Reflection Equation Research Articles

Zernike like functions on spherical cap: principle and applications in optical surface fitting and graphics rendering

Zernike circular polynomials (ZCP) are widely used in optical testing, fabrication and adaptive optics. However, their ability to characterize information on spherical cap is often limited by aperture angle, with highly curved surface being particularly challenging. In this study, we propose a simple and systematic process to derive Zernike like functions that are applicable to all types of spherical cap. The analytical expressions of three function sets are calculated using Gram-Schmidt algorithm. They are hemispherical harmonics (HSH), Zernike spherical function (ZSF) and longitudinal spherical function (LSF). HSH satisfies Laplacian equation and composes a subset of spherical harmonics (SH). ZSF and LSF can be applied to arbitrary spherical cap and their orthogonality is invariant to aperture. The achieved functions, with their complete and orthogonal performance, can serve as essential tools for surface characterization required for a wide range of applications like large-angle lenses description in illumination design, aberration analysis in high aperture systems, human cornea measurement fitting, geomagnetic field modelling, etc. Moreover, they are important for graphics rendering in virtual reality and games by solving the reflectance equation efficiently.

Computing the Moore-Penrose Inverse for Bidiagonal Matrices

The Moore-Penrose inverse is the most popular type of matrix generalized inverses which has many applications both in matrix theory and numerical linear algebra. It is well known that the Moore-Penrose inverse can be found via singular value decomposition. In this regard, there is the most effective algorithm which consists of two stages. In the first stage, through the use of the Householder reflections, an initial matrix is reduced to the upper bidiagonal form (the Golub-Kahan bidiagonalization algorithm). The second stage is known in scientific literature as the Golub-Reinsch algorithm. This is an iterative procedure which with the help of the Givens rotations generates a sequence of bidiagonal matrices converging to a diagonal form. This allows to obtain an iterative approximation to the singular value decomposition of the bidiagonal matrix. The principal intention of the present paper is to develop a method which can be considered as an alternative to the Golub-Reinsch iterative algorithm. Realizing the approach proposed in the study, the following two main results have been achieved. First, we obtain explicit expressions for the entries of the Moore-Penrose inverse of bidigonal matrices. Secondly, based on the closed form formulas, we get a finite recursive numerical algorithm of optimal computational complexity. Thus, we can compute the Moore-Penrose inverse of bidiagonal matrices without using the singular value decomposition.

Quantum circuits synthesis using Householder transformations

The synthesis of a quantum circuit consists in decomposing a unitary matrix into a series of elementary operations. In this paper, we propose a circuit synthesis method based on the QR factorization via Householder transformations. We provide a two-step algorithm: during the first step we exploit the specific structure of a quantum operator to compute its QR factorization, then the factorized matrix is used to produce a quantum circuit. We analyze several costs (circuit size and computational time) and compare them to existing techniques from the literature. For a final quantum circuit twice as large as the one obtained by the best existing method, we accelerate the computation by orders of magnitude.

The center of the reflection equation algebra via quantum minors

We give simple formulas for the elements ck appearing in a quantum Cayley-Hamilton formula for the reflection equation algebra (RE algebra) associated to the quantum group Uq(glN), answering a question of Kolb and Stokman. The ck's are certain canonical generators of the center of the RE algebra, and hence of Uq(glN) itself; they have been described by Reshetikhin using graphical calculus, by Nazarov-Tarasov using quantum Yangians, and by Gurevich, Pyatov and Saponov using quantum Schur functions; however no explicit formulas for these elements were previously known.As byproducts, we prove a quantum Girard-Newton identity relating the ck's to the so-called quantum power traces, and we give a new presentation for the quantum group Uq(glN), as a localization of the RE algebra along certain principal minors.

Analysis of Bragg fiber waveguides having a defect layer for biosensing application

A novel hollow core Bragg fiber waveguides having a defect layer are proposed and analyzed theoretically for sensing application. Matching the electric and magnetic fields at various interfaces, a relation between fields of first layer with final layer has been stabilized; hence, the equations for reflectance and transmittance of proposed structure are derived. Due to periodicity of concentric cylindrical structure, a perfect photonic band gap is observed in considered wavelength range. The presence of defect layer in cylindrical periodic structure shows a peak corresponding to defect mode in perfect photonic band gap region. The full width at half maxima of this peak depends on the periodicity of the cladding layers. Also, the spectral position and shift of peak of defect mode depend on the angle of incidence of light, refractive index of core material and design wavelength of structure. Therefore, it is more appropriate to consider this peak of defect mode as sensing signal instead of considering whole photonic band gap as sensing signal.

