About Hydrodynamic Limit of Some Exclusion Processes via Functional Integration
|
|
- Mildred Lester
- 6 years ago
- Views:
Transcription
1 Abou Hydrodynamc Lm of Some Excluson Processes va Funconal Inegraon Guy Fayolle Cyrl Furlehner Absrac Ths arcle consders some classes of models dealng wh he dynamcs of dscree curves subjeced o sochasc deformaons. I urns ou ha he problems of neres can be se n erms of neracng excluson processes, he ulmae goal beng o derve hydrodynamc lms afer proper scalngs. A seemngly new mehod s proposed, whch reles on he analyss of specfc paral dfferenal operaors, nvolvng varaonal calculus and funconal negraon: ndeed, he varables are he values of some funcons a gven pons, he number of whch ends o become nfne, whch requres he consrucon of generalzed measures. Sarng from a dealed analyss of he asep sysem on he orus Z/Z, we clam ha he argumens a pror work n hgher dmensons (abc, mul-ype excluson processes, ec, leadng o syems of coupled paral dfferenal equaons of Burgers ype. Keywords Cauchy problem, excluson process, funconal negraon, hydrodynamc lm, marngale, weak soluon. 1 Prelmnares Inerplay beween dscree and connuous descrpon s a recurren queson n sascal physcs, whch n some cases can be answered que rgorously va probablsc mehods. In he conex of reacon-dffuson sysems, hs s anamoun o sudyng flud or hydrodynamcs lms. umber of approaches have been proposed, n parcular n he framework of excluson processes, see e.g. [15,[4 [18, [14 and references heren. As far as flud or hydrodynamc lms are a sake, mos of hese mehods have n common o IRIA - Domane de Voluceau, Rocquencour BP Le Chesnay Cedex - France. Conac: Guy.Fayolle@nra.fr 1
2 2 2 Model defnon be lmed o sysems for whch he saonary saes are gven n closed produc forms, or a leas for whch he nvaran measure for fne (he sze of he sysem s explcly known. For nsance, asep wh open boundary can be descrbed n erms of marx produc form (a sor of a non-commuave produc form and he connuous lms can be undersood by means of brownan brdges (see [5. We propose o address hs queson from he followng dfferen pon of vew: sarng from dscree sample pahs subjeced o sochasc deformaons, he ulmae goal s o undersand he naure of he lm curves when ncreases o nfny. How do hese curves evolve wh me, and whch lmng process do hey represen as goes o nfny (equlbrum curves? Followng [1, 11, 12, we plan o gve some paral answers o hese problems n a seres of forhcomng papers. The mehod proposed n he presen sudy s appled n deal o he asep model. The mahemacal kernel reles on he analyss of specfc paral dfferenal equaons nvolvng varaonal calculus. A usual sequence of emprcal measures s shown o converge n probably o a deermnsc measure, whch s he unque weak soluon of a Cauchy problem. Here varables are n fac he values of some funcon a gven pons, and her number becomes nfne. In our opnon, he approach presens some new feaures, and very lkely exend o hgher dmensons, namely mul-ype excluson processes. A fuure concern wll be o esablsh a complee herarchy of sysems of hydrodynamc equaons, he sudy of whch should allow us o descrbe non-gbbs saes. All hese quesons form also he maer of ongong works. 2 Model defnon 2.1 A general sochasc clock model Consder an orened sample pah of a planar random walk n R 2, conssng of seps (or lnks of equal sze. Each sep can have n dscree possble orenaons, drawn from he se of angles wh some gven orgn {θ k = 2kπ n, k =,..., n 1}. The sochasc dynamcs n force consss n dsplacng one sngle pon a a me whou breakng he pah, whle keepng all lnks whn he se of admssble orenaons. In hs operaon, wo lnks are smulaneously dsplaced, wha consrans que srongly he possble dynamcal rules Consrucng a relaed connuous-me Markov chan Jumps are produced by ndependen exponenal evens.
3 2.2 Examples 3 Perodc boundary condons wll be assumed, hs pon beng no a crucal resrcon. Dynamcal rules are gven by a se of reacons beween consecuve lnks, an equvalen formulaon beng possble n erms of random grammar. Wh each lnk s assocaed a ype,.e. a leer of an alphabe. Hence, for any n, we can defne whch laer on wll be somemes referred o as a local exchange process. For [1, and k [1, n, le X k represen a lnk of ype k a se. Then we can defne he followng se of reacons. X k X l λ kl +1 XX l +1, k k = 1,..., n, l k + n λ lk 2, (2.1 X k X k+n/2 γ k +1 X k+n/2+1 +1, k = 1,..., n. δ k+1 X k+1 The red equaons does exs only for even n, because of he exsence of folds [wo consecuve lnks wh oppose drecons, whch yeld a rcher dynamcs. X k can also be vewed as a bnary random varable descrbng he occupaon of se by a leer of ype k. Hence, he sae space of he sysem s represened by he array η def = {X k, = 1,..., ; k = 1,..., n}. 2.2 Examples (1 The smple excluson process The frs elemenary and mos suded example s he smple excluson process: hs model, afer mappng parcles ono lnks, corresponds o a onedmensonal flucuang nerface. Here we ake a bnary alphabe and, leng X 1 = τ and X 2 = τ, he se of reacons smply rewres τ τ λ λ + ττ, where λ ± are he ranson raes for he jump of a parcle o he rgh or o he lef. (2 The rangular lace and he ABC model Here he evoluon of he random walk s resrced o he rangular lace. Each lnk (or sep of he walk s eher 1, e 2π/3 or e 4π/3, and que naurally
4 4 3 Hydrodynamcs for he basc asymmerc excluson process (ASEP wll be sad o be of ype A, B or C, respecvely. Ths corresponds o he so-called ABC model, snce here s a codng by means of a 3-leer alphabe. The se of ransons (or reacons s gven by AB p BA, BC q CB, CA r AC, (2.2 p + q + r + where here s a pror no symmery, bu we wll mpose perodc boundary condons on he sample pahs. Ths model was frs nroduced n [8 n he conex of parcles wh excluson, and for some cases correspondng o he reversbly of he process, a Gbbs form for he nvaran measure was gven n [9 3 Hydrodynamcs for he basc asymmerc excluson process (ASEP As menoned above, we am a obanng hydrodynamc equaons for a class of excluson models. The mehod, alhough relyng on classcal powerful ools (marngales, relave compacness of measures, funconal analyss, has some new feaures whch should hopefully prove fruful n oher conexs. The essence of he approach s n fac conaned n he analyss of he popular asep model, presened below. We noe he dffculy o fnd n he exsng leraure a complee sudy encompassng varous specal cases (symmery, oal or weak asymmery, ec. Consder ses labelled from 1 o, formng a dscree closed curve n he plane, so ha he numberng of ses s mplcly aken modulo,.e. on ( def he dscree orus G = Z/Z. In hgher dmenson, say on he lace Z k, he relaed se of ses would be drawn on he orus (Z/Z k. We gaher below some noaonal maeral vald hroughou hs paper. R (resp. R + sands for he real (resp. posve real lne. C k [, 1 s he collecon of all real-valued, k-connuously dfferenable funcons defned on he nerval [, 1, and M s he space of all fne posve measures on he orus G def = [, 1. For S an arbrary merc space, P(S s he se of probably measures on S, and D S [, T s he space of rgh connuous funcons z : [, S wh lef lms and z. C (K s he space of nfnely dfferenable funcons wh compac suppor ncluded n K S, and we shall wre C[T o denoe he subse of funcons φ(x, C ([, 1 [, T vanshng a = T.
