Matrices Of Order Research Articles

A family of subfactors arising from a pair of complex Hadamard matrices

In this paper, we define a new equivalence relation ‘[Formula: see text]’ on the set of all Hadamard inequivalent complex Hadamard matrices of order [Formula: see text] and show that pairs [Formula: see text] of equivalent matrices [Formula: see text] produce an infinite family of potentially new subfactors of the hyperfinite type [Formula: see text] factor [Formula: see text]. All these subfactors are irreducible with the Jones index [Formula: see text], including all possibilities. We also show that this family contains infinitely many infinite-depth subfactors. As an application, we compute the Connes–Størmer relative entropy and the angle between the pair [Formula: see text] of spin model subfactors arising from the pair [Formula: see text] of equivalent matrices. On the other hand, pairs [Formula: see text] of inequivalent matrices [Formula: see text] lead to subalgebras of [Formula: see text] with infinite Pimsner–Popa index.

Inequivalent Hadamard Matrices of Order 40 using Block Designs

This article presents the classification of Hadamard matrices of order 40 with automorphisms of order 19. We determine the 2-(19,9,8) and 2-(20,10,9) designs with automorphisms of order 19. These designs are embedded into 2-(39,19,9) designs which are then extended to 3-(40,20,9) designs. We get 33 non-isomorphic 2-(39,19,9) designs out of which 18 designs gives non-isomorphic 3-(40,20,9) designs and hence 18 inequivalent Hadamard matrices of order 40 with automorphisms of order 19.

Novel Steganographic Method Based on Hermitian Positive Definite Matrix and Weighted Moore–Penrose Inverses

In this paper, we describe the concept of a new data-hiding technique for steganography in RGB images where a secret message is embedded in the blue layer of specific bytes. For increasing security, bytes are chosen randomly using a random square Hermitian positive definite matrix, which is a stego-key. The proposed solution represents a very strong key since the number of variants of positive definite matrices of order 8 is huge. Implementing the proposed steganographic method consists of splitting a color image into its R, G, and B channels and implementing two segments, which take place in several phases. The first segment refers to embedding a secret message in the carrier (image or text) based on the unique absolute elements values of the Hermitian positive definite matrix. The second segment refers to extracting a hidden message based on a stego-key generated based on the Hermitian positive definite matrix elements. The objective of the data-hiding technique using a Hermitian positive definite matrix is to embed confidential or sensitive data within cover media (such as images, audio, or video) securely and imperceptibly; by doing so, the hidden data remain confidential and tamper-resistant while the cover media’s visual or auditory quality is maintained.

Rota–Baxter Operators of Weight Zero on Upper-Triangular Matrices of Order Three

Rota–Baxter Operators of Weight Zero on Upper-Triangular Matrices of Order Three

On the trace-zero doubly stochastic matrices of order 5

We propose a graph theoretic approach to determine the trace of the product of two permutation matrices through a weighted digraph representation for a pair of permutation matrices. Consequently, we derive trace-zero doubly stochastic (DS) matrices of order 5 whose k-th power is also a trace-zero DS matrix for k∈{2,3,4,5}. Then, we determine necessary conditions for the coefficients of a generic polynomial of degree 5 to be realizable as the characteristic polynomial of a trace-zero DS matrix of order 5. Finally, we approximate the eigenvalue region of trace-zero DS matrices of order 5.

Two Novel Multidimensional Affine Variations of the Hill Cipher

Two novel symmetric multidimensional affine nested variations of the Hill Cipher are presented. The Hill Cipher is a block polygraphic substitution encryption scheme based on a linear transformation of plaintext characters into ciphertext characters. In the time since Hill first published his encryption scheme, variations, modifications, and improvements of theoretical and practical importance have been published every year indicating that the Hill Cipher is an active area of cryptography research. The first variation presented in this paper incorporated invertible key matrices of orders 2, 4, and 8 such that the matrix values of the <i>2×2</i> matrix rotate positions with each block of characters in a similar manner to the rotating letter wheels of a German Enigma Encoder, then results of the <i>2×2</i> key matrices output are passed to <i>4×4</i> key matrices, and <i>8x8</i> key matrix, <i>4×4</i> key matrices, and rotative-value <i>2×2</i> key matrices. The second variation is configured with invertible key matrices of orders 4, 8, and 16 without rotation of matrix values in a similar manner to the first variation. In both variations, plaintext characters of each block are operated on by exclusive-or (XOR) vectors prior to multiplication with the matrices to create the affine ciphers. Strengths, weaknesses, and other considerations are provided in the discussion. Two proposals are also argued with rationale for a more robust character set for encryption and the increase in modulus that the character set allows, and the possible advantages and disadvantages of affine XOR vectors.

On coupled semilinear evolution systems: Techniques on fractional powers of 4×4$4\times 4$ matrices and applications

AbstractIn this paper, we provide several techniques to explicitly calculate fractional powers of operator matrices focusing on creating a theory that can be applied to distinct situations. To illustrate the abstract results developed, we consider its application in systems of coupled reaction–diffusion equations and in (strongly damped) wave equations. We also discuss how these techniques can be applied to higher order matrices and we specifically calculate the fractional powers of a operator matrix associated to a weakly coupled system of wave equation. In addition, we deal with the applicability of this analysis with respect to solvability, stabilization, regularity, smooth dynamics, and connection with evolutionary classic equation and its fractional counterpart.

