ë2ë P. Erdíos, Problems and results in additive number theory, in Colloque ë4ë P. Erdíos, Graph Theory and Probability II., Canad. J. Math.

Size: px
Start display at page:

Download "ë2ë P. Erdíos, Problems and results in additive number theory, in Colloque ë4ë P. Erdíos, Graph Theory and Probability II., Canad. J. Math."

Transcription

1 ëë P. Erdíos, Probles ad results i additive uber theory, i Colloque sur la Thçeorie des Nobres ècbrmè, Bruxelles, 1955, ë3ë P. Erdíos, Graph Theory ad Probability, Caad. J. Math. 11 è1959è, ë4ë P. Erdíos, Graph Theory ad Probability II., Caad. J. Math. 13 è1961è, ë5ë P. Erdíos, O circuits ad subgraphs of chroatic graphs, Matheatika 9è196è, ë6ë P. Erdíos, O a cobiatorial proble I., Nordisk. Mat. Tidskr. 11 è1963è, ë7ë P. Erdíos, O a cobiatorial proble II., Acta. Math. Acad. Sci. Hugar. 15 è1964è, ë8ë P. Erdíos ad J. W. Moo, O sets of cosistet arcs i a touraet, Caad. Math. Bull. 8 è1965è, ë9ë P. Erdíos ad A. Rçeyi, O the evolutio of rado graphs, Mat. Kutatço It. Kíozl. 5 è1960è, ë10ë P. Erdíos ad J. Specer, Ibalaces i k-coloratios, Networks 1 è197è, ë11ë P. Erdíos ad G. Szekeres, A cobiatorial proble i geoetry, Copositio Math. è1935è, ë1ë W. F. de la Vega, O the axial cardiality of a cosistet set of arcs i a rado touraet, J. Cobiatorial Theory, Series B 35 è1983è,

2 7 1965: Urakable Touraets Let T be a touraet with players 1;...; èeach pair play oe gae ad there are o tiesè ad ç a rakig of the players, techically a perutatio o f1;...;g. Call gae fi; jg a oupset if i beats j ad çèiè éçèjè; a upset if i beats j but çèjè é çèiè. The æt f èt;çè is the uber of oupsets ius the uber of upsets. Oe ight have thought - i preprobabilistic days! - that every touraet T had a rakig ç with a reasoably good æt. With J.W. Moo, Erdíos ë?ë easily destroyed that cojecture. Theore. There is a T so that for all ç fèt;çè ç 3= èl è 1= Thus, for exaple, there are touraets so that uder ay rakig at least 49è of the gaes are upsets. Erdíos ad Moo take the rado touraet, for each pair fi,jg oe ëæips a fair coi" to see who wis the gae. For ay æxed ç each gae is equally likely to be upset or oupset, ad the diæeret gaes are idepedet. Thus fèt;çè ç S, where = ad S is the uber of heads ius the uber of tails i æips of a fair coi. Large deviatio theory gives PrëS éæë ée, æ Oe ow uses very large deviatios. Set æ = 3= èl è 1= so that the above probability is less tha, é!,1. This supersall probability is used because there are! possible ç. Now with positive probability o ç have fèt;çè éæ.thus there is a T with o ç havig fèt;çè éæ. The use of extree large deviatios has becoe a aistay of the Probabilistic Method. But I have a ore persoal reaso for cocludig with this exaple. Let gèè be the least iteger so that every touraet T o players has a rakig ç with fèt;çè ç gèè. The gèè ç 3= èl è 1=. Erdíos ad Moo showed gèè éc, leavig ope the asyptotics of gèè. I y doctoral dissertatio I showed gèè éc 1 3= ad later èbut see delavega ë?ë for the ëbook proof" è that gèè éc 3=. Though at the tie I was but a æ Paul respoded with his characteristic opeess ad soo ë?ë I had a Erdíos uber of oe. Thigs have't bee the sae sice. Refereces ë1ë P. Erdíos, Soe rearks o the theory of graphs, Bull. Aer. Math. Soc. 53 è1947è,

