Some Useful Results for Spherical and General Displacements
|
|
- Verity Freeman
- 5 years ago
- Views:
Transcription
1 E 5 Fall 997 V. Kumar Some Useful Results for Spherial an General Displaements. Spherial Displaements.. Eulers heorem We have seen that a spherial isplaement or a pure rotation is esribe by a 3 3 rotation matrix. oring to Eulers theorem, "ny isplaement of a rigi boy suh that a point on the rigi boy, say, remains fixe, is equivalent to a rotation about a fixe axis through the point." We will start with a general spherial isplaement an show Euler s theorem is vali. Consier a boy fixe frame that is isplae from {F} to {}. he isplaement of any point on the rigi boy an be written as: Rp where p an are the position vetors before an after isplaement respetively, an we are eleting the supersripts an subsripts assoiate with F R. et us fin the eigenvalues an eigenvetors of R by writing: Rp λp he harateristi equation is: R λi axis of rotation u eigenspae spanne by x an x _ φ φ Rv v Rv v
2 Figure he real eigenvetor, u, an the orthogonal eigenspae for a rotation matrix If the elements of R are enote by R ij, the above equation an be written as: b g 3 λ + λ R + R + R33 λbr R R R g + br R R R g + br R R R gh + R Beause R is a rotation matrix, R. lso, beause R R R R R R R R R R R R R R R R 33 his simplifies the harateristi equation: b g b g λ 3 + λ R + R + R33 λ R + R + R33 + Fatoring the left han sie, we get: b g b 33 g λ λ λ R + R + R + h R, We see that λ is an eigenvalue of R. In other wors, there exists a real vetor, u, suh that all points on the line, p αu remain fixe (invariant) uner the transformation R. hus R is a rotation that leaves this line or axis fixe. Define φ so that b g () R + R + R33 hen the remaining eigenvalues are: iφ iφ λ e + i, λ e i hus the eigenvalues an eigenvetors are: λ λ e i, p x, say φ e i, p x () φ λ 3 3, p u. --
3 where x is the omplex onjugate of x. Sine R is orthogonal, x an x are perpeniular to u an span the plane perpeniular to the axis of rotation. real basis for this plane an be onstrute using the vetors: v b i x + xg, v bx xg, (3) b g e j iφ iφ Rv λx + λx e x + e x v + v b g e j i i iφ iφ Rv λx λx e x e x v + v s shown in Figure, the effet of the transformation, R, is to rotate vetors in the plane spanne by v an v through an angle φ about u, while vetors along u are invariant. hus the transformation R rotates the rigi boy about u through an angle φ. his proves Eulers theorem. here is a anonial representation of any rotation matrix whih allows us to view it as a rotation through an angle φ about the z axis. If we onsier the tria of orthonormal vetors, {v, v, u}, we an easily show that R an be written in the form: R Λ (4) where, v v u, an Λ. his is essentially a similarity transformation. If we view the rotation from a new frame, {F }, whose orientation is given by the oorinate transformation, F R F, it is lear that in this new frame, the isplaement is a rotation about the z axis through an angle φ. ote that we oul also have hosen to onstrut our anonial representation of a rotation by viewing it as a rotation about the x or the y axis. -3-
4 .. Representations for finite rotations We have seen that any rotation or spherial isplaement an be esribe by a rotation matrix. However, beause the rotation matrix is orthogonal, its nine elements are not inepenent of eah other. Speifially, they satisfy six inepenent equations: R + R + R 3 R + R + R 3 R + R + R R R + R R + R R 3 3 R R + R R + R R R R + R R + R R herefore there are only three inepenent parameters that ompletely esribe a given rotation an the rotation matrix is a very reunant representation. In orer to evelop more ompat representations, we pursue two types of representations. he first relies on eomposing a given rotation into three finite suessive rotations an these three rotation angles, alle Euler angles, ompletely esribe the given rotation. he seon is erive from the results of the previous setion an expliitly ientifies the axis of rotation an the angle of rotation. he Euler angles are efine in stanar kinematis an ynamis texts an are not isusse here. Instea, we isuss axis-angle representations in some etail. In this setion we efine Euler parameters whih provie another ompat representation of the axis an angle of rotation. We also evelop a simple result that allow us to obtain the rotation matrix from the axis an angle of rotation. he axis-angle representation Consier the rotation of a rigi boy about u through an angle φ as shown in Figure. Consier the triangle in the figure. We an write: or, p a + a Rp p a a
5 axis of rotation u (a) φ -p p θ (b) () φ +p -p a p a -p Figure Rotation of p about u through φ. he vetor is relate to p by the rotation matrix, R p. Sine the tail of an p are oinient at, we an raw a one whose apex is suh that both vetors are generators of the one emanating from. From the geometry, -5-
