner
More informationConvergence of Random Walks
Chapter 16 Convergence of Ranom Walks This lecture examines the convergence of ranom walks to the Wiener process. This is very important both physically an statistically, an illustrates the utility of
More informationOn the enumeration of partitions with summands in arithmetic progression
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 8 (003), Pages 149 159 On the enumeration of partitions with summans in arithmetic progression M. A. Nyblom C. Evans Department of Mathematics an Statistics
More informationSharp Thresholds. Zachary Hamaker. March 15, 2010
Sharp Threshols Zachary Hamaker March 15, 2010 Abstract The Kolmogorov Zero-One law states that for tail events on infinite-imensional probability spaces, the probability must be either zero or one. Behavior
More informationFLUCTUATIONS IN THE NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS. 1. Introduction
FLUCTUATIONS IN THE NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS ALINA BUCUR, CHANTAL DAVID, BROOKE FEIGON, MATILDE LALÍN 1 Introuction In this note, we stuy the fluctuations in the number
More informationTopic 7: Convergence of Random Variables
Topic 7: Convergence of Ranom Variables Course 003, 2016 Page 0 The Inference Problem So far, our starting point has been a given probability space (S, F, P). We now look at how to generate information
More informationChromatic number for a generalization of Cartesian product graphs
Chromatic number for a generalization of Cartesian prouct graphs Daniel Král Douglas B. West Abstract Let G be a class of graphs. The -fol gri over G, enote G, is the family of graphs obtaine from -imensional
More informationLATTICE-BASED D-OPTIMUM DESIGN FOR FOURIER REGRESSION
The Annals of Statistics 1997, Vol. 25, No. 6, 2313 2327 LATTICE-BASED D-OPTIMUM DESIGN FOR FOURIER REGRESSION By Eva Riccomagno, 1 Rainer Schwabe 2 an Henry P. Wynn 1 University of Warwick, Technische
More informationALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS
ALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS MARK SCHACHNER Abstract. When consiere as an algebraic space, the set of arithmetic functions equippe with the operations of pointwise aition an
More informationWitt#5: Around the integrality criterion 9.93 [version 1.1 (21 April 2013), not completed, not proofread]
Witt vectors. Part 1 Michiel Hazewinkel Sienotes by Darij Grinberg Witt#5: Aroun the integrality criterion 9.93 [version 1.1 21 April 2013, not complete, not proofrea In [1, section 9.93, Hazewinkel states
More informationn 1 conv(ai) 0. ( 8. 1 ) we get u1 = u2 = = ur. Hence the common value of all the Uj Tverberg's Tl1eorem
8.3 Tverberg's Tl1eorem 203 hence Uj E cone(aj ) Above we have erive L;=l 'Pi (uj ) = 0, an so by ( 8. 1 ) we get u1 = u2 = = ur. Hence the common value of all the Uj belongs to n;=l cone(aj ). It remains
More informationTwo formulas for the Euler ϕ-function
Two formulas for the Euler ϕ-function Robert Frieman A multiplication formula for ϕ(n) The first formula we want to prove is the following: Theorem 1. If n 1 an n 2 are relatively prime positive integers,
More informationMultiplicative properties of sets of residues
Multiplicative properties of sets of resiues C. Pomerance Hanover an A. Schinzel Warszawa Abstract: Given a natural number n, we ask whether every set of resiues mo n of carinality at least n/2 contains
More informationOn combinatorial approaches to compressed sensing
On combinatorial approaches to compresse sensing Abolreza Abolhosseini Moghaam an Hayer Raha Department of Electrical an Computer Engineering, Michigan State University, East Lansing, MI, U.S. Emails:{abolhos,raha}@msu.eu
More informationCombinatorica 9(1)(1989) A New Lower Bound for Snake-in-the-Box Codes. Jerzy Wojciechowski. AMS subject classification 1980: 05 C 35, 94 B 25
Combinatorica 9(1)(1989)91 99 A New Lower Boun for Snake-in-the-Box Coes Jerzy Wojciechowski Department of Pure Mathematics an Mathematical Statistics, University of Cambrige, 16 Mill Lane, Cambrige, CB2
More informationA new proof of the sharpness of the phase transition for Bernoulli percolation on Z d
A new proof of the sharpness of the phase transition for Bernoulli percolation on Z Hugo Duminil-Copin an Vincent Tassion October 8, 205 Abstract We provie a new proof of the sharpness of the phase transition
More informationQuantum mechanical approaches to the virial
Quantum mechanical approaches to the virial S.LeBohec Department of Physics an Astronomy, University of Utah, Salt Lae City, UT 84112, USA Date: June 30 th 2015 In this note, we approach the virial from
More informationOptimization Notes. Note: Any material in red you will need to have memorized verbatim (more or less) for tests, quizzes, and the final exam.