‘From Rubber to Oil Palm’: Livelihood Structural Transformation of Local and Transmigrant Farmer Households in Minangkabau

This study aims to analyze the transformation of livelihood structures in local and transmigrant farmer households that occur due to the entry of oil palm. Oil palm has become a new agricultural commodity that it is believed to provide better income for farmers. This research was conducted with a quantitative and qualitative approach. Quantitative data collection was carried out through a survey of 63 farm households. Meanwhile, qualitative data collection was carried out through in-depth interviews. The results of the study indicate that the transformation of rubber commodities to oil palm in general supports the economy of farmer households, which are income increases, livelihood diversity, and welfare increases. In addition, the transformation also has an impact on consumptive and materialistic lifestyle changes in farm households and the formation of farmer household typologies based on post-transformation livelihoods.

Reflection matrices associated with an Onsager coideal of and

We determine the intertwiners of a family of Onsager coideal subalgebras of the quantum affine algebra in the fundamental representations and in the spin representations. They reproduce the reflection K matrices obtained recently by the matrix product construction connected to the three dimensional integrability. In particular the present approach provides the first proof of the reflection equation for the non type A cases.

Iterative solution and preconditioning for the tangent plane scheme in computational micromagnetics

The tangent plane scheme is a time-marching scheme for the numerical solution of the nonlinear parabolic Landau–Lifshitz–Gilbert equation, which describes the time evolution of ferromagnetic configurations. Exploiting the geometric structure of the equation, the tangent plane scheme requires only the solution of one linear variational form per time-step, which is posed in the discrete tangent space determined by the nodal values of the current magnetization. We develop an effective solution strategy for the arising constrained linear systems, which is based on appropriate Householder reflections. We derive possible preconditioners, which are (essentially) independent of the time-step, and prove linear convergence of the preconditioned GMRES algorithm. Numerical experiments underpin the theoretical findings.

A topological origin of quantum symmetric pairs

It is well known that braided monoidal categories are the categorical algebras of the little two-dimensional disks operad. We introduce involutive little disks operads, which are {mathbb {Z}}_2-orbifold versions of the little disks operads. We classify their categorical algebras and describe these explicitly in terms of a finite list of functors, natural isomorphisms and coherence equations. In dimension two, the categorical algebras are braided monoidal categories with an anti-involution together with a pointed module category carrying a universal solution to the (twisted) reflection equation. Examples of these are obtained from the categories of representations of a ribbon Hopf algebra with an involution and a quasi-triangular coideal subalgebra, such as a quantum group and a quantum symmetric pair coideal subalgebra.

Bethe Subalgebras in Braided Yangians and Gaudin-Type Models

In \cite{GS1} the notion of braided Yangians of Reflection Equation type was introduced. Each of these algebras is associated with an involutive or Hecke symmetry $R$. Besides, the quantum analogs of certain symmetric polynomials (elementary symmetric ones, power sums) were suggested. In the present paper we show that these quantum symmetric polynomials commute with each other and consequently generate a commutative Bethe subalgebra. As an application, we get some Gaudin-type models and the corresponding Bethe subalgebras.

Household disaster management capacities in disaster prone II area of Mt. Slamet.

