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The purpose of this article is to introduce the readers, especially the researchers, to some topics connected with functions of matrix argument, scaling models, distributions of products and ratios, Bayesian structures, symmetric products and symmetric ratios of matrices, scalar and matrix-variate fractional integrals, functions of matrix argument through entropy optimization, singular matrix-variate gamma and beta functions etc which are currently active so that interested readers can get into these classes of problems for their current research or teaching. Let X be a p × q, p = q matrix of rank p in the real domain. If the function f(X), associated with X, is a function of XX', where a prime denotes the transpose, then such a function appears in a number of different disciplines. This paper examines the recent developments in such matrix-variate functions when X is in the real or complex domain. Connections to Bayes procedures, quantum physics, scalar and matrix texture models in communication and engineering problems, fractional integrals, distributions of symmetric products and symmetric ratios of matrices, singular matrix-variate gamma and beta functions and other related areas are pointed out. Only an overview of the current research in these topics with some illustrative examples are given in this paper. Since the material is summarized from the author's own works, most of the references are author's own papers, and hence similarity index, similarity with author's own works, may be high. The materials also cover some current results which are being published. [ABSTRACT FROM AUTHOR]

In this paper, we present a statistical approach to evaluate the relationship between variables observed in a two-factors experiment. We consider a three-level model with covariance structure Σ ⊗ Ψ 1 ⊗ Ψ 2 , where Σ is an arbitrary positive definite covariance matrix, and Ψ 1 and Ψ 2 are both correlation matrices with a compound symmetric structure corresponding to two different factors. The Rao's score test is used to test the hypotheses that observations grouped by one or two factors are uncorrelated. We analyze a fermentation process to illustrate the results. Supplementary materials accompanying this paper appear online. [ABSTRACT FROM AUTHOR]

Although Selberg-type single positive definite symmetric matrices gamma and beta integrals have been evaluated by several authors, see e.g., Askey and Richards [1], Gupta and Kabe [2, 4], Mathai [8], and elsewhere in the vast multivariate statistical analysis literature. However, several other types of Selberg-type integrals appear to have been neglected in the literature. Thus e.g., Selberg-type integrals associated with inverse Wishart densities, inverse multivariate beta densities, their noncentral counterparts, etc, have not been explored as yet. The present paper records Selberg-type generalized quadratic forms gamma and beta integrals. Our methodology is based on hypercomplex (HC) multivariate normal distribution theory, Kabe [6]. [ABSTRACT FROM AUTHOR]

Frieze patterns have been introduced by Coxeter [Acta Arith. 18 (1971), pp. 297–310] in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Çanakçı and Jørgensen [Adv. in Appl. Math. 131 (2021), Paper No. 102253], are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals. To any frieze pattern one can associate a symmetric matrix using a triangular fundamental domain of the frieze pattern in the upper and lower half of the matrix and putting zeroes on the diagonal. Broline, Crowe and Isaacs [Geometriae Dedicata 3 (1974), pp. 171–176] have found a formula for the determinants of these matrices and their work has later been generalized in various directions by other authors. These frieze pattern determinants are the main focus of our paper. As our main result we show that this determinant behaves well with respect to gluing weak friezes: the determinant is the product of the determinants for the pieces glued, up to a scalar factor coming from the gluing diagonal. Then we give several applications of this result, showing that formulas from the literature, obtained by Broline-Crowe-Isaacs, Baur-Marsh [J. Combin. Theory Ser. A 119 (2012), pp. 1110–1122], Bessenrodt-Holm-Jørgensen [J. Combin. Theory Ser. A 123 (2014), pp. 30–42] and Maldonado [ Frieze matrices and infinite frieze patterns with coefficients , Preprint, arXiv: 2207.04120 , 2022] can all be obtained as consequences of our result. [ABSTRACT FROM AUTHOR]

The rational field Q is highly desired in many applications. Algorithms using the rational number field Q algebraic number fields use only integer arithmetics and are easy to implement. Therefore, studying and designing systems and expansions with coefficients in Q or algebraic number fields is particularly interesting. This paper discusses constructing quasi-tight framelets with symmetry over an algebraic field. Compared to tight framelets, quasi-tight framelets have very similar structures but much more flexibility in construction. Several recent papers have explored the structure of quasi-tight framelets. The construction of symmetric quasi-tight framelets directly applies the generalized spectral factorization of 2 × 2 matrices of Laurent polynomials with specific symmetry structures. We adequately formulate the latter problem and establish the necessary and sufficient conditions for such a factorization over a general subfield F of C , including algebraic number fields as particular cases. Our proofs of the main results are constructive and thus serve as a guideline for construction. We provide several examples to demonstrate our main results. [ABSTRACT FROM AUTHOR]

