On randomly generated non-trivially intersecting hypergraphs
|
|
- Kathleen Tyler
- 5 years ago
- Views:
Transcription
1 O adomly geeated o-tivially itesectig hypegaphs Balázs Patkós Submitted: May 5, 009; Accepted: Feb, 010; Published: Feb 8, 010 Mathematics Subject Classificatio: 05C65, 05D05, 05D40 Abstact We popose two pocedues to choose membes of sequetially at adom to fom a o-tivially itesectig hypegaph. I both cases we show what is the limitig pobability that if = c 1/3 with c c, the the pocess esults i a Hilto-Mile-type hypegaph. 1 Itoductio I 1961, Edős, Ko ad Rado [5] poved that if, the the edge set E of a itesectig -uifom hypegaph with vetex set V ad V = caot have lage size tha 1 1, moeove if <, the the oly hypegaphs with that may edges ae of the fom {e V : v e} fo some fixed v V. I the past almost five decades, the aea of itesectio theoems has bee widely studied, but adomized vesios of the Edős-Ko-Rado theoem have oly attacted the attetio of eseaches ecetly. Thee ae maily two appoaches to the adomized poblem. Balogh, Bohma ad Mubayi [] cosideed the poblem of fidig the lagest itesectig hypegaph i the pobability space G, p of all labeled -uifom hypegaphs o vetices whee evey hypeedge appeas adomly ad idepedetly with pobability p = p. I this pape, we follow the appoach of Bohma et al. [3], [4]. They cosideed the followig pocess to geeate a itesectig hypegaph by selectig edges sequetially ad adomly. Choose Radom Itesectig System Choose e 1 uifomly at adom. Give Fi = {e 1,..., e i } let AF i = {e : e / F i, 1 j i : e e j }. Choose e i+1 uifomly at adom fom AF i. The pocedue halts whe AF i = ad F = F i is the output by the pocedue. Depatmet of Compute Sciece, The Uivesity of Memphis, Memphis, TN, 3815, USA. Suppoted by NSF Gat #: CCF bpatkos@memphis.edu, patkos@eyi.hu the electoic joual of combiatoics , #R6 1
2 Theoem 1.1 Bohma et al. [3] Let E, deote the evet that F = 1 1. The if = c 1/3, lim PE, = 1 if c c 3 if c c 0 if c. Theoem 1.1 states that the pobability that the esultig hypegaph will be tivially itesectig i.e. all of its edges will shae a commo elemet with pobability tedig to 1 i othe wods, with high pobablity, w.h.p. povided = o 1/3. I this pape we will be iteested i two pocesses that geeate o-tivially itesectig hypegaphs fo this age of. Befoe itoducig the actual pocesses, let us state the theoem of Hilto ad Mile that detemies the size of the lagest o-tivially itesectig hypegaph. Theoem 1. Hilto, Mile [6] Let F be a o-tivially itesectig hypegaph with 3, + 1. The F The hypegaphs achievig that size ae i fo ay -subset F ad x \ F the hypegaph F HM = {F } {G : x G, F G }, ii if = 3, the fo ay 3-subset S the hypegaph F = {F 3 : F S }. We will call the hypegaphs descibed i i HM-type hypegaphs, while hypegaphs F fo which thee exists a 3-subset S of such that F cosists of all -subsets of with F S will be called -3 hypegaphs eve if > 3 the atual geealizatios of hypegaphs of the fom of ii. We ow itoduce the two pocesses we will be iteested i. I some sese they ae the opposite of each othe as the fist pocess assues as ealy as possible i.e. whe pickig the thid edge e 3 that it poduces a o-tivially itesectig hypegaph while the secod oe is the same as the oigial pocess of Bohma et al. as log as it is possible that the pocess esults a o-tivially itesectig hypegaph. The mai value of the fist model is that the esults coceig this model allows us to calculate the pobability that the oigial model of Bohma et al. poduces a HM-type hypegaph whe = Θ 1/3, while the secod model seems to be the model that ca be obtaied with the least modificatio to the oigial such that it esults a o-tivially itesectig hypegaph fo all values of ad. Hee ae the fomal defiitios. The Thid Roud Pocess Choose e 1 uifomly at adom. Give Fi = {e 1,..., e i } if i let AF i = {e : e / Fi, 1 j i : e e j } while fo i = let AF = {e : e / F, e e j j = 1,, e e 1 e = }. Choose e i+1 uifomly at adom fom AF i. The pocedue halts whe AF i = ad F = F i is the output by the pocedue. Note that by Lemma 7 i [3] if O /3 = = ω 1/3, the w.h.p. F 3 of the oigial pocess of Bohma et al. is o-tivially itesectig ad thus the two pocesses ae the the electoic joual of combiatoics , #R6
3 same w.h.p. The pobability of a evet E i the Thid Roud Pocess will be deoted by P 3R E. The Put-Off Pocess Choose e 1 uifomly at adom. Give Fi = {e 1,..., e i } let { } AF i = e : e / F i ; G o-tivially itesectig with {e} F i G, i.e. AF i is the set of all edges that ca be added to F i such that {e} F i ca be exteded to a o-tivially itesectig hypegaph. Choose e i+1 uifomly at adom fom AF i. The pocedue halts whe AF i = ad F = F i is the output by the pocedue. Note agai that by Lemmas 7 ad 8 i [3] if = ω 1/3, the w.h.p aleady F log of the oigial pocess of Bohma et al. is o-tivially itesectig ad thus the two pocesses ae the same. The pobability of a evet E i the Put-Off Pocess will be deoted by P PO E. If the pobability of a evet E is the same i the two models o the same boud applies fo it i both models, the we will deote this pobability by P 3R,PO E. The pobability of a evet E i the oigial pocess will be deoted by P INT E. To fomulate the mai esults of the pape we eed to itoduce the followig evets: E HM stads fo the evet that the pocess outputs a HM-type hypegaph while E deotes the evet that the output is a -3 hypegaph. Theoem 1.3 If ω1 = = c 1/3, the lim P 3RE HM = Theoem 1.4 If 3 is a fixed costat, the 1 lim P 3RE HM = 1 1 Theoem 1.5 If = c 1/3, the lim P POE HM = 1 if c c 3 /3 if c c 0 if c. 3, lim P 3RE = 1 1+c 3 + c3 Coollay 1.6 If = c 1/3 with c c, the P INT E HM = 1 if c 0 1 if c 1+c 3 /3 c 0 if c. 1+c 3 c3 1 + c c 3 / The est of the pape is ogaized as follows: i the ext sectio we itoduce some evets that will be useful i the poofs ad estate some of the lemmas of [3]. I Sectio 3, we pove Theoem 1.3 ad Coollay 1.6, Sectio 4 cotais the poof of Theoem 1.4 ad Sectio 5 cotais the poof of Theoem 1.5. the electoic joual of combiatoics , #R6 3
4 Defiitios ad Lemmas fom [3] g We will wite g = of g = ωf to deote the fact that lim = 0 f =, while g = Of g = Ωf will mea that thee exists a lim g f positive umbe K such that g < K g > K fo all iteges ad g = Θf f f deotes the fact that both g = Of ad g = Ωf hold. Thoughout the pape log stads fo the logaithm i the atual base e. We will use the followig well-kow iequalities: fo ay x we have 1 + x e x ad if x teds to 0, the 1 + x = expx + Ox. Biomial coefficiets will be bouded by a b b a b ea b b. Fially, fo biomial adom vaiables we have the followig fact see e.g. [1]. Fact.1 If X is a adom vaiable with X Bi, p, the we have P X p > δp e δ p/3. I paticula, fo ay costat c with 0 < c < 1 we have P X p > cp = exp Ωp. We call a hypegaph with i edges a i-sta if the paiwise itesectios of the edges ae the same ad have oe elemet which we will call the keel of the i-sta. A hypegaph of 3 edges e 1, e, e 3 is a tiagle if 3 i=1 e i = ad e i e j = 1 fo all 1 i < j 3. The base of a tiagle is the 3-set {e i e j : 1 i < j 3}. A hypegaph is a suflowe if the itesectio of ay two of its edges ae the same which is the keel of the suflowe. A hypegaph H of 3 edges is a -tiagle if H ca be patitioed ito 3 suflowes each of edges with keel size such that ay 3 edges take fom diffeet suflowes fom a tiagle with the same base. A