What case should you bring to court to alter precedent in your favour - a strong or a weak case?

Size: px
Start display at page:

Download "What case should you bring to court to alter precedent in your favour - a strong or a weak case?"

Transcription

1 What cas should you bring to court to altr prcdnt in your favour - a strong or a wak cas? Hnrik Borchgrvink Dpartmnt of conomics, Univrsity of Oslo, PB 95 Blindrn, 37 Oslo, Norway Sptmbr, 9 Abstract Taking a cas to court in an unsttld lgal fild yilds nw prcdnt which affcts both subsqunt court cass and out of court sttlmnts. A long-run playr will sk to bring to court th cass that giv him th bst possibl prcdnt changs. This papr builds a modl to study th optimal cas to bring to court. Th modl shows that th optimal cas is compltly diffrnt undr two diffrnt, plausibl lgal rgims. JEL classification: K4 Kywords: Prcdnt; Slction for trial Introduction Evolution of prcdnt is affctd by th slction of cass prsntd to th court (s.g. Hadfild (99)). My papr studis what cas a disputant should bring to court to affct prcdnt in his favor. I show that th optimal cas to bring to court is compltly diffrnt undr two diffrnt, plausibl rgims. What cas a disputant would prfr bringing to court to affct prcdnt is so far not thoroughly studid in th litratur. Thanks to Ton Ogndal, Halvor Mhlum and Endr Stavang h.h.c.borchgrvink@con.uio.no, tl: , fax: This is surprising givn th larg litratur

2 on fficincy in volution of prcdnt in common law, s.g. Gnnaioli and Shlifr (7). My papr argus that a long-run playr can affct prcdnt volution, also in th long run, by slcting which cass to bring to court. Th argumnt rquirs an agnt who can control slction of court cass and who hav long-run intrst in affcting lgal prcdnt. Thr ar at last two typs who fulfill ths rquirmnts, stat authoritis who can choos in which cass whr thy will prss chargs, and firms who ar financially strong nough to bar unwantd cass through sttlmnt offrs. Th modl is prsntd through th cas of a firm. Th firm wants to minimiz th xpctd damag paymnts in th futur. Th firm is dfndant and choos whthr to sttl or tak th cas to court. Plaintiffs ar assumd on-shot playrs unintrstd in prcdnt, typically an individual. Prcdnt volution is modld as a procss whr nw prcdnt covrs a prviously uncovrd shar of th population of possibl futur cass. A cas covrd by prcdnt is commonly known to gt a crtain judgmnt (liabl or not liabl) with crtainty if it is takn to court. Th modl shows that of th cass not covrd by prcdnt, th optimal cas for th dfndant to bring to court to altr prcdnt in his favor is a strong cas (i.. wak vidnc against dfndant) in a lgal rgim whr judgs constantly adjust thir prsonal dcision standard for th cass uncovrd by prcdnt as prcdnt volvs. Th optimal cas is, on th contrary, a wak cas (strong vidnc against dfndant) in a lgal rgim whr judgs do not adjust thir idiosyncratic dcision standard as prcdnt volvs. Th rason for th optimal cas bing compltly diffrnt undr th two rgims is: Whn judgs do not adjust thir dcision standard thr is only th dirct ffct of th nw prcdnt covring mor cass. Whn judgs adjust thir dcision standard, thr is in addition an updating of th population of dcision standards for cass uncovrd by prcdnt. Th rsult holds for a broad rang of possibl distributions of th dcision standards of judgs. First, I build a modl and study what cas dfndant will bring in th vry first cas in court in a lgal fild without prcdnt. Thn, I us th rsults of th modl to discuss incntivs in th long run whn prcdnt can b changd mor than onc.

3 Th modl. St-up Th firm facs an xognous stram of cass against him,.g. suits for damags. For simplicity, assum thr ar on cas ach priod. Th disputd amount is J in all possibl cass. A cas has an vidnc lvl [, ]. Th cass ar indpndntly, uniformly distributd ovr this intrval. Highr mans a wors cas for dfndant (bttr for plaintiff). Th vidnc lvl is obsrvabl and vrifiabl... Prcdnt tchnology Th court judgmnt in th first cas brought to court,, binds all cass if was found not liabl, i.. dfndant won th cas, or binds all cass if was found liabl. This prcdnt tchnology is vry simpl, but capturs th natur of a cas law rgim with th star dcisis principl: Th law is nothing but th st of judgmnts mad in th past. For cass not covrd by prcdnt thr is a commonly known probability distribution dscribing th probability of a liabl judgmnt in court for ach vidnc lvl. Figur illustrats for th cas whr was found not liabl. Th cas whn is found liabl is paralll. Figur : Illustration of prcdnt and rmaining lgal uncrtainty. Two diffrnt lgal rgims I considr two diffrnt, plausibl lgal rgims. Thy diffr in how a nw prcdnt ffcts th liability probability distribution ovr th rmaining intrval whr thr is no 3

4 prcdnt. Th two mchanisms considrd rprsnt, arguably, th main two basically diffrnt mchanisms to considr in this stting. On rsmbls truncation and th othr rsmbls cnsoring. Th following is common for both rgims: Th judg who will dcid a cas takn to court is drawn from th population of judgs. If th cas is not covrd by prcdnt, th judg dcids th cas liabl if th vidnc lvl is largr than his idiosyncratic dcision standard. For a givn prcdnt ach judg has a givn idiosyncratic dcision standard. Dfin g( ) as th pdf of judgs idiosyncratic dcision standards bfor any prcdnt. Th support of g( ) is [, ]... Rlativists rgim: Rlativ dcision standards Each judg is ithr laissz-fair, and want all cass to b not liabl, or strict and want all cass to b liabl. Howvr, th judgs hav diffrnt idiosyncratic (political) cost of confronting th prcdnt by giving a cas rlativly clos to a liabl judgmnt or a cas rlativly clos to a not liabl judgmnt. Aftr th first prcdnt is mad, or is rplacd with th first cas. That is, th support of th distribution of judgs dcision standards is updatd to th updatd vidnc intrval without prcdnt,, if th first cas in court is found not liabl, and, if it is found liabl. Aftr th first prcdnt, th probability of a liabl judgmnt in court for any nw cas is thn as xprssd blow and illustratd in Figur and Figur 3. was found not liabl: F (;, ) = G(;, ) = g( )d < G(;, ) = g( )d < was found liabl: F (;, ) =.. Absolutists rgim: Absolut dcision standards In this rgim, judgs dcision standards do not chang with th nw prcdnt. In othr words, th support of th distribution of judgs dcision standards is [, ] irrspctiv of prcdnt. A judg acts in accordanc to his idiosyncratic dcision standard whn not 4

5 Rlativists rgim F(;,) Figur : Probability of a liabl judgmnt undr Rlativists rgim aftr a not liabl first cas. Rlativists rgim F(;,) Figur 3: Probability of a liabl judgmnt undr Rlativists rgim aftr a liabl first cas. 5