6 p sinθ u p a u p Sine a is perpeniular to u an p, a bu pg. he length an also be expresse as p bp ug u. herefore, b g hb g a p p u u. Substituting a an a in (.6) yiels If we let U b g. u u, this equation yiels: b g (5) Rp p + uu p + u p u u u 3 3 u R I + uu + U his is alle the Rorigues formula, a result that is attribute to Euler, exell, an Rorigues. It esribes the rotation matrix in terms of the angle an axis of the rotation. o obtain the angle, φ, an axis of rotation, u, from the rotation matrix, we an easily erive the following formulas: b g (6) R + R + R33 U R R h (7) here are many solutions for the angle of rotation beause the inverse osine funtion is multivalue. If we fin φ from (6) restriting the range of the inverse osine funtion to the interval [,π], we an fin the axis of rotation from (7) provie φ is not either or π. If R I, φ an the axis of rotation is not well efine. If the trae of the rotation matrix is -, φπ, an from the Euler-exell formula: R uu I -6-
7 from whih u an be solve. For every rotation of angle φ about the axis u, we an also obtain an equivalent axis-angle representation with a rotation -φ about the axis -u. lso for every solution (u, φ), we have other solutions (u, φ+kπ) for all integer values of k. Euler parameters We now turn to a representation base on half the rotation angle an the axis of rotation. Define the four Euler parameters: φ φ φ φ os, ux sin, uy sin, 3 uz sin (8) ote his is the equation of a unit sphere (or hypersphere) in R 4, also 3 alle the projetive three sphere. ny point on this sphere orrespons to a unique axis an angle of rotation an therefore a unique rotation matrix. However, there are two points (iametrially opposite) on the sphere that give the same axis an angle. an - enote the same rotation. For example, a rotation of 9 o egrees about the x-axis yiels F H,,, I K an a rotation of 7 o about the same axis yiels F HG,,, although they are really the same rotation. If is partitione as follows: 3 I K J it an be easily seen that (.7) an be written as: i R I + + D (9) -7-
8 where D 3 3. his is the rotation matrix expresse in terms of Euler parameters.. Spatial isplaements an transformations.. Chasles heorem ne of the most funamental results in spatial kinematis is a theorem that is usually attribute to Chasles (83), although ozzi an Cauhy are reite with earlier results that are similar. he theorem states: "he most general rigi boy isplaement an be proue by a translation along a line followe (or preee) by a rotation about that line." Beause this isplaement is reminisent of the isplaement of a srew, it is alle a srew isplaement, an the line or axis is alle the srew axis. We prove this theorem for the planar ase an then for the spatial ase. For a planar isplaement, one an fin a fixe point of the isplaement. his is a point that is left unhange by the isplaement. Consier a general 3 3 homogeneous transfer matrix: R R, osθ sinθ, sinθ osθ x If R is not the ientity matrix, there is one fixe point on the rigi boy for any isplaement (in ontrast to spatial isplaements) alle the pole (or the instantaneous enter) of the isplaement. If is the position vetor of the pole, R+ whih allows us to fin the pole: I R y b g (). -8-
9 his point orrespons to the eigenvetor of the matrix for a unit eigenvalue. hus any planar isplaement an be onsiere as either a pure rotation about a point alle the pole (if (R-I) is not singular) or a pure translation in the iretion given by (if RI). Sine pure rotations an pure translations are speial ases of a srew motion, Chasles theorem is prove for the planar ase. If we onsier a general 4 4 homogeneous transfer matrix,, it has four eigenvalues two of whih are equal to. (he other two are omplex onjugate with a magnitue of ). However there are no real eigenvetors orresponing to λ. his implies that a general isplaement has no fixe points. o prove Chasles theorem for spatial isplaements, onsier the following similarity transformation of the matrix : Λ R R R + ow hoose as we i for the anonial representation of the rotation matrix in Equation (4) so that it onsists of the eigenvetors of R: v v u () he 3 3 orthogonal matrix part of Λ reues to the rotation matrix orresponing to a rotation about the z axis: R If we efine, the 3 translation vetor in Λ an be simplifie as follows: -9-
10 R + R I + h x x y y z + z where. If the top submatrix of ( R-I) is nonsingular, we an always fin suh that x y + x y In other wors, we an always solve the first two equations of ( R-I) - for the first two omponents of as we i in Equation () an let the thir omponent be zero. In this ase, Λ has the form: Λ k where k is given by the thir omponent of the vetor. In this ase, the isplaement an be esribe by a rotation about the z axis through an angle φ an a onurrent translation along the z axis through a istane k. his is a srew isplaement an proves Chasles theorem for a general isplaement. If the top submatrix of ( R-I) is singular, then RI. In this speial ase, Λ is a pure translation given by the vetor. his is a speial ase of the srew isplaement. hus we have prove Chasles theorem for this speial ase as well. he geometri interpretation is shown in Figure 3. We are intereste in the isplaement that takes frame {G } into frame {G }. he matrix G E () (3) --
11 represents the oorinate transformation from {E } to {G }. In the referene frame {E }, the isplaement is a pure rotation about the z axis through an angle φ, aommpanie by a onurrent translation along the same axis of khφ. hus, E Λ E straightforwar similarity transformation onfirms: k. G G E E G E E G G E Λ G E (4).. Determination of the srew axis from the homogeneous transformation matrix Given the rotation matrix, R, we an fin u, a unit vetor along the srew axis, an φ, the angle of rotation about the srew axis from Equations (6-7). In orer to ompletely speify the srew isplaement, we want to fin at the position vetor of at least one point on the srew axis an the translation of a point on the rigi boy along the srew axis. --