hypegaph of edges e 1 1, e 1,..., e1, e is a -double-boom if i=1 j=1 ej i = 1, e 1 i e i = fo all 1 i ad ej i ej i = 1 fo ay i i. We call i=1 j=1 ej i the keel of the double-boom. The subhypegaph of a -double-boom cosistig of the d + edges e 1 1, e 1,..., e1 d, e d, e1 d+1,..., e1 is a d-patial -double-boom. The elemets ot idetical to the keel that belog to e 1 i e i ae called the semi-keels of the d-patial -double-boom ad the sets e 1 j without the keel d + 1 j ae called the loely figes of the d-patial -double-boom. The followig two tivial popositios show what itesectig subhypegaphs of F j assue that the output of the pocess will be -3-hypegaph o a HM-type hypegaph. Popositio. If a itesectig hypegaph H cotais a -tiagle, the thee is oly oe maximal itesectig hypegaph H cotaiig H ad H is a -3-hypegaph. Popositio.3 If a -set f does ot cotai the keel x of a d-patial -doubleboom B, but meets all sets i B, the f must cotai all semi-keels of B ad meet each loely fige of B i exactly oe elemet. I paticula, the oly -set meetig all sets of a -double-boom ot cotaiig the keel is the set of all semi-keels ad thus the electoic joual of combiatoics , #R6 4
5 Figue 1: A -tiagle with = 3. if a itesectig hypegaph H cotais a -double boom, the thee is oly oe otivially itesectig hypegaph H that cotais H ad H is a HM-type hypegaph. Let A i be the evet that F i is a i-sta. Let A j, deote the evet that F j cotais a -sta ad thee exists at most 1 edge e F j ot cotaiig the keel of the -sta. I paticula, A, = A. Let A j, deote the evet that F j cotais a -double boom ad thee exists at most 1 edge e F j ot cotaiig the keel of the -double boom. Let H deote the evet that e 3 cotais all of e 1 e as well as at least oe vetex fom e 1 \ e e \ e 1. Let deote the evet that F 3 is a tiagle. Let j,l deote the evet that F j cotais a l-tiagle ad all edges i F j meet the base of this l-tiagle i at least elemets. Let B j deote the evet that e F j e. Let C j,1 deote the evet that F j+1 is a j-sta with a tasvesal, a set meetig all sets of the sta i 1 elemet which is diffeet fom the keel of the sta. the electoic joual of combiatoics , #R6 5
6 Figue : A -double boom with = 4. Let C l,j,1 deote the evet that F l cotais a j-sta T ad thee is exactly oe edge e F l ot cotaiig the keel of T ad e is a tasvesal of T. I paticula, C j+1,j,1 = C j,1. Let C l,j,1 deote the evet that F l cotais a subhypegaph H of a -double boom B with EH = j ad thee is oly oe edge e F l ot cotaiig the keel of H ad e is the set of semi-keels of B. Let D j deote the evet that thee exists a x such that thee is at most oe edge e F j that does ot cotai x. Fo ay evet E, the complemet of the evet is deoted by E. We fiish this sectio by statig some of the lemmas fom [3] that we will use i the poofs of Theoem 1.3 ad Theoem 1.5. Lemma.4 Lemma 1 i [3] If = o 1/, the w.h.p. A holds. Lemma.5 Lemma i [3] If = o 1/, the P INT A 3 = 1 o o1. the electoic joual of combiatoics , #R6 6
7 Lemma.6 Lemma 3 i [3] If = o /5 ad m = O 1/ /, the m PA m A 3 = exp + o1. 4 Lemma.7 Lemma 4 i [3] If = o 1/, the PH A = o1. Lemma.8 Lemma 7 i [3] If ω 1/3 = = o /3, the PB 3 = o1. 3 The Thid Roud Model I. I this sectio we pove Theoem 1.3 ad Coollay 1.6. Fist we give a outlie of the poof, the we poceed with lemmas coespodig to the diffeet cases of Theoem 1.3 ad at the ed of the sectio we show how to deduce Theoem 1.3 fom these lemmas ad how Coollay 1.6 follows fom Theoem 1.3 ad Theoem 1.1. Outlie of the poof : We will use Popositio. ad Popositio.3 to calculate the pobability of the evets E ad E HM, while to pove that E HM does ot hold w.h.p. if = ω 1/3 we will show that fo evey vetex x thee exist at least edges i F i oe of them cotaiig x, i.e. D i does ot hold. The latte will be doe by Lemma 3.4 ad Lemma 3.5. To show the emegece of a -double boom we will pove i Lemma 3.3 that it follows fom the ealy appeaace of a 3-sta of which the pobability is calculated i Lemma 3.. Ou fist lemma states that if = o 1/, the F 3 is a tiagle w.h.p. Lemma 3.1 I the Thid Roud Model, if = o 1/, the holds w.h.p. Poof P 3R A 1 +1 = O = O exp Togethe with Lemma.4, this poves the statemet. 3 j + 1 j = o1. 4 j=0 Lemma 3., fo the Thid Roud Model, is the equivalet of Lemma.5 i [3] fo the Itesectio Model. It gives the pobability that F 4 cotais a 3-sta. Lemma 3. I the Thid Roud Model, if = o 1/, the P 3R C 3,1 = 1 o the electoic joual of combiatoics , #R6 7
8 Poof. If S is the base of F 3, the the keel of the 3-sta i F 4 ca oly be a elemet of S. Thus the umbe of sets that ca exted F 3 to F 4 i such a way that C 3,1 should hold is Let Ni deote the umbe of sets f i AF 3 with f S = i i = 0, 1,, 3. Evey set f with f S belogs to AF 3, sets belogig to AF 3 with f S = 1 must meet oe edge of F 3 outside S, while sets disjoit fom S that belog to AF 3 must meet all thee edges i F 3 outside S. Theefoe we have the followig bouds o N i : 3 3 N = 3 3, N 3 =, N 1 3, N By the assumptio = o 1/ we have c 1 c expo 1 fo ay costats c 1, c, ad thus the lowe ad uppe bouds o N 0 ad N 1 ae of the same ode of magitude. Hece we obtai P 3R C 3,1 = i=0 N i = o1 1 = o o1 3 Lemma 3.3 states that if F j cotais a 3-sta fo some small eough j, the F will cotai a -double boom w.h.p. which by Popositio.3 assues that the pocess outputs a HM-type hypegaph. Lemma 3.3 If = O 1/3 ad j log, the P 3R l : A l, C j,3,1 = 1 o1. Poof. Suppose C j,3,1 holds fo some j with j j log. The the umbe of sets i AF j cotaiig the keel of a 3-sta S i F j is 1 1 M = j as they all must meet the tasvesal t of S aleady i F j. Clealy, we have 1 j + 1 M, the electoic joual of combiatoics , #R6 8
9 as fo the lowe boud we eumeated the -sets cotaiig the keel ad exactly oe elemet of t, while fo he uppe boud we couted times the umbe of -sets cotaiig the keel ad oe fixed elemet of t. The umbe of sets i AF j ot cotaiig the keel of S is at most , whee the fist tem of the sum stads fo the sets i AF j that meet all elemets of S outside t ad thus we have to make sue that they meet t as well, while the secod tem stads fo the othe sets. Thus the pobability that the adom pocess picks a edge ot cotaiig the keel is at most j = O Remembe that D k deotes the evet that thee is a vetex x which is cotaied i all but at most oe edge of F k, thus as = O 1/3, we obtai that P PO,3R D 1/7 C j,3,1 = 1 o1. Fo i j let α i deote the maximum umbe k such that thee exist k edges i F i that fom a subhypegaph of a -double boom of which the semi-keels ae elemets of t, i paticula α i = implies the existece of a -double boom. Let us itoduce the followig adom vaiables: { 1 if αi α Z i = i+1 o α i = 0 othewise. The umbe of edges that would make α i gow if α i < is at least α i αi 1. The total umbe of edges i AF i is at most = O 1 as = O 1/3. Thus fo j i 1/7 we have αi α i 1 P 3R,PO Z i = 1 D 1/7, C j,3,1 = Ω αi = Ω αi = Ω as = O 1/3. Note that if α i =, the by defiitio PZ i = 1 = 1, thus ay lowe boud obtaied i the α i < case is valid i this case, too. Let us coside cases: Case I = o 1/15 By, we have P 3R Z i = 1 D 1/7, C j,3 Ω1/, thus 1/7 P 3R Z i < D 1/7, C j,3,1 < PBi 1/7, Ω1/ < 0 i=j the electoic joual of combiatoics , #R6 9