Disaster prone II in Mt. Slamet, Indonesia presents the highest risk for human settlement. To live in this natural disaster-prone area, specific household characteristics are essential. Household capitals and transformation in process and structure were supported by the disaster management framework. However, households in disaster prone II area had limited assets and were required to identify factors influencing disaster management. To study the factors influencing household disaster management capacities, this research, using the sample measurement of Becker and Hursh-Cesar, collected data of 538 households spread across five villages in the disaster prone II area of Mt. Slamet. Sequential mixed methodology combining both qualitative and quantitative research methods were used: samples in the Rukun-Warga-level area were collected by a two-stage stratified random sampling, and to choose the sample of households systematic random sampling was employed. Path analysis through Stata was carried out to analyse the direct and indirect factors supporting disaster management capacity, and multicollinearity was tested before path analysis. This research found direct and indirect effects of household characteristics and household capitals on disaster management. This could be influenced by the transformation in process and the structure of the local government. The quantitative result has been confirmed by the result of the qualitative methodology. Social capital owned by households in disaster-prone area supports disaster management practices. The household relationship and networking access has been strongly supported by disaster management capacities. Disaster management capacities of households in disaster prone II areas could be improved by both internal and external factors. Internal factors include supporting the household members’ health and increasing the size of land and vehicle owning. Meanwhile, external factors has been applied by the policy published by government as to improve the social and cultural belief of households.

Actions of skew braces and set-theoretic solutions of the reflection equation

AbstractA skew brace, as introduced by L. Guarnieri and L. Vendramin, is a set with two group structures interacting in a particular way. When one of the group structures is abelian, one gets back the notion of brace as introduced by W. Rump. Skew braces can be used to construct solutions of the quantum Yang–Baxter equation. In this article, we introduce a notion of action of a skew brace, and show how it leads to solutions of the closely associated reflection equation.

Boundary matrices for the higher spin six vertex model

In this paper we consider solutions to the reflection equation related to the higher spin stochastic six vertex model. The corresponding higher spin R-matrix is associated with the affine quantum algebra Uq(sl(2)ˆ). The explicit formulas for boundary K-matrices for spins s=1/2,1 are well known. We derive difference equations for the generating function of matrix elements of the K-matrix for any spin s and solve them in terms of hypergeometric functions. As a result we derive the explicit formula for matrix elements of the K-matrix for arbitrary spin. In the lower- and upper- triangular cases, the K-matrix simplifies and reduces to simple products of q-Pochhammer symbols.

Integrable Matrix Product States from boundary integrability

We consider integrable Matrix Product States (MPS) in integrable spin chains and show that they correspond to “operator valued” solutions of the so-called twisted Boundary Yang-Baxter (or reflection) equation. We argue that the integrability condition is equivalent to a new linear intertwiner relation, which we call the “square root relation”, because it involves half of the steps of the reflection equation. It is then shown that the square root relation leads to the full Boundary Yang-Baxter equations. We provide explicit solutions in a number of cases characterized by special symmetries. These correspond to the “symmetric pairs” (SU(N),SO(N)) and (SO(N),SO(D)\otimes⊗SO(N-D)), where in each pair the first and second elements are the symmetry groups of the spin chain and the integrable state, respectively. These solutions can be considered as explicit representations of the corresponding twisted Yangians, that are new in a number of cases. Examples include certain concrete MPS relevant for the computation of one-point functions in defect AdS/CFT.

Computation of the vertical and horizontal polarizations of brightness temperature of flat surface water over the Persian Gulf at the L-Band

The brightness temperature is the principle parameter detected by passive microwave radiometers. The radio frequency interference at L-band radiometers (e.g., SMOS, Aquarius and SMAP) impacts the quality of brightness temperature detection in many parts of the world such as the Middle East and the Persian Gulf. In the present work, vertical and horizontal polarizations of brightness temperature over flat surface water of Persian Gulf at L-band were calculated by using physical computations and an empirical model. For this purpose, Rayleigh–Jeans radiation law and Fresnel reflection equations were used and complex permittivity was calculated by Blanch and Aguasca model. Input data for the model calculations are temperature and salinity that were provided from World Ocean Atlas 2013. The calculations showed that the brightness temperature distribution in the Persian Gulf experiences significant spatial and seasonal variations. At nadir incidence angle, the vertical and horizontal components of brightness temperature over the Persian Gulf vary in the range 90.5–96.5 °k and 85.5–90.2 °k, respectively. At off-nadir incidence angle, the temporal variability pattern of brightness temperature horizontal polarization is similar to its vertical counterpart but the difference between them significantly increases.

