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Within the context of fewest-switch surface hopping (FSSH) dynamics, one often wishes to remove the angular component of the derivative coupling between states J and K . In a previous set of papers, Shu et al. [J. Phys. Chem. Lett. 11, 1135–1140 (2020)] posited one approach for such a removal based on direct projection, while we isolated a second approach by constructing and differentiating a rotationally invariant basis. Unfortunately, neither approach was able to demonstrate a one-electron operator O ̂ whose matrix element J O ̂ K was the angular component of the derivative coupling. Here, we show that a one-electron operator can, in fact, be constructed efficiently in a semi-local fashion. The present results yield physical insight into designing new surface hopping algorithms and are of immediate use for FSSH calculations. [ABSTRACT FROM AUTHOR]

Exact methods for the exponentiation of matrices of dimension N can be computationally expensive in terms of execution time (N3) and memory requirements (N2), not to mention numerical precision issues. A matrix often exponentiated in the natural sciences is the rate matrix. Here, we explore five methods to exponentiate rate matrices, some of which apply more broadly to other matrix types. Three of the methods leverage a mathematical analogy between computing matrix elements of a matrix exponential process and computing transition probabilities of a dynamical process (technically a Markov jump process, MJP, typically simulated using Gillespie). In doing so, we identify a novel MJP-based method relying on restricting the number of "trajectory" jumps that incurs improved computational scaling. We then discuss this method's downstream implications on mixing properties of Monte Carlo posterior samplers. We also benchmark two other methods of matrix exponentiation valid for any matrix (beyond rate matrices and, more generally, positive definite matrices) related to solving differential equations: Runge–Kutta integrators and Krylov subspace methods. Under conditions where both the largest matrix element and the number of non-vanishing elements scale linearly with N—reasonable conditions for rate matrices often exponentiated—computational time scaling with the most competitive methods (Krylov and one of the MJP-based methods) reduces to N2 with total memory requirements of N. [ABSTRACT FROM AUTHOR]

Numerical difficulties associated with computing matrix elements of operators between Hartree–Fock–Bogoliubov (HFB) wavefunctions have plagued the development of HFB-based many-body theories for decades. The problem arises from divisions by zero in the standard formulation of the nonorthogonal Wick's theorem in the limit of vanishing HFB overlap. In this Communication, we present a robust formulation of Wick's theorem that stays well-behaved regardless of whether the HFB states are orthogonal or not. This new formulation ensures cancellation between the zeros of the overlap and the poles of the Pfaffian, which appears naturally in fermionic systems. Our formula explicitly eliminates self-interaction, which otherwise causes additional numerical challenges. A computationally efficient version of our formalism enables robust symmetry-projected HFB calculations with the same computational cost as mean-field theories. Moreover, we avoid potentially diverging normalization factors by introducing a robust normalization procedure. The resulting formalism treats even and odd number of particles on equal footing and reduces to Hartree–Fock as a natural limit. As proof of concept, we present a numerically stable and accurate solution to a Jordan–Wigner-transformed Hamiltonian, whose singularities motivated the present work. Our robust formulation of Wick's theorem is a most promising development for methods using quasiparticle vacuum states. [ABSTRACT FROM AUTHOR]

In this paper, we present an innovative approach for assessing the stability of linear time-invariant (LTI) systems. The key innovations in our methodology include the calculation of Markov parameters, the extension to nonlinear continuous-time systems, and significantly improved computational efficiency. Our novel approach leverages Markov parameters derived from the characteristic polynomial of the system. Unlike traditional methods, we employ the Leverrier-Takeno algorithm to efficiently determine the coefficients of the characteristic polynomial for state-space representation. The systematic calculation of Markov parameters using a specialized Maclaurin series expansion enhances our method's precision. One of our major achievements is the rigorous verification of stability through Hankel matrices, ensuring positive definiteness using the Cholesky decomposition algorithm. In addition, we introduce a second stability criterion based on iterated polynomial long divisions, although we acknowledge its limitations in handling high-order systems due to computational inefficiency. Our pioneering contributions extend beyond LTI systems. We adapt our innovative techniques to analyze the local stability of nonlinear continuous-time systems by introducing the concept of Jacobian matrices and employing the indirect Lyapunov method. This expansion broadens the applicability of our approach to a wider range of real-world systems. Notably, our approach exhibits remarkable computational efficiency. According to CPU-time analysis, our first stability criterion outperforms the classical Routh–Hurwitz method by more than 12 times for dynamic systems with orders ranging from 2 to 180. This significant reduction in computation time positions our method as a valuable option for real-time stability analysis applications. [ABSTRACT FROM AUTHOR]

