Reflection Equation Research Articles

Lax pair formulation for the open boundary Osp(1∣2) spin chain

Based on the Lax pair formulation, we study the integrable conditions of the Osp(1∣2) spin chain with open boundaries. We consider both the non-graded and graded versions of the Osp(1∣2) chain. The Lax pair operators M ± for the boundaries can be induced by the Lax operator M j for the bulk of the system. They correspond to the reflection equations (RE) and the Yang–Baxter equation, respectively. We further calculate the boundary K-matrices for both the non-graded and graded versions of the model with open boundaries. The double row monodromy matrix and transfer matrix of the spin chain have also been constructed. The K-matrices obtained from the Lax pair formulation are consistent with those from Sklyanin’s RE. This construction is another way to prove the quantum integrability of the Osp(1∣2) chain. We find that the Lax pair formulation has advantages in dealing with the boundary terms of the supersymmetric model.

Estimation of in situ stresses from PP-wave azimuthal seismic data in fracture-induced anisotropic media

Horizontally transverse isotropy (HTI) induced by vertical or subvertical aligned fractures is common for unconventional fractured porous shale oil or gas reservoirs. Compared with the unfractured rocks, the seismic response characteristics of PP-wave azimuthal amplitudes are usually disturbed by the fractures and the in situ stresses. Knowledge of fracture properties, as well as in situ stresses, is required to optimize horizontal well planning and hydraulic fracturing during production, and the seismic inversion for in situ stresses from the PP-wave azimuthal amplitude data in fracture-induced anisotropic media is an essential step. Using the linear-slip theory and the effective stress law, we derive the fluid-saturated effective elastic stiffness tensors parameterized by background elastic moduli, the effective stress coefficient of isotropic host rocks, fluid modulus, porosity, and fracture parameters based on the anisotropic Gassmann’s fluid substitution equation. Combining the perturbations in saturated stiffness tensors and scattering theory, we formulate the reflection coefficient equation of PP-wave data as a function of background porosity-related stress parameter and two (i.e., normal and shear) fracture weakness parameters. Following Bayes’ rule, we estimate the porosity-related stress parameter and fracture weaknesses using the inversion method of azimuthal Fourier coefficients. Finally, we compute the effective horizontal and vertical in situ stresses using the estimated elastic moduli, porosity-related stress parameter, and fracture weaknesses. Synthetic and real data sets demonstrate that our proposed inversion approach based on the derived reflection equation provides us another way to obtain reasonable estimates of in situ stresses in a complex-fractured porous shale reservoir.

Non-parabolic conical rotations

In this paper, we determine non-parabolic conical motions that occur on any given ellipse or hyperbola without using affine transformations. To achieve this aim, first, we define a generalized inner product whose circle is the given ellipse or hyperbola, and then determine elliptical and hyperbolic versions of skew-symmetric and orthogonal matrices using the associated inner product. Finally, we generate elliptical and hyperbolic versions of rotation and reflection matrices using the famous Rodrigues, Cayley, and Householder transformations. For each of the generalized formulas, we give numerical examples.

Finansallaşma ve Gelir Eşitsizliği: Türkiye ve Avrupa Birliği Üzerine Dinamik Panel Veri Analizi

Finansallaşma; finansal aktörlerin ve piyasaların öncülüğünde hanehalkı ve firmalar, ekonomiler ve devletlerin yapısal dönüşümüyle sonuçlanan bir süreci ifade etmektedir. Finansallaşmanın gelir eşitsizliği üzerindeki etkileri, finansal ve finansal olmayan kurumlar ve hanehalkı sektörü bağlamında farklı vekil değişkenler kullanılarak analiz edilebilmektedir. Bu çalışma, Avrupa Birliği ve Türkiye’de gelir eşitsizliği ve finansallaşma arasındaki ilişkiye odaklanmaktadır. Bu bağlamda, finansal kurumlar, finans dışı kurumlar ve hanehalkı sektörünün finansallaşma eğiliminin gelir eşitsizliği üzerindeki etkileri araştırılmıştır. 2000-2020 dönemine ilişkin olarak Arellano ve Bover/ Blundell ve Bond Sistem Genelleştirilmiş Momentler Tahmincisi (SYS-GMM) doğrultusunda dinamik panel veri analizi gerçekleştirilmiştir. Elde edilen sonuçlara göre, finansal sektörü temsil eden vekil değişken olan bankalar tarafından özel sektöre verilen yurt içi kredilerin GSYH içindeki payının artması ve hanehalkı sektörünü temsil eden vekil değişken olan hanehalkına ve hanehalkını destekleyen kar amacı taşımayan kuruluşlara tüm sektörler (finans ve finans dışı) tarafından verilen kredilerin GSYH içindeki payının artması gelir eşitsizliğini arttırırken, finans dışı sektöre ve hanehalkı sektörüne verilen kredilerin GSYH içindeki payının artması ise gelir eşitsizliğini gecikmeli olarak azaltmaktadır.