3 Proof: Color æ radoly. Each A i has probability 1, of beig oochroatic, the probability soe A i is oochroatic is the at ost 1, é 1 so with positive probability oa i is oochroatic. Take that colorig. I 1964 Erdíos ë?ëshowed this result was close to best possible. Theore. There exists a faily A with = c which is ot -colorable. Here Erdíos turs the origial probability arguet iside out. Before the sets were æxed ad the colorig was rado, ow, essetially, the colorig is æxed ad the sets are rado. He sets æ = f1;...;ug with u a paraeter to be optiized later. Let A 1 ;...;A be rado -sets of æ. Fix a colorig ç with a red poits ad b = u, a blue poits. As A i is rado PrëçèA i è costatë =, a, u æ, + b æ æ, æ u= ç, u æ, x The secod iequality, which follows fro the covexity of æ, idicates that it is the equicolorigs that are the ost troublesoe. As the A i are idepedet Prëo A i oochroaticë ç "1, æ u=,, u æ è Now suppose u ", æ u= 1,, u æ è é 1 The expected uber of ç with o A i oochroatic is less tha oe. Therefore there is a choice of A 1 ;...;A for which o such ç exists, i.e., A is ot -colorable. Solvig, oe ay take l = d çe, l ç1, èu= è èè u, uæ, Estiatig, lè1, æè ç æ this is roughly cu = u= æ. This leads to a iterestig calculatio proble èas do ay probles ivolvig the Probabilistic Method!è í æd u so as to. The aswer turs out to be u ç = at which value ç èe l è,. Erdíos has deæed èè as the least for which there is a faily of - sets which caot be -colored. His results give æè è=èè =Oè è. Beck has iproved the lower boud to æè 1=3 è but the actual asyptotics of èè reai elusive. 9

4 cojecture that if every, say, =èl è vertices could be 3-colored the G could be 4-colored. This theore disproves that cojecture. We exaie the rado graph G ç Gè; pè with p = c=. Asithe 1957 paper PrëæèGè ç xë é è!è1, pè èxè h i x é èe=xèe,pèx,1è= x Whe c is large ad, say, x =10èl cè=c, the bracketed quatity is less tha oe so the etire quatity isoè1è ad a.s. æègè ç x ad so çègè ç c=è10 l cè. Give k Erdíos ay ow siply select c so that, with p = c=, çègè éka.s. Now for the local colorig. If soe set of ç æ vertices caot be 3- colored the there is a iial such set S with, say, jsj = i ç æ. I the restrictio Gj S every vertex v ust have degree at least 3 - otherwise oe could 3-color S,fvg by iiality ad the color v diæeretly fro its eighbors. Thus Gj S has at least 3i= edges. The probability ofg havig such as is bouded by!è æx, i! " æx e è i p 3i= ç ç ei 3 ç 3= ç c ç 3= è i 3i= i i=4 i=4, a, eployig the useful iequality bæ ç ea æ b. Pickig æ = æècè sall the b bracketed ter is always less tha oe, the etire su is oè1è, a.s. o such S exists, ad a.s. every æ vertices ay be 3-colored. Erdíos's ouetal study with Alfred Rçeyi ëo the Evolutio of Rado Graphs" ë?ë had bee copleted oly a few years before. The behavior of the basic graph fuctios such aschroatic ad clique uber were fairly well uderstood throughout the evolutio. The arguet for local colorig required a ëew idea" but the basic fraework was already i place è4: Colorig Hypergraphs Let A 1 ;...;A be -sets i a arbitrary uiverse æ. The faily A = fa 1 ;...; A g is -colorable èerdíos used the ter ëproperty B"è if there is a -colorig of the uderlyig poits æ so that o set A i is oochroatic. I 1963 Erdíos gave perhaps the quickest deostratio of the Probabilistic Method. Theoreë?ë: If é,1 the A is -colorable. 8

5 Each dètè has Bioial Distributio Bèx; pè ad so expectatio xp = æèl è so that oe ca get fairly easily EëZë =æè l è. Note this is the sae order as x. It is deæitely ot easy to show that for appropriate A; c èerdíos takes c = A,1= ad A largeè Zé 1, x with high probability. The requireet ëwith high probability" is quite severe. But ote, at least, that this is a pure probability stateet. Lets accept it ad ove o. Call a pair fi; jg çs soiled if it lies i a triagle with third vertex outside of S. At ost Z pairs are soiled so with high probability at least 1, x pairs are usoiled. Now we expose the edges of G iside S. If ay of the usoiled pairs are i G the G is good ad so the failure probability at ost è1, pè 1 è x è ée,æèpx è = o x!,1 1 A ad so G is good with high probability. Soud coplicated. Well, it is coplicated ad it is siultaeously a powerful applicatio of the Probabilistic Method ad a techical tour de force. The story has a coda: the Lovçasz Local Lea, developed i the id-1970s, gave a ew sieve ethod for showig that a set of bad evets could siultaeously ot hold. This author applied it to the rado graph Gè; pè with p = c,1= with the bad evets beig the existece of the various potetial triagles ad the idepedece of the various x-sets. The coditios of the Local Lea ade for soe calculatios but it was relatively straightforward to duplicate this result. Still, the ideas behid this proof, the subtle extesio of the Deletio Method otio, are too beautiful to be forgotte : No Local Colorig With his 1957 paper previously discussed Erdíos had already show that chroatic uber caot be cosidered siply a local pheoeo. With this result he puts the ail i the coæ. Theoreë?ë. For ay k ç 3 there is a æé0 so that the followig holds for all suæcietly large : There exists a graph G o vertices which caot be k-colored ad yet the restrictio of G to ay æ vertex subgraph ca be 3-colored. Ofte probabilistic theore are best uderstood as egative results, as couterexaples to atural cojectures. A priori, for exaple, oe ight 7