We study the image of SU (n) under the diagonal product map. Using only elementary tools of linear algebra, analysis and general topology, we find the analytical formula for the boundary of this image and find all special unitary matrices whose diagonal product lies on that boundary. [ABSTRACT FROM AUTHOR]

The Four point condition (4PC henceforth) is a well known condition characterising distances in trees T. Let w , x , y , z be four vertices in T and let d x , y denote the distance between vertices x , y in T. The 4PC condition says that among the three terms d w , x + d y , z , d w , y + d x , z and d w , z + d x , y the maximum value equals the second maximum value. We define an ( n 2 ) × ( n 2 ) sized matrix Max4PC T from a tree T where the rows and columns are indexed by size-2 subsets. The entry of Max4PC T corresponding to the row indexed by { w , x } and column { y , z } is the maximum value among the three terms d w , x + d y , z , d w , y + d x , z and d w , z + d x , y. In this work, we determine basic properties of this matrix like rank, give an algorithm that outputs a family of bases, and find the determinant of Max4PC T when restricted to our basis. We further determine the inertia and the Smith Normal Form (SNF) of Max4PC T. [ABSTRACT FROM AUTHOR]

Haagerup's proof of the non commutative little Grothendieck inequality raises some questions on the commutative little inequality, and it offers a new result on scalar matrices with non negative entries. The theory of completely bounded maps may be used to show that the commutative Grothendieck inequality follows from the little commutative inequality, and that this passage may be given a geometric form as a relation between a pair of compact convex sets of positive matrices, which, in turn, characterizes the little constant k G C . [ABSTRACT FROM AUTHOR]

We develop a version of quantum mechanics that can handle nonassociative algebras of observables and which reduces to standard quantum theory in the traditional associative setting. Our algebraic approach is naturally probabilistic and is based on using the universal enveloping algebra of a general nonassociative algebra to introduce a generalized notion of associative composition product. We formulate properties of states together with notions of trace, and use them to develop Gel'fand–Naimark–Segal constructions. We describe Heisenberg and Schrödinger pictures of completely positive dynamics, and we illustrate our formalism on the explicit examples of finite-dimensional matrix Jordan algebras as well as the octonion algebra. [ABSTRACT FROM AUTHOR]

A new approach to the transformations of the matrices of the fractional linear systems with desired eigenvalues is proposed. Conditions for the existence of the solution to the transformation problem of the linear system to its asymptotically stable controllable and observable canonical forms with desired eigenvalues are given and illustrated by numerical examples of fractional linear systems. [ABSTRACT FROM AUTHOR]

In this paper, we develop an eigenvector-assisted estimation framework for a collection of signal-plus-noise matrix models arising in high-dimensional statistics and many applications. The framework is built upon a novel asymptotically unbiased estimating equation using the leading eigenvectors of the data matrix. However, the estimator obtained by directly solving the estimating equation could be numerically unstable in practice and lacks robustness against model misspecification. We propose to use the quasi-posterior distribution by exponentiating a criterion function whose maximizer coincides with the estimating equation estimator. The proposed framework can incorporate heteroskedastic variance information, but does not require the complete specification of the sampling distribution and is also robust to the potential misspecification of the distribution of the noise matrix. Computationally, the quasi-posterior distribution can be obtained via a Markov chain Monte Carlo sampler, which exhibits superior numerical stability over some of the existing optimization-based estimators and is straightforward for uncertainty quantification. Under mild regularity conditions, we establish the large sample properties of the quasi-posterior distributions. In particular, the quasi-posterior credible sets have the correct frequentist nominal coverage probability provided that the criterion function is carefully selected. The validity and usefulness of the proposed framework are demonstrated through the analysis of synthetic datasets and the real-world ENZYMES network datasets. [ABSTRACT FROM AUTHOR]

We characterize the matrix-weighted Triebel–Lizorkin spaces and Besov spaces by Peetre maximal function and approximation. Using these characterizations, we obtain the boundedness of pseudo-differential operators with symbol in Hörmander's class on matrix weighted Besov and Triebel–Lizorkin spaces. [ABSTRACT FROM AUTHOR]

