Pair Of Matrices Research Articles

Differentiable learning of matricized DNFs and its application to Boolean networks

Boolean networks (BNs) are well-studied models of genomic regulation in biology where nodes are genes and their state transition is controlled by Boolean functions. We propose to learn Boolean functions as Boolean formulas in disjunctive normal form (DNFs) by an explainable neural network Mat_DNF and apply it to learning BNs. Directly expressing DNFs as a pair of binary matrices, we learn them using a single layer NN by minimizing a logically inspired non-negative cost function to zero. As a result, every parameter in the network has a clear meaning of representing a conjunction or literal in the learned DNF. Also we can prove that learning DNFs by the proposed approach is equivalent to inferring interpolants in logic between the positive and negative data. We applied our approach to learning three literature-curated BNs and confirmed its effectiveness. We also examine how generalization occurs when learning data is scarce. In doing so, we introduce two new operations that can improve accuracy, or equivalently generalizability for scarce data. The first one is to append a noise vector to the input learning vector. The second one is to continue learning even after learning error becomes zero. The first one is explainable by the second one. These two operations help us choose a learnable DNF, i.e., a root of the cost function, to achieve high generalizability.

Pairs of Symmetric and Skew-Symmetric Toeplitz Matrices with Identical Squares

Certain sets consisting of pairs of symmetric and skew-symmetric Toeplitz matrices with identical squares are described.

Length Function and Simultaneous Triangularization of Matrix Pairs

The paper interrelates the simultaneous triangularization problem for matrix pairs with the Paz problem and known results on the length of the matrix algebra. The length function is applied to the Al’pin–Koreshkov algorithm, and it is demonstrated how its multiplicative complexity can be reduced. An asymptotically superior procedure for verifying the simultaneous triangularizability of a pair of complex matrices is provided. The procedure is based on results on the lengths of upper triangular matrix algebras. Also the definition of the hereditary length of an algebra is introduced, and the problem of computing the hereditary lengths of matrix algebras is discussed.

UAV image stitching by estimating orthograph with RGB cameras

In the field of image stitching, cases with large camera optical center movement and large parallax have been the virgin territory of research. The goal of image stitching is to overcome the parallax and stitch a natural image. We look into this problem in the context of ultra-low altitude flight of a UAV. We model the 3D world in this scenario and quickly estimate orthographic projection by pairs of homography matrices. Our stitching method can achieve precise alignment since it takes parallax well into consideration. The stitching results are natural and the extra time consumed is short.

Linear maps that preserve parts of the spectrum on pairs of similar matrices

In this paper, we characterize linear bijective maps $\varphi$ on the space of all $n \times n$ matrices over an algebraically closed field $\mathbb{F}$ having the property that the spectrum of $\varphi (A)$ and $\varphi (B)$ have at least one common eigenvalue for each similar matrices $A$ and $B$. Using this result, we characterize linear bijective maps having the property that the spectrum of $\varphi (A)$ and $\varphi (B)$ have common elements for each matrices $A$ and $B$ having the same spectrum. As a corollary, we also characterize linear bijective maps preserving the equality of the spectrum.

Enhanced Fuzzy Fault Estimation of Discrete-Time Nonlinear Systems via a New Real-Time Gain-Scheduling Mechanism.

The problem of enhancing the robust performance of nonlinear fault estimation (FE) is addressed by proposing a novel real-time gain-scheduling mechanism for discrete-time Takagi-Sugeno fuzzy systems. The real-time status of the operating point for the considered nonlinear plant is characterized by using these available normalized fuzzy weighting functions at both the current and the past instants of time. To achieve this, the developed fuzzy real-time gain-scheduling mechanism produces different switching modes by introducing key tunable parameters. Thus, a pair of exclusive FE gain matrices is designed for each switching mode on the strength of time-varying balanced matrices developed in this study, respectively. Since the implementation of more FE gain matrices can be scheduled according to the real-time status of the operating point at each sampling instant, the robust performance of nonlinear FE will be enhanced over the previous methods to a great extent. Finally, considerable numerical comparisons are implemented in order to illustrate that the proposed method is much superior to those existing ones reported in the literature.