5 5 For = 1,...,, le A ( ( and B ( ( be bnary random varables represenng respecvely a parcule or a hole a se, so ha, owng o he excluson consran, A ( ( + B ( ( = 1, for all 1. Thus { A ( ( def = ( A ( (,..., A ( (, } s a Markov process. Ω ( F ( wll denoe he generaor of he Markov process A ( (, and = σ ( A ( (s, s s he assocaed naural flraon. Our purpose s o analyze he sequence of emprcal random measures µ ( = 1 A ( G ( (δ, (3.1 when, afer a convenen scalng of he parameers of he generaor Ω (. The probably dsrbuon assocaed wh he pah of he Markov process µ (, [, T, for some fxed T, wll be smply denoed by Q (. As usual, one can embed G ( n G, so ha a pon G ( corresponds o he pon / n G. Hence, n vew of (3.1, s que naural o le he sequence Q ( be defned on a unque space D M [, T, whch becomes a polsh space (.e. complee and separable va he usual Skorokod opology, as soon as M s self Polsh (see e.g. [7, chaper 3. Whou furher commen, M s assumed o be endowed wh he vague produc opology, as a consequence of he famous Banach-Alaoglo and Tychonoff heorems (see e.g. [17, 13. Choose wo arbrary funcons φ a, φ b C[T and defne he followng realvalued posve measure [ 1 ( ( Z ( [φ a, φ b def = exp, A ( ( + φ b, B ( (, (3.2 G ( φ a vewed as a funconal of φ a, φ b. Snce A ( ( + B ( ( = 1, for 1, he ransform Z ( s essenally a funconal of he sole funcon φ a φ b, up o a consan unformly bounded n. everheless, wll appear laer ha we need 2 ndependen funcons. For he sake of brevy, he explc dependence on, or φ, of quanes lke A ( (, B ( (, Z ( [φ a, φ b, wll frequenly be omed, wherever he meanng remans clear from he conex: for nsance, we smply wre A, B or Z (. Also Z ( sands for he process {Z (, }. A sandard powerful mehod o prove he convergence (n a sense o be specfed laer of he sequence of probably measures nroduced n (3.1 consss frs n showng s relave compacness, and hen n verfyng he concdence of all possble lm pons (see e.g. [14. Moreover here, by he
6 6 3 Hydrodynamcs for he basc asymmerc excluson process (ASEP choce of he funcons φ a, φ b, suffces o prove hese wo properes for he sequence of projeced measures defned on D R [, T and correspondng o he processes {Z ( [φ a, φ b, }. Le us now nroduce quanes whch, as far as scalng s concerned, are crucal n order o oban meanngful hydrodynamc equaons. λ( def = λab ( + λ ba (, 2 (3.3 µ( def = λ ab ( λ ba (, where he dependence of he raes on s explcly menoned. Theorem 3.1. Le he sysem (3.3 have a gven asympoc expanson of he form, for large, λ( def = λ 2 + o( 2, (3.4 µ( def = µ + o(, where λ and µ are fxed consans. [As for he scalng assumpon (3.4, he random measure log Z ( s a funconal of he underlyng Markov process, n whch he me has been speeded up by a facor 2 and he space shrunk by 1. Assume also he sequence of nal emprcal measures log Z (, aken a me =, converges n probably o some deermnsc measure wh a gven densy ρ(x,, so ha, n probably, ( lm log Z = 1 for any par of funcons φ a, φ b C[T. [ρ(x, φ a (x, + (1 ρ(x, φ b (x, dx, (3.5 Then, for every >, he sequence of random measures µ ( converges n probably, as, o a deermnsc measure havng a densy ρ(x, wh respec o he Lebesgue measure, whch s he unque weak soluon of he Cauchy problem T 1 [ ( φ(x, ρ(x, + λ 2 φ(x, x 2 µρ(x, ( 1 ρ(x, φ(x, x where (3.6 holds for any funcon φ C[T. + 1 ρ(x, φ(x, dx =, dxd (3.6
7 3.1 Exsence of lm pons: sequenal compacness 7 If, moreover, one assumes he exsence of 2 ρ(x,, for ρ(x, gven, hen x 2 (3.6 reduces o a classcal Burgers equaon ρ(x, = λ 2 ρ(x, ρ(x, x 2 + µ[1 2ρ(x,. x Proof. The proof s conaned n he nex hree subsecons. 3.1 Exsence of lm pons: sequenal compacness As usual n problems dealng wh convergence of sequences of probably measures, our very sarng pon wll be o esablsh he weak relave compacness of he se {log Z (, 1}. Some of he probablsc argumens employed n hs paragraph are classcal and can be found n he leraure, e.g. [18, 14, alhough for slghly dfferen or smpler models. Leng φ a, φ b be wo arbrary funcons n C[T, we refer o equaon (3.2. Usng he exponenal form of Z ( and Lemma [A1-5.1 n [14 (see also chaper 3 n [7 for relaed calculus, one can easly check ha he wo followng random processes U ( V ( def = Z ( Z ( def = (U ( 2 ( ( Ω ( [Z ( s Ω ( [(Z ( s are bounded {F ( }-marngales, where [ φa (, θ ( Seng now def = 1 ψ xy ψ xy (, we have where λ ( xy (, G ( A ( def = φ x φ y = ψ yx, + θ ( s Z s ( ds, ( Z ( Ω ( [Z ( ds (3.8 ( + φ ( b, ( def + 1 ( = ψ xy, ψ xy,, ( 1 def = λ xy ( L ( def = [ exp Ω ( [Z ( λ( G ( ψ xy = L ( ab (, A B +1 + s B ( s (. (3.9 (, 1, xy = ab or ba, Z (, (3.1 λ ( ba (, B A +1. (3.11
8 8 3 Hydrodynamcs for he basc asymmerc excluson process (ASEP On he oher hand, a sraghforward calculaon n equaon (3.11 allows o rewre (3.8 n he form V ( where he process R ( R ( = G ( = (U ( 2 (Z ( s s srcly posve and gven by [ λ ( ab (, 2 A B +1 + λ ab ( 2 R s ( ds, (3.12 ( [ λ ba (, 2 λ ba ( B A +1. The negral erm n (3.12 s nohng else bu he ncreasng process assocaed wh Doob s decomposon of he submarngale (U ( 2. The folllowng esmaes are crucal. Lemma 3.2. L ( = O(1, (3.13 R ( = O ( 1. (3.14 Proof. We wll derve (3.13 by esmang he rgh-hand sde member of equaon (3.11. From now on, for he sake of shorness, he frs and second paral dervaves of ψ(z, wh respec o z wll be denoed respecvely by ψ (z, and ψ (z,. ( Clearly, ψ xy, = 1 xy( ψ, + O ( 1. Then, akng a second order 2 expanson of he exponenal funcon and usng equaons (3.3 and (3.4, we can rewre (3.11 as L ( = µ( + λ( G ( [ A + A +1 2 G ( (A A +1 ψ ab A A +1 ψ ab (, (, ( 1 + O. (3.15 The frs sum on he rgh n (3.15 s unformly bounded by a consan dependng on ψ. Indeed A 1, and ψ s of bounded varaon snce ψ C[T. As for he second sum comng n (3.15, we have ( (A A +1 ψ ab, = [ ( + 1 ( A +1 ψ ab, ψ ab,. G ( G ( Then he dscree Laplacan ψ ab ( + 1, ψ ab (, ψ ab ( + 2, 2ψ ab ( + 1, +ψ ab (,
9 3.1 Exsence of lm pons: sequenal compacness 9 adms of he smple expanson ( + 1 ( ψ ab, ψ ab, = 1 ( ( 1 2 ψ ab, + O 2. (3.16 By (3.4, λ( = λ 2 + o( 2, so ha (3.16 mples λ( G ( (A A +1 ψ ab (, = G ( λa ( +1 ( 1 ψ ab, + o = O(1, (3.17 whch concludes he proof of (3.13. The compuaon of R ( (3.14 can be obaned va smlar argumens, remarkng ha leadng o R ( [φ a, φ b = L ( [2φ a, 2φ b 2L ( [φ a, φ b. To show he relave compacness of he famly Z (, whch from he separably and he compleeness of he underlyng spaces s here equvalen o ghness, we proceed as n [14 by means of he followng useful creron. Proposon 3.3 (Aldous s ghness creron, see [1. A sequence {X ( } of random elemens of D R [, T s gh (.e. he dsrbuons of he {X ( } are gh f he wo followng condons hold: ( lm a where X ( def = sup X (. T lm sup P [ X ( a =, (3.18 ( For each ɛ, η, here posve numbers δ and, such ha, f δ δ and, and f τ s an arbrary soppng me wh τ + δ T, hen P [ X ( τ+δ X τ ( ɛ η. (3.19 oe ha condon (3.18 s always necessary for ghness. We shall now apply Lemma 3.2 o equaons (3.7 and (3.12, he role of X ( n Proposon 3.3 beng played by Z (. The random varables Z ( and θ ( are clearly unformly bounded, so ha condon (3.18 s mmedae. To check condon (3.19, rewre (3.7 as +δ Z ( +δ Z ( = U ( +δ U ( + (L ( s + θ s ( Z s ( ds. (3.2
10 1 3 Hydrodynamcs for he basc asymmerc excluson process (ASEP For [, T, he negral erm n (3.2 s bounded n modulus by Kδ (K beng a consan unformly bounded n and ψ, and sasfes (3.19, whenever s replaced by an arbrary soppng me. We are lef wh he analyss of U (. Bu, from (3.12, (3.14 and Doob s nequaly for submarngales, we have E [ (U ( +δ U ( 2 [ +δ = E (Z s ( P [ sup T U ( ɛ 4 ɛ 2 E [ T (Z ( s 2 R ( s 2 R ( s ds ds Cδ, 4CT ɛ 2, (3.21 where C s a posve consan dependng only on ψ. Thus U ( n probably, as. Ths las propery, ogeher wh (3.7, (3.2 and assumpon (3.5, yeld (3.19 and he announced (weak relave compacness of he sequence Z (. Hence, he sequence of probably measures Q (, defned on D M [, T and correspondng o he process µ (, s also relavely compac: hs s a consequence of classcal projecon heorems (see for nsance Theorem n [13. We are now n a poson o esablsh a furher mporan propery. Le Q he lm pon of some arbrary subsequence Q (k, as k, def and Z = lm k Z ( k. Then he suppor of Q s a se of sample pahs absoluely connuous wh respec o he Lebesgue measure. Indeed, he applcaon µ sup T log Z s connuous and we have he mmedae bound sup log Z T 1 [ φ a (x, + φ b (x, dx, whch holds for all ψ a, ψ b C 2 [, 1. Hence, by weak convergence, any lm pon Z has he form [ 1 Z [φ a, φ b = exp [ρ(x, φ a (x, + (1 ρ(x, φ b (x, dx, (3.22 where ρ(x, denoes he lm densy [whch a pror s a random quany of he sequence of emprcal measures µ ( k nroduced n ( A funconal negral operaor o characerze lm pons Ths s somehow he Gordan kno of he problem. Relyng on he above weak compacness propery, our nex resul shows ha any arbrary lm pon Q s concenraed on a se of rajecores whch are weak soluons of an negral equaon.
