Fitting affine and orthogonal transformations between two sets of points
|
|
- Willis Dalton
- 5 years ago
- Views:
Transcription
1 Mathematica Communications 9(2004), Fitting affine and orthogona transformations between two sets of points Hemuth Späth Abstract. Let two point sets P and Q be given in R n. We determine a transation and an affine transformation or an isometry such that the image of Q approximates P as best as possibe in the east squares sense. Key words: fitting inear transformations, east squares AMS subject cassifications: 65D10 Received May 28, 2003 Accepted February 20, Introduction Let the two sets of points P and Q by given by p i =(p 1i,...,p ni ) T, q i =(a 1i,...,a ni ) T (i =1,...,m). (1) We are ooking for some matrix and some transation vector A =(a jk ) j,k=1,...,n (2) t =(t 1,...,t n ) T (3) such that p i Aq i + t (i =1,...,m). (4) Asthe number of unknownsisn 2 +n, werequirem>n 2 +n. One possibe objective function to be minimized is the sum of squared Eucidean distances, i. e. S(A, t) = p i Aq i t 2 2. (5) The first soution method wi consist of a generaization of the east square fitting by a straight ine (n = 1). The second method is an iteration method that coud aso be appied but shoud not. Department of Mathematics, University of Odenburg, Postfach 2 503, D Odenburg, Germany, e-mai: spaeth@mathematik.uni-odenburg.de
2 28 H. Späth If A isorthogona, we wi have the n(n +1)/2 side conditions A T A = I, i. e. a T j a k = δ jk (j k) (6) where a j (j =1,...,n) are the coumnsof A. It wi turn out in Section 3 that it is easier to manage them impicity and that the mentioned second method can be used here. Other methods are discussed in [1, 2, 4, 5]. 2. Fitting affine transformations The necessary and because of inearity aso sufficient conditions for (5) to be minimized are S = 2 (E jk q i ) T (p i Aq i t) =0 (j, k =1,...,n) (7) a jk S t = 2 (p i Aq i t) =0 (8) In (7) we set E jk = A a jk,i.e.then n matrix that contains1 in the j-th row and the k-th coumn and 0 esewhere. As (E jk q i ) T =(0,...,0,q ki, 0,...,0), (9) where q ki isin the j-the position, the conditions (7) wi read ( m ) ( m ) ( m ) q 1i q ki a j q ni q ki a jn + q ki t j = q ki p ji (10) (j, k =1,...,n). Additionay, (8) gives ( m ) ( m ) q 1i a j q ni a jn + mt j = p ji (11) (j =1,...,n). In order to integrate (10) and (11) we introduce q n+1,i =1 (i =1,...,m) (12) and put q i =(q 1i,...,q ni,q n+1,i ) T (i =1,...,m), (13) Q = q i q T i, (14) ãa j =(a j1,...,a jn,t j ) T, (15) c j =( c j1,..., c j,n+1 ) T with c jk = q ki p ji (k =1,...,n+1). (16)
3 Fitting transformations 29 Then (10) and (11) can be integrated within Qãa j = c j (j =1,...,n). (17) Note that Q iscommon to a n systems of inear equations with n + 1 unknowns, i. e. you need just one LU-decomposition to sove a systems. Q wi normay be positivey definite for m>n 2 + n and thusnonsinguar. Exampe 1. For Q we used m =20points q i R 3 given by Q = Then we used A = , t = to produce p i = Aq i + t (i =1,...,20). In this case the minima soution of (5) must give S(A, t) =0. Of course, this was done, and A and t were retrieved. Exampe 2. We disturbed a ot of coefficients of p i and q i randomy by ±1 to receive Q = P = We received A = , t = and S(A, t) = Now we wi mention an iterative descent method that, it is true, does not exceed the straightforward and direct soution of (13) but wi aso be abe to be appied in the next section. We can write (8) as t = 1 m (p i Aq i ) (18) and (7) as (E jk q i ) T Aq i = (E jk q i ) T (p i t) (19) We consider the foowing agorithm:
4 30 H. Späth Step 0: Give some starting vaue A (0) for A, e.g.a (0) = I. Setw =0. Step 1: Use (18) to cacuate t (w+1) = 1 m (p i A (w) q i ). (20) Step 2: Use (19) to cacuate A (w+1),i.e.tosove (E jk q i ) T A (w+1) q i = (E jk q i ) T (p i t (w+1) ). (21) Step 3: STOP, if some convergence criterion is fufied, otherwise set w := w +1 and go back to Step 1. Thisaternating method givesa descent within every iteration. For the above two exampes it converged to 6 correct digits within 6 and 9 iterations, respectivey, and gave the same soution as (17). 3. Fitting orthogona transformations Orthogona transformations are affine transformations with the property (6). Thus to the objective function (5) one coud add the side conditions (6). Instead of trying to sove a noninear system after introducing the Lagrangian function we wi describe a more effective way. It is we-known [3] that every orthogona matrix A of size n n can be written asa product A = R 1 R N B, (22) where R = R(ϕ jk ) ( =1,...,N = and where R(ϕ jk )=(r is ) i,s=1,...,n hasthe eements (n 1)n ) (23) 2 r jj = r kk =cos(ϕ jk ) r jk = r kj = sin(ϕ jk ) (24) r ii =1 (i j, k), r is = 0 ese. The indices uniquey correspond to pairs (j, k) with j < k asindicated in the foowing tabe: 1 2 n 1 n n+1 2n 32n 2 N (j, k) (1, 2) (1, 3) (1,n)(2, 3) (2, 4) (2,n) (3, 4) (n 1,n)
5 Fitting transformations 31 Finay, B = I (identity) for det A = +1 [3, 6] and B =diag(1, 1,...,1, 1) for det A = 1 [3]. The anges ϕ ik are caed the Euer rotation anges. We wi aso write ϕ =(ϕ 1,...,ϕ N ) T With thisnotation the objective function (5) to be minimized here reads S(ϕ, t) = p i R 1 R 2 R N Bq i t 2 2. (25) The necessary conditions for a minimum are S m t = 2 (p i R 1 R N Bq i t) =0, (26) S ϕ = 2 i?1 (p i R 1 R N Bq i t) T R 1 R 1 R R +1 R N Bq i = 0 (27) where the matrix hasthe coefficients ( =1,...,N), R = R ϕ jk (28) r jj = r kk = sin(ϕ jk ), r jk = r kj = cos(ϕ jk), (29) r is = 0 ese. Note that the matrix C = R T R =(c is ) i,s=1,...,n (30) hasthe eements c jk = c kj = 1, c is = 0 ese. (31) Now(26)canbewrittenas t = t(ϕ) = 1 (p i R 1 R N Bq i ) m (32) and (27) as (p i t) T R 1 R R N q i = q T i RT N R T 1 R 1 R R N q i = = s T i RT R s i (33) s T i C s i i=2
6 32 H. Späth where s i = R +1 R N Bq i. But because of (31) the right-hand side of (33) is zero. Thus(27) reducesto Putting (p i t) T R 1 R 1 R R +1 R N Bq i =0. (34) u ()T i = (p i t) T R 1 R 1, v () i = R +1 R n Bq i, (i =1,...,n; =1,...,N) (35) (34) reads Using (29) this resuts in u ()T i R v () i = 0 (36) g jk sin(ϕ jk ) f jk cos(ϕ jk )=0, (37) where f jk = g jk = v () ij u() ik v() ik u() ij, v () ij u() ij + v() ik u() ik. (38) Thus ϕ ik = ϕ isgiven by ϕ ik =atan ( fjk A minimum of S w.r.t. ϕ jk isgiven by (39) if g jk ). (39) S ϕ 2 ik = g jk cos(ϕ jk )+f jk sin(ϕ jk ) > 0. (40) Otherwise ϕ jk hasto be repaced by ϕ jk +π. Additionay, in order to avoid negative anges, we repace ϕ jk by ϕ jk +2π. Atogether we can now design the foowing descent agorithm: Step 1: Give starting vaues ϕ (0) ( =1,...,N), e.g. ϕ (0) = 0. Decide whether to use B = I or B =diag(1,...,1, 1). Set w =0. Step 2: Step 3: Cacuate R (w) = R(ϕ (w) )( =1,...,N) corresponding to (23) and (24). Set F := R (w) 2 R (w) N and G := I.