Let a = (a i) i ≥ 1 be a sequence in a field F , and f : F × F → F be a function such that f (a i , a i) ≠ 0 for all i ≥ 1. For any tournament T over [ n ] , consider the n × n symmetric matrix M T with zero diagonal whose (i , j) th entry (for i < j) is f (a i , a j) if i → j in T , and f (a j , a i) if j → i in T. It is known [3] that if T is a uniformly random tournament over [ n ] , then rank (M T) ≥ (1 2 − o (1)) n with high probability when char (F) ≠ 2 and f is a linear function. In this paper, we investigate the other extremal question: how low can the ranks of such matrices be? We work with sequences a that take only two distinct values, so the rank of any such n × n matrix is at least n / 2. First, we show that the rank of any such matrix depends on whether an associated bipartite graph has certain eigenvalues of high multiplicity. Using this, we show that if f is linear, then there are real matrices M T (f ; a) of rank at most n 2 + O (1). For rational matrices, we show that for each ε > 0 we can find a sequence a (ε) for which there are matrices M T (f ; a) of rank at most (1 2 + ε) n + O (1). These matrices are constructed from symmetric designs, and we also use them to produce bisection-closed families of size greater than ⌊ 3 n / 2 ⌋ − 2 for n ≤ 15 , which improves the previously best known bound given in [5]. [ABSTRACT FROM AUTHOR]

Free spectrahedra are dimension free solution sets to linear matrix inequalities of the form \[ L_A(X)=I_d{\otimes}I_n+A_1{\otimes}X_1+A_2{\otimes}X_2+\dots+A_g{\otimes}X_g\succeq 0, \]LA(X)=Id⊗In+A1⊗X1+A2⊗X2+⋯+Ag⊗Xg⪰0, where the $ A_i $ Ai and $ X_i $ Xi are symmetric matrices and the $ X_i $ Xi have any size $ n\times n $ n×n. Free spectrahedra are ubiquitous in systems engineering, operator algebras, and the theory of matrix convex sets. Matrix and free extreme points of free spectrahedra are particularly important. We present theoretical, algorithmic, and experimental results illuminating basic properties of extreme points. For example, though many authors have studied matrix and free extreme points, it has until now been unknown if these two types of extreme points are actually different. This paper settles that issue.We also present and analyze several algorithms. Namely, we perfect an algorithm for computing an expansion of an element of a free spectrahedron in terms of free extreme points. We also give algorithms for testing if a point is matrix extreme and for computing matrix extreme points that are not free extreme. [ABSTRACT FROM AUTHOR]

This paper investigates the spectral properties of block circulant matrices with high-order symmetric (or Hermitian) blocks. We analyze cases with dependent or sparse independent entries within these blocks. Additionally, we analyze the distribution of singular values for the product of independent circulant matrices with non-Hermitian blocks. [ABSTRACT FROM AUTHOR]

In this paper, we address the extremal inverse eigenvalue problem for pentadiagonal matrices. We provide sufficient conditions for their existence and realizability through new constructions that consider spectral data of its leading principal submatrices. Finally, we present some examples generated from the algorithmic procedures derived from our results. [ABSTRACT FROM AUTHOR]

The symmetric nonnegative inverse eigenvalue problem (SNIEP) is the problem of characterizing all possible spectra of entry-wise nonnegative symmetric matrices of given dimension. A list of real numbers is said to be symmetrically realizable if it is the spectrum of some nonnegative symmetric matrix. One of the most general sufficient conditions for realizability is the so-called C-realizability, which amounts to some kind of compensation between the positive and negative entries of the list of real numbers whose realizability one is trying to decide. A combinatorial characterization of C-realizable lists with zero trace was given in [11]. In this paper we make use of a recursive method for constructing simmetrically realizable lists due to Ellard and Šmigoc [3] to extend this combinatorial characterization of C-realizability to general lists with nonnegative trace. One consequence of this characterization is that the set of nonnegative C-realizable lists is a union of polyhedral cones whose faces are described by equations involving only linear combinations with coefficients 1 and −1 of the entries in the list. Another remarkable consequence is the monotonicity of C-realizability, i.e., the operation of increasing any positive entry of a C-realizable list preserves C-realizability. [ABSTRACT FROM AUTHOR]

In this paper, we analytically derive closed-form expressions for the tangency portfolio weights: the fully invested portfolio that maximizes the expected return over the risk-free rate, relative to the volatility of the portfolio return. We explicitly derive this portfolio from a range of underlying return models and show examples where it coincides with different well-known smart beta products. Specifically, we find the closed-form expression for the tangency portfolio weights for a return model with compound symmetric correlation matrix. We also deduce the tangency portfolio weights for the CAPM return model and illustrate in a case study that the estimated tangency portfolio weights may distinctly deviate from the market value weighted portfolio. Furthermore, we show that depending on the return model, the tangency portfolio weights may take a diverse set of shapes; from very diversified to highly concentrated portfolios. [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n, where n is a positive integer. A spanning subgraph F of G is called a path-factor if every component of F is a path of order at least 2. A P ≥ k -factor means a path-factor in which every component admits order at least k ( k ≥ 2 ). The distance matrix D (G) of G is an n × n real symmetric matrix whose (i, j)-entry is the distance between the vertices v i and v j . The distance signless Laplacian matrix Q (G) of G is defined by Q (G) = T r (G) + D (G) , where Tr(G) is the diagonal matrix of the vertex transmissions in G. The largest eigenvalue η 1 (G) of Q (G) is called the distance signless Laplacian spectral radius of G. In this paper, we aim to present a distance signless Laplacian spectral radius condition to guarantee the existence of a P ≥ 2 -factor in a graph and claim that the following statements are true: (i) G admits a P ≥ 2 -factor for n ≥ 4 and n ≠ 7 if η 1 (G) < θ (n) , where θ (n) is the largest root of the equation x 3 - (5 n - 3) x 2 + (8 n 2 - 23 n + 48) x - 4 n 3 + 22 n 2 - 74 n + 80 = 0 ; (ii) G admits a P ≥ 2 -factor for n = 7 if η 1 (G) < 25 + 161 2 . [ABSTRACT FROM AUTHOR]

