δ N formalism for curvature perturbations
|
|
- Tamsyn Stafford
- 5 years ago
- Views:
Transcription
1 δ N formalsm for curvaure perurbaons from nflaon Msao Sasak Yukawa Insue (YITP) Kyoo Unversy
2 . Inroducon. Lnear perurbaon heory merc perurbaon & me slcng δn N formalsm 3. Nonlnear exenson on superhorzon scales graden expanson, conservaon law local Fredmann equaon 4. Nonlnear ΔN N formula ΔN N for slowroll nflaon dagrammac mehod for ΔN IR dvergence ssue 5. Summary
3 . Inroducon Sandard (sngle-feld, slowroll) nflaon predcs scale- nvaran Gaussan curvaure perurbaons. CMB (WMAP) s conssen wh he predcon. Lnear perurbaon heory seems o be vald.
4 So, why boher dong more research on nflaon? Because observaonal daa does no exclude oher models. Tensor perurbaons have no been deeced ye. T/S ~ ? or smaller? In fac, nflaon may no be so smple. mul-feld, non-slowroll, exra-dm dm s, srng heory PLANCK, CMBpol, may deec non-gaussany Ψ=Ψ gauss gauss + f NL NL Ψ gauss gauss + ; f NL 5? Nonlnear backreacon on superhorzon scales? Re-consder he dynamcs on super-horzon scales
5 . Lnear perurbaon heory Σ(+d) Σ() = R + ( A) () j ds d a ( ) δj Hj dx dx x = cons. dτ properme along x = cons.: dτ = ( + A) d curvaure perurbaon on Σ(): Ρ expanson (Hubble parameer): Bardeen 80, Mukhanov 8, Kodama & MS 84,. merc on a spaally fla background ( H ) ( H ) scalar ensor 4 = ΔR a (3) (3) expanson (Hubble parameer): H% H ( A) j j (g 0j =0 for smplcy) R = = j E ransverse-raceless (3) = + R + Δ E 3
6 Choce of me-slcng comovng slcng maer-based slces unform densy slcng μ comovng slcng T = φ φ( ) unform densy slcng T 0 ρ = ρ( ) ( = for a scalar ) 0 feld 0 unform Hubble slcng geomercal slces fla slcng (3) Newon (shear-free) slcng (3) H% = H( ) HA+ R + Δ E 3 = (3) 4 R = ΔR=0 R = 0 a scalar ( 3) H j 0 0 raceless j δj Δ E = E = 3 comovng = unform ρ = unform H on superhorzon scales E 0 = 0
7 δn N formalsm n lnear heory MS & Sewar 96 e-foldng number perurbaon beween Σ() ) and Σ( fn ): ( ) ( ) ; fn Hd % Hd fn fn δ N τ τ background (3) E d R = 3 R R ( ) ( ) ( ) fn O ε fn = + Δ + Σ 0 ( fn ) Σ 0 () δn(, fn ) x =cons. N 0 (, fn ) Σ ( fn ), Ρ( fn ) Σ (), Ρ() δn=0 f boh Σ() ) and Σ( fn ) are chosen o be fla (Ρ=0).
8 Choose Σ() ) = fla (Ρ=0)( and Σ( fn ) = comovng: Σ( fn ), Ρ C ( fn ) δ N ( ; ) = R ( ) fn C fn x =cons. Σ(), Ρ()=0 curvaure perurbaon on comovng slce (suffx C for comovng) The gauge-nvaran varable ζ used n he leraure s relaed o Ρ C as ζ = -Ρ C or ζ = Ρ C By defnon, δn(; fn ) s -ndependen
9 Example: slow-roll nflaon sngle-feld nflaon, no exra degree of freedom Ρ C becomes consan soon afer horzon-crossng (=( h ): log L ( ; ) R ( ) R ( ) δ N = = h fn C fn C h Ρ C = cons. L=H - = h nflaon = fn log a
10 Also δn = H( h ) δ F C, where δ F C s he me dfference beween he comovng and fla slces a = h. δ F C Σ C ( h ) : comovng Σ F ( h ) : fla (, x ) ( ) φ δ φ + = ( ) F h F C C h δφ=0, Ρ=Ρ C δφ & φ δ Ρ=0, δφ=δφ F F + h F C = 0 R H dφ / d ( ) = δn( ; ) = δφ ( ) C fn h fn F h = dn F dφ δφ ( ) h δn formula dn = Hd Sarobnsky 85 Only he knowledge of he background evoluon s necessary o calculae Ρ C ( fn ).