10 as 1/7 = ω by the assumptio = o 1/15. Case II = ω 1/16 By, we obtai P 3R Z i = 1 D 1/7, C j,3,1, α i / = Θ1, thus P 3R 1/7 Z i < 1/0 D 1/7, C j,3,1 < PBi 1/7, Θ1 < 1/0 0 i=j as 1/0 < / by the assumptio = ω 1/16. Fo ay subhypegaph of a -double boom thee exists a set of at least half the edges that ae paiwise disjoit apat fom the keel, thus if α i 1/0, the the umbe of - sets that do ot cotai the keel but meet all edges of F i is at most 1 1/0 1/0. 1/0 As befoe, if thee is oly oe edge i F i ot cotaiig x, the the umbe of -sets i AF i cotaiig x is j Hece, we have 1/0 11/0 1/0 P 3R D C 1/7, 1/0,1 1 1/0 0 as = o 1/ ǫ. O the othe had, just as i we have P 3R Z i = 1 D, C j,3,1 = Ω αi = Ω1/ ad thus P 3R Z i < D, C j,3,1 PBi, Ω1/ < 0 i=j as / = ω sice = O 1/3. Lemma 3.4 assets that if ω 1/3 = = o 1/ log 1/10, the all vetices ae cotaied i at most edges of F 4 ad theefoe the esultig hypegaph of the pocess caot be HM-type. Lemma 3.4 If ω 1/3 = = o 1/ log 1/10, the P 3R,PO D 4 = o1. the electoic joual of combiatoics , #R6 10
11 Poof. We coside cases: Case I ω 1/3 = = o 1/ I this case, Lemma 3.1 states that holds w.h.p., ad the computatio i Lemma 3. shows that e 4 is disjoit fom the base of the tiagle of F 3 w.h.p. Case II ω 1/ log 1 = = o 1/ log 1/10 Fist ote that 1/ 1/ 3 3 1/ exp O Usig 3 ad witig E fo the evet e 1 e 1/ we have P 3R E 1/ 1/ 1/ 1 = e 1/ 1/ 1/. 3 exp O, as the deomiato bouds fom below the umbe of -sets meetig e 1 i exactly 1 elemet, while the eumeato is a uppe boud o the umbe of -sets meetig e 1 i at least 1/ elemets. This boud teds to 0 as 3/ / teds to 0. Futhemoe, still usig 3 ad witig E 3 fo the evet e 1 e e 3 1/, we have 1/ P 3R E 3 E 1/ 1/ 1/ = e 1/ 1/ 1/ exp O, which teds to 0 fo the same easo as the pevious boud. Now ote that by the defiitio of the Thid Roud pocess e 1 e e 3 = ad thus D 4 is equivalet to e 4 1 i<j 3 e i e j. As E, E 3 imply e 1 \ e e 3, e \ e 1 e 3, e 3 \ e 1 e 1/, we have 1/ P 3R D 4 E, E 3 1/ 3 3 = 3 5/ exp O = O 6/5 5/4 log 3 0, whee fo the last equality we used the assumptio ω 1/ log 1 = = o 1/ log 1/10 to obtai exp O = O 1/5 ad 5/ = Ω 5/4 log 5/. Lemma 3.5 is the equivalet of Lemma 8 i [4] ad the poofs ae almost idetical. Lemma 3.5 If = ω 1/ ad log m exp, the 3 P 3R,PO D m = o1. Poof. Pick the fist 3 edges e 1, e, e 3 accodig to ay of the pocesses ad the coside m elemets of \ {e1, e, e 3 } beig chose at adom without eplacemet. The pobability that these m + 3 edges fail to fom a itesectig family is at most m 3m + m exp 1 exp, 3 the electoic joual of combiatoics , #R6 11
12 which teds to 0 as = ω 1/. We kow that the Put-Off pocess ad the Radom Itesectig Hypegaph pocess is w.h.p. the same if = ω 1/3 ad the Thid Roud pocess is the same as the Radom Itesectig Hypegaph pocess fom the fouth oud by defiitio. Thus coditioig o e 1, e, e 3, the distibutio of F m+3 will be the same as pickig m distict -sets uifomly at adom if we futhe coditio o the evet of pobability 1 o1 that the adomly picked sets togethe with the fist 3 edges fom a itesectig hypegaph. Thus we obtai that P 3R,PO D m 1 exp + o1 + m m 1 = O exp + o1 + m m 1 m O exp + o1 + m m. 3 3 Poof of Theoem 1.3. We coside seveal cases. Let ω1 = = c 1/3. If c teds to 0, the fom Lemma 3.1 ad Lemma 3. it follows that C 3,1 holds w.h.p ad the Lemma 3.3 togethe with Popositio.3 fiishes the poof of this case. If c c, the Lemma 3.1 states that holds almost suely. Accodig to Lemma 3. the pobability that C 3,1 holds is 1 1+c o1 ad the poof of Lemma 3. shows that if C 3,1 does ot hold, the o does D 4 thus E HM caot happe. Agai, Lemma 3.3 togethe with Popositio.3 fiishes the poof of this case. Fially, if c teds to ifiity the fo ω 1/3 = = o 1/ log 1/10 Lemma 3.4 while fo ω 1/ = / Lemma 3.5 poves that the pobability of E HM is o1. Poof of Coollay 1.6: Bohma et al. i [3] pove that coditioed o the evet A 3, a tivially itesectig family is the output of the oigial pocess w.h.p. Lemma.4 ad Lemma.7 give that coditioed o the evet that A3 does ot hold, we have e F 3 e = with pobability tedig to 1, that is, e 3 is chose accodig to the ule of the Thid Roud Pocess. Thus the followig equality holds: lim P INTE HM = 1 lim P INT A 3 lim P 3R E HM. Lemma.5 ad Theoem 1.3 complete the poof. 4 The Thid Roud Model II. costat I this sectio we coside the Thid Roud Model whe is a fixed costat ad we pove Theoem 1.4. We will use oe of the lemmas poved i the pevious sectio ad we will also eed ew oes. Lemma 4.1 states that F log cotais eithe a 3-sta o a -tiagle w.h.p. but ot both. the electoic joual of combiatoics , #R6 1
13 Lemma 4.1 If 3 is a costat, the 3 1 P 3R log, = 1 + o1, P 3R C log,3,1 = o1, 1 P 3R log, C log,3,1 = o1. Poof. Let S be the base of F 3 if holds ad fo ay 3 j log ad i = 0, 1,, 3 let AF j,i deote the -sets i AF j that meet S i i elemets ad let M j,i = AF j,i. We will pove that fo ay 3 j log the pocess picks a edge eithe fom AF j,1 o fom AF j, w.h.p. Futhemoe, if fo some 3 j log the pocess picks a edge fom AF j,1, the C log,3,1 holds w.h.p., while if this is ot the case i.e. the pocess picks a edge fom AF j, fo all 3 j log, the log, holds w.h.p. We will eed some bouds o M j,i. Clealy, we have 3 6 M j,3, M j, Fo the fist iequality we couted all -sets cotaiig S, while fo the secod iequality we used that if a -set f AF j is disjoit fom S, the f must meet e 1, e, e 3 outside S. Thus as is costat we have M j,0, M j,3 = O 3. Let j deote the evet that fo all edges e of F j we have e S. Clealy 3 holds. If j holds, the evey edge f with f S = belogs to AF j, theefoe we have M j, 3 3 j = Θ. By compaig this to the bouds o M j,0 ad M j,3 we obtai that if thee is a j log such that e j+1 / AF j,, the e j+1 AF j,1 w.h.p. We claim that if j holds, the all edges of F j ae paiwise disjoit outside S w.h.p. Ideed, fo ay 3 j j the umbe of -sets f AF j that meet e Fj e \ S ad f S = is at most log = O 3 log as we ca choose i 3 ways which elemets of S belog to f, which othe elemet of e Fj e \ S belogs to f ad e Fj e\s log as j log. Thus the pobability that e j +1 meets e Fj e\s fo some j log is Olog /. Thus if j deotes the evet that j holds ad the edges i F j ae paiwise disjoit outside e Fj e\s, the we have just see that j implies j w.h.p. To calculate the pobability that thee is a j log fo which e j+1 AF j,1 ad to have moe isight o the pocess we itoduce some moe otatios. Let S j = {s S : d j s = j 1} ad h j = S j, i.e. S j is the set of elemets which ae cotaied i all but at most oe edge of F j ad theefoe they still might become the keel of a possible HM-type extesio of F j. As holds, we have S 3 = S ad h 3 = 3. Note that if fo some j log the evet j holds ad we have h j = 0, the j, holds. Let us suppose that j holds ad let us coside e j+1. We distiguish seveal possibilities as we aleady uled out the occuece of a edge fom AF j,0 AF j,3 w.h.p., we omit these possibilities: the electoic joual of combiatoics , #R6 13
