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


 Asher Barber
 2 years ago
 Views:
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 kcoloratios, 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 3colored the G could be 4colored. 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 3color 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 3colored. 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 id1970s, 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 xsets. 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 kcolored ad yet the restrictio of G to ay æ vertex subgraph ca be 3colored. 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 xset 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 xsets 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 BorelCatelli 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
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 informationarxiv: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 informationRADICALS AND SOLVING QUADRATIC EQUATIONS
RADICALS AND SOLVING QUADRATIC EQUATIONS Evaluate Roots Overview of Objectives, studets should be able to:. Evaluate roots a. Siplify expressios of the for a b. Siplify expressios of the for a. Evaluate
More informationCHAPTER 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 informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo 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 informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, 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= 1. n n 2 )= n n 2 σ2 = σ2
SAMLE STATISTICS A rado saple of size fro a distributio f(x is a set of rado variables x 1,x,,x which are idepedetly ad idetically distributed with x i f(x for all i Thus, the joit pdf of the rado saple
More informationA 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 informationDiscrete 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 informationrepresented by 4! different arrangements of boxes, divide by 4! to get ways
Problem Set #6 solutios A juggler colors idetical jugglig balls red, white, ad blue (a I how may ways ca this be doe if each color is used at least oce? Let us preemptively color oe ball i each color,
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationIn 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 informationCycloidal Areas without Calculus
TOM M APOSTOL ad MAMIKON A MNATSAKANIAN Cycloidal Areas without Calculus Itroductio For ceturies atheaticias have bee iterested i curves that ca be costructed by siple echaical istruets Aog these curves
More informationThe Binomial Multi Section Transformer
4/15/21 The Bioial Multisectio Matchig Trasforer.doc 1/17 The Bioial Multi Sectio Trasforer Recall that a ultisectio atchig etwork ca be described usig the theory of sall reflectios as: where: Γ ( ω
More informationIrreducible 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 information5 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 informationAuxiliary Linear Program
Auxiliary Liear Progra L: LP i stadard for: ax j=1 s.t. j =1 L aux : Auxiliary LP: c j x j a ij x j b i for i=1, 2,..., x j 0 for j=1, 2,..., ax x 0 s.t. j =1 L aux is bouded ad feasible. a ij x j x 0
More informationConvexity, 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 information1 The Binomial Theorem: Another Approach
The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationChapter 3. Compound Interest. Section 2 Compound and Continuous Compound Interest. Solution. Example
Chapter 3 Matheatics of Fiace Sectio 2 Copoud ad Cotiuous Copoud Iterest Copoud Iterest Ulike siple iterest, copoud iterest o a aout accuulates at a faster rate tha siple iterest. The basic idea is that
More informationSoving 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 informationNonlife insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
Nolife 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 informationCase 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 informationWeek 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 informationSECTION 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 informationThe 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 informationIncremental 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 informationDUBLIN INSTITUTE OF TECHNOLOGY KEVIN STREET, DUBLIN 8. Probability Based Learning: Introduction to Probability REVISION QUESTIONS *** SOLUTIONS ***
DUBLIN INSTITUTE OF TECHNOLOGY KEVIN STREET, DUBLIN 8 Probability Based Learig: Itroductio to Probability REVISION QUESTIONS *** *** MACHINE LEARNING AT DIT Dr. Joh Kelleher Dr. Bria Mac Namee *** ***
More information1. Solving simple equations 2. Evaluation and transposition of formulae
Algebra Matheatics Worksheet This is oe of a series of worksheets desiged to hel you icrease your cofidece i hadlig Matheatics. This worksheet cotais both theory ad eercises which cover:. Solvig sile
More informationThe 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 informationMATH 361 Homework 9. Royden Royden Royden
MATH 61 Homework 9 Royde..9 First, we show that for ay subset E of the real umbers, E c + y = E + y) c traslatig the complemet is equivalet to the complemet of the traslated set). Without loss of geerality,
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationLecture 7: Borel Sets and Lebesgue Measure
EE50: Probability Foudatios for Electrical Egieers JulyNovember 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture,
More informationExample 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 informationSequences 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 informationA Gentle Introduction to Algorithms: Part II
A Getle Itroductio to Algorithms: Part II Cotets of Part I:. Merge: (to merge two sorted lists ito a sigle sorted list.) 2. Bubble Sort 3. Merge Sort: 4. The BigO, BigΘ, BigΩ otatios: asymptotic bouds