11 for a mul-componen scalar: δ N for a (for slowroll nflaon) N a R = δn = δφ a φ ( ) ( ) C fn F h a MS & Sewar 96 N.B. Ρ C (=ζ) ) s no longer consan n me: & F C ( ) H φ R = δφ me varyng even on & φ superhorzon scales R ( ) = N δφ = N C fn F Furher exenson o non-slowroll case s possble, f general slow-roll condon s sasfed a horzon-crossng. Lee, MS, Sewar, Tanaka & Yokoyama 05 & φ O( ), O( ), O( ),..., H = φ& φ ξ ξ ξ ξ H = & & H & = = φ φ ( h ) ( π ) H a N N a φ
12 3. Nonlnear exenson On superhorzon scales, graden expanson s vald: x Q = Q : HQ; H : Gρ Belnsk e al. 70, Toma 7, Salopek & Bond 90, Ths s a consequence of causaly: L»H - lgh cone H - A lowes order, no sgnal propagaes n spaal drecons. Feld equaons reduce o ODE s
13 merc on superhorzon scales graden expanson: ε, ε = expanson parameer merc: α j de %, ( )( j j + β + β ) ds = N d + e % γ dx d dx d ( ) γ j = β = O ε he only non-rval assumpon conans GW (~ ensor) modes (, ) ( ) ( x = ln a +, x ); : curvaure perurbaon α ψ ψ fducal `background e.g., choose ψ ( *,0) = 0
14 Energy momenum ensor: ( ) μν μ ν μν μ ν μν T = ρu u + p g + u u ; u T = d μ α ρ+ μu + = = + dτ N u assumpon: v = O 0 ( ε ) u u μ n μ = O(ε) (absence of vorcy mode) Local Hubble parameer: μ μ H% μn = μu + O 3 3 n μ dx μ μ μ ( ) 0 ; ( ρ p u 3 O ) μ ε = Nd ( ε ) ν 0 normal o = cons. A leadng order, local Hubble parameer on any slcng s equvalen o expanson rae of maer flow. ~ So, hereafer, we redefne H o be H% u μ μ 3 u μ n μ =cons.
15 Local Fredmann equaon 8 G H% (, x ) π = ρ (, x ) + O( ε ) 3 d 3H( p) 0 d ρ + % τ ρ + = x : comovng (Lagrangean) coordnaes. dτ = d : proper me along maer flow exacly he same as he background equaons. separae unverse unform ρ slce = unform Hubble slce = comovng slce as n he case of lnear heory no modfcaons/backreacon due o super-hubble perurbaons. cf. Hraa & Seljak 05 Noh & Hwang 05
16 4. Nonlnear ΔN formula energy conservaon: (applcable o each ndependen maer componen) ρ ( ) a ( O ε = ) α = & + + ψ = H% O ε 3 + a N + ( ρ p) e-foldng number: ( ), ; % N N x H d = d 3 ρ + where x =cons. s a comovng worldlne. Ths defnon apples o any choce of me-slcng. ψ where (, x ) ψ (, x ) =ΔN (, ; x ) (, ; ) ( x N, ; x ) ΔN ln ρ P x a ( ) a ( )
17 ΔN - formula Lyh & Wands 03, Malk, Lyh & MS 04, Lyh & Rodrguez 05, Langlos & Vernzz 05 Le us ake slcng such ha Σ() ) s fla a = [ Σ F ( ) ] and unform densy/unform H/comovng a = [ Σ C ( ) ] : ( fla slce: Σ() ) on whch ψ = 0 e α = a() ) Σ C ( ) : unform densy ρ ( )=cons. N(, ;x ) ΔN F Σ C ( ) : unform densy Σ F ( ) : fla N (, ; x) = N0(, ) +ΔN F a ( ) N (, ) = ln beween Σ ( ) and Σ 0 C C a ( ) ρ ( )=cons. ψ ( )=0 ( )
18 Then (, ) (, ) ( x, x ) Δ N = ψ ψ = ψ F x C suffx C for comovng/unform ρ/unform H where ΔΝ s equal o e-foldng number from Σ F F ( ) o Σ C ( ): Σ ( ) ρ Σ ( ) ρ Δ = + 3 ρ+ P 3 ρ+ P C C N F d d ΣF( ) Σ ( ) C = 3 Σ Σ C F x ( ) ( ) ρ + ρ P x d For slow-roll nflaon n lnear heory, hs reduces o N ( ) ( ) a ψ C( ) R C( ) = δn ; = H δf C = δφ ( ) a F a φ