Let f = f (x 1 , ... , x m) be a multilinear polynomial over a field F. An F -algebra A is said to be f -zpd (f -zero product determined) if every m -linear functional φ : A m → F which preserves zeros of f is of the form φ (a 1 , ... , a m) = τ (f (a 1 , ... , a m)) for some linear functional τ on A. We are primarily interested in the question whether the matrix algebra M d (F) is f -zpd. While the answer is negative in general, we provide several families of polynomials for which it is positive. We also consider a related problem on the form of a multilinear polynomial g = g (x 1 , ... , x m) with the property that every zero of f in M d (F) m is a zero of g. Under the assumption that m < 2 d − 3 , we show that g and f are linearly dependent. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, further characterizations of selfadjoint operators on Hilbert spaces are established. Thus the recent results concerning two-sided removal and cancellation laws associated with a Hermitian matrix (Central University of Finance and Economics, DOI: https://doi.org/10.21203/rs.3.rs-1104901/v1) are extended to the infinite dimensional setting with proofs based on operator matrices. [ABSTRACT FROM AUTHOR]

Vertices in the graph of a square matrix over a field may be classified as to how their removal changes the geometric multiplicity of an identified eigenvalue. There are three possibilities: $ +1 $ + 1 (Parter); no change (neutral); and $ -1 $ − 1 (downer). When the graph is a tree, the 'downer branch mechanism' distinguishes the Parter vertices. Here, we discover how this mechanism generalizes for general graphs, both for Hermitian matrices and general matrices. Then, we apply the new ideas, both to classify pendent edges in general graphs, and to understand the existence of 2-downer edge cycles in general graphs, when there is a 2-downer edge. This is a further explanation of why such edges cannot occur in trees. [ABSTRACT FROM AUTHOR]

The paper treats the modeling of stationary multivariate stochastic processes via frequency domain, and extends the notion of generalized autocovariance function, given by Proietti and Luati (2015) for univariate time series, to the multivariate setting. The generalized autocovariance matrices are defined for stationary multivariate stochastic processes as the Fourier transform of the power transformation of the spectral density matrix. Then we prove the consistency and derive the asymptotic distribution of frequency domain non-parametric estimators of the generalized autocovariance matrices, based on the power transformation of the periodogram matrix. Generalized autocovariance matrices are used to construct white noise hypothesis testing, to discriminate stochastic processes, and to introduce a generalized Yule–Walker estimator for the spectrum. A so-called λ–squared distance between two multivariate stochastic processes is also defined by using their generalized autocovariance matrices, and it serves for clustering time series and estimation by feature matching. Another use is in discriminant analysis. [ABSTRACT FROM AUTHOR]

In this paper, we study the q -distance matrix for a distance-regular graph and show that the q -distance matrix of a distance-regular graph with classical parameters (D , q , α , β) has exactly three distinct eigenvalues, of which one is zero. Moreover, we study distance-regular graphs whose q -distance matrix has exactly one positive eigenvalue. [ABSTRACT FROM AUTHOR]

The first order eigenvalue perturbation theory for structure-preserving perturbations of real or complex symplectic matrices started in [21] is continued and completed. While in [21] all eigenvalues were considered that qualitatively behave similarly under structure-preserving and structure-ignoring perturbations, the focus of this paper is on one particular case when the behavior under structure-preserving perturbations differs significantly from the one under structure-ignoring perturbations. This case occurs for the exceptional eigenvalues ±1 and is caused by restrictions in the possible Jordan structure of symplectic matrices. The main result gives asymptotic expansions of specific perturbed eigenvalues that are not yet covered by the results from [21]. While the main theorem requires that the rank of the perturbation is equal to the rank of its first order term, it is also discussed what can be said if this hypothesis is not satisfied. [ABSTRACT FROM AUTHOR]

Except for the empty graph, we show that the orthogonal matrix X of the adjacency matrix A determines that adjacency matrix completely, but not always uniquely. The proof relies on interesting properties of the Hadamard product Ξ = X ∘ X. As a consequence of the theory, we show that irregular co-eigenvector graphs exist only if the number of nodes N ≥ 6. Co-eigenvector graphs possess the same orthogonal eigenvector matrix X , but different eigenvalues of the adjacency matrix. Co-eigenvector graphs are the dual of co-spectral graphs, that share all eigenvalues of the adjacency matrix, but possess a different orthogonal eigenvector matrix. We deduce general properties of co-eigenvector graph and start to enumerate all co-eigenvector graphs on N = 6 and N = 7 nodes. Finally, we list many open problems. [ABSTRACT FROM AUTHOR]