Let r and s be positive integers such that r ⩾ 2. Let U and V be vector spaces over fields F and K , respectively, such that dim ⁡ U ⩾ 3 and F has at least r + 1 elements. In this paper, we characterize surjective maps ψ : S r U → S s V preserving adjacency in both directions on symmetric tensors of finite order, which generalizes Hua's fundamental theorem of geometry of symmetric matrices. We give examples showing the indispensability of the assumptions dim ⁡ U ⩾ 3 and the cardinality | F | ⩾ r + 1 in our result. [ABSTRACT FROM AUTHOR]

Matrix factorization is an inference problem that has acquired importance due to its vast range of applications that go from dictionary learning to recommendation systems and machine learning with deep networks. The study of its fundamental statistical limits represents a true challenge, and despite a decade-long history of efforts in the community, there is still no closed formula able to describe its optimal performances in the case where the rank of the matrix scales linearly with its size. In the present paper, we study this extensive rank problem, extending the alternative 'decimation' procedure that was recently introduced by Camilli and Mézard, and carry out a thorough study of its performance. Decimation aims at recovering one column/line of the factors at a time, by mapping the problem into a sequence of neural network models of associative memory at a tunable temperature. The main advantage of decimation is that its theoretical performance can be studied using statistical physics techniques. In this paper we provide the general analysis applying to a large class of compactly supported priors and we show that the replica symmetric free entropy of the neural network models takes a universal form in the low temperature limit. We then study the decimation phase diagrams in two concrete cases, the sparse Ising prior and the uniform prior, for which we show that matrix factorization is theoretically possible when the ration P / N of rank to variable is below a certain threshold. This suggest that a possible route to solving algorithmically the matrix factorization problem could be to find efficient decimation algorithms; in this respect, we show that simple simulated annealing, though being effective for limited signal sizes, is not scalable. [ABSTRACT FROM AUTHOR]

Estimation of variance components is essential in data analysis based on a linear mixed model. Among different estimation methods, the most commonly used in practice are likelihood-based approaches: maximum likelihood and restricted maximum likelihood. These methods do not guarantee positive-unbiased estimators, however. The method of moments ensures unbiasedness but may yield negative estimates, which can cause problems when estimating fixed effects by generalized least squares. The paper develops an algorithm to derive strictly positive unbiased estimators with minimum variance in a well-defined class. The key idea is to take the method-of-moments estimator, given by a quadratic form of a symmetric matrix A , as a starting point and modify it using the class of square non-singular regularization matrices Q while preserving unbiasedness in addition to ensuring positivity. Different subclasses of structured Q are possible for convenience instead of all possible Q matrices. A search algorithm then finds a local or global optimal matrix A depending on Q and the corresponding optimal variance component estimator by minimizing the variance, a function of the unknown variance components and kurtosis parameters in the model. Availability of an optimal matrix A means that the theoretical and numerical properties of the estimator can be conveniently studied, also in small samples. The proposed method further allows finding matrices A leading to quadratic forms closely approximating the corresponding numerical values of the likelihood-based estimates. The paper investigates the dependence of variance functions on the unknown model parameters. Using two illustrative examples, Examples I and II, the paper also illustrates the use of the matrix Q and the determination of the kurtosis parameters in the search for the optimal variance component estimators by keeping the bias zero and the variance trim for the bias–variance trade-offs. [ABSTRACT FROM AUTHOR]

The parameterized block splitting (PBS) is a convergent and efficient iterative method to solve the large complex symmetric linear systems. In this paper, by using PBS iterative technique, the Newton equation is approximately solved, then we establish the modified Newton-PBS iterative method to solve the complex nonlinear systems whose Jacobian matrices are large, sparse, and complex symmetric. Subsequently, the local convergence analysis are explored under appropriate conditions. Ultimately, we apply the new method and several known methods to experimental numerical examples, and experimental results verify the superiority and efficiency of our new method. Especially, in terms of CPU time and iteration steps, our method is obviously better. [ABSTRACT FROM AUTHOR]

In this paper, we advocate a new technique to determine inner bounds for the extreme eigenvalues of real symmetric matrices. Our method involves the matrix elements and compares favourably with existing methods. We also show how the bounds can be optimized. [ABSTRACT FROM AUTHOR]