11 3.2 A funconal negral operaor o characerze lm pons 11 ( The man dea s o consder for a whle he 2 quanes φ a, (, φ b,, 1, as ordnary free varables, whch for he sake of shorness wll be denoed respecvely by x ( and y (. Wh hs approach, he problem of he hydrodynamc lm wll appear o be mosly of an analycal naure. Le [ ( x ( α xy ( (, = λ ab ( exp [ ( y ( α yx ( (, = λ ba ( exp +1 x( +1 y( + y ( + x ( y ( +1 x ( +1 1, 1. Then, usng (3.7, (3.9, (3.1, (3.11 and he defnon of Z (, we oban mmedaely he followng funconal paral dfferenal equaon (FPDE where L ( d(z ( U ( d s he operaor L ( [h def = 2 α ( G ( xy (, = L ( [Z ( 2 h x ( y ( +1 + θ ( Z ( + α yx ( (, More precsely, nroducng he famly of cylnder ses wh, ( h y ( x ( +1 (p def V = [ Φ, Φ p, p = 1, 2..., (3.24 Φ def = sup ( φa (z,, φ b (z,, (3.25 (z, [,1 [,T we see mmedaely ha L ( acs on a subspace of C (V (2, snce Z ( s analyc wh respec o he coordnaes {φ a (.,, φ b (., }, for each fne (hngs wll be made more precse n secon eedless o say ha L ( s no of parabolc ype, as he quadrac form assocaed wh he second order dervave erms s clearly non defne (see e.g. [6. In addon, equaon (3.23 s a well defned sochasc FPDE, as all underlyng probably spaces emanae from famles of neracng Posson processes. ow, mgh be worh accounng for he use of he word funconal above, and for he reason of solang he hrd erm on he rgh n (3.23. Indeed, by (3.9, θ ( s a funconal nvolvng also a paral dervave of Z ( wh respec o, as we can wre. θ ( Z ( = 1 G ( [ Z ( x ( x ( + Z ( y ( y ( = ( Z. (3.26
12 12 3 Hydrodynamcs for he basc asymmerc excluson process (ASEP The las equaly n (3.26 mgh look somewha formal, bu wll come ou more clearly n Secon oe also, by he same argumen whch led o (3.22, ha we have lm k θ( k = 1 [ ρ(x, φ a (x, + (1 ρ(x, φ b (x, dx. (3.27 Our essenal agendum s o prove ha any lm pon of he sequence of random measures µ = lm weak k µ( k sasfes an negral equaon correspondng o a weak soluon (or dsrbuonal n Schwarz sense of a Cauchy ype operaor. To overcome he chef dffculy, namely he behavour of he lm sum n (3.23, we propose a seemngly new approach, whch s anamoun o analyzng he famly of second order lnear paral dfferenal operaors L ( along he sequence k. As brefly emphaszed n he remark a he end of hs secon, a brue force analyss of (3.23 would lead o a dead end. Indeed an mporan prelmnary sep consss n exracng he juce of he esmaes obaned n Lemma 3.2, o rewre he operaor L ( n erms of only prncpal varables, up o quanes of order O( 1. Ths s summarzed n he nex lemma. Lemma 3.4. The followng FPDE holds. where A ( d(z ( U ( d def = A ( [Z ( + θ ( Z ( ( 1 + O, (3.28 s vewed as an operaor wh doman C (V ( such ha A ( [g def = ( [ µψ ab, 1 2 G ( + λ ψ ab G ( (, g x ( +1 ( g x (, + g x ( +1 2 g x ( x ( +1 (3.29 he erm O ( 1 beng n modulus unformly bounded by C, where C denoes a consan dependng only on he quany sup {ψ(x,, ψ (x,, ψ (x, }. (x, [,1 [,T Proof. The resul follows by elemenary algebrac manpulaons from equaons (3.4, (3.7, (3.1, (3.15, (3.17, and deals wll be omed.
13 3.2 A funconal negral operaor o characerze lm pons 13 Takng Lemma 3.4 as a sarng pon, we skech ou below n Secons and he man lnes of our analycal approach, whch ndeed can be brefly summarzed by means of some keywords. Couplng, aken here n Skohorod s conex. Regularzaon and funconal negraon. Regularzaon refers o he fundamenal mehod used n he heory of dsrbuons o approxmae eher funcons or dsrbuons. We shall apply o he FPDE (3.28, consderng and he values of he funcon φ a (., (aken a pons of he orus as + 1 ordnary varables. Then, passng o he lm as, we nroduce convenen funconal negrals ogeher wh varaonal dervaves. Ths mgh lkely exend o much wder sysems, alhough hs asseron could ceranly be debaed Inerm reducon o an almos sure convergence seng Ths can be acheved by means of he exended Skohorod couplng (or ransfer heorem (see Corollary 6.12 n [13, whch n bref says ha, f a sequence of real random varables (ξ k s such ha lm k f k (ξ k = f(ξ converges n dsrbuon, hen here exs a probably space V and a new random sequence ξ k, such ha ξ L k = ξk and lm k f k ( ξ k = f(ξ, almos surely n V, wh ξ = L ξ. Here hs heorem wll be appled o he famly Z ( k, whch hus gves rse o a new sequence, named Y ( k n he sequel. Clearly hs sep s n no way oblgaory, bu raher a maer of ase. Indeed, one could sll keep on workng n a weak convergence conex, wh Alexandrov s pormaneau heorem (see e.g. [7 whenever needed Consderng (3.29 as a paral dfferenal operaor wh consan coeffcens For each fne, we consder he quanes ψ ab ( (,, ψ ab,, = 1,...,, as consan parameers, whle he x ( s wll be aken as free varables from a varaonal calculus pon of vew. Ths s clearly feasble, rememberng ha, by defnon, ψ ab = φ a φ b, for all φ a, φ b C[T. Hereafer, wll be vewed as an exogeneous mue varable, no parcpang concreely n he proposed varaonal approach. Then, accordng o he noaon nroduced n secon 3.2.1, we can rewre
14 14 3 Hydrodynamcs for he basc asymmerc excluson process (ASEP (3.28 n he form d(y ( U ( d def = A ( [Y ( + θ ( Y ( ( 1 + O, (3.3 where θ ( s sll gven by (3.9, keepng n mnd ha all random varables n (3.3 are defned wh respec o hs new (hough unspecfed probably space nroduced n secon above. In parcular, from he ghness proved n secon 3.1, leng along some subsequence k, we have lm Y ( k [φ a, φ b a.s. Y [φ a, φ b. k Analyss of he FDPE (3.3 The dea now s o propose a regularzaon procedure, whch consss n carryng ou he convoluon of (3.3 wh a suably chosen es funcon, nong ha Y ( nowhere vanshes and s unformly bounded. As usual, he convoluon f g of wo negrable funcons f, g C (V (, wh V ( gven n (3.24, wll be defned by (f g(u = f(u vg(vdv. V ( Le ω be he funcon of he real varable z defned by exp ( 1 ω(z def z f z <, = f z, where wll be convenen o wre ω (z def = dω(z dz def and ω (z def = d2 ω(z. dz 2 Seng x ( = (x ( 1, x( 2,..., ( x(, wh x( = φ a,, 1, we nroduce he followng famly of posve es funcons χ ( ε C (V (, ( 1 χ ε ( ( x ( = ω (x ( 2 ε, 2 ε. (3.31 =1 oe: I s worh keepng n mnd ha, as ofen as possble, he me varable wll be omed n mos of he mahemacal quanes, e.g. x ( (. Indeed, as menoned before, plays n some sense he role of a parameer. From (3.3, follows mmedaely ha ( d(y ( U ( χ ε ( d ( x ( = ( A ( + ( (θ ( [Y ( Y ( χ ( ( x ( ε ( χ ( ( x ( 1 + O. ε (3.32