MATH 2250 Calculus I Date: October 5, 2017 Eric Perkerson Optimization Notes 1 Chapter 4 Note: Any material in re you will nee to have memorize verbatim (more or less) for tests, quizzes, an the final
More information6 General properties of an autonomous system of two first order ODE
6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x
More informationAgmon Kolmogorov Inequalities on l 2 (Z d )
Journal of Mathematics Research; Vol. 6, No. ; 04 ISSN 96-9795 E-ISSN 96-9809 Publishe by Canaian Center of Science an Eucation Agmon Kolmogorov Inequalities on l (Z ) Arman Sahovic Mathematics Department,
More informationarxiv: v1 [math.co] 29 May 2009
arxiv:0905.4913v1 [math.co] 29 May 2009 simple Havel-Hakimi type algorithm to realize graphical egree sequences of irecte graphs Péter L. Erős an István Miklós. Rényi Institute of Mathematics, Hungarian
More informationSolutions to Math 41 Second Exam November 4, 2010
Solutions to Math 41 Secon Exam November 4, 2010 1. (13 points) Differentiate, using the metho of your choice. (a) p(t) = ln(sec t + tan t) + log 2 (2 + t) (4 points) Using the rule for the erivative of
More informationAsymptotic determination of edge-bandwidth of multidimensional grids and Hamming graphs
Asymptotic etermination of ege-banwith of multiimensional gris an Hamming graphs Reza Akhtar Tao Jiang Zevi Miller. Revise on May 7, 007 Abstract The ege-banwith B (G) of a graph G is the banwith of the
More informationMultiplicative properties of sets of residues
Multiplicative properties of sets of resiues C. Pomerance Hanover an A. Schinzel Warszawa Abstract: We conjecture that for each natural number n, every set of resiues mo n of carinality at least n/2 contains
More informationMultiplicative properties of sets of residues
Multiplicative properties of sets of resiues C. Pomerance Hanover an A. Schinzel Warszawa Abstract: We conjecture that for each natural number n, every set of resiues mo n of carinality at least n/2 contains
More informationLower Bounds for the Smoothed Number of Pareto optimal Solutions
Lower Bouns for the Smoothe Number of Pareto optimal Solutions Tobias Brunsch an Heiko Röglin Department of Computer Science, University of Bonn, Germany brunsch@cs.uni-bonn.e, heiko@roeglin.org Abstract.
More informationQubit channels that achieve capacity with two states
Qubit channels that achieve capacity with two states Dominic W. Berry Department of Physics, The University of Queenslan, Brisbane, Queenslan 4072, Australia Receive 22 December 2004; publishe 22 March
More informationPseudo-Free Families of Finite Computational Elementary Abelian p-groups
Pseuo-Free Families of Finite Computational Elementary Abelian p-groups Mikhail Anokhin Information Security Institute, Lomonosov University, Moscow, Russia anokhin@mccme.ru Abstract We initiate the stuy
More informationarxiv: v2 [math.st] 29 Oct 2015
EXPONENTIAL RANDOM SIMPLICIAL COMPLEXES KONSTANTIN ZUEV, OR EISENBERG, AND DMITRI KRIOUKOV arxiv:1502.05032v2 [math.st] 29 Oct 2015 Abstract. Exponential ranom graph moels have attracte significant research
More informationChapter 5. Factorization of Integers
Chapter 5 Factorization of Integers 51 Definition: For a, b Z we say that a ivies b (or that a is a factor of b, or that b is a multiple of a, an we write a b, when b = ak for some k Z 52 Theorem: (Basic
More informationu!i = a T u = 0. Then S satisfies
Deterministic Conitions for Subspace Ientifiability from Incomplete Sampling Daniel L Pimentel-Alarcón, Nigel Boston, Robert D Nowak University of Wisconsin-Maison Abstract Consier an r-imensional subspace
More informationGeneralized Tractability for Multivariate Problems
Generalize Tractability for Multivariate Problems Part II: Linear Tensor Prouct Problems, Linear Information, an Unrestricte Tractability Michael Gnewuch Department of Computer Science, University of Kiel,
More informationNEERAJ KUMAR AND K. SENTHIL KUMAR. 1. Introduction. Theorem 1 motivate us to ask the following:
