3D Rotations Matrices
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Quantum Information
Quantum Information J. A. Jones Michaelmas Term 2010 Contents 1 Dirac Notation 3 1.1 Hilbert Space . 3 1.2 Dirac notation . 4 1.3 Operators . 5 1.4 Vectors and matrices . 6 1.5 Eigenvalues and eigenvectors . 8 1.6 Hermitian operators . 9 1.7 Commutators . 10 1.8 Unitary operators . 11 1.9 Operator exponentials . 11 1.10 Physical systems . 12 1.11 Time-dependent Hamiltonians . 13 1.12 Global phases . 13 2 Quantum bits and quantum gates 15 2.1 The Bloch sphere . 16 2.2 Density matrices . 16 2.3 Propagators and Pauli matrices . 18 2.4 Quantum logic gates . 18 2.5 Gate notation . 21 2.6 Quantum networks . 21 2.7 Initialization and measurement . 23 2.8 Experimental methods . 24 3 An atom in a laser field 25 3.1 Time-dependent systems . 25 3.2 Sudden jumps . 26 3.3 Oscillating fields . 27 3.4 Time-dependent perturbation theory . 29 3.5 Rabi flopping and Fermi's Golden Rule . 30 3.6 Raman transitions . 32 3.7 Rabi flopping as a quantum gate . 32 3.8 Ramsey fringes . 33 3.9 Measurement and initialisation . 34 1 CONTENTS 2 4 Spins in magnetic fields 35 4.1 The nuclear spin Hamiltonian . 35 4.2 The rotating frame . 36 4.3 On-resonance excitation . 38 4.4 Excitation phases . 38 4.5 Off-resonance excitation . 39 4.6 Practicalities . 40 4.7 The vector model . 40 4.8 Spin echoes . 41 4.9 Measurement and initialisation . 42 5 Photon techniques 43 5.1 Spatial encoding . -
MATH 237 Differential Equations and Computer Methods
Queen’s University Mathematics and Engineering and Mathematics and Statistics MATH 237 Differential Equations and Computer Methods Supplemental Course Notes Serdar Y¨uksel November 19, 2010 This document is a collection of supplemental lecture notes used for Math 237: Differential Equations and Computer Methods. Serdar Y¨uksel Contents 1 Introduction to Differential Equations 7 1.1 Introduction: ................................... ....... 7 1.2 Classification of Differential Equations . ............... 7 1.2.1 OrdinaryDifferentialEquations. .......... 8 1.2.2 PartialDifferentialEquations . .......... 8 1.2.3 Homogeneous Differential Equations . .......... 8 1.2.4 N-thorderDifferentialEquations . ......... 8 1.2.5 LinearDifferentialEquations . ......... 8 1.3 Solutions of Differential equations . .............. 9 1.4 DirectionFields................................. ........ 10 1.5 Fundamental Questions on First-Order Differential Equations............... 10 2 First-Order Ordinary Differential Equations 11 2.1 ExactDifferentialEquations. ........... 11 2.2 MethodofIntegratingFactors. ........... 12 2.3 SeparableDifferentialEquations . ............ 13 2.4 Differential Equations with Homogenous Coefficients . ................ 13 2.5 First-Order Linear Differential Equations . .............. 14 2.6 Applications.................................... ....... 14 3 Higher-Order Ordinary Linear Differential Equations 15 3.1 Higher-OrderDifferentialEquations . ............ 15 3.1.1 LinearIndependence . ...... 16 3.1.2 WronskianofasetofSolutions . ........ 16 3.1.3 Non-HomogeneousProblem -
The Exponential of a Matrix