15 3.2 A funconal negral operaor o characerze lm pons 15 ow, by (3.29 and accordng o he saemen made n Secon 3.2.2, he frs erm n he rgh-hand sde member of (3.32 can be negraed by pars, so ha, for any fne and ε suffcenly small, ( (Ã( A ( [Y ( χ ε ( ( x ( = [χ ( ε Y ( ( x (, (3.33 where Ã( s by defnon he adjon operaor of A ( n he Lagrange sense. Here he doman of à ( consss of all funcons h ( of he form [ 1 h ( = exp dσ a ( (xv [φ a (x, + dσ ( (xv [φ b (x,, where V : C[T R + sands for an arbrary analyc funcon; σ ( a and σ ( b are arbrary dscree probably measures on G (. Clearly χ ε ( belongs o he doman of à (. Under he assumpons made n Secon 3.2.2, a drec negraon by pars n (3.29 yelds he formula à ( [h = λ G ( µψ ab G ( ψ ab ( [ (, 1 h 2 x ( (, h x ( +1 where Ã( has been defned n (3.33. Le, for each φ φ a C[T, χ ε (φ def = lm χ( ε, ( 1 ( x ( = ω b + h x ( h x ( x ( +1 (3.34 φ 2 (x, dx ε 2, (3.35 where he negral n (3.35 s readly obaned as he lm of he Remann sum n (3.31. Lemma 3.5. For each φ(x, C[T, he followng lm holds unformly. lm à ( [χ ε ( ( x ( = 1 [ µψ ab (x, K(φ, x, + λψ ab (x, H(φ, x, dx, (3.36 wh ( 1 H(φ, z, = 2φ(z, ω φ 2 (u, du ε, 2 ( 1 K(φ, z, = H(φ, z, + 4φ 2 (z, ω φ 2 (u, du ε. 2 (3.37
16 16 3 Hydrodynamcs for he basc asymmerc excluson process (ASEP Proof. For fxed, he funcon Ã( [χ ( ε has paral dervaves of any order wh respec o he coordnaes x (, = 1,...,. In parcular, we ge from (3.31, for each G (, χ ( ε x ( 2 χ ( ε x ( x ( +1 ( ( x ( = 2 x( 1 ω ( x ( = 4 x( x ( +1 ω =1 ( 1 ( x ( =1 2 ε 2 ( x (, 2 ε 2 So we are agan lef wh Remann sums, whch, for any connuous funcon k(x,, yeld a once and lm k G ( (, χ ( ε x ( ( x ( = ( 1 1 2ω φ 2 (x, dx ε 2 k(x, φ(x, dx, lm G ( k ( 2, χ ( ε x ( x ( +1 ( x ( = ( 1 1 4ω φ 2 (x, dx ε 2 k(x, φ 2 (x, dx. Keepng n mnd ha, for all, he vecor x ( (from s very defnon mus be a dscrezaon of some funcon elemen n C[T, he las mporan sep o derve (3.36 requres o gve a precse meanng o he lm lm χ ( ε ( x ( d x (, V ( allowng o carry ou funconal negraon and varaonal dfferenaon. In hs respec, le us emphasze (f necessary a all! ha he usual consrucons of measures and negrals do no apply n general when he doman of negraon s an nfne-dmensonal space of funcons or mappngs, all he more because a complee axomac for funconal negraon does no really exs; ndeed each case requres he consrucon of ad hoc generalzed measures, see promeasures n [2, 3 or quas-measures n [16. These quesons prove o be of mporance n varous problems relaed o heorecal physcs. In our case sudy, negraon over pahs belongng o C[T needs o be properly consruced. The pon s o defne, as, a volume elemen denoed by δ(φ. We shall no do here, bu hs could be acheved by mmckng classcal.
17 3.2 A funconal negral operaor o characerze lm pons 17 fundamenal approaches, see e.g. [2, 3, 16. The man ool s he mporan F. Resz represenaon heorem, whch o every posve lnear funconal le correspond a unque posve measure. ow from equaon (3.35 s permssble o nroduce he normalzed es funconal χ ε (φ = χ ε(φ D, where D s chosen o ensure C ([ Φ, Φ χ ε (φ δ(φ = 1, and Φ s gven by (3.25, so ha D = C ([ Φ, Φ ω ( 1 φ 2 (x, dx ε 2 δ(φ. From Skohorod s couplng heorem, Y does sasfy an equaon of he form (3.22. Hence, we can wre he followng funconal dervaves (whch are planly of a Radon-ykodym naure Y φ = ρ(., Y, 2 Y φ 2 = ρ2 (., Y. (3.38 ow everyhng s n order o complee he puzzle, accordng o he followng seps. 1. Frs, usng (3.33, rewre (3.32 as ( d(y ( U ( χ ε ( d (Ã( ( x ( = + ( (θ ( [χ ( ε Y ( Y ( ( x ( ( χ ( ( x ( 1 + O. ε ( Le n (3.39 and hen replace χ ε (φ by χ ε (φ, rememberng ha by (3.21 U ( = O(1/ unformly. 3. Carry ou wo funconal negraon by pars n equaon (3.36 by makng use of ( Fnally, negrae on [, T, le ε and swch back o he orgnal probably space, where Z ( [φ a, φ b, by secon 3.2.1, converges n dsrbuon o Z [φ a, φ b : hs yelds exacly he announced Cauchy problem (3.6.
18 18 References Hence, he famly of random measures µ ( converges n dsrbuon o a deermnsc measure µ, whch n hs pecular case mples also convergence n probably. 3.3 Unqueness The problem of unqueness of weak soluons of he Cauchy problem (3.6 for nonlnear equaon s n fac already solved n he leraure. For a wde bblography on he subjec, we refer he reader for nsance o [6. The proof of Theorem 3.1 s concluded 4 Conjecure for he n-speces model We wll sae a conjecure abou hydrodynamc equaons for he n-speces model, brefly nroduced n secon 2.1, n he so-called equdffusve case, precsely descrbed hereafer. Defnon 4.1. The n-speces sysem s sad o be equdffusve whenever here exss a consan λ, such ha, for all pars (k, l, Then, leng λ kl ( lm 2 = λ. [ kl def λ kl α = lm log ( λ lk, ( we asser he followng hydrodynamc sysem holds. ρ k = β 2 ρ k x 2 + ( α lk ρ k ρ l, k = 1,..., n. x l k The dea s o apply he funconal approach presened n hs paper: hs s he subjec maer of an ongong work. References [1 P. Bllngsley. Convergence of Probably Measures. Wley Seres n Probably and Sascs. John Wley & Sons Inc., 2 edon, [2. Bourbak. Inegraon, Chapre IX. Hermann, Pars, 1969.
19 References 19 [3 P. Carer and C. DeW-Moree. A rgorous mahemacal foundaon of funconal negraon. In Funconal Inegraon : Bascs and Applcaons, volume 361 of ATO ASI - Seres B: Physcs, pages 1 5. Plenum Press, [4 A. De Mas and E. Presu. Mahemacal Mehods for Hydrodynamc Lms, volume 151 of Lecure oes n Mahemacs. Sprnger-Verlag, [5 B. Derrda, M. Evans, V. Hakm, and V. Pasquer. Exac soluon for 1d asymmerc excluson model usng a marx formulaon. J. Phys. A: Mah. Gen., 26: , [6 Y. Egorov and M. Shubn, edors. Paral Dfferenal Equaons, volume I-II-III of Encyclopeda of Mahemacal Scences. Sprnger Verlag, [7 S. Eher and T. Kurz. Markov Processes, Characerzaon and Convergence. John Wley & Sons, [8 M. Evans, D. P. Foser, C. Godrèche, and D. Mukamel. Sponaneous symmery breakng n a one dmensonal drven dffusve sysem. Phys. Rev. Le., 74:28 211, [9 M. Evans, Y. Kafr, M. Koduvely, and D. Mukamel. Phase Separaon and Coarsenng n one-dmensonal Drven Dffusve Sysems. Phys. Rev. E., 58:2764, [1 G. Fayolle and C. Furlehner. Dynamcal Wndngs of Random Walks and Excluson Models. Par I: Thermodynamc lm n Z 2. Journal of Sascal Physcs, 114(1/2:229 26, January 24. [11 G. Fayolle and C. Furlehner. Sochasc deformaons of sample pahs of random walks and excluson models. In Mahemacs and compuer scence. III, Trends Mah., pages Brkhäuser, Basel, 24. [12 G. Fayolle and C. Furlehner. Sochasc Dynamcs of Dscree Curves and Mul-Type Excluson Processes. Journal of Sascal Physcs, 127(5: , 27. [13 O. Kallenberg. Foundaons of Modern Probably. Sprnger, 2 edon, 22. [14 C. Kpns and C. Landm. Scalng lms of Ineracng Parcles Sysems. Sprnger-Verlag, [15 T. M. Lgge. Sochasc Ineracng Sysems: Conac, Voer and Excluson Processes, volume 324 of Grundlehren der mahemaschen Wssenschafen. Sprnger, 1999.