For a smooth affine algebra R of dimension d ≥ 3 over a field k and an invertible alternating matrix χ of rank 2 n , the group S p (χ) of invertible matrices of rank 2 n over R which are symplectic with respect to χ acts on the right on the set U m 2 n (R) of unimodular rows of length 2 n over R. In this paper, we prove that S p (χ) acts transitively on U m 2 n (R) if k is algebraically closed, d ! ∈ k × and 2 n ≥ d. [ABSTRACT FROM AUTHOR]

The paper provides a characterization of linear operators preserving column majorization of the (0, 1)-vectors. In addition, such operators are characterized explicitly in the case where they are given by special matrices, namely, by (±1)-matrices of order not exceeding 10 or (0,±1)-matrices of order not exceeding 5. Several related combinatorial matrix theory results also are proved. [ABSTRACT FROM AUTHOR]

The paper is devoted to the problem of determining the number of subalgebras in the matrix algebra that are generated by nonderogatory matrices and are distinct up to similarity. A similarity criterion in terms of isomorphism of quotient algebras of the algebra of polynomials for matrix algebras of the type indicated is established. The existence of infinite fields for which the number of distinct algebras is infinite for all values of the matrix order is proved. For algebraically closed fields, the field of reals, and finite fields of sufficiently large cardinalities, the number of distinct algebras generated by nonderogatory matrices is determined as a function of the matrix order. [ABSTRACT FROM AUTHOR]

The paper suggests and illustrates a simple unified approach to deriving upper bounds for the dominant eigenvalues of the so-called PF matrices (or matrices with the Perron–Frobenius property) from those for the Perron root of a nonnegative matrix. [ABSTRACT FROM AUTHOR]

Initially, fuzzy sets and intuitionistic fuzzy sets were used to address real-world problems with imprecise data. Eventually, the notion of the hesitant fuzzy set was formulated to handle decision makers' reluctance to accept asymmetric information. However, in certain scenarios, asymmetric information is gathered in terms of a possible range of acceptance and nonacceptance by players rather than specific values. Furthermore, decision makers exhibit some hesitancy regarding this information. In such a situation, all the aforementioned expansions of fuzzy sets are unable to accurately represent the scenario. The purpose of this article is to present asymmetric information situations in which the range of choices takes into account the hesitancy of players in accepting or not accepting information. To illustrate these problems, we develop matrix games that consider the payoffs of interval-valued intuitionistic hesitant fuzzy elements (IIHFEs). Dealing with these types of fuzzy programming problems requires a significant amount of effort. To solve these matrix games, we formulate two interval-valued intuitionistic hesitant fuzzy programming problems. Preserving the hesitant nature of the payoffs to determine the optimal strategies, these two problems are transformed into two nonlinear programming problems. This transformation involves using mathematical operations for IIHFEs. Here, we construct a novel aggregation operator of IIHFEs, viz., min-max operators of IIHFEs. This operator is suitable for applying the developed methodology. The cogency and applicability of the proposed methodology are verified through a numerical example based on the situation of conflict between hackers and defenders to prevent damage to cybersecurity. To validate the superiority of the proposed model along with the computed results, we provide comparisons with the existing models. [ABSTRACT FROM AUTHOR]

Aiming at addressing the security and efficiency challenges during image transmission, an efficient image cryptosystem utilizing difference matrix and genetic algorithm is proposed in this paper. A difference matrix is a typical combinatorial structure that exhibits properties of discretization and approximate uniformity. It can serve as a pseudo-random sequence, offering various scrambling techniques while occupying a small storage space. The genetic algorithm generates multiple ciphertext images with strong randomness through local crossover and mutation operations, then obtains high-quality ciphertext images through multiple iterations using the optimal preservation strategy. The whole encryption process is divided into three stages: first, the difference matrix is generated; second, it is utilized for initial encryption to ensure that the resulting ciphertext image has relatively good initial randomness; finally, multiple rounds of local genetic operations are used to optimize the output. The proposed cryptosystem is demonstrated to be effective and robust through simulation experiments and statistical analyses, highlighting its superiority over other existing algorithms. [ABSTRACT FROM AUTHOR]