12 {G } φ {E } hφ {E } {F} {G } u Figure 3 he isplaement of a boy fixe frame from {G } to {G }, the anonial representation of this isplaement from {E } to {E }, an the orresponing srew axis. We proee aoring to the evelopment in the previous subsetion. First, fin an appropriate proper orthogonal matrix suh that its thir olumn is u. ne may hoose to be the matrix efine in Equation () but this requires the omputation of the eigenvetors of R an is unneessary. If we let p enote the projetion of the vetor on a plane perpeniular to u, p - (. u) u (5) then we an hoose v w u (6) where, --
13 p v w u v, p an p is the magnitue of p or the norm p x os sin y φ φ p b g b g v w. ow Equation () simplifies to: ote that we have use the fat.w. hus, we get the position vetor of a point on the srew axis: p p sinφ, v w os + os b φg φ b g e j (7) he isplaement of a point on the axis is given by the thir omponent of : ku. (8) an the pith of the srew, h, is given by h k φ (9) 3. Referenes [] Bottema,. an Roth, B., heoretial Kinematis. Dover ubliations, 99. [] Bran,., Vetor an ensor nalysis, John Wiley, 947. [3] Hunt, K.H., Kinemati Geometry of ehanisms, Clarenon ress, xfor, 978. [4] Carthy. J.., Introution to heoretial Kinematis,.I.. ress, 99. [5] aul, R., Robot anipulators, athematis, rogramming an Control, he I ress, Cambrige,
Math 225B: Differential Geometry, Homework 6
ath 225B: Differential Geometry, Homework 6 Ian Coley February 13, 214 Problem 8.7. Let ω be a 1-form on a manifol. Suppose that ω = for every lose urve in. Show that ω is exat. We laim that this onition
More informationChapter 2: One-dimensional Steady State Conduction
1 Chapter : One-imensional Steay State Conution.1 Eamples of One-imensional Conution Eample.1: Plate with Energy Generation an Variable Conutivity Sine k is variable it must remain insie the ifferentiation
More informationSensitivity Analysis of Resonant Circuits
1 Sensitivity Analysis of Resonant Ciruits Olivier Buu Abstrat We use first-orer perturbation theory to provie a loal linear relation between the iruit parameters an the poles of an RLC network. The sensitivity
More informationThe numbers inside a matrix are called the elements or entries of the matrix.
Chapter Review of Matries. Definitions A matrix is a retangular array of numers of the form a a a 3 a n a a a 3 a n a 3 a 3 a 33 a 3n..... a m a m a m3 a mn We usually use apital letters (for example,
More informationGLOBAL EDITION. Calculus. Briggs Cochran Gillett SECOND EDITION. William Briggs Lyle Cochran Bernard Gillett
GOBA EDITION Briggs Cohran Gillett Calulus SECOND EDITION William Briggs le Cohran Bernar Gillett ( (, ) (, ) (, Q ), Q ) (, ) ( Q, ) / 5 /4 5 5 /6 7 /6 ( Q, 5 5 /4 ) 4 4 / 7 / (, ) 9 / (, ) 6 / 5 / (Q,
More informationAsymptotic behavior of solutions to wave equations with a memory condition at the boundary
Eletroni Journal of Differential Equations, Vol. 2(2), No. 73, pp.. ISSN: 72-669. URL: http://eje.math.swt.eu or http://eje.math.unt.eu ftp eje.math.swt.eu (login: ftp) Asymptoti behavior of solutions
More information1 - a 1 - b 1 - c a) 1 b) 2 c) -1 d) The projection of OP on a unit vector OQ equals thrice the area of parallelogram OPRQ.
Regter Number MODEL EXAMINATION PART III - MATHEMATICS [ENGLISH VERSION] Time : Hrs. Ma. Marks : 00 SECTION - A 0 = 0 Note :- (i) All questions are ompulsory. (ii) Eah question arries one mark. (iii) Choose
More informationComputing 2-Walks in Cubic Time
Computing 2-Walks in Cubi Time Anreas Shmi Max Plank Institute for Informatis Jens M. Shmit Tehnishe Universität Ilmenau Abstrat A 2-walk of a graph is a walk visiting every vertex at least one an at most
More informationImplementing the Law of Sines to solve SAS triangles
Implementing the Law of Sines to solve SAS triangles June 8, 009 Konstantine Zelator Dept. of Math an Computer Siene Rhoe Islan College 600 Mount Pleasant Avenue Proviene, RI 0908 U.S.A. e-mail : kzelator@ri.eu
More informationTwo Dimensional Principal Component Analysis for Online Tamil Character Recognition
Two Dimensional Prinipal Component Analysis for Online Tamil Charater Reognition Suresh Sunaram, A G Ramarishnan Inian Institute of Siene,Bangalore, Inia suresh@ee.iis.ernet.in, ramiag@ee.iis.ernet.in
More informationProblem set 6 for the course Theoretical Optics Sample Solutions
Karlsruher Institut für Tehnologie KIT) Institut für theoretishe Festkörperphysik SS01 Prof. Dr. G. Shön, Dr. R. Frank 15.06.01 http://www.tfp.kit.eu/stuium-lehre.php Tutorial: Group 1, Name: Group, Group
More informationZero-Free Region for ζ(s) and PNT
Contents Zero-Free Region for ζs an PN att Rosenzweig Chebyshev heory ellin ransforms an Perron s Formula Zero-Free Region of Zeta Funtion 6. Jensen s Inequality..........................................
More informationA Primer on the Statistics of Longest Increasing Subsequences and Quantum States
A Primer on the Statistis of Longest Inreasing Subsequenes an Quantum States Ryan O Donnell John Wright Abstrat We give an introution to the statistis of quantum states, with a fous on reent results giving
More informationLinear Capacity Scaling in Wireless Networks: Beyond Physical Limits?