20 2 References [16 E. V. Maïkov. τ-smooh funconals. Tans. Moscow Mah. Soc., 2:1 4, [17 W. Rudn. Funconal Analyss. Inernaonal Seres n Pure and Appled Mahemacs. McGraw-Hll, 2 edon, [18 H. Spohn. Large Scale Dynamcs of Ineracng Parcles. Sprnger, 1991.
Part II CONTINUOUS TIME STOCHASTIC PROCESSES
Par II CONTINUOUS TIME STOCHASTIC PROCESSES 4 Chaper 4 For an advanced analyss of he properes of he Wener process, see: Revus D and Yor M: Connuous marngales and Brownan Moon Karazas I and Shreve S E:
More information( ) () we define the interaction representation by the unitary transformation () = ()
Hgher Order Perurbaon Theory Mchael Fowler 3/7/6 The neracon Represenaon Recall ha n he frs par of hs course sequence, we dscussed he chrödnger and Hesenberg represenaons of quanum mechancs here n he chrödnger
More informationV.Abramov - FURTHER ANALYSIS OF CONFIDENCE INTERVALS FOR LARGE CLIENT/SERVER COMPUTER NETWORKS
R&RATA # Vol.) 8, March FURTHER AALYSIS OF COFIDECE ITERVALS FOR LARGE CLIET/SERVER COMPUTER ETWORKS Vyacheslav Abramov School of Mahemacal Scences, Monash Unversy, Buldng 8, Level 4, Clayon Campus, Wellngon
More informationOn One Analytic Method of. Constructing Program Controls
Appled Mahemacal Scences, Vol. 9, 05, no. 8, 409-407 HIKARI Ld, www.m-hkar.com hp://dx.do.org/0.988/ams.05.54349 On One Analyc Mehod of Consrucng Program Conrols A. N. Kvko, S. V. Chsyakov and Yu. E. Balyna
More informationExistence and Uniqueness Results for Random Impulsive Integro-Differential Equation
Global Journal of Pure and Appled Mahemacs. ISSN 973-768 Volume 4, Number 6 (8), pp. 89-87 Research Inda Publcaons hp://www.rpublcaon.com Exsence and Unqueness Resuls for Random Impulsve Inegro-Dfferenal
More informationRelative controllability of nonlinear systems with delays in control
Relave conrollably o nonlnear sysems wh delays n conrol Jerzy Klamka Insue o Conrol Engneerng, Slesan Techncal Unversy, 44- Glwce, Poland. phone/ax : 48 32 37227, {jklamka}@a.polsl.glwce.pl Keywor: Conrollably.
More informationGENERATING CERTAIN QUINTIC IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS. Youngwoo Ahn and Kitae Kim
Korean J. Mah. 19 (2011), No. 3, pp. 263 272 GENERATING CERTAIN QUINTIC IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS Youngwoo Ahn and Kae Km Absrac. In he paper [1], an explc correspondence beween ceran
More informationCS286.2 Lecture 14: Quantum de Finetti Theorems II
CS286.2 Lecure 14: Quanum de Fne Theorems II Scrbe: Mara Okounkova 1 Saemen of he heorem Recall he las saemen of he quanum de Fne heorem from he prevous lecure. Theorem 1 Quanum de Fne). Le ρ Dens C 2
More informationSolution in semi infinite diffusion couples (error function analysis)
Soluon n sem nfne dffuson couples (error funcon analyss) Le us consder now he sem nfne dffuson couple of wo blocks wh concenraon of and I means ha, n a A- bnary sysem, s bondng beween wo blocks made of
More information. The geometric multiplicity is dim[ker( λi. number of linearly independent eigenvectors associated with this eigenvalue.
Lnear Algebra Lecure # Noes We connue wh he dscusson of egenvalues, egenvecors, and dagonalzably of marces We wan o know, n parcular wha condons wll assure ha a marx can be dagonalzed and wha he obsrucons
More informationLinear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Lnear Response Theory: The connecon beween QFT and expermens 3.1. Basc conceps and deas Q: ow do we measure he conducvy of a meal? A: we frs nroduce a weak elecrc feld E, and hen measure
More informationMechanics Physics 151
Mechancs Physcs 5 Lecure 9 Hamlonan Equaons of Moon (Chaper 8) Wha We Dd Las Tme Consruced Hamlonan formalsm H ( q, p, ) = q p L( q, q, ) H p = q H q = p H = L Equvalen o Lagrangan formalsm Smpler, bu
More informationMechanics Physics 151
Mechancs Physcs 5 Lecure 9 Hamlonan Equaons of Moon (Chaper 8) Wha We Dd Las Tme Consruced Hamlonan formalsm Hqp (,,) = qp Lqq (,,) H p = q H q = p H L = Equvalen o Lagrangan formalsm Smpler, bu wce as
More information. The geometric multiplicity is dim[ker( λi. A )], i.e. the number of linearly independent eigenvectors associated with this eigenvalue.
Mah E-b Lecure #0 Noes We connue wh he dscusson of egenvalues, egenvecors, and dagonalzably of marces We wan o know, n parcular wha condons wll assure ha a marx can be dagonalzed and wha he obsrucons are
More informationJohn Geweke a and Gianni Amisano b a Departments of Economics and Statistics, University of Iowa, USA b European Central Bank, Frankfurt, Germany
Herarchcal Markov Normal Mxure models wh Applcaons o Fnancal Asse Reurns Appendx: Proofs of Theorems and Condonal Poseror Dsrbuons John Geweke a and Gann Amsano b a Deparmens of Economcs and Sascs, Unversy
More informationMechanics Physics 151
Mechancs Physcs 5 Lecure 0 Canoncal Transformaons (Chaper 9) Wha We Dd Las Tme Hamlon s Prncple n he Hamlonan formalsm Dervaon was smple δi δ Addonal end-pon consrans pq H( q, p, ) d 0 δ q ( ) δq ( ) δ
More informationON THE WEAK LIMITS OF SMOOTH MAPS FOR THE DIRICHLET ENERGY BETWEEN MANIFOLDS
ON THE WEA LIMITS OF SMOOTH MAPS FOR THE DIRICHLET ENERGY BETWEEN MANIFOLDS FENGBO HANG Absrac. We denfy all he weak sequenal lms of smooh maps n W (M N). In parcular, hs mples a necessary su cen opologcal
More informationHEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD
Journal of Appled Mahemacs and Compuaonal Mechancs 3, (), 45-5 HEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD Sansław Kukla, Urszula Sedlecka Insue of Mahemacs,
More informationCH.3. COMPATIBILITY EQUATIONS. Continuum Mechanics Course (MMC) - ETSECCPB - UPC
CH.3. COMPATIBILITY EQUATIONS Connuum Mechancs Course (MMC) - ETSECCPB - UPC Overvew Compably Condons Compably Equaons of a Poenal Vecor Feld Compably Condons for Infnesmal Srans Inegraon of he Infnesmal
More informationLecture 18: The Laplace Transform (See Sections and 14.7 in Boas)
Lecure 8: The Lalace Transform (See Secons 88- and 47 n Boas) Recall ha our bg-cure goal s he analyss of he dfferenal equaon, ax bx cx F, where we emloy varous exansons for he drvng funcon F deendng on
More informationHow about the more general "linear" scalar functions of scalars (i.e., a 1st degree polynomial of the following form with a constant term )?