5-28-2012 The Exponential of a Matrix The solution to the exponential growth equation dx kt = kx is given by x = c e . dt 0 It is natural to ask whether you can solve a constant coefficient linear system ′ ~x = A~x in a similar way. If a solution to the system is to have the same form as the growth equation solution, it should look like At ~x = e ~x0. The first thing I need to do is to make sense of the matrix exponential eAt. The Taylor series for ez is ∞ n z z e = . n! n=0 X It converges absolutely for all z. It A is an n × n matrix with real entries, define ∞ n n At t A e = . n! n=0 X The powers An make sense, since A is a square matrix. It is possible to show that this series converges for all t and every matrix A. Differentiating the series term-by-term, ∞ ∞ ∞ ∞ n−1 n n−1 n n−1 n−1 m m d At t A t A t A t A At e = n = = A = A = Ae . dt n! (n − 1)! (n − 1)! m! n=0 n=1 n=1 m=0 X X X X At ′ This shows that e solves the differential equation ~x = A~x. The initial condition vector ~x(0) = ~x0 yields the particular solution At ~x = e ~x0. This works, because e0·A = I (by setting t = 0 in the power series). Another familiar property of ordinary exponentials holds for the matrix exponential: If A and B com- mute (that is, AB = BA), then A B A B e e = e + . -
Molecular Symmetry
Molecular Symmetry Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy) – used with group theory to predict vibrational spectra for the identification of molecular shape, and as a tool for understanding electronic structure and bonding. Symmetrical : implies the species possesses a number of indistinguishable configurations. 1 Group Theory : mathematical treatment of symmetry. symmetry operation – an operation performed on an object which leaves it in a configuration that is indistinguishable from, and superimposable on, the original configuration. symmetry elements – the points, lines, or planes to which a symmetry operation is carried out. Element Operation Symbol Identity Identity E Symmetry plane Reflection in the plane σ Inversion center Inversion of a point x,y,z to -x,-y,-z i Proper axis Rotation by (360/n)° Cn 1. Rotation by (360/n)° Improper axis S 2. Reflection in plane perpendicular to rotation axis n Proper axes of rotation (C n) Rotation with respect to a line (axis of rotation). •Cn is a rotation of (360/n)°. •C2 = 180° rotation, C 3 = 120° rotation, C 4 = 90° rotation, C 5 = 72° rotation, C 6 = 60° rotation… •Each rotation brings you to an indistinguishable state from the original. However, rotation by 90° about the same axis does not give back the identical molecule. XeF 4 is square planar. Therefore H 2O does NOT possess It has four different C 2 axes. a C 4 symmetry axis. A C 4 axis out of the page is called the principle axis because it has the largest n . By convention, the principle axis is in the z-direction 2 3 Reflection through a planes of symmetry (mirror plane) If reflection of all parts of a molecule through a plane produced an indistinguishable configuration, the symmetry element is called a mirror plane or plane of symmetry . -
(II) I Main Topics a Rotations Using a Stereonet B
GG303 Lecture 11 9/4/01 1 ROTATIONS (II) I Main Topics A Rotations using a stereonet B Comments on rotations using stereonets and matrices C Uses of rotation in geology (and engineering) II II Rotations using a stereonet A Best uses 1 Rotation axis is vertical 2 Rotation axis is horizontal (e.g., to restore tilted beds) B Construction technique 1 Find orientation of rotation axis 2 Find angle of rotation and rotate pole to plane (or a linear feature) along a small circle perpendicular to the rotation axis. a For a horizontal rotation axis the small circle is vertical b For a vertical rotation axis the small circle is horizontal 3 WARNING: DON'T rotate a plane by rotating its dip direction vector; this doesn't work (see handout) IIIComments on rotations using stereonets and matrices Advantages Disadvantages Stereonets Good for visualization Relatively slow Can bring into field Need good stereonets, paper Matrices Speed and flexibility Computer really required to Good for multiple rotations cut down on errors A General comments about stereonets vs. rotation matrices 1 With stereonets, an object is usually considered to be rotated and the coordinate axes are held fixed. 