Approximation and Fast Calculation of Non-local Boundary Conditions for the Time-dependent Schrödinger Equation
|
|
- Jonah Hunter
- 5 years ago
- Views:
Transcription
1 Approximation and Fast Cacuation of Non-oca Boundary Conditions for the Time-dependent Schrödinger Equation Anton Arnod, Matthias Ehrhardt 2, and Ivan Sofronov 3 Universität Münster, Institut für Numerische Mathematik, Einsteinstr. 62, D-4849 Münster, Germany ( 2 Technische Universität Berin, Institut für Mathematik, Str. des 7. Juni 36, D-623 Berin, Germany ( 3 Kedysh Institute of Appied Mathematics, Russian Academy of Sciences, Moscow, Russia (sofronov@spp.kedysh.ru) Summary. We present a way to efficienty treat the we-known transparent boundary conditions for the Schrödinger equation. Our approach is based on two ideas: firsty, to derive a discrete transparent boundary condition (DTBC) based on the Crank-Nicoson finite difference scheme for the governing equation. And, secondy, to approximate the discrete convoution kerne of DTBC by sum-of-exponentias for a rapid recursive cacuation of the convoution. We iustrate the efficiency of the proposed method on severa exampes. A much more detaied version of this artice can be found in Arnod et a. [23]. Introduction Discrete transparent boundary conditions for the discrete D Schrödinger equation ir(ψ j,n+ ψ j,n ) = 2 (ψ j,n+ + ψ j,n ) wv j,n+ 2 (ψ j,n+ + ψ j,n ), () where 2 ψ j = ψ j+ 2ψ j + ψ j, R = 4 x 2 / t, w = 2 x 2, V j,n+ := 2 V (x j, t n+ ), x j = j x, j Z; and V (x, t) = V 2 = const. for x ; V (x, t) = V + = const. for x X, t, ψ(x, ) = ψ I (x), with supp ψ I [, X], were introduced in Arnod [998]. The DTBC at e.g. the eft boundary point j = reads, cf. Thm. 3.8 in Ehrhardt and Arnod [2]: ψ,n s ψ,n = n k= s n kψ,k ψ,n, n. (2) The convoution kerne {s n } can be obtained by expicity cacuating the inverse Z transform of the function ŝ(z) := z+ ˆ z (z), where ˆ (z) = iζ ± ζ(ζ + 2i), ζ = R z 2 z+ + i x2 V (choose sign such that ˆ (z) > ).
2 42 Anton Arnod, Matthias Ehrhardt, and Ivan Sofronov Using (2) in numerica simuations permits to avoid any boundary refections and it renders the fuy discrete scheme unconditionay stabe, ike the Crank-Nicoson scheme () for the whoe-space probem. However, the numerica effort to evauate the DTBC increases ineary in t and it can sharpy raise the tota computationa costs. A strategy to overcome this drawback is the key issue of this paper. 2 Approximation by Sums of Exponentias The convoution coefficients s n appearing in the DTBC (2) can either be obtained from (engthy) expicit formuas or evauated numericay: s n ρ n N N k= ŝ(ρeiϕ k )e inϕ k, n =,,...,N. Here ϕ k = 2πk/N, and ρ > is a reguarization parameter. For the question of choosing ρ we refer the reader to Arnod et a. [23] and the references therein. Our fast method to cacuate the discrete convoution in (2) is based on approximating these coefficients s n by the foowing ansatz (sum of exponentias): s n s n := { sn, n =,...,ν, L = b q n, n = ν, ν +,..., where L, ν IN are fixed numbers. In order to find the appropriate constants {b, q }, we fix L and ν in (3) (e.g. ν = 2), and consider the Padé approximation P L (x) Q L(x) for the forma power series: f(x) := s ν + s ν+ x + s ν+2 x , x. Theorem. Let the poynomia Q L (x) have L simpe roots q with q >, =,...,L. Then s n = L = (3) b q n, n = ν, ν +,..., (4) where b := P L (q ) Q L (q ) qν, =,...,L. (5) Remark. A our practica cacuations confirm that the assumption of Theorem hods for any desired L, athough we cannot prove this. Remark 2. According to the definition of the Padé agorithm the first 2L+ν coefficients are reproduced exacty: s n = s n for n = ν, ν +,...,2L + ν. For the remaining s n with n > 2L + ν, the foowing estimate hods: s n s n = O(n 3 2). A typica graph of s n and s n s n versus n for L = 2 is shown in Fig. (note the different scaing for both graphs).