6 ever sice. We have already spoke of his 1947 paper o Rèk; kè. I his 1961 paper Erdíos ë?ë proves Rè3;kè éc k l k The upper boud Rè3;kè=Oèk èwas already apparet fro the origial Szekeres proof so the gap was relatively sall. Oly i 1994 was the correct order Rè3;kè=æè k l k è æally show. Erdíos shows that there is a graph o vertices with o triagle ad o idepedet set of size x where x = da 1= l e, ad A is a large absolute costat. This gives Rè3;xè éfro which the origial stateet follows easily. We'll igore A i our iforal discussio. He takes a rado graph Gè; pè with p = c,1=. The probability that soe x-set is idepedet is at ost è x! i x è1, pè xèx,1è= é he,pèx,1è= which isvery sall. Ufortuately this G will have lots èæè 3= èè of triagles. Oe eeds to reove a edge fro each triagle without akig ay of the x-sets idepedet. The Erdíos ethod ay be thought of algorithically. Order the edges e 1 ;...;e of G ç Gè; pè arbitrarily. Cosider the sequetially ad reject e i if it would ake a triagle with the edges previously accepted, otherwise accept e i. The graph G, so created is certaily triaglefree. What about the sets of x vertices. Call a set S of x vertices good èi G, ot G, èifit cotais a edge e which caot be exteded to a triagle with third vertex outside of S. Suppose S is good ad let e be such a edge. The S caot be idepedet ig,. If e is accepted we're clearly OK. The oly way e could be rejected is if e is part of a triagle e; e 1 ;e where the other edges have already bee accepted. But the e 1 ;e ust èas S is goodè lie i S ad agai S is ot idepedet. Call S bad if it is't good. Erdíos shows that alost always there are o bad S. Lets, say soethig occurs with high probability if its failure probability isoè xæ è,1 è. It suæces to show that a give S = f1;...;xg is good with high probability. This is the core of the arguet. We expose èto use oder teriologyè G i two phases. First we exaied the pairs fs; tg with s S; t 6 S. For each t 6 S let dètè be the uber of edges to S. Set è! dètè Z = X t6s 6

7 u was the X è, u!è, æ,, u! i=1 l, l u é è +1èè, è!u è,, éu, æ! è, æ é é è, æ è!u 1, u! é è,! æ!è, æ,, u! 1,, u,! é u e,u = Now the uber of possible choices for the u poits is è! é u éu u ad so the uber of graphs without the desired property is è,! èè, æ!! u 3 e,1+æ,ç = o as desired. Today, with large deviatio results assued beforehad, the proof ca be give i oe relatively leisurely page. May cosider this oe of the ost pleasig applicatios of the Probabilistic Method as the result sees ot to call for probability i the slightest ad earlier attepts had bee etirely costructive. The further use of large deviatios ad the itroductio of the Deletio Method greatly advaced the Probabilistic Method. Ad, ost iportat, the theore gives a iportat truth about graphs. I a rough sese the truth is a egative oe: chroatic uber caot be deteried by local cosideratios oly. é : Rasey Rè3;kè Rasey Theory was oe of Paul Erdíos's earliest iterests. The ivolveet ca be dated back to the witer of 193è33. Workig o a proble of Esther Klei, Erdíos proved his faous result that i every sequece of + 1 real ubers there is a ootoe subsequece of legth +1. At the sae tie, ad for the sae proble, George Szekeres rediscovered Rasey's Theore. Both arguets appeared i their 1935 joit paperë?ë. Bouds o the various Rasey fuctios, particularly the fuctio Rèl; kè, have fasciated Erdíos 5

