An intuitive approach to group representation theory

In this paper, a new global camera pose estimation algorithm WTLS-IRLS is proposed, which can effectively solve the global rotation when there are outliers. Firstly, according to the relationship between the rotation vector and the rotation matrix, we simplify the product operation of the rotation matrix into the subtraction operation of the rotation vector, which reduces the complexity of the algorithm. Secondly, the weighted total least squares (WTLS) and the iteratively reweighted least squares (IRLS) are used to average relative rotations. As the initialization of IRLS, WTLS provides a good initial guess by correcting the linearization equation and adding weight information to the relative rotations. IRLS continues to add weight information to the relative rotation matrices to optimize the global rotations. We demonstrate the performance of our approach by a number of large-scale data sets, the results show that our method has been greatly improved in efficiency, accuracy and iteration. In order to verify the correctness of our proposed method, we completed the complete reconstruction process, the experimental results show that our proposed WTLS-IRLS rotation averaging algorithm can obtain dense point clouds with more three-dimensional points.

Experimental full field investigations of resonant vibrations for piezoceramic plates by an optical interferometry method

A novel micro-electromechanical resonator based on in-plane mode was presented for the measurements of fluid density and viscosity. With the purpose of enhancing the quality factor of micro resonator, double vibrators were connected by a center beam to not only drive the resonant structure in in-plane vibration mode but also reduce the vibration stress to be concentrated in the fixed terminal. Finite element analysis and numerical simulation were conducted to analyze and optimize the resonator structure. Based on the simulated analysis, its quality factor reached highly to 96 in ethanol with the resonant frequency of approximately 34 kHz. And the stress amplitude up to MPa on the center beam had strong relationship with both of the fluid characteristics and output voltage, which benefited to improve the measurement accuracy of fluid density and viscosity based on the piezoresistive principle.

In this paper, a linear lumped-element equivalent circuit model (ECM) for ultrasonic laterally transduced electrostatic bulk-mode air-coupled resonant micro-electro-mechanical systems (MEMS) is described. A single-crystal silicon (SCS) square plate with T-shaped tethers is considered as the geometry of interest with a one-sided electrostatic actuation. This type of sensor can be used for sensitive mass sensing of airborne particles and possesses a large active surface with in-plane vibration modes in the ultrasonic frequency range. Firstly, the eigensolutions and eigenvectors of the problem are obtained using analytical equations and compared with finite-element modeling (FEM) solutions. Secondly, using modal analysis, the number of degrees of freedom is reduced and individual solutions are provided for each vibration mode, leading to various effective masses, stiffnesses and dampings. The first order Taylor expansion of both the electrical current equation and the electrostatic force applied on the resonator allows one to obtain expressions for the additional stiffness and the electro-mechanical transformation coefficient linked to the membrane actuation. Based on theses results, single-input single output (SISO) equivalent circuits are established using electro-mechanical and Butterworth-Van Dyke (BVD) approaches. Electrical admittance simulations resulting from different in-plane vibration modes are proven to be in excellent agreement with FEM simulations. Finally, a numerical mass sensing application is described to evaluate the relevance of both the model and the resonator design to act as a microbalance. The proposed model can be used to design, predict, analyze and optimize the behavior of highly sensitive air-coupled ultrasonic bulk-mode SCS MEMS for various physical applications.

Dispersed fluorescence spectra of jet-cooled adenine were obtained by exciting the vibronic bands observed in the fluorescence excitation spectrum from an earlier study (Kim et al., J. Chem. Phys., 2000, 113, 10 051). The dispersed fluorescence spectra reveal sufficiently well-resolved vibrational bands of the ground electronic state. The excitation of the C′ band at 36 062 cm−1 and the D′ band at 36 105 cm−1 results in emissions that are associated with out-of-plane and in-plane vibrational modes in the ground state, respectively, which suggests that these bands belong to different electronic states, as first proposed by us and later supported by others. A vibrational analysis yielded an assignment of the C′ band as an out-of-plane vibrational mode of the nπ* state and the D′ band as the 0–0 band of the ππ* state. Combination bands of the out-of-plane and in-plane vibrational modes are observed in the dispersed fluorescence spectrum of the C′ band, which seems to support that there is indeed a strong vibronic coupling between the nπ* and ππ* excited states of adenine.

