Math 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs
|
|
- Mervin Jenkins
- 5 years ago
- Views:
Transcription
1 Math 30: The Edős-Stone-Simonovitz Theoem and Extemal Numbes fo Bipatite Gaphs May Radcliffe The Edős-Stone-Simonovitz Theoem Recall, in class we poved Tuán s Gaph Theoem, namely Theoem Tuán s Theoem Let k 3 Then exn, K k That is to say, if G is a gaph having at least On the othe hand, thee exist gaphs having k n k k n k k n k + edges, then G contains K k as a subgaph edges that do not contain K k as a subgaph Moeove, we showed that the extemal gaphs that is, the gaphs achieving the bound ae the complete balanced k -patite gaphs shown in Figue As these will come up again, we shall denote this gaph by T k n, called a Tuán gaph Ou goal fo the next few theoems is to extend this esult as best we can to othe classes of gaphs Fist, let s think about what was special about the Tuán gaphs These gaphs wok fo ou situation because we can goup the vetices up into k independent sets, and the Pigeonhole Pinciple allows us to say that if a K k wee pesent, it would have to have two vetices in one of these independent sets which is, of couse, impossible So pehaps, then, what is special about Tuán gaphs is the independent sets That is to say, it is cetainly clea that any gaph whose vetex set can be witten as a union of k independent sets can contain no copy of K k Pehaps this is the key to unlocking extensions of this extemal numbe So when can a gaph be witten as a union of k independent sets? Lemma Let G be a gaph on n vetices The vetex set of G can be patitioned into k independent sets if and only if G can be popely coloed with at most k colos The poof of this lemma is essentially tivial Note that if we popely colo the vetices of G, then any two vetices having the same colo must be independent Taking the patition as exactly the colo classes will yield the esult, and vice vesa: given a patition as above, we may colo each independent set with one colo to obtain a k -coloing Fundamentally, this is the poblem with finding a K k inside of T k n: the Tuán gaph is k - coloable, so any subgaph of T k n is also k -coloable But K k is not This leads to the following immediate genealization: Theoem Edős-Stone-Simonovitz Theoem Let H be a gaph with chomatic numbe χh k 3 Then exn, H k n k of edges in T k n + o n ; ie, the extemal numbe fo H is asymptotically equal to the numbe Futhemoe, it should be clea that the extemal gaphs will be the Tuán gaphs If that is not clea, pove it as an execise Use induction
2 V k- V V V 3 Figue : The Tuán Gaph Hee, we assume that V i and V j ae sets of vetices, whose sizes diffe by at most fo any i, j, and the solid lines between sets of vetices indicate that all edges ae pesent This gaph is a complete k -patite gaph, in that we have all edges between patition sets, and is balanced in the sense that the sizes of the vetex patition ae balanced The poof of this theoem is messy, so befoe we get into the details, let s sketch it out Since we ae looking at an asymptotic esult, we actually need to show that fo any ɛ > 0, thee exists some N sufficiently lage that if n > N, then exn, H k n k + ɛn Since ɛ is abitaily small, we essentially have witten that exn, H k n k + o n, just as we wanted So hee will be ou stategy: Choose some small ɛ Look at a gaph having n vetices and at least k n k + ɛn edges 3 Show that if n is lage enough, we can find, as a subgaph hee, the complete k-patite gaph having patite sets of size at least t fo any t We will specify late Actually, we will show it contains a copy of T k kt as a subgaph which is the same thing 4 Notice that since H is k-coloable, as long as t is lage then the size of each colo class, we can embed h inside T k kt, by embedding each colo class in one of the patite sets, and then using whicheve edges we need 5 Realize that these steps ae enough to pove the statement: we have shown that in any gaph G having at least k n k + ɛn with n lage enough, we can find a copy of H by lazily picking t V H 6 WIN THE GAME!!!! Cetainly, the difficulty of this stategy lies entiely in Step 3 This will be the bulk of the wok in poving the theoem Since this pat of the poof is haiy, we shall split it up into some lemmas Lemma Fix ɛ > 0, and let G be a gaph on n vetices having at least k n k + ɛn edges Then G contains a subgaph D having at least m ɛk n vetices, such that the degee of evey vetex in D k is at least k + ɛ m + Poof We shall poduce D using the following algoithm:
3 Initialize: D 0 G, n 0 n Given D i, n i : let v V D i be a vetex in D i having degee k k + ɛ n i 3 Define D i+ D i \{v}; that is, D i+ is obtained fom D i by emoving the vetex v Define n i+ V D i+ n i n i + 4 Repeat steps and 3 until no vetex v satisfies 5 Output D i Clealy, this algoithm will teminate eventually, since eventually we will eithe have a gaph whose vetices all have a lage degee, o a gaph with no vetices We d eally like it to be the fist thing, so let s conside how long the algoithm will un fo To do so, let s look caefully at the i th step Notice that in the i + st k step, we emove at most k + ɛ n i edges fom D i Hence, the subgaph D i is obtained by emoving at most i k k + ɛ n j j0 i k k + ɛ n j j0 k i k + ɛ n j j0 k k + ɛ ii ni k nn k + ɛ ni n n i n n i k n k + ɛ n i + n n i since i n n i edges fom G Also, the total numbe of edges in D i is at most n i n i Theefoe, the total numbe of edges in G is at most n i k n + k + ɛ n i + n n i On the othe hand, the hypothesis was that G contains at least k n k + ɛn edges Theefoe, it must be that k n k + ɛn n i k n + k + ɛ n i + n n i Theefoe, this pocess must stop wheneve the above inequality fails to hold That is, the pocess teminates when k n + ɛn > n i k n k + k + ɛ n i + n n i k n k + ɛ k k ɛ n k k + ɛ > n i k k ɛ n i k k + ɛ k k n ɛ n k + ɛ > n i k ɛ n i k + ɛ Notice, the left hand side of this inequality is just a constant when G is fixed, which it is, and the left hand side is a paabola that deceases when n i deceases If you use the quadatic fomula, you will find that this inequality is tue so long as n i < ɛk n That is to say, afte at most n ɛk steps, the algoithm must teminate, and thus the esult holds 3
4 A k- A A P Figue : The set P defined in the poof of Lemma 3 Note that if we conside a vetex in P, it has a lot of edges namely, t edges into each set A i Howeve, the edges need not always be to the same vetices Lemma 3 Let ɛ > 0, k, and let G be a gaph on n such that evey vetex of G has degee at least k k + ɛ n Fix t > 0 Then if n is sufficiently lage, T k kt is a subgaph of G Poof We shall wok by induction on k Note that the case of k is tivial, since T t is the empty gaph on t vetices Notice that k k k Suppose the esult is known fo k fo any choice of t Let s t ɛ, and hence G has sufficiently many edges to apply the induction hypothesis k Choose n lage enough that we can find a copy of T k k s in G Label the patite sets as A, A,, A k, and let A A A A k Let W be the set of vetices in V G that do not appea in A Define P {v W Nv A i t fo evey i}; that is, P is the set of vetices that have at least t edges into each A i This is illustated in Figue Notice that ou goal now is to constuct a set B k P, such that fo each i, thee is a subset B i A i with evey possible edge between B k and B i, and B i t This will be exactly a T k kt We have esticted to P, since these ae the only possible vetices to be included in such a set B k See Figue 3 Most of the est of the poof comes down to counting Fist, let us count the numbe of nonedges between W and A Fist, if a vetex v W is not in P, then thee exists some i such that the numbe of edges fom v to A i is at most t Hence, the numbe of nonedges between W and A is at least W P s t + W P s t k On the othe hand, the degee of evey vetex in G is at least k + ɛ n, and hence given any vetex k v, the numbe of nonedges involving v is at most n k + ɛ n n k ɛ Summing ove all vetices in A, we thus obtain that the numbe of nonedges between W and A is at most sk n k ɛ Taking these two bounds togethe yields sk n k ɛ W P s t 4
5 B A B A A k- B k- B k P Figue 3: The goal of the poof: to find a size t subset B i in each A i, and a set B k in P, with all the edges between B k and each B i Notice that we aleady have all the edges between B i and B j when i j and both i, j < k, since they ae subsets of the A i sets Recalling that s t ɛ and W n sk, this yields tk n k ɛ n sk P t ɛt Solving this inequality fo P and using lots of algeba, we obtain ɛk P n sk ɛ The impotant thing hee is that P is monotonically inceasing with n, even as A stays the same Hence, we may choose P as lage as we like Moeove, note that by the pigeonhole pinciple, as long as P is lage enough, we will be able to find a set B k in P satisfying the desied popeties Hence, by taking n sufficiently lage, we can poduce such B i, and the esult is demonstated Note that these two lemmas immediately pove the Edős-Stone-Simonovitz theoem, as we can fist find a good subgaph D in G by Lemma, and then find a T k kt inside D and hence inside G using Lemma 3 Once we have a T k kt as a subgaph of G, applying the obsevation made in the poof sketch then yields the esult Extemal numbes fo bipatite gaphs Note that the Edős-Stone-Simonovits Theoem woks fo any gaph H whose chomatic numbe is at least 3 This leaves out the entie class of gaphs having chomatic numbe exactly two Recall the following: Lemma 4 A gaph G has chomatic numbe if and only if G is bipatite Hence, we ae left asking, how do we compute extemal numbes fo bipatite gaphs? 5