lmcd Lnear ransformaon of a vecor he deas presened here are que general hey go beyond he radonal mar-vecor ype seen n lnear algebra Furhermore, hey do no deal wh bass and are equally vald for any se of
More informationOrdinary Differential Equations in Neuroscience with Matlab examples. Aim 1- Gain understanding of how to set up and solve ODE s
Ordnary Dfferenal Equaons n Neuroscence wh Malab eamples. Am - Gan undersandng of how o se up and solve ODE s Am Undersand how o se up an solve a smple eample of he Hebb rule n D Our goal a end of class
More informationP R = P 0. The system is shown on the next figure:
TPG460 Reservor Smulaon 08 page of INTRODUCTION TO RESERVOIR SIMULATION Analycal and numercal soluons of smple one-dmensonal, one-phase flow equaons As an nroducon o reservor smulaon, we wll revew he smples
More informationNATIONAL UNIVERSITY OF SINGAPORE PC5202 ADVANCED STATISTICAL MECHANICS. (Semester II: AY ) Time Allowed: 2 Hours
NATONAL UNVERSTY OF SNGAPORE PC5 ADVANCED STATSTCAL MECHANCS (Semeser : AY 1-13) Tme Allowed: Hours NSTRUCTONS TO CANDDATES 1. Ths examnaon paper conans 5 quesons and comprses 4 prned pages.. Answer all
More informationSOME NOISELESS CODING THEOREMS OF INACCURACY MEASURE OF ORDER α AND TYPE β
SARAJEVO JOURNAL OF MATHEMATICS Vol.3 (15) (2007), 137 143 SOME NOISELESS CODING THEOREMS OF INACCURACY MEASURE OF ORDER α AND TYPE β M. A. K. BAIG AND RAYEES AHMAD DAR Absrac. In hs paper, we propose
More information( t) Outline of program: BGC1: Survival and event history analysis Oslo, March-May Recapitulation. The additive regression model
BGC1: Survval and even hsory analyss Oslo, March-May 212 Monday May 7h and Tuesday May 8h The addve regresson model Ørnulf Borgan Deparmen of Mahemacs Unversy of Oslo Oulne of program: Recapulaon Counng
More informationDynamic Team Decision Theory. EECS 558 Project Shrutivandana Sharma and David Shuman December 10, 2005
Dynamc Team Decson Theory EECS 558 Proec Shruvandana Sharma and Davd Shuman December 0, 005 Oulne Inroducon o Team Decson Theory Decomposon of he Dynamc Team Decson Problem Equvalence of Sac and Dynamc
More informationSurvival Analysis and Reliability. A Note on the Mean Residual Life Function of a Parallel System
Communcaons n Sascs Theory and Mehods, 34: 475 484, 2005 Copyrgh Taylor & Francs, Inc. ISSN: 0361-0926 prn/1532-415x onlne DOI: 10.1081/STA-200047430 Survval Analyss and Relably A Noe on he Mean Resdual
More information2.1 Constitutive Theory
Secon.. Consuve Theory.. Consuve Equaons Governng Equaons The equaons governng he behavour of maerals are (n he spaal form) dρ v & ρ + ρdv v = + ρ = Conservaon of Mass (..a) d x σ j dv dvσ + b = ρ v& +
More informationLet s treat the problem of the response of a system to an applied external force. Again,
Page 33 QUANTUM LNEAR RESPONSE FUNCTON Le s rea he problem of he response of a sysem o an appled exernal force. Agan, H() H f () A H + V () Exernal agen acng on nernal varable Hamlonan for equlbrum sysem
More informationEpistemic Game Theory: Online Appendix
Epsemc Game Theory: Onlne Appendx Edde Dekel Lucano Pomao Marcano Snscalch July 18, 2014 Prelmnares Fx a fne ype srucure T I, S, T, β I and a probably µ S T. Le T µ I, S, T µ, βµ I be a ype srucure ha
More informatione-journal Reliability: Theory& Applications No 2 (Vol.2) Vyacheslav Abramov
June 7 e-ournal Relably: Theory& Applcaons No (Vol. CONFIDENCE INTERVALS ASSOCIATED WITH PERFORMANCE ANALYSIS OF SYMMETRIC LARGE CLOSED CLIENT/SERVER COMPUTER NETWORKS Absrac Vyacheslav Abramov School
More informationIn the complete model, these slopes are ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL. (! i+1 -! i ) + [(!") i+1,q - [(!
ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL The frs hng o es n wo-way ANOVA: Is here neracon? "No neracon" means: The man effecs model would f. Ths n urn means: In he neracon plo (wh A on he horzonal
More informationFI 3103 Quantum Physics
/9/4 FI 33 Quanum Physcs Aleander A. Iskandar Physcs of Magnesm and Phooncs Research Grou Insu Teknolog Bandung Basc Conces n Quanum Physcs Probably and Eecaon Value Hesenberg Uncerany Prncle Wave Funcon
More informationVariants of Pegasos. December 11, 2009
Inroducon Varans of Pegasos SooWoong Ryu bshboy@sanford.edu December, 009 Youngsoo Cho yc344@sanford.edu Developng a new SVM algorhm s ongong research opc. Among many exng SVM algorhms, we wll focus on
More informationDensity Matrix Description of NMR BCMB/CHEM 8190
Densy Marx Descrpon of NMR BCMBCHEM 89 Operaors n Marx Noaon Alernae approach o second order specra: ask abou x magnezaon nsead of energes and ranson probables. If we say wh one bass se, properes vary
More information[ ] 2. [ ]3 + (Δx i + Δx i 1 ) / 2. Δx i-1 Δx i Δx i+1. TPG4160 Reservoir Simulation 2018 Lecture note 3. page 1 of 5
TPG460 Reservor Smulaon 08 page of 5 DISCRETIZATIO OF THE FOW EQUATIOS As we already have seen, fne dfference appromaons of he paral dervaves appearng n he flow equaons may be obaned from Taylor seres
More informationTesting a new idea to solve the P = NP problem with mathematical induction
Tesng a new dea o solve he P = NP problem wh mahemacal nducon Bacground P and NP are wo classes (ses) of languages n Compuer Scence An open problem s wheher P = NP Ths paper ess a new dea o compare he
More informationNotes on the stability of dynamic systems and the use of Eigen Values.
Noes on he sabl of dnamc ssems and he use of Egen Values. Source: Macro II course noes, Dr. Davd Bessler s Tme Seres course noes, zarads (999) Ineremporal Macroeconomcs chaper 4 & Techncal ppend, and Hamlon
More informationTHE PREDICTION OF COMPETITIVE ENVIRONMENT IN BUSINESS
THE PREICTION OF COMPETITIVE ENVIRONMENT IN BUSINESS INTROUCTION The wo dmensonal paral dfferenal equaons of second order can be used for he smulaon of compeve envronmen n busness The arcle presens he
More informationApproximate Analytic Solution of (2+1) - Dimensional Zakharov-Kuznetsov(Zk) Equations Using Homotopy
Arcle Inernaonal Journal of Modern Mahemacal Scences, 4, (): - Inernaonal Journal of Modern Mahemacal Scences Journal homepage: www.modernscenfcpress.com/journals/jmms.aspx ISSN: 66-86X Florda, USA Approxmae
More informationMethod of upper lower solutions for nonlinear system of fractional differential equations and applications
Malaya Journal of Maemak, Vol. 6, No. 3, 467-472, 218 hps://do.org/1.26637/mjm63/1 Mehod of upper lower soluons for nonlnear sysem of fraconal dfferenal equaons and applcaons D.B. Dhagude1 *, N.B. Jadhav2
More informationDEEP UNFOLDING FOR MULTICHANNEL SOURCE SEPARATION SUPPLEMENTARY MATERIAL
DEEP UNFOLDING FOR MULTICHANNEL SOURCE SEPARATION SUPPLEMENTARY MATERIAL Sco Wsdom, John Hershey 2, Jonahan Le Roux 2, and Shnj Waanabe 2 Deparmen o Elecrcal Engneerng, Unversy o Washngon, Seale, WA, USA
More informationA NEW TECHNIQUE FOR SOLVING THE 1-D BURGERS EQUATION
S19 A NEW TECHNIQUE FOR SOLVING THE 1-D BURGERS EQUATION by Xaojun YANG a,b, Yugu YANG a*, Carlo CATTANI c, and Mngzheng ZHU b a Sae Key Laboraory for Geomechancs and Deep Underground Engneerng, Chna Unversy
More information@FMI c Kyung Moon Sa Co.