8 uber ad arbitrarily high girth í i.e. o sall cycles. To ay graph theorists this seeed alost paradoxical. A graph with high girth would locally look like a tree ad trees ca easily be colored with two colors. What reaso could force such a graph to have high chroatic uber? As we'll see, there is a global reaso: çègè ç =æègè. To show çègè is large oe ëoly" has to show the oexistece of large idepedet sets. Erdíos ë?ë proved the existece of such graphs by probabilistic eas. Fix l; k, a graph is wated with çègè élad o cycles of size ç k. Fix æé 1 k, set p = æ,1 ad cosider G ç Gè; pè as!1. There are sall cycles, the expected uber of cycles of size ç k is kx i=3 èè i i pi = kx i=3 Oèèpè i è=oèè as kæ é 1. So alost surely the uber of edges i sall cycles is oèè. Also æx positive çéæ=. Set bu = 1,ç c. A set of u vertices will cotai, o average, ç ç u p= =æè æ è edges where æ =1+æ, ç é1. Further, the uber of such edges is give by a Bioial Distributio. Applyig large deviatio results, the probability of the u poits havig fewer tha half their expected uber of edges is e,cç.asæé1 this is saller tha expoetial, so oè, è so that alost surely every u poits have at least ç= edges. We eed oly that ç= é. Now Erdíos itroduces what is ow called the Deletio Method. This rado graph G alost surely has oly oèè edges i sall cycles ad every u vertices have at least edges. Take a speciæc graph G with these properties. Delete all the edges i sall cycles givig a graph G,. The certaily G, has o sall cycles. As fewer tha edges have bee deleted every u vertices of G,, which had ore tha edges i G, still have a edge. Thus the idepedece uber æèg, è ç u. But çèg, è ç æèg, è ç u ç ç As ca be arbitrarily large oe ca ow ake çèg, è ç k, copletig the proof. The use of coutig arguets becae a typographical ightare. Erdíos cosidered all graphs with precisely edges where = b 1+æ c. He eeded that alost all of the had the property that every u vertices èu as aboveè had ore tha. The uber of graphs failig that for a give set of size 4

9 evets beig utually idepedet over itegers x, settig ç l x ç 1= p x = K x where K is a large absolute costat. èfor the æitely ay x for which this is greater tha oe siply place x S.è Now fèè becoes a rado variable. For each xéywith x + y = let I xy be the idicator rado variable for x; y S. The we ay express fèè = P I xy. Fro Liearity of Expectatio Eëfèèë = X EëI xy ë= X p x p y ç K 0 l by a straightforward calculatio. Lets write ç = çèè = Eëf èèë. deviatio result. Oe shows, say, that The key igrediet is ow a large Prëfèè é 1 çë ée,cç Prëfèè é çë ée,cç where c is a positive absolute costat, ot depedet o; K or ç. This akes ituitive sese: as f èè is the su of utually idepedet rare idicator rado variables it should be roughly a Poisso distributio ad such large deviatio bouds hold for the Poisso. Now pick K so large that K 0 is so large that cç é l. Call a failure if either fèè é ç or fèè éç=. Each has probability less tha, failure probability. By the Borel-Catelli Lea èas P, covergesè alost surely there are oly a æite uber of failures ad so alost surely this rado S has the desired properties. While the origial Erdíos proof was couched i diæeret, coutig, laguage the use of large deviatio bouds ca be clearly see ad, o this cout aloe, this paper arks a otable advace i the Probabilistic Method : High Girth, High Chroatic Nuber Tutte was the ærst to show the existece of graphs with arbitrarily high chroatic uber ad o triagles, this was exteded by Kelly to arbitrarily high chroatic uber ad o cycles of sizes three, four or æve. A atural questio occured í could graphs be foud with arbitrarily high chroatic 3