NOTE ON VANISHING POWER SUMS OF ROOTS OF UNITY NEERAJ KUMAR AND K. SENTHIL KUMAR Abstract. For xe positive integers m an l, we give a complete list of integers n for which their exist mth complex roots
More informationA Sketch of Menshikov s Theorem
A Sketch of Menshikov s Theorem Thomas Bao March 14, 2010 Abstract Let Λ be an infinite, locally finite oriente multi-graph with C Λ finite an strongly connecte, an let p
More informationRegular tree languages definable in FO and in FO mod
Regular tree languages efinable in FO an in FO mo Michael Beneikt Luc Segoufin Abstract We consier regular languages of labele trees. We give an effective characterization of the regular languages over
More informationTractability results for weighted Banach spaces of smooth functions
Tractability results for weighte Banach spaces of smooth functions Markus Weimar Mathematisches Institut, Universität Jena Ernst-Abbe-Platz 2, 07740 Jena, Germany email: markus.weimar@uni-jena.e March
More informationLinear First-Order Equations
5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)
More informationBisecting Sparse Random Graphs
Bisecting Sparse Ranom Graphs Malwina J. Luczak,, Colin McDiarmi Mathematical Institute, University of Oxfor, Oxfor OX 3LB, Unite Kingom; e-mail: luczak@maths.ox.ac.uk Department of Statistics, University
More informationarxiv: v1 [math.mg] 10 Apr 2018
ON THE VOLUME BOUND IN THE DVORETZKY ROGERS LEMMA FERENC FODOR, MÁRTON NASZÓDI, AND TAMÁS ZARNÓCZ arxiv:1804.03444v1 [math.mg] 10 Apr 2018 Abstract. The classical Dvoretzky Rogers lemma provies a eterministic
More informationSome properties of random staircase tableaux
Some properties of ranom staircase tableaux Sanrine Dasse Hartaut Pawe l Hitczenko Downloae /4/7 to 744940 Reistribution subject to SIAM license or copyright; see http://wwwsiamorg/journals/ojsaphp Abstract
More informationLower bounds on Locality Sensitive Hashing
Lower bouns on Locality Sensitive Hashing Rajeev Motwani Assaf Naor Rina Panigrahy Abstract Given a metric space (X, X ), c 1, r > 0, an p, q [0, 1], a istribution over mappings H : X N is calle a (r,
More informationIMFM Institute of Mathematics, Physics and Mechanics Jadranska 19, 1000 Ljubljana, Slovenia. Preprint series Vol. 50 (2012), 1173 ISSN
IMFM Institute of Mathematics, Physics an Mechanics Jaransa 19, 1000 Ljubljana, Slovenia Preprint series Vol. 50 (2012), 1173 ISSN 2232-2094 PARITY INDEX OF BINARY WORDS AND POWERS OF PRIME WORDS Alesanar
More informationarxiv: v1 [math.co] 31 Mar 2008
On the maximum size of a (k,l)-sum-free subset of an abelian group arxiv:080386v1 [mathco] 31 Mar 2008 Béla Bajnok Department of Mathematics, Gettysburg College Gettysburg, PA 17325-186 USA E-mail: bbajnok@gettysburgeu
More informationTime-of-Arrival Estimation in Non-Line-Of-Sight Environments
2 Conference on Information Sciences an Systems, The Johns Hopkins University, March 2, 2 Time-of-Arrival Estimation in Non-Line-Of-Sight Environments Sinan Gezici, Hisashi Kobayashi an H. Vincent Poor
More informationCONTAINMENT GAME PLAYED ON RANDOM GRAPHS: ANOTHER ZIG-ZAG THEOREM
CONTAINMENT GAME PLAYED ON RANDOM GRAPHS: ANOTHER ZIG-ZAG THEOREM PAWE L PRA LAT Abstract. We consier a variant of the game of Cops an Robbers, calle Containment, in which cops move from ege to ajacent
More information19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control
19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior
More informationIntroduction to the Vlasov-Poisson system
Introuction to the Vlasov-Poisson system Simone Calogero 1 The Vlasov equation Consier a particle with mass m > 0. Let x(t) R 3 enote the position of the particle at time t R an v(t) = ẋ(t) = x(t)/t its
More informationLECTURE NOTES ON DVORETZKY S THEOREM
LECTURE NOTES ON DVORETZKY S THEOREM STEVEN HEILMAN Abstract. We present the first half of the paper [S]. In particular, the results below, unless otherwise state, shoul be attribute to G. Schechtman.