2 With rotation matrices, an object is usually considered to be held fixed and the coordinate axes are rotated. B To return tilted bedding to horizontal choose a rotation axis that coincides with the direction of strike. The angle of rotation is the negative of the dip of the bedding (right-hand rule! ) if the bedding is to be rotated back to horizontal and positive if the axes are to be rotated to the plane of the tilted bedding. -
Bearing Rigidity Theory in SE(3) Giulia Michieletto, Angelo Cenedese, Antonio Franchi
Bearing Rigidity Theory in SE(3) Giulia Michieletto, Angelo Cenedese, Antonio Franchi To cite this version: Giulia Michieletto, Angelo Cenedese, Antonio Franchi. Bearing Rigidity Theory in SE(3). 55th IEEE Conference on Decision and Control, Dec 2016, Las Vegas, United States. hal-01371084 HAL Id: hal-01371084 https://hal.archives-ouvertes.fr/hal-01371084 Submitted on 23 Sep 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Preprint version, final version at http://ieeexplore.ieee.org/ 55th IEEE Conference on Decision and Control. Las Vegas, NV, 2016 Bearing Rigidity Theory in SE(3) Giulia Michieletto, Angelo Cenedese, and Antonio Franchi Abstract— Rigidity theory has recently emerged as an ef- determines the infinitesimal rigidity properties of the system, ficient tool in the control field of coordinated multi-agent providing a necessary and sufficient condition. In such a systems, such as multi-robot formations and UAVs swarms, that context, a framework is generally represented by means of are characterized by sensing, communication and movement capabilities. This work aims at describing the rigidity properties the bar-and-joint model where agents are represented as for frameworks embedded in the three-dimensional Special points joined by bars whose fixed lengths enforce the inter- Euclidean space SE(3) wherein each agent has 6DoF. -
Chapter 5 ANGULAR MOMENTUM and ROTATIONS
Chapter 5 ANGULAR MOMENTUM AND ROTATIONS In classical mechanics the total angular momentum L~ of an isolated system about any …xed point is conserved. The existence of a conserved vector L~ associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian com- ponents Li of the total angular momentum operator L~ of an isolated system are also constants of the motion. The di¤erent components of L~ are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one an- other. Thus, the vector operator L~ is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another. -
Euler Quaternions
Four different ways to represent rotation my head is spinning... The space of rotations SO( 3) = {R ∈ R3×3 | RRT = I,det(R) = +1} Special orthogonal group(3): Why det( R ) = ± 1 ? − = − Rotations preserve distance: Rp1 Rp2 p1 p2 Rotations preserve orientation: ( ) × ( ) = ( × ) Rp1 Rp2 R p1 p2 The space of rotations SO( 3) = {R ∈ R3×3 | RRT = I,det(R) = +1} Special orthogonal group(3): Why it’s a group: • Closed under multiplication: if ∈ ( ) then ∈ ( ) R1, R2 SO 3 R1R2 SO 3 • Has an identity: ∃ ∈ ( ) = I SO 3 s.t. IR1 R1 • Has a unique inverse… • Is associative… Why orthogonal: • vectors in matrix are orthogonal Why it’s special: det( R ) = + 1 , NOT det(R) = ±1 Right hand coordinate system Possible rotation representations You need at least three numbers to represent an arbitrary rotation in SO(3) (Euler theorem). Some three-number representations: • ZYZ Euler angles • ZYX Euler angles (roll, pitch, yaw) • Axis angle One four-number representation: • quaternions ZYZ Euler Angles φ = θ rzyz ψ φ − φ cos sin 0 To get from A to B: φ = φ φ Rz ( ) sin cos 0 1. Rotate φ about z axis 0 0 1 θ θ 2. Then rotate θ about y axis cos 0 sin θ = ψ Ry ( ) 0 1 0 3. Then rotate about z axis − sinθ 0 cosθ ψ − ψ cos sin 0 ψ = ψ ψ Rz ( ) sin cos 0 0 0 1 ZYZ Euler Angles φ θ ψ Remember that R z ( ) R y ( ) R z ( ) encode the desired rotation in the pre- rotation reference frame: φ = pre−rotation Rz ( ) Rpost−rotation Therefore, the sequence of rotations is concatentated as follows: (φ θ ψ ) = φ θ ψ Rzyz , , Rz ( )Ry ( )Rz ( ) φ − φ θ θ ψ − ψ cos sin 0 cos 0 sin cos sin 0 (φ θ ψ ) = φ φ ψ ψ Rzyz , , sin cos 0 0 1 0 sin cos 0 0 0 1− sinθ 0 cosθ 0 0 1 − − − cφ cθ cψ sφ sψ cφ cθ sψ sφ cψ cφ sθ (φ θ ψ ) = + − + Rzyz , , sφ cθ cψ cφ sψ sφ cθ sψ cφ cψ sφ sθ − sθ cψ sθ sψ cθ ZYX Euler Angles (roll, pitch, yaw) φ − φ cos sin 0 To get from A to B: φ = φ φ Rz ( ) sin cos 0 1. -
For Deep Rotation Learning with Uncertainty
A Smooth Representation of Belief over SO(3) for Deep Rotation Learning with Uncertainty Valentin Peretroukhin,1;3 Matthew Giamou,1 David M. Rosen,2 W. Nicholas Greene,3 Nicholas Roy,3 and Jonathan Kelly1 1Institute for Aerospace Studies, University of Toronto; 2Laboratory for Information and Decision Systems, 3Computer Science & Artificial Intelligence Laboratory, Massachusetts Institute of Technology Abstract—Accurate rotation estimation is at the heart of unit robot perception tasks such as visual odometry and object pose quaternions ✓ ✓ q = q<latexit sha1_base64="BMaQsvdb47NYnqSxvFbzzwnHZhA=">AAACLXicbVDLSsQwFE19W9+6dBMcBBEcWh+o4EJw41LBGYWZKmnmjhMmbTrJrTiUfodb/QK/xoUgbv0NM50ivg4ETs65N/fmhIkUBj3v1RkZHRufmJyadmdm5+YXFpeW60almkONK6n0VcgMSBFDDQVKuEo0sCiUcBl2Twb+5R1oI1R8gf0EgojdxqItOEMrBc06cFQ66+XXm/RmseJVvQL0L/FLUiElzm6WHLfZUjyNIEYumTEN30swyJhGwSXkbjM1kDDeZbfQsDRmEZggK7bO6bpVWrSttD0x0kL93pGxyJh+FNrKiGHH/PYG4n9eI8X2QZCJOEkRYj4c1E4lRUUHEdCW0PbXsm8J41rYXSnvMM042qB+TLlPmDaQU7dZvJb1UmHD3JIM4X7LdJRGnqKp2lvuFukdFqBDsr9bkkP/K736dtXfqe6d71aOj8ocp8gqWSMbxCf75JickjNSI5z0yAN5JE/Os/PivDnvw9IRp+xZIT/gfHwCW8eo/g==</latexit> ⇤ ✓ estimation. Deep neural networks have provided a new way to <latexit sha1_base64="1dHo/MB41p5RxdHfWHI4xsZNFIs=">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</latexit> -
The Exponential Function for Matrices
The exponential function for matrices Matrix exponentials provide a concise way of describing the solutions to systems of homoge- neous linear differential equations that parallels the use of ordinary exponentials to solve simple differential equations of the form y0 = λ y. For square matrices the exponential function can be defined by the same sort of infinite series used in calculus courses, but some work is needed in order to justify the construction of such an infinite sum. Therefore we begin with some material needed to prove that certain infinite sums of matrices can be defined in a mathematically sound manner and have reasonable properties. Limits and infinite series of matrices Limits of vector valued sequences in Rn can be defined and manipulated much like limits of scalar valued sequences, the key adjustment being that distances between real numbers that are expressed in the form js−tj are replaced by distances between vectors expressed in the form jx−yj. 1 Similarly, one can talk about convergence of a vector valued infinite series n=0 vn in terms of n the convergence of the sequence of partial sums sn = i=0 vk. As in the case of ordinary infinite series, the best form of convergence is absolute convergence, which correspondsP to the convergence 1 P of the real valued infinite series jvnj with nonnegative terms. A fundamental theorem states n=0 1 that a vector valued infinite series converges if the auxiliary series jvnj does, and there is P n=0 a generalization of the standard M-test: If jvnj ≤ Mn for all n where Mn converges, then P n vn also converges. -