10 Erdíos used a coutig arguetabove, i the ore oder laguage we would speak of the rado graph G ç Gè; pè with p = 1. The probability that G cotais a coplete graph of order k is less tha è! N,kèk,1è= é N k k k!,kèk,1è= é 1 ècalculatios as i the origial paperè ad so the probability that G or G 0 cotais a coplete graph is less tha oe so that with positive probability G does't have this property ad therefore there exists a G as desired. Erdíos has related that after lecturig o his result the probabilist J. Doob rearked ëwell, thats very ice but it really is a coutig arguet." For this result the proofs are early idetical, the probabilistic proof havig the ior advatage of avoidig the aoyig N èn,1è= factors. Erdíos writes iterchagably i the two styles. As the ethodology has progressed the probabilistic ideas have becoe ore subtle ad today it is quite rare to see a paper writte i the coutig style. We'll take the liberty of traslatig Erdíos's later results ito the ore oder style. The gap betwee k= ad 4 k for Rèk; kè reais oe of the ost vexig probles i Rasey Theory ad i the Probabilistic Method. All iproveets sice this 1947 paper have bee oly to saller order ters so that eve today li Rèk; kè 1=k could be aywhere fro p to 4, iclusive. Eve the existece of the liit has ot bee show! 1955: Sido Cojecture Let S be a set of positive itegers. Deæe fèè =f S èè as the uber of represetatios = x + y where x; y are distict eleets of S. We call S a basis if fèè é 0 for all suæcietly large. Sido, i the early 1930s, asked if there existed ëthi" bases, i particular he asked if for all positive æ there existed a basis with fèè =Oè æ è. Erdíos heard of this proble at that tie ad relates that he told Sido that he thought he could get a solutio i ëa few days". It took soewhat loger. I 1941 Erdíos ad Turça ade the stroger cojecture that there exists a basis with f èè bouded fro above by a absolute costat í a cojecture that reais ope today. I 1955 Erdíos ë?ë resolved the Sido cojecture with the followig stroger result. Theore: There exists S with fèè = æèl è. The proof is probabilistic. Deæe a rado set by Prëx Së =p x, the

11 THE ERD í OS EXISTENCE ARGUMENT Joel Specer The Probabilistic Method is ow a stadard tool i the cobiatorial toolbox but such was ot always the case. The developet of this ethodology was for ay years early etirely due to oe a: Paul Erdíos. Here we reexaie soe of his critical early papers. We begi, as all with kowledge of the æeld would expect, with the 1947 paper ë?ë givig a lower boud o the Rasey fuctio Rèk; kè. There is the a curious gap ècertaily ot reæected i Erdíos's overall atheatical publicatiosè ad our reaiig papers all were published i a sigle te year spa fro 1955 to : Rasey Rèk; kè Let us repeat the key paragraph early verbati. Erdíos deæes Rèk; lèas the least iteger so that give ay graph G of ç Rèk; lèvertices the either G cotais a coplete graph of order k or the copleet G 0 cotais a coplete graph of order l. Theore. Let k ç 3 The k= érèk; kè ç è! k, é 4 k,1 k, 1 Proof. The secod iequality was proved by Szekeres thus we oly cosider the ærst oe. Let N ç =. Clearly the uber of graphs of N vertices equals N èn,1è=.èwe cosider the vertices of the graph as distiguishable.è The uber of diæeret graphs cotaiig a coplete graph of order k is less tha è! N èn,1è= k é N k N èn,1è= kèk,1è= k! é N èn,1è= kèk,1è= sice by a siple calculatio for N ç k= ad k ç 3 N k ék! kèk,1è= But it follows iediately fro è*è that there exists a graph such that either it or its copleetary graph cotais a coplete subgraph of order k, which copletes the proof of the Theore. è*è 1

ECONOMICS. Calculating loan interest no. 3.758

ECONOMICS. Calculating loan interest no. 3.758 F A M & A N H S E E S EONOMS alculatig loa iterest o. 3.758 y Nora L. Dalsted ad Paul H. Gutierrez Quick Facts... The aual percetage rate provides a coo basis to copare iterest charges associated with

More information

arxiv:0903.5136v2 [math.pr] 13 Oct 2009

arxiv:0903.5136v2 [math.pr] 13 Oct 2009 First passage percolatio o rado graphs with fiite ea degrees Shakar Bhaidi Reco va der Hofstad Gerard Hooghiestra October 3, 2009 arxiv:0903.536v2 [ath.pr 3 Oct 2009 Abstract We study first passage percolatio

More information

CHAPTER 4: NET PRESENT VALUE

CHAPTER 4: NET PRESENT VALUE EMBA 807 Corporate Fiace Dr. Rodey Boehe CHAPTER 4: NET PRESENT VALUE (Assiged probles are, 2, 7, 8,, 6, 23, 25, 28, 29, 3, 33, 36, 4, 42, 46, 50, ad 52) The title of this chapter ay be Net Preset Value,

More information

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

More information

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

More information

A probabilistic proof of a binomial identity

A probabilistic proof of a binomial identity A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

More information

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13 EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

More information

Department of Computer Science, University of Otago

Department of Computer Science, University of Otago Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

The Binomial Multi- Section Transformer

The Binomial Multi- Section Transformer 4/15/21 The Bioial Multisectio Matchig Trasforer.doc 1/17 The Bioial Multi- Sectio Trasforer Recall that a ulti-sectio atchig etwork ca be described usig the theory of sall reflectios as: where: Γ ( ω