6 bound by prcdnt. A judg is bound by prcdnt whn prcdnt forcs him to giv a cas th opposit judgmnt of that of his own dcision standard. It follows that for a cas uncovrd by prcdnt, th probability of gtting a judg who finds th cas liabl, or not liabl, is th sam as bfor th first prcdnt. Aftr th first prcdnt, th probability of a liabl judgmnt in court for any nw cas is thn as xprssd blow and illustratd in Figur 4 and Figur 5. was found not liabl: F (;, ) = G(E) = g( )d < Absolutists rgim F(;,) Figur 4: Probability of a liabl judgmnt undr Absolutists rgim aftr a not liabl first cas. G(E) = g( )d < was found liabl: F (;, i ) = 6

7 Absolutists rgim F(;,) Figur 5: Probability of a liabl judgmnt undr Absolutists rgim aftr a liabl first cas...3 Symmtry For simplicity, I assum that g( ) is symmtric, so G(;, ) is symmtric undr both rgims. Undr Rlativists rgim G(.) is thn symmtric also aftr th first prcdnt. For th Rlativists rgim, symmtry implis that thr ar as many laissz-fair judgs as strict judgs and th cost distribution within ach group is idntical. For th Absolutists rgim, symmtry implis that th idiosyncratic dcision standards ar symmtrically distributd around th mid-point =...4 Comparison of th ffct of nw prcdnt undr th two rgims Undr Absolutists rgim thr is only th dirct prcdnt ffct from th cass that bcam covrd by th nw prcdnt. Undr Rlativists rgim, howvr, thr is an additional ffct ovr th cass still not covrd, du to th updating of th distribution of th judgs dcision standards. Th two ffcts undr Rlativists rgim work in th sam dirction: If is found not liabl, th dirct ffct is th mony dfndant savs for all cass blow and qual to. In addition, dfndant will pay lss for all cass with vidnc This is not a vry rstrictiv assumption as on is fr to dsign th scal of th support of g(.), [, ]. 7

8 lvl in th intrval,, bcaus ach judg s dcision standard is movd closr to as th vidnc intrval uncovrd by prcdnt is shortnd from blow. Similarly if is found liabl. 3 Tak to court or sttl? Dfndant has two objctivs whn a cas mrgs and h has to choos btwn taking th cas to court or sttling it: Minimiz paymnts to plaintiff in th particular cas and improv futur prcdnt. To focus on th prcdnt objctiv, th amount paid in sttlmnts is assumd qual to th xpctd damag pay if th cas was takn to court: F (;, ) J. F (.) is assumd common knowldg. Court costs ar ignord for simplicity. For th objctiv of minimizing damag paymnts to plaintiff in th particular cas in hand, th risk nutral dfndant is thus indiffrnt btwn court cas and sttlmnt. Plaintiff is assumd to accpt th sttlmnt amount, or at last ɛ mor: Plaintiff is by assumption not abl to xtract any of th valu of a potntial prcdnt chang from th dfndant. 3 Consquntly, dfndant controls th slction of cass to th court. For th objctiv of improving prcdnt th choic btwn sttlmnt and court cas mattrs. Futur damag paymnts in both sttlmnts and court cass ar changd if th cas is takn to court, confr th prsntation of th two rgims. Consquntly, dfndant can affct prcdnt through th slction of cass to court. If dfndant could not control whthr a cas gos to court or not, i. if som random cas wr takn to court to mak prcdnt, th prcdnt chang would also b random and thus x ant in ithr litigants favor bcaus th distribution of judgs is assumd symmtric. 3. Two priods Assum thr ar only two priods: Bfor and aftr th vry first cas. This simplification nabls m to dscrib problm of th dfndant without assuming particular forms of th judg population function g( ). Dfndant will sk to minimiz xpctd damag Avoiding futur court costs is clarly anothr incntiv for taking cass to court to obtain a prcdnt undr which sttlmnts can b mad in futur, s.g. Robson and Skaprdas (8). This incntiv is not th focus of my papr. 3 In othr words, dfndant has all bargaining powr. 8

9 paymnt in th scond priod by optimally choosing sttlmnt or court in th cas in th first priod. Th xpctd damag pay this priod and nxt priod undr th two options ar xprssd blow. Dfndant will choos th lowst of thm to minimiz his paymnts. For ach possibl cas nxt priod,, dfndant will pay F ( ;, ) J if is sttld outsid court and F ( ;, ) J or F ( ;, ) J if is takn to court. Th possibl cass nxt priod ar continuously and uniformly distributd ovr [, ], so th wight to b put on ach cas is th sam. Consquntly, whn comparing th two options sttl and tak to court, it is sufficint to hav βj F (.)d as xpctd damag pay for th cas which appars nxt priod. Sttl: S = F ( ;, ) J + βj F (;, )d () Tak to court: C = F ( ;, )[J + βj F (;, )d] + ( F ( ;, ))[ + βj F (;, )d] () This minimization problm can b simplifid in th following way. First, paymnt today, F ( ;, ) J, which is qual undr th two options, can b subtractd from both. Thn th xprssions only contain xpctd damag pay nxt priod. Th minimization problm is thn: min[ F (;, )d, F ( ;, )[ F (;, )d] + ( F ( ;, ))[ F (;, )d]] (3) In othr words, th option, sttl or tak to court, which minimizs damag paymnts is th on whr th xpctd siz of th ara undr F (.) is smallst. 3. Rlativists rgim Considr th cas and in Figur 6. If dfndant sttls th cas, th xpctd damag pay nxt priod is th sam as undr prvailing prcdnt bfor th cas matrializd. This xpctd damag pay is rprsntd by th ara undr th graph from to. If 9

10 dfndant taks th cas to th court and loss, th nw prcdnt is,. Th corrsponding nw damag pay is th ara undr th graph from to. This nw prcdnt implis an incras in xpctd pay nxt priod. In othr words, dfndant loss th ara L compard to th prvailing prcdnt. Similarly, if dfndant wins th cas, th nw prcdnt is, and dfndant wins th ara W. Th probability-wightd sum of L and W is what dfndant compars to no chang in prcdnt whn choosing sttl or tak to court. Rlativists rgim, Expctd damag paymnt as intgrals of F(.) J F(.),Damag paymnt L W Figur 6: Rlativists rgim. Th thr possibl prcdnts nxt priod. 3.3 Absolutists rgim Considr th cas in Figur 7 for illustration undr absolutists rgim. If dfndant loss th cas, th nw prcdnt is, and dfndant loss th ara L, bcaus only th judgs with idiosyncratic dcision standard btwn and will b unbound by prcdnt. Similarly, if dfndant wins th cas, h wins th ara W. Dfndant compars th probability-wightd sum of L and W to no prcdnt chang whn choosing sttl or tak to court.