14 1. e j+1 AF j,1, e j+1 S / S j, that is, the oly commo elemet x of e j+1 ad S is ot cotaied i at least edges of F j. Thee ae at most 3 h j 5 3 = Θ 3 possibilities fo such a e j+1, as we ca pick e j+1 S i 3 h j ways ad e j+1 must meet all the edges ot cotaiig x thee ae at least of them outside S whee those edges ae paiwise disjoit. Thus the pobability of pickig such a edge is O1/.. e j+1 AF j,1, e j+1 S S j ad e j+1 does ot ceate a 3-sta. We ca pick the oly elemet x of e j+1 S j i h j ways. We kow that e j+1 must meet the edge of F 3 that does ot cotai x outside S ad to ot ceate a 3-sta e j+1 must meet at least oe of the edges of F 3 cotaiig x outside S as well. These sets ae paiwise disjoit outside S, thus thee ae at most h j = Θ 3 possibilities agai, thus the pobability of this to happe is O1/. 3. e j+1 AF j,1, e j+1 S S j ad e j+1 ceates a 3-sta, i.e. C j+1,3,1 holds. Numbe of possibilities: agai, we ca pick the oly elemet x of e j+1 S j i h j ways ad e j+1 must meet the edge of F 3 that does ot cotai x outside S. Thus thee ae at most h j 4 possibilities ad if ej+1 does ot meet ay of the edges of F 3 cotaiig x outside S, the a 3-sta is ceated, thus thee ae at least h j 3+3 to choose e j+1. Theefoe the umbe of possibilities is h j 1 + Θ1/. 4. e j+1 AF j,, h j+1 = h j, that is, the oly elemet x of S which is ot i e j+1 does ot belog to S j. To pick x we have 3 h j possibilities ad the fact that the othe elemets of S do belog to e j+1 assues that e j+1 itesects all edges i F j, thus the umbe of possible e j+1 s is 3 h j 1 + Θ1/. 5. e j+1 AF j,, h j+1 = h j 1. The oly diffeece to the pevious case is that ow we have to pick x fom S j ad thus the umbe of possible e j+1 s is h j 3 1+Θ1/. The pobability that the fist two possibilities happe at least oce fo some j log is O log thus eithe 3 happes o the pocess always picks a edge accodig to 4 o 5. The pobability that possibility 4 happes with h j < 3 at least log times while 3 does ot occu at all is O3 log. Thus we obtai that fo some j log eithe possibility 3 happes o we will have h j = 0 which is equivalet to j,. The pobability that if h j > 0, the possibility 5 happes befoe possibility 3 is h 3 j h 3 j + hj 1 + O1/ log = o1. 1 Thus the pobability that j, holds fo some j log is o1, 1 the electoic joual of combiatoics , #R6 14
15 while the pobability that C j,3,1 holds fo some j log is o1. 1 Note that Lemma 3.3 implies PC log,3,1 C j,3,1 = 1 o1 i fact, it states that C j,3,1 implies the appeaace of a -double boom, which is much moe tha C log,3,1. Also, P 3R j +1, j, = O1/ fo ay j j log as the umbe of -sets f meetig all edges of F j ad itesectig S i oe elemet is at most = Θ 3 pick {x} = S f i 3 ways ad f must meet the edges of the -tiagle, assued by j,, that do ot cotai x thus P 3R log, j, = O log. Lemma 4.1 assets that, i the case of costat, F log cotais eithe a 3-sta o a -tiagle w.h.p. By Lemma 3.3, i the fome case F j cotais a -double boom w.h.p. fo sufficietly lage j ad thus by Popositio.3 assues that the pocess esults a HM-type hypegaph. The ext lemma states that i the latte case F log cotais a -tiagle w.h.p. ad thus by Popositio. assues that the pocess outputs a -3 hypegaph. Lemma 4. If 3 is a costat, the P 3R log, log, = 1 o1. Poof. With the otatio of Lemma 4.1, the evet log, implies that fo ay j log we have M j,0 6 M j,1 M j, Futhemoe, if all edges i F j itesect the base S of the -tiagle cotaied i F log i at least elemets, the j M j, 3. Thus the pobability, that F log will cotai a edge e with e S is O log. Let β j deote the lagest itege k such that F j cotais a subhypegaph of a - tiagle with k edges, i paticula β j = 3 if ad oly if F j cotais a -tiagle. Let us itoduce the followig adom vaiable { 1 if βj α W j = j+1 o β j = 3 0 othewise. If β j < 3 ad if all edges i F j itesect the base S of the -tiagle cotaied i F log i at least elemets, the P 3R W j = 1 βj 3 3 i=0 M j,i 1/3 o1. the electoic joual of combiatoics , #R6 15
16 Theefoe log P 3R β log < 3 log, = P 3R j=log W j < 3 log, log PBilog, 1/3 o1 < 3 + O. Poof of Theoem 1.4. Lemma 3.1 assues that holds w.h.p. Lemma 4.1 ad Lemma 4. togethe with Popositio. ad Popositio.3 poves Theoem The Put-Off Model I this sectio, we pove Theoem 1.5. The poof is simila to that of Theoem 1.3 as agai fo a lage eough j we will pove the existece of a -double boom i F j to assue that E HM happes. The mai diffeece betwee the models is that while i the Thid Roud Model the set of semi-keels of the -double boom is aleady detemied afte e 4 is picked i the Put-Off Model thee might be lots of possibilities fo this set eve late. Let us defie j 1 = mi{ 1 1/4, } ad j = mi{ 1 1/, 3 }. Ou fist lemma states that if F 3 is a 3-sta, the w.h.p. the keel of this 3-sta is cotaied i all edges of F j. Lemma 5.1 If = O 1/3, the P PO B j A 3 = 1 o1. Poof. Obseve that if B j holds, the the umbe of sets i AF j cotaiig a elemet of e F j e is at least 1 j. The umbe of -sets cotaiig a fixed elemet x ad meetig a fixed -set f AF j i exactly oe elemet with x / f is 1 ad eve if all of them belog to F j the the othes ae i AF j. Suppose fist that = o 1/4. The by Lemma.6 we may assume that A holds ad thus fo ay j the umbe of sets i AF j ot cotaiig the keel of F j is at most 1. Thus we have P PO B j+1 B j, A j If is costat, the multiplyig the last atio by j 1 1/ still gives a boud that teds to 0. If teds to ifiity, the holds as = o1/ ǫ. Suppose ow that = ω 1/5 ad thus j = 3. The by Lemma.6 we kow that A 1/5 holds w.h.p. I this case fo ay j with 1/5 j 3 = j the umbe of sets i the electoic joual of combiatoics , #R6 16
17 AF j ot cotaiig the keel of F 1/5 is at most 1 1/5 1/5 1 1/5 ad thus we have 1/5 1/5 P PO B j+1 B j, A 3 11/5 1/ /5 1/5 1 j 1 1 1/5 1/5 1 j=0 1/5 1 j 1 j = 1/5 exp O. Theefoe P PO B 3 A 3 3 1/5 exp O + P PO A 1/5 A 3 0. Lemma 5. assets that if the edges of F 3 fom a 3-sta, the F j1 cotais a -sta ad its keel belogs to all but at most oe edge of F j1. Lemma 5. If = O 1/3, the Poof. We coside two cases. Case I = o 1/4 P PO A j 1, A 3 = 1 o1. I this case Lemma.6 shows that P PO A A 3 = 1 o1 ad we ae doe as A = A, ad A, B 3 implies A j, fo ay 3 j. Case II ω 1/5 = = O 1/3 I this case j 1 =. Usig Lemma.6 ad Lemma 5.1 we ca assume that A 1/5 ad B 3 hold. Let x deote the keel of F 1/5. Let us defie T = {t : x / t, t ei i, 1 i 1/5 }. Clealy, we have 1/5 1 T 1 1/5. 1/5 Fo ay t T ad j 1/5 let ν t,j deote the maximum umbe k such that thee exists a k-sta i F j such that t is a tasvesal of the sets of the k-sta. Clealy, the evet A j, holds if ad oly if thee is a t T with ν j,t =. Let us itoduce the followig adom vaiables: { 1 if νj,t ν X j,t = j+1,t o ν j,t = o t / AF j F j 0 othewise. Let us wite futhemoe X t = j= X 1/5 j,t. Obseve that if X t fo all t T, the holds. Ideed, by the defiitio of the Put-Off Pocess thee must always be a A, the electoic joual of combiatoics , #R6 17
18 t T which belogs AF j F j ad fo this t the fact X t shows that thee exists a -sta i F 1/3 of which t is a tasvesal. Let us boud fom below the pobability P PO X j,t = 1 fo ay t ad j 1/5. If t / AF j F j ad ν j,t < as othewise X j,t = 1 fo sue, the let us fix a ν j,t -sta B showig this. The keel of B must be x. The umbe of -sets cotaiig x but othewise disjoit fom b B b that meet t i exactly oe elemet is νj,t 1 1 ν j,t. O the othe had, fo ay j 1/5 we have AF j {e : x e} T. Sice T 1 1/5 1/ /5 1, we have AFj 1 1 ad thus 1 P PO X j,t = Theefoe we have = Ω 1 P PO t T : X t T P = Ω 1 1 = Ω. Bi, Ω expo log exp Ω 0. Lemma 5.3 states that if F j1 cotais a -sta, the F j cotais a -double boom w.h.p. ad thus with Popositio.3 assues that the pocess outputs a HM-type hypegaph. Lemma 5.3 If = O 1/3, the we have P PO A j, B 3, A, = 1 o1. Poof. The poof is simila to that of Case II i the poof of Lemma 5.. Fo ay j j 1 let µ j deote the lagest itege d such that F j cotais a d-patial -double boom. Sice a -sta is a 0-patial -double boom, µ j is well-defied. Let us itoduce the followig adom vaiables: { 1 if µj µ Y j = j+1 o µ j = 0 othewise. Let us wite futhemoe Y = j j=j 1 Y j. Let us shape the uppe boud o AF j fom Lemma 5. by cosideig a µ j -patial -double boom B with keel x. Evey set f i AF j must meet b B b\{x} as othewise {f} F j would cotai a +1-sta which is impossible to exted to a o-tivially itesectig hypegaph. Thus the umbe of sets the electoic joual of combiatoics , #R6 18