3 Efficient Non-oca Boundary Conditions for the Schrödinger Equation 43 s n and error s n s ~ n s n 5 error s n s ~ n n Fig.. Convoution coefficients s n (eft axis, dashed ine) and error s n s n of the convoution coefficients (right axis); x = /6, t = 2 5, V (L = 2). 3 The Transformation Rue A nice property of the considered approach consists of the foowing: once the approximate convoution coefficients { s n } are cacuated for particuar discretization parameters { x, t, V }, it is easy to transform them into appropriate coefficients for any other discretization. We sha confine this discussion to the case ν = 2: Transformation rue 3. For ν = 2, et the rationa function ˆ s(z) = s + s z + L = b q z q z (6) be the Z transform of the convoution kerne { s n } n= from (3), where { s n } is assumed to be an approximation to a DTBC for the equation () with a given set { x, t, V }. Then, for another set { x, t, V }, one can take the approximation where ˆ s (z) := s + s z + L b q = z q z, (7) q := q ā b a q b, b := b aā b b + q q (a q b)(q ā b), (8) + q
4 44 Anton Arnod, Matthias Ehrhardt, and Ivan Sofronov a := 2 x2 t + 2 x2 t + i( x 2 V x 2 V ), (9) b := 2 x2 t 2 x2 t i( x 2 V x 2 V ). () s, s are the exact convoution coefficients for the parameters { x, t, V }. Whie the Padé agorithm provides a method to cacuate approximate convoution coefficients s n for fixed parameters { x, t, V }, the Transformation rue yieds the natura ink between different parameter sets { x, t, V } (and L fixed). Exampe. For L = we cacuated the coefficients {b, q } with the parameters x =, t =, V = and then used the Transformation 3. to cacuate the coefficients {b, q } for the parameters x = /6, t = 2 5, V = 45. Fig. 2 shows that the resuting convoution coefficients s n are in this exampe even better approximations to the exact coefficients s n than the coefficients s n, which are obtained directy from the Padé agorithm discussed in Theorem. Hence, the numerica soution of the corresponding Schrödinger equation is aso more accurate (cf. Fig. 5). The Mape code that was used to cacuate the coefficients q, b in the approximation (3) incuding the expicit formuas in Transformation rue 3. can be downoaded from the authors homepages. ~ 7 x error s n s 3 n, s n s~ * n n Fig. 2. Approximation error of the approximate convoution coefficients for ν = 2, x = /6, t = 2 5, V = 45: The error of s n (- - -) obtained from the transformation rue and the error of s n ( ) obtained from a direct Padé approximation of the exact coefficients s n.
5 Efficient Non-oca Boundary Conditions for the Schrödinger Equation 45 4 Fast Evauation of the Discrete Convoution Given the approximation (3) of the discrete convoution kerne appearing in the DTBC (2), the convoution n ν C (n) (u) := u k s n k, s n = k= L = b q n, q >, () of a discrete function u k, k =, 2,..., can be cacuated efficienty by recurrence formuas, cf. Sofronov [998]: Theorem 2. The function C (n) (u) from () for n ν + is represented by where C (n) (u) = q C (n ) C (n) (u) = L = C (n) (u), (2) (u) + b q ν u n ν for n ν +, C (ν) (u). (3) This recursion drasticay reduces the computationa effort of evauating DT- BCs for ong time computations (n ): O(L n) instead of O(n 2 ) arithmetic operations. 5 Numerica Exampes In this section we sha present two exampes to compare the numerica resuts from using our approach of the approximated DTBC, i.e. the sum-of-exponentias-ansatz (3) (with ν = 2) to the soution using the exact DTBC (2). Exampe 2. As an exampe, we consider () on x with V = V + =, and initia data ψ I (x) = exp(i5x 3(x.5) 2 ). The time evoution of the approximate soution ψ a (x, t) using the approximated DTBC with convoution coefficients { s n } and L =, L = 2, respectivey, is shown in Fig. 3 (observe the viewing ange). Whie one can observe some refected wave when using the approximated DTBC with L =, there are amost no refections visibe when using the approximated DTBC with L = 2. The goa is to investigate the ong time stabiity behaviour of the approximated DTBC with the sum-of-exponentias ansatz. The reference soution ψ ref with x = /4, t = 2 5 is obtained by using exact DTBCs (2) at the ends x = and x =. We vary the parameter L =, 2, 3, 4 in (3), find the corresponding approximate DTBCs, and show the reative error of the approximate soution, i.e. ψa ψ ref L2 (t) ψ I L2. The resut up to n = 5
6 46 Anton Arnod, Matthias Ehrhardt, and Ivan Sofronov..2.8 ψ(x,t) t.8 x..2.8 ψ(x,t) t.8 x Fig. 3. Time evoution of ψa (x, t) : The approximate convoution coefficients consisting of L = discrete exponentias give rise to a refected wave (upper figure). Using L = 2 discrete exponentias make refections amost invisibe (ower figure). is shown in Fig. 4. Larger vaues of L ceary yied more accurate coefficients and hence a more accurate soution ψa. Fig. 4 aso shows the discretization kψ ψ k (t) error, i.e. refkψi kanl L2, where ψan, the anaytic soution of this exampe is 2 expicity computabe. Exampe 3. The second exampe considers () on [, 2] with zero potentia in the interior (V (x) for < x < 2) and V (x) 45 outside the computa-
7 Efficient Non-oca Boundary Conditions for the Schrödinger Equation 47 reative L 2 Norm of refected part for different number of coefficients 2 ψ ref L = L = 2 L = 3 L = 4 3 ψ a ψ ref L 2 / ψ I L t Fig. 4. Error of the approximate soution ψ a(t) with approximate convoution coefficients consisting of L =, 2, 3, 4 discrete exponentias and discretization error. ψ ref is the reative error of ψ ref. The error-peak between t =. and t =.2 corresponds to the first refected wave. tiona domain. The initia data is taken as [ψ I (x) = exp(ix 3(x ) 2 )], and this wave packet is partiay refected at the boundaries. We use the rather coarse space discretization x = /6, the time step t = 2 5, and the exact DTBC (2). The vaue of the potentia is chosen such that at time t =.8, i.e. after 4 time steps 75% of the mass ( ψ(., t) 2 2 ) has eft the domain. Fig. 5 shows the time decay of the discrete 2 -norm ψ(., t) 2 and the tempora evoution of the error e L (., t) 2 := ψ a (., t) ψ ref (., t) 2 / ψ I L2 when using an approximated DTBC with L = 2, 3, 4. Additionay, we cacuated for L = 2 the coefficients {b, q } for the normaized parameters x =, t =, V = and then used the Transformation rue 3. to cacuate the coefficients {b, q } for the desired parameters. 6 Concusion For numerica simuations of the Schrödinger equation one has to introduce artificia (preferabe transparent) boundary conditions in order to confine the cacuation to a finite region. Such TBCs are non-oca in time (of convoution form). Hence, the numerica costs (just) for evauating these BCs grow quadraticay in time. And for ong-time cacuations it can easiy outweigh the costs for soving the PDE inside the computationa domain. Here, we presented an efficient method to overcome this probem. We construct approximate DTBCs that are of a sum-of-exponentia form and hence