11 Absolutists rgim, Expctd damag paymnt as intgrals of F(.) J F(.),Damag paymnt L W Figur 7: Absolutists rgim. Th thr possibl prcdnts nxt priod. 3.4 Dfndant s choic I will dscrib for which intrvals dfndant will want to tak th cas to court and not. A cas, < <, for which xpctd chang in damag pay nxt priod is zro, is calld ẽ. In th following I considr only symmtric distributions g(.) with no or on local maximum point, i.. g (.) for and g (.) for Rlativists rgim Undr Rlativists rgim it is asy to driv a gnral xprssion for th xpctd damag pay nxt priod. Dfndant pays J for cass if is found not liabl and J for cass if is found liabl. For cass in th intrval uncovrd by prcdnt, dfndant pays on avrag J, bcaus G(.) is symmtric. Thrfor xpctd damag pay nxt priod can b xprssd: is found not liabl F (;, )d = + ( ) = ( ) (4)

12 is found liabl F (;, )d = ( ) + ( ) (5) = Expctd chang in xpctd damag pay nxt priod from taking to court can thrfor b xprssd: F ( ;, ) F (;, )d + ( F ( ;, )) F (;, )d F (;, )d = F ( ;, )( ) + ( F ( ;, )) ( ) = (F ( ;, ) ) (6) Proposition Undr Rlativists rgim, i) Expctd chang in damag pay is zro for =, = and = ẽ =, ii) For uniform judg distribution g( ) xpctd chang in damag pay is zro for all, iii) For symmtric judg distribution g( ) with a mod, xpctd chang in damag pay is in dfndant s favor for cass < ẽ and in dfndant s disfavor for > ẽ. Proof. i) F (;, ) = and F (;, ) = mak Equation 6 zro. Bcaus F (;, ) is symmtric by assumption, F ( ;, ) =, so Equation 6 is zro for = ẽ =. ii) For uniform F (;, ), F (;, ) = for all, so Equation 6 is zro for all. iii) A symmtric F (;, ) with a mod yilds F (;, ) < and Equation 6< for < <, and F (;, ) > and Equation 6> for < <. Th rsult for symmtric judg distribution with a mod has th following intuition: If is clos to, losing th cas implis a vry bad prcdnt for dfndant, but th probability of losing is vry small. Th probability of winning th cas is larg and th rsulting prcdnt improvmnt is not only stablishing zro damag pay in th small intrval blow, but also to pay a littl lss in xpctation for all cass abov. Thrfor, dfndant has incntivs to bring to court cass blow ẽ =. In othr

13 words, undr Rlativists rgim, dfndant will bring to court only if it is mor likly that h wins th cas than loss it. For uniform distribution, th rsult that dfndant is indiffrnt btwn sttling and bringing to court any cas, is a consqunc of th xtrm symmtry of th uniform distribution. Figur 8 dpicts th situation of uniform distribution for an <. Subscript d in Figur 8 rfrs to th part of th chang which is du to dirct prcdnt covrag and u th part that is du to updating of th distribution of judgs. Expctd chang in damag paymnt is qual to zro du to symmtry: F ( )(L d + L u ) ( F ( ))(W d + W u ) = ( ) ( ) = (7) Rlativists rgim, Expctd damag paymnt as intgrals of F(.) J Ld F(.) F( ) Lu Wd Wu Figur 8: Rlativists rgim. Uniform judg distribution. From Figur 8 and Equation 7 it is asy to s what happns whn judg distribution is changd from uniform to a symmtric with mod: Th straight lins in Figur 8 bcom S-shapd. Not that L = L d + L u and W = W d + W u will b unchangd bcaus also th distribution with mod is symmtric. Howvr, F ( ) will dcras and thus ( F ( )) incrass. Consquntly, th vrsion of Equation 7 for distribution with mod will b ngativ, which mans that xpctd damag pay from taking to court is rducd. Figur 9 xmplifis using th triangl distribution. 3

14 Rlativists rgim, Triangl distribution J Rlativists rgim, Triangl distribution F(;,) Expctd chang in damag pay (a) (b) Figur 9: Rlativists rgim with triangl judg distribution: (a) Th cdf, (b) Expctd chang in damag pay [CHECK th triang with mov in not from /-8. Insrt mtring on y-axis on all thr] From Figur 8 and Equation 7 it is also asy to anticipat th rsult undr Absolutists rgim which is tratd in nxt subsction. Undr Absolutists rgim thr is no updating ffct, so L u = W u =. For Equation 7, that implis subtracting F ( )L u and adding ( F ( ))W u. Bcaus L u = W u, having sam ground lin and hight, whil F ( ) < ( F ( )), th modifid vrsion of Equation 7 will bcom positiv. That is, undr Absolutists rgim and uniform judg distribution xpctd chang in damag pay from taking < to court is positiv. In othr words, th opposit rsult of Rlativists rgim and symmtric judg distribution with mod Absolutists rgim Undr Absolutists rgim, an xprssion for th xpctd damag pay can b drivd in th following way. Bcaus th distribution of judgs g( ) is symmtric and [, ], th ara undr F (;, ) is qual to ( ) =. Aftr an has mad prcdnt, th rmaining ara is subtractd th ara cut off by th prcdnt. If was found liabl, th full pay intrval must b addd: is found not liabl F (;, )d (8) 4

15 is found liabl F (;, )d + (9) Givn ths xprssions for xpctd damag pay, th xpctd chang in damag pay from taking cas to court can b writtn: F ( ;, ) ( F (;, ))d ( F ( ;, )) F (;, )d () Proposition Undr Absolutists rgim, i) Expctd chang in damag pay is zro for =, = and = ẽ =, ii) For uniform g(.), xpctd chang in damag pay is in dfndant s favor for cass > ẽ and in dfndant s disfavor for < ẽ. iii) For symmtric judg distributions with mod, xpctd prcdnt chang is in dfndant s disfavor for cass clos to,, and in dfndant s favor for cass clos to,. Proof. i) F (;, ) = and F (;, ) = mak Equation zro. Bcaus F (;, ) is symmtric by assumption, F ( ;, ) =, so Equation is zro for = ẽ =. ii) For uniform distribution, Equation is: ( ) ( ) = ( 3 + ) which is > for < < and < for < <. iii) First, considr a vry small. Lt F ( ) = ɛ. Lt F (;, )d = δ ɛ. Thn Equation can b writtn ɛ( ( δ)) ( ɛ)δ = ɛ( + δ δ ) which is > as also δ is smallr than ɛ. ɛ Scond, considr th paralll cas for an qually clos to. Thn Equation can b writtn ( ɛ)δ ɛ( ( δ)) = ɛ( + δ + δ ) which is < for th sam rason. ɛ Th rsult for symmtric judg distribution with a mod has th following intuition: If is clos to, although th probability of winning th cas is vry high, th prcdnt gain from winning is vry small, sinc is clos to. Th vry larg prcdnt loss from losing th cas is dominating vn though probability of losing is vry low. For an clos to it is th othr way around. This rsult is valid for all judg distributions with a mod. Figur xmplifis using th triangl distribution. Howvr, xpctd chang in damag paymnt for around, has no gnral answr valid for all symmtric judg distributions with a mod. Of cours, distributions sufficintly clos to th uniform distribution will hav th sam rsult as uniform distribution, 5

16 Absolutists rgim, Triangl distribution J Absolutists rgim, Triangl distribution F(;,) Expctd chang in damag pay (a) (b) Figur : Absolutists rgim with triangl judg distribution: (a) Th cdf, (b) Expctd chang in damag pay i.. xpctd chang in damag pay changs sign only for =. Figur shows that th triangl distribution is sufficintly similar to th uniform distribution in this rspct. Howvr, th othr xtrm, whn th cumulativ dnsity function is vry stp for =, xpctd chang in damag pay will also incras for. In Figur (a), such a distribution is for simplicity xmplifid with a thr-pics-linar cdf. Th corrsponding xpctd chang in damag pay has two mor points whr xpctd chang in damag pay changs sign, s Figur (b). Not that also for this xtrm distribution for th bulk of th intrval [, ] it is still so that dfndant will want to bring cass h xpcts to los. Absolutists rgim Absolutists rgim F(;,) Expctd chang in damag pay (a) (b) Figur : Illustration of a judg distribution with stp cdf: (a) Th cdf, (b) Expctd chang in damag pay 6