19 ΔN N for slowroll nflaon In slowroll nflaon, all decayng mode soluons of he (mul-componen) nflaon feld φ de ou. Nonlnear ΔN for mul-componen nflaon : A A A Δ N = N φ + δφ N φ = n ( ) ( ) n N n! L A A An φ φ φ MS & Tanaka 98, Lyh & Rodrguez 05 If φ s slow rollng when he scale of our neres leaves he horzon, N s only a funcon of φ (ndep of dφ/d,, apar from rval dep. on me fn from whch N s measured), no maer how complcaed he subsequen evoluon would be. A δφ δφ A L δφ where δφ =δφδφ F (on fla slce) a horzon-crossng. (δφδφ F may conan non-gaussany from subhorzon neracons) cf. Maldacena 03, Wenberg 05,... A n
20 Dagrammac mehod for nonlnear Δ N n DN ζ =Δ = AA L A n A A An N δφ δφ L δφ ; N AA L A n! A φ A φ An L φ n N n basc -p funcon: A B AB δφ ( x) δφ ( y) = h ( φ) G0( x y) feld space merc δφ s assumed o be Gaussan for non-gaussan δφ,, here wll be basc n-p funcons conneced n-p funcon of ζ: Byrnes, Koyama, MS & Wands 07 -p funcon A AB ζ( x) ζ ( y) = N A N G0( x y) + N ABN G0( x y) c! x y N N G x y A A 3! A A A A + x y + x y! B B 3! BC BC ABC 3 + ABC 0( ) +L +
21 3-p funcon A B ζ( x) ζ( y) ζ ( z) = N N N G ( x y) G ( y z) + perm. x A A y B + c AB 0 0 AB CA + N N N G ( x y) G ( y z) G ( z x) BC A BC + N N ABC N G0( x y) G0( y z) + perm.! +L x A C + perm. +! x A C z! y z B B B x A A + perm. +! y z B C B C
22 IR dvergence problem Loop dagrams lke x A C AB CA N N A C BC N G0( x y) G0( y zg ) 0( z x) y z B B n he m-p funcon gve rse o IR dvergence n he (m-)( )-specrum f P(k)~k n 4 wh n. eg,, he above dagram gves Boubekeur & Lyh 05 Pk (, k, k) : δ ( k+ k + k) d p P( p) P( k+ p ) P( k p ) cuoff-dependen! Is hs IR cuoff physcally observable? (real space 3-p 3 fcn s perfecly regular f G 0 (x)) s regular.)
23 8. Summary Superhorzon scale perurbaons can never affec local (horzon-sze) dynamcs,, hence never cause backreacon. nonlneary on superhorzon scales are always local. However, nonlocal nonlneary (non-gaussany) may appear due o quanum neracons on subhorzon scales. cf. Wenberg 06 There exss a nonlnear generalzaon of δ N formula whch s useful n evaluang non-gaussany from nflaon. dagrammac mehod can by sysemacally appled. IR dvergence from loop dagrams needs furher consderaon.
Let 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 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 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 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 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 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 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 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 informationLecture 9: Dynamic Properties
Shor Course on Molecular Dynamcs Smulaon Lecure 9: Dynamc Properes Professor A. Marn Purdue Unversy Hgh Level Course Oulne 1. MD Bascs. Poenal Energy Funcons 3. Inegraon Algorhms 4. Temperaure Conrol 5.