More informationSection 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 information4.1 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 informationTHE 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 informationRecursion 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 informationMath Discrete Math Combinatorics MULTIPLICATION PRINCIPLE:
Math 355  Discrete Math 4.14.4 Combiatorics Notes MULTIPLICATION PRINCIPLE: If there m ways to do somethig ad ways to do aother thig the there are m ways to do both. I the laguage of set theory: Let
More informationUC 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 informationChapter 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 informationx(x 1)(x 2)... (x k + 1) = [x] k n+m 1
1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,
More informationCHAPTER 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 informationThe Field of Complex Numbers
The Field of Complex Numbers S. F. Ellermeyer The costructio of the system of complex umbers begis by appedig to the system of real umbers a umber which we call i with the property that i = 1. (Note that
More information0,1 is an accumulation
Sectio 5.4 1 Accumulatio Poits Sectio 5.4 BolzaoWeierstrass ad HeieBorel Theorems Purpose of Sectio: To itroduce the cocept of a accumulatio poit of a set, ad state ad prove two major theorems of real
More informationMath 475, Problem Set #6: Solutions
Math 475, Problem Set #6: Solutios A (a) For each poit (a, b) with a, b oegative itegers satisfyig ab 8, cout the paths from (0,0) to (a, b) where the legal steps from (i, j) are to (i 2, j), (i, j 2),
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measuretheoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More informationA CHARACTERIZATION OF MINIMAL ZEROSEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A27 A CHARACTERIZATION OF MINIMAL ZEROSEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS Scott T. Chapma 1 Triity Uiversity, Departmet
More informationStandard Errors and Confidence Intervals
Stadard Errors ad Cofidece Itervals Itroductio I the documet Data Descriptio, Populatios ad the Normal Distributio a sample had bee obtaied from the populatio of heights of 5yearold boys. If we assume
More information5: 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 informationGENERATING A FRACTAL SQUARE
GENERATING A FRACTAL SQUARE I 194 the Swedish mathematicia Helge vo Koch(187194 itroduced oe o the earliest ow ractals, amely, the Koch Sowlae. It is a closed cotiuous curve with discotiuities i its derivative
More information3. 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 informationAn example of nonquenched convergence in the conditional central limit theorem for partial sums of a linear process
A example of oqueched 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 informationLecture 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 BruMikowski iequality for boxes. Today we ll go over the
More information2.3. GEOMETRIC SERIES
6 CHAPTER INFINITE SERIES GEOMETRIC SERIES Oe of the most importat types of ifiite series are geometric series A geometric series is simply the sum of a geometric sequece, Fortuately, geometric series
More informationDetermining 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 informationCS103X: 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 sixfigure salaries i whole dollar amouts are there that cotai at least
More informationAsymptotic 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 informationORDERS OF GROWTH KEITH CONRAD
ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus
More informationFactors 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 informationCS103A 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 informationN04/5/MATHL/HP2/ENG/TZ0/XX MATHEMATICS HIGHER LEVEL PAPER 2. Thursday 4 November 2004 (morning) 3 hours INSTRUCTIONS TO CANDIDATES
c IB MATHEMATICS HIGHER LEVEL PAPER DIPLOMA PROGRAMME PROGRAMME DU DIPLÔME DU BI PROGRAMA DEL DIPLOMA DEL BI N/5/MATHL/HP/ENG/TZ/XX 887 Thursday November (morig) hours INSTRUCTIONS TO CANDIDATES! Do ot
More informationSAMPLE 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 informationConfidence Intervals for One Mean with Tolerance Probability
Chapter 421 Cofidece Itervals for Oe Mea with Tolerace Probability Itroductio This procedure calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) with
More information7. 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 informationDistributed 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 informationElementary 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 informationOutput 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 informationA Comparison of Hypothesis Testing Methods for the Mean of a LogNormal Distribution
World Applied Scieces Joural (6): 845849 ISS 88495 IDOSI Publicatios A Copariso of Hypothesis Testig ethods for the ea of a ogoral Distributio 3 F. egahdari K. Abdollahezhad ad A.A. Jafari Islaic Azad
More information3. Continuous Random Variables
Statistics ad probability: 31 3. Cotiuous Radom Variables A cotiuous radom variable is a radom variable which ca take values measured o a cotiuous scale e.g. weights, stregths, times or legths. For ay
More informationNotes 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 information8.5 Alternating infinite series
65 8.5 Alteratig ifiite series I the previous two sectios we cosidered oly series with positive terms. I this sectio we cosider series with both positive ad egative terms which alterate: positive, egative,
More informationLecture 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 informationg x is a generator polynomial and generates a cyclic code.