Large-scale symmetric and asymmetric matrices have emerged in predicting the relationship between genes and diseases. The emergence of large-scale matrices increases the computational complexity of the problem. Therefore, using low-rank matrices instead of original symmetric and asymmetric matrices can greatly reduce computational complexity. In this paper, we propose an approximation conjugate gradient method for solving the low-rank matrix recovery problem, i.e., the low-rank matrix is obtained to replace the original symmetric and asymmetric matrices such that the approximation error is the smallest. The conjugate gradient search direction is given through matrix addition and matrix multiplication. The new conjugate gradient update parameter is given by the F-norm of matrix and the trace inner product of matrices. The conjugate gradient generated by the algorithm avoids SVD decomposition. The backtracking linear search is used so that the approximation conjugate gradient direction is computed only once, which ensures that the objective function decreases monotonically. The global convergence and local superlinear convergence of the algorithm are given. The numerical results are reported and show the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the Pascal matrices presented in [9] as a functional for several variables. The entries of these matrices are defined on the set R ⊆ Z ≥ 0 n with multidimensional binomial coefficients along with several variables. In the following, we will investigate the algebraic properties of these matrices in terms of products, powers, and inverses of the matrices and also some factorization formulas using this method. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the construction of symmetric matrix whose corresponding graph is connected and unicyclic using some pre-assigned spectral data. Spectral data for the problem consist of the smallest and the largest eigenvalues of each leading principal submatrices. Inverse eigenvalue problem (IEP) with this set of spectral data is generally known as the extremal IEP. We use a standard scheme of labeling the vertices of the graph, which helps in getting a simple relation between the characteristic polynomials of each leading principal submatrix. Sufficient condition for the existence of the solution is obtained. The proof is constructive, hence provides an algorithmic procedure for finding the required matrix. Furthermore, we provide the condition under which the same problem is solvable when two particular entries of the required matrix satisfy a linear relation. [ABSTRACT FROM AUTHOR]

In this paper, we study parameter deformations of matrix valued orthogonal polynomials. These deformations are built on the use of certain matrix valued operators which are symmetric with respect to the matrix valued inner product defined by the orthogonality weight. We show that the recurrence coefficients associated with these operators satisfy generalizations of the non-Abelian lattice equations. We provide a Lax pair formulation for these equations, and an example of deformed Hermite-type matrix valued polynomials is discussed in detail. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph on n vertices, and let G σ be an orientation of G with skew adjacency matrix S (G σ). Let d i be the degree of the vertex v i in G. The skew Randić matrix of G σ is the n × n real skew symmetric matrix R S (G σ) = [ (R S) i j ] , where (R S) i j = − (R S) j i = (d i d j) − 1 2 if (v i , v j) is an arc of G σ , and (R S) i j = (R S) j i = 0 otherwise. The skew spectral radius ρ S (G σ) and the skew Randić spectral radius ρ R S (G σ) of G σ are defined as the spectral radius of S (G σ) and R S (G σ) respectively. In this paper we give upper bounds for the skew spectral radius and skew Randić spectral radius of general random oriented graphs. [ABSTRACT FROM AUTHOR]

An efficient iteration method is provided in this paper for solving a class of nonlinear systems whose Jacobian matrices are complex and symmetric. The modified Newton-NDSS method is developed and applied to the class of nonlinear systems by adopting the modified Newton method as the outer solver and a new double-step splitting (NDSS) iteration scheme as the inner solver. Additionally, we theoretically analyze the local convergent properties of the new method under the weaker Hölder conditions. Lastly, the new method is compared numerically with some existing ones and the numerical experiments solving the nonlinear equations demonstrate the superiority of the Newton-NDSS method. [ABSTRACT FROM AUTHOR]

Nowadays, deep representations have gained significant attention due to their outstanding performance in a wide range of tasks. However, the interpretability of deep representations in specific applications poses a significant challenge. For instances where the generated quantity matrices exhibit symmetry, this paper introduces a variant of deep matrix factorization (deep MF) called deep L p smooth symmetric non-negative matrix factorization (DSSNMF), which aims to improve the extraction of clustering structures inherent in complex hierarchical and graphical representations in high-dimensional datasets by improving the sparsity of the factor matrices. We successfully applied DSSNMF to synthetic datasets as well as datasets related to post-traumatic stress disorder (PTSD) to extract several hierarchical communities. Specifically, we identified non-disjoint communities within the partial correlation networks of PTSD psychiatric symptoms, resulting in highly meaningful clinical interpretations. Numerical experiments demonstrate the promising applications of DSSNMF in fields like network analysis and medicine. [ABSTRACT FROM AUTHOR]

The main focus of this paper is the study of efficient multigrid methods for large linear systems with a particular saddle-point structure. Indeed, when the system matrix is symmetric, but indefinite, the variational convergence theory that is usually used to prove multigrid convergence cannot be directly applied. However, different algebraic approaches analyze properly preconditioned saddle-point problems, proving convergence of the Two-Grid method. In particular, this is efficient when the blocks of the coefficient matrix possess a Toeplitz or circulant structure. Indeed, it is possible to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle-point problem in terms of the associated symbols. In this paper, we propose a symbol based convergence analysis for problems that have a hidden block Toeplitz structure. Then, they can be investigated focusing on the properties of the associated generating function f , which consequently is a matrix-valued function with dimension depending on the block size of the problem. As numerical tests we focus on the matrix sequence stemming from the finite element approximation of the Stokes problem. We show the efficiency of the methods studying the hidden 9-by-9 block multilevel structure of the obtained matrix sequence. Moreover we propose an efficient algebraic multigrid method with convergence rate independent of the matrix size. Finally, we present several numerical tests comparing the results with state-of-the-art strategies. [ABSTRACT FROM AUTHOR]