6 The answe, unfotunately, is a lot of we don t know We do have the following bound fo any bipatite gaphs: Theoem 3 Let H be a bipatite gaph having v H vetices Then thee exists a constant c depending on H such that exn, H cn /v H To pove this bound, we shall make use of the following binomial identity While we will not pove this hee, you should be able to pove this youself if you know how to use Janson s inequality you can pove that n is convex in n Lemma 5 Let a, a,, a N be positive integes, with aveage A N ai Then a i N A That is to say, eplacing each a i with a copy of A only inceases the sum Poof Fo simplicity let s wite v H Notice that if K, is a subgaph of G, then so is H, since we may embed H just as we did in the Tuá n gaphs fo the poof of the Edős-Stone-Simonovits Theoem Now, suppose that fo each v V G, we take S v to be a set of neighbos of v Note that if thee exists a set of vetices, say A, such that fo all u, v A, S u S v, then we can fom a copy of K, inside of v having one patite set as A, and the othe patite set as S v fo any v A Hence, it must be that this constuction is impossible Theefoe, v V G dv n, since no set in [n] can be epesented in the sum o moe times Applying the bound fom the pevious Lemma, and setting eg equal to the numbe of edges in G, we thus obtain dv v V G n n dv eg/n n n n n Note that n n, and eg/n eg n use Stiling s fomula, fo example Theefoe, we have eg n n n Solving fo eg, we obtain eg / n / Note that the initial faction is independent of n, but does depend upon H; it is this faction that we take to be c in the statement of the theoem At this point, you might think Awesome, we did it! We got a bound! DONE And you e ight to celebate, this is useful, sot of But imagine if H has only a few vetices, say 3 The exponent on n in this case is n /3 n 5/3 But the only bipatite gaph on 3 vetices is the path of length, and we know that the extemal numbe fo this path is n, which is a fa, fa cy fom n 5/3 In fact, thee is NO bipatite gaph containing a cycle fo which exn, H is known We know a few of them asymptotically, but we don t know the constants Fo an example, let s conside the fist nontivial case: H C 4 By applying ou theoem caefully actually, by edoing the analysis in this somewhat special case, we obtain exn, C 4 n3/ + n 6
7 This uppe bound is asymptotically tight in SOME cases Specifically, it is known analysis complicated that exn, C 4 n3/ + on 3/, in the case that n p + p + fo some pime p I know, this n looks idiculous It comes fom analyzing cetain gaphs built out of finite fields, which ae always of ode p k fo a pime p This constuction is due to Füedi, and I d be happy to show it to you in office hous And in fact, in a constuction due to Klein, we have that exn, C 4 Θn 3/, that is, we ae asymptotically on the ight ode hee And while it is conjectued that the constant / is coect, we cannot pove it fo any n othe than some fancy foms involving pimes like the above 7
ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS. D.A. Mojdeh and B. Samadi
Opuscula Math. 37, no. 3 (017), 447 456 http://dx.doi.og/10.7494/opmath.017.37.3.447 Opuscula Mathematica ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS D.A. Mojdeh and B. Samadi Communicated
More informationA Bijective Approach to the Permutational Power of a Priority Queue
A Bijective Appoach to the Pemutational Powe of a Pioity Queue Ia M. Gessel Kuang-Yeh Wang Depatment of Mathematics Bandeis Univesity Waltham, MA 02254-9110 Abstact A pioity queue tansfoms an input pemutation
More informationThe Chromatic Villainy of Complete Multipartite Graphs
Rocheste Institute of Technology RIT Schola Wos Theses Thesis/Dissetation Collections 8--08 The Chomatic Villainy of Complete Multipatite Gaphs Anna Raleigh an9@it.edu Follow this and additional wos at:
More informationarxiv: v1 [math.co] 6 Mar 2008
An uppe bound fo the numbe of pefect matchings in gaphs Shmuel Fiedland axiv:0803.0864v [math.co] 6 Ma 2008 Depatment of Mathematics, Statistics, and Compute Science, Univesity of Illinois at Chicago Chicago,
More informationSolution to HW 3, Ma 1a Fall 2016
Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.
More informationSUFFICIENT CONDITIONS FOR MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED GRAPHS DEPENDING ON THE CLIQUE NUMBER
Discussiones Mathematicae Gaph Theoy 39 (019) 567 573 doi:10.7151/dmgt.096 SUFFICIENT CONDITIONS FOR MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED GRAPHS DEPENDING ON THE CLIQUE NUMBER Lutz Volkmann
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Septembe 5, 011 Abstact To study how balanced o unbalanced a maximal intesecting
More informationChapter 3: Theory of Modular Arithmetic 38
Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences
More informationFractional Zero Forcing via Three-color Forcing Games
Factional Zeo Focing via Thee-colo Focing Games Leslie Hogben Kevin F. Palmowski David E. Robeson Michael Young May 13, 2015 Abstact An -fold analogue of the positive semidefinite zeo focing pocess that
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Mach 6, 013 Abstact To study how balanced o unbalanced a maximal intesecting
More informationAn intersection theorem for four sets
An intesection theoem fo fou sets Dhuv Mubayi Novembe 22, 2006 Abstact Fix integes n, 4 and let F denote a family of -sets of an n-element set Suppose that fo evey fou distinct A, B, C, D F with A B C
More informationAuchmuty High School Mathematics Department Advanced Higher Notes Teacher Version
The Binomial Theoem Factoials Auchmuty High School Mathematics Depatment The calculations,, 6 etc. often appea in mathematics. They ae called factoials and have been given the notation n!. e.g. 6! 6!!!!!
