Reflection Equation Research Articles

Spectral Properties and Optimality for Elementary Matrices

A generalization of the Householder transformation, renamed as elementary matrix by A.S. Householder: Unitary transformation of a nonsymmetric matrix, J. ACM, 5(4), 339–342, 1958, was introduced by LaBudde (Math Comput 17(84):433–437, 1963) as a tool to obtain a tridiagonal matrix similar to a given square matrix. Some of the free parameters of the transformation can be chosen to attain better numerical properties. In this work, we study the spectral properties of the transformation. We also propose a special choice for free coefficients of that transformation to minimize its condition number. The transformation with such suitable choice of parameters is called optimal.

Dual pairs of quantum moment maps and doubles of Hopf algebras

Given a dual pair of topological Hopf algebras A,A⁎, under mild conditions there exists a natural associative algebra homomorphism D(A)→H(A) between the corresponding Drinfeld double D(A) and Heisenberg double H(A). We construct this homomorphism using a pair of commuting quantum moment maps, and then use it to provide a homomorphism of certain reflection equation algebras. We also explain how the quantization of the Grothendieck–Springer resolution arises in this context.

Diagonal reduction algebra and the reflection equation

We describe the diagonal reduction algebra D(gl n ) of the Lie algebra gl n in the R-matrix formalism. As a byproduct we present two families of central elements and the braided bialgebra structure of D(gl n ).

Generalized Yangians and their Poisson counterparts

By a generalized Yangian we mean a Yangian-like algebra of one of two classes. One of these classes consists of the so-called braided Yangians, introduced in our previous paper. The braided Yangians are in a sense similar to the reflection equation algebra. The generalized Yangians of second class, called the Yangians of RTT type, are defined by the same formulae as the usual Yangians are but with other quantum $R$-matrices. If such an $R$-matrix is the simplest trigonometrical $R$-matrix, the corresponding Yangian of RTT type is the so-called q-Yangian. We claim that each generalized Yangian is a deformation of the commutative algebra ${\rm Sym}(gl(m)[t^{-1}])$ provided that the corresponding $R$-matrix is a deformation of the flip. Also, we exhibit the corresponding Poisson brackets.

Seismic inversion for geologic fractures and fractured media

Amplitude variation with offset (AVO) inversion attempts to use the available surface seismic data to estimate the density, P-wave velocity, and S-wave velocity of the earth model. Under linear slip interface theory, synthetic seismograms for models with fractures prove that fractures are also reflection generators. Consequently, observed reflections are not necessarily due to lithologic variations only, but they could be due in part to the effect of fractures. To obtain approximate equations for AVO inversion for fractured media, denoted by AVO with fracture (AVOF), we derived new equations for PP-wave reflection and transmission coefficients that are based on nonwelded contact boundary conditions. In particular, along with the fracture compliances, azimuth has also been taken into account in the equations because the fractures can have any orientation. The new approximate AVOF equations for a horizontally fractured medium with impedance contrast are developed by simplifying the equations for the new PP-wave reflection and transmission coefficients. In the new approximate AVOF equations, the reflection coefficients are divided into a welded contact part (a conventional impedance contrast part) and a nonwelded contact part (a fracture part). This makes the equations flexible enough to separately invert for the rock properties of the fracture and the background medium in the case of a fractured medium with impedance contrast. The new approximate AVOF equations state that fractures could cause the seismic reflectivity to be frequency dependent, and that the fractures not only influence the wave amplitude but also change the wave phase. The linear least-squares and nonlinear conjugate gradient inversion algorithms are applied to estimate the elastic reflectivity using the new approximate AVOF equations. The inverted results for seismic data for a horizontally fractured medium with impedance contrast are evaluated to find a more accurate delineation of the subsurface rock properties.

