Optimal Expected Rank in a Two-Sided Secretary Problem
|
|
- Amber McKenzie
- 6 years ago
- Views:
Transcription
1 OPERATIOS RESEARCH Vol. 55, o. 5, Septembe Octobe 007, pp issn X eissn infoms doi 0.87/ope IFORMS Optimal Expected Rank in a Two-Sided Secetay Poblem Kimmo Eiksson, Jonas Sjöstand, Pontus Stimling Depatment of Mathematics and Physics, Mäladalen Univesity, SE-7 3 Västeås, Sweden {kimmo.eiksson@mdh.se, jonas.sjostand@mdh.se, pontus.stimling@mdh.se} In a two-sided vesion of the famous secetay poblem, employes seach fo a secetay at the same time as secetaies seach fo an employe. obody accepts being put on hold, and nobody is willing to take pat in moe than inteviews. Pefeences ae independent, and agents seek to optimize the expected ank of the patne they obtain among the potential patnes. We find that in any subgame pefect equilibium, the expected ank gows as the squae oot of wheeas it tends to a constant in the oiginal secetay poblem). We also compute how much agents can gain by coopeation. Subject classifications: games/goup decisions: stategic secetay poblem; dynamic pogamming/optimal contol: optimal stopping. Aea of eview: Optimization. Histoy: Received June 005; evisions eceived Mach 006, June 006; accepted June 006. Published online in Aticles in Advance. July 0, Intoduction Two-sided matching has been studied intensely in the last few decades. This aea of eseach has been motivated by impotant eal-life poblems of matching employes with job seekes, o colleges with potential students, o men with women. In game theoy, the standad appoach is to study stable matchings Roth and Sotomayo 990). Matchings ae stable if no pai of agents would pefe to leave thei cuent patnes fo each othe. Accoding to theoy, a stable two-sided matching can always be found assuming that all pefeences ae known. Howeve, thee ae many situations whee agents do not know thei own pefeences fom the beginning. Fo example, an employe may base his pefeences on inteviews with applicants. If thee ae too many candidates, the employe will not be able to inteview moe than a few of them. Job seekes may have altenative employes to inteview with, and the employe may not be able to put a easonably good candidate on hold fo long, while seaching fo a bette candidate. In these types of situations whee only a small potion of pefeences will eve be evealed, it does not make sense to speak about the best oveall matching. Instead, we focus on how agents optimize thei seach stategies. A simple vesion of this poblem involves a single employe willing to inteview at most secetaial candidates, who all want the job but will not be put on hold. This is the famous secetay poblem in optimal stopping theoy. Fo the histoy of this poblem, see Feguson 989.) The employe knows that if he wee to inteview all candidates, he could ank ode them. Howeve, he must stop at some point and hie this peson without having seen the emaining candidates. When to stop depends on what measue the employe is tying to optimize; we will assume this measue to be the expected ank of the hied secetay among all candidates. Lindley 96) showed that as gows, the optimal expected ank tends to a constant slightly less than fou. Thus, even with a line of thousands of candidates, the employe has a stopping stategy so that he can expect to obtain the fouth anked candidate. The exact fomula fo this constant was poved by Chow et al. 964). In this pape, we extend thei methods to solve Lindley s poblem fo the moe ealistic two-sided case whee both employes and candidates ae choosy Rittaud 005). We find that in this case, the optimal stopping stategy leads to agents obtaining patnes with an expected ank of among the potential patnes any agent is willing to evaluate. Fo example, if evey employe and job seeke is willing to take pat in at most 00 inteviews, then they can expect to obtain a patne at the tenth best pecentile. Of couse, ou esults depend on details of the maket. We make the necessay assumptions simple to have a welldefined and solvable poblem. All agents ae ational and intelligent, i.e., assume othe agents to be ational as well. Pefeences ae independent; the favoite of one agent is no moe likely than anyone else to be favoed by anothe agent. Agents ae eithe coopeative o noncoopeative: if eveyone is coopeative it gives a highe on aveage payoff fo all. We compae the espective outcomes and obtain a measue of the cost to society fo noncoopeation in this kind of game. 9
2 9 Opeations Reseach 555), pp. 9 93, 007 IFORMS In the following, we will adhee to the popula tadition in two-sided matching of descibing the model in tems of mutual mate selection of males and females cf. Roth and Sotomayo 990). Indeed, mate seach models elated to ous ae aleady studied in theoetical biology, although not in connection with Lindley s poblem. Fo a suvey of such models, see Alpen et al. 005.) The outline of this pape is as follows. In, we descibe the fomal model and summaize ou esults. Sections 3 and 4 descibe the solutions to the coopeative and the noncoopeative vesions of the poblem. In 5, we discuss aspects of the model, the method, and the esults. Thee appendices contain technical pats of the poofs.. Model, Appoach, and Summay of Results We assume a lage univese of men and women who will seach fo a mate fo a maximum of peiods. In each peiod, evey agent will date an agent of the opposite sex. If they both agee on maying, they leave the game. Othewise, they poceed to the next peiod and will neve date each othe again. In the th peiod, agents must may. The pefeences of agents ae geneated by the following pocess. Fom the viewpoint of any agent in peiod, they ank thei cuent date elative to the patnes aleady obseved. The ank of the th patne is a andom vaiable dawn fom a unifom distibution on the set of possible anks to ). Let us call this the unifomity assumption. All these andom vaiables ae assumed to be independent, meaning that how agent A anks agent B caies no infomation about how A anks futue dates, no about the anks of any othe agent. Let us call this the independence assumption. This implies that no conclusions can be dawn fom an agent s histoy of being ejected o poposed to. As in Lindley s 96) vaiant of the secetay poblem, we assume that each agent wants to minimize the expected ank of the mate among the patnes the agent would meet if she completed all peiods. Although the actual set of mates that an agent would meet in the emaining peiods is not known, we can easily compute the expected final ank fo a mate who is anked R among the patnes obseved up to peiod. Accoding to the unifomity and independence assumptions, the expected ank of this mate afte one moe date is R + R R + + R = R whee the fist tem coesponds to the case whee the next date is wose anked than R and the second tem coesponds to when it is bette anked. It follows immediately by epeating the same computation fo peiods + + ) that the expected final ank of the cuent mate is R = + + R A stategy in this two-sided secetay game is a ule that says whethe to popose maiage in peiod to a date of obseved ank R. Remembe that the agent cannot be cetain that the date will agee on maying. If agents on the othe side always agee, then we would be back in the one-sided secetay poblem. Given all agents stategies, we can define R end as the expected final ank of a cetain agent enteing peiod. In the last peiod, eveyone can expect to obtain an aveage mate: R end = + /. Agents want to choose stategies that minimize the expected final ank at the stat of the game: R end... Appoach and Summay of Results Denote pobability and expected value by P and E, espectively. Fom the definition of the model, it is evident that the following fundamental ecuence govens the expected final ank when an agent entes peiod : R end = Pmay + ER may + + Pnot may R end + ) whee R denotes the ank of the date in peiod among the patnes obseved up to that point. This ecuence will be the