More informationON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},
ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability
More informationQuasi-Randomness and the Distribution of Copies of a Fixed Graph
Quasi-Randomness and the Distibution of Copies of a Fixed Gaph Asaf Shapia Abstact We show that if a gaph G has the popety that all subsets of vetices of size n/4 contain the coect numbe of tiangles one
More informationMethod for Approximating Irrational Numbers
Method fo Appoximating Iational Numbes Eic Reichwein Depatment of Physics Univesity of Califonia, Santa Cuz June 6, 0 Abstact I will put foth an algoithm fo poducing inceasingly accuate ational appoximations
More informationA proof of the binomial theorem
A poof of the binomial theoem If n is a natual numbe, let n! denote the poduct of the numbes,2,3,,n. So! =, 2! = 2 = 2, 3! = 2 3 = 6, 4! = 2 3 4 = 24 and so on. We also let 0! =. If n is a non-negative
More information3.1 Random variables
3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated
More informationDeterministic vs Non-deterministic Graph Property Testing
Deteministic vs Non-deteministic Gaph Popety Testing Lio Gishboline Asaf Shapia Abstact A gaph popety P is said to be testable if one can check whethe a gaph is close o fa fom satisfying P using few andom
More informationStanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012
Stanfod Univesity CS59Q: Quantum Computing Handout 8 Luca Tevisan Octobe 8, 0 Lectue 8 In which we use the quantum Fouie tansfom to solve the peiod-finding poblem. The Peiod Finding Poblem Let f : {0,...,
More informationWhen two numbers are written as the product of their prime factors, they are in factored form.
10 1 Study Guide Pages 420 425 Factos Because 3 4 12, we say that 3 and 4 ae factos of 12. In othe wods, factos ae the numbes you multiply to get a poduct. Since 2 6 12, 2 and 6 ae also factos of 12. The
More informationAdditive Approximation for Edge-Deletion Problems
Additive Appoximation fo Edge-Deletion Poblems Noga Alon Asaf Shapia Benny Sudakov Abstact A gaph popety is monotone if it is closed unde emoval of vetices and edges. In this pape we conside the following
More informationThe Erdős-Hajnal conjecture for rainbow triangles
The Edős-Hajnal conjectue fo ainbow tiangles Jacob Fox Andey Ginshpun János Pach Abstact We pove that evey 3-coloing of the edges of the complete gaph on n vetices without a ainbow tiangle contains a set
More informationChromatic number and spectral radius
Linea Algeba and its Applications 426 2007) 810 814 www.elsevie.com/locate/laa Chomatic numbe and spectal adius Vladimi Nikifoov Depatment of Mathematical Sciences, Univesity of Memphis, Memphis, TN 38152,
More informationDo Managers Do Good With Other People s Money? Online Appendix
Do Manages Do Good With Othe People s Money? Online Appendix Ing-Haw Cheng Haison Hong Kelly Shue Abstact This is the Online Appendix fo Cheng, Hong and Shue 2013) containing details of the model. Datmouth
More informationTurán Numbers of Vertex-disjoint Cliques in r- Partite Graphs
Univesity of Wyoming Wyoming Scholas Repositoy Honos Theses AY 16/17 Undegaduate Honos Theses Sping 5-1-017 Tuán Numbes of Vetex-disjoint Cliques in - Patite Gaphs Anna Schenfisch Univesity of Wyoming,
More information9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic.
Chapte 9 Pimitive Roots 9.1 The multiplicative goup of a finite fld Theoem 9.1. The multiplicative goup F of a finite fld is cyclic. Remak: In paticula, if p is a pime then (Z/p) is cyclic. In fact, this
More information10/04/18. P [P(x)] 1 negl(n).