Diagonalization of complex symmetric matrices: Generalized Householder reflections, iterative deflation and implicit shifts

We describe a matrix diagonalization algorithm for complex symmetric (not Hermitian) matrices, A̲=A̲T, which is based on a two-step algorithm involving generalized Householder reflections based on the indefinite inner product 〈u̲,v̲〉∗=∑iuivi. This inner product is linear in both arguments and avoids complex conjugation. The complex symmetric input matrix is transformed to tridiagonal form using generalized Householder transformations (first step). An iterative, generalized QL decomposition of the tridiagonal matrix employing an implicit shift converges toward diagonal form (second step). The QL algorithm employs iterative deflation techniques when a machine-precision zero is encountered “prematurely” on the super-/sub-diagonal. The algorithm allows for a reliable and computationally efficient computation of resonance and antiresonance energies which emerge from complex-scaled Hamiltonians, and for the numerical determination of the real energy eigenvalues of pseudo-Hermitian and PT-symmetric Hamilton matrices. Numerical reference values are provided. Program summaryProgram Title: HTDQLSProgram Files doi:http://dx.doi.org/10.17632/x24wjxtrsg.1Licensing provisions: GPLv3Programming language: Fortran 90 using fixed form notationNature of problem: Calculating the eigenvalues and optionally the eigenvectors of complex symmetric (non-Hermitian), densely populated matrices.Solution method: The complex symmetric (not Hermitian) input matrix is diagonalized in two steps. First step: The matrix is tridiagonalized via a series of (n−2) generalized Householder reflections, where n is the rank of the input matrix. Second step: The tridiagonal matrix is diagonalized via a generalization of the “chasing the bulge” technique, which is an iterative process utilizing an implicitly shifted initial rotation followed by (n−2) Givens rotations. This technique is an implementation of QL factorization, and converges roughly as [(λi−σi)∕(λi+1−σi)]j where λi is the eigenvalue located in the (i,i) position of the final diagonal matrix and the eigenvalues are ordered (|λ1|<|λ2|<…<|λn|), and j is the iteration. The “educated guess” σi for the eigenvalue λi is obtained from the analytic determination of the eigenvalues of (k×k)-submatrices of A̲, in the vicinity of the ith element, where k=0,1,2,3 (here, k=0 means that the implicit shift vanishes, σi=0). The routine optionally calculates the rotation matrix Z̲, such that Z̲−1A̲Z̲=D̲ where A̲ is the input matrix and D̲ is the diagonal matrix containing the eigenvalues. The ith column of Z̲ then is the eigenvector of A̲ corresponding to the eigenvalue found at the element D̲(i,i), in the ith position on the diagonal of the matrix D̲.Unusual features:For simplicity, the “wrapper” program which contains an example application and the HTDQLS routine are distributed in the same file.

Weaving for tourists in Chinchero, Peru

This article examines the effects that ‘performing’ weaving for the tourists is having on the organization of textile production for sale in the Peruvian Quechua-speaking town of Chinchero. Recent dynamics of tourism in town have encouraged the transformation of households into weaving centres for textile exhibitions, a transformation that has substantially altered the relations between weavers and weaving, as well as with the production mode in a context of hierarchical control. Analysis shows how artisan centres that have been set up to demonstrate weaving techniques to tourists do not work as a location of production, so that production needs to be organized externally. In the wake of this transition and of the social changes introduced, household economies have been forced to redefine the terms of their articulation with the wider community organization. In relation to this point, the article explores the tensions that arise when non-family members are taken to work within the household.

Sparse kernel minimum squared error using Householder transformation and givens rotation

Two obvious limitations exist for baseline kernel minimum squared error (KMSE): lack of sparseness of the solution and the ill-posed problem. Previous sparse methods for KMSE have overcome the second limitation using a regularization strategy, which introduces an increase in the computational cost to determine the regularization parameter. Hence, in this paper, a constructive sparse algorithm for KMSE (CS-KMSE) and its improved version (ICS-KMSE) are proposed which will simultaneously address the two limitations described above. CS-KMSE chooses the training samples that incur the largest reductions on the objective function as the significant nodes on the basis of the Householder transformation. In contrast with CS-KMSE, there is an additional replacement mechanism using Givens rotation in ICS-KMSE, which results in ICS-KMSE giving better performance than CS-KMSE in terms of sparseness. CS-KMSE and ICS-KMSE do not require the regularization parameter at all before they begin to choose significant nodes, which is beneficial since it saves on the model selection time. More importantly, CS-KMSE and ICS-KMSE terminate their procedures with an early stopping strategy that acts as an implicit regularization term, which avoids overfitting and curbs the sparse level on the solution of the baseline KMSE. Finally, in comparison with other algorithms, both ICS-KMSE and CS-KMSE have superior sparseness, and extensive comparisons confirm their effectiveness and feasibility.