7 Fitting transformations 33 Step 4: Corresponding to (32) cacuate t (w) = 1 m (p i R (w) 1 Fq i ) Step 5: do =1,N end g := 0; f := 0; if = N then F := I do i =1,m u T := (p i t (w) ) T G v := Fq i g := g + v j u k v k u j f := f + v j u j + v k u k end i ϕ =atan(g/f) if (g cos(ϕ)+f sin(ϕ)) 0thenϕ := ϕ + π if (ϕ <0) then ϕ := ϕ +2π ϕ (w+1) := ϕ; R (w+1) if ( <n)then end if := R(ϕ) G := GR (w+1) ; F := R (w)t +1 F Step 6: Set w := w +1. If some convergence criterion is not fufied, then go back to Step 3; otherwise set t := t (w) and A := R (w) B and stop. 1 R (w) N Convergence of thisagorithm to a goba minimum cannot be proved. However, empiricay, the goba minimum wasawaysattained independenty of the starting vauesfor ϕ (0). We wi give some exampes where we aways used ϕ (0) =0( = 1,...,N). Starting with a set Q of m =20pointsa i R 4,namey Q = and anges ϕ = ( =1,...,6) and a transation t =( 1, 0, 1, 2) T we produced p i = R 1 R 6 q i + t (i =1,...,m= 20). Exampe 3. Using those data after 206 iterations we got S = (as expected) and t =( , , , ) T ; instead of ϕ = ( =1,...,6) weendedupwithϕ 1 =1.0000, ϕ 2 =5.1416, ϕ 3 =.1416, ϕ 4 =5.4248, ϕ 5 =
8 34 H. Späth , ϕ 6 = This resut is right because with (ϕ 1,...,ϕ 6 ) T (among others) (ϕ 1,ϕ 2 + π, π ϕ 3, 3π ϕ 4,ϕ 5 π, ϕ 6 ) T is aso a goba minimum. Exampe 4. We used the same data as before but dropped a components of p i from d 1.d 2 d 3 d 4 d 5 d 6 to d 1.d 2 thus introducing sma errors. After 202 iterations we received S = , t =(.9644,.0459,.9469, ) T and ϕ = (.9614, ,.1431, , , ) T. As expected, these resuts do not differ very much from those of Exampe 3. Exampe 5. We further dropped the components of p i from d 1.d 2 to integer vaues d 1 getting P = After 258 iterations we got S = , t =(.5893,.5366,.6593, ) T and ϕ =(1.3334, ,.1224, , , ) T. Of course, these resuts differ more than before. Varying the starting vaues ϕ (0) we aways received the same resuts, i.e. hopefuy a goba minimum. References [1] J. C. Gower, Mutivariabe anaysis: Ordination, mutidimensiona scaing and aied topics, in: Statistics, Handbook of Appicabe Mathematics, Vo. VI, (E. H. Loyd, Ed.), Wiey, Chichester, 1984, [2] R. J. Hanson, M. J. Norris, Anaysis of measurements based on the singuar vaue decomposition, SIAM J. Sci. Statist. Comput. 2(1981), [3] F. Neiss, Determinanten und Matrizen, 2. Aufage, Springer, [4] P. H. Schönemann, R. M. Carro, Fitting one matrix to another under choice of a centra diation and a rigid motion, Psychometrika35(1970), [5] I. Söderkvist, P.-Å. Wedin, On condition numbers and agorithms for determining a rigid body movement, BIT 34(1994), [6] N. JA. Vienkin, Specia Functions and the Theory of Group Representations, AMS, Providence 1968.