Universal K-matrices for quantum Kac-Moody algebras

We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra H H endowed with a universal K-matrix, i.e., a universal solution of a generalized reflection equation, yielding an action of cylindrical braid groups on tensor products of its representations. We prove that new examples of such universal K-matrices arise from quantum symmetric pairs of Kac-Moody type and depend upon the choice of a pair of generalized Satake diagrams. In finite type, this yields a refinement of a result obtained by Balagović and Kolb, producing a family of non-equivalent solutions interpolating between the quasi-K-matrix originally due to Bao and Wang and the full universal K-matrix. Finally, we prove that this construction yields formal solutions of the generalized reflection equation with a spectral parameter in the case of finite-dimensional representations over the quantum affine algebra U q L s l 2 U_qL\mathfrak {sl}_{2} .

Matrix Capelli identities related to reflection equation algebra

By using the notion of quantum double we introduce analogs of partial derivatives on a reflection equation algebra, associated with a Hecke symmetry of GLN type. We construct the matrix L=MD, where M is the generating matrix of the reflection equation algebra and D is the matrix composed of the quantum partial derivatives and prove that the matrices M, D and L satisfy a matrix identity, called the matrix Capelli one. Upon applying quantum trace, it becomes a scalar relation, which is a far-reaching generalization of the classical Capelli identity. Also, we get a generalization of the some higher Capelli identities proved by A. Okounkov in [6].

Log-Canonical Coordinates for Symplectic Groupoid and Cluster Algebras

Abstract Using Fock–Goncharov higher Teichmüller space variables we derive log-canonical coordinate representation for entries of general symplectic leaves of the $\mathcal A_n$ groupoid of upper-triangular matrices and, in a more general setting, of higher-dimensional symplectic leaves for algebras governed by the reflection equation with the trigonometric $R$-matrix. The obtained results are in a perfect agreement with the previously obtained Poisson and quantum representations of groupoid variables for $\mathcal A_3$ and $\mathcal A_4$ in terms of geodesic functions for Riemann surfaces with holes. We realize braid-group transformations for $\mathcal A_n$ via sequences of cluster mutations in the special $\mathcal A_n$-quiver. We prove the groupoid relations for normalized quantum transport matrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit. We prove the quantum algebraic relations of transport matrices for arbitrary (cyclic or acyclic) directed planar network.Dedicated to the memory of a great mathematician and person, Boris Dubrovin.

Generalized Householder Transformations.

The Householder transformation, allowing a rewrite of probabilities into expectations of dichotomic observables, is generalized in terms of its spectral decomposition. The dichotomy is modulated by allowing more than one negative eigenvalue or by abandoning binaries altogether, yielding generalized operator-valued arguments for contextuality. We also discuss a form of contextuality by the variation of the functional relations of the operators, in particular by additivity.

Bragg fiber with a doubly defect layer-based tunable multiband optical filter

A Bragg fiber waveguide having double-defect layers in its periodic cladding region is investigated and optimized for inline tunable multi-bandpass filter applications. The proposed waveguide is classified in terms of a symmetric waveguide and an asymmetric waveguide depending upon the refractive index of the defect layers. The required reflectance and confinement loss equations are obtained using the boundary matching technique and the transfer matrix method. Due to the presence of the double-defect layer, two sharp passbands in the bandgap region are observed. The separation of the passbands and their intensities can be controlled by introducing the number of unit cells between the defect layers. Our analysis shows that the passband’s intensity, bandwidth, and central wavelength can also be tuned by changing the refractive index of the core material. In addition, the central wavelength for both defect modes are blue shifted with the increase of the core refractive index. In all of our considered cases, the higher tunability with a core refractive index variation in the right defect mode is 25.38 nm/RIU for the symmetric cases having defect layer refractive indices of 1.45. Similarly, the higher tunability with a core refractive index variation in the left defect mode is 16.9 nm/RIU for the asymmetric cases having a refractive index of 1.45 in the first defect layer and 3.42 in the second defect layer. Hence, our proposed waveguide can work as a tunable multi-bandpass inline filter with a narrow bandwidth.