Mastemath, Sping 208 Into to Lattice lgs & Cypto Lectue 0 0/04/8 Lectues: D. Dadush, L. Ducas Scibe: K. de Boe Intoduction In this lectue, we will teat two main pats. Duing the fist pat we continue the
More informationOn decompositions of complete multipartite graphs into the union of two even cycles
On decompositions of complete multipatite gaphs into the union of two even cycles A. Su, J. Buchanan, R. C. Bunge, S. I. El-Zanati, E. Pelttai, G. Rasmuson, E. Spaks, S. Tagais Depatment of Mathematics
More informationJournal of Inequalities in Pure and Applied Mathematics
Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics
More informationarxiv: v1 [math.co] 4 May 2017
On The Numbe Of Unlabeled Bipatite Gaphs Abdullah Atmaca and A Yavuz Ouç axiv:7050800v [mathco] 4 May 207 Abstact This pape solves a poblem that was stated by M A Haison in 973 [] This poblem, that has
More informationThe Substring Search Problem
The Substing Seach Poblem One algoithm which is used in a vaiety of applications is the family of substing seach algoithms. These algoithms allow a use to detemine if, given two chaacte stings, one is
More informationChapter Eight Notes N P U1C8S4-6
Chapte Eight Notes N P UC8S-6 Name Peiod Section 8.: Tigonometic Identities An identit is, b definition, an equation that is alwas tue thoughout its domain. B tue thoughout its domain, that is to sa that
More informationarxiv: v1 [math.co] 2 Feb 2018
A VERSION OF THE LOEBL-KOMLÓS-SÓS CONJECTURE FOR SKEWED TREES TEREZA KLIMOŠOVÁ, DIANA PIGUET, AND VÁCLAV ROZHOŇ axiv:1802.00679v1 [math.co] 2 Feb 2018 Abstact. Loebl, Komlós, and Sós conjectued that any
More informationNew problems in universal algebraic geometry illustrated by boolean equations
New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic
More informationMatrix Colorings of P 4 -sparse Graphs
Diplomabeit Matix Coloings of P 4 -spase Gaphs Chistoph Hannnebaue Januay 23, 2010 Beteue: Pof. D. Winfied Hochstättle FenUnivesität in Hagen Fakultät fü Mathematik und Infomatik Contents Intoduction iii
More informationUpper Bounds for Tura n Numbers. Alexander Sidorenko
jounal of combinatoial theoy, Seies A 77, 134147 (1997) aticle no. TA962739 Uppe Bounds fo Tua n Numbes Alexande Sidoenko Couant Institute of Mathematical Sciences, New Yok Univesity, 251 Mece Steet, New
More informationCOLLAPSING WALLS THEOREM
COLLAPSING WALLS THEOREM IGOR PAK AND ROM PINCHASI Abstact. Let P R 3 be a pyamid with the base a convex polygon Q. We show that when othe faces ae collapsed (otated aound the edges onto the plane spanned
More informationworking pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50
woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,
More informationBrief summary of functional analysis APPM 5440 Fall 2014 Applied Analysis
Bief summay of functional analysis APPM 5440 Fall 014 Applied Analysis Stephen Becke, stephen.becke@coloado.edu Standad theoems. When necessay, I used Royden s and Keyzsig s books as a efeence. Vesion
More information(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2.
Paabola Volume 5, Issue (017) Solutions 151 1540 Q151 Take any fou consecutive whole numbes, multiply them togethe and add 1. Make a conjectue and pove it! The esulting numbe can, fo instance, be expessed
More information1 Explicit Explore or Exploit (E 3 ) Algorithm
2.997 Decision-Making in Lage-Scale Systems Mach 3 MIT, Sping 2004 Handout #2 Lectue Note 9 Explicit Exploe o Exploit (E 3 ) Algoithm Last lectue, we studied the Q-leaning algoithm: [ ] Q t+ (x t, a t
More informationOn the integration of the equations of hydrodynamics
Uebe die Integation de hydodynamischen Gleichungen J f eine u angew Math 56 (859) -0 On the integation of the equations of hydodynamics (By A Clebsch at Calsuhe) Tanslated by D H Delphenich In a pevious
More informationThe Congestion of n-cube Layout on a Rectangular Grid S.L. Bezrukov J.D. Chavez y L.H. Harper z M. Rottger U.-P. Schroeder Abstract We consider the pr
The Congestion of n-cube Layout on a Rectangula Gid S.L. Bezukov J.D. Chavez y L.H. Hape z M. Rottge U.-P. Schoede Abstact We conside the poblem of embedding the n-dimensional cube into a ectangula gid
More informationOn a quantity that is analogous to potential and a theorem that relates to it
Su une quantité analogue au potential et su un théoème y elatif C R Acad Sci 7 (87) 34-39 On a quantity that is analogous to potential and a theoem that elates to it By R CLAUSIUS Tanslated by D H Delphenich
More informationPhysics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
More informationLecture 8 - Gauss s Law
Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign.