The Symmetric and Antipersymmetric Solutions of the Matrix Equation A 1 X 1 B 1 + A 2 X 2 B A l X l B l = C and Its Optimal Approximation
The Symmetric Antipersymmetric Soutions of the Matrix Equation A 1 X 1 B 1 + A 2 X 2 B 2 + + A X B C Its Optima Approximation Ying Zhang Member IAENG Abstract A matrix A (a ij) R n n is said to be symmetric
More informationProduct Cosines of Angles between Subspaces
Product Cosines of Anges between Subspaces Jianming Miao and Adi Ben-Israe Juy, 993 Dedicated to Professor C.R. Rao on his 75th birthday Let Abstract cos{l,m} : i cos θ i, denote the product of the cosines
More informationAlgorithms to solve massively under-defined systems of multivariate quadratic equations
Agorithms to sove massivey under-defined systems of mutivariate quadratic equations Yasufumi Hashimoto Abstract It is we known that the probem to sove a set of randomy chosen mutivariate quadratic equations
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 informationCombining reaction kinetics to the multi-phase Gibbs energy calculation
7 th European Symposium on Computer Aided Process Engineering ESCAPE7 V. Pesu and P.S. Agachi (Editors) 2007 Esevier B.V. A rights reserved. Combining reaction inetics to the muti-phase Gibbs energy cacuation
More informationMARKOV CHAINS AND MARKOV DECISION THEORY. Contents
MARKOV CHAINS AND MARKOV DECISION THEORY ARINDRIMA DATTA Abstract. In this paper, we begin with a forma introduction to probabiity and expain the concept of random variabes and stochastic processes. After
More informationImproving the Reliability of a Series-Parallel System Using Modified Weibull Distribution
Internationa Mathematica Forum, Vo. 12, 217, no. 6, 257-269 HIKARI Ltd, www.m-hikari.com https://doi.org/1.12988/imf.217.611155 Improving the Reiabiity of a Series-Parae System Using Modified Weibu Distribution
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 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 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 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 informationWavelet Galerkin Solution for Boundary Value Problems
Internationa Journa of Engineering Research and Deveopment e-issn: 2278-67X, p-issn: 2278-8X, www.ijerd.com Voume, Issue 5 (May 24), PP.2-3 Waveet Gaerkin Soution for Boundary Vaue Probems D. Pate, M.K.
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 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 informationMath 124B January 17, 2012
Math 124B January 17, 212 Viktor Grigoryan 3 Fu Fourier series We saw in previous ectures how the Dirichet and Neumann boundary conditions ead to respectivey sine and cosine Fourier series of the initia
More informationBALANCING REGULAR MATRIX PENCILS
BALANCING REGULAR MATRIX PENCILS DAMIEN LEMONNIER AND PAUL VAN DOOREN Abstract. In this paper we present a new diagona baancing technique for reguar matrix pencis λb A, which aims at reducing the sensitivity
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 informationGaussian Curvature in a p-orbital, Hydrogen-like Atoms
Advanced Studies in Theoretica Physics Vo. 9, 015, no. 6, 81-85 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/astp.015.5115 Gaussian Curvature in a p-orbita, Hydrogen-ike Atoms Sandro-Jose Berrio-Guzman
More informationPartial permutation decoding for MacDonald codes
Partia permutation decoding for MacDonad codes J.D. Key Department of Mathematics and Appied Mathematics University of the Western Cape 7535 Bevie, South Africa P. Seneviratne Department of Mathematics
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 informationSeparation of Variables and a Spherical Shell with Surface Charge
Separation of Variabes and a Spherica She with Surface Charge In cass we worked out the eectrostatic potentia due to a spherica she of radius R with a surface charge density σθ = σ cos θ. This cacuation
More informationLecture 6: Moderately Large Deflection Theory of Beams
Structura Mechanics 2.8 Lecture 6 Semester Yr Lecture 6: Moderatey Large Defection Theory of Beams 6.1 Genera Formuation Compare to the cassica theory of beams with infinitesima deformation, the moderatey
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 informationA proposed nonparametric mixture density estimation using B-spline functions
A proposed nonparametric mixture density estimation using B-spine functions Atizez Hadrich a,b, Mourad Zribi a, Afif Masmoudi b a Laboratoire d Informatique Signa et Image de a Côte d Opae (LISIC-EA 4491),
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 informationHigher dimensional PDEs and multidimensional eigenvalue problems
Higher dimensiona PEs and mutidimensiona eigenvaue probems 1 Probems with three independent variabes Consider the prototypica equations u t = u (iffusion) u tt = u (W ave) u zz = u (Lapace) where u = u
More information(This is a sample cover image for this issue. The actual cover is not yet available at this time.)