More information

Irreducible polynomials with consecutive zero coefficients

Irreducible polynomials with consecutive zero coefficients Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem

More information

5 Boolean Decision Trees (February 11)

5 Boolean Decision Trees (February 11) 5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected

More information

Convexity, Inequalities, and Norms

Convexity, Inequalities, and Norms Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for

More information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

Soving Recurrence Relations

Soving Recurrence Relations Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

More information

Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy

More information

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5

More information

The Stable Marriage Problem

The Stable Marriage Problem The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Case Study. Normal and t Distributions. Density Plot. Normal Distributions Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca

More information

The second difference is the sequence of differences of the first difference sequence, 2

The second difference is the sequence of differences of the first difference sequence, 2 Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for

More information

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample

More information

Incremental calculation of weighted mean and variance

Incremental calculation of weighted mean and variance Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

More information

Sequences II. Chapter 3. 3.1 Convergent Sequences

Sequences II. Chapter 3. 3.1 Convergent Sequences Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,

More information

MARTINGALES AND A BASIC APPLICATION

MARTINGALES AND A BASIC APPLICATION MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this

More information

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006 Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam

More information

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here). BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

More information

Section 11.3: The Integral Test

Section 11.3: The Integral Test Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

More information

Recursion and Recurrences

Recursion and Recurrences Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,

More information

CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

More information

4.1 Sigma Notation and Riemann Sums

4.1 Sigma Notation and Riemann Sums 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

More information

Chapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity

More information

GENERATING A FRACTAL SQUARE

GENERATING A FRACTAL SQUARE GENERATING A FRACTAL SQUARE I 194 the Swedish mathematicia Helge vo Koch(187-194 itroduced oe o the earliest ow ractals, amely, the Koch Sowlae. It is a closed cotiuous curve with discotiuities i its derivative

More information

Factors of sums of powers of binomial coefficients

Factors of sums of powers of binomial coefficients ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the

More information

3. Covariance and Correlation

3. Covariance and Correlation Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics

More information

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009) 18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the

More information

Determining the sample size

Determining the sample size Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

More information

CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homework 4 Solutions CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

More information

5: Introduction to Estimation

5: Introduction to Estimation 5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample

More information

An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process

An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process A example of o-queched covergece i the coditioal cetral limit theorem for partial sums of a liear process Dalibor Volý ad Michael Woodroofe Abstract A causal liear processes X,X 0,X is costructed for which

More information

Notes on exponential generating functions and structures.

Notes on exponential generating functions and structures. Notes o expoetial geeratig fuctios ad structures. 1. The cocept of a structure. Cosider the followig coutig problems: (1) to fid for each the umber of partitios of a -elemet set, (2) to fid for each the

More information

Lecture 2: Karger s Min Cut Algorithm

Lecture 2: Karger s Min Cut Algorithm priceto uiv. F 3 cos 5: Advaced Algorithm Desig Lecture : Karger s Mi Cut Algorithm Lecturer: Sajeev Arora Scribe:Sajeev Today s topic is simple but gorgeous: Karger s mi cut algorithm ad its extesio.

More information

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

More information

Asymptotic Growth of Functions

Asymptotic Growth of Functions CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

More information

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

More information

7. Sample Covariance and Correlation

7. Sample Covariance and Correlation 1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y

More information

Output Analysis (2, Chapters 10 &11 Law)

Output Analysis (2, Chapters 10 &11 Law) B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

More information

A Comparison of Hypothesis Testing Methods for the Mean of a Log-Normal Distribution

A Comparison of Hypothesis Testing Methods for the Mean of a Log-Normal Distribution World Applied Scieces Joural (6): 845-849 ISS 88-495 IDOSI Publicatios A Copariso of Hypothesis Testig ethods for the ea of a og-oral Distributio 3 F. egahdari K. Abdollahezhad ad A.A. Jafari Islaic Azad

More information

Distributed Storage Allocations for Optimal Delay

Distributed Storage Allocations for Optimal Delay Distributed Storage Allocatios for Optial Delay Derek Leog Departet of Electrical Egieerig Califoria Istitute of echology Pasadea, Califoria 925, USA derekleog@caltechedu Alexadros G Diakis Departet of

More information

Lecture 4: Cheeger s Inequality

Lecture 4: Cheeger s Inequality Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a d-regular

More information

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,

More information

GSR: A Global Stripe-based Redistribution Approach to Accelerate RAID-5 Scaling

GSR: A Global Stripe-based Redistribution Approach to Accelerate RAID-5 Scaling : A Global -based Redistributio Approach to Accelerate RAID-5 Scalig Chetao Wu ad Xubi He Departet of Electrical & Coputer Egieerig Virgiia Coowealth Uiversity {wuc4,xhe2}@vcu.edu Abstract Uder the severe