17 4 Long run Thr ar implications from th two-priod modl for th outcom undr infinitly many priods. First, rdfin prcdnt to opn for continud prcdnt changs. Lt prcdnt b dfind by a pair l, h whr l is th so far highst vidnc lvl that has bn found not liabl in court and h is th so far lowst vidnc lvl that has bn found liabl in court. 4. Rlativists rgim Th probability of a liabl judgmnt in court for a givn cas i and prcdnt l, h undr this rgim is thn as xprssd blow and illustratd in Figur. i l i F ( i ; l, h) = G( i ; l, h) = g( l h l )d l < E < h l h i F(;l,h), Rlativists rgim J F(;l,h) l h Figur : Probability of a liabl judgmnt undr rlativists rgim Bcaus th population of judgs dcision standards ar constantly updatd to b symmtrically distributd ovr th vidnc intrval not covrd by prcdnt, l, h, dfndant will always hav incntivs to tak som cass to court, confr Proposition. Consquntly, prcdnt will in th long run convrg to covr all cass, unlss dfndant 7

18 has som costs of bringing cass to court that at som point will prvnt him from bringing mor cass bcaus th gain from nw prcdnt dcrass whn th vidnc intrval not covrd by prcdnt dcrass. Howvr, dfndant will not tak any cas with i < (l + h) to court. Dfndant will hav incntivs to wait for cass clos to l. H will wait longr, th mor wight h puts on futur damag paymnt, i.. whn β is high. Again this is du to th constant updating of th dcision standards. Th risk of losing and paying mor in th futur is lowr whn prcdnt is volving with smallr stps. To illustrat, F (l + δ; l, h) > F (l + δ ; l, h) + F (l + δ; l + δ, h), whr th vidnc intrval δ is covrd in on stp on th lft hand sid, and in two qual stps on th right hand sid. Givn that dfndant wins in both options in th inquality abov, h pays mor for th scond cas undr th right hand sid option, but if dfndant has sufficintly high β this is dominatd by lowr xpctd pay in futur bcaus of th lowr risk of losing th intrval (l + δ, h) for all futur. 4.. Absolutists rgim Th probability of a liabl judgmnt in court for a givn cas i and prcdnt l, h undr Absolutists rgim is as xprssd blow and illustratd in Figur 3. i l i F ( i ; l, h) = G( i ) = g( )d l < i < h h i If dfndant loss a cas i that h taks to court, h is lowrd to h = i. Consquntly, thr is thn lss to los in subsqunt cass brought to court. Whn thr is lss to los from taking a cas to th court, incntivs for taking cass to court incrass, so dfndant will dfinitly kp on taking som cass to court. Similarly, if dfndant wins a cas takn to court, thr will b lss to win in th futur, so incntivs to bring cass to court dclins. Nvrthlss, thr will always b som cass clos to h in th uncovrd lgal intrval which lowrs xpctd damag pay in futur. Similarly, cass clos to l will always hav xpctd incras in damag pay if takn to court. This follows of part iii) in Proposition from th two-priod modl. This part is valid for any prcdnt l, h: Equation for a cas i l, h bcoms: 8

19 F(;l,h), Absolutists rgim J F(;l,h) l h Figur 3: Probability of a liabl judgmnt undr absolutists rgim h F ( i ; l, h) i ( F (; l, h))d ( F ( i ; l, h)) i l F (; l, h)d () Equation is > for th cas of ( i = l + ɛ; l, h) and < for ( i = h ɛ; l, h) bcaus, in both cass, on of th intgrals bcoms infinitsimally small for infinitsimally small ɛ. In othr words, as long as thr ar cass not covrd by prcdnt, thr xists cass which dfndant will want to tak to th court. Bcaus thr is no updating of th population of judgs, th probability of losing dos nithr dclin nor incras by ltting prcdnt volv in small stps rathr than larg stps - contrary to undr Rlativists rgim. Howvr, also undr Absolutists rgim, dfndant has incntivs to not tak to court any cas that in xpctation improvs prcdnt. Dfndant will hav incntivs to wait for cass clos to h: In cass clos to h, dfndant gains most if h is lucky and wins, and h dos not los much if h loss. If dfndant in stad brings to court a cas furthr away from h (but still with xpctd rduction in damag pay), h incrass th probability of winning th cas. Howvr, if h loss it, h has for all futur lost th intrval btwn this cas and th formr h. It follows that th mor patint dfndant is and th longr th horizon, it is mor likly that dfndant only will bring to court cass clos to h, i.. waiting for a surprisingly 9

20 good prcdnt. 5 Discussion 5. Spd of volution Undr both rgims, thr will for any prcdnt l, h b cass that dfndant wants to bring to court. Thrfor, in th long run, both rgims will convrg to full prcdnt covrag of th vidnc intrval. Howvr, as th uncrtain lgal intrval shrinks with nw prcdnt, both th valu of nw prcdnt and th probability of gtting a cas insid th uncrtain lgal intrval shrink. In othr words, prcdnt volution will b slowr ovr tim. In particular, if a cas clos to h is won undr Absolutists rgim, thr will b a vry small intrval lft to win in futur prcdnt improvmnts and th intrval whr dfndant has incntiv to bring cass will b vn smallr, so furthr prcdnt volution will b vry slow. Not that if a cost of taking cass to court rlativ to sttlmnt is addd to th modl, thr could b uncrtain lgal intrval lft in th long run. 5. Th paralll cas of an authority Th rsults of th modl ar translatabl to th cas of an authority,.g. th antitrust authority, contmplating which cas to bring to court in an unsttld lgal fild. Bcaus such an authority acts as plaintiff whil th cas of a firm, as prsntd in th modl, acts as dfndant, th main translation will b to chang signs. In addition, th concivably mor complx objctiv function of a stat authority will mak th pictur mor complicatd. Not that in th cas of an authority suing a privat lgal ntity, thr is no nd to assum asymmtry btwn th two litigants with rspct to intrst in prcdnt. Evn if also th privat lgal ntity is intrstd in prcdnt, it is th authority who both chooss which cass to prss chargs in and whthr thy will accpt a sttlmnt offr. For th sam rason, it is nithr ncssary to assum that ithr party has th ntir bargaining powr in sttlmnts.

21 6 Conclusion I hav built a modl to answr whthr a dfndant with long trm intrst in prcdnt has incntivs to bring a wak or a strong cas to court whn altring prcdnt in his favor is th objctiv. Th main rsult of th modl is that th optimal kind of cas to bring to court hings on th lgal systm, in particular whthr or not judgs adjust thir idiosyncratic dcision standard as prcdnt volvs. Th analysis shows that a dfndant will want to bring cass h xpcts to win whn judgs adjust thir dcision standard, whras if judgs do not adjust thir idiosyncratic dcision standard th dfndant will bring cass h xpcts to los in a gambl for a surprisingly good prcdnt. In th vry long run, thr will b no lgal uncrtainty lft undr both rgims, but th long-run prcdnt will b path dpndnt and thus affctd by th slction of cass that has bn prsntd to th court. Rfrncs Gnnaioli, N., Shlifr, A., 7. Th volution of common law. Journal of Political Economy 5 (), Hadfild, G., 99. Bias in th volution of lgal ruls. Gorgtown Law Journal 8, Robson, A., Skaprdas, S., 8. Costly nforcmnt of proprty rights and th coas thorm. Economic Thory 36 (), 9 8.