Linear Capaity Saling in Wireless Networks: Beyon Physial Limits? Ayfer Özgür, Olivier Lévêque EPFL, Switzerlan {ayfer.ozgur, olivier.leveque}@epfl.h Davi Tse University of California at Berkeley tse@ees.berkeley.eu
More informationSupplementary Materials for A universal data based method for reconstructing complex networks with binary-state dynamics
Supplementary Materials for A universal ata ase metho for reonstruting omplex networks with inary-state ynamis Jingwen Li, Zhesi Shen, Wen-Xu Wang, Celso Greogi, an Ying-Cheng Lai 1 Computation etails
More informationForce Reconstruction for Nonlinear Structures in Time Domain
Fore Reonstrution for Nonlinear Strutures in ime Domain Jie Liu 1, Bing Li 2, Meng Li 3, an Huihui Miao 4 1,2,3,4 State Key Laboratory for Manufaturing Systems Engineering, Xi an Jiaotong niversity, Xi
More informationThe optimization of kinematical response of gear transmission
Proeeings of the 7 WSEAS Int. Conferene on Ciruits, Systems, Signal an Teleommuniations, Gol Coast, Australia, January 7-9, 7 The optimization of inematial response of gear transmission VINCENZO NIOLA
More informationA Characterization of Wavelet Convergence in Sobolev Spaces
A Charaterization of Wavelet Convergene in Sobolev Spaes Mark A. Kon 1 oston University Louise Arakelian Raphael Howard University Dediated to Prof. Robert Carroll on the oasion of his 70th birthday. Abstrat
More informationAdvanced Computational Fluid Dynamics AA215A Lecture 4
Advaned Computational Fluid Dynamis AA5A Leture 4 Antony Jameson Winter Quarter,, Stanford, CA Abstrat Leture 4 overs analysis of the equations of gas dynamis Contents Analysis of the equations of gas
More information1 sin 2 r = 1 n 2 sin 2 i
Physis 505 Fall 005 Homework Assignment #11 Solutions Textbook problems: Ch. 7: 7.3, 7.5, 7.8, 7.16 7.3 Two plane semi-infinite slabs of the same uniform, isotropi, nonpermeable, lossless dieletri with
More informationJournal of Mathematical Analysis and Applications
J. Math. Anal. Appl. 371 (010) 759 763 Contents lists available at SieneDiret Journal of Mathematial Analysis an Appliations www.elsevier.om/loate/jmaa Singular Sturm omparison theorems Dov Aharonov, Uri
More informationA EUCLIDEAN ALTERNATIVE TO MINKOWSKI SPACETIME DIAGRAM.
A EUCLIDEAN ALTERNATIVE TO MINKOWSKI SPACETIME DIAGRAM. S. Kanagaraj Eulidean Relativity s.kana.raj@gmail.om (1 August 009) Abstrat By re-interpreting the speial relativity (SR) postulates based on Eulidean
More informationPerformance Evaluation of atall Building with Damped Outriggers Ping TAN
Performane Evaluation of atall Builing with Dampe Outriggers Ping TAN Earthquake Engineering Researh an Test Center Guangzhou University, Guangzhou, China OUTLINES RESEARCH BACKGROUND IMPROVED ANALYTICAL
More informationFast Evaluation of Canonical Oscillatory Integrals
Appl. Math. Inf. Si. 6, No., 45-51 (01) 45 Applie Mathematis & Information Sienes An International Journal 01 NSP Natural Sienes Publishing Cor. Fast Evaluation of Canonial Osillatory Integrals Ying Liu
More informationCSIR-UGC NET/JRF JUNE - 6 PHYSICAL SCIENCES OOKLET - [A] PART. The raius of onvergene of the Taylor series epansion of the funtion (). The value of the ontour integral the anti-lokwise iretion, is 4z e
More informationIEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 3, MARCH A DS CDMA system is said to be approximately synchronized if the modulated
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 3, MARCH 2008 1339 two ases we get infinite lasses of DPM. The most important result is the onstrution of DPM from ternary vetors of lengths at least
More informationExtended Spectral Nonlinear Conjugate Gradient methods for solving unconstrained problems
International Journal of All Researh Euation an Sientifi Methos IJARESM ISSN: 55-6 Volume Issue 5 May-0 Extene Spetral Nonlinear Conjuate Graient methos for solvin unonstraine problems Dr Basim A Hassan
More informationAn Integer Solution of Fractional Programming Problem
Gen. Math. Notes, Vol. 4, No., June 0, pp. -9 ISSN 9-784; Copyright ICSRS Publiation, 0 www.i-srs.org Available free online at http://www.geman.in An Integer Solution of Frational Programming Problem S.C.