19 i AF j cotaiig x is at most. The umbe of sets i AFj ot cotaiig x is at most 1 as A j, holds. Thus AF j O the othe had, by Popositio.3 evey t AF j with x / t cotais all the semikeels of B ad meets all the loely figes of B. Note that thee is at least oe such set by the defiitio of the Put-Off Pocess, say t j. Obseve that ay -set e cotaiig x with t j e = 1, e b B b = ad t j e belogig to a loely fige of B is i AF j, futhemoe if such a set is chose to be e j+1 by the Put-Off pocess, the µ j+1 = µ j + 1. Thus we obtai P PO Y j = 1 µ j as = O 1/3. Now we have 1 P PO A j, B j, A j 1, P PO Y < B j, A j 1, P 1 Ω 1 1 = Ω, 1 Bi j j 1, Ω < 0, as if is costat, the 1/ is costat ad j j 1, while if = ω1, the j j 1 = 3. Poof of Theoem 1.5. Let c = < /. If c teds to 0, the Lemma.5, Lemma 5.1, Lemma 5. ad Lemma 5.3 assues that fo a suitably chose j F j cotais a -double-boom w.h.p. ad thus, by Popositio.3, E HM holds w.h.p. If c teds to c, the Lemma.5 states that the pobability that A 3 holds teds to 1 i which case, just as fo = ω 1/3, E 1+c 3 HM holds. Lemma.7 assues that B 3 holds with pobability tedig to 1 1. As if B 1+c 3 3 holds, the the Thid Roud Model ad the Put-Off Model coicide ad thus by Theoem 1.3 we obtai that P PO E HM B 3 = 1 P 3R E HM 1 + o1 teds to. Hece P 1 1+c 3 /3 POE HM teds to as 1+c 3 1+c 3 1+c 3 /3 claimed i the theoem. If c teds to ifiity, the ote that fo ω 1/3 = = o /3 a cosequece of Lemma.8 is that w.h.p the Thid Roud Model ad the Put-Off Model ae the same, while Lemma 3.5 poves Theoem 1.5 fo ω 1/ = < /. Ackowledgemet. I would like to thak the aoymous efeee fo his/he caeful eadig ad valuable commets. Refeeces [1] N. Alo; J. Spece, The Pobabilistic Method, Wiley-Itesciece Seies i Discete Mathematics ad Optimizatio. Wiley-Itesciece, New Yok, secod editio, 000. the electoic joual of combiatoics , #R6 19
20 [] J. Balogh; T. Bohma; D. Mubayi, Edos-Ko-Rado i Radom Hypegaphs, to appea i Combiatoics, Pobability ad Computig. [3] T. Bohma; C. Coope; A. Fieze; R. Mati; M. Ruszikó, O adomly geeated itesectig hypegaphs, The Electoic Joual of Combiatoics, R 9. [4] T. Bohma; A. Fieze; R. Mati; M. Ruszikó; C. Smyth, Radomly geeated itesectig hypegaphs. II. Radom Stuctues Algoithms , [5] P. Edős; C. Ko; R. Rado, Itesectio theoems fo systems of fiite sets. Quat. J. Math. Oxfod Se [6] A.J.W. Hilto; E.C. Mile, Some itesectio theoems fo systems of fiite sets. Quat. J. Math. Oxfod Se the electoic joual of combiatoics , #R6 0
A NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS
Discussioes Mathematicae Gaph Theoy 28 (2008 335 343 A NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS Athoy Boato Depatmet of Mathematics Wilfid Lauie Uivesity Wateloo, ON, Caada, N2L 3C5 e-mail: aboato@oges.com
More informationA note on random minimum length spanning trees
A ote o adom miimum legth spaig tees Ala Fieze Miklós Ruszikó Lubos Thoma Depatmet of Mathematical Scieces Caegie Mello Uivesity Pittsbugh PA15213, USA ala@adom.math.cmu.edu, usziko@luta.sztaki.hu, thoma@qwes.math.cmu.edu
More informationLower Bounds for Cover-Free Families
Loe Bouds fo Cove-Fee Families Ali Z. Abdi Covet of Nazaeth High School Gade, Abas 7, Haifa Nade H. Bshouty Dept. of Compute Sciece Techio, Haifa, 3000 Apil, 05 Abstact Let F be a set of blocks of a t-set
More information= 5! 3! 2! = 5! 3! (5 3)!. In general, the number of different groups of r items out of n items (when the order is ignored) is given by n!
0 Combiatoial Aalysis Copyight by Deiz Kalı 4 Combiatios Questio 4 What is the diffeece betwee the followig questio i How may 3-lette wods ca you wite usig the lettes A, B, C, D, E ii How may 3-elemet
More informationCounting Functions and Subsets
CHAPTER 1 Coutig Fuctios ad Subsets This chapte of the otes is based o Chapte 12 of PJE See PJE p144 Hee ad below, the efeeces to the PJEccles book ae give as PJE The goal of this shot chapte is to itoduce
More informationDANIEL YAQUBI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD
MIXED -STIRLING NUMERS OF THE SEOND KIND DANIEL YAQUI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD Abstact The Stilig umbe of the secod id { } couts the umbe of ways to patitio a set of labeled balls ito
More informationMATH Midterm Solutions
MATH 2113 - Midtem Solutios Febuay 18 1. A bag of mables cotais 4 which ae ed, 4 which ae blue ad 4 which ae gee. a How may mables must be chose fom the bag to guaatee that thee ae the same colou? We ca
More informationBINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a
BINOMIAL THEOREM hapte 8 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4 5y, etc., ae all biomial epessios. 8.. Biomial theoem If
More informationThe Pigeonhole Principle 3.4 Binomial Coefficients
Discete M athematic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Ageda Ch 3. The Pigeohole Piciple
More informationBINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a
8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4, etc., ae all biomial 5y epessios. 8.. Biomial theoem BINOMIAL THEOREM If a ad b ae
More informationLecture 6: October 16, 2017
Ifomatio ad Codig Theoy Autum 207 Lectue: Madhu Tulsiai Lectue 6: Octobe 6, 207 The Method of Types Fo this lectue, we will take U to be a fiite uivese U, ad use x (x, x 2,..., x to deote a sequece of
More informationCHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method
CHAPTER 5 : SERIES 5.1 Seies 5. The Sum of a Seies 5..1 Sum of Powe of Positive Iteges 5.. Sum of Seies of Patial Factio 5..3 Diffeece Method 5.3 Test of covegece 5.3.1 Divegece Test 5.3. Itegal Test 5.3.3
More informationStrong Result for Level Crossings of Random Polynomials
IOSR Joual of haacy ad Biological Scieces (IOSR-JBS) e-issn:78-8, p-issn:19-7676 Volue 11, Issue Ve III (ay - Ju16), 1-18 wwwiosjoualsog Stog Result fo Level Cossigs of Rado olyoials 1 DKisha, AK asigh
More informationSums of Involving the Harmonic Numbers and the Binomial Coefficients
Ameica Joual of Computatioal Mathematics 5 5 96-5 Published Olie Jue 5 i SciRes. http://www.scip.og/oual/acm http://dx.doi.og/.46/acm.5.58 Sums of Ivolvig the amoic Numbes ad the Biomial Coefficiets Wuyugaowa
More informationBy the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences
Chapte : Theoy of Modula Aithmetic 8 Sectio D Chiese Remaide Theoem By the ed of this sectio you will be able to pove the Chiese Remaide Theoem apply this theoem to solve simultaeous liea cogueces The
More informationSOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES
#A17 INTEGERS 9 2009), 181-190 SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES Deick M. Keiste Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA 16802 dmk5075@psu.edu James A. Selles Depatmet
More informationa) The average (mean) of the two fractions is halfway between them: b) The answer is yes. Assume without loss of generality that p < r.