Solution to the reflection equation related to the ι quantum group of type AII

A solution to the reflection equation associated to a coideal subalgebra of of type AII in the symmetric tensor representations is presented. If parameters of the coideal subalgebra are suitably chosen, the K matrix does not depend on the quantum parameter q and still agrees with a solution in [8] at q = 0.

DOORWAY models in the inverse problem for a complex vibronic analogue of the fermi resonance

We solve the inverse problem for the complex Fermi resonance or its vibronic analogue, and to this end we use the matrix XEXt, where E=diag(\Ek\) is a diagonal matrix, Ek are the energies of the observed "conglomerate" of lines, and the intensities of these lines Ik determine the first row of the matrix X, (X1k)2=Ik k=1,2,...,n,n≥3. Hamiltonian matrix of the direct model, HDIR, whose parameters are the energies of pre-diagonalized "dark" states, Ai, and the matrix elements of their coupling to the "bright" state, Bi, (i=1,2,...,n-1), is obtained after the diagonalization of the XEXt block, which belongs to the "dark" states. We show that Hamiltonian matrix with the single doorway state (DW), HDW1, can be obtained from the matrices HDIR or XEXt by first step of the Householder triangularization, i.e. by similarity transformation with a reflection matrix constructed by quantities Bi or Di=(XEXt)1,i+1. For the energy of the first DW1 state, g1, and the matrix element of its coupling to the "bright" state, w1, the use of the Householder transformation gives: g1=Sigman-1i=1B2iAi/(Sigman-1j=1B2j)= Sigmank=1E3kIk/Sigmanl=1E2lIl, |w1|=(Sigman-1i=1B2i)1/2= Sigmank=1E2kIk)1/2. In similar way, using the Householder transformation, the Hamiltonians for the models with several doorway states, HDW2,HDW3,...,HDW(n-1), are successively obtained. The Hamiltonian of the DW(n-2) model is represented by a symmetric tridiagonal matrix HDW(n-1), its diagonal elements gi determine the energies of the DW1-,DW2-,...,DW(n-1) states, and the off-diagonal elements wi determine the corresponding coupling between them. Keywords: vibronic coupling, complex vibronic analogue of the Fermi resonance, inverse problem, Householder transformation, doorway models.

DOORWAY-модели в обратной задаче для сложного вибронного аналога резонанса Ферми

We solve the inverse problem for the complex Fermi resonance or its vibronic analogue, and to this end we use the matrix XEXt, where E = diag({Ek}) is a diagonal matrix, Ek are the energies of the observed “conglomerate” of lines, and the intensities of these lines Ik determine the first row of the matrix X, (X1k)2 = Ik, k = 1, 2, … , n, n  3. Hamiltonian matrix of the direct model, HDIR, whose parameters are the energies of pre-diagonalized “dark” states, Ai, and the matrix elements of their coupling to the “bright” state, Bi, (i = 1, 2, … , n – 1), is obtained after the diagonalization of the XEXt block, which belongs to the “dark” states. We show that Hamiltonian matrix with the single doorway state (DW), HDW1, can be obtained from the matrices HDIR or XEXt by first step of the Householder triangularization, i.e. by similarity transformation with a reflection matrix constructed by quantities Bi or Di = (XEXt)1,i+1. For the energy of the first DW1 state, g1, and the matrix element of its coupling to the “bright” state, w1, the use of the Householder transformation gives: g1=Sigman-1i=1B2iAi/(Sigman-1j=1B2j)= Sigmank=1E3kIk/Sigmanl=1E2lIl, |w1|=(Sigman-1i=1B2i)1/2= Sigmank=1E2kIk)1/2. In similar way, using the Householder transformation, the Hamiltonians for the models with several doorway states, HDW2, HDW3,…, HDW(n-1), are successively obtained. The Hamiltonian of the DW(n-2) model is represented by a symmetric tridiagonal matrix HDW(n-1), its diagonal elements gi determine the energies of the DW1, DW2,…, DW(n-1) states, and the off-diagonal elements wi determine the corresponding coupling between them.