A vector may be acted upon, simultaneously or successively, by two kinds of displacement operators. One of them is a rotation operator and the other is a translation operator. This chapter is devoted to the rotation of vectors and the rotation operators that rotate vectors. The vector equation is obtained as the Rodrigues formula. The matrix equation is written in terms of the rotation matrix, which is the matrix representation of the rotation operator in the selected reference frame. The chapter presents the salient mathematical properties of the rotation matrices that can be used conveniently in the symbolic manipulations concerning rotational kinematics. Demonstrative examples are also included.

It is shown that the Lauricella functions FD in n variables transform as basis vectors corresponding to irreducible representations of the Lie algebra sl(n+3,C). Group representation theory can then be applied to derive addition theorems, transformation formulas, and generating functions for the FD. It is clear from this analysis that the use of SL(m,C) symmetry in atomic and elementary particle physics will lead inevitably to the remarkable functions FD.

For a real semisimple group of matrices, every unitary principal series representation splits into inequivalent irreducible representations [5]. This multiplicity-one result was conjectured [7] in 1971 because one expected in general and knew in some special cases that all reducibility was accounted for by canonical geometric constructions, such as the imbedding of a space of analytic functions on the disc in the space of functions on the circle by passage to boundary values. We shall give an example to show that the corresponding multiplicity-one statement is false for a semisimple group of matrices defined over a locally compact, totally disconnected, nondiscrete field of characteristic 0. This example is summarized in §2, and its properties are verified in §5. It is ultimately motivated by the work of Langlands [11] on classification of irreducible admissible representations. Langlands [12] was able to give a formulation of the results of [8] that suggests that straightforward generalization of the multiplicityone theorem to other fields is not likely to succeed. An exposition of [12] is given in [9], and the way in which this work motivates our example is explained in § 6. Verification of the properties of our example depends on a suitable development of intertwining operators for split groups. Most of such a development has been carried out by Sally [15] and Winarsky [18], Some small modifications and elaborations of their work are the subject of § 3 and 4. The work in this paper grew out of conversations at the American Mathematical Society Summer Institute in 1977 with Langlands, Lusztig, Schiffmann, Shelstad, and Wallach. The work by I. Muller [13] on intertwining operators was also of influence; Muller developed a variation on Winarsky's work and was able to push through analogs of the results of [8]. In addition, she came close to discovering the example of § 2. We thank all these people* for their help. The results in this paper were announced in a talk at Hiroshima University on August 30, 1977, and later in Notices of the American Mathematical Society 25 (1978), abstract 753-G6, in connection with a meeting of the American Mathematical Society.

A systematic method is presented for constructing the irreducible multiplier representations (ray representations) of a class of finite groups corresponding to a given factor system. The method consists in first identifying a normal subgroup of prime index, classifying its irreducible multiplier representations into orbits and then inducing the required representations from these orbits. The proposed technique finds useful application in solid state physics where irreducible multiplier representations of the point group underlying the group of the wave vector are frequently required.

The electronic adiabatic-to-diabatic transformation matrix and the Wigner rotation matrix

Multiplicity One Theorems for Generalized Gelfand-Graev Representations of Semisimple Lie Groups and Whittaker Models for the Discrete Series