Epress, a Iteratioal Joural of Multi Discipliary Research ISSN: 2348 2052, Vol, Issue 6, Jue 204 Available at: wwwepressjouralco Abstract ENUMERATION OF CYCLIC CODES OVER GF (5) By Flora Mati Ruji Departet
More informationLecture Notes CMSC 251
We have this messy summatio to solve though First observe that the value remais costat throughout the sum, ad so we ca pull it out frot Also ote that we ca write 3 i / i ad (3/) i T () = log 3 (log ) 1
More informationThe geometric series and the ratio test
The geometric series ad the ratio test Today we are goig to develop aother test for covergece based o the iterplay betwee the it compariso test we developed last time ad the geometric series. A ote about
More informationif A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S,
Lecture 5: Borel Sets Topologically, the Borel sets i a topological space are the σalgebra geerated by the ope sets. Oe ca build up the Borel sets from the ope sets by iteratig the operatios of complemetatio
More informationSequences, Series and Convergence with the TI 92. Roger G. Brown Monash University
Sequeces, Series ad Covergece with the TI 92. Roger G. Brow Moash Uiversity email: rgbrow@deaki.edu.au Itroductio. Studets erollig i calculus at Moash Uiversity, like may other calculus courses, are itroduced
More informationHypothesis 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 informationLecture 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 dregular
More informationHere 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 informationTHE 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 informationB1. Fourier Analysis of Discrete Time Signals
B. Fourier Aalysis of Discrete Time Sigals Objectives Itroduce discrete time periodic sigals Defie the Discrete Fourier Series (DFS) expasio of periodic sigals Defie the Discrete Fourier Trasform (DFT)
More informationTrading 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 informationSolving 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 CK12
More informationClass 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 informationThe Computational Rise and Fall of Fairness
Proceedigs of the TwetyEighth 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 informationwhere: 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 informationEconomics 140A Confidence Intervals and Hypothesis Testing
Ecoomics 140A Cofidece Itervals ad Hypothesis Testig Obtaiig a estimate of a parameter is ot the al purpose of statistical iferece because it is highly ulikely that the populatio value of a parameter is
More informationDivide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016
CS 6, Lecture 3 Divide ad Coquer, Solvig Recurreces, Iteger Multiplicatio Scribe: Juliaa Cook (05, V Williams Date: April 6, 06 Itroductio Today we will cotiue to talk about divide ad coquer, ad go ito
More informationProperties 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 informationGSR: A Global Stripebased Redistribution Approach to Accelerate RAID5 Scaling
: A Global based Redistributio Approach to Accelerate RAID5 Scalig Chetao Wu ad Xubi He Departet of Electrical & Coputer Egieerig Virgiia Coowealth Uiversity {wuc4,xhe2}@vcu.edu Abstract Uder the severe
More informationTime 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 informationRiemann Sums y = f (x)
Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, oegative fuctio o the closed iterval [a, b] Fid
More informationBuilding 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 informationARITHMETIC AND GEOMETRIC PROGRESSIONS
Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More information