Innovative coupling of s-stage one-step and spectral methods for non-smooth solutions of nonlinear problems

The behavior of nonlinear dynamical systems arising in mathematical physics through numerical tools is a challenging task for researchers. In this context, an efficient semi-spectral method is proposed and applied to observe the robust solutions for the mathematical physics problems. Firstly, the space variable is approximated by the Vieta-Lucas polynomials and then the s-stage one-step method is applied to discretize the temporal variable which transfers the problem in the form Cn+1=Cn+Δtϕ(x,t,Cn,F(un)). Novel operational matrices of integer order are developed to replace the spatial derivative terms presented in the discussed problem. Related theorems are included in the study to validate the approach mathematically. The proposed semi-spectral schemes convert the considered nonlinear problem to a system of linear algebraic equations which is easier to tackle. We also accomplish an investigation on the error bound and convergence to confirm the mathematical formulation of the computational algorithm. To show the accuracy and effectiveness of the suggested computational method numerous test problems, such as the advection-diffusion problem, generalized Burger-Huxley, sine-Gordon, and modified KdV–Burgers’ equations are considered. An inclusive comparative examination demonstrates the currently suggested computational method in terms of credibility, accuracy, and reliability. Moreover, the coupling of the spectral method with the fourth-order Runge-Kutta method seems outstanding to handle the nonlinear problem to examine the precise smooth and non-smooth solutions of physical problems. The computational order of convergence (COC) is computed numerically through numerous simulations of the proposed schemes. It is found that the proposed schemes are in exponential order of convergence in the spatial direction and the COC in the temporal direction validates the studies in the literature.

On the classification of skew Hadamard matrices of order $$\varvec{36}$$ and related structures

On the classification of skew Hadamard matrices of order $$\varvec{36}$$ and related structures

Algebraic curves associated with centrosymmetric matrices of orders up to 6

In this paper the boundary generating curves and numerical ranges of centrosymmetric matrices of orders up to 6 are characterized in terms of the matrices entries. These results extend previous ones concerning Kac-Sylvester matrices. The classification of all the possible boundary generating curves for centrosymmetric matrices of higher dimensions remains open.

Multi-Unitary Complex Hadamard Matrices

We analyze the set of real and complex Hadamard matrices with additional symmetry constrains. In particular, we link the problem of existence of maximally entangled multipartite states of [Formula: see text] subsystems with [Formula: see text] levels each to the set of complex Hadamard matrices of order [Formula: see text]. To this end, we investigate possible subsets of such matrices which are, dual, strongly dual ([Formula: see text] or [Formula: see text]), two-unitary ([Formula: see text] and [Formula: see text] are unitary), or [Formula: see text]-unitary. Here [Formula: see text] denotes reshuffling of a matrix [Formula: see text] describing a bipartite system, and [Formula: see text] its partial transpose. Such matrices find several applications in quantum many-body theory, tensor networks and classification of multipartite quantum entanglement and imply a broad class of analytically solvable quantum models in [Formula: see text] dimensions.

On the generalized Apostol-Kolodner differential equation of the second-order

This paper concerns another approach for solving the matrix differential equations of the second order $X''(t)=AX'(t)+BX(t)$, where $A$, $B$ are square matrices of order ${d\times d}$. Such approach is based on some properties of matrix square root and the linear difference equation. We establish various new results and explicit formulas for the solutions of this type of matrix differential equations. Moreover, because of the non-commutativity condition between $A$ and $B$, the matrix fundamental system will play an important role. Finally, our results and their robustness, are validated by providing some examples and applications.

Two-unitary complex Hadamard matrices of order 36

Abstract A family of two-unitary complex Hadamard matrices (CHMs) of size 36 stemming from a particular matrix is constructed. Every matrix in this orbit remains unitary after operations of partial transpose and reshuffling which makes it a distinguished subset of CHM. It provides a novel solution to the quantum version of the Euler problem, in which each field of the Graeco-Latin square of size six contains a symmetric superposition of all 36 officers with phases being multiples of sixth root of unity. This simplifies previously known solutions as all amplitudes of the superposition are equal and the set of phases consists of six elements only. Multidimensional parameterization allows for more flexibility in a potential experimental realization.

Accurate complex Jacobi rotations

This note shows how to compute, to high relative accuracy under mild assumptions, complex Jacobi rotations for diagonalization of Hermitian matrices of order two, using the correctly rounded functions cr_hypot and cr_rsqrt, proposed for standardization in the C programming language as recommended by the IEEE-754 floating-point standard. The rounding to nearest (ties to even) and the non-stop arithmetic are assumed. The numerical examples compare the observed with theoretical bounds on the relative errors in the rotations’ elements, and show that the maximal observed departure of the rotations’ determinants from unity is smaller than that of the transformations computed by LAPACK.