(This is a sampe cover image for this issue The actua cover is not yet avaiabe at this time) This artice appeared in a journa pubished by Esevier The attached copy is furnished to the author for interna
More informationFRST Multivariate Statistics. Multivariate Discriminant Analysis (MDA)
1 FRST 531 -- Mutivariate Statistics Mutivariate Discriminant Anaysis (MDA) Purpose: 1. To predict which group (Y) an observation beongs to based on the characteristics of p predictor (X) variabes, using
More informationProblem set 6 The Perron Frobenius theorem.
Probem set 6 The Perron Frobenius theorem. Math 22a4 Oct 2 204, Due Oct.28 In a future probem set I want to discuss some criteria which aow us to concude that that the ground state of a sef-adjoint operator
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 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 informationHaar Decomposition and Reconstruction Algorithms
Jim Lambers MAT 773 Fa Semester 018-19 Lecture 15 and 16 Notes These notes correspond to Sections 4.3 and 4.4 in the text. Haar Decomposition and Reconstruction Agorithms Decomposition Suppose we approximate
More informationLECTURE 10. The world of pendula
LECTURE 0 The word of pendua For the next few ectures we are going to ook at the word of the pane penduum (Figure 0.). In a previous probem set we showed that we coud use the Euer- Lagrange method to derive
More informationASummaryofGaussianProcesses Coryn A.L. Bailer-Jones
ASummaryofGaussianProcesses Coryn A.L. Baier-Jones Cavendish Laboratory University of Cambridge caj@mrao.cam.ac.uk Introduction A genera prediction probem can be posed as foows. We consider that the variabe
More informationAkaike Information Criterion for ANOVA Model with a Simple Order Restriction
Akaike Information Criterion for ANOVA Mode with a Simpe Order Restriction Yu Inatsu * Department of Mathematics, Graduate Schoo of Science, Hiroshima University ABSTRACT In this paper, we consider Akaike
More informationTheory of Generalized k-difference Operator and Its Application in Number Theory
Internationa Journa of Mathematica Anaysis Vo. 9, 2015, no. 19, 955-964 HIKARI Ltd, www.m-hiari.com http://dx.doi.org/10.12988/ijma.2015.5389 Theory of Generaized -Difference Operator and Its Appication
More informationMatrices and Determinants
Matrices and Determinants Teaching-Learning Points A matri is an ordered rectanguar arra (arrangement) of numbers and encosed b capita bracket [ ]. These numbers are caed eements of the matri. Matri is
More informationA Solution to the 4-bit Parity Problem with a Single Quaternary Neuron
Neura Information Processing - Letters and Reviews Vo. 5, No. 2, November 2004 LETTER A Soution to the 4-bit Parity Probem with a Singe Quaternary Neuron Tohru Nitta Nationa Institute of Advanced Industria
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 informationInteractive Fuzzy Programming for Two-level Nonlinear Integer Programming Problems through Genetic Algorithms
Md. Abu Kaam Azad et a./asia Paciic Management Review (5) (), 7-77 Interactive Fuzzy Programming or Two-eve Noninear Integer Programming Probems through Genetic Agorithms Abstract Md. Abu Kaam Azad a,*,
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 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 informationLECTURE NOTES 9 TRACELESS SYMMETRIC TENSOR APPROACH TO LEGENDRE POLYNOMIALS AND SPHERICAL HARMONICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Eectromagnetism II October 7, 202 Prof. Aan Guth LECTURE NOTES 9 TRACELESS SYMMETRIC TENSOR APPROACH TO LEGENDRE POLYNOMIALS AND SPHERICAL
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 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 informationAlberto Maydeu Olivares Instituto de Empresa Marketing Dept. C/Maria de Molina Madrid Spain
CORRECTIONS TO CLASSICAL PROCEDURES FOR ESTIMATING THURSTONE S CASE V MODEL FOR RANKING DATA Aberto Maydeu Oivares Instituto de Empresa Marketing Dept. C/Maria de Moina -5 28006 Madrid Spain Aberto.Maydeu@ie.edu
More informationTheory and implementation behind: Universal surface creation - smallest unitcell