More informationProof of SPNs as Mixture of Trees
A Proof of SPNs as Mixture of Trees Theorem 1. If T is an inuce SPN from a complete an ecomposable SPN S, then T is a tree that is complete an ecomposable. Proof. Argue by contraiction that T is not a
More informationTEMPORAL AND TIME-FREQUENCY CORRELATION-BASED BLIND SOURCE SEPARATION METHODS. Yannick DEVILLE
TEMPORAL AND TIME-FREQUENCY CORRELATION-BASED BLIND SOURCE SEPARATION METHODS Yannick DEVILLE Université Paul Sabatier Laboratoire Acoustique, Métrologie, Instrumentation Bât. 3RB2, 8 Route e Narbonne,
More informationMcMaster University. Advanced Optimization Laboratory. Title: The Central Path Visits all the Vertices of the Klee-Minty Cube.
McMaster University Avance Optimization Laboratory Title: The Central Path Visits all the Vertices of the Klee-Minty Cube Authors: Antoine Deza, Eissa Nematollahi, Reza Peyghami an Tamás Terlaky AvOl-Report
More informationUnit #6 - Families of Functions, Taylor Polynomials, l Hopital s Rule
Unit # - Families of Functions, Taylor Polynomials, l Hopital s Rule Some problems an solutions selecte or aapte from Hughes-Hallett Calculus. Critical Points. Consier the function f) = 54 +. b) a) Fin
More informationarxiv: v2 [math.pr] 27 Nov 2018
Range an spee of rotor wals on trees arxiv:15.57v [math.pr] 7 Nov 1 Wilfrie Huss an Ecaterina Sava-Huss November, 1 Abstract We prove a law of large numbers for the range of rotor wals with ranom initial
More informationPerfect Matchings in Õ(n1.5 ) Time in Regular Bipartite Graphs
Perfect Matchings in Õ(n1.5 ) Time in Regular Bipartite Graphs Ashish Goel Michael Kapralov Sanjeev Khanna Abstract We consier the well-stuie problem of fining a perfect matching in -regular bipartite
More informationAssignment #3: Mathematical Induction
Math 3AH, Fall 011 Section 10045 Assignment #3: Mathematical Inuction Directions: This assignment is ue no later than Monay, November 8, 011, at the beginning of class. Late assignments will not be grae.
More informationUnit vectors with non-negative inner products
Unit vectors with non-negative inner proucts Bos, A.; Seiel, J.J. Publishe: 01/01/1980 Document Version Publisher s PDF, also known as Version of Recor (inclues final page, issue an volume numbers) Please
More informationTHE USE OF KIRCHOFF S CURRENT LAW AND CUT-SET EQUATIONS IN THE ANALYSIS OF BRIDGES AND TRUSSES
Session TH US O KIRCHO S CURRNT LAW AND CUT-ST QUATIONS IN TH ANALYSIS O BRIDGS AND TRUSSS Ravi P. Ramachanran an V. Ramachanran. Department of lectrical an Computer ngineering, Rowan University, Glassboro,
More informationHilbert functions and Betti numbers of reverse lexicographic ideals in the exterior algebra
Turk J Math 36 (2012), 366 375. c TÜBİTAK oi:10.3906/mat-1102-21 Hilbert functions an Betti numbers of reverse lexicographic ieals in the exterior algebra Marilena Crupi, Carmela Ferró Abstract Let K be
More informationMultiple Rank-1 Lattices as Sampling Schemes for Multivariate Trigonometric Polynomials
Multiple Rank-1 Lattices as Sampling Schemes for Multivariate Trigonometric Polynomials Lutz Kämmerer June 23, 2015 We present a new sampling metho that allows the unique reconstruction of (sparse) multivariate
More informationCounting Lattice Points in Polytopes: The Ehrhart Theory
3 Counting Lattice Points in Polytopes: The Ehrhart Theory Ubi materia, ibi geometria. Johannes Kepler (1571 1630) Given the profusion of examples that gave rise to the polynomial behavior of the integer-point
More information. Using a multinomial model gives us the following equation for P d. , with respect to same length term sequences.