basis fo ou analysis. In any given peiod, agents face the following basic game. Two agents, Playe, and Playe meet. Thee exist some utilities, u and u, fo Playe of stopping with Playe and vice vesa. These ae based on thei anking of each othe. These values ae pivate infomation. If the playes do not both agee to stop in this peiod, they must go on seaching. By the popeties of the model and assuming that both agents continue to play optimally, they can then expect the same utility c = R end +. See Table. The basic game can be tackled eithe coopeatively o noncoopeatively by the playes. We will discuss below what this will entail and what esults we obtain in the diffeent cases. oncoopeation. The optimal noncoopeative stategy is fo Playe i to accept only if u i c fo i =. In the th and last peiod, evey agent who is still in the game will accept maiage because it is always pefeable to not maying at all). In the next to last peiod, an agent will want to may if his o he cuent date is at least as good as what he o she would expect to obtain in the last peiod. Solutions to games obtained in this way by backwad induction ae called subgame pefect equilibia Selten 975). If at some point an agent is indiffeent between maying a Table. Payoff matix fo the basic game in a given peiod. Accept Don t accept Accept u u cc Don t accept cc cc
3 Opeations Reseach 555), pp. 9 93, 007 IFORMS 93 mate o poceeding to the next peiod, we get multiple subgame pefect equilibia. Howeve, the expected final ank is the same in both cases, so all subgame pefect equilibia yield the same value of R end. Fo simplicity, we theefoe efe to the paticula subgame pefect stategy whee agents always may if indiffeent as the optimal stategy, which is the same fo all agents in this subgame pefect equilibium. In 4, we will pove the following esult. Theoem. In a subgame pefect equilibium of the noncoopeative two-sided secetay poblem, lim R end = Coopeation. If utilities ae tansfeable and the agents tuthfully communicate the values of u and u to each othe, then they can do bette by ageeing to stop wheneve u + u > c. Because tansfeable utilities and tustwothiness ae vey stong assumptions, we call this stategy stongly coopeative. Howeve, even if utilities ae not tansfeable and communication is absent, it is still possible fo the agents to benefit fom coopeation. The ageement should be fo Playe i to accept wheneve u i ĉ fo some theshold ĉ<c chosen to optimize the expected payoff, i.e., maximizing Pu ĉ Pu ĉ Eu u ĉ + c Pu ĉ Pu ĉ fo Playe and analogously fo Playe. We call this stategy weakly coopeative. In 3, we analyze the two coopeative vesions of the model. The weakly coopeative scenaio is completely solved. Theoem. Unde the optimal stategy of the weakly coopeative two-sided secetay game, R lim end = 7/3 09 We pesent evidence in the stongly coopeative scenaio fo the following conjectue. Conjectue. Unde the optimal stategy of the stongly coopeative two-sided secetay game, lim R end = 3/4 087 Social cost of noncoopeation. This game illustates the game-theoetic concept of a social dilemma. Befoe the game, all agents would like to agee on the coopeative stategy and pomise to be somewhat moe geneous in accepting maiage poposals. Howeve, when an individual agent finds heself in a position whee the stategy calls on he to may a date that she finds below he expectations, she is tempted to eject this patne to optimize he own good instead. If all agents do that, then the expected outcome is wose fo all; accoding to Theoems and, the expected change is about 8% fo the wose. Compaed to the conjectued outcome of the stongly coopeative game, the cost of noncoopeation is about 3%. 3. The Optimal Coopeative Stategies Let us assume that the agents agee on a stategy, i.e., a set of intege thesholds s s s with the ule that maiage must be poposed in peiod if the obseved ank does not exceed the theshold s. Consequently, the pobability that an agent will popose in peiod is s /, due to the unifomity assumption. Hence, the pobability of an ageement to may is Pmay = s ) and the expected obseved ank of the patne, given that an agent poposes, is ER may = s + Plugging these expessions into the fundamental ecuence ) yields the ecuence fo the coopeative poblem: s ) R end = s ) ) s + R end + ) with bounday condition R end = + /. We want to minimize R end. All factos and tems ae nonnegative, so fo each fom to, we simply want to agee on the theshold s that will minimize R end. To find this theshold, we diffeentiate the expession with espect to s and find the geate of the two oots to be t = 3 ) + + R end + 3) The smalle oot is zeo. ow set s to eithe t o t, depending on which value minimizes R end, with the exception that s =t if t < 0. Because R end + + /, we have /3 <t /3 4) which guaantees that 0 s as equied. Fo example, we have t = /3 5) which means that in the next-to-the-last peiod an agent will popose if the cuent date is among the best two thids of all patnes the agent has seen. Fo futue use, let us also investigate when the theshold s becomes zeo, i.e., when an agent will not may anyone. Lemma. The theshold s = 0 if and only if t /3.
4 94 Opeations Reseach 555), pp. 9 93, 007 IFORMS Poof. It is staightfowad to veify fom ) that zeo is the nonnegative intege value of s that minimizes R end if and only if R end + + / +, which is equivalent to t /3. Appoximating s by t, we can ewite Equation ) using 3) into a ecuence puely in R end, which fo educes to a simple appoximative ecuence: R end R end + R end = + / R end + 3 If we now appoximate the diffeence R end + R end by the deivative, we obtain a simple diffeential equation with the exact solution R end = + 7/3 + 4 /. If this cude appoximation would wok all the way down to the fist peiod, then an agent s expected final ank when enteing the game would be R end 7/3. This tuns out to be tue, but we do not know how to pove it diectly fom the quality of the appoximation. Instead, we analyze the ecuence in detail, using the same appoach as Chow et al. 964), although we will encounte diffeent technical difficulties. Recall that s is eithe t o t. Ou fist measue is to define = s t, so that. ow ewite the ecuence ) as a ecuence in t by eplacing s with t + and eplacing R end and R end + accoding to 3). Afte extensive cancellations, we obtain t = t3 /3 + + t + 3 /3 /3 6) + Because and 3, we have an uppe bound of t T t def = t3 /3 + + t + /3 /3 7) + Similaly, the inequalities 0 and 3 givealowe bound of t t def = t3 /3 + t /3 /3 8) + To pove explicit lowe and uppe bounds fo t we need to examine the functions T t and t. Lemma. Fo sufficiently lage, the following holds:if /3 x /3 and x<y< 3/ +, then T x < T y and x < y. Poof. Diffeentiating the cubic polynomial T t with espect to t eveals that T t has a local minimum at t = + and a local maximum at t = +. This means that T t is an inceasing function in the inteval + +. Thus, if y +, it follows diectly that T x<t y. The case when y> + emains. In this case, because T t is deceasing in the inteval + 3/ +,wehave T y > T 3/ + ) = 6/4 + o while T x T /3 = 46/8 + o Fotunately, 6/4 > 46/8, which shows that T x < T y fo lage. Fo t, the easoning is completely analogous. Lemma 3. Fo sufficiently lage, the uppe bound t 3/ + 3 holds fo all in the inteval. Poof. By backwads induction. The lemma holds fo = because by 5), t = /3 3/ + 3 fo all. ow assume that the lemma holds fo a given >, and let 3/ + y = 3 By the induction hypothesis, t y. Because > which is lage, it follow fom Lemma with x = t, togethe with 4), and the fact that y< 3/ +, that T t T y. Combined with 7), this yields 3/ + t T t T ) 3 To conclude the induction step, we only need to pove that 3/ + T ) 3/ wheneve fo sufficiently lage. This can be veified though Stum sequences and seveal steps of computations in Maple see Appendix A). The lowe bound follows a simila patten. Lemma 4. Fo sufficiently lage, the lowe bound t 3/ + 3 holds fo all in the inteval.