Teory and impementation beind: Universa surface creation - smaest unitce Bjare Brin Buus, Jaob Howat & Tomas Bigaard September 15, 218 1 Construction of surface sabs Te aim for tis part of te project is
More informationPHYS 110B - HW #1 Fall 2005, Solutions by David Pace Equations referenced as Eq. # are from Griffiths Problem statements are paraphrased
PHYS 110B - HW #1 Fa 2005, Soutions by David Pace Equations referenced as Eq. # are from Griffiths Probem statements are paraphrased [1.] Probem 6.8 from Griffiths A ong cyinder has radius R and a magnetization
More informationTHINKING IN PYRAMIDS
ECS 178 Course Notes THINKING IN PYRAMIDS Kenneth I. Joy Institute for Data Anaysis and Visuaization Department of Computer Science University of Caifornia, Davis Overview It is frequenty usefu to think
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 information18-660: Numerical Methods for Engineering Design and Optimization
8-660: Numerica Methods for Engineering esign and Optimization in i epartment of ECE Carnegie Meon University Pittsburgh, PA 523 Side Overview Conjugate Gradient Method (Part 4) Pre-conditioning Noninear
More informationModel Solutions (week 4)
CIV-E16 (17) Engineering Computation and Simuation 1 Home Exercise 6.3 Mode Soutions (week 4) Construct the inear Lagrange basis functions (noda vaues as degrees of freedom) of the ine segment reference
More informationAssignment 7 Due Tuessday, March 29, 2016
Math 45 / AMCS 55 Dr. DeTurck Assignment 7 Due Tuessday, March 9, 6 Topics for this week Convergence of Fourier series; Lapace s equation and harmonic functions: basic properties, compuations on rectanges
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 informationA Two-Parameter Trigonometric Series
A Two-Parameter Trigonometric Series Xiang-Qian Chang Xiang.Chang@bos.mcphs.edu Massachusetts Coege of Pharmacy and Heath Sciences Boston MA 25 Let us consider the foowing dua questions. A. Givenafunction
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 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 informationLobontiu: System Dynamics for Engineering Students Website Chapter 3 1. z b z
Chapter W3 Mechanica Systems II Introduction This companion website chapter anayzes the foowing topics in connection to the printed-book Chapter 3: Lumped-parameter inertia fractions of basic compiant
More informationLECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Eectromagnetism II October, 202 Prof. Aan Guth LECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS
More informationTRANSFORMATION OF REAL SPHERICAL HARMONICS UNDER ROTATIONS
Vo. 39 (008) ACTA PHYSICA POLONICA B No 8 TRANSFORMATION OF REAL SPHERICAL HARMONICS UNDER ROTATIONS Zbigniew Romanowski Interdiscipinary Centre for Materias Modeing Pawińskiego 5a, 0-106 Warsaw, Poand
More informationBP neural network-based sports performance prediction model applied research
Avaiabe onine www.jocpr.com Journa of Chemica and Pharmaceutica Research, 204, 6(7:93-936 Research Artice ISSN : 0975-7384 CODEN(USA : JCPRC5 BP neura networ-based sports performance prediction mode appied
More informationResearch Article On the Lower Bound for the Number of Real Roots of a Random Algebraic Equation
Appied Mathematics and Stochastic Anaysis Voume 007, Artice ID 74191, 8 pages doi:10.1155/007/74191 Research Artice On the Lower Bound for the Number of Rea Roots of a Random Agebraic Equation Takashi
More informationCE601-Structura Anaysis I UNIT-IV SOPE-DEFECTION METHOD 1. What are the assumptions made in sope-defection method? (i) Between each pair of the supports the beam section is constant. (ii) The joint in
More informationDiscrete Bernoulli s Formula and its Applications Arising from Generalized Difference Operator
Int. Journa of Math. Anaysis, Vo. 7, 2013, no. 5, 229-240 Discrete Bernoui s Formua and its Appications Arising from Generaized Difference Operator G. Britto Antony Xavier 1 Department of Mathematics,
More informationALGORITHMIC SUMMATION OF RECIPROCALS OF PRODUCTS OF FIBONACCI NUMBERS. F. = I j. ^ = 1 ^ -, and K w = ^. 0) n=l r n «=1 -*/!