Householder Transformation in the Inverse Problem for a Complex Vibronic Analog of the Fermi Resonance

Householder Transformation in the Inverse Problem for a Complex Vibronic Analog of the Fermi Resonance

Solving Block Low-Rank Matrix Eigenvalue Problems

To solve large-scale matrix eigenvalue problems (EVPs), a two-step tridiagonalization method using the block Householder transformation (HT) is often employed. Although the method based on dense matrix arithmetic requires a memory storage of O(N2) and an arithmetic operations of O(N3), in this study, these were reduced by approximating the method using block low-rank (BLR-) matrices. A special block HT for BLR-matrices and a two-step tridiagonalization method using it are proposed to solve an EVP with a real symmetric BLR-matrix. In the proposed block HT, block Householder vectors are also formed using BLR-matrices. It is demonstrated how the block size m in the BLR-matrix should be determined and confirmed that the memory and arithmetic complexities of the proposed method were O(N5/3) and O(N7/3), respectively, for typical cases when using an appropriate block size m ∝ N1/3. In numerical experiments of a string free vibration problem with known analytical solutions, for large eigenvalues, the calculated eigenvalues using the proposed method converge toward the analytical ones in accordance with the theoretical convergence curves. Owing to the reduced complexity, an EVP of a matrix was solved with about N =300,000, which is significantly larger than the limit of the conventional method with dense matrices, within a reasonable amount of time on a CPU core. For the calculation time, the proposed method was faster than the conventional method when the matrix size N was larger than a few tens of thousands.

Efficient Biometric Identification on the Cloud With Privacy Preservation Guarantee

Benefited from its reliability and convenience, biometric identification has become one of the most popular authentication technologies. Due to the sensitivity of biometric data, various privacy-preserving biometric identification protocols have been proposed. However, the low computational efficiency or the security vulnerabilities of these protocols limit their wide deployment in practice. To further improve the efficiency and enhance the security, in this paper, we propose two new privacy-preserving biometric identification outsourcing protocols. One mainly utilizes the efficient Householder transformation and permutation technique to realize the high-efficiency intention under the known candidate attack model. The other initializes a novel random split technique and combines it with the invertible linear transformation to achieve a higher security requirement under the known-plaintext attack model. Also, we argue the security of our proposed two protocols with a strict theoretical analysis and, by comparing them with the prior existing works, comprehensively evaluate their efficiency.

Density stability estimation method from pre-stack seismic data

As one of the most important elastic parameters in seismic exploration, density plays an important role in lithology identification and hydrocarbon indicator. Based on the approximate Zoeppritz equation, the density term is usually estimated using AVO/AVA (Amplitude variation with offset/incident angle) inversion. However, the small contribution of density to the reflection coefficient or the strong correlation with other parameters in the approximate equation renders the density estimation tough. By analyzing the contribution of density to reflection coefficient in different approximate equations, Aki-Richards is finally selected as the inversion equation, in which the density term has a higher contribution to the reflectivity. The conventional inversion approach is to invert several parameters simultaneously on the basis of approximate equation. The same convergence criterion is used for the estimation of the three parameters, but the density estimation error fails to converge to the minimum due to its small sensitivity, which makes density inversion unstable. Furthermore, the conventional inversion methods involve the inverse of large matrix, which also reduces the stability of density estimation. Therefore, a new form of Aki-Richards approximate reflection equation is derived on the assumption that the inverse of coefficient matrix exists. The reflectivity of elastic parameter is independently expressed as the weighted superposition of seismic reflection coefficients at different incident angles. Then utilizing well logging data and seismic reflection coefficient estimated by elastic impedance inversion as constraints, the inverse of coefficient matrix in inversion equation is estimated directly, which avoids inverse of large matrix and improves the precision of density estimation. In physical, the weighted coefficients directly reflect the contribution of elastic parameters reflectivity to seismic data. Finally, different experimental examples show that proposed method can stably estimate density term.

A reduction algorithm for reconstructing periodic Jacobi matrices in Minkowski spaces

The periodic Jacobi inverse eigenvalue problem concerns the reconstruction of a periodic Jacobi matrix from prescribed spectral data. In Minkowski spaces, with a given signature operator H=diag(1,1,…,1,−1), the corresponding matrix is a periodic pseudo-Jacobi matrix. The inverse eigenvalue problem for such matrices consists in the reconstruction of pseudo-Jacobi matrices, with the same order and signature operator H. In this paper we solve this problem by applying Sylvester’s identity and Householder transformation. The solution number and the corresponding reconstruction algorithm are here exhibited, and illustrative numerical examples are given. Comparing this approach with the known Lanczos algorithm for reconstructing pseudo-Jacobi matrices, our method is shown to be more stable and effective.