Decomposable pairs and construction of multiwavelets

We study the construction of refinable function vectors and multiwavelets from the perspective of decomposable pairs. There exists a pair of matrices known as the decomposable pair, associated with a refinement mask of the form H(ξ)=12∑k=0lHke−iξk, which gives the spectral information about H(ξ). We show that the existence of a solution to a matrix refinement equation is related to the dimension of 1-eigenspace of the square matrix in the decomposable pair. We also show that the existence of a refinement mask with sum rules of order 1 is related to the dimension of (−1)-eigenspace of the square matrix in the decomposable pair. A systematic procedure is developed to create refinement masks satisfying the sum rules of order 1 using decomposable pairs, such that the corresponding matrix refinement equation has a solution. Several examples are provided to illustrate the theoretical results obtained.

Schur decomposition of several matrices

Schur decompositions and the corresponding Schur forms of a single matrix, a pair of matrices, or a collection of matrices associated with the periodic eigenvalue problem are frequently used and studied. These forms are upper-triangular complex matrices or quasi-upper-triangular real matrices that are equivalent to the original matrices via unitary or, respectively, orthogonal transformations. In general, for theoretical and numerical purposes we often need to reduce, by admissible transformations, a collection of matrices to the Schur form. Unfortunately, such a reduction is not always possible. In this paper we describe all collections of complex (real) matrices that can be reduced to the Schur form by the corresponding unitary (orthogonal) transformations and explain how such a reduction can be done. We prove that this class consists of the collections of matrices associated with pseudoforest graphs. In other words, we describe when the Schur form of a collection of matrices exists and how to find it.

The Missing Label of $$\mathfrak {su}_3$$ and Its Symmetry

We present explicit formulas for the operators providing missing labels for the tensor product of two irreducible representations of $$\mathfrak {su}_3$$ . The result is seen as a particular representation of the diagonal centraliser of $$\mathfrak {su}_3$$ through a pair of tridiagonal matrices. Using these explicit formulas, we investigate the symmetry of this missing label problem and we find a symmetry group of order 144 larger than what can be expected from the natural symmetries. Several realisations of this symmetry group are given, including an interpretation as a subgroup of the Weyl group of type $$E_6$$ , which appeared in an earlier work as the symmetry group of the diagonal centraliser. Using the combinatorics of the root system of type $$E_6$$ , we provide a family of representations of the diagonal centraliser by infinite tridiagonal matrices, from which all the finite-dimensional representations affording the missing label can be extracted. Besides, some connections with the Hahn algebra, Heun–Hahn operators and Bethe ansatz are discussed along with some similarities with the well-known symmetries of the Clebsch–Gordan coefficients.

The discrepancy between min–max statistics of Gaussian and Gaussian-subordinated matrices

We compute quantitative bounds for measuring the discrepancy between the distribution of two min–max statistics involving either one pair of Gaussian random matrices, or one Gaussian and one Gaussian-subordinated random matrix. In the fully Gaussian setup, our approach allows us to recover quantitative versions of well-known inequalities by Gordon (1985, 1987, 1992), thus generalizing the quantitative version of the Sudakov–Fernique inequality deduced in Chatterjee (2005). On the other hand, the Gaussian-subordinated case yields generalizations of estimates by Chernozhukov et al. (2015) and Koike (2019). As an application, we establish fourth moment bounds for matrices of multiple stochastic Wiener–Itô integrals, that we illustrate with an example having a statistical flavor.

FIR-SIMO channel order determination by invariant-signal-subspace matching

We present a novel and computationally simple solution to the problem of determining the order of finite impulse response (FIR) single-input multiple-output (SIMO) channel. The solution is applicable to ideal and realistic wireless channels, with both white and colored noise. It is based on a non-asymptotic goodness-of-fit metric, referred to as invariant-signal-subspace matching (ISSM), aimed at matching pairs of invariant-signal-subspaces, created by exploiting the shift-invariance property of the block-Toeplitz data matrix. A pair of projection matrices corresponding to these subspaces is created for every potential FIR-SIMO order, and the order is then determined as that for which this metric is minimized. Simulation results are included, demonstrating the performance of the proposed solution for both ideal and realistic channels, with both white and colored noise.