8 48 Anton Arnod, Matthias Ehrhardt, and Ivan Sofronov.5 ψ(.,t) 2, e L (.,t) 2.4 DTBC L = 2 L = 2 (trafo) L = 3 L = t Fig. 5. Time evoution within the potentia we (V = 45) of ψ(., t) 2 and of the errors e L(., t) 2 that are due to approximated DTBCs with L = 2, 3, 4. L = 2 (trafo) uses coefficients cacuated by the Transformation rue 3.. ony invove a ineary growing numerica effort. Moreover, these BCs yied very accurate soutions, and it was shown in Arnod et a. [23] that the resuting initia-boundary vaue scheme is conditionay 2 -stabe on [, T] as t (e.g. for < t < t and x = x = const.). Acknowedgement. The first two authors were partiay supported by the IHP Network HPRN-CT from the EU and the DFG under Grant-No. AR 277/3-. The second author was supported by the DFG Research Center Mathematics for key technoogies (FZT 86) in Berin. The third author was partiay supported by RFBR-Grant No and by Saarand University. References A. Arnod. Numericay absorbing boundary conditions for quantum evoution equations. VLSI Design, 6:33 39, 998. A. Arnod, M. Ehrhardt, and I. Sofronov. Discrete transparent boundary conditions for the Schrödinger equation: Fast cacuation, approximation, and stabiity. Comm. Math. Sci., :5 556, 23. M. Ehrhardt and A. Arnod. Discrete transparent boundary conditions for the Schrödinger equation. Riv. Mat. Univ. Parma, 6:57 8, 2. I. Sofronov. Artificia boundary conditions of absoute transparency for twoand threedimensiona externa time dependent scattering probems. Euro. J. App. Math., 9:56 588, 998.
An approximate method for solving the inverse scattering problem with fixed-energy data
J. Inv. I-Posed Probems, Vo. 7, No. 6, pp. 561 571 (1999) c VSP 1999 An approximate method for soving the inverse scattering probem with fixed-energy data A. G. Ramm and W. Scheid Received May 12, 1999
More informationV.B The Cluster Expansion
V.B The Custer Expansion For short range interactions, speciay with a hard core, it is much better to repace the expansion parameter V( q ) by f( q ) = exp ( βv( q )), which is obtained by summing over
More informationV.B The Cluster Expansion
V.B The Custer Expansion For short range interactions, speciay with a hard core, it is much better to repace the expansion parameter V( q ) by f(q ) = exp ( βv( q )) 1, which is obtained by summing over
More informationA Brief Introduction to Markov Chains and Hidden Markov Models
A Brief Introduction to Markov Chains and Hidden Markov Modes Aen B MacKenzie Notes for December 1, 3, &8, 2015 Discrete-Time Markov Chains You may reca that when we first introduced random processes,
More informationDIGITAL FILTER DESIGN OF IIR FILTERS USING REAL VALUED GENETIC ALGORITHM
DIGITAL FILTER DESIGN OF IIR FILTERS USING REAL VALUED GENETIC ALGORITHM MIKAEL NILSSON, MATTIAS DAHL AND INGVAR CLAESSON Bekinge Institute of Technoogy Department of Teecommunications and Signa Processing
More informationFirst-Order Corrections to Gutzwiller s Trace Formula for Systems with Discrete Symmetries
c 26 Noninear Phenomena in Compex Systems First-Order Corrections to Gutzwier s Trace Formua for Systems with Discrete Symmetries Hoger Cartarius, Jörg Main, and Günter Wunner Institut für Theoretische
More informationSupporting Information for Suppressing Klein tunneling in graphene using a one-dimensional array of localized scatterers
Supporting Information for Suppressing Kein tunneing in graphene using a one-dimensiona array of ocaized scatterers Jamie D Was, and Danie Hadad Department of Chemistry, University of Miami, Cora Gabes,
More informationExplicit overall risk minimization transductive bound
1 Expicit overa risk minimization transductive bound Sergio Decherchi, Paoo Gastado, Sandro Ridea, Rodofo Zunino Dept. of Biophysica and Eectronic Engineering (DIBE), Genoa University Via Opera Pia 11a,
More informationMath 124B January 31, 2012
Math 124B January 31, 212 Viktor Grigoryan 7 Inhomogeneous boundary vaue probems Having studied the theory of Fourier series, with which we successfuy soved boundary vaue probems for the homogeneous heat
More informationMATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES
MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES Separation of variabes is a method to sove certain PDEs which have a warped product structure. First, on R n, a inear PDE of order m is
More informationIntroduction to Simulation - Lecture 13. Convergence of Multistep Methods. Jacob White. Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy
Introduction to Simuation - Lecture 13 Convergence of Mutistep Methods Jacob White Thans to Deepa Ramaswamy, Micha Rewiensi, and Karen Veroy Outine Sma Timestep issues for Mutistep Methods Loca truncation
More informationIntroduction to Simulation - Lecture 14. Multistep Methods II. Jacob White. Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy
Introduction to Simuation - Lecture 14 Mutistep Methods II Jacob White Thans to Deepa Ramaswamy, Micha Rewiensi, and Karen Veroy Outine Sma Timestep issues for Mutistep Methods Reminder about LTE minimization
More information6 Wave Equation on an Interval: Separation of Variables
6 Wave Equation on an Interva: Separation of Variabes 6.1 Dirichet Boundary Conditions Ref: Strauss, Chapter 4 We now use the separation of variabes technique to study the wave equation on a finite interva.