Approximating the Exponential from a Lie Algebra to a Lie Group
MATHEMATICS OF COMPUTATION Volume 69, Number 232, Pages 1457{1480 S 0025-5718(00)01223-0 Article electronically published on March 15, 2000 APPROXIMATING THE EXPONENTIAL FROM A LIE ALGEBRA TO A LIE GROUP ELENA CELLEDONI AND ARIEH ISERLES 0 Abstract. Consider a differential equation y = A(t; y)y; y(0) = y0 with + y0 2 GandA : R × G ! g,whereg is a Lie algebra of the matricial Lie group G. Every B 2 g canbemappedtoGbythematrixexponentialmap exp (tB)witht 2 R. Most numerical methods for solving ordinary differential equations (ODEs) on Lie groups are based on the idea of representing the approximation yn of + the exact solution y(tn), tn 2 R , by means of exact exponentials of suitable elements of the Lie algebra, applied to the initial value y0. This ensures that yn 2 G. When the exponential is difficult to compute exactly, as is the case when the dimension is large, an approximation of exp (tB) plays an important role in the numerical solution of ODEs on Lie groups. In some cases rational or poly- nomial approximants are unsuitable and we consider alternative techniques, whereby exp (tB) is approximated by a product of simpler exponentials. In this paper we present some ideas based on the use of the Strang splitting for the approximation of matrix exponentials. Several cases of g and G are considered, in tandem with general theory. Order conditions are discussed, and a number of numerical experiments conclude the paper. 1. Introduction Consider the differential equation (1.1) y0 = A(t; y)y; y(0) 2 G; with A : R+ × G ! g; where G is a matricial Lie group and g is the underlying Lie algebra. -
The Structuring of Pattern Repeats in the Paracas Necropolis Embroideries
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Textile Society of America Symposium Proceedings Textile Society of America 1988 Orientation And Symmetry: The Structuring Of Pattern Repeats In The Paracas Necropolis Embroideries Mary Frame University of British Columbia Follow this and additional works at: https://digitalcommons.unl.edu/tsaconf Part of the Art and Design Commons Frame, Mary, "Orientation And Symmetry: The Structuring Of Pattern Repeats In The Paracas Necropolis Embroideries" (1988). Textile Society of America Symposium Proceedings. 637. https://digitalcommons.unl.edu/tsaconf/637 This Article is brought to you for free and open access by the Textile Society of America at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Textile Society of America Symposium Proceedings by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 136 RESEARCH-IN-PROGRESS REPORT ORIENTATION AND SYMMETRY: THE STRUCTURING OF PATTERN REPEATS IN THE PARACAS NECROPOLIS EMBROIDERIES MARY FRAME RESEARCH ASSOCIATE, INSTITUTE OF ANDEAN STUDIES 4611 Marine Drive West Vancouver, B.C. V7W 2P1 The most extensive Peruvian fabric remains come from the archeological site of Paracas Necropolis on the South Coast of Peru. Preserved by the dry desert conditions, this cache of 429 mummy bundles, excavated in 1926-27, provides an unparal- leled opportunity for comparing the range and nature of varia- tions in similar fabrics which are securely related in time and space. The bundles are thought to span the time period from 500-200 B.C. The most numerous and notable fabrics are embroideries: garments that have been classified as mantles, tunics, wrap- around skirts, loincloths, turbans and ponchos.