5 Opeations Reseach 555), pp. 9 93, 007 IFORMS 95 Poof. Following the poof of the pevious lemma, using 8) instead, we only need to pove that 3/ ) 3/ wheneve fo sufficiently lage. Again, this can be veified though Stum sequences and seveal steps of computations in Maple see Appendix A). Fo a peiod such that t < /3, we have s = 0by Lemma, so no patnes ae accepted and the expected final ank is the same as in the next peiod: R end = R end +. Define cit to be the geatest with t < /3 o let cit = 0 if thee is no such ). Then, R end = R end = =R end cit +. Lemma 5. cit / 3/7 and t cit /3 as. Poof. Let = 3/7 /3 and = 3/7. Fom Lemma 3 with =, we obtain t < /3 fo sufficiently lage. Fom Lemma 4 with =, we obtain t /3 fo sufficiently lage. We conclude that cit < and hence cit / 3/7 as. Once again using Lemma 3 and Lemma 4, we obtain t /3 and t /3, and hence t cit /3 as. We can now finish the poof of Theoem. By 3), ) 3 R end + = t + + / + 9) Thus, we have R end = R end cit + = 3/t cit + + / cit + = 3/t cit + + / cit / + / 3/ /3 + 3/7 = 7/3 3.. The Stongly Coopeative Case In the stongly coopeative case, both agents accept maiage if thei aveage ank of each othe is bette than the ank they can expect by poceeding to the next peiod. This will yield the best possible aveage outcome fo agents. We conjectue that this outcome satisfies R end / 3/4. We will motivate this conjectue, as well as discuss why the method fo the weakly coopeative case does not wok completely in the stongly coopeative case. Suppose that two agents on a date agee to may if the sum of thei obseved anks is at most s + fo some theshold value s to be optimized. Fom the bounday condition, it is easy to deduce that the optimal thesholds will satisfy s fo evey peiod. Hence, out of the possible combinations of anks in peiod, the numbe of combinations with a sum of at most s + is+ + +s = s s + /. We obtain the pobability of maiage to be Pmay = s s + Given that two agents agee to may i.e., given that thei ank sum is at most s + ), the expected ank of each date is ER may = s + s + +s s s + / = s + 3 The fundamental ecuence ) now takes the fom R end = s s + s s ) s + R end + with the invaiant bounday condition R end = +/. Diffeentiating the ight-hand expession with espect to s, we find the smalle of the two oots to be negative while the lage oot is t = + + R end + ) R end R end ++/3 = + + R end whee is an eo tem that tends to zeo as + / + R end + gows lage. This eo tem makes it difficult to poceed with the same method as in the pevious section. In addition to a tem to account fo the ounding, we now must handle the two tems and + to account fo the above eo in t, espectively, t +. The algeba becomes moe complicated but is still wokable. Unfotunately, the appoximations needed to deive uppe and lowe bounds of the exact ecuence become so cude that the bounds do not convege. Howeve, we can just ignoe the small eo tems to deive the same kind of appoximative ecuence as befoe. We obtain R end R end R end + 3 R end = + / Again, the coesponding diffeential equation is solvable, and the asymptotics of the solution is R end / 3/4. This is the theoetical suppot fo ou conjectue. Futhemoe, it is possible to compute R end / fo lage by compute, and these values indeed seem to appoach 3/4 087.
6 96 Opeations Reseach 555), pp. 9 93, 007 IFORMS 4. The Optimal oncoopeative Stategy The thesholds that define the subgame pefect equilibium stategy in the noncoopeative case ae detemined by what agents can expect to achieve if they decline a poposal. An agent should agee to may a mate of ank R in peiod if the expected final ank achieved by maying now does not exceed the expected final ank in case of not maying: + + R R end + The theshold s should be the lagest intege value of R satisfying the above inequality, so that s =t with def t = + + R end + Fo example, we have t = 0) which means that in the next-to-the-last peiod we shall popose if and only if ou cuent date is among the best half of all patnes we have seen including the median if we have seen an odd numbe of patnes). To find the subgame pefect equilibium, we assume that all othe agents eason in the same way, so that we have Pmay = s / as in the pevious section. Consequently, we can just plug this new value of s into the ecuence ). The same appoximations now yield the ecuence R end R end + R end = + / + R end + 3 whose appoximating diffeential equation has the solution R end = + / + 4. If we could tust this appoximation, we would obtain R end. The poof of this esult will mimic the poof of the pevious section, with one new difficulty appeaing at one stage. Fo convenience, we euse the symbols T and fom the pevious section fo the coesponding functions in this new context. Fist, we ewite the ecuence ) as t = s s + + s t t + = ) Fo any, define = t s. Then, 0 <, and ) becomes t = t t + + t t ) + We now see that t + t + t 0. If we add this numbe to the numeato of ), we obtain the uppe bound t T t def = t3 + t + t 3) + Similaly, if we subtact t + t + t + 0 fom the numeato of ), we obtain the lowe bound t t def = t3 + t + t 4) + Lemma 6. Let f be any eal function with lim f= Fo sufficiently lage, the uppe bound t holds fo all in the inteval f. Poof. By backwads induction. The lemma holds fo = because t = / + 4 fo all. ow assume that the lemma holds fo a given >f. It is easy to see that T t is an inceasing function in the inteval 0 t /3. Thus, by the induction hypothesis and the facts that t > 0 and + / + 3 < /3, we obtain + t T t T ) + 3 whee the fist inequality stems fom 3). Hence, to conclude the induction step, we only need to pove that ) + T holds fo all in the inteval f fo sufficiently lage. This can be veified though seveal steps of computations in Maple see Appendix B). The lowe bound is tickie. We have not been able to find a lowe bound fo which the induction step woks in the whole inteval that we need, but by adding a constant tem of 0.48 to the denominato, we obtain a lowe bound that woks fom and downwads. Lemma 7. Fo sufficiently lage, the lowe bound t holds fo all in the inteval +. Poof. Fo lage, we can compute an appoximation of t by pefoming steps of the ecuence, keeping only the most significant tem in each step: If t k = a k + o, wehaves k = a k + o too, so
7 Opeations Reseach 555), pp. 9 93, 007 IFORMS 97 ou ecuence ) gives t k = a 3 k + a k/ + o. Computation shows that t = Again, we poceed by backwads induction. Assume that the lemma holds fo a given > +. It is easy to see that t is an inceasing function in the inteval 0 t /3. Thus, by the induction hypothesis and the facts that t > 0 and +/ < /3, we obtain + t t ) whee the fist inequality stems fom 4). It emains fo us to pove that ) fo lage. Again, seveal steps in Maple veify this inequality see Appendix B). We can now finish the poof of Theoem in much the same way as we poved Theoem. In the case t <, we always have s = 0, so no patnes ae eve accepted. Let cit be the geatest with t < o put cit = 0 if thee is no such ), so that R end = R end = =R end cit +. Lemma 7 gives that, fo lage, t > so cit < +. Lemma 6 with f= /3 gives, fo lage, t / /3 / /3 + / /3 / /3 +3 = 5/6 + o 5/6 / + o / < so cit + /3. Because /3 cit + < +, Lemma 6 with f= / /3 gives that t cit + cit ++ cit + cit as. Then, s cit + =, and by ), we get t cit. By definition of t,wehaver end cit + = t cit + / cit +. Thus, we have R end = R end cit + = + cit + t cit as because t cit and / /3 cit < Discussion We will biefly discuss thee points about ou model, method, and esults. A unifying esult? We have computed the asymptotic behavio of the expected final ank of two vesions of the two-sided secetay poblem. In both cases, the answe was suggested immediately fom an appoximating diffeential equation, but to pove the coectness of the answes, we needed a much moe detailed analysis equiing seveal technical steps specific to each case. Pehaps some unifying appoximation esult of the following kind can be found: Given a ecuence t = t + bt 3 / + Q 3 t 3 + Q t + Q t + Q 0 t = c whee b and c ae constants and Q i fo i = 0 3, ae ational functions satisfying cetain conditions. The appoximating diffeential equation t = bt 3 / t= c satisfies that t/t as. Although confident that some geneal esult of this type must hold, we have not been able to see how to sufficiently bound the eo of the appoximation. We have consideed vaious subtleties of ou poofs fo specific cases. These indicate that a geneal esult might be difficult to obtain. Anothe way of looking at the noncoopeative poblem? An unexpected consequence of adopting the optimal noncoopeative stategy is that maiages will be appoximately unifomly distibuted ove the peiods. Fo evey agent who entes peiod, the pobability p of maiage and leaving the game in this peiod is p = s /. In the solution to the noncoopeative case, we obtained s / +, and hence p / +. Thus, the initial pobability of maying in peiod is p p p p = Seeing this simple solution, we cannot help but wonde if thee is some a pioi agument showing that the optimal stategy must be one that makes optimal use of all the peiods. A model with eplacement. Suppose that the total numbe of agents, say U, is finite and not vey lage compaed to ). Then, we would have to allow fo the possibility that the same agents may come to date seveal times. We can think of it as unmaied dates being put back in the pool. Such a model would be much like the game studied in the laboatoy with human subjects by Eiksson and Stimling 005). How fa does this vaiation with eplacement take us fom the model studied in this pape? In the model with eplacement, you gain some infomation about the pool of
8 98 Opeations Reseach 555), pp. 9 93, 007 IFORMS emaining agents each time you do not may a date. This infomation may not change decisions vey much, but it fundamentally changes the complexity of the poblem. Futhemoe, the initial assumption of independence of pefeences will be compomised as the game poceeds, with an inceasing bias towad emaining agents not liking each othe vey much. Hence, the unifomity assumption will not hold, so we cannot even define the expected final ank of a date. Moe conceptual wok is needed to make the model with eplacement well defined. We leave it as an open poblem. Appendix A. Lemmas fo Theoem Hee we pove two lemmas that ae needed in the poof of Theoem. Lemma 8. Let T be the function defined in 7). Fo sufficiently lage, the inequality 3/ + T ) 3/ holds fo all. Poof. Afte the substitution z =, ou desied inequality tansfoms to 3/ + T ) z 3 3/ + A) z + 3 and the inteval tansfoms to z, which implies z We will pove that, fo sufficiently lage, inequality A) holds fo z see Figue A.). A lage implies a lage, so the lemma will follow. Evaluating T, inequality A) becomes gz def 3/ + = z + 3 3/ + 3 z 3 3/ + ++ z + 0 ) 3 3 Fist, investigating the case of z =, we obtain g = o )+ 3 3 ) Figue A.. Lage numbe We show that inequality A) holds in a egion like this. z > lage numbe which is positive fo lage. Because gz is continuous, it suffices to show that fo lage, gz 0 fo z. The zeos of gz ae the same as the zeos of 3/ + z + ) 3 [ 3/ z 3/ z 3 ) 3 3 ) + ] 3 We multiply this by z + z 6 and obtain a polynomial of degee 8 in z: q z def = 5z / z / / / + 5 ) z / + 3 7/ / + 4 / + 3/ / z / / 4 4 7/ / / +z / / / / z / / / / 4 z / + 5 5/ z 9683 / We must show that fo lage, q z 0 fo z. Using a compute pogam like Maple, this can be done z
9 Opeations Reseach 555), pp. 9 93, 007 IFORMS 99 accoding to the following pocedue: Compute the fouth deivative q 4 z = a 4 z 4 + a 3 z 3 + a z + a z + a 0. Using Stum sequences, veify that fo lage, the polynomial q 4 z has no eal oots in the inteval z. Check that q 4 >0 fo lage, so that q 4 z > 0 eveywhee in the inteval z. Check that q, q, q, and q ae positive fo lage. This shows that fo sufficiently lage, q z > 0 in the inteval z. Lemma 9. Let be the function defined in 8). Fo sufficiently lage, the inequality 3/ ) 3/ holds fo all. Poof. Afte the substitution z =, ou inequality tansfoms to 3/ z + ) 3/ 3 z + + A) 3 and the inteval tansfoms to z, which implies z< Evaluating in A) and multiplying by z + 3 z + + >0 yields the equivalent inequality p z def = 8 6 z 3 4z z + z + 4z + z + ) z + z + + ) z 3 +3z +9z+5 4z+ z + ) 84z + 3 z + + ) 0 whee p z is clealy a cubic polynomial in. Let us fist show that the leading coefficient of p z is positive, i.e., z + 4z + z + > z 3 + 4z + z + Afte squaing, expanding, and collecting, we obtain 4z 3 + 9z + 4z>0, which is tue fo all z. Using Stum sequences, we show that thee is a ẑ such that fo all z>ẑ, the polynomial p z has no eal oots geate than z, see Appendix C). Because the leading coefficient of p z is positive, we conclude that p z > 0 wheneve ẑ<z<. It emains to pove positivity of p z fo z ẑ. Fo each z, let + z be the lagest absolute value of the thee complex oots of the polynomial p z. Because + z is a continuous function of z, it attains a supemum on evey compact inteval, so we can define ˆ = sup + z z ẑ see Figue A.). Figue A.. ˆ The gaph of + z. We know that p z > 0 above the cuve.) z > ˆ zˆ Consequently, fo any >ˆ, it is tue that p z > 0 wheneve z<. A lage implies a lage because n, so the lemma follows. Appendix B. Lemmas fo Theoem Hee we pove the two lemmas that ae needed in the poof of Theoem. Lemma 0. Let T be the function defined in 3), and let f be any eal function with lim f=. Then, fo sufficiently lage, the inequality ) + T holds fo all f. Poof. Afte the substitution z = + 3, ou inequality tansfoms to ) + T + B3) z z + and the condition tansfoms to z. We will pove that inequality B3) holds fo all z and sufficiently lage. A lage implies a lage because f, so the lemma will follow. Evaluating T in B3) yields gz def = + z z 3 z z >0 + We see that [ ) g = 3 + 5/ ) + 3 3/ + ) ] z