More informationMATH Non-Euclidean Geometry Exercise Set #8 Solutions
MATH 68-9 Non-Euliean Geometry Exerise Set #8 Let ( ab, :, ) Show that ( ab, :, ) an ( a b) to fin ( a, : b,, ) ( a, : b,, ) an ( a, : b, ) Sine ( ab, :, ) while Likewise,, we have ( a, : b, ) ( ab, :,
More informationSURFACE WAVES OF NON-RAYLEIGH TYPE
SURFACE WAVES OF NON-RAYLEIGH TYPE by SERGEY V. KUZNETSOV Institute for Problems in Mehanis Prosp. Vernadskogo, 0, Mosow, 75 Russia e-mail: sv@kuznetsov.msk.ru Abstrat. Existene of surfae waves of non-rayleigh
More informationConformal Mapping among Orthogonal, Symmetric, and Skew-Symmetric Matrices
AAS 03-190 Conformal Mapping among Orthogonal, Symmetri, and Skew-Symmetri Matries Daniele Mortari Department of Aerospae Engineering, Texas A&M University, College Station, TX 77843-3141 Abstrat This
More informationarxiv: v1 [quant-ph] 30 Jan 2019
Universal logial gates with onstant overhea: instantaneous Dehn twists for hyperoli quantum oes Ali Lavasani, Guanyu Zhu, an Maissam Barkeshli Department of Physis, Conense Matter Theory Center, University
More informationOn the Reverse Problem of Fechnerian Scaling
On the Reverse Prolem of Fehnerian Saling Ehtiar N. Dzhafarov Astrat Fehnerian Saling imposes metris on two sets of stimuli relate to eah other y a isrimination funtion sujet to Regular Minimality. The
More informationl. For adjacent fringes, m dsin m
Test 3 Pratie Problems Ch 4 Wave Nature of Light ) Double Slit A parallel beam of light from a He-Ne laser, with a wavelength of 656 nm, falls on two very narrow slits that are 0.050 mm apart. How far
More informationOptimal Distributed Estimation Fusion with Transformed Data
Optimal Distribute Estimation Fusion with Transforme Data Zhansheng Duan X. Rong Li Department of Eletrial Engineering University of New Orleans New Orleans LA 70148 U.S.A. Email: {zuanxli@uno.eu Abstrat
More informationExpressiveness of the Interval Logics of Allen s Relations on the Class of all Linear Orders: Complete Classification
Proeeings of the Twenty-Seon International Joint Conferene on Artifiial Intelligene Expressiveness of the Interval Logis of Allen s Relations on the Class of all Linear Orers: Complete Classifiation Dario
More informationMillennium Relativity Acceleration Composition. The Relativistic Relationship between Acceleration and Uniform Motion
Millennium Relativity Aeleration Composition he Relativisti Relationship between Aeleration and niform Motion Copyright 003 Joseph A. Rybzyk Abstrat he relativisti priniples developed throughout the six
More informationPseudo-Differential Operators Involving Fractional Fourier Cosine (Sine) Transform
ilomat 31:6 17, 1791 181 DOI 1.98/IL176791P Publishe b ault of Sienes an Mathematis, Universit of Niš, Serbia Available at: http://www.pmf.ni.a.rs/filomat Pseuo-Differential Operators Involving rational
More informationGEOMETRIC AND STOCHASTIC ERROR MINIMISATION IN MOTION TRACKING. Karteek Alahari, Sujit Kuthirummal, C. V. Jawahar, P. J. Narayanan
GEOMETRIC AND STOCHASTIC ERROR MINIMISATION IN MOTION TRACKING Karteek Alahari, Sujit Kuthirummal, C. V. Jawahar, P. J. Narayanan Centre for Visual Information Tehnology International Institute of Information
More informationarxiv: v1 [math-ph] 19 Apr 2009
arxiv:0904.933v1 [math-ph] 19 Apr 009 The relativisti mehanis in a nonholonomi setting: A unifie approah to partiles with non-zero mass an massless partiles. Olga Krupková an Jana Musilová Deember 008
More informationStrauss PDEs 2e: Section Exercise 3 Page 1 of 13. u tt c 2 u xx = cos x. ( 2 t c 2 2 x)u = cos x. v = ( t c x )u
Strauss PDEs e: Setion 3.4 - Exerise 3 Page 1 of 13 Exerise 3 Solve u tt = u xx + os x, u(x, ) = sin x, u t (x, ) = 1 + x. Solution Solution by Operator Fatorization Bring u xx to the other side. Write
More informationThe Computational Complexity of the Unrooted Subtree Prune and Regraft Distance. Technical Report CS
The Computational Complexit of the Unroote ubtree rune an egraft Distane Glenn Hike Frank Dehne Anrew au-chaplin Christian Blouin Tehnial eport C-006-06 Jul, 006 Fault of Computer iene 6050 Universit Ave.,
More informationNew Equation of Motion of an Electron: the Covariance of Self-action
New Equation of Motion of an Eletron: the Covariane of Self-ation Xiaowen Tong Sihuan University Abstrat It is well known that our knowlege about the raiation reation of an eletron in lassial eletroynamis
More informationBrazilian Journal of Physics, vol. 29, no. 1, March,
Brazilian Journal of hysis, vol. 29, no., Marh, 999 79 Computational Methos Inspire by Tsallis Statistis: Monte Carlo an Moleular Dynamis Algorithms for the Simulation of Classial an Quantum Systems John
More informationGeneralized resolution for orthogonal arrays
Submitted to he Annals of Statistis Generalized resolution for orthogonal arrays y Ulrike Grömping and Hongquan u 2 euth University of Applied Sienes erlin and University of alifornia Los Angeles he generalized
More informationPN Code Tracking Loops
Wireless Information Transmission System Lab. PN Coe Traking Loops Institute of Communiations Engineering National Sun Yat-sen University Introution Coe synhronization is generally arrie out in two steps
More informationBOOLEAN GRÖBNER BASIS REDUCTIONS ON FINITE FIELD DATAPATH CIRCUITS
BOOLEAN GRÖBNER BASIS REDUCTIONS ON FINITE FIELD DATAPATH CIRCUITS USING THE UNATE CUBE SET ALGEBRA Utkarsh Gupta, Priyank Kalla, Senior Member, IEEE, Vikas Rao Abstrat Reent evelopments in formal verifiation
More information18.06 Problem Set 4 Due Wednesday, Oct. 11, 2006 at 4:00 p.m. in 2-106
8. Problem Set Due Wenesy, Ot., t : p.m. in - Problem Mony / Consier the eight vetors 5, 5, 5,..., () List ll of the one-element, linerly epenent sets forme from these. (b) Wht re the two-element, linerly
More informationMost results in this section are stated without proof.