More information6 PROBABILITY GENERATING FUNCTIONS
6 PROBABILITY GENERATING FUNCTIONS Cetain deivations pesented in this couse have been somewhat heavy on algeba. Fo example, detemining the expectation of the Binomial distibution (page 5.1 tuned out to
More informationarxiv: v1 [math.nt] 12 May 2017
SEQUENCES OF CONSECUTIVE HAPPY NUMBERS IN NEGATIVE BASES HELEN G. GRUNDMAN AND PAMELA E. HARRIS axiv:1705.04648v1 [math.nt] 12 May 2017 ABSTRACT. Fo b 2 and e 2, let S e,b : Z Z 0 be the function taking
More informationSupplementary information Efficient Enumeration of Monocyclic Chemical Graphs with Given Path Frequencies
Supplementay infomation Efficient Enumeation of Monocyclic Chemical Gaphs with Given Path Fequencies Masaki Suzuki, Hioshi Nagamochi Gaduate School of Infomatics, Kyoto Univesity {m suzuki,nag}@amp.i.kyoto-u.ac.jp
More informationEM Boundary Value Problems
EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do
More informationMarkscheme May 2017 Calculus Higher level Paper 3
M7/5/MATHL/HP3/ENG/TZ0/SE/M Makscheme May 07 Calculus Highe level Pape 3 pages M7/5/MATHL/HP3/ENG/TZ0/SE/M This makscheme is the popety of the Intenational Baccalaueate and must not be epoduced o distibuted
More informationProbablistically Checkable Proofs
Lectue 12 Pobablistically Checkable Poofs May 13, 2004 Lectue: Paul Beame Notes: Chis Re 12.1 Pobablisitically Checkable Poofs Oveview We know that IP = PSPACE. This means thee is an inteactive potocol
More informationSection 8.2 Polar Coordinates
Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal
More informationPermutations and Combinations
Pemutations and Combinations Mach 11, 2005 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication Pinciple
More informationQIP Course 10: Quantum Factorization Algorithm (Part 3)
QIP Couse 10: Quantum Factoization Algoithm (Pat 3 Ryutaoh Matsumoto Nagoya Univesity, Japan Send you comments to yutaoh.matsumoto@nagoya-u.jp Septembe 2018 @ Tokyo Tech. Matsumoto (Nagoya U. QIP Couse
More informationA generalization of the Bernstein polynomials
A genealization of the Benstein polynomials Halil Ouç and Geoge M Phillips Mathematical Institute, Univesity of St Andews, Noth Haugh, St Andews, Fife KY16 9SS, Scotland Dedicated to Philip J Davis This
More informationClassical Worm algorithms (WA)
Classical Wom algoithms (WA) WA was oiginally intoduced fo quantum statistical models by Pokof ev, Svistunov and Tupitsyn (997), and late genealized to classical models by Pokof ev and Svistunov (200).
More informationModified Linear Programming and Class 0 Bounds for Graph Pebbling
Modified Linea Pogamming and Class 0 Bounds fo Gaph Pebbling Daniel W. Canston Luke Postle Chenxiao Xue Cal Yege August 8, 05 Abstact Given a configuation of pebbles on the vetices of a connected gaph
More informationNew lower bounds for the independence number of sparse graphs and hypergraphs
New lowe bounds fo the independence numbe of spase gaphs and hypegaphs Kunal Dutta, Dhuv Mubayi, and C.R. Subamanian May 23, 202 Abstact We obtain new lowe bounds fo the independence numbe of K -fee gaphs
More informationUnobserved Correlation in Ascending Auctions: Example And Extensions
Unobseved Coelation in Ascending Auctions: Example And Extensions Daniel Quint Univesity of Wisconsin Novembe 2009 Intoduction In pivate-value ascending auctions, the winning bidde s willingness to pay
More informationTHE JEU DE TAQUIN ON THE SHIFTED RIM HOOK TABLEAUX. Jaejin Lee
Koean J. Math. 23 (2015), No. 3, pp. 427 438 http://dx.doi.og/10.11568/kjm.2015.23.3.427 THE JEU DE TAQUIN ON THE SHIFTED RIM HOOK TABLEAUX Jaejin Lee Abstact. The Schensted algoithm fist descibed by Robinson
More informationThen the number of elements of S of weight n is exactly the number of compositions of n into k parts.
Geneating Function In a geneal combinatoial poblem, we have a univee S of object, and we want to count the numbe of object with a cetain popety. Fo example, if S i the et of all gaph, we might want to
More informationF-IF Logistic Growth Model, Abstract Version
F-IF Logistic Gowth Model, Abstact Vesion Alignments to Content Standads: F-IFB4 Task An impotant example of a model often used in biology o ecology to model population gowth is called the logistic gowth
More informationNOTE. Some New Bounds for Cover-Free Families
Jounal of Combinatoial Theoy, Seies A 90, 224234 (2000) doi:10.1006jcta.1999.3036, available online at http:.idealibay.com on NOTE Some Ne Bounds fo Cove-Fee Families D. R. Stinson 1 and R. Wei Depatment
More informationWhat to Expect on the Placement Exam
What to Epect on the Placement Eam Placement into: MTH o MTH 44 05 05 The ACCUPLACER placement eam is an adaptive test ceated by the College Boad Educational Testing Sevice. This document was ceated to
More informationResearch Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function
Abstact and Applied Analysis Volume 011, Aticle ID 697547, 7 pages doi:10.1155/011/697547 Reseach Aticle On Alze and Qiu s Conjectue fo Complete Elliptic Integal and Invese Hypebolic Tangent Function Yu-Ming
More informationPROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr.