Solutios to MAML Olympiad Level 00. Factioated a) The aveage (mea) of the two factios is halfway betwee them: p ps+ q ps+ q + q s qs qs b) The aswe is yes. Assume without loss of geeality that p
More informationCh 3.4 Binomial Coefficients. Pascal's Identit y and Triangle. Chapter 3.2 & 3.4. South China University of Technology
Disc ete Mathem atic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Pigeohole Piciple Suppose that a
More informationUsing Counting Techniques to Determine Probabilities
Kowledge ticle: obability ad Statistics Usig outig Techiques to Detemie obabilities Tee Diagams ad the Fudametal outig iciple impotat aspect of pobability theoy is the ability to detemie the total umbe
More informationStrong Result for Level Crossings of Random Polynomials. Dipty Rani Dhal, Dr. P. K. Mishra. Department of Mathematics, CET, BPUT, BBSR, ODISHA, INDIA
Iteatioal Joual of Reseach i Egieeig ad aageet Techology (IJRET) olue Issue July 5 Available at http://wwwijetco/ Stog Result fo Level Cossigs of Rado olyoials Dipty Rai Dhal D K isha Depatet of atheatics
More informationON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS
Joual of Pue ad Alied Mathematics: Advaces ad Alicatios Volume 0 Numbe 03 Pages 5-58 ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS ALI H HAKAMI Deatmet of Mathematics
More informationMath 7409 Homework 2 Fall from which we can calculate the cycle index of the action of S 5 on pairs of vertices as
Math 7409 Hoewok 2 Fall 2010 1. Eueate the equivalece classes of siple gaphs o 5 vetices by usig the patte ivetoy as a guide. The cycle idex of S 5 actig o 5 vetices is 1 x 5 120 1 10 x 3 1 x 2 15 x 1
More information( ) 1 Comparison Functions. α is strictly increasing since ( r) ( r ) α = for any positive real number c. = 0. It is said to belong to
Compaiso Fuctios I this lesso, we study stability popeties of the oautoomous system = f t, x The difficulty is that ay solutio of this system statig at x( t ) depeds o both t ad t = x Thee ae thee special
More informationAuchmuty High School Mathematics Department Sequences & Series Notes Teacher Version
equeces ad eies Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio A sequece takes the fom,,7,0,, while 7 0 is a seies. Thee ae two types of sequece/seies aithmetic ad geometic.
More informationTHE ANALYTIC LARGE SIEVE
THE ANALYTIC LAGE SIEVE 1. The aalytic lage sieve I the last lectue we saw how to apply the aalytic lage sieve to deive a aithmetic fomulatio of the lage sieve, which we applied to the poblem of boudig
More informationEVALUATION OF SUMS INVOLVING GAUSSIAN q-binomial COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS
EVALUATION OF SUMS INVOLVING GAUSSIAN -BINOMIAL COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS EMRAH KILIÇ AND HELMUT PRODINGER Abstact We coside sums of the Gaussia -biomial coefficiets with a paametic atioal
More informationConditional Convergence of Infinite Products
Coditioal Covegece of Ifiite Poducts William F. Tech Ameica Mathematical Mothly 106 1999), 646-651 I this aticle we evisit the classical subject of ifiite poducts. Fo stadad defiitios ad theoems o this
More informationFinite q-identities related to well-known theorems of Euler and Gauss. Johann Cigler
Fiite -idetities elated to well-ow theoems of Eule ad Gauss Joha Cigle Faultät fü Mathemati Uivesität Wie A-9 Wie, Nodbegstaße 5 email: oha.cigle@uivie.ac.at Abstact We give geealizatios of a fiite vesio
More informationGeneralized Near Rough Probability. in Topological Spaces
It J Cotemp Math Scieces, Vol 6, 20, o 23, 099-0 Geealized Nea Rough Pobability i Topological Spaces M E Abd El-Mosef a, A M ozae a ad R A Abu-Gdaii b a Depatmet of Mathematics, Faculty of Sciece Tata
More informationUsing Difference Equations to Generalize Results for Periodic Nested Radicals
Usig Diffeece Equatios to Geealize Results fo Peiodic Nested Radicals Chis Lyd Uivesity of Rhode Islad, Depatmet of Mathematics South Kigsto, Rhode Islad 2 2 2 2 2 2 2 π = + + +... Vieta (593) 2 2 2 =
More informationELEMENTARY AND COMPOUND EVENTS PROBABILITY
Euopea Joual of Basic ad Applied Scieces Vol. 5 No., 08 ELEMENTARY AND COMPOUND EVENTS PROBABILITY William W.S. Che Depatmet of Statistics The Geoge Washigto Uivesity Washigto D.C. 003 E-mail: williamwsche@gmail.com
More informationSome Integral Mean Estimates for Polynomials
Iteatioal Mathematical Foum, Vol. 8, 23, o., 5-5 HIKARI Ltd, www.m-hikai.com Some Itegal Mea Estimates fo Polyomials Abdullah Mi, Bilal Ahmad Da ad Q. M. Dawood Depatmet of Mathematics, Uivesity of Kashmi
More informationHypergraph Independent Sets
Uivesity of Nebasa - Licol DigitalCommos@Uivesity of Nebasa - Licol Faculty Publicatios, Depatmet of Mathematics Mathematics, Depatmet of 2013 Hypegaph Idepedet Sets Joatha Cutle Motclai State Uivesity,
More informationMATH /19: problems for supervision in week 08 SOLUTIONS
MATH10101 2018/19: poblems fo supevisio i week 08 Q1. Let A be a set. SOLUTIONS (i Pove that the fuctio c: P(A P(A, defied by c(x A \ X, is bijective. (ii Let ow A be fiite, A. Use (i to show that fo each
More informationLecture 3 : Concentration and Correlation
Lectue 3 : Cocetatio ad Coelatio 1. Talagad s iequality 2. Covegece i distibutio 3. Coelatio iequalities 1. Talagad s iequality Cetifiable fuctios Let g : R N be a fuctio. The a fuctio f : 1 2 Ω Ω L Ω
More informationOn a Problem of Littlewood
Ž. JOURAL OF MATHEMATICAL AALYSIS AD APPLICATIOS 199, 403 408 1996 ARTICLE O. 0149 O a Poblem of Littlewood Host Alze Mosbache Stasse 10, 51545 Waldbol, Gemay Submitted by J. L. Bee Received May 19, 1995
More informationDisjoint Sets { 9} { 1} { 11} Disjoint Sets (cont) Operations. Disjoint Sets (cont) Disjoint Sets (cont) n elements
Disjoit Sets elemets { x, x, } X =, K Opeatios x Patitioed ito k sets (disjoit sets S, S,, K Fid-Set(x - etu set cotaiig x Uio(x,y - make a ew set by combiig the sets cotaiig x ad y (destoyig them S k
More informationProgression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P.