10 930 Opeations Reseach 555), pp. 9 93, 007 IFORMS which is positive fo lage. Because gz is continuous, it suffices to show that if is lage, gz 0 holds fo all z. The zeos of gz ae the same as the zeos of ) + z + [ ] + + z 3 z z We multiply this by + z + z 6 and obtain a polynomial of degee 6 in z: q z def = / 4 3) z / / z 5 +4 / / 4 7/ / z / / / z / / 5 4 7/ / 4 z / / / z 6 +6 / / / + 3 We must show that if is lage, q z 0 holds fo all z. Using a compute pogam like Maple, this is just a matte of veification: Compute the fouth deivative q 4 z = a z + a z + a 0. Veify that a /a a 0 /a < 0 fo lage. This implies that q 4 z has no eal oots. Check that q 4 0>0 fo lage so that q 4 z > 0 eveywhee. This is immediately evident fom the expession fo q z above. Check that q, q, q, and q ae positive fo lage. This shows that fo sufficiently lage, the inequality q z > 0 holds fo all z. Lemma. Let be the function defined in 4). Fo sufficiently lage, the inequality ) holds fo all in the inteval +. Poof. Let = 048. Afte the substitution + 3 +, ou inequality tansfoms to z = ) + z z + + B4) The inteval + tansfoms to 5 z + 3 B5) Evaluating in B4) and multiplying by z 3 z + + > 0 yields p z def = z z + z 3 + z ) + z z + + ) z z + z + + ) 0 a quadatic polynomial in. Let us fist show that the leading coefficient of p z is positive, i.e., z z + > z 3 + z Afte squaing, expanding, and collecting, we get 4z 3 3z z + > 0. The zeos of this cubic polynomial in z ae z 059, z 0559, and z 3 500, all of which ae less than 5 +. Thee ae always eal oots to p z because its constant tem is negative fo z 5 +. Fo each z, let + z be the geate of the two oots of p z. We can, of couse, wite down an explicit fomula fo + z, and with a compute pogam like Maple, it is easy to check that lim z + z z =. Thus, fo any >0, thee is a ẑ such that p z > 0 when >z+ +and z>ẑ. We choose = /4. In the inteesting inteval B5), we have z + + / /4 <, whee the last inequality follows fom + 3 < 5/4, which is tue if 5. In the inteval ẑ /4 <z+ + 3, we know that p z > 0, so we only have to woy about the inteval 5+ z ẑ /4. Because + z is a continuous function, we can define ˆ = sup + z 5+ z ẑ /4. Then, p z > 0 in the inteval B5) fo all >ˆ. A lage implies a lage because +, so the lemma follows. Appendix C. Stum Sequences Given an n-degee polynomial Px with eal coefficients, the Stum sequence of Px is a finite sequence of polynomials P 0 x P xp n x defined ecusively by P 0 x = Px P x = P x P k x = emp k x P k x fo k n Hee empx Qx is the emainde when Px ae divided by Qx. Let Va 0 a a n be the numbe of sign vaiations in the list a 0 a a n, i.e., Va 0 a a n = i a i a i+ < 0 The following famous theoem counts the eal oots of Px. Fo a poof, we efe to Pestel and Delzell 00). Theoem 3 Stum s Theoem). If a<b ae eal numbes which ae not oots of Px, then the numbe of eal
11 Opeations Reseach 555), pp. 9 93, 007 IFORMS 93 oots of Px in the inteval a b is VP 0 ap n a VP 0 bp n b A geneal cubic polynomial Px= bx 3 + cx + dx + e has the Stum sequence P 0 x = bx 3 + cx + dx + e P x = 3bx + cx + d P x = 9b c 6bdx + cd 9be P 3 x = 9b 4bd3 + c d + 8bcde 4c 3 e 7b e 43bd c povided that the denominatos do not vanish. ow let Px be the polynomial p z in the poof of Lemma 9, identifying x with. Staightfowad computation in Maple yields lim P iz =+ and z lim lim P ib =+ z b fo 0 i 3. Theefoe, lim lim VP 0zP n z VP 0 bp n b z b = 0 0 = 0 and Stum s theoem poves that thee is a ẑ such that if z>ẑ, then p z has no eal oots geate than z. ow conside a geneal fouth-degee polynomial Px = ax 4 + bx 3 + cx + dx + e with Stum sequence P 0 xp 4 x. With Maple, we can compute these polynomials explicitly, but thee is no point in displaying them hee. Let Px be the polynomial q z in the poof of Lemma 8, identifying x with z. Fo 0 i 4, define a i = lim P i b i = lim P i Staightfowad computation in Maple yields a 0 a a a 3 a 4 = b 0 b b b 3 b 4 = Theefoe, lim VP 0P n VP 0 P n = =0 and Stum s theoem poves that fo sufficiently lage, q z has no eal oots in the inteval. Acknowledgments This eseach was patially suppoted by the Swedish Reseach Council and the Euopean Commission s IHRP Pogamme, Algebaic Combinatoics in Euope gant HPR- CT ). Refeences Alpen, S., I. Katantzi, D. Reynies Mathematical models of mutual mate choice. Recent Reseach Developments in Expeimental and Theoetical Biology. Tanswold Reseach etwok, Tivandum, Keala, India, Chow, Y. S., S. Moiguti, H. Robbins, S. M. Samuels Optimal selection based on elative ank the secetay poblem ). Isael J. Math Eiksson, K., P. Stimling How unstable ae matchings fom decentalized mate seach? Pepint. Feguson, T Who solved the secetay poblem? Statist. Sci Lindley, D. 96. Dynamic pogamming and decision theoy. Appl. Statist Pestel, A., C.. Delzell. 00. Positive Polynomials Fom Hilbet s 7th Poblem to Real Algeba. Spinge-Velag, Belin, Gemany. Rittaud, B Quand les assistantes choisissent leu paton. La Recheche Roth, A. E., M. A. O. Sotomayo Two-Sided Matching. Cambidge Univesity Pess, ew Yok. Selten, R Reexamination of the pefectness concept fo equilibium points in extensive games. Intenat. J. Game Theoy
Multiple Criteria Secretary Problem: A New Approach
J. Stat. Appl. Po. 3, o., 9-38 (04 9 Jounal of Statistics Applications & Pobability An Intenational Jounal http://dx.doi.og/0.785/jsap/0303 Multiple Citeia Secetay Poblem: A ew Appoach Alaka Padhye, and
More informationA Bijective Approach to the Permutational Power of a Priority Queue
A Bijective Appoach to the Pemutational Powe of a Pioity Queue Ia M. Gessel Kuang-Yeh Wang Depatment of Mathematics Bandeis Univesity Waltham, MA 02254-9110 Abstact A pioity queue tansfoms an input pemutation
More information3.1 Random variables
3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated
More informationEM Boundary Value Problems
EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do
More informationGoodness-of-fit for composite hypotheses.
Section 11 Goodness-of-fit fo composite hypotheses. Example. Let us conside a Matlab example. Let us geneate 50 obsevations fom N(1, 2): X=nomnd(1,2,50,1); Then, unning a chi-squaed goodness-of-fit test
More informationMath 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs
Math 30: The Edős-Stone-Simonovitz Theoem and Extemal Numbes fo Bipatite Gaphs May Radcliffe The Edős-Stone-Simonovitz Theoem Recall, in class we poved Tuán s Gaph Theoem, namely Theoem Tuán s Theoem Let
More informationWhen two numbers are written as the product of their prime factors, they are in factored form.
10 1 Study Guide Pages 420 425 Factos Because 3 4 12, we say that 3 and 4 ae factos of 12. In othe wods, factos ae the numbes you multiply to get a poduct. Since 2 6 12, 2 and 6 ae also factos of 12. The
More information4/18/2005. Statistical Learning Theory
Statistical Leaning Theoy Statistical Leaning Theoy A model of supevised leaning consists of: a Envionment - Supplying a vecto x with a fixed but unknown pdf F x (x b Teache. It povides a desied esponse
More informationStanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012
Stanfod Univesity CS59Q: Quantum Computing Handout 8 Luca Tevisan Octobe 8, 0 Lectue 8 In which we use the quantum Fouie tansfom to solve the peiod-finding poblem. The Peiod Finding Poblem Let f : {0,...,
More informationUnobserved Correlation in Ascending Auctions: Example And Extensions
Unobseved Coelation in Ascending Auctions: Example And Extensions Daniel Quint Univesity of Wisconsin Novembe 2009 Intoduction In pivate-value ascending auctions, the winning bidde s willingness to pay
More informationSolution to HW 3, Ma 1a Fall 2016
Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.
More informationExploration of the three-person duel
Exploation of the thee-peson duel Andy Paish 15 August 2006 1 The duel Pictue a duel: two shootes facing one anothe, taking tuns fiing at one anothe, each with a fixed pobability of hitting his opponent.