MODEL OF A MATRIX CROSSCORRELATION FUNCTION OF THE PROBING AND REFLECTED VECTOR SIGNALS FOR A CONCEPTUAL DESIGN OF A SYNTHETIC APERTURE RADAR ON AN AERIAL CARRIER

In order to simulate the process of design development in full on computer models, including virtual tests of the synthetic aperture radar on an air carrier in model media, the study develops a structural scheme of the conceptual design of the synthetic aperture radar on an air carrier. The scheme is invariant with respect to the type of an air carrier with a synthetic aperture radar: an aircraft, a helicopter, an unmanned aerial vehicle and similar ones: an air carrier "enters" it by only an automatic control system, a model of trajectory instabilities and a spectrum of frequencies of elastic oscillations of its design. To perform a computer simulation of radar systems with full polarization sensing, a model of a matrix cross-correlation function of probing and reflected vector signals is proposed. As a model of the scattering object, a set of independent point reflectors distributed over space and generally having different rates of motion is accepted. The reflected signal is a sum of elementary signals, their form completely repeats the shape of the emitted signal, and the amplitude, the phase and polarization are respectively determined by the coordinate, velocity and polarization parameters of elementary reflectors forming a spatially extended object. Taking into account the developed models for the formation of the vector sounding signal and the matrix response function of the distributed radar object, a block-diagram of the model of the matrix cross-correlation function of the emitted and reflected vector signals is proposed. A block-diagram is the basis for the development of an algorithm and a program for computer modeling of the primary signal processing in a radar station with full polarization sensing.

Matrix product solution to the reflection equation associated with a coideal subalgebra of $$\varvec{U_q(A^{(1)}_{n-1})}$$ U q ( A n - 1 ( 1 ) )

We present a new solution to the reflection equation associated with a coideal subalgebra of $U_q(A^{(1)}_{n-1})$ in the symmetric tensor representations and their dual. Elements of the $K$ matrix are expressed by a matrix product formula involving terminating $q$-hypergeometric series in $q$-boson generators. At $q=0$, our result reproduces a known set-theoretical solution to the reflection equation connected to the crystal base theory.

Reflections Over the Dual Ring—Applications to Kinematic Synthesis

Least-square problems arise in multiple application areas. The numerical algorithm intended to compute offline the minimum (Euclidian)-norm approximation to an overdetermined system of linear equations, the core of least squares, is based on Householder reflections. It is self-understood, in the application of this algorithm, that the coefficient matrix is dimensionally homogeneous, i.e., all its entries bear the same physical units. Not all applications lead to such matrices, a case in point being parameter identification in mechanical systems involving rigid bodies, whereby both rotation and translation occur; the former being dimensionless and the latter bearing units of length. Because of this feature, dual numbers have found extensive applications in these fields, as they allow the analyst to include translations within the same relations applicable to rotations, on dualization2 of the rotation equations, as occurring in the geometric, kinematic, or dynamic analyses of mechanical systems. After recalling the basic background on dual numbers and introducing reflection matrices defined over the dual ring, we obtain the dual version of Householder reflections applicable to the offline implementation of parameter identification. For the online parameter identification, recursive least squares are to be applied. We provide also the dual version of recursive least squares. Numerical examples are included to illustrate the underlying principles and algorithms.

On diagonal solutions of the reflection equation

We study solutions of the reflection equation associated with the quantum affine algebra and obtain diagonal K-operators in terms of the Cartan elements of a quotient of Uq(gl(N)). We also consider intertwining relations for these K-operators and find an augmented q-Onsager algebra like symmetry behind them.

AN ALTERNATIVE MATRIX TRANSFORMATION TO THE F TEST STATISTIC FOR CLUSTERED DATA

Abstract For the regression analysis of clustered data, the error of cluster data violates the independence assumption. Consequently, the test statistic based on the ordinary least square method leads to incorrect inferences. To overcome this issue, the transformation is required to apply to the observations. In this paper we propose an alternative matrix transformation that adjusts the intra-cluster correlation with Householder matrix and apply it to the F test statistic based on generalized least squares procedures for the regression coefficients hypothesis. By Monte Carlo simulations of the balanced and unbalanced data, it is found that the F test statistic based on generalized least squares procedures with Adjusted Householder transformation performs well in terms of the type I error rate and power of the test.