Mediation analysis assesses whether an exposure directly produces changes in cognitive behavior or is influenced by intermediate "mediators". Electroencephalographic (EEG) spectral measurements have been previously used as effective mediators representing diverse aspects of brain function. However, it has been necessary to collapse EEG measures onto a single scalar using standard mediation methods. In this article, we overcome this limitation and examine EEG frequency‐resolved functional connectivity measures as a mediator using the full EEG cross‐spectral tensor (CST). Since CST samples do not exist in Euclidean space but in the Riemannian manifold of positive‐definite tensors, we transform the problem, allowing for the use of classic multivariate statistics. Toward this end, we map the data from the original manifold space to the Euclidean tangent space, eliminating redundant information to conform to a "compressed CST." The resulting object is a matrix with rows corresponding to frequencies and columns to cross spectra between channels. We have developed a novel matrix mediation approach that leverages a nuclear norm regularization to determine the matrix‐valued regression parameters. Furthermore, we introduced a global test for the overall CST mediation and a test to determine specific channels and frequencies driving the mediation. We validated the method through simulations and applied it to our well‐studied 50+‐year Barbados Nutrition Study dataset by comparing EEGs collected in school‐age children (5–11 years) who were malnourished in the first year of life with those of healthy classmate controls. We hypothesized that the CST mediates the effect of malnutrition on cognitive performance. We can now explicitly pinpoint the frequencies (delta, theta, alpha, and beta bands) and regions (frontal, central, and occipital) in which functional connectivity was altered in previously malnourished children, an improvement to prior studies. Understanding the specific networks impacted by a history of postnatal malnutrition could pave the way for developing more targeted and personalized therapeutic interventions. Our methods offer a versatile framework applicable to mediation studies encompassing matrix and Hermitian 3D tensor mediators alongside scalar exposures and outcomes, facilitating comprehensive analyses across diverse research domains. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in a special class of integer matrices, namely the power-product matrix, defined with two positive integers n and d. Each matrix element is computed by a power-product of two weak compositions of d into n parts. The power-product matrix has several interesting applications such as the power-sum representation of polynomials and the difference-of-convex-sums-of-squares decomposition of polynomials. We investigate some properties of this matrix including: nonsingularity, sparsity and determinant. Based on techniques in enumerative combinatorics, we prove that the power-product matrix is nonsingular and the number of nonzero entries can be computed exactly. This matrix shows sparse structure which is a good feature in numerical computation of its inverse required in some applications. Special attention is devoted to the computation of the determinant for n = 2 whose explicit formulation is obtained. [ABSTRACT FROM AUTHOR]

For a simple signed graph G with the adjacency matrix A and net degree matrix $ D^{\pm } $ D ± , the net Laplacian matrix is $ L^{\pm }=D^{\pm }-A $ L ± = D ± − A. We introduce a new oriented incidence matrix $ N^{\pm } $ N ± which can keep track of the sign as well as the orientation of each edge of G. Also $ L^{\pm }=N^{\pm }(N^{\pm })^T $ L ± = N ± (N ± ) T . Using this decomposition, we find the number of both positive and negative spanning trees of G in terms of the principal minors of $ L^{\pm } $ L ± generalizing the Matrix Tree Theorem for an unsigned graph. We present similar results for the signless net Laplacian matrix $ Q^{\pm }=D^{\pm }+A $ Q ± = D ± + A along with a combinatorial formula for its determinant. [ABSTRACT FROM AUTHOR]

Let m, n be integers such that 1

The looking-down mode of space airship bistatic radars faces complex sea–land clutter, and the mode of wide-range surveillance and the over-sight detection of the satellite platform generates a low SNR and range–Doppler ambiguity. The method traditionally used involves the transmission of multiple Pulse Repetition Frequencies (PRFs) and correlating them to solve the ambiguity. However, with a low SNR, the traditional disambiguation fails due to the large number of false alarms and target omissions. In order to solve this problem, a new algorithm for multi-target joint detection and range–Doppler disambiguation based on an ambiguity matrix is presented. Firstly, all possible state values corresponding to the ambiguous sequence are filled into the ambiguity matrix one by one. Secondly, the state values in the matrix cell are divided into several groups of subsequences according to the PRF. By disambiguating multiple sets of subsequences, performing subsequence fusion, and then undertaking point aggregation, the targets can be effectively detected in scenarios with a strong clutter rate, the false alarms can be suppressed, and the disambiguation of the range and Doppler is completed. The simulation shows that the proposed algorithm has the strong ability to detect targets and perform ambiguity resolution in the scenario of a multi-target and multi-false alarm. [ABSTRACT FROM AUTHOR]