More information

Class Meeting # 16: The Fourier Transform on R n

Class Meeting # 16: The Fourier Transform on R n MATH 18.152 COUSE NOTES - CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,

More information

Hypothesis testing. Null and alternative hypotheses

Hypothesis testing. Null and alternative hypotheses Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

More information

Trading the randomness - Designing an optimal trading strategy under a drifted random walk price model

Trading the randomness - Designing an optimal trading strategy under a drifted random walk price model Tradig the radomess - Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore

More information

The Computational Rise and Fall of Fairness

The Computational Rise and Fall of Fairness Proceedigs of the Twety-Eighth AAAI Coferece o Artificial Itelligece The Coputatioal Rise ad Fall of Fairess Joh P Dickerso Caregie Mello Uiversity dickerso@cscuedu Joatha Golda Caregie Mello Uiversity

More information

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The

More information

Solving Inequalities

Solving Inequalities Solvig Iequalities Say Thaks to the Authors Click http://www.ck12.org/saythaks (No sig i required) To access a customizable versio of this book, as well as other iteractive cotet, visit www.ck12.org CK-12

More information

Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown

Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown Z-TEST / Z-STATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large T-TEST / T-STATISTIC: used to test hypotheses about

More information

Elementary Theory of Russian Roulette

Elementary Theory of Russian Roulette Elemetary Theory of Russia Roulette -iterestig patters of fractios- Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some

More information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties of MLE: consistency, asymptotic normality. Fisher information. Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

More information

3 Basic Definitions of Probability Theory

3 Basic Definitions of Probability Theory 3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio

More information

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria

More information

Building Blocks Problem Related to Harmonic Series

Building Blocks Problem Related to Harmonic Series TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite

More information

Integer programming solution methods. Exactly where on this line this optimal solution lies we do not know, but it must be somewhere!

Integer programming solution methods. Exactly where on this line this optimal solution lies we do not know, but it must be somewhere! Iteger prograig solutio ethods J E Beasley Itroductio Suppose that we have soe proble istace of a cobiatorial optiisatio proble ad further suppose that it is a iiisatio proble. If, as i Figure 1, we draw

More information

THE HEIGHT OF q-binary SEARCH TREES

THE HEIGHT OF q-binary SEARCH TREES THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average

More information

1. C. The formula for the confidence interval for a population mean is: x t, which was

1. C. The formula for the confidence interval for a population mean is: x t, which was s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value

More information

On Generalized Tian Ji s Horse Racing Strategy

On Generalized Tian Ji s Horse Racing Strategy Source: Iterdiscipliary Sciece Reviews, Vol. 37, o., pp. 87-93, ; DOI:.79/3888Z.4 O Geeralized Tia Ji s Horse Racig Strategy Jia-Ju SHU School of echaical & Aerospace gieerig, ayag Techological Uiversity,

More information

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means) CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

More information

4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then

4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or p-series (the Compariso Test), but of

More information

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

More information

MINIMUM VERTEX DEGREE THRESHOLD FOR LOOSE HAMILTON CYCLES IN 3-UNIFORM HYPERGRAPHS

MINIMUM VERTEX DEGREE THRESHOLD FOR LOOSE HAMILTON CYCLES IN 3-UNIFORM HYPERGRAPHS MINIMUM VERTEX DEGREE THRESHOLD FOR LOOSE HAMILTON CYCLES IN 3-UNIFORM HYPERGRAPHS JIE HAN AND YI ZHAO Abstract. We show that for sufficiently large n, every 3-unifor hypergraph on n vertices with iniu

More information

Overview of some probability distributions.

Overview of some probability distributions. Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability

More information

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find 1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.

More information

Learning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr.

Learning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr. Algorithms ad Data Structures Algorithm efficiecy Learig outcomes Able to carry out simple asymptotic aalysisof algorithms Prof. Dr. Qi Xi 2 Time Complexity Aalysis How fast is the algorithm? Code the

More information

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio

More information

Confidence Intervals for One Mean

Confidence Intervals for One Mean Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

More information

Hypergeometric Distributions

Hypergeometric Distributions 7.4 Hypergeometric Distributios Whe choosig the startig lie-up for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you

More information

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.