Diagonally and antidiagonally symmetric alternating sign matrices of odd order

We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang–Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbins from the mid 1980's that the total number of (2n+1)×(2n+1) DASASMs is ∏i=0n(3i)!(n+i)!, and a conjecture of Stroganov from 2008 that the ratio between the numbers of (2n+1)×(2n+1) DASASMs with central entry −1 and 1 is n/(n+1). Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980's, that for odd-order DASASMs is the last to have been proved.

Hardware Implementation of Novel Symbol Detection Algorithm for 4G LTE Downlink Receiver

This paper presents novel very large scale of integration implementation of symbol detection based on modified householder transformation QR decomposition algorithm for 4th generation long term evolution downlink receiver. QR decomposition of a channel matrix H is decomposition of matrix H into an orthogonal matrix Q and an upper triangular matrix R. Pre-processing modules based on QRD makes the symbol detection in signal processing easier. Implementing symbol detection with QRD in multiple input multiple output-orthogonal frequency division multiplexing based LTE downlink receiver helps to reduce the complexity of that receiver. Because of simplicity and numerical stability, the householder QRD algorithm is considered and modified to reduce the computation time and hardware area of the QRD block compared to the existing householder algorithm. In order to show the efficiency, the existing and proposed modified householder transformation algorithms are designed, simulated using Xilinx ISE 14.1 and synthesized in FPGA Virtex6 xc6vlx550tl-1Lff1759 device.

Model Parameterization and P-wave AVA Direct Inversion for Young’s Impedance

AVA inversion is an important tool for elastic parameters estimation to guide the lithology prediction and “sweet spot” identification of hydrocarbon reservoirs. The product of the Young’s modulus and density (named as Young’s impedance in this study) is known as an effective lithology and brittleness indicator of unconventional hydrocarbon reservoirs. Density is difficult to predict from seismic data, which renders the estimation of the Young’s impedance inaccurate in conventional approaches. In this study, a pragmatic seismic AVA inversion approach with only P-wave pre-stack seismic data is proposed to estimate the Young’s impedance to avoid the uncertainty brought by density. First, based on the linearized P-wave approximate reflectivity equation in terms of P-wave and S-wave moduli, the P-wave approximate reflectivity equation in terms of the Young’s impedance is derived according to the relationship between P-wave modulus, S-wave modulus, Young’s modulus and Poisson ratio. This equation is further compared to the exact Zoeppritz equation and the linearized P-wave approximate reflectivity equation in terms of P- and S-wave velocities and density, which illustrates that this equation is accurate enough to be used for AVA inversion when the incident angle is within the critical angle. Parameter sensitivity analysis illustrates that the high correlation between the Young’s impedance and density render the estimation of the Young’s impedance difficult. Therefore, a de-correlation scheme is used in the pragmatic AVA inversion with Bayesian inference to estimate Young’s impedance only with pre-stack P-wave seismic data. Synthetic examples demonstrate that the proposed approach is able to predict the Young’s impedance stably even with moderate noise and the field data examples verify the effectiveness of the proposed approach in Young’s impedance estimation and “sweet spots” evaluation.

Dependence of surface plasmon-phonon-polariton in 4 H-SiC on free carrier concentration

In this paper, we present a thorough study of the characteristics of the surface modes that result from coupling between plasmon electronic oscillation modes, phonon modes, and electromagnetic modes. The Fourier transform of p-polarized reflectivity measurements were carried out on different 4 H-SiC epilayers differing in their free carrier concentration. The reflectivity measurements were performed with appropriate care to record reflectivity spectra averaged over a wide range of incidence angles. The complex infrared dielectric functions of the measured samples were determined by correcting the values obtained from the conventional Kramers-Kronig conversion technique with reference to Fresnel equations for reflectivity. The obtained dielectric functions were used to compute the effect of the free carrier concentration and the resulting plasmon electronic oscillation on the dispersion spectrum, lifetime, mean propagation length of the bulk, and surface plasmon-phonon-polariton modes. The effect of the free carrier concentration on the temporal coherence of surface plasmon-phonon-polariton is investigated, showing a potential practical method for enhancing the temporal coherence of SiC based thermal sources.