Pogessio Sequece & Seies A set of umbes whose domai is a eal umbe is called a SEQUENCE ad sum of the sequece is called a SERIES. If a, a, a, a 4,., a, is a sequece, the the expessio a + a + a + a 4 + a
More informationConsider unordered sample of size r. This sample can be used to make r! Ordered samples (r! permutations). unordered sample
Uodeed Samples without Replacemet oside populatio of elemets a a... a. y uodeed aagemet of elemets is called a uodeed sample of size. Two uodeed samples ae diffeet oly if oe cotais a elemet ot cotaied
More informationThe Multivariate-t distribution and the Simes Inequality. Abstract. Sarkar (1998) showed that certain positively dependent (MTP 2 ) random variables
The Multivaiate-t distibutio ad the Simes Iequality by Hey W. Block 1, Saat K. Saka 2, Thomas H. Savits 1 ad Jie Wag 3 Uivesity of ittsbugh 1,Temple Uivesity 2,Gad Valley State Uivesity 3 Abstact. Saka
More informationThe number of r element subsets of a set with n r elements
Popositio: is The umbe of elemet subsets of a set with elemets Poof: Each such subset aises whe we pick a fist elemet followed by a secod elemet up to a th elemet The umbe of such choices is P But this
More informationOn ARMA(1,q) models with bounded and periodically correlated solutions
Reseach Repot HSC/03/3 O ARMA(,q) models with bouded ad peiodically coelated solutios Aleksade Weo,2 ad Agieszka Wy oma ska,2 Hugo Steihaus Cete, Woc aw Uivesity of Techology 2 Istitute of Mathematics,
More informationTechnical Report: Bessel Filter Analysis
Sasa Mahmoodi 1 Techical Repot: Bessel Filte Aalysis 1 School of Electoics ad Compute Sciece, Buildig 1, Southampto Uivesity, Southampto, S17 1BJ, UK, Email: sm3@ecs.soto.ac.uk I this techical epot, we
More informationIDENTITIES FOR THE NUMBER OF STANDARD YOUNG TABLEAUX IN SOME (k, l)-hooks
Sémiaie Lothaigie de Combiatoie 63 (010), Aticle B63c IDENTITIES FOR THE NUMBER OF STANDARD YOUNG TABLEAUX IN SOME (k, l)-hooks A. REGEV Abstact. Closed fomulas ae kow fo S(k,0;), the umbe of stadad Youg
More informationFIXED POINT AND HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES
IJRRAS 6 () July 0 www.apapess.com/volumes/vol6issue/ijrras_6.pdf FIXED POINT AND HYERS-UAM-RASSIAS STABIITY OF A QUADRATIC FUNCTIONA EQUATION IN BANACH SPACES E. Movahedia Behbaha Khatam Al-Abia Uivesity
More informationLecture 24: Observability and Constructibility
ectue 24: Obsevability ad Costuctibility 7 Obsevability ad Costuctibility Motivatio: State feedback laws deped o a kowledge of the cuet state. I some systems, xt () ca be measued diectly, e.g., positio
More informationICS141: Discrete Mathematics for Computer Science I
Uivesity of Hawaii ICS141: Discete Mathematics fo Compute Sciece I Dept. Ifomatio & Compute Sci., Uivesity of Hawaii Ja Stelovsy based o slides by D. Bae ad D. Still Oigials by D. M. P. Fa ad D. J.L. Goss
More informationWe are interested in the problem sending messages over a noisy channel. channel noise is behave nicely.
Chapte 31 Shao s theoem By Saiel Ha-Peled, Novembe 28, 2018 1 Vesio: 0.1 This has bee a ovel about some people who wee puished etiely too much fo what they did. They wated to have a good time, but they
More information( ) ( ) ( ) ( ) Solved Examples. JEE Main/Boards = The total number of terms in the expansion are 8.
Mathematics. Solved Eamples JEE Mai/Boads Eample : Fid the coefficiet of y i c y y Sol: By usig fomula of fidig geeal tem we ca easily get coefficiet of y. I the biomial epasio, ( ) th tem is c T ( y )
More informationIt is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.
MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study real-valued fuctios defied
More informationTaylor Transformations into G 2
Iteatioal Mathematical Foum, 5,, o. 43, - 3 Taylo Tasfomatios ito Mulatu Lemma Savaah State Uivesity Savaah, a 344, USA Lemmam@savstate.edu Abstact. Though out this pape, we assume that
More informationSHIFTED HARMONIC SUMS OF ORDER TWO
Commu Koea Math Soc 9 0, No, pp 39 55 http://dxdoiog/03/ckms0939 SHIFTED HARMONIC SUMS OF ORDER TWO Athoy Sofo Abstact We develop a set of idetities fo Eule type sums I paticula we ivestigate poducts of
More informationDisjoint Systems. Abstract
Disjoit Systems Noga Alo ad Bey Sudaov Departmet of Mathematics Raymod ad Beverly Sacler Faculty of Exact Scieces Tel Aviv Uiversity, Tel Aviv, Israel Abstract A disjoit system of type (,,, ) is a collectio
More informationMATHS FOR ENGINEERS ALGEBRA TUTORIAL 8 MATHEMATICAL PROGRESSIONS AND SERIES
MATHS FOR ENGINEERS ALGEBRA TUTORIAL 8 MATHEMATICAL PROGRESSIONS AND SERIES O completio of this ttoial yo shold be able to do the followig. Eplai aithmetical ad geometic pogessios. Eplai factoial otatio
More informationRange Symmetric Matrices in Minkowski Space
BULLETIN of the Bull. alaysia ath. Sc. Soc. (Secod Seies) 3 (000) 45-5 LYSIN THETICL SCIENCES SOCIETY Rae Symmetic atices i ikowski Space.R. EENKSHI Depatmet of athematics, amalai Uivesity, amalaiaa 608
More informationMultivector Functions
I: J. Math. Aal. ad Appl., ol. 24, No. 3, c Academic Pess (968) 467 473. Multivecto Fuctios David Hestees I a pevious pape [], the fudametals of diffeetial ad itegal calculus o Euclidea -space wee expessed
More informationEXAMPLES. Leader in CBSE Coaching. Solutions of BINOMIAL THEOREM A.V.T.E. by AVTE (avte.in) Class XI
avtei EXAMPLES Solutios of AVTE by AVTE (avtei) lass XI Leade i BSE oachig 1 avtei SHORT ANSWER TYPE 1 Fid the th tem i the epasio of 1 We have T 1 1 1 1 1 1 1 1 1 1 Epad the followig (1 + ) 4 Put 1 y
More informationProbabilities of hitting a convex hull
Pobabilities of hittig a covex hull Zhexia Liu ad Xiagfeg Yag Liköpig Uivesity Post Pit N.B.: Whe citig this wok, cite the oigial aticle. Oigial Publicatio: Zhexia Liu ad Xiagfeg Yag, Pobabilities of hittig
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationLecture 2. The Lovász Local Lemma
Staford Uiversity Sprig 208 Math 233A: No-costructive methods i combiatorics Istructor: Ja Vodrák Lecture date: Jauary 0, 208 Origial scribe: Apoorva Khare Lecture 2. The Lovász Local Lemma 2. Itroductio
More informationINVERSE CAUCHY PROBLEMS FOR NONLINEAR FRACTIONAL PARABOLIC EQUATIONS IN HILBERT SPACE
IJAS 6 (3 Febuay www.apapess.com/volumes/vol6issue3/ijas_6_3_.pdf INVESE CAUCH POBLEMS FO NONLINEA FACTIONAL PAABOLIC EQUATIONS IN HILBET SPACE Mahmoud M. El-Boai Faculty of Sciece Aleadia Uivesit Aleadia
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
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 informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so 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 informationOn Random Line Segments in the Unit Square
O Radom Lie Segmets i the Uit Square Thomas A. Courtade Departmet of Electrical Egieerig Uiversity of Califoria Los Ageles, Califoria 90095 Email: tacourta@ee.ucla.edu I. INTRODUCTION Let Q = [0, 1] [0,
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit theorems Throughout this sectio we will assume a probability space (Ω, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationEinstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,
MB BINOMIAL THEOREM Biomial Epessio : A algebaic epessio which cotais two dissimila tems is called biomial epessio Fo eample :,,, etc / ( ) Statemet of Biomial theoem : If, R ad N, the : ( + ) = a b +
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 informationMath 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs
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
More informationGeneralized Fibonacci-Lucas Sequence
Tuish Joual of Aalysis ad Numbe Theoy, 4, Vol, No 6, -7 Available olie at http://pubssciepubcom/tjat//6/ Sciece ad Educatio Publishig DOI:6/tjat--6- Geealized Fiboacci-Lucas Sequece Bijeda Sigh, Ompaash
More informationAdvanced Physical Geodesy
Supplemetal Notes Review of g Tems i Moitz s Aalytic Cotiuatio Method. Advaced hysical Geodesy GS887 Chistophe Jekeli Geodetic Sciece The Ohio State Uivesity 5 South Oval Mall Columbus, OH 4 7 The followig
More informationComplementary Dual Subfield Linear Codes Over Finite Fields
1 Complemetay Dual Subfield Liea Codes Ove Fiite Fields Kiagai Booiyoma ad Somphog Jitma,1 Depatmet of Mathematics, Faculty of Sciece, Silpao Uivesity, Naho Pathom 73000, hailad e-mail : ai_b_555@hotmail.com
More informationHitting time results for Maker-Breaker games
Hittig time esults fo Make-Beake games Exteded Abstact Soy Be-Shimo Asaf Febe Da Hefetz Michael Kivelevich Abstact We aalyze classical Make-Beake games played o the edge set of a adomly geeated gaph G.