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 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 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 informationPanel Data Regression Models
Panel Daa Regresson Models Wha s Panel Daa? () Mulple dmensoned Dmensons, e.g., cross-secon and me node-o-node (c) Pongsa Pornchawseskul, Faculy of Economcs, Chulalongkorn Unversy (c) Pongsa Pornchawseskul,
More informationNon-gaussianity in axion N-flation models
Non-gaussanty n axon N-flaton models Soo A Km Kyung Hee Unversty Based on arxv:1005.4410 by SAK, Andrew R. Lddle and Davd Seery (Sussex), and earler papers by SAK and Lddle. COSMO/CosPA 2010 @ Unversty
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 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 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 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 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 informationNormal Random Variable and its discriminant functions
Noral Rando Varable and s dscrnan funcons Oulne Noral Rando Varable Properes Dscrnan funcons Why Noral Rando Varables? Analycally racable Works well when observaon coes for a corruped snle prooype 3 The
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( 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 informationern.ch
COSMOLOGY PHYS 3039 THE EARLY UNIVERSE Par II hp://cms.web.ce ern.ch Gampaolo Psano - Jodrell Bank Cenre for Asrophyscs The Unversy of Mancheser - Aprl 013 hp://www.jb.man.ac.uk/~gp/ gampaolo.psano@mancheser.ac.uk
More informationResponse of MDOF systems
Response of MDOF syses Degree of freedo DOF: he nu nuber of ndependen coordnaes requred o deerne copleely he posons of all pars of a syse a any nsan of e. wo DOF syses hree DOF syses he noral ode analyss
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 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 informationAdvanced Machine Learning & Perception
Advanced Machne Learnng & Percepon Insrucor: Tony Jebara SVM Feaure & Kernel Selecon SVM Eensons Feaure Selecon (Flerng and Wrappng) SVM Feaure Selecon SVM Kernel Selecon SVM Eensons Classfcaon Feaure/Kernel
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 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 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 informationStochastic Maxwell Equations in Photonic Crystal Modeling and Simulations
Sochasc Maxwell Equaons n Phoonc Crsal Modelng and Smulaons Hao-Mn Zhou School of Mah Georga Insue of Technolog Jon work wh: Al Adb ECE Majd Bade ECE Shu-Nee Chow Mah IPAM UCLA Aprl 14-18 2008 Parall suppored
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 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 informationBayes rule for a classification problem INF Discriminant functions for the normal density. Euclidean distance. Mahalanobis distance
INF 43 3.. Repeon Anne Solberg (anne@f.uo.no Bayes rule for a classfcaon problem Suppose we have J, =,...J classes. s he class label for a pxel, and x s he observed feaure vecor. We can use Bayes rule
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 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 informationBorn Oppenheimer Approximation and Beyond
L Born Oppenhemer Approxmaon and Beyond aro Barba A*dex Char Professor maro.barba@unv amu.fr Ax arselle Unversé, nsu de Chme Radcalare LGHT AD Adabac x dabac x nonadabac LGHT AD From Gree dabaos: o be
More informationEvading non-gaussianity consistency in single field inflation
Lorentz Center 7 Aug, 013 Evading non-gaussianity consistency in single field inflation Misao Sasaki Yukawa Institute for Theoretical Physic (YITP) Kyoto University M.H. Namjoo, H. Firouzjahi& MS, EPL101
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 informationOnline Appendix for. Strategic safety stocks in supply chains with evolving forecasts
Onlne Appendx for Sraegc safey socs n supply chans wh evolvng forecass Tor Schoenmeyr Sephen C. Graves Opsolar, Inc. 332 Hunwood Avenue Hayward, CA 94544 A. P. Sloan School of Managemen Massachuses Insue
More informationMisao Sasaki YITP, Kyoto University. 29 June, 2009 ICG, Portsmouth
Misao Sasaki YITP, Kyoto University 9 June, 009 ICG, Portsmouth contents 1. Inflation and curvature perturbations δn formalism. Origin of non-gaussianity subhorizon or superhorizon scales 3. Non-Gaussianity
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 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 informationH = d d q 1 d d q N d d p 1 d d p N exp
8333: Sacal Mechanc I roblem Se # 7 Soluon Fall 3 Canoncal Enemble Non-harmonc Ga: The Hamlonan for a ga of N non neracng parcle n a d dmenonal box ha he form H A p a The paron funcon gven by ZN T d d
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 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 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 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 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 informationTHERMODYNAMICS 1. The First Law and Other Basic Concepts (part 2)
Company LOGO THERMODYNAMICS The Frs Law and Oher Basc Conceps (par ) Deparmen of Chemcal Engneerng, Semarang Sae Unversy Dhon Harano S.T., M.T., M.Sc. Have you ever cooked? Equlbrum Equlbrum (con.) Equlbrum
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 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 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 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 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 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 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 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 informationBernoulli process with 282 ky periodicity is detected in the R-N reversals of the earth s magnetic field
Submed o: Suden Essay Awards n Magnecs Bernoull process wh 8 ky perodcy s deeced n he R-N reversals of he earh s magnec feld Jozsef Gara Deparmen of Earh Scences Florda Inernaonal Unversy Unversy Park,
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 informationMulti-Fuel and Mixed-Mode IC Engine Combustion Simulation with a Detailed Chemistry Based Progress Variable Library Approach
Mul-Fuel and Med-Mode IC Engne Combuson Smulaon wh a Dealed Chemsry Based Progress Varable Lbrary Approach Conens Inroducon Approach Resuls Conclusons 2 Inroducon New Combuson Model- PVM-MF New Legslaons
More informationGeneralized double sinh-gordon equation: Symmetry reductions, exact solutions and conservation laws
IJS (05) 9A: 89-96 Iranan Journal of Scence & echnology hp://ss.shrazu.ac.r Generalzed double snh-gordon equaon: Symmery reducons eac soluons and conservaon laws G. Magalawe B. Muaeea and C. M. Khalque
More informationLecture Notes 4. Univariate Forecasting and the Time Series Properties of Dynamic Economic Models
Tme Seres Seven N. Durlauf Unversy of Wsconsn Lecure Noes 4. Unvarae Forecasng and he Tme Seres Properes of Dynamc Economc Models Ths se of noes presens does hree hngs. Frs, formulas are developed o descrbe
More informationNonlinearity versus Perturbation Theory in Quantum Mechanics
Nonlneary versus Perurbaon Theory n Quanum Mechancs he parcle-parcle Coulomb neracon Glber Rensch Scence Insue, Unversy o Iceland, Dunhaga 3, IS-107 Reykjavk, Iceland There are bascally wo "smple" (.e.