Annals of Fuzzy Mahemacs and Informacs Volume 8, No. 2, (Augus 2014), pp. 245 257 ISSN: 2093 9310 (prn verson) ISSN: 2287 6235 (elecronc verson) hp://www.afm.or.kr @FMI c Kyung Moon Sa Co. hp://www.kyungmoon.com
More informationM. Y. Adamu Mathematical Sciences Programme, AbubakarTafawaBalewa University, Bauchi, Nigeria
IOSR Journal of Mahemacs (IOSR-JM e-issn: 78-578, p-issn: 9-765X. Volume 0, Issue 4 Ver. IV (Jul-Aug. 04, PP 40-44 Mulple SolonSoluons for a (+-dmensonalhroa-sasuma shallow waer wave equaon UsngPanlevé-Bӓclund
More informationComparison of Differences between Power Means 1
In. Journal of Mah. Analyss, Vol. 7, 203, no., 5-55 Comparson of Dfferences beween Power Means Chang-An Tan, Guanghua Sh and Fe Zuo College of Mahemacs and Informaon Scence Henan Normal Unversy, 453007,
More informationExistence of Time Periodic Solutions for the Ginzburg-Landau Equations. model of superconductivity
Journal of Mahemacal Analyss and Applcaons 3, 3944 999 Arcle ID jmaa.999.683, avalable onlne a hp:www.dealbrary.com on Exsence of me Perodc Soluons for he Gnzburg-Landau Equaons of Superconducvy Bxang
More informationDensity Matrix Description of NMR BCMB/CHEM 8190
Densy Marx Descrpon of NMR BCMBCHEM 89 Operaors n Marx Noaon If we say wh one bass se, properes vary only because of changes n he coeffcens weghng each bass se funcon x = h< Ix > - hs s how we calculae
More informationChapter Lagrangian Interpolation
Chaper 5.4 agrangan Inerpolaon Afer readng hs chaper you should be able o:. dere agrangan mehod of nerpolaon. sole problems usng agrangan mehod of nerpolaon and. use agrangan nerpolans o fnd deraes and
More informationThe Finite Element Method for the Analysis of Non-Linear and Dynamic Systems
Swss Federal Insue of Page 1 The Fne Elemen Mehod for he Analyss of Non-Lnear and Dynamc Sysems Prof. Dr. Mchael Havbro Faber Dr. Nebojsa Mojslovc Swss Federal Insue of ETH Zurch, Swzerland Mehod of Fne
More informationCS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 4
CS434a/54a: Paern Recognon Prof. Olga Veksler Lecure 4 Oulne Normal Random Varable Properes Dscrmnan funcons Why Normal Random Varables? Analycally racable Works well when observaon comes form a corruped
More information2/20/2013. EE 101 Midterm 2 Review
//3 EE Mderm eew //3 Volage-mplfer Model The npu ressance s he equalen ressance see when lookng no he npu ermnals of he amplfer. o s he oupu ressance. I causes he oupu olage o decrease as he load ressance
More informationLi An-Ping. Beijing , P.R.China
A New Type of Cpher: DICING_csb L An-Png Bejng 100085, P.R.Chna apl0001@sna.com Absrac: In hs paper, we wll propose a new ype of cpher named DICING_csb, whch s derved from our prevous sream cpher DICING.
More informationOn computing differential transform of nonlinear non-autonomous functions and its applications
On compung dfferenal ransform of nonlnear non-auonomous funcons and s applcaons Essam. R. El-Zahar, and Abdelhalm Ebad Deparmen of Mahemacs, Faculy of Scences and Humanes, Prnce Saam Bn Abdulazz Unversy,
More informationAppendix H: Rarefaction and extrapolation of Hill numbers for incidence data
Anne Chao Ncholas J Goell C seh lzabeh L ander K Ma Rober K Colwell and Aaron M llson 03 Rarefacon and erapolaon wh ll numbers: a framewor for samplng and esmaon n speces dversy sudes cology Monographs
More informationGraduate Macroeconomics 2 Problem set 5. - Solutions
Graduae Macroeconomcs 2 Problem se. - Soluons Queson 1 To answer hs queson we need he frms frs order condons and he equaon ha deermnes he number of frms n equlbrum. The frms frs order condons are: F K
More informationVolatility Interpolation
Volaly Inerpolaon Prelmnary Verson March 00 Jesper Andreasen and Bran Huge Danse Mares, Copenhagen wan.daddy@danseban.com brno@danseban.com Elecronc copy avalable a: hp://ssrn.com/absrac=69497 Inro Local
More informationSampling Procedure of the Sum of two Binary Markov Process Realizations
Samplng Procedure of he Sum of wo Bnary Markov Process Realzaons YURY GORITSKIY Dep. of Mahemacal Modelng of Moscow Power Insue (Techncal Unversy), Moscow, RUSSIA, E-mal: gorsky@yandex.ru VLADIMIR KAZAKOV
More informationPerformance Analysis for a Network having Standby Redundant Unit with Waiting in Repair
TECHNI Inernaonal Journal of Compung Scence Communcaon Technologes VOL.5 NO. July 22 (ISSN 974-3375 erformance nalyss for a Nework havng Sby edundan Un wh ang n epar Jendra Sngh 2 abns orwal 2 Deparmen
More informationTrack Properities of Normal Chain
In. J. Conemp. Mah. Scences, Vol. 8, 213, no. 4, 163-171 HIKARI Ld, www.m-har.com rac Propes of Normal Chan L Chen School of Mahemacs and Sascs, Zhengzhou Normal Unversy Zhengzhou Cy, Hennan Provnce, 4544,
More informationScattering at an Interface: Oblique Incidence
Course Insrucor Dr. Raymond C. Rumpf Offce: A 337 Phone: (915) 747 6958 E Mal: rcrumpf@uep.edu EE 4347 Appled Elecromagnecs Topc 3g Scaerng a an Inerface: Oblque Incdence Scaerng These Oblque noes may
More informationThe Pricing of Basket Options: A Weak Convergence Approach
The Prcng of Baske Opons: A Weak Convergence Approach Ljun Bo Yongjn Wang Absrac We consder a lm prce of baske opons n a large porfolo where he dynamcs of baske asses s descrbed as a CEV jump dffuson sysem.
More informationOn the numerical treatment ofthenonlinear partial differentialequation of fractional order
IOSR Journal of Mahemacs (IOSR-JM) e-iss: 2278-5728, p-iss: 239-765X. Volume 2, Issue 6 Ver. I (ov. - Dec.26), PP 28-37 www.osrjournals.org On he numercal reamen ofhenonlnear paral dfferenalequaon of fraconal
More informationChapter 6: AC Circuits
Chaper 6: AC Crcus Chaper 6: Oulne Phasors and he AC Seady Sae AC Crcus A sable, lnear crcu operang n he seady sae wh snusodal excaon (.e., snusodal seady sae. Complee response forced response naural response.
More information2 Aggregate demand in partial equilibrium static framework
Unversy of Mnnesoa 8107 Macroeconomc Theory, Sprng 2009, Mn 1 Fabrzo Perr Lecure 1. Aggregaon 1 Inroducon Probably so far n he macro sequence you have deal drecly wh represenave consumers and represenave
More informationEP2200 Queuing theory and teletraffic systems. 3rd lecture Markov chains Birth-death process - Poisson process. Viktoria Fodor KTH EES
EP Queung heory and eleraffc sysems 3rd lecure Marov chans Brh-deah rocess - Posson rocess Vora Fodor KTH EES Oulne for oday Marov rocesses Connuous-me Marov-chans Grah and marx reresenaon Transen and
More informationLecture 2 M/G/1 queues. M/G/1-queue
Lecure M/G/ queues M/G/-queue Posson arrval process Arbrary servce me dsrbuon Sngle server To deermne he sae of he sysem a me, we mus now The number of cusomers n he sysems N() Tme ha he cusomer currenly
More informationOutline. Probabilistic Model Learning. Probabilistic Model Learning. Probabilistic Model for Time-series Data: Hidden Markov Model
Probablsc Model for Tme-seres Daa: Hdden Markov Model Hrosh Mamsuka Bonformacs Cener Kyoo Unversy Oulne Three Problems for probablsc models n machne learnng. Compung lkelhood 2. Learnng 3. Parsng (predcon
More informationarxiv: v1 [cs.sy] 2 Sep 2014
Noname manuscrp No. wll be nsered by he edor Sgnalng for Decenralzed Roung n a Queueng Nework Y Ouyang Demoshens Tenekezs Receved: dae / Acceped: dae arxv:409.0887v [cs.sy] Sep 04 Absrac A dscree-me decenralzed
More informationDepartment of Economics University of Toronto
Deparmen of Economcs Unversy of Torono ECO408F M.A. Economercs Lecure Noes on Heeroskedascy Heeroskedascy o Ths lecure nvolves lookng a modfcaons we need o make o deal wh he regresson model when some of
More informationSELFSIMILAR PROCESSES WITH STATIONARY INCREMENTS IN THE SECOND WIENER CHAOS
POBABILITY AD MATEMATICAL STATISTICS Vol., Fasc., pp. SELFSIMILA POCESSES WIT STATIOAY ICEMETS I TE SECOD WIEE CAOS BY M. M A E J I M A YOKOAMA AD C. A. T U D O LILLE Absrac. We sudy selfsmlar processes
More informationLecture 6: Learning for Control (Generalised Linear Regression)
Lecure 6: Learnng for Conrol (Generalsed Lnear Regresson) Conens: Lnear Mehods for Regresson Leas Squares, Gauss Markov heorem Recursve Leas Squares Lecure 6: RLSC - Prof. Sehu Vjayakumar Lnear Regresson
More information3. OVERVIEW OF NUMERICAL METHODS
3 OVERVIEW OF NUMERICAL METHODS 3 Inroducory remarks Ths chaper summarzes hose numercal echnques whose knowledge s ndspensable for he undersandng of he dfferen dscree elemen mehods: he Newon-Raphson-mehod,
More informationFirst-order piecewise-linear dynamic circuits
Frs-order pecewse-lnear dynamc crcus. Fndng he soluon We wll sudy rs-order dynamc crcus composed o a nonlnear resse one-por, ermnaed eher by a lnear capacor or a lnear nducor (see Fg.. Nonlnear resse one-por
More informationDiscrete Time Approximation and Monte-Carlo Simulation of Backward Stochastic Differential Equations
Dscree Tme Approxmaon and Mone-Carlo Smulaon of Backward Sochasc Dfferenal Equaons Bruno Bouchard Unversé Pars VI, PMA, and CREST Pars, France bouchard@ccrjusseufr Nzar Touz CREST Pars, France ouz@ensaefr
More informationJ i-1 i. J i i+1. Numerical integration of the diffusion equation (I) Finite difference method. Spatial Discretization. Internal nodes.