More informationMethod for Approximating Irrational Numbers
Method fo Appoximating Iational Numbes Eic Reichwein Depatment of Physics Univesity of Califonia, Santa Cuz June 6, 0 Abstact I will put foth an algoithm fo poducing inceasingly accuate ational appoximations
More informationMath 124B February 02, 2012
Math 24B Febuay 02, 202 Vikto Gigoyan 8 Laplace s equation: popeties We have aleady encounteed Laplace s equation in the context of stationay heat conduction and wave phenomena. Recall that in two spatial
More informationPearson s Chi-Square Test Modifications for Comparison of Unweighted and Weighted Histograms and Two Weighted Histograms
Peason s Chi-Squae Test Modifications fo Compaison of Unweighted and Weighted Histogams and Two Weighted Histogams Univesity of Akueyi, Bogi, v/noduslód, IS-6 Akueyi, Iceland E-mail: nikolai@unak.is Two
More informationNew problems in universal algebraic geometry illustrated by boolean equations
New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic
More informationSurveillance Points in High Dimensional Spaces
Société de Calcul Mathématique SA Tools fo decision help since 995 Suveillance Points in High Dimensional Spaces by Benad Beauzamy Januay 06 Abstact Let us conside any compute softwae, elying upon a lage
More informationMATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE. We consider second order constant coefficient scalar linear PDEs on R n. These have the form
MATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE ANDRAS VASY We conside second ode constant coefficient scala linea PDEs on R n. These have the fom Lu = f L = a ij xi xj + b i xi + c i whee a ij b i and
More informationON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},
ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability
More informationThe Substring Search Problem
The Substing Seach Poblem One algoithm which is used in a vaiety of applications is the family of substing seach algoithms. These algoithms allow a use to detemine if, given two chaacte stings, one is
More informationA Multivariate Normal Law for Turing s Formulae
A Multivaiate Nomal Law fo Tuing s Fomulae Zhiyi Zhang Depatment of Mathematics and Statistics Univesity of Noth Caolina at Chalotte Chalotte, NC 28223 Abstact This pape establishes a sufficient condition
More information6 PROBABILITY GENERATING FUNCTIONS
6 PROBABILITY GENERATING FUNCTIONS Cetain deivations pesented in this couse have been somewhat heavy on algeba. Fo example, detemining the expectation of the Binomial distibution (page 5.1 tuned out to
More informationof the contestants play as Falco, and 1 6
JHMT 05 Algeba Test Solutions 4 Febuay 05. In a Supe Smash Bothes tounament, of the contestants play as Fox, 3 of the contestants play as Falco, and 6 of the contestants play as Peach. Given that thee
More informationCentral Coverage Bayes Prediction Intervals for the Generalized Pareto Distribution
Statistics Reseach Lettes Vol. Iss., Novembe Cental Coveage Bayes Pediction Intevals fo the Genealized Paeto Distibution Gyan Pakash Depatment of Community Medicine S. N. Medical College, Aga, U. P., India
More informationOn the integration of the equations of hydrodynamics
Uebe die Integation de hydodynamischen Gleichungen J f eine u angew Math 56 (859) -0 On the integation of the equations of hydodynamics (By A Clebsch at Calsuhe) Tanslated by D H Delphenich In a pevious
More informationJournal of Inequalities in Pure and Applied Mathematics
Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics
More informationMarkscheme May 2017 Calculus Higher level Paper 3
M7/5/MATHL/HP3/ENG/TZ0/SE/M Makscheme May 07 Calculus Highe level Pape 3 pages M7/5/MATHL/HP3/ENG/TZ0/SE/M This makscheme is the popety of the Intenational Baccalaueate and must not be epoduced o distibuted
More informationNotes on McCall s Model of Job Search. Timothy J. Kehoe March if job offer has been accepted. b if searching
Notes on McCall s Model of Job Seach Timothy J Kehoe Mach Fv ( ) pob( v), [, ] Choice: accept age offe o eceive b and seach again next peiod An unemployed oke solves hee max E t t y t y t if job offe has
More informationAn upper bound on the number of high-dimensional permutations
An uppe bound on the numbe of high-dimensional pemutations Nathan Linial Zu Luia Abstact What is the highe-dimensional analog of a pemutation? If we think of a pemutation as given by a pemutation matix,
More informationAuchmuty High School Mathematics Department Advanced Higher Notes Teacher Version
The Binomial Theoem Factoials Auchmuty High School Mathematics Depatment The calculations,, 6 etc. often appea in mathematics. They ae called factoials and have been given the notation n!. e.g. 6! 6!!!!!
More informationC/CS/Phys C191 Shor s order (period) finding algorithm and factoring 11/12/14 Fall 2014 Lecture 22
C/CS/Phys C9 Sho s ode (peiod) finding algoithm and factoing /2/4 Fall 204 Lectue 22 With a fast algoithm fo the uantum Fouie Tansfom in hand, it is clea that many useful applications should be possible.
More informationRelating Branching Program Size and. Formula Size over the Full Binary Basis. FB Informatik, LS II, Univ. Dortmund, Dortmund, Germany
Relating Banching Pogam Size and omula Size ove the ull Binay Basis Matin Saueho y Ingo Wegene y Ralph Wechne z y B Infomatik, LS II, Univ. Dotmund, 44 Dotmund, Gemany z ankfut, Gemany sauehof/wegene@ls.cs.uni-dotmund.de
More informationResearch Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function
Abstact and Applied Analysis Volume 011, Aticle ID 697547, 7 pages doi:10.1155/011/697547 Reseach Aticle On Alze and Qiu s Conjectue fo Complete Elliptic Integal and Invese Hypebolic Tangent Function Yu-Ming
More informationarxiv: v1 [math.co] 4 May 2017
On The Numbe Of Unlabeled Bipatite Gaphs Abdullah Atmaca and A Yavuz Ouç axiv:7050800v [mathco] 4 May 207 Abstact This pape solves a poblem that was stated by M A Haison in 973 [] This poblem, that has
More informationBerkeley Math Circle AIME Preparation March 5, 2013
Algeba Toolkit Rules of Thumb. Make sue that you can pove all fomulas you use. This is even bette than memoizing the fomulas. Although it is best to memoize, as well. Stive fo elegant, economical methods.
More informationChapter 3: Theory of Modular Arithmetic 38
Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences
More informationF-IF Logistic Growth Model, Abstract Version
F-IF Logistic Gowth Model, Abstact Vesion Alignments to Content Standads: F-IFB4 Task An impotant example of a model often used in biology o ecology to model population gowth is called the logistic gowth
More informationac p Answers to questions for The New Introduction to Geographical Economics, 2 nd edition Chapter 3 The core model of geographical economics
Answes to questions fo The New ntoduction to Geogaphical Economics, nd edition Chapte 3 The coe model of geogaphical economics Question 3. Fom intoductoy mico-economics we know that the condition fo pofit
More informationA Comparison and Contrast of Some Methods for Sample Quartiles
A Compaison and Contast of Some Methods fo Sample Quatiles Anwa H. Joade and aja M. Latif King Fahd Univesity of Petoleum & Mineals ABSTACT A emainde epesentation of the sample size n = 4m ( =, 1, 2, 3)
More information(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2.
Paabola Volume 5, Issue (017) Solutions 151 1540 Q151 Take any fou consecutive whole numbes, multiply them togethe and add 1. Make a conjectue and pove it! The esulting numbe can, fo instance, be expessed
More informationOn decompositions of complete multipartite graphs into the union of two even cycles
On decompositions of complete multipatite gaphs into the union of two even cycles A. Su, J. Buchanan, R. C. Bunge, S. I. El-Zanati, E. Pelttai, G. Rasmuson, E. Spaks, S. Tagais Depatment of Mathematics
More informationEncapsulation theory: the transformation equations of absolute information hiding.
1 Encapsulation theoy: the tansfomation equations of absolute infomation hiding. Edmund Kiwan * www.edmundkiwan.com Abstact This pape descibes how the potential coupling of a set vaies as the set is tansfomed,
More informationMATH 415, WEEK 3: Parameter-Dependence and Bifurcations
MATH 415, WEEK 3: Paamete-Dependence and Bifucations 1 A Note on Paamete Dependence We should pause to make a bief note about the ole played in the study of dynamical systems by the system s paametes.
More informationANA BERRIZBEITIA, LUIS A. MEDINA, ALEXANDER C. MOLL, VICTOR H. MOLL, AND LAINE NOBLE
THE p-adic VALUATION OF STIRLING NUMBERS ANA BERRIZBEITIA, LUIS A. MEDINA, ALEXANDER C. MOLL, VICTOR H. MOLL, AND LAINE NOBLE Abstact. Let p > 2 be a pime. The p-adic valuation of Stiling numbes of the
More informationA generalization of the Bernstein polynomials
A genealization of the Benstein polynomials Halil Ouç and Geoge M Phillips Mathematical Institute, Univesity of St Andews, Noth Haugh, St Andews, Fife KY16 9SS, Scotland Dedicated to Philip J Davis This
More informationAbsorption Rate into a Small Sphere for a Diffusing Particle Confined in a Large Sphere
Applied Mathematics, 06, 7, 709-70 Published Online Apil 06 in SciRes. http://www.scip.og/jounal/am http://dx.doi.og/0.46/am.06.77065 Absoption Rate into a Small Sphee fo a Diffusing Paticle Confined in
More information10/04/18. P [P(x)] 1 negl(n).