The inverse eigenvalue problem of a graph G is the problem of characterizing all lists of eigenvalues of real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of G. The strong spectral property is a powerful tool in this problem, which identifies matrices whose entries can be perturbed while controlling the pattern and preserving the eigenvalues. The Matrix Liberation Lemma introduced by Barrett et al. in 2020 advances the notion to a more general setting. In this paper we revisit the Matrix Liberation Lemma and prove an equivalent statement, that reduces some of the technical difficulties in applying the result. We test our method on matrices of the form M = A ⊕ B and show how this new approach supplements the results that can be obtained from the strong spectral property only. While extending this notion to the direct sums of graphs, we discover a surprising connection with the zero forcing game on Cartesian products of graphs. Throughout the paper we apply our results to resolve a selection of open cases for the inverse eigenvalue problem of a graph on six vertices. [ABSTRACT FROM AUTHOR]

Due to properties such as interpolation, smoothness, and spline connections, Hermite subdivision schemes employ fast iterative algorithms for geometrically modeling curves/surfaces in CAGD and for building Hermite wavelets in numerical PDEs. In this paper, we introduce a notion of generalized Hermite (dyadic) subdivision schemes and then we characterize their convergence, smoothness and underlying matrix masks with or without interpolation properties. We also introduce the notion of linear-phase moments for achieving the polynomial-interpolation property. For any given integer m ∈ N , we constructively prove that there always exist convergent smooth generalized Hermite subdivision schemes with linear-phase moments such that their basis vector functions are spline functions in C m (R d) and have linearly independent integer shifts. As by-products, our results resolve convergence, smoothness, and existence of Lagrange, Hermite, or Birkhoff subdivision schemes. Even in dimension one our results significantly generalize and extend many known results on extensively studied univariate Hermite subdivision schemes. To illustrate the theoretical results in this paper, we provide examples of convergent generalized Hermite subdivision schemes with symmetric matrix masks having short support and smooth basis vector functions with or without interpolation property. [ABSTRACT FROM AUTHOR]

Similar to cells, bacteria, and other micro-organisms, synthetic chemically active colloids can harness the energy from their environment through a surface chemical reaction and use the energy to self-propel in fluidic environments. In this paper, we study the chemo-mechanical coupling that leads to the self-propulsion of chemically active colloids. The coupling between chemical reactions and momentum transport is a consequence of Onsager reciprocal relations. They state that the velocity and the surface reaction rate are related to mechanical and chemical affinities through a symmetric matrix. A consequence of Onsager reciprocal relations is that if a chemical reaction drives the motion of the colloid, then an external force generates a reaction rate. Here, we investigate Onsager reciprocal relations for a spherical active colloid that catalyzes a reversible surface chemical reaction between two species. We solve the relevant transport equations using a perturbation expansion and numerical simulations to demonstrate the validity of reciprocal relations around the equilibrium. Our results are consistent with previous studies and highlight the key role of solute advection in preserving the symmetry of the Onsager matrix. Finally, we show that Onsager reciprocal relations break down around a nonequilibrium steady state, which has implications for the thermal fluctuations of the active colloids used in experiments. [ABSTRACT FROM AUTHOR]

Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations. They can be formulated such that the system matrix is symmetric, but indefinite, so the variational convergence theory that is usually used to prove multigrid convergence cannot be applied. In a 2016 paper in Numerische Mathematik Notay has presented a different algebraic approach that analyzes properly preconditioned saddle point problems, proving convergence of the two-grid method. The present paper analyzes saddle point problems where the blocks are circulant within this framework. It contains sufficient conditions for convergence and optimal parameters for the preconditioning of the unilevel and multilevel saddle point problem and for the point smoother that is used. The analysis is based on the generating symbols of the circulant blocks. Further, it is shown that the structure can be kept on the coarse level, allowing for a recursive application of the approach in a W- or V-cycle and studying the "level independency" property. Numerical results demonstrate the efficiency of the proposed method in the circulant and the Toeplitz case. [ABSTRACT FROM AUTHOR]

In this paper we consider n coupled Schrödinger equations − d 2 y d t 2 + u (ω t) y = E y , y ∈ R n , where E = diag ( λ 1 2 , ... , λ n 2 ) is a diagonal matrix, u(ωt) is a real analytic quasi-periodic symmetric matrix. If the basic frequencies ω = (1, α), where α is irrational, it is proved that for most of sufficiently large λj, j = 1, ..., n, all the solutions of n coupled Schrödinger equations are bounded. Furthermore, if the basic frequencies satisfy that 0 ≤ β(α) < r, where β (α) = lim sup n → ∞ ln q n + 1 q n , qn is the sequence of denominations of the best rational approximations for α ∈ R \ Q , r is the initial analytic radius, we obtain the existence of n pairs of conjugate quasi-periodic solutions for most of sufficiently large λj, j = 1, ..., n. [ABSTRACT FROM AUTHOR]