POBLM S # SOLUIONS by obet A. DiStasio J. Q. he Bon-Oppenheime appoximation is the standad way of appoximating the gound state of a molecula system. Wite down the conditions that detemine the tonic and
More informationProblem Set #10 Math 471 Real Analysis Assignment: Chapter 8 #2, 3, 6, 8
Poblem Set #0 Math 47 Real Analysis Assignment: Chate 8 #2, 3, 6, 8 Clayton J. Lungstum Decembe, 202 xecise 8.2 Pove the convese of Hölde s inequality fo = and =. Show also that fo eal-valued f / L ),
More informationf h = u, h g = v, we have u + v = f g. So, we wish
Answes to Homewok 4, Math 4111 (1) Pove that the following examples fom class ae indeed metic spaces. You only need to veify the tiangle inequality. (a) Let C be the set of continuous functions fom [0,
More informationConspiracy and Information Flow in the Take-Grant Protection Model
Conspiacy and Infomation Flow in the Take-Gant Potection Model Matt Bishop Depatment of Compute Science Univesity of Califonia at Davis Davis, CA 95616-8562 ABSTRACT The Take Gant Potection Model is a
More informationA solution to a problem of Grünbaum and Motzkin and of Erdős and Purdy about bichromatic configurations of points in the plane
A solution to a poblem of Günbaum and Motzkin and of Edős and Pudy about bichomatic configuations of points in the plane Rom Pinchasi July 29, 2012 Abstact Let P be a set of n blue points in the plane,
More informationA Crash Course in (2 2) Matrices
A Cash Couse in ( ) Matices Seveal weeks woth of matix algeba in an hou (Relax, we will only stuy the simplest case, that of matices) Review topics: What is a matix (pl matices)? A matix is a ectangula
More informationLargest and smallest minimal percolating sets in trees
Lagest and smallest minimal pecolating sets in tees Eic Riedl Havad Univesity Depatment of Mathematics ebiedl@math.havad.edu Submitted: Sep 2, 2010; Accepted: Ma 21, 2012; Published: Ma 31, 2012 Abstact
More informationExtremal problems on ordered and convex geometric hypergraphs
Extemal poblems on odeed and convex geometic hypegaphs Zoltán Füedi Tao Jiang Alexand Kostochka Dhuv Mubayi Jacques Vestaëte July 16, 2018 Abstact An odeed hypegaph is a hypegaph whose vetex set is linealy
More informationAlgebra. Substitution in algebra. 3 Find the value of the following expressions if u = 4, k = 7 and t = 9.
lgeba Substitution in algeba Remembe... In an algebaic expession, lettes ae used as substitutes fo numbes. Example Find the value of the following expessions if s =. a) s + + = = s + + = = Example Find
More informationFall 2014 Randomized Algorithms Oct 8, Lecture 3
Fall 204 Randomized Algoithms Oct 8, 204 Lectue 3 Pof. Fiedich Eisenband Scibes: Floian Tamè In this lectue we will be concened with linea pogamming, in paticula Clakson s Las Vegas algoithm []. The main
More informationIntroduction Common Divisors. Discrete Mathematics Andrei Bulatov
Intoduction Common Divisos Discete Mathematics Andei Bulatov Discete Mathematics Common Divisos 3- Pevious Lectue Integes Division, popeties of divisibility The division algoithm Repesentation of numbes
More informationConvergence Dynamics of Resource-Homogeneous Congestion Games: Technical Report
1 Convegence Dynamics of Resouce-Homogeneous Congestion Games: Technical Repot Richad Southwell and Jianwei Huang Abstact Many esouce shaing scenaios can be modeled using congestion games A nice popety
More informationAsymptotically Lacunary Statistical Equivalent Sequence Spaces Defined by Ideal Convergence and an Orlicz Function
"Science Stays Tue Hee" Jounal of Mathematics and Statistical Science, 335-35 Science Signpost Publishing Asymptotically Lacunay Statistical Equivalent Sequence Spaces Defined by Ideal Convegence and an
More informationTHE CONE THEOREM JOEL A. TROPP. Abstract. We prove a fixed point theorem for functions which are positive with respect to a cone in a Banach space.
THE ONE THEOEM JOEL A. TOPP Abstact. We pove a fixed point theoem fo functions which ae positive with espect to a cone in a Banach space. 1. Definitions Definition 1. Let X be a eal Banach space. A subset
More informationPushdown Automata (PDAs)
CHAPTER 2 Context-Fee Languages Contents Context-Fee Gammas definitions, examples, designing, ambiguity, Chomsky nomal fom Pushdown Automata definitions, examples, euivalence with context-fee gammas Non-Context-Fee
More informationCompactly Supported Radial Basis Functions
Chapte 4 Compactly Suppoted Radial Basis Functions As we saw ealie, compactly suppoted functions Φ that ae tuly stictly conditionally positive definite of ode m > do not exist The compact suppot automatically
More informationLecture 28: Convergence of Random Variables and Related Theorems
EE50: Pobability Foundations fo Electical Enginees July-Novembe 205 Lectue 28: Convegence of Random Vaiables and Related Theoems Lectue:. Kishna Jagannathan Scibe: Gopal, Sudhasan, Ajay, Swamy, Kolla An
More informationEQUI-PARTITIONING OF HIGHER-DIMENSIONAL HYPER-RECTANGULAR GRID GRAPHS
EQUI-PARTITIONING OF HIGHER-DIMENSIONAL HYPER-RECTANGULAR GRID GRAPHS ATHULA GUNAWARDENA AND ROBERT R MEYER Abstact A d-dimensional gid gaph G is the gaph on a finite subset in the intege lattice Z d in
More information3.6 Applied Optimization
.6 Applied Optimization Section.6 Notes Page In this section we will be looking at wod poblems whee it asks us to maimize o minimize something. Fo all the poblems in this section you will be taking the
More informationC/CS/Phys C191 Shor s order (period) finding algorithm and factoring 11/12/14 Fall 2014 Lecture 22
C/CS/Phys C9 Sho s ode (peiod) finding algoithm and factoing /2/4 Fall 204 Lectue 22 With a fast algoithm fo the uantum Fouie Tansfom in hand, it is clea that many useful applications should be possible.