Leture 8 Level 4 v2 he Expliit formula. Most results in this setion are stated without proof. Reall that we have shown that ζ (s has only one pole, a simple one at s =. It has trivial zeros at the negative
More informationarxiv: v1 [math.sg] 27 Mar 2019
HOMOLOGICALLY TRIVIAL SYMPLECTIC CYCLIC ACTIONS NEED NOT EXTEND TO HAMILTONIAN CIRCLE ACTIONS RIVER CHIANG AND LIAT KESSLER arxiv:903.568v [math.sg] 27 Mar 209 Abstrat. We give examples of sympleti ations
More informationMcCreight s Suffix Tree Construction Algorithm. Milko Izamski B.Sc. Informatics Instructor: Barbara König
1. Introution MCreight s Suffix Tree Constrution Algorithm Milko Izamski B.S. Informatis Instrutor: Barbara König The main goal of MCreight s algorithm is to buil a suffix tree in linear time. This is
More informationLabeling Workflow Views with Fine-Grained Dependencies
Labeling Workflow Views with Fine-Graine Depenenies Zhuowei Bao Department of omputer an Information iene University of Pennsylvania Philaelphia, P 1914, U zhuowei@is.upenn.eu usan B. Davison Department
More informationv = fy c u = fx c z c The Pinhole Camera Model Camera Projection Models
The Pinhole Camera Model Camera Projetion Models We will introdue dierent amera projetion models that relate the loation o an image point to the oordinates o the orresponding 3D points. The projetion models
More informationOn Predictive Density Estimation for Location Families under Integrated Absolute Error Loss
On Preitive Density Estimation for Loation Families uner Integrate Absolute Error Loss Tatsuya Kubokawa a, Éri Marhanb, William E. Strawerman a Department of Eonomis, University of Tokyo, 7-3- Hongo, Bunkyo-ku,
More informationDynamic Progressive Buckling of Square Tubes
THE 7TH CONFERENCE ON THEORETICAL. AND ALIED MECHANICS Tainan,Taiwan,R.O.C., - Deeber Dynai rogressive Bukling of Square Tubes Chih-Cheng Yang Departent of Autoation Engineering Kao Yuan Institute of Tehnology
More informationOptimal Design of Fault-Tolerant Petri Net Controllers
Optial Design of Fault-Tolerant Petri Net ontrollers Yizhi Qu, Lingxi Li, Yaobin hen, an Yaping Dai Abstrat This paper proposes an approah for the optial esign of fault-tolerant Petri net ontrollers Given
More informationFree rotation of a rigid body 1 D. E. Soper 2 University of Oregon Physics 611, Theoretical Mechanics 5 November 2012
Free rotation of a rigi boy 1 D. E. Soper 2 University of Oregon Physics 611, Theoretical Mechanics 5 November 2012 1 Introuction In this section, we escribe the motion of a rigi boy that is free to rotate
More informationSturm-Liouville Theory
LECTURE 5 Sturm-Liouville Theory In the three preceing lectures I emonstrate the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series are just the tip of the iceberg of the theory
More informationReview of Force, Stress, and Strain Tensors
Review of Fore, Stress, and Strain Tensors.1 The Fore Vetor Fores an be grouped into two broad ategories: surfae fores and body fores. Surfae fores are those that at over a surfae (as the name implies),
More informationMULTIPLE-INPUT MULTIPLE-OUTPUT (MIMO) is. Spatial Degrees of Freedom of Large Distributed MIMO Systems and Wireless Ad Hoc Networks
22 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 3, NO. 2, FEBRUARY 23 Spatial Degrees of Freeom of Large Distribute MIMO Systems an Wireless A Ho Networks Ayfer Özgür, Member, IEEE, OlivierLévêque,
More informationImprovement on spherical symmetry in two-dimensional cylindrical coordinates for a class of control volume Lagrangian schemes
Improvement on spherial symmetry in two-imensional ylinrial oorinates for a lass of ontrol volume Lagrangian shemes Juan Cheng an Chi-Wang Shu 2 Abstrat In [4], Maire evelope a lass of ell-entere Lagrangian
More informationLECTURE 2 Geometrical Properties of Rod Cross Sections (Part 2) 1 Moments of Inertia Transformation with Parallel Transfer of Axes.
V. DEMENKO MECHNCS OF MTERLS 05 LECTURE Geometrial Properties of Rod Cross Setions (Part ) Moments of nertia Transformation with Parallel Transfer of xes. Parallel-xes Theorems S Given: a b = S = 0. z
More informationQuantum Mechanics: Wheeler: Physics 6210
Quantum Mehanis: Wheeler: Physis 60 Problems some modified from Sakurai, hapter. W. S..: The Pauli matries, σ i, are a triple of matries, σ, σ i = σ, σ, σ 3 given by σ = σ = σ 3 = i i Let stand for the
More informationMOLECULAR ORBITAL THEORY- PART I
5.6 Physial Chemistry Leture #24-25 MOLECULAR ORBITAL THEORY- PART I At this point, we have nearly ompleted our rash-ourse introdution to quantum mehanis and we re finally ready to deal with moleules.