More information

Modified Line Search Method for Global Optimization

Modified Line Search Method for Global Optimization Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o

More information

Time Value of Money. First some technical stuff. HP10B II users

Time Value of Money. First some technical stuff. HP10B II users Time Value of Moey Basis for the course Power of compoud iterest $3,600 each year ito a 401(k) pla yields $2,390,000 i 40 years First some techical stuff You will use your fiacial calculator i every sigle

More information

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This

More information

Ant Colony Algorithm Based Scheduling for Handling Software Project Delay

Ant Colony Algorithm Based Scheduling for Handling Software Project Delay At Coloy Algorith Based Schedulig for Hadlig Software Project Delay Wei Zhag 1,2, Yu Yag 3, Juchao Xiao 4, Xiao Liu 5, Muhaad Ali Babar 6 1 School of Coputer Sciece ad Techology, Ahui Uiversity, Hefei,

More information

Entropy of bi-capacities

Entropy of bi-capacities Etropy of bi-capacities Iva Kojadiovic LINA CNRS FRE 2729 Site école polytechique de l uiv. de Nates Rue Christia Pauc 44306 Nates, Frace iva.kojadiovic@uiv-ates.fr Jea-Luc Marichal Applied Mathematics

More information

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern. 5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers

More information

Ramsey-type theorems with forbidden subgraphs

Ramsey-type theorems with forbidden subgraphs Ramsey-type theorems with forbidde subgraphs Noga Alo Jáos Pach József Solymosi Abstract A graph is called H-free if it cotais o iduced copy of H. We discuss the followig questio raised by Erdős ad Hajal.

More information

Supply Chain Network Design with Preferential Tariff under Economic Partnership Agreement

Supply Chain Network Design with Preferential Tariff under Economic Partnership Agreement roceedigs of the 2014 Iteratioal oferece o Idustrial Egieerig ad Oeratios Maageet Bali, Idoesia, Jauary 7 9, 2014 Suly hai Network Desig with referetial ariff uder Ecooic artershi greeet eichi Fuaki Yokohaa

More information

2-3 The Remainder and Factor Theorems

2-3 The Remainder and Factor Theorems - The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x

More information

Chapter 7 Methods of Finding Estimators

Chapter 7 Methods of Finding Estimators Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

More information

Chapter 5: Inner Product Spaces

Chapter 5: Inner Product Spaces Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples

More information

On the L p -conjecture for locally compact groups

On the L p -conjecture for locally compact groups Arch. Math. 89 (2007), 237 242 c 2007 Birkhäuser Verlag Basel/Switzerlad 0003/889X/030237-6, ublished olie 2007-08-0 DOI 0.007/s0003-007-993-x Archiv der Mathematik O the L -cojecture for locally comact

More information

Solving Logarithms and Exponential Equations

Solving Logarithms and Exponential Equations Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:

More information

Exploratory Data Analysis

Exploratory Data Analysis 1 Exploratory Data Aalysis Exploratory data aalysis is ofte the rst step i a statistical aalysis, for it helps uderstadig the mai features of the particular sample that a aalyst is usig. Itelliget descriptios

More information

THE ABRACADABRA PROBLEM

THE ABRACADABRA PROBLEM THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected

More information

Numerical Analysis for Characterization of a Salty Water Meter

Numerical Analysis for Characterization of a Salty Water Meter Nuerical Aalysis for Characterizatio of a Salty Water Meter José Erique Salias Carrillo Departaeto de Ciecias Básicas Istituto Tecológico de Tehuacá Bolio Arago Perdoo Departaeto de Mecatróica Istituto

More information

Project Deliverables. CS 361, Lecture 28. Outline. Project Deliverables. Administrative. Project Comments

Project Deliverables. CS 361, Lecture 28. Outline. Project Deliverables. Administrative. Project Comments Project Deliverables CS 361, Lecture 28 Jared Saia Uiversity of New Mexico Each Group should tur i oe group project cosistig of: About 6-12 pages of text (ca be loger with appedix) 6-12 figures (please

More information

Sequences and Series

Sequences and Series CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

More information

19 Another Look at Differentiability in Quadratic Mean

19 Another Look at Differentiability in Quadratic Mean 19 Aother Look at Differetiability i Quadratic Mea David Pollard 1 ABSTRACT This ote revisits the delightfully subtle itercoectios betwee three ideas: differetiability, i a L 2 sese, of the square-root

More information

Unit 20 Hypotheses Testing

Unit 20 Hypotheses Testing Uit 2 Hypotheses Testig Objectives: To uderstad how to formulate a ull hypothesis ad a alterative hypothesis about a populatio proportio, ad how to choose a sigificace level To uderstad how to collect

More information

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5 Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

More information