Classification of the pairs of matrices of fixed Jordan types and representations of bundles of semichains

We study the problem of classifying the pairs of matrices over an algebraically closed field with some restrictions on their Jordan canonical forms using earlier results of the first author. All tame and wild, polynomial and non-polynomial growth cases are described.

Pseudo Similarity of Neutrosophic Fuzzy matrices

In this paper, first we shall define Pseudo Similarity for Neutrosophic Fuzzy Matrices and prove that Pseudo Similarity relation on pair of Neutrosophic Fuzzy Matrices. Also, we derive some relation between Pseudo Similarity and Idempotent matrices. Finally, we give in varies inverse of Neutrosophic Fuzzy Matrices.

On the Bott index of unitary matrices on a finite torus

This article reviews the foundations of the theory of the Bott index of a pair of unitary matrices in the context of condensed matter theory, as developed by Hastings and Loring (J. Math. Phys. 51, 015214 (2010), Ann. Phys. 326, 1699 (2011)), providing a novel proof of the equality with the Chern number. The Bott index is defined for a pair of unitary matrices, then extended to a pair of invertible matrices and homotopic invariance of the index is proven. An insulator defined on a lattice on a two-torus, that is a rectangular lattice with periodic boundary conditions, is considered and a pair of quasi-unitary matrices associated to this physical system are introduced. It is shown that their Bott index is well defined and the connection with the transverse conductance, the Chern number, is established proving the equality of the two quantities, in certain units.

Nonautonomous three soliton interactions in an inhomogeneous optical fiber: Application to soliton switching devices

We investigate the interactional behavior of three nonautonomous solitons in an optical fiber with inhomogeneous nature which can be described by nonlinear Schrödinger equation with nonautonomous term. A pair of matrices constructed with the aid of AKNS procedure for nonautonomous NLS equation with variable coefficients. Based on the Lax pair, three soliton solutions are generated by employing Darboux method. We show the switchable characteristics of nonautonomous three solitons by typical graphical illustration. The remarkable results presented in this study can be used to design the optical logic gate devices for optical computing.

Precision replication of all types of optical surfaces—scientific and technological basis for the radical transformation of modern optical production

Subject of study. The developments and proposed innovations of the Scientific and Production Association “State Institute of Applied Optics” in the field of advancement of precision replication technology of all types of optical surfaces (planar, spherical, aspheric, and freeform) aimed at ensuring the possibility of series and mass production of the so-called combined optical elements (COEs) and the large-scale fabrication of objectives operating in a wide spectral range for various purposes based on these elements are discussed. Aim of study. This study is aimed at the introduction of new proposals for the radical transformation of modern optical production based on advanced precision replication technology of all types of optical surfaces of lenses and mirrors to the Russian and foreign scientific and technological community and the executives and process engineers of optical production facilities. Method. The technology for shaping the working surfaces of optical elements of all types (lenses, mirrors, and prisms) is based on precision replication of their working surfaces using highly precise and certified master details in relatively thin polymer layers on substrates made of glass or other optical materials. The use of cold-curing resins, predominantly photopolymerizable compositions, is advisable. Main results. We demonstrate that a radical solution to the problem of establishing the series and mass production of optical elements with working surfaces of any type can be based on the precision replication technology using cold-curable polymer materials. The COEs obtained using this method comprise a glass base with one or two relatively thin polymer layers that have outer surfaces of a specified shape. Using low-shrinkage cold-curing polymer materials ensures the absence of thermal deformations of the COEs and their high reproducibility in each batch. Such elements have high mechanical, thermal, and optical properties of glass optics, and the cost of their series production is manyfold lower. We established that (when the technological regimen is followed) the replicated surfaces are identical to the working surface of the original master detail in terms of surface shape (N, ΔN), surface finish class (P), and roughness parameters (Rz, Ra). Moreover, the storability and resistance to environmental factors of the COEs are practically equal to those of their purely glass counterparts, which is confirmed by the results of multiple complex tests of different COE types over a period of several years. A hierarchic system of master details comprising reference, control, and working matrices is proposed. The reference matrices must always be produced in pairs of convex and concave matrices (similar to spherical test plates). Practical significance. Owing to a rapid increase in the demand in optoelectronic instrumentation engineering for objectives for different applications in a wide spectral range (smartphones, tablets, digital cameras, video projectors, analytical instruments, thermal imaging equipment, video surveillance and security systems, and spaceborne optical surveillance instruments), there has been an urgent need for a significant increase in the production capacity and cost reduction of the manufacturing of basic optical elements. Precision replication technologies provide the fundamental and practical possibility for solving this problem.