More informationarxiv:nlin/ v2 [nlin.cd] 30 Jan 2006
expansions in semicassica theories for systems with smooth potentias and discrete symmetries Hoger Cartarius, Jörg Main, and Günter Wunner arxiv:nin/0510051v [nin.cd] 30 Jan 006 1. Institut für Theoretische
More informationAn explicit Jordan Decomposition of Companion matrices
An expicit Jordan Decomposition of Companion matrices Fermín S V Bazán Departamento de Matemática CFM UFSC 88040-900 Forianópois SC E-mai: fermin@mtmufscbr S Gratton CERFACS 42 Av Gaspard Coriois 31057
More informationLecture Note 3: Stationary Iterative Methods
MATH 5330: Computationa Methods of Linear Agebra Lecture Note 3: Stationary Iterative Methods Xianyi Zeng Department of Mathematica Sciences, UTEP Stationary Iterative Methods The Gaussian eimination (or
More informationIntegrating Factor Methods as Exponential Integrators
Integrating Factor Methods as Exponentia Integrators Borisav V. Minchev Department of Mathematica Science, NTNU, 7491 Trondheim, Norway Borko.Minchev@ii.uib.no Abstract. Recenty a ot of effort has been
More informationPublished in: Proceedings of the Twenty Second Nordic Seminar on Computational Mechanics
Aaborg Universitet An Efficient Formuation of the Easto-pastic Constitutive Matrix on Yied Surface Corners Causen, Johan Christian; Andersen, Lars Vabbersgaard; Damkide, Lars Pubished in: Proceedings of
More information2-loop additive mass renormalization with clover fermions and Symanzik improved gluons
2-oop additive mass renormaization with cover fermions and Symanzik improved guons Apostoos Skouroupathis Department of Physics, University of Cyprus, Nicosia, CY-1678, Cyprus E-mai: php4as01@ucy.ac.cy
More informationSmall generators of function fields
Journa de Théorie des Nombres de Bordeaux 00 (XXXX), 000 000 Sma generators of function fieds par Martin Widmer Résumé. Soit K/k une extension finie d un corps goba, donc K contient un éément primitif
More informationTwo Kinds of Parabolic Equation algorithms in the Computational Electromagnetics
Avaiabe onine at www.sciencedirect.com Procedia Engineering 9 (0) 45 49 0 Internationa Workshop on Information and Eectronics Engineering (IWIEE) Two Kinds of Paraboic Equation agorithms in the Computationa
More informationC. Fourier Sine Series Overview
12 PHILIP D. LOEWEN C. Fourier Sine Series Overview Let some constant > be given. The symboic form of the FSS Eigenvaue probem combines an ordinary differentia equation (ODE) on the interva (, ) with a
More informationRapid and Stable Determination of Rotation Matrices between Spherical Harmonics by Direct Recursion
Chemistry Pubications Chemistry 11-1999 Rapid and Stabe Determination of Rotation Matrices between Spherica Harmonics by Direct Recursion Cheo Ho Choi Iowa State University Joseph Ivanic Iowa State University
More informationCopyright information to be inserted by the Publishers. Unsplitting BGK-type Schemes for the Shallow. Water Equations KUN XU
Copyright information to be inserted by the Pubishers Unspitting BGK-type Schemes for the Shaow Water Equations KUN XU Mathematics Department, Hong Kong University of Science and Technoogy, Cear Water
More informationOn the Computation of (2-2) Three- Center Slater-Type Orbital Integrals of 1/r 12 Using Fourier-Transform-Based Analytical Formulas
On the Computation of (2-2) Three- Center Sater-Type Orbita Integras of /r 2 Using Fourier-Transform-Based Anaytica Formuas DANKO ANTOLOVIC, HARRIS J. SILVERSTONE 2 Pervasive Technoogy Labs, Indiana University,
More informationCS229 Lecture notes. Andrew Ng
CS229 Lecture notes Andrew Ng Part IX The EM agorithm In the previous set of notes, we taked about the EM agorithm as appied to fitting a mixture of Gaussians. In this set of notes, we give a broader view
More informationAn Algorithm for Pruning Redundant Modules in Min-Max Modular Network
An Agorithm for Pruning Redundant Modues in Min-Max Moduar Network Hui-Cheng Lian and Bao-Liang Lu Department of Computer Science and Engineering, Shanghai Jiao Tong University 1954 Hua Shan Rd., Shanghai
More informationNumerical solution of the Schrödinger equation on unbounded domains
Numerical solution of the Schrödinger equation on unbounded domains Maike Schulte Institute for Numerical and Applied Mathematics Westfälische Wilhelms-Universität Münster Cetraro, September 6 OUTLINE
More information4 Separation of Variables
4 Separation of Variabes In this chapter we describe a cassica technique for constructing forma soutions to inear boundary vaue probems. The soution of three cassica (paraboic, hyperboic and eiptic) PDE
More informationHYDROGEN ATOM SELECTION RULES TRANSITION RATES
DOING PHYSICS WITH MATLAB QUANTUM PHYSICS Ian Cooper Schoo of Physics, University of Sydney ian.cooper@sydney.edu.au HYDROGEN ATOM SELECTION RULES TRANSITION RATES DOWNLOAD DIRECTORY FOR MATLAB SCRIPTS
More informationUniprocessor Feasibility of Sporadic Tasks with Constrained Deadlines is Strongly conp-complete