In the field of radar target detection, the conventional approach is to employ the range profile energy accumulation method for detecting extended targets. However, this method becomes ineffective when dealing with non-stationary and non-uniform radar clutter scenarios, as well as long-distance targets with weak radar cross sections (RCSs). In such cases, the signal-to-noise ratio (SNR) of the target echo is severely degraded, rendering the energy accumulation detection algorithm unreliable. To address this issue, this paper presents a new extended target detection method based on the maximum eigenvalue of the Hermitian matrix. This method utilizes a detection model that incorporates observed data and employs the likelihood ratio test (LRT) theory to derive the maximum eigenvalue detector at low SNR. Specifically, the detector constructs a matrix using a sliding window block with the available data and then computes the maximum eigenvalue of the covariance matrix. Subsequently, the maximum eigenvalue matrix is transformed into a one-dimensional eigenvalue image, enabling extended target detection through analogy with the energy accumulation detection method. Furthermore, this paper analyzes the proposed extended target detection method from both theoretical and experimental perspectives, validating it through field-measured data. The results obtained from the measured data demonstrate that the method effectively enhances the SNR in low SNR conditions, thereby improving target detection performance. Additionally, the method exhibits robustness across different scattering center targets. [ABSTRACT FROM AUTHOR]

In this paper, the clique polynomial method (CPM) is proposed for the numerical solution of Singular Lane-Emden type equations. A new operational matrix of integration concerning clique polynomials of the complete graph has been generated in its generalized representation. This operational matrix is applied to differential equations and is transformed into a system of algebraic equations that can be solved efficiently with the help of Newton's iterative solver. The efficiency of the developed method is revealed by considering illustrative examples, and the obtained results are compared favorably with the corresponding exact solution and errors. [ABSTRACT FROM AUTHOR]

The concept of weighted numerical radius has been defined recently. In this article, we obtain several upper bounds for the weighted numerical radius of operators and 2 × 2 operator matrices which generalize and improve some well-known famous inequalities for the classical numerical radius. The article also derives an upper bound for the weighted numerical radius of the Aluthge transformation, T ~ of an operator T ∈ B (H) , where T ~ = | T | 1 / 2 U | T | 1 / 2 , and T = U | T | is the Canonical Polar decomposition of T. [ABSTRACT FROM AUTHOR]

Modeling fluid flow in fractured reservoirs can be complicated, not only because of the permeability differences between fractures and matrix, but also due to the complex network of fractures. If the exact fracture geometry is known, accurate flow models can be obtained by discrete fracture matrix (DFM) models. However, implementing the DFM model for densely fractured reservoirs is practically infeasible because of the computational cost and time limitations. To overcome this problem, dual/multi-continuum models such as dual-porosity (DP), multiple interacting continua (MINC), and extended MINC (E-MINC) have been developed. In these upscaling methods, fracture-to-matrix exchange flows are calculated using shape factors, volume fractions, and transmissibility exchange parameters. The main bottleneck of dual/multi-continuum models is in estimating parameters for exchange flow in complex fracture networks. In this study, the E-MINC model parameters are improved by using a K-means clustering algorithm in Python, taking the equi-dimensional DFM (ED-DFM) model as a reference solution. The E-MINC module is developed under MATLAB LiveLink for COMSOL. Pressure levels for the determination of volume fractions and transmissibility of interacting continua are optimized using a Python scikit-learn K-means clustering algorithm. It is shown that for bar-type fractures, both the E-MINC K-means model and DP model with the Vermeulen equation provide better results than DP models having constant shape factors. For a naturally fractured domain, the E-MINC model with K-means clustering provides better results than the E-MINC model with uniformly distributed iso-pressure levels. The transient results of E-MINC K-means and ED-DFM models on both small- and large-scale domains are very similar. [ABSTRACT FROM AUTHOR]

In this paper, we construct the super Virasoro algebra with an arbitrary conformal dimension Δ from the generalized ℛ (p , q) -deformed quantum algebra and investigate the ℛ (p , q) -deformed super Virasoro algebra with the particular conformal dimension Δ = 1. Furthermore, we perform the ℛ (p , q) -conformal Virasoro n -algebra, the ℛ (p , q) -conformal super Virasoro n -algebra (n -even) and discuss a toy model for the ℛ (p , q) -conformal Virasoro constraints and ℛ (p , q) -conformal super Virasoro constraints. Besides, we generalized the notion of the ℛ (p , q) -elliptic Hermitian matrix model with an arbitrary conformal dimension Δ. Finally, we deduce relevant particular cases generated by quantum algebras known in the literature. [ABSTRACT FROM AUTHOR]