More informationBandlimited channel. Intersymbol interference (ISI) This non-ideal communication channel is also called dispersive channel
Inersymol nererence ISI ISI s a sgnal-dependen orm o nererence ha arses ecause o devaons n he requency response o a channel rom he deal channel. Example: Bandlmed channel Tme Doman Bandlmed channel Frequency
More informationFall 2009 Social Sciences 7418 University of Wisconsin-Madison. Problem Set 2 Answers (4) (6) di = D (10)
Publc Affars 974 Menze D. Chnn Fall 2009 Socal Scences 7418 Unversy of Wsconsn-Madson Problem Se 2 Answers Due n lecure on Thursday, November 12. " Box n" your answers o he algebrac quesons. 1. Consder
More informationarxiv: v3 [physics.optics] 14 May 2012
Elecromagnec energy-momenum n dspersve meda T. G. Phlbn School of Physcs and Asronomy, Unversy of S Andrews, Norh Haugh, S Andrews, Ffe, KY16 9SS, Scoland, Uned Kngdom arxv:1008.3336v3 [physcs.opcs] 1
More informationLarge Primordial Non- Gaussianity from early Universe. Kazuya Koyama University of Portsmouth
Large Primordial Non- Gaussianity from early Universe Kazuya Koyama University of Portsmouth Primordial curvature perturbations Proved by CMB anisotropies nearly scale invariant n s = 0.960 ± 0.013 nearly
More informationChapter 6 DETECTION AND ESTIMATION: Model of digital communication system. Fundamental issues in digital communications are
Chaper 6 DEECIO AD EIMAIO: Fundamenal ssues n dgal communcaons are. Deecon and. Esmaon Deecon heory: I deals wh he desgn and evaluaon of decson makng processor ha observes he receved sgnal and guesses
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 informationPrimordial non Gaussianity from modulated trapping. David Langlois (APC)
Primordial non Gaussianity from modulated trapping David Langlois (APC) Outline 1. Par&cle produc&on during infla&on 2. Consequences for the power spectrum 3. Modulaton and primordial perturba&ons 4. Non
More informationDIFFERENTIAL GEOMETRY AND MODERN COSMOLOGY WITH FRACTIONALY DIFFERENTIATED LAGRANGIAN FUNCTION AND FRACTIONAL DECAYING FORCE TERM
DIFFERENTIAL GEOMETRY AND MODERN COSMOLOGY WITH FRACTIONALY DIFFERENTIATED LAGRANGIAN FUNCTION AND FRACTIONAL DECAYING FORCE TERM EL-NABULSI AHMAD RAMI Plasma Applcaon Laboraory, Deparmen of Nuclear and
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 informationTSS = SST + SSE An orthogonal partition of the total SS
ANOVA: Topc 4. Orhogonal conrass [ST&D p. 183] H 0 : µ 1 = µ =... = µ H 1 : The mean of a leas one reamen group s dfferen To es hs hypohess, a basc ANOVA allocaes he varaon among reamen means (SST) equally
More informationPHYS 1443 Section 001 Lecture #4
PHYS 1443 Secon 001 Lecure #4 Monda, June 5, 006 Moon n Two Dmensons Moon under consan acceleraon Projecle Moon Mamum ranges and heghs Reerence Frames and relae moon Newon s Laws o Moon Force Newon s Law
More informationChapter 6 Conservation Equations for Multiphase- Multicomponent Flow Through Porous Media
Chaper 6 Conservaon Equaons for Mulphase- Mulcomponen Flow Through Porous Meda The mass conservaon equaons wll appear repeaedly n many dfferen forms when dfferen dsplacemen processes are consdered. The
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 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 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 informationHomework 8: Rigid Body Dynamics Due Friday April 21, 2017
EN40: Dynacs and Vbraons Hoework 8: gd Body Dynacs Due Frday Aprl 1, 017 School of Engneerng Brown Unversy 1. The earh s roaon rae has been esaed o decrease so as o ncrease he lengh of a day a a rae of