The range of combined matrices and doubly quasi-stochastic matrices of order 3

The combined matrix of a nonsingular matrix A is the matrix C(A)=A∘A−t, where ∘ means the Hadamard (entrywise) product. Since the sum of the entries of each row and each column of C(A) is one, when the combined matrix is a real matrix, it is a doubly quasi-stochastic matrix. In this work, doubly quasi-stochastic combined matrices are studied. The characterization of all real matrices such that their combined matrix is doubly quasi-stochastic of order 3 is obtained, and the study of the class of totally positive matrices is given.

A Heuristic Method for Solving Polynomial Matrix Equations

We propose a heuristic method to solve polynomial matrix equations of the type ∑k=1makXk=B, where ak are scalar coefficients and X and B are square matrices of order n. The method is based on the decomposition of the B matrix as a linear combination of the identity matrix and an idempotent, involutive, or nilpotent matrix. We prove that this decomposition is always possible when n=2. Moreover, in some cases we can compute solutions when we have an infinite number of them (singular solutions). This method has been coded in MATLAB and has been compared to other methods found in the existing literature, such as the diagonalization and the interpolation methods. It turns out that the proposed method is considerably faster than the latter methods. Furthermore, the proposed method can calculate solutions when diagonalization and interpolation methods fail or calculate singular solutions when these methods are not capable of doing so.

Construction of bases in space M(2,R) in computer mathematics system

In finite-dimensional linear spaces, the question of constructing a basis is related to matrix computations, the labor intensity of which increases not only with increasing dimensionality of spaces, but also with taking into account the nature of this space. In this context, we consider an alternative solution of this question in the space of square matrices of the 2nd order in the system of computer mathematics, which has a very rapid development today.

Texture analysis of 18F-FDG PET/CT and CECT: Prediction of refractoriness of Hodgkin lymphoma with mediastinal bulk involvement.

To recognize patients at high risk of refractory disease, the identification of novel prognostic parameters improving stratification of newly diagnosed Hodgkin Lymphoma (HL) is still needed. This study investigates the potential value of metabolic and texture features, extracted from baseline 18F-FDG Positron Emission Tomography/Computed Tomography (PET) and Contrast-Enhanced Computed Tomography scan (CECT), together with clinical data, in predicting first-line therapy refractoriness (R) of classical HL (cHL) with mediastinal bulk involvement. We reviewed 69 cHL patients who underwent staging PET and CECT. Lesion segmentation and texture parameter extraction were performed using the freeware software LIFEx 6.3. The prognostic significance of clinical and imaging features was evaluated in relation to the development of refractory disease. Receiver operating characteristic curve, Cox proportional hazard regression and Kaplan-Meier analyses were performed to examine the potential independent predictors and to evaluate their prognostic value. Among clinical characteristics, only stage according to the German Hodgkin Group (GHSG) classification system significantly differed between R and not-R. Among CECT variables, only parameters derived from second order matrices (gray-level co-occurrence matrix (GLCM) and gray-level run length matrix (GLRLM) demonstrated significant prognostic power. Among PET variables, SUVmean, several variables derived from first (histograms, shape), and second order analyses (GLCM, GLRLM, NGLDM) exhibited significant predictive power. Such variables obtained accuracies greater than 70% at receiver operating characteristic analysis and their PFS curves resulted statistically significant in predicting refractoriness. At multivariate analysis, only HISTO_EntropyPET extracted from PET (HISTO_EntropyPET ) and GHSG stage resulted as significant independent predictors. Their combination identified 4 patient groups with significantly different PFS curves, with worst prognosis in patients with higher HISTO_EntropyPET values, regardless of the stage. Imaging radiomics may provide a reference for prognostic evaluation of patients with mediastinal bulky cHL. The best prognostic value in the prediction of R versus not-R disease was reached by combining HISTO_EntropyPET with GHSG stage.

Hadamard Matrices of Orders 60 and 64 with Automorphisms of Orders 29 and 31

A classification of Hadamard matrices of order $2p+2$ with an automorphism of order $p$ is given for $p=29$ and $31$. The ternary self-dual codes spanned by the newly found Hadamard matrices of order 60 with an automorphism of order 29 are computed, as well as the binary doubly even self-dual codes of length 120 with generator matrices defined by related Hadamard designs. Several new ternary near-extremal self-dual codes, as well as binary near-extremal doubly even self-dual codes with previously unknown weight enumerators are found.

О дифференцированиях линейных алгебр специального типа

In this work, Lie algebras of differentiation of linear algebra, the op­eration of multiplication in which is defined using a linear form and two fixed elements of the main field are studied. In the first part of the work, a definition of differentiation of linear algebra is given, a system of linear homogeneous equations is obtained, which is satisfied by the components of arbitrary differentiation. An embedding of the Lie algebra of differenti­ations into the Lie algebra of square matrices of order n over the field P is constructed. This made it possible to give an upper bound for the dimen­sion of the Lie algebra of derivations. It has been proven that the dimen­sion of the algebra of differentiation of the algebras under study is equal to n2 – n, where n is the dimension of the algebra. Next we give a result on the maximum dimension of the Lie algebra of derivations of a linear alge­bra with identity. Based on the above facts, it is proven that the algebras under study cannot have a unit.