4. Money cannot be neutral in the short-run the neutrality of money is exclusively a medium run phenomenon.

4. Money cannot be neutral in the short-run the neutrality of money is exclusively a medium run phenomenon. PART I TRUE/FALSE/UNCERTAIN (5 points ach) 1. Lik xpansionary montary policy, xpansionary fiscal policy rturns output in th mdium run to its natural lvl, and incrass prics. Thrfor, fiscal policy is also

More information

A Propagating Wave Packet Group Velocity Dispersion

A Propagating Wave Packet Group Velocity Dispersion Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to

More information

The van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012

The van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012 Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor

More information

22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.

22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches. Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M

More information

Coupled Pendulums. Two normal modes.

Coupled Pendulums. Two normal modes. Tim Dpndnt Two Stat Problm Coupld Pndulums Wak spring Two normal mods. No friction. No air rsistanc. Prfct Spring Start Swinging Som tim latr - swings with full amplitud. stationary M +n L M +m Elctron

More information

1 Minimum Cut Problem

1 Minimum Cut Problem CS 6 Lctur 6 Min Cut and argr s Algorithm Scribs: Png Hui How (05), Virginia Dat: May 4, 06 Minimum Cut Problm Today, w introduc th minimum cut problm. This problm has many motivations, on of which coms

More information

Estimation of apparent fraction defective: A mathematical approach

Estimation of apparent fraction defective: A mathematical approach Availabl onlin at www.plagiarsarchlibrary.com Plagia Rsarch Library Advancs in Applid Scinc Rsarch, 011, (): 84-89 ISSN: 0976-8610 CODEN (USA): AASRFC Estimation of apparnt fraction dfctiv: A mathmatical

More information

Supplementary Materials

Supplementary Materials 6 Supplmntary Matrials APPENDIX A PHYSICAL INTERPRETATION OF FUEL-RATE-SPEED FUNCTION A truck running on a road with grad/slop θ positiv if moving up and ngativ if moving down facs thr rsistancs: arodynamic

More information

That is, we start with a general matrix: And end with a simpler matrix:

That is, we start with a general matrix: And end with a simpler matrix: DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss

More information

Exam 1. It is important that you clearly show your work and mark the final answer clearly, closed book, closed notes, no calculator.

Exam 1. It is important that you clearly show your work and mark the final answer clearly, closed book, closed notes, no calculator. Exam N a m : _ S O L U T I O N P U I D : I n s t r u c t i o n s : It is important that you clarly show your work and mark th final answr clarly, closd book, closd nots, no calculator. T i m : h o u r

More information

u x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula

u x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula 7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting

More information

What are those βs anyway? Understanding Design Matrix & Odds ratios

What are those βs anyway? Understanding Design Matrix & Odds ratios Ral paramtr stimat WILD 750 - Wildlif Population Analysis of 6 What ar thos βs anyway? Undrsting Dsign Matrix & Odds ratios Rfrncs Hosmr D.W.. Lmshow. 000. Applid logistic rgrssion. John Wily & ons Inc.

More information

Ch. 24 Molecular Reaction Dynamics 1. Collision Theory

Ch. 24 Molecular Reaction Dynamics 1. Collision Theory Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic

More information

Addition of angular momentum

Addition of angular momentum Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th

More information

Alpha and beta decay equation practice

Alpha and beta decay equation practice Alpha and bta dcay quation practic Introduction Alpha and bta particls may b rprsntd in quations in svral diffrnt ways. Diffrnt xam boards hav thir own prfrnc. For xampl: Alpha Bta α β alpha bta Dspit

More information

Partial Derivatives: Suppose that z = f(x, y) is a function of two variables.

Partial Derivatives: Suppose that z = f(x, y) is a function of two variables. Chaptr Functions o Two Variabls Applid Calculus 61 Sction : Calculus o Functions o Two Variabls Now that ou hav som amiliarit with unctions o two variabls it s tim to start appling calculus to hlp us solv

More information

Two Products Manufacturer s Production Decisions with Carbon Constraint

Two Products Manufacturer s Production Decisions with Carbon Constraint Managmnt Scinc and Enginring Vol 7 No 3 pp 3-34 DOI:3968/jms9335X374 ISSN 93-34 [Print] ISSN 93-35X [Onlin] wwwcscanadant wwwcscanadaorg Two Products Manufacturr s Production Dcisions with Carbon Constraint

More information

Addition of angular momentum

Addition of angular momentum Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat

More information

First derivative analysis

First derivative analysis Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points

More information

Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.

Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J. Probability and Stochastic Procsss: A Frindly Introduction for Elctrical and Computr Enginrs Roy D. Yats and David J. Goodman Problm Solutions : Yats and Goodman,4.3. 4.3.4 4.3. 4.4. 4.4.4 4.4.6 4.. 4..7

More information

Why is a E&M nature of light not sufficient to explain experiments?

Why is a E&M nature of light not sufficient to explain experiments? 1 Th wird world of photons Why is a E&M natur of light not sufficint to xplain xprimnts? Do photons xist? Som quantum proprtis of photons 2 Black body radiation Stfan s law: Enrgy/ ara/ tim = Win s displacmnt

More information

Brief Introduction to Statistical Mechanics

Brief Introduction to Statistical Mechanics Brif Introduction to Statistical Mchanics. Purpos: Ths nots ar intndd to provid a vry quick introduction to Statistical Mchanics. Th fild is of cours far mor vast than could b containd in ths fw pags.

More information

Fourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.

Fourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation. Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform

More information

COMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.

COMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH. C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH

More information

The Ramsey Model. Reading: Firms. Households. Behavior of Households and Firms. Romer, Chapter 2-A;

The Ramsey Model. Reading: Firms. Households. Behavior of Households and Firms. Romer, Chapter 2-A; Th Ramsy Modl Rading: Romr, Chaptr 2-A; Dvlopd by Ramsy (1928), latr dvlopd furthr by Cass (1965) and Koopmans (1965). Similar to th Solow modl: labor and knowldg grow at xognous rats. Important diffrnc:

More information

LINEAR DELAY DIFFERENTIAL EQUATION WITH A POSITIVE AND A NEGATIVE TERM

LINEAR DELAY DIFFERENTIAL EQUATION WITH A POSITIVE AND A NEGATIVE TERM Elctronic Journal of Diffrntial Equations, Vol. 2003(2003), No. 92, pp. 1 6. ISSN: 1072-6691. URL: http://jd.math.swt.du or http://jd.math.unt.du ftp jd.math.swt.du (login: ftp) LINEAR DELAY DIFFERENTIAL

More information

Differentiation of Exponential Functions

Differentiation of Exponential Functions Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!