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 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 informationThe topology and signature of the regulatory interactions predict the expression pattern of the segment polarity genes in Drosophila m elanogaster
The opology and sgnaure of he regulaory neracons predc he expresson paern of he segmen polary genes n Drosophla m elanogaser Hans Ohmer and Réka Alber Deparmen of Mahemacs Unversy of Mnnesoa Complex bologcal
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 informationNational Exams December 2015 NOTES: 04-BS-13, Biology. 3 hours duration
Naonal Exams December 205 04-BS-3 Bology 3 hours duraon NOTES: f doub exss as o he nerpreaon of any queson he canddae s urged o subm wh he answer paper a clear saemen of any assumpons made 2 Ths s a CLOSED
More informationCooling of a hot metal forging. , dt dt
Tranen Conducon Uneady Analy - Lumped Thermal Capacy Model Performed when; Hea ranfer whn a yem produced a unform emperaure drbuon n he yem (mall emperaure graden). The emperaure change whn he yem condered
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 informationMachine Learning Linear Regression
Machne Learnng Lnear Regresson Lesson 3 Lnear Regresson Bascs of Regresson Leas Squares esmaon Polynomal Regresson Bass funcons Regresson model Regularzed Regresson Sascal Regresson Mamum Lkelhood (ML)
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 informationRobustness Experiments with Two Variance Components
Naonal Insue of Sandards and Technology (NIST) Informaon Technology Laboraory (ITL) Sascal Engneerng Dvson (SED) Robusness Expermens wh Two Varance Componens by Ana Ivelsse Avlés avles@ns.gov Conference
More informationControl Systems. Mathematical Modeling of Control Systems.
Conrol Syem Mahemacal Modelng of Conrol Syem chbum@eoulech.ac.kr Oulne Mahemacal model and model ype. Tranfer funcon model Syem pole and zero Chbum Lee -Seoulech Conrol Syem Mahemacal Model Model are key
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 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 informationTranscription: Messenger RNA, mrna, is produced and transported to Ribosomes
Quanave Cenral Dogma I Reference hp//book.bonumbers.org Inaon ranscrpon RNA polymerase and ranscrpon Facor (F) s bnds o promoer regon of DNA ranscrpon Meenger RNA, mrna, s produced and ranspored o Rbosomes
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 informationFiltrage particulaire et suivi multi-pistes Carine Hue Jean-Pierre Le Cadre and Patrick Pérez
Chaînes de Markov cachées e flrage parculare 2-22 anver 2002 Flrage parculare e suv mul-pses Carne Hue Jean-Perre Le Cadre and Parck Pérez Conex Applcaons: Sgnal processng: arge rackng bearngs-onl rackng
More informationDynamic Team Decision Theory
Dynamc Team Decson Theory EECS 558 Proec Repor Shruvandana Sharma and Davd Shuman December, 005 I. Inroducon Whle he sochasc conrol problem feaures one decson maker acng over me, many complex conrolled
More informationSingle-loop System Reliability-Based Design & Topology Optimization (SRBDO/SRBTO): A Matrix-based System Reliability (MSR) Method
10 h US Naonal Congress on Compuaonal Mechancs Columbus, Oho 16-19, 2009 Sngle-loop Sysem Relably-Based Desgn & Topology Opmzaon (SRBDO/SRBTO): A Marx-based Sysem Relably (MSR) Mehod Tam Nguyen, Junho
More informationA Model for the Expansion of the Universe
Issue 2 April PROGRESS IN PHYSICS Volume 0 204 A Model for he Expansion of he Universe Nilon Penha Silva Deparmeno de Física Reired Professor, Universidade Federal de Minas Gerais, Belo Horizone, MG, Brazil
More information