А Gpu-based Orthogonal Matrix Factorization Algorithm that Produces a Two-Diagonal Shape


 
 
 With the development of the Big Data sphere, as well as those fields of study that we can relate to artificial intelligence, the need for fast and efficient computing has become one of the most important tasks nowadays. That is why in the recent decade, graphics processing unit computations have been actively developing to provide an ability for scientists and developers to use thousands of cores GPUs have in order to perform intensive computations. The goal of this research is to implement orthogonal decomposition of a matrix by applying a series of Householder transformations in Java language using JCuda library to conduct a research on its benefits. Several related papers were examined. Malaschonok and Savchenko in their work have introduced an improved version of QR algorithm for this purpose [4] and achieved better results, however Householder algorithm is more promising for GPUs according to another team of researchers – Lahabar and Narayanan [6]. However, they were using Float numbers, while we are using Double, and apart from that we are working on a new BigDecimal type for CUDA. Apart from that, there is still no solution for handling huge matrices where errors in calculations might occur.
 The algorithm of orthogonal matrix decomposition, which is the first part of SVD algorithm, is researched and implemented in this work. The implementation of matrix bidiagonalization and calculation of orthogonal factors by the Hausholder method in the jCUDA environment on a graphics processor is presented, and the algorithm for the central processor for comparisons is also implemented. Research of the received results where we experimentally measured acceleration of calculations with the use of the graphic processor in comparison with the implementation on the central processor are carried out. We show a speedup up to 53 times compared to CPU implementation on a big matrix size, specifically 2048, and even better results when using more advanced GPUs. At the same time, we still experience bigger errors in calculations while using graphic processing units due to synchronization problems. We compared execution on different platforms (Windows 10 and Arch Linux) and discovered that they are almost the same, taking the computation speed into account. The results have shown that on GPU we can achieve better performance, however there are more implementation difficulties with this approach.
 
 

Social and labor aspects of livestock development of livestock in agricultural enterprises

The subject of scientific research is the scientific and practical principles of development of the labor market and the market of livestock products in Ukraine. The purpose of the article is to substantiate the areas of formal employment and increase livestock production in agricultural enterprises. The data of the State Statistics Service of Ukraine, normative-legal acts of Ukraine and the following methods of scientific research are used: abstract-logical; system approach; monographic; statistical and economic; balance. Based on the analysis of meat and milk consumption by the population of Ukraine, it is proved that there is a violation of the balanced diet of the population, which is one of the main tasks of state social policy. Its structure has a large shortage of meat products (35 % of norm); dairy products (46 % of the norm), etc. In the studied period of 2014-2019, the production of livestock products, in particular beef and veal, significantly decreased. The scenario-empirical calculation of the increase in formal employment in agriculture due to the increase in domestic production of livestock products to ensure a rational food balance of the country and the formation of export supplies showed the following. It is possible to involve more than 900 thousand workers in agricultural enterprises. These indicators correspond to the need to increase the number of jobs in rural areas (about 500 thousand unemployed and more than 600 thousand self-employed people in households alone). It is proved that to ensure a socially oriented direction of livestock development in agricultural enterprises it is necessary to increase the efficiency of livestock production to ensure short-term employment and wages, stability of employment, development of rural areas and communities. Based on the analysis of economic indicators, livestock and cows in different types of agricultural producers, the content of employment in animal husbandry, it was possible to substantiate social and labor imperatives and the relevant organizational and economic principles of livestock development in large, medium and small agribusiness. The development of the latter is especially important due to the transformation of efficient rural households into farms and small agribusiness enterprises. Key words: rural population, employment, labor market, territorial community, development, cattle breeding, agricultural enterprise, farm.

Householder Dice: A Matrix-Free Algorithm for Simulating Dynamics on Gaussian and Random Orthogonal Ensembles

This paper proposes a new algorithm, named Householder Dice (HD), for simulating dynamics on dense random matrix ensembles with rotational invariance. Examples include the Gaussian ensemble, the Haar-distributed random orthogonal ensemble, and their complex-valued counterparts. A “direct” approach to the simulation, where one first generates a dense n×n matrix from the ensemble, requires at least O(n2) resource in space and time. The HD algorithm overcomes this O(n2) bottleneck by using the principle of deferred decisions: rather than fixing the entire random matrix in advance, it lets the randomness unfold with the dynamics. At the heart of this matrix-free algorithm is an adaptive and recursive construction of (random) Householder reflectors. These orthogonal transformations exploit the group symmetry of the matrix ensembles, while simultaneously maintaining the statistical correlations induced by the dynamics. The memory and computation costs of the HD algorithm are O(nT) and O(nT2), respectively, with T being the number of iterations. When T ≪ n, which is nearly always the case in practice, the new algorithm leads to significant reductions in runtime and memory footprint. Numerical results demonstrate the promise of the HD algorithm as a new computational tool in the study of high-dimensional random systems.