An n-by-n real symmetric matrix $ H=[h_{i,j}] $ H = [ h i , j ] is said to be a Hankel matrix if $ h_{i,j}=h_{i-1,j+1} $ h i , j = h i − 1 , j + 1 , for each $ i=2,\ldots,n $ i = 2 , ... , n and $ j=1,\ldots,n-1 $ j = 1 , ... , n − 1. The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks when a list $ \Lambda =\{\lambda _{1},\ldots,\lambda _{n}\} $ Λ = { λ 1 , ... , λ n } of real numbers is the spectrum of an n-by-n symmetric nonnegative matrix H. In this paper, we search for conditions on the list $ \Lambda =\{\lambda _{1},\ldots,\lambda _{n}\} $ Λ = { λ 1 , ... , λ n } for the matrix H to be Hankel. For n = 3, sufficient conditions are established. In particular, a necessary and sufficient condition is obtained if Λ is a list of three nonnegative numbers. Also, if $ \sum _{i=1}^{n}\lambda _{i}=0 $ ∑ i = 1 n λ i = 0 , we give conditions for realizability by a Hankel matrix. Finally, we present a special type of list that can serve as the spectrum of a Hankel nonnegative matrix with positive trace. Several of our results are constructive and provide a Hankel realizing matrix. [ABSTRACT FROM AUTHOR]

The spectral radius ρ (A) is the maximum absolute value of the eigenvalues of a matrix A. In this paper, we establish some relationship between the spectral radius of a symmetric matrix and its principal submatrices, i.e., if A is partitioned as a 2 × 2 block matrix A = ( 0 A 12 A 21 A 22 ) , then ρ (A) 2 ≤ ρ 2 2 + θ ⁎ , where θ ⁎ is the largest real root of the equation μ 2 = (x − ν) 2 (ρ 2 2 + x) and ρ 2 = ρ (A 22) , μ = ρ (A 12 A 22 A 21) , ν = ρ (A 12 A 21). Furthermore, the results are used to obtain several upper bounds of the spectral radius of graphs, which strengthen or improve some known results. [ABSTRACT FROM AUTHOR]

The study of regular incidence structures, such as projective planes and symmetric block designs, is a well established topic in discrete mathematics. The work of Bruck, Ryser and Chowla in the mid-twentieth century applied the Hasse-Minkowski local-global theory for quadratic forms to derive non-existence results for certain design parameters. Several combinatorialists have provided alternative proofs of this result, replacing conceptual arguments with algorithmic ones. In this paper, we show that the methods required are purely linear-algebraic and are no more difficult conceptually than the theory of the Jordan Canonical Form. Computationally, they are rather easier. We conclude with some classical and recent applications to design theory, including a novel application to the decomposition of incidence matrices of symmetric designs. [ABSTRACT FROM AUTHOR]

This paper studies a priori and regularity estimates of Evans-Krylov type in Hölder spaces for fully nonlinear uniformly elliptic and parabolic equations of second order when the operator fails to be concave or convex in the space of symmetric matrices. In particular, it is assumed that either the level sets are convex or the operator is concave, convex or close to a linear function near infinity. As a byproduct, these results imply polynomial Liouville theorems for entire solutions of elliptic equations and for ancient solutions to parabolic problems. [ABSTRACT FROM AUTHOR]

Let S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I be the determinantal ideal of maximal minors or Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S-module structures of Ext^j_S(S/I^t, S) and Ext^j_S(S/Pf^t, S), from which we get the degrees of generators of these Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H^j_\mathfrak {m}(S/I^t). [ABSTRACT FROM AUTHOR]

( R , S )-(skew) symmetric matrices have numerous applications in civil engineering, information theory, numerical analysis, etc. In this paper, we deal with the ( R , S )-(skew) symmetric solutions to the quaternion matrix equation A X B = C . We use a real representation A τ to obtain the necessary and sufficient conditions for A X B = C to have ( R , S )-(skew) symmetric solutions and derive the solutions when it is consistent. We also derive the least-squares ( R , S )-(skew) symmetric solution to the above matrix equation. [ABSTRACT FROM AUTHOR]

The paper is a survey and concerns with infinite symmetric block Jacobi matrices J with m × m-matrix entries. We discuss several results on general block Jacobi matrices to be either self-adjoint or have maximal as well as intermediate deficiency indices. We also discuss several conditions for J to have discrete spectrum. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with plane wave discontinuous Galerkin (PWDG) methods for time-harmonic Maxwell equations in three-dimensional anisotropic media, for which the coefficients of the equations are piecewise constant symmetric matrices, where each constant symmetric matrix is defined on a medium (subdomain). By using suitable scaling transformations and coordinate (complex) transformations on every subdomain, the original Maxwell equation in anisotropic media is transformed into a Maxwell equation in isotropic media occupying a union domain of specific subdomains of complex Euclidean space. Based on these transformations, we define anisotropic plane wave basis functions and discretize the considered Maxwell equations by PWDG method with the proposed plane wave basis functions. We derive error estimates of the resulting approximate solutions, and further introduce a practically feasible local hp-refinement algorithm, which substantially improves accuracies of the approximate solutions. Numerical results indicate that the approximate solutions generated by the proposed PWDG methods possess high accuracy for the case of strong discontinuity media. [ABSTRACT FROM AUTHOR]