3D multi-directional sensor with pyramid mirror and structured light

Structured-light-based 3D multi-directional sensors are widely used due to their flexible field of view (FOV), which is a significant benefit. Nevertheless, existing 3D multi-directional sensors based on linear lasers are only capable of measuring a slice of a 3D scene at low resolution. In addition, other 3D multi-directional sensors are capable of measuring 3D scenes by shooting 2D structured light patterns in different directions through curved mirrors. However, the projected and captured structured light patterns are blurred due to the optical aberration property of curved mirrors, resulting in inaccurate 3D reconstruction. This paper proposes a 3D, high-resolution, high-accuracy, real-time multi-directional sensor that utilizes pyramid mirrors to decompose the FOVs into different directions and reduce the optical aberration phenomenon. The mathematical model of the 3D sensor is presented, and the FOV is analyzed. A dual-frequency phase-shifting fringe pattern is used for 3D reconstruction in real time. Moreover, the sensor is calibrated using a planar circle calibration gauge and the Householder reflection constraint. Finally, the experimental results demonstrate that the measurement error of the 3D multi-directional sensor is less than 1mm and 0.03rad, thus verifying the method's feasibility.

Euler-Richardson method preconditioned by weakly stochastic matrix algebras: a potential contribution to Pagerank computation

Let S be a column stochastic matrix with at least one full row. Then S describes a Pagerank-like random walk since the computation of the Perron vector x of S can be tackled by solving a suitable M-matrix linear system Mx = y, where M = I − τ A, A is a column stochastic matrix and τ is a positive coefficient smaller than one. The Pagerank centrality index on graphs is a relevant example where these two formulations appear. Previous investigations have shown that the Euler- Richardson (ER) method can be considered in order to approach the Pagerank computation problem by means of preconditioning strategies. In this work, it is observed indeed that the classical power method can be embedded into the ER scheme, through a suitable simple preconditioner. Therefore, a new preconditioner is proposed based on fast Householder transformations and the concept of low complexity weakly stochastic algebras, which gives rise to an effective alternative to the power method for large-scale sparse problems. Detailed mathematical reasonings for this choice are given and the convergence properties discussed. Numerical tests performed on real-world datasets are presented, showing the advantages given by the use of the proposed Householder-Richardson method.

Derivatives in noncommutative calculus and deformation property of quantum algebras

The aim of the paper is twofold. First, we introduce analogs of (partial) derivatives on certain noncommutative algebras, including some enveloping algebras and their “braided counterparts” – the so-called modified Reflection Equation algebras. With the use of the mentioned derivatives we construct an analog of the de Rham complex on these algebras. Second, we discuss deformation property of some quantum algebras and show that contrary to a commonly held view, in the so-called q-Witt algebra there is no analog of the PBW property. In this connection, we discuss different forms of the Jacobi condition related to quadratic-linear algebras.

Polynomial amplitude versus azimuth inversion in horizontally transverse isotropic media, as tested on fractured coal seams in the Surat Basin

A new technique of amplitude versus azimuth (AVAZ) seismic inversion in horizontally transverse isotropic (HTI) media is presented. AVAZ is an effective method of characterising anisotropic variation within individual reflectors as well as characterising fractures. The compressional wave reflectivity equation in HTI media has been reformulated into a parabolic form that allows for fast and efficient inversion. The isotropic component of the azimuthal reflectivity has been separated precisely from the anisotropic component and the anisotropic component has been decoupled exactly into its constituent elliptic and anelliptic components. The exact isotropic, elliptic and anelliptic amplitude versus offset (AVO) gradient equations in HTI media are presented herein and the amount of error associated with previous approximations is also defined under the assumption of weak anisotropy. A method of calculating Thomsen’s weak anisotropy parameters using these AVO gradient terms is then outlined. Compared with the elliptic method, there is reduced error incorporated in the new AVAZ method and the error relationships of this method are compared with the Fourier method. Data from an open file 3D wide azimuth seismic survey in the Surat Basin were inverted to demonstrate the effectiveness of the techniques which are presented herein. Seismic amplitudes from six azimuthal stacks were extracted over two horizons and inverted around a well where full-wave sonic and density logs were acquired. For both horizons, the error between the inverted anisotropy parameters from seismic and the inverted anisotropy parameters from wire line logs were found to be less than 5% for both horizons.