More information

Math 34A. Final Review

Math 34A. Final Review Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right

More information

Lecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields

Lecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration

More information

dr Bartłomiej Rokicki Chair of Macroeconomics and International Trade Theory Faculty of Economic Sciences, University of Warsaw

dr Bartłomiej Rokicki Chair of Macroeconomics and International Trade Theory Faculty of Economic Sciences, University of Warsaw dr Bartłomij Rokicki Chair of Macroconomics and Intrnational Trad Thory Faculty of Economic Scincs, Univrsity of Warsaw dr Bartłomij Rokicki Opn Economy Macroconomics Small opn conomy. Main assumptions

More information

cycle that does not cross any edges (including its own), then it has at least

cycle that does not cross any edges (including its own), then it has at least W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th

More information

EXST Regression Techniques Page 1

EXST Regression Techniques Page 1 EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy

More information

ME 321 Kinematics and Dynamics of Machines S. Lambert Winter 2002

ME 321 Kinematics and Dynamics of Machines S. Lambert Winter 2002 3.4 Forc Analysis of Linkas An undrstandin of forc analysis of linkas is rquird to: Dtrmin th raction forcs on pins, tc. as a consqunc of a spcifid motion (don t undrstimat th sinificanc of dynamic or

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!

More information

Search sequence databases 3 10/25/2016

Search sequence databases 3 10/25/2016 Sarch squnc databass 3 10/25/2016 Etrm valu distribution Ø Suppos X is a random variabl with probability dnsity function p(, w sampl a larg numbr S of indpndnt valus of X from this distribution for an

More information

Hydrogen Atom and One Electron Ions

Hydrogen Atom and One Electron Ions Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial

More information

Brief Notes on the Fermi-Dirac and Bose-Einstein Distributions, Bose-Einstein Condensates and Degenerate Fermi Gases Last Update: 28 th December 2008

Brief Notes on the Fermi-Dirac and Bose-Einstein Distributions, Bose-Einstein Condensates and Degenerate Fermi Gases Last Update: 28 th December 2008 Brif ots on th Frmi-Dirac and Bos-Einstin Distributions, Bos-Einstin Condnsats and Dgnrat Frmi Gass Last Updat: 8 th Dcmbr 8 (A)Basics of Statistical Thrmodynamics Th Gibbs Factor A systm is assumd to

More information

Limiting value of higher Mahler measure

Limiting value of higher Mahler measure Limiting valu of highr Mahlr masur Arunabha Biswas a, Chris Monico a, a Dpartmnt of Mathmatics & Statistics, Txas Tch Univrsity, Lubbock, TX 7949, USA Abstract W considr th k-highr Mahlr masur m k P )

More information

5.80 Small-Molecule Spectroscopy and Dynamics

5.80 Small-Molecule Spectroscopy and Dynamics MIT OpnCoursWar http://ocw.mit.du 5.80 Small-Molcul Spctroscopy and Dynamics Fall 008 For information about citing ths matrials or our Trms of Us, visit: http://ocw.mit.du/trms. Lctur # 3 Supplmnt Contnts

More information

Construction of asymmetric orthogonal arrays of strength three via a replacement method

Construction of asymmetric orthogonal arrays of strength three via a replacement method isid/ms/26/2 Fbruary, 26 http://www.isid.ac.in/ statmath/indx.php?modul=prprint Construction of asymmtric orthogonal arrays of strngth thr via a rplacmnt mthod Tian-fang Zhang, Qiaoling Dng and Alok Dy

More information

Propositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018

Propositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018 Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric Rodríguz-Carbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013 18.782 Introduction to Arithmtic Gomtry Fall 2013 Lctur #20 11/14/2013 20.1 Dgr thorm for morphisms of curvs Lt us rstat th thorm givn at th nd of th last lctur, which w will now prov. Thorm 20.1. Lt φ:

More information

The pn junction: 2 Current vs Voltage (IV) characteristics

The pn junction: 2 Current vs Voltage (IV) characteristics Th pn junction: Currnt vs Voltag (V) charactristics Considr a pn junction in quilibrium with no applid xtrnal voltag: o th V E F E F V p-typ Dpltion rgion n-typ Elctron movmnt across th junction: 1. n

More information

Mor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration

More information

The graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the

The graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th

More information

Economics 201b Spring 2010 Solutions to Problem Set 3 John Zhu

Economics 201b Spring 2010 Solutions to Problem Set 3 John Zhu Economics 20b Spring 200 Solutions to Problm St 3 John Zhu. Not in th 200 vrsion of Profssor Andrson s ctur 4 Nots, th charactrization of th firm in a Robinson Cruso conomy is that it maximizs profit ovr

More information

Answer Homework 5 PHA5127 Fall 1999 Jeff Stark

Answer Homework 5 PHA5127 Fall 1999 Jeff Stark Answr omwork 5 PA527 Fall 999 Jff Stark A patint is bing tratd with Drug X in a clinical stting. Upon admiion, an IV bolus dos of 000mg was givn which yildd an initial concntration of 5.56 µg/ml. A fw

More information

Extraction of Doping Density Distributions from C-V Curves

Extraction of Doping Density Distributions from C-V Curves Extraction of Doping Dnsity Distributions from C-V Curvs Hartmut F.-W. Sadrozinski SCIPP, Univ. California Santa Cruz, Santa Cruz, CA 9564 USA 1. Connction btwn C, N, V Start with Poisson quation d V =

More information

(Upside-Down o Direct Rotation) β - Numbers

(Upside-Down o Direct Rotation) β - Numbers Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 DOI: 10593/jajms0140400 (Upsid-Down o Dirct Rotation) β - Numbrs Ammar Sddiq Mahmood 1, Shukriyah Sabir Ali,* 1 Dpartmnt of Mathmatics, Collg

More information

Cramér-Rao Inequality: Let f(x; θ) be a probability density function with continuous parameter

Cramér-Rao Inequality: Let f(x; θ) be a probability density function with continuous parameter WHEN THE CRAMÉR-RAO INEQUALITY PROVIDES NO INFORMATION STEVEN J. MILLER Abstract. W invstigat a on-paramtr family of probability dnsitis (rlatd to th Parto distribution, which dscribs many natural phnomna)

More information

1.2 Faraday s law A changing magnetic field induces an electric field. Their relation is given by:

1.2 Faraday s law A changing magnetic field induces an electric field. Their relation is given by: Elctromagntic Induction. Lorntz forc on moving charg Point charg moving at vlocity v, F qv B () For a sction of lctric currnt I in a thin wir dl is Idl, th forc is df Idl B () Elctromotiv forc f s any

More information

2. Laser physics - basics

2. Laser physics - basics . Lasr physics - basics Spontanous and stimulatd procsss Einstin A and B cofficints Rat quation analysis Gain saturation What is a lasr? LASER: Light Amplification by Stimulatd Emission of Radiation "light"

More information

Problem Set 6 Solutions

Problem Set 6 Solutions 6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr

More information

REGISTER!!! The Farmer and the Seeds (a parable of scientific reasoning) Class Updates. The Farmer and the Seeds. The Farmer and the Seeds

REGISTER!!! The Farmer and the Seeds (a parable of scientific reasoning) Class Updates. The Farmer and the Seeds. The Farmer and the Seeds How dos light intract with mattr? And what dos (this say about) mattr? REGISTER!!! If Schrödingr s Cat walks into a forst, and no on is around to obsrv it, is h rally in th forst? sourc unknown Phys 1010