In this paper, we consider a coupled Brusselator model of chemical reactions, for which no symmetry for the coupling matrices is assumed. We show that the model can undergo a Hopf bifurcation, and consequently periodic solutions can arise when the dispersal rates are large. Moreover, the effect of the coupling matrices on the Hopf bifurcation value is considered for a special case. [ABSTRACT FROM AUTHOR]

Given a family of nearly commuting symmetric matrices, we consider the task of computing an orthogonal matrix that nearly diagonalizes every matrix in the family. In this paper, we propose and analyze randomized joint diagonalization (RJD) for performing this task. RJD applies a standard eigenvalue solver to random linear combinations of the matrices. Unlike existing optimization-based methods, RJD is simple to implement and leverages existing high-quality linear algebra software packages. Our main novel contribution is to prove robust recovery: Given a family that is e-near to a commuting family, RJD jointly diagonalizes this family, with high probability, up to an error of norm O (epsilon). We also discuss how the algorithm can be further improved by deflation techniques and demonstrate its state-of-the-art performance by numerical experiments with synthetic and real-world data. [ABSTRACT FROM AUTHOR]

The fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulting numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by an integral operator with zero boundary condition. Second, because the solutions of fractional diffusion equations are usually singular near the boundary, we use a fractional interpolation function in the region near the boundary and use a classical interpolation function in other intervals. Then, we apply a finite difference scheme to the discrete fractional Laplacian operator and fractional diffusion equation with the above fractional interpolation function and classical interpolation function. Moreover, it is found that the differential matrix of the above scheme is a symmetric matrix and strictly row-wise diagonally dominant in special fractional interpolation functions. Third, we show a finite volume scheme for a discrete fractional diffusion equation by fractional interpolation function and classical interpolation function and analyze the properties of the differential matrix. Finally, the numerical experiments are given, and we verify the correctness of the theoretical results and the efficiency of the schemes. [ABSTRACT FROM AUTHOR]

In this paper, we have studied various types of Monte Carlo methods along with the Power method to evaluate the maximum and minimum eigenvalue of a linear system of equations. We have studied how the accuracy of the maximum eigenvalue depends on the parameters, (moves in Markova chain), (no of Markova chain), (accelerating parameter), and a parameter m (the power applied on the resolving matrix). We have applied these methods to the randomly chosen symmetric matrices. We have also made comparisons for the different matrices of different orders depending on the parameters by using the Monte Carlo methods. We are using Matlab 2020R for the calculation. [ABSTRACT FROM AUTHOR]

In this paper we show the existence of the minimal solution to the multidimensional Lambert–Euler inversion, a multidimensional generalization of [ - e - 1 , 0) branch of Lambert W function W 0 (x) . Specifically, for a given nonnegative irreducible symmetric matrix V ∈ R k × k and a vector u ∈ (0 , ∞) k , we show that, if the system of equations y j exp { - e j T V y } = u j ∀ j = 1 , ... , k , has at least one solution, it must have a minimal solution y ∗ , where the minimum is achieved in all coordinates y j simultaneously. Moreover, such y ∗ is the unique solution satisfying ρ V D [ y j ∗ ] ≤ 1 , where D [ y j ∗ ] = diag (y j ∗) is the diagonal matrix with entries y j ∗ and ρ denotes the spectral radius. Our main application is in the analysis of the vector-multiplicative coalescent process. It is a coalescent process with k types of particles and k-dimensional vector-valued cluster weights representing the composition of a cluster by particle types. The clusters merge according to the vector-multiplicative kernel K (x , y) = x T V y . First, we derive some new combinatorial results, and use them to solve the corresponding modified Smoluchowski equations obtained as a hydrodynamic limit of vector-multiplicative coalescent. Then, we use multidimensional Lambert–Euler inversion to establish gelation and find a closed form expression for the gelation time. We also find the asymptotic length of the minimal spanning tree for a broad range of graphs equipped with random edge lengths. [ABSTRACT FROM AUTHOR]

The present paper deals with the external identification of a reciprocal, special passive, 2 n -port network under measurement uncertainties. In the present context, the multiport model is represented by an admittance matrix and the condition that the network is 'reciprocal special passive' refers to the assumption that the real part of the admittance matrix is symmetric and positive-definite. The key point is to reformulate the identification problem as a matrix optimization program over the matrix manifold S + (2 n) × S (2 n) . The optimization problem requires a least-squares criterion function designed to cope with over-determinacy due to the incoherent data pairs whose cardinality exceeds the problem's number of degrees of freedom. The present paper also proposes a numerical solution to such an optimization problem based on the Riemannian-gradient steepest descent method. The numerical results show that the proposed method is effective as long as reasonable measurement error levels and problem sizes are being dealt with. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a Helmholtz-type decomposition for the space of square integrable, symmetric-matrix-valued functions analogous to the standard Helmholtz decomposition for vector fields. This decomposition provides a better understanding of the strain constraint space, which is important to the Navier–Stokes regularity problem. In particular, we give a full characterization the orthogonal complement of the strain constraint space and investigate the geometry of the eigenvalue distribution of matrices in the strain constraint space. [ABSTRACT FROM AUTHOR]