Uniprocessor Feasibiity of Sporadic Tasks with Constrained Deadines is Strongy conp-compete Pontus Ekberg and Wang Yi Uppsaa University, Sweden Emai: {pontus.ekberg yi}@it.uu.se Abstract Deciding the feasibiity
More informationLegendre Polynomials - Lecture 8
Legendre Poynomias - Lecture 8 Introduction In spherica coordinates the separation of variabes for the function of the poar ange resuts in Legendre s equation when the soution is independent of the azimutha
More informationHigh-order approximations to the Mie series for electromagnetic scattering in three dimensions
Proceedings of the 9th WSEAS Internationa Conference on Appied Mathematics Istanbu Turkey May 27-29 2006 (pp199-204) High-order approximations to the Mie series for eectromagnetic scattering in three dimensions
More informationMore Scattering: the Partial Wave Expansion
More Scattering: the Partia Wave Expansion Michae Fower /7/8 Pane Waves and Partia Waves We are considering the soution to Schrödinger s equation for scattering of an incoming pane wave in the z-direction
More informationResearch on liquid sloshing performance in vane type tank under microgravity
IOP Conference Series: Materias Science and Engineering PAPER OPEN ACCESS Research on iquid soshing performance in vane type tan under microgravity Reated content - Numerica simuation of fuid fow in the
More informationXSAT of linear CNF formulas
XSAT of inear CN formuas Bernd R. Schuh Dr. Bernd Schuh, D-50968 Kön, Germany; bernd.schuh@netcoogne.de eywords: compexity, XSAT, exact inear formua, -reguarity, -uniformity, NPcompeteness Abstract. Open
More informationInternational Journal of Mass Spectrometry
Internationa Journa of Mass Spectrometry 280 (2009) 179 183 Contents ists avaiabe at ScienceDirect Internationa Journa of Mass Spectrometry journa homepage: www.esevier.com/ocate/ijms Stark mixing by ion-rydberg
More informationVALIDATED CONTINUATION FOR EQUILIBRIA OF PDES
SIAM J. NUMER. ANAL. Vo. 0, No. 0, pp. 000 000 c 200X Society for Industria and Appied Mathematics VALIDATED CONTINUATION FOR EQUILIBRIA OF PDES SARAH DAY, JEAN-PHILIPPE LESSARD, AND KONSTANTIN MISCHAIKOW
More informationResearch Article Solution of Point Reactor Neutron Kinetics Equations with Temperature Feedback by Singularly Perturbed Method
Science and Technoogy of Nucear Instaations Voume 213, Artice ID 261327, 6 pages http://dx.doi.org/1.1155/213/261327 Research Artice Soution of Point Reactor Neutron Kinetics Equations with Temperature
More informationIdentification of macro and micro parameters in solidification model
BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES Vo. 55, No. 1, 27 Identification of macro and micro parameters in soidification mode B. MOCHNACKI 1 and E. MAJCHRZAK 2,1 1 Czestochowa University
More informationThe Group Structure on a Smooth Tropical Cubic
The Group Structure on a Smooth Tropica Cubic Ethan Lake Apri 20, 2015 Abstract Just as in in cassica agebraic geometry, it is possibe to define a group aw on a smooth tropica cubic curve. In this note,
More information6.434J/16.391J Statistics for Engineers and Scientists May 4 MIT, Spring 2006 Handout #17. Solution 7
6.434J/16.391J Statistics for Engineers and Scientists May 4 MIT, Spring 2006 Handout #17 Soution 7 Probem 1: Generating Random Variabes Each part of this probem requires impementation in MATLAB. For the
More informationA Novel Learning Method for Elman Neural Network Using Local Search
Neura Information Processing Letters and Reviews Vo. 11, No. 8, August 2007 LETTER A Nove Learning Method for Eman Neura Networ Using Loca Search Facuty of Engineering, Toyama University, Gofuu 3190 Toyama
More informationVALIDATED CONTINUATION FOR EQUILIBRIA OF PDES
VALIDATED CONTINUATION FOR EQUILIBRIA OF PDES SARAH DAY, JEAN-PHILIPPE LESSARD, AND KONSTANTIN MISCHAIKOW Abstract. One of the most efficient methods for determining the equiibria of a continuous parameterized
More informationPhysics 127c: Statistical Mechanics. Fermi Liquid Theory: Collective Modes. Boltzmann Equation. The quasiparticle energy including interactions
Physics 27c: Statistica Mechanics Fermi Liquid Theory: Coective Modes Botzmann Equation The quasipartice energy incuding interactions ε p,σ = ε p + f(p, p ; σ, σ )δn p,σ, () p,σ with ε p ε F + v F (p p
More informationc 2007 Society for Industrial and Applied Mathematics
SIAM REVIEW Vo. 49,No. 1,pp. 111 1 c 7 Society for Industria and Appied Mathematics Domino Waves C. J. Efthimiou M. D. Johnson Abstract. Motivated by a proposa of Daykin [Probem 71-19*, SIAM Rev., 13 (1971),
More informationarxiv: v1 [hep-lat] 23 Nov 2017
arxiv:1711.08830v1 [hep-at] 23 Nov 2017 Tetraquark resonances computed with static attice QCD potentias and scattering theory Pedro Bicudo 1,, Marco Cardoso 1, Antje Peters 2, Martin Pfaumer 2, and Marc
More informationDavid Eigen. MA112 Final Paper. May 10, 2002
David Eigen MA112 Fina Paper May 1, 22 The Schrodinger equation describes the position of an eectron as a wave. The wave function Ψ(t, x is interpreted as a probabiity density for the position of the eectron.