The Analytic Hierarchy Process (AHP) is a multi-criteria decision-making method that relies on constructing a pairwise comparison matrix (PCM) based on decision-makers' (DMs) judgments regarding alternatives and criteria. The objective is to obtain weights and ultimately rank and select alternatives. However, in some cases, DMs may not provide judgments due to a lack of experience, knowledge, or reluctance to express opinions on the subject, resulting in missing pairs in the PCM. Existing techniques for filling these missing pairs have inherent limitations. This paper demonstrates the application of the pymissingAHP algorithm, implemented in Python and available at https://bit.ly/3UFdqSZ, to address missing pairs using mono and multiobjective approaches. The pymissingAHP algorithm employs a Genetic Algorithm (GA) featuring a specialized encoding to address this issue. Our approach can handle mono or multiple objective scenarios, which involve minimizing the consistency index and maintaining the ranking of weights as defined by experts. Additionally, the algorithm accommodates fuzzy numbers within the AHP framework (FAHP). PCMs containing numerous missing values may yield multiple solutions and not accurately reflect experts' opinions. Although our approach can solve entirely depleted PCMs, obtaining as many comparisons as possible is recommended to ensure a faithful representation of expert opinions for the decision-makers. The pymissingAHP algorithm provides a significant advantage: the capacity to seek solutions that address single or multiple objectives, utilizing continuous or discrete values, and additionally, the ability to solve FAHP problems. [ABSTRACT FROM AUTHOR]

A signed network Σ = (G , σ) with the underlying graph G = (V , E) , used as a mathematical model for analyzing social networks, has each edge in E with a weight 1 or − 1 assigned by the signature function σ. In this paper, we deal with two types of Detour Distance Laplacian (DDL) matrices for signed networks. We characterize balance in signed social networks using these matrices and we compute the DDL spectrum of certain unbalanced signed networks, as balanced signed networks behave like unsigned ones. [ABSTRACT FROM AUTHOR]

We study order units in the real group ring and the augmentation ideal, as well as in matrix algebras. We identify an infinite family of order units in the powers of the augmentation ideal, that includes the Laplacian, and show that these order units are naturally obtained via cohomological operations from natural diagonal order units in matrix algebras. [ABSTRACT FROM AUTHOR]

The B-operators (abbreviation for Brownian-type operators) are upper triangular 2\times 2 block matrix operators that satisfy certain algebraic constraints. The purpose of this paper is to characterize the weak, the strong and the uniform stability of B-operators, respectively. This is achieved by giving equivalent conditions for the convergence of powers of a B-operator in each of the corresponding topologies. A more subtle characterization is obtained for B-operators with subnormal (2,2) entry. The issue of the strong stability of the adjoint of a B-operator is also discussed. [ABSTRACT FROM AUTHOR]

We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the fundamental theorem of algebra for quaternions to polynomials with two monomials in the leading form, while showing that it fails for three. [ABSTRACT FROM AUTHOR]

For solving large sparse systems of linear equations, we construct a paradigm of two‐step matrix splitting iteration methods and analyze its convergence property for the nonsingular and the positive‐definite matrix class. This two‐step matrix splitting iteration paradigm adopts only one single splitting of the coefficient matrix, together with several arbitrary iteration parameters. Hence, it can be constructed easily in actual applications, and can also recover a number of representatives of the existing two‐step matrix splitting iteration methods. This result provides systematic treatment for the two‐step matrix splitting iteration methods, establishes rigorous theory for their asymptotic convergence, and enriches algorithmic family of the linear iteration solvers, for the iterative solutions of large sparse linear systems. [ABSTRACT FROM AUTHOR]

We consider cyclic algebras, Milnor's symbol algebras, and certain graded algebra structures on them. We classify these gradings with respect to both isomorphism and equivalence relations. Some of them induce gradings on matrix algebras, which we also classify. As an application, we obtain the classification of all group gradings on the algebra M p (F) , where p is a prime number and F is an arbitrary field. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic zero and let m ≥ 2 be an integer. In this paper, we prove that if a group grading on U T m (F) admits a graded involution then this grading is a coarsening of a Z ⌊ m 2 ⌋ -grading on U T m (F) and the graded involution is equivalent to the reflection or symplectic involution on U T m (F) , this grading is called the finest grading on U T m (F). Furthermore, if m ≤ 4 the algebra U T m (F) with the finest grading satisfies no non-trivial monomial identities. For the finest grading, a finite basis for the (Z ⌊ m 2 ⌋ , ⁎) -identities is exhibited with the reflection and symplectic involutions and the asymptotic growth of the (Z ⌊ m 2 ⌋ , ⁎) -codimensions is determined. As a consequence, we prove that for any G -grading on U T m (F) and any graded involution the (G , ⁎) -exponent is m. Finally, we study the algebra U T 3 (F). For this algebra, there are, up to equivalence, two non-trivial gradings that admit a graded involution: the canonical Z -grading and the Z 2 -grading induced by (0 , 1 , 0). We determine a basis for the (Z 2 , ⁎) -identities and we compute the codimension sequence for the (Z 2 , ⁎) -graded identities for U T 3 (F). [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe the irreducible modules for the Jordan algebra of 2 × 2 matrices over an algebraically closed field of characteristic different from 2, 3 and 5 in the class of the commutative power-associative algebras. All irreducible non-unital modules, and irreducible unital modules up to dimension three are classified, namely we find seven non-parameterized and five families of parameterized modules of dimension three. For every k ≥ 2 , an irreducible module of dimension 3k is also constructed. [ABSTRACT FROM AUTHOR]