More informationSQUARE ROOTS AND AND DIRECTIONS
SQUARE ROOS AND AND DIRECIONS We onstrut a lattie-like point set in the Eulidean plane that eluidates the relationship between the loal statistis of the frational parts of n and diretions in a shifted
More informationDirectional Coupler. 4-port Network
Diretional Coupler 4-port Network 3 4 A diretional oupler is a 4-port network exhibiting: All ports mathed on the referene load (i.e. S =S =S 33 =S 44 =0) Two pair of ports unoupled (i.e. the orresponding
More informationEcon 455 Answers - Problem Set Consider a small country (Belgium) with the following demand and supply curves for corn:
Spring 004 Eon 455 Harvey Lapan Eon 455 Answers - Problem Set 4 1. Consier a small ountry (Belgium with the ollowing eman an supply urves or orn: Supply = 4P s ; Deman = 1000 Assume Belgium an import steel
More informationConstraint-free Analog Placement with Topological Symmetry Structure
Constraint-free Analog Plaement with Topologial Symmetry Struture Qing DONG Department of Information an Meia Sienes University of Kitakyushu Wakamatsu, Kitakyushu, Fukuoka, 808-0135, Japan e-mail: ongqing@env.kitakyu-u.a.jp
More informationNelson Pinto-Neto 331 funtion is given in terms of the pointer basis states, an why we o not see superpositions of marosopi objets. In this way, lassi
330 Brazilian Journal of Physis, vol. 30, no. 2, June, 2000 Quantum Cosmology: How to Interpret an Obtain Results Nelson Pinto-Neto Centro Brasileiro e Pesquisas F sias, Rua Dr. Xavier Sigau 150, Ura 22290-180,
More informationEuler and Hamilton Paths
Euler an Hamilton Paths The town of Königserg, Prussia (now know as Kaliningra an part of the Russian repuli), was ivie into four setion y ranhes of the Pregel River. These four setions C A D B Figure:
More informationModern Physics I Solutions to Homework 4 Handout
Moern Physis I Solutions to Homework 4 Hanout TA: Alvaro Núñez an33@sires.nyu.eu New York University, Department of Physis, 4 Washington Pl., New York, NY 0003. Bernstein, Fishbane, Gasiorowiz: Chapter
More informationHe s Semi-Inverse Method and Ansatz Approach to look for Topological and Non-Topological Solutions Generalized Nonlinear Schrödinger Equation
Quant. Phys. Lett. 3, No. 2, 23-27 2014) 23 Quantum Physis Letters An International Journal http://x.oi.org/10.12785/qpl/030202 He s Semi-Inverse Metho an Ansatz Approah to look for Topologial an Non-Topologial
More informationRank, Trace, Determinant, Transpose an Inverse of a Matrix Let A be an n n square matrix: A = a11 a1 a1n a1 a an a n1 a n a nn nn where is the jth col
Review of Linear Algebra { E18 Hanout Vectors an Their Inner Proucts Let X an Y be two vectors: an Their inner prouct is ene as X =[x1; ;x n ] T Y =[y1; ;y n ] T (X; Y ) = X T Y = x k y k k=1 where T an
More informationthe following action R of T on T n+1 : for each θ T, R θ : T n+1 T n+1 is defined by stated, we assume that all the curves in this paper are defined
How should a snake turn on ie: A ase study of the asymptoti isoholonomi problem Jianghai Hu, Slobodan N. Simić, and Shankar Sastry Department of Eletrial Engineering and Computer Sienes University of California
More informationThe Sokhotski-Plemelj Formula
hysics 25 Winter 208 The Sokhotski-lemelj Formula. The Sokhotski-lemelj formula The Sokhotski-lemelj formula is a relation between the following generalize functions (also calle istributions), ±iǫ = iπ(),
More informationGradient Elasticity Theory for Mode III Fracture in Functionally Graded Materials Part II: Crack Parallel to the Material Gradation
Youn-Sha Chan Department of Computer an Mathematial Sienes, University of Houston-Downtown, One Main Street, Houston, TX 77 Glauio H. Paulino Department of Civil an Environmental Engineering, University
More informationMaximum Entropy and Exponential Families
Maximum Entropy and Exponential Families April 9, 209 Abstrat The goal of this note is to derive the exponential form of probability distribution from more basi onsiderations, in partiular Entropy. It
More informationJUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson
JUST THE MATHS UNIT NUMBER 10.2 DIFFERENTIATION 2 (Rates of change) by A.J.Hobson 10.2.1 Introuction 10.2.2 Average rates of change 10.2.3 Instantaneous rates of change 10.2.4 Derivatives 10.2.5 Exercises
More informationKinematics of Rotations: A Summary
A Kinematics of Rotations: A Summary The purpose of this appenix is to outline proofs of some results in the realm of kinematics of rotations that were invoke in the preceing chapters. Further etails are
More informationCombinatorial remarks on two-dimensional Languages
Combinatorial remarks on two-imensional Languages Franesa De Carli To ite this version: Franesa De Carli. Combinatorial remarks on two-imensional Languages. Mathematis [math]. Université e Savoie 2009.
More informationRemark 4.1 Unlike Lyapunov theorems, LaSalle s theorem does not require the function V ( x ) to be positive definite.
Leture Remark 4.1 Unlike Lyapunov theorems, LaSalle s theorem does not require the funtion V ( x ) to be positive definite. ost often, our interest will be to show that x( t) as t. For that we will need
More informationSimple FIR Digital Filters. Simple FIR Digital Filters. Simple Digital Filters. Simple FIR Digital Filters. Simple FIR Digital Filters
Simple Digital Filters Later in the ourse we shall review various methods of designing frequeny-seletive filters satisfying presribed speifiations We now desribe several low-order FIR and IIR digital filters
More informationMath 220A - Fall 2002 Homework 8 Solutions
Math A - Fall Homework 8 Solutions 1. Consider u tt u = x R 3, t > u(x, ) = φ(x) u t (x, ) = ψ(x). Suppose φ, ψ are supported in the annular region a < x < b. (a) Find the time T 1 > suh that u(x, t) is
More informationOn the sustainability of collusion in Bertrand supergames with discrete pricing and nonlinear demand
PRA unih Personal RePE Arhive On the sustainability of ollusion in Bertran supergames with isrete priing an nonlinear eman Paul R. Zimmerman US Feeral Trae Commission 25. January 2010 Online at http://mpra.ub.uni-muenhen.e/20249/
More informationMulti-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics.
Multi-sale Gounov-type metho for ell-entere isrete Lagrangian hyroynamis. Pierre-Henri Maire, Bonifae Nkonga To ite this version: Pierre-Henri Maire, Bonifae Nkonga. Multi-sale Gounov-type metho for ell-entere
More informationMath Implicit Differentiation. We have discovered (and proved) formulas for finding derivatives of functions like
Math 400 3.5 Implicit Differentiation Name We have iscovere (an prove) formulas for fining erivatives of functions like f x x 3x 4x. 3 This amounts to fining y for 3 y x 3x 4x. Notice that in this case,
More informationMath Review for Physical Chemistry
Chemistry 362 Spring 27 Dr. Jean M. Stanar January 25, 27 Math Review for Physical Chemistry I. Algebra an Trigonometry A. Logarithms an Exponentials General rules for logarithms These rules, except where
More information19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control
19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior
More informationComplexity of Regularization RBF Networks
Complexity of Regularization RBF Networks Mark A Kon Department of Mathematis and Statistis Boston University Boston, MA 02215 mkon@buedu Leszek Plaskota Institute of Applied Mathematis University of Warsaw
More informationSampler-B. Secondary Mathematics Assessment. Sampler 521-B
Sampler-B Seonary Mathematis Assessment Sampler 51-B Instrutions for Stuents Desription This sample test inlues 15 Selete Response an 5 Construte Response questions. Eah Selete Response has a value of
More informationPDE Notes, Lecture #11
PDE Notes, Lecture # from Professor Jalal Shatah s Lectures Febuary 9th, 2009 Sobolev Spaces Recall that for u L loc we can efine the weak erivative Du by Du, φ := udφ φ C0 If v L loc such that Du, φ =
More informationHomework 3 - Solutions
Homework 3 - Solutions The Transpose an Partial Transpose. 1 Let { 1, 2,, } be an orthonormal basis for C. The transpose map efine with respect to this basis is a superoperator Γ that acts on an operator
More informationOptimal torque control of permanent magnet synchronous machines using magnetic equivalent circuits
This oument ontains a post-print version of the paper Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits authore by W. Kemmetmüller, D. Faustner, an A. Kugi an
More informationSystems & Control Letters
Systems & ontrol Letters ( ) ontents lists available at ScienceDirect Systems & ontrol Letters journal homepage: www.elsevier.com/locate/sysconle A converse to the eterministic separation principle Jochen
More informationExam 1 Study Guide. Differentiation and Anti-differentiation Rules from Calculus I
Exm Stuy Guie Mth 26 - Clulus II, Fll 205 The following is list of importnt onepts from eh setion tht will be teste on exm. This is not omplete list of the mteril tht you shoul know for the ourse, but
More informationOrientational ordering in lipid monolayers: A two-dimensional model of rigid rods grafted onto a lattice
Orientational orering in lipi monolayers: A two-imensional moel of rigi ros grafte onto a lattie M. Sheringer, ) F Hilfer, an K. Biner nstitut ftir Physik, Johannes Gutenberg-Universittit Maim, Postfah
More informationR13 SET - 1 PART-A. is analytic. c) Write the test statistic for the differences of means of two large samples. about z =1.
R3 SET - II B. Teh I Semester Regular Examinations, Jan - 5 COMPLEX VARIABLES AND STATISTICAL METHODS (Eletrial and Eletronis Engineering) Time: 3 hours Max. Marks: 7 Note:. Question Paper onsists of two
More informationDetermination the Invert Level of a Stilling Basin to Control Hydraulic Jump
Global Avane Researh Journal of Agriultural Siene Vol. (4) pp. 074-079, June, 0 Available online http://garj.org/garjas/inex.htm Copyright 0 Global Avane Researh Journals Full Length Researh Paper Determination
More informationCSE 5311 Notes 18: NP-Completeness
SE 53 Notes 8: NP-ompleteness (Last upate 7//3 8:3 PM) ELEMENTRY ONEPTS Satisfiability: ( p q) ( p q ) ( p q) ( p q ) Is there an assignment? (Deision Problem) Similar to ebugging a logi iruit - Is there
More informationIntercepts To find the y-intercept (b, fixed value or starting value), set x = 0 and solve for y. To find the x-intercept, set y = 0 and solve for x.
Units 4 an 5: Linear Relations partial variation iret variation Points on a Coorinate Gri (x-oorinate, y-oorinate) origin is (0, 0) "run, then jump" Interepts To fin the y-interept (, fixe value or starting
More information