More informationORTHOGONAL MULTI-WAVELETS FROM MATRIX FACTORIZATION
J. Korean Math. Soc. 46 2009, No. 2, pp. 281 294 ORHOGONAL MLI-WAVELES FROM MARIX FACORIZAION Hongying Xiao Abstract. Accuracy of the scaing function is very crucia in waveet theory, or correspondingy,
More informationhttps://doi.org/ /epjconf/
HOW TO APPLY THE OPTIMAL ESTIMATION METHOD TO YOUR LIDAR MEASUREMENTS FOR IMPROVED RETRIEVALS OF TEMPERATURE AND COMPOSITION R. J. Sica 1,2,*, A. Haefee 2,1, A. Jaai 1, S. Gamage 1 and G. Farhani 1 1 Department
More informationA NOTE ON QUASI-STATIONARY DISTRIBUTIONS OF BIRTH-DEATH PROCESSES AND THE SIS LOGISTIC EPIDEMIC
(January 8, 2003) A NOTE ON QUASI-STATIONARY DISTRIBUTIONS OF BIRTH-DEATH PROCESSES AND THE SIS LOGISTIC EPIDEMIC DAMIAN CLANCY, University of Liverpoo PHILIP K. POLLETT, University of Queensand Abstract
More informationExpectation-Maximization for Estimating Parameters for a Mixture of Poissons
Expectation-Maximization for Estimating Parameters for a Mixture of Poissons Brandon Maone Department of Computer Science University of Hesini February 18, 2014 Abstract This document derives, in excrutiating
More informationPhysics 505 Fall Homework Assignment #4 Solutions
Physics 505 Fa 2005 Homework Assignment #4 Soutions Textbook probems: Ch. 3: 3.4, 3.6, 3.9, 3.0 3.4 The surface of a hoow conducting sphere of inner radius a is divided into an even number of equa segments
More information$, (2.1) n="# #. (2.2)
Chapter. Eectrostatic II Notes: Most of the materia presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartoo, Chap... Mathematica Considerations.. The Fourier series and the Fourier
More informationPRIME TWISTS OF ELLIPTIC CURVES
PRIME TWISTS OF ELLIPTIC CURVES DANIEL KRIZ AND CHAO LI Abstract. For certain eiptic curves E/Q with E(Q)[2] = Z/2Z, we prove a criterion for prime twists of E to have anaytic rank 0 or 1, based on a mod
More informationON THE REPRESENTATION OF OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS
SIAM J. NUMER. ANAL. c 1992 Society for Industria Appied Mathematics Vo. 6, No. 6, pp. 1716-1740, December 1992 011 ON THE REPRESENTATION OF OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS G. BEYLKIN
More information14 Separation of Variables Method
14 Separation of Variabes Method Consider, for exampe, the Dirichet probem u t = Du xx < x u(x, ) = f(x) < x < u(, t) = = u(, t) t > Let u(x, t) = T (t)φ(x); now substitute into the equation: dt
More information4 1-D Boundary Value Problems Heat Equation
4 -D Boundary Vaue Probems Heat Equation The main purpose of this chapter is to study boundary vaue probems for the heat equation on a finite rod a x b. u t (x, t = ku xx (x, t, a < x < b, t > u(x, = ϕ(x
More informationFactorization of Cyclotomic Polynomials with Quadratic Radicals in the Coefficients
Advances in Pure Mathematics, 07, 7, 47-506 http://www.scirp.org/journa/apm ISSN Onine: 60-0384 ISSN Print: 60-0368 Factoriation of Cycotomic Poynomias with Quadratic Radicas in the Coefficients Afred
More informationarxiv: v1 [math-ph] 4 Feb 2013
Fast and Accurate Computation of Exact Nonrefecting Boundary Condition for Maxwe s Equations Xiaodan Zhao and Li-Lian Wang Division of Mathematica Sciences Nanyang Technoogica University, Singapore, 637371
More informationWidth of Percolation Transition in Complex Networks
APS/23-QED Width of Percoation Transition in Compex Networs Tomer Kaisy, and Reuven Cohen 2 Minerva Center and Department of Physics, Bar-Ian University, 52900 Ramat-Gan, Israe 2 Department of Computer
More informationWave Equation Dirichlet Boundary Conditions
Wave Equation Dirichet Boundary Conditions u tt x, t = c u xx x, t, < x 1 u, t =, u, t = ux, = fx u t x, = gx Look for simpe soutions in the form ux, t = XxT t Substituting into 13 and dividing
More informationChapter 5. Wave equation. 5.1 Physical derivation
Chapter 5 Wave equation In this chapter, we discuss the wave equation u tt a 2 u = f, (5.1) where a > is a constant. We wi discover that soutions of the wave equation behave in a different way comparing
More informationSEMINAR 2. PENDULUMS. V = mgl cos θ. (2) L = T V = 1 2 ml2 θ2 + mgl cos θ, (3) d dt ml2 θ2 + mgl sin θ = 0, (4) θ + g l
Probem 7. Simpe Penduum SEMINAR. PENDULUMS A simpe penduum means a mass m suspended by a string weightess rigid rod of ength so that it can swing in a pane. The y-axis is directed down, x-axis is directed
More informationA. Distribution of the test statistic
A. Distribution of the test statistic In the sequentia test, we first compute the test statistic from a mini-batch of size m. If a decision cannot be made with this statistic, we keep increasing the mini-batch
More informationPath planning with PH G2 splines in R2
Path panning with PH G2 spines in R2 Laurent Gajny, Richard Béarée, Eric Nyiri, Oivier Gibaru To cite this version: Laurent Gajny, Richard Béarée, Eric Nyiri, Oivier Gibaru. Path panning with PH G2 spines
More informationOptimum Design Method of Viscous Dampers in Building Frames Using Calibration Model
The 4 th Word Conference on Earthquake Engineering October -7, 8, Beijing, China Optimum Design Method of iscous Dampers in Buiding Frames sing Caibration Mode M. Yamakawa, Y. Nagano, Y. ee 3, K. etani
More informationComponentwise Determination of the Interval Hull Solution for Linear Interval Parameter Systems