S 63 Lecture 8 2/2/26 Lecturer Lillian Lee Scribes Peter Babinski, Davi Lin Basic Language Moeling Approach I. Special ase of LM-base Approach a. Recap of Formulas an Terms b. Fixing θ? c. About that Multinomial
More informationCalculus of Variations
16.323 Lecture 5 Calculus of Variations Calculus of Variations Most books cover this material well, but Kirk Chapter 4 oes a particularly nice job. x(t) x* x*+ αδx (1) x*- αδx (1) αδx (1) αδx (1) t f t
More informationON BEAUVILLE STRUCTURES FOR PSL(2, q)
ON BEAUVILLE STRUCTURES FOR PSL(, q) SHELLY GARION Abstract. We characterize Beauville surfaces of unmixe type with group either PSL(, p e ) or PGL(, p e ), thus extening previous results of Bauer, Catanese
More informationSYNCHRONOUS SEQUENTIAL CIRCUITS
CHAPTER SYNCHRONOUS SEUENTIAL CIRCUITS Registers an counters, two very common synchronous sequential circuits, are introuce in this chapter. Register is a igital circuit for storing information. Contents
More informationEquilibrium in Queues Under Unknown Service Times and Service Value
University of Pennsylvania ScholarlyCommons Finance Papers Wharton Faculty Research 1-2014 Equilibrium in Queues Uner Unknown Service Times an Service Value Laurens Debo Senthil K. Veeraraghavan University
More information1 dx. where is a large constant, i.e., 1, (7.6) and Px is of the order of unity. Indeed, if px is given by (7.5), the inequality (7.
Lectures Nine an Ten The WKB Approximation The WKB metho is a powerful tool to obtain solutions for many physical problems It is generally applicable to problems of wave propagation in which the frequency
More informationRobust Forward Algorithms via PAC-Bayes and Laplace Distributions. ω Q. Pr (y(ω x) < 0) = Pr A k
A Proof of Lemma 2 B Proof of Lemma 3 Proof: Since the support of LL istributions is R, two such istributions are equivalent absolutely continuous with respect to each other an the ivergence is well-efine
More informationOn the Enumeration of Double-Base Chains with Applications to Elliptic Curve Cryptography
On the Enumeration of Double-Base Chains with Applications to Elliptic Curve Cryptography Christophe Doche Department of Computing Macquarie University, Australia christophe.oche@mq.eu.au. Abstract. The
More informationThe total derivative. Chapter Lagrangian and Eulerian approaches
Chapter 5 The total erivative 51 Lagrangian an Eulerian approaches The representation of a flui through scalar or vector fiels means that each physical quantity uner consieration is escribe as a function
More informationMulti-View Clustering via Canonical Correlation Analysis
Technical Report TTI-TR-2008-5 Multi-View Clustering via Canonical Correlation Analysis Kamalika Chauhuri UC San Diego Sham M. Kakae Toyota Technological Institute at Chicago ABSTRACT Clustering ata in
More informationensembles When working with density operators, we can use this connection to define a generalized Bloch vector: v x Tr x, v y Tr y
Ph195a lecture notes, 1/3/01 Density operators for spin- 1 ensembles So far in our iscussion of spin- 1 systems, we have restricte our attention to the case of pure states an Hamiltonian evolution. Toay
More informationClosed and Open Loop Optimal Control of Buffer and Energy of a Wireless Device
Close an Open Loop Optimal Control of Buffer an Energy of a Wireless Device V. S. Borkar School of Technology an Computer Science TIFR, umbai, Inia. borkar@tifr.res.in A. A. Kherani B. J. Prabhu INRIA
More informationLeaving Randomness to Nature: d-dimensional Product Codes through the lens of Generalized-LDPC codes