Relaxed Fault Estimation of Discrete-Time Nonlinear System Based on a New Multi-Instant Real-Time Scheduling Fuzzy Observer

Relaxed fault estimation (FE) criteria of a class of discrete-time nonlinear plants are proposed based on a new multi-instant real-time scheduling fuzzy FE observer. Compared with the existing methods given in the recent literature, the newly developed fuzzy observer is capable of providing much freedom for reducing the conservatism of designing feasible FE criteria. At each sampling point, the information of adjacent groups of normalized fuzzy weighting functions can be updated and utilized for deciding the activated mode of our developed fuzzy FE observer. Based on this, an exclusive pair of gain matrices can be given by introducing a set of well-designed free matrices into the designed FE criteria for every activated mode. Moreover, it can be found that the proposed result will include the previous one as a special case if all of those free matrices are removed. Finally, the well-known nonlinear truck–trailer plant with actuator fault is utilized to validate the superiority of our approach over the existing ones reported in the literature.

Pfaffian Pairs and Parities: Counting on Linear Matroid Intersection and Parity Problems

Spanning trees are a representative example of linear matroid bases that are efficiently countable. Perfect matchings of Pfaffian bipartite graphs are a countable example of common bases of two matrices. Generalizing these two, Webb Counting Bases, Ph.D. thesis, University of Waterloo, 2004 introduced the notion of Pfaffian pairs as a pair of matrices for which counting of their common bases is tractable via the Cauchy--Binet formula. This paper studies counting on linear matroid problems extending Webb's work. We first introduce “Pfaffian parities” as an extension of Pfaffian pairs to the linear matroid parity problem, which is a common generalization of the linear matroid intersection problem and the matching problem. We show that a large number of efficiently countable discrete structures are interpretable as special cases of Pfaffian pairs and parities. Our study then turns to algorithmic aspects. We observe that the fastest randomized algorithms for the linear matroid intersection and parity problems by Harvey SIAM J. Comput., 39 (2009), pp. 679--702 and Cheung, Lau, and Leung ACM Trans. Algorithms, 10 (2014), pp. 1--26 can be derandomized for Pfaffian pairs and parities. We further present polynomial-time algorithms to count the number of minimum-weight solutions on weighted Pfaffian pairs and parities. Our algorithms make use of Frank's weight-splitting lemma for the weighted matroid intersection problem and the algebraic optimality criterion of the weighted linear matroid parity problem given by Iwata and Kobayashi SIAM J. Comput., 51 (2002) pp. STOC17-238--STOC17-280.

A method for computing a few eigenpairs of large generalized eigenvalue problems

Many methods for the computation of selected eigenpairs of generalized eigenproblems for matrix pairs use a shift-and-invert technique. When applied to large-scale problems, this requires the solution of large linear systems of equations. This paper proposes an application of an Arnoldi method described in Voss (2004) [21] to the computation of a few extreme eigenpairs of a matrix pair. An advantage of this approach, when compared to methods that use the shift-and-invert technique, is that no large systems of equations have to be solved. We compare this approach to using a technique for simultaneously reducing a pair of large matrices to a pair of small matrices by a generalized Arnoldi process described in Li and Ye (2003) [12] and Hoffnung et al. (2006) [6]. The latter technique does not require the solution of large linear systems of equations either. Computed examples show the proposed method to yield approximations of the desired eigenpairs of higher accuracy when using about the same amount of computer storage space.

THE MATRIX LINEAR UNILATERAL AND BILATERAL EQUATIONS

In this article the method of solving matrix linear equations over commutative Bezout domains by means of standard form of a pair of matrices with respect to generalized equivalence is found. The criterions of uniqueness of particular solutions of matrix linear equations are determined. The formulas of general solutions of matrix linear equations AX + BY = C and AX + YB = C are deduced.