Componentwise Determination of the Interva Hu Soution for Linear Interva Parameter Systems L. V. Koev Dept. of Theoretica Eectrotechnics, Facuty of Automatics, Technica University of Sofia, 1000 Sofia,
More informationMultilayer Kerceptron
Mutiayer Kerceptron Zotán Szabó, András Lőrincz Department of Information Systems, Facuty of Informatics Eötvös Loránd University Pázmány Péter sétány 1/C H-1117, Budapest, Hungary e-mai: szzoi@csetehu,
More informationNUMERICAL STUDY OF A QUANTUM-DIFFUSIVE SPIN MODEL FOR TWO-DIMENSIONAL ELECTRON GASES
COMMUN. MATH. SCI. Vo. 13, No. 6, pp. 1347 1378 c 2015 Internationa Press NUMERICAL STUDY OF A QUANTUM-DIFFUSIVE SPIN MODEL FOR TWO-DIMENSIONAL ELECTRON GASES LUIGI BARLETTI, FLORIAN MÉHATS, CLAUDIA NEGULESCU,
More informationOn a geometrical approach in contact mechanics
Institut für Mechanik On a geometrica approach in contact mechanics Aexander Konyukhov, Kar Schweizerhof Universität Karsruhe, Institut für Mechanik Institut für Mechanik Kaiserstr. 12, Geb. 20.30 76128
More informationPaper presented at the Workshop on Space Charge Physics in High Intensity Hadron Rings, sponsored by Brookhaven National Laboratory, May 4-7,1998
Paper presented at the Workshop on Space Charge Physics in High ntensity Hadron Rings, sponsored by Brookhaven Nationa Laboratory, May 4-7,998 Noninear Sef Consistent High Resoution Beam Hao Agorithm in
More informationSmoothness equivalence properties of univariate subdivision schemes and their projection analogues
Numerische Mathematik manuscript No. (wi be inserted by the editor) Smoothness equivaence properties of univariate subdivision schemes and their projection anaogues Phiipp Grohs TU Graz Institute of Geometry
More informationThe EM Algorithm applied to determining new limit points of Mahler measures
Contro and Cybernetics vo. 39 (2010) No. 4 The EM Agorithm appied to determining new imit points of Maher measures by Souad E Otmani, Georges Rhin and Jean-Marc Sac-Épée Université Pau Veraine-Metz, LMAM,
More informationTIME DEPENDENT TEMPERATURE DISTRIBUTION MODEL IN LAYERED HUMAN DERMAL PART
VOL. 8, No. II, DECEMBER, 0, pp 66-76 TIME DEPENDENT TEMPERATURE DISTRIBUTION MODEL IN LAYERED HUMAN DERMAL PART Saraswati Acharya*, D. B. Gurung, V. P. Saxena Department of Natura Sciences (Mathematics),
More informationFormulas for Angular-Momentum Barrier Factors Version II
BNL PREPRINT BNL-QGS-06-101 brfactor1.tex Formuas for Anguar-Momentum Barrier Factors Version II S. U. Chung Physics Department, Brookhaven Nationa Laboratory, Upton, NY 11973 March 19, 2015 abstract A
More informationCoupling of LWR and phase transition models at boundary
Couping of LW and phase transition modes at boundary Mauro Garaveo Dipartimento di Matematica e Appicazioni, Università di Miano Bicocca, via. Cozzi 53, 20125 Miano Itay. Benedetto Piccoi Department of
More informationPhysics 235 Chapter 8. Chapter 8 Central-Force Motion
Physics 35 Chapter 8 Chapter 8 Centra-Force Motion In this Chapter we wi use the theory we have discussed in Chapter 6 and 7 and appy it to very important probems in physics, in which we study the motion
More informationNIKOS FRANTZIKINAKIS. N n N where (Φ N) N N is any Følner sequence
SOME OPE PROBLEMS O MULTIPLE ERGODIC AVERAGES IKOS FRATZIKIAKIS. Probems reated to poynomia sequences In this section we give a ist of probems reated to the study of mutipe ergodic averages invoving iterates
More informationNumerical simulation of javelin best throwing angle based on biomechanical model
ISSN : 0974-7435 Voume 8 Issue 8 Numerica simuation of javein best throwing ange based on biomechanica mode Xia Zeng*, Xiongwei Zuo Department of Physica Education, Changsha Medica University, Changsha
More informationThe distribution of the number of nodes in the relative interior of the typical I-segment in homogeneous planar anisotropic STIT Tessellations
Comment.Math.Univ.Caroin. 51,3(21) 53 512 53 The distribution of the number of nodes in the reative interior of the typica I-segment in homogeneous panar anisotropic STIT Tesseations Christoph Thäe Abstract.
More information1D Heat Propagation Problems
Chapter 1 1D Heat Propagation Probems If the ambient space of the heat conduction has ony one dimension, the Fourier equation reduces to the foowing for an homogeneous body cρ T t = T λ 2 + Q, 1.1) x2
More informationData Search Algorithms based on Quantum Walk
Data Search Agorithms based on Quantum Wak Masataka Fujisaki, Hiromi Miyajima, oritaka Shigei Abstract For searching any item in an unsorted database with items, a cassica computer takes O() steps but
More informationarxiv:hep-ph/ v1 19 Dec 1997
arxiv:hep-ph/972449v 9 Dec 997 A new method for the cacuation of massive mutioop diagrams K. Knecht, H. Verschede Department of Mathematica Physics and Astronomy, University of Ghent, Krijgsaan 28, 9 Ghent,
More informationPhysics 505 Fall 2007 Homework Assignment #5 Solutions. Textbook problems: Ch. 3: 3.13, 3.17, 3.26, 3.27
Physics 55 Fa 7 Homework Assignment #5 Soutions Textook proems: Ch. 3: 3.3, 3.7, 3.6, 3.7 3.3 Sove for the potentia in Proem 3., using the appropriate Green function otained in the text, and verify that
More informationDual Integral Equations and Singular Integral. Equations for Helmholtz Equation
Int.. Contemp. Math. Sciences, Vo. 4, 9, no. 34, 1695-1699 Dua Integra Equations and Singuar Integra Equations for Hemhotz Equation Naser A. Hoshan Department of Mathematics TafiaTechnica University P.O.