umercal negraon of he dffuson equaon (I) Fne dfference mehod. Spaal screaon. Inernal nodes. R L V For hermal conducon le s dscree he spaal doman no small fne spans, =,,: Balance of parcles for an nernal
More informationAn introduction to Support Vector Machine
An nroducon o Suppor Vecor Machne 報告者 : 黃立德 References: Smon Haykn, "Neural Neworks: a comprehensve foundaon, second edon, 999, Chaper 2,6 Nello Chrsann, John Shawe-Tayer, An Inroducon o Suppor Vecor Machnes,
More informationESTIMATIONS OF RESIDUAL LIFETIME OF ALTERNATING PROCESS. COMMON APPROACH TO ESTIMATIONS OF RESIDUAL LIFETIME
Srucural relably. The heory and pracce Chumakov I.A., Chepurko V.A., Anonov A.V. ESTIMATIONS OF RESIDUAL LIFETIME OF ALTERNATING PROCESS. COMMON APPROACH TO ESTIMATIONS OF RESIDUAL LIFETIME The paper descrbes
More informationA NUMERICAL SCHEME FOR BSDES. BY JIANFENG ZHANG University of Southern California, Los Angeles
The Annals of Appled Probably 24, Vol. 14, No. 1, 459 488 Insue of Mahemacal Sascs, 24 A NUMERICAL SCHEME FOR BSDES BY JIANFENG ZHANG Unversy of Souhern Calforna, Los Angeles In hs paper we propose a numercal
More informationNew M-Estimator Objective Function. in Simultaneous Equations Model. (A Comparative Study)
Inernaonal Mahemacal Forum, Vol. 8, 3, no., 7 - HIKARI Ld, www.m-hkar.com hp://dx.do.org/.988/mf.3.3488 New M-Esmaor Objecve Funcon n Smulaneous Equaons Model (A Comparave Sudy) Ahmed H. Youssef Professor
More informationOnline Supplement for Dynamic Multi-Technology. Production-Inventory Problem with Emissions Trading
Onlne Supplemen for Dynamc Mul-Technology Producon-Invenory Problem wh Emssons Tradng by We Zhang Zhongsheng Hua Yu Xa and Baofeng Huo Proof of Lemma For any ( qr ) Θ s easy o verfy ha he lnear programmng
More informationMotion in Two Dimensions
Phys 1 Chaper 4 Moon n Two Dmensons adzyubenko@csub.edu hp://www.csub.edu/~adzyubenko 005, 014 A. Dzyubenko 004 Brooks/Cole 1 Dsplacemen as a Vecor The poson of an objec s descrbed by s poson ecor, r The
More informationUNIVERSITAT AUTÒNOMA DE BARCELONA MARCH 2017 EXAMINATION
INTERNATIONAL TRADE T. J. KEHOE UNIVERSITAT AUTÒNOMA DE BARCELONA MARCH 27 EXAMINATION Please answer wo of he hree quesons. You can consul class noes, workng papers, and arcles whle you are workng on he
More informationAdvanced Macroeconomics II: Exchange economy
Advanced Macroeconomcs II: Exchange economy Krzyszof Makarsk 1 Smple deermnsc dynamc model. 1.1 Inroducon Inroducon Smple deermnsc dynamc model. Defnons of equlbrum: Arrow-Debreu Sequenal Recursve Equvalence
More informationPHYS 705: Classical Mechanics. Canonical Transformation
PHYS 705: Classcal Mechancs Canoncal Transformaon Canoncal Varables and Hamlonan Formalsm As we have seen, n he Hamlonan Formulaon of Mechancs,, are ndeenden varables n hase sace on eual foong The Hamlon
More informationLecture VI Regression
Lecure VI Regresson (Lnear Mehods for Regresson) Conens: Lnear Mehods for Regresson Leas Squares, Gauss Markov heorem Recursve Leas Squares Lecure VI: MLSC - Dr. Sehu Vjayakumar Lnear Regresson Model M
More informationMEEN Handout 4a ELEMENTS OF ANALYTICAL MECHANICS
MEEN 67 - Handou 4a ELEMENTS OF ANALYTICAL MECHANICS Newon's laws (Euler's fundamenal prncples of moon) are formulaed for a sngle parcle and easly exended o sysems of parcles and rgd bodes. In descrbng
More informationFTCS Solution to the Heat Equation
FTCS Soluon o he Hea Equaon ME 448/548 Noes Gerald Reckenwald Porland Sae Unversy Deparmen of Mechancal Engneerng gerry@pdxedu ME 448/548: FTCS Soluon o he Hea Equaon Overvew Use he forward fne d erence
More informationEcon107 Applied Econometrics Topic 5: Specification: Choosing Independent Variables (Studenmund, Chapter 6)
Econ7 Appled Economercs Topc 5: Specfcaon: Choosng Independen Varables (Sudenmund, Chaper 6 Specfcaon errors ha we wll deal wh: wrong ndependen varable; wrong funconal form. Ths lecure deals wh wrong ndependen
More informationShould Exact Index Numbers have Standard Errors? Theory and Application to Asian Growth
Should Exac Index umbers have Sandard Errors? Theory and Applcaon o Asan Growh Rober C. Feensra Marshall B. Rensdorf ovember 003 Proof of Proposon APPEDIX () Frs, we wll derve he convenonal Sao-Vara prce
More informationComb Filters. Comb Filters
The smple flers dscussed so far are characered eher by a sngle passband and/or a sngle sopband There are applcaons where flers wh mulple passbands and sopbands are requred Thecomb fler s an example of
More information2 Aggregate demand in partial equilibrium static framework
Unversy of Mnnesoa 8107 Macroeconomc Theory, Sprng 2012, Mn 1 Fabrzo Perr Lecure 1. Aggregaon 1 Inroducon Probably so far n he macro sequence you have deal drecly wh represenave consumers and represenave
More informationTime-interval analysis of β decay. V. Horvat and J. C. Hardy
Tme-nerval analyss of β decay V. Horva and J. C. Hardy Work on he even analyss of β decay [1] connued and resuled n he developmen of a novel mehod of bea-decay me-nerval analyss ha produces hghly accurae
More information10. A.C CIRCUITS. Theoretically current grows to maximum value after infinite time. But practically it grows to maximum after 5τ. Decay of current :
. A. IUITS Synopss : GOWTH OF UNT IN IUIT : d. When swch S s closed a =; = d. A me, curren = e 3. The consan / has dmensons of me and s called he nducve me consan ( τ ) of he crcu. 4. = τ; =.63, n one
More informationarxiv: v1 [math.pr] 6 Mar 2019
Local law and Tracy Wdom lm for sparse sochasc block models Jong Yun Hwang J Oon Lee Wooseok Yang arxv:1903.02179v1 mah.pr 6 Mar 2019 March 7, 2019 Absrac We consder he specral properes of sparse sochasc
More information( ) [ ] MAP Decision Rule
Announcemens Bayes Decson Theory wh Normal Dsrbuons HW0 due oday HW o be assgned soon Proec descrpon posed Bomercs CSE 90 Lecure 4 CSE90, Sprng 04 CSE90, Sprng 04 Key Probables 4 ω class label X feaure
More informationMotion of Wavepackets in Non-Hermitian. Quantum Mechanics
Moon of Wavepaces n Non-Herman Quanum Mechancs Nmrod Moseyev Deparmen of Chemsry and Mnerva Cener for Non-lnear Physcs of Complex Sysems, Technon-Israel Insue of Technology www.echnon echnon.ac..ac.l\~nmrod
More informationChapter 2 Linear dynamic analysis of a structural system
Chaper Lnear dynamc analyss of a srucural sysem. Dynamc equlbrum he dynamc equlbrum analyss of a srucure s he mos general case ha can be suded as akes no accoun all he forces acng on. When he exernal loads
More informationF-Tests and Analysis of Variance (ANOVA) in the Simple Linear Regression Model. 1. Introduction
ECOOMICS 35* -- OTE 9 ECO 35* -- OTE 9 F-Tess and Analyss of Varance (AOVA n he Smple Lnear Regresson Model Inroducon The smple lnear regresson model s gven by he followng populaon regresson equaon, or
More information