Volume reflection and channeling of ultrarelativistic protons in germanium bent single crystals

The paper devoted to investigation of volume reflection and channeling processes of ultrarela- tivistic positive charged particles moving in germanium single crystals. We demonstrate that the choice of atomic potential on the basis of Hartree-Fock method and correct choice of Debye tem- perature allow us to describe the above mentioned processes in a good agreement with the recent experiments. Moreover, the presented in the paper universal form of equations for volume reflection gives true description of the process at a wide range of particle energies. Standing on this study we make predictions for mean angle reflection (as a function of bending radius) of positive and negative particles for germanium (110) and (111) crystallographic planes.

Efficient Realization of Householder Transform Through Algorithm-Architecture Co-Design for Acceleration of QR Factorization

We present efficient realization of Householder Transform (HT) based QR factorization through algorithm-architecture co-design where we achieve performance improvement of 3-90x in-terms of Gflops/watt over state-of-the-art multicore, General Purpose Graphics Processing Units (GPGPUs), Field Programmable Gate Arrays (FPGAs), and ClearSpeed CSX700. Theoretical and experimental analysis of classical HT is performed for opportunities to exhibit higher degree of parallelism where parallelism is quantified as a number of parallel operations per level in the Directed Acyclic Graph (DAG) of the transform. Based on theoretical analysis of classical HT, an opportunity re-arrange computations in the classical HT is identified that results in Modified HT (MHT) where it is shown that MHT exhibits 1.33x times higher parallelism than classical HT. Experiments in off-the-shelf multicore and General Purpose Graphics Processing Units (GPGPUs) for HT and MHT suggest that MHT is capable of achieving slightly better or equal performance compared to classical HT based QR factorization realizations in the optimized software packages for Dense Linear Algebra (DLA). We implement MHT on a customized platform for Dense Linear Algebra (DLA) and show that MHT achieves 1.3x better performance than native implementation of classical HT on the same accelerator. For custom realization of HT and MHT based QR factorization, we also identify macro operations in the DAGs of HT and MHT that are realized on a Reconfigurable Data-path (RDP). We also observe that due to re-arrangement in the computations in MHT, custom realization of MHT is capable of achieving 12% better performance improvement over multicore and GPGPUs than the performance improvement reported by General Matrix Multiplication (GEMM) over highly tuned DLA software packages for multicore and GPGPUs which is counter-intuitive.

Fast Orthonormal Sparsifying Transforms Based on Householder Reflectors

Dictionary learning is the task of determining a data-dependent transform that yields a sparse representation of some observed data. The dictionary learning problem is non-convex, and usually solved via computationally complex iterative algorithms. Furthermore, the resulting transforms obtained generally lack structure that permits their fast application to data. To address this issue, this paper develops a framework for learning orthonormal dictionaries which are built from products of a few Householder reflectors. Two algorithms are proposed to learn the reflector coefficients: one that considers a sequential update of the reflectors and one with a simultaneous update of all reflectors that imposes an additional internal orthogonal constraint. The proposed methods have low computational complexity and are shown to converge to local minimum points which can be described in terms of the spectral properties of the matrices involved. The resulting dictionaries balance between the computational complexity and the quality of the sparse representations by controlling the number of Householder reflectors in their product. Simulations of the proposed algorithms are shown in the image processing setting where well-known fast transforms are available for comparisons. The proposed algorithms have favorable reconstruction error and the advantage of a fast implementation relative to the classical, unstructured, dictionaries.

Algebraic approach to form factors in the complex sinh-Gordon theory

We study form factors of the quantum complex sinh-Gordon theory in the algebraic approach. In the case of exponential fields the form factors can be obtained from the known form factors of the $Z_N$-symmetric Ising model. The algebraic construction also provides an Ansatz for form factors of descendant operators. We obtain generating functions of such form factors and establish their main properties: the cluster factorization and reflection equations.