ALGORITHMIC SUMMATIO OF RECIPROCALS OF PRODUCTS OF FIBOACCI UMBERS Staney Rabinowitz MathPro Press, 2 Vine Brook Road, Westford, MA 0886 staney@tiac.net (Submitted May 997). ITRODUCTIO There is no known
More informationResearch Article Numerical Range of Two Operators in Semi-Inner Product Spaces
Abstract and Appied Anaysis Voume 01, Artice ID 846396, 13 pages doi:10.1155/01/846396 Research Artice Numerica Range of Two Operators in Semi-Inner Product Spaces N. K. Sahu, 1 C. Nahak, 1 and S. Nanda
More informationAppendix A: MATLAB commands for neural networks
Appendix A: MATLAB commands for neura networks 132 Appendix A: MATLAB commands for neura networks p=importdata('pn.xs'); t=importdata('tn.xs'); [pn,meanp,stdp,tn,meant,stdt]=prestd(p,t); for m=1:10 net=newff(minmax(pn),[m,1],{'tansig','purein'},'trainm');
More informationTwo-Stage Least Squares as Minimum Distance
Two-Stage Least Squares as Minimum Distance Frank Windmeijer Discussion Paper 17 / 683 7 June 2017 Department of Economics University of Bristo Priory Road Compex Bristo BS8 1TU United Kingdom Two-Stage
More informationIntroduction. Figure 1 W8LC Line Array, box and horn element. Highlighted section modelled.
imuation of the acoustic fied produced by cavities using the Boundary Eement Rayeigh Integra Method () and its appication to a horn oudspeaer. tephen Kirup East Lancashire Institute, Due treet, Bacburn,
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 informationIntroduction to PDEs and Numerical Methods Tutorial 10. Finite Element Analysis
Patzhater für Bid, Bid auf Titefoie hinter das Logo einsetzen Introduction to PDEs and Numerica Methods Tutoria. Finite Eement Anaysis Dr. Noemi Friedman, 3..2 FROM STRONG FORM TO WEAK FORM inhomogeneous
More informationLecture Notes for Math 251: ODE and PDE. Lecture 32: 10.2 Fourier Series
Lecture Notes for Math 251: ODE and PDE. Lecture 32: 1.2 Fourier Series Shawn D. Ryan Spring 212 Last Time: We studied the heat equation and the method of Separation of Variabes. We then used Separation
More informationIntroduction to LMTO method
1 Introduction to MTO method 24 February 2011; V172 P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method Ab initio Eectronic Structure Cacuations
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 informationaddenda and errata addenda and errata
addenda and errata Journa of Appied Crystaography ISSN 16-5767 addenda and errata Eastic strain and stress determination by Rietved refinement: generaized treatment for textured poycrystas for a Laue casses.
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 informationCONJUGATE GRADIENT WITH SUBSPACE OPTIMIZATION
CONJUGATE GRADIENT WITH SUBSPACE OPTIMIZATION SAHAR KARIMI AND STEPHEN VAVASIS Abstract. In this paper we present a variant of the conjugate gradient (CG) agorithm in which we invoke a subspace minimization
More informationRestricted weak type on maximal linear and multilinear integral maps.