More information

Network Congestion Games

Network Congestion Games Ntwork Congstion Gams Assistant Profssor Tas A&M Univrsity Collg Station, TX TX Dallas Collg Station Austin Houston Bst rout dpnds on othrs Ntwork Congstion Gams Travl tim incrass with congstion Highway

More information

Classical Magnetic Dipole

Classical Magnetic Dipole Lctur 18 1 Classical Magntic Dipol In gnral, a particl of mass m and charg q (not ncssarily a point charg), w hav q g L m whr g is calld th gyromagntic ratio, which accounts for th ffcts of non-point charg

More information

Einstein Equations for Tetrad Fields

Einstein Equations for Tetrad Fields Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for

More information

Homotopy perturbation technique

Homotopy perturbation technique Comput. Mthods Appl. Mch. Engrg. 178 (1999) 257±262 www.lsvir.com/locat/cma Homotopy prturbation tchniqu Ji-Huan H 1 Shanghai Univrsity, Shanghai Institut of Applid Mathmatics and Mchanics, Shanghai 272,

More information

Higher order derivatives

Higher order derivatives Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of

More information

GEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES. Eduard N. Klenov* Rostov-on-Don, Russia

GEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES. Eduard N. Klenov* Rostov-on-Don, Russia GEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES Eduard N. Klnov* Rostov-on-Don, Russia Th articl considrs phnomnal gomtry figurs bing th carrirs of valu spctra for th pairs of th rmaining additiv

More information

u 3 = u 3 (x 1, x 2, x 3 )

u 3 = u 3 (x 1, x 2, x 3 ) Lctur 23: Curvilinar Coordinats (RHB 8.0 It is oftn convnint to work with variabls othr than th Cartsian coordinats x i ( = x, y, z. For xampl in Lctur 5 w mt sphrical polar and cylindrical polar coordinats.

More information

Chapter 13 Aggregate Supply

Chapter 13 Aggregate Supply Chaptr 13 Aggrgat Supply 0 1 Larning Objctivs thr modls of aggrgat supply in which output dpnds positivly on th pric lvl in th short run th short-run tradoff btwn inflation and unmploymnt known as th Phillips

More information

Random Access Techniques: ALOHA (cont.)

Random Access Techniques: ALOHA (cont.) Random Accss Tchniqus: ALOHA (cont.) 1 Exampl [ Aloha avoiding collision ] A pur ALOHA ntwork transmits a 200-bit fram on a shard channl Of 200 kbps at tim. What is th rquirmnt to mak this fram collision

More information

Basic Polyhedral theory

Basic Polyhedral theory Basic Polyhdral thory Th st P = { A b} is calld a polyhdron. Lmma 1. Eithr th systm A = b, b 0, 0 has a solution or thr is a vctorπ such that π A 0, πb < 0 Thr cass, if solution in top row dos not ist

More information

Collisions between electrons and ions

Collisions between electrons and ions DRAFT 1 Collisions btwn lctrons and ions Flix I. Parra Rudolf Pirls Cntr for Thortical Physics, Unirsity of Oxford, Oxford OX1 NP, UK This rsion is of 8 May 217 1. Introduction Th Fokkr-Planck collision

More information

EEO 401 Digital Signal Processing Prof. Mark Fowler

EEO 401 Digital Signal Processing Prof. Mark Fowler EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1

More information

A Prey-Predator Model with an Alternative Food for the Predator, Harvesting of Both the Species and with A Gestation Period for Interaction

A Prey-Predator Model with an Alternative Food for the Predator, Harvesting of Both the Species and with A Gestation Period for Interaction Int. J. Opn Problms Compt. Math., Vol., o., Jun 008 A Pry-Prdator Modl with an Altrnativ Food for th Prdator, Harvsting of Both th Spcis and with A Gstation Priod for Intraction K. L. arayan and. CH. P.

More information

Introduction - the economics of incomplete information

Introduction - the economics of incomplete information Introdction - th conomics of incomplt information Backgrond: Noclassical thory of labor spply: No nmploymnt, individals ithr mployd or nonparticipants. Altrnativs: Job sarch Workrs hav incomplt info on

More information

SECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.

SECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero. SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain

More information

The Open Economy in the Short Run

The Open Economy in the Short Run Economics 442 Mnzi D. Chinn Spring 208 Social Scincs 748 Univrsity of Wisconsin-Madison Th Opn Economy in th Short Run This st of nots outlins th IS-LM modl of th opn conomy. First, it covrs an accounting

More information

Part 7: Capacitance And Capacitors

Part 7: Capacitance And Capacitors Part 7: apacitanc And apacitors 7. Elctric harg And Elctric Filds onsidr a pair of flat, conducting plats, arrangd paralll to ach othr (as in figur 7.) and sparatd by an insulator, which may simply b air.

More information

CS 361 Meeting 12 10/3/18

CS 361 Meeting 12 10/3/18 CS 36 Mting 2 /3/8 Announcmnts. Homwork 4 is du Friday. If Friday is Mountain Day, homwork should b turnd in at my offic or th dpartmnt offic bfor 4. 2. Homwork 5 will b availabl ovr th wknd. 3. Our midtrm

More information

10. The Discrete-Time Fourier Transform (DTFT)

10. The Discrete-Time Fourier Transform (DTFT) Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w

More information

VALUING SURRENDER OPTIONS IN KOREAN INTEREST INDEXED ANNUITIES

VALUING SURRENDER OPTIONS IN KOREAN INTEREST INDEXED ANNUITIES VALUING SURRENDER OPTIONS IN KOREAN INTEREST INDEXED ANNUITIES Changi Kim* * Dr. Changi Kim is Lcturr at Actuarial Studis Faculty of Commrc & Economics Th Univrsity of Nw South Wals Sydny NSW 2052 Australia.

More information

ph People Grade Level: basic Duration: minutes Setting: classroom or field site

ph People Grade Level: basic Duration: minutes Setting: classroom or field site ph Popl Adaptd from: Whr Ar th Frogs? in Projct WET: Curriculum & Activity Guid. Bozman: Th Watrcours and th Council for Environmntal Education, 1995. ph Grad Lvl: basic Duration: 10 15 minuts Stting:

More information

CPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming

CPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming CPSC 665 : An Algorithmist s Toolkit Lctur 4 : 21 Jan 2015 Lcturr: Sushant Sachdva Linar Programming Scrib: Rasmus Kyng 1. Introduction An optimization problm rquirs us to find th minimum or maximum) of

More information

Principles of Humidity Dalton s law

Principles of Humidity Dalton s law Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid

More information

4.2 Design of Sections for Flexure

4.2 Design of Sections for Flexure 4. Dsign of Sctions for Flxur This sction covrs th following topics Prliminary Dsign Final Dsign for Typ 1 Mmbrs Spcial Cas Calculation of Momnt Dmand For simply supportd prstrssd bams, th maximum momnt

More information

Linear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let

Linear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let It is impossibl to dsign an IIR transfr function with an xact linar-phas It is always possibl to dsign an FIR transfr function with an xact linar-phas rspons W now dvlop th forms of th linarphas FIR transfr