More informationA Tutorial on Multiple Integrals (for Natural Sciences / Computer Sciences Tripos Part IA Maths)
A Tutoial on Multiple Integals (fo Natual Sciences / Compute Sciences Tipos Pat IA Maths) Coections to D Ian Rud (http://people.ds.cam.ac.uk/ia/contact.html) please. This tutoial gives some bief eamples
More informationRelated Rates - the Basics
Related Rates - the Basics In this section we exploe the way we can use deivatives to find the velocity at which things ae changing ove time. Up to now we have been finding the deivative to compae the
More information18.06 Problem Set 4 Solution
8.6 Poblem Set 4 Solution Total: points Section 3.5. Poblem 2: (Recommended) Find the lagest possible numbe of independent vectos among ) ) ) v = v 4 = v 5 = v 6 = v 2 = v 3 =. Solution (4 points): Since
More informationANA BERRIZBEITIA, LUIS A. MEDINA, ALEXANDER C. MOLL, VICTOR H. MOLL, AND LAINE NOBLE
THE p-adic VALUATION OF STIRLING NUMBERS ANA BERRIZBEITIA, LUIS A. MEDINA, ALEXANDER C. MOLL, VICTOR H. MOLL, AND LAINE NOBLE Abstact. Let p > 2 be a pime. The p-adic valuation of Stiling numbes of the
More informationINTRODUCTION. 2. Vectors in Physics 1
INTRODUCTION Vectos ae used in physics to extend the study of motion fom one dimension to two dimensions Vectos ae indispensable when a physical quantity has a diection associated with it As an example,
More informationradians). Figure 2.1 Figure 2.2 (a) quadrant I angle (b) quadrant II angle is in standard position Terminal side Terminal side Terminal side
. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an
More informationLab 10: Newton s Second Law in Rotation
Lab 10: Newton s Second Law in Rotation We can descibe the motion of objects that otate (i.e. spin on an axis, like a popelle o a doo) using the same definitions, adapted fo otational motion, that we have
More informationSurveillance Points in High Dimensional Spaces
Société de Calcul Mathématique SA Tools fo decision help since 995 Suveillance Points in High Dimensional Spaces by Benad Beauzamy Januay 06 Abstact Let us conside any compute softwae, elying upon a lage
More informationRECTIFYING THE CIRCUMFERENCE WITH GEOGEBRA
ECTIFYING THE CICUMFEENCE WITH GEOGEBA A. Matín Dinnbie, G. Matín González and Anthony C.M. O 1 Intoducction The elation between the cicumfeence and the adius of a cicle is one of the most impotant concepts
More informationLecture 18: Graph Isomorphisms
INFR11102: Computational Complexity 22/11/2018 Lectue: Heng Guo Lectue 18: Gaph Isomophisms 1 An Athu-Melin potocol fo GNI Last time we gave a simple inteactive potocol fo GNI with pivate coins. We will
More informationTANTON S TAKE ON CONTINUOUS COMPOUND INTEREST
CURRICULUM ISPIRATIOS: www.maa.og/ci www.theglobalmathpoject.og IOVATIVE CURRICULUM OLIE EXPERIECES: www.gdaymath.com TATO TIDBITS: www.jamestanton.com TATO S TAKE O COTIUOUS COMPOUD ITEREST DECEMBER 208
More informationGraphs of Sine and Cosine Functions
Gaphs of Sine and Cosine Functions In pevious sections, we defined the tigonometic o cicula functions in tems of the movement of a point aound the cicumfeence of a unit cicle, o the angle fomed by the
More informationA Short Combinatorial Proof of Derangement Identity arxiv: v1 [math.co] 13 Nov Introduction
A Shot Combinatoial Poof of Deangement Identity axiv:1711.04537v1 [math.co] 13 Nov 2017 Ivica Matinjak Faculty of Science, Univesity of Zageb Bijenička cesta 32, HR-10000 Zageb, Coatia and Dajana Stanić
More informationASTR415: Problem Set #6
ASTR45: Poblem Set #6 Cuan D. Muhlbege Univesity of Mayland (Dated: May 7, 27) Using existing implementations of the leapfog and Runge-Kutta methods fo solving coupled odinay diffeential equations, seveal
More information