Restricted weak type on maxima inear and mutiinear integra maps. Oscar Basco Abstract It is shown that mutiinear operators of the form T (f 1,..., f k )(x) = R K(x, y n 1,..., y k )f 1 (y 1 )...f k (y
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 informationMAT 167: Advanced Linear Algebra
< Proem 1 (15 pts) MAT 167: Advanced Linear Agera Fina Exam Soutions (a) (5 pts) State the definition of a unitary matrix and expain the difference etween an orthogona matrix and an unitary matrix. Soution:
More informationEstablishment of Weak Conditions for Darboux- Goursat-Beudon Theorem
Georgia Southern University Digita Commons@Georgia Southern Mathematica Sciences Facuty Pubications Department of Mathematica Sciences 2009 Estabishment of Weak Conditions for Darboux- Goursat-Beudon Theorem
More information12.2. Maxima and Minima. Introduction. Prerequisites. Learning Outcomes
Maima and Minima 1. Introduction In this Section we anayse curves in the oca neighbourhood of a stationary point and, from this anaysis, deduce necessary conditions satisfied by oca maima and oca minima.
More informationEffect of transport ratio on source term in determination of surface emission coefficient
Internationa Journa of heoretica & Appied Sciences, (): 74-78(9) ISSN : 975-78 Effect of transport ratio on source term in determination of surface emission coefficient Sanjeev Kumar and Apna Mishra epartment
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 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 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 informationFRIEZE GROUPS IN R 2
FRIEZE GROUPS IN R 2 MAXWELL STOLARSKI Abstract. Focusing on the Eucidean pane under the Pythagorean Metric, our goa is to cassify the frieze groups, discrete subgroups of the set of isometries of the
More informationStochastic Variational Inference with Gradient Linearization
Stochastic Variationa Inference with Gradient Linearization Suppementa Materia Tobias Pötz * Anne S Wannenwetsch Stefan Roth Department of Computer Science, TU Darmstadt Preface In this suppementa materia,
More informationConvolutional Networks 2: Training, deep convolutional networks
Convoutiona Networks 2: Training, deep convoutiona networks Hakan Bien Machine Learning Practica MLP Lecture 8 30 October / 6 November 2018 MLP Lecture 8 / 30 October / 6 November 2018 Convoutiona Networks
More informationPost-buckling behaviour of a slender beam in a circular tube, under axial load
Computationa Metho and Experimenta Measurements XIII 547 Post-bucking behaviour of a sender beam in a circuar tube, under axia oad M. Gh. Munteanu & A. Barraco Transivania University of Brasov, Romania
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 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 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 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 informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This artice appeared in a journa pubished by Esevier. The attached copy is furnished to the author for interna non-commercia research and education use, incuding for instruction at the authors institution
More informationWorst Case Analysis of the Analog Circuits
Proceedings of the 11th WSEAS Internationa Conference on CIRCUITS, Agios Nikoaos, Crete Isand, Greece, Juy 3-5, 7 9 Worst Case Anaysis of the Anaog Circuits ELENA NICULESCU*, DORINA-MIOARA PURCARU* and
More informationCS 331: Artificial Intelligence Propositional Logic 2. Review of Last Time
CS 33 Artificia Inteigence Propositiona Logic 2 Review of Last Time = means ogicay foows - i means can be derived from If your inference agorithm derives ony things that foow ogicay from the KB, the inference
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 informationANALOG OF HEAT EQUATION FOR GAUSSIAN MEASURE OF A BALL IN HILBERT SPACE GYULA PAP
ANALOG OF HEAT EQUATION FOR GAUSSIAN MEASURE OF A BALL IN HILBERT SPACE GYULA PAP ABSTRACT. If µ is a Gaussian measure on a Hibert space with mean a and covariance operator T, and r is a} fixed positive
More informationLecture 9: Submatrices and Some Special Matrices
Lecture 9: Submatrices and Some Special Matrices Winfried Just Department of Mathematics, Ohio University February 2, 2018 Submatrices A submatrix B of a given matrix A is any matrix that can be obtained
More information