Given a matrix A\in \mathrm {GL}_d(\mathbb {Z}). We study the pseudorandomness of vectors \mathbf {u}_n generated by a linear recurrence relation of the form \begin{equation*} \mathbf {u}_{n+1} \equiv A \mathbf {u}_n \pmod {p^t}, \qquad n = 0, 1, \ldots, \end{equation*} modulo p^t with a fixed prime p and sufficiently large integer t \geqslant 1. We study such sequences over very short segments of length which has not been accessible via previously used methods. Our technique is based on the method of N. M. Korobov [Mat. Sb. (N.S.) 89(131) (1972), pp. 654–670, 672] of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. Ford [Proc. London Math. Soc. (3) 85 (2002), pp. 565–633]. This is combined with some ideas from the work of I. E. Shparlinski [Proc. Voronezh State Pedagogical Inst., 197 (1978), 74–85 (in Russian)] which allows us to construct polynomial representations of the coordinates of \mathbf {u}_n and control the p-adic orders of their coefficients in polynomial representations. [ABSTRACT FROM AUTHOR]

The concept of the Fan product of several M-matrices is introduced. Furthermore, two new lower bounds of the minimum eigenvalue of the Fan product of several M-matrices are proposed. These obtained new lower bounds generalize and improve some earlier findings. One example is presented to illustrate the precision of the given lower bounds. [ABSTRACT FROM AUTHOR]

In this study, a novel unified stability criterion is first proposed for general fractional-order systems with time delay when the fractional order is from 0 to 1. Such a new unified criterion has the advantage of having an initiative link with the fractional orders. A further advantage is that the corresponding asymptotic stability theorem, derived from the proposed criterion used to analyze the asymptotic stability, is only slightly affected by the change of the fractional order. In addition, the unified stability criterion is applied to general multi-dimensional nonlinear fractional-order systems with time delays, the corresponding asymptotic stability criterion is applied by combining the vector Lyapunov function with the M-matrix method. Compared with the traditional stability criterion, the unified stability criterion is slightly influenced by the changing fractional order and large time delays. The reliability and effectiveness of the novel uniform stability criterion were verified through three representative examples. [ABSTRACT FROM AUTHOR]

We investigate the symmetries of the so-called generalized extended Cantero–Moral–Velázquez (CMV) matrices. It is well-documented that problems involving reflection symmetries of standard extended CMV matrices can be subtle. We show how to deal with this in an elegant fashion by passing to the class of generalized extended CMV matrices via explicit diagonal unitaries in the spirit of Cantero–Grünbaum–Moral–Velázquez. As an application of these ideas, we construct an explicit family of almost-periodic CMV matrices, which we call the mosaic unitary almost-Mathieu operator, and prove the occurrence of exact mobility edges. That is, we show the existence of energies that separate spectral regions with absolutely continuous and pure point spectrum and exactly calculate them. [ABSTRACT FROM AUTHOR]

This paper proposes explicit solutions for the algebraic Riccati matrix equation. For single‐input systems in controllable canonical form, the explicit Hermitian solutions of the non‐homogeneous Riccati equation are obtained using the entries of the system matrix, the closed‐loop system matrix, and the weighting matrix. The unknown entries of the closed‐loop system matrix are solved by scalar quadratic equations. For a homogeneous Riccati equation with a zero weighting matrix, the explicit solutions are proposed analytically in terms of the system eigenvalues. The advantages of the explicit solutions are threefold: first, if the system is controllable, the solution is directly given and the invariant subspaces of the Hamiltonian matrix are not required; second, if the system is near singularity, the explicit solution has higher numerical precision compared with the solution computed by numerical algorithms; third, for a real system in the controllable canonical form, the non‐negativity can be analysed for the explicit almost stabilizing solution. [ABSTRACT FROM AUTHOR]