Leaving Ranomness to Nature: -Dimensional Prouct Coes through the lens of Generalize-LDPC coes Tavor Baharav, Kannan Ramchanran Dept. of Electrical Engineering an Computer Sciences, U.C. Berkeley {tavorb,
More informationComputing Exact Confidence Coefficients of Simultaneous Confidence Intervals for Multinomial Proportions and their Functions
Working Paper 2013:5 Department of Statistics Computing Exact Confience Coefficients of Simultaneous Confience Intervals for Multinomial Proportions an their Functions Shaobo Jin Working Paper 2013:5
More informationLecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations
Lecture XII Abstract We introuce the Laplace equation in spherical coorinates an apply the metho of separation of variables to solve it. This will generate three linear orinary secon orer ifferential equations:
More informationLenny Jones Department of Mathematics, Shippensburg University, Shippensburg, Pennsylvania Daniel White
#A10 INTEGERS 1A (01): John Selfrige Memorial Issue SIERPIŃSKI NUMBERS IN IMAGINARY QUADRATIC FIELDS Lenny Jones Deartment of Mathematics, Shiensburg University, Shiensburg, Pennsylvania lkjone@shi.eu
More informationAssignment 1. g i (x 1,..., x n ) dx i = 0. i=1
Assignment 1 Golstein 1.4 The equations of motion for the rolling isk are special cases of general linear ifferential equations of constraint of the form g i (x 1,..., x n x i = 0. i=1 A constraint conition
More informationLecture 2 Lagrangian formulation of classical mechanics Mechanics
Lecture Lagrangian formulation of classical mechanics 70.00 Mechanics Principle of stationary action MATH-GA To specify a motion uniquely in classical mechanics, it suffices to give, at some time t 0,
More informationOn the Aloha throughput-fairness tradeoff
On the Aloha throughput-fairness traeoff 1 Nan Xie, Member, IEEE, an Steven Weber, Senior Member, IEEE Abstract arxiv:1605.01557v1 [cs.it] 5 May 2016 A well-known inner boun of the stability region of
More informationMark J. Machina CARDINAL PROPERTIES OF "LOCAL UTILITY FUNCTIONS"
Mark J. Machina CARDINAL PROPERTIES OF "LOCAL UTILITY FUNCTIONS" This paper outlines the carinal properties of "local utility functions" of the type use by Allen [1985], Chew [1983], Chew an MacCrimmon
More informationProbabilistic Analysis of Power Assignments
Probabilistic Analysis of Power Assignments Maurits e Graaf 1,2 an Boo Manthey 1 1 University of Twente, Department of Applie Mathematics, Enschee, Netherlans m.egraaf/b.manthey@utwente.nl 2 Thales Neerlan
More informationA New Vulnerable Class of Exponents in RSA
A ew Vulnerable Class of Exponents in RSA Aberrahmane itaj Laboratoire e Mathématiues icolas Oresme Campus II, Boulevar u Maréchal Juin BP 586, 4032 Caen Ceex, France. nitaj@math.unicaen.fr http://www.math.unicaen.fr/~nitaj
More informationThe average number of spanning trees in sparse graphs with given degrees
The average number of spanning trees in sparse graphs with given egrees Catherine Greenhill School of Mathematics an Statistics UNSW Australia Syney NSW 05, Australia cgreenhill@unsweuau Matthew Kwan Department
More informationTOEPLITZ AND POSITIVE SEMIDEFINITE COMPLETION PROBLEM FOR CYCLE GRAPH
English NUMERICAL MATHEMATICS Vol14, No1 Series A Journal of Chinese Universities Feb 2005 TOEPLITZ AND POSITIVE SEMIDEFINITE COMPLETION PROBLEM FOR CYCLE GRAPH He Ming( Λ) Michael K Ng(Ξ ) Abstract We
More informationLecture Introduction. 2 Examples of Measure Concentration. 3 The Johnson-Lindenstrauss Lemma. CS-621 Theory Gems November 28, 2012
CS-6 Theory Gems November 8, 0 Lecture Lecturer: Alesaner Mąry Scribes: Alhussein Fawzi, Dorina Thanou Introuction Toay, we will briefly iscuss an important technique in probability theory measure concentration
More information