Mastemath, Sping 208 Into to Lattice lgs & Cypto Lectue 0 0/04/8 Lectues: D. Dadush, L. Ducas Scibe: K. de Boe Intoduction In this lectue, we will teat two main pats. Duing the fist pat we continue the
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Septembe 5, 011 Abstact To study how balanced o unbalanced a maximal intesecting
More informationLiquid gas interface under hydrostatic pressure
Advances in Fluid Mechanics IX 5 Liquid gas inteface unde hydostatic pessue A. Gajewski Bialystok Univesity of Technology, Faculty of Civil Engineeing and Envionmental Engineeing, Depatment of Heat Engineeing,
More informationJENSEN S INEQUALITY FOR DISTRIBUTIONS POSSESSING HIGHER MOMENTS, WITH APPLICATION TO SHARP BOUNDS FOR LAPLACE-STIELTJES TRANSFORMS
J. Austal. Math. Soc. Se. B 40(1998), 80 85 JENSEN S INEQUALITY FO DISTIBUTIONS POSSESSING HIGHE MOMENTS, WITH APPLICATION TO SHAP BOUNDS FO LAPLACE-STIELTJES TANSFOMS B. GULJAŠ 1,C.E.M.PEACE 2 and J.
More informationApplication of Parseval s Theorem on Evaluating Some Definite Integrals
Tukish Jounal of Analysis and Numbe Theoy, 4, Vol., No., -5 Available online at http://pubs.sciepub.com/tjant/// Science and Education Publishing DOI:.69/tjant--- Application of Paseval s Theoem on Evaluating
More informationAlternative Tests for the Poisson Distribution
Chiang Mai J Sci 015; 4() : 774-78 http://epgsciencecmuacth/ejounal/ Contibuted Pape Altenative Tests fo the Poisson Distibution Manad Khamkong*[a] and Pachitjianut Siipanich [b] [a] Depatment of Statistics,
More informationUsing Laplace Transform to Evaluate Improper Integrals Chii-Huei Yu
Available at https://edupediapublicationsog/jounals Volume 3 Issue 4 Febuay 216 Using Laplace Tansfom to Evaluate Impope Integals Chii-Huei Yu Depatment of Infomation Technology, Nan Jeon Univesity of
More informationworking pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50
woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,
More informationUniversal Gravitation
Chapte 1 Univesal Gavitation Pactice Poblem Solutions Student Textbook page 580 1. Conceptualize the Poblem - The law of univesal gavitation applies to this poblem. The gavitational foce, F g, between
More informationLikelihood vs. Information in Aligning Biopolymer Sequences. UCSD Technical Report CS Timothy L. Bailey
Likelihood vs. Infomation in Aligning Biopolyme Sequences UCSD Technical Repot CS93-318 Timothy L. Bailey Depatment of Compute Science and Engineeing Univesity of Califonia, San Diego 1 Febuay, 1993 ABSTRACT:
More informationGraphs of Sine and Cosine Functions
Gaphs of Sine and Cosine Functions In pevious sections, we defined the tigonometic o cicula functions in tems of the movement of a point aound the cicumfeence of a unit cicle, o the angle fomed by the
More informationPROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr.
POBLM S # SOLUIONS by obet A. DiStasio J. Q. he Bon-Oppenheime appoximation is the standad way of appoximating the gound state of a molecula system. Wite down the conditions that detemine the tonic and
More informationAn Application of Fuzzy Linear System of Equations in Economic Sciences
Austalian Jounal of Basic and Applied Sciences, 5(7): 7-14, 2011 ISSN 1991-8178 An Application of Fuzzy Linea System of Equations in Economic Sciences 1 S.H. Nassei, 2 M. Abdi and 3 B. Khabii 1 Depatment
More informationInternet Appendix for A Bayesian Approach to Real Options: The Case of Distinguishing Between Temporary and Permanent Shocks
Intenet Appendix fo A Bayesian Appoach to Real Options: The Case of Distinguishing Between Tempoay and Pemanent Shocks Steven R. Genadie Gaduate School of Business, Stanfod Univesity Andey Malenko Gaduate
More informationBoundary Layers and Singular Perturbation Lectures 16 and 17 Boundary Layers and Singular Perturbation. x% 0 Ω0æ + Kx% 1 Ω0æ + ` : 0. (9.
Lectues 16 and 17 Bounday Layes and Singula Petubation A Regula Petubation In some physical poblems, the solution is dependent on a paamete K. When the paamete K is vey small, it is natual to expect that
More informationChapter 2: Introduction to Implicit Equations
Habeman MTH 11 Section V: Paametic and Implicit Equations Chapte : Intoduction to Implicit Equations When we descibe cuves on the coodinate plane with algebaic equations, we can define the elationship
More informationMacro Theory B. The Permanent Income Hypothesis
Maco Theoy B The Pemanent Income Hypothesis Ofe Setty The Eitan Beglas School of Economics - Tel Aviv Univesity May 15, 2015 1 1 Motivation 1.1 An econometic check We want to build an empiical model with
More information7.2. Coulomb s Law. The Electric Force
Coulomb s aw Recall that chaged objects attact some objects and epel othes at a distance, without making any contact with those objects Electic foce,, o the foce acting between two chaged objects, is somewhat
More informationA NEW VARIABLE STIFFNESS SPRING USING A PRESTRESSED MECHANISM
Poceedings of the ASME 2010 Intenational Design Engineeing Technical Confeences & Computes and Infomation in Engineeing Confeence IDETC/CIE 2010 August 15-18, 2010, Monteal, Quebec, Canada DETC2010-28496
More informationTo Feel a Force Chapter 7 Static equilibrium - torque and friction
To eel a oce Chapte 7 Chapte 7: Static fiction, toque and static equilibium A. Review of foce vectos Between the eath and a small mass, gavitational foces of equal magnitude and opposite diection act on
More informationLong-range stress re-distribution resulting from damage in heterogeneous media
Long-ange stess e-distibution esulting fom damage in heteogeneous media Y.L.Bai (1), F.J.Ke (1,2), M.F.Xia (1,3) X.H.Zhang (1) and Z.K. Jia (1) (1) State Key Laboatoy fo Non-linea Mechanics (LNM), Institute
More informationQuasi-Randomness and the Distribution of Copies of a Fixed Graph
Quasi-Randomness and the Distibution of Copies of a Fixed Gaph Asaf Shapia Abstact We show that if a gaph G has the popety that all subsets of vetices of size n/4 contain the coect numbe of tiangles one
More informationKOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS
Jounal of Applied Analysis Vol. 14, No. 1 2008), pp. 43 52 KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS L. KOCZAN and P. ZAPRAWA Received Mach 12, 2007 and, in evised fom,
More informationTHE CONE THEOREM JOEL A. TROPP. Abstract. We prove a fixed point theorem for functions which are positive with respect to a cone in a Banach space.