A quaternion method for georeferencing of satellite linear array stereo images is proposed in this paper. In this method, quaternion is used to describe the exterior orientation parameters (EOPs) of the satellite linear array stereo images, and gain the EOPs of any image line through quaternion attitude kinematics equation. Then the georeferencing results can be obtained after solving the collinearity equations with ground control points (GCPs). We validate this method through experiment with SPOT images. Experimental results indicate that quaternion attitude kinematics equation can be used for georeferencing of satellite linear array stereo images, the accuracy is as much as or even better than that of the conventional method, and the quaternion EOPs can overcome much of correlation. This method can successfully used in satellite linear array stereo images, and it can also effectively avoid a great deal of computation of trigonometric functions and the potential unstable factors. It provides another technique for georeferencing of satellite linear array stereo images. Quaternion algebra was invented by W.R.Hamilton in 1843, and it can be used to represent space orientation and rotations of space vectors very expediently. Quaternion can avoid the singularity may aroused by using Euler angles to describe attitude. So from the very beginning quaternion played a special role in the study of rotations in 3-dimensional vector spaces. For the pass few years, many photogrammetry scholars pay an attention to quaternion for their ascendant capability, and many photogrammetry algorithms are presented for processing close-range and aerial images. Jiang (2) proposed an algorithm of space resection based on unit quaternion; experiment indicated that this algorithm can obtain the EOPs independent on initial values. Wang (3) used Jiang's method in bundle block adjustment of aerial images. Guan (4) also presented a quaternion space resection which is similarity to triangular pyramid, and this method is not suit to block aerotriangulation. Liu (5) used quaternion to model the eccentric angle of IMU, and introduced it into the orientation image adjustment of airborne three-line CCD image supported by GPS/IMU. Experimental shows that the adjustment accuracy of quaternion is as much as or even better than that of the conventional method, and can avoid a great deal of computation of trigonometric functions. Zhao (6) presented a solution of 3D space similarity transformation model adjustment algorithm, which is based on quaternion rotation matrix. Some tests show that quaternion rotation matrix used in photogrammetry has a very clear superiority. Ji (7) detailedly studied the application of quaternion in aerial photogrammetry, and relative orientation and bundle block adjustment tests were done using actual aerial images. Peng (8) used quaternion for the general digital camera calibration, experimental results show that inner orientation and distortion parameters can be easily obtained using quaternion even when there were larger rotation angle between ground coordinate system and virtual photo coordinate. Tan (9) discussed the construction of rotation matrix in close- rang photogrammetry, and compared with Euler angle for the advantages and disadvantages. The test proved that the quaternion method needs fewer times in iteration and has steady result in calculation. Jiang (10) also tried to use quaternion in exterior orientation of linear array stereo images. All these show that quaternion can be used in photogrammetry very well.

A ring-shaped low-frequency resonator operating in the (2,1) in-plane vibration mode was designed and fabricated by anodic bonding of a 9-μm-thick single-crystal silicon layer to a glass substrate. Although the gap between the ring and the electrodes was relatively large (900 nm), a high quality factor of 4212 was realized at 1.609 MHz. Further, the motional resistance was 1.853 MΩ. In addition, the resonant frequency could be electrically tuned by varying the dc bias of the drive electrodes at a rate of 21.5 ppm/V. Thus, this resonator is expected to replace the conventional quartz resonators for use at low frequencies of a few MHz.

ABSTRACTMagnetoelectric properties of crystals having an inhomogeneous spin density in the unit cell of the crystal lattice considered. The quadrupole moment and the toroidal vector adopted as an indication of a heterogeneity. It is assumed that magnetoelectric effect in crystals is linearly related to these characteristics. Using the theory of group representations, a list of black-white groups obtained, in which magnetoelectric effect is absent. The method of finding the nonzero component of polarization and magnetization vectors developed for all groups of magnetoelectrics, when external electric and magnetic fields were applied. The list of possible phase transitions with the formation of polarization and magnetization in the crystal obtained.

We show how to perform exact diagonalizations of SU(N) Fermi-Hubbard models on L-site clusters separately in each irreducible representation (irrep) of SU(N). Using the representation theory of the unitary group U(L), we demonstrate that a convenient orthonormal basis, on which matrix elements of the Hamiltonian are very simple, is given by the set of semistandard Young tableaux (or, equivalently, the Gelfand-Tsetlin patterns) corresponding to the targeted irrep. As an application of this color factorization, we study the robustness of some SU(N) phases predicted in the Heisenberg limit upon decreasing the on-site interaction U on various lattices of size L≤12 and for 2≤N≤6. In particular, we show that a long-range color ordered phase emerges for intermediate U for N=4 at filling 1/4 on the triangular lattice.