More informationStrauss PDEs 2e: Section Exercise 2 Page 1 of 12. For problem (1), complete the calculation of the series in case j(t) = 0 and h(t) = e t.
Strauss PDEs e: Section 5.6 - Exercise Page 1 of 1 Exercise For probem (1, compete the cacuation of the series in case j(t = and h(t = e t. Soution With j(t = and h(t = e t, probem (1 on page 147 becomes
More informationHigh Accuracy Split-Step Finite Difference Method for Schrödinger-KdV Equations
Commun. Theor. Phys. 70 208 43 422 Vo. 70, No. 4, October, 208 High Accuracy Spit-Step Finite Difference Method for Schrödinger-KdV Equations Feng Liao 廖锋, and Lu-Ming Zhang 张鲁明 2 Schoo of Mathematics
More informationMultigrid Method for Elliptic Control Problems
J OHANNES KEPLER UNIVERSITÄT LINZ Netzwerk f ür Forschung, L ehre und Praxis Mutigrid Method for Eiptic Contro Probems MASTERARBEIT zur Erangung des akademischen Grades MASTER OF SCIENCE in der Studienrichtung
More informationTechnische Universität Chemnitz
Technische Universität Chemnitz Sonderforschungsbereich 393 Numerische Simuation auf massiv paraeen Rechnern Zhanav T. Some choices of moments of renabe function and appications Preprint SFB393/03-11 Preprint-Reihe
More informationProceedings of the 2012 Winter Simulation Conference C. Laroque, J. Himmelspach, R. Pasupathy, O. Rose, and A. M. Uhrmacher, eds.
Proceedings of the 2012 Winter Simuation Conference C. aroque, J. Himmespach, R. Pasupathy, O. Rose, and A. M. Uhrmacher, eds. TIGHT BOUNDS FOR AMERICAN OPTIONS VIA MUTIEVE MONTE CARO Denis Beomestny Duisburg-Essen
More informationNumerical Methods and Comparison for the Dirac Equation in the Nonrelativistic Limit Regime
J Sci Comput (217) 71:194 1134 DOI 1.17/s1915-16-333-3 Numerica Methods and Comparison for the Dirac Equation in the Nonreativistic Limit Regime Weizhu Bao 1 Yongyong Cai 2,3 Xiaowei Jia 1 Qingin Tang
More informationMat 1501 lecture notes, penultimate installment
Mat 1501 ecture notes, penutimate instament 1. bounded variation: functions of a singe variabe optiona) I beieve that we wi not actuay use the materia in this section the point is mainy to motivate the
More information2M2. Fourier Series Prof Bill Lionheart
M. Fourier Series Prof Bi Lionheart 1. The Fourier series of the periodic function f(x) with period has the form f(x) = a 0 + ( a n cos πnx + b n sin πnx ). Here the rea numbers a n, b n are caed the Fourier
More informationCASCADIC MULTILEVEL METHODS FOR FAST NONSYMMETRIC BLUR- AND NOISE-REMOVAL. Dedicated to Richard S. Varga on the occasion of his 80th birthday.
CASCADIC MULTILEVEL METHODS FOR FAST NONSYMMETRIC BLUR- AND NOISE-REMOVAL S. MORIGI, L. REICHEL, AND F. SGALLARI Dedicated to Richard S. Varga on the occasion of his 80th birthday. Abstract. Image deburring
More informationNumerical Simulation for Optimizing Temperature Gradients during Single Crystal Casting Process
ISIJ Internationa Vo 54 (2014) No 2 pp 254 258 Numerica Simuation for Optimizing Temperature Gradients during Singe Crysta Casting Process Aeksandr Aeksandrovich INOZEMTSEV 1) Aeksandra Sergeevna DUBROVSKAYA
More informationarxiv: v4 [math.na] 25 Aug 2014
BIT manuscript No. (wi be inserted by the editor) A muti-eve spectra deferred correction method Robert Speck Danie Ruprecht Matthew Emmett Michae Minion Matthias Boten Rof Krause arxiv:1307.1312v4 [math.na]
More informationarxiv: v1 [math.ca] 6 Mar 2017
Indefinite Integras of Spherica Besse Functions MIT-CTP/487 arxiv:703.0648v [math.ca] 6 Mar 07 Joyon K. Boomfied,, Stephen H. P. Face,, and Zander Moss, Center for Theoretica Physics, Laboratory for Nucear
More informationThroughput Optimal Scheduling for Wireless Downlinks with Reconfiguration Delay
Throughput Optima Scheduing for Wireess Downinks with Reconfiguration Deay Vineeth Baa Sukumaran vineethbs@gmai.com Department of Avionics Indian Institute of Space Science and Technoogy. Abstract We consider
More informationEfficient Generation of Random Bits from Finite State Markov Chains
Efficient Generation of Random Bits from Finite State Markov Chains Hongchao Zhou and Jehoshua Bruck, Feow, IEEE Abstract The probem of random number generation from an uncorreated random source (of unknown
More information