THE ONE THEOEM JOEL A. TOPP Abstact. We pove a fixed point theoem fo functions which ae positive with espect to a cone in a Banach space. 1. Definitions Definition 1. Let X be a eal Banach space. A subset
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Mach 6, 013 Abstact To study how balanced o unbalanced a maximal intesecting
More informationASTR415: Problem Set #6
ASTR45: Poblem Set #6 Cuan D. Muhlbege Univesity of Mayland (Dated: May 7, 27) Using existing implementations of the leapfog and Runge-Kutta methods fo solving coupled odinay diffeential equations, seveal
More informationA proof of the binomial theorem
A poof of the binomial theoem If n is a natual numbe, let n! denote the poduct of the numbes,2,3,,n. So! =, 2! = 2 = 2, 3! = 2 3 = 6, 4! = 2 3 4 = 24 and so on. We also let 0! =. If n is a non-negative
More informationarxiv: v1 [math.co] 1 Apr 2011
Weight enumeation of codes fom finite spaces Relinde Juius Octobe 23, 2018 axiv:1104.0172v1 [math.co] 1 Ap 2011 Abstact We study the genealized and extended weight enumeato of the - ay Simplex code and
More informationSuggested Solutions to Homework #4 Econ 511b (Part I), Spring 2004
Suggested Solutions to Homewok #4 Econ 5b (Pat I), Sping 2004. Conside a neoclassical gowth model with valued leisue. The (epesentative) consume values steams of consumption and leisue accoding to P t=0
More informationOn a quantity that is analogous to potential and a theorem that relates to it
Su une quantité analogue au potential et su un théoème y elatif C R Acad Sci 7 (87) 34-39 On a quantity that is analogous to potential and a theoem that elates to it By R CLAUSIUS Tanslated by D H Delphenich
More informationON SPARSELY SCHEMMEL TOTIENT NUMBERS. Colin Defant 1 Department of Mathematics, University of Florida, Gainesville, Florida
#A8 INTEGERS 5 (205) ON SPARSEL SCHEMMEL TOTIENT NUMBERS Colin Defant Depatment of Mathematics, Univesity of Floida, Gainesville, Floida cdefant@ufl.edu Received: 7/30/4, Revised: 2/23/4, Accepted: 4/26/5,
More informationCentripetal Force OBJECTIVE INTRODUCTION APPARATUS THEORY
Centipetal Foce OBJECTIVE To veify that a mass moving in cicula motion expeiences a foce diected towad the cente of its cicula path. To detemine how the mass, velocity, and adius affect a paticle's centipetal
More informationPsychometric Methods: Theory into Practice Larry R. Price
ERRATA Psychometic Methods: Theoy into Pactice Lay R. Pice Eos wee made in Equations 3.5a and 3.5b, Figue 3., equations and text on pages 76 80, and Table 9.1. Vesions of the elevant pages that include
More informationExceptional regular singular points of second-order ODEs. 1. Solving second-order ODEs
(May 14, 2011 Exceptional egula singula points of second-ode ODEs Paul Gaett gaett@math.umn.edu http://www.math.umn.edu/ gaett/ 1. Solving second-ode ODEs 2. Examples 3. Convegence Fobenius method fo solving
More informationDetermining solar characteristics using planetary data
Detemining sola chaacteistics using planetay data Intoduction The Sun is a G-type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this investigation
More informationOLYMON. Produced by the Canadian Mathematical Society and the Department of Mathematics of the University of Toronto. Issue 9:2.
OLYMON Poduced by the Canadian Mathematical Society and the Depatment of Mathematics of the Univesity of Toonto Please send you solution to Pofesso EJ Babeau Depatment of Mathematics Univesity of Toonto
More information15 Solving the Laplace equation by Fourier method
5 Solving the Laplace equation by Fouie method I aleady intoduced two o thee dimensional heat equation, when I deived it, ecall that it taes the fom u t = α 2 u + F, (5.) whee u: [0, ) D R, D R is the
More informationPulse Neutron Neutron (PNN) tool logging for porosity Some theoretical aspects
Pulse Neuton Neuton (PNN) tool logging fo poosity Some theoetical aspects Intoduction Pehaps the most citicism of Pulse Neuton Neuon (PNN) logging methods has been chage that PNN is to sensitive to the
More informationCSCE 478/878 Lecture 4: Experimental Design and Analysis. Stephen Scott. 3 Building a tree on the training set Introduction. Outline.
In Homewok, you ae (supposedly) Choosing a data set 2 Extacting a test set of size > 3 3 Building a tee on the taining set 4 Testing on the test set 5 Repoting the accuacy (Adapted fom Ethem Alpaydin and
More informationarxiv: v1 [math.co] 6 Mar 2008
An uppe bound fo the numbe of pefect matchings in gaphs Shmuel Fiedland axiv:0803.0864v [math.co] 6 Ma 2008 Depatment of Mathematics, Statistics, and Compute Science, Univesity of Illinois at Chicago Chicago,
More informationFresnel Diffraction. monchromatic light source
Fesnel Diffaction Equipment Helium-Neon lase (632.8 nm) on 2 axis tanslation stage, Concave lens (focal length 3.80 cm) mounted on slide holde, iis mounted on slide holde, m optical bench, micoscope slide
More informationNOTE. Some New Bounds for Cover-Free Families
Jounal of Combinatoial Theoy, Seies A 90, 224234 (2000) doi:10.1006jcta.1999.3036, available online at http:.idealibay.com on NOTE Some Ne Bounds fo Cove-Fee Families D. R. Stinson 1 and R. Wei Depatment
More informationA Converse to Low-Rank Matrix Completion
A Convese to Low-Rank Matix Completion Daniel L. Pimentel-Alacón, Robet D. Nowak Univesity of Wisconsin-Madison Abstact In many pactical applications, one is given a subset Ω of the enties in a d N data
More informationMODULE 5a and 5b (Stewart, Sections 12.2, 12.3) INTRO: In MATH 1114 vectors were written either as rows (a1, a2,..., an) or as columns a 1 a. ...
MODULE 5a and 5b (Stewat, Sections 2.2, 2.3) INTRO: In MATH 4 vectos wee witten eithe as ows (a, a2,..., an) o as columns a a 2... a n and the set of all such vectos of fixed length n was called the vecto
More informationCALCULUS II Vectors. Paul Dawkins
CALCULUS II Vectos Paul Dawkins Table of Contents Peface... ii Vectos... 3 Intoduction... 3 Vectos The Basics... 4 Vecto Aithmetic... 8 Dot Poduct... 13 Coss Poduct... 21 2007 Paul Dawkins i http://tutoial.math.lama.edu/tems.aspx
More information2 x 8 2 x 2 SKILLS Determine whether the given value is a solution of the. equation. (a) x 2 (b) x 4. (a) x 2 (b) x 4 (a) x 4 (b) x 8
5 CHAPTER Fundamentals When solving equations that involve absolute values, we usually take cases. EXAMPLE An Absolute Value Equation Solve the equation 0 x 5 0 3. SOLUTION By the definition of absolute
More informationOSCILLATIONS AND GRAVITATION
1. SIMPLE HARMONIC MOTION Simple hamonic motion is any motion that is equivalent to a single component of unifom cicula motion. In this situation the velocity is always geatest in the middle of the motion,
More informationLET a random variable x follows the two - parameter
INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING ISSN: 2231-5330, VOL. 5, NO. 1, 2015 19 Shinkage Bayesian Appoach in Item - Failue Gamma Data In Pesence of Pio Point Guess Value Gyan Pakash
More informationHua Xu 3 and Hiroaki Mukaidani 33. The University of Tsukuba, Otsuka. Hiroshima City University, 3-4-1, Ozuka-Higashi
he inea Quadatic Dynamic Game fo Discete-ime Descipto Systems Hua Xu 3 and Hioai Muaidani 33 3 Gaduate School of Systems Management he Univesity of suuba, 3-9- Otsua Bunyo-u, oyo -0, Japan xuhua@gssm.otsua.tsuuba.ac.jp
More informationBifurcation Analysis for the Delay Logistic Equation with Two Delays
IOSR Jounal of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume, Issue 5 Ve. IV (Sep. - Oct. 05), PP 53-58 www.iosjounals.og Bifucation Analysis fo the Delay Logistic Equation with Two Delays
More informationFractional Zero Forcing via Three-color Forcing Games
Factional Zeo Focing via Thee-colo Focing Games Leslie Hogben Kevin F. Palmowski David E. Robeson Michael Young May 13, 2015 Abstact An -fold analogue of the positive semidefinite zeo focing pocess that
More informationLecture 28: Convergence of Random Variables and Related Theorems
EE50: Pobability Foundations fo Electical Enginees July-Novembe 205 Lectue 28: Convegence of Random Vaiables and Related Theoems Lectue:. Kishna Jagannathan Scibe: Gopal, Sudhasan, Ajay, Swamy, Kolla An
More informationScattering in Three Dimensions
Scatteing in Thee Dimensions Scatteing expeiments ae an impotant souce of infomation about quantum systems, anging in enegy fom vey low enegy chemical eactions to the highest possible enegies at the LHC.
More informationOn the global uniform asymptotic stability of time-varying dynamical systems
Stud. Univ. Babeş-Bolyai Math. 59014), No. 1, 57 67 On the global unifom asymptotic stability of time-vaying dynamical systems Zaineb HajSalem, Mohamed Ali Hammami and Mohamed Mabouk Abstact. The objective
More information