More informationGeneralization of Horadam s Sequence
Tuish Joual of Aalysis ad Nube Theoy 6 Vol No 3-7 Available olie at http://pubssciepubco/tjat///5 Sciece ad Educatio Publishig DOI:69/tjat---5 Geealizatio of Hoada s Sequece CN Phadte * YS Valaulia Depatet
More informationOn Some Generalizations via Multinomial Coefficients
Bitish Joual of Applied Sciece & Techology 71: 1-13, 01, Aticle objast0111 ISSN: 31-0843 SCIENCEDOMAIN iteatioal wwwsciecedomaiog O Some Geealizatios via Multiomial Coefficiets Mahid M Magotaum 1 ad Najma
More information4. PERMUTATIONS AND COMBINATIONS
4. PERMUTATIONS AND COMBINATIONS PREVIOUS EAMCET BITS 1. The umbe of ways i which 13 gold cois ca be distibuted amog thee pesos such that each oe gets at least two gold cois is [EAMCET-000] 1) 3 ) 4 3)
More informationOn composite conformal mapping of an annulus to a plane with two holes
O composite cofomal mappig of a aulus to a plae with two holes Mila Batista (July 07) Abstact I the aticle we coside the composite cofomal map which maps aulus to ifiite egio with symmetic hole ad ealy
More information9.7 Pascal s Formula and the Binomial Theorem
592 Chapte 9 Coutig ad Pobability Example 971 Values of 97 Pascal s Fomula ad the Biomial Theoem I m vey well acquaited, too, with mattes mathematical, I udestad equatios both the simple ad quadatical
More informationGeneralizations and analogues of the Nesbitt s inequality
OCTOGON MATHEMATICAL MAGAZINE Vol 17, No1, Apil 2009, pp 215-220 ISSN 1222-5657, ISBN 978-973-88255-5-0, wwwhetfaluo/octogo 215 Geealiatios ad aalogues of the Nesbitt s iequalit Fuhua Wei ad Shahe Wu 19
More informationTHE ANALYSIS OF SOME MODELS FOR CLAIM PROCESSING IN INSURANCE COMPANIES
Please cite this atle as: Mhal Matalyck Tacaa Romaiuk The aalysis of some models fo claim pocessig i isuace compaies Scietif Reseach of the Istitute of Mathemats ad Compute Sciece 004 Volume 3 Issue pages
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 informationBernoulli, poly-bernoulli, and Cauchy polynomials in terms of Stirling and r-stirling numbers
Novembe 4, 2016 Beoulli, oly-beoulli, ad Cauchy olyomials i tems of Stilig ad -Stilig umbes Khisto N. Boyadzhiev Deatmet of Mathematics ad Statistics, Ohio Nothe Uivesity, Ada, OH 45810, USA -boyadzhiev@ou.edu
More informationSteiner Hyper Wiener Index A. Babu 1, J. Baskar Babujee 2 Department of mathematics, Anna University MIT Campus, Chennai-44, India.
Steie Hype Wiee Idex A. Babu 1, J. Baska Babujee Depatmet of mathematics, Aa Uivesity MIT Campus, Cheai-44, Idia. Abstact Fo a coected gaph G Hype Wiee Idex is defied as WW G = 1 {u,v} V(G) d u, v + d
More informationChapter 0. Review of set theory. 0.1 Sets
Chapter 0 Review of set theory Set theory plays a cetral role i the theory of probability. Thus, we will ope this course with a quick review of those otios of set theory which will be used repeatedly.
More informationPROGRESSION AND SERIES
INTRODUCTION PROGRESSION AND SERIES A gemet of umbes {,,,,, } ccodig to some well defied ule o set of ules is clled sequece Moe pecisely, we my defie sequece s fuctio whose domi is some subset of set of
More informationApplication to Random Graphs
A Applicatio to Radom Graphs Brachig processes have a umber of iterestig ad importat applicatios. We shall cosider oe of the most famous of them, the Erdős-Réyi radom graph theory. 1 Defiitio A.1. Let
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 informationAt the end of this topic, students should be able to understand the meaning of finite and infinite sequences and series, and use the notation u
Natioal Jio College Mathematics Depatmet 00 Natioal Jio College 00 H Mathematics (Seio High ) Seqeces ad Seies (Lecte Notes) Topic : Seqeces ad Seies Objectives: At the ed of this topic, stdets shold be
More informationLESSON 15: COMPOUND INTEREST
High School: Expoeial Fuctios LESSON 15: COMPOUND INTEREST 1. You have see this fomula fo compoud ieest. Paamete P is the picipal amou (the moey you stat with). Paamete is the ieest ate pe yea expessed
More informationOn the maximum of r-stirling numbers
Advaces i Applied Mathematics 4 2008) 293 306 www.elsevie.com/locate/yaama O the maximum of -Stilig umbes Istvá Mező Depatmet of Algeba ad Numbe Theoy, Istitute of Mathematics, Uivesity of Debece, Hugay
More informationA Proof of Birkhoff s Ergodic Theorem
A Proof of Birkhoff s Ergodic Theorem Joseph Hora September 2, 205 Itroductio I Fall 203, I was learig the basics of ergodic theory, ad I came across this theorem. Oe of my supervisors, Athoy Quas, showed
More informationON CERTAIN CLASS OF ANALYTIC FUNCTIONS
ON CERTAIN CLASS OF ANALYTIC FUNCTIONS Nailah Abdul Rahma Al Diha Mathematics Depatmet Gils College of Educatio PO Box 60 Riyadh 567 Saudi Aabia Received Febuay 005 accepted Septembe 005 Commuicated by
More informationThis web appendix outlines sketch of proofs in Sections 3 5 of the paper. In this appendix we will use the following notations: c i. j=1.
Web Appedix: Supplemetay Mateials fo Two-fold Nested Desigs: Thei Aalysis ad oectio with Nopaametic ANOVA by Shu-Mi Liao ad Michael G. Akitas This web appedix outlies sketch of poofs i Sectios 3 5 of the
More informationRandomized Algorithms I, Spring 2018, Department of Computer Science, University of Helsinki Homework 1: Solutions (Discussed January 25, 2018)
Radomized Algorithms I, Sprig 08, Departmet of Computer Sciece, Uiversity of Helsiki Homework : Solutios Discussed Jauary 5, 08). Exercise.: Cosider the followig balls-ad-bi game. We start with oe black
More informationExpected Norms of Zero-One Polynomials
DRAFT: Caad. Math. Bull. July 4, 08 :5 File: borwei80 pp. Page Sheet of Caad. Math. Bull. Vol. XX (Y, ZZZZ pp. 0 0 Expected Norms of Zero-Oe Polyomials Peter Borwei, Kwok-Kwog Stephe Choi, ad Idris Mercer
More informationGreatest term (numerically) in the expansion of (1 + x) Method 1 Let T
BINOMIAL THEOREM_SYNOPSIS Geatest tem (umeically) i the epasio of ( + ) Method Let T ( The th tem) be the geatest tem. Fid T, T, T fom the give epasio. Put T T T ad. Th will give a iequality fom whee value
More informationAlmost intersecting families of sets
Almost itersectig families of sets Dáiel Gerber a Natha Lemos b Cory Palmer a Balázs Patkós a, Vajk Szécsi b a Hugaria Academy of Scieces, Alfréd Réyi Istitute of Mathematics, P.O.B. 17, Budapest H-1364,
More informationRECIPROCAL POWER SUMS. Anthony Sofo Victoria University, Melbourne City, Australia.
#A39 INTEGERS () RECIPROCAL POWER SUMS Athoy Sofo Victoia Uivesity, Melboue City, Austalia. athoy.sofo@vu.edu.au Received: /8/, Acceted: 6//, Published: 6/5/ Abstact I this ae we give a alteative oof ad
More informationKEY. Math 334 Midterm II Fall 2007 section 004 Instructor: Scott Glasgow
KEY Math 334 Midtem II Fall 7 sectio 4 Istucto: Scott Glasgow Please do NOT wite o this exam. No cedit will be give fo such wok. Rathe wite i a blue book, o o you ow pape, pefeably egieeig pape. Wite you
More information