The study of emergent behavior of swarms is of great interest for applied sciences. One of the most fundamental questions for self-organizing swarms is whether the swarms disperse or remain in a spatially cohesive configuration. In the paper we study dissipativity properties and spatial cohesiveness of the swarm of self-propelled particles governed by the model rk=-pk(|rk|)rk-∑mak,mrm, where rk∈Rd, k=1,...,n, and A={ak,m} is a symmetric positive-semidefinie matrix. The self-propulsion term is assumed to be continuously differentiable and to grow faster than 1/z, that is, pk(z)z→∞ as z→∞. We establish that the velocity and acceleration of the particles are ultimately bounded. We show that when ker(A) is trivial, the positions of the particles are also ultimately bounded. For systems with ker(A)≠{0}, we show that, while the system might infinitely drift away from its initial location, the particles remain within a bounded distance from the generalized center of mass of the system, which geometrically coincides with the weighted average of agent positions. The weights are determined by the coefficients of the projection matrix onto ker(A). We also discuss the ultimate boundedness for systems with bounded coupling, including the Morse potential systems, and systems governed by power-law potentials with strong repulsion properties. We show that the former systems are ultimately bounded in the velocity-acceleration domain, whereas the models based on the power-law potentials are not. [ABSTRACT FROM AUTHOR]

Given a tree T , let q (T) be the minimum number of distinct eigenvalues in a symmetric matrix whose underlying graph is T. It is well known that q (T) ≥ d (T) + 1 , where d (T) is the diameter of T , and a tree T is said to be diminimal if q (T) = d (T) + 1. In this paper, we present families of diminimal trees of any fixed diameter. Our proof is constructive, allowing us to compute, for any diminimal tree T of diameter d in these families, a symmetric matrix M with underlying graph T whose spectrum has exactly d + 1 distinct eigenvalues. [ABSTRACT FROM AUTHOR]

The accurate modeling of triply and doubly degenerate seam together while computing the coupled multiple potential energy surfaces globally remains a challenging task due to the unavailability of a suitable model function taking care of this aspect. In this paper, we present such a function capable of predicting many‐fold electronic degeneracies simultaneously as well as the features found away from the vicinity of such degeneracies if present in the coupled surfaces disregarding the symmetry requirements. The basis of the proposed model comes from the analytical diagonalization of a 3×3 real valued diagonally symmetric diabatic matrix and thus, the resulting three symbolic Eigen valued functions form the basis of model to be used in this work. From the analytical model, an attempt was made to provide an analytical expression for the adiabatic‐diabatic transformation angle. Few applications were considered starting from the simple physical model systems to actual realistic molecule illustrating how the present model function are being applied to them to show the presence of degeneracies among the three electronic states. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to analyze the Moore-Penrose pseudo-inversion of symmetric real matrices with application in the graph theory. We introduce a novel concept of positively and negatively pseudo-inverse matrices and graphs. We also give sufficient conditions on the elements of a block symmetric matrix yielding an explicit form of its Moore-Penrose pseudo-inversion. Using the explicit form of the pseudo-inverse matrix we can construct pseudo-inverse graphs for a class of graphs which are constructed from the original graph by adding pendent vertices or pendant paths. [ABSTRACT FROM AUTHOR]

Non-homogeneous real-order systems often arise in the construction of equations arising in many inequality systems. This paper considers non-homogeneous linear systems attached to random initial time and associated with Caputo real-order operators and time-dependent elements in standard and block matrix forms. Is it possible to find the convergence and estimate bounds of such systems? We use the idea of comparison methodology along with establishing the nonnegativity of such systems when the coefficient matrix is Metzler and the non-homogeneous term becomes nonnegative. Using the hypothesis on the existence of a constant Metzler matrix via construction, we prove that non-trivial solutions to such non-homogeneous systems converge to 0 under some reasonable conditions. We use the generalized Laplace transform technique as a tool to establish some reasonable conditions. In short, we prove that, if every root of a characteristic equation associated with such a constant Metzler matrix lies in the open left half complex plane and the non-homogeneous term approaches 0 as time tends to ∞ , then the non-trivial solution to the original system must be globally asymptotically convergent to 0. This result is then extended to the non-homogeneous real-order systems associated with coefficient block matrix form. We do not know the bounds and rate decay of the solutions to such systems. As a typical result, we prove that, if there is a constant symmetric Metzler matrix Δ M + with maximum eigenvalue λ max negative and the matrix Δ M + - λ max I is negative semi-definite, then there exist different measures for bounds of solutions to such systems. A particular case of the above-mentioned result gives rise to the new concept of γ -Mittag–Leffler asymptotic stability in the absence of a non-homogeneous input term and it thus provides γ -Mittag–Leffler decay rate. We provide an application to an electrical circuit system that illustrates the novel significance of some proposed theoretical results. [ABSTRACT FROM AUTHOR]