More information

Homework #3. 1 x. dx. It therefore follows that a sum of the

Homework #3. 1 x. dx. It therefore follows that a sum of the Danil Cannon CS 62 / Luan March 5, 2009 Homwork # 1. Th natural logarithm is dfind by ln n = n 1 dx. It thrfor follows that a sum of th 1 x sam addnd ovr th sam intrval should b both asymptotically uppr-

More information

Continuous probability distributions

Continuous probability distributions Continuous probability distributions Many continuous probability distributions, including: Uniform Normal Gamma Eponntial Chi-Squard Lognormal Wibull EGR 5 Ch. 6 Uniform distribution Simplst charactrizd

More information

Computing and Communications -- Network Coding

Computing and Communications -- Network Coding 89 90 98 00 Computing and Communications -- Ntwork Coding Dr. Zhiyong Chn Institut of Wirlss Communications Tchnology Shanghai Jiao Tong Univrsity China Lctur 5- Nov. 05 0 Classical Information Thory Sourc

More information

Equidistribution and Weyl s criterion

Equidistribution and Weyl s criterion Euidistribution and Wyl s critrion by Brad Hannigan-Daly W introduc th ida of a sunc of numbrs bing uidistributd (mod ), and w stat and prov a thorm of Hrmann Wyl which charactrizs such suncs. W also discuss

More information

Forces. Quantum ElectroDynamics. α = = We have now:

Forces. Quantum ElectroDynamics. α = = We have now: W hav now: Forcs Considrd th gnral proprtis of forcs mdiatd by xchang (Yukawa potntial); Examind consrvation laws which ar obyd by (som) forcs. W will nxt look at thr forcs in mor dtail: Elctromagntic

More information

SCHUR S THEOREM REU SUMMER 2005

SCHUR S THEOREM REU SUMMER 2005 SCHUR S THEOREM REU SUMMER 2005 1. Combinatorial aroach Prhas th first rsult in th subjct blongs to I. Schur and dats back to 1916. On of his motivation was to study th local vrsion of th famous quation

More information

Precise Masses of particles

Precise Masses of particles /1/15 Physics 1 April 1, 15 Ovrviw of topic Th constitunts and structur of nucli Radioactivity Half-lif and Radioactiv dating Nuclar Binding Enrgy Nuclar Fission Nuclar Fusion Practical Applications of

More information

1 Isoparametric Concept

1 Isoparametric Concept UNIVERSITY OF CALIFORNIA BERKELEY Dpartmnt of Civil Enginring Spring 06 Structural Enginring, Mchanics and Matrials Profssor: S. Govindj Nots on D isoparamtric lmnts Isoparamtric Concpt Th isoparamtric

More information

6.1 Integration by Parts and Present Value. Copyright Cengage Learning. All rights reserved.

6.1 Integration by Parts and Present Value. Copyright Cengage Learning. All rights reserved. 6.1 Intgration by Parts and Prsnt Valu Copyright Cngag Larning. All rights rsrvd. Warm-Up: Find f () 1. F() = ln(+1). F() = 3 3. F() =. F() = ln ( 1) 5. F() = 6. F() = - Objctivs, Day #1 Studnts will b

More information

There is an arbitrary overall complex phase that could be added to A, but since this makes no difference we set it to zero and choose A real.

There is an arbitrary overall complex phase that could be added to A, but since this makes no difference we set it to zero and choose A real. Midtrm #, Physics 37A, Spring 07. Writ your rsponss blow or on xtra pags. Show your work, and tak car to xplain what you ar doing; partial crdit will b givn for incomplt answrs that dmonstrat som concptual

More information

Application of Vague Soft Sets in students evaluation

Application of Vague Soft Sets in students evaluation Availabl onlin at www.plagiarsarchlibrary.com Advancs in Applid Scinc Rsarch, 0, (6):48-43 ISSN: 0976-860 CODEN (USA): AASRFC Application of Vagu Soft Sts in studnts valuation B. Chtia*and P. K. Das Dpartmnt

More information

PHYS-333: Problem set #2 Solutions

PHYS-333: Problem set #2 Solutions PHYS-333: Problm st #2 Solutions Vrsion of March 5, 2016. 1. Visual binary 15 points): Ovr a priod of 10 yars, two stars sparatd by an angl of 1 arcsc ar obsrvd to mov through a full circl about a point

More information

Chapter 8: Electron Configurations and Periodicity

Chapter 8: Electron Configurations and Periodicity Elctron Spin & th Pauli Exclusion Principl Chaptr 8: Elctron Configurations and Priodicity 3 quantum numbrs (n, l, ml) dfin th nrgy, siz, shap, and spatial orintation of ach atomic orbital. To xplain how

More information

Chapter 14 Aggregate Supply and the Short-run Tradeoff Between Inflation and Unemployment

Chapter 14 Aggregate Supply and the Short-run Tradeoff Between Inflation and Unemployment Chaptr 14 Aggrgat Supply and th Short-run Tradoff Btwn Inflation and Unmploymnt Modifid by Yun Wang Eco 3203 Intrmdiat Macroconomics Florida Intrnational Univrsity Summr 2017 2016 Worth Publishrs, all

More information

orbiting electron turns out to be wrong even though it Unfortunately, the classical visualization of the

orbiting electron turns out to be wrong even though it Unfortunately, the classical visualization of the Lctur 22-1 Byond Bohr Modl Unfortunatly, th classical visualization of th orbiting lctron turns out to b wrong vn though it still givs us a simpl way to think of th atom. Quantum Mchanics is ndd to truly

More information

Contemporary, atomic, nuclear, and particle physics

Contemporary, atomic, nuclear, and particle physics Contmporary, atomic, nuclar, and particl physics 1 Blackbody radiation as a thrmal quilibrium condition (in vacuum this is th only hat loss) Exampl-1 black plan surfac at a constant high tmpratur T h is

More information

First order differential equation Linear equation; Method of integrating factors

First order differential equation Linear equation; Method of integrating factors First orr iffrntial quation Linar quation; Mtho of intgrating factors Exampl 1: Rwrit th lft han si as th rivativ of th prouct of y an som function by prouct rul irctly. Solving th first orr iffrntial

More information

Differential Equations

Differential Equations UNIT I Diffrntial Equations.0 INTRODUCTION W li in a world of intrrlatd changing ntitis. Th locit of a falling bod changs with distanc, th position of th arth changs with tim, th ara of a circl changs

More information

SCALING OF SYNCHROTRON RADIATION WITH MULTIPOLE ORDER. J. C. Sprott

SCALING OF SYNCHROTRON RADIATION WITH MULTIPOLE ORDER. J. C. Sprott SCALING OF SYNCHROTRON RADIATION WITH MULTIPOLE ORDER J. C. Sprott PLP 821 Novmbr 1979 Plasma Studis Univrsity of Wisconsin Ths PLP Rports ar informal and prliminary and as such may contain rrors not yt

More information

The Transmission Line Wave Equation

The Transmission Line Wave Equation 1//5 Th Transmission Lin Wav Equation.doc 1/6 Th Transmission Lin Wav Equation Q: So, what functions I (z) and V (z) do satisfy both tlgraphr s quations?? A: